continued to encourage philosophical pursuits; and now, by directing the mensuration of the Meridian at the most distant points,—within the Arctic Circle, and under the Equator,—it procured accurate data for determining the true figure of our Planet. It was the more precise measure of a degree of latitude ascertained by Picard, that had at last enabled Newton to subject to the test of calculation his grand idea of extending the power of gravitation from the earth's surface, after the simplest law of decrease, to the orbit of the moon. But the satisfaction was reserved for the successors of our immortal chief, to see his determination of the oblate form of the earth, arising from the mutual attraction of all its particles, modified by the influence of centrifugal force, fully confirmed by a close investigation of the results of the skilful and laborious observations brought home by the French astronomers. Nothing, indeed, contributed so much to exalt the character of the Newtonian Philosophy on the Continent as those scientific expeditions; while the credit of the system of Descartes, which bears no such scrutiny, rapidly declined. Mathematicians, invited to a rich field of discovery, were eager to examine the Principia through all its details; and the progress which the New Calculus had made, provided them with instruments for this dissection. The profound author had left indeed several parts unfinished; of some of the more difficult problems he had merely sketched the solution; and he had commonly supplied the defects of his analytical procedure by the exercise of wonderful sagacity and penetration.

Maupertuis was the first mathematician of any note that ventured publicly in France to espouse the Newtonian Philosophy. But perhaps no writer contributed more effectually to promote its diffusion and popular reception than Voltaire, whose universal talent ranged through the sphere of human knowledge, and whose easy and simple style gave transparency to his conceptions. The French, at length awakened from the Cartesian dreams, and directed into the path of inductive philosophy, again put forth their

inventive powers. The fine genius of Clairaut and D'Alembert early soared to the highest distinction in scientific research. Both these illustrious men showed, from their infancy, the strongest disposition for mathematical studies, which they cultivated with the most ardent and persevering application, and displayed in their profound investigations all the rich and varied stores of original invention.

Clairaut began his labours with a capital extension of the Theory of Curve Lines. But, in his riper years, he directed the whole force of his ingenuity and analytical skill to explore the depths of Physical Astronomy. He deduced the figure of the earth strictly from the principle of universal attraction, with the admission only of the very simplest conditions. But not content with mere speculation, he sought the practical solution of that problem, and therefore eagerly joined the Academicians who were dispatched to measure a degree of latitude within the Arctic Circle. After achieving this grand operation, he resumed with vigour the examination of the Newtonian System, and concentrated his utmost exertions in resolving the arduous problem of the three bodies, or in determining the influence of a planet or the sun to modify the motion and orbit of another planet or satellite. The Integral Calculus furnishes yet no direct or absolute solution. It became necessary to proceed by a train of successive approximations, simplifying and condensing the computation by the judicious rejection of the small excrescences.¹ But the first result was most perplexing, and seemed to betray an evident imperfection in the great Law of Gravitation. Not discouraged, however, by this repulse, the persevering analyst pushed his calculations still farther, and collecting the minor terms of the series, at last arrived at a conclusion entirely conformable with observation, and thus established beyond all dispute the harmony of the Theory of Attraction.

But Clairaut did not rest satisfied with speculative conclusions, however beautiful; he sought to embody his formulae in real numbers; and availing himself of the aid of some expert calcu-

¹ The question especially selected was the annual change of the position of the axis of the moon's elliptical path about the earth, occasioned by solar attraction.

lators, he soon produced a set of Lunar Tables, much more accurate than had been yet attained. Encouraged by such eminent success, he now directed his inquiries to the more distant objects of our system. The comet, whose return was predicted by the sagacity of Halley, did not appear at the time assigned, and astronomers began to feel uneasy at their disappointment. From this state of inquietude they were soon relieved by the soaring investigation of Clairaut, who found that the disturbing influence of Jupiter and Saturn would retard for several months the advance of the distant stranger, and computed with surprising nearness, from his imperfect data, the time of the comet's actual appearance. But excessive application and the solicitations of society preyed on a languid frame, and premature death tore away that brilliant genius in the midst of his career of triumph and applause.

D'Alembert, who rose to still higher celebrity, and contributed to distinguish the character of the age in which he lived, closely followed in this research, though by a different path; encountered similar difficulties, obtained the same partial result, but finally arrived at a like satisfactory conclusion. He carried his acuteness and penetration into all the abstruse departments of physics, and marked his progress by originality of conception. The success of this most ingenious philosopher would have been more complete, if he had confined his views to the pursuits of abstract science. But he wanted that patience to which Newton, with innate modesty, ascribed all his advantages. In discussing the most arduous questions, D'Alembert advanced generally by new paths, and displayed great resources and much versatility of talent. He would seldom retouch his formulæ, or seek to mould them into simplicity and elegance, and never submitted to the labour of reducing them to actual application. Ambitious to excel in literature as in science, he transferred into his miscellaneous compositions the same strict logic and nice discrimination which guided his analytical researches, and likewise contracted a corresponding style, remarkable for the qualities of precision and clearness, but possessing no warmth or elevation.

The last of the three illustrious men who, by separate roads, arrived at the same conclusions, and thus concurred in fixing the true system of the world, was Leonard Euler, born near Bâle in Switzerland, and educated under the Bernoullis, but who, patronised by foreign courts, passed a very long and most laborious life at St Petersburg and Berlin. He was indisputably the greatest analyst that has ever appeared, displaying infinite address, perspicuity, and elegance in his mode of investigation, and pursuing or transforming the most intricate calculations with such astonishing readiness and rapidity, as if they seemed only mere pastimes to recreate his invention. But the supremacy of Euler was confined to his unrivalled skill in applying analysis; he attained little eminence in philosophy and general science, and showed no relish for the charms of literature. His improvements and discoveries, however, during a life of assiduous and unremitting labour in every branch of the Calculus, form a monument of the most stupendous magnitude.

Our island, after the decease of Maclaurin, produced none to compete with the great mathematicians of the Continent, except Thomas Simpson, whose native talent had struggled through indigence and a neglected education. He solved with commendable neatness and brevity several of the difficult questions of Physical Astronomy; but he was deficient in taste and method, and only followed tardily and at a distance those masters in science. For a long period afterwards the inventive genius of England appeared to slumber. The learned were content with merely commenting on the Principia, but rarely borrowing a few scattered lights from abroad. The current of investigation was diverted into other channels, or absorbed among humbler objects. In the meanwhile, a new and brilliant science, beyond the dominion of Mechanical Philosophy, had been gradually forming, to which our experimenters contributed their full share. Electricity captivated its numerous cultivators by surprising and splendid displays; but though it engaged the imagination, it afforded little exercise to the judgment, and was not fitted to call forth the higher mental energies. The application of English talent was now

mainly directed to the improvement and extension of the mechanical arts, though perhaps few nations in Europe have less availed themselves of the results of abstract science towards aiding and correcting the operations of practice.

In the career of the sublimer sciences, the Continent for a long time afterwards maintained its ascendancy, which was secured by the very superior skill displayed in managing the integral calculus, with its improved and refined notation. But a succession of minor discoveries continued to expand and consolidate every department of Natural Philosophy. Daniel Bernoulli, the most amiable, if not the most ingenious of that shining family, embraced with candour the doctrine of Newton, and likewise evinced, in all his physical researches, a considerable share of the sagacity and singular address which so eminently distinguished that great master. The attention of Lambert was diverted to a wide variety of pursuits, and his original and excursive mind shed new lights on every subject it explored. He conjoined analytical skill with the talent of experimental research. But unfortunately he contented himself in operating with rude instruments, and commonly trusted to the probability of rectifying such imperfect results by the help of combined calculations. If Mayer had not the same reach and versatility of genius, he possessed that inciting ardour and unconquerable perseverance which enabled his discrimination to erect a durable monument. Adopting the clear formulæ of Euler for the several elements of the moon's motions, he deduced the indeterminate co-efficients from a strict and most laborious comparison of Bradley's observations, and by thus poising and adjusting the numerical quantities, he framed a body of Lunar Tables, which has long been regarded as a standard of excellence. It is thus that England has generally supplied the means which rendered the conclusions of the Continental mathematicians really available. The reflecting instrument of Hadley, the achromatic glass of Dolland, and the dividing machine of Ramsden, have, in succession, mightily contributed to the progress of practical astronomy. The numerous observations of Bradley and Maskelyne furnished the correct data which guided Lagrange, Laplace, and Gauss,

through all their profound researches, to the discovery of the cyclical and reciprocating motions of the heavenly bodies. The various disturbing causes incessantly in operation are, after the lapse of certain vast periods, again renewed and repeated in the same order of succession, and thus preserve the fine harmony, and maintain the permanent stability, of the Universal System.

The British mathematicians had long neglected the cultivation of the Higher Analysis, and were perhaps the more disposed to overlook its pre-eminent advantages, from observing the course of their brethren on the Continent, who, on various occasions, after magnificent displays of the powers of calculation, but either from their incomplete integrations or the defective statement of the physical principles, arrived merely at the same imperfect conclusions that had been discovered before by much simpler means. Our island has at last resumed its proper station in the loftiest departments of science. Emancipated from the trammels of a narrow notation, the more aspiring votaries pursue with ardour the most refined calculus; while they guide its application and avoid its abuse by the infusion of that spirit of purity and elegance derived from the discipline of the ancient geometry. Among the illustrious few whom foreign nations would adopt with distinction, we may cite the gifted individual who, rivalling the fame of Lagrange and Laplace, has, in his beautiful solution of the problem of the Attraction of Spheroids, struck out a new and direct path, which completely throws into the shade their very laborious and perplexed trains of investigation.

Having traced the great outline, it only remains to sketch the plan of this Dissertation. I shall arrange Mathematical and Physical Science under the two great heads—of Pure or Speculative Science—and that of Applied or Practical Science; each of these again to be classed in subordinate divisions. Pure Mathematical Science includes Geometry, Arithmetic, Algebra, and the Higher Calculus. Its applications are very numerous and diversified; but without pursuing the details, our attention will be mainly directed to the progress of general and pervading principles.

Physical Science occupies a confined space till it receives the accession of Mathematics, with which it becomes always the more blended in proportion as it improves and expands. Pure Physics at present appears limited to Magnetism and Electricity, which have yet drawn scarcely any aid from the disquisitions of Geometry. The infusion of Mathematics into Physics, and the application of Physical Science to the Practical Arts, led to the most wonderful results,

which have aroused the ingenuity and vastly multiplied the productive industry of the eighteenth century. Applicate Physical Science embraces Dynamics, Hydrostatics, Pneumatics, Optics, Electricity, Magnetism, the Doctrine of Heat, Meteorology, Geography, and Astronomy. In discussing these important subjects, separately or inclusively, incidental lights will be reflected on General Mechanics and the Elements of Naval Architecture.

SECTION I.

SPECULATIVE MATHEMATICS.

1. GEOMETRY.

PURE Geometry is in strictness limited to the mere equality of lines, angles, and spaces, whether superficial or solid. It owes its main extension to the principle of comparison or the doctrine of proportion, which is really but an application of Arithmetic, the idea of Number being transferred to Quantity or Magnitude by a process of subdivision. Such concert has produced the most perfect of abstract sciences, and erected the noblest monument of the genius and invention of the ancient Greeks. That acute people nearly completed the Elements of Geometry; and carrying their speculations beyond the properties of the Circle, they investigated the Sections of the Cone, and traced the character of some of the Higher Curves. The demonstrations left by the Greek Geometers are models of accuracy, clearness, and elegance—admirably calculated for training the minds of youth to habits of close reasoning and luminous arrangement. The circumspection of those great instructors of mankind in distinguishing the several cases, and marking the limitations of a proposition, though frequently bordering on prolixity, might serve to warn the rapid cultivator of Algebra against indulging the tendency to hasty generalization which has given rise to the various paradoxes, and even palpable absurdities, that still

disfigure the excellence of the modern art. The method of Geometrical Analysis, which investigates the construction of a problem, by remounting from its conditions along a chain of dependence to some known property, affords decidedly the best exercise and initiatory discipline for the student in Mathematics. It imbues the mind with a taste and elegance which insensibly extend their influence over the culture of the other sciences. The most curious and difficult portion, however, of the Greek Geometry, has unfortunately been transmitted to us in a mutilated and imperfect state, which has often tortured the skill of commentators and mathematicians to restore it. Soon after the revival of letters, the principal works of the Greek Geometers were translated in Italy by Commandine; but of Apollonius' Conics several books are wanting, and some parts of the Collections of Pappus exhibit only detached fragments. Near the close of the sixteenth century, and early in the seventeenth, Vieta, from a few scattered hints, restored the lost Tract on Tangencies, Fermat framed some beautiful separate demonstrations, and Snellius reproduced the Plane Loci, but in a tasteless shape. Soon afterwards Viviani, the surviving disciple of Galileo, supplied the fifth book of Apollonius, and with such remarkable

success, that on comparing his production with an Arabic version just then discovered, he appeared to have surpassed his original. Huygens¹ afterwards gave, in the purest taste, some specimens of the Ancient Geometry. A Collection of the Mathematical Treatises of the Greeks respecting the Art of War was now published in a magnificent folio from the Royal Press at Paris. But a more extensive undertaking was planned by our oriental traveller Bernard, to print at Oxford a complete series of the Greek Geometers, filling up the blanks from the inspection of Arabic Manuscripts. In pursuance of this scheme, Dr David Gregory² edited Euclid, and

Dr Halley³ Apollonius, while he restored likewise the Tracts on the Section of Ratio and of Space. After an interval of fourscore years, Torelli's⁴ elegant edition of Archimedes, purchased in Italy, has issued from the same press. Every lover of science would rejoice to see a portion of those ample funds that have been provided at Oxford for the encouragement of such expensive works, appropriated to the republication of Pappus' Mathematical Collections, of which several manuscripts exist far more complete than the copy printed by Commandine.

The relish for the Ancient Geometry has been

¹ Born at the Hague in 1629, and son of the Lord of Zuylichen, Secretary to the Prince of Orange; completed his mathematical studies at Leyden under Schooten, and printed a beautiful tract on the Circle and Hyperbola in 1651. But after visiting different countries, he published in 1658 his original and immortal work entitled Horologium Oscillatorum. Having successfully applied the pendulum and the spiral spring to regulate the motions of clocks and watches, he was anxious to accommodate those instruments to the finding of the longitude at sea. For that purpose he visited England, where he was treated with distinction; but returning in 1663 through France, he was induced by the Minister Colbert to accept of a large pension, and fix his abode at Paris. There he resided till his health became impaired, and in 1681 he retired to the calmer enjoyment of his native country. The same pursuits, however, engaged his attention, till his death at the Hague on the 8th of June 1695. He was one of the clearest writers and most elegant geometers of modern times, and his powers of invention have seldom been surpassed. The finished works of Huygens have been collected at two several times into three quarto volumes.

² Nephew to the famous James Gregory, born at Aberdeen in 1661, completed his education at Edinburgh, and was appointed Professor of Mathematics in that University in 1683, the office having been suffered to remain vacant for eight years after the death of his uncle. In 1691 he had sufficient interest to obtain the Savilian professorship at Oxford, and had the honorary degree of Doctor in Physic conferred on him by the University. He published his Elements of Astronomy in 1702, and in the following year brought out his Edition of Euclid. He had made some progress in preparing the Conics of Apollonius, but fell a sacrifice in 1710 to an attack of malignant small-pox, at Maidenhead, where he chanced to stop at the Inn, on his return from a visit to Bath. He possessed some learning, but his genius was of a very inferior order to that of his uncle.

³ Edmund Halley, born in London October 29. 1656, the son of a substantial citizen—educated at St Paul's school, and sent to Oxford in 1673—sailed for the island of St Helena in November 1676, and returned with his catalogue of fixed stars after an absence of exactly two years—elected immediately fellow of the Royal Society, and deputed by that learned body in 1679 to visit Hevelius, at Dantzic, and examine his observatory—spent the years 1680 and 1681 in France and Italy. In 1684, having turned his attention to Kepler's problem, he tried, as some other mathematicians about this time had done, to derive it from a graduating central force, but was unable to find a geometrical demonstration; and not obtaining any help as he expected in this investigation from Hooke or Wren, he had recourse to Newton, who astonished him at Cambridge by the store of his grand discoveries, condensed into eight general propositions. He overcame the scruples of the modest philosopher, and prevailed with him to arrange the materials of the Principia, of which he superintended the publication in 1687, having written the preface and some elegant recommendatory verses. Halley now gave a geometrical construction of the higher equations, computed the effects of evaporation in the Mediterranean, and formed tables of life annuities. For the purpose of improving Nautical Science, he had the command of the Paramour Pink, with which he sailed from England on the 24th November 1693, traversed the Atlantic, and crossed the Equinoctial Line; but his crew growing sick and mutinous, he was obliged to return in the following June. Invested with fuller powers, he set sail again in September, and spent twelve months in exploring both hemispheres, and during the year after his return, he delineated and published his famous magnetical chart. He was next employed on a survey of the British Channel, and then sent by the English Government to assist her ally the Emperor of Germany in forming a harbour at the bottom of the Adriatic. On his return he was appointed, in November 1703, Savilian Professor of Geometry at Oxford, having been thwarted before in a similar canvass by clerical influence. He now set about recovering the works of Apollonius; studied as much Arabic as enabled him to translate the tract on the Section of Ratio, and he restored the other tract on the Section of Space from the hints left by Pappus. These pieces, in a small octavo volume, appeared in 1706; but four years afterwards came forth, in a splendid folio, his edition of the Conics, with the eighth book restored, and the additional treatise of Serenus. In 1719 Halley was appointed astronomer royal, and resided at Greenwich during the rest of his life, devoting his advanced age to the careful and assiduous observation of the Heavens. He completed even his projected task of embracing a lunar period of eighteen years. A paralytic disorder seized him in 1737, from which he partially recovered, but his strength declined insensibly, and he expired on the 14th of January 1742. Few philosophers have contributed more largely to the advancement of useful knowledge. Ingenuity, ardour, indefatigable perseverance, learning, and general information, were possessed by Dr Halley in a most eminent degree; and having mingled in the active scenes of life, he had the rare advantage of conjoining the love of study with the habits of social intercourse.

⁴ Born in 1721 at Verona, where he died in 1781. He studied at Padua, and became a great linguist, a good mathematician, and an excellent critical scholar. Being in easy circumstances, he devoted his whole time to literary pursuits, and carried his admiration of the Ancient Geometry almost to a pitch of bigotry.

longer preserved in Italy and in England than over the rest of Europe. But no person ever cultivated that fine science with more assiduity, perseverance, and success, than our countryman Dr Robert Simson¹ of Glasgow, the learned and critical editor of Euclid's Elements. In this department he concentrated his whole efforts, and appears, from his familiarity with the ancient mode of demonstration, to have inhaled the very spirit of the Greeks. In 1749, he published his Restoration of the books of Apollonius on Plane Loci, which by its fulness and peculiar elegance leaves scarcely a shadow of regret for the loss of the original. Simson pursued his researches in the Ancient Analysis through a long life, and not only restored various fragments, but threw light on some very difficult and abstruse questions connected with it. A posthumous volume, printed in 1776, at the expence of Earl Stanhope, besides many fine geo-

metrical speculations, contains the first satisfactory exposition of Porisms, of which the definition advanced by Pappus had been commonly regarded as an incomprehensible enigma. But from the unvarying tenor of his studies, the Scotch professor became a rigorist for the Ancient Analysis, and rejected with disdain the most obvious improvements in the form of exhibition. It deserves remark, that mathematical demonstration, being addressed to the eye rather than to the ear, must attain its greatest perspicuity when the successive steps of reasoning are seized at a glance. This is effected by adopting the symbols of Algebraic Notation, the most concise and perfect of all written characters; nor is the beauty and logical accuracy of the procedure in any degree impaired by such a transparent covering.

Dr Matthew Stewart² of Edinburgh, who had been the pupil of Dr Simson, and possessed a

¹ Born at Kirtonhall in Ayrshire on 14th October 1687, studied at Glasgow, and made such progress in elementary geometry, that at the early age of 22 he had an offer of the professorship of mathematics, which was immediately expected to become vacant in that University. Feeling his deficiency, however, he obtained leave of a year's absence, which he spent in London under the tuition of Humphry Ditton, and was admitted to the chair on the 20th of November 1711. His time seems afterwards to have been mostly spent in discharging the duties of his office, the intervals being allotted to the solution of geometrical problems, the perusal of the older mathematicians, and to miscellaneous reading. He led the life of a recluse, and all his steps were formal and methodical; yet his disposition was amiable, and he indulged at stated times in easy conviviality. He sent two papers, geometrical and algebraical, on Indeterminate Problems, to the Royal Society in 1723 and 1753, gave his restoration of the Loci Plani in 1745, published his Conic Sections in 1755, produced the Latin edition of Euclid's Elements in a quarto volume in 1756, which he compressed in 1760 into an English octavo, to which he annexed the data in 1762. It contains only the first six books of the original, with the eleventh and twelfth, the rest of the books being omitted as of little consequence in the present state of science. Perhaps the selection should have been carried farther. This edition is correct and creditable to the compiler, and has obtained prodigious success. Nay, the very Scotticisms with which it abounds appear now to be adopted at Oxford and Cambridge as the appropriate diction of the Ancient Geometry. Simson quite idolized his original. He had a fine taste for geometry, some talents of invention, and considerable attainments as a scholar, but without any great force of intellect. In his latter years, from excessive veneration of the Greek Geometry, he not only viewed the Cartesian method with aversion, but began to regard the Fluxionary Calculus with mistrust and suspicion. He became contritus professor in 1761, and died on the 1st of October 1768.

² Born in 1717 at Rothsay, in the Isle of Bute, of which place his father was minister; studied seven years at Glasgow, where he distinguished himself; to gain further instruction, in 1741 he removed to Edinburgh, where he cultivated the society of the celebrated Maclaurin, while he corresponded with Simson, his old master. He now prosecuted geometrical studies with ardour, and put forth his uncommon powers of invention; and after he became minister at Roseneath, he found leisure in that seclusion to continue his favourite pursuits. On the occasion of a vacancy of the mathematical chair at Edinburgh by the death of Maclaurin, he was induced to draw out the substance of his most profound geometrical investigations, which he printed under the title of General Theorems, in a small volume, about the close of 1746. This publication secured his election, and after some delay he was appointed professor in September 1747. He now enjoyed a situation most congenial to his taste, and favourable to the exercise of his rare talents. Ambitious to apply his beloved geometry to unravel questions which were believed to demand all the resources of algebraic art, he discovered a solution of Kepler's problem at once simple and direct, and greatly surpassing in beauty the more laboured efforts of calculation. This investigation was inserted in the second volume of the Transactions of the Edinburgh Society, which appeared in 1756. Encouraged by such success, he five years afterwards produced his Physical and Mathematical Tracts; a very ingenious and elegant work, which would have been clearer had the author only admitted an abbreviation by the simpler algebraic symbols. About this time the results of the observations of the Transit of Venus, which had been expected to give the true distance of the Sun, were found to be unsatisfactory and discordant. Stewart therefore published in 1763, as a sequel to his Tracts, a theoretical solution of the problem; in which he was directed by Geometry to the exact motion of the Lunar Apogee, a question that, since the analysis of Newton, had perplexed the greatest mathematicians. But he was not equally felicitous in Determining the Solar Distance, which required the inversion of the problem, and involved some deceitful and precarious compensations of error. This pamphlet was his last production, having printed only a few months before an octavo volume under the title of Propositiones More Veterum Demonstratae. These propositions, however, derive their value merely as exercises of the method of conducting geometrical analysis and synthesis. But his lamp of genius was already nearly extinguished. Dr Stewart fell into a state of bad health, and having devolved the charge of teaching on his son, then a youth of the highest promise, he retired in 1772 to a small paternal property in Ayrshire, where he chiefly spent the rest of his days, and died on the 23d of January 1785.

much richer invention, was likewise an able and zealous promoter of the Greek Geometry, which he directed besides to the investigation of certain difficult parts of Physical Astronomy. But though managed with sufficient address, the instrument he employed was hardly fitted for exploring the more abstruse and recondite problems, which often require all the concentrated powers of the Modern Analysis. The conclusions at which he arrived display much elegance, though merely approximative, and devoid of the precision that is indispensable in the present advanced state of Astronomical Science.

I need not stop to notice the attempts of other English mathematicians, to restore some fragments of Apollonius. After the principal demonstration was obtained, there could be little difficulty in evolving its different phases, and modifying it to the several cases. Playfair's elucidation of the nature of Porisms is entitled to higher distinction.

It has often been matter of surprise, that the Greeks should have spent so much ingenuity, and set such a high value on the Geometrical Construction of Problems. But the application of Geometry served them in some measure the purpose of calculation, and became a sort of substitute for the tedious and laborious operations of their imperfect system of Arithmetic. Accustomed, as we are, to the extreme facility of computing by help of the Arabic ciphers, we can form no adequate conception of the toil of working with alphabetic numerals, though the Greeks had made some capital improvements in their system of notation. They could extract laboriously the Square Root of a number, but never attained the extraction of the Cube Root, which it appears was first discovered many centuries afterwards by the Arabians. Hence undoubtedly the solicitude of the Greeks to solve, by a Geometrical, or even a Mechani-

cal Construction, the famous Delian Problem, or the Duplication of the Cube.

It would seem that, in finding the square root, the Greek mathematicians must have employed certain methods of abbreviation which are not explained. Archimedes, in his famous Quadrature of the Circle, to which he approximated by measuring the successive inscribed and circumscribing polygons, having occasion repeatedly to extract the square root, expressed the value by fractions, and yet with such felicity as always to adopt the lowest integral numerator and denominator. He thus discovered that the circumference of a circle is less than $3\backslash frac\{1\}\{2\}$ times and greater than $3\backslash frac\{1\}\{3\}$ times its diameter, or that their ratio lies between the ratios 7 to 22 and of 71 to 223.¹ This approximation within the limits of the 2000th and 4000th parts, might be sufficient for ordinary practice. But Apollonius and Ptolemy afterwards approached a hundred times nearer. Yet no farther advances were made in solving that important problem, which required the most refined address, till the lapse of near a thousand years, when the Arabians became possessed of the denary system of notation, and carried the expression for the circumference of a circle to ten decimal places. But this elaborate result was imperfectly known to the revivers of science in Europe; and Vieta, Adrianus Romanus, and others, exerted their ingenuity and patience in extending the earlier solution. The simplest and most elegant is that of 113 to 355, an approximation differing scarcely by the ten millionth part from the truth, which was discovered about the year 1585 by the elder Adrian Metius, a military engineer in Holland. His countrymen Van Keulen and Snellius, in the next century, pushed the expression for the circumference of the circle to 35 figures. But the progress of the higher analysis opened more easy and rapid modes of approximation. In the early part of the eighteenth century, Sharp² and Ma-

¹ This ratio, it may be observed, is easily derived from the limits assigned by Archimedes, for $71\; +\; 7\; \backslash times\; 6\; =\; 113$, and $223\; +\; 22\; \backslash times\; 6\; =\; 355$.

² Abraham Sharp, born at Little Horton, in the West Riding of Yorkshire, in 1651, was apprenticed to a merchant at Manchester: but preferring the study of mathematics, he supported himself by teaching a school in Liverpool. Here he engaged as amanuensis to Flamsteed, and became his assistant when the Royal Observatory was erected at Greenwich in 1676. His services were most eminent in every department. He constructed optical and astronomical instruments, observed stars, calculated tables, and delineated celestial charts. His peculiar neatness of execution is displayed in a small geometrical work which he published in 1717. About that time he seems to have retired to his native village, where he led the life of a hermit, but esteemed for his quiet and beneficent disposition. He died in 1742.

chin¹ computed in England the quadrature of the circle to 75 and 100 places of decimals, and Lagny in France advanced to 128 figures. But these labours were outdone by Vega,² an officer of artillery in the Austrian service, who amused his leisure during a campaign, in employing one of Euler's formulae to derive the length of a quadrant from the tangents of fractional arcs, carried the expression to 140 decimal places. This was the luxury of calculation; and though no doubt superfluous, it might at least convince any judicious person of the impossibility of stating the ratio of the diameter to the circumference in finite terms. Yet the squaring of the circle is a problem which has at all times fascinated the attention and bewildered the reason of many superficial or antiquated students in Geometry. The incom-

mensurability of the circle, which James Gregory had attempted to prove in 1661, was finally demonstrated a century afterwards by Lambert, from an ingenious transformation of the known series for the quadrantal arc in terms of its tangent or the radius. The same ingenious mathematician likewise proposed several neat geometrical constructions, for approximating to the length of arcs of a circle still more nearly than the methods given by Fermat, Gregory, Huygens, and others.

Elementary Plane Geometry rests on the combined properties of the straight line and the circle. Many important additions have been made to the digest of Euclid; and several eminent mathematicians of the eighteenth century have corrected, simplified, and essentially improved the only valuable portion of the large work.³ Some

¹ John Machin, elected professor of Astronomy in Gresham College in 1713, and became secretary of the Royal Society. He had the reputation of being an able mathematician, yet his essay on the Laws of the Moon's Motion, in which he attempted to rectify the Principia, was but a superficial performance. He died in 1751.

² This excellent mathematician, who published in 1794 a complete collection of Logarithmic Tables and Analytical Formulae, was eight years afterwards robbed and barbarously murdered by a miller, in whose house he lodged, and his body thrown into the Danube.

³ It may be sufficient perhaps to notice the Elements of Geometry by Thomas Simpson, and the similar treatises in the French language by Clairaut and Legendre. These productions all unite clearness with precision; and excluding whatever appears superfluous, they still comprise the whole series of connected propositions. Simpson's unpretending volume is neat and very brief, yet sufficiently perspicuous. The Geometry of Clairaut is still shorter, but has an air of originality, being designed to show the road of induction, or to guide the learner through his efforts at the solution of geometric problems, to discover the great elementary truths. He followed the same plan, and with more complete success, in his excellent Elements of Algebra. But the Geometry by Legendre claims much higher merit, and is perhaps the best on the whole that has yet appeared. It were vain, however, to expect perfection; the steps of his demonstrations are sometimes incomplete, and his entire separation of the problems is at least a very questionable improvement. The notes he has added are valuable, but appear disproportionate to the text, and certainly too profound for beginners. This elegant geometer attempted to found the comparison of triangles on abstract considerations derived from the common theory of functions. But such reasonings a priori are fallacious, involving unperceived some metaphysical assumptions.

The works now mentioned might suffice for the instruction of practical or professional men; but the pursuit of a liberal education aspires to greater attainments. The main object is to sharpen the faculty of perception, and invigorate by due exercise the tone of the intellectual powers. For contributing to that effect, the fulness and circumspection of the ancient mode of demonstration are admirably calculated. It seemed, therefore, an estimable task to select the scattered wrecks of the Greek Analysis, and dispose them into a form accessible to ordinary students. The beauty of the propositions concerning Loci was particularly striking.

It would be preposterous, however, to hold up the Elements of Euclid as a standard of instruction in Geometry for the present day. They were composed before the invention of Trigonometry, and probably designed chiefly as an introduction to the Pythagorean Philosophy. Hence the large portion of them devoted to the relations of numbers, and the properties of the regular polygons and solids. By common consent, therefore, the greater part of the system is now laid aside, and the other books are commonly altered or curtailed in practice. Many trivial propositions occur in the third book; several of the fourth have little interest; and it may abate that extravagant praise which several mathematicians have lavished on the Doctrine of Proportion contained in the fifth book, to know that it really cannot be taught. But the language employed in that celebrated compilation, for want of appropriate technical terms, is often vague and indistinct. The word magic, for instance, has no less than three different significations; and other examples of a like confusion might be easily cited. In some cases, the demonstrations of Euclid are imperfect or inconclusive. Thus, the reasoning in the twenty-fourth proposition of the first book applies only to the particular position of the figure; an objection first started by Thomas Simpson, to the great annoyance of his critical namesake. But similar oversight was made in the demonstration of the seventh proposition of the same book, which fails if the figure be changed. Both these propositions are only of consequence as auxiliaries in the train of combination. But of the fundamental property of parallel lines, the demonstration rests on the mere assumption of an intricate axiom. In a few cases the reasoning is unnecessarily complicated, from the peculiar conceptions of the author. Thus, the noted fifth proposition of the first book is of that description. Its demonstration implies the reversed application, at least mentally, of the isosceles triangle; but Euclid evades or disguises the process, by producing the sides of the triangle, and forming two interwoven triangles, which are virtually adapted by inversion, and the annexed triangles being then taken away, the equality of the angles at the base hence follows. But the last is the only step wanted, and the other two were evidently superfluous. This stumbling proposition has been called the Pons Asinorum; and no wonder that the beginner should feel puzzled at seeing such a parade of argument end in so plain a result.

remarkable propositions, derived from sources beyond the usual scope of Geometry, are yet assimilated with that science. Of such accessions, one of the most beautiful is the late curious and unexpected discovery of Gauss, that, besides the ordinary regular polygons, a numerous class more complex, including the next polygon of seventeen sides, are capable of being inscribed within a circle, by a mere geometrical construction.¹

In solving the common problems, the straight line and the circle are generally combined, which requires the application of both the ruler and the compasses. Schooten effected by the ruler alone several simple constructions, of use particularly in castrametation. But the late Mascheroni² of Bergamo, in a small work which appeared in 1795, made a beautiful addition to Elementary Geometry, by the solution of a variety of problems, especially those concerning the inscription of the regular polygons, with the help exclusively of compasses. Such speculations, however, are in a great measure extraneous to the science.

The theory of Lines of the Second Order forms one of the finest speculations of the more advanced geometry. These curves were derived from the Section of the Cone; but the clearest and most philosophical way of treating them is by considering their description on a plane. The distinguishing property of the foci was known to Apollonius, but the advantage of its application overlooked. The Parabola forms the intermediate transition of the Ellipse to the Hyperbola, the remote focus stretching farther out, till it vanishes into indefinite space, again to re-appear on the opposite side. The features of the Parabola are therefore not strictly included in the phases of the Ellipse and Hyperbola. The more comprehensive relation of the distances of any point in the curve from a focus

and the directrix is mentioned by Pappus, but the beauty of that locus was not perceived till Boscovich, in 1752, deduced the properties of those lines from it in a string of corollaries, more ingenious than elegant. This essay was compressed and methodized by Thomas Newton in 1794; and in the same year Walker, a respectable mathematician, unaware of what had been already done, produced a work on a similar plan, of which the first part occupied a quarto volume, composed indeed after the manner of the ancients, but so exceedingly prolix in diction as to have very few readers.

An elegant mode of investigating curves of the second or higher orders, is to consider them as generated by the conditional intersection of angles or lines turning about fixed points or poles. The property had been stated generally in the Principia, but was expanded in 1720 by Maclaurin in his Geometria Organica. The intersections of Polar Radiants were discussed by Brackenridge in 1733. The subject has been again revived, and prosecuted with great ingenuity and research, by the celebrated Carnot, under the denomination of Transversals.

No part of Geometry has been more improved in the course of the last century than Trigonometry, which is not only simplified but much extended in its application. The theory of planes and solids has likewise been cultivated with eminent success. As the position of a point on a plane is assigned by referring it to two co-ordinate lines, so the place of a point in space may be determined by its distances from three planes which are mutually perpendicular. The properties of a line of single curvature may be derived from the equation of its co-ordinates; the properties of a curved surface can likewise be deduced from its triple shades, or the perpendicular projections on the three planes. Such is the process employed by Monge³ in his Descriptive

¹ Gauss found that the expression for the multiple cosine is always decomposable into binomial factors, and hence concluded that any regular polygon, the number of whose sides is a prime, and denoted by $2^n\; +\; 1$, may be inscribed in a circle by mere Elementary Geometry. These numbers form the series 3, 5, 17, 257, 65537, &c. of which the first two only were known before.

² Born in 1759; died at Paris, July 14, 1800. His tract on the Compasses happening to fall into the hands of General Bonaparte during his first triumphant campaign in Italy, this extraordinary man was so struck with its ingenuity, that, on his return to Paris, he communicated, in conversation, some of the propositions to the members of the Institute. The book was immediately translated into French, and the author promoted to a place of trust and emolument in the Italian Republic.

³ Gaspard Monge, born in 1746 at Beaune, and educated at the Jesuits' seminary at Lyons. His uncommon talent for drawing procured him admission into the Military School at Mezieres, where he soon became assistant teacher. Disgusted

Geometry, which may be considered as the completion of the analytical method of Descartes. It furnishes direct and general solutions of several important problems, especially of those relating to the intersection of planes and the con-

stitution of solids. Descriptive Geometry comprehends also the theory of Perspective, which was founded by Ubaldi, and long afterwards recast and simplified by Brook Taylor, and rendered still more practical by Lambert.

2. ARITHMETIC.

To express large numbers by continued additions being impossible, it soon became necessary to arrange them on a scale of ascending progression. The simplest mode was evidently to repeat the same root of the scale; and as men in the earlier periods of society used to reckon with the ten fingers of both hands, they were led by Nature herself to frame the Denary System of Numeration. If they had likewise formed marks for a Digital Notation, they would have completed the bases of Arithmetic. But unfortunately the Alphabetic Characters had been introduced before distinct symbols were contrived to represent numbers. The letters designed merely for written language, came hence among all nations to be employed besides in numeral notation, though hardly manageable, and ill adapted to any regular system of arrangement. In the application of those characters the ancient Greeks showed great ingenuity, by distinguishing them into three classes, appropriated to the ascending progression of units, tens, and hundreds. In this way they reached the term of a thousand, and by employing as an auxiliary the capital M, they could indicate myriads or ten thousands. The triple series of thousands up to a million were more easily denoted, however, by placing a dash under the successive sections of letters. But Archimedes indicated a mode of obtaining almost un-

bounded extension of this system, by decomposing the elements of the notation into periods of ascending myriads. Apollonius simplified the plan, by adopting a thousand as the root of the progressive scale. Had he proceeded only a little farther, and rejected the letters, except the first class denoting digits, he would have rendered his notation perfect, and have achieved one of the most useful and prolific discoveries ever made. Ptolemy afterwards advanced much nearer to this ultimate object, by employing, in his Trigonometrical Tables, the descending Sexagesimal Scale, suggested by the subdivision of the circumference of a circle into degrees, of which sixty, or the root of this scale, correspond nearly to the length of the radius. It would seem, however, that, during the interval between Archimedes and Ptolemy, the Greek Arithmetic had received some essential improvements in practice; for the Sicilian geometer, by a very tedious and operose process, of which he gives merely the results, was content with stating the ratio of the diameter to the circumference of a circle as intermediate between that of 7 to 22 and of 71 to 223, differing therefore by a 1200th part from the truth; whereas the chords computed by the astronomer of Alexandria for every half degree of the semicircle are correct to the last place, and consequently never deviate by a 300,000th part of the whole.

with the tedious and operose methods of calculation then practised among engineers, he sought to shorten the road by recurring to general principles and the aid of Geometry. Thinking closely on that subject, he matured a regular system, which has changed the theory of planes and solids. The ardour of his pursuits was infused into his pupils, and the frequent communications he made to learned societies rapidly extended his fame. In 1780 he was conjoined with Bessut as acting professor of Hydrodynamics at the Louvre, a place erected by the patriotic minister Turgot, which required him to reside at Paris only during half the year. But three years afterwards, on being appointed to succeed Bezout as Examiner of the Marine, he fixed his permanent abode in the capital, and engaged in the experiments and discussions of the philosophers. He became a warm partisan of the Revolution, and contributed his utmost efforts to promote the various plans adopted by its leaders. But he deserved praise for the great concern he took in founding the Polytechnic School, an institution eventually productive of the very best effects. Having accompanied Bonaparte to Egypt, he returned with the fruit of his observations to share in the fortunes of that wonderful man. He resumed with lustre the place of professor, and continued to delight the numerous pupils by his kind attention and the clearness and expansion of his oral discourses. But on the second return of the Bourbons in 1815, he felt the weight of their vengeance. The Institute was remodelled, and the Polytechnic School suppressed. This sad reverse preyed on his spirits, and produced alienation of mind; in which melancholy state he languished for some time, and expired on the 26th July 1818.

The Romans, who cared only about objects of vulgar ambition, overlooked the refined mode of Greek Notation, and remained satisfied with their own very clumsy and involved system, which could with difficulty represent a large number, but was absolutely disqualified for serving in any way the purposes of calculation. They were obliged therefore to have recourse to counters, and by help of the Abacus, or decimal board, they performed the ordinary operations of summing accounts. This humble expedient was practised in Europe till the seventeenth century, and is still used by the traders throughout the Chinese Empire, its application being there facilitated by the prevalence of the decimal subdivision of weights, measures, and coins.

The discovery of the Denary Notation, so beautiful and simple in its application, is commonly referred to India, though neither the place nor the date of its origin has been ascertained. It was unknown to the Arabian astronomers till near the close of the eleventh century of our era, and even at this period its expressions appear sometimes intermingled with the alphabetic numerals. But the advantages of the system were soon perceived; and the Tables of Sines and Tangents calculated by those laborious mathematicians are now found, from examination of the manuscripts belonging to the University of Leyden, to be correct to the tenth decimal figure. The Arabians transmitted the new system of notation to the flourishing colony of their countrymen in Spain, whence it was slowly communicated to the several Christian States over Europe. This change was effected chiefly by the calendars or perpetual almanacs compiled in the Moorish seminaries, and eagerly purchased by the various monasteries and convents throughout Christendom. Yet the Arabic ciphers appear to have remained unknown in every part of Europe beyond Spain, before the middle of the fourteenth century; nor in mercantile transactions were they commonly adopted till near two centuries later. It is a singular circumstance that the advantages of employing the descending progression in the nicer calculations were not sooner perceived, especially after the use which Ptolemy had made of sexagesimals. Stevinus

was the first who distinctly introduced the practice of decimal fractions in 1585, though Regiomontanus had made a great step towards that improvement, which Ramus even indirectly employed. To count downwards might seem as easy as to reckon upwards. But the mode of denoting the ranks of decimals was then most cumbrous, the successive numerals, like the indices in Algebra, being inclosed in small circles. Bayer in 1619 proposed to substitute for these complex marks an accent repeated. It was our illustrious countryman Napier, however, that brought the notation of decimals to its ultimate simplicity, having proposed in his Rhaphodologia, printed ten years earlier, to reject entirely the marks placed over the fractions, and merely to set a point at the end of the units. But his sublime invention of Logarithms about this epoch eclipsed every minor improvement, and as far transcended the denary notation, as this had surpassed the numeral system of the Greeks.

Various speculations have been framed regarding the properties of the different arithmetical scales. Leibnitz fancied important advantages to accrue from the adoption of the Binary Scale, which operates with extreme facility, and requires only a single character besides the zero. This progression mounts so slowly, however, that it cannot express a large number without employing a multitude of terms. But the Duodenary Notation, which is partially admitted in the uncial subdivisions, would evidently answer the best for general practice. It proceeds faster than the Denary Scale, and is less subject to be affected by fractions, since its root has no fewer than four divisors, while ten admits only of two. Still these advantages would not compensate for its want of conformity with the train of ordinary language. The Decimal Arithmetic would soon generally supersede the use of every other kind of fractions, if it were likewise in practice combined with an extensive decimal subdivision of weights and measures.

The curious or mystical properties of numbers have at all times fixed the attention of mankind. Square numbers, denoting the sides of a right-angled triangle, were sought for by the Pythagoreans, who gave a very simple rule

to compute them.¹ Perfect numbers, or such as may be composed by the addition of their aliquot parts, could not fail to excite admiration; and Euclid produced an elegant theorem for discovering them. It is remarkable that these numbers should appear frequent at first in the series, and afterwards occur only at wide intervals. Thus, below 10,000, there are four perfect numbers; but the last of the succeeding four includes no fewer than nineteen ciphers.² The moderns have advanced farther, and imagined what are called Amicable numbers, on all which subjects Euler has shown his ingenuity and unrivalled analytical skill.

It is less a matter of speculation, while it is of considerable consequence in the practice of calculation, to assign the Prime Numbers, or such as admit of no division, and to resolve the Composite Numbers into their several factors. No general method has yet been devised for the investigation of Primes, though by the researches of Bachelier and Fermat in the seventeenth century, and of Euler, Lagrange, Legendre, and Gauss, in the eighteenth, they are now reduced to certain constant forms, which may facilitate their discovery. By an extension of the method called the Sieve of Eratosthenes, a sort of mechanical process, the prime numbers are separated from the ordinary progression. In this way several authors have constructed tables of Primes, and of Composite Numbers, with their various factors or divisors. Lindenau proceeded to a hundred thousand, but Cherac has lately carried the enumeration as high as ten millions.

The formation of Magic Squares, by which the numbers in their natural order are disposed in cells, so that each column gives an equal sum, remounts to high antiquity. They have been regarded in the east as charms or talismans, of great potency in the tide of human affairs. The

digital squares from 3 to 9 were dedicated to the seven stars, including the sun and moon, which in the astrological visions were believed to rule the successive days of the week.³ The people of Siam had learned a very simple method of filling up those squares; but some European Mathematicians have since amused themselves with such arithmetical curiosities. Stifels, in his Arithmetica Integra, incidentally treated of Magic Squares, yet without naming them. He was immediately followed in a small arithmetical treatise by Adam Riese; but the French seem to have directed most attention to those scientific recreations. From the latter part of the 17th till beyond the middle of the 18th century, a succession of ingenious persons were at pains to improve and extend the construction. Frenicle, Poignard, Delahire, Ons-en-Bray, and Rallier des Ourmes, are entitled to particular notice.

The construction of Magic Squares belongs to the class of problems styled Indeterminate, which depend on variable combinations, and therefore admit of very numerous answers. Thus, Frenicle, who successfully studied this subject near the beginning of the last century, showed that the first sixteen of the natural numbers could be formed into a Magic Square in not fewer than 880 different ways. Analysts have since tried to narrow those changes, and augment the difficulty of the investigation, by annexing other conditions. Dr Franklin produced, by dint of perseverance, what he called a Magic Square of Squares, and a Magical Circle. De la Hire⁴ treated the formation of Magic Squares with such copious detail as to leave very little to be done by his successors. Yet several eminent mathematicians have occasionally reverted to this amusing subject. Even the great Euler has not disdained to direct his transcendent skill in algebraic analysis to the solution of

¹ See notes to the Elements of Geometry. Euler has demonstrated that every number whatever consists of not more than four squares. An example may be given of the composition of numbers between 10 and 20: $10\; =\; 9\; +\; 1$, $11\; =\; 9\; +\; 1\; +\; 1$, $12\; =\; 9\; +\; 1\; +\; 1\; +\; 1$, $13\; =\; 9\; +\; 4$, $14\; =\; 9\; +\; 4\; +\; 1$, $15\; =\; 9\; +\; 4\; +\; 1\; +\; 1$, $16\; =\; 16\; +\; 1$, $17\; =\; 16\; +\; 1\; +\; 1$, $18\; =\; 16\; +\; 1\; +\; 1\; +\; 1$, $19\; =\; 16\; +\; 1\; +\; 1\; +\; 1\; +\; 1$, $20\; =\; 16\; +\; 4$.

² It may gratify the curious to see the series 6, 26, 496, 8128, 33550336, 8589869056, 13743895328, and 2305813008139952128. These numbers are obtained by multiplying the 1, 2, 4, 6, 12, 16, 18, and 30 powers of 2 by its 2, 3, 5, 7, 13, 17, 19, and 31 powers diminished by unit.

³ The first in modern times who noticed Magic Squares was Agrippa of Nettesheim, who gave in his books de Occulta Philosophia, printed in 1535, the Sigilla Jovis, Martis, Solis, Veneris, Mercurii, and Lunæ.

⁴ Born at Paris in 1640, and died in 1718, a man of some attainments, and great application—well acquainted with the details of surveying, and of practical astronomy.

problems connected with the theory of Magic Squares.¹

Another portion of Arithmetic, not merely curious, but of material importance in the application of that science, has been much cultivated during the eighteenth century. If both the terms of a proper fraction be divided by the numerator, there will result an equivalent fraction, having unit for its numerator, and an integer, conjoined with another fraction of smaller terms, for its denominator. But this new fraction may be again broken down in the same way, and the process of decomposition renewed successively, till a fraction with an integral denominator is at length obtained. This mode of resolution is therefore the very same as what is employed in finding a common measure or divisor, only the several quotients are here preserved. When the original fraction is expressed by rational numbers, its decomposition must always terminate; but if the numerator and denominator be mutually incommensurable, the process of evolving their elements will never draw to a conclusion. The fractions which thence arise may simply repeat, or they may circulate at short intervals or periods, or they may succeed each other without any sort of order whatever. But it is evident that such progressive subdivision must approach constantly nearer to the true value, and that the resulting fractions, if successively recombined, will exhibit a series of approximations.

The fractions of this kind were first noticed by Lord Brounker,² who proposed one of them to express the ratio of the diameter to the circumference of a circle, which Dr Wallis³ after-

wards investigated. Huygens perceived the utility of employing such fractions to represent approximate ratios, and in this way he found convenient small numbers to represent the teeth of the wheels and pinions of his Planetarium. But the subject appears to have lain neglected till 1737, when Euler explored it with his usual depth and felicity, in a distinct memoir published in the Petersburg Transactions. He may consequently be regarded as the real founder of the Theory of Continued Fractions, to which indeed he gave the appellation they have since retained. It was a favourite speculation of that great analyst, who resumed the subject at different times, and widely extended its application. Euler showed how to convert any quadratic equation into a Continued Fraction, and demonstrated that the expression for an irrational square root consists of terms which either repeat or circulate in regular periods. A cube root may likewise be expressed by a Continued Fraction, but its terms can have no sort of order in their sequence. As these Fractions, in approaching to their ultimate value, oscillate continually between excess and defect, they are always capable of being changed into Converging Series, affected by the alternate signs of plus and minus, and the transformation may be reversed. By this method the very ingenious Lambert converted the series of Gregory and Leibnitz for the length of a quadrant into a Continued Fraction, whose terms have a regular and incessant circulation, and hence proved (what had only been presumed before) that the Circle is incommensurable. But he was enabled to proceed still farther, and to establish, that even the square of the number

¹ Cornelius Capito printed, in 1767, at Glückstadt, a short treatise on Magic Squares, in which he proposed to vary their construction no fewer than a million times. But the latest and complete work on this curious subject was compiled by the late Professor Mollweide of Leipzig, the ingenious editor of Killig's valuable Mathematisches Wörterbuch.

² Born in 1620, and died in 1684; an able mathematician, but an unprincipled courtier.

³ This very remarkable man was born November 23, 1616, at Ashford in Kent—educated at Cambridge—became chaplain to Lady Vere in 1643, and was then led to cultivate the art of deciphering, in which he acquired such skill, as to be afterwards frequently employed by the Long Parliament in examining intercepted dispatches. Through that interest he was appointed to a sequestered church in London, and chosen in 1644 one of the scribes to the Westminster Assembly of Divines. But his austere occupation did not divert him from the nobler pursuits of philosophy and science; and being appointed by the Parliamentary commissioners in 1649, Savilian Professor of Geometry at Oxford, he continued regularly to discharge the duties of that office. He had afterwards some hot disputes with Hobbes, who, though a most ingenious metaphysician, unfortunately mistook his own blunders for discoveries in Mathematics. Wallis maintained a large correspondence with men of science, and assiduously contributed to the progress of philosophical discovery during that active period. The moderation of his character assured him the possession of the places he held at the Restoration; and though he found leisure to compose a very learned philosophical grammar of the English language, and sometimes indulged his early taste for controversial divinity, he devoted the greater part of a long life to the laborious cultivation and extension of the various departments of Mathematical Science. His works are voluminous, and display great erudition, talents for deep research, and originality of conception. He died on the 28th October 1703, and was succeeded in the chair by Dr Halley.

expressing the circumference is likewise irrational.¹

Lagrange has since recommended and explained the application of Continued Fractions to the solution of numerical equations. These fractions may be readily changed into Recurring Series,² which, in the case of small roots, are often beautifully simple and remarkably convergent. But sometimes they betray the opposite tendency to diverge; and the great object of Lagrange was to remove that defect. Euler, in several of his last memoirs and opuscles, directed them to indeterminate problems, to the finding of maxima and minima, and to the development and integration of certain analytical formulae. Nothing can exceed the beauty of some of these speculations.

It is a singular fact, that though Continued Fractions had their origin in this island, they were afterwards almost wholly overlooked by our mathematicians. Scarcely thirty years have passed away since the attention of the curious was again recalled to them. But an acquaintance with the transformation of those fractions would prove extremely useful to practical men, by furnishing a variety of short approximations, which are easily remembered and readily applied. Ordinary computations in mensuration, in the comparison of different weights and measures, and in several standard proportions, might thus be much abbreviated without any sensible inaccuracy. The method proposed by Lambert for decomposing fractions into progressions that

have their denominators formed by successive multipliers, could be employed in a similar way; but its application is somewhat difficult and intricate.³

To facilitate the operations of arithmetic, tables have been digested of the powers and products of numbers. The most astonishing work of that kind, including all the multipliers from one to one thousand, was produced by Hervart, Chancellor of Bavaria, in 1610, only four years before the invention of Logarithms, the want of which it could in some faint degree have supplied. But tables of products in a less unwieldy shape are still occasionally used for abridging the labour of calculation. A notable improvement has been lately introduced by the substitution of Quarter Squares, grounded on the known property that the product of any two numbers is equal to the fourth part of the excess of the square of their sum above the square of their difference. A single page might comprise as many of those numbers as, by simple addition and subtraction, would exhibit an extent of products equal to those contained in Hervart's enormous folio.⁴

It would be a great improvement in elementary education to introduce into the schools a more extended multiplication table, including at least all the products of the numbers from 1 to 25. A very useful practice has been adopted lately in the north of Germany, to accustom learners to the readiness of mental calculation.⁵

¹ The area of a circle having unit for its diameter is expressed by the alternating series $1\; -\; \backslash frac\{1\}\{2\}\; +\; \backslash frac\{1\}\{4\}\; -\; \backslash frac\{1\}\{8\}\; +\; \backslash frac\{1\}\{16\}$, &c. which was converted by Euler into the continued fraction

of the odd numbers. This extension never stops, and consequently the fraction is not reducible to finite terms. The square of the expression runs into a similar infinitude.

² It deserves to be mentioned that Girard first gave the simple recurring series which expresses the parts of a line divided into extreme and mean ratio. It is formed by the continued addition of the two preceding terms. Thus 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, &c. The square of each number in this series differs from the product of the two numbers which inclose it, only by unit alternately in excess and defect: Thus $8^2\; =\; 64\; =\; 5\; \backslash times\; 13\; -\; 1$, $13^2\; =\; 169\; =\; 8\; \backslash times\; 21\; +\; 1$.

³ For an explanation of Lambert's method of decomposing fractions, see Philosophy of Arithmetic.

⁴ A large table of multiplication, from 1 to 10,000, and a specimen of quarter squares, extending to a million, may be seen in the treatise quoted above.

⁵ It is called Kopfrechnung or calculating by the head. Three works are current on this subject, those of Biermann, Heuss, and Köhler—published between 1790 and 1816.

ALGEBRA derives its immense superiority over the ancient analysis from the very complete system of notation which it has at length attained. Each step of an investigation being now registered in the clearest and most precise manner, the mind is relieved from the fatigue of carrying forward the whole of a continuous chain of reasoning. It can rest at any place, and again resume the process of deduction with the greatest facility. But this perfection of pictured language has been the result of a series of slow and successive improvements. Diophantus, who, soon after the Christian era, composed a large treatise on the resolution of certain arithmetical problems, a portion of which work is preserved, made the first attempt towards an abbreviated form of analysis. The symbols he used were, however, exceedingly few, being the initials or terminations of the ordinary words, only somewhat modified. It deserves remark, that Diophantus viewing a number as composed from another by the addition or subtraction of a third, he termed this last one abundant or deficient, and gave a rule for the multiplication of those affected numbers, similar to what is now applied to the signs plus and minus. His treatise passed into the hands of the Arabians, who studied the numerical properties with much ardour, but made no advances in refining or simplifying the form of notation. From them again this higher species of arithmetic was transplanted into Italy, probably during the fourteenth century, by Leonard, a merchant of Pisa, who had travelled long in the East. A series of works on Arithmetic and Geometry were published between the years 1470 and 1487 by Pacciolo, a Minorite Friar, who in 1494 brought out his Arte Maggiore, or what he says was vulgarly called the Regola de la

Cosa, or Algebra and Almucabala, the first appellation derived from the Italian Cosa, denoting the Thing sought, and the other borrowed from the Arabic words expressing Resolution and Composition. After the appearance of this elementary digest, Algebra was cultivated in Italy, during the first half of the sixteenth century, with great industry and success. Ferro, Tartalea, and especially Cardan, made rapid advances in the new science. This ingenious though very singular person not only gave the solution of Cubic Equations which bears his name, but discovered the leading properties of equations in general, distinguishing their roots into the true and the fictitious; and besides improved the notation by employing frequently the letters of the alphabet. The Italians, however, still used contracted words for symbols, and the initials p. and m. for the signs of plus and minus. In 1572, Bombelli of Bologna, retaining this embarrassed mode of writing, composed a regular summary of Algebra, which he enriched with Ferrari's rule for resolving biquadratic equations. In the meanwhile the knowledge of the analytic art had penetrated into Germany, where it received its capital improvement in notation from the systematic genius of that people. The change was chiefly effected by the industry of Stifels,¹ a Protestant minister and zealous follower of Luther, who in 1544 published his Arithmetica Integra. In that remarkable work he first introduced the symbols $+$ and $-$ for plus and minus, and the character $\backslash sqrt\{\backslash quad\}$ (or a contracted R) for Radix or root; and he represented unknown quantities by the capital letters A, B, C, &c. and intimated the successive powers and their reciprocals by an ascending and descending series of exponents. Nor was his merit confined to algebraic writing: the en-

¹ Born at Eslingen, in Saxony, in 1509, and died at Jena in 1567. Though a profound and inventive mathematician, he seems to have imbibed all the wild enthusiasm of that convulsed period. Captivated perhaps by the wonderful properties of numbers, he fancied, as other ingenious persons have since done, that he could interpret the visions of the Apocalypse, and foretell the end of the world. He was so imprudent as to place that awful dissolution very near hand. Early in the morning of the day predicted in the year 1553, he assembled his trembling flock in a wide open field, where he endeavoured to season their minds for the tremendous change by fervid prayers and pathetic exhortations. The sky was lowering, the darkness thickened, a portentous silence prevailed, and the preacher rolled his thunders with overpowering energy. But the clouds soon passed away, the sun shone forth in his wonted splendour, and all nature smiled. The populace recovered their agitated spirits; and now breathing rage and disdain, they chased the unlucky prophet home with volleys of stones.

thusiasm of the age seems to have roused the spirit of invention in him; and Stifels anticipated some of the later discoveries, pointed out the nature of Logarithms, explained the properties of figurate numbers, and showed how the co-efficients of the powers of a binomial quantity may be derived from the columns of what has been since called the arithmetical triangle. Scheubel a few years afterwards pursued the same path, yet without adding any thing material to the science. In this state it made its way at last into England, where Recorde,¹ an ingenious though unfortunate man, printed the Cossic Art at London in 1553. He was the first to propose the sign = for equality, but made no other advances; and during a period of most active enterprise, till the close of the century, Algebra was not cultivated at all in this country.

In the meanwhile France continued the pursuit of the mathematical sciences. Ramus had revived those abstract studies; but Vieta, who succeeded him, and flourished between the years 1570 and 1600, rose by his numerous discoveries to much higher eminence; a lawyer by profession, yet a man of great learning, and gifted with profound and original genius, conjoined with the most indefatigable application. Vieta introduced the literal or specious Algebra, and thus rendered its procedure quite general, by employing the Roman capitals always to denote numbers, the vowels being appropriated to unknown quantities. He likewise traced various distinctions, and framed several significant terms, which are still retained. But while he improved the analytical symbols, he greatly extended the theory of equations, and cultivated with success the prolific and important subject of angular sections.

Flanders at this period equally displayed the spirit of invention. Stevin of Bruges, born about the middle of the sixteenth century, was an engineer eminently skilled both in the theory

and practice of his profession, possessing great original powers of mind. He reduced Statics and Hydrostatics to their simplest principles; and in his Arithmetic or Algebra, printed in 1585, he extended the range of calculation by several fine improvements and discoveries. Marking the unknown quantity with a small circle, he denoted its power by inserting the index; and was enabled, by prefixing a fraction, to represent also the roots. But, besides improving and simplifying the symbols, he enriched the analytic art by his inventions, and gave a general method for the resolution of numerical equations. The works of Stevin were, in 1625, collected and expanded by his countryman Girard,² a man likewise of most original conception, who eagerly promoted the objects of science. Four years afterwards this editor produced a small tract exclusively his own, and full of new and ingenious deductions. He there gave a very complete theory of equations, distinguishing them into their several orders, and proving that they had always a corresponding number of roots; he showed how these roots are successively combined in forming the co-efficients of the several terms; and finally anticipated the remarkable rule discovered by Newton, to find the sums of their different powers. Girard was possessed of fancy as well as invention; and his fondness for philological speculation led him to frame new terms, and to adopt certain modes of expression which are not always strictly logical. Though he stated well the contrast of the signs plus and minus, in reference to mere geometrical position, he first introduced the very inaccurate phrases of greater and less than nothing, and began the unfortunate appellation of impossible quantities.

It is indeed the reproach of modern analysis to be clothed in such loose and figurative language, which has created mysticism, paradox, and misconception. The Algebraist, confident in the accuracy of his results, whenever they

¹ Robert Recorde, born of a good family in Wales, about the year 1506—studied at Oxford, and elected a fellow there in 1531. He embraced the medical profession, but taught mathematics at both universities, and afterwards in London, where between the years 1551 and 1557 he published several elementary treatises on Geometry and Algebra, with quaint titles. But Recorde was unfortunate, and having been thrown into the Fleet for debt, he died a prisoner in 1558.

² Albert Girard published his edition of Stevin's Arithmetic in 1625, and his own discoveries in Algebra in a small quarto at Amsterdam in 1629. He died in 1633, and his widow the year after put forth his complete collection of Stevin's works.

become significant, hastens through the successive steps to a conclusion, without stopping to mark the conditions and restrictions implicated in the problem. This rapidity of operation, though in many respects most advantageous, yet affords less mental exercise than the cautious and guarded procedure of the Greeks. It will not be deemed foreign to the scope of this discourse, to remark, that the signs plus and minus confer no distinctive character, but merely indicate that the number to which they are prefixed, is to be annexed to some other number, or disjoined from it. The terms additive and subtractive would express correctly their whole import. The rules for the involution of those signs are derived from the consideration of the properties of compounds, arising from the addition or multiplication, for instance, of the binomials $a\; \backslash pm\; b$ and $c\; \backslash pm\; d$. The operation proceeds by detail; but the real meaning of the notation may be suspended, and it becomes significant only after the several members have been recombined. A number, in strict language, is altogether devoid of quality, and can be reckoned neither positive or affirmative, nor negative, which designations are accidents, and not attributes.¹ In proposing these terms, Vieta did not, therefore, discriminate the precise nature of the symbols; and his powerful example has continued to in-

fect the language and darken the conceptions of algebraists. A disposition has also prevailed in modern times, of hastening to general conclusions, although the data be limited or imperfect. Such careless deductions are but awkwardly amended, by the adoption of expedients more like the fictions of lawyers than the reasonings of sound logicians. The introduction of equal and impossible roots of equations served only to restrict the ordinary rules, which had been made too general, representing the number of roots as always equal to the index of the highest power. The involution or repeated multiplication of binomials will produce the successive orders of expressions, which pass into equations on the supposition that any one of them vanishes or has its parts mutually balanced. But the converse of this proposition will not always hold true. That every compound expression is resolvable into as many binomial factors as the index of its highest power signifies. Several forms even of the quadratic or cubic expressions resist all binomial decomposition. But it is a property demonstrated, that every higher expression whatever may be resolved into binomial or trinomial factors, or into simple or quadratic elements. Impossible quantities are thus merely the symbolical exhibition of the binomial factors of a quadratic or trinomial expression which is irreducible; and

¹ This distinction might perhaps have satisfied the scruples of the late venerable Baron Maseres, who wrote an express treatise against the abuse of the Negative Sign, which he came to view with a sort of aversion as the main source of the incorrect language and vague conception so prevalent among algebraists. Though not quite entitled to the rank of a discoverer, that excellent person deserves a place in the history of Mathematics, for his valuable contributions, and his zealous and unwearied exertions to promote accurate science. He was an able geometer, a profound constitutional lawyer, a man of sound and most extensive learning, and of very general information. But he possessed the higher qualities of our nature, and combined liberality of sentiment and unbending integrity with the feelings of a kind, generous, and social disposition. His Elements of Trigonometry was the completest treatise in English at the period of its appearance. He afterwards improved the solutions of cubic and biquadratic equations, and illustrated the methods of approximating to the resolution of equations in general. But the most important service he rendered to mathematical science consists in reprinting at great expense, and chiefly for distribution, the series of original authors in logarithms, with ample annotations, in six large quarto volumes. From his anxiety to be perspicuous, he was apt to fall into the opposite extreme of tiresome prolixity.

The grandfather of Maseres was an officer in the French guards, born in the district of Bearn, and a Huguenot, who, on the revocation of the Edict of Nantes, retired to Holland, and transferring his services to the Prince of Orange, accompanied the Preserver of our liberties into England, and fought by his side in Ireland. The grandson having embraced the profession of the law, was soon after the peace of 1763, probably on account of his intimate acquaintance with French language and customs, appointed Attorney General of Quebec, where he resided till 1776. At leaving his charge, he visited Boston and New York, while that agitation was fermenting which burst into open revolt. Soon after his return to England, he published in successive volumes his Canadian Freeholder, in which he refuted the despotic maxims of Lord Mansfield, and earnestly recommended conciliatory measures with our American colonies. But it was a time of infatuation and disaster. Maseres by his firmness gave mortal offence to the courtiers, and never obtained any promotion, farther than the small sinecure office of Cursitor Baron of Exchequer. But fortune had abundantly provided for all his wants, and he enjoyed unbroken health, and the renovating pleasures of study and social intercourse. The Baron had composed a full and learned treatise on Life Annuities, and matured a plan for securing small pensions from Government to such of the poorer classes as should make certain contributions during the vigour of their days. This he embodied in a Bill which passed the Commons, but was lost in the House of Lords, through the influence of the Bishops, who seem to have considered pauperism as a right appendage to their splendid ecclesiastical establishment. He died at Reigate in May 1824, at the very advanced age of 93.

the mark of impossibility is removed, either by a change of the signs, or by a recombination that restores those factors to their primitive binomial form. Such notation may indicate the limits of a problem, and seems to originate in neglecting the previous statement of the limitations.¹ In reference to Geometry, the impossible expressions intimate the transition of the circle into the equilateral hyperbola, or of arcs into logarithms.

The publication of the Geometry of Descartes in 1637 is justly deemed an epoch in the history of analytical science. The capacious mind of the author seized on all the preceding discoveries, and moulded them with his original inventions into a regular and comprehensive system. He gave rapidity to the writing of Algebra by the introduction of small characters, and rendered the distinction more palpable between known and unknown quantities, by the appropriation of the initial and final letters of the alphabet. He adopted the notation of the integral exponents or indices, as reduced to its simplest form by Herigon; and this apparently very slight improvement led, in the sequel, to the most important results. Though the index was at first merely a contraction for the repeating of the same letter in the involution of powers, yet it acquired a most extensive import when it came to be treated abstractly as a number. The prefixing of the subtractive sign changed the expression into its reciprocal, and the substituting of a fraction converted it into the symbol of evolution or the extraction of roots. Nothing can better illustrate the efficiency of a systematic notation, as an instrument for enabling the intellect to pursue and generalize its deductions, than the successive modifications which analogy suggested in the

marking of exponents, during the short period from Descartes to Newton.

But the French philosopher effected a revolution in scientific procedure, by applying (what his countryman Vieta had only partially attempted before) the symbols and calculations of Algebra to the solution of geometrical problems. By referring curve lines to co-ordinates or mutual perpendiculars, he expressed their relations by equations, of which he distinguished the rank and composition. He showed that a biquadratic equation is, by the help of indeterminate co-efficients, resolved into two quadratics, and may be constructed by combining a circle with a conic section. The construction of a cubic equation was somewhat simpler, though derived from the same principles. But Descartes proceeded still farther, and represented what are now called curves of double curvature, by reducing them, by a series of perpendiculars, to some plane of projection. These were all important advances in mathematical speculation.

Attempts have been made to apply the Cartesian method even to Elementary Geometry, which is in fact to convert a clear and simple train of reasoning into a sort of hard mechanical process of calculation. This change, were it attainable, would be the reverse of an improvement. It would extinguish that fine study which affords the best exercise and discipline of the intellectual powers. The wonderful dexterity and readiness which men of such teeming invention as the Bernoullis, Euler, and Lagrange generally displayed in managing algebraic analysis, appear to have seduced their admirers into an over-estimate of its real advantages. The constructions derived from algebra very seldom reach the purity and elegant simplicity of the geometrical methods. In those cases where Geometry

¹ A very simple problem will illustrate this: Suppose it were required, from a given point, to draw a tangent to a given circle. The construction would be, to join the centre with the given point, and upon the connecting line describe a circle to cut the former in the points of contact. It is evident, that if the given point occupied the extremity of the diameter, there would be only one point of contact; and if it lay within the circle, there could be none. This construction hence intimates sufficiently the conditions of the problem; and yet it fails in the case where the circle described merges in the given point, and consequently ceases to determine the position of the tangent. The algebraical solution, if rightly interpreted, gives similar information. Let $r$ denote the radius of the given circle, and $a$ the distance from its centre to the given point; then $\backslash sqrt\{a^2\; -\; r^2\}$ will express the length of the tangent, which only requires to be inflected to the given circumference. But if the given point fall on that circumference, then $a\; =\; r$ and the origin and termination of the tangent coincide, leaving its position undetermined. Again, if the given point lie within the circle, $a$ will be less than $r$, and consequently the subtractive portion $-r^2$ will predominate over the other portion $a^2$, and the compound radical $\backslash sqrt\{a^2\; -\; r^2\}$ become impossible. It serves to show, however, that by a change of condition the problem would be rendered soluble.

is most felicitous, they betray the features of a clumsy and artificial combination. The application of Algebra should therefore be reserved for the most arduous and complicated problems in the Higher Geometry.

In the latter part of the seventeenth, and during the course of the eighteenth century, Algebra still continued to advance; and though it underwent no revolution, it acquired greater perfection in all its details. The construction of cubic and biquadratic equations, which Descartes had effected by combining a circle with a parabola, was afterwards improved; but Newton proposed the Conchoid as the curve best adapted for the practical solution of such problems. Leibnitz succeeded in conquering the irreducible case of Cardan, by a bold application of the binomial theorem, having converted the cubic roots of the compound expressions into two series, with alternate imaginary terms, which are extinguished by their conjunction. In examining generally the nature of cubic equations, their solution came afterwards to be referred to the trisection of a Circular or Hyperbolic Sector, and was therefore accomplished by the application of a table either of Sines or of Logarithms. The theory of the Section of Angles and of Ratios, founded by the ingenuity of Cotes, furnished a clue to the resolution of certain forms of the

Higher Equations. But though Euler solved biquadratics in a different way from that of Descartes, all attempts to find directly the roots of the fifth and the superior orders of equations had totally failed. It became necessary therefore to have recourse to the approximative methods of resolving equations. Successive improvements in the process of calculation were made by Raphson, Halley, Lagny, and Taylor; so that little seemed wanting for any practical end.

During the last century likewise the method of series, so various and extensive in its application, was cultivated with the greatest success. About the years 1714 and 1718, Montmort¹ and James Bernoulli employed it in the investigation of the Laws of Chance. But De Moivre,² a French refugee, and a man of learning and profound science, carried these researches much farther, having devoted his time professionally to the calculation of probabilities, and of the values of life-annuities. In 1730, he published an original work, which, besides other useful inventions, explained the properties of a Recurring Series, that always repeats the same succession of co-efficients in distinct sequences. About the same time, Stirling³ brought out a complete treatise on Series, in which he advanced by a different road to consider their convergence, interpolation, and summation. This ingenious

¹ A man of some rank, and an able mathematician, born in 1678 at Paris, where he died in 1719.

² Abraham De Moivre, born in 1667 at Vitri in Champagne. Being a Protestant, he was obliged at the age of eighteen to seek shelter in England after the revocation of the Edict of Nantes. Having shown an early taste for Mathematics, he continued to prosecute the study as a profession. He supported himself creditably in exile, by giving lessons and reading public lectures. His analytical discoveries extended his fame, and his good conduct insured him respect. He embraced the Newtonian Philosophy, and made some improvements on it. He generalized Cotes' famous theorem of the section of a circle, but turned his attention chiefly to the method of series, which he applied successfully to the doctrine of annuities and chances. After his reputation for such calculations had been established, it is said that he spent a great part of the day at Slaughter's Coffee-House, in St Martin's Lane, where he was ready to answer any question of that nature propounded to him, for the fee of one guinea. His principal work on the Doctrine of Chances was published in 1738, but the enlarged and improved edition bears date 1756. De Moivre lived to a great age, but in his advanced years he was subject to obstinate fits of sleeping, and in that torpid state he remained for at least several weeks, till death closed the scene in November 1754.

³ James Stirling, born about the year 1690 in Stirlingshire, where his father owned a small property. He was educated at Glasgow, and sent by that University on Snell's Foundation to Balliol College in Oxford. During his retreat there he applied himself so diligently to the study of Mathematics, that he printed in 1717 a small tract on Lines of the Third Order, with new solutions of one or two difficult problems by the fluxionary calculus. He then repaired to London, and becoming acquainted with the Venetian Minister, he accepted an invitation to settle at Venice, where he resided several years, and taught Mathematics. The vicinity of Padua gave him an opportunity of acquiring the friendship of Nicholas Bernoulli, nephew of the two elder Bernoullis, who was Mathematical professor in that University. During his stay at Venice, Stirling contrived to gain access to the manufacture of glass plates, and escaping at some personal risk, he is said to have transferred the secret to England. He now conducted a mathematical or nautical school on Tower Hill, while he maintained a correspondence with the philosophers both abroad and at home. Here he published his great work on the Differential Method and Series in 1730. After tilling in his academy several years, he was induced to leave London, and undertake the direction of the Mines at Leadhills in Scotland. In that elevated district near Sanquhar he resided during the rest of his life, and by his skill and activity he greatly improved the operations of extracting the lead ore. He now held a profitable employment, but his high mathematical fame would have secured him the honour of succeeding Maclaurin in 1746, if he had not at that unhappy period been tainted with Jacobite principles. In his latter years he seems to have confined his attention to practical concerns, and died at Leadhills in 1772.

mathematician followed no general procedure, but showed great felicity and address in transforming one series into another. The subject was at different times partially extended by Simpson, Maclaurin, Landen, Lorgna, Hutton, Waring, Pfaff, and Kramp. Euler handled Series with his usual fulness, perspicuity, and penetration; and yet Lagrange and Laplace, in surveying that field, have gleaned fresh discoveries, which the latter of these illustrious men applied most happily to develop the Theory of Probabilities.

The composition of equations was at the same time investigated with more address and precision. The rule for distinguishing the positive and negative roots from the alternating signs, which Descartes had merely stated, now received a strict demonstration by Segner. The mode of finding the impossible roots, or rather the quadratic factors, of an equation, which Newton probably had inferred from mere induction, was traced by Maclaurin to its real source. D'Alembert proved that every irrational expression is reducible to the form $A^2\; \backslash pm\; \backslash sqrt\{B^2\}$; or, in other words, that every compound expression of a higher order may be resolved either into simple or quadratic factors. This is the most important conclusion perhaps relating to the nature of equations. But the general theory was never so clearly and completely discussed as by Gauss, in a small work published by him in 1779. Various methods, and several of them elegant, have been proposed by Euler and others for simplifying and expediting the approximation to the roots of equations. When a question involved the powers and products of more than one unknown quantity, it exercised the skill of algebraists to separate them; but Bezout¹ has

brought the mode of this elimination into a regular system, which removes all the difficulties. Among the promoters of Algebraical Science may be ranked our countryman De Waring,² a profound analyst, but unfortunately an obscure and confused writer. His Miscellanea Analytica and other detached pieces discover uncommon penetration and originality of conception, and though neglected at home, they have, notwithstanding their repulsive form, been duly appreciated by the great mathematicians on the Continent.

The latest improvement that Algebra has received consists in the Combinatorial Analysis, which may be viewed as an important extension of the principles of the binomial theorem. Vieta traced the rudiments of the doctrine, which was successively enlarged by Mersenne, Guldinus, Schooten, Pascal, Wallis, James Bernoulli, De Moivre, and Euler. But it has received its greatest expansion in Germany, and chiefly from the laborious and persevering researches of Hindenburg, who first published a tract on this subject in 1779, and resumed the discussion in the years 1793, 1794, and 1795. He was followed in the same line of investigation by Bureckhardt, Rothe, De Prassi, and Pfaff; and this interesting branch of analysis now forms a part of the algebraical course pursued in the German States. It throws much light on the theory of equations, renders more general the method of series, and facilitates the calculation of chances.

In the application of Algebra to Geometry, Euler, pursuing the route marked out in 1727 by Frederick Mayer, has given such extension to the properties of Angular Sections, as to create almost a new science, bearing the appellation of the Arithmetic of Series. Vieta prepared the

¹ Stephen Bezout, born at Nemours on the 31st March 1730—led accidentally to the study of Geometry, and fired by the perusal of the Éloges of Fontenelle. Having soon distinguished himself by proficiency in Mathematical Science, he was appointed Examiner to the Navy in 1763, and Examiner of the Artillery Department 1768. In the discharge of his duty in those offices, he gained the love and respect of the youth by his impartiality, kindness, and solicitude for their advancement. He published The General Theory of Equations in 1779, and gave further proofs of his original inventive powers in other occasional Memoirs. But he has owed his celebrity chiefly to the excellent Course of Mathematics compiled for professional education, which is highly esteemed and generally adopted on the Continent. The constitution of Bezout was prematurely exhausted, and he died of a malignant fever on the 27th of September 1783.

² Edward Waring, born near Shrewsbury in 1736. He studied at Cambridge, where he distinguished himself so much in abstruse calculations, that he was elected, after a hard contest, Lucasian professor of Mathematics in January 1760. He led the life of a recluse student, and though a man of great worth, he was unfitted by his extreme diffidence for the general intercourse of society. His latter years appear to have been spent at his place of nativity, where he died on the 15th August 1798. Waring had not entirely confined his application to mathematical research; he printed a metaphysical tract of considerable merit, which was distributed among his friends; and it deserves mention, that he there distinctly brought forward the philosophical view of Causation which is now very generally embraced.

way, and the Bernoullis had advanced far in the research, but it was reserved for their great disciple to expand and convert their conclusions into a compact body of doctrine. This important branch of Analysis has continued to be much cultivated in Germany, under the more expressive name of Goniometry, which was first applied by Lagny in the Parisian Transactions for 1724, to an ingenious method proposed by him for measuring angles.¹ Pfaff has lately enlarged it with the properties of the multiple tangents, and his countryman Mollweide has likewise made some valuable additions. The store of formulae now collected is of the utmost utility in transforming and reducing into practical operation the different series adopted by Physical Astronomy.²

Algebra has also been applied with advantage to Goniometry generally, as comprehending the solution of the cases both in Plane and Spherical Trigonometry. This was early attempted by Girard, but carried to its completion by Euler and Lagrange. Some concise and elegant formulae have been deduced, which facilitate and diversify the practice of computation. In short, if Algebra were purged of the vitious language and inaccurate conceptions that pervade it, and which were early introduced by the confidence of rapid and careless calculators, it might at last claim the character of a perfect science.

The number of elementary treatises of Algebra which have been produced in various languages during the currency of the eighteenth century, appears quite incredible. They are almost ephemeral, and assume every feature and dimension,

superficial or profound, from the size of a primer to the magnitude of a lexicon. A very few of those works claim the highest praise, and may retain their hold of education. The Algebra of Euler is in various respects a most remarkable production. That illustrious analyst, when totally deprived of sight in his advanced age, dictated it in the German language to a young domestic, whom he trained for an amanuensis. He was obliged therefore to be plain, distinct, and perspicuous; and these qualities he combined with richness of invention. The second volume had an air of originality, which made it peculiarly interesting. It treated of Diophantine Problems, and the resolution of Indeterminate Equations,³ and was afterwards expanded in the French version by the masterly annotations of Lagrange. In our own language, MacLaurin's Elements of Algebra, though a posthumous work, is perhaps the ablest on the whole, and the most complete. The bulky volumes of Dr Sanderson,⁴ which were likewise printed after the death of the author, are commendable chiefly as the production of an ingenious person, afflicted from infancy with the calamity of total blindness; they have little claims to depth, originality, or logical precision, but possess the merit of being eminently clear, methodical, and copious even to diffusion. The Algebra of Thomas Simpson is a work of an opposite description—brief, condensed, and marked with traits of invention. Passing over the numerous smaller treatises which have run through the schools, the compilation of Emerson⁵ deserves notice. It forms part of the Cyclomathesis, or

¹ This consists in performing by the help of compasses a repeated decomposition, similar to what is effected in the reduction of a common to a continued fraction. There being in practice generally a balance of errors, the numerical relation hence derived approximates to great precision. The method may be applied to the comparison either of arcs or of straight lines.

² The collection of formulae by Hirsch deserves particular commendation.

³ A very simple and regular method of solving such questions is given in the second volume of the Transactions of the Royal Society of Edinburgh.

⁴ Nicholas Sanderson, born at Thurleston in Yorkshire in 1682. When only twelve months old, he lost both eyeballs by small-pox; but showing early capacity, he was sent to the free-school at Penniston, where he made great proficiency in the knowledge of the classics, and soon understood the works of the ancient Geometers when read to him in the original Greek. He afterwards profited by the kind instructions of some mathematical friends, and found himself so far advanced in science, that at the age of 25 he repaired to Cambridge, and established his fame in that university by the warmth and splendour of his lectures on the Newtonian Philosophy. He was encouraged by the gentleness of Whiston, who saw no rival; and, on the ejection of that amiable enthusiast in 1711 for heresy or dogmatism, succeeded to the Lucasian Professorship of Mathematics, the duties of which he continued henceforth to perform with zeal and assiduity. The faculties of touch and hearing he possessed in a wonderful degree; and he was a lively companion, breathing a free and open disposition. But sedentary habits impaired his vigorous constitution, and he died of a mortification in his limbs on the 19th of April 1739.

⁵ William Emerson, born in June 1701 at Hurworth, a village near Darlington—the son of a schoolmaster, who taught him Mathematics, and left him a small property, which he occupied and cultivated. He was a person of strong intellect

series of elementary works in all the departments of Mathematics and Mechanical Philosophy, executed with ability by that singular man; exhibiting a mass of valuable matter, clothed in a slovenly style, and digested without method, or any regard to taste or elegance. Nothing is

more wanted for the purpose of education than a classical treatise on Algebra, which, avoiding all vague terms and hasty analogies, should unfold the principles with simplicity and rigid accuracy, and follow the train of induction with close and philosophical circumspection.

4. THE HIGHER CALCULUS.

LEIBNITZ, at his death, left the Calculus which he had framed in a most flourishing condition. The superiority of its algorithm, joined to the ardour of his disciples and successors the Bernoullis, gave it the complete possession of the Continent. But in England, the Method of Fluxions, now greatly in arrear, was cultivated only by a very few aspirants. Its inventor had vacillated about the notation he should adopt, and the unfortunate marking by points appeared the first time in one of the volumes of Dr Wallis' works, printed in 1699.¹ Newton had published nothing separately on the subject, till the Tract on Fluxions, drawn from the recesses of his closet, made its appearance in 1704; which, though full of ingenuity, was then decidedly imperfect and insufficient. But Cotes² and Taylor³ laboured successfully in extending the analytical discoveries of their great master. The former, one of the brightest mathematicians whom England has produced, died at an early age, after achiev-

ing, however, some fine analytical discoveries. Besides producing the beautiful geometrical theorem on which is founded the application of Binomial Factors, he constructed an ample Table of Fluents or Integrals, which were solved by the decomposition of fractions, by Logarithms or Circular Arcs. Other ingenious problems are introduced in his posthumous volume entitled Harmonia Mensurarum, published in 1722 by his relative and successor Dr Smith; but for want of the author's revision, this profound tract is unluckily so concise and obscure as to need a commentary. Some of the propositions were neatly demonstrated by John Bernoulli; but De Moivre considerably extended the doctrine, which was afterwards reduced into a systematic form by Walmsley.

Dr Brook Taylor was a man of rich invention and elegant accomplishments, who supported with dignity the reputation of English science. He was indeed our only mathematician that,

and considerable talent of invention; but affected singularity, and indulged coarse boorish habits. Having a mechanical turn, he constructed his own instruments, and delineated all the figures he wanted. His best work, perhaps, was the treatise of Increments, but his System of Mechanics has had the greatest currency among all ranks of students. In the decline of life he suffered much from the cruel attacks of gravel, and died after a lingering illness on the 20th May 1782.

¹ The earliest mode used by Newton for denoting a Fluxion seems to have been a zero prefixed to the variable quantity. Thus, the fluxion of $x^n$ was written $\backslash dot\{x\}^n$ or $x^n\; \backslash dot\{x\}$. It is evident therefore that the great inventor must have then had nearly the same conception of the origin of Infinitesimals as his rival Leibnitz.

² Roger Cotes, born July 10, 1682, at Burbach in Leicestershire, of which his father was rector. Having shown an early and decided inclination to Mathematics, he was encouraged by his uncle, the Rev. John Smith, who carried the boy to his house in Lincolnshire, and carefully instructed him in the principles of science. Hence he removed to St Paul's school, where he made great progress in classical attainments, but still found leisure to cultivate his favourite studies. Thus accomplished, he was sent to Trinity College at Cambridge, and advanced through all the gradations with unrivalled distinction. In January 1706, he was elected unanimously to the chair of Astronomy and Experimental Philosophy, which had been just founded by Archdeacon Plume. He took orders in 1713, and at the desire of Dr Bentley superintended the printing of the second edition of the Principia, to which he furnished a learned and ingenious preface. His genius and uncommon talents had raised the highest expectations, when a putrid fever snatched him away in the full tide of vigour, on the 5th of June 1716. Newton in his latter days used often to exclaim with a modest feeling of regret, "If Cotes had lived we should have known something." This very promising philosopher was perhaps rather fastidious; for, except a short but ingenious essay on Logarithms, he produced nothing of decided importance during his lifetime. The papers left by him in an imperfect and unfinished state were collected and published with some annotations by his cousin and successor Dr Robert Smith.

³ Born in affluent circumstances at Edmonston in Middlesex in 1695; studied at Cambridge, where he distinguished himself, and passed through the successive gradations with the greatest applause. The mathematical and physical sciences were his favourite pursuits, but he embraced general learning and cultivated the fine arts, being especially fond of painting and music. Fortune enabled him to indulge his taste. He held the office of Secretary to the Royal Society for a few years, and afterwards visited the Continent. His mind was most acute and inventive; but, though methodical on the whole, he was at no pains to unfold his original conceptions. He died at an early age in 1731.

after the retreat of Newton, could safely enter the lists with the Bernoullis. His most original work, the Methodus Incrementorum, published in 1715, unfolded the elements of the increase and decrease of a variable quantity or function. It was a fine extension of the Differential or Fluxionary Calculus, and constituted a new and important branch of the Higher Analysis, which has since received an appropriate algorithm, and acquired the appellation of Finite Differences. The Method of Increments was applied to the summation of series, and a variety of difficult problems, of which it afforded easy and rapid solutions. It furnished the first investigation of the motions of a vibrating cord, showing that the harmonic curve is a Trochoid or Prolate Cycloid. Among other improvements, it contained the celebrated theorem for the expansion of a magnitude, arising from the combination of its successive orders of differentials, which has deservedly retained the name of its ingenious author. This elegant formula is of extensive use in almost every analytical inquiry, and even performs the inverse process of Fluxions, by exhibiting in many cases at once the resulting fluent or integral. But the notation employed was imperfect, consisting merely of accents instead of points; and the Tract, like other works of Taylor, affected a degree of brevity which borders on obscurity. It was therefore elucidated by Nicole, an eminent French mathematician, in a series of able dissertations, between the years 1717 and 1727. The Taylorian Theorem itself has been successively modified, transformed, and extended by Maclaurin, Lagrange, and Laplace, whose names are attached to their several formulæ.

The Italian mathematicians contributed materially to the progress of the Higher Calculus. The Integration of a class of Differential Equations was proposed by Count Riccati, and solved in different ways by himself and the sons and nephews of the first Bernoullis. Manfredi¹ in 1707 gave a skilful solution of differential equations of the first degree; and in 1722, Vincent,

the son of James Riccati, pursued likewise the same road. A curious kind of problems was started by Fagnano,² another Italian Count, to determine those Elliptical or Hyperbolic Arcs, which have their difference expressed by an algebraical quantity. This led Euler, in 1756, to invent a very comprehensive method of determining particular integrals, that Lagrange improved in 1766 and 1769, and which he himself again simplified in 1778. Our ingenious countryman Landen has chanced to fall into a similar train, having in 1775 converted the formula for the rectification of the Hyperbola into another which includes two arcs of an Ellipse, together with an algebraical quantity. The rectification of the Ellipse has been since improved by Legendre and Ivory.

To obtain the differential of any integral expression, is always practicable; the great difficulty consists in reversing the problem, and finding the integral which corresponds to a given differential equation. Such equation must be rendered complete before the integration is effected. It was hence a question of much importance, to save unavailing efforts, by determining the conditions necessarily required for integration. Euler, as usual, was the first to discover the rule in 1736; but, only three years afterwards, it was without any communication produced by Fontaine and Clairaut. At a later period, Euler extended the conditions of integrability to the higher orders of Differential Equations, which Condorcet demonstrated with equal simplicity and elegance.

The Modern Analysis thus constantly advancing, received a capital extension about the middle of the last century, by what is termed the Calculus of Partial Differences, which applies with singular felicity to the solution of the most arduous and recondite physical problems. It would be difficult to communicate any distinct conception of this subtle doctrine to the uninitiated, but the object proposed may be stated generally as, the method of finding the function of several

¹ Born at Bologna in 1674—appointed professor of Mathematics there in 1698, and astronomer to the Institute of Bologna in 1711. He died in 1739, having produced several excellent works, in which his taste, learning, and science appear conspicuous.

² Born about 1690 at Senegaglia in the Roman State, where he died in 1760. His mathematical researches were published in two volumes quarto, at Pesaro, in 1750. He devoted much attention to the remarkable properties of the curves called Lemniscata.

variable quantities, from the relation merely of the Differential Co-efficients contained in their Complete Differential. The first specimen of this sort of Integration was given by Euler in 1734; but D'Alembert expanded the process in his Discourse on the General Cause of Winds, which appeared in 1749. It was likewise employed by both these illustrious mathematicians in the rigorous solution of the problem of the vibrating musical string, the position of any point in the harmonic curve depending evidently on the interval of time, as well as on the relation between the absciss and ordinate. But Euler left nothing unfinished. In 1762 he resumed the subject of Partial Differences, and gave a complete explication of its principles, embodying the mode of calculation by an appropriate Algorithm, in a memoir of the Petersburg Acts, entitled, Investigatio functionum ex data differentialium conditione. But still further progress was made by him in the third volume of his immortal work on the Differential and Integral Calculus, which came out in 1770. Euler, having then advanced to differential equations of the second order, digested the whole doctrine of Partial Differences into a clear and systematic form. Yet several important additions and improvements have been since contributed to this intricate subject by Lagrange and Laplace, by Nieuport and Trembley.

The last great accession to the Higher Analysis, and somewhat resembling the method of Partial Differences, is the Calculus of Variations. Of this fine theory, the first trace was shown by the rapid genius of Leibnitz, in his mode of differentiating a curve whose equation itself is supposed to undergo the minutest alteration. The controversies of the Bernoullis concerning Isoperimeters and Lines of the Swiftest Descent, augmented greatly the various resources of that kind of Analysis. But Euler, combining their discoveries, produced a new and direct mode of finding such Maxima and Minima, and compressed the solutions into a systematic form, in a distinct treatise published at Lausanne in 1744. This elaborate performance led the way for the simpler and most elegant and comprehensive doctrine created by the early genius of Lagrange, and invested by him with a commodious notation.

Euler, who was far superior to any feeling of jealousy, readily embraced this improvement, and bestowed on it the denomination of the Calculus of Variations, of which he explained the principles and application, after the most complete manner, in a tract appended to the third volume of his great work. The distinction between Differentials and Variations is rather subtle, yet may be easily conceived by considering the nature of the Parabola. When an ordinate shifts into a proximate position, both it and the corresponding absciss acquire Differentials; but if the Parameter suffer the minutest alteration, the whole trace of the curve will vary, and the infinitesimal mutation which the ordinate thence undergoes is termed its Variation. The algorithm adopted to denote Variations consists of the Greek letters instead of the Roman or Italic, which have been appropriated to the ordinary Differentials. The modes of Integration used in both forms are fortunately convertible.

The Integration of Differential Equations has at length, perhaps, nearly attained that degree of perfection of which it is susceptible. A multitude of expedients are devised for effecting the process in particular cases, but no general and direct method has been yet found. To integrate a differential equation of the first order composed of two variable quantities, is the problem originally known by the title of the Inverse Method of Tangents, and is commonly solved by the separation of the indeterminate members, which brings it to a question of Quadratures. Some equations consist of parts which, though not separately integrable, will yet admit of integration when combined. Others are integrated by the inverse method of factors; but the process by successive approximations may be regarded as in most cases the preferable mode. The skill of the analyst is chiefly displayed in the evolution of series the most converging.

Of the Theory of the Infinitesimal Calculus, various modifications have been offered during the course of the eighteenth century, which help to elucidate the subject by their contrast, and may be regarded as fine speculations, though they should lead to no material improvements in the practice. The notion of flowing quantities,

first proposed by Newton, and from which he framed the terms Fluxions and Fluents, appears, on the whole, very clear and satisfactory; nor should the metaphysical objection of introducing ideas of motion into Geometry have much weight. Maclaurin was induced, however, by such cavilling, to devote half a volume to an able but superfluous discussion of this question. As a refinement on the ancient process of Exhaustions, the noted method of Prime and Ultimate Ratios, or of the Relations of Vanishing or Evanescent Quantities, which Newton preferred in the Principia, deserves the highest praise for accuracy of conception. It has been justly commended by D'Alembert, who expounded it copiously, and adopted it as the basis of the Higher Calculus. The same doctrine was likewise elucidated by our acute countryman Robins,¹ who did not scruple to seize the occasion of making a coarse attack on the great Euler. Landen,² one of those men so frequent in England whose talents surmount their narrow education, produced, in 1758, a new form of the Fluxionary Calculus, under the title of Residual Analysis, which, though framed with little elegance, may be deemed, on the whole, an improvement on the method of ultimate ratios.³ To confer more consequence on his innovation, he contrived likewise a set of symbols, and applied his algorithm to the solution of different problems. But it never obtained any currency, and soon fell into oblivion.

The method of Analytical or Derivative Functions, which has since acquired such celebrity, was darkly anticipated by Dr Waring, one of the profoundest but most obscure of algebraical writers. This elegant theory was concisely sketched by Lagrange in 1772, and enlarged thirty years afterwards into a distinct work; the same subject having been treated very fully and somewhat differently by Arbogast in 1800. It is grounded chiefly on the inversion of Taylor's theorem, the co-efficients of the several terms of a Derived or Expanded Function involving the successive orders of its Differentials. But Lagrange deduces the formula from considerations purely analytical, and endeavours thence to erect a science on strict logical principles, entirely disengaged from ideas of Infinitesimals or of Vanishing Ratios. It may however be doubted whether, with all his ingenuity, he has attained that object, or gained, indeed, any real advantage. A mathematician so deeply imbued as that sublime genius with the spirit of calculation, would almost spontaneously regard a process of analysis in the same light as a train of legitimate reasoning. However satisfactory may seem the origin of Derivative Functions, if we examine it closely, we shall probably find the demonstration to rest merely on the strength of analogy, which surely is not the most conclusive sort of argument. But though the method of Derivations should not possess that logical superiority over the Fluxionary or Differential Calculus,

¹ Benjamin Robins, born of Quaker parents at Bath in 1707. He discovered early and powerful talents, which he improved with intense ardour, and soon emancipated himself from the trammels of his cold and narrow sect. Mathematics were his favourite study, but he enriched his mind by a course of extensive reading and the pursuit of general information. He became a successful teacher of those sciences in London, and turned his attention to their practical application. Having cultivated also the art of writing, he distinguished himself by several controversial pieces in science, and even assisted by his pen the opposition to Walpole's administration. He was now so well known as a clever writer as to be employed in correcting Walter's account of Anson's Voyage round the World; but finding the performance very poor, he re-composed the whole. The appearance of this celebrated work in 1748 secured the patronage of the Admiral, and he was appointed Engineer General to the East India Company. He sailed from England at Christmas 1749, and on his arrival in India began immediately the fortifications at Madras; but he fell a sacrifice to that baneful climate on the 29th July 1751. His various mathematical works are stamped with originality of conception, and composed in a clear, neat, and forcible style.

² John Landen, born at Peckirk in Northamptonshire in January 1719. Bred up to business, he showed an early talent for Mathematics, which he strenuously cultivated at his leisure hours. In 1762 he was appointed agent to Earl Fitzwilliam, an employment which he held till within two years of his death. Besides his separate publications, he communicated to the Royal Society, at different times, valuable papers on the most difficult parts of mathematical and physical science, all of them distinguished by depth, ingenuity, and powerful invention. It is only to be regretted that, perhaps for want of the collision of society, he indulged a dogmatic and pugnacious temper. He died near his native spot on the 15th of January 1790.

³ An example will show how Landen's operation might be simplified. Let it be required to find the ratio of the differential of $x^n$ to that of $x$. Assume $v$ a proximate value or state of the variable $x$; then by division $\backslash frac\{x^n\; -\; v^n\}\{x\; -\; v\}\; =\; x^\{n-1\}$ $+\; x^\{n-2\}\; v\; +\; \backslash dots\; +\; x\; v^\{n-2\}\; +\; v^\{n-1\}$, the number of terms being $n$. But it is evident that the closer $x$ comes to $v$, the nearer will each of these terms approximate to $x^\{n-1\}$. Wherefore, at the coincidence of $x$ with $v$, the differential of $x^n$ divided by the differential of $x$ is $x^\{n-1\}$, and consequently $dx^n\; =\; x^\{n-1\}\; dx$.

lus which its author fondly supposed, yet is the invention entitled to the highest praise for its beautiful perspicuity and its ready and most extensive application. We have only to regret that it has required a new system of characters, when the ordinary notation has become so fa-

miliar, and attained so great perfection. Such mutations, like the diversity of languages, may be deemed a serious evil, since they divert the attention to the mere accessories of learning, and retard or obstruct the acquisition of real knowledge.¹

SECTION II.

APPLICATE SCIENCE.

1. DYNAMICS.

THIS most important science may be considered as grounded on the Composition of Forces. Although the ancients could not fail to remark that the effect of two oblique forces is equivalent to the action of some intermediate force, yet the principle in its simplest form was first distinctly stated, after the revival of letters, by the famous painter Leonardo da Vinci,² who being likewise a skilful mechanic and engineer, was acquainted at that early period with the right mode of conducting experimental research. He showed the action of two perpendicular forces against a point to be the same as what is denoted by the diagonal of a rectangle, of which

they represent the sides. It would hence be easy to derive the general proposition of oblique forces; but this beautiful property was suffered to remain buried amidst an unpublished chaos of ingenious hints and multifarious projects. Near a century more elapsed, till it was re-discovered in its full extent by Stevinus, a Flemish engineer, who applied it successfully to the explication of the common mechanical powers. About the same time, Galileo, exploring the acceleration of falling bodies, employed the principle of the composition of forces in determining the paths described by Projectiles. Another century nearly passed away before Huygens enrich-

¹ As an introduction to this study, the essay of Robins on Prime and Ultimate Ratio may be read with profit. A very clear exposition of the principles will be found in the Principiorum Calculi Differentialis et Integralis Expositio Elementaria, a thin quarto published by L'Huillier of Geneva, at Tübingen, in 1795. But the most complete and perspicuous treatise of the Higher Analysis unquestionably is Euler's great work, comprised in seven volumes quarto, printed at different times between the years 1748 and 1797, and consisting of the Introductio ad Analyses Infinitorum in two volumes, one volume of the Calculus Differentialis, three volumes of the Calculus Integralis, to which have been added two posthumous volumes. The treatise composed in Italian in two small quarto volumes, by the late Professor Paoli, though entitled only Elements of Algebra, is of a more comprehensive import, and remarkably clear and elegant. The German language has very full and able articles on every part of the Higher Calculus, in Klügel's Mathematical Dictionary. The large French work of Lacroix is valuable for its contents, but deficient in clearness and elegance. His Abridgement seems very obscure and unsatisfactory. The neatest and most concise treatise on the Higher Calculus in the French language is that of Bossut, in two octavo volumes. In English, the Treatise of Fluxions by Simpson may still be studied with profit. Of Maclaurin's large work, the first volume contains an excellent account of the ancient method of Exhaustions, and the second volume is valuable for its physical disquisitions. Our later publications on Fluxions are numerous, but have not acquired such a character as might entitle them to any particular notice.

² So called from the name of the place where he was born near Florence in 1452, being the natural son of a notary. Bred in an age and country so transcendent in art, he soon distinguished himself by originality of conception, harmony of design, and elegance of execution. But having likewise ardently cultivated the Mechanical Sciences, he was appointed director of an Academy of Architecture and Engineering by the Duke of Milan, in which capacity he constructed the canal of Mortesana, to supply that city with water. But the political convulsions of that period drove him back to Florence, where he resumed his former profession. Near the close of his days he accepted from Francis I. an invitation to France, where he languished a few months, and expired in the arms of that monarch in 1520. His numerous sketches of projects and machines, with his observations (written backwards), after having been carried to Paris, are now deposited in the Ambrosian Library at Milan, or among the collections at Florence. A digested abstract of these would form a curious monument.

ed Dynamics with his elegant theorems of the measure of Centrifugal Force in the case of circular revolutions. But Newton had already, and without receiving any communication, extended Centrifugal Forces to curve lines in general, and had thence deduced the elliptical orbits of the planets. The Dutch philosopher gave a further expansion to Dynamics, by his beautiful theory of the oscillation of pendulums.

John Bernoulli¹ simplified the science of Equilibrium or Statics, by proposing the principle of Virtual Velocities, which Varignon adapted to the several mechanical powers. This consists in the fundamental property, that any system at rest being supposed to suffer a minute derangement, if the velocity of each body, estimated in the same direction, be multiplied into its mass, all those collected products will extinguish each other. Virtual velocities might be therefore viewed as an extension of the principle of the lever.

But the same inventive mathematician afterwards produced another radical property, which he styled the Conservation of Living Forces, and which had been suggested by the ingenious hypothesis employed by Huygens for discerning the centre of oscillation. The phrase expressed that permanence, through all the gradual changes

of any system of connected bodies, in the aggregate of the products of their masses into the squares of their velocities. This important principle abridges the solution of various dynamical problems, and was embraced by Daniel Bernoulli, the son of its propounder, as the basis of his able and complete theory of Hydrodynamics, published in 1738.²

Herman,³ a disciple and countryman of the Bernoullis, and imbued with the taste of the Leibnizian school, published, in 1716, under the title of Phoronomia, the first regular treatise on Dynamics, drawn up in the analytical or algebraical form. It was clear and compact, but incomplete, and not distinguished by much novelty of conception.

The task of composing a work on Dynamics, full and original in every part, devolved on Euler, who exerted all the resources of his penetrating genius, aided by intense and indefatigable application. This capital performance came out in 1736, in two quarto volumes, entitled Mechanics, or the Science of Motion, and exhibited the completest and most elaborate body of analytical investigation that had been yet produced.

James,⁴ the elder Bernoulli, in his mode of treating the problem of the Centre of Oscilla-

¹ John Bernoulli was born at Bâle on the 7th of August 1667, being consequently above twelve years younger than his brother James. He possessed equal genius, but had a keener and more pertinacious temper. Designed by his father to be bred a merchant, inclination led him to the culture of letters, which, from the example and instructions of his brother, he afterwards exchanged for the pursuit of mathematics. The force of his inventive faculty he soon displayed in the resolution of those arduous questions which then agitated the scientific world. The spirit of chivalry had about this period passed over to the learned, and the practice of sending mutual challenges incited application and fomented the talent of discovery. Along with Huygens and Leibnitz, he was the first in solving the problem proposed by his brother, to determine the nature of the Catenarian Curve, or the flexure formed by an equable chain suspended from both ends. Having completed the usual course of education, and turned his thoughts in 1693 to the medical profession, he took a degree in physic. He now travelled into France, and spent some time at the country-seat of the Marquis de l'Hôpital, whom he taught the New Calculus. Proceeding next to Holland, he was appointed in 1695 Professor of Medicine and Mathematics (for these sciences were strangely conjoined) in the University of Groningen. The flame he there acquired by his lectures and writings appears to have awakened the jealousy or bigotry of the Calvinistic clergy, who brought a serious charge against him, for impugning the doctrine of the resurrection in a Thesis maintained by one of his scholars, and composed or corrected by himself, which represented the body as in a state of continual mutation, not only the liquid, but even the solid parts being successively absorbed and renewed, and no particle of the composition remaining after a very short term of years. The coolness of the magistrates, however, effectually silenced the ignorant clamour of the Synod, and Bernoulli was pressed to remain in Holland, with flattering offers of promotion. But Bernoulli preferred his native city, whither he was spontaneously invited to succeed his brother. In this honourable station he spent the rest of a long life in extending, by the activity of his genius, every branch of Mathematical and Physical Science. He corresponded with all the foreign societies, and the numerous prizes he bore away proved a source of income. His treatise on seamanship came out in 1714, and his dissertation on the elliptical figure of the planets appeared in 1730. But his various philosophical papers were carefully collected into four quarto volumes, and published in 1742 at Geneva and Lausanne. He died full of years and of glory on the 1st of January 1748.

² These two properties—Virtual Velocities and Conservation of Living Forces—are easily and directly proved from the elementary principle of the composition of forces. See Elements of Natural Philosophy, p. 134, 135.

³ Born at Bâle in 1678. Having been six years Professor of Mathematics at Padua, he accepted an invitation in 1724 to Petersburg; but not long afterwards returning home, he became Professor of Morals and Public Law in his native city, where he died in 1733.

⁴ James Bernoulli, the first of a most illustrious race of mathematicians, was born on the 27th December 1654, at Bâle, and educated in the seminary of that frontier city. His father wished him to follow the profession of divinity, but the im-

tion, struck out a more direct and ingenious procedure, by which the forces exerted are resolved into separate portions, and an equality obtained between their accelerating and retarding influence. This fine conception, after the lapse of forty years, appears to have prompted D'Alembert¹ to the discovery of the simple and general principle on which he framed his Treatise of Dynamics in 1743. In every system of bodies acting mutually, their several movements at any instant of time may be decomposed into two portions, one which is retained in the next instant, and the other spent; and since an equilibrium must obtain among the lost motions, an expression is hence derived for the motions that are preserved. The most intricate questions in Dynamics were thus reduced to mere statical problems, and solved constantly in the same easy and uniform way. Maclaurin's method of expounding forces by co-ordinates facilitated still further the application of this principle, which D'Alembert in 1744 and 1752 extended like-

wise to Hydrodynamics, comprehending both the motion and the resistance of fluids.

The slightest incident has often led to the most interesting discoveries. Few would suppose that the spinning of a common top involves the most difficult conditions in Dynamics, and serves to explain all the intricacies of the planetary vertiginous motions. Segner, a celebrated professor of mathematics at Göttingen, and afterwards at Halle, and a man of original and independent mind, published, in 1755, a short dissertation, with the modest title of Specimen Theorice Turbinum, wherein he showed that every body having a determinate figure, which after combined impulsions is abandoned in free space, will, besides its progressive motion, perform simultaneously, and without the smallest interference, a constant and uniform revolution about each of three principal axes mutually perpendicular, and passing through the centre of gravity. These axes of rotation possess some beautiful but abstruse properties, which the

pulse of native genius burst through constraint, and hurried him into the ardent pursuit of geometry and astronomy, which he cultivated in private. Alluding to this circumstance, he chose for his device Phœton driving the chariot of the Sun, with the motto, "Invito patre sidera verso." He began his travels for information in 1676, visited Geneva, traversed France, and eagerly sought the society of men of learning and science. In 1680 he descended to the Netherlands and Holland, passed over to England, and frequented the philosophical meetings in London. After his return home in 1682, he gave a course of mechanical lectures, in which he exhibited the new discoveries. Being appointed Professor of Mathematics in 1687, the fame of his vast attainments drew crowds of strangers to the academy. His genius was a torch of illumination, and the most elegant simplicity distinguished all his profound inventions. In conjunction with his younger brother, he not only found out the secret of the Differential Calculus, on which Leibnitz had given a very short and obscure essay in the Leipsic Acts, but unfolded the principles with such superior talent and address, that the German philosopher had the generosity to regard him as entitled to share in the honour of the great discovery. He pursued the New Calculus with extreme ardour, and applied it most happily to a variety of arduous investigations: He greatly improved the Method of Series, extended the Theory of Curve Lines, approximated to the rectification of the Parabola, and discovered many beautiful properties of cycloids, epicycloids, and spirals. With the logarithmic spiral, which he discovered to renovate itself by evolution, his fancy was so much struck, that, in imitation of Archimedes, but in allusion to the sublime prospect of resurrection, when he should "shuffle off this mortal coil," and rise like Phoenix from her ashes, he directed with a fine enthusiasm this curve to be inscribed on his tomb, with the Ovidian line, "Eadem mutata vicinior."

But such mighty strides required proportional exertion of intellect, and this intense and unremitting application undermined the constitution of Bernoulli, and carried him to a premature grave. He sunk under a slow fever on the 16th August 1703. He had nearly finished his great work on the doctrine of series, and the calculation of annuities and chances, entitled De Arte Conjectandi, which was published in 1713. His other miscellaneous productions were not collected till 1744, when they came out in two quarto volumes at Geneva.

¹ Born at Paris 16th November 1717, the fruit of an illicit amour with a lady of high rank, who exposed her infant; but the father, listening to the calls of nature, settled a competent annuity. He was educated by the Jansenists in the Collège de Quatre Nations, where he gave early tokens of capacity and genius. After finishing the usual course, he returned to the family of his nurse, with whom he lived forty years in great simplicity, cherishing his independence, and devoting himself wholly to mathematical pursuits and the cultivation of general literature. Such was the progress he made, that he was admitted, at the age of 24, a member of the Academy of Sciences, and two years afterwards he produced his Treatise on Dynamics. The germs of the Calculus of Partial Differences appeared in his Dissertation on Winds, which obtained the prize from the Academy of Berlin in 1746. His new dynamical principle he applied to the investigation of the Earth's vertiginous motion in 1749, and to the theory of the resistance of fluids in 1752. About this time he engaged with Diderot in the compilation of the Encyclopédie, wrote the famous introductory discourse, and furnished several capital articles, especially in mathematical science. But he now sought to distinguish himself likewise in philosophy and literature, and published a variety of tracts, which gave occasion to controversy and violent opposition. But this hostility carried his reputation to a higher pitch. He corresponded with the great Frederick of Prussia, but refused the flattering offers of that Monarch and of the Empress Catharine of Russia. The literary honours he enjoyed at home were more agreeable to his taste. Having become secretary to both Academies, he was esteemed the head of the philosophical body at Paris. His works are numerous in various departments. He was a close, accurate, and original thinker; and his style partakes of the same qualities, neat, hard, and precise. He died on the 29th October 1783.

great Euler investigated with that profound talent and luminous method which distinguish all his productions. The curious discovery of Segner drew likewise the attention of D'Alembert and Lagrange, and deserves to be regarded as one of the most important additions ever made to the science of Dynamics. It has contributed signal-ly to the advancement of physical astronomy, and thrown a clear light on the theory of the nutation of the earth's axis, and of the precession of the equinoxes.

Dynamics might seem to have nearly reached its perfection, when Lagrange in 1788, by combining the principle of D'Alembert with that of virtual velocities, converted the whole into an absolute analytical science. His procedure was to refer the efforts of every particle of a moving system to three mutual perpendiculars, and thence derive three several differential equations, which being integrated, would give the final solution of the problem. But to discover a general form of integration is the great difficulty which still remains. All attempts to remove this obstacle have hitherto failed, and we are reluctantly obliged to remain satisfied with merely partial and approximative methods. It must be confessed that the subtleties of the Higher Analysis have been often displayed with very little effect, and that the most celebrated mathematicians, in resolving some arduous questions, especially those involving physical considerations, have still not advanced beyond the vague and imperfect results which sagacity had with very slender aid already attained. Analysts affect too much the air of generalizing, and deceive themselves with expectations which are never destined to be realized. It is thus that Nature appears to set limits to the soaring of human genius.

Statics and Dynamics, though really grounded on observation, depend on principles so simple, and apparently so congruous to reason,

that they are often derived from mere abstract considerations, or the consonance of geometrical analogies. But the constituent properties of bodies lie beyond the reach of anticipation, and can be explored only by experiment. Every body may be divided into parts, and each of these parts again subdivided; nor has any limit been yet found to this repeated process of decomposition. Each separate portion likewise retains all the properties of the mass from which it was detached. Hence the corpuscular elements, or the materials that form bodies, have precisely the same character as the compounds themselves. But though imagination represents an interminable series of subdivisions, such cannot be the actual constitution of nature, which is always defined by number and measure. We may therefore infer the existence of certain ultimate portions or atoms, endowed with the very fewest qualities, but which by their various combinations constitute the particles of matter, or form the corpuscular composition of bodies.

Every body whatever can have its dimensions contracted or dilated by the application of force; and the extent of this effect is limited only by our circumscribed power. The atoms must hence be held together by some mutual appetite, whose intensity varies with the intermediate distance. This tendency is evidently repulsive within a certain interval, and attractive beyond it. But the essence of matter consists in impenetrability, and therefore the repulsive force must increase from approximation above any finite measure. The inherent power, thus variously changing at near intervals, will merge at remote distances into the great law of attraction.

Such is a brief outline of the reasoning by which the ingenious Boscovich¹ in 1759 supported his beautiful Theory of the Constitution of the Universe. It was partly suggested perhaps by the modification first proposed by Clairaut on the Newtonian System, which assumed

¹ Roger Joseph Boscovich, born in the small republic of Ragusa on the 11th of May 1711; received his elementary instruction under the Jesuits, who noticed the promising talents of the youth, and sent him to complete his education in their college at Rome. He soon attained great eminence in erudition and science; and became Professor of Mathematics and Astronomy successively at Rome, Pavia, and Milan. In 1753 he performed the mensuration of a degree in the Papal States, and during the intervals of that laborious occupation he composed in three octavo volumes a treatise of elementary mathematics, remarkable for its simplicity and elegance. He was afterwards employed in several public negotiations, and the claims of his native state brought him to London, where he spent some time, mingling in the societies of the

the attractive power to be composed of a small portion of the inverse cube of the distance, joined to the ordinary term of the inverse square. But the idea of the Ragusan Philosopher was far more general, exhibiting this power as what algebraists call a function of the distance. He rendered the conception still clearer by means of a primordial curve stretching to indefinite distance, of which the ordinates on either side of the axis indicate the corresponding attractive or repulsive force. The extended horizontal branch includes the solar attraction, the alternate convolutions of the curve mark the proximate changes of cohesion, and the final descending branch, by its continual approach to a perpendicular, intimates the insuperable repulsion which prevents the collapse of matter. The intermediate alternations of the curve are unknown; but as in their progress they cross the extended absciss from above or below, they mark the limits of instability or stability. In this way the different constitution of solid, liquid, or gaseous substances may be represented. The atoms being likewise variously grouped, must by their blended action produce that immense diversity of effects which animate the spectacle of the external world.

It is to be regretted that Boscovich obscured his fine theory by an infusion of scholastic metaphysics: He maintained that those atoms or physical points had no magnitude, or differed from mathematical points only in being endowed with primary force. But this difficulty, which proceeds rather from our modes of conception than from the actual state of things, might be obviated, by supposing the atoms to have real dimensions, though far smaller than any assigned measure. The primordial curve can likewise be presumed to advance, not by insensible gradations, but by a succession of most minute steps. This slight modification would accord with the tendency of atoms to collect into certain regular

groups, a property which seems deducible from the principles of crystallography and the theory of definite proportions. We thus catch a glimpse of the recondite composition of corpuscular elements.

The Boscovichian Theory reduces the investigation of the properties of bodies to the utmost simplicity, and may be regarded as a very happy extension of the great law of attraction. Why it has not been received with more favour, especially on the Continent, might provoke some surprise. Being probably considered as too speculative, it has never been studied with the attention it deserved. Purged of its antiquated metaphysics and crude chemical notions, it would form the best introduction to general physical science.

A portion of that beautiful theory, however, is now, in a modified shape, very commonly received, especially by the continental philosophers, who view the constituent molecules as held together by their mutual attraction opposed to the repulsive energy among the attached particles of heat. It is a supposition which readily explains the general properties of bodies, and serves to elucidate their structure and internal operations.

The Boscovichian System assumed the general principle, that every substance whatever is capable of contraction and dilatation. But this was not admitted at the time in its full extent, the noted experiment of the Florentine Academicians seeming to have established the absolute incompressibility of water. The conclusion however was too hastily drawn; for though the liquid included within a hollow sphere of gold resisted the blows of a hammer, and burst the shell, yet the celerity with which it spirted through the crevice, only displayed the elastic force resulting, as Bellegradi rightly observed, from an expansive effort to recover its previous condition. But the actual condensation of water and other liquids under pressure was first shown

learned, and composed his Poem on Eclipses, which exhibits a neat view of the Newtonian Philosophy. Thence he went with an embassy to Vienna, where he observed the transit of Venus in 1769. On the suppression of the Jesuits by Pope Ganganelli in 1772, he was invited to France, and appointed Director of Optics. But the Parisian philosophers derided his school of theology, and undervalued his scientific talents; and after a residence of ten years, he returned in disgust to Italy, and printed in 1784 his Opera in five quarto volumes at Bassano. Chagrin and vexation in the decline of life preyed on his sensitive mind; and he sunk by degrees into a deep melancholy, which ended in utter and hopeless insanity. But from this deplorable state he was relieved by death on the 13th of February 1787. Boscovich possessed a fine geometrical taste, joined to considerable powers of invention in a variety of subjects.

experimentally in 1762 by our ingenious countryman Mr Canton,¹ who measured the effect by means of a sort of large open thermometer, containing the fluid in a very capacious glass ball, which terminated in a long capillary tube. Professor Zimmerman of Brunswick in 1779 carried this condensation much farther, having introduced water into the cavity of a brass cannon, and compressed it by a force exceeding the weight of 300 atmospheres. Still the popular and elementary treatises, neglecting such decisive facts, continued to repeat with complacency the paradoxical assertion, that water retains the same volume under every degree of compression. Nor

was the public recalled from its careless acquiescence, till the mention of some experiments of enormous compression performed at great depths in the Atlantic Ocean. But the celebrated Oerstedt of Copenhagen has lately made an elegant improvement of Canton's method, by which the condensing power exerted on water by a progressive pressure as high as thirty atmospheres, is most easily and accurately measured. By a further extension of the apparatus, the contractility of various solid substances could likewise be readily ascertained, which might lead to the detection of facts interesting in the economy of Nature.

2. HYDROSTATICS AND PNEUMATICS.

THE theory of the Pressure and Equilibrium of Fluids is readily deduced from the principles of Statics, joined to the consideration of the internal mobility of the fluid particles, or their absolute indifference to maintaining figure. But the ordinary demonstration seems incomplete, without taking likewise into view that Compressibility, which belongs in a more eminent degree to fluid than even to solid substances. The particles retreat from the action of pressure, contracting on all sides, till the repulsion occasioned by their mutual approach becomes a sufficient countervailing force, diffusible in every direction. Hydrostatics is therefore a complete science. The conditions of equilibrium had been discovered by Archimedes, but Stevinus traced the effects of fluid pressure, which Pascal afterwards explained.

Hydrostatics, or rather Hydrodynamics, which treats of the Motion of Fluids, is a subject of far more difficult investigation. Torricelli gave the first traces of this theory, which exercised the sagacity of Newton, and yet required considerable correction. The fundamental problem is to find the discharge of water through a small hole in the bottom of a cylinder filled with the fluid. This efflux amounts only to about five eighth parts of the first emission; and to explain the discrepancy, the immortal author of the Principia had recourse to the supposition of a cataract, or funnel-shaped conoid, by which the various streamlets bend their course towards the orifice. But the theory was not investigated in a more rigorous and systematic manner, till half a century afterwards, when Daniel Bernoulli² published his important treatise of Hydrodynamics.

¹ John Canton, one of our most diligent, skilful, and accurate experimental philosophers, was born at Stroud in Gloucestershire in 1718. His father, a broad-cloth weaver, bred him to the same trade; but the youth, incited by the lessons of an elementary teacher, had from his earliest years imbibed a taste for mathematical studies, and continued to devote every leisure moment to those fascinating pursuits. The ingenuity he showed in constructing sun-dials drew the notice of his more intelligent neighbours, who kindly fostered his efforts, and lent him instructive books. Nothing has contributed more in England, not only to spread general information, but to awaken latent genius, than the circulation of popular compendiums and dictionaries of science. With these aids the ardent application of Canton made such rapid and striking progress, that his patrons counselled him to quit the loom and repair to the metropolis, the great mart of talent. In 1738 he was admitted assistant in a mathematical school in Spital Square, and afterwards became partner and successor in the concern, which he conducted during the rest of his life. Without aspiring to the higher walks of science, he was active in the search of knowledge. He devised new experiments, and carefully repeated those in vogue; thus extending our acquaintance with Electricity, Magnetism, and some parts of Physical Chemistry. But his career was short, for he died of a dropsy in 1772.

² Daniel, son of John Bernoulli, born at Groningen 9th February 1700, conjoined the hereditary talent with a mild and balanced disposition. He was only five years old when his father returned to Bâle, where he prosecuted his education, and enjoyed the peculiar advantages of paternal aid, and the inciting emulation of his brother Nicholas. Being destined for the profession of physic, he sought to improve his medical attainments by resorting to the schools of Italy. There he published in 1724 his Exercitationes Mathematicæ, and already acquired such reputation that he was pressed to become director of an academy projected at Genoa. This offer, however, he declined, but soon afterwards accepted, along with his

The subject was next discussed in 1744 by D'Alembert, and with the depth and originality which distinguish all his productions. This celebrated philosopher, eight years thereafter, advanced to the arduous investigation of the Resistance of Fluids, and displayed the resources of his invention, in evolving a variety of ingenious formulæ. Euler likewise, by a series of repeated progressive essays between the years 1755 and 1772, successively simplified the theory of Hydrodynamics, and reduced the doctrine into a strict analytical form, the whole comprised in two differential equations of the second order. Lagrange followed next, and discussed in 1781 the chief difficulties which encumbered the theory; and five years thereafter, he simplified still further the mode of investigation in his Mécanique Analytique.

But it is rather mortifying to confess, that these refined speculations are of little avail in the practice of the science. It is indeed impossible, by any effort of analysis, to embrace all the physical conditions concerned in such an intricate system of internal corpuscular motions. Assumptions become necessary for the sake of abbreviation, and the process of integration itself is seldom if ever complete. The results drawn from such laborious researches are, after all this parade of rigid accuracy, to be viewed as only near approximations, which could often be attained by much easier means. It would therefore be preferable in many cases to study the physical relations more closely, avoiding the abuse of abstract calculation, and to rest satisfied with arriving at the conclusions by a sort of balance of errors. In the sober application

of analysis, Daniel Bernoulli was singularly happy, guiding his steps always by the light of experiment and observation. And, since the result of the most elaborate investigation is very seldom better than a simple approximation, it seems more judicious in such researches to lay the chief stress on the estimate of physical principles. After the most strenuous efforts of genius displayed during the eighteenth century, the theory of the motions of fluids has really not arrived at more precise conclusions than those first assigned by the penetration of Newton. Nay, so far from reaching any higher degree of perfection, some of the recondite speculations of Lagrange and Poisson, those particularly relating to the extent and celerity of waves, and the general undulatory commotions in fluids, involve consequences which are palpably at variance with the known phenomena of Nature. In such complicated investigations, the safest mode of proceeding is to follow the example of astronomers, who determine the co-efficients of the several analytical formulæ, by the comparison of observations. The simplest and most elementary theory is hence rendered subservient to practical use.

The principles of Hydrostatics constitute the grounds of Naval Architecture, which embraces the theory of the Construction and Sailing of Ships. To this very important branch of science, Euler turned his mighty talents, and produced in 1749 a large and elaborate work, containing a series of ingenious propositions, but which unfortunately are not the best adapted to actual practice. About the same time the celebrated Bouguer,¹ combining geometrical skill with the stores of experimental knowledge, ex-

brother, a more flattering invitation from Peter the Great of Russia, to hold a prominent station in the Institute then founded in the new capital. Nicholas soon sunk under the severity of that northern climate, but Daniel remained till 1733, when he was appointed to the professorship of physics at Bâle. In quitting Russia, he did not forget to recommend for successor his countryman Euler, who, though seven years younger, had been associated in some degree as his fellow-student. Bernoulli during the rest of his life pursued at home his philosophical investigations with unwearied assiduity and eminent success. He gained or shared most of the prizes given by the different learned bodies over Europe. In 1734 he divided with his father the honour of solving the very difficult problem of the inclination of the orbits of the planets. But his great rival was Euler, nor did their frequent collision ever cool the warmth of mutual friendship. Bernoulli long enjoyed the high respect due to his talents, his virtues, and many amiable qualities. He died on the 17th March 1782.

¹ Peter Bouguer, born on the 10th of February 1698, at Crouse in Lower Brittany, where his father was royal professor of Hydrography, and had shown abilities by publishing an esteemed treatise of Navigation. Under such guidance he imbibed a very early taste for mathematics, and having distinguished himself at the Jesuits' college of Vannes, he was, after a strict public examination, found qualified to succeed his father when only fifteen years old. Notwithstanding his immature age, he discharged the duties of his office with dignity and success. But he found leisure also to pursue his private studies, and in 1727 he gained a prize of the Parisian Academy for a paper on the Masting of Ships, and obtained other prizes in the following years. His Treatise on the Gradation of Light appeared in 1729, and established his high reputation. His celebrity always increasing, he was appointed in 1733 to join the famous scientific expedition sent out to

mined the subject in its details, and reduced the theory into a form remarkably simple and elegant. When a body floats on the surface of water, it is held in equilibrium by the action of two equal and opposite perpendicular forces passing through its centre of gravity, and the centre of buoyancy or that of the fluid displaced by immersion. Its weight draws it downwards, while the buoyant effort presses it directly upwards in the line of support, the vertical position of those centres being essential to the stability of flotation. The centre of buoyancy does not remain constant, but shifts with the inclination of the natant body. The sustaining force may be conceived to act at any point in its direction, and consequently where the line of support crosses the axis. On this point of concurrence, Bouguer bestowed the appropriate name of Metacentre, as the limiting position which determines the conditions of flotation, there being stability whenever it stands above the centre of gravity, but the contrary if it falls below. This useful theorem he had invented in 1746; and nine years afterwards was published his investigation of the several motions of ships under the impulse of wind. That inquiry rested too much, however, on the ordinary resistance of fluids, which needs material corrections. It is not very difficult to measure the primary shock of the fluid particles; but, since they accumulate on the front of the body while it advances, the most arduous task is to estimate the force expended, in turning them aside into the general mass, and restoring the diffuse quiescence. To include such multiplied conditions, which are still essential towards a correct and complete solution of the problem, would evidently transcend the powers of the most complex analysis. The utmost that can be effected is only an approximation, which may be easily obtained by adopting a simplified hypothesis, and correcting the terms from observation.

The most complete set of experiments on the resistance of water in narrow canals was made by Bossut between the years 1766 and 1775, and forms the groundwork of his valuable theoretical and practical treatise on Hydrodynamics. It hence appears that water has its celerity reduced to the tenth part, by flowing through a smooth leaden pipe whose length is four thousand times its diameter, or that every particle of the fluid has its motion extinguished and successively renewed ten times during the passage along the extended cavity, by grazing against sides and again relapsing into the current. If in this case, therefore, the whole pressure were distinguished into an hundred equal parts, one of these alone generates the velocity, the remaining ninety-nine parts being expended in merely surmounting the impediments to the flow. A similar consideration was made the basis of Practical Hydrodynamics by Dubuat, who in 1786 gave simple and easy formulæ, for determining the discharge of pipes, canals, and rivers.

Smeaton, our celebrated engineer, performed a series of experiments by means of small models in 1759, on the action of water and wind against the floats or sails of mills. The results obtained on such a narrow scale have, notwithstanding their unavoidable imperfection, been adopted into general practice. Other later observations made by Banks have likewise gained credit among our artisans. The most skilful experiments however of that kind were instituted by Eytelwein of Berlin in 1799. But the recent observations made at Fahlun in Sweden are decidedly the completest and most scientific which have been yet performed.

The usual method of investigating the resistance of Fluids is to estimate the momentary shock or impact of their particles against the anterior surface of the penetrating body. But a more complete and elegant solution may be derived, from considering the pressure caused by

measure a degree of the meridian under the equator. He spent ten years on the lofty chain of the Andes, prosecuting with unabated ardour a variety of new, ingenious, and important researches. On his return to Europe, he poured forth a flood of most interesting information in most branches of physical science. His last works related to the construction and sailing of ships. Incessant labour and intense application had worn out his constitution, and he expired on the 15th of August 1768. Few persons have contributed so largely to the promotion of Natural Philosophy as Bouguer, who united the rare qualities of an able mathematician and a skilful and accurate experimenter. Like studious persons little conversant with the world, he is alleged, however, to have had the misfortune of a jealous and irritable temper.

the incessant accumulation, and therefore condensation, of the medium in the track of the passage through it. There is likewise another retarding force occasioned by the constant dilatation of the fluid in the rear of the advancing body. In the case of the gaseous media, this impediment often rises to a great amount. When the velocity of penetration, for instance, through common air exceeds 1350 feet each second, a vacuum is formed behind the missile, which thence suffers a retardation equal to the weight of a whole atmosphere. This consequence was first remarked by Robins in the flight of cannon-balls, and it completely deranges the parabolic theory. He was mistaken however in restricting the effect to such extremely swift motions; for the same consideration applies in every degree of rapidity, the condensation and corresponding dilatation of the air close to the anterior and the posterior surfaces of the ball being proportional to the square of its celerity. Hence the measure of retardation is always affected by the shape of the rear, as well as of the front of the projectile.

Robins employed an ingenious method of determining the impulse of balls, by firing them against a heavy loaded pendulum, whose deflection correctly indicated the momentum thus communicated; and by intercepting the flight at different distances, it was easy to discover the loss of velocity, and consequently the resistance opposed by the air. Borda, assuming a much lower scale of velocities, followed a different mode in the valuable set of experiments on the retardation arising from fluids which he performed between the years 1763 and 1769. He particularly noticed the influence of the figure of the hinder part of a moving body in modifying the resistance of the medium. Dr Hutton adopted the Ballistic Pendulum of Robins in his most extensive series of experiments executed with great circumspection on Woolwich Common during the years 1790 and 1791. It comprehends the whole range of velocities, from 5 feet in a second to 2000. The resistance appeared to increase in a

ratio faster than the square of the velocity, and therefore Hutton proposed to increase the index 2 by a small variable addition. This modification however is not conformable to the physical principles. Coulomb has shown, by his very delicate Balance of Torsion, that, in the case of extremely slow motions, the resistance of fluids is proportional to the velocity simply. The complete expression of resistance therefore includes probably, besides the second power, likewise the first and third, of which the co-efficients are to be found by a scrupulous comparison of the results. More precise experiments are still wanted, to improve the science of Gunnery.

The flow of air through a small orifice suffers nearly the same loss as water, or delivers only about five eighth parts of the measure indicated by theory. But, in passing through long trains of pipes, the several gases appear, from some recent though very limited experiments of Girard, to encounter proportionally still greater obstruction; insomuch that a stream of air would require a pressure exceeding tenfold its emergent force, to enable it to work its way through a pipe whose length is 180 times the diameter.¹ It is a question hitherto overlooked, but of real importance in the economical distribution of pipes for conveying coal or oil gas.

Water and other liquids, when heated, are observed to flow more freely from a very narrow aperture or through a long capillary bore, owing evidently to increased fluidity or the greater internal mobility of their particles. From a very slender syphon pure water near boiling will drop six times faster than on the verge of freezing. But air seems not affected in the same way: If it be forced by a certain pressure through a long capillary tube of glass laid horizontally in a water bath, the current, so far from accelerating, will become retarded by the application of heat. This discrepancy is occasioned by the dilatation of the fluid, which enfeebles the rate of discharge.

The progress of chemistry during the eighteenth century has brought us acquainted with

¹ The general complexion of the fact had been accidentally noticed before at Wilkins' great iron-works in Wales, where the nozzle of the bellows of a blast furnace being inserted into the one end of a train of large pipes, the efflux from the other end scarcely affected the flame of a candle.

other kinds of air or gases besides the atmospheric. But their mechanical properties are not so easily determined. The very inferior density of hydrogen gas was first discovered in 1766 by Cavendish, who made it ten times lighter than common air. By some late experimenters it has been represented as even sixteen times lighter, when carried to the highest state of purity. This conclusion, however, appears very doubtful, since it involves hypothetical considerations. It is very difficult to detach the latent moisture and to weigh accurately the dried gases contained in large glass balloons. This experiment, though quite practicable, remains still to be performed; and there is reason to suspect that the alleged lightness of hydrogen gas has been much exaggerated. But Pneumatic theory suggests a method of ascertaining the relative density of any gas, which is at once most ready, and susceptible of very considerable accuracy. It is only to observe the time of a given discharge of the fluids, from a small orifice, and under the action of a certain pressure; when the density will, in like circumstances, be inversely as the square of the number of seconds elapsed.¹ But the celerity of the flow may be discovered indirectly by another mode, which is practised with the greatest ease and nicety by any person possessing a fine musical ear. He needs only to send the gas through a detached organ pipe, and distinguish the precise note which it yields, or find another pipe in unison with it; the inverse subduplicate ratio of these lengths will express the relative densities of the fluids. This suggestion deserves to be pursued.

When air is thrown by pressure through a small round hole, either into free space or into a close vessel with a wider exit, it must evidently spread in diverging streamlets, and suffer a certain degree of rarefaction. The radiating discharge of a fluid thus involves a principle,² which explains a number of curious facts. Water rushing in a foaming current along a pipe or confined channel, leaves a partial va-

cuity at the sides, from which a small inserted syphon would draw a stagnant liquid to a considerable height. This was particularly noticed in 1795 by Venturi, who termed the property Lateral Action. On the same principle depends the suspension and play of a small ball above a jet of air from hydraulic bellows or a condensing engine. Hence likewise an explication of a seeming paradox in the action of fluids, lately considered by the ingenious Hachette. If a pipe bent directly downwards swell below into a cone with an horizontal base, encircled by a narrow rim, and pierced with a central hole, a strong current of air issuing through that aperture, but having a circular plate fitting closely with a weight appended, so far from blowing away the water, will draw it forcibly and support the load. The sheet of air between the opposite surfaces being kept rarefied by its diverging streamlets, the external atmosphere presses strongly upwards. The experiment is reducible to the simplest form; for a common tobacco-pipe held inverted, will by mere blowing be made to suspend a card from its bowl. The safety-valve therefore of a high pressure engine may not always afford the expected protection; under peculiar circumstances it will remain attached at a very small distance, allowing only a partial escape of steam in very thin diverging streamlets.

The like effect is produced in the expanding discharge of water from a narrow circular outlet. Here the streamlets dividing from the centre, draw in air and keep it rarefied by their rapid divergence. The adhesion fails with a rectangular aperture, the streamlets then proceeding in parallel lines. On a modification of the same principle depends an elegant experiment of Ampere. If water be projected horizontally from the bottom of a tall vessel through a vertical slit, this able philosopher observed that the sharp parabolic stream will at a certain distance form a sort of node, and sink into a broad arch. The different retardations of the exterior streamlets appear in this case to determine the change of appearance.

¹ These methods of experimenting present themselves to a mathematician, but perhaps the first intimation of them was given in the Experimental Inquiry into the Nature and Propagation of Heat, p. 534. Since a delicate musical ear can distinguish even the quarter of a note, the error of observation in the second mode should not exceed the 64th part, which seems a nearer approximation than is attainable in any other way.

² See the article METEOROLOGY in this Encyclopedia.

Brook Taylor had shown that a musical chord during its vibrations has the swelling outline of a trochoid. But Euler and D'Alembert drew a more general conclusion, and proved that the curve might consist of several intermediate branches, subject merely to certain conditions depending on the Theory of Partial Differences. The solution of the problem gave occasion to a controversy between those illustrious philosophers, the former maintaining that the portions of the harmonic curve might be discrete, and the latter, in conformity with the great principle of Leibnitz, contending for their necessary continuity. But Daniel Bernoulli, avoiding the mazes of intricate analysis and visionary metaphysics, gave a clear investigation based on experiment. He established from induction that a vibrating string naturally divides itself into aliquot parts, which perform independently their several oscillations during each sweep of the whole chord. Applying the same kind of reasoning to wind-instruments, he showed that a cylinder of air which, by its tremor darting in the cavity of a tube with the rapidity of sound, produces its fundamental note, divides, likewise according to the force of intonation, into halves, thirds, fourths, or fifths, which segments yield at the same time their melting subordinate notes. Hence a musical tone is never single, but consists in the union and concord of certain elementary sounds. And such is the origin of Music, which receives its harmonious composition from the balance and concert of Nature. The theory of Rameau, which D'Alembert took the pains to expound, is reducible to the same principle. Those acute subordinate notes rise at times above the fundamental, or outlive it. Hence the shrill expiring note of a deep-sounding bell, and hence also the narrow compass of the French horn, which yields merely the successive natural tones according to the force of the blast.

These natural tones are derived from the mutual relations of the simple series of numbers 2, 3, 4, and 5. The ratio of 1 to 2 is evidently compounded of the ratio of 2 to 3 and that of 3 to 4, and these ratios mark the sweetest con-

cords. The interval between the $\backslash frac\{3\}\{4\}$ and $\backslash frac\{4\}\{5\}$ is $\backslash frac\{5\}\{6\}$, and between $\backslash frac\{3\}\{4\}$ and $\backslash frac\{4\}\{5\}$ it is $\backslash frac\{1\}\{2\}$; which fractions indicate a tone and a semi-tone, the latter differing from unit only by about half the difference of the former. But $\backslash frac\{3\}\{4\}$ may be split into $\backslash frac\{5\}\{6\}$, $\backslash frac\{1\}\{2\}$, and $\backslash frac\{1\}\{3\}$; and hence $\backslash frac\{5\}\{6\}$ and $\backslash frac\{1\}\{2\}$, though not quite equal, are both deemed tones, the former being called the Major and the latter the Minor. The fraction $\backslash frac\{3\}\{4\}$ is therefore composed of $\backslash frac\{5\}\{6\}$, $\backslash frac\{1\}\{2\}$, $\backslash frac\{1\}\{3\}$, and $\backslash frac\{1\}\{6\}$; and thus seven divisions, consisting of three tones major, two tones minor, and two semi-tones, fill up the whole extent of the fraction $\backslash frac\{1\}\{2\}$, which is hence termed the octave. It is impossible to express an equal subdivision of ratios by small numbers, but twelve semi-tones may be interpolated with tolerable nearness. Speculative authors have endeavoured to adjust or temper the scale, and for this purpose they have sometimes proposed the application of logarithms. Wallis illustrated the Theory of Music with his ingenuity and stores of erudition. In the earlier part of the subsequent century Malcolm and Smith produced learned treatises on the subject. But Euler himself, having at different times resumed the investigation, composed an ample treatise on the General Principles of Music.

It is curious to compare the different degrees of sensibility possessed by the organs of seeing and of hearing. The eye extends its perception over a range of intensity from one to beyond a billion, or from the feeblest glimmer of twilight to the dazzling glare of the meridian sun. But it loses its grasp unless the impression lasts nearly the tenth of a second on the sensorium. Hence a momentary gleam of light, as in Electricity, may elude observation, while a brand whirled quickly round appears a continued circle of fire.¹ The discrimination of the human ear has a more limited extent: It comprises only eight octaves, from the gravest to the sharpest note, or from those produced by only 16 vibrations in a second to those excited by a succession of 8192 vibrations in that interval. Such extreme rapidity might seem far beyond the reach of our faculties; but it is the nature of a musical note to repeat its action till an-

¹ This remark might serve to elucidate the more recondite Electrical Theory. A variety of curious appearances are explained from the same principle. Hence likewise the ingenious toy lately brought forward and called by Dr Paris the Thermatrop.

other arises; and this divided duration, short as it may be, is sufficient to impress the auditory nerve.

The application of the Higher Calculus to unfold the Motions of the Gaseous Fluids has exercised the skill and ingenuity of the greatest analysts, without producing however any material results. But what is really valuable has been obtained, by the simplest theory guiding experimental research. One of the finest corollaries drawn from the principles of Pneumatics is the method of ascertaining the heights of mountains by Barometrical observations. Pascal was the first to propose this nice problem, after the success of the famous experiment performed at his suggestion on the summit of the Puy de Dome. But the essential element of the solution—the relation between the pressure and the density of the air—had not been yet discovered. Twenty years elapsed till Richard Townley, assisting at some of the experiments which Boyle was making to refute the miserable objections of Father Linus, perceived that simple law, which the English philosopher confirmed by other more extended experiments on the compression and dilatation of air. But in 1676, Mariotte, a French experimenter, endued with greater penetration, and possessing some geometrical skill, published a work on the Atmosphere, of high merit for the time, and stamped with originality. He had instituted a clear set of experiments for investigating, and stated the result in explicit terms, that the Density of the Air is always proportional to its Compression. From this law he sought to derive the graduated rarefaction in the upper regions of the Atmosphere. He thought the variation of a line by the barometer near the level of the sea should answer to an ascent of 10$\backslash frac\{1\}\{2\}$ toises, which is the same in English measures as the tenth part of an inch for 75$\backslash frac\{1\}\{2\}$ feet. He then proceeded to find in succession the thickness of the strata of air corresponding to 4032 twelfth parts of a line in the whole mercurial column of 28 inches.

These strata he remarked might be found by Logarithms; but contenting himself with a rough computation, he adopted an arithmetical instead of the geometrical progression. Hence he reckoned the air to be 4032 times attenuated at an elevation of 15 leagues or 43$\backslash frac\{7\}\{8\}$ miles, which is a tolerable approximation, the true height being only 41$\backslash frac\{1\}\{2\}$ miles.

Mariotte had therefore no very distinct conception of the great property that at equal ascents in the atmosphere the density of the air diminishes in a continued proportion. This beautiful theorem appears to have been first discovered and demonstrated about the year 1662 by Huygens, who never took the trouble, however, to publish it.¹ But in 1685, Halley gave an elegant geometrical demonstration of the property, founded on the relations of the segments included between the hyperbola and its asymptotes. For the sake of round numbers, he assumed 30 inches for the standard mercurial column, air at the surface having 800 times less density than water; and he made the elevation proportional to the difference of the logarithms of these columns, the interval between 30 and 29 inches corresponding to 900 feet. This rule, drawn directly from theoretical considerations, was found to apply with tolerable accuracy to an observation made a few years afterwards on the top of Snowdon. Newton generalized the problem in his Principia, by taking into the estimate the decrease of gravity in receding from the centre of the earth, and arrived at the conclusion, that the densities of the higher strata of the atmosphere form a geometrical progression corresponding to the altitudes disposed in harmonic proportion. But this correction may be deemed superfluous in most cases of barometrical measurement.

It was only wanted, therefore, to rectify the method of Halley, by trying it with observations made at great altitudes. But, for more than half a century afterwards, the subject of barometrical measurement was entirely neglected in Eng-

¹ The demonstration, with its date, is given on a Dutch Almanac, now preserved in the Library of the University of Leyden, which, among other valuable notices, contains various microscopical observations, and a judicious parallel between Descartes and Bacon, whom he censures for his ignorance of Geometry, his violent opposition to the Copernican System, and his general inattention to the progress of physical and astronomical science.

land; and, instead of being improved on the Continent, it had relapsed into confusion, and became involved in hypotheses fostered by the lingering influence of the Cartesian philosophy. Bouguer returned to the right path, and availed himself of the opportunity afforded in surveying the stupendous Cordilleras, to compare theory with observations on the grandest scale. His investigation was not published, however, till 1753, when he gave the very simple rule which is the ground of all the modern practice: That the difference between the logarithms of the barometrical columns, reckoning as integers the first four figures, and deducting a thirtieth part, will express the altitude in TOISES. To accommodate the result to English measures, it is only required to add, instead of subtracting, the thirtieth part of the logarithmic difference, to denote fathoms. But still there were considerable discrepancies which the ingenious observer endeavoured to remove by some plausible suppositions respecting the elasticity of the air. To find the density of that fugacious fluid, he proposed an experiment which marks originality of conception: It was to note the relaxation of a pendulum or the rate of its diminished sweeps at different elevations in the atmosphere.

The chief cause of the perplexity was owing to the variable influence of heat on the density of the air. No further improvement could therefore be made till that effect had been ascertained, and the difference of temperature between the stations in every case measured by the application of an accurate thermometer. De Luc has the merit of reviving the subject, and of pursuing his researches with ardour, skill, and perseverance. Geneva, his native city, from its position near lofty mountains, afforded the greatest facility for conducting extensive observations. He began his excursions among the Alps in 1754, and soon perceived the necessity of improving the barometer, and rendering it more portable. By carefully boiling the mercury within the tube, he was enabled to expel any particles of air or of moisture which adhered to its sides; and having adjusted the level of the basin, he affixed to the instrument a scale marked with fine subdivisions. But instead of following the decimal division, and distinguishing

the inch into a thousand equal parts, he unfortunately took an irregular sequence, by adopting lines or twelfth parts, which he broke down again into sixteenths.

The construction of the thermometer engaged his particular attention. He tried alcohol, and various refined oils, and discovered their great defects. He was therefore induced to prefer mercury, which, besides other advantages, he found, by the methods practised by Renaldini and Brook Taylor, to possess the essential property of expanding very nearly equally with equal accessions of heat. Substituting that mineral fluid for the dilute spirit of wine, he merely corrected the divisions of Reaumur, reckoning the arbitrary number of eighty degrees from freezing to boiling water. Corresponding to each of these degrees, he determined the expansion of air by heat to be the 215th part, which answers to a 269th on the centesimal scale.

The rules which Deluc inferred from numerous experiments and observations were published in 1772 in his able but very diffuse work entitled Recherches sur les Modifications de l'Atmosphère. Two years afterwards they were reduced to English measures, and simplified by Maskelyne and Horsley; and the problem of barometrical measurement appears then to have excited much attention in our island. General Roy computed and observed the heights of mountains in Wales, and Sir George Shuckburgh at the same time, and for a similar purpose, ascended the lofty summits of the Alps. They had the advantage of employing barometers constructed by the celebrated Ramsden; and their conclusions, published in 1777, accordingly very nearly agreed, making the difference between the logarithms of the mercurial columns to express the elevation in fathoms at a temperature only the fraction of a degree below the freezing point. Since that time little more improvement has been effected, though several circumstances that modify the solution of the problem, especially the influence of humidity, might require to be reconsidered. Laplace indeed has introduced some niceties which appear hardly suited to such imperfect data, and render his formula extremely complex and inelegant. Till the modifications of the atmos-

phere be measured with much greater precision, we may rest satisfied with a simple and rapid approximation. Nay, the application of logarithms might in most cases be superseded, by employing a very easy proportion.¹

The mutual appetency of the elemental atoms includes both the cohesion of the particles of a body to each other and their adhesion to a different substance; and since this cohesive force belongs alike to solids and to fluids, it only preserves their volume, and does not essentially maintain the firmness or figure of a body. Hence the small detached portions of liquids run into globules or drops, and hence also they may be retained in a vessel at some height above the brim. In very narrow glass tubes, water and other liquids will indeed stand considerably higher than the level. This curious fact was first distinctly noticed by Aggiunto, one of the leading members of the Academy del Cimento, who died at an early age. But the completest series of experiments on Capillary Action was made by our countryman Hauksbee. He proved that it is not affected by atmospheric pressure, and succeeds equally well in vacuo; he showed the ascent of water and other liquids between proximate glass plates, and compared it with the rise in narrow tubes; and he ascertained the elevation to be always inversely as the width of the bore or the separation of the plates. He found the same property belongs to plates of marble and brass, and remarked the ascension of water in an open barometer tube crammed with fine ashes. Dr Brook Taylor likewise performed several ingenious experiments on this subject: Having joined two plates of glass at their vertical sides so as to form a sharp wedge, he dipped it in a sheet of water, and observed the liquid to rise and form a rectangular hyperbola; thus clearly exhibiting the relation of the ascent to the interval between the proximate surfaces. But he pursued the phenomena of corpuscular attraction still farther, and measured the adhesion of a disc of glass to water and

mercury. Seventy years afterwards this inquiry was resumed by the celebrated Guyton-Morveau, who endeavoured to ground upon it the chemical theory of Elective Attraction. The attempt however was fallacious, because the force required to detach a disc of glass, marble, or metal, from a surface of water or mercury, is not a single effort, but combines the adhesion of the liquid particles to the solid with their cohesion to each other.

In 1718 Dr Jurin, being led to examine the phenomena of Capillary Action, proposed a theory for their explication which seemed at least plausible. He rightly ascribed the rise of water in the cavity of the tube to the close attraction of the internal surface of the glass, though he did not perceive the way in which that force must act. He fancied the suspension of the slender column of liquid to be caused by the attraction of the ring of glass immediately above the summit. But such an assumption was quite illusory, for the ring below that limit would evidently exert an equal force in the opposite direction, and thus extinguish the influence of the former. This singular oversight very long however escaped remark, and Jurin's hypothesis became popular and commonly adopted.

Clairaut, discussing the equilibrium of fluids in his famous work on the Figure of the Earth, incidentally examined the property of Capillary Attraction, of which he gave a profound but incomplete analytical investigation. The acute Segner took up the subject in 1754, and gave a different solution, distinguished by its depth, ingenuity, and general accuracy. Assuming as a principle, that the attractive energy is confined to a mere exterior film of the liquid, he found the curve of the upper surface to be what is called the Lintearia, or the cavity of an inflated sail formed by an uniform tension. The results he obtained were perfectly accordant with the phenomena, except the figure of a drop, in the determination of which he had, from overlooking the double curvature, committed a small error.

Nearly half a century more elapsed till any further attempt deserving notice was made to im-

¹ As the sum of the mercurial columns is to their difference, so is the constant number 52,000 to the approximate height. This number is the more easily remembered from the division of the year into weeks. For a condensed explication of the subject of Barometrical Measurement, see Elements of Geometry, 4th edition, p. 454-456.

prove the Theory of Capillary Action. In 1802, a short dissertation on the subject, drawn up for a particular occasion, and cast into a popular form, was inserted in the Philosophical Magazine.¹ It set out from the simplest principles, refuted the notion of Jurin, proved that the attraction of the inner surface of the glass must be exerted laterally, and showed from the nature of fluids how such a force would produce a vertical effort in the liquid column. The author had designed to expand the outline into a strict analytical investigation, that should embrace the whole range of the phenomena, but deferred the task on account of the more interesting objects which happened then to engage his attention. This unpretending essay, though little appreciated at the time, appears however to have recalled the attention of philosophers to a subject so long neglected. In less than two years the late lamented Dr Thomas Young resumed the investigation of Capillary Action after the manner of Segner, and obtained a very complete solution, but which required, besides, the admission of a repulsive force among the particles of the liquid at a certain small distance. At the mention of Dr Young's name the historian must pause. None of our countrymen has approached more nearly to the character of the celebrated Dr Brook Taylor. Possessing the same ingenuity, extensive learning, varied accomplishments, and profound science, he combined likewise a concise, hard, and sometimes obscure mode of stating his reasonings and calculations. He manifested some chagrin at seeing Laplace, within a twelvemonth after, digress from the train of his Mécanique Céleste, to produce an analytical investigation of the phenomena of Capillary Action, so closely resembling his own in the general conclusions. The illustrious French Philosopher, overlooking the previous advances that had been made, and little solicitous about tracing physical principles,

trusted to his consummate skill in the process of calculation, and involved the subject in a maze of intricate and abstruse formulae. The reflections drawn from Dr Young on that occasion were extremely judicious, and deserve to be held up to view at present, when such a false taste prevails as threatens to involve the science of our island in specious mysticism. "It must be confessed that, in this country, the cultivation of the higher branches of the Mathematics, and the invention of new methods of calculation, cannot be too much recommended to the generality of those who apply themselves to Natural Philosophy; but it is equally true, on the other hand, that the first mathematicians on the Continent have exerted great ingenuity in involving the plainest truths of mechanics in the intricacies of Algebraical formulas, and in some instances have even lost sight of the real state of an investigation, by attending only to the symbols, which they have employed for expressing its steps." (Lectures on Natural Philosophy, vol. II. p. 670.)

Laplace's intricate formula for Capillary Action has been since unravelled by the acute discrimination of Mr Ivory, who disjoined it into two separate portions; the one depending on the adhesion of the watery film to the inside of the tube, and the other resulting from half the cohesion of the particles of the liquid to each other. But our ingenious countryman deduced these elements of the complete force from the simplest physical principles, availing himself of the property of equable diffusion of pressure through the mass of a fluid. The same investigation gave the measure and limits of depression observed in mercury and some other liquids. The treatise on the Elevation of Fluids which first appeared in the Supplement to this Encyclopædia² was a masterly production, which fulfilled every requisite that could be desired.

¹ In a memorandum bearing the date of December 1798, the principle is thus briefly stated: "The attraction of water to glass, being directed perpendicular to the surface, produces, in consequence of the laws of fluids, a lateral motion, in the same manner as would the pressure of a vertical column equal to that force." It then goes on to investigate generally the curvature of the surface of the water, and gives the simple formulae $\backslash frac\{2\}\{3\}d$ and $\backslash frac\{1\}\{3\}d$, for the ascent of that liquid between glass plates or within the bore of glass tubes, the mutual distance or diameter being denoted in inches by $d$.

² Supplement to the fourth, fifth, and sixth editions, vol. IV. p. 309.

ELECTRICITY has, from very small beginnings, risen in the space of little more than half a century to the rank of a Science, superficial indeed, but full of interest and splendour. Without engaging the attention of profound philosophers, it owes its progress chiefly to the ardent application of ingenious experimentalists. In this department Hauksbee,¹ a diligent and skilful operator, made some considerable advances soon after the commencement of the eighteenth century. He marked the circumstances of electrical attraction and repulsion, and observed the production of light by friction both in air and in vacuo. But to him succeeded a more fortunate explorer of Nature—Stephen Gray—a pensioner of the Charter House, a man of reclusive and peculiar habits, and no great reach of mind, yet allured to the pursuit by curiosity and the gleams of fancy. The apparatus he employed was very humble, suited to his slender means. About the year 1720 he gave a catalogue of bodies, which show electricity on being rubbed by the hand, including most substances, except the metals which he could not excite. But in 1732 he was led by a chain of accidents to discover the conducting property inherent in certain bodies, by which they are distinguished from such as may become electrical. The fundamental facts thus elicited were in 1739 arranged perspicuously by Desaguliers, who framed the interchangeable epithets Conductors and Non-electrics, and Non-conductors and Electrics, and proposed the term Insulation to denote the interrupted communication of electrical virtue. In the meanwhile Dufay,² keeper of the King's Garden at Paris, availing himself of these discoveries, advanced in the investigation with a keener and more philosophical spirit. Between the years 1733 and 1739 he detected two opposite kinds of electricity, which he called the Vitreous and the Resinous, the former being excited by rub-

bing glass, crystal, and other kindred substances, and the latter by the friction of amber, lac, rosin, &c.; and he gave a clear view of the phenomena of electrical attraction and repulsion, by demonstrating that bodies similarly electrified mutually repel, while those dissimilarly electrified attract each other.

Electricity began about this period to draw general notice, and accordingly more efficient means were employed for the exhibition of its properties. The friction of a glass tube or stick of sealing-wax by the hand was abandoned for the adoption of a large wheel to whirl swiftly a globe or cylinder under the pressure of a cushion. Otto Güericke had used for the same purpose a globe of sulphur, and Hauksbee one of glass; but the advantage of applying a cushion was not immediately perceived. One or more gun-barrels suspended horizontally by silk cords, and having small bundles of linen threads fastened to the nearer ends, formed the prime conductors. Such was the clumsy machine then constructed in Germany, and thence introduced into France and England. The curious found amusement in drawing sparks and firing inflammable substances; and experiments of that sort were ingeniously varied and multiplied. The analogy of Lightning to the Electrical Flash could not fail to be remarked, though their absolute identity was not yet proved. Recourse was now had to electric agency, for the mitigation or cure of certain chronic disorders; and with that view principally were different substances subjected to its immediate influence.

The attempt to electrify water in a phial suspended by a hook from the prime conductor, gave occasion at this time to a discovery which constitutes an epoch in the annals of science. The experiment appears to have been originally performed in Poland, but was repeated in November 1745 by Cuneus and Lallemand at Ley-

¹ Francis Hauksbee. Of this ingenious man, the best experimenter of his time, it is mortifying that so little should be recorded. Neither the date of his birth nor of his death is known; but he flourished between the years 1709 and 1731, and appears to have been Curator to the Royal Society of London.

² Born in 1698 at Paris, where he died in 1739. He was son of an officer in the French Guards, and had spent his youth in the military profession, which he quitted for the study of chemistry and botany.

den, and described by Musschenbroeck. It hence acquired the appellation of the Leyden Phial or Jar, and consists of a jar or plate of glass covered or coated on both sides, to near the top or edge, by a metallic leaf or other conducting surface, which renders it capable of holding a very high charge of accumulated electricity. By this wonderful invention, a power was obtained incomparably greater than any former exhibition of electrical influence. The shock or convulsive agitation accompanying the discharge of a loaded jar through the nervous system, at first inspired terror, and still continues to excite surprise and astonishment. The swiftness of electrical communication thus displayed seemed to exceed the rapidity of thought itself. Nolle, ¹ a popular lecturer on Experimental Philosophy at Paris, sent the shock with instantaneous effect through a whole regiment of guards; and Watson, ² an ingenious physician in London, could discover no interval of time in its transmission by a circuit of about six miles, formed partly by a wire, but mostly by the course of the river Thames.

Various attempts were now made to explain the electrical phenomena. The ordinary class of philosophers had so long indulged the belief of ethereal media, that they were easily induced to ascribe the powers of electricity to the agency of such a subtle fluid, which might seem at length revealed to our observation. Dufay showed that the electrical machine draws its supply from the ground, and Watson advanced a step farther, in assuming that every substance naturally holds a certain share of the fluid, which in a charged jar is redundant on the one side and deficient on the other, or distributed in a positive and negative state. The credit of inventing this Theory has been assigned to the celebrated Franklin, ³ who might probably without communication have hit

on the same idea, and is certainly entitled to the merit of expounding it in a clear and interesting manner. The shrewd American, without possessing any superlative talents, or the advantages of education, had, by industry, judgment, and perseverance, raised himself to a conspicuous station in society. Having some tincture of science, and a taste for experiments, he was peculiarly fortunate in choosing a subject of inquiry which so well accorded with the pitch of his acquirements. He had repeatedly visited England, and after his return to America he maintained a constant interchange of letters with his friend Peter Collinson, a wealthy grocer in London, but a zealous botanist, who valued learning, and held a regular and extensive correspondence over Europe. Through this channel of communication, was Franklin early apprized of whatever chanced to engage attention in the scientific world; and his observations were in their turn quickly transmitted across the Atlantic, and conveyed from England to France and Germany. Having carefully studied the important art of writing, he had attained a style remarkably simple and perspicuous, which gave the best effect to all his compositions. His explication of the Leyden Jar, from the redundancy and defect of a single fluid, was favourably received both at home and abroad; though the hypothesis of a vitreous and a resinous fluid, proposed by Dufay, and generally preferred on the Continent, appeared to explain the principal facts with equal readiness and facility.

But it was reserved for the American philosopher to complete a grander discovery, which, though unreasonably extolled, is the foundation of his permanent fame. The similarity of Electricity and Lightning already struck several experimenters; and the Abbé Nolle had in 1746 drawn a parallel, in which he compared the con-

¹ John Anthony Nolle, born at Pimpre, in the district of Noyon, on the 19th of November 1700. From that obscure retreat he was drawn by his growing reputation; and his amiable character secured the support and encouragement of zealous friends. He had the advantage of visiting England and Italy, and those journeys procured him the favour of the King of Sardinia, and afterwards the patronage of the Royal Family of France. Enjoying some appointments at Paris, he for many years gave regular courses of Natural Philosophy, and by his popular and eloquent lectures, and his skill in preparing and exhibiting the illustrative experiments, he spread a general taste for science. His elementary works, without aiming at originality, are remarkably neat and clear. He died much regretted on the 24th of April 1770.

² William Watson, born in London in 1715. Bred an apothecary; and being very prosperous, he turned physician. He was active and ingenious, with some talents, and still greater pretensions. By his ardour he extended considerably the knowledge of Electricity. He died in 1787, having the year before obtained the honour of knighthood.

³ Benjamin Franklin, born at Boston in 1706, and died at Philadelphia on the 17th of April 1790. His political career is well known, and his literary progress has been sufficiently appreciated.

glomeration of thunder-clouds to the prime conductor of an electrical machine. Winkler next, and with more decided arguments, contended for the identity of those powers. Franklin, following the same train of speculation, enumerated in a clear and methodical order the different circumstances of resemblance. But it was necessary to pass the circle of mere probabilities, and he proposed as the final proof to have recourse to direct observation. His suggestion was, to erect on the top of some eminence a tapering iron rod 40 feet high, which he conceived would attract electricity from the charged thunder-clouds. Preparations were accordingly made in France for trying that bold experiment. In the spring of 1752, Dalibard, a distinguished botanist, had an iron rod 40 feet in height tied with silk cords to a post in the neighbourhood of Paris, the lower end being protected from rain by a sort of sentry-box. The apparatus was, during his absence, intrusted to the charge of a resolute carpenter, who watching the first appearance of a thunder-storm on the 10th of May, ran to the spot, drew sparks from the rod, and, assisted by the curate of the village, actually charged an electric jar. The demonstration of the nature of Lightning was thus rendered complete; but an experiment so wonderful deserved repetition. It was eagerly performed, under the direction of Buffon, in the Royal Garden at Paris; and, during the months of July and August, it was tried with the same results near London, where Canton succeeded in detecting atmospheric electricity by means of a common fishing-rod.

Intelligence now came from America, that Franklin had performed his experiment in a finer style. Not having the opportunity of an eminence in the flat country around Philadelphia, he imagined the expedient of employing a boy's kite, to gain a great elevation in the air. To the end of the hempen string, from which hung a small key, he fastened a piece of silk cord, for holding in his hand. On the 15th of June in the afternoon, while a storm was gathering, he walked into the fields, and, assisted by his son, he launched the kite, when, to his

inexpressible delight, he saw the loose fibres stretch out from the string, heard a snapping noise, and observed the pendant key not only to attract light substances, but to give sparks on approaching his knuckle. These were all decisive marks of electrical action, and answered his most sanguine expectations. The success of this trial encouraged him to entertain the daring project of turning aside the stroke of heaven, and of guarding our edifices from the ravages of thunder, by the erection of lofty conductors. These views, combined with amusing experiments and lively speculations, were explained at great length, and in a very easy and pleasing manner, in the series of letters to Collinson, and through his friendship printed at London in 1755, and immediately translated into the French language, and circulated with zeal and rapidity over Europe.¹

Notwithstanding the enthusiasm excited by the dispersion of that work, the scheme of annexing conductors to buildings was yet slowly adopted. The first one erected was by Watson, at his villa near London, in 1762; two years afterwards, they appeared in Germany; and not till the year 1776 were they applied to protect some cathedrals in Bavaria and Italy. They have spread since in all directions, especially on the Continent, flowing, it might seem, with the tide of Franklin's political celebrity.

But his famed hypothesis, if examined strictly, will be found to rest merely on the extension of vague analogical considerations. If a pointed wire sensibly draw electricity at the distance of a foot from the prime conductor, through what a wide range, it is argued, must a long tapering rod shoot its influence in the vast magazine of the atmosphere! It may be shown, however, that the point acts only on the electrified air which streams from the machine, and by repelling again laterally those affluent particles, facilitates their continued flow. The slightest impediment to the motion of the aerial currents reduces the action of the point; while, on the contrary, a knob or blunt termination may acquire the same influence as a pointed end, if the flow of electrified

¹ It deserves to be noted, that the Royal Society, which rarely extends its patronage to untried merit, had refused these ingenious letters a place in their Transactions.

air towards it be maintained, either by rendering the ball very hot, or by sticking to it a small lighted taper. The idea of stealing the lightning from the thunder-cloud, and silently disarming the fulminating agent, is therefore utterly chimerical. To produce such an effect, the whole body of succumbent air must discharge its electric store, by actually coming in successive contact with the end of the rod. And, in what reasonable length of time could that extended communication be accomplished? But it is equally futile to suppose, that the terminating of the rod by a point could have the slightest influence, in tempting or leading the thunder-stroke towards any particular spot. Yet such was the main argument used in the famous dispute which once agitated the scientific world, with regard to the comparative advantages of terminating the conductors with points or with knobs.

But though any attempt to avert the course of thunder seem preposterous, it is of extreme importance to consider whether the intervention of a proper conductor may not mitigate, or even render innocuous, the fulminating stroke. The coruscation of lightning resembles most the electric spark, its tortuous path being marked by the violent divulsion of the aerial track. The rapidity of transit might indeed be regarded as instantaneous; but Helvig has lately endeavoured, in Germany, to measure it by help of the Camera Lucida, and estimates it, from probable conjecture, at 8 or 10 miles in a second, or above 40 times swifter than sound. According to the ingenious Gay-Lussac, lightning often darts at once more than three miles in a rectilinear path.

The ordinary classing of conductors into perfect or imperfect, is very loose and unsatisfactory. Their true distinction consists in the time they require, however short this may be, for the transmission of electrical energy. From a set of peculiar experiments, it would appear that the copper transmits this virtue more than a thousand times quicker than water, which again conveys it several thousand times faster than dry stone. But all the particles of the conductor are thrown during the passage into a state of vehement re-

pulsion, and, consequently, the disruptive effect must be proportioned to the interval of action. A copper tube, or ribband, of even moderate dimensions, if well spread over the roof and continued into the ground, may hence be sufficient to protect a house from thunder, by reducing exceedingly the duration of the repulsive efforts. Iron, which is generally employed, conducts twenty times slower than copper; and lead is still worse adapted for the purpose, since it conducts about a hundred times slower. The electrical energy being conveyed chiefly along the surface of bodies, sheet-copper should decidedly be preferred for the substance of a conductor.¹

Thunder-rods of the ordinary construction have been now tried seventy years, and the accidents from lightning seem just as numerous and indiscriminate as before. The public is gradually losing its confidence, therefore, in the efficacy of those vaunted protectors; but, in spite of such warnings, philosophers are most unwilling to descend from their proud eminence. When any sinister event occurs, they are solicitous to find excuses and parry objections. They recommend the conductors to be planted nearer, carried higher, and perhaps armed with a cluster of points. It may require a few more of such fatal accidents as the late explosion, in France, of a protected powder-magazine, to demolish finally the Franklinian hypothesis.

But the injury caused by the stroke of lightning is, in this climate, much less considerable, after all, than has been generally believed. The loss of human life by thunder-storms over Europe appears not to exceed annually, perhaps, one individual in three millions. The damage to property from such appalling discharges is likewise comparatively small. The ravages inflicted on the Continent by showers of hail are of a more formidable kind. Those frozen masses are often large, and fall with such terrible force as in the space of a few hours to tear in pieces the vines and batter down the fields of corn over wide and fertile districts. It was an object therefore of still greater moment to mitigate, if possible, the fury of such rageful storms.

¹ On this important subject see a paper on Electrical Points and Conductors, in the first number of Professor JAMESON'S Philosophical Journal, published in 1823.

About the year 1776, several speculative experimenters, conceiving electricity promoted congelation, and therefore the conversion of the drops of rain into solid icicles, proposed erecting thunder-rods to prevent the formation of hail. It was even suggested, in aid of those protectors, to kindle fires on the high grounds, and to shake the atmosphere by the discharge of mortars. But Heinrich showed the futility of all such expedients, in a dissertation printed in the Bavarian Transactions for 1785; and, except the notice of a curious experiment by Seiferheld in 1790, the question seems to have sunk into oblivion. The attention of the public, however, was again for a moment recalled in 1800, by the offer of a prize from the Physical Society at Berlin; but the successful candidates, Wrede and Weiss, proved that hailstones had no connexion with Electricity, and that the various expedients tried for averting them were entirely fallacious. The subject seemed deservedly forgotten for twenty years, but has been lately revived again by the periodical folly of mankind. La Postolle, apparently a zealous visionary, proposed in France with the utmost confidence, as a complete safeguard against the ravages of hail, the erection of a tall wooden pole with straw ropes hanging loosely from the top of it to the ground. He was followed in 1823 by Thollard, another projector equally sanguine. But as the world is governed by names, the powerful talisman or lofty thatched pole received the sounding, though not very classical, appellation of Paragrelle or Paragrandino. Certificates of its efficacy flowed in from all parts; the storm-clouds were seen by veracious witnesses collecting over those poles, and rolling their hailstones harmless along the straw to the ground. The cultivators of the mountain districts on the confines of France, Switzerland, Italy, and Germany, seemed enchanted by the discovery, and whole fields were planted with those Paragrelles,

which, like embattled spears, dared the front of heaven. The question relative to the utility of Hail-Protectors was now referred to the Institute of France; but that learned body, though it clung to the charm of the Thunder-Rod or Paratonnerre, rejected with scorn the pretensions of the Paragrelle. But the faith of the believers was not so easily shaken, and a succession of angry and querulous pamphlets for a while kept up the dispute. Yet we may infer that the confidence in the efficacy of such notable protectors is fast declining, since an old auxiliary—the agitation of the air by the firing of mortars—has been recalled to their aid.¹

In the latter part of the eighteenth century, Electricity was enriched by a continual accession of curious facts and experiments. New instruments were devised, of nicer construction, to measure the intensity of electrical attractions and repulsions. The catalogue of electrics themselves was not only extended, but rendered more definite, by distinguishing the opposite qualities of the substances rubbed. Some steps have likewise been made towards a more precise theory. If a ball electrified vitreously be approached to an insulated cylinder, the nearer end will assume the resinous electricity, while the remoter extremity will indicate the vitreous; the limit of neutrality always advancing to the approximating ball. This capital experiment appears to furnish the true explication of the Leyden Jar. But it involves a more extensive principle: Pressure may be considered as only close apposition, and friction is evidently a case of repeated pressure. Hence the action of the Electrical Machine itself, and hence likewise the theory of that beautiful contrivance, the Electrophorus, first announced by Wilke of Stockholm in 1762, but named and fully described in 1776 by Volta² of Como, to whom electrical science was afterwards so much in-

¹ Orioli, professor of Natural Philosophy at Bologna, seems not disposed to acquiesce in the decision of the Academy of Sciences at Paris. He appeals to the liberality and opulence of the French nation, and gravely recommends it to plant paragrelles, armed with metallic wires about 40 feet high, and not more than 500 feet apart, over its wide territory; and farther proposes that the experiment should be continued for ten years, under the inspection of some competent agricultural board. If the sum of 100,000 livres has been raised at his suggestion for that project in the small province of Bologna, what might not be expected from the munificence of such a kingdom as France?

² Alexander Volta, descended of an ancient family in the north of Italy, and born in 1745. He embraced the clerical profession; but, devoting himself to the pursuits of natural science, he was appointed to the chair of physics at Pavia in 1774. Electricity engaged his attention, and the observations of Galvani opened to him that brilliant career of discovery.

debted for its extension. The Vindicating Electricity of Beccaria and the Returning Stroke of Lord Mahon depend on the same property. Of a like nature is the accumulation of Electricity by the Doubler and other instruments of that sort.

It appears that the electrical state of bodies suffers a modification from every change of their chemical or mechanical constitution. Such alterations are detected by delicate Electrometers, of which the best perhaps may be the gold-leaf one proposed by Bennet in 1787. The origin of atmospheric Electricity is finely illustrated by that sensible instrument, which shows an evolution or an absorption during the condensation of vapour or the transition of water into steam. Hence a cloud suddenly collected from the conglomerating vapours is vitreously charged, and flashes its lightnings, which, by the succession of the air in their devious lengthened tracks, conveyed to our ears with the tardiness of sound, occasion the prolonged rolling noise of thunder.

The power of benumbing the touch, which belongs to a certain fish of the ray kind, thence called the Torpedo, and very frequent in the Mediterranean, had been remarked from the earliest times. This singular property was now suspected to be owing to electrical influence; and the zeal of Walsh in 1773 converted the conjecture into demonstration, showing by satisfactory experiments that the animal could send its shock only through conducting substances. But the power of stunning its prey is possessed in a much higher degree by a large species of eel, the Silurus Electricus, which was originally brought from Surinam, and abounds in the pools and sluggish streams of the hot region of Venezuela. From a healthy specimen exhibited in London, vivid sparks were drawn in a darkened room. But from a rapid emission of shocks, these animals suffer great prostration of strength; and the celebrated Humboldt gives an amusing account of the method of catching them in New Spain, by driving into the waters where they

resort, a number of horses, which, though stunned by the multiplied successions, yet generally recover and withstand those attacks, till the eels themselves become quite exhausted, and are dragged out helpless. It appears from dissection, that the Silurus Electricus is furnished with a very peculiar and complex nervous apparatus, which has been fancifully compared to an electrical battery. But we are entitled only to infer, that the animal is endowed with a faculty of modifying, by sudden compression or otherwise, its internal constitution, and consequently its electrical condition.

Electrical agency, by effecting new combinations among substances, has contributed essentially to the advancement of Chemistry. Cavendish discovered the composition of nitric acid in 1788, by passing successive sparks through a mixture composed of certain portions of the oxygenous and azotic gases. By a similar process, he found atmospheric air to consist of the same elements, only combined in a different proportion. But his fine discovery of the composition of water was afterwards confirmed, by concentrating the effort of electrical repulsion, in the experiment that Dr Pearson, assisted by Cuthbertson, an expert electrician, performed in 1799, which evolved those gaseous components. This satisfactory experiment has been since greatly simplified, by one of those happy miniature contrivances, in which the late very ingenious Dr Wollaston so much excelled.

Electricity can be transmitted through the several gases; and the light which it then extricates is brighter in proportion to the condensed state of the medium. As the air becomes rarer, the projected spark assumes a spreading lambent appearance, through all the gradations of colour, from white to yellow, orange, and purple, vanishing into the faintest violet. The most copious display of purpureine gleams appears when the air is rarefied about 1000 times, which corresponds to an altitude of thirty-five miles in the atmosphere. Such may be the proper region of the diffuse and tremulous coruscations of the

which will render his name immortal. Volta was particularly distinguished by the liberal and discerning patronage of Napoleon. During the fervour of reform and revolution, he laid aside the ecclesiastical habits, and married; but, in the decline of life, the early impressions regained their ascendancy, and compunction for the breach of the vow of celibacy preyed on his spirits, and undermined his health. He died on the 6th of March 1826.

Aurora Borealis, which is decidedly an electrical phenomenon. If the rarefaction be pushed farther, the luminous appearance grows always fainter, till it becomes extinct. Accordingly, Morgan¹ found by a very careful experiment that an electrical charge is not conducted through a Torricellian vacuum,—an important discovery, since it shows that Electricity, like Heat, can exist only in a state of combination with its recipient substances.

The close of the eighteenth century was distinguished by the accession of a new branch of Electrical Science, more brilliant and astonishing than even the parent stock. It originated in a fortunate incident which occurred in the year 1790. Galvani, whose name it bears, professor of anatomy at Bologna, remarked, in the course of his demonstrations, that the limbs of a dissected frog were strongly convulsed at every spark which one of his pupils happened to draw from the prime conductor of an electrical machine standing in the immediate vicinity.² Being thus led to consider the subject, he made several curious experiments, and published a Dissertation on Animal Electricity, which engaged very general attention. The femoral muscles of a frog, bared of their integuments, but left connected with the trunk of nerves, were found to serve as a most delicate sort of electrometer. With this aid, it was easy to trace the faintest vestiges of electrical influence, and to contrast the properties of various conductors. The very weakest chemical solutions, the mere contact of different metals, nay the apposition of animal fibres, were all found in their several degrees to develop electricity. But the simplest mode of exciting it is, by the mutual application of small plates or discs of copper and zinc. Dr Robison made a capital improvement, in proposing a pillar of those discs, like a rouleau of half-crowns, to augment by their combination the intensity of effect. This

happy idea seems to have been overlooked, when Volta in 1800 invented his famous Pile; the most energetic instrument of all electrico-chemical analysis, and commencing deservedly a new epoch in physical science. By Crookshanks it was rendered far more commodious, in being converted into the Galvanic Trough; which, again enlarged into Batteries sometimes of enormous extent or dimensions, has conducted Davy, Berzelius, and others, to the most splendid and wonderful discoveries.

The Voltaic Pile is only a modification of the Electrical Battery; but its peculiar action may assume two distinct features. It either exerts the slowness and duration of repulsive force, or displays the most intense concentration of that power. Hence the opposite effects produced by very large single plates, and by a very numerous series of small ones. Light and heat are most copiously projected from their recipients by the former, while chemical decomposition is effected with the greatest energy by the latter. But it would require much patient and profound investigation, to discover the working of such recondite principles.

The original design of this discourse was to come no lower than the early part of the present century, and to avoid discussing the merits of contemporaries. But I cannot resist the pleasure of noticing the signal advance which Electricity has lately made. Its close connexion, if not perfect identity, with Magnetism had been long suspected, and was even adopted by several ingenious theorists. This affinity found a most zealous supporter in Ritter; but the fancy and mysticism blended with his opinions had begun before the year 1818 to weaken their influence. In this state of uncertainty, Professor Oerstedt of Copenhagen, happening, in the course of his lectures during the winter of 1819–20, to show his pupils the intense heat excited in a small wire of platinum, laid horizontally and nearly

¹ A very ingenious person, who died young, being nephew to Dr Price, and brother of the able Actuary of the Equitable Assurance Company.

² The Germans lay claim to the origin of Galvanism. Sulze, about the middle of the eighteenth century, had, in his work on Taste, noticed the singular impression made on the tongue by the contact of two distinct metals. But Zimmerman has lately produced a passage from the Bibliotheca Naturæ, a book published at Leipzig in 1752 by Swammerdam, in which the author mentions his having observed the convulsions of the muscle of a frog held against a glass tube by a silver wire pendant from a ring of brass. Such facts are curious, and deserve attention; but every honourable mind must pity or scorn that invidious spirit with which some unhappy jackals hunt after imperfect and neglected anticipations, with a view of detracting from the merit of full discovery.

in the direction from east to west, to join the conductors from the copper and zinc plates of a Galvanic Battery, thought of placing under it a small compass, and the needle was observed instantly to turn aside as if it had been drawn by another magnet. This surprising fact was not much heeded by him at the time; but having afterwards carefully traced the conditions of the experiment, he published near the close of the year 1820 his great discovery, which awakened the public attention, and gave rise to numerous speculations that frolic in the giddy maze of electric and magnetic currents.

It must indeed be confessed, that after all the progress which Electricity and its younger branch Galvanism have made, the hypotheses commonly received are exceedingly vague and unphilosophical. In cultivating these attractive sciences, experimenters would seem to satisfy themselves with the exercise of a looser and humbler species of reasoning. It is rather amusing to observe the complacency with which some ingenious persons describe the play and vagaries of an Electrical Current, whose existence was never proved. We are acquainted only with electric attraction and repulsion, and with the transmission of electric influence: All beyond these elementary principles, rests on

hasty conjecture. Instead of adopting one or two fluids, it were safer to suspend the assumption of any. We can perceive no distinctive marks of the operation of a fluid, which is often confounded with the mere luminous track occasioned by the particles of Light disengaged from the substance of the conductor; the colour of emission being modified by the peculiar character and intensity of the retaining force.

The Theories proposed by Æpinus and Cavendish are entitled, however, to the praise of great ingenuity, and may serve to connect with elegance the chain of principal facts. The latter most accurate philosopher likewise stated the immense disproportion in the celerity of different conductors, though he did not explain the grounds of his conclusions. Coulomb agreed with him in limiting electrical diffusion to the surface of bodies; and the Balance of Torsion showed the intensity of attractive and repulsive power to be inversely as the square of the distance. These were real discoveries, deduced from nice and cautious observation; but his countryman Poisson has since exercised profound skill in the play of analysis, by attempts to explore the hypothetical influence of Electricity, without having arrived, however, at any conclusion that is not obvious or of no value.¹

4. MAGNETISM.

NEARLY allied to Electricity is the science of Magnetism. The property of attracting iron possessed by a certain stone or metallic ore, was known from the remotest times; but the directive power, or disposition to turn always towards the north, that most wonderful property which guides the modern navigator over the dark and desert expanse of ocean, lay hid through a long succession of ages. This remarkable substance derived its name among the Greeks from Magnesia, a district of Macedonia, where it was chiefly found.

The Magnetic Compass, with the art of distillation, which was never practised by the ancient Greeks or Romans, seems to have been

discovered in Upper Asia, and thence communicated by their Tartarian conquerors to the Chinese. From them again, the knowledge of the invention spread gradually over the East. The Crusaders, during the occupation of their bloody conquests in those regions, had leisure to admire the arts acquired by their more civilized rivals. Having their curiosity thus awakened, they appear, about the latter part of the twelfth century, to have imported into Europe the Compass, along with the substance which, mistaking it for Natron, they called Salt Petre, and of which they had learned the deflagrating property. That invaluable instrument was at first very rudely formed, consisting merely of a piece of

¹ Such abuse of a noble science would have merited the censure of the Dancied:
Or set on Analytic ground to prance,
Show all his paces, not a step advance.

the native mineral fixed to a broad cork, and set to float in a dish of water. An artist of the opulent town of Amalfi, the great emporium of the East, and seated on the shore of Calabria, in the direct route of the Crusaders, improved the construction, and marked the north point by a Fleur-de-Lis, the armorial bearing of the kingdom of Naples. From its directive property, it was now called in English the Loadstone or Leading Stone. About a century afterwards, the method of communicating magnetism by the touch was probably discovered, the needle or small bar of steel so treated being then applied to a card suspended on a pivot. The Germans bisected successively the eight cardinal divisions, which had satisfied the Romans and the Chinese, into sixteen and thirty-two points, to which they gave those compound names which are still retained. About this period, when observations were not very precise, the needle was judged to turn nearly towards the north; but Columbus, in his first voyage of discovery, found it to decline from the meridian as he advanced on the Atlantic; and this apparent change of the laws of nature occurred under circumstances which would have appalled a less determined commander. The variation of the compass, however, was distinctly noted in the year 1500 by Cabot, another celebrated Italian navigator.

Magnetism made little further progress, till Dr Gilbert, the founder of experimental science in England, explored the subject by a course of patient and skilful investigation. To this eminent philosopher we are indebted for the discovery of the few connecting principles. Every magnet, whether natural or artificial, has its powers concentrated in two opposite points, termed the north and south poles; and the similar poles of separate magnets repel each other, while their dissimilar poles exert a mutual attraction. When a piece of soft iron is approximated to a magnet, it becomes itself a magnet, the nearest end assuming an opposite polarity, and therefore being constantly attracted. If a long iron bar be held in a position nearly vertical, its lower extremity is always found to manifest the properties of a north pole; and, from this induced power, Gilbert legitimately inferred the magnetism of our globe. He likewise imi-

tated its structure, by fashioning a magnet into a small sphere or terrella, and hence illustrated the declination of the needle, as well as its dip, or the position which it takes when, after being poised freely on its centre of gravity, it receives the magnetic virtue,—a property which had been first noticed by his countryman Robert Norman in 1576. But something more was required to explain completely the directive property of the needle. If a magnetized sewing needle be set to swim on the surface of water or quicksilver, it will not advance towards the north, but readily traverse in that direction. In fact, while the one end is attracted, the other is equally repelled, by the vast magnetic power of the earth concentrated below the Arctic Region, but having really the same quality as what was first named the South Pole of the needle. Those antagonist forces will have no influence therefore to draw it forwards; but if it be turned aside, they will combine their oblique action to bring it back into its meridional position.

Gilbert's original work was republished at Ferrara in 1629, with a commentary by Cabæus, a Jesuit; and fourteen years afterwards another member of that learned fraternity, Kircher, a man of singular talent and immense erudition, produced at Cologne a full treatise on Magnetism, which contained little, however, of sound doctrine, but abundance of fanciful speculation.

Hooke remarked the debilitating effect of heat on the power of the magnet. Newton appears to have sometimes amused himself with magnetical experiments, but did not bestow much thought on the subject: He was disposed to consider the force exerted as reciprocally proportional to the cube of the distance.

The celebrated Halley, who, by his ingenuity, learning, zeal, and enterprise, contributed so largely to the promotion of physical science, now turned his attention to the subject of terrestrial magnetism. In 1683, and more distinctly in 1692, he endeavoured to explain the declination of the needle and its variations, by supposing the Earth to be a hollow sphere with two opposite magnetic poles, but having another solid sphere, of analogous polarity, which revolved slowly within it. From this bold hypothesis, of two fixed, combined with two mov-

able poles, he sought to calculate the changes of internal constitution that are continually going forward. At a time when the method of finding the longitude at sea was extremely imperfect, he proposed the variation of the compass observed in different latitudes, as an easy way of obtaining at least an approximation to that important problem. With this view it was necessary to collect numerous observations made at remote points on the ocean; and in 1699 Halley obtained the command of two small sloops of war, with the rank of post-captain, the better to insure subordination among the crews. Thus equipped and empowered, he traversed the vast Atlantic, diligently exploring both hemispheres, till he was arrested by the icy barriers. On his return in the following year, enriched with a store of various information, he published his Magnetical Chart, in which the limits corresponding on the surface of the globe to every five degrees of declination, were marked by certain curve lines formed by connecting the points of observation. This was the model of all the charts of a similar nature which have been since constructed. One of the most noted was produced by Mountain and Dodson in 1744, and again improved in 1756.¹ Wilke laid down another in 1772, and Lambert sketched out a third in 1776. But the most accurate and complete magnetic chart that has yet appeared, was published by Churchman in 1794. The value of such delineations, however, is unfortunately diminished by the errors proceeding from the local attraction of the magnetic needle, which were quite overlooked at that period.

Between the years 1712 and 1725, numerous experiments were made by Hauksbee, Brook Taylor, and Musschenbroeck, to determine the relation of the intensity of magnetic force to the distance of its action; but though the power appeared to decrease most rapidly with its remoteness, no satisfactory conclusion was obtained. The cause of this failure must be imputed to the intermixing, in the statement of the results, no

fewer than four distinct forces, emanating from the several poles. Taylor, the most acute of those observers, reckoned the magnetic force, if exerted very near, to be inversely as the square, but when more remote, as the cube, of the distance. It was reserved for Coulomb,² sixty years afterwards, to discover the true law of magnetic attraction and repulsion, by means of his delicate Balance of Torsion, though the same deduction had been previously announced by the ingenious Lambert. The earlier experimenters had sought to ascertain those forces by the loads required to effect a separation, or to counterbalance their action. Graham proposed the more precise method of computing the magnetic forces, from the number of vibrations performed in a given time by the needle; but his suggestion was overlooked, to be invented again, and generally adopted.

The properties of the magnet appear mysterious, though reducible to a few primary facts. But to discover the great pervading principle, still baffles the ingenuity and penetration of the most ardent research. The supposition of a subtle permeating fluid is very generally embraced, though it rather darkens than elucidates the subject. Yet it seemed to receive countenance, from the curve lines marked out by iron-filings when strewed on a sheet of paper or a plate of glass laid over one or more magnetic bars. On a hasty glance, these traces might be regarded as indicating the circulation of an invisible fluid. Euler, whose strength lay in the command of analysis, went so far as to imagine that the pores of the magnet were furnished with a sort of valves, which permitted the entrance of the current and prevented its return. But this amusing experiment is most easily and satisfactorily explained, from the composition of the attractive or repulsive forces centred in the magnetic poles. The curves delineated by the filamentous chains, when two dissimilar poles are exerted, resemble elliptical arcs, but seem hyperbolic ones, if two similar poles unite their action. The magnetic curve is distinguished by some

¹ These laborious compilers in 1757 gave a table of no fewer than 50,000 observations arranged corresponding to the years 1700, 1710, 1720, 1730, 1744, and 1756. They found it impossible, however, to reduce all the changes to calculation.

² Charles Augustin Coulomb, a most accurate and ingenious experimenter, was born at Angoulême in 1736, and died at Paris on the 23d of August 1806. Receiving his education in that capital, he embraced the military profession, and was sent as engineer to the island of Martinique. He afterwards held various employments, and might have risen to the highest distinction, if he had been as compliant with the times as some others of his countrymen.

remarkable properties, and one of the most beautiful may be cited: It is, that if from the same points in the axis, tangents be drawn to them, the several points of contact will range in the circumference of a circle.¹

Æpinus adopted the hypothesis of an active fluid, to explain the phenomena both of Magnetism and Electricity, and endeavoured to reduce its operations to great simplicity. Brugman and others have entertained similar notions. But admitting the ingenuity of such conceptions, they enable us merely to shift the difficulties. We must imagine the constitution of the unknown fluid, while the properties of the magnet itself are obvious to the senses. Sound logic, therefore, dissuades us from indulging in dreams hardly more instructive than the occult qualities of the Schoolmen. The true business of the philosopher, though not flattering to his vanity, is merely to ascertain, arrange, and condense the leading facts.

Ingenuity and patience were for many years exercised in improving the art of constructing artificial loadstones, or of communicating magnetism by different modes of Touch. The general procedure depends on the magnetic power induced by apposition, though it is often affected by very slight circumstances, and always requires dextrous manipulation to insure success. In France, Reaumur and Duhamel gave circumstantial directions; but our countrymen Knight, Mitchell, and Canton, acquired peculiar address in magnetizing; and Æpinus gave an improved method, suggested by his theory, for communicating magnetism by the Double Touch, the two rubbing bars being held each reclining about half a right angle from the vertical position.

It was proved that the magnetic virtue resides near the surface, and therefore hollow magnets were proposed, though perhaps never actually tried. But thin bars seemed to answer best for attaching under the card of the mariner's compass. Coulomb remarked that a steel wire is, by twisting, made capable of being nine times more strongly magnetized. It is singular that cast steel is unfit for a magnet, and that the smallest admixture of antimony destroys the po-

larity of iron. In the years 1786 and 1787, Cavallo tried the magnetism of various substances, by setting them to float on a very clean surface of mercury, on which they turned nimbly by the smallest force. In this way, he found that a trace of iron which no chemical test could detect was yet capable of sensibly affecting the needle. Nickel is strongly magnetical, but seems enfeebled by the addition of cobalt. Yet brass, though naturally passive, becomes susceptible of magnetism by hammering, and loses this power again when heated to near redness. But a more numerous and far more precise collection of experiments of that kind was made during the course of twenty years, between 1784 and 1804, by Coulomb. This most acute and accurate experimenter found every substance almost to be susceptible of magnetism. To examine the property, he formed needles about three-eighths of an inch in length, suspended by fine silk lines; and these obeyed the magnet, though composed of gold, silver, copper, lead, and tin, nay of bone and chalk. But this seeming universal diffusion of the property might still be owing to the presence of iron, however much attenuated. When bees-wax had incorporated with it a portion of iron-filings equal only to the 130,000th part of its weight, it was yet sensibly affected by the magnet. Nor is it improbable that nickel, as once believed, may be only a refractory ore of iron, still resisting chemical decomposition, though deriving from this source its magnetic virtue.

The variation of the needle has been accurately observed in Europe during nearly two centuries. It seems to be continually increasing, though in a most irregular manner. In the year 1657, it stood directly north at London; and it held the same meridional position in 1660 at Paris. But during the remainder of the century it changed towards the west, at the rate of 11' annually at London, but only 6' at Paris. From 1700 to 1725 the yearly increase of variation at London was only 9', from 1725 to 1750 it rose to 14'; but from 1750 to 1775 it returned to 9' again, and from 1775 to 1800 it declined to 6'. For the next 25 years the

¹ See Analysis of Curve Lines, near the end of the volume.

rate of augmentation has been scarcely $\backslash frac\{2\}\{3\}$, and the change at Paris is still smaller. This rapid decline of the annual change appeared to intimate the needle's approach to the limit of westerly digression, from which it would slowly return in the contrary direction. Such expectation, however, is not yet realized. In 1817 the needle was observed at Paris to make indeed a slight retreat; but, in the following year, it moved forward again, and continues to advance, though slowly, to the west. Its relaxation in London is less apparent.¹

But the variation of the needle, besides its annual progress, is liable to other alternating changes, amounting sometimes to a considerable quantity. Its diurnal deviation was first remarked by Graham in 1722, and ascertained with more precision in 1750 by Wargentini of Stockholm. But Canton investigated the subject with great ability and perseverance, in a very extended series of observations begun in 1756. He found the needle to travel westwards from about nine o'clock in the morning till two in the afternoon, when it remained for a while stationary, and again slowly returned, regaining its station during the night, or before the early dawn. In the morning, its movements were less regular. The extent of those digressions was found to depend on the season of the year; near the end of December, the diurnal aberration was only 7', but it mounted to 13$\backslash frac\{1\}\{2\}$' about the middle of June.

Canton, with great probability, referred these curious facts to the influence of heat on the magnetic forces. Though the direction of the needle is mainly determined by the internal magnetism of the globe, it must also be affected in some degree by the local attraction of the ferruginous particles united to mineral substances, and profusely scattered on the surface. If we suppose their joint power inclines to the east of the magnetic meridian, it will evidently act more feebly in drawing the needle aside during the warm portion of the day, and in the height of summer. This ingenious explication seems to be corroborated by the observations made in

1794, on the island of Sumatra, by Macdonald, who found the diurnal aberration there not to exceed 3'. But the alternation of heat and cold in that equatorial climate is likewise more limited.

The Dip of the Needle is much less changeable than its Declination. It is more difficult to observe, but appears to have been diminishing during the last hundred years, at the very slow rate of a minute annually. The quantity of Dip, or the depression of a poised needle below the horizontal position, depends in every place over the surface of the earth nearly on the latitude. It would be very desirable to render the observations with the Dipping Needle more certain and delicate. Coulomb proposed an ingenious mode of computation, from the load requisite to bring it into an horizontal position compared with the time of its vibrating in that plane.

Albert Euler, son of the great analyst, censured Halley's magnetical hypothesis, and proposed in 1766 a simple theory deduced from observations, and requiring the assumption of only two poles, distinct however from those of the terrestrial axis. The North Pole he would place in the latitude of 75° 7', and the South Pole in the latitude of 60° 37'; assigning to the former a western longitude of 91° 54', and to the latter only 38° 36'.

Biot and Humboldt, from a comparison of later and ampler observations, assigned for both the magnetic poles the opposite latitudes of 79° 1'; the longitude of the northern one being 27° 37', and that of the southern one 205° 12' west from Greenwich. The plane perpendicular to the magnetic axis intersects the equator at an angle of 10° 59', and in west longitudes 117° 37' and 300° 27'.

But deductions of this kind are liable to a source of inaccuracy which had long escaped observation. The bolts and other pieces of iron in the frame of a ship, ranging mostly before the binacle, very sensibly derange the bearings of the compass by their local attraction. This deviation seems to have been first remarked by

¹ The celebrated astronomer Burekhardt, by combining together a number of observations, has deduced a formula for the variation of the needle at Paris. Admitting its accuracy, the maximum will be attained in 1837, and reach to 24° 26', the period of magnetical revolution being completed in the space of 860 years.

Bayly, who accompanied Captain Cook as astronomer in the two last voyages of that great navigator. But it was accurately investigated by Lieutenant Flinders in 1795. The error thus caused is often as much as 10 degrees; and, owing to the oblique influence of terrestrial magnetism, it attains to a much greater amount in the high latitudes. When the vessel's keel lies in the magnetic meridian, the needle is not disturbed, and the divergence becomes greater in the transverse position; in every other position, the sine of the deflexion has a certain ratio to the sine of the angle of the course. A single experiment will determine that ratio, and hence the rectification is easily applied. But Barlow has proposed a very simple contrivance, to avoid the trouble of computation: It is to fix, on the deck and immediately behind the binacle, a thin iron plate, which by its proximity may counteract the diffuse attraction along the forepart of the ship. This adjustment being once attained, must suit every situation.

But the action of terrestrial magnetism is evidently the result of its intensity combined with its direction. Though the attractive force augments in approaching to the magnetic pole, yet the Dip of the Needle rapidly increases, insomuch that the late Arctic voyagers found it in Baffin's Bay only three degrees from the vertical position. The efficacy of the directive force was hence reduced about nineteen times, and the compass betrayed a deviation amounting to $70^\{\backslash circ\}$ or $80^\{\backslash circ\}$.

The study of Magnetism, so long neglected, has lately been revived with splendid success. Conjoined with observations of the length of the pendulum in different latitudes, experiments have been likewise made on the oscillation of the needle, which indicates the directive power of the great internal magnet. From a comparison of these, it appears that the intensity of the terrestrial magnetism is doubled in the ascent from the equator to the western limit of Baffin's Bay. No person has shown such ardour in pursuing the investigation of this subject as Professor Hansteen of Christiania, who not only with infinite labour collected and digested the multifarious facts, but has undertaken distant journeys for the purpose of rectifying former observations. From all these combinations, he

was enabled to detect the position of the magnetic poles of the Earth, and trace the curves of variation on its surface. But the results undergo correction as often as fresh data are procured. Hansteen infers, from experiments he made between the years 1819 and 1826 on the oscillation of the needle, that the magnetic intensity has been decreasing annually at Christiania, London, and Paris, by the 235th, the 725th, and the 1020th parts respectively. This diversified effect he ascribes to the revolution of the Siberian pole.

The remarkable discovery of Oerstedt has greatly enlarged the field of magnetic influence. A wire of any kind of metal being laid horizontally and at right angles to the magnetic meridian, to connect the opposite conductors of a Galvanic Battery, a needle either below or above it is drawn considerably to the one side or the other. Instead of bewildering the imagination with the vagaries of invisible streams, a sufficient explication of the phenomenon may be deduced from two leading principles:—1. Magnetism is in some proportion diffused through all metallic substances, owing either to their peculiar constitution or the universal dissemination of ferruginous molecules: 2. The cross wire, from its position with regard to the Terrestrial Magnet, acquires induced magnetism, but extending transversely; the under side having the virtue of a north pole, and the upper side that of a south pole. The copious infusion of that virtue is occasioned probably by the duration of the internal tremor, excited by intense electrical action, and analogous to the effects on a bar of iron or steel subjected to hammering, twisting, heating, or the fulminating shock. Hence are easily explained the diversified phases of attraction, rotation, or impressed magnetism.

Arago proved that continuous electrical sparks or discharges operate the same combinations as the Galvanic Battery. But his penetrating ingenuity soon discovered the means of augmenting prodigiously the intensity of the magnetic action, by coiling the conjugate wire into a cylindrical spiral, and thus forming a sort of magazine of transverse inferior and superior needles which unite their influence. Schwegger, pursuing the same ideas in Germany, has produced

a very simple and convenient instrument, for detecting the smallest traces of magnetic virtue.

Magnetism has again received a beautiful accession from the keen glance of Arago, whose mind embraces and enriches every department of science. If a needle be finely suspended over a thick circular plate of copper made to revolve horizontally, it is drawn aside from the meridian in the direction of the motion; and when this becomes very rapid, it will even follow the tide of circumvolution. The effect is nowise occasioned by the whirlwind raised, for the experiment succeeds best when the needle, pendant from a silk line, is inclosed within a glass case that has a bottom of mica or thin glass resting immediately over the plate of copper. The power is augmented by the proximity of the needle and the thickness of the revolving disc,—an evident proof of the attraction exerted by the accumulated ferruginous molecules.

The simplest and most satisfactory way of explaining this curious phenomenon seems derived from the principle, that, though magnetism acts instantaneously, a certain portion of time, however small, is required for the infusion or communication of the magnetic virtue. The circular plate acquires this property from the mere apposition of the needle; its axis with reversed poles lying when at rest exactly under the pointer. But if the plate be quickly turned,

the axis, not being instantly impressed, will be carried forward, and must consequently draw after it the needle, or even involve it in a continued rotation. On the other hand, if the copper remain fixed while the needle is made to vibrate, the axis of induced magnetism must always hang in the rear, and therefore retard, or tend to diminish, and soon extinguish the oscillations. Hence an important practical inference, that the needle of a ship's compass should traverse near the bottom of a thick copper box, in order to correct that excessive sensibility which is so inconvenient in ordinary seamanship.

Magnetism has been gaining accessions likewise in detail. Seebeck of Berlin proved that various substances have their magnetical state affected by the unequal distribution of heat. Becquerel described an instrument which, by neutralizing the terrestrial influence, shows the faintest shades of magnetism, and distinguishes the slowest electrical conductors. His countryman Rousseau, advancing by the same path, gave the construction of the Sideroscope, which detects the very feeblest traces of iron in bodies. This curious combination of magnetic needles has been improved and successfully applied by Leballif, who devotes the moments of respite from the drudgery of a laborious office to the ardent and unwearied pursuit of experimental science.

5. OPTICS.

THE science of Optics, next to that of Astronomy, is the noblest creation of human genius. No branch of knowledge so far transcends ordinary notions, and none has more essentially contributed to augment our perceptive powers or expand the range of observation. The sense of Touch, diffused with various intensity over the surface of the body, connects us with the proximate objects of an external world. Taste is akin to Touch, and makes us acquainted with those solvent properties of food which stimulate the animal frame. The sense of Smell is not confined to mere contiguity, but extends its information to the substances which emit from a distance their peculiar odorous effluvia. Hearing carries our sensations still farther, and intimates

the operation and the direction of those distant changes which excite the tremulous commotion of sound in our atmosphere. But the faculty of Vision is of a much higher order. It soars above the grossness of matter, transports us above this sublunary scene, and holds commerce with the skies. The other senses have gained no help from instruments, if we except the partial advantages which Hearing has derived from the speaking and ear trumpets. But Sight was refreshed by the application of Spectacles, and its powers of exploration have been advanced in a most astonishing degree by the tardy subsequent discovery of the Microscope and the Telescope. The wonders of atomic Nature seem now laid open, and all the glories of the remotest celes-

tial forms are brought under immediate observation. The medium of such intercourse is that pure empyrean stream of Light, which approaches the nearest to our conception of Spiritual Essence.

The chief properties of the rays of Light consist in Reflexion and Refraction. With the former the ancients were sufficiently acquainted, but they had only a vague notion of the connexion between the angles of Incidence and Refraction. It is somewhat singular that both the Telescope and the Microscope should have been several years invented before the law on which their construction depends, or the constant relation between the sines of those angles, was detected. This important discovery was made by Snellius, but simplified and first published in 1637 by Descartes, who greatly reformed optical science. The progressive motion of Light was next beautifully deduced from the annual anticipation and retardation of the eclipses of Jupiter's satellites, by the ingenuity of Rømer, a young Danish astronomer, who had been invited to France in 1672, and there liberally entertained for nine years, till he was recalled to his native country and loaded with lucrative and honourable appointments. One might suppose that the aberration of the fixed stars, or their apparent circular shifting in the heavens, would have been hence anticipated, as a simple inference from the motion of Light combined with the revolution of the Earth in its orbit. Such a consequence had been foreseen by the acute mind of Descartes, who therefore adopted the instantaneous propulsion of light as a fundamental principle in his Dioptrics. Were it otherwise, he maintained the stars would not appear in their true positions, and the sun would be seen eclipsed long after his conjunction with the moon. These considerations,

however, were quite overlooked, when Bradley in 1729 established his fine discovery within close limits, but on the very same grounds.¹

Astronomers had been obliged to abandon their attempts to construct telescopes of much higher magnifying powers, and to contract the apertures, and consequently to reduce the measure of illumination in order to procure distinct vision. Colour was still imagined to depend merely on the various mixture of light and shade. But Newton's grand discovery of the decomposition of the Solar Beam, first communicated to the Royal Society in 1671, and further explained by his correspondence during the three following years, changed the whole aspect of the science. The ingenuity in devising the simple and beautiful experiments with combined prisms, is not more admirable, than the force and clearness of the reasoning which deduced from them such splendid conclusions. The mild spirit of the philosopher was wounded by the attacks of petulance and contradiction; and he quietly remained, in his closet, for the space of upwards of thirty years, the sequel of those invaluable experiments, and only consented to publish the Treatise of Optics after his reputation had finally triumphed over all opposition. This resolution is much to be regretted, as it not only postponed the diffusion of genuine science, but gave occasion to the blending of some speculations which the discerning author would probably not have published in the high meridian of his intellect. Though the experimental procedure on which the work rests had been strictly inductive, yet was the composition cast in a synthetical form,—broken into a series of propositions, with their preliminary apparatus of Axioms and Definitions; which commonly involve assumptions, and so far from giving more precision to our reasonings, imperceptibly lead to the admis-

¹ In a letter of Descartes, bearing the date of 1634, he notices an experiment proposed by a Dutch correspondent, to prove that Light takes a sensible time to arrive at the eye from a distance. This was to wave a flaming torch at night, and observe its reflexion from a mirror removed only a quarter of a league. The French philosopher with his friends repeated the experiment several times, and could not discern the smallest interval between each bending of the flame and its reflexion. But even admitting the lapse of the 24th part of an arterial pulse to intervene during the short passage to the mirror and back again, or allowing the velocity of 46 English miles in a second, it would completely derange Astronomical Observations. The sun would not appear in his real position, but in that which he occupied when the rays of light which reach us began their journey. A solar eclipse would therefore be seen, not at the moment of conjunction, but an hour afterwards, the Light having to travel from the Earth to the edge of the Moon and back again, or to pass and repass the distance of 50 semidiameters, each of 3000 leagues.

sion of errors. The very term Refrangibility, which Newton, under the guise of a definition, applied to the rays of light, became afterwards a source of misconception. It evidently converted the refraction of light into a general property belonging peculiarly to the rays themselves, and therefore independent altogether of the quality or nature of the refracting medium. This might appear only a slight distinction, but it probably contributed to the hasty and inaccurate experimental inference, that in every case the refraction of the mean ray by a prism determines the corresponding refractions of the extreme rays, or that the solar spectrum is always distinguished into the same proportions of coloured spaces. Since all material action, however, is reciprocal, the refractive process cannot be referred to the single agency of the particles of Light, but must result from their mutual attraction to the substance of the transparent medium. It is not the lot of humanity to reach perfection, and the most gifted of mortals will at times betray the weakness of our nature. The subdivision of white light into seven component rays, and the distinction of the spectrum into coloured spaces adapted exactly to the intervals in the diatonic scale of music, were no doubt fine illusions which reflected the mysticism of the age. But it must now be admitted that the primary colours melt into each other by imperceptible shades, and that the spectra painted by different prisms differ widely in their relative extent, and exhibit even a diversified partition of spaces.

Certain it is that the philosophers on the Continent generally refused to admit the accuracy of the conclusions of the Treatise of Optics, and Mariotte, the most ingenious and skilful of the French experimentalists, was unsuccessful in his attempts at the prismatic decomposition of the sun's rays. His failure must no doubt be attributed to the influence of prejudice, and the stiffness of advanced age; but finally to silence all such opposition, it was judged expedient in 1716 that Desaguliers should repeat and somewhat vary the original experiments before a committee of the Royal Society at London.

Notwithstanding these few blemishes, arising merely from hasty generalization, the Newtonian

Theory of Light and Colours is undoubtedly one of the noblest efforts of human invention. It quickly modified the projects of opticians. The imperfection of the Telescope was hence found to proceed only from undue mixture of colour in the image, occasioned by unequal refraction, which no change in the forms of the glasses could remedy. The Reflexion of Light alone offered any chance of improvement. Newton had constructed a small Catoptrical Instrument with his own hands; but for many years afterwards no artist appeared that possessed sufficient skill to imitate and improve it. The ingenious Mr Hadley in 1723 was the first who succeeded in getting a Reflecting Telescope made for his private use; and we may presume that the great inventor himself, before the close of life, enjoyed the satisfaction of seeing his early ideas realized. Such was not the fortune of James Gregory, whose Reflector, more complex indeed, but likewise more commodious, had remained a mere speculation, till about this time it was manufactured and came to acquire high reputation. James Short particularly distinguished himself by the finished execution of those instruments, and for a long time he maintained the superiority of the English artists. Every improvement in the construction of the telescope contributed to the accuracy of astronomical observations.

Hadley not only gave an impulse to practical astronomy, but soon enriched its stores by that invaluable instrument the Quadrant of Reflexion, which has ultimately produced a complete revolution in the art of observing. A similar instrument was proposed by Newton as far back as the year 1669, in consequence of an unsuccessful attempt of Hooke; but the description had been laid aside and neglected till 1731, when the announcement of Hadley's invention recalled it from oblivion. The mode of finding an altitude hitherto practised at sea was liable to much uncertainty, since it required to direct the eye first to the horizon and next to the disc of the sun, though the instrument might have its position altered during this interval by the motion of the ship. The difficulty was now obviated by employing a movable index, which brought the sun's image to touch the boundary of the horizon. It is a very happy application of the simple prin-

ciple in Catoptries, namely, that a ray of light which has suffered two reflexions deviates from its course by an angle just double of the mutual inclination of the mirrors. The Quadrant was afterwards enlarged into a Sextant, it had telescopic sights affixed to it, and received its final simplification in this country from the dividing engine of Ramsden. This delicate instrument has come since into almost universal use in measuring angles; and though originally contrived only to observe the latitude at sea, it is now employed with the greatest facility in ascertaining the angular distance of the moon from the sun or a star, and thence supplying correct data for computing by the help of improved lunar tables the longitude itself.

Meanwhile the Sextant underwent a radical transformation on the Continent, being changed into an entire Circle. This was effected by the celebrated Mayer, whose very narrow circumstances drew forth all the resources of his invention. He sought to remedy the imperfection of the common Graphometer or Circumferentor, by multiplying the angle observed, and thus blending and extenuating the errors of subdivision. By this simple contrivance not only were the grosser inaccuracies obviated, but the instrument was rendered even superior to others of the finest construction. Mayer next conjoined this principle with that of Hadley, and produced the Repeating Circle, which, having afterwards received some further improvements from Borda, is now employed universally on the Continent in Goniometrical Surveys, and in the practice of Navigation and Astronomy. This instrument being complex, however, in its construction, and tedious and operose in its application, seems after many trials not to be gaining ground in England. Its chief advantage consists in reducing the errors occasioned by imperfect workmanship,

which a more skilful execution might nearly preclude.

The middle of the eighteenth century was distinguished by a capital correction of the Newtonian principle of the proportional refraction of the several rays of light, which has led to the most important practical results. It had been concluded that no combined unequal refractions could ever form a white image, because the same refractive powers which might neutralize the colours must likewise bend the converging or diverging rays into exactly parallel directions. But, dissatisfied with this theory, Euler in 1747 endeavoured to prove the possibility of destroying the coloured margin of a focal image, by imitating the structure of the eye, which he considered as a perfect optical instrument. He proposed to construct a convex lens, by joining two meniscus glasses holding water in their cavity; and expected from such a combination not only to correct the spherical aberration, but to prevent the border of colour. All his attempts, however, to obtain distinct vision, were unsuccessful. Yet he still persisted in holding it to be a Law of Nature, that not the refractions of the extreme rays, but the indices of their powers, are constantly proportional; and from this mathematical or metaphysical assumption, he strictly deduced his calculations. These speculations of Euler made some noise in the scientific world, and shook the confidence abroad in the accuracy of Newton's experimental conclusions. About this time John Dolland,¹ bred a silk-weaver, the trade of his father, a French Protestant refugee, being fond of mathematical studies, had chosen to embrace, in partnership with his son, the profession of an Optician, as more congenial to his taste. Having already acquired reputation for ingenuity and skill, he communicated in 1752,

¹ John Dolland, born in Spitalfields in June 1706. Left by the early death of his father in straitened circumstances, he was obliged to toil for the support of the family. But his thirst after useful knowledge led him to devote every spare moment to private study, and even to encroach on the hours of repose. By such severe application he made considerable proficiency in geometry and algebra, in optics and astronomy, and added to these attainments a tolerable acquaintance with the ancient languages. He became anxious to follow a profession better fitted to his genius, and in 1752 he joined his son Peter, who having quitted silk-weaving, had successfully commenced the business of optician. The discovery of achromatic glasses in 1758 established his reputation, and afforded a prospect of the large fortune afterwards reaped by the industry and perseverance of his successor. But the elder Dolland had only begun to taste the sweets of prosperity: while engaged intensely in the perusal of a new memoir of Clairaut, he was, on the 30th November 1761, struck with a fit of apoplexy, which in the space of a few hours hurried him to the grave.

to the Royal Society of London, a short paper, showing that the principle advanced by Euler was discordant with the property deducible from the Newtonian experiments. But it did not thence follow that either of these must be the true Law of Nature; and accordingly Klingenshierna, an ingenious and learned Swedish philosopher, proved that the hypothesis espoused by Dolland would not stand a rigorous analytical investigation. The English artist was therefore compelled to adopt the only sure and decisive mode of settling the question, the unbiased appeal to experiment. He soon found that his preconceptions were inaccurate, and that refraction may subsist without any fringe of colour. He formed a wedge or hollow prism with two thin rectangular pieces of plate-glass, joined at the edges and cemented to planes of brass at the ends, and having filled the cavity with distilled water, he inverted in it an acute-angled glass prism, and looking through this compound medium, he gradually widened the angle of the glass plates, till an object placed in front appeared free from any coloured border, but considerably depressed. The refraction of the water had therefore predominated over the opposite refraction of the glass, without expanding at the same time the prismatic colours. This simple and well-devised experiment established therefore, in contradiction to Newton, the important principle, that the length of the spectrum, or the dispersion of the extreme rays, is not always proportional to the mean refraction, but depends on the constitution of the diaphanous medium. The result already indicated an improved construction of the telescope, by substituting a compound object-glass, inclosing water, nearly in the manner attempted by Euler. But Dolland rejected this imperfect expedient, and sought to produce an unalterable combination with different kinds of glass: Having struck into the right path, he pushed forward to a great discovery. He tried to select among the different kinds of glass such as might effect his purpose; and after a long and perplexing research, he finally preferred the combination of crown and flint glass; the refractive power of the former being to that of the latter as two to three, while their opposite dispersions are equal.

A convex lens of crown glass of the focal length of two feet, conjoined with a concave lens of flint glass having a virtual focus at the distance of three feet, must hence form a colourless image at the distance of six feet. This capital discovery, achieved in 1757, enabled Dolland to construct refracting telescopes of much larger aperture and wider field than before, and presenting the image with such a pure brilliancy as to entitle them to the name of Achromatic. It was likewise possible to correct the spherical aberration by modifying the coalescent curvatures. But the discharge of extraneous colour is attained most completely by forming a triple object-glass, composed of a concave lens of flint glass inclosed on both sides by two convex lenses of crown glass.

The theory of Achromatic Telescopes was now complete, but it still required patience and address to carry it into successful execution. Dolland, and his son afterwards, took incredible pains in choosing their samples of glass, and acquired such exquisite skill in combining them, as to set all rivalship at defiance. The Continental Mathematicians were eager in examining the structure of those achromatic lenses, and in applying their superlative powers of calculation to define the proper forms. Klingenshierna and Boscovich, but especially D'Alembert and Euler, distinguished themselves by their most elaborate and profound investigations in this new branch of Dioptrics. Yet it may be questioned whether artists have derived any real help in practice from such vast and profuse displays of analytical research. England, following a more tentative procedure, continued exclusively for half a century to supply the world with Achromatic Glasses.

The correction of colour obtained by the ingenuity and perseverance of Dolland consisted thus in the blending of a spectrum with another of the same reversed length, but caused by an inferior refraction. Though the extreme boundaries were necessarily white, it did not however follow that the intermediate portions should be absolutely colourless. On the contrary, it seems ascertained that not only the entire expansion of the spectrum, but the relative extent of its several coloured spaces, depend on the peculiar quality of the refracting medium.

A redoubled combination of lenses might therefore mingle and destroy the secondary colours scattered through the middle range. But the light would be necessarily enfeebled by suffering those repeated refractions.

Several experimenters, in their attempts to improve the achromatic telescope, had proposed the interposition of fluids between the pieces of the compound object-glass. But the late Dr Robert Blair instituted the most elaborate investigation on this subject about the year 1787. He found all the essential oils, but especially certain metallic solutions in muriatic acid, to possess the greatest dispersive powers. A most complicated arrangement of those fluids with different lenses was first tried, and next abandoned for a less intricate combination of a triple muriatic solution, inclosed in glass shells, and invested on both sides by a semiconvex and a meniscus of crown glass. This compound appeared to produce the effect desired, not only discharging from the image the extreme fringes of red and violet, but excluding also the intermediate streaks of green and yellow. Unfortunately the liquid by degrees lost its transparency, either from the change of its own constitution or from the slow but continual corrosion of the surface of the glass. A similar attempt has been lately revived, but we fear with no better prospect of success. No confidence can be placed in the permanent transparency of any fluid medium.

Although the prismatic colours follow the same invariable order, they yet expand in very different proportions, according to the nature of the refracting substance. It is impossible, from the closest examination of the solar spectrum in a dark room, to distinguish the limits of the several coloured spaces, which appear to melt away by insensible gradations. The extreme boundaries of illumination are still more undefined. The red seems to graduate near the edge into a dull brown, while the violet spreads out by lengthening shades into blackness. Those verges are no doubt more extended in our climate, owing to the greater profusion of lateral rays sent from the white portion of the sky which encircles the sun's disc.

Three primary colours—red, green, and violet—are commonly supposed to be sufficient by their various admixture to generate all the rest.

Thus, red mingled in different proportions with green produces orange or yellow, and green suffused with violet gives blue; while red blended with violet forms crimson, that fine brilliant hue which is not emergent in the solar spectrum. In this pencilled expansion, we may thus reckon three principal colours, or more completely four, five, or six, with innumerable shades. If the whole extent of the spectrum produced by a prism of flint glass be reckoned 50 parts, the spaces occupied by the unfolded colours will be nearly represented in this series: red 5, orange 4, yellow 7, green 9, blue 10, and violet 15; or if the orange be merged in red, and the yellow in green, the expanse of the four resulting colours will be, red 9, green 16, blue 10, and violet 15. These successive spaces are all considerably different, it must be confessed, from the fanciful subdivision of the diatonic scale of music attributed to the range of the spectrum formed by water or crown glass.

It is often assumed that the numbers of the component rays of light are proportional to the breadths of the several coloured spaces of the spectrum. But the most ordinary experience contradicts that supposition; for the eye is offended by the excessive glare of red and orange, is relieved by the softness of green, and feels languid under the feeble action of blue or violet. Whatever may be the distinctive properties of the coloured particles themselves, they are evidently collected by the prism in groups of very different densities. It is hardly possible to compare the various intensities of such heterogeneous things as colours with any degree of precision; but when highly condensed, the impression they make on the eye is nearly the same as that of white light itself. On this principle the late ingenious Fraunhofer, the great improver of achromatic glasses, was enabled in 1814 to estimate nearly the illuminating powers of the prismatic rays, by examining them as concentrated in the field of a large telescope of a theodolite. He found the brightest spot on the spectrum of flint glass to lie on the verge of the yellow next the orange, and about 11 parts of its whole extension of 50, from the extremity of the red. But the most remarkable discovery he made was, that though the coloured spaces appear not parted by

any distinct boundaries, yet they are broken and subdivided by numerous white and black lines, or even dark stripes. He reckoned altogether above 600 lines; a few occur in the red, but they are multiplied in the orange, the yellow, the green, and the blue. A stripe, opened by a fine white line, divides the red; other stripes emerge at intervals between the orange and the blue; and two very broad approximating bars cross the violet. Other glass prisms, and even those filled with liquids, gave similar appearances, which were therefore not accidental, but the constant results of some law of nature. The powers of refraction thus advance not by insensible gradations, but seem to ascend with irregular bounds. This inference bears some analogy to the Newtonian hypothesis of the easy fits of reflexion and transmission.

The same judicious philosopher, combining eminent practical skill with deep and accurate science, succeeded by unwearied application in achieving the grandest improvement effected in the construction of achromatic glasses since the time of Dolland. Fraunhofer conducted personally the whole train of a large establishment; he directed the preparation and fusion of the materials, selected the proper pieces for grinding, and prescribed the due forms. The glass produced under the patronage of the liberal Bavarian government greatly surpassed in quality every other kind: It had a translucent purity, and a uniform consistency. Fraunhofer constructed with it the most perfect object-lenses, of more than eight times any former dimension, and exceeding nine inches in diameter. He had proposed to attain an aperture of ten inches or even more, and would unquestionably have succeeded, if death had not prematurely stopt his ardent career. The power and brilliancy of such magnificent telescopes transcended all conception.¹

If the rays of the coloured spaces of the spectrum be so widely diversified in their illuminating energy, they differ no less in the property

of exciting or communicating heat. Common language betrays in this respect their distinctive characters. Red and orange are termed warm colours, green temperate, and blue and violet are said to be cold. The first that endeavoured to measure the heating powers of the coloured rays of the spectrum was the Abbé Rochon, who in the summer of 1776 employed for that purpose an air thermometer and a prism of flint glass. But though he used some precautions, and repeated his experiments, the results were not very nice or consistent. Without venturing to compare the calorific intensities of the extreme rays, he yet reckoned those of a bright red as about eight times more powerful than such as gave the liveliest violet, and considered the bordering orange as the hottest spot in the spectrum. But in this country the differential thermometer, modified as a photometer, being an instrument susceptible of incomparably more precision, was applied to the spectrum of flint glass in 1798. Dividing the whole extent into four equal spaces, the calorific energies of blue, green, yellow, and red might be represented with tolerable exactness by the series of square numbers 1, 4, 9, and 16. Two years afterwards the famous Dr Herschel tried to measure the impressions made by the coloured rays on the small bulb of a mercurial thermometer, and arrived at the paradoxical conclusion, that the hottest part lies even on the outside of the red, and at a little distance beyond the extreme termination of the spectrum. It was thence inferred that there are dark rays which give only heat and not illumination, and travel in company with the solar beams, though less subject to refraction. This bold hypothesis was for a time regarded with wonder and applause; but the delicate observations of Berard, by the help of an Heliostate to direct the incident rays and give them a steady effect, soon demolished the fabric. A large circular prism or the outer ring of a huge burning glass has been since employed to mark with nicety the limit of the

¹ The Bavarian flint-glass appears free from those wavy lines or streaks which still impair the best English specimens, and displays besides one-third more of dispersive power. The famous telescope constructed for the University of Dorpat has an aperture of 9½ inches, with a focal length of 14 feet 3 inches English measure: it turns with a double parallactic motion, guided by the constant revolution of a centrifugal pendulum. This wonderful production of art, pouring a flood of light on the image, magnifies 200 and occasionally 600 times, and has enabled Professor Struve to augment prodigiously the catalogue of double and changing stars.

greatest calorific effect, which always occurs within the red space, though nearer its exterior border. The notion of dark rays of light, which enveloped the science in mystery, stands now therefore without any proof, and is utterly discountenanced by sound philosophy.

The improvements made in the application of Optics to the measuring of angles, since the invention of the repeating circle, relate principally to the micrometer. Bouguer proposed his Heliometer, or double object-glass micrometer, in 1748; but Short and Dolland, probably from a different suggestion, simplified the contrivance in 1754, by dividing a single object-glass into two equal portions. Beccaria had shown that rock crystal, like Iceland spar, has the power of double refraction; and Rochon availed himself of this property to construct in 1777 a very delicate micrometer, though Boscovich about the same time appears to have conceived a similar idea. Dr Maskelyne had more than a year before pursued a path little different, his micrometer consisting of movable glass prisms.

The French have ingeniously directed the principles of Optics to the improvement of Light-Houses. In this useful pursuit the celebrated Buffon led the way. Endeavouring to realize the performance ascribed to Archimedes, he disposed 400 mirrors, each of them half a foot square, into a frame, so as to collect the sun's rays into a single focus, and by the reflexion from this large surface in 1747 he actually set fire to wood at the distance of 70 yards. The experiment was next inverted, and a burner placed in the focus, to have its rays thrown parallel, and produce remote illumination. But for this purpose a much smaller reflector was sufficient, composed of bits of mirror planted in a spherical cavity. The final improvement consisted in hammering thin plated copper into a parabolic shape. These powerful reflectors, combined with Argand's lamp, were about thirty years since introduced from France into our light-houses.

Buffon tried likewise to concentrate the power

of refracted rays. The ancient engravers had employed spheres of glass or crystal to assist vision; but the substitution of lenses, which at the close of the thirteenth century served for spectacles, was an important advance. This figure greatly reduced the thickness of the diaphanous mass, and therefore facilitated proportionally the passage of light. But the progress of improvement seemed completed in the plan proposed by the French Naturalist to obtain the exterior surfaces of a lens by combining a number of circular prismatic segments. From the difficulty of the construction, however, this project was long retarded, and afterwards forgotten. The knowledge of such attempts was not wanted to direct the ingenuity of Fresnel¹ into the right path. He calculated the curvature of the successive concentric segments which would produce a correct focus, and advanced the construction to a regular system, on which Soleil, an expert optician at Paris, has founded a manufacture. This compound lens has displaced the reflectors from the French light-houses, and must soon, from its very superior power of concentration, become generally adopted.

The eighteenth century created a new branch of optical science, destined to measure or compare the intensities of different lights, and therefore termed Photometry. The first notions of this curious subject were given by Marie, a French Capuchin, in a small tract printed in 1700. Though this ingenious person erred egregiously in the mathematical deductions, he had probably the merit of setting to work the superior talents and skill of his countryman Bouguer, who published in 1729 an original treatise on the gradation of light. He there sets out from the obvious principle that light darting in straight lines must become dilated or attenuated in the ratio of the square of the distance from the radiant source. But the eye with a little training can with tolerable accuracy distinguish in a dark room when two surfaces presented at once are equally illuminated, and consequently the relative powers of the lights may be readily computed from the distances required

¹ The insidious advance of a consumption has arrested his career of discovery, and drawn that inventive and amiable philosopher to a premature grave.

to produce such a congruous appearance. Bouguer compared the light of a candle with the dazzling brilliancy of the sun and the soft radiance of the moon; and found that the intensity diminishes from the centre to the margin of the solar disc, but that the gradation is inverted in our satellite. He discovered also experimentally the diminution of the sun's rays in traversing the atmosphere with different angles of obliquity. The same able experimenter ascertained the loss of light, as occasioned under different circumstances, by reflexion and refraction. He then held the office of Royal Hydrographer at Croisic in Lower Brittany; but after his return from the laborious and memorable scientific expedition to Peru, he resumed his early pursuits, and prepared an enlarged edition of his optical work, which was edited by Lacaille in 1760, two years after his death. At the same time came out the systematic treatise which Lambert entitled Photometria, designed as the sequel to a small tract which this ardent and inquisitive philosopher had printed two years before on the remarkable properties of the route of light through the air. This production is very complete in its mathematical structure; but the experiments on which it rests are generally not so well devised or so nicely performed as those of Bouguer. The ingenious author seems to confide too much in a sort of filtration and adjustment of facts, as obtained by the aid of calculation. Lambert, in his preamble, admits the imperfection of those observations, owing to the infirmity and variable condition of the eye, and regrets that he had no instrument like the ther-

mometer to guide the sense of vision.¹ This desideratum was happily supplied before the close of the century; and the Photometer constructed on the principle of the minute differences of temperature is not only very delicate, but exempt from the remotest chance of error. The method of shadows proposed by Count Rumford in 1796, may be reckoned an improvement on Bouguer's procedure, but it is liable to the insuperable objection of depending altogether on the patient attention of the observer and his uncertain sharpness of sight.

The present century opened with the most auspicious prospects, and was soon distinguished by one of those great and unexpected discoveries which form an epoch in the progress of science. Such may be regarded the fine detection of the polarized rays of light by Malus, a French officer of Engineers, and one of the most ardent disciples of the Polytechnic School; which in so short a period drew together and put into rapid motion a far greater mass of mathematical and physical talent than any of the older seminaries in Europe. The double refracting property of Iceland Spar, or the crystallized carbonate of lime, noticed by Bartholinus and accurately investigated by Huygens, had likewise engaged the penetration of Newton, who concluded that the ray which suffers the unusual or extraordinary refraction must have its opposite sides affected by some virtue like magnetism which gives them a tendency to polarity.² This curious and acute remark appears to have lain neglected near a hundred years among the crowd of ingenious speculations started in the optical queries. Ma-

¹ Optandum certe est, ut excogitaretur Photometrum thermometro analogum, quod luminis expositum ejus intensitatem atque claritatem indicaret. He subjoins, Enimvero ipsius oculis ejus silet exemplar, quippe pupillae apertura luminis sequitur magnitudinem ac claritatem, et utrique sese accommodat. At magnopere dubitandum artem in hoc negotio naturam posse imitari. (Photometria, sive de Mensura et Gradibus Luminis, Colorum, et Umbrae, p. 7.)

² The whole passage deserves transcribing. "The unusual refraction of island crystal looks very much as if it were performed by some kind of attractive virtue lodged in certain sides both of the rays and of the particles of the crystal. For were it not for some kind of disposition, or virtue, lodged in some sides of the particles of the crystal, and not in their other sides, and which inclines and bends the rays towards the coast of unusual refraction, the rays which fall perpendicularly on the crystal would not be refracted towards that coast rather than towards any other coast, both at their incidence and at their emergence, so as to emerge perpendicularly, by a contrary situation of the coast of unusual refraction, at the second surface; the crystal acting upon the rays after they have passed through it and are emerging into the air, or, if you please, into a vacuum. And since the crystal by this disposition or virtue does not act upon the rays, unless when one of their sides of unusual refraction looks towards that coast, this argues a virtue or disposition in those sides of the rays, which answers to and sympathizes with that virtue or disposition of the crystal, as the poles of two magnets answer to one another. And as magnetism may be intended or remitted, and is found only in the magnet and in iron; so this virtue, of refracting the perpendicular rays, is greater in island crystal, less in crystal of the rock, and is not yet found in other bodies. I do not say that this virtue is magnetical; it seems to be of another kind: I only say, that whatever it be, it is difficult to conceive how the rays of light, unless they be bodies, can have a permanent virtue in two of their sides, which is not in their other sides; and this without any regard to their position to the space, or the medium, through which they pass." (Optica, book III. query 29.)

lus had early turned his attention to the more difficult problems in optics, and after his return, with a shattered constitution from an absence of several years spent in foreign service, obtaining more congenial employment at home, he resumed with enthusiasm his favourite studies, and devoted to them every moment he could spare. While intensely occupied with the investigation of the phenomena of double refraction, a fortunate accident occurred, which disclosed to him a new and splendid field of contemplation. In one of his frequent visits to the Observatory, during his residence at Paris, he chanced in the summer of 1809 to be struck with the brilliant reflexion of the setting sun from one of the windows of the Luxembourg Palace, on looking at the appearance through a prism of rock crystal which he slowly turned round, and remarked with surprise that one of the images changed regularly to brightness from obscurity. Next morning he repeated this

observation with the same results, and soon found that light reflected at a certain angle from the surface of glass acquires the same character as the extraordinary ray in the double refracting prism. Water showed a similar disposition, but at a different angle of incidence. It was not difficult to trace the law through the various reflecting surfaces; but Malus extended his researches farther, and pursued them with rapidity and success. But, in the midst of this career, he was prematurely carried away from his friends and the philosophic world, by a lingering disease, on the 23d of February 1812. The subject of Polarity has been since carried forward by several eminent experimenters both abroad and at home, particularly by Arago, Biot, and Fresnel, and by Brewster, Seebeck, and Herschel; but it still wants the simplicity and evidence which always mark the perfection of science.

6. DOCTRINE OF HEAT.

HEAT is the great principle of all internal motion. Its various changes and gradations determine the growth of plants and the expansion of animated beings. The knowledge of the production of fire was the earliest of human discoveries, and already distinguished our species in its lowest condition from the brute tenants of the forest. The application of that element has most essentially contributed to the advancement of the arts and the general progress of society. It has contributed prodigiously to the increase of population, by converting the crude produce of the soil into nutritive and wholesome aliments. Applied in a higher degree, it has aided industry in fashioning the various utensils necessary for the comforts of life; but urged to the greatest intensity, it has enabled man to fuse the metallic ores, and forge those efficient tools by which he controls the powers of nature, and renders them subservient to his convenience. What a vast interval in the range of progression from the rude savage, that laboriously kindles two sticks by rubbing them, to the experienced engineer, who combines all the deductions of philosophy in wielding and directing the resistless force of steam!

The Doctrine of Heat has, in the course of the eighteenth century, been advanced to the rank of a science. Its transfusion through the mechanical arts has communicated a grand movement to society, and wonderfully augmented our national wealth and resources. That a subject so nearly concerned with the wants and comforts of human life should have remained during the lapse of ages in a state of mere infancy, might furnish matter for grave reflection. The science is essentially experimental, while the ingenuity of the ancients was expended in framing loose and airy visions. Fire they held to be the Fourth Element, which, by its extreme levity, soared to the highest place in the Heavens, and spread its lambent ethereal essence over the boundless regions of space. Being regarded of pure divine origin, the sparks of the celestial flame were believed to impart animation to the beings of this nether world. But the Heat which commonly pervades the terrestrial bodies, and feeds our culinary fires, was considered by the sages of antiquity as variable in its constitution, and of a lower and perishable nature.

Heat was viewed by Aristotle as a peculiar element, composed of minute particles in continual agitation. Bacon, after a very prolix and confused exemplification of his formal method of pursuing induction, arrived at a conclusion nearly similar, though less intelligible, that Heat consists in a certain expansive motion. But other philosophers, and especially the alchemists, entertained juster notions of the subject. They conceived Heat to be a material substance, of extreme subtlety, diffused in various proportions internally through all bodies. Unfortunately those theorists were not very consistent in their speculations. The different applications of fire they considered as not merely varying in degree, but quite distinct in kind. The heat of a furnace and a stove, that of a sand and a water bath, and that of putrefaction and fermentation, seemed in their apprehension all separate species; and animal heat was regarded by the Arabian physicians as a finer essence, endued with invigorating and restoring virtue.

So long as such notions prevailed, it is evident that no real advances could be made in the knowledge of the properties of Heat. The first thing required was to find a method of ascertaining its intensity; and for this invention we are indebted to Sanctorio, Professor of Physic in the University of Padua, the celebrated discoverer of Insensible Perspiration, who, near the end of the sixteenth century, laboured with ardour and success in improving Medical Science by the application of Mechanics. Hero, of the Alexandrian School, had anciently described a curious machine, which, by the alternate expansion and contraction of included air during day and night, produced certain reciprocating motions; the Paduan physician seized the principle, and constructed an instrument on a small scale, chiefly adapted to measure the temperature of the human body during fevers. It consisted of a pretty large ball of glass, terminating in a long narrow stem, which was inverted and plunged perpendicularly in water or coloured liquid. For the sake of convenience, the ball was sometimes flattened on the one side, and the stem rendered more compact by serpentine involutions. With this instrument Sanctorio tried to measure the heat of the rays shot from

the sun, and even that of the lunar beams, which were fancied then to have a gelid influence.

The Air Thermometer, as it is now called, was about twenty years afterwards, though probably without any communication, reproduced by Drebbel of Alcmær in North Holland, who carried it, with other ingenious contrivances, into England during the latter part of the reign of James I. The instrument being observed to be affected by the changes of weather, was hence called a Weather Glass. In fact, it indicated only the elasticity of the air, which depends on the blended operation of two separate causes, the degree of temperature, and the varying force of atmospheric pressure. But the latter influence was yet unknown, nor till the great discovery of the Barometer by Torricelli could it be distinctly traced. The insufficiency of the Air Thermometer was then perceived, and another fluid substituted, which sensibly expands by heat while secluded from the compression of the atmosphere. The Florentine Academicians, about the year 1655, first constructed an instrument of that kind, consisting as before of a glass ball, but filled with spirit of wine, and having its stem of a smaller bore, sealed hermetically, or melted at the top by the flame of a blow-pipe. Italy had thus the honour of inventing both those instruments, the Thermometer and Barometer, the most important auxiliaries of modern science. Three different forms of Thermometers, constructed for the Academy del Cimento, were copied and circulated over Europe. But the great object now was to procure an exact correspondence, by fixing on some standard scale. The Italians began the divisions at the cellar heat, supposing this to be uniform; and they marked other points by plunging the ball in certain chemical mixtures. Boyle proposed the freezing of the oil of aniseed for the lower point; and Halley recommended the ebullition of alcohol as the limit of the scale. The great advantage of deriving a point from the congelation of water was not at first perceived. Many observers believed it to vary under different circumstances, being deceived by the fact that water will bear a greater cold before it shoots into crystals, though the temperature at which ice or snow begins to

melt is always the same. Guericke was the first who started from the thawing point, in dividing the scale of a large spirit of wine thermometer. To regulate the divisions, two different ways occurred; either to assume a higher point, or to distinguish into small portions the capacity of the ball. The latter was the mode adopted by Newton, who recreated his mind with chemical experiments. Employing linseed oil, which expands largely, and bears a great heat, he subdivided the capacity of the glass into 10,000 equal parts; and beginning from the point of thawing ice, he found the liquid to expand 256 at the heat of the human body, 725 at boiling water, and 1516 at melting tin; but preferring smaller numbers, he assumed 12 for blood heat, and reduced the other numbers in the same proportion to 34 and 71. This method was quite philosophical, but of very difficult execution. A correct and ingenious process for determining the scale of heat was proposed in 1694 by Renaldini, a learned Italian mathematician. Taking the freezing and boiling of water for the extremes, he found the intermediate points from a mixture of ten parts, composed progressively of melting snow and water just ceased from ebullition, the ingredients being 1 and 9, 2 and 8, 3 and 7, &c. About twenty years afterwards, Brook Taylor and Hauksbee applied that ingenious mode to ascertain whether the expansions of certain fluids are exactly proportional to the corresponding accessions of heat.

It had been proposed by Roemer to fill thermometers with quicksilver, and Halley adopted the suggestion. But instruments of this kind being observed to disagree with the common spirit of wine thermometers, the cause of such discrepancy was now detected. Alcohol swells with increasing progression on the application of heat; while mercury shows the valuable property of expanding with an uniform advance.

After the point of thawing snow had been fixed, the degrees of the scale were generally derived from the decimal subdivision of the capacity of the ball. But this absolute bulk is not the true basis of the scale, since the expansion indicated by the thermometer must be the excess of

the dilatation of the liquid above that of the glass which contains it. The mensuration of the ball hence became troublesome and liable to much uncertainty.

The first who succeeded in constructing thermometers with adequate skill was Fahrenheit. This ingenious man had been a merchant at Dantzic, and through misfortune failed in business; but having a good taste for mechanics, he removed into Holland, and settled as a philosophical instrument maker at Amsterdam. He began with spirit of wine thermometers, which he formed much smaller and neater than had been attempted before. But he soon preferred quicksilver, and having found it to expand from freezing water to blood heat, about 60 parts in 10,000, he assumed the number 64, and obtained the degrees by repeated bisections. In this practice he was confirmed, on observing what he considered as extreme cold, to descend just through half that space, or 32 degrees. From a mixture of water, ice, and sal ammoniac, the scale commenced; 32 degrees were allotted for the interval to ice-water, and 64 more for the ascent to blood heat. But he afterwards enlarged the range, and assumed another point from the limit of boiling water, which he placed at the 212th degree in the mean state of the atmosphere, though liable to some variation from the change of barometric pressure. Such was now his confidence in the delicacy of the construction, that he proposed the thermometer as an instrument for ascertaining the heights of mountains from the depressed temperature of boiling water; a very simple method, which has been lately revived by the reverend Mr Wollaston.

The thermometers manufactured by Fahrenheit being remarkably neat and convenient, were, notwithstanding their arbitrary subdivision, widely dispersed over Germany, and introduced into Britain by the practitioners in physic, who at that period usually resorted to Holland for medical instruction. But most strangely in France a very unwieldy and inaccurate instrument maintained its ascendancy. The thermometer of Reaumur had a ball near four inches in diameter, which was filled with spirit of wine diluted with water. The scale began at the term of congelation obtained by an artificial process, but

the highest point was marked by plunging the instrument into boiling water. But it is evident that the liquid could never reach this degree of heat, but would stop several degrees lower at the limit of its own ebullition. Besides, so large a mass receiving the impression very slowly, would not indicate with any precision the temperature of the surrounding medium. Reckoning the ball to contain 1000 parts of the fluid, the whole expansion was found to be $87\backslash frac\{1\}\{2\}$, and Reaumur therefore took the round number 80 for the extent of his scale. The same artificial subdivision is still very generally retained, though transferred to the mercurial thermometer, which is exempt from the glaring defects and errors of the original instrument.

Provided with Fahrenheit's thermometer, the philosophers of Holland and Germany made some progress in the science of heat, while France appeared to rest satisfied with mere speculation. But experimenters were too eager in hastening towards general conclusions. While the theory of gravitation possessed the charm of novelty, it seemed natural to suppose that heat is distributed among bodies according to their density or mass. This inference, however, is often very remote from the fact. Boerhaave found that a pound of mercury at $200^\{\backslash circ\}$, so far from communicating to a pound of water at $32^\{\backslash circ\}$ half the excess of heat, or $84^\{\backslash circ\}$, as another pound of water at that high temperature would have done, gave only an addition of $4^\{\backslash circ\}$; so that twenty pounds of mercury had scarcely as much heating influence as a single pound of water. It would have required, according to his experiment, even a greater bulk of mercury by one-half more, to produce the same calorific effect as water. Had he performed the process with due precaution, he might have observed that hot water exerts as much influence as mercury of 32 times its weight or $2\backslash frac\{1\}\{2\}$ times its volume. But this celebrated chemist, satisfied with gross results, and mistrusting perhaps his own experiments, hastened to the conclusion that heat is distributed among bodies in

proportion to their bulks, or maintains an equal diffusion through space. This principle was likewise adopted by Musschenbroeck, and the simplicity of the proposition seems to have procured it a very general reception.

Wolffius, who about this time was spatiating over the vast field of physical and moral science, and forming a new philosophical sect in Germany, taught that bodies had two sorts of interstices, the one filled with air and the other charged with fire. He attempted to explain why iron acquires heat sooner than wood, and why the solution in nitre occasions cold; but neglecting to follow experiments, he produced only vague and shallow speculation. Equally visionary, but enlivened by French vivacity, was Mairau's Dissertation on Ice, published in 1740.

The most judicious writer that had yet appeared on the subject of heat was Dr Martine¹ of St Andrews, who studied medicine on the Continent, and, like the accomplished physicians of that period, cultivated learning and general science. His acute Essays, published in the years 1739 and 1740, not only corrected the different thermometric scales, but enriched philosophy by several well-devised and original experiments. Unfortunately the career of this promising genius was very short. Having in the pursuit of his profession accompanied Admiral Vernon in the fatal expedition against Carthage, he perished by a malignant fever. Martine did not consider with sufficient attention the unequal distribution of heat among bodies, but he carefully investigated the variable rates of heating and cooling of different substances. He noticed especially the facility with which mercury, in comparison of water, receives or parts with its heat. He conceived the process of cooling to be commonly retarded by a sort of covering or atmosphere of warm air closely encompassing the body; and he endeavoured to distinguish two sources of the dispersion of heat. Had such investigations been steadily prosecuted, they must have led to interesting results.

¹ He was the son of George Martine, a small Fife proprietor, who having officiated as chamberlain to Archbishop Sharp, wrote a credulous and bigoted account of the See of St Andrews. The youth displayed the principles of the defunct Episcopacy, by heading a riot of some students at the College of St Andrews, on the breaking out of the feeble rebellion in 1715.

About this time, the practice of making observations with the thermometer had become pretty general in England, in consequence of the diffusion of the very neat and accurate instruments manufactured by Fahrenheit and Prins, at Amsterdam, which were likewise imitated in London by Wilson, who afterwards established a flourishing type-foundry at Glasgow, and held the professorship of Practical Astronomy. These thermometers were filled with quicksilver, and only four or five inches long, the stem clasped by a folded paper scale, engraved with successively bisected divisions, and the whole inclosed in a thin glass case. But the most philosophical construction of the thermometer was now proposed by the learned Celsius, who filled the chair of Astronomy in the University of Upsal, and had borne an active share in the operation of determining a degree of latitude on the verge of the Arctic Circle. He started from the point of congelation, and reckoned just a hundred divisions upwards to that of ebullition. This simple scale, commonly used in Sweden and Denmark, is the same as that which was afterwards adopted by the French under the name of centigrade, in their complete and elegant system of decimal weights and measures. It is singular that Deluc, in rectifying Reaumur's thermometer, should have retained such an artificial subdivision as that by fourscore. But this philosopher had the merit of ascertaining that mercury dilates almost uniformly with equal additions of heat, while the expansion of the spirit of wine is variable, and ascends with an accelerating progression. A thermometer formed of alcohol, whether diluted or not, is hence exceedingly inaccurate, unless it be regulated by the comparison with a mercurial one.

The practice of using thermometers filled with quicksilver led to the curious discovery of the congelation of that dense liquid. Professor Gmelin, who had been sent with a party of naturalists to explore the remote regions of the Russian empire, observed in Siberia, on the 16th of January 1735, the mercury to sink in the thermometric tube to a point which corresponds to the 120th degree below the zero of Fahrenheit's scale, while it changed into a solid mass. But the artificial freezing of mercury was first per-

formed by Braun, Professor of Philosophy at Petersburg, in the month of December 1759, when the natural cold seemed to be $34^\{\backslash circ\}$ under the beginning of the scale. Having plunged the bulb of a thermometer into a mixture of pounded ice and sal ammoniac, the quicksilver appeared to sink progressively to $350^\{\backslash circ\}$, and then to consolidate. But such enormous cold exceeded the bounds of credibility, and Æpinus soon discovered the source of this illusion. Employing tubes of a wider bore, he observed the quicksilver, drawing after it a concavesurface, to retreat as it congealed, and sink into the rest of the liquid mass. The semi-metal must therefore have suffered a large contraction in the act of freezing, and become much denser. Yet twenty years elapsed before the point of mercurial congelation was correctly ascertained. On the 16th of January 1780, Von Elterlein of Vytegra, a town of Northern Russia, in the latitude of $61^\{\backslash circ\}$, found that some pure quicksilver exposed in a porcelain tea-cup during the night had become solid, but began to melt again when the thermometer rose to 40 degrees below the zero of Fahrenheit's scale. This limit was more decisively fixed, at the suggestion of Dr Black, by Hutchins, then governor of Hudson's Bay, who repeatedly froze mercury during the winter of 1781-2, by the application of artificial cold. In rigorous climates, or for measuring very depressed temperatures, it hence becomes necessary to employ thermometers filled with alcohol, which has never been made to freeze.

The cold occasioned by the evaporation of water and other liquids in the more genial regions had been known and turned to domestic use from the earliest ages. In the northern countries, that property, being less apparent, was commonly disregarded; but the thermometer now detected changes which might elude ordinary observation. It was soon remarked that, after dipping the bulb in water or alcohol, the quicksilver constantly sunk in the stem. About the year 1750, Dr Cullen, then Lecturer in Chemistry at Glasgow, in repeating some experiments with a pupil, remarked that a thermometer with a wetted bulb showed a much greater depression of temperature under the exhausted receiver of an air-pump. But he did not pursue the observation farther, or attempt

to investigate the theory of evaporation. This interesting subject remained long afterwards involved in deep obscurity.

The temperature of water exposed to heat appears to rest stationary at the two distinct limits of freezing and boiling, which serve to mark the points of the thermometric scale. Though the fact had been long well known, it was first examined with attention in 1760 and the three following years by the celebrated Dr Black, who had already distinguished himself by unfolding the nature and properties of fixed air. He rightly inferred that a certain additional share of heat must at each station unite with what is already lodged in the substance of the ice or the water; but not aware that he was really framing an hypothesis, he termed the former latent and the latter sensible heat. Strictly speaking, however, all heat is latent or concealed from the senses, and never exhibited in a detached form. The thermometer indicates merely the heat contained in its own bulb, and not necessarily that of the body with which this communicates. "But heat is evidently not passive; it is an expansive fluid, which dilates in consequence of the repulsion subsisting among its own particles; and it would spread indefinitely through space, if it were not fixed or retained by the counterbalancing attraction of the substances which absorb it. Were each corpuscle to exert the same action, this universal fluid would be disseminated among bodies, exactly in proportion to their respective quantities of matter. The mutual adhesion depends, however, on the density of the substance, modified by its degree of inherent disposition to combine. A sort of affinity or specific attraction for heat is thus produced, evidently proceeding from the peculiar nature of each body. To trace its immediate origin is not more possible perhaps than to discover the source of other physical properties. Yet there appears some tendency towards a general principle: the particles of heat, like those of all expansive fluids, have their repulsion diminished in proportion to their mutual distance; while the molecules of the containing substance suffer the corresponding decrease of attraction after a slower ratio than the

spaces of internal separation. Heat has therefore a narrower range of density than the bodies with which it combines. It holds a sort of middle station, and is distributed according to the quantity of matter, joined to the consideration of the space which this occupies; that is, it obeys some compounded relation of the weight and the bulk. Hence the denser bodies receive a proportionally smaller share of heat. Thus, a pound of metal contains less heat than one of stone; this, less than an equal weight of liquid; and this last, still less than a pound of any species of gas."

"When two bodies are united chemically, the compound has an attractive force generally different from that of the mean result. Hence a corresponding portion of heat is, during the act of coalescence, either absorbed or evolved. Thus, water, on being joined to sulphuric acid, occasions an extrication of heat, because the diluted acid exerts less power of adhesion than did its ingredients. And, for an opposite reason, the muriate of ammonia, in dissolving, is attended by an absorption of heat, or an apparent production of cold."

"Every substance capable of assuming different states of constitution betrays likewise analogous variations of attractive force. When a solid body melts into a fluid, and thence passes into vapour, each transit is marked by an augmentation of that force, and is therefore accompanied with a corresponding absorption of heat; during which process, the temperature must evidently remain stationary. Thus, a lump of ice transported intensely cold into a close apartment, will grow warmer by regular gradations, till it begins to thaw, and then the farther accumulation of heat will appear to be suspended; and if the water so formed be poured into a covered pot, and set over a steady fire, the temperature will again rise uniformly, till it reaches the limit of boiling, when the act of conversion to steam will henceforth absorb the whole affluent heat; yet the temperature will mount still higher if the escape of the vapour be prevented, but which soon acquires such prodigious elasticity as to burst whatever obstacle can be opposed to it."¹

¹ Experimental Inquiry into the Nature and Propagation of Heat, pp. 529-531.

The first that made any advance towards such views of the distribution of heat was Irvine, who had been assistant to Black at Glasgow. Unfortunately he seems to have introduced the term Capacity, which involves a crude notion, as if heat were merely passive, filling up the interior pores or internal vacuities of bodies. But, a few years afterwards, Wilcke of Stockholm, directed, he says, by the views of his colleague Klingenstierna, a mathematician and philosopher of the first order, examined the subject philosophically, and performed several accurate experiments. Thus, he found that water at 98° of Celsius, poured upon an equal weight of snow, had its temperature reduced to 13°, and that three parts of water at 40°, joined to one of snow, produced the common temperature of 12°. These trials gave both the same result, or 73° of Celsius, corresponding to 129.6° on Fahrenheit's scale, for the increase of heat required to convert snow into water.¹ A similar conclusion was obtained by filling equal tin vessels with equal parts of snow and snow-water, and placing these within a larger one full of boiling water, and set over a fire; the moment the chill water rose to 72°, the snow had all melted.

In 1780 Lavoisier, assisted by Laplace, and employing his Calorimeter, ascertained the heat evolved from different bodies during their various changes, from the several quantities of ice which it melted. This seemed to be a very simple and correct mode of investigation; yet was it liable to considerable inaccuracy, from the large portion of water which still remained adhering to the mass of pounded ice. But the French chemists rather confused the subject, by mixing up incongruous principles. Embracing the theory of Capacity, they likewise tacitly adopted the hypothesis of sensible and latent heat, only changing these terms by a sort of appropriation into free and absolute, while the poverty of their language suggested for that fluid such an imperfect etymon as Caloric.

The best series of experiments on the distribution of heat among different bodies was

performed before the year 1784 by Professor Gadolin of Abo, who, rejecting the notion of Capacity, introduced the unexceptionable expression, Specific Heat. One of the most beautiful consequences derived from this theory, was the determination of the absolute zero or lowest point in the scale of Heat. Two different ways of solving the problem led to nearly the same results: 1. From the measure of Heat absorbed in the melting of ice, and 2. From the quantity evolved in the mixture of certain liquids. Thus, it being found that the heat contained in water is to that of ice as 10 to 9, and that 135 degrees are required for the equilibrium of the liquid constitution, it follows that the water just formed must actually contain 1350 degrees of heat, or that the real scale would descend to the enormous depth of 1315° below the beginning of Fahrenheit's divisions. Again, the concentrated sulphuric acid holding only six tenth parts of the heat of water, the arithmetical mean of a mixture of one part of it by weight with three of water would evidently be nine tenths; but suppose the specific heat of that compound were only eight tenths, while the thermometer rose on the affusion 169°, or from 32° to 201°, the whole extension below the point of congelation would be 8° × 169°, or 1352°. This experiment is susceptible of greater accuracy than the former, and deserves to be repeated in different proportions. Considerable discrepancies have hitherto affected such observations, arising probably from the want of sufficient precaution in the manner of conducting them.

Dr Crawford followed nearly the same train, and was enabled, by employing finer instruments, to obtain a greater degree of precision. He took, on the whole, a philosophical view of the subject, but sacrificed his better judgment to popular opinion, by retaining also the hypothesis of Latent Heat, "the ideas associated with which have spread a cloud of mystery and paradox most unfavourable to the progress of real science." It is to be regretted that a similar confusion of ideas should still pervade the writings of so

¹ In the first experiment the hot water lost 93° — 13°, or 85°, while the snow water gained only 13°, or had absorbed 72°; in the second experiment the hot water had transferred 3 (40 — 12) or 84°, and consequently 84° — 12°, or 72°, was spent in liquefying the snow. If every precaution were used, the measure of heat absorbed would amount to 75° of Celsius, or 135° by Fahrenheit's scale.

many of the chemical philosophers. Crawford had expected much accuracy from thermometers with large bulbs and minute subdivisions; but these advantages were more than overbalanced by the want of delicacy to transient impressions. On a revision of the experiments from which he sought to ground his theory of animal heat, he had occasion to change the numerical results.

About this time shone forth the brilliant and original genius of Scheele, who not only expanded the boundaries of chemistry by his various inventions, but enriched other departments of physical science. In particular, he bestowed a keen glance on the subject of the emission of Heat.

That Heat, and even the opposite influence, impression of Cold, admits of Reflexion, like light, is a property long known, and had been distinctly announced by Baptista Porta, in the sixth edition of his Natural Magic, printed about the year 1590. This remarkable experiment was more particularly described in 1632 by Cavalleri in his tract on the Burning Speculum: it was repeated and varied in 1658 by the members of the Academy del Cimento, who received the impression on the bulb of an air thermometer. The concentration of cold in the focus of a metallic reflector was again tried, during the severe winter of 1740, by the academicians of Petersburg, though with rather doubtful results, owing to their ignorance of the true circumstances on which the success depends. But, after an apparent oblivion of near forty years, the experiment was revived with striking effect at Geneva, and continues to hold a conspicuous place in every course of physical lectures.

Mariotte, as far back as the year 1682, had made a capital remark, which some late discoveries have revived and explained. That able experimenter, in concentrating the heat of a fire by help of a burning mirror,¹ found the effect to cease when a piece of glass was interposed

behind the focus. A similar experiment was about the same time made by Hooke, though not published till very long afterwards. This ingenious philosopher showed that a glass mirror which reflects light will not repel heat. But these interesting facts lay buried for a century amidst a multitude of scattered observations, and seem to have first attracted the notice of Sanssure and Pietet, both of whom, and especially the latter, carried the investigation to some extent. They still wanted the connecting principle, which should embody and reanimate the mass.

Scheele pursued a similar path, but was little versant in the history of science, and had none of the advantages of an improved apparatus. Endowed with original and inventive powers, he followed the current of his own conceptions, and consulted always the book of Nature. An examination of the circumstances disclosed in the management of stoves and chemical furnaces, induced him to consider Heat or Fire as existing under two distinct modifications; one which rises and spreads by a successive transfer among the ambient substances, and the other which streams immediately from its source in rectilinear directions. The latter he designated by the phrase Radiant Heat, which has since become a favourite appellation. If this expression indicated merely the tendency of Heat, under certain circumstances, to disperse in straight lines, it would be quite unexceptionable. But in technical language, the Radiation of Caloric associates an obscure idea of a subtle fluid invested with vague or uncertain properties.

The common thermometer comprises, between freezing and boiling mercury, the space of scarcely 700 degrees; which is evidently but a small portion of the vast range of heat. It was most desirable, therefore, to have an instrument of greater extent, adapted especially to the ascending scale. The Pyrometer, composed of metallic

¹This must have been a metallic reflector: an ordinary mirror, or one of silvered glass, will scarcely collect any heat at all. Indeed the art of applying a metallic plate to the surface of glass is a modern invention. The earliest method was the Dutch, which consisted in pouring melted lead intensely hot over a sheet of glass softened nearly to fusion. But the adhesion was afterwards more easily obtained in the ordinary way, by applying tinfoil to the glass, and dextrously flooding it with mercury, to form an amalgam. It is from this shining metallic coating that light is chiefly reflected, having penetrated and again escaped through the substance of the glass. But with regard to Heat, a very small reverberation takes place at the anterior surface of the glass, and the effect is exactly the same if the silvering be rubbed off. It would prevent much confusion were chemical writers to appropriate the term speculum to metallic reflectors.

bars, answered, though very imperfectly, that purpose. But the elegant contrivance of Wedgwood, the celebrated improver of English pottery, promised the greatest advantages, both to science and the chemical arts. It consisted merely of a small cylindrical piece of finely prepared clay, which bears the most intense heat, and yet regularly contracts in the fire. Assuming that the intensity of heat must cause a proportional contraction, he measured this by sliding the thermometric piece within a tapered groove, its width at the one end being to that of the other as 5 to 3, and its whole length divided into 240 equal parts. By comparing the dilatation of silver, it was found that each of these divisions answers to 130 degrees of Fahrenheit's scale. Hence the melting points of silver, gold, and cast-iron, and those of the fusion of flint-glass and of Chinese porcelain, being respectively 28, 32, 130, 114, and 156, by Wedgwood, would correspond to the enormous numbers 4717°, 5237°, 17977°, 15897°, and 21257°.

These results, however, appear far to transcend the truth. The principle of the instrument is totally different from that of the thermometer. It does not mark the transient impression of heat, but the change or permanent effect produced on the bit of clay by baking, or the exposing it during a certain time to the action of the furnace. Hence the very ingenious method of discovering the degrees of heat which the Chinese and the ancient Etruscans employed in their potteries, by observing the moment when a specimen of their ware subjected to an increasing fire began to contract anew. That contraction is undoubtedly caused by the expulsion of a part of the water held so obstinately by the clay, the repulsive force excited by the accession of heat controlling the adhesion of the moisture. It may be inferred that the clay attracts the remains of latent humidity with a force always increasing; but the repulsion infused by the conversion of the expelled particles into steam ascends nearly in a geometrical progression with equal accessions of heat. A nearer approximation to the scale of temperature might probably be derived from the logarithms of the very large numbers stated above.

It remains however still an important desideratum to construct a thermometer that shall accurately measure the higher degrees of heat.

The fusible metal, composed of lead, tin, and bismuth, has been suggested as an expansible substance, which may be heated to near the fusion of copper; but, in congealing again, it unfortunately swells suddenly, and would burst its glass ball. A combination of metallic bars has likewise been proposed, though no plan for effecting this has yet been devised, sufficiently simple and reducible to practice.

Metallic thermometers have indeed been constructed, which indicate with great delicacy the ordinary temperature, and all the sudden changes of heat. These consist generally of thin slips of steel and brass, united together, and formed into circles or spirals. The most elegant instruments of this sort were executed at Paris, by the late ingenious artist Breguet.

It was an object of much greater consequence to construct a register-thermometer, which should mark the greatest heat and cold during the absence of the observer. This had been attempted by different experimenters in the course of the eighteenth century, but was first carried to any tolerable perfection about the year 1780, by Mr Six of Canterbury. These registers consisted of spirit of wine thermometers, with compound reverted stems, partially filled with quicksilver, which carried small bits of steel to indicate the limits of ascent. With such instruments the inventor made some curious observations on local heat, or on the differences of temperature at moderate altitudes during the alternations of day and night. He particularly remarked the influence of a clear sky in chilling the ground.

The simplest and most commodious register-thermometers, however, were a few years afterwards contrived by Dr Rutherford, an ingenious physician, who resided on his property in the neighbourhood of Kinross. They turn on a pivot, the stems being placed nearly horizontal, and only reclined occasionally for adjustment after the heat of the day. The minimum thermometer is filled with alcohol, and the maximum one with mercury, the tubes of both being rather wide, with bits of enamel swelled at each end, which are drawn down the slanting tube by the retreat of the alcohol, or pushed upwards by the expansion of the mercury. Rutherford's thermometer was first happily applied by his neighbour Dr Coventry, the well-known

Professor of Agriculture in the University of Edinburgh, in ascertaining the progress of heat into the ground, and its effects on the germination of the seeds of the cerealia. Some curious facts have thus been detected regarding the influence of an herbaceous covering to retard the penetration of frost.

The thermometer soon afterwards received a most important modification, which renders that instrument extremely sensible to the alterations of local heat. This improvement consisted in adapting the instrument to measure, not the actual temperature of any spot, but its occasional change or variation. The extent of the scale being limited, the dimensions of the degrees could be proportionally enlarged.

The construction of the Differential Thermometer was early suggested by a close analysis of the phenomena of evaporation. It is well known that a wet sponge suspended in the air loses gradually its water, and during this dissipation it continues colder than the encircling medium. The depressed temperature was commonly attributed to a process of rapid evaporation, which seemed to result from the combined action of various causes,—the heat of the air, its state of dryness or rarefaction, but especially the swiftness of its circulation. Halley had, near the close of the seventeenth century, advanced the rational hypothesis, that moisture is exhaled from the surface of water by a sort of chemical solution in the atmosphere. The same opinion was about the middle of the following century proposed and expounded by Le Roy of Montpellier; but though supported by concurring analogies, it might appear perhaps too simple for the general taste. The fanciful notion, first started by Derham, that evaporation is occasioned by very minute vesicles or hollow spherules of water, charged with highly attenuated air, which rise and float in the atmosphere, proved more acceptable to a wondering public. The vesicular theory afterwards obtained a wide currency from the philosophical reputation of Saussure, who considered it as countenanced by some dubious optical appearances. But the mere subdivision of fluid matter could produce no change of temperature, and the clearest facts are opposed to the notion of a passive dispersion of aqueous corpuscles.

An absorption of heat always accompanies the passage of a fluid from the liquid to the gaseous form, which seems to require the same surplus, in whatever way the conversion has been produced. If a body of air be confined over a surface of water, it will soon become charged with moisture, and then there is no more evaporation. But if the damp air be suffered to escape and disperse itself, the process will be perpetually renewed. The corresponding abstraction or transfer of heat must consequently proceed without intermission. Although the dissipation of moisture and of heat from a humid surface exposed to a current of air continues incessant, the coldness thence resulting is not, however, likewise progressive. A wet sponge or a porous vessel full of water will, when placed in the same situation, gradually arrive at a certain depression of temperature; nor will this limit alter without a change of circumstances. It follows, therefore, that the exhaling surface, after it has become colder than the ambient medium, must begin to receive as well as to discharge heat. Every thin shell of air which successively comes to touch the humid surface must be cooled down to this standard, and thus deliver its excess of heat, while it absorbs another portion spent in vapourizing the moisture it dissolves. As the process of evaporation goes on, the heat communicated at each repeated contact of the fugacious medium must soon come to be precisely equal to what is again abstracted from the exhaling liquid in the act of saturation. When this equality of interchange is attained, the resulting temperature must have reached its lowest point of depression, where it will continue stationary. But the heat deposited by each successive shell of air must evidently be proportional to that depression, which will hence indicate the portion of humidity that passes into the gaseous state of solution; or the difference of temperature induced will furnish an accurate measure of the dryness or distance from saturation of the circulating air. To obtain, therefore, a perfect Hygrometer, it was only required to construct a Thermometer that should mark distinctly small differences of heat. In the first application of this principle, made as early as the year 1791, the water was evaporated from a thin porous earthen cup, like

a truncated egg-shell; and the thermometer had rather a large bulb, with a very narrow bore, and bearing a scale of only a few degrees subdivided into tenth parts. The tube was not sealed, but had a wider cup of glass cemented a little below the top, containing some mercury, which united with that of the stem when the instrument was inverted, and separated the moment it was dipped into the evaporating cup, leaving the mercurial thread to contract downwards from the end. This arrangement was found, with due care and attention, to answer satisfactorily. It was more than three years afterwards, when the severity of the season suggested the idea of making the ball of the thermometer itself the evaporating surface, by covering it with a crust of ice.

The next advance was at all times to invest the ball with a thick coat of tissue paper or soft lint, that should hold a sufficient charge of water. To find the depression of temperature which marks the dryness of the air, nothing was wanted but to compare it with another thermometer having a naked bulb. This purpose appeared to be most easily effected, by joining the stems of two air-thermometers; and thus was fortunately produced the DIFFERENTIAL THERMOMETER, which under a variety of forms is capable of so many curious and important applications. Its primary object was to serve for an Hygrometer, but the instrument was soon found, when placed out of doors, to be sensibly deranged by the diffuse light of the sky. This circumstance led to its modification as a Photometer, the covering being removed from one of the balls, and the other ball blown of black enamel. The addition of a pellucid glass case now became necessary, to screen the action of light from the consuming and irregular influence of wind. The luminous rays pass without interruption through the clear ball, but are by the dark ball absorbed and converted into heat, which continues to accumulate and to raise the temperature, till the corresponding augmented dispersion of heat by the ambient air comes at length to equal the continued accessions from the afflux of light.

In arranging the instrument as an Hygrometer, it was hence requisite to neutralize the photometrical influence, by covering the evaporat-

ing ball with coloured silk, and blowing glass or enamel of the same shade for the naked ball. A free exposure to the atmosphere is in this case essential. The sweep of the wind may accelerate the equilibrium, but cannot derange its limit; for the same portion of air which abstracts vapour and the constituent heat, likewise deposits its excess of temperature on the chilled surface of the wet ball. The interposition of a screen will not in the smallest degree affect the indication of this Hygrometer,—a result at complete variance with the notions generally received. To regulate the scale of the instrument, the tenth part of the centesimal subdivision was adopted, or the distance from freezing to boiling was distinguished into a thousand degrees.

By successive improvements in its construction, the Differential Thermometer was rendered simple, precise, and elegant. But the instrument still betrayed imperfection, the limit of equilibrium or the zero of the scale visibly sinking in the course of several months. This was owing to the diminished elasticity of air in immediate contact with the liquid lodged below one of the balls, which continued to absorb oxygen, and deposit its colouring matter. But early in 1800 every source of error was removed, by adopting, as the movable fluid, concentrated sulphuric acid tinged with carmine, which remains unaltered for any length of time, though exposed to the full glare of the sun.

The precision of the Differential Thermometer now fitted it for a variety of nice inquiries. One of the most important was to ascertain the alteration of the charge of heat in air, corresponding to a change of the elasticity or volume of this fluid. When the variation was small, the allowance of two centesimal degrees for each hundredth part of the difference of density seemed to be a very near approximation. Thus, if a mass of air be condensed one fiftieth part, it will have its temperature raised one degree, or give out one degree of heat; on the contrary, a like expansion of air occasions an equal absorption of that element, or the appearance of cold. But when the difference of elasticity is considerable, some modification becomes requisite; and a very simple formula was found to connect the experimental results. Let this elasticity change from 1 to $e$, then the heat

evolved or absorbed is expressed in centesimal degrees by $25^\{\backslash circ\}\; \backslash left(\; e\; -\; \backslash frac\{1\}\{e\}\; \backslash right)$ or $\backslash frac\{25^\{\backslash circ\}\}\{e\}\; (e^2\; -\; 1)$. A similar mode of investigation determined the relative shares of heat contained in the artificial gases and in common air.

This principle elucidates the decrease of temperature in the higher strata of the atmosphere, while the formula exhibits the scale of gradation. Hence, knowing the mean temperature of the ground at any part of the globe, it is easy to compute the elasticity and corresponding elevation which would reduce that to the zero, and mark the limit of perpetual congelation. A curve is thus traced, bending at first with a gentle declivity on either side from the equator, and then descending more rapidly and turning with a contrary flexure, grazing along the surface at the pole. The application of the instrument led more immediately to the solution of another important physical problem,—to determine the proportion of humidity or of volatilized water which air is capable of holding in solution at different temperatures. A very simple law was derived,—that the quantity of moisture dissolved constantly doubles at every 15 degrees in the rise of temperature. Thus, air at the freezing point may contain the two hundredth part of its weight of moisture, but at 15° it will hold the 100th, at 30° the 50th, at 45° the 25th part; so that, if it were safe to push the progression to its extremity, the air could unite with rather more than half its weight of steam at the limit of ordinary ebullition.

Similar conclusions were derived from the waste by exhalation at different temperatures. A large hollow ball of thin metal, covered with a close coat of linen, and filled with boiling water, being suspended from the end of a fine beam, and 10 grains of water at successive intervals repeatedly sprinkled over the surface, the times of dissipation were carefully noted, and the corresponding solvent powers of air hence inferred.

The Hygrometer likewise determined the relative portions of Heat contained in water and air. It is evident, from the theory before explained, that were the heat diffused according to the densities of these fluids, each millesimal degree of moisture would equal the 6000th part of the air which held it. But adopting

the capacity or measure assigned to air by Dr Crawford (then esteemed the most accurate), each hygrometric degree would correspond to the 3460th part of the weight of the solvent. Both of these proportions were found, however, to differ widely from the truth. A wet card being tied to the instrument, it was let down into the centre of a large glass globe with a narrow neck, containing 4000 cubic inches of air, and suspended through a hole in the glass cover from a fine balance; the progress of humification and the concomitant loss of weight were at short intervals of time nicely observed. From this comparison it resulted, that the moisture sufficient to mark an hygrometric degree is only the 20,000th part of the air with which it unites. Hence the atmospheric fluid, instead of being charged with nearly twice as much heat as water, as had been generally presumed, holds only three-tenths of that quantity. The proportion afterwards given, in some elaborate experiments of Petit and Dulong, was rather smaller, or 26 parts in 100; but the former result is simpler, and probably more exact.

The principles thus established serve at once to confirm and illustrate the ingenious Theory of Rain, which the keen sagacity of Dr James Hutton had elicited from observation about the year 1786. The commixing of two bodies of damp air that have different temperatures, must always occasion a precipitation of moisture more or less copious; for, since the solvent power augments in a geometrical progression, the loss of it which the warmer air suffers will constantly exceed what is gained by the heating of the colder portion of the damp mass.

A curious conclusion followed from those data: Supposing the vast canopy of air, by some sudden change of internal constitution, at once to discharge its whole watery store, this precipitate would form a sheet of scarcely five inches thick over the surface of the globe. During the course of a year, our atmosphere must therefore deposit five or even ten times all the moisture it held in solution. To explain the actual phenomena, it is hence requisite that this restless medium should change unceasingly from a state of dryness to humidity. Such alternations are effected by the system of winds, which sweep variously over the land and the

ocean. But to complete Dr Hutton's Theory of Rain, it becomes necessary to have recourse to the operation of opposite humid currents of different temperatures, which bring their grazing surfaces into rapid intermixture over a given spot, where the aqueous precipitate is copiously discharged.

A further application of the Hygrometer detected the influence of rarefaction in augmenting the dryness of air. The change is rendered very perceptible, by introducing that instrument in its quiescent state under the receiver of a pneumatic machine. On working the pump, the hygrometric action which had ceased is instantly renewed, and urged constantly forward in proportion as the process of exhaustion advances. The dryness thus induced amounts to 50 degrees, every time the rarefaction is doubled. If air had the same temperature at all elevations, it would exhibit a regular gradation of dryness. But at each ascent of 2800 feet, though the air is constitutionally 7 degrees drier, it likewise becomes 5 degrees colder. This continued diminution of temperature predominates at first over the progressive arid quality of the medium, till the apparent moisture has become extreme, beyond which limit the attenuated air continues to spread drier and more pellucid. The Curve of Greatest Humidity, forming the proper range of clouds, rises about two miles above the Line of Perpetual Congelation. The pervading clearness of the upper region, without which the surface of our Earth would have been wrapped in perpetual darkness, is entirely owing to the property of rarefaction augmenting the solvent power of the air.

From the theory of the Hygrometer, it is obvious that the instrument marks the dryness of the ambient air only at the reduced temperature of the wetted ball. To find the true dryness of the atmosphere, a correction is to be applied for the superior solvent power of its actual temperature. With this view auxiliary tables are constructed, which give also the coldness of extreme humidity, or what has been

termed the Dewing Point. But it is liable to considerable uncertainty. Placed within a metallic case, the wetted Hygrometer will gain the zero of the scale; but when covered with a glass shade, it will stand at 5 or even 10 degrees of dryness, the vitreous surface, aided by heat, abstracting the humidity from the inclosed air, and preventing it from attaining the state of saturation.

The Differential Thermometer was now directed to a more extensive and important subject of inquiry,—the mode of the propagation of heat among various bodies. Considering it as a subtle diffusible fluid, a nice investigation of the phenomena might disclose the nature of its connexion with other substances. Ascending by a regular gradation, bodies may be distinguished into solid, liquid, and gaseous. Suppose a cylinder of silver, cased in down, had the one end heated always to the same degree, the heat would advance and discharge itself from the other. If the igneous fluid were merely passive, like a current of water, the elevation of temperature and the inverse length of the cylinder would follow the square of the velocity of discharge, and not the simple ratio, as actually observed. Heat is therefore never passive, but powerfully attracted by its recipient. No portion of the cylinder can receive or deposit its heat, without simultaneously suffering a corresponding expansion or contraction. The heat is thus transferred from stage to stage, by a series of concatenated dilatations and contractions; and these alternate oscillations, which are quicker in some substances than in others, create the only impediments to the flow, that would otherwise be almost instantaneous. The elasticity of heat must be prodigious, since its weight eludes all examination. Considering its extreme tenuity, the expansive energy which it displays can only be assimilated to the mutual repulsion of the particles of Light. Every appearance, indeed, seems to indicate that Heat is merely Light under a latent and combined form.¹

In the case of liquid substances, the commu-

¹ "The only fluid we know, combining enormous elasticity with extreme diffusion, is LIGHT itself, which when embodied constitutes Elemental Heat, or Fire. It is elicited from every substance by percussion or compression, by electrical agency or chemical affinity. With every species of light our vision is equally perfect, and consequently the luminous particles must

nication of Heat is more complicated. Suppose Heat were applied to the upper surface of a vessel containing water, its influence would gradually descend, precisely in the same manner as if the whole mass had been consolidated; for the warmer and dilated portions, retaining their several positions, would continue to float in successive strata graduating downwards. But if the bottom of the vessel were subjected to the same process of heating, the water, as fast as it grew warmer and specifically lighter, would rise and diffuse its acquired heat. The communication of this influence would hence become rapid, and spread itself in horizontal strata. But every renewal of the sentient surface must likewise promote the dispersion of heat through a fluid mass. Such is the expenditure of heat from a hot ball plunged in the current; but in cold still water, the heat being dispersed by two distinct processes, the ball at first cools in a much faster ratio than the difference of temperature, but ultimately approximates to that ratio. The loss of heat is, however, the same, whatever be the nature of the surface encircled by the water, whether of metal or glass, or cased in linen. Water moving in a current at the rate of only a mile an hour, will abstract four times as much heat as if it had remained stagnant.

But when a hot ball is suspended in air, another element of dispersion now comes into play. The air in contact becoming heated and rarefied, ascends continually and scatters the heat. But another portion of heat is discharged from the cooling surface by a rapid rectilinear projection. This emanation is best observed in substituting a cubical vessel with different sides of glass, or of polished or papered metal. It is most powerful in front, diminishing laterally as the cosine of the angle of declination. From a surface of linen the propulsion of heat is ten times, and from that of glass eight times greater, than from a bright polished metallic surface. This peculiar dissemination of heat, which from a glass ball amounts to nearly half of the

whole discharge, is not performed by the actual flow of the heated air, though it requires the agency of a gaseous medium. The discharge can be collected at any distance by a metallic reflector, and thrown concentrated on the naked ball of the Differential Thermometer. But the intervention of a large sheet of tin-foil obstructs the whole effect, while a very fine screen of gauze, which would stop the passage of streamlets of hot air, scarcely impedes the operation. It is impossible, therefore, to resist the conclusion, that what has been vaguely termed the radiation of heat, consists in a certain propulsive effort or internal tremor, excited in the gaseous medium, analogous to the undulatory propagation of sound. The air can be shown to have a closer contact with glass than with metal, and hence the former surface exerts a superior energy of pulsation. The interval in the case of glass appears to be about the 500th part of an inch. A metallic ball, coated with the finest gold-beater's skin, of only the 3000th part of an inch in thickness, will have its pulsatory energy augmented from 1 to 6; and by the addition of other films, this dispersive power will successively rise to 10.

In air rarefied 200 times, the abductive power from the glass balls is reduced from 6 to 1, while the peculiar discharge of heat at the naked surface is depressed from 7 to 5, and that at the gilt surface from 1 to $\backslash frac\{2\}\{3\}$; the naked ball emitting now $6\backslash frac\{1\}\{2\}$ parts of heat, and the gilt one only $2\backslash frac\{1\}\{2\}$. The effects are changed in a different gaseous medium. Thus, the same balls, with a vitreous and metallic surface, would discharge 31 and 25 parts of heat if immersed in hydrogen, both of them losing 24 parts by the powerful abduction of that gas. But were the medium rarefied about 200 times, the quantities of heat emitted from the naked and the gilt ball would be reduced to 13 and $8\backslash frac\{1\}\{2\}$.

The principle thus unfolded applies with mathematical precision, and illumines an extensive train of phenomena. It likewise suggests

in all cases dart forward with the same celerity, or travel at the rate of about 200,000 miles in a second. But since atmospheric air is projected into a vacuum with the velocity of only a quarter of a mile each second, the motion of Light is thus 800,000 times more rapid. Wherefore the propulsive force of Light compared with that of Air is expressed by the square of this number, or 640 billions." In the state of combination with Air, since it must have the same elasticity as that fluid, Light must hence be 544,000,000,000,000,000 times rarer than water. See Elements of Natural Philosophy, vol. i. p. 452.

various improvements in the practical arts, and in general economy. One of the finest modifications of the Differential Thermometer converts it into the Pyroscope, by merely gilding richly one of its balls. By this alteration it is fitted to measure nicely the hot or cold pulses, and hence to determine the relative powers of fuel, and the efficacy of variously constructed chimneys. The access of heat is almost entirely repelled from the gilt ball, while its accumulation on the naked ball is regulated by its subsequent tendency to disperse through the ambient air. The action of wind must evidently lower the effect, insomuch that a current at the rate of only eight miles an hour will reduce it to the half. To fit the Pyroscope for observations out of doors, it became therefore necessary to shelter it from the sweep of aerial streams. This protection was obtained by adopting the pendant form of the Differential Thermometer, inclosing the lower ball within a larger one of silver, and inserting the upper one in the focus of an oblong spheroidal cup, cut over at the remoter focus. It was thus converted into an Aethroscope, which indicates the cold pulses darted at all times from the higher strata of a clear atmosphere, but more copious in summer than in winter, and stronger during the day than the night. The information afforded by that delicate instrument completely elucidates the successive steps towards a correct Theory of the formation of Dew, by Musschenbroeck, Dufay, Six, Wilson, Bernard Prevost, and finally the late ingenious Dr Wells.

In analyzing the process of evaporation, the cold induced on the humid surface was attributed solely to the quickened transfer of the contiguous portions of the ambient medium. "But the conterminous air must besides communicate heat to the water by pulsation; and consequently the balance of temperature would be liable to incidental variations, if moisture, with its embodied heat, were not likewise abstracted by some corresponding process. And such is the harmonious adaptation of these elements. The discharge of vapour appears to be subject precisely to the same conditions as the emission of heat, and in both cases the proximity of a vitreous or a metallic surface produces effects which are entirely similar." This beau-

tiful property was established by decisive experiments, which render the theory of the Hygro-meter absolutely complete. But the performance of that instrument received further confirmation and clear elucidation from the contrast of the Atmometer, which indicates with great accuracy the quantity of exhalation from a given surface in a certain time.

A train of observations performed by this Hygro-meter in 1810 led to the method of producing artificial congelation on a large scale. Having ascertained the increased power of aqueous solution which air acquires as it grows thinner, it was next suggested to combine the action of a vigorous absorbent with the transient dryness created within a receiver by rarefaction. "On introducing a broad surface of concentrated sulphuric acid, this substance was found to superadd its peculiar attraction for moisture to the ordinary effects resulting from the progress of exhaustion; and, what was still more important, it continued to support, with scarcely diminished energy, the dryness thus created. The attenuated air was not suffered, as before, to grow charged with humidity; but each portion of this medium, as fast as it became saturated by touching the evaporating porous dish, transported its vapours to the acid, and was thence sent back denuded of the load, and fitted again to renew its attack. By this perpetual circulation, therefore, between the exhaling and the absorbing surface, the diffuse residuum of air is maintained constantly at the same state of dryness. Heat is hence abstracted in proportion to this invigorated evaporation. If air be rarefied only 50 times, a depression of temperature will be produced, amounting to 80 or even 100 degrees of Fahrenheit's scale." It is hence easy in the hottest weather, and in every climate, to freeze a body of water, and keep it in a congealed state, till it gradually wastes away by a continual but invisible process of evaporation. On a large scale the operation would be conducted the most profitably; insomuch that a proper system of air-pumps, put in motion by a steam-engine of only six horse power, might in the climate of London produce at the rate of a ton of pure ice every day. Other absorbent substances besides sulphuric acid could be em-

ployed, though not so advantageously; such as pounded trap-rock slightly roasted, parched oatmeal, and dry sole-leather. When water is excluded, the dryness created within the exhausted receiver may be turned to different useful purposes. This process is now commonly employed, especially on the Continent, in the nicer chemical analyses. It can likewise be directed to delicate preparations, which the action of heat might derange.

The application of these various refined instruments forms a sort of aera in the progress of philosophical apparatus, and has contributed to introduce into physico-chemical researches a precision and delicacy unattempted before. What seems wanted at present to complete our knowledge of Heat, is not the vague repetition of experiments already carefully performed, but a nice investigation of several unexplored properties, directed with scrupulous accuracy and on a large scale.

The systematic application of Heat has, during the eighteenth century, produced a complete revolution in the mechanical arts. Air, steam, and water are often employed as carriers of Heat, but with very different energies. These fluids being all raised to the pitch of boiling water, while a current of air will transmit only one degree of Heat, a like current of steam will communicate 12 degrees; but a stream of water with the same celerity and transverse section will, from its density and high charge, communicate no less than 2800 degrees of Heat.

The alternate expansion and contraction of fluids by heating and cooling evolves prodigious powers. It may be shown that, with the same expense of Heat, air itself treated in this way exerts about three times as much force as what is obtained from the successive formation and condensation of steam. If such reciprocal changes were impressed on air as rapidly as on steam, the application of that medium would soon supersede the most efficient of all engines.

The expansive force of steam was finely displayed by the Æolipile, an instrument constructed in the Alexandrian School. After the revival of science, it had been employed to set in motion a variety of toys; but no efficient application of steam was achieved in the seventeenth century.

It seems idle to examine the pretensions of such projectors as the Marquis of Worcester or Sir Samuel Morland, who caught at any shadowy schemes, to retrieve their ruined fortunes.

The person who first considered the properties of steam philosophically was Papin, a physician born at Blois, who visited England in 1680, and became curator to the Royal Society. The Digestor, which he constructed next year, exemplified the enormous elastic force that steam acquires when heated and confined. As a source of power he proposed atmospheric pressure, by forming a vacuum either by the inflammation of gun-powder or the condensation of steam. Precluded by the revocation of the Edict of Nantes from returning to France, he accepted in 1687 an invitation from the Prince of Hesse to a chair in the University of Marburg. There he extended his plans, but found no helps to carry them into execution. The superior skill of English artists transferred the palm of invention. Newcomen, a blacksmith, and Cawley, a glazier, natives of Dartmouth in Devonshire, by their united studies and labours, produced the Atmospheric or Fire Engine in 1705; but, to prevent opposition, they associated in their patent a naval officer, Savery, who had a few years before contrived a very imperfect machine, which acted however by alternate expansion and condensation. Accident suggested the injection of cold water within the steam-vessel, instead of external affusion. Beighton simplified the mode of opening and shutting the valves, and improved the whole system of machinery; and in the course of half a century, the Fire Engine was adopted for raising water in all the coal-mines.

At this epoch the genius of Watt, guided by sound judgment and urged by unremitting application, effected in less than forty years a complete change in the powers of mechanism. His steady purpose was to reform the principles of the steam-engine, and reduce its composition to mathematical precision. He began with performing the condensation in a vessel distinct from the working cylinder; and he next excluded atmospheric pressure, and merely employed the alternate action of expanding and contracting steam. But to pursue his career of invention might demand a separate disquisition.

THE varying spectacle of the heavenly bodies has attracted the curiosity of mankind since the very dawn of civilisation. In the genial regions of the East, where societies were first collected, the rural inhabitants, accustomed to enjoy the freshness of the nights in the open fields, contemplated with wonder the sparkling radiance of a pure canopy, remarked the groups of stars, and learned by degrees to trace their successive changes, and to connect these with the periodic motions of the great luminaries. The rudeness of such early observations was compensated by the number of the included revolutions. The occasion of an eclipse or an occultation served, by disclosing the coincidence of different periods, to rectify the elements of the planetary movements. Astronomy has grown up by a slow accumulation of facts, continued through a long tract of ages. It had acquired considerable accuracy in practice, before it received any tolerable lights from theory. But the art of observing has in modern times acquired amazing precision, from the very improved construction of instruments, and the extended methods of calculation. The register of a single year may now furnish more complete data than the aggregate observations of a whole century in remote ages.

The great cultivators of astronomy are divided into two classes,—accurate and inventive observers,—and profound and original theorists. Of those illustrious men who, in the lapse of three thousand years, have devoted their labours to so noble a purpose, we may distinguish a few that have stood pre-eminently above their compeers;—as observers, Hipparchus, Ptolemy, Tycho Brahe, Hevelius, Cassini, Flamsteed, La Caille, Bradley, Maskelyne, and Piazzi;—and as theorists, Copernicus, Kepler, Newton, and after some interval Clairaut, D'Alembert, and Euler, and in our own times the associated names of Lagrange and Laplace. Astronomy is not only the sublimest of all the sciences, but has at last reached the highest pitch of perfection, and constitutes by far the grandest monument of human assiduity and genius.

The progress which Astronomy has made since the decease of Newton may be referred to five separate heads; 1. The investigation of the figure of the earth, and of the other planets; 2. The determination of the anomalies of the moon's motions, and their application to the finding of the longitude at sea; 3. The analysis of the prolonged effects resulting from the mutual disturbing influence of the planets; 4. The improved theory and observation of comets; and, 5. The telescopic discoveries of new planets, and the vast expansion of the catalogue of stars.

1. Investigation of the Figure of the Earth and of the other Planets.—It was easy, by combining the rotation with the length of the radius of the Earth, to compute that every body at the equator must lose the 289th part of its weight, from the action of centrifugal force, which, being proportional to the distance from the centre, will exert only half such influence over the whole mass; and therefore supposing this to be homogeneous, and assuming with Huygens that gravitation is all directed to a central point, and equal at every distance, it thence followed that our planet is an oblate spheroid, of which the Equatorial is to the Polar diameter as 578 to 577. But Newton had already explored the question with consummate penetration; and, setting out from the great principle of an attraction subsisting among all the particles of matter, had, on the hypothesis of homogeneous fluidity, shown that the equatorial column has its pressure at the centre diminished by the elliptical defect of the sphere, while the action of the perpendicular column is augmented from the opposite cause; and thus a greater inequality than the mere influence of centrifugal force becomes requisite to produce equilibrium, assigning 230 to 229 as the ratio of the diameter to the axis. This solution, however, was merely tentative, the imperfection of the calculus being compensated by a refined sagacity, which noted only the leading steps.

The explication of the System of the World from the Theory of Attraction, embraced with

indolent acquiescence by the countrymen of Newton, was either opposed or viewed merely as an ingenious speculation by most of the philosophers on the Continent.¹ But a controversy now arose, which brought the accuracy of these principles to the test of observation, and secured their final triumph. It is evident that the length of a degree of the meridian must be proportional to the distance of the centre of mutual concurrence of the plummets from its extremities, or to the radius of external curvature. In the case of an oblate spheroid, therefore, those degrees would regularly enlarge from the equator to the pole. But the Trigonometrical Mensuration of France, begun by Picard in 1675, and completed in 1716 by Cassini, forming the most extensive survey yet attempted, was found leading to an opposite conclusion in the meridional arc, the degrees on the north of Paris, instead of lengthening, appearing to contract about a 430th part below those on the south, which would indicate an oblong figure, having its perpendicular diameters as 96 to 95. This perplexing result served to keep alive for some time longer the controversy on the Continent between the followers of Descartes and the growing partisans of the Newtonian Philosophy. In the meanwhile, a clock made by the famous Graham, and carried out to Jamaica in 1732 by Colin Campbell, was ascertained by astronomical observations at Black-river, in the latitude of 18°, allowing for the influence of heat on the pendulum, to go 1' 58" slower every day than in London. This quantity of variation Bradley found to exceed what would suit the figure calculated by Newton, and he was therefore disposed to attribute the effects to some diminution of density in the equatorial regions. James Stirling comprised the attractions of homogeneous spheroids under two beautiful theorems, which were published, though without demonstrations, in the Philosophical Transactions for 1735. About the same

time Bouguer noticed some restrictions essential to the rigorous solution of the problem; but now Maclaurin took a most commanding view of the subject, and produced, first in his famous Dissertation on the Tides, and again more fully in his Treatise of Fluxions, printed in 1742, a geometrical investigation, distinguished by its originality, clearness, and superior elegance. This illustrious mathematician established, from the law of gravitation, the stability of a fluid spheroid revolving about its axis in a certain time, proved that every point within the mass would be pressed equally on all sides, and showed that even the superadded attractions of the sun and moon could not disturb the internal equilibrium of our Earth. The conclusions applied likewise to the other planets, and embraced the theory of the tides. But his profound investigation likewise disclosed several general theorems, which have greatly simplified and extended the science of Hydrodynamics.

The most refined theory, thus confirming the Newtonian figure of the Earth, again recalled the attention of philosophers to the actual survey of its surface. The French triangulation being resumed in 1740, an error was detected in the measurement of the base, and another not less considerable in the observation of the meridional arc; but to rectify the various details required the labour of several years, nor did the work terminate before 1754. In the extent of above eight degrees, their lengths appeared regularly to increase, from Perpignan to Dunkirk, by about the 1730th part; marking evidently the oblateness of the spheroid.

But it was most desirable to resolve the question on a larger scale; and the Academy of Paris, urged in 1733 by Condamine to send a party to measure a degree under the Equator, adopted his project with zeal, and fortunately obtained the sanction of government. After all the preparations were made, the Academicians, Con-

¹ In a letter written from Basle on the 22d November 1729 by Nicholas Bernoulli, to his friend James Stirling at London, with whom he had become acquainted several years before, when the former was professor at Padua, and the latter a teacher of Mathematics in Venice, he confesses, with regard to Machin's attempt to illustrate the Lunar Theory, that he had not yet taken the trouble to read the third book of the Principia. Pauca quidem in eo intelligo, quia nullam adhuc operam colloqui in lectione tertii libri Principiorum D. Newtoni. Stirling, in his answer, dated in September 1730, frankly remarks, that "it is somewhat strange, that though the Principles have been published above 40 years, nobody has read further than the two first books, although they be barely speculations, and were written for no other reason but that the third might be understood." (From a collection of Stirling's papers in the possession of Mr Irving, one of the Judges of the Court of Session, under the title of Lord Newton.)

damine, Bouguer, and Godin, in May 1735 sailed from Rochelle to Peru, where they joined Juan and Ulloa, two naval officers deputed by the king of Spain. The whole company assembled in the month of June in the following year at Quito, almost under the Equator, and arranged their plan of operations. They chose a valley of the Cordilleras running above 200 miles southwards from that city, and inclosed on both sides by the loftiest ranges of the Andes. Divided into two troops, they carried a series of triangles along the flanks and summits of those stupendous mountains covered with eternal snows, and connected the reticulation with the base measured below. In pursuing their laborious task, the observers had to encounter the greatest difficulties, and to suffer privations of every kind, owing to the severity of the climate and the total want of accommodation, joined to the continual impediments occasioned by the superstition of the natives and the pilfering habits of the Indians. But resolution and perseverance at length overcame the multiplied obstacles, though no fewer than eight years were consumed in completing with elaborate care and accuracy the different operations undertaken. Condamine was a man of an ardent and active mind, and persuasive address; Godin, an excellent observer; but Bouguer combined the character of a deep and original thinker, with the talent of invention, unwearied application, and an extended acquaintance in the various pursuits of philosophy and science. The Academicians were ably assisted by their Spanish associates, though each performed separately his own calculations. It was reserved, however, for Bouguer to give the complete narrative of all the various operations, in his Treatise on the Figure of the Earth, not published till 1749, but one of the most valuable scientific works that has ever appeared. Besides amply discussing the principal question, it investigated incidentally the effects of local attraction on the plummet, the length of the pendulum, the theory of refractions, the rules of barometrical measurements, the limits of perpetual congelation, the intensity of the solar rays at different elevations, and the celerity of sound as affected by heat. He concluded that the Earth

is not only an oblate spheroid, but so considerably flattened as to have its equatorial diameter to its axis in the ratio of 179 to 178; and he thence inferred that the central parts must be four or five times denser than the superficial crust.

In the mean time, it being foreseen that the mensuration of a degree in Peru would be necessarily retarded, the celebrated Maupertuis, a person of agreeable attainments, though not very profound in science, prevailed with the French minister Maurepas to dispatch another company, which he reluctantly consented to lead, for a similar purpose to the Arctic Circle. His associates were Monnier, Camus, Outhier, and Clairaut, already distinguished by the precocity of his fine mathematical genius. They arrived at Stockholm in June 1736, and were joined by Celsius, the professor of astronomy at Upsal, who had brought from London Graham's zenith sector and transit instruments. Having settled their general plan, they proceeded to the bottom of the Gulf of Bothnia, and selected Tornea for the principal station. From that small capital they stretched a chain of triangles along the wooded heights to the top of Kittis, a distance however not exceeding 60 miles. Working with great spirit and activity, and having no difficulties to encounter but the severity of the climate, they observed the angles from the several stations during the remainder of the summer, determined the celestial arc in autumn, and measured the base along the frozen surface of the river Muonio in the early part of winter. The length of a degree of latitude at the Arctic Circle was found to be 57,419 toises, or 349 toises longer than the corresponding measure at Paris, which gave the ratio of 178 to 177 for that of the Equatorial and Polar diameters, almost the same as the result of the more extensive operations afterwards in Peru. Maupertuis, having with such expedition ascertained the oblateness of the Earth, and assigned a depression still greater than had been computed by Newton, returned triumphant to Paris in the spring of 1737. But this arctic triangulation, betraying evident marks of haste, became suspected afterwards of inaccuracy; and, at the suggestion of Melanderhjelm, the Swedish academy, about the

beginning of the present century, sent Svanberg, with proper assistants, to resume the operations, who not only rectified the former observations, but carried the mensuration about 40 miles farther north.

It was then determined that the length of a degree in Lapland is only 57,209 toises, which, compared with Cassini's corrected measure in France, reduces the oblateness to the 290th part of the Earth's axis. But a more accurate result is obtained from the comparison of the distant observations made under the Equator and at the Arctic Circle, the perpendicular diameters of the terrestrial spheroid being in the ratio of 302 to 301. Still greater nicety has been attempted by Littrow, who, on re-computing the degree in Lapland, reduced it to 57,168 toises, which gives an oblateness of only $\backslash frac\{1\}\{312\}$.

Other determinations of a degree of latitude have been made at different parts of the Earth's surface, the most remarkable of which may be noticed. La Caille, one of the best observers, and the most accurate and laborious of astronomers, rectified the French triangulation during the years 1739 and 1740; nor did the extreme severity of that memorable winter prevent his carrying forward these operations over the lofty mountains of Auvergne. He proved decidedly that the degrees of the meridian regularly increase from the south to the north of France. Some years afterwards he went to the Cape of Good Hope, chiefly for the purpose of framing a catalogue of the stars of the Southern Constellations; but his ardour and extreme activity enabled him to accumulate other valuable celestial observations. Having performed the task he had proposed, and the ship not arriving to carry him home, La Caille was tempted to employ the time thus afforded in measuring by

his single exertions a degree of latitude. This he found, on the parallel of 34°, to be 57,037 toises, and therefore of the same length with a degree as far north as 49° in France; which seems to indicate not only a flatter outline, but a difference of internal constitution in the southern hemisphere. The accuracy of the measurement itself, however, has been strongly suspected.

In 1750 and the two following years, Boscovich and his associate La Maire measured an arc of two degrees in the Papal States, and fixed the length on the parallel of 43° at 56,979. Beccaria traced a single degree in Piedmont, and found the opposite attractions of the lofty sides of the Alps had such influence in deflecting the plummet, as to render the northern half of the measure a 69th part longer than the southern.

After a pause of near forty years, the mensuration of the Earth was revived with increasing ardour and on a grander scale. The object first contemplated was merely to settle the difference of longitude between the observatories of Greenwich and Paris, by connecting them with a series of triangles. General Roy directed the English survey in 1784, while similar operations were conducted by Cassini in France. The Revolution soon followed, and prepared the public mind for the reception of every bold project. It was now proposed to adopt a universal and permanent system of weights and measures drawn from nature, the unit of linear extent being the thousandth part of a centesimal minute, or the ten millionth part of the quadrantal arc from the Pole to the Equator. To obtain this standard, it had been resolved to carry a chain of triangles over the whole extent, from Dunkirk to Barcelona; which was performed during the most eventful period of general excitement and commotion, by the profound sagacity, perseverance, and undaunted zeal of Delambre,¹ joined to

¹ Born at Amiens on the 19th September 1749. Being designed for the church, he studied the ancient languages in the Gymnasium of his native city, first under the Jesuits, and afterwards under the tuition of the poet Delille, to whom he became warmly attached. At a ripe age, he was invited to the study of astronomy by Lalande, and soon adopted as his favourite pupil. In 1785 he began the vast series of his scientific labours, which occupied the rest of a long life. Without aiming at originality, Delambre produced very complete and valuable works, pursued his immense calculations with scrupulous nicety and incessant application, and distinguished himself as a most accurate, skilful, and indefatigable observer. He sought precision merely, and was rather indifferent about the elegance of his formule or composition. Yet he was an excellent Greek scholar, well acquainted with the principal modern languages, and indeed a man of general erudition. Of an amiable disposition, with fixed principles of integrity joined to simple habits, he passed through all the storms of the Revolution with a blameless character. He obtained his full share of the honours and distinction which France so liberally bestows on those eminent in science, and died universally regretted, on the 19th of August 1822.

Mechain.¹ After completing this operation, they measured a base of seven miles in length near Paris, and another of verification, somewhat shorter, on the road from Perpignan to Narbonne. The register of all the observations was in the spring of 1799 submitted to a sort of congress of scientific men assembled in the French capital, who having examined and repeated the various calculations, and compared the celestial arc with the mensuration in Peru, adopted the 334th for the oblateness of the earth, and determined the distance of the pole from the equator to be 5,130,740 toises.

Mechain, anxious however to extend the meridian as far as the Balearic Isles, again repaired to Spain, and conducted a series of triangles over the savage heights from Barcelona to Tortosa, when, exhausted by incessant fatigues, he caught an epidemic fever, which proved fatal in September 1805. The operations were continued by his young but very able successors, Biot and Arago, who terminated the meridional arc at the isle of Formentera in 1807. This extension of the chain scarcely altered in the slightest manner the former conclusion.

The English survey, being designed merely for the construction of an accurate set of maps, suffered some interruption by the death of General Roy, but was resumed with spirit in 1793, under the skilful direction of Colonel Mudge, who carried the triangulation as far as the north of Scotland; and since his decease in 1820, the work has been transferred to Ireland by his persevering assistant and successor Colonel Colby. The method of observing is now improved, and

all the refinements of modern art have been exhausted in the mensuration of a new base.

Those operations have no doubt proceeded very slowly, and yet the results of the first series were perplexing, the three degrees measured from Dunmore to Clifton successively contracting instead of lengthening, and therefore indicating, like the earlier observations of Cassini, an oblong figure of the Earth. The whole arc agrees exactly with the last triangulation of France, in giving an oblateness of only the 334th part; but whether the intermediate discrepancies should be imputed to an error of azimuth, or to the probable derangement from local attraction, still remains undecided.

In 1802 and 1803 Colonel Lambton measured a degree of the meridian in the East Indies, and found, at the latitudes of $9^\{\backslash circ\}\; 35\text{'}$, $12^\{\backslash circ\}\; 3\text{'}$, and $16^\{\backslash circ\}\; 35\text{'}$, the lengths to rise successively to 60,477, 60,490, and 60,512 fathoms. Hence Delambre deduced the oblateness of the terrestrial spheroid to be the 206th part. But Rodriguez, an able Spanish mathematician, who had already criticised the observations of Mudge, detected various mistakes in Lambton's calculations, which being rectified, reduced the depression of the Earth to the 320th part.

Since the spheroidal figure of our planet is evidently subject to great irregularities, we are led to inquire, whether it be a solid of revolution and have true circles for its parallels of latitude. Cassini and Maraldi seem to have been the first to solve this interesting question. They found, in 1734 and 1735, a degree of longitude at Paris and at Strasburg to be 36,676 and 37,745 toises,

¹ Born at Laon 16th August 1744, the son of an unprosperous architect, who could hardly educate him in the same profession. The youth supported himself, and even aided his father, by teaching mathematics, and made such proficiency in astronomy as to attract the notice of Lalande, by whose warm and active patronage he soon obtained the office of hydrographer to the Marine at Versailles. Though he had not the talent of invention, he became an expert, patient, and correct calculator. He was, besides, a very nice and accurate observer, and applied himself particularly to the search of comets, and the computation of their orbits. In 1784 he was joined to the commission for adjusting the difference of longitude between the observatories of Paris and Greenwich; and when the Constituent Assembly had resolved to adopt a grand metrical system founded on the length of the meridional arc, he was appointed to measure the southern portion of it from Barcelona to Rodez. He conducted his operations with great celerity and success; but, inspecting a new engine constructed by a friend at Barcelona, he met with a dreadful accident, which had well nigh proved fatal. After a slow convalescence he resumed his labours; but every thing now combined to dishearten him,—the war with Spain, the ravages of the yellow fever, and the news of the reign of terror in France. After moderation had revived, Mechain with much reluctance was prevailed on to come to Paris; but strongly pressed the expediency of extending the measurement two degrees farther into Spain, and insisted on carrying on that work himself. This pertinacity was not explained till after his death, when it appeared that he had concealed a discrepancy occurring in the rectification of his observations, which, without acknowledging the error, he no doubt hoped to remove by subsequent operations. After experiencing many difficulties, he vigorously renewed his labours in Valencia; but his mind was depressed by anxiety for his reputation, and alarms at the spread of contagious disease. This mental prostration favoured its attack, and he expired of yellow fever in his retreat at Castellon de la Plana, on the 29th September 1805.

and therefore considerably less than what belongs to a perfect sphere. But the most accurate observation of this kind was made by La Caille in 1740, from the top of a hill near Aix, to a signal on a high tower above the Rhone, about 96 miles remote, where he caused ten pounds of gunpowder to be fired, and found by the instantaneous blaze the difference of time between the two meridians, and then measured trigonometrically along the plain of Arles the intermediate distance, which gave 41,358 toises for a degree of longitude, or 260 toises less than a globular form would require. Lambton estimated the degree of longitude on the parallel of $12^\{\backslash circ\}\; 32\text{'}$ to be 61,061 fathoms, or 57,294 toises; but his observations, and calculations seem not entitled to much confidence. The latest and most accurate measurement of a perpendicular to the meridian was performed in 1823, near Geneva, by Colonel Brousseau, of the French engineers, in connexion with the late ingenious Professor Pictet, while the skilful astronomers Plana and Carlini completed a similar operation on the plains of Italy as far as Padua. The first, being compared with the arc from Greenwich to Formentera, gave a 271st part for the oblateness of the Earth; but, when conjoined with the Italian prolongation, it reduced the quantity of depression to a 292d part of the equatorial diameter. It must be confessed, however, that the accuracy of these conclusions has been contested by some of the ablest calculators.

From a skilful combination of the five principal measurements, in Peru, India, France, England, and Lapland, Mr Ivory has, by the method of the least squares, deduced $\backslash frac\{1\}\{309\}$ for the oblateness of the Earth, and shown that the corresponding elliptical meridian agrees with the best observations. The result would have been still more satisfactory, had the English survey comprised the whole extent of the island. But a very small portion of the arc is computed, and the fruits of twenty years' expensive operations are not yet communicated to the public. Meanwhile, other surveys, conducted after the most scientific manner, are now far advanced; in Lower Saxony by Gauss, in Holstein by Schumacher, in Swabia by Bohnenberger,

and in Russia and Finland by Struve. These continental observations must throw much light on the form and constitution of our planet.

Local attraction may deflect the plummet from its vertical position, but it can have little effect on its general gravitation. The vibration of a pendulum, therefore, promises a readier and surer method of discovering the figure of the Earth. The difficulty is to ascertain the length with extreme precision, or the distance between the point of suspension and the centre of oscillation. The pendulum constructed by Borda seemed to require little correction. It consisted of a slender iron wire about 13 feet long, terminated by a small cap of copper, to which was nicely fitted a ball of platinum about an inch and half in diameter. The vibrations were performed in two seconds and continued for twelve hours, the number being reckoned from the distant coincidences with the pendulum of an astronomical clock. Mathieu has computed, from six observations made with this instrument along the meridian from Dunkirk to Formentera, the depression of the terrestrial spheroid to be $\backslash frac\{1\}\{298.2\}$. But from a general comparison of all the various measures over the surface of the Earth, Laplace reckoned this quantity only $\backslash frac\{1\}\{335.78\}$. It has been alleged, indeed, that the illustrious author had committed a mistake in the taking out of a logarithm, which being corrected would give $\backslash frac\{1\}\{319\}$.

The Convertible Pendulum constructed by Captain Kater, on the beautiful principle first demonstrated by Huygens, that the point of suspension and the centre of oscillation are interchangeable, seemed to promise greater accuracy, and was proposed by the inventor as furnishing an invariable standard of linear measure. So sanguine were his expectations, indeed, that he believed it could not occasion an error of the 400,000th part of the whole. But the observations which he made with this pendulum, at different points in an arc of ten degrees extending to Shetland, display no such harmony, for the resulting

oblateness of the Earth varies from the extremes, between $\backslash frac\{1\}\{397.5\}$ and $\backslash frac\{1\}\{229.6\}$. It has been customary to attribute the discrepancies to the influence of local attraction, but they are frequently too considerable for the most exaggerated hypothesis to explain, and must be referred either to the inaccuracy of the observer or to the imperfection of the instrument employed. Calculators, in their endeavours to harmonize such observations, can seldom proceed dispassionately; they are apt to modify some of the data, and arbitrarily to reject others as anomalous or liable to suspicion. Parry's experiments with the convertible pendulum at London and Melville Island gave an oblateness of $\backslash frac\{1\}\{312.6\}$. The numerous observations made by Freycinet and Duperrey, in their voyages of discovery, indicated, for both the northern and southern hemispheres, a depression between $\backslash frac\{1\}\{280\}$ and $\backslash frac\{1\}\{282\}$. Captain Sabine concluded, from his own combined observations, that the quantity is $\backslash frac\{1\}\{288.4\}$. But Mr Ivory, whose decision is entitled to the greatest confidence, has, from a critical examination of those data, concluded the ellipticity to be $\backslash frac\{1\}\{300\}$; and found the same figure to agree both with the measures along the meridian and those perpendicular to it. To arrive at so satisfactory a result, however, it was necessary to exclude nearly one-sixth of all the observations, those made near the equator being the most discordant. Such aberrations leave considerable distrust, which is not removed by inspecting the later observations. But it were idle to affect extreme precision, where the discrepancy amounts sometimes to a six thousandth part. The convertible pendulum appears liable to different sources of error, and though it may furnish an approximation to the determination of the terrestrial spheroid, it can hardly be expected to reach the accuracy and certainty requisite for a metrical standard.

We may now revert to the analytical investigation of the figure of the Earth. Clairaut had given the elements of the solution previous to his set-DISS. V.

ting out on the expedition to the Polar Circle. But after his return from that memorable achievement, he resumed the disquisition; and adopting the geometrical discoveries of Maclaurin, he produced the completest analysis of the problem, in his very ingenious work which came out in 1743. He determined the equilibrium of a revolving spheroid, when composed of concentric layers of different densities, assuming the deviation from the spherical figure to be very small. Clairaut showed that the particles of the same density must always range in distinct strata, to which the resulting force that urges them is perpendicular; and found the general equations, simplified afterwards by Euler, for assigning the stability of the fluid mass. But it was not until 1784 that Legendre gave a direct analytical solution of the problem, in the case of a homogeneous spheroid approaching nearly to a sphere, which required the single condition that gravity should act perpendicularly at the surface. Laplace afterwards simplified and completed this investigation, by extending it to elliptical strata of different densities. The solution, however, was not rigorous, and merely exhibited an approximation, by leaving out the cubes and higher powers of the series. The numerical results yet vacillate between the proportions announced by Huygens and Newton. It is indeed remarkable, that a subject which has exercised the efforts of sublime genius, should have yet derived so little real elucidation from the stores of the higher calculus. The theorems of Maclaurin still far transcend the most elaborate displays of analytical research. But his illustrious countryman Ivory, undismayed by such formidable difficulties, has lately revived the discussion, and, examining more narrowly the physical conditions of the problem, and thus restricting the differential equations, he has with felicitous address succeeded in conducting his analytical procedure to a definite result. Britain has therefore the honour both of originating the discovery of the true figure of the Earth, and of completing its demonstration.

But the Earth's oblate shape is likewise deducible from certain small mutations in the orbit of the moon. The attraction exerted by our satellite on the protuberant matter at the Equator occa-

sions the Nutation of the terrestrial axis; and this action being reciprocal, the moon suffers a corresponding alteration both in latitude and longitude. Such is the wonderful precision attained in astronomical observations, that Burg, one of the most expert calculators in our day, computed at the request of Laplace, from very numerous data, the ellipticity of the Earth to be respectively $\backslash frac\{1\}\{304.6\}$ and $\backslash frac\{1\}\{305\}$. The motions of the lunar nodes and perigee have been shown by the same profound philosopher to indicate a similar conclusion. The form of the Earth is hence flatter than that of a homogeneous spheroid; and this difference appears to arise from the increasing density of the internal mass caused by superincumbent pressure. A like character belongs to Jupiter, the largest planet in our system; for his axes, which, from the combined action of gravity and centrifugal force, should have the ratio of 41 to 36, are observed to be only as 14 to 13, or 41 to 38, thus evincing the great compression of the internal mass.

The discussions relating to the figure of the Earth led to an examination of the Theory of the Tides, the great outlines of which had been likewise traced by Newton from the Law of Gravitation. The Academy of Sciences at Paris selected the complete investigation of this difficult subject for the prize offered in 1740; and never did three more illustrious competitors contend for the honour of an award, which was shared among Maclaurin, Euler, and Daniel Bernoulli. The dissertation of our countryman, as already observed, was pre-eminently distinguished by the new and beautiful propositions which it contained. Supposing the Earth were at rest and covered with a shell of water, any remote body, attracting the anterior surface with greater force than the lateral mass, and this again with greater force than the posterior surface, would evidently cause the fluid to rise on the opposite sides and form a prolate spheroid, having its longer axis directed to the disturbing agent. But this influence was proved by Newton to be inversely as the cube of the distance. Though the sun has above twenty million times more matter than the moon, yet being four hundred times more remote, he therefore exerts three times less influence in raising

the tides. While the ocean swells out both under and opposite to the sun, it turns another trebly more protuberant spheroid towards the moon. These elevations, differently combined, produce the variable heaving of the waters which constitutes the general tide. The conditions of equilibrium are easily determined; but it is a most arduous research to distinguish the several effects of the rotation of the earth and the revolution of the moon, in retarding and modifying the oscillations of the irregular aqueous expanse. Such was the problem which so long exercised the genius and penetration of Laplace. He began the consideration of this intricate subject in 1774, and resumed his investigation repeatedly afterwards. Having obtained a general expression for the oscillations of the ocean, he distinguished these into three separate swells, obeying different periods of succession. The tides of the first class depend on the motions of the sun and moon, the variations of the distances, and the change of their declinations. The stream tide occurs at the interval of a day and a half after the conjunction or opposition, the lunar swell at first preceding the solar, but falling back at the next two returns, till at the third accession it combines with the latter. The decrease of consecutive tides is about a third faster at the conjunctions which happen in the equinox than at those in the solstice, and nearly twice as rapid at the quadratures that occur in the equinox as at those in the solstice. Similar consequences result from the alterations in the right ascension and the declination of the sun and moon. All these conclusions of a refined theory are singularly confirmed by a critical and very laborious examination of the observations made for a series of years at the port of Brest. The second sort of oscillations is occasioned by the diurnal change of the elevation and depression of the sun; so that in midsummer, when the declination is greater, the stream tide at Brest rises seven inches higher in the morning than the following tide towards evening. This divaricating tide has no visible influence however on our shores; and to explain the equal swell of the waters during the day and the night, Laplace refers it to the retardation from a current of a certain uniform depth, which he estimated at

four leagues, or eleven English miles, but has since greatly reduced it, without venturing to assign the true measure.¹ The third kind of tides, depending on the greater revolutions of the moon, have much longer periods; but their influence, however small, is traced through the observations. The immense calculations of Bouvard have detected every varying phasis of the law of Universal Attraction; but a closer approximation is still wanted to unfold separately all the terms of the disturbing forces. Some of the minute shades are indicated by a fine application of the Doctrine of Chances, and a similar process renders highly probable the existence of corresponding fluctuations in the mass of our atmosphere.

2. Mathematicians were at length prepared for investigating dispassionately the application of the great principle of attraction to all the various motions of the heavenly bodies. It had explained with beautiful simplicity the revolutions of the planets about the Sun, and of the satellites around their primaries. But to distinguish its operation in the anomalies of the general system was a most arduous undertaking. Astronomers indeed had already, from a diligent and skilful comparison of distant eclipses, detected the principal irregularities of the Lunar Motions. Hipparchus and Ptolemy discovered what are called The Equation of the Centre and the Evection; and after an interval of fourteen centuries, Tycho and Kepler added the Variation and Annual Equation. Newton not only expounded all these anomalies by the simple law of gravitation, but, probably without any helps from observation, discovered six more auxiliary equations. The Theory of the Moon, which crowns his immortal Principia, is a production of genius, sagacity, and invention, almost superhuman. He ascends with admirable order from the easier to the more difficult problems, reducing them always to greater simplicity; he pursues his approximations with consummate address, and

seldom passing the clear bounds of geometry, or entangling his demonstrations in the labyrinth of Algebraical formulae, he advances with elegance and apparently without effort to the disclosure of the most recondite truths. But it must be confessed that, in his eagle flight, he was satisfied with taking a general glance of the objects, and seldom stooped to mark the details or investigate the grounds of his calculations. He no doubt managed the research by estimating the different elements of perturbation, and supplying, as in other cases, the deficiency of the analytical process by a selection of circumstances, and the nice balancing of errors. The illustrious author himself candidly admitted the imperfection of his Lunar Theory; but what seems truly astonishing is, that in such a novel and arduous attempt he should have been guilty of so very few mistakes or omissions. The English commentators contributed very little to its extension and improvement. Machin, professor of Astronomy in Gresham College, secretary of the Royal Society, and esteemed an expert calculator, speaks of it with flippancy, and seeks only to annex a vague hypothesis, which leads to an arbitrary geometrical construction; "especially," says he, "since the greatest part of the Theory of the Moon is laid down without any proof, and since those propositions relating to the Moon's motion which are demonstrated in the Principia do generally depend upon calculations very intricate and abstruse, the truth of which is not easily examined, even by those that are most skilful; and which, however, might be easily deduced from other principles and hints of calculations which he has not produced."² He admits, indeed, that Theory gives only half the motion of the Lunar Apogee.

The first who improved and expanded the Newtonian Theory of the Moon was Calandrin, professor of Mathematics at Geneva, who superintended the printing of the Jesuits' edition of the Principia in 1739 and 1742. He investigated by a direct method the principal lunar equa-

¹ In the last edition of the Mécanique Céleste he seems to have silently abandoned his calculation altogether, contenting himself with adopting the conjecture of the geographer Varenus, that the depth of the sea must bear some proportion to the altitude of the shore.

² The Laws of the Moon's Motion according to Gravity, a very short tract appended to Motte's Translation of the Principia which came out in 1729.

tions, and likewise the smaller inequalities which Newton had left undemonstrated. He revised the investigation of the motion of the apses, but was mortified to find his calculations gave only half the quantity derived from observation. The honour of confirming the Newtonian Theory of the Moon was reserved, however, for our own countrymen. Dr Stewart, the successor of Maclaurin in the University of Edinburgh, discovered the true motion of the line of apses by a simple and beautiful geometrical procedure, in which his inventive penetration happily supplied the flexibility of the modern calculus. About the same time Walmsley,¹ an English Benedictine Monk, who afterwards attained the rank of Catholic Bishop and Apostolic Vicar, but had been compelled by religious and political bigotry to reap the advantages of a foreign education, produced in 1749, at the early age of 27, a correct analytical investigation of the motion of the Lunar Apogee, which he extended and completed in 1758.

In the meanwhile, the profoundest mathematicians were directed to the right points of attack, and incited to exert all their penetration in exploring the influence of the System of Gravitation, by the prizes proposed by the learned associations on the Continent. Clairaut began his examination of the Lunar Theory in 1743. At first he was content with merely studying the Newtonian procedure, and converting it into analytical expressions; but as he became more familiar with the subject, he pushed his investigation further, and in 1747 comprised all the subordinate motions of the Moon under the famous general problem of the Three Bodies. But, after a prodigious exertion of ingenuity and perseverance, it was mortifying to find his solution assign for the variation of the Lunar Apogee only half the measure established by observation. Euler and D'Alembert, nearly about the same time, arrived at a similar conclusion. The followers of Newton were surprised and mortified at the result, while the adherents of the Cartesian system already began

to exult in the prospect of the immediate fall of such a towering rival. In this dilemma, Clairaut, to reconcile the result of analysis with the actual phenomena, proposed, without abandoning the great principle of attraction, to modify its reciprocal gradation of intensity, by annexing to the square of the distance a small subsidiary term depending on its cube. This correction might be sufficient, from the proximity to the earth, for adjusting the Lunar motions, while it could have no sensible influence, he conceived, in affecting that duplicate ratio which directs the remote revolutions of the planets about the sun. But Buffon, who began his career with the study of mathematics, showed, from the properties of the roots of equations, that such a modification of the law of gravity involved what appeared an absurd consequence, that a body would be attracted equally at different distances. Clairaut was not shaken by this argument, which he considered as only metaphysical; yet so much discussion had been provoked, that, anxious to remove all doubts from the subject, he resumed his investigation, and pursued it with incredible labour and resolution. Carrying the approximation much farther, by computing the values of the higher terms of the series, and reiterating the process, he found the combined integrals to give exactly the double of the former result. This satisfactory conclusion, which confirmed the simple law of gravity, and restored the harmony of the universal system, was announced early in 1749. The bare mention of the fact was enough for the inventive powers of Euler, who, by quite a different procedure, soon obtained the true variation of the Lunar Apogee. If D'Alembert needed some previous information to put him in the right train, he pushed his calculations to wider extent, and approached still nearer the absolute effect. It would seem that, during this eager competition, the advances of Stewart and Walmsley were overlooked. But their countryman Thomas Simpson ventured, though considerably in the rear, to travel partly over the same ground of inquiry.

This remarkable coincidence among the first

¹ Dr Charles Walmsley, born in 1722. In his latter years he resided at Bath, and took to the study of mystic theology, having written on the Revelations, and explained the Vision of Ezekiel. His papers were burnt by an antipapish mob in 1780, and he died in 1797.

mathematicians of the age extinguished for ever any lurking suspicion of the Law of Universal Attraction. The great object of scientific research was henceforth to direct its application to the celestial motions and the improvement of practical astronomy. The rectification of the Lunar Tables, now become of such importance in the practice of navigation, was the fruit of those arduous calculations. Clairaut bestowed intense application, and digested his results into a clear form; he skilfully employed some data furnished to him by La Caille for adjusting the auxiliary equations, and thus produced Tables of the Moon's motions distinguished by their very superior accuracy. D'Alembert proceeded more slowly and neglected such aids, so that his Lunar Tables never obtained estimation with astronomers. Euler excelled both his illustrious rivals in the wonderful command of the powers of calculation; he worked with ease and rapidity, and his fertile invention continually supplied new resources and suggested other paths of advance. He suspended for a few months the further improvement of the Lunar Theory, and at the invitation of the Academy of Sciences at Paris, analyzed with all the fulness of research the influence of the mutual disturbance of the planets Jupiter and Saturn. But aiming at still higher perfection, after the interval of a few years he resumed this investigation, simplifying and greatly extending it. In the mean time he had computed a set of Lunar Tables, which he disposed in a clear method, having discovered the co-efficients of the several equations from theory alone.

About this time Mayer of Württemberg began to distinguish himself as one of the ablest and most ingenious astronomers that has appeared in any age. In 1751 he had been appointed director of the observatory at Göttingen; and in this situation, encircled by foreign troops, and exposed to the danger of powder magazines, he yet laboured with such intense and enthusiastic ardour as to shorten his days. His efforts were mainly directed to the improvement of the Tables of the motions of the Sun and Moon. These elements he derived from a discussion of numerous observations of eclipses and occultations, and he borrowed little from theory, though

he preferred the arrangement of the elements adopted by Euler. He was the first who employed the method of conditional equations to find the true values of the co-efficients. His Tables were inserted in the Göttingen Transactions; and after the most sedulous correction, he sent them in 1755 to London, for the patronage of the Board of Longitude. At his death in 1762 he left two copies, greatly improved, one of which his widow transmitted to that scientific body. After long and protracted deliberations, the modified prize of L.3000 was at last awarded to his family, with a present of L.300 to Euler, for his excellent formulas. But another more complete copy having been afterwards presented, the Board of Longitude bestowed an additional reward of L.2000 at the instance of Dr Maskelyne, who zealously undertook the charge of editing those Tables in 1770.

The exertions of the continental mathematicians were now successfully directed to the investigation of the disturbing influence or mutual perturbations of the larger and nearer planets. Euler in 1747 sent to the Academy of Sciences at Paris a most ingenious memoir on the derangement of Saturn's motion, occasioned by the superior attraction of Jupiter. It was not only the first solution of the problem, but the simplest and most direct, referring the forces exerted to three perpendicular co-ordinates. He then discovered the beautiful principle in celestial mechanics, that there exist really no secular equations, but that all the deviations from the regular course are strictly periodical, and return always in the same order, though separated at vast intervals. Notwithstanding the exuberance of his analytical resources, he was yet obliged to omit the smaller quantities, and to adopt certain admissible suppositions, in order to shorten the immensity of the calculation. But being again recalled to the consideration of the subject, he produced, four years afterwards, another dissertation, which surpassed all his former efforts, and obtained the double prize of the Academy. The great analyst new-modelled his investigation, rendered the process much simpler and clearer, pursued the approximations farther, and arrived at more accurate results. He found that the mean motions of Jupiter and Saturn are equally subject

to a very slow increase or diminution, which alternates however in the lapse of 15,000 years. But Euler was induced to push those researches still farther, and gained the prizes proposed by the Academy for the years 1754 and 1756, by his theory of the inequalities in the Earth's motion, caused by the planets. The three methods of investigation that he proposed were all of them quite different; and nothing seemed more astonishing than the facility with which his prolific invention struck out new paths. He discovered four small anomalies to result from their combined attractions, though it was scarcely possible, for want of proper data, to assign the precise measures of those aberrations. He ventured, however, to estimate the mean progression of the aphelia at 12 seconds annually, and the diminution of the obliquity of the ecliptic, which some astronomers still doubted, at 49 seconds in a century. He found that the eccentricities of the aphelia of Jupiter and Saturn are periodical, and complete their cycle in the space of 30,000 years. By an inverted application of the same principles, it was possible, by computing the co-efficients of the formulæ from actual observations, to determine the masses of those planets which are not accompanied by satellites. Euler hence found that Mars contains rather less matter than the Earth, and Venus only about the half, but which by a subsequent calculation he nearly doubled. The process was too much involved to secure entire confidence in the results; and the rapidity and extent of his calculations had led to occasional mistakes.

The same subject was in 1757 discussed by Clairaut, in a clear and concise manner. By comparing his formulæ with the accurate observations of La Caille, he determined very nearly the masses of the principal planets, and showed that the greatest effect of their accumulated influence in deranging the Earth's motion can amount only to about a minute. His estimate of the attraction of Venus has been confirmed by later and more elaborate calculations.

In 1749 D'Alembert investigated rigorously the effects arising from the Moon's attracting the

spheroidal prominence of the Earth. By the transformation of his general expressions, he found the Precession, or conical motion of the terrestrial axis about the poles of the ecliptic, to be 50 seconds annually, and its Nutation or alternate vibration on the same plane only 18 seconds during the period of the revolution of the Lunar Nodes. Comparing this quantity with observation, he concluded that at the surface of the Earth the attraction of the Sun is to that of the Moon as 3 to 7, which makes the satellite to have only the 70th part of the mass of our planet.

The powers of calculation were now turned to the erratic class of the celestial bodies, so long the objects of superstitious terror, but which Newton had likewise subjected to the great law of gravitation. While the planets revolve in ellipses approaching to circles, and lying nearly in the plane of the ecliptic, the comets describe elliptical orbits with very different inclinations, and so extremely elongated as to resemble parabolas through a considerable part of their course. Being very small, they are seen for a short space only in the vicinity of the Sun, but become quite invisible during their distant excursive journey beyond perhaps the boundaries of our planetary system. The periodic time of a comet, depending on the length of the transverse axis of its ellipse, can seldom be determined with any sort of accuracy. The few observations that can be made in the transient interval of its apparition are scarcely sufficient to assign its mean motion and the places of its nodes and perihelion. Newton, while he proposed the parabolic theory as sufficiently correct in the visible portion of the cometary track, gave two elegant constructions for discovering the elements of the curve from three proximate observations. Halley applied those principles to the laborious computation of twenty-four remarkable comets. But his attention was more especially fixed on the nearest of them, which had been observed in 1531, 1607, and 1682, and seemed to be the same with one noticed by chroniclers in 1080, 1155, 1230, 1305, 1381, and 1456,¹ and hence

¹ This would seem to be the comet whose appearance after the capture of Constantinople by the Turks spread terror through Christendom, and which pope Calixtus III. so devoutly excised in the same anathema with these dreaded infidels.

performing its revolutions in about 75½ years; he therefore ventured to predict its return about the end of 1758 or the beginning of 1759. In arriving at this conclusion he was equally fortunate and circumspect; he found that the comet must suffer great derangements in its passage through our planetary system, from the attractions of Jupiter and Saturn, which he endeavoured to calculate, though with very doubtful success. The time of the expected return approaching, excited intense curiosity in the philosophical world; and Clairaut was induced to apply his formulae in the investigation of the progress of the comet, which, after immense labour in calculation, assisted by several expert computers, he announced at a public meeting of the Academy of Sciences on the 14th of November 1758. He found that the last revolution would be retarded about 618 days longer than the former, not reaching the perihelion till the middle of April 1759. But on revising his computation, he reduced that term to the 4th of April, exceeding that of observation by 12 days only, a discrepancy owing probably to the influence of the remote planet Uranus, which was not yet discovered. The comet was first seen by a peasant in Saxony on Christmas-day, but soon became the admiration of Paris, and procured for Clairaut the enthusiasm of popular applause. This remarkable erratic star, though visible only for a few months, had journeyed five years within the orbit of Saturn, though it never stretched its excursion to twice

the distance of Uranus. It approached so near to the Earth as would have disturbed her motions, if its mass had not been extremely small.

The success and popularity of Clairaut drew some peevish reflections from D'Alembert, whose formulae, if he had taken the trouble to pursue their application, would have led to similar results. Euler, whose researches on Comets were still more extensive and diversified, viewed this triumph of ardent exertion with the calmness and magnanimity of a sage. But Clairaut, eager to complete a work in which he had gathered so many laurels, next proceeded to calculate the disturbing influence of Jupiter and Saturn on the place of the nodes of the comet of 1682 and 1759, which has an inverted motion. Newton had shown that the perturbations in the planetary system always advance the perihelion and retract the nodes; but the case here was just reversed, and the quantity of recession thence determined agreed most exactly with observation.

So gleamed the last efforts of a philosopher who happily united the highest vigour of intellect with suavity of manners and elegance of conversation, but whose days were shortened by excessive labour, and the seductions of refined society, which incessantly courted and caressed his presence. Clairaut died on the 17th May 1765, at the age of 52, deeply regretted by a very numerous circle of friends, and leaving imperishable monuments of genius.¹

¹ A tribute of warm respect is paid to the memory of Clairaut by the celebrated but unfortunate Bailly, who thus eloquently pleads the cause of abstract science:—

"Le portrait de M. Clairaut seroit celui du véritable Géomètre. Un Géomètre est un homme qui entreprend de trouver la vérité; et cette recherche est toujours pénible, dans les sciences comme dans la morale. Profondeur de vue, justesse de jugement, imagination vive, voilà les qualités du Géomètre: profondeur de vue, pour appercevoir toutes les conséquences d'un principe, cette immense postérité d'un même père; justesse de jugement, pour distinguer entr'elles les traits de famille, et pour remonter de ces conséquences isolées au principe dont elles dépendent. Mais ce qui donne cette profondeur, ce qui exerce ce jugement, c'est l'imagination; non celle qui se joue à la surface des choses, qui les anime de ses couleurs, qui y répand l'éclat, la vie, et le mouvement, mais une imagination qui agit au-dedans des corps, comme celle-ci au-dedans. Elle se peint leur constitution intime; elle la change et la dépouille à volonté; elle fait, pour ainsi dire, l'anatomie des choses, et ne leur laisse que les organes des effets qu'elle veut expliquer. L'une accumule pour embellir, l'autre divise pour connaître. L'imagination, qui pénètre ainsi la nature, vaut bien celle qui tente de la parer; moins brillante que l'enchanteresse qui nous amuse, elle a autant de puissance et plus de fidélité. Quand l'imagination a tout montré, les difficultés et les moyens, le Géomètre peut aller en avant; et s'il est parti d'un principe incontestable, qui rende sa solution certaine, on lui reconnoît un esprit sage; ce principe le plus simple, offre-t-il la voie la plus courte, il a l'élégance de son art; et enfin il en a le génie, s'il atteint une vérité grande, utile, et long-tems séparée des vérités connues!

"Aucune de ces qualités n'a manqué à M. Clairaut; les preuves sont de l'histoire de la Géométrie, les succès seuls sont de notre ressort. L'Astronomie lui doit des progrès difficiles; nous le jugeons ici par ce qui intéresse les hommes, et sur ce qu'il a fait d'utile. La théorie de la lune restée imparfaite dans les mains de Newton, le cours des comètes calculé, leur retour prédit, en rendant compte des causes qui le retardent ou qui le précipitent; voilà ce qui restoit à faire depuis Newton, depuis Halley; et voilà ce que M. Clairaut a fait. Cela étoit difficile, puisque deux grands hommes y ont été arrêtés; cela étoit utile, parceque la connaissance des mouvemens de la lune amenera la perfection de la géographie et de la navigation, parceque la prédiction du retour des comètes caractérisera notre siècle et fera sa gloire. Le principal mérite

But the Theory of Comets still remained incomplete. Those excursive bodies, in traversing our system, often suffer such derangement from the influence of the proximate planets, that the most select observations are insufficient to determine with any sort of precision their elliptical orbits. The famous comet of 1759 was calculated by Euler and Lexell to perform its revolution in a space between 449 and 519 years, while Pingré assigned it the period of 1231 years. In some cases the observations have indicated an hyperbolic orbit. Conti, following the method of Gauss, found the comet of 1811 to revolve in 3056.3 years; but by a second computation he reduced the time to 2301 years. The profound and experienced calculator Bessel lately gave the comet of 1807 a period of 1953.2 years, which he next brought down to 1483.3. The attraction however of the Earth would increase this to 1713.5 years, while the influence of Jupiter would reduce it again to 1543.1 years. Although the process of calculation be now greatly improved, the return of a new comet cannot be safely predicted. Our expectations can acquire confidence only from a comparison with former appearances.

Though the comets suffer such derangement from the action of the larger planets, they have no sensible influence on our system. They must therefore be extremely small, consisting of a dark nucleus, invested with a cloudy or hazy excrecence, and generally provided with very long sweeping tails. They have never disturbed our tides; though, having sometimes approached within the third part of the distance of the Moon, they would consequently with the same mass exert twenty-seven times greater deranging force. But their passage was then so rapid as not to allow the accumulation of impulse required to heave the wide expanse of ocean.

Comets are distinguished by the elements of

their orbits, derived only from observations made near the perihelion. Newton gave an elegant method of discovering those paths, by a geometrical construction grounded on the parabolic theory, and embracing three distinct observations. But this solution is liable to much uncertainty in its application, and Boscovich has shown how in most cases it merges in a porismatic or indefinite result. Mathematicians have therefore employed the resources of analysis in the resolution of this problem. Euler, besides the three consecutive observations, adopted a fourth one more remote, and thus obtained an accurate but very tedious and involved process. Lambert derived a different mode of solution from some general properties which he discovered of the Conic Sections in 1761. He employed calculation combined with a graphical process, and sought to abridge the labour, by help of a table of the descents of comets, or the motions which they would perform if their parabolic paths collapsed into straight lines. Thomas Simpson, the only person in our island who at this time appeared to emulate the Continental mathematicians, likewise produced ingenious disquisitions on the Comets, and framed similar tables. Hennert, Tempelhoff, and Sejour, afterwards pursued the same subject, which has since been discussed with great ability by Lagrange, Laplace, and Legendre. Other methods for discovering the orbits of comets have been more recently proposed by Delambre, Olbers, and Gauss; the last of which may be considered as the most elegant and complete. But all these analytical modes of solution are still very tedious and complicated; insomuch that the humbler procedure by the way of Trial and Error, or what is called False Position, is found in practice to be on the whole the shortest and most satisfactory.¹

While the sublimer geometry thus gradually disclosed the various recondite anomalies which

de M. Clairaut fut le talent des applications: malgré son génie, il n'étoit point rebuté des détails: il pensoit qu'une vérité de pratique étoit préférable à celles qui restent ensevelies dans vingt pages d'analyse; aussi n'a-t-il fait que des choses utiles. Son nom a été connu, porté partout, et sera répété dans les âges. Si nous avons osé le louer, c'est que l'Astronomie lui doit de la reconnaissance, c'est que nous sommes au moment où la postérité commence pour lui. La vérité peut élever une voix franche et libre, et nous ne sommes que son organe." (Histoire de l'Astronomie Moderne, tome III. pp. 197, 198.)

¹ The most accessible work that has recently appeared on this very difficult subject is the Essay on Comets, which gained the first of Dr. Fellowes' Prizes in the University of Edinburgh, and which reflects such credit on its youthful author, David Milne, Esq. advocate.

affect the planetary motions, and yet maintain the balanced harmony of the universe, a genius of the first order arose, and placed himself at once on the summit beside the chief mathematicians of the age. In 1759 Lagrange, when only 23 years old, gave a new extension to the Integral Calculus, under a form which has obtained the appellation of the Calculus of Variations, and applied this with singular address to the rigorous solution of the very difficult problems of the propagation of sound, and the vibration of a musical chord. His early progress was hailed by Euler, and encouraged by D'Alembert. No mathematician has ever displayed a completer command of the Calculus than Lagrange, who rivals Euler himself in clearness, elegance, and fertility of invention, and eminently surpasses that great master of analysis in the justness and expansion of his philosophical views.

The next effort of Lagrange obtained him the prize offered by the Academy of Sciences at Paris in 1764, for his Discourse on the Libration of the Moon, in which he most satisfactorily explained, from the theory of Attraction, the cause of the Moon's presenting always nearly the same face towards the Earth. He arrived directly at the same general equations as D'Alembert, by a happy combination of the dynamical principle of that philosopher with the property of virtual velocities, which was the germ of his capital work on Analytical Mechanics. This subject he again resumed in 1780, and discussed it in the fullest and most accurate manner, having succeeded in completing the integration of the chief equations.

The theory of Jupiter's Satellites is not only an object interesting in speculative science, but of great importance in the practice of astronomical observation for finding the longitude. Bailly, applying merely the formulæ of Clairaut, obtained an imperfect solution of their periodic motions. In 1766 the genius of Lagrange embraced this subject in its full extent, by introducing into his equations not only the attractive force of the Sun, but the mutual attractions of the Satellites themselves. He neglected no term which might sensibly affect the results, and advanced with caution and address by successive

approximations. His investigation was a model of analytical research; yet, though capable of extensive application, it did not descend into all the practical details.

The Lunar Theory still presented difficulties which it required the utmost efforts of genius to abridge and partially remove. In 1768 D'Alembert simplified the evolution of the disturbing forces, by projecting orthographically the moon's orbit on the plane of the ecliptic. Euler, in conjunction with his son Albert, now gave the completest solution of the general problem that had yet appeared; but having discovered no indication of a secular equation affecting the Lunar motions, they were inclined to doubt its existence. This arduous discussion was repeated in 1772, on which occasion Lagrange shared the prize with the Eulers. Two years afterwards he resumed the subject, and closely examined the different conditions annexed to the programme. There are five elements of a planet's motion liable to the influence of Disturbing Forces: 1. The Great Axis of the primary ellipse; 2. The Eccentricity of the orbit; 3. The Inclination of its plane; 4. The position of its Nodes; and, 5. The direction of its line of Apsides. In 1776 Lagrange demonstrated that the great axis, on which depends the period of revolution, has no term involving the lapse of time, and therefore cannot be affected by any Secular Equations, which constantly increase or diminish. But with regard to the mean motions of Jupiter and Saturn, Lagrange and Euler came to opposite conclusions, the one exhibiting a continual acceleration, and the other a like retardation.

Laplace, who has attained still higher celebrity, now appeared on the scene. Displaying the same faculty of invention, he possessed nearly equal skill in the management of the Calculus, without reaching, however, the clearness, simplicity, and elegance of his illustrious rival. But he ranged over a wider field of discovery, and exerted greater and more persevering industry, and pressed forward with a loftier feeling of ambition.

In 1773 Laplace, following the steps of Lagrange, adopted a new mode of investigation, and pushed his calculations farther. Instructed by the successive advances of his predecessor, he

proved in 1775 that the mean motions of Jupiter and Saturn have no secular equations, but perform all their changes within certain long periods, which hence reconciles the discordant conclusions of Euler and Lagrange. Having found that the variation of eccentricity of Jupiter's orbit must cause a corresponding alteration in the motion of the Satellites, Laplace transferred the same idea to the perturbations of our Moon, and thus discovered the true theory of her secular equation, or rather of that vast cycle in which the lunar revolutions are alternately accelerated and retarded. This very slow but gradual diminution of the moon's periodic motion, which the intermediate observations of the Arabian Astronomers, between Hipparchus and Bradley, have incontestably established, was at one time referred to the resistance from an ethereal fluid imagined to occupy the celestial spaces. Laplace at first conceived that the retardation might be explained on the hypothesis of a progressive transmission of the power of Attraction, and computed that it would require 8000 times the celerity of Light, which travels 200,000 miles in a second, to produce this effect. But his subsequent inquiries having shown that Gravity must dart its influence more than 50 million times faster than Light, we may therefore safely consider it as quite instantaneous. This conclusion is highly important in a philosophical view, since it sets for ever at rest the various speculative attempts to explain the cause of Attraction by the agency of certain mechanical intermedia, and demonstrates it to be a primordial and ultimate principle ordained by the Wisdom of the Supreme Architect.¹

In the meanwhile Lagrange had in successive dissertations investigated the secular variations of the planets by a new and direct process, which he conducted with incomparable address and elegance. Nor did he confine his researches merely to theory, but applied the formulae to the motions of the five planets then known. In a subsequent investigation he most skilfully se-

parated the terms of the equation which exhibit the secular deviations, from those which represent the periodical changes.

The soaring rivals appear alternately to surpass each other. Laplace, continuing his researches, at last discovered, that the secular equation of the Moon affecting her mean motion and that of her perigee and of her nodes in the ratio of 4, 12, and 3, is produced by the slow variation of the solar attraction occasioned by the changing eccentricity of the Earth's orbit resulting from the influence of the larger planets, though they cannot alter the great axis which determines the mean periodic revolution. Lagrange then showed that the same results were deducible from his general formulae. In 1785 Laplace resumed his investigation of the motions of Jupiter and Saturn; and suspecting that Euler and Lagrange had not carried their approximation sufficiently far, he pushed his calculations from the second to the fourth powers of the eccentricities of their orbits, and proved that those planets can have no secular equations. But, having remarked that their mean periods are commensurable and very nearly as 2 to 5, he found their reciprocal acceleration and retardation to follow the same ratio. The cycle began in 1560, and comprehends 929 years; so that at the epoch of 1750 Saturn had his period shortened 48' 44", while that of Jupiter was lengthened by 19' 28". In 1788 he discovered two curious laws that connect the periods of Jupiter's Satellites, and gave a complete theory of their motions, which served as the basis of Delambre's excellent Tables.

After an interval of several years, during which Lagrange had totally suspended his mathematical studies, he returned to his early pursuits with all the freshness of youthful invention, and all the vigour of matured and improved intellect. In 1808 he gave a most general solution of the problem of Disturbing Forces, and by a wonderful effort of sagacity he reduced his equations into a form of the utmost simplicity and elegance.²

¹ It is rather singular that Laplace should on several occasions betray a disposition towards materialism, while his investigations point evidently to an opposite inference.

² Since all observations are liable to incidental inaccuracies, it requires great address to balance the opposite errors. The ordinary way is to take the arithmetical mean; but this mode confounds equally the remote and the approximate indications. Cotes, in his Estimate of the Errors of Observation, proposed an ingenious and more accurate mode of investigation,

The eighteenth century was equally distinguished by the progress of Practical Astronomy. Observatories were greatly multiplied and widely dispersed over the surface of the globe, the art of making observations was brought to higher perfection, and instruments of more delicate construction were successively introduced. Besides the great problems of the figure of the Earth and the nutation of her axis, we may enumerate four capital improvements: The measure of celestial refraction—the determination of the parallaxes, and consequently the mean distances of the Moon and Sun—the discovery of new planets and satellites—and the enlarged survey of the heavens, comprising the groups and various modifications of the fixed stars.

1. Celestial Refraction. A ray of light passing obliquely from a rarer into a denser medium, is bent or refracted towards the perpendicular. The light emitted by the heavenly bodies, when it enters our atmosphere and descends to the surface through a succession of condensing strata, must evidently describe an incurved path; and the particles which reach the eye and produce vision, pursuing the direction of the terminal tangent, will give to a star an apparent elevation. The portion of the trajectory traced in approaching the surface of the Earth may be considered as an arc of a circle having its radius about six times greater. The quantity of refraction is hence nearly proportional to the tangent of the altitude.

Hawksbee had in 1710 found by a nice experiment, that the refractive power of air compared with that of water is in the ratio of its density; and this law was ascertained by Biot and Arago about a century afterwards to be general in our atmosphere. Hence refraction of that medium increased with the elevation of the barometer or the depression of the thermometer.

The effect of cold was early remarked by the Dutch Arctic voyagers; but Bouguer first accurately observed the diminished agency of attenuated air during his sojourn on the summits of the Andes. Yet without demonstration Bradley gave a simple rule for computing the celestial refraction, and Mayer adopted another nearly similar. Thomas Simpson, in a very ingenious dissertation, derived a formula which is substantially the same. The subject of refraction has likewise been discussed by Lambert, Kramp, Fontana, and more recently by Laplace, Gauss, Bessel, Young, and Ivory. The formula of Laplace is complicated and inelegant, while Ivory's method seems clear and simple.

2. The Lunar and Solar Parallaxes. The nearer celestial bodies are seen from the surface of the Earth in a position somewhat different than if viewed from the centre. This deviation, termed parallax, is obviously greatest at the horizon, and diminishes constantly in approaching to the zenith. To ascertain parallax with any precision, it is therefore required to institute observations at distant stations. It was chiefly for this purpose that La Caille selected the Cape of Good Hope, where he determined the mean parallax of the moon to be $57^\{\backslash circ\}\; 39\text{'}$. But the parallax of the sun, being so small a quantity, is much more difficult to find. Kepler had taken it for a minute, Halley reckoned it at $25\text{'}$, but subsequent astronomers had generally reduced the estimate to 10 seconds. To determine this element with accuracy, Halley proposed a very ingenious method from the next transit of Venus, by measuring the acceleration and retardation of the time of her passage over the disc of the sun as viewed from remote points on the surface of the globe. He could not expect his life to be prolonged till that event, but he warmly exhorted his successors to prepare them-

which, reduced to its simplest form, consisted in projecting the several data on a plane, and assigning their common centre of gravity for the true result. Euler disposed the observations into conditional equations with indeterminate co-efficients, which he traced out by successive eliminations. Mayer adopted the same plan, and employed it most extensively in the construction of his valuable Tables. But this procedure being very laborious, Legendre proposed to abridge it by the method of the least squares, which had likewise occurred to Gauss, being generally used by him in the reduction of observations. It was a fine application of the Doctrine of Chances. Various algebraical investigations of the principle have been given; but they are commonly very intricate and abstruse.

Mathematicians, in threading the labyrinths of analysis, seem to have overlooked a most simple and luminous demonstration furnished by the Ancient Geometry. From an elegant proposition of the Loci Plani, it follows as a corollary, that the sum of the squares of the distances of any number of points from their centre of gravity is a minimum; which therefore merges in the solution of Cotes.

selves for observing it on the 5th of June 1761. Astronomers were accordingly dispatched by the several maritime powers of Europe to all the stations that were considered as the most eligible and accessible. Owing to various accidents, however, the results did not answer the expectations raised. The stations had not been always the best chosen; some of the most expert observers did not reach their proper destination, others were obstructed in their operations by the state of the weather, and many difficulties generally occurred which had not been provided for. From a comparison of the collected observations, Pingré deduced a parallax of $10\backslash frac\{1\}\{2\}\text{'}\text{'}$, while Short made it only $8\backslash frac\{1\}\{2\}\text{'}\text{'}$. Such a discrepancy was mortifying, and astronomers looked forward with impatience to the succeeding transit of Venus, which was fortunately to happen within the space of eight years, though such an occurrence would not take place again till the 8th of December 1874. In the meanwhile Dr M. Stewart revived an idea of Machin, who, from a rude computation grounded on the motion of the Moon's Nodes, estimated the Solar parallax at $8\text{'}\text{'}$. He preferred however the change of the apogee, or of the direction of the principal axis, which is affected in some degree by the sun's distance. By the application of the Greek Geometry alone, Stewart had with profound ingenuity achieved the solution of a problem which so long baffled the address of the great masters of the modern calculus; but in pursuing his deductions, the passion for purity and elegance of demonstration led him to hazard so many simplifications, as to render the conclusion, amidst a balance of errors, very doubtful and precarious. He reduced the parallax to $6.9\text{'}\text{'}$, which from its smallness excited considerable surprise.

The uncertainty regarding the sun's real distance was finally removed by the skilful and numerous observations of the Transit of the 3d of June 1769. The several results differed scarcely the quarter of a second, and their concurrence fixed the parallax at $8.6\text{'}\text{'}$. This likewise agrees with the theoretical calculations of Laplace from the Lunar anomalies. But Bessel, having with immense labour combined and carefully recomputed the original observations, has recently detected a small correction, which makes

the parallax to be only $8.575\text{'}\text{'}$, and consequently the mean distance of the sun 95,158,440 English miles.

3. Discovery of New Planets and Satellites. It was an inveterate opinion, derived from the ancient Pythagoreans, that the number of the great celestial bodies must of necessity be six, the first perfect number, or one which contains all its subdivisions. When Galileo directed his tube to the heavens and detected some of the satellites of Jupiter, this notion retarded the assent of the learned to his discovery; but after Saturn was found to be accompanied likewise by satellites, speculative philosophers sought to extend the catalogue of revolving stars to 28, the next perfect number. In this expectation, however, they were disappointed, and the Solar System received no further accession for the space of a century. The discovery of a New Planet was reserved for our own times.

Herschel, a musician residing at Bath, though a native of Hanover, which he had left in early youth, devoted his leisure to the construction and improvement of reflecting telescopes, with which he continued ardently to survey the heavens. His zeal and assiduity had already drawn the notice of astronomers, when he announced to Dr Maskelyne, that, on the night of the 13th of March 1781, he observed a shifting star, which from its smallness he judged to be a Comet, though it was distinguished neither by a nebulousity nor a tail. The motion of the star, however, was so slow as to require distant observations to ascertain its path. It was for several months presumed to be a Comet; but the hypothesis of a parabolic orbit led to very discordant results. The president Saron, an expert and obliging calculator, was the first who conceived it to be a Planet, having inferred from the few observations communicated to him, that it described a circle with a radius of about twelve times the mean distance of the Earth from the Sun. Lexell removed all doubt, and before the close of the year he computed the elements of the New Planet with considerable accuracy, making the great axis of its orbit 19 times greater than that of the earth, and the period of its revolution 84 years. Bradley, mistaking it for a fixed star, had observed it on the 3d December

1753; and it was again seen by Mayer on the 23d September 1756.

Herschel proposed, out of gratitude to his Royal Patron, to call the planet he had found by the barbarous appellation of Georgium Sidus; but the classical name of Uranus, which Bode afterwards applied, is almost universally adopted. Animated by this happy omen, he prosecuted his astronomical observations with unwearied zeal and ardour, and continued during the remainder of a long life to enrich science with a succession of splendid discoveries. But Herschel also detected the satellites that accompany his planet, amounting to six, which revolve in a plane nearly perpendicular to its orbit, and contrary to the order of the signs. Both these primary and secondary bodies obey, in the relation of their motions and distances, the great law of Kepler. The same conformity obtains in the revolutions of the satellites of Saturn, which he increased to seven; and thus every step in the progress of astronomy gives additional solidity to the grand system of attraction.

Some German philosophers, indulging their curious fancy, have traced among the distances of the planets an analogy, which, though not strictly accurate, yet approximates so near the truth, that if not really founded in nature, it may at least assist the memory. Kepler, whose ardent imagination inflamed all his speculations, believed that the harmony of our system wanted a planet between Mars and Jupiter. A similar notion was entertained by the ingenious Lambert, who further supposed it might be dark and invisible. Reckoning from Mercury, the first of planets, and assuming as the unit of measures its interval from the sun, the intervals of the rest will be expressed by the binary progression 2, 4, 8, 16, &c.; and consequently the distances of the series of planets from the sun may be represented by the numbers 1, 2, 3, 5, 9, 17, 33, 65, &c. While 3 denotes the distance of the Earth, and 17 and 33 indicate the distances of Jupiter and Saturn, the remote Uranus has come to occupy the place marked by 65. But a planet at the distance 9, between 5 and 17, the distances of Mars and Jupiter, was still wanting to complete the regular sequence. Bode slightly modified these proportions, and

attained greater accuracy, at some expense, however, of simplicity. Taking the number 4 to express the interval of Mercury, he multiplied the terms of the binary progression by 3. The distances of the planets from the Sun are hence denoted by the series 4, 7, 10, 16, 28, 52, 100, 196, &c. which numbers are convertible into English miles by multiplying by 9½ millions. The dark or deficient planet, corresponding to 28 or the distance of 266 millions of miles, is now supplied by the discovery of four very small stars, which have been fancifully conjectured to be only fragments discovered from the principal, while other portions, still unobserved, are whirling through space. The detection of those singular planets distinguishes the commencement of the nineteenth century. Piazzi discovered Ceres at Palermo on the 1st of January 1801, Olbers at Bremen found Pallas on the 28th of March 1802, and his countryman Harding added Juno on the 2d of September 1804, and Vesta on the 29th of March 1807. These asteroides, as they have been called by Herschel, differ from other planets not only by their diminutive size, but by the remarkable inclination of their orbits to the plane of the ecliptic, which, however, they intersect nearly in the same nodes.

This wonderful extension of the Solar System was chiefly due to the zeal and industry of the German and other confederated Astronomers, who, at the instance of the spirited Baron Zach, had divided the heavens into different sections, and occupied themselves in surveying their several allotments. The catalogue of fixed stars was thus prodigiously augmented, and many peculiarities in their character and configuration detected. In the meanwhile Herschel, with the superior aid of his powerful reflecting telescopes, observed the numerous clusters of nebulosities, and distinguished many of the changing and double stars, which, though suns of other systems, yet appear connected, and may probably circulate about their common centre of gravity. Assuming that the instrument he used could enable him to penetrate 497 times farther than Sirius, he reckoned 116,000 stars to pass in a quarter of an hour over the field of view, which subtended an angle of only 15'. If we compute from such a narrow zone, the whole

celestial vault must display, within the range of telescopic vision, the stupendous number of more than five billions of fixed stars. Imagination is bewildered in the immensity of such prospects. But a sober retrospect of the progress of astronomy would aid our conception of the structure and harmonious adjustment of the universe.

"To generate a circular description, it is requisite that the body should have the precise celerity due to a fall through half that radius. But, projected with inferior celerity, it would, about the same axis, trace elliptical orbits more or less compressed; the foci mutually retiring towards the extremities, while the conjugate diameter contracts. When the primary impulse becomes extinct, the elongated ellipse merges in a straight line. On the other hand, if the projectile motion should exceed the limit of circular velocity, a new series of curves will arise. But when the acceleration is directly as the distance, on passing the limit of celerity, the foci will first unite in the centre, and the axis suddenly turning at right angles, the foci will now gradually dispart in this transverse position. As the velocity of impulse is successively augmented, the ellipse will assume all the degrees of oblateness, till it finally vanishes in two parallel lines.

"If the centripetal force be inversely as the square of the distance, the moment the primary impulse transcends the limit due to a circular trajectory, the orbit would suddenly change into an equilateral hyperbola; the farther focus, which had come to coalesce with the attractive one, now flying to the opposite side beyond the diameter. When the celerity of projection receives a farther increase, the incurvation at the vertex will be proportionally diminished. With an extreme impulse, the body would shoot off in a straight line perpendicular to the axis.

"It thus appears, that a circular revolution, which the ancients so fondly contemplated as the perfection of the celestial movements, is incompatible with the stability of the universe. The most absolute precision of impulse would have been necessary, and the very slightest subsequent addition of celerity, from the incidental influ-

ence of those disturbing forces which are incessantly in operation, would at once have transformed the circle into a hyperbola, and have carried the planet away for ever into the boundless expanse of heaven. In viewing the grand phenomena of nature, our admiration is drawn to those conservatory principles which, in shorter or longer periods, correct every occasional deviation from the general balance of the system.

"It is curious, however, to remark how very nearly the planetary orbits approximate to circles. In that of our earth, the two axes differ only by the 7086th part. In the trajectory of Mars, this difference amounts to the 231st part. It was accordingly the greater eccentricity of that orbit which led Kepler to detect its elliptical form. The group of small kindred planets lately discovered revolve in curves still more elongated, the diameters of those of Juno and Pallas being nearly in the ratio of 30 to 29. These singular bodies might seem to rank between the ordinary planets and the comets, which wander in ellipses of extreme elongation, scarcely distinguished from parabolas, during a great part of their visible track. In another circumstance, too, the analogy obtains; for while the larger planets deviate not more than 2 or 3 degrees from the plane of the ecliptic, Pallas crosses it at an angle of 35°, and the paths of the comets have every possible inclination. It remains to be discovered, whether diversified bodies, travelling in the celestial spaces, may not fill up more completely that prolonged gradation of existence, which appears so conspicuous in other parts of nature. Supposing the projection of the planets to be the result of some more general law, those which had an excess of impulse would totally escape our range of observation, being swept away in hyperbolas into boundless space, there perhaps to form other stellar systems."¹

The ingenious Lambert produced some curious speculations respecting the celestial bodies. He presumed that the various planetary systems are all connected, and revolve about a common centre, which Galileo had conjectured to be situated in Orion or Sirius. The planets, therefore,

¹ Elements of Natural Philosophy, pp. 123-125.

do not strictly describe ellipses, but a series of convoluted epicycloids. He showed that comets, in passing the nearest planets, might have their orbits changed into parabolas or even hyperbolas, and hence journeying from system to system, would perform the tour of the universe. Those erratic bodies he peopled with a race of contemplative astronomers, who, enjoying such peculiar advantages, could survey the whole extent of the starry frame.

Lambert remarks, that, between the 16th and 17th centuries of our era, we have been visited by 40 comets that have not appeared again. These must, from their brilliancy, have reached the earth's orbit, and perhaps come within the verge of Venus and Mercury. But the very elongated cometary paths might descend in various directions without interfering with the planets. Calculating on the principle of chances, he found that no fewer than 500,000 comets might have the aphelia of their ellipses situate in the orbit of Saturn. But if we extend this computation to the orbit of Uranus, we may reckon up two millions of comets, as probably the number of those erratic attendants of our system. Mr Milne carries the estimate still higher. Assuming a thousand years for the average period of the revolutions of the comets, it would require to multiply by ten the 140 which have been observed within the earth's orbit during the last hundred years, giving 1400 for the whole of such near visitors. But Uranus being twenty times more distant, the product of the cube of this, or 8000 into 1400, exhibits 11,200,000 as the approximate number of all the comets which range within the known extent of our system.

The periodical change of the inclination of the ecliptic is at last established, though the quantity of variation and the length of its cycle have not been absolutely determined. Schubert, in his treatise of Practical Astronomy, makes it to fluctuate between $18^\{\backslash circ\}$ and $29^\{\backslash circ\}$, altering in the space of 65,000 years from $27^\{\backslash circ\}\; 45\text{'}$ to $20^\{\backslash circ\}\; 43\text{'}$; the maximum having occurred at the remote period of 36,300 years. But Delambre and Piazzi, whose judgment and scientific attainments rank much higher, have reduced this aberration within the moderate limits of $1^\{\backslash circ\}\; 20\text{'}$. The greater

obliquity of this ecliptic in ancient times, by elevating the sun more in summer, and depressing him in winter, must have proportionally increased the diversity of the seasons. But still the mean temperature of the earth has continued for ages exactly the same; for the momentum of rotation at first impressed remaining necessarily invariable, the smallest rise of heat, expanding the revolving mass, must have retarded its motion, and consequently lengthened somewhat the day. Any decrease of heat, on the contrary, would accelerate the terrestrial rotation. Now, the length of day has certainly not altered a single second of time since Hipparchus observed eclipses 3000 years ago; and it therefore follows, that during this long period the mean temperature of the earth has not varied the fraction of a degree. The constant accession of heat from the sun must hence be consumed by some process yet unexplored. Unless this heat were again to resume the form of light, it could not be darted from the upper atmosphere through the boundless void. Since the waters of the ocean are colder in proportion to their depth, might we not suppose the bottom to be frozen, and all the surplus heat to be absorbed in melting the subaqueous ice?

Astronomical considerations likewise elucidate the internal structure of our globe. The mean density of the mass is about double that of its exterior crust. But if the general materials were the same, and the law of compression subsisted in any degree, the resulting condensation would be inconceivably greater. Air compressed into the fiftieth part of its volume was lately found in France to have its elasticity fifty times augmented; and had it continued to contract at this rate, it would from its own incumbent weight have acquired the density of water at the depth of 34 miles. But water itself would have its density doubled at the depth of 93 miles, and even attain the density of quicksilver at a depth of 362 miles. In descending therefore towards the centre, through the space of about 4000 miles, the condensation of ordinary materials would surpass the utmost powers of conception. "It seems therefore to follow conclusively, that our planet must have a very widely cavernous structure, and that we tread on a crust or shell,

whose thickness bears a very small proportion to the diameter of its sphere. But since an absolute void is inadmissible, the vast subterranean cavity must be filled with some very diffusive medium, of astonishing elasticity or internal repulsion among its molecules. The only fluid we know possessing that character is LIGHT itself, which when embodied constitutes Elemental Heat or Fire. The great concavity may thus be filled with the purest ethereal essence,—Light in its most concentrated state, shining with intense refulgence and overpowering splendour.¹

It was the firm opinion of the ancients that our atmosphere rises as high as the Moon, the whole space below being doomed to change, decay, and mortality; and this notion still tinctures the language of poetry. But Kepler first reduced the extent of atmosphere within moderate limits. Astronomers having observed that twilight commonly closes after the sun has sunk 18 degrees below the horizon, it was hence easy to compute that the highest portion of air which from the west reflects the departing ray must have an altitude of about 44 miles. This conclusion, however, seems invalidated by the remark of Lambert, that twilight does not absolutely cease, but is succeeded during the night by a series of decreasing crepuscles, formed by a double, a triple, or even a quadruple reflection. These secondary lights may therefore be produced by the repeated gleaming from the sky at much lower elevations.

A wider limitation of the atmosphere is derived from a rigid principle. It is evident that the particles of air, as they expand from the axis of rotation, will augment their centrifugal tendencies, while their attraction to the centre will diminish in a duplicate ratio. At a certain distance, therefore, those antagonist forces will produce a mutual equilibrium. This balance must occur in the equatorial region at the height of 6.6 semidiameters, or about 26,000 miles, when the centrifugal force is increased, and the power of gravitation is reduced to about $\backslash frac\{1\}\{43\backslash frac\{1\}\{2\}\}$.

The late Dr Wollaston arrived at nearly a si-

milar conclusion, from an hypothesis respecting the ultimate molecules of the atmosphere. But a different consideration will bring us within much narrower limits. If we assume the depression of the absolute zero to be 750 centesimal degrees, and adopt the formula already stated for the relation of the density and temperature of air, it will follow that the tenuity of the highest stratum of the atmosphere cannot exceed $50\; \backslash times\; 750$ or 37,500 times, which corresponds to an altitude of about 51$\backslash frac\{1\}\{2\}$ miles. This boundary nearly coincides with the approximation of Kepler, and likewise agrees sufficiently with the estimated range of the Aurora Borealis. It thus appears from concurring probabilities, that our atmosphere extends not above 50 miles, the higher regions perhaps being occupied by hydrogen gas transfused with phosphorescent matter.

The perfection of astronomical instruments has afforded the prospect of being able to determine the Annual Parallax, and consequently the distance of the fixed stars; but the quantity of deviation is so small as to have hitherto eluded the closest observation. It cannot amount to a single second in the most conspicuous and probably the nearest of the stars. These luminous bodies must therefore be more distant at least two hundred thousand times than the measure of the diameter of the earth. The light emitted from such neighbouring suns, though it flies with enormous rapidity, must yet travel more than six thousand years before it approaches the confines of our system.

But scattered over the immensity of space, there may exist bodies which, by their magnitude and predominant attraction, retain or recall the rays of light, and are lost in solitude and darkness. Had the celerity of the luminous particles not exceeded four hundred miles in a second, we should never have enjoyed the cheering beams of the Sun. They would have been arrested in their journey, and drawn back to their source, before they reached the orbit of Mercury. But a star similar to our Sun, and having a diameter 63 times greater, would entirely overpower the impetus of light.

¹ Elements of Natural Philosophy, pp. 449-453.

Many of the smaller stars are found to have proper motions, some of them to an extent of several degrees. Mayer attempted to class the shifting stars according to their places in the heavens, and remarked that, while those situate in two opposite quarters appeared nearly stationary, such as had a lateral position varied the most. He therefore inferred that our Sun and the whole train of his attendants are carried forward in the line joining those opposite points. Herschel seized this ingenious idea, and combining his own observations, he concluded that the translation of the planetary system is directed to the constellation of Hercules. But the critical examination of Bessel has proved that no such regular transfer would reconcile the various discrepancies of the shifting stars.

The Double and Multiple Stars have lately engaged the attention of observers. Their catalogue has been prodigiously augmented by the ardour of South and the younger Herschel, and

by the unwearied assiduity of Struve at Dorpat, and of Inghirami at Florence. The Double Stars assume every hue, but generally the contrasted or complementary colours; a circumstance which seems to betray the influence of ocular deception. Their proper motions result probably from a circulation around their common centre of gravity. The Multiple Stars may possibly derive their peculiar aberrations in certain cases from the revolution about huge invisible Suns. All these mutable Stars are extremely remote, descending so low as the twentieth degree of magnitude, and can therefore be distinguished only by the penetration of the most powerful telescopes. Imagination is utterly bewildered by the shadowy visions which flicker along the horizon of Illimitable Space. But each successive observation reveals more clearly the extension of that exquisite harmony which the great law of attraction maintains through the countless systems of Worlds.

END OF VOLUME FIRST.

The following is a list of the names of the several persons who have been High Sheriffs of the County of Essex, from the year 1189 to the year 1215, and also of the several persons who have been Justices of the Peace of the County of Essex, from the year 1189 to the year 1215.

The names of the High Sheriffs are as follows:

The names of the Justices of the Peace are as follows:

The following is a list of the names of the several persons who have been High Sheriffs of the County of Essex, from the year 1189 to the year 1215, and also of the several persons who have been Justices of the Peace of the County of Essex, from the year 1189 to the year 1215.

The names of the High Sheriffs are as follows:

The names of the Justices of the Peace are as follows:

1215

DISSERTATION SIXTH:

EXHIBITING A GENERAL VIEW

OF THE

PROGRESS OF MATHEMATICAL AND PHYSICAL SCIENCE,

PRINCIPALLY FROM 1775 TO 1850.

By JAMES DAVID FORBES, D.C.L., F.R.S., SEC. R.S. ED.

PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF EDINBURGH,
AND CORRESPONDING MEMBER OF THE INSTITUTE OF FRANCE.

Adv. Phil.

DISSERTATION SIXTH

CONTAINING A GENERAL VIEW

OF THE

PROGRESS OF MATHEMATICAL AND PHYSICAL SCIENCE

PRECEDED FROM THE 15TH TO THE

BY JAMES DAVID BOWLES, D.C.L., F.R.S., &c. &c. &c.

PRESENTED TO THE ROYAL SOCIETY OF LONDON,
BY THE AUTHOR, ON THE 15TH OF NOVEMBER, 1818.

LONDON.

CONTENTS OF THE SIXTH DISSERTATION.

CHAPTER I.—INTRODUCTORY.

CHAPTER II.—PHYSICAL ASTRONOMY AND ANALYTICAL MECHANICS.

CHAPTER III.—ASTRONOMY.

CHAPTER IV.—MECHANICS OF SOLID AND FLUID BODIES, CIVIL ENGINEERING, AND ACOUSTICS.

§ 1. THOMAS YOUNG.—The Undulatory Theory of Light.—Its History from the time of Hooke and Huygens—The Law of Interference—Its application to Diffraction—to the Rainbow—and to other subjects.—The Theory of Polarization referred to another section.....page 893

§ 2. MALUS.—Discovery of the Polarization of Light by Reflection.—Early History of Double Refraction and Polarization.....page 901

§ 3. FRESNEL.—The Undulatory Theory of Light continued.—Diffraction.—Transverse Vibrations; Young.—Polarization and Double Refraction explained.—Lighthouse Illumination.....page 903

§ 4. ARAGO.—Short Account of his Scientific Career—He discovers the Colours of Polarized Light—Laws and Theory of Depolarization; M. Biot; Young; Fresnel.—Non-interference of oppositely Polarized Rays—Rotatory Action of Quartz.—M. Foucault's Experiment on the Velocity of Light.....page 907

§ 5. SIR DAVID BRUWSTER.—Progress of Experimental Optics—Laws of Polarization—Double Refraction produced by Heat and Compression—Discovery of Bixal Crystals—Laws of Metallic Reflection—Absorption of Light; and Lines of the Solar Spectrum; FRAUNHOFER.—Seebeck; M. BIOT.....page 911

§ 6. MR AIRY, SIR WILLIAM R. HAMILTON, and Professors LILOYD and MACCULLAGH.—Confirmation of Fresnel's Theory—Investigation of the Wave Surface completed; Conical Refraction.—M. CAUCHY—Mechanical Theory of Elastic Media, and of Ordinary and Metallic Refraction; M. Jamin.—Theory of Dispersion; Professor Powell.....page 917

§ 7. RITTER.—Chemical Rays of the Spectrum.—NIRPCE; DAGUERRE; Mr TALBOT. Art of Heliography or Photography—Daguerreotype—Calotype.—Professor STOKES. Chemical Rays rendered visible—Fluorescence.....page 921

§ 1. BLACK.—Latent and Specific Heat.—Irvine.—Hutton.—Doctrines of Heat applied to some Natural Phenomena.....page 925

§ 2. CAVENDISH.—His Singular Character and Attainments—Eminent Chemical Discoveries—Observations on Heat and on other Branches of Physics.—LAVOISIER.—The Calorimeter—Theory of Combustion and of Oxidation.....page 928

§ 3. DALTON.—Theory of Gases and Vapours—Law of Expansion by Heat—Atomic Theory of Chemistry.—GAY LUSAC.....page 933

§ 4. RUMFORD.—Economical applications of Heat—Point of Maximum Density of Water; Hope.—Friction as a source of Heat. Theory that Heat is convertible into Mechanical Energy; Mr Joule.....page 940

§ 5. SIR JOHN LESLIE.—Establishment of certain Laws of Radiant Heat.—Pictet.—Prevost.....page 942

§ 6. FOURIER.—Mathematical Theory of the Conduction of Heat—Lambert; Poisson.—Temperature of the Earth and of Space.....page 946

§ 7. DULONG.—The Law of Cooling.—Progress of the Science of Radiant Heat between Leslie's and Melloni's Discoveries; transmission of Radiant Heat through Glass. Herschel; De la Roche; Professor Powell.—Theory of Dew; Wells.....page 952

§ 8. MELLONI.—Recent History of Radiant Heat—Transmission and Refraction of Heat; Properties of Heat analogous to Colour—Experiments in Great Britain on the Polarization and Double Refraction of Heat.....page 955

§ 9. M. REONNAULT.—Numerical Laws of Expansion by Heat; Rudberg.—Vaporization; Dulong.—Latent Heat; Hygrometry.....page 957

§ 1. GALVANI.—Discovery of Galvanism; Proper Animal Electricity.—The subject revived by Nobili.—MM. Matteucci and Du Bois-Reymond.....page 958

§ 2. VOLTA.—Progress of Discovery in Common and Atmospheric Electricity—The Electro-motive Theory—Voltaic Pile—Chemical Analogies and Decomposition Fabbroni; Nicholson and Carlisle.....page 961

§ 3. SIR HUMPHRY DAVY.—Progress of Voltaic Electricity—Electro-Chemistry; Berzelius.—Davy's Invention of the Safety-Lamp.—WOLLASTON; his Electrical and other Observations. Contrast of his Character with that of Davy.....page 966

§ 4. ØRSTED.—AMPÈRE.—Discovery of Electro-Magnetism—Electro-Dynamic Theory—Discovery of Terrestrial Electricity; SEEbeck. The Galvanometer of Schweigger and Nobili.....page 973

§ 5. DR FARADAY.—Progress of the Theory of Electro-Chemical Decomposition—Volta-Electric Induction—Magneto-Electricity—Diamagnetism—Optical Changes induced by Magnetism.—Professor Plücker—Magneto-optic Action.....page 977

§ 6. OHM.—DANIELL.—Mr WHEATSTONE.—M. JACOBI.—Laws of Electrical Conduction;—Constant Battery;—Applications of Electricity to Telegraphs—Clocks—Motive Engines—the Electrotypewriter.....page 982

§ 7. CAVENDISH.—COULOMB.—On the Distribution of Statical Electricity, and on the Mathematical Theory of the same.—POISSON.—Mathematical Theory of Statical Electricity and of Magnetism generalized. Green; Professor William Thomson.....page 987

§ 8. PROFESSOR HANSTEEN.—Baron A. VON HUMBOLDT.—GAUSS.—Major-General SABINE.—Captain Sir J. C. ROSS.—Progress of our Knowledge of Terrestrial Magnetism in the Present Century.....page 990

CONSULTING INDEX

TO THE

PRINCIPAL NAMES AND SUBJECTS* MENTIONED IN THE FOLLOWING DISSERTATION.

The References are to the Articles or Paragraphs, not to the Pages.

Abel, 98.
Absorption of Light, 536.
Acceleration of the Moon, Secular, 62.
Acoustics, 433, &c.
Adams, Mr, 127, &c.
Airy, Mr, 82, 115, 228, &c.; 421, 549, 879.
Ampère, 792, 794, &c.
Arago, 166, 500, 792.
Asteroids, 282. List of, 283, and Additions, p. 993.
Astronomy, Physical, 41, &c.; Practical, 149, &c.; 218, &c.; Sidereal, 172, &c.; 269, &c.
Atomic Theory, Dalton's, 618.
Attraction of Spheroids, 99, &c.; of Mountains, 154, &c.
Attractions, Theorems about, 99, &c.; 877, 905.
Babbage, Mr, 377.
Baily, 158, 242, &c.
Balloon ascents, 630, and note.
Barlow, Mr, 879.
Berzelius, 763, 767, and note.
Bessel, 221, 310.
Biaxal Crystals, 520; Theory of, 493.
Biot, M., 545, 662.
Black, 318, 583.
Block Machinery, 370.
Borda, 237, &c.
Bowditch, 126.
Brewster, Sir David, 519, &c.
Bridges, Suspension, 361; Wooden, 356;
Girder, 358; Tubular, 359, &c.
Briakley, 307, and note.
Brunel, Sir M. I., 366.
Cogniard de la Tour, M., 441.
Calculating Engine, 377.
Calorimeter, 58, 607.
Calotype, The, 571.
Canal Navigation, 422.
Capillary Attraction, 88, 432.
Catenary, 353.
Cauchy, M., 556.
Cavendish, 156, 593, &c.; 871.
Cavendish Experiment, 156, &c.
Chances, 83.
Chemical Subjects connected with Natural Philosophy, 582, 596, 618, &c.; 763, &c.; 811.
Chemical action of Light, 562.
Chemical Equivalents, 628.
Chemical Theory of Galvanism, 751, 763, 811.
Chladni, 434.
Civil Engineering, 312-431.
Clairaut, 235.
Clock, Electric, 231, 862.
Coloration of Heat, 713.
Combustion, 608, 775.
Comets, 262; Halley's, 263; Encke's, 267; Gambart's or Biela's, 276; Comets of 1811 and 1843, 281.
Conduction of Heat, 661, &c.
Conduction of Electricity, 813, 842.
Cornish Steam Engines, 323.
Conical Refraction, 551.
Cooling, Law of, 654, 694.
Coulomb, 339, 413, 873.
Daguerre, 569.
Dalton, 610, 721, note.
Daniel, 852.
Davy, Sir H., 758, &c.; 792, 808.
Declination of the Needle, 881.
Delambre, 164, &c.
De la Roche, 704.
Density of the Earth, 154, &c.
Depolarization, 505.
Detrusion, 346.
Diamagnetism, 823.
Diffraction of Light, 458, 486, &c.
Dip, Magnetic, 882, note; 883.
Discontinuous functions, 31, 685.
Dispersion of Light, 559.
Double Refraction, 475, &c.
Du Bois-Reymond, M., 739.
Dubuat, 410.
Dulong, 693, 721.
Earth's Figure, 163, 234, 249, &c.; Density, 154, &c.; Rotation, 268.
Earth, Proper Heat of the, 675.
Economical application of Heat, 634.
Elastic Solids, 342, &c.
Electricity, 728, &c.; Animal, 734, 738, 872; Atmospheric, 743; Mathematical Theory of, 869, &c.; Distribution of, 871, 874.
Electric Telegraph and Clock, 856, &c.
Electro-Chemistry, 751, 754, 763, 811.
Electrodynamical Machines, 886.
Electro-Magnetism, 788, &c.
Electrotypes, 868.
Ellipticity of the Earth, 256.
Ellis, Mr Leslie, 20, note; 86.
Emanation of Radiant Heat, 651, 667.
Encke, M. 262, &c.; Encke's Comet, 267.
English Arc of Meridian, 187.
Erman, M. Ad., 885.
Evaporation, 613, 744.
Everest, Colonel, 250.
Expansion of the Gases, 617, 721, and note.
Fabroni, 751.
Fairbairn, Mr W., 363.
Faraday, Dr, 798, 807, &c.
Fechner, 851.
Figure of the Earth, 163, &c.; 234, &c. 249, &c.; Results, 256.
Flame, 770.
Flexure of Beams, 347, &c.
Fluids, Friction and Resistance of, 410.
Fluorescence, 579.
Foucault, M., 258, 514.
Fourier, 663, &c.
Fourneyron's Turbine, 427.
Fraunhofer, 538.
French Arc of Meridian, 165.
French experimenters, Skill of, 720.
Fresnel, 468, 485, &c.
Frog, Electricity of the, 734.
Galle, M., 157.
Galloway, 212.
Galvani, 728, &c.
Galvanism, Discovery of, 732.
Gases and Vapours, Dalton's Theory of, 613, 615, 617.
Gauss, 48, 856, 895.
Guy-Lussac, 617, 625, 630.
Green, 558 and note, 878.
Greenwich Observatory, 150, 159, 227.
Gregory, Duncan, 31, note.
Halley, 882.
Hamilton, Sir W. R., 550, 552, note.
Hansen, M., 57, note; 122, &c., 261.
Hansteen, Professor, 881, &c.
Heat, 582, &c.; Economical Applications of, 634; Hypothesis on the Nature of, 639; Radiant, 641, &c.; 692, &c.; Mathematical Theory of, 651, &c.; Motion in a Sphere, 673; Solar, 673.
Heliometer, 310.
Henderson, 304.
Herschel, Caroline, 269, note.
Herschel, Sir J. F. W., 294, 549.
Herschel, Sir William, 172, &c.; 702.
Hind, Mr, 292.
Historic periods in Science, 2, &c.
Hodgkinson, Mr, 352, 353, 363.
Hook, 452.
Hope, 636.

* For the Grounds of Selection of these, and on the Use of the Index, see Articles (13) and (23).

Humboldt, Baron A. von, 734, 892, &c.
Hutton, 591.
Huygens, 453, 475. The Principle of, 454.
Hydrodynamics, 408, &c.
Hygrometry, 614, 727.
Indian Arc of Meridian, 250.
Induction of Electric Currents, 818.
Inequalities, Secular and Periodic, 47.
Intensity, Magnetic, 885, 911.
Integration, 31, 97.
Interference of Light, 460; of Chemical Rays, 565.
Isothermal Lines, 894.
Irvine, 586, 589.
Ivory, 103.
Jacobi, C. G. I., 98.
Jacobi, Mr M. H., 848, 868.
Jamieson, 557.
Joule, Mr, 640.
Jupiter and Saturn, Theory of, 65.
Kelland, Professor, 421, 690.
Kater, 240.
Lagrange, 20, note; 41, &c., 90.
Lalande, 162.
Lambert, 661.
Lamilton, 250.
Laplace, 51, 60, &c., 199, 235, note, 433.
Lassell, Mr, 284.
Latent Heat, 584, &c., 599, 724.
Lavoisier, 605.
Legendre, 85, 95, &c.
Leite, Sir John, 641, &c.
Leverrier, M., 127, &c.
Libration of the Moon, 57.
Lines of the Spectrum, 538.
Lighthouses, 495.
Lloyd, Dr., 550, &c., 900, 910.
Locomotive Engine, 383, &c.
Lubbock, Sir J. W., 79, 117.
Luminous Waves, Length of, 463.
Lunar Observations, 152, 229.
Lunar Theory, 61, &c., 118, &c., 229.
Maccullagh, 553.
Magneto-crystalline Force, 832.
Magneto-optic Force, 831.
Magnetism, Terrestrial, 880, &c.; Theory of, 903.
Magneto-electricity, 819.
Magnifying Power, 179, 180.
Malus, 474.
Maskelyne, 150.
Mathematics, Pure, 20, 24, 28, &c.
Mathematical Theory of Heat, 661, &c.; of Electricity, 869, &c.; of Magnetism, 903.
Mattucci, 739.
"Mécanique Céleste," 89.
Mécanique, 166.
Mechanical Arts, 11, 32, &c.
Mechanical Notation, 378.
Mechanics, 312, &c.
Milloni, 707.
Metallic Polarization, 534.
Mitchell, 86, note, 156, 340.
Milky Way, 201, &c.
Miller, Patrick, 325.
Modulus of Elasticity, 345.
Natural Philosophy, its Connection with Mathematics, 24; with the Mechanical Arts, 32, &c.
Nautical Almanac, 153.
Navier, M., 354.
Nebulae, 190, 302.
Nebular Theory, 198, &c.
Neptune, Discovery of, 127, &c.; Elements of, 143.
Nicholson and Carlisle, 754.
Niepe, Nicephore, 569.
Nobili, 708, 738, 802.
Occult, 786, &c., 803, &c.
Obers, 161.
Ohm, 840.
Optics, 444, &c.
Parallax of Stars, 188, 306, &c.
Parameters, Variation of, 44.
Pendulum Observations, 233; M. Foucault's Pendulum, 258.
Perturbations, Planetary, 46.
Peters, M., 311.
Photography, 567.
Pictet, 648.
Pile of Volts, 750, 752.
Plana, M., 118, &c.
Planets, New small, 282, &c.
Playfair, 16, note.
Plenitude of Stars, 203.
Plücker, Professor, 831.
Poisson, M., 113.
Poisson, M., 112, &c., 689.
Polarization of Light, 474, &c., 525, &c.; affected by Magnetic Actions, 834; of Heat, 717; of Chemical Rays, 565; Circular, 491; by Metals, 534.
Poles, Terrestrial Magnetic, 882, 884.
Poncelet, M., 431.
Pond, 227, 307.
Potential, 877, 905.
Powell, Professor, 560, 705.
Previst, 649.
Probabilities, 83.
Proper Motions of Stars, 209, &c.
Pulkowa, Observatory of, 253.
Quartz, Optical Properties of, 511.
Quaternions, 552, note.
Radiant Heat, 641, &c., 692, &c.
Railways, History of, 398.
Rainbow, Theory of the, 465.
Reflection and Refraction of Light, 490.
Refrangibility of Heat, 714.
Refractions, Astronomical, 107.
Reynauld, M., 719, &c.
Rheostat, 848.
Ritter, 563.
Rivers, Theory of, 410.
Robison, 329, &c., 749.
Ross, Sir J. C., 884, 910.
Rosse, Earl of, 299.
Rotation of the Earth, 258.
Royal Institution, 637.
Rumford, 632, &c.; Rumford Prize, 637.
Russell, Mr J. S., 421.
Russo-Scandinavian Arc of Meridian, 254.
Sakéne, Major-General, 238, 246, 909.
Safety Lamp, 393, 770.
Satellites, New, 284.
Savart, 441.
Schehallien Experiment, 155.
Scheutz, M., 380.
Schwerer, 802.
Sebeck, 512, 544, 801.
Ships, Magnetism of, 879.
"Sirène," 441.
Slide-Rest, and Planing Machine, 372.
Solar Heat, 679.
Somerville, Mrs., 566 and note.
Sound, Theory of, 433.
Specific Heat, 588, 599, 607, 727.
Spectrum, Lines of the, 538; Heat of the, 702, 715.
Stability of the Solar System, 50.
Stars, Double, 187, 291, 295; Proper Motions of, 209, 292; Parallax of, 188, 306; Brightness of, 297; Variable, 298.
Steam, Elastic Force of, 699.
Steam Carriage, 384.
Steam Engine, 317, &c.
Steam Navigation, 325.
Stephenson, George, 392, &c.
Stephenson, Mr R., 355, &c.
Stereoscope, 581.
Stokes, Professor, 235, note; 247, 416, 579, 691.
Struve, M. W., 209, note; 252, 290, &c.
Sun, The, 185.
Suspension Bridges, 351.
Tait, Mr Fox, 571.
Telegraph, Electric, 856.
Telescopes, 178, &c., 299, &c.
Telford, 351.
Thermo-electricity, 801.
Thermo-multiplier, 709.
Thomson, Professor W., 878.
Tides, 69, &c.
Torpedo, 872.
Torsion Balance, 157, 340, 874.
Transverse Vibrations, 488.
Tredgold, 352.
Trevithick, 385, &c.
Turbine, 425.
Tunnel, The Thames, 373.
Undulatory Theory of Light, 452, &c.
Uranus, Discovery of, 183; Perturbations of, 132, &c.
Variable Stars, 298.
Variation or Declination of the Needle, 881.
Variation of Parameters, 44.
Venturi, 413.
Venus and the Earth, long Inequality of, 115.
Vision, 471, 681.
Volta, 740, &c.
Volta-electric Induction, 817.
Voltaic Pile, 749.
Waves, Theory of, 419, &c.
Water, Composition of, 597; Maximum Density of, 636.
Water Wheels, 426.
Watt, James, 312, &c., 597.
Weber, The M.M., 420, 856, 899.
Wells, 706.
Wheatstone, Mr, 581, 853, &c.
Whewell, Dr., 19, 79.
Wilson, P., 185.
Wollaston, 476, 538, 564, 780, &c.
Young, Thomas, 80, 342, &c., 445, &c., 488, 506, 780.

DISSERTATION SIXTH.

MATHEMATICAL AND PHYSICAL SCIENCE.

CHAPTER I.

INTRODUCTORY.

§ 1. On the Plan of this Dissertation.

THE year 1850 may be said to complete the Third century of modern scientific progress, or the Fourth if we include its earliest dawn. To each of these ages of discovery may be assigned a peculiarity in the character of its improvements, and even in the methods which conduced to that improvement.

Between 1450 and 1550 (a period so distinguished in letters and the arts), some great truths in physics and mathematics had presented themselves to a few precocious minds, yet they had not received any public acknowledgment, nor perhaps an adequate demonstration. Algebra then first became a science. Leonardo da Vinci made the earliest steps since the time of Archimedes, in rational mechanics, and Copernicus almost at the close of this period promulgated the true system of the world.

But the next centenary (1550-1650) was the first of true scientific activity. Its characteristic feature was the vindication of observation and experiment as the prime essentials to the increase of natural knowledge, with the consequent repudiation of the dogmas of the schoolmen, and the baseless methods of a priori reasoning. The men of science formed a goodly array at this stirring time; and signal were their triumphs. Galileo was beyond all comparison the glory of his age. His sagacity, his knowledge, his versatility of talent, his ingenuity as an inventor, his success in prosecuting his discoveries, and his zeal and eloquence in making known their importance, gave him an enviable pre-eminence even amidst a mighty generation. Bacon laid down the canons of a new method

in philosophy which Gilbert and Kepler, as well as Galileo, had already acted on. Napier and Descartes prepared for the general application of mathematics in the coming struggle.

The hundred years which next succeeded (1650-1750) saw the triumphant application of mathematics to Mechanics and Physics, and the establishment of the greatest mechanical theory of any age, that of Gravitation. The preparatory labours of a hundred and fifty years were brought, chiefly by the unparalleled sagacity and genius of Newton, to a speedy and dazzling climax. His success brought numerous and worthy labourers into the field, but they found enough to do in gathering in the harvest which he had prepared for them.

If we look for the distinguishing characteristic of the centenary period just elapsed (1750-1850), we find it in this,—that it has drawn far more largely upon Experiment as a means of arriving at truth than had previously been done. By a natural conversion of the process, the knowledge thus acquired has been applied with more freedom and boldness to the exigencies of mankind, and to the farther investigation of the secrets of nature. If we compare the now extensive subjects of Heat, Electricity, and Magnetism, with the mere rudiments of these sciences as understood in 1750, or if we think of the astonishing revival of physical and experimental Optics (which had well nigh slumbered for more than a century) during the too short lives of Young and Fresnel, we shall be disposed to admit

the former part of the statement; and when we recollect that the same period has given birth to the steam-engine of Watt, with its application to shipping and railways,—to the gigantic telescopes of Herschel and Lord Rosse, wonderful as works of art as well as instruments of sublime discovery,—to the electric telegraph, and to the tubular bridge,—we shall be ready to grant the last part of the proposition, that science and art have been more indissolubly united than at any previous period.

(6.) Limits of this Dissertation.
The Dissertation of Professor Playfair closes with the period of Newton; that of Sir John Leslie, professedly devoted to the history of the eighteenth century, embraces some matters which belong more properly to those which preceded and followed it. After considering how I might best carry out the plan of these essays, I have adopted the period from about the year 1775 to 1850 as the general limit of my review. We may imagine this period, of three quarters of a century preceding the present time, to be divided into three lesser intervals of 25 years each, which have also some peculiar features of their own.

(7.) Character of the close of the 18th century.
From 1775 to 1800 many branches of science still continued in the comparatively inert state which characterized a great part of the eighteenth century. There were, however, two or three notable exceptions. One was the continued successful solution of the outstanding difficulties of the Theory of Gravity applied to the moon and planets, a task in which the continental mathematicians, and of these, in chief, Lagrange and Laplace, had no rivals, or even coadjutors, on this side of the channel: Another was the foundation of Sidereal Astronomy by Sir William Herschel; and the last was the commencement of a system of Chemical Philosophy based on new and important experiments, and including the laws of heat in combination with matter, which at that period very naturally ranged themselves within the province of the chemist. In this department two British and one foreign name stand conspicuous, Black, Cavadish, and Lavoisier. I do not of course mean to affirm that other branches of science were not cultivated with success within the exact period of which we speak. Electricity, for instance, first statical, afterwards that of the pile, had a share in the discoveries and speculations of the time. But these were rather the mere extension of what had previously been thought of; or the first dawn of future important results, whose development fills a large space in the succeeding story. Volta and his inventions belong rather to the nineteenth than to the eighteenth century.

(8.) Character of the period 1800-1825.
The first quarter of the present century attained a higher and more universal celebrity. Scarcely a branch of physical science but received important and even capital additions. Physical Astronomy indeed, no longer filled so large a space in the page of discovery, simply because the exhaustive labours of the geometers of the former period had brought it to a stage of perfection nearly co-ordinate with the means

of observation, and because, by the publication of the Mécanique Céleste, Laplace had rendered available and precise the masses of scattered research accumulated by the labours of a century since the close of Newton's career of discovery. It was in some sense a new book of "Principia,"—not, indeed, the work of one, but of many; nor of a few years, but of two generations at least. Still there it was, a great monument of successful toil, which, like its prototype, was for many years to be studied, even by minds of the highest order, rather than to be enlarged.

But the other branches of Natural Philosophy were now to make a stride, such as perhaps no preceding time had witnessed. The science of Optics was speedily expanded almost twofold, both in its facts and in its doctrines. Galvanic Electricity disclosed a series of phenomena not less brilliant and unexpected in themselves than important from the new light they threw on the still dawning science of chemistry, and from the power of the tool which they placed in the hands of philosophers. Before the first quarter of the present century closed, the important and long suspected connection between Electricity and Magnetism was revealed, and its immediate consequences had been traced out with almost unparalleled ingenuity and expedition. The basis of the science of Radiant Heat, slightly anticipated by the philosophers of the eighteenth and even the seventeenth centuries (Lambert and Mariotte), was finally laid in a distinct form, assigning to the agent, heat, an independent position dissociated from grosser matter, such as light had long enjoyed. Astronomy, though enriched on the very first night of the new century by the discovery of a small planet, the herald of so many more of the same class, made perhaps less signal progress; but Chemistry, besides the aid it received from the invention of the pile, had a triumph peculiarly its own in the addition of the comprehensive doctrine of Definite Proportions, destined to throw at some later time a steady light on the vexed question of the constitution of matter. The great number of scientific names of the first order of merit concerned in these numerous discoveries marks the extraordinary fertility of the period. They are imperfectly comprehended in the following list: Young, Malus, Sir David Brewster, Fresnel, and Arago; Volta, Dalton, Davy, and Oersted; Prevost, Leslie, and Fourier; Gauss, Ivory, Olbers, Bessel, and Encke.

Of the twenty-five years just elapsed, it is not so easy to speak with precision. The voice of criticism may be fairly uttered with that reserve which every one must feel in speaking of his immediate contemporaries. Yet it may perhaps be stated without just cause either of offence or regret, that it has not on the whole been characterized by the full maturity of so many commanding minds. Of the great discoverers of the former period, several survived and continued their efficient labours during no small por-

tion of the latter; and a few happily still remain to claim the respect and veneration of their disciples and successors. But the vast steps so recently made in Optics, in Electricity, in Magnetism, in Thermotics, and in Chemical principles, tended of necessity to call forth such an amount of laborious detail in the defining and connecting of facts and laws, and the deductions of the theories started to explain them, as seemed to render fresh and striking originality somewhat hopeless, whilst they occasioned a vast amount of useful employment to minds of every order of talent. The undulatory Theory of Light, nobly blocked out by the massive labours of Young and Fresnel, has afforded still unexhausted material to the mathematician on the one hand, and to the experimentalist on the other; and ably have they fulfilled the double task, adding at the same time discoveries whose importance and difficulty would have made them still more prominent, had they not been the legitimate consequences of a still greater discovery already in our possession. Nearly the same might have been said for the sciences of Electricity, Electro-magnetism, and Electro-chemistry, had not the comparative newness of the whole doctrine of these sciences, and the suddenness of their first rise, and, perhaps still more, the appearance of a philosopher of the very highest merit, Mr Faraday, who fortunately attached himself to this special department, made the last thirty years an almost unbroken period of discovery. Radiant Heat, too, has been successfully advanced by labours comparable perhaps to those which marked its first rise as a science, and some other topics connected with heat have risen into great and practical consequence. Astronomy has been prosecuted with a systematic assiduity and success, especially at the British and Russian national observatories, which yields to that of no former period, whilst physical astronomy has been cultivated by methods of still improved analysis, and has achieved one triumph which France need not grudge to England, nor England to France,—so signal as to be placed by common consent in a position superior to any since the first publication of the theory of gravitation, more than a century and a half before. This was the prediction of the position in space of a planet whose existence was unknown except by the disturbance which it produced in the movements of another. Terrestrial Magnetism has, for the first time, aspired to the rank of an exact science. In an illustrious philosopher of Germany, it has found its Kepler; and the combination of national efforts in collecting reliable data from the remotest corners of the globe is characteristic of the practical energy of the age. Pure Chemistry has been cultivated with extraordinary assiduity; but though some general principles have emerged, none are comparable, from their importance, to the discovery of Dalton. To cite, then, at present, but a few names, amongst the most conspicuous benefac-

tors of science of the last, or contemporary period, are MM. Airy, Cauchy, Hamilton, and MacCullagh; MM. Faraday, Melloni, and Gauss; Sir John Herschel, M. Struve, and Lord Rosse; MM. Plana, Poisson, Leverrier, and Adams; MM. Mitscherlich, Liebig, and Dumas.

It seems to me impossible to exclude from a review, however slight, of contemporary progress in the exact sciences, the advances which have accrued to them both directly, and, as it were, reflexively, by the astonishing progress of the Mechanical Arts. The causes, indeed, which called them forth are somewhat different from those which are active in more abstract, though scarcely more difficult, studies. Increasing national wealth, numbers, and enterprise, are stimulants unlike the laurels, or even the golden medals of academies, and the quiet applause of a few studious men. But the result is not less real, and the advance of knowledge scarcely more indirect. The masterpieces of civil engineering—the Steam Engine, the Locomotive Engine, and the Tubular Bridge—are only experiments on the powers of nature on a gigantic scale, and are not to be compassed without inductive skill as remarkable and as truly philosophic as any effort which the man of science exerts, save only the origination of great theories, of which one or two in a hundred years may be considered as a liberal allowance. Whilst then we claim for Watt a place amongst the eminent contributors to the progress of science in the eighteenth century, we must reserve a similar one for the Stephensons and the Brunels of the present: and whilst we are proud of the changes wrought by the increase of knowledge during the last twenty-five years on the face of society, we must recollect that these very changes, and the inventions which have occasioned them, have stamped perhaps the most characteristic feature—its intense practicalness—on the science itself of the same period. (11.)

Having thus briefly reviewed the course of discovery since the latter portion of the eighteenth century, I proceed in the succeeding chapters to attempt to sketch it more in detail, dividing the sciences into this groups, and in each of these endeavouring to present a lively view of its progress by connecting it with the individual career of the eminent men who have most contributed thereto, and introducing collaterally the chief results obtained by their contemporaries. In this manner I hope, on the one hand, to escape the formality of a history of science, and the meagre detail which our limits would prescribe to so vast a subject; and on the other to be enabled to impress upon the reader (as seems to be the design of these Essays) the leading facts and features of discovery in every age, together with the intellectual characteristics of the greatest minds which contributed to it. (12.)

It is with no overweening confidence that I lay the

result of this attempt before the reader. During the lengthened period of composition of this Dissertation—protracted by indisposition and untoward circumstances of different kinds, I have had abundant leisure to reflect on the advantages and disadvantages of a plan which I had sketched in the previous paragraphs, at the very opening of my task. I am aware that a rigid criticism awaits every attempt like the present. I am aware also, that it is far easier to detect real faults, especially of omission, than to make sufficient allowance for the exceeding difficulty and delicacy of the undertaking. I have but one ground of confidence, and that is so strong, that I trust it will enable me calmly to meet every just critical reflection. I am conscious of having written in a spirit of absolute impartiality whether as regards persons or subjects, and that I have exercised to the full amount of my opportunities what powers of judgment I possess. I have striven to speak judicially and historically whether of friend or stranger, the dead or the living, Englishman or Foreigner. What I have felt the most constant effort, has been the needful exclusion of meritorious names, far more numerous than those especially included and dwelt upon in these pages. But this has appeared to me the cardinal point of my whole plan. The labourers in science have been in these latter days so numerous, that had I noticed, even briefly, every one who had made a real step in science, my pages must have been crowded by names and titles of books. Even with the extension of bulk to which this essay has gradually and unavoidably grown (nearly double of its projected amount), the reader would rather have been bewildered than led by the perusal of such a catalogue. Besides, since such a brief historical synopsis forms very generally an introduction to the several articles of the Encyclopædia, to repeat it all here would have been but a tedious redundancy. No one conversant with such matters will imagine that I have saved myself any labour by this particularity of selection. On the contrary, it would have cost no effort to enumerate under each subject the living or recently deceased authors upon it who are best known; such a detail must have left a vague and shadowy impression on the mind of the general reader, and when regard is paid to the necessary limits of the essay, and the multitude of technical details and technical words which there is no space to define and illustrate, it is plain that the perusal must have been rendered as dry and unpalatable to those who seek general and elementary yet clear ideas, as it would have been tantalizing and unsatisfactory to the accomplished student, or to the man of science in search of particular historical details.

(14.) The end at which I have aimed is to select the more striking land-marks of progress in each subject in each age, and endeavour to connect them with the character and position of all the more eminent dis-

coverers, thus conveying to the general reader sufficient information on the limited number of particular subjects discussed, and interesting him not only in the science but in the individuals. Then, by a few slighter touches only, and the mention of some secondary names, to connect with one another these brighter periods of eminent progress, in which every country and every age feels a just pride.

That by many I shall be considered to have dwelt too much on some eras of invention, and to have omitted others not less important, is a difference of judgment which it is impossible not to anticipate, but equally impossible to prevent. I will only add that I have endeavoured to extend my impartiality to subjects as well as to persons; and that I have not intentionally dwelt longer on the topics of my own predilection than on those naturally considered by other persons equally or more important. Many subjects as well as persons familiarly known to me are scarcely, if at all, mentioned in these pages. To leave some definite and vivid impressions, selected solely for their importance, on the mind of my reader, has been the great object constantly before me during the composition of this work.

It will be seen from the preceding paragraphs, that I have deviated in some respects from the scheme of my two distinguished predecessors in the composition of the Dissertations on Science, Professor Playfair and Sir John Leslie. The essay of the former, which is the more finished and methodical, is admirably adapted to the period of the history of science of which he had principally to treat; the period, namely, of Galileo, Bacon, Newton, and Leibnitz. But the amount of material was smaller, and the principle of selection was also much simpler. The positive science of that age might almost be reduced to two heads, Astronomy, including its mechanical principles, and Optics. It was an age not more distinguished for the Truths it disclosed, than for the invention and right appreciation of the Methods of Discovery. Inductive Logic, and Mathematical Logic applied for the first time to dynamics, very justly claimed a place in a dissertation on the progress of science, in a period when these preliminary doctrines and discoveries were the stepping stones by which even the basement story of the Temple of Nature could alone be reached. The Philosophy of Bacon and the discovery of Fluxions, occupied therefore, with much reason, a large portion of Professor Playfair's beautiful Dissertation; and it is impossible to regret that an intellect so admirably qualified for tracing and displaying the intimate and historical connection of branches of knowledge so varied in their principles and character, should have been thus congenially employed, to the delight and edification of readers of every degree of acquirement from the highest to the humblest.¹

¹ In mentioning the name of my distinguished predecessor in the Chair of Natural Philosophy in the University of Edinburgh, I willingly take the opportunity of noticing, in a few words, his peculiar merits, to which the Dissertation contained in

(17.) Extent and variety of materials. But the extended domain of the science of the last hundred years enhances vastly the difficulty of the subjects succinctly handled as one by Mr Playfair. The mechanical and experimental sciences alone constitute a body of knowledge so large that it is a responsibility sufficient for one person to attempt to grasp them all, and to set forth in order the steps of progress and improvement which have been so rapid, and even so startling. Since some of these have scarcely as yet been historically digested, and the broad features of contemporary discovery have not been gradually separated by the judgment of an impartial posterity from those slighter though praiseworthy details, which lapse of time and advance of knowledge will throw into the shadows of distance,—this difficult and most laborious task falls principally upon the reviewer. The length and breadth of the subject of Natural Philosophy, and the cumbrous and scattered depositories of knowledge in which its records must be sought, combine to render not only the undertaking an arduous one, but the result of it a good deal more bulky than might be desired, or than was easily possible, in dealing with the glorious, but compact, history of Newton's age. It might be compared to the difference between writing a history of the Jews or Romans and that of the whole of modern Europe.

(18.) Logic of induction. The mere magnitude of the undertaking, then, might well excuse me from entering upon the cognate, but exceedingly distinct, subjects of the Logic of Inductive Discovery and the progress of the Pure

Mathematics. But an equally sound reason might be found in my consciousness of inadequacy to undertake, whatever had been the dimensions of my work, a threefold scheme of such magnitude and difficulty. I do not think that any one person could be found to treat the whole as it ought to be treated, and I am certain that I am not that person.

One attempt—a bold and successful one—has (19.) been made, in our own day, to unite two of the Writings of Dr Whewell and Mr Stuart Mill. three departments,—I mean the History and the Philosophy of the Inductive Sciences. An English philosopher of wonderful versatility, industry, and power has erected a permanent monument to his reputation in a voluminous work bearing the preceding title.¹ A slight inspection of that work will show how impracticable and self-destructive a plan it would have been to attempt anything like a systematic abridgment of such a mass of facts and speculations within our present limits. Mr J. Stuart Mill has also published a work bearing on the origin of our scientific knowledge, diametrically opposed in principle to the preceding one, yet marked by great ability.² Such disquisitions belong more properly to the philosophy of the human mind than of physics. After all, be it remembered that whatever has been learnt or discussed concerning the means of arriving at truth in Natural Science, it is not pretended that we have recently become possessed of any canons or rules of discovery superseding those fundamental principles of observation and experiment so well laid down by Bacon, and practised both

the present volume will bear an enduring testimony. Playfair's original contributions to science were not so marked and considerable as to justify me in including his name in the comparatively brief catalogue of discoverers chronicled in the succeeding of Professor pages; but his efforts are, nevertheless, deserving of notice, and indirectly were perhaps hardly less beneficial. He was a most patient and admiring student of the greatest mathematical writers of his time, and, when we consider the singularly backward state of that science in Great Britain about the end of the last and commencement of the present century, it was of no slight importance to find a man placed in the position of a public instructor able and willing to direct attention to the splendid achievements of the continental mathematicians. By his lectures both on Mathematics and Natural Philosophy—by his luminous articles in the Edinburgh Review—by some of his original papers in the Transactions of the Royal Society of Edinburgh—he contributed to this useful end, and would have done so still farther had he been enabled to complete the Dissertation which he so ably commenced. He had an excellent mathematical capacity, and mathematical taste, rather than power. His explanations, even of matters of inherent difficulty, are perspicuous and popular, qualities possessed by few of his contemporaries. His style has been pronounced by the highest authorities to be a model of clearness and eloquence. He was extensively read in subjects of metaphysics and morals, as well as of pure science; and by a combination of talent rare, I am inclined to say, in a high degree, his taste, though eminently mathematical, was also directed, with signal success (at first through his intimate friendship for Dr Hutton), to the very opposite studies of Geology and Physical Geography, which may be said to have been the subjects of his predilection during the last twenty years of his life. Nor were these labours of the closet merely; he was far more intimately versed in the mineral structure of the earth, from observation, than any except a few professed geologists; and he exceeded them all in the ability with which he expounded and maintained the striking doctrines of the Huttonian theory. Though professedly the "illustrator" of the principles specifically but obscurely laid down by Hutton, he certainly added much of his own. There is no reason to doubt that Playfair first apprehended the moving power of glaciers as geological agents in modifying the surface of Alpine countries, a matter which has of late been so earnestly discussed by the ablest geologists.

What adds to the singularity of the combination of tastes and talents to which I have referred is, that he appears to have had the slightest possible taste for that art of experiment which he eloquently advocated, with Bacon, as the grand distinction of modern science. I may be wrong in stating it broadly, but I do not now recollect a single experimental novelty, much less discovery, which we owe to Playfair, I mean in the department of Natural Philosophy; for we cannot include barometrical measurements under this head, of which, indeed, it was the mathematical theory, and not the application to practice, which chiefly occupied him. The same was the case in Astronomy, which, of the mechanical sciences, interested him most. In two capacities he will be remembered,—first, as the able, eloquent, and generally impartial and accurate Historian of Science; secondly, as the promoter, to so great a degree as to be considered a second founder, of modern Dynamical Geology. He was much beloved in private life, and was singularly free from the tendency to carping criticism and personal prejudices sometimes, unfortunately, found in men of letters. He was the intimate associate of Jeffrey and the other founders of the Edinburgh Review. His character has been drawn in three words by Sir James Mackintosh, and as happily contrasted with that of his illustrious friend:—"Playfair and Jeffrey; the first a person very remarkable for understanding, calmness, and simplicity, the second more lively, fertile, and brilliant than any Scotchman of letters" (Life of Sir James Mackintosh, II. 251).

¹ Whewell's History and Philosophy of the Inductive Sciences, 5 vols. 8vo.

² Mill's Logic, 2 vols. 8vo.

before him and since by Galileo, Newton, and their disciples.

(20.)
Pure mathematics—
their progress and technicality.

With regard to Pure Mathematics, and their progress during the last seventy years, to the difficulty arising from the extent to which the review must have enlarged this Essay, and the enormous and disproportionate labour it must have cost (a labour far greater to the present writer than to one by taste and habit more addicted to the study of merely abstract Analysis), a conclusive argument against their systematic introduction into this historical sketch is to be found in the very nature of these modern improvements. All sciences—but especially the abstract sciences—tend to become more intensely technical the farther they are pursued. These especially are incapable of popular treatment, although in their applications to physical science they occasionally admit of it in a very remarkable manner. The progress of analysis cannot be even enunciated or expressed but in the language of analysis, and the History becomes almost a Treatise, or, if not a Treatise, something nearly as technical. It is partly for this reason that the history of the Pure Mathematics has so seldom been even attempted to be written.¹ Mathematicians have, since the time of D'Alembert, been noted for being more ready themselves to publish than to become acquainted with what others have done; and one consequence of this has been the formation of a mathematical literature, able, profound, and original, but cumbrous, fragmentary, and full of repetitions.² Besides, the seventeenth century had attained the vantage-ground of those grand and striking improvements in methods to which no subsequent improvements, however real and ingenious, can by possibility compare. We shall never have inventions comparable, in universality and importance, to the application of Algebra to Geometry, and the discovery of Fluxions. These also admit of being at least partly explained in language not obtrusively technical, and have been so explained by the facile pen of Playfair; but all subsequent discoveries have been but enlargements and improvements on these primary and distinguishing ones; and before the date at which our present discourse properly opens, even the larger generalizations of Newton's fertile calculus—the method, namely, of Variations, and the integration of partial differential equations, had been

established, by the genius of Euler and Lagrange, on an impregnable basis.

The intense, and praiseworthy, and successful labours of their followers have been, then, chiefly devoted to the occupation of the fields of conquest thus summarily opened; or, rather, to storming, one by one, fortresses still unoccupied, after the main resisting army had been first routed in the open field. To quit metaphor, the efforts of mathematicians have for many years been chiefly applied to rendering possible the solution of problems involving quantities which actually occurred in the course of the rapid simultaneous advances of physical science. They are in a manner inseparable from the branches of physics in aid of which they were originally called forth, and will therefore be most properly noticed, however briefly, in the chapters of this Dissertation where their application is considered. Some farther observations on this subject will be found in the immediately succeeding section of the present Essay.

In reviewing the progress of science—physical science in particular—during the last seventy or eighty years, I have thought it advisable not to subdivide the subjects too minutely, and, following nearly the arrangements of Dr Whewell's excellent treatise, already quoted,³ I shall treat, in successive chapters, of Analytical Mechanics including Physical Astronomy as their loftiest and most successful application; of Astronomy as a science of observation; of Mechanics, with reference to the intimate constitution of matter, including Hydrodynamics, Acoustics, and Civil Engineering; of Optics, or Light; of Heat, including the Daltonian theory of the gases and chemical elements; and, finally, of the large and comprehensive science of Electro-magnetism, including ordinary and Voltaic Electricity, Terrestrial Magnetism, and Diamagnetism the discovery of Dr Faraday.

The arrangement of the chapters is thus strictly Methodical; but in the subdivision into sections, I have allowed the Biographical principle to predominate, thus giving as much as possible a historical character to the whole, and endeavouring to introduce the reader to the intellectual acquaintance of the eminent men who are selected for notice on the principles which have been already detailed. In some

¹ Specimens of what a history of pure mathematics would be, and must be, are to be found in the able "Reports" of Dr Peacock and Mr Leslie Ellis, in the Transactions of the British Association for 1833 and for 1846. A glance at these profound and very technical essays will show the impossibility of a popular mode of treatment, whilst the difficulty and labour of producing such summaries may be argued from their exceeding rarity in this or any other language.

² The celebrated Lagrange, in his later years, contrasting the mathematical works of his own generation with those which he studied when a youth, is said to have observed:—"I pity the young mathematicians who have so many theories to wade through. If I were to begin, I would not study; these large quartos frighten me too much."—(Thomson's Annals, vol. iv.) And it is stated that whilst his own most abstruse investigations were conducted in Paris, he kept the perusal of M. Gauss's writings for the tranquil retirement of the country,—a distinction intelligible enough between the intense effort of invention more than sustained by the vis viva of genius which prompts it, and the strain required to master the dead weight of reasoning imposed upon the mind by the discoveries of another. Dr Young, in his biography of Lagrange, observes upon the voluminous mathematical literature of his time, that "unless something be done to check the useless accumulation of weighty materials, the fabric of science will sink in a few ages under its own insupportable bulk." The fact is that a large proportion of the mathematical writings of even eminent authors are in a few years forgotten, or only casually consulted on some matter of history.

³ History and Philosophy of the Inductive Sciences.

(22.)
Subdivision of Physico-Mathematical Sciences.

(23.)
Arrangement partly methodical, partly biographical.

instances, however, it has been necessary to introduce the same individual into two or three different sections, and even into different chapters, when his pursuits have been in very various branches of science. This has been avoided, however, as much as possible, and a sacrifice has occasionally been made of the methodical order of the subjects, so as to combine in one view all that has made an eminent philosopher illustrious. Such little sacrifices of arrangement are

incidental to the way of treating the whole subject; and it may be hoped that its practical advantages, in the eye of the general reader, will be found to compensate for its defects as they may appear to the more rigorous student. For the sake of the latter especially, a short but comprehensive index of Names and Terms is prefixed, by which, I believe, it will be easy to trace all that is said of any one person or subject in any of the chapters.¹

§ 2. On the relations between Mathematics and Natural Philosophy, and between the latter and the Mechanical Arts.

(24.) Connection of Mathematics, physics, and mechanical arts.

The object of this Dissertation has little in common with an attempt formally to subdivide human knowledge into compartments, and to assign their boundaries with metaphysical exactness. It is chiefly in their practical bearing on one another that they must be considered. If one science, like Mathematics, furnish the only sure step towards the understanding or the enlargement of another, as Astronomy or Optics, a practical link is constructed between them, which renders the progress of the one not independent of the progress of the other. The intimate and reciprocal connection thus subsisting between Mathematics and Physics is to be found in almost an equal degree between Pure Physics and the Mechanical Arts, of which we take Civil Engineering to represent the department most cognate to that of Natural Philosophy, of which this Dissertation more especially treats.

(25.) Boundaries of Science and Art undefined.

The history of the last seventy or eighty years enforces this conclusion. The boundaries of Science and Art are as undefinable as those of "fact" and "theory," or those which separate the kingdoms of nature from one another. There are arts which can hardly be called scientific, and there are others which have contributed more to the original stock of knowledge than they ever drew from it. These last are like the shoots of those tropical plants which at first are mere buds upon the trunk, and are nourished solely by its juices, but which, when they reach the ground, plant themselves there, and become not only the props and stays of the parent stem, but supply it from an ever-increasing area with the sap which they originally borrowed.

(26.) The more closely we examine the subject, the more are we satisfied that it is impossible to teach science

rightly without teaching its applications; and that the limit to which we are to do so is a limit depending solely on the judgment of the teacher, and on the special purpose of the lesson. But the progress of science is a lesson learnt from the great book of experience; and if we are to feel the force of its teachings, we must consult, not one, but many of its pages. Looking to the history of science since 1750, but especially during the present century, it is quite impossible not to admit how large a share the sciences of application have had in moulding the direction of men's thoughts and speculations, and in enabling, nay, compelling them to realize certain abstract notions far from easy of conception. Instances of this are to be found in the force of percussion, the co-existence of vibrations in air and other substances, and such notions of body as we derive from practical efforts of continually-increasing boldness to extend the scale of our constructions.

(27.) Practical character of the last 100 years.

The analogy of the relation between Mathematics and Physics, and of the latter to civil Engineering, is so close that the three subjects might almost be represented as three terms of a continued proportion. What the second is to the first may be affirmed of the third relatively to the second. Physics may exist, at least to a limited extent, without a mathematical basis, as the art of construction long preceded a knowledge of the principles on which it is founded. But as knowledge advances it extends in both directions towards speculation and towards practical applications, but most towards the applications. This Bacon well understood, and he has consistently maintained, that knowledge, to be profitable to its cultivators, must also be fruitful to mankind. And all the history of science since Bacon's day has read this

¹ I have borrowed but sparingly, in the following pages, from the existing compilations on the history of science. Indeed, a writer who intends to make a subject his own by a well-considered, fundamental plan of treating it, will use such works principally as a guide to his own further inquiries, and to assist him in selecting the topics worthy of fullest discussion. In this respect Dr Whewell's excellent writings, already cited, have been of great use to me; and in the particular department of Astronomy, I have often referred to Mr Grant's valuable History of that science, as well as to the writings of Delambre, and the very elaborate Historical Essay by M. Gautier, on the problem of the Three Bodies, which is not, I think, noticed by Mr Grant, but which contains a most elaborate history of the researches of Lagrange and Laplace. In optics, I have consulted the systematic treatises of Dr Young, Sir D. Brewster, Sir J. Herschel, Dr Lloyd, M. Moigno, and M. Radicke; and so of other subjects. Gehler's Physikalisches Wörterbuch, Fechner's Reptorium, and that of Dove, afford a vast amount of historical information. The Transactions and Proceedings of the various societies, British and Foreign, have of course furnished a great part of my information. Such strictly biographical details as I have made use of, have in general been very carefully taken from the best accessible authority.

lesson more and more loudly in the ears of mankind. The era of Newton and Leibnitz was grandly distinguished by the continually increasing applications of mathematics to physics, whereof Newton was the great teacher; the century 1750-1850, whilst profiting by the lessons of the past, has added almost a new one in the eminently practical character of its science, and in the no less scientific character of its practice. The result of these gradual modifications of human knowledge has not been in the slightest degree injurious to the real progress of the more abstract ingredient of the mixed sciences. Did mathematics ever flourish more vigorously than under Newton? has pure physical science had greater triumphs than in the era of Volta, Watt, and Young? It was precisely because the new application of mathematics stimulated their growth, because abstract relations of quantity were vivified by concrete solutions of physical problems, that a new geometry arose. Dynamics could hardly be said to exist as a science without the invention of Fluxions as a language by which its conditions and results might be expressed; and from that time onwards, the necessities of the natural philosopher have been the prime sources of inspiration to the geometer, while the subjects have become so blended that a mere discoverer in mathematics has become a singularity. It would be hardly possible to point out any mathematician of the highest class since Newton, or but a few of the second class, who have not contributed almost as much to physical science as they have to analysis. Of purely mathematical discoveries, the great majority have been called forth by the immediate necessity arising from some problem requiring solution in astronomy, mechanics, optics, or heat. Lagrange's method of Variations of arbitrary Constants in Integration, the artifices for the computation of attractions by Laplace's coefficients; the introduction of the method of factorials by Kramp in his solution of the problem of refraction, and numberless improvements in the Theory of Definite Integrals by Fourier and his successors, sufficiently warrant the statement, and show how richly the physical sciences have repaid to the purely mathematical ones the debt which they originally owed. One other conclusion may be drawn from these and parallel facts. It is that the combinations arising out of external phenomena are more suggestive of the possible relations of number and quantity than is the most unlimited stretch of fancy and imagination; and I believe it will be conceded that, with few exceptions, theorems of the greatest value and beauty have been more frequently discovered during the attempt to solve some physical or at least geometrical problem, than in comprehensive yet indefinite attempts to generalize the relations of abstract magnitude.

(28.) These views are strikingly confirmed by the his-

torical fact of the paucity of pure mathematicians, and of distinct mathematical treatises of a strictly original character in an age distinguished by the diffusion of mathematical knowledge, and in countries (like France) most celebrated for its triumphs. There are not, perhaps, much more than half a dozen really great mathematicians of the last seventy years, who have not left treatises more numerous and more distinguished on physical science, treated mathematically, than on pure mathematics. Among the exceptions which more immediately occur, are Monge, Legendre, and Abel. And of distinct treatises, whilst we have the Mécanique Analytique, the Mécanique Céleste, the Théorie de la Chaleur, and numberless others, containing precious mathematical developments, in connection with the applications which suggested them, the purely mathematical memoirs of the same period are to be sought chiefly in the form of detached essays, in the ponderous volumes of Academy Transactions.

One point in the History of Mathematics has especial interest for the English reader, and as such may be touched upon here with reference to the progress of science for the last three quarters of a century. The national pride of England in the triumphs of Newton impelled her ablest mathematicians to attempt to carry forward the synthetic methods which he had chiefly used, at least in his published works, to the more arduous and intricate questions of Mechanics and Astronomy which presented themselves for solution in the course of the 18th century. MacLaurin was almost the last Englishman of that period whose mathematical writings came into direct competition with the rising schools of Germany and France. The labours of Matthew Stewart, and Simpson were mostly geometrical; those of Landen and Waring, though profound, created little general impression; and, gradually, the extent and difficulty of the foreign mathematics, increased by the use of the Leibnitzian notation of differentials which was absolutely unfamiliar in England, deterred almost every one even from perusing the writings of Clairaut and D'Alembert, Lagrange and Laplace. Of the continental mathematicians, Euler was probably the best known, owing to the lucidity of his writings and their eminently practical tendencies. Some idea may be formed of the negation of mathematical talent in Britain during the later portion of the last century, when we find D'Alembert declaring, in 1769, that if an Englishman is to be elected one of the eight foreign associates of the Academy of Sciences, he will vote for Earl Stanhope as the best mathematician there, as he believes, not having read any of his works! If the choice was to be free, he should prefer M. de Lagrange!!¹ A more cutting, though unintentional satire on the state of Mathematics in this country could not have been written.

¹ Letters of eminent persons, addressed to David Hume, edited by Mr Burton, p. 215.

(30.) At Cambridge, Edinburgh, and Dublin.

The commencement of a better era originated, early in the present century, with Woodhouse at Cambridge, and Playfair in Edinburgh, by both of whom the continental methods were introduced into the studies of their respective universities; whilst Ivory, a native of Scotland, was the first to challenge, by his writings, a place in the list of great living mathematicians. The systematic form of the Mécanique Céleste rendered the subject more accessible than were the countless memoirs by men of the highest name, which then filled the Transactions of Paris, Turin, Berlin, and St Petersburg. But the notation of differentials, which could alone break down the barrier between the British and foreign mathematicians, was first introduced at Cambridge by the efforts of Sir John Herschel and Dean Peacock about 1816, soon after which the translation of Lacroix's Differential Calculus, which they superintended, came into use as an university textbook. From this time the works of foreign mathematicians began to be more generally read, particularly the writings of Laplace and Poisson; and within ten or a dozen years subsequently, a few active and undaunted men, chiefly of the Cambridge school, such as Mr Airy and Sir John Lubbock, grappled with the outstanding difficulties of physical astronomy, whilst a larger number applied themselves to the most difficult parts of pure analysis, and acquired great dexterity in its use in the solution of geometrical and mechanical problems. Such, for example, were Mr Babbage, Mr De Morgan, Mr Murphy, and Mr Green; and at Dublin Sir William R. Hamilton and Mr MacCullagh, whose names will occur in other parts of this Dissertation.

(31.) Improvements in integration.

No new calculus or great general method in analysis has resulted from these persevering labours, whether of British or foreign mathematicians, but an increased facility and power in applying the existing resources of mathematics to the solution of large classes of problems previously intractable, or resolved only indirectly or by approximation. The Integral Calculus, in particular, affords an almost boundless field for research, and each branch of science in succession—not only Physical Astronomy, but Optics, Heat, Electricity, and Civil Engineering—has offered problems of great importance, which awaited only the skill of the pure mathematician to resolve in a practical and finite form.¹ Every year, and every civilized community, contribute to these real improvements. The principle of discontinuity, con-

spicuously introduced into the doctrine of the conduction of Heat in consequence of the abrupt variation of physical circumstances at the boundary of the conducting body, enters largely into the speculations of mathematicians of the present century; and the doctrine of definite integrals so intimately connected with it has received a proportional extension.

Next, analytical geometry has acquired a very great enlargement and by attention principally to symmetry

in the arrangement of the results, solutions otherwise the most intricate are obtained with facility and directness. Of this we shall find examples in our history of the Undulatory Theory of Light. Lastly,

notwithstanding the pre-eminently practical character of the mathematics of the last age, speculative geometers and analysts have found time to discuss the metaphysics of their respective sciences, both as regards the foundations of the Differential Calculus and as to the use of imaginary and other symbols in Algebra. An almost new branch of abstract science (though faintly foreshadowed by Leibnitz) has come into existence—the separation of symbols of operation from symbols of quantity, and the treatment of the former like ordinary algebraic magnitudes. In some cases remarkable simplicity is thus introduced into the solution of problems, although perhaps few mathematicians would choose to depend implicitly upon the method in untried cases. Sir John Herschel and the late Mr Gregory² were amongst the most active introducers of this new algebra, but few of the more eminent living British or foreign mathematicians have failed to contribute their share to this more metaphysical department of analysis.

I shall now attempt to consider more particularly the reciprocal relations of pure physical science and the mechanical arts.

This is evidently a very intimate one. The discoveries of pure physics (such as Astronomy, Acoustics, Magnetism), are the results of either observation or experiment, and they consist in generalizations, by means of which a multitude of facts are reduced under one simple expression of a more general fact or principle. But instruments often very complicated are necessary for observation and for experiment; as telescopes in astronomy, organs in acoustics, properly magnetized and suspended steel bars in magnetism. Art is required to construct these. The highest possible degree of science, and the utmost

¹ For example the Lucasian Professor at Cambridge, Mr Stokes, has effected two previously impracticable integrations, one occurring in the theory of the rainbow, the other in that of railway girder bridges.

² Mr Duncan Gregory, a promising mathematician who died 23d February 1844, at the early age of 30, was the youngest Mr Duncan son of Dr James Gregory, the late distinguished Professor of Medicine at Edinburgh. His name deserves a passing record, not only Gregory, from the influence he exercised on the progress of the English mathematics of his time, but as having revived the dormant character for this peculiar kind of talent, so long connected with the family of Gregory. He was, in fact, the lineal descendant of the inventor of the reflecting telescope. Mr Gregory was the first editor of the Cambridge Mathematical Journal, and author of an excellent book of Examples in the Differential and Integral Calculus, both of which have exercised a beneficial influence on the progress of science in England. He also wrote several original memoirs on the subjects referred to in the text. Mr Leslie Ellis, a man of congenial ability, has written a short but pleasing biography of his friend.

mechanical skill, are both indispensable to make a telescope. A telescope then, though a mechanical invention, inasmuch as it is a deduction from physical principles, not an addition to them, is yet a deduction so ingenious—so far from obvious—so impossible to be conceived accidentally or by a loose thinker, and so easily made the instrument of innumerable discoveries of the highest novelty and grandeur, that no historian of science ever thought of omitting the invention of the telescope, or of giving it an importance inferior to that of a discovery, containing as it did the germ of so many discoveries. In like manner a theory of optics might be written in which a telescope need never be mentioned; but would not the pedantry of such a work be obvious, or would any reasonable person wish to learn science after such a fashion?

The clear co-ordination of the parts of an invention towards the attainment of a given result, with a due regard to natural laws, and the properties of the substances used, constitutes the merit of the invention; and this merit may be irrespective of the precise importance of the end of the invention, which may be intended to promote science, or commerce, or convenience, or even to satisfy mere curiosity. A telescope would have been a telescope still, could we have imagined it invented with no other object than a deer-stalker's sport; and so contrivances which in their origin and application seem remote from scientific uses, constitute nevertheless real steps in the progress of knowledge. The steam-engine is one striking example. Originally devised with an exclusively commercial object—the extrication of the Cornish mines from subterranean water, it became in the hands of Watt, first an instrument for experiments on the relation of heat to matter; next, in its improved form, a beautiful exemplification of these laws, and an enduring monument of the sagacity and skill of its author, as well as the most important inorganic agent which exists in modifying the social condition of the entire globe. Finally, to illustrate the position from which we started, it becomes the instrument of fresh discoveries. This very engine has a theory to be worked out, probably unimaginable even by its sagacious author—its operation as an agent for obtaining power from matter by the application of heat, is shown to be in all probability a single case of a more general law, including all kinds of machines and all sorts of matter; and this more general theory of heat as a motive power leads, once more, to new practical deductions, to the conditions under which such machines may be most usefully constructed and employed.¹

(35.) Every instrument, every construction, which is and generally an inductive process. founded on a theory, and in which a certain complication of conditions is required to produce a certain

result; a telescope, for example, or a steam-engine, or a bridge, is, in the first instance at least, an experiment. Few inventions are so simple and straightforward in their plan, are so independent of the seemingly capricious behaviour of matter under untried circumstances, or depend so entirely on physical laws thoroughly understood, that the inventor can await, without the pang—at least of impatience—if not of anxiety, the moment of the realization, by actual trial, of his hopes and his calculations. We can all readily imagine the throb of anxiety with which Galileo pointed his glasses for the first time to the moon—with which Watt saw the cylinder of his model exhausted, and the piston descend under the action of his separate condenser—and Stephenson, the stupendous iron tube at Conway resting for the first time straight as a ramrod on its two piers—these are moments of anxiety and of triumph, which place the inventor of a machine and the architect of a structure on a par with the discoverer of a planet, or with the author of a theory. "Whenever an original mind produces new combinations of thought and feeling," says Sir James Mackintosh, with equal impartiality and truth, "whether its means be words or colours, or marble or sound, or command over the mighty agents of nature; whether the result be an epic poem, or a statue, or a steam-engine, we must equally reverence those transcendent faculties to which we give the name of genius."² It is almost needless to add the caution, that such praise is only applicable when the invention is such as to call forth the qualities which distinguish the Philosopher. It is not the mere command over the agents of nature which challenges our admiration, it is the foresight, the patience, the conceptive faculty, the clear-sighted and confident anticipations of what will be the results of natural laws acting in given circumstances, these circumstances being in some essential particulars new. Merely to adopt known contrivances, where experience has already anticipated the result, may exercise judgment, but hardly genius; and to make contrivances in which the result depends rather upon laws of geometry than of physics, hardly come within the scope of these remarks.

Watt's Parallel Motion, perhaps the most ingenious of his inventions, would not have made a great reputation; nor does the endless variety of machines used in the arts, as in spinning, printing, and paper-making, stand higher. It is when the inventor places Matter in new relations to Force, or derives power from new sources, or teaches Light or Electricity to act under new conditions, that he becomes really a Mechanical Philosopher.

It is not given to man to endue matter with new properties, or to prescribe the laws under which his inventions are to take effect. A new motive power,

¹ See Carnot, Puissance Motrice du Feu, and the writings of M. Clapeyron, Professor W. Thomson, &c.

² Speech at a meeting for erecting a monument to Watt, in Arago's Eloge of Watt.

a new form of construction, are experiments on the resources of nature under new conditions.

(38.)
Solutions of mechanical as of mathematical problems sometimes manifold.

In fact, even in comparatively simple cases, we cannot set forces to act on matter, or dispose matter so as to resist force, without doing not only what we intend to do, but also a great deal more. Man may put powers in motion which he is unable to control; and whilst he calculates confidently upon the effects of such and such dispositions of force or resistance, he may overlook consequences equally necessary, because resulting from laws of nature which are either unknown to him, or the magnitude of which he had overlooked, in considering those only which he required. A complicated mechanical contrivance may be compared to the mathematical solution of a problem. It represents commonly a great deal more than is meant to be derived from it. It may represent several distinct results, some possible, some impossible, and of the former only one, it may be, congruous to the real conditions of the problem proposed. In mechanics, the laws of nature are as impatient of control as the laws of quantity in geometry, and the engineer may find, too late, that nature has solved his problem differently from what he expected. But even when successful, it is to be presumed that his own contrivance contains within it results unforeseen by himself. If he is wise he will become a student in his own workshop. The material contrivances are indeed his own, but the powers which they awaken or distribute are beyond his control. Even if his reading of the equation be strictly correct, there may remain in the background others no less important.

(39.) The considerations here imperfectly laid before the reader are intended to justify the introduction of certain practical topics into the present Dissertation, which, though many readers will see their insertion without surprise, or would have been sorry to find them omitted, others possibly may think more or less independent of, and separable from a scheme already sufficiently extensive and intricate, if confined to mere subjects of scientific doctrine, to the exclusion of its applications. My chief reason for including such subjects as the steam-engine, the strength of materials and some great examples of

construction, and the electric telegraph, is that these important practical improvements are both historically and logically interwoven with the progress of pure or abstract physics. They have besides impressed upon the character of scientific discovery of the last hundred years a peculiar stamp which it would have been absurd to ignore while endeavouring, within a moderate compass, and in the plainest language, to convey a vivid though comprehensive sketch of the advancement of Natural Philosophy during this and the preceding, or rather two preceding generations.

It is not to be imagined that the difficulty of the problems which occupy the speculative philosopher, or the comprehensiveness of mind required for their solution, diminishes in any degree as we descend from the regions of pure science to the walks of everyday life—from the vast periods and majestic motions which astronomy enables us to explain and predict, to the common details of the workshop and the railway. In fact, the former are to be regarded as the simpler investigations, whilst our terrestrial agents have their effects modified by the diversified states of aggregation and various mechanical properties of matter, and by the numerous modifications of force arising from heat, electricity, or magnetism, to which it may be exposed. We have as yet made but an insignificant advance towards that complete system of Natural Philosophy of which Newton's will form but one section, in which all the properties of matter and their consequences shall be as well understood as the particular property of gravity is at present. Many of these are to be learned by daily observation of the effects which occur in the ordinary progress of civilization amongst us. We are continually performing experiments on a great scale and on purely commercial principles, which no individual philosopher or merely scientific society could have ventured to attempt. And in the midst of these appeals to experience, unexpected results are frequently occurring which send us back once more to the study of first principles, which, indeed, while they confound the empiric, do but establish the reputation of the philosophic engineer, who seldom fails to turn them to good account, both in his theory and practice.

CHAPTER II.

PHYSICAL ASTRONOMY AND ANALYTICAL MECHANICS.

§ 1. LAGRANGE.—Variation of Parameters—Application to Physical Astronomy. The Stability of the Planetary System; Laplace; Poisson. Moon's Libration.

(41.)
Lagrange.

The period of Lagrange's most celebrated labours extends so far back into the preceding century, that having been already mentioned in Sir John Leslie's

Dissertation, it might have been excusable, with so great a mass of matter before us, to have passed them over without farther notice. But they are so

intimately connected with the most salient points of the history of physical astronomy, down even to the present time, and are so interwoven with the discoveries of Laplace, and represent altogether so much of the substantive character of the progress of the age, that I have thought it necessary to devote a small space to the recital of a few of the most prominent of them, having regard to the intellectual portraiture of the man as one of the most pre-eminent and successful reasoners of his class who have ever done honour to their race. I shall repeat as little as possible what has been said elsewhere, and confine myself to only two or three topics.

(42.) Joseph Louis Lagrange was born at Turin in 1736; his birth and education; he died at Paris 10th April 1813. His first paper was written at the age of 17 or 18, and his end was accelerated by the unremitting ardour of his labours at the age of 77. He was consequently an original author during sixty years; and for the greater part of this period he, together with Laplace, monopolized the greatest discoveries connected with analysis and physical astronomy, and exercised an almost undisputed authority in the more recondite sciences. Euler, Lagrange, and Laplace, by a singular coincidence, lived to the respective ages of 90, 77, and 79, and all retained their activity nearly to the last. They produced, by the continuity and friendly rivalry of their labours, carried to an extent in each case which only astonishing physical vigour united to astonishing mental aptitude could have produced, during almost a century, an impression on the progress of science altogether remarkable. This coincidence was the more happy, because physical astronomy was exactly in that predicament when nothing less than such a combination of intelligence and intensity of application systematically urged, could have carried Newton's theory through the difficulties which at that time beset it—difficulties which left the Principia for so many years alone, and far in advance of the general intelligence of the age.

(43.) The pregnant suggestions of Euler were developed and applied by Lagrange, and the triumphs of Lagrange—nay, even his occasional failures—were the immediate precursors of some of Laplace's happiest efforts.

Amongst the former we reckon the method of the variation of parameters, expounded to a certain point by Euler, though, as in many other cases, his results were vitiated by the haste and inaccuracy of his calculations. That Lagrange borrowed the idea from Euler cannot admit of a doubt, any more than that he was indebted to him for the principles of the Calculus of Variations. Lagrange, with customary truthfulness, even to his latest days, always spoke of Euler as his best instructor and model, and as the chief of modern mathematicians, Newton only excepted. We know that he so regarded him in the case of the calculus of variations which he studied in Euler's "Methodus invenendi lineas curvas, &c.," during the first two years of his application to the higher mathematics; whilst Euler, with equal candour, acknowledged the transcendent genius of the rising geometer, forcing its way where he himself had failed.

The method of the Variation of the arbitrary constants or Parameters, though it may be regarded in one point of view as a merely analytical artifice for effecting integrations, is in reality a conception purely geometrical, first introduced by Newton² under the name of "revolving orbits," and applied by him to the explication of the conception (to use a recently introduced phrase) of the lunar inequalities. Neither the moon nor any planet really describes a mathematical ellipse (in consequence of the mutual perturbations of the heavenly bodies). They describe curves of double curvature in space, of which we could form no intelligible idea, except by referring them to the very approximate type of the ellipse, of which the eccentricity, line of apsides, inclination, &c., are continually varying, not only from one revolution to another, but throughout every part of a revolution. This representation is not only convenient, but strictly accurate. At each instant the moon or planet is describing a portion of an ellipse, which may be called the instantaneous ellipse, and which may be defined as the particular ellipse which the body would go on to describe if it were at that instant freed from all perturbation, and allowed to complete a revolution under the single influence of its acquired motion and the central force. To take

Lagrange's early studies. ¹ The following is a list of the books he then read, taken from a paper probably little known, which appeared soon after the death of Lagrange in the Moniteur newspaper, and which was translated in Thomson's Annals of Philosophy, vol. iv. He first read Euclid's Elements, Clairaut's Algebra; then, in less than two years, and in the following order, Agnesi's Analytical Institutions, Euler's Analysis of Infinites, John Bernoulli's Lectures, Euler's Mechanics, the two first books of Newton's Principia, D'Alembert's Dynamics, and Bougainville's Integral Calculus, Euler's Differential Calculus and Methodus Inveniendi—a pretty course of mathematical reading for a youth between 17 and 19.

From the same paper we abridge a few practical directions given by Lagrange for the study of mathematics, which, if tolerably obvious, are interesting from the extraordinary genius of the man, and from his singular reticence on subjects of a personal nature. "I never," he said, "studied more than one book at a time; if good, I read it to the end." "I did not perplex myself with difficulties, but returned to them twenty times if necessary. This failing, I examined another author." "I considered reading large treatises of pure analysis quite useless. We ought to devote our time and labour chiefly to the applications." Thus he read Euler's Mechanics when he had acquired a very slight knowledge of the differential and integral calculus. "I always read with my pen in my hand, developing the calculations, and exercising myself on the questions."

"From the very beginning of my career, I endeavoured to make myself master of certain subjects, that I might have an opportunity of inventing improvements; and I always, as far as possible, made theories to myself of the essential points, in order to fix them more completely in my mind, to render them my own, and to accustom myself to composition." "Finally, I every day assigned myself a task for the next. I learned this custom from the King of Prussia."

² This Lagrange himself points out in his Mécanique Analytique.

a single example; the planet Uranus has not yet completed one revolution since the time of its discovery in 1781, yet its observed path differed so much from a true elliptic arc (even when we allow for the perturbation of Jupiter and Saturn), that the orbit which satisfied the observations from 1781 to 1800 would not satisfy those from 1800 to 1820; and since 1820 a new orbit had to be computed for every few years, so great were the variations of the instantaneous from any permanent ellipse. These variations led to the discovery of the planet Neptune.

To adapt the notion of the perpetual variation of the elliptic elements to analytical calculation, and to ascribe to each planet its influence in perturbing the elliptic motion of the others, was the great problem mainly solved by Lagrange. In the planetary theory, where the perturbations are all very small, on account of the excessive preponderance of the mass of the sun, the motion of each planet may be considered as under the separate disturbing influence of every other, and the whole perturbation is the sum of the separate perturbations.

Now, these perturbations of elliptic motion may be divided into two great classes, which Lagrange first, in 1782, included in a common analysis, which expressed the disturbed elements of planetary motion by two sets of terms: those which include the relative positions (or configuration) of the disturbing and disturbed planets being the one set, and those which included only the masses and elements being the other. The former are called periodic, the latter secular inequalities. The distinction is important, since, after a sufficiently long time, two planets (suppose the Earth and Mars) will have been presented to one another in space in every conceivable position of which, by the form and position of their orbits, they are susceptible, a like recurrence of configurations will recommence, and like perturbations will result. Such influences, though running through long periods, will be evidently recurring. But there is another class of disturbances, which may in thought be entirely separated from the former, being the ultimate or average effect of the influence of one planet on another, arising, not from the position of the planets in their orbits at any one time, but from the position of the orbits themselves. Thus in a single revolution (and on account of the independent eccentricities of the orbits during many successive revolutions) of Mars and the Earth, the attraction of the former on the latter sometimes conspires with the sun's attraction, sometimes opposes it, sometimes urges the Earth forward in its path, and sometimes pulls it back, producing numerous periodic inequalities; but it is quite evident that, in the long run, the attraction of Mars on the Earth tends to pull it away from the sun, and to diminish the effect of the solar attraction—in fact, to increase the length of our year; and that this influence will be precisely the same if we take the average of a great many re-

volutions now, and compare them with a similar average hereafter, provided that the orbits undergo no permanent change. This, therefore, though not strictly an inequality, because the length of the year is permanently changed by it, shows an average effect independent of the configuration of the planets. An example of a true secular inequality is the revolution of the line of apsides or major axis of any orbit, by the influence of the disturbing forces of the planets, whether interior or exterior to the one considered.

Few of these secular inequalities have been detected by observation throughout the entire records of Astronomy. It is known, however, that the apsides of the planetary orbits (at least in the case of the old planets) all progress, with the exception of those of Venus, which retrograde, and that the inclination of all is at present diminishing. The eccentricity of the Earth's orbit is decreasing at the rate of 40 miles per annum. The exclusive dependence of the secular inequalities on the orbits, not on the places of the planets, may be well illustrated by a method actually employed by Gauss for computing them (though it does not appear to be attended by any special advantage). He conceives the orbit of the disturbing planet to be strewed with attractive matter, whose thickness at any point is inversely as the planet's velocity there, or directly as the time of its sojourn in any small length of the orbit.

The method of variations of the elements is evidently most applicable to the determination of Secular Perturbations; for to compute by means of it the ordinary inequalities involves an apparently unnecessary labour. The place of a planet is completely determined by three co-ordinates—its longitude, latitude, and radius vector; whilst the elements of the orbit are six in number, and when found, a further calculation must be made to find the co-ordinates of position. The more direct method of deducing the co-ordinates at once from the conditions of perturbation, was generally followed until 1808, when Lagrange and Laplace almost simultaneously devised methods of using the variation of the elements with directness and despatch in the calculation of Planetary Perturbations. In the estimation of the second and higher orders of disturbance, it has even the advantage in these respects over the other method.

Stability and Permanence of the Solar System.—After it had been clearly recognised, principally by the labours of Lagrange, that the elements of the planetary orbits are in a condition of perpetual change, it came to be a most interesting question how far such variations were likely to be continuous, and ultimately so great as to modify altogether the forms of the orbits, and even endanger the separate existence of the planets. This is a question which has excited a very general, as well as scientific interest. It is evident that the variations of the different elements are not all equally important in affecting the permanence of an orbit. The five properly orbital ele-

ments (the sixth being the longitude of the planet on its orbit at a given time) may conveniently be considered thus: 1st, the major axis, which, for one and the same system, involves the periodic time or mean motion; 2d, the eccentricity and position of the line of apsides; 3d, the inclination and position of the line of nodes. Of these, the stability primarily depends upon the first. If the major axis and mean period increase or diminish without limit, the planets will diverge into infinite space, or rush after myriads of ages to utter annihilation in the burning embrace of the sun. The latter alternative was the popular belief about the middle of last century, and was maintained by the grave authority of Euler; whilst Darwin, in his florid but picturesque language, described the order and beauty of the planetary system as but a little more permanent than the glowing ornaments of the gay parterre.¹ The principal reason for this conclusion, and its refutation, will be mentioned in the next section.

(51.) Laplace's share in the discovery. The first person who perceived the probable stability of the major axes and mean motions was not Lagrange but Laplace, who, in a paper published in 1773, gave a demonstration, the sufficiency of which has not been doubted, that the major axes are invariable, so far as the influence of the principal terms of the disturbances are concerned, that is as far as terms containing the cubes of the eccentricities inclusive, and the first powers of the perturbing masses. Nor does Laplace appear to have doubted that the mutual distinction of the terms, including secular changes, was not accidental, but would extend also to the farther approximations. Lagrange, however, in a celebrated though short memoir of 1776, demonstrated the truth of the conclusion for the higher powers of quantities contained in the perturbations of the first order, and that by methods peculiarly comprehensive and elegant, which he farther extended in 1781 to the other five orbital elements, showing the periodicity within certain narrow limits of the eccentricity and inclination, the only elements, except the major axis, whose variations menace the stability of the system. Yet it is quite impossible to separate completely the names of Lagrange and Laplace in the effectual demonstration of this important truth, the former as frequently in-

dicating the means of overcoming the more purely mathematical difficulties, as the latter was suggestive and far-sighted in anticipating their application to the peculiarities of our system.

Laplace discovered (1784) two remarkable theorems which limit the whole amount of the eccentricities and inclinations of the orbits of the planetary system, showing that if once small, they must ever remain so; and, in particular, that the most massive planets of the system (Jupiter and Saturn) must also undergo the most trifling variation in these respects. In the case of the small planets between Mars and Jupiter, a wider range may occur (as indeed we practically find to be the case), without endangering the permanency of the whole. It also follows that these variations, though "secular," are practically "periodic;" that is, that the eccentricities and inclinations oscillate about certain mean values and within extremely narrow limits, the periods of these oscillations being also of vast duration. Concerning such changes, theory is our only guide. The whole duration of astronomical records can barely reveal the existence of two or three of them, and tells us absolutely nothing of their remoter consequences. Lagrange calculated the superior limits of the eccentricities of the larger planets, and M. Leverrier has recently, by more accurate methods, obtained results nearly coincident. According to him, the maximum eccentricity of the Earth's orbit is 0.07775, the minimum 0.003314, so that it can never be quite a circle. It is now diminishing, and will continue (according to the same geometer) to do so for 24,000 years, when it will begin to increase. The inclinations of the Earth's orbit to its equator, and also to a fixed plane, are confined within definite limits which are not perhaps very perfectly known.

The motions of the apsides and nodes of the orbits which gradually complete the entire circumference have manifestly no tendency to affect the stability of the system. The grand cycle of the Earth's perihelion will only be completed in 110,000 years. It coincided with the vernal equinox 4089 years before Christ, a period (as Laplace remarks) nearly coincident with that assigned by chronologists to the creation.

These results may be considered as among the most astonishing with which science brings us ac-

1 "Roll on ye stars! exult in youthful prime,
Mark with bright curves the printless steps of Time,
Near and more near your beamy cars approach,
And lessening orbs on lessening orbs encroach;
Flowers of the sky! ye too to age must yield,
Frail as your silken sisters of the field!
Star after star from heaven's high arch shall rush,
Suns sink on suns, and systems systems crush,
Headlong, extinct, to one dark centre fall,
And Death and Night and Chaos mingle all!
Till o'er the wreck, emerging from the storm,
Immortal Nature lifts her changeful form,
Mounts from her funeral pyre on wings of flame,
And soars and shines, another and the same."

Darwin's Botanic Garden, Canto iv., line 357.

quainted. The range of insight which man has acquired into the past and future history of the universe throughout periods, compared to which, the whole existence of his species is but a span, enhances our admiration of the reasoning power which can attain to knowledge so high and excellent. And the sublimity of the contemplation is increased when we recollect that these recondite truths are all consequences of a law so simple as that of gravity. Observation will reveal only to a late posterity the secular modifications of the planetary orbits which geometry now predicts to us. Some of the ellipses will elongate, whilst others tend to become circles; their planes will vary in inclination, but ultimately be stayed within the limit which human sagacity had predicted myriads of years before. "These," says a French analyst, "are the pendulums of eternity, which beat ages whilst ours beat seconds." And amidst all these variations, subject to law and to impassable limits, the Major Axes of the orbits preserve a steadfast uniformity, or are subject only to transient fluctuations; and thus permanence arises in the midst of change, and the perfection of the system is demonstrated by the very nature of the disturbances which seemed at one time inevitably to limit its duration.

(55.) Poisson's addition to the theory.
It remains to add, in closing this interesting discussion, that Lagrange himself had not quitted the field before his able disciple and follower, Poisson, pursued the inquiry of the stability of the system, and the permanency of the major axes particularly, to a degree of approximation not before attempted. He included the perturbations of the second order, or those which arise by correcting the elements for the disturbances first found, and including the effects of the correction in the modification of the perturbations themselves. These also are subject to the same laws as found by Laplace and Lagrange for lower degrees of approximation; and as MM. de Pontecoulant and Leverrier have confirmed the result (at least for all the larger planets of the system), we may conclude it to be a truth as firmly established as any negative fact can be, that our system is arranged for a duration apparently indefinite; that if the planets cease to roll, and the sun and moon to do their office in enlightening the world, it must be in all

probability by an interposition of Almighty power, as direct and immediate as the creative energy by which they were launched into space, and (our earth at least) peopled with successive races of animated beings.¹

We have, in the commencement of this section, (56.) disclaimed the intention of entering at large upon the history of Lagrange's discoveries. They fell more properly under the scope of the preceding Dissertation, and an able summary and enumeration of his writings by no less competent a person than Dr Thomas Young will be found in the alphabetical part of this Encyclopædia. I will only add, that while scarcely a topic in physical astronomy, or in pure mathematics, failed to receive important additions from his pen, his memoirs on the Libration of the Moon, his solution of several problems of Sound and vibrating strings, and his methods of computing the perturbations of Comets, are amongst his contributions to science, most vividly remembered and most justly admired as models of analytical ability. He himself is stated to have preferred, amongst all his papers, one in the Turin Memoirs of 1784, on the Integral Calculus.²

(57.) With reference to the Lunar Libration, Lagrange confirmed the singular conclusion of Newton, that the moon is a spheroid, having three unequal axes, the longest of which is always approximately directed to the earth, and the shortest is her axis of rotation. In consequence of this, the moon, of necessity, revolves on her axis in the exact time that she circulates round the earth (supposing that at any time these periods were nearly, though not absolutely, coincident), and is subject (as Newton had divined) to a species of oscillation upon her axis, owing to the line of the earth's attraction not always coinciding (in consequence of the moon's irregular motion in longitude) with the moon's greatest diameter. This constitutes a physical libration, as inequalities in longitude, by enabling us to see more or less of the lunar hemisphere diametrically opposed to us, constitute an optical libration, or apparent to-and-fro motion on her axis. In this investigation Lagrange first used the combination of D'Alembert's Principle with that of Virtual Velocities, afterwards fully expanded in the Mécanique Analytique.³

Lagrange was happy in passing his days with (58.)

¹ No reasonable doubt exists as to the stability of the planetary system to which our earth belongs, as it is at present constituted. To what extent the laws of order which we observe in it might be transgressed with impunity, it is more difficult to say. The investigations of Laplace and Lagrange assume the motion of the planets in one direction, and their moderate eccentricities and inclinations, as conditions of the guarantee of stability. But it does not appear by any means certain that all these conditions are essential, and consequently the argument which has been sometimes employed, that the concurrence of many independent circumstances were requisite to the stability of the system, is at least incomplete. Compare Laplace, Système du Monde, edit. 1824, vol. ii., p. 29, and Herschel's Outlines of Astronomy, art. (669).

² It is No. 17 in the enumeration of his papers in the article LAGRANGE in the Encyclopædia.

³ Very recently (1854), M. Hansen of Gotha, a most eminent living authority, has somewhat modified the received opinion M. Hansen respecting the moon's figure. He finds that the presumed ellipticity of the moon in the direction of the radius of her orbit is on the not justified by observations, which ought to show a slight variation in her horizontal diameter when the libration presents to moon's us our satellite in a slightly varied aspect. And he infers from an elaborate investigation of the lunar observations, that her figure, centre of figure does not coincide with her centre of gravity, but lies about 31 English miles nearer to us than the latter. M. Hansen adds that the existence of such a protuberance of the moon's body relatively to the centre of gravity on the side which we can alone view, would account for the apparent absence of water and air, which may abound upon the opposite side.—Astronomical Society's Notices, vol. xv., p. 13.

Private character of Lagrange. the tranquillity of a philosopher. He was respected and rewarded alike by kings and democrats—he was honoured and promoted in three great states, Sardinia, France, and Prussia. Though patronized by the despotic Frederick, and lodged in her palace by the gentle queen of Louis XVI., he escaped the misfortunes of almost every one of his contemporaries, including Laplace, Lavoisier, and Delambre; he retained his scientific appointments throughout all the frenzy of the French Revolution. His mildness of disposition and disinterested devotion to science, more than the European celebrity of his name, contributed to this result. He was equally fortunate in his scientific relations. Euler, D'Alembert, and Laplace, whilst they were emphatically his rivals, were also his sincere friends. If he ever felt jealousy,

it was perhaps towards those who, he thought, attained too easily by circumstances to a high reputation: Monge seems to have been of this number. It is remarkable that for a series of years Lagrange diverted his mind altogether from mathematics, and studied chemistry, natural history, and even metaphysics. His reply is well known, when asked how he liked the first of these sciences; "Oh," said he, "I find it on trial as easy as algebra." It may be doubted whether in our own day he would have given as favourable an opinion!

He was unassuming in conversation, and disliked speaking of himself. His commonest answer was "I don't know." He was happy in his domestic relations, and died universally honoured and regretted, 10th April 1813.

§ 2. LAPLACE.—Lunar Theory Improved.—Great Inequality of Jupiter and Saturn.—Theory of the Tides.—Young; Dr Whewell; Mr Airy.—Theory of Probabilities.—Character of Laplace as a Physicist and Author.

(60.)
Laplace. PIERRE SIMON LAPLACE has generally, and not without reason, been considered as a sort of exemplar or type of the highest class of mathematical natural philosophers of this, or rather the immediately preceding age. The causes of this, and the degree in which it is warranted, we shall endeavour to state towards the close of this section. In the meantime, finding it quite impossible within our prescribed limits to notice, ever so briefly, all his more material investigations, we shall select three or four marked by their originality and general interest. Such are, 1. His improvements of the lunar theory. 2. His discovery of the cause of the great inequality of Jupiter and Saturn's motions. 3. His theory of the tides. 4. His work on probabilities. 5. We shall consider his character as a general physicist, and as a writer.

(61.)
Improvements in the Lunar Theory. I. First, then, we are to speak of the improvements of the lunar theory effected by him. The application of Newton's own principles to the perfecting of the theory of the moon's motion has been related in Sir John Leslie's Dissertation, and so far as the labours of Clairaut, D'Alembert, Euler, and Mayer, are concerned, belongs distinctly to the middle portion of the last century. The errors of Mayer's tables little exceeded one minute of space, which was twice more accurate than in Halley's time. With one important exception, the main outstanding differences between theory and observation had disappeared. The eclipses recorded in the Arabic and Chaldean annals could not (as Halley first observed) be correctly explained by the motion of the moon as given by recent tables. At length it became admitted that the mean motion of the moon has been accelerated from century to century by a minute

quantity, which, in the lapse of thousands of years, has become recognisable. It amounts to this, that the moon comes to the meridian two hours sooner than she would have done had her present period remained invariable from the earliest astronomical records of eclipses. It is at once evident how delicate a test this must be of changes otherwise imperceptible. The effect on the dimension of the moon's orbit may be thus expressed, that at each lunation she approaches nearer to the earth than during the last by one-fourteenth of an inch! thus describing a spiral of almost infinitely slow convergence. The minuteness of the effect may be illustrated by the shortening of the pendulum of a clock by an amount absolutely insensible, which yet, after days and weeks, will alter by many seconds the time shown by the hands.

After several unsuccessful speculations as to the cause of this anomaly, Laplace, in 1787, thus satisfactorily accounts for it:—It is well known that the sun's attraction on the earth and moon lessens, on the whole, the tendency of the latter to the former, and lengthens permanently the lunar period. But, so far as this effect is uniform, it does not directly appear. The effect is greater, however, when the earth is near the sun than when it is farther off. The lunations are therefore longer in winter (when the earth is in perihelion) than in summer. This is called the annual equation, and the amount is very sensible for this reason, that (as may be easily seen) the perturbing force varies inversely as the cube of the sun's distance. Now, though the earth's mean distance from the sun has not varied in the lapse of ages, the eccentricity of the earth's orbit has been diminishing from the earliest historic times, and the average inverse cube of the distance has

been also slowly increasing. The result is that the moon's motion has been continually accelerated. Now, we have in the last section referred by anticipation to this acceleration as having led to the belief that the moon must at last fall to the earth. Laplace's discovery, however, shows that the acceleration has a limit, depending on that of the eccentricity of the earth's orbit, which having reached its minimum, the lunar mean motion will begin to be retarded, and will continue so through a vast cycle of ages, and so on alternately. Theory enables us to assign, with considerable accuracy, the amount of the acceleration, which is now about $10\text{'}\text{'}$ of longitude in a century.¹

Besides this very satisfactory discovery, Laplace investigated three of the lunar inequalities in a manner leading to curious and unexpected results. Two of these depend on the spheroidal figure of the earth. The nutation of the earth's axis, which is due to the attraction of the moon on the protuberant equatorial parts of the earth, is exactly reproduced by the equivalence of action and reaction in the movements of the lunar orbit, only less perceptible in degree on account of the length of the leverage at which they are effected. The inequality of the moon's motion in latitude may be used to determine the degree of compression of our globe at the poles. Laplace deduced from the Greenwich observations of the moon the fraction $\backslash frac\{1\}\{30000\}$, and, from a relative inequality in longitude, $\backslash frac\{1\}\{30000\}$; a coincidence really astonishing, not only as between themselves, but also when compared with the mean result of laborious investigations by actual measurement of the earth's surface. The other result we referred to was the determination from the lunar theory of the solar parallax,—in other words, the distance of the earth from the sun, which enters into the expression of a certain inequality of the moon's motion in longitude. From the observed amount of this inequality, Laplace obtained a value of the solar parallax exactly coincident with that obtained with so much labour on occasion of the transit of Venus in 1769. Strange and admirable result (as Laplace himself remarks), that the astronomer, immured in his observatory, and watching our satellite through his telescopes, and reading the result by the aid of mathematical analysis and the theory of gravitation, should be able to determine the figure of our earth, and its distance from the sun, with perhaps quite as great accuracy as by any direct measurements. Truly the wonders of fact exceed those of fiction, and the divinations of true science may match the pretensions of her counterfeit, astrology.

In conclusion of this subject, I regret that space (64.) does not allow me to advert particularly to Laplace's Theory of Jupiter's remarkable success in accounting for some singular peculiarities in the system of Jupiter's satellites, arising from, and partly occasioning, an exact commensurability in the periods of some of them (which Sir John Herschel has lately observed to hold also in the Saturnian system in a somewhat different manner), a case which we have seen to be especially excluded in the instance of the planets, and which has been pronounced by a very competent judge (Mr Airy) to be "the most curious and complicated system that has ever been reduced to calculation." It ought to be stated, however, that Laplace's discoveries were based upon a previous and highly original investigation of Lagrange.

II. In the second place, we shall briefly state the (65.) nature of Laplace's happy explanation of a great inequality of the solar system, to which, like the fact of the lunar acceleration, especial attention had been and is called by the sagacity of Halley, and which, like it, resisting all the efforts of geometers to interpret, threatened the credibility of the Newtonian theory of gravity. We are therefore to look upon this step as something more than a solution of a difficult problem; it was a new, peculiar, and unsuspected combination of circumstances on which it depended, and the solution afforded a key in all time coming to difficulties depending upon a like cause.

Halley had ascertained, that by comparing modern with the most ancient observations of Jupiter and Saturn, the mean motion of the former planet had been (66.) accelerated, and that of the latter retarded. Lambert remarked subsequently that, if we confine ourselves to modern observations alone, an opposite change would appear to be in progress. The amount of the error of the tables was so considerable (amounting to $20\text{'}\text{'}$ or more in the middle of the eighteenth century, and capable, in fact, of becoming much larger), as to have been (along with the apsidal motion of the lunar orbit) one of the first subjects of anxiety and speculation to geometers, when the Newtonian theory came fairly into discussion. For nearly forty years this stubborn inequality was vainly attempted to be accounted for by Euler, Clairaut, D'Alembert, Lagrange, and by Laplace himself, before the latter hit upon the true cause of the anomaly. It was long, and naturally, believed to be a properly secular inequality, arising from the average mutual effects of the planets Jupiter and Saturn, though Lambert's remark rendered this less probable. It was in the course of the consequent research that Laplace proved that the mutual action of the two planets could produce

¹ We here add that very recently Mr Adams has discovered that Laplace, and also his followers, in confining their attention to the radial effect of the sun's interference with the lunar motions, as affected by the eccentricity of the earth's orbit, have unwarrantably assumed that the area described by the moon a unit of time is invariable. He finds, on the contrary, tangential perturbations depending on the same cause, and sensibly modifying the amount of secular mean motion deduced from theory.—(Philosophical Transactions, 1853.)

no permanent alteration of the mean motion of either; a conclusion which, as we have seen in the last section, he afterwards generalized for the planetary system. Several other memoirs by Lagrange and himself followed; and when the question became thus narrowed to periodic perturbations only, Laplace, with characteristic ardour and resolution, determined to search out every term which could affect the result; an irksome task, less congenial to the generalizing spirit of Lagrange. He had already noticed, in his memoir of 1773, that Euler and Lagrange had, in their researches on this very subject, omitted terms, multiplied by sines and cosines of very small angles, which yet might, in the process of integration, become considerable by the largeness of the coefficients. Eleven years later he detected, in the expansion of the mutual perturbation of Jupiter and Saturn, terms of this kind. The coefficient (or maximum value of the term) is in this case divided by the square¹ of the same quantity which renders the angle under the sine or cosine small. These terms were indeed likewise multiplied by the cubes of the eccentricity, or like small quantities; but notwithstanding this, by reason of the small divisor just mentioned, they were capable of attaining a formidable magnitude; in the case of Jupiter, to 21', and of Saturn, to 48' or 49'. That so small a force should produce so large an effect is due to the very long period of the most considerable portion of this inequality, which, in fact, led to its being confounded with perturbations properly secular. The period of complete recurrence of the effects is about 920 years; and during half this time the motion of one planet is being constantly accelerated, and that of the other retarded; during the other half the action is reversed. An effect continually increasing or diminishing for so long a time, and between the two most massive of the planetary bodies, is evidently liable to become considerable. The maximum displacement of Jupiter and Saturn Laplace found by calculation to have occurred in 1560, explaining the peculiarity above mentioned in the comparison of ancient with modern observations.

When we look to the physical cause of the largeness of these particular perturbative terms, it is found to be this;—that the period of revolution of Jupiter compared to that of Saturn, is almost as the numbers 2 and 5;—in other words to the near commensurability of the mean motions. Were they exactly in proportion to these numbers, formidable and permanent changes would possibly result in the orbits. As it is, the planets come into conjunction when Jupiter has completed 5 revolutions, and about $\backslash frac\{1\}\{2\}$th more; Saturn 2 revolutions, and $\backslash frac\{1\}\{4\}$th more. Consequently the point of conjunction travels round the circumference after about 44 conjunctions have occurred, which requires nearly 2700 years. But a little considera-

tion will show that conjunctions occur successively at three nearly equidistant points of the circumference; consequently the two planets will have been presented to one another in every possible variety of configuration, when the point of conjunction has travelled one-third round the circumference, that is in about 900 years.

The effect of this great improvement in the Theory of Jupiter and Saturn was, that the most ancient observations were completely reconciled with the modern, and the modern with one another; the errors of the tables were immediately reduced to one-tenth of their former amount, and soon after to much less. (68.)

III. The third topic which I must shortly discuss in connection with the career of Laplace, is the Theory of the Tides. (69.)

The Newtonian Theory of the Tides has been explained in Mr Playfair's Dissertation, but its progress during the 18th century has not been adverted to in the continuation by Sir John Leslie. It will be sufficient to state here, that it was pursued into its consequences with ability and success by Daniel Bernoulli, who in 1740 shared a prize of the French Academy of Sciences on this subject, along with Euler, MacLaurin, and Cavalleri, a Jesuit, the last a supporter of the Cartesian vortices. It was, perhaps, the concluding honour paid to that once popular theory. (70.)

The Tidal Theory of Newton and Bernoulli presumes the earth to be at rest; and also the waters of the ocean to be at rest, and at every moment in a state of equilibrium between the force of gravity, tending to the earth's centre, and the lesser forces tending towards the sun and moon. That a theory, founded on suppositions so far from the truth (not to mention the irregular distribution of sea and land on the earth's surface), should in any manner or degree represent correctly what happens, may be matter of just surprise. The leading phenomena are however tolerably consistent with it; the dependence of the great tides on the moon's position with respect to the meridian of the port; the spring and neap tides when the sun's action and that of the moon conspire with or oppose one another; the priming and lagging of the tides depending on the displacement of the vertex of the compound ellipsoid due to the combined effect of the sun's and moon's attraction, depending therefore on the moon's elongation from the sun; the effects of the moon being in the nearer or remoter part of her orbit; all these facts are indicated by the Equilibrium Theory (as it has been termed), and are also results of observation. The theory, however, does not give the true depth of tide, nor (except in casual instances) does the time of high and low water coincide with theory; besides many minor imperfections. (71.)

Laplace had the singular boldness to attempt the solution of a problem, which is more one of hydro- (72.)

¹ In consequence of a double integration in respect of the time.

Laplace's Dynamical Theory. dynamics than of astronomy, and to estimate all the causes of movement of the particles of a heavy fluid, surrounding a spheroidal rotating nucleus exposed to the attractions of the sun and moon. This he did in a series of memoirs, more systematically condensed in the Traité de Mécanique Céleste, and it may safely be affirmed that no other mathematician of his day was equal to the labours and disappointments of an investigation attended with every species of difficulty, in which each result must be attained by a combination of general sagacity with mathematical rigour, and for the verification of which observations were yet in a great measure wanting. The Theory of the Tides was, upon the whole, the most arduous and complicated problem which could well be conceived, in a branch of science (hydrodynamics) hitherto remarkably little successful in predicting the results of the most simple and arbitrarily selected experiments.

(73.) Its difficulty. That Laplace has been in a measure successful in such an undertaking must be considered the highest test of his genius, especially in reducing his mathematics to practical application; but the result has been a treatise so profound and obscure (I mean as regards the tide theory), that very few persons have attempted to master its difficulties. Mr Airy, the present astronomer royal, has done a great service to men of science, and to that far wider community whom the laws of the tides nearly interest, by giving a connected and tolerably elementary view of Laplace's investigation, which he states confidently to be "the most obscure of the Mécanique Céleste."

(74.) Its incompleteness. In this theory the figure of the ocean at any moment is considered as a dynamical problem; and that figure as a momentary state arising from the internal movements of the fluid itself, as well as from the variation of the external forces. The resulting differential equations, expressing the attractions of the sun, moon, and earth, the rotatory movement of the earth, and the pressure of the water itself in motion, are abundantly complex, and the solutions only partial and imperfect. The inferences from these solutions, too, partake not only of their imperfection, but, since they take no cognisance of the irregular distribution of land and water, present cases almost impossible to verify by observation. Some of the results are indeed so paradoxical, that without better evidence of their truth we do not further allude to them.

(75.) Three classes of Tides. The tidal effects are divided by Laplace into three classes; the distinction of which, however, cannot be called a discovery of his. The first class are independent of the earth's rotation, and are practically insignificant. The second class includes the diurnal tide occurring once in about 24 hours. Concerning

it, Laplace draws this conclusion, that its rise and fall (not, however, its horizontal motion) are insensible if the depth of the ocean is uniform;¹ and being practically insensible in most latitudes, we have thence an argument of more or less weight for a general tendency to uniformity in the depth of the sea. The third class of tides are the ordinary semi-diurnal tides. They afford, as Newton acutely perceived, the most direct and attainable measure of the relative attractions of the sun and moon. We have the sum of these attractions at the conjunctions and oppositions of the luminaries, and the difference when they are 90° apart; and a higher maximum when both bodies are without latitude. From the observation of the tides on the Severn near Bristol, Newton computed the relative action of the moon and sun to be as 4.48 to 1; but this value is much too great, and gave far too large a relative mass to the moon. The result in the harbour of Brest, from observations made under Laplace's direction, is about 2.90 to 1; and the moon's mass $\backslash frac\{1\}\{2\}$th that of the earth, agreeing almost identically with that deduced from the nutation of the earth's axis caused by her attraction. Observations at London and Liverpool reduced by Sir John Lubbock and Dr Whewell give about 2.66 to 1.²

From the general theory of Laplace, the following results have been deduced with confidence:—(1.) Laplace's results. That the stability of the ocean is secure, whilst the density of the ocean is inferior to that of the earth generally; which it is about five or six times. (2.) That the phenomenon of precession is not modified by the fluid covering of the globe.

In the application of his theory to special cases, Laplace is compelled to have recourse to an assumption entirely arbitrary—namely, that the periodic fluctuations, however otherwise modified by circumstances, recur in the same periods as the causes to which they are due. In this manner he conciliates the results of observation with his theory, which the latter would have been altogether incompetent to predict.

The general merits of Laplace's theory we will sum up in the words of Mr Airy, who, of all his successors, has probably most attentively studied it:—"If putting from our thoughts the details of the investigation, we consider its general plan and objects, we must allow it to be one of the most splendid works of the greatest mathematician of the past age. To appreciate this, the reader must consider—first, the boldness of the writer who, having a clear understanding of the gross imperfection of the methods of his predecessors, had also the courage deliberately to take up the problem on grounds fundamentally correct (however it might be limited by suppositions afterwards introduced); secondly, the general diffi-

¹ Dr Young asserts that the conclusion will not hold unless the depth be also evanescent. Laplace has shown in his Fifth Book that the disappearance of diurnal tides will take place only when the nucleus is completely covered.

² These ratios, however, being found to depend upon the configuration of the coast or estuary, cannot be used directly to determine the relative action of the sun and moon. See Phil. Trans., 1815, p. 42.

culty of treating the motions of fluids; thirdly, the peculiar difficulty of treating the motions when the fluids cover an area which is not plane but convex; and, fourthly, the sagacity of perceiving that it was necessary to consider the earth as a revolving body, and the skill of correctly introducing this consideration. The last point alone, in our opinion, gives a greater claim for reputation than the boasted explanation of the long inequality of Jupiter and Saturn."¹ We must, however, qualify this eulogy by adding, in the words of the same writer, that Laplace's theory, though based on sounder principles than the equilibrium theory, "has far too little regard to the actual state of the earth to serve for the explanation of even the principal phenomena of the tides." It is, in fact, like many other productions of the same age and school, a great display of ingenuity and mathematical skill, which hardly yields a single result worthy of confidence, or agreeing with nature, except by the abandonment of its deductive rigour, or a concealed induction backwards from the phenomena to be accounted for. The same amount of skill and resource which Mr Airy has shown in adapting it to his own views, and to recent observations, would probably have sufficed to construct a theory from the foundation. By others the attempt seems to have been abandoned as hopeless.

Since our limits will not permit us to return to the subject of tides, we shall here briefly state the progress of the subject since the time of Laplace. The chief steps have consisted in co-ordinating the results of observation and analyzing them into their partial phenomena, by the help of Newton's and Bernoulli's theory. This labour has been greatly advanced by Dr Whewell, and also by Sir John Lubbock. The former has constructed maps of "cotidal lines," which, indicating the relative time of high water in different parts of the globe, give us a graphic conception of the course and propagation of the tidal wave. The tides of the Eastern Pacific are but little known; but a vast wave advances northwards between Australia and Africa, diverted or retarded by the obstacles it meets with in the Indian Archipelago. Another (and to us the most important) branch sets from south to north up the vast canal of the Atlantic, where it is gradually complicated by local tides, having their origin in the wide expanse between Africa and the Gulf of Mexico. The two sets of waves sometimes reinforce, sometimes oppose, one another; they are prolonged to the western shores of England and Norway, where the tidal impulse arrives 24 hours after it passed the Cape of Good Hope. It is propagated most rapidly at a distance from coasts, and is retarded in narrow and shallow straits. It sends offshoots into every bay and strait, always greatly retarded in point of time (ap-

parently by friction), but often increased in elevation by concentration of the effect in a gradually narrowing channel, as we see in the exaggerated tides of the river Amazon, the Severn, and the Bay of Fundy. The same place may be the seat of several tides at once, which may increase or destroy one another; thus, a small tide is propagated through the Straits of Dover as far as the Dutch coast, where it only arrives simultaneously with the principal wave, which has made the entire tour of Great Britain.

As regards the progress of theory, Dr Thomas Young, whose character as one of the greatest philosophers of the past age we shall have to consider in another chapter, next after Laplace grappled with the difficulties of this arduous subject. Employing mathematical methods of inferior power but greater directness, and taking into account causes of local action which Laplace had not ventured to include in his analysis, he gradually matured a theory adequate to represent many of the results of experience, of which Laplace gives no account.

He distinguishes the results of the forced and free oscillations of the sea; the former resulting from the direct action of the sun and moon combined with the rotation of the earth, and whose periods of rise and fall are determined solely by those external causes (external, I mean, to the mass of the ocean); the free waves, on the contrary, derived from the former, are transmitted with velocities depending on the mechanism of the fluid itself, on its depth, and on the resistances arising from friction to which those motions are exposed. These all-important modifications of the dynamical theory of the Tides were deduced by Dr Young from the general theory of oscillations and resistances, and from the laws of fluids detected by Dubuat,² and he applied them with no ordinary skill to the solution of the problems of tides in oceans, estuaries, and rivers. It is an extraordinary fact, and not without significance, in the history of science, that these researches of Young, published anonymously in the Supplement to the Sixth Edition of the Encyclopedia Britannica, and in the Seventh Edition of this work, and likewise in several journals and reviews, so generally escaped notice as to have been almost unknown till Dr Peacock, in his recently published "Life and Miscellaneous Works of Dr Thomas Young" has fortunately recalled attention to their existence and their important results.

In doing so, Dr Peacock has communicated with Mr Airy, whose very valuable article on Tides and Waves has been above referred to (78), and has ascertained from him that Dr Young's researches had escaped his notice when he undertook that elaborate recension of Laplace's theory, and made those important additions to it to which I have

¹ Encyclopaedia Metropolitana, "Tides and Waves," art. 117.

² See Chapter IV., Section 6, where we shall return to some portion of this subject.

called attention.¹ It is very satisfactory to find that, by their independent and very different modes of analysis, Mr Airy and Dr Young have arrived at results generally coincident. It is in the essay of the former that most readers will now seek for an acquaintance with Laplace's abstruse investigations, whilst they will find in it the bearing of experiments more recent than the time of Young on the propagation of waves in canals, the theory of Mr Airy, beginning as it were at the opposite end from that of Laplace, and offering far more points of contact with actual observation, particularly in the Tides of Rivers and Estuaries. The theory of Young will naturally be best studied in his own article Tides, in this Encyclopædia.

(83.) IV. In the fourth place, we connect the name of Laplace with the progress, during the period we are considering, of the curious doctrine of probability, or the laws of chance and expectation. These he discussed in two works, the Théorie Analytique and the Essai Philosophique sur les Probabilités—the first the most mathematically profound, the last the most popular and elegant, account of the subject which has yet been given. Nearly all mathematicians are agreed on two points—first, in considering this the "most subtle" and "difficult to handle" of all the applications of their science, involving a perpetual recurrence to contingencies, and to elements of the argument easily left out of account, and in which, more than in any other, it is dangerous to let sleep for a moment the severely reasoning faculty, or permit it to be lulled to security amidst the maze of symbolic transformation. In truth, from experience, I am disposed to receive with doubt the solution of even a tolerably simple problem of chances, unless two competent persons at least have concurred in verifying it. Secondly, Mathematicians are agreed in considering Laplace's Théorie nearly, if not quite, the ablest specimen of mathematical writing of his age, notwithstanding a degree of obscurity and repetition in addition to the inherent abstruseness of the subject, which render it, in the opinion of one of the most learned and extensively read of our pure mathematicians,² "by very much the most difficult mathematical work he ever met with."

(84.) A single paragraph has been devoted to the subject of probabilities in Sir John Leslie's Dissertation, relating to its earlier history; and the subject was so popular during the last century, that there was scarcely an eminent mathematician who did not add something to its practical development; so that Laplace may be considered rather to have enlarged widely its applications by means of his almost unexampled power in handling the calculus, than to have improved or established its first principles, or even

applied it to classes of problems altogether new. We find that most of the principles of the Calculus were established by James Bernoulli, in the earliest part of the eighteenth century, who gave the first History of application of the Binomial Theorem to determine the probability of a particular combination of a given number of things occurring, in preference to all the other equally possible combinations. Stirling discovered a curious theorem for approximating to the continued product of the arithmetical series of numbers carried to any extent, which perpetually occurs in such calculations. Demoiivre carried out Halley's application of it to the laws of mortality. Condorcet applied it to moral questions; Mitchell to natural phenomena, considered as the results of accident or design; Lagrange to errors of observation. The chief applications of the Theory of Probability are the following:—1. To chances known a priori, as that of throwing two given numbers with dice, the whole range of possibilities being known with precision. 2. The calculation of the expectation of future events on a great or average scale, deduced from the past course of events observed also on a great or average scale. Of this description are the calculations of life assurance, first tabulated by Halley. 3. To find the most probable result of a number of independent observations and problems of a like kind. 4. To the proof of causation as opposed to accident or "random," derived from existing combinations of facts. 5. To the probability of testimony, and the confidence due to legal decisions. None of these inquiries are peculiar to Laplace, or originated with him. We select, however, for a brief notice (which must be confined to a few sentences) the third and fourth of these applications.

The chances are enormous against the most experienced marksman's hitting the bull's eye of a target. But if he make many shots in succession, the balls will be lodged round about the spot at which he aimed, and they will be fewer in each successive ring of equal area drawn round the mark. The degree of their scattering will depend upon the skill of the marksman; but in all cases the most probable result will be, that the point aimed at is the centre of gravity of the shots. This may be shown to be equivalent to saying that the most probable result of any number of equally reliable observations is that which will make the sum of the squares of the outstanding errors a minimum. This rule was conjecturally proposed by Legendre in 1806. A demonstration of its truth was first published by Laplace. It is of great practical use in deducing the results of complex observations, such as those of Astronomy, and generally in combining "equations of condition" more numerous than the quantities whose value is sought to be extracted from them. In very many cases, however, a graphical

¹ See note to p. 262 of the Second Volume of "Young's Miscellaneous Works," by Peacock.

² Professor De Morgan in Encyclopædia Metropolitana.

method of interpolation will serve the same purpose, as well as save a very tedious calculation. Perhaps too much importance has been assigned to the "probable mean error" of a single observation deduced from many individual errors by the same theory. If a man has but one shot at a target, it is perfectly uncertain (by hypothesis) whether that shot be one of his best, or one of his worst, or one of his middling ones. But as there are more middling hits than very bad or very good ones, there is a certain distance which it will be safe to bet (i. e. for which the probability is $\backslash frac\{1\}\{2\}$), that he will not exceed; though it would be a strange occurrence indeed, if he exactly struck the ring in question. This is all that is meant by the phrase "probable error;" it is an entirely artificial number, which serves to give a sort of numerical value of the skill of the performer, but is otherwise of no importance. Its application has sometimes been strangely mistaken. Even the "rule of least squares" is often misapplied, and empirical laws altogether false have been deduced from it; for it is rare in practice that the chances of error of observation of a varying quantity are the same throughout the limits of observation.

But those applications of the doctrine of probability which pretend to give us a measure of our belief in the constancy of natural laws, in the confidence due to testimony and to the teachings of history, in the proofs of design in cosmical arrangements,—are inquiries which, from their connection with metaphysics, religion, and morals, have had a higher interest for mankind at large than ordinary problems about cards and dice. To these Laplace paid peculiar attention; and the reputation of his name tended to create in others a belief that the analysis he so powerfully wielded could communicate a portion of its certainty to the data subjected to it, and gave a currency to many of his conclusions to which we believe them by no means entitled. Professor Playfair, one of the most ardent of Laplace's admirers, has recorded (in a criticism in the Edinburgh Review, of the works we are considering) his total dissent from Laplace's doctrine that the transmission from age to age of the historic record of a fact diminishes its credibility in a geometrical ratio. But a Cambridge mathematician and speculative philosopher of singular penetration, Mr Leslie Ellis, has most formally assailed¹ the principle of nearly all Laplace's estimates of our expectation of events arising from causes unknown or assumed to be so, such, for instance, as that a common cause determined the revolution of all the planets in one direction. The subject of the metaphysics of probability evidently requires a complete reconsideration; and, owing to the singular subtlety of

the matter, it is one which few persons are competent to handle. The state of health of Mr Ellis leaves us little hope of his resuming the inquiry; but two eminent mathematicians, Mr De Morgan and Mr Boole, have published considerable works chiefly bearing upon it.

As an implement bearing upon discovery in science, the Calculus of Probabilities has as yet been of little service. Whilst Laplace tries to indicate how it guided his researches connected with planetary irregularities, every one sees at a glance that, with the data before him, common sense must have outstripped analysis. Laplace has called the doctrines of probability "good sense reduced to calculation." What is to be feared is, that the calculation should outstrip the good sense.

V. Fifthly, We are to consider Laplace's character as a general physicist and as a writer.

In the former respect he stands in a higher position than that usually attained by eminent analysts. In fact, if we compare him with any of his own generation, we find him not only better acquainted with physical principles, and more scrupulous in taking account of them in his mathematical discussions, but even possessing skill and interest in experiments. The Calorimeter for measuring the capacities of bodies for heat was the joint invention of Lavoisier and himself; at least their memoir does not assign to either a predominant share in it,² and their determinations of the expansions of the metals by heat seem also to have been made in common. His happy discovery of the chief or sole cause of the discrepancy between the theoretical and observed velocity of sound (due to the heat developed by compression) would alone have given him a just reputation, so anxiously had the matter been debated, and so much was it involved in a purely mathematical intricacy. Even in his great work on physical astronomy he takes a peculiar pleasure in embracing topics of terrestrial physics. We there find discussed the theory of barometrical measurements, the question of atmospheric tides, the laws of capillary attraction, and the constitution of the gases. As to the first of these topics he made a practical improvement on the formulæ of his predecessors, so that his rules are in fact still in use. As regards capillary attraction, he was materially anticipated by Young, who evidently considered his principles to have been pirated; yet his theory, though obscured by a display of redundant mathematics, was a real improvement. His theory of tides, and that of atmospheric refractions, though closely connected with physical astronomy, were in fact not less so with the doctrines of hydro-

¹ Cambridge Transactions, Vol. VIII. The writer of these pages has also given his reasons for dissenting from the arguments of Michell (which have been sanctioned by the authority of Laplace) on an astronomical question as discussed by the Theory of Probabilities. See Philosophical Magazine for December 1850.

² Dr Black, the discoverer of latent heat (who was probably well-informed), states in a letter to Watt that he believes Laplace to have been the inventor. See Correspondence of Watt on the Decomposition of Water, p. 66.

dynamics and optics; and we shall find very few important branches of general physics in which he has not left some permanent record of his interest, in the course of his career of fifty-five years of anxious devotion to science.

(89.)
His Traité de Mécanique Céleste.

His largest and most systematic work, the Traité de Mécanique Céleste, in five quarto volumes, was not only most ably executed, but exceedingly well timed. The applications of analysis to physical astronomy had been accumulating for nearly a century. Hundreds of memoirs relating to them, dispersed through many volumes in different languages, written with varying ability, in various stages of scientific progress, and with differing notations, presented a mass of reading almost beyond the reach of the most resolute student. Laplace undertook to digest the whole into one body of doctrine, composed throughout on a uniform plan, with the best mathematical aids which were known at the commencement of this century. And though improvements and discoveries have been since made, the methods and most of the results of the Mécanique Céleste remain, with little variation, the preferable ones of our own time. As a work of labour, it may compete with the Principia of Newton; as an original work, it is of course immeasurably inferior. Its principles are, in fact, the same with those laid down in that immortal code, and its deductions are collected (as we have said) from the writings of Clairaut, D'Alembert, Euler, and Lagrange, as well as from the previous memoirs of the author himself. Laplace has been too sparing of his citations and acknowledgments, and a consequence of this literary avarice has been that he is sometimes considered as more of a compiler and less of a discoverer than is justly his due. For however ill he could have dispensed with the skilful preparations of his illustrious rivals and contemporaries, his pre-eminent sagacity furnished on several occasions the key-stone of the arch which imparted at once strength and completeness to the fabric. We have seen in the last section that though the credit of the theorems respecting the stability of the solar system is very generally attributed to Lagrange, who, indeed, principally furnished the methods, and gave great generality to the results, yet the capital discovery of the invariability of the major axes of the planetary orbits is due to Laplace. It was he, again, who removed from the theory of gravity the two greatest and most impracticable difficulties with which it had ever been assailed—the anomaly of the lunar accelerations, and the great inequalities of Jupiter and Saturn, and by so doing rendered it almost infinitely improbable that any future discrepancy should more than temporarily embarrass a theory which had tri-

umphed in succession over such formidable causes of doubt. True that Lagrange, in his memoir of 1783, had come within a single step of the first of these discoveries, and, by a process of exclusion, had almost forced attention in the right direction respecting the latter; still Laplace seized the prize in both cases, after a fair, prolonged, and arduous struggle. Now these three discoveries were the greatest in physical astronomy between those of D'Alembert and Clairaut on the precession of the equinoxes, the motion of the lunar apse, and the periodicity of comets,—and that of Leverrier and Adams on the perturbations of Uranus about a century later.

The universal testimony of mathematicians is to the effect that Lagrange was unrivalled as a pure analyst, in his power of generalization, and in the with Laplace inherent elegance of his methods; that Laplace, with nearly equal power in using the calculus, had more sagacity in its mechanical and astronomical applications, or rather, perhaps, we should say, in directing it to the discrimination of causes, and the revelation of consequences.

In other respects he differed far more widely from his illustrious compeer. He rather courted popularity, and was pleased at being considered worthy of political distinction. For a short time he was one of Napoleon's ministers; but the Emperor, it is said, was more satisfied with his mathematical than with his diplomatic talents. He had none of the shyness of Lagrange, nor his repugnance to general society. He received with affability and kindness those who were introduced to him, and his attentions were afterwards recollected with gratitude by rising men of science abroad. He had a villa at Arcueil, adjoining that of Berthollet, and was one of the original members of the "Société d'Arcueil," to whose memoirs he contributed. He exercised a powerful influence and as a member of the Academy of Sciences, of which for a time he acted as dictator, and he was not very tolerant of views in science opposed to his own. The undulatory theory of light he always opposed, and was mainly determined in doing so by the facility with which the attraction of luminous corpuscles could be subjected to calculation.

The weak point of his scientific character was one so natural, and perhaps so inseparable from his prevailing studies, that it is not fair to criticise it too severely. This was a love of analytical display in treating questions which it rather embarrassed than illustrated; and generally, a disposition to overrate the sphere of mathematical discovery. This he had in common with Euler, to whom he was very superior in physical attainments and sagacity. His language, and that of his eulogists,¹ often amounts to the assumption that

¹ For instance Arago says (speaking of the invariability of the major axes): "Enfin par la toute-puissance d'une formule mathématique, le monde matériel se trouva raffermi sur ses fondements" (Annales, 1844, p. 304). This is indeed the idolatry of mathematics. Many examples may be found in Laplace's writings on Probability; which occasioned Mr Ellis to say of him, that "to Laplace all the lessons of History were merely confirmations of the 'resultats de calcul.'" To the same effect was the note of Napoleon, "that Laplace carried into the art of government the principles of the infinitesimal calculus."

the marvellous power of analysis, in unravelling intricate consequences of admitted or assumed laws, could supply deficiencies of primary conceptions of the laws of nature, or could teach men fundamental truths in natural science.

(93.)
His Système du Monde. To add to the remarkable list of Laplace's endowments one more; he was a perspicuous and elegant writer. His Système du Monde contains a popular

exposition of astronomy in theory and practice entirely original in its plan and execution, and though frequently imitated, it is still perhaps the first of its class.

Laplace was born in Normandy 23d March 1749, and died the 5th May 1827, leaving in the Academy he had so long honoured no one within many degrees of his ability in the same peculiar walk of science. (94.)
Date of birth and death.

§ 3. LEGENDRE.—IVORY.—Theory of Integration; Elliptic Transcendents (Abel, Jacobi). The Attraction of Spheroids, and Theory of the Earth's Figure. Atmospheric Refractions.

(95.)
Legendre. ADRIEN MARIE LEGENDRE was born in France in 1752, and died in 1833 (10th January); he was consequently three years younger than Laplace, and survived him by nearly six years. He formed, therefore, an integral part of that constellation of mathematical talent of which Paris was for more than two generations the main centre. Like his illustrious compeers Lagrange and Laplace, he laboured with enthusiasm all the days of his life, and like them was engaged in editing and improving his works down almost to the day of his death, at the ripe age of fourscore.

(96.)
Distinguished as an able mathematician. The mathematical career of Legendre was less splendid than that of the other two whom we have just mentioned. He did not possess the wonderful powers of generalization of Lagrange, and he wanted the flexibility of mind, and the general physical knowledge, of Laplace. Legendre was very strictly a mathematician; and he has been exceeded by none in the unquenchable zeal with which he pursued subjects of a dry and even repulsive character, often till he had hunted them down by sheer force of application, or, to adopt the metaphor applied to him by Lagrange, until he carried, sword in hand, the stronghold which he besieged.

(97.)
His researches on Elliptic Functions. No more striking proof can be given of these statements than the unflinching pertinacity with which, during nearly fifty years (1786–1833), he studied and improved the theory of Integration, applicable to those cases frequently occurring, which involve the higher powers of the independent variable, and which do not usually admit of finite expression. Two large works, the Exercices du Calcul Integral (1811), and the Traité des Fonctions Elliptiques (1827–32),—the latter in good measure a republication of the materials of the former,—bear testimony to his diligence; and these works were almost entirely original, and contained tables of most laboured construction, calculated by himself. Hardly any mathematician entered into competition or co-operation with him until his labours were drawing to a close, when, with a liberality worthy of all commendation, he welcomed the

analytical discoveries of Abel and Jacobi, which were to give an unlooked-for extension to his own. These methods of integration, and their reference to certain properties of the Lemniscate and the Ellipse, originated in the early part of the last century with Fagnano and Euler. Legendre took up the subject exactly where Euler left it, and finally reduced the large class of expressions to which his methods are applicable to three standard forms or integrals in which the independent variable is always expressed by a circular function, and to which a numerical approximation may always be made by means of the tables calculated by himself.

ABEL, who succeeded in generalizing Legendre's methods to a far greater extent, was a native of Norway, born in 1802 (25th August),¹ and died at the premature age of twenty-six (1829, 26th April). His principal memoir was presented to the Institute when he was only twenty-four years old; and, to use the language of Mr Leslie Ellis, "when the resources of the integral calculus were apparently exhausted, Abel was enabled to pass into new fields of research by bringing it into intimate connection with a new branch of analysis, namely the Theory of Equations. The manner in which this was done shows that he was not unworthy to follow in the path of Euler and of Lagrange."² Legendre's eulogy of Abel was concise:—"Quelle tête celle du jeune Norvégien!" It is less agreeable to add that the life of Abel was perhaps shortened by poverty and care. Though ultimately befriended by Legendre, Poisson, and others, his first visit to Paris (in 1826) occasioned nothing but disappointment, and his great memoir (no unusual lot, for the same happened to Fresnel) lay hopelessly lost amidst the papers of the Institute for fifteen years. Much, however, to their credit, the geometers above mentioned at length addressed the King of Sweden on behalf of the rare genius his dominions contained; but in vain. Abel died neglected, unable even to print his researches, which were tardily given to the world in a collected form, at the expense of the government which refused to support

¹ There is some discrepancy, as to the year of his birth, but I believe this to be correct.

² Report on the recent progress of Analysis. British Association Report for 1846.

him when alive. The French Academy, which had buried his memoir in their most inaccessible "archives," decreed too late the unprecedented honour of a posthumous medal to his mother. Jacobi, the friendly rival of Abel in his discoveries, died recently, at a mature though not advanced age, at Königsberg, where he was professor.

(99.) Legendre's researches on the attraction of Ellipsoids. But to return to the labours of Legendre. The theory of the figure and attraction of the earth and of other planets naturally divides itself into two parts—(1.) the law of attraction of an ellipsoid on a material point without or within it; (2.) the figure of equilibrium of a fluid subjected to no forces but the mutual attractions of its particles, and the centrifugal force due to its rotation. The latter of these problems is still imperfectly solved, the former completely so, and that mainly in consequence of the labours of Legendre and Ivory.¹

(100.) Legendre's researches on the attraction of Ellipsoids. Though the services of Legendre are well known and admitted, the superior address of Laplace in the applications of mathematics has occasioned his receiving the credit of what in some instances belonged to the former.

(101.) History of the subject. Maclaurin, by an exercise of synthetic skill not exceeded since the death of Newton, had demonstrated the attraction of an ellipsoid of revolution upon a material point anywhere within it or on its surface, as well as for an exterior point in the prolongation of its axis or in the plane of its equator. The same problem was afterwards analytically solved by D'Alembert and Lagrange. In 1782, Legendre, by a profound and complicated analysis, obtained an expression, by means of series, for the attraction of an exterior particle generally, and he was the first to imagine and employ those artifices of calculation known usually by the name of "Laplace's functions." Laplace made a step towards the simplification of the expression of exterior attractions, but the complete solution was reserved for Mr Ivory, as I shall mention below.

(102.) Other works. The other labours of Legendre need not be specified here. He co-operated in the trigonometrical survey of France, and gave the formula, known by his name, for approximately reducing a spherical to a plane triangle. He also wrote on the orbits of comets, and on the method of least squares.

(103.) Ivory. JAMES IVORY, the most considerable British mathematician of his time, or that had appeared since Maclaurin, was born at Dundee in 1765, and studied at St Andrews along with Sir John Leslie. The most active period of his life was passed as mathematical professor at the Military College of Marlow (afterwards removed to Sandhurst). He was essentially a self-taught mathematician, and spent much of his time in retirement. He fathomed in private the profoundest writings of the most learned continental mathemati-

cians, and, at a period when but few Englishmen were able to understand those difficult works, he showed his capacity of adding to their value by original contributions, not unworthy of the first analysts. We pass over his earlier contributions connected with mathematics and astronomy, several of which are contained in the Transactions of the Royal Society of Edinburgh, and proceed to his most celebrated paper, published in the Philosophical Transactions for 1809, in which he completely and definitely resolves the problem of attractions for every class of ellipsoidal bodies. After what has been stated above as to the position of the problem as treated by Legendre, a few words will explain the precise import of Ivory's Theorem, one of the most celebrated mathematical results of that time.

We have seen that the attraction of an ellipsoid on a point within or at its surface had been assigned by Maclaurin. The theorem in question enables us at once to reduce the case of an exterior attracted point to that of a point on the surface of the ellipsoid. Suppose an ellipsoid having the same eccentricity, and with the principal sections parallel to the first, but whose surface passes through the given exterior point. Let points on the surface whose co-ordinates parallel to the three axes of the respective solids are proportional to those axes be called corresponding points; then the attraction parallel to each axis which one of these bodies exerts on a point situated on the surface of the other is to the attraction of the latter on the "corresponding point" of the surface of the former, as the product of the two other axes of the first ellipsoid is to the product of the two other axes of the second. By this means the attraction on an exterior point is accurately expressed in terms of the attraction on an interior point, which is known. It is fair to add that both Legendre and Laplace had some glimpses of the principle, though they failed to apply it to the direct solution of the problem, and between the publication of their Memoirs and that of Mr Ivory there elapsed nearly a quarter of a century.

Besides this paper, Ivory contributed many others on the subject of the attraction of spheroids and the theory of the figure of the earth, during a period of nearly thirty years; several of these were controversial, and did not add materially to the progress of the subject; others are considered as masterpieces of analytical skill. One of the last subjects which occupied his attention was the possible equilibrium of a spheroid with three unequal axes, which Jacobi had discovered.

Between the labours of Ivory and those of Legendre a great analogy subsists; for the doctrine of elliptic integrals also occupied the attention of the former.

But next to the theory of attractions, that of Atmospheric Refraction most seriously engaged Ivory's attention. Its great importance in astronomy, and

¹ See the articles ATTRACTION and FIGURE OF THE EARTH (the former by Ivory) in this Encyclopedia.

the curious mathematical difficulties which it presents, renders it very interesting to analysts. Laplace had applied to it his method of Generating Functions; Kramp had introduced into his (now scarce) treatise the almost new Calculus of Factorials; and others, like Bessel and Atkinson, had skilfully combined theory and observation for the construction of useful tables. One of the most curious results of recent enquiries into this subject is, that Sir Isaac Newton's table of refractions (Phil. Trans., 1721) must have been founded on a profound consideration of the problem, such as no one else thought of till a much later period, and is so numerically exact as to agree closely with the later tables, Kramp's for example.¹

(108.) Mr Ivory attained the age of seventy-seven, dying
His death.

on the 21st September 1842. Probably his unceasing devotion to a confined and abstruse topic of enquiry, reacting on a sensitive frame, rendered him in some degree irritable and unsocial. He was not altogether responsible for this; but students of science should recollect that diversity of occupations and interests is subservient not only to bodily health, but also to mental equanimity and vigour.

The historian of science dwells with a special interest on the results of Ivory's labours, when we recall the singular destitution of higher mathematical talent which had reigned in this country for so long a period, and which left us not only no position in the great struggle going on abroad for the advancement of physical astronomy, but scarcely even the rank of intelligent spectators. (109.) His eminent position as a British Mathematician.

§ 4. Progress of Physical Astronomy since the publication of the Mécanique Céleste.—POISSON.—Theory of Rotation (Poisson).—MR AIRY.—The Solar Theory.—MM. PLANA and HANSEN.—The Lunar Theory.—Physical Astronomy in America.

(110.) The more that any theory of a mathematical kind, like that of Gravitation, advances to perfection, the less reason have we to expect great and striking results in the prosecution of it, and the more intense and continuous is the labour in matters of detail necessary to make any advance at all.
Increasing difficulty of physical astronomy.

(111.) As regards the general and popular view of the subject, we might pass at once from the epoch of Lagrange and Laplace to that of Leverrier and Adams. Notwithstanding, however, the necessity of extreme compression, I must devote one short section to mentioning the chief labours in connection with physical astronomy of four eminent men mentioned in our title, who may fitly be considered as the immediate successors of Laplace. Two of them will be again referred to in this discourse.
Its progress since the publication of the Mécanique Céleste.

(112.) SIMEON DENIS POISSON, born in 1781, may be said truly to have been brought up at the feet of Lagrange and Laplace. He was their pupil in the first and brightest years of the Polytechnic School, where he was especially noticed by the former. He had the distinguished privilege of being literally their fellow-worker, his early memoirs having reference to their labours, and stimulating the still vigorous mind of Lagrange to the production, in his latest years, of several memoirs, which have been considered worthy of his best days. I refer more particularly to Poisson's proof that the stability of the planetary system holds when perturbations of the second order are taken into account, as has been stated in the first section of this chapter, Art. (55.) This was in 1808. Soon after, following out another of Lagrange's admirable generalizations of his theory of Arbitrary

Constants, he embraced in a common series of formulating the result of those mechanical laws which regulate the rotation of bodies, together with those concerned in their translation in space. This important subject (rotation) continued at intervals to engage the attention of Poisson, not only as regards the motions of the heavenly bodies on their axes, but also as a branch of common mechanics. The basis of this intricate doctrine was laid by Huygens; Euler, in a celebrated and original work, gave it a general and analytical form; D'Alembert solved by it the problems of precession and nutation; Laplace demonstrated the constancy of the time of the earth's rotation round its axis. This last problem was more fully discussed by Poisson, who showed by theory that neither can the earth ever rotate round an axis different from its present one, nor can the time of its rotation vary in consequence of any external attractions to which it is subject. These two matters are of the utmost moment; the first prevents the latitude of places from varying, and also renders impossible the extensive flooding of dry land by the waters of the ocean, which would be the evident consequence of such a change; the second assures us that the grand unit of reckoning in all ages, the basis of astronomical chronology and of physical astronomy generally, the length of the mean solar day, has not varied, and never will perceptibly vary under the action of known forces. Laplace had long before proved, by a comparison of ancient eclipses with modern observations, that, practically, the length of the day had not varied in 2000 years. It appears, indeed, that since the earliest recorded Chaldean
on Rotation;
invariability of the mean solar day.

¹ See Blot in Connaissance des Temps, 1839; and Bally's Life of Flamsteed.

eclipse (that of B.C. 720), the rotatory velocity has not altered by one ten-millionth part.

(113.) M. Poinso
on Rotat-
tion.
An eminent contemporary and rival of Poisson, M. Poinso, has added many elegant propositions to the Theory of Rotation. M. Poinso is also the author of the Theory of Statical Couples, which now forms part of all elementary treatises on mechanics.

(114.) Other
writings of
Poisson.
Poisson wrote many other papers on subjects of physical astronomy, such, for example, as the Lunar Theory, but they did not lead, on the whole, to striking conclusions. In fact, he allowed himself to be diverted from this his most natural calling, by the ambition of constructing a system of Physics mainly founded on the applications of analysis. Some bulky volumes of this series appeared, especially those on Capillary Attraction, and on the Theory of Heat. The author here shows himself as a profound analyst, but adds little to our knowledge either of principles or of important results. A similar criticism may be applied to his theory of Elastic Substances, and to his doctrine of Waves. His papers on Magnetism and Electricity will be mentioned elsewhere, but their character is somewhat similar. In Optics he was attached to the Newtonian theory. In Mechanics not requiring assumptions as to the properties of matter, he was very successful. He was eminently a solver of hard problems: his investigation of the whole circumstances of motion of a projectile in air deserves notice. Indeed every branch of mechanics received his attention, and the number of his printed papers is said to exceed three hundred. On the whole, he will, perhaps, be most generally and favourably known by the excellent Treatise on Mechanics which he wrote for the use of advanced students. He was an eminent and diligent Professor, and his whole life was one of almost unremitting study. "La vie c'est le travail" was his reply, when urged to consult his health by reposing from his labours: and he actually died in the discharge of his duty as examiner of the Polytechnic School. This occurred on the 25th April 1840, in the fifty-ninth year of his age.

(115.) Mr Airy on
the pertur-
bation of
the Earth
by Venus.
Mr AIRY on the perturbation by Venus of the Earth's motion.—We here detach from what we shall elsewhere have to record of the eminent services rendered to the cause of astronomy by the present Astronomer Royal of England, a notice of his chief discovery in physical astronomy, the more remarkable from being almost the only improvement in the theory of the planetary motions, as applicable to the tables, which had proceeded from an English mathematician for a very long period. Sir James South called attention, in 1826, to a small but well marked deviation of the Sun's place from that given by Delambre's solar theory (the Sun's place or the Earth's are of course convertible terms). Mr Airy, then Lucasian professor at Cambridge, instituted, in 1827, a more extensive comparison with the Greenwich Observa-

tions, and attributed the error chiefly to the assumption of erroneous masses for Venus and Mars. But prolonged study satisfied him that it arose from a "long inequality," arising from the mutual action of the Earth and Venus, similar in its nature to that mentioned in Art. (66), as detected by Laplace in the case of Jupiter and Saturn. It arises from the circumstance that eight times the orbital period of the Earth is almost equivalent to thirteen times that of Venus. Terms of the series expressive of the mutual action of the planets, which are divided by thirteen times the mean motion of the former minus eight times the mean motion of the latter (which is a very small quantity), may consequently become considerable; still more, those involving the square of this quantity. Such terms belong to the fifth and higher orders of the eccentricity, and would consequently be very minute indeed, but for the casual magnitude of their coefficient. The labour of tracing out and calculating the effects of these important terms from the vast mass of algebraic developments is enormously great,—much greater than in the corresponding case of Jupiter and Saturn. As the calculations are not very likely to be repeated, Mr Airy took extraordinary precautions in their verification. The period of the inequality depends, first, as has been explained in Art. (67), upon the period required to carry the point of conjunction of the two planets completely round the circumference. This period is no less than 270 years. The perturbation is of course mutual, and affects the place of Venus as well as of the Earth.

Mr Airy is also the author of investigations connected with the Figure of the Earth, and the Theory of the Tides [see Art. (82)]; and he has published valuable Tracts on Physical Astronomy for the use of students. (116.)

Sir John Lubbock's name deserves mention here (117.) as having devoted his energies, about the same time as having devoted his energies, about the same time with Mr Airy, to intricate and laborious researches connected with the least inviting parts of physical astronomy, and striven to redeem England from the reproach of indifference or incapacity in respect of such inquiries. The chief object of his numerous memoirs (in the Philosophical Transactions for 1830, and following years) is to express, in a more convenient and exact manner than had been in use, the complicated series indicative of the perturbations as well in the Lunar as in the Planetary Theory.

The Lunar Theory: MM. PLANA and HANSEN.—(118.) MM. Plana and Hansen, an astronomer and analyst of the greatest and Hanmerit, who fortunately still does honour to the native city of Lagrange (Turin), though the author of a great number of Memoirs in the Transactions of the Turin Academy, on different points of Applied Mathematics, is, and will be, best known by his elaborate and learned treatise on the Theory of the Moon. This extends to three very bulky quarto volumes, which are, so to speak, one mass of symbols. Nothing can

give a more impressive idea of the condition at which Physical Astronomy has now arrived, than a glance at this mountain of intellectual labour,—especially as to the intricacy, inexpressible by words, of the motions of our own satellite. Here we have a work, not much smaller in appearance than the whole Mécanique Céleste, devoted to this one object. It is not superfluous to add, that this is no chimerical undertaking—no curious puzzle—no learned trifling; the Lunar Theory is the grand basis of the Art of Navigation. The real and main use to mankind of our companion planet is a discovery of these latter ages: her cheerful and beneficial light, which all appreciate, and all enjoy, may almost be termed a secondary boon.

(119.) Great work of the former.
The merit of M. Plana is not so much that of an original geometer, outstripping the theories of Laplace and Lagrange, as of a most intrepid and skilful calculator, who has contrived to place in complete order the whole mathematical and many of the arithmetical steps of the solution of one vast problem, extending to some 2000 quarto pages. The calculations he has made unaided! The details are arranged in so lucid a manner, as to court enquiry from those interested in verifying them; and though the readers of so abstruse and indeed repulsive a work must be few indeed, it has already proved of essential service in the way which was intended—the improvement of the lunar tables. The approximations by series are in all cases carried to the fifth powers of small quantities, and, in some instances, to the seventh powers. Sir John Herschel has expressed in forcible and picturesque language the nature of what is wanting to the completeness of Laplace's investigations and which M. Plana has supplied, as regards the theory of the moon's motions:—"In the Mécanique Céleste, we admire the elegance displayed in the alternate interlinking and development of formulæ, and exult in the power of the analytical methods used; but when we come to the statement of numerical results, we quail before the vast task of filling in those distant steps; and while cloud rolls after cloud in majesty and darkness, we feel our dependence on the conclusions attained to partake of superstitious trust, or of amicable confidence, rather than of clear and demonstrative conviction."¹ The courage of M. Plana did not "quail" before the serried ranks of symbolic legions. He attacked them first, and finally became their commander. But more than this; on the same high authority, "his analysis is always graceful, his combinations well considered, and his conceptions of the ultimate results to be expected from them perfectly just, and justified by the results when obtained."

I may here mention that M. Plana, in conjunction with M. Carlini of Milan, undertook the calculation

of the moon's motions from theory alone (that is, with only the fundamental constants required to be given by observation), in compliance with a programme issued by the French Academy of Sciences alone, on the proposition of Laplace; and that a French geometer, M. Danoiseau, on the same occasion, produced an independent investigation with very elaborate and valuable tables founded thereon. (121.)

M. Plana is the author of many other more circumscribed researches on important points in physical astronomy, in geodesy, and in mathematical physics.

M. HANSEN, a German astronomer and analyst of great merit, has made the most recent considerable improvement in the theory of the moon. We will first, however, allude to an invention which is applicable to the whole theory of perturbation. (122.)

We have seen, in the first section of this chapter, that the effects of perturbation may be considered either as applicable directly to the three co-ordinates of the perturbed body, or to the variation of the elements of the orbit considered as instantaneously varying; that the former method has the advantage of being in most cases, and especially in first approximations, most direct; the latter is most applicable to secular inequalities, and has a great recommendation in the exact physical conceptions on which it is founded. M. Hansen has proposed a third method, a refinement, in fact, on that of Lagrange, in which, by an assumption purely mathematical and arbitrary, he throws the effect of perturbation entirely on the element of time, so that with the time so altered, and invariable elliptic elements, the conditions of perturbation may be satisfied, and the true place of the body may result from the calculation. Such is the general nature of the conception, which, to be carried out, requires the introduction of subsidiary terms, which serve to correct the latitude and radius vector. The advantages which are understood to follow from this highly artificial mode of proceeding are stated to be (1.) that the series expressing the perturbation of the co-ordinates are more convergent than in the other methods, and likewise the coefficients of the terms to be retained are more easily calculated; (2.) M. Hansen considers that his method enables him to ascertain with certainty those terms which, when fully calculated, will affect the result in a sensible manner. The inventor has applied his methods both to the Lunar and to the Planetary theory. (123.)

We have said that physical astronomy is indebted to M. Hansen for a notable improvement in the theory of the Moon,—the discovery, in fact, of two inequalities of long period, the existence of which had been more than suspected from observation, but which were (124.)

¹ Astronomical Society's Monthly Notices, vol. v., p. 37. The last expressions may astonish some persons, but experienced analytical calculators agree in the same view. Mr Alry (probably the most competent authority in Britain) states the same thing in many passages of his writings, to the effect that the evidence for conclusions so obtained is rather that of moral than of mathematical certainty.

not accounted for. In the infancy of the lunar theory, Euler had predicted that it would always be impossible, on account of the perturbing forces of the planets, to predict the Moon's place within 30"; and it was a quantity of about this measure which remained outstanding after all the resources of analysis seemed exhausted. M. Poisson and Sir J. Lubbock showed that the anomaly could not be due to the solar action, nor yet to the irregularity of the Earth's figure. The planetary attractions, then, alone remained. M. Hansen discovered two independent inequalities due to the action of Venus. One of these is an indirect (or, as it is sometimes called, reflected) effect depending on the change of form of the Earth's orbit by the attraction of Venus, which, of course, modifies slightly the solar perturbation of the Moon's longitude. It is, in fact, a secondary consequence of long inequality of Venus and the Earth investigated by Mr Airy [Art. (115)], and has the same period, namely, about 240 years. Its greatest amount is 23".2. The other inequality discovered by Hansen is of a still more curious and complicated kind, which goes on increasing for 2000 lunations, when it attains a maximum value of 27".4 in longitude (although the perturbation of the radius vector does not exceed 10 feet), after which it diminishes for an equal space of time.

By these discoveries, the movements of our refractory satellite may be considered to be, after an unprecedented amount of labour, accounted for by theory almost or quite within the limits of the present accuracy of observation.¹ M. Hansen has recently been engaged in perfecting the practical details of the lunar theory, in accordance with the extensive reductions of the Greenwich observations which will be mentioned in the next chapter. He has also given the first complete theory of "Foucault's pendulum," also to be mentioned hereafter. Altogether, he stands amongst the most eminent analytical astronomers of the present day.

We have in this section selected (though not by design) representatives of the four great intellectual communities of Europe, engaged in the mighty task of perfecting the theories of physical astronomy. Astronomy Poisson for France, Airy for England, Plana for Italy, Hansen for Germany. It would be easy, of course, to add the names of many others engaged in similar works, and scarcely less deserving of notice. Some of these will find a place in other chapters, and we have yet one section of this chapter to devote to the history of a discovery of rare interest, in which France and England have a joint share. We may in Europe, be allowed to mention the names of M. Damoiseau and M. Pontecoulant in France; the former known by his excellent lunar tables deduced from theory, the latter for his calculations of cometary perturbation, and his compendious treatise on physical astronomy, based on the Mécanique Céleste; in Italy, MM. Carlini and Santini. But in Germany these studies have been perhaps most systematically pursued. MM. Gauss and Enecke have only not been included in this section because we find it more suitable to our plan to associate the name of the former with his theory of Terrestrial Magnetism, and the latter with the recent history of Comets. Bessel likewise was a physical as well as first-rate practical astronomer. I will here only add that these and in severe and arduous studies have at length been effectually cultivated beyond the limits of Europe. Mr Bowditch (born 1773, died 1838), a private gentleman of the United States, undertook the gigantic labour of translating and illustrating, with a complete commentary in which every difficulty is considered, and every step of analysis supplied, the Mécanique Céleste of Laplace. Since his death, a younger race of American mathematicians has taken up the great problems of physical astronomy, amongst whom may be mentioned Mr Walker and Mr Peirce. The latter gentleman has recently (1853) published lunar tables, embracing the latest researches of theory.

§ 5. M. LEVERRIER—MR. ADAMS.—The inverse method of Perturbations. Prediction of the place and orbit of Neptune from the motions of Uranus.

We have now to chronicle a discovery which, by general consent, stands first in the achievements of science, not only in the period now under review, but even in the long and eventful series of years which have elapsed since Newton established the doctrine of universal gravitation.

The discovery of which we speak was no less than the proof of the existence of a planet beyond the recognised boundary of our system, merely as an inference from the perturbed motion of the outmost planet Uranus; a proof, not general or abstract, but

particular and specific: "Look, on such a night, and in such a direction, and there you will see (by the telescope) a star, small indeed, but with a distinguishable disk,—that is the planet which has made Uranus move so unsteadily in its orbit;"—so spoke the mathematician; and the zealous astronomer, to whom the call was especially addressed, pointing his glass to the sky, discovered at once, that is, the same evening, a body answering almost precisely in position, as well as in brilliancy, to the oracular announcement.

¹ M. Hansen has also discovered the course of a small inequality of the Moon's latitude, detected from observation by Mr Airy. His theory of the Moon's figure has been referred to in a note to Art. (57.)

Since the publication of the Principia, or rather we should say since the great theory contained in that work had fully attracted the sympathies of thoughtful and able men, nearly the whole science of physical astronomy consisted in the solution of one vast and intricate problem, which has been called the "Problem of the Three Bodies." To this were bent the powers of Clairaut, Euler, Lagrange, Laplace, Plana, Hansen, and so many more. "Let three bodies be placed and move in a given manner in space and attract one another by the Newtonian law, to determine the motions as affected by their mutual influence." The problem solved independently by the two analysts whose names stand at the head of this section is this, "Given two bodies (the Sun and Uranus) and their relative motion, to find at any moment the position of a third body whose attraction shall be required to account for those motions." To have solved this new and far more difficult problem (under certain limitations) is a triumph altogether unlike in kind to any of the other brilliant successes of which we have had to speak in the preceding pages.

The great intricacy of the problem is not perhaps at the first moment fully apparent. The perturbation of the known planet (Uranus) is not the effect, either in direction or in amount, of the attraction of the body sought, either at the instant, or at any previous instant. It is an accumulated effect arising from the totality of the mutual influences of the two planets during a long space of time, and under a variety of circumstances, which circumstances it is the aim of the solution to discover. But, more than this, we do not even know the quantity or direction of the perturbative action at any moment for which the cause is sought, for we do not know the purely elliptic elements of Uranus. His motion has, by hypothesis, been always troubled by this exterior planet, and Uranus has been so short a time known and observed (only accurately since 1781) that the motion has not yet been cleared of ordinary inequalities (due to the action of the unseen body), which, therefore, are inextricably mixed up with the elliptic elements. It is absolutely necessary, therefore, to suppose not only the elements of the new planet to be unknown, but also the elements of Uranus to be severally affected by unknown errors. This nearly doubles the unknown quantities to be found.

We shall now glance at the history of these unexplained perturbations, and at the rise and growth of the idea of their being explicable by the influence of an unseen body.

After Sir William Herschel had discovered, in

1781, the planet which he called Georgium Sidus, (afterwards named Herschel, and finally, by general consent, Uranus,) it was easy, by a few observations, to ascertain its approximate orbit and distance from the sun. But the extreme slowness of its motion (it will not have completed a single revolution before 1861) made it impossible to determine its elements with precision, until it had been discovered that the records of astronomy contained about twenty observations of this body before its planetary nature was discovered; being, in fact, registered places of fixed stars where no stars exist, concurring in brightness and position with the circumstances of Uranus at those remoter periods; the earliest of these observations was one of Flamsteed, in 1690. The first person who constructed tables of the planet was Delambre; but even at that early period, it was remarked, that the modern were not satisfactorily reconcilable with the ancient observations; and, finally, Delambre included of the latter only one, by Mayer, which he did, he tells us, "out of pure respect," although it certainly rendered the tables less exact and less durable. The longer the planet was observed, and the greater the care that was expended in analysing and combining the observations, the more clearly it appeared that the tables had only an empirical character, satisfying observations for a few years before and after the time for which they had been constructed; and that, in particular, as the nineteenth century advanced, the deviations from elliptic regularity became more and more intolerable, till the "ancient" observations were at length totally given up. This was the state of matters in 1821, after M. Bouvard of the Observatory of Paris, a most able calculator, had exhausted every resource in improving the Tables of Uranus. A few years more gave that patient astronomer the mortification of seeing his tables as obsolete as those of his predecessors, and numerous surmises were circulated as to possible explanations of the anomaly: a failure in the law of gravity, cometary perturbation, a resisting medium, and, finally, the presence of an unseen planet, were amongst these guesses. The last and most plausible of these hypotheses occurred to many; amongst others, to Mr Hussey in England, M. Bouvard in France, and later to M. Bessel in Germany.¹ The first of these astronomers actually consulted Mr Airy, in 1834, on the possibility of predicting the place of the perturbing planet from theory, and then discovering it by observation. Mrs Somerville, in 1836, gave a precise expression to the same idea.² From this time the subject could not be lost sight of. The errors of the tables which in 1821 were insensible, increased in 1830 to 15" or 20", and in 1845

¹ Clairaut, nearly a century before, in calculating the return of Halley's comet, hinted at the possible perturbation due to a planet superior to Saturn.

² In her Connection of the Physical Sciences.

amounted to two minutes of longitude. M. Bouvard to his latest years, perhaps his latest hours,¹ cherished the hope of extricating this theory from its difficulties. He also engaged his nephew M. Eugène Bouvard in the same career, who appears to have followed it with much zeal and intelligence, and in 1845 constructed new tables of Uranus. But by this time two geometers had separately and independently undertaken the problem, with the determination of finding, if possible, a physical solution of all this perplexity. The earliest in point of date was Mr Adams, a young graduate of Cambridge; the other was M. Leverrier of Paris, whose attention was directed to the subject by M. Arago. As the researches of M. Leverrier, though second in point of time, occasioned the actual recognition of the planet, and thus stamped the correctness of the solution with success, we shall consider them in the first instance.

(133.) M. Leverrier's investigations;
M. LEVERRIER is, we believe, a native of St Lo in Normandy, a province which has been singularly productive of eminent men (Laplace and Fresnel were of the number). With no advantages, but the reverse, he won a high position at entering the polytechnic school, which he constantly maintained. He at first, we believe, attached himself to chemistry, but his taste for physical astronomy was soon developed, and was advanced entirely by his private efforts. It is a peculiarity of the mode of cultivating the sciences in Paris, that such abstruse and difficult studies are not merely engaged in temporarily for purposes of academical distinction, but that they actually become a "carrière" or calling, and are pursued in that methodical manner for which the French are distinguished. In 1845, when he commenced the careful examination of the theory of Uranus, M. Leverrier was already favourably known by his researches on comets, and on the orbit of Mercury, but especially by immense calculations, connected with the secular inequalities of the planets, by which his ability and hardihood in computation had been thoroughly exercised. He began his new enquiry with the method and intrepidity of calculation which distinguish him. He revised with the most minute care the observations of Uranus, and computed afresh every sensible perturbation which theory recognised as arising from known planets. This done, and having compared the most probable orbit with observations which he collected from authentic sources, and especially from the Greenwich observations which were communicated to him for this purpose, the result was, that even confining himself to observations since 1781, arranged in eleven convenient groups (each resulting from many observed places), and attributing to each group the

largest error which could be in reason allowed, and even admitting that all these errors were in the direction most favourable to the assumption, it was still impossible to account for more than one-fourth part of the observed discordances.

M. Leverrier then assumed that a perturbing planet existed beyond the orbit of Uranus, and at nearly double its distance from the sun, in conformity with the empirical law, (usually attributed to Bode the German astronomer,) which expresses with general accuracy, thus far, the arrangement of the planetary system. The law is, that the distances of the planetary orbits from Mercury are successively doubled. This assumption—(it was absolutely necessary to assume some distance to begin with)—was ingeniously confirmed by other considerations. (134.)

Leaving the perturbations in latitude out of account, he now considered each error of Uranus in longitude as the expression of a perturbation due to the action of the unknown planet, and capable therefore of algebraic expression in terms of the elements of that planet, namely its excentricity, longitude of perihelion, epoch in its orbit, and mass; but, as we have already remarked, the first three of these elements must be considered as incorrectly assumed for Uranus itself, as well as the mean distance of that planet, and, therefore, there are four unknown corrections for its elliptic elements, making in all eight quantities to be eliminated from the discordances of theory and observation. So complex an elimination cannot be directly effected; and even if it could, the result could not be depended on, as the possible error of each observation involves a fresh and important source of doubt in the conclusion. M. Leverrier proceeded, by a series of gradually restricted assumptions, to find within what limits the more important elements might be made to vary without producing effects incompatible with observation, and his attention was at first confined to the approximate mean longitude of the planet. He obtained a result after a prodigious amount of tentative calculation. The excentricity and position of the perihelion were then inferred. On the 1st June 1846 he announced to the Academy of Sciences that the true longitude of the expected planet for 1st January 1847 was 325°, with a probable error of 10°. This result was immediately published in the Comptes Rendus. (135.) how conducted:

Between the 1st June and 31st August 1846, when his third memoir on the perturbations of Uranus appeared, M. Leverrier busied himself in obtaining a farther approximation to the elements and place of the suspected planet. He now assumed the correction of the mean distance amongst the other quantities to be sought. By a fresh calculation he

¹ M. Bouvard was born in 1767. He performed almost all of the numerical calculations required by Laplace in his great work, and was associated with that eminent man by the most friendly ties. He "ceased to calculate and to live" 7th June 1843.

deduced a complete list of elements as regards longitude; and diminished the mean distance considerably. The true longitude by this calculation differed only $1\backslash frac\{1\}\{2\}^\backslash circ$ from his previous estimation. He finds for the mean distance from the Sun 36 times that of our Earth (that of Uranus being 19), the period 217 years, and the mass $5\backslash frac\{1\}\{3\}$ of the Sun's. Assuming the density of the planet to be the same as that of Uranus, he conjectures that the apparent diameter will be $3^\backslash circ\; 3\text{'}$; that it will therefore have a visible disk sufficient to distinguish it from a fixed star, and that its brilliancy should equal that of a star of the 8th or 9th magnitude. As the result of these suppositions, he found that the whole errors in the places of Uranus from 1781 to 1844 were reconciled within quantities amounting but in one instance to $5\text{'}\text{'}$; that the ancient observations of the 18th century were reconciled within $7\text{'}\text{'}$ or $8\text{'}\text{'}$; and the oldest observation of all, that of Flamsteed in 1690, had an outstanding error of $20\text{'}\text{'}$, a quantity very far from excessive, considering the state of astronomy at that period.

(137.) No one who read at the time the abstract of this remarkable paper in the Comptes Rendus failed to be struck with it, not only as regarded the weighty matter, thus publicly announced, but also on account of the calm and well-grounded conviction which the author manifested in the truth of his bold conclusions, and the definite manner in which he gives the challenge to practical astronomers to verify or disprove them. "Since Copernicus declared" (according to the prevalent tradition) "that when means should be discovered for improving the vision, it would be found that Venus had phases like the moon, nothing," writes Mr Airy, "so bold, and so justifiably bold, has been uttered in astronomical prediction." M. Leverrier had hastened his calculations in anticipation of the approaching opposition of the new planet in the autumn of 1846, but it is very doubtful whether astronomers would have made the discovery at that time, but for his personal application to M. Galle, then assistant-astronomer at Berlin, where a powerful refractor suitable for the search existed. So ardent a conviction in a manner compelled the proof which the geometer claimed, and M. Galle, whose intelligence and zeal are well known, pointed his telescope to the sky the very evening that M. Leverrier's letter reached him. Fortunately provided with a newly published star map by Brenicker of that region of the heavens, which was not at that time diffused generally amongst European observatories, he detected that same night (the 23d September 1846) a star-like body of the 8th magnitude, not noted in the star chart, therefore a wandering body, having a manifest disk from $2\backslash frac\{1\}\{2\}\text{'}\text{'}$ to $3\text{'}\text{'}$ in diameter, and distant only fifty-four minutes of a degree from the predicted place.

(138.) It will be remarked that the discovery in question

was anticipated and completed in France and Germany alone; England had no direct participation. We must now, however, state briefly what occurred there of a similar character at the same time, and even earlier.

Mr JOHN COUCH ADAMS, when a student at St John's College, Cambridge, in 1841, formed the design of detecting the position of a perturbing planet which should account for the anomalous motions of Uranus. He made a preliminary essay on the problem in 1843, assuming the distance of the suspected body from the Sun to be double that of Uranus. I learn from good authority that he obtained a place for the unseen planet not very different from that which he finally adopted. Early in 1844 he obtained from Greenwich the valuable series of places of Uranus which were afterwards in like manner applied for by M. Leverrier. In September 1845, he communicated to Professor Challis the elements of the new planet's orbit (neglecting the inclination) and an ephemeris of its geocentric place; and in October he transmitted the elements also to the Astronomer Royal. This, it will be observed, was soon after M. Leverrier's attention was first directed to the subject, and nine months previous to his announcement of the locality where the new body should be sought. Mr Adams afterwards repeated his calculation with a mean distance $\backslash frac\{1\}\{3\}$th less than before, and considered himself warranted, by the improvement thus produced on the residual errors, in inferring that a farther considerable diminution of the mean distance would satisfy the observations still better. This was communicated to Mr Airy on the 2d September 1846; subsequently therefore to the publication of M. Leverrier's Elements. Mr Adams, in communicating his results (at a later time) to the Astronomical Society, with characteristic modesty says, "I mention these dates merely to show that my results were arrived at independently, and previously to the publication of M. Leverrier, and not with the intention of interfering with his just claims to the honors of the discovery, for there is no doubt that his researches were first published to the world and led to the actual discovery of the planet by Dr Galle."

And such is no doubt the fact. The priority of Mr Adams in the mathematical investigation is as certain as that the researches of M. Leverrier alone produced the discovery of Neptune. Even the search for the planet which took place in August and September 1846 at Cambridge by Professor Challis, was not occasioned by Mr Adams' researches only; it was the near coincidence of the longitude assigned by Mr Adams in the previous October with that published by M. Leverrier in June 1846 which induced Mr Airy to suggest this investigation of the heavens, and to offer (if need were) himself to bear the expense. Had the planet been discovered¹ at Cambridge

¹ The planet was indeed seen at Cambridge by Mr Challis, for it was recorded more than once amongst the numerous fixed stars whose places were taken down in the progress of the search; but as the comparisons of the "sweeps" were not made at the time, the discovery was anticipated by M. Galle.

before it was at Berlin, M. Leverrier must still have had a share in the credit of success.

(141.) Remarks on the history.

It is perhaps to be regretted that Mr Adams had not given his whole investigation to the world, or at least published his results, so as to avouch the confidence which he felt in his own prediction, and to throw upon practical astronomers generally the responsibility of its verification. Had he done so in 1845, it is possible that the planet might have been discovered at the opposition of that year; but it is at least certain that M. Leverrier's claims to priority as regards the discovery of Neptune would have been effectually anticipated. But it is only just to our countryman to recollect the difference of his age and position. M. Leverrier was at the time about 35 years of age, and was a candidate for the substantial benefits as well as for the honour of a membership of the Academy of Sciences. Mr Adams must have been nine or ten years younger at the period of this discovery, a circumstance which enhances our admiration at the achievement, whilst it gives an additional grace to the modest conduct of the author.

(142.) I have endeavoured to state correctly (with due regard to the limits of this essay) the main facts of the most curious case of double discovery which, perhaps, the history of science presents; and happily as to the facts, down to the minutest detail, no discrepancy of opinion ever existed. Different minds will, with perfect truth, attach more or less distinction to the two illustrious rivals,—neither of whom has for an in-

stant lowered the dignity of his position by one ungenerous expression,—but that the absolute merit of both is of the very highest character is on all hands admitted. "The names of M. Leverrier and Mr Adams," said Sir John Herschel, addressing the Astronomical Society, "which Genius and Destiny have joined, I shall by no means put asunder; nor will they ever be pronounced apart so long as language shall celebrate the triumphs of Science in her sublimest walks."

But before closing, I must briefly state how far the orbit of the planet Neptune, when discovered, realized the provisions of theory. A fortunate circumstance rendered it easy to obtain at an early period a correct knowledge of the elements. It seems that the planet Neptune was observed by Lalande at Paris on the 8th and 10th May 1795, and entered as a fixed star, notwithstanding a distinct change of place between the observations, actually corresponding to what the planet should have had.¹ But such an oversight had been made by Lemonnier in the case of Uranus. By means of this observation of fifty years back, the orbit was easily computed. It is a singular and startling fact, that, except as regards the longitude on the orbit, the other elements computed from observation were somewhat widely different from those assigned by M. Leverrier and Mr Adams. We shall present them in a tabular view. The value of the mass in the last column is calculated from the elongation and period of a satellite of Neptune discovered by Mr Lassell.

(144.) The differences of theory and observation are so striking as to have occasioned surprise to many persons that, with data so erroneous, the perturbations of Uranus or the longitude of Neptune at the epoch of discovery should have been obtained even approximately.

(145.) How an assumption of erroneous elements led to the discovery;

The fact, however, is this:—that the mutual perturbations of Uranus and Neptune are sensible for only a small portion of the joint orbits when the planets are nearly in conjunction. The conjunction (when the mutual distance is least and the attraction strongest) took place in 1822. Now the places of Uranus from 1690 (the first observation) until 1800 can be sufficiently well represented by elliptic elements. The perturbation of Neptune became sen-

sible therefore only twenty years before conjunction. In this time Neptune describes (really) only $\backslash frac\{1\}\{8\}$ of a circumference, or 45°, and relatively to the motion of Uranus about the same. It is evident then that even a considerable error in the period of Neptune would scarcely sensibly affect the law of perturbation during twenty years, and that the approximate determination of the place of the perturbing planet about the time of conjunction will not be much affected by the error of that assumption. Again, as to the error of mean distance, we may observe, that since the mutual action of the planets is confined within such (comparatively) narrow limits of space and time, though we might anticipate a tolerable approximation to the interval between the bodies

¹ One of the observations was suppressed in the publication, and only discovered on searching Lalande's MS.

² This was the hypothesis upon which Mr Adams made his second or corrected calculation of elements. Nevertheless, he inferred from that calculation that the mean distance might with much probability be reduced to 33.4.

³ Pierce. Struve's mass is 1/300.

at that time, it would be unreasonable to expect a correct determination of the form and ellipticity of the orbit of Neptune, such as might be looked for if the perturbations were sensible through the entire orbits; and in fact, by varying the position of the perihelion and the amount of excentricity, we may, for an assumed mean distance, obtain any value whatever for the interval of the two planets at a particular time. We have seen the origin of the false assumption of mean distance on the part both of M. Leverrier and Mr Adams, and we find that the mathematical solution corrects to a great extent the error of that assumption by giving a correspondingly incorrect position of the perihelion, and also an exaggerated measure of the excentricity, by which two circumstances the planets of both mathematicians would have had, near the time of conjunction, a distance from the sun of only 32 or 33 radii of the earth's orbit, the true distance being about $\backslash frac\{1\}\{2\}$ part less. This still remaining error was palliated, and evidently might for a time have been completely masked, by assuming a mass of Neptune proportionally too great, as indeed the table we have given shows was the case.

(146.) That the discovery of Neptune took place at the time when it did was no accident. The conjunction of Uranus and Neptune, when alone the perturbation of the elliptic elements is perceptible, is a rare phenomenon, occurring but once in about 172 years. The last conjunction previous to 1822 was in 1649: we have seen that the attention of astronomers was importunately called to the subject by the irregularities in the motions of Uranus at the first conjunction succeeding its discovery.

(147.) We are indebted to Professor Peirce and Mr

Walker of the United States for many useful investigations connected with the orbits of Uranus and Neptune; but it is to be wished that the theory were completely re-examined, and also the problem of the inverse method of perturbations which has now become a systematic portion of physical astronomy. The near commensurability of the two periods presents a peculiar case of perturbation similar to the long inequality of Jupiter and Saturn; but as its period is no less than 4047 years,¹ although the coefficient of the inequality is considerable, it will not perceptibly alter the motions of either planet except in a long course of years.

It will not be supposed that either M. Leverrier or Mr Adams could, after such a memorable triumph, abandon the pursuits of physical astronomy. The first has continued his researches on the orbits of comets and on the perturbations of the solar system; and having been recently appointed to succeed M. Arago in the direction of the Paris Observatory, we cannot doubt that he will infuse new life into its management. Mr Adams has made the discovery of some important oversights in the details of the lunar theory. One of these, with reference to Laplace's Theory of the Secular Acceleration of the Moon's mean Motion, has been referred to in a note to Art. (62). Though Mr Adams occupies no public post, and though he has declined the honours of a title, his continued residence at Cambridge must influence very beneficially the studies of the place, where some of his many friends have founded in memory of his achievements a perpetual prize for the advancement of physical astronomy, which is denominated the Adams Prize.²

CHAPTER III.

ASTRONOMY.

§ 1 MASKELYNE—DELAMBRE.—Progress of Practical Astronomy from 1770 till 1810—Of the Lunar Theory deduced from Observation—The Density and Figure of the Globe. Cavendish; Baily. Trigonometrical Surveys.

(149.) As examples of those astronomers who most contributed in the period of which we now principally speak (1770–1810) to the progress of exact observation,

whether of the motions of the heavenly bodies or of the figure and density of our own planet, I have selected Maskelyne and Delambre. Their characters

¹ According to the calculations of M. Peirson (Camb. Trans. vol. ix.).

² It has been our business to condense the history of the discovery of Neptune within a compass proportioned to the general scheme of this Dissertation. The reader who desires further details will consult M. Leverrier's Memoir on the Perturbations of the Herschel planet (Uranus) in the Connaissance des Temps for 1849 (also published separately), and in the Comptes Rendus for 1846; Mr Adams' papers in the Nautical Almanac for 1851, and the Astronomical Society's Memoirs, vol. xvi.; Mr Airy's singularly curious and impartial Historical Account in the 7th vol. of the Astronomical Society's Monthly Notices; M. von Lindenau's History in Schumacher's Nachrichten (Ergänzungsheft) (favourable to M. Leverrier); Grant's History of Physical Astronomy (advocating principally the claims of Mr Adams). Sir J. Herschel's Outlines of Astronomy contain a very valuable description of the scientific part of the question.

were both distinguished by modesty, simplicity, and love of truth. Of all their contemporaries of eminence, few escaped so happily from the unprofitable strife of rivalry and personal disputes, and none exhibited a more impartial desire for the advancement of the science to which they were devotedly attached. Perhaps neither was a man of lofty talent, yet they did not fail to secure deserved respect in their own day, and far more gratitude from the posterity whom they essentially benefited than falls commonly to the share of men of higher pretensions in this respect. Both their names are important links in the history of astronomy.

(150.) NEVIL MASKELYNE, born in 1732, and educated at Maskelyne; Cambridge, was early attached to astronomical pursuits. After various minor services connected chiefly with navigation and the discovery of the longitude, he attained the honourable post of Astronomer Royal, which he filled from 1765 till 1811 with distinguished success. His immediate predecessor was Dr Bliss, who held the office for but three years and without distinction. Practically, Maskelyne may be said to have succeeded Bradley, probably the greatest astronomer whom England has yet seen, whose discoveries have been recorded in the last Dissertation, and whose observations, through a variety of circumstances, were destined rather for the benefit of the nineteenth century than for his own. Maskelyne wisely recollected that the observatory was mainly founded for the improvement of navigation,¹ and one of his earliest labours was the establishment in 1767 of the "Nautical Almanac," a work based on the best astronomical observations and of the highest service to seamen. "During Maskelyne's long tenure of office he was entirely devoted to its duties, making himself all the most delicate observations, particularly those of the moon, and rarely quitting the observatory except to attend the meetings of the Royal Society. The perfect method and continuity of his observations give to them a great value, especially for the correction of the Lunar Tables, in which respect they are indeed without a parallel. But the regularity of their publication was not their least merit. Four large folio volumes include the patient labours of a life (for he had but one assistant). Delambre in his character of Maskelyne says, that if through some catastrophe the whole materials of science should be lost, except these volumes, they would suffice to reconstruct entirely the edifice of modern astronomy."² In fact, on Maskelyne's accession, the only methodical publication which had issued from Greenwich Observatory was the "Historia Celestis" of Flamsteed; Halley's, Bradley's, and Bliss's observations remained in manuscript. A like fate attended most

of the foreign astronomical observations; labours the most irksome and conscientious lie buried in piles of MS. useless to science, and which therefore might almost as well have never been made. This is particularly the case with the Parisian observations (as the impartial Delambre records with pain), commenced even before the time of Flamsteed; but which, owing to this cause principally, have remained even to the present day (and it is forty years since Delambre wrote his patriotic protest) without contributing materially to advance astronomy. Let it then be recorded to the honour of Maskelyne, that this important step of regular and full publication at the public expense was entirely due to him. His places of the sun, moon, and planets, were the foundation of the improved theories of physical astronomy, then more ably cultivated on the Continent than in Britain, and of the tables also chiefly furnished by German and French computers.

Maskelyne's more important contributions to science may be briefly stated under these two heads; (151.) —The determination of the Lunar orbit from observation, and its application to navigation; and the determination of the local attraction of Schehallien and of the density of the earth.

I. The determination of the Lunar Orbit from Observation and its application to Navigation.—Though (152.) Maskelyne owes (as we have already said) much to Maskelyne; Lunar observations. Maskelyne in the exact determination of the places of the sun, planets, and the most conspicuous fixed stars, the comparison of the Lunar place with the tables was by far the most arduous and the most important of his undertakings. It was the most arduous, because, from the extreme complexity of the moon's motions, every part of its orbit must be narrowly watched, requiring the astronomer's presence at his instruments at all possible hours during the course of a lunation, these motions being subject to important changes which recur every nineteen years, besides others of yet far longer duration. It was Maskelyne's good fortune, and at the same time the reward of his perseverance, to watch three revolutions of the lunar nodes. He did not content himself, however, with making observations; he contributed in every possible way to the improvement of the tables of the moon's motion. He was in constant communication with Mayer, one of the ablest astronomers of his day, and he directed the calculation of Bradley's observations by Mason, for the further improvement of Mayer's Tables. Maskelyne's own observations, to the number of at least 5000, were used by Bürg in his excellent tables of the moon which have even yet been hardly surpassed; but their full value has only been tested recently by the

¹ In the warrant appointing Flamsteed to be the Royal "Astronomical Observer," his duty is declared to be "to rectify the tables of the motions of the heavens and the fixed stars, so as to find out the so much desired longitude at sea, for perfecting the art of navigation."

² From an article on Greenwich Observatory in the Edinburgh Review, written by the author of this Dissertation.

systematic reduction of all the lunar observations of Maskelyne and Pond, and a comparison with Damoiseau's tables, under the direction of the present Astronomer Royal, Mr Airy.

(153.) But, for the application of the method of lunar distances to navigation, farther aid than the construction of good tables was required. This Maskelyne provided by obtaining the regular publication of the Nautical Almanac, superintended by himself, and containing the distances of the moon from the principal fixed stars at predetermined hours for the meridian of Greenwich, a comparison of which with the distance observed by means of the sextant in any part of the world enabled the seaman (after proper reductions) to infer the exact Greenwich time of the observation, and thence, by comparison with the local time obtained by the usual methods, to obtain his longitude. To this, long admitted to be the best practical solution of the celebrated problem of "the longitude at sea," Maskelyne contributed probably more than any other person. His "Lunar Distances" were reprinted in the French Almanac (Connaissance des Temps) for a considerable number of years.

(154.) II. The determination of the Attraction of Schehallien, and of the Earth's Density.—The deviation of the plumb-line from the vertical by the neighbourhood of a mountain had been pointed out by Newton¹ as a direct consequence, and also as a test, of the principle that gravity resides in every part of the earth as well as in the earth as a whole. Bouguer had the merit of pointing out the form in which the experiment might be made, and of making the trial, though in a rude and insufficient manner, in the Peruvian Andes in 1738. He observed the effect of the mountain on the south side only, but at two stations unequally distant from its centre of attraction. The numerical result being (as the author himself admitted) without value, Maskelyne proposed to the Royal Society in 1772 to repeat the observation on some British mountain. A "Committee of Attraction" was named, which, besides Maskelyne, included Cavendish, Franklin, and Horsley. Cavendish, as might have been expected, took an earnest part in it. The search for a suitable hill was confided to Mr Charles Mason in 1773. Skiddaw and the Yorkshire Hills were first thought of, but finally Schehallien in Perthshire was preferred.² Thither Maskelyne himself proceeded in 1774, with his assistant Burrows, and by these two, with the aid of a local land-surveyor, the labour of the astronomical and

chief geodetical operations, including the measurement of two base lines, was effected between the 30th June and the 24th October, notwithstanding the hindrances of a most unfavourable season.

The distance between the two stations obtained with Ramsden's 9-inch theodolite, was 4364.4 feet, which in the latitude of Schehallien corresponds to $42^\{\backslash circ\}94\text{'}$ of latitude. The observed difference of latitude by 337 observations with Sisson's 10-feet zenith sector was $54^\{\backslash circ\}6.2\text{'}$. The excess, or $11^\{\backslash circ\}6\text{'}$, is the double attraction of the hill drawing the plumb-line towards itself at the two stations. The sine of this angle, or $\backslash frac\{11^\{\backslash circ\}6\text{'}\}\{180^\{\backslash circ\}\}$, represents the actual ratio between the double attraction of the hill and the attraction of the earth. But by the computation of the attraction which the hill ought to exert, from its figure, as determined by Maskelyne's gauges, were its density the same as that of the globe generally, this ratio should amount to $\backslash frac\{1\}\{3333\}$, which can only be accounted for by assuming the earth to be denser on the average than the hill of Schehallien in the proportion of 17804 to 9933. This deduction was made by Dr Hutton by means of a troublesome calculation of the summation of the attractive effects of a number of vertical prisms into which the hill was imagined to be divided. The artifices of calculation were, however, due to Cavendish (who it will be recollected was on the "Committee of Attractions,") as Mr Airy ascertained from his manuscripts. A careful lithological survey of the hill enabled Professor Playfair to deduce the probable mean specific gravity of the globe to be between 4.56 and 4.87, which was somewhat greater than Dr Hutton assumed it.

This is the proper place to mention an experiment on the density of the Earth perhaps still more remarkable, devised by the Rev. Mr Michell, who constructed the apparatus, but first put in practice by Mr Cavendish in 1797-8. It consisted in measuring the force of gravitation between two spheres of such small size that they could be moved by the hand nearer to or farther from one another. The essential part of the invention was to contrive a balance so delicate as to measure the almost inappreciable tendency of such small bodies to unite. Newton had shown that the attraction at the surface of any sphere is directly as its radius, which he observed must always be incomparably smaller than their tendency towards the earth, that is, their weight. In the largest and heaviest masses with which it has hitherto been found practicable to operate, this tendency

¹ De Mundi Systemate, § 22. Newton, in a very remarkable passage of the Third Book of the Principia (Prop. X.), conjectures that "the quantity of matter in the earth may be five or six times greater than if the whole were composed of water."

² A laughable mistake of Zach in his account of the Schehallien experiment (in his Attraction des Montagnes) is commented on by Playfair in the Edinburgh Review. In a note to the word Schehallien, Zach says, "Montagne appelée dans le pays en langue Erse Maiden pap, qui veut dire orage perpétuel." It is needless to add that these two alleged synonyms are different interpretations given by Gaelic scholars of the word. "From this inaccuracy," adds his reviewer, "his residence in London ought to have delivered him, for though he could not learn there what was Erse, he might have learned what was English."

³ Zach obtained the same result exactly by including all the observations, as Maskelyne had provisionally obtained by using only those stars on which he most depended.

amounts to only a very minute fraction of a grain. How could such quantities be accurately estimated, so as not only to leave no doubt of the phenomenon of gravity thus acting on the small scale, but to deduce its amount, and hence to weigh the globe?

(157.)
The Torsion Balance.

Mitchell imagined for this purpose the balance of torsion, which was re-invented by Coulomb (who probably first executed it) for the purpose of measuring electrical forces. Mitchell's apparatus came first into the hands of Wollaston, then of Cavendish, who made the experiment. He used a very light rod of deal, six feet long, suspended by a fine silver or copper wire, forty inches long, within a wooden case to defend it from currents of air. At each end of the lever was hung a ball of lead, two inches in diameter, and by a simple contrivance a pair of leaden spheres, weighing, together, 348 pounds, could be brought simultaneously into the neighbourhood of the balls (but outside the case), on opposite sides, so that their attractions might concur to swing the suspended lever out of the position of repose which it had previously taken up, under the action of the slight twisting force of the silver wire. A new position of rest was thus established, the small balls being pulled as much one way by the attraction of the spheres as they were urged in the opposite direction by the torsion of the wire. The position of repose being observed from a distance by a telescope (to avoid disturbance from the heat of the observer's body), the great spheres were then changed in position so as to act upon the opposite sides of the small balls, from what they formerly did. The deflection and new stable position would be as much on the other side of the zero, and the arc described would be an accurate measure of the double deflection. The force of torsion for $1^\{\backslash circ\}$ of deflection is known by the time of oscillation of the lever and balls when free, and as the forces are exactly as the angles, the force corresponding to any displacement becomes known. Cavendish conducted the experiment with his usual patience, judgment, and success; he found the joint attraction of the small balls and large spheres to be about $\backslash frac\{1\}\{435\}$ of a grain, their centres being 8.85 inches apart, and he thence computed the density of the Earth to be 5.48 times that of water. Cavendish's paper is as usual a model of precision, lucidity, and conciseness. The attraction of the fixed parts of the apparatus is calculated and allowed for.

(158.)
Baily's result.

It would be difficult to mention in the whole range of physics a more beautiful and more important experiment. It has been repeated since by Reich of Freiberg and Baily of London. The former obtained 5.44, the latter 5.66 for the Earth's specific gravity;

this last result being worthy of much confidence from the extraordinary care taken to avoid errors and to obtain independent values of the quantities sought.¹

Maskelyne continued his zeal for the promotion of astronomy to the last. He superintended the publication of 45 annual volumes of the Nautical Almanac. He left the whole of the observatory work in perfect order, and the greater part printed. He had the well-earned satisfaction of finding his observations in request in every civilized country, the bases of the most useful tables, and the tests of the most advanced theories. He cultivated the friendly correspondence of astronomers in every country. Not given to change, he preserved the instruments and chief methods of the immortal Bradley; but sufficiently alive to the necessity of progress in the sciences, he introduced many simple but practical improvements in the art of observation. Even in his last years, satisfied that the celebrated quadrant of Bird was no longer the best instrument for its purpose, and was besides sensibly deteriorated by use, he adopted the circle instead (then recently come into notice, though first used more than a century before by Römer), and directed the construction of that by Troughton, though it was not placed at Greenwich until after his death, which occurred in 1811, in the 79th year of his age. His biographer Delambre mentions, that a considerable number of his posthumous memoirs were put into the hands of Professor Vince for publication. They have not, however, appeared.

Practical astronomy was on the Continent far behind its state in England at the period of which we speak. The various national observatories contributed comparatively little to the progress of science; but there were of course exceptions, a few of which we will here briefly notice.

The discovery of the four small planets² Ceres, Pallas, Juno, and Vesta, the first by Piazzi, the second and fourth by Olbers, the third by Harding, gave celebrity to those astronomers, of whom Piazzi and Olbers were farther distinguished by many important labours. To the latter we owe the discovery of several comets, and one of the best methods of calculating their orbits from observation. He was a person of much simplicity of manner, made his observations with the most unpretending means in the attic of his house, and died at a great age generally respected. His firmest friend was Baron von Zach, who had no slight share in the discovery of the new planets, owing to his having instigated the association of twenty-four astronomers, chiefly in Germany, for the express pur-

¹ The experiments of Cavendish are related in the "Philosophical Transactions" for 1798; those of Reich in a separate small work, entitled "Versuche über die mittlere Dichtigkeit der Erde," Freiberg, 1838; those of Baily in the "Memoirs of the Astronomical Society," vol. xiv. A still more recent series of experiments by Reich gives 5.58, a close approximation to Baily's result.—(Phil. Mag. March 1853.) From observations with a pendulum in Harton coal-pit in 1854, Mr. Airy has obtained a density higher than any of the above.

² In the Fifth Dissertation, page 789, Harding instead of Olbers is named as the discoverer of Vesta.

pose of ascertaining whether no planetary body filled the void between Mars and Jupiter. To Piazzi, of Palermo, we owe a most excellent catalogue of fixed stars from observations with a moveable circle of four feet radius by Ramsden. Oriani of Milan was likewise one of the best informed practical astronomers of his time.

In France, after the death of the celebrated Lalande, perhaps Lalande (who was exactly Maskelyne's contemporary) was the most active astronomer. To him and his nephew we owe a very valuable catalogue of 50,000 stars, lately edited by the British Association. But practical astronomy was seriously neglected in France generally. The national observatory was feebly superintended by the later members of the Cassini family; of the French expedition under Maupertuis to measure the length of a degree in Lapland, the Abbé Outhier alone, it is said, knew how to use a quadrant, and the celebrated Lagrange was as ill informed until instructed by Lalande.¹

(163.) French are astronomers at the close of the last, and at the commencement of the present century, were in carrying out the measurement of the arc of the meridian from Dunkirk to the Balearic Isles, with the more immediate object of fixing the length of the mètre, but contributing to the solution of far more considerable problems connected with the FIGURE OF THE EARTH. We connect this labour with the respectable name of Delambre, who was more intimately associated with it than perhaps any other person, though united with such eminent men as Méchain, Biot, and Arago.

(164.) DELAMBRE was the pupil of Lalande, who used to say that his disciple was his best work. He first obtained distinction as a computer of tables. Those of the motions of the Sun, Jupiter, Saturn, and Uranus, and of the satellites of Jupiter were deservedly prized, and some of them are still the best of their class. He was a man in whom the love of truth and accuracy was conspicuous. Learned and patient, he spared no pains in acquiring knowledge, and in using it to the best purpose. As a calculator he was eminent. Physical astronomy he did not cultivate, except with a view to compare its deductions with facts. He was intimately conversant with all properly astronomical methods and formulae. He knew the history of every problem, and the details and modifications of every astronomical instrument. He has embodied the results of this vast industry in a series of works (forming six quarto volumes) on the history of his favourite science, which are without a parallel for fulness and impartiality. He laboured as conscientiously to ascribe the due credit to Hipparchus and Ptolemy as to hold an equal scale between the merits of French and British astronomers.

His critical knowledge of the ancient languages (for he could speak Greek with fluency) was not more remarkable than his complete freedom from national prejudices. Both attributes qualified him pre-eminently for the office of an historian. He published also a large treatise on astronomy, and numerous memoirs on practical subjects in the Connaissance des Temps between 1788 and 1817.

Of his original labours the measurement of the French Arc of the meridian, of which he has given a full account in his Base du Système Métrique Décimal, is the most important. As some account of this undertaking has been given in Sir John Leslie's Dissertation, I shall state concisely a few particulars not there mentioned. Not the least singular feature of this gigantic work was the political crisis under which it was conducted. So early as August 1790, the French Constituent Assembly, on the motion of Talleyrand, desired the king to write to the English government, to represent the advantage of the two nations uniting to adopt a common unit or standard of weight and measure, which it was proposed should be done by a joint committee of the Royal Society and the French Academy.² This application was probably never made, at least nothing came of it; but the Academy named their own committee, who, after discussing three sorts of natural standards,—the length of the pendulum in lat. 45° (first proposed by Huygens in his Horologium Oscillatorium), the length of a quadrant of the equator, and that of a quadrant of the meridian from the equator to the pole assumed to be elliptic,—adopted the latter, and this labour was committed to Méchain for the southern part, from Rodez to Barcelona (170,000 toises), and to Delambre for the northern, from Rodez to Dunkirk (380,000 toises). The southern arc was afterwards extended to Formentera in the Balearic Isles, and the whole length of the arc was found astronomically to be 12° 22' 12". 6. Two bases were measured (both by Delambre), one at Perpignan of 6006 toises, the other at Melun of 6076 toises (each about 7.3 miles). When the length of the former was computed by triangulation from the latter, the difference of the observed and inferred amount is said to have been only ten or eleven inches.

But difficulties greater than physical obstacles presented themselves to the completion of this vast work. The excited state of the public mind during the most frenzied period of the French Revolution rendered the simplest operations matters of suspicion. The nearer to Paris that the survey was carried on, the greater were the precautions necessary. Instruments were seized, assistants arrested, and night signals had to be totally dispensed with. "The coolness and intrepidity of Delambre, added to unexampled patience,

¹ Both anecdotes are told by Moll, who had them from Delambre.

² The two governments and their respective learned societies had already co-operated in 1784 for a survey to connect Paris and Greenwich Observatories.

were the principal means of extricating him from his difficulties: but his danger was often imminent, and he appears to have sometimes heard the dreadful words which, as an eloquent author has expressed it, were the last sounds that vibrated in the ear of many an unhappy victim." The operations were actually suspended for a time by a decree of Robespierre and his colleagues, who deposed Delambre, along with Laplace, Lavoisier, Borda, and others, from the Commission of Weights and Measures, as being deficient in "republican virtues and their hatred of kings." They were, however, resumed, and Delambre had finished his share of the work long before his colleague Méchain, whose shorter task was conducted amidst a people rude and uneducated, indeed, yet far more to be trusted than were then those of the north. Méchain was apparently wayward and impracticable, somewhat too aged for so great a work, yet a really good astronomer. The want of agreement to within 3" of two sets of observations for latitude at Barcelona, the southern end of the arc at that time, led him to the suppression of one of them, and he was tormented ever after by the consciousness of the evasion, which deprived him of the tranquillity necessary to resume and complete his work, which was done chiefly by Delambre after vexatious delays.¹ The error, which may be said to have cost Méchain his life, was probably owing to the instrument employed on this survey, the repeating circle of Borda, only fourteen inches diameter, with a rather weak telescope. The opinion generally entertained in Britain is, that the repeating circle was quite inadequate to the prodigious accuracy required of it, especially in the determination of latitudes. The errors of mere division are often trivial compared to those inherent in other parts of an instrument. Of these a deficiency of optical power, and the want of absolute security of the clamps, upon which the entire success of the principle of repetition depends, are amongst the most obvious. The arc was finally prolonged from Barcelona to Formentera by Biot and Arago in 1806. The conclusion of the survey was not destitute of the adventurous character of its commencement. The French astronomers ran many risks, underwent much suffering, and Arago narrowly escaped finishing his days in the dungeons of Spain.

The English survey carried on by Roy and Mudge has been also noticed in the previous Dissertation. The arc from Dunmore to Burleigh Moor amounts to 3° 57' 13"·1, the measured length to 1442953 feet. An arc of parallel was also measured from Dover to Falmouth. We shall say something of its later progress in the concluding part of this essay, but we have still to regret the postponed publication of the British Arc of the Meridian, which we have no reason to doubt will bear a favourable comparison with the work of Delambre. The practical

appliances, the three-feet theodolite of Ramsden for horizontal angles, and the eight-feet zenith sector of the same artist for latitudes, were unequalled instruments, and contrasted in almost every respect with the light and portable apparatus of the French. By means of the former the spherical excess of terrestrial triangles was first observed as a fact. The results of the French and British arcs taken separately concurred in showing a local curvature in this part of the world altogether anomalous, the degrees rather shortening in the northern part of each arc. This fact, which must be imputed either to large local attractions, giving errors of several seconds in the determination of latitudes, or (less probably) to a local departure in our quarter of the world from the general or mean figure of the earth, sufficiently shows the futility of the proposed method of determining a natural, recoverable standard of length. When combined with the measures of Bouguer in South America and Lambton in India, and the revised arc (measured in the beginning of this century) of Swanberg and Melanderhjelm in Lapland, the French and English measures give a general ellipticity somewhat under $\backslash frac\{1\}\{230\}$, which is probably as near the truth as local inequalities admit of the determination being made.

To Delambre was confided the drawing up of the trigonometric formulæ used in the calculations of the survey, which were published in a separate work; De Prony conducting the laborious calculation of an altogether new set of logarithmic tables, with the aid of an immense staff of computers, the results of whose labour (still in MS.) are preserved at Paris in 17 folio volumes. Delambre carried his personal exertions so far as to compute his own triangles—which were also independently calculated by Legendre, Van Swinden, and Tralles.

As an acknowledgment of his merit, the highest indeed in their power to bestow, the Institute of France decreed to him in 1810 one of the Decennial Prizes instituted by Napoleon. But the Emperor, though professing to be the warm encourager of science, suffered some meaner motive to interfere, and refused to ratify the decision. "Ce fut," writes Dupin, "un pas dans la route qui le menait à sa chute." After the siege of Paris in 1814 Delambre wrote a characteristic letter to his friend Moll. The tranquil spirit which had braved the horrors of the Revolution was not to be moved by the sounds of the artillery of the allied armies. In spite of the cannonade which he heard from his library, he laboured from eight in the morning until midnight; and, conscious of rectitude, he feared little the revolution of circumstances, which changing dynasties might call forth. "Labour," he says, "occupies all my time and all my faculties."

As Secretary to the Academy of Sciences for the

¹ The history is minutely given by Delambre himself in his Biography of Méchain.—Astron. du XVIII. Siècle.

His impartiality—
his death.

Mathematical Sciences, he had to make frequent reports on the progress of those progressive branches of knowledge, and to contribute memoirs of the most eminent academicians, both native and foreign. Both these duties he executed with his customary labour and fidelity. His private character was as amiable as his public career was distinguished. The tendency of his writings was constantly to enforce the histori-

cal credibility of the Scriptures, especially with regard to the most ancient astronomical records. The value of his testimony is increased by his unusual skill in philology. He died 19th August 1822, aged 73.

We reserve some farther remarks on the geodetic results of this period, and especially on the pendulum observations with which they were accompanied, for the fourth section of this chapter. (171.)

§ 2. SIR WILLIAM HERSCHEL.—History of Sidereal and Telescopic Astronomy to 1820. Herschel as an Optician.—Planet Uranus—Solar Spots—Orbits of Double Stars—Nebulæ—The Milky Way—Sun's Motion in Space.

(172.)
Sir Wm.
Herschel;

The eighteenth century was not, generally speaking, distinguished by original observations. The exceptions stand out with all the brighter lustre. Amongst these, the discoveries of Sir William Herschel occupy perhaps the foremost place. Arago has affirmed that Slough is the spot on the earth's surface signalized by the most numerous discoveries, and in a certain sense this is strictly true. No one, before or since (with possibly the exception of Mr Hind) has added so many new bodies to the known planetary system, and this at a time when such discoveries were rarer, more unexpected, and more difficult than now. His researches on the fixed stars are not of a nature to have their importance numerically estimated.

(173.)
his exten-
sive works;

Sir William Herschel's career was one of the longest and of the most sustained labour in the history of science. To have contributed papers—often several in one year—to the Philosophical Transactions for thirty-nine consecutive years, from 1780 to 1818, with but two exceptions, is a feat of astonishing perseverance; but if we recollect that many of these papers contain announcements of capital discoveries, that every one of them is stored with original matter, and that the author had already passed his fortieth year when he commenced the production of this astronomical library, we cannot withhold a tribute of the warmest admiration.

(174.)
early his-
tory.

England cannot claim Herschel as her own, except by adoption. He was born at Hanover in 1738, and was one of a numerous family who supported themselves chiefly by their musical talents. William Herschel, the third son, came to England in 1759 with his elder brother, and after struggling with many difficulties, found himself in comparatively comfortable circumstances as an organist at Bath. In 1774 he had executed a reflecting telescope with his own hands, and soon acquired so much dexterity as to construct instruments of ten and twenty feet in focal

length. In 1780 he contributed his first paper on the variable star in Cetus to the Royal Society, and the following year (13th March 1781) discovered the erratic body, which he at first took for a comet, but which proved to be a planet exterior to Saturn; the first addition therefore to the number of the primary planets since a period of an immemorial antiquity. So fortunate a success made the name of Herschel speedily famous, and he was effectively befriended by George III., who brought him to live near Windsor, and gave him a pension.¹

From this brief sketch it will appear how great were the obstacles which Herschel had to vanquish before he became a man of science, and that, besides the claims to distinction already enumerated, his knowledge and his skill were acquired in spite of every disadvantage. (175.)

Practical Astronomy naturally divides itself into two great branches, that which depends upon the use of the telescope merely; on the telescope and micrometer; and that which determines the absolute places of the heavenly bodies, and requires the aids of divided instruments and a well furnished observatory. In the century to which he belonged Herschel is the type of the telescopic observer, Bradley of the instrumental. The discoveries of Aberration and of Nutation by the latter may stand a comparison with any in the history of science, but the resolution of Nebulæ and the proof of the mutual connection of stars in binary systems are not less distinguished and original. Herschel would have gained a great reputation as an optician, merely by the wonderful improvement which he effected in the dimensions and magnifying power of telescopes, and by the skill with which he applied them to celestial observations. He would have stood still higher as an astronomer had he been merely the first observer of a new planet and of eight secondary ones, as well as of several comets, and the author of many nice

¹ It may be doubted whether any other similar annuity was given at that period on scientific grounds alone, and it is difficult to estimate the amount of benefit thus conferred on astronomy, for nothing short of the entire devotion of a lifetime could have produced such results as we owe to Herschel. His proposal to name his planet after the Sovereign was a very natural expression of his gratitude at a period when no rule whatever existed on the subject. See an interesting letter from Herschel to Sir Joseph Banks (who was also a warm friend) in Weld's History of the Royal Society, ii. 148, note. Herschel was elected into the Royal Society 6th Dec. 1781.

observations on the physical appearance and constitution of the sun and planets; but his great glory was in sidereal astronomy, of which he laid almost the foundations, by means of discoveries which he fortunately lived long enough to see confirmed and enlarged beyond his reasonable hopes.

In this brief sketch of Herschel's achievements (which in his own department may be said to embrace those of his age, for he had hardly an imitator, and certainly no rival) we shall, in the first place, condense into small compass what is to be said on the two former heads—his merit as an optician and as a planetary observer.

Herschel as magnificient. To construct good telescopes is itself no easy art, especially if they are to be carried to a size and perfection previously unattained. To ascertain the best composition, whether of glass or metal, to melt and to cast it in the right way, is one branch of the art which may be called generally the chemical part. To fashion the lens or mirror correctly by grinding, and to fit it for optical use by giving it an exquisite polish, is a second step requiring a peculiar mechanical skill and perseverance. To mount the telescope effectively is another and entirely different problem in mechanics; and all these the amateur astronomer must be prepared to accomplish with his own hands, unless he command the services of practical opticians to an extent rarely to be bought. Amateurs have indeed seldom succeeded except with reflecting instruments, and Herschel acted very wisely in devoting himself to their improvement at a time when the success of Dollond with achromatic telescopes had rendered the Newtonian form unfashionable. The real secret of Herschel's success was his astonishing perseverance; his determination was to obtain telescopes of twenty feet focal length or more, and of a perfection equal or superior to that of the small ones then in use. He himself relates, that whilst at Bath he had constructed 200 specula of seven feet focus, 150 of ten feet, and about 80 of twenty feet; a proof of extraordinary resolution in a man of limited means and engaged in a laborious profession. By making so great a number he could select those having the most perfect figure, especially before he had contrived a method of obtaining mechanically a parabolic form, almost with certainty. This method was not divulged. He was justified in keeping his secret, whilst he made a handsome income from the manufacture of telescopes for sale. Lord Rosse and Mr Lassell have had the merit of publishing their methods, which by their results would appear to be at least equal to Herschel's.

Magnifying and illuminating power of telescopes. The usefulness of telescopes depends on two distinct qualities—magnifying power and illuminating power. The former may be gained indefinitely by

diminishing the focal distance of the eye-glass; the latter can be had only by increasing the diameter of the object-glass or speculum, so as to collect the rays which fall upon a circle of large diameter from every point of the object examined. But in order that the whole light may be effective, the magnifying power of the eye-glass must be sufficient to condense the emergent pencil within the diameter of the pupil of the eye. The proportion in which these two qualifications are requisite to obtain the best results in astronomy is a matter involving nice questions both of theory and practice. Sir William Herschel did more than any one who preceded or followed him to solve them. On the whole, the illuminating power is perhaps the most important, whilst it is the most difficult of attainment; but in truth each class of telescopic objects have their own rule in this respect. Brilliant objects, such as Venus, and moderately bright stars, do not require large apertures; very feeble objects, as the remoter satellites and the nebulae, require indispensably great illumination. Five satellites of Saturn were seen by Sir W. Herschel in his great telescope by the naked eye alone, that is, without the advantage of the magnifying power of a lens. On the other hand, Sir John Herschel tells us that the satellites of Uranus cannot be clearly made out without a magnifying power of at least 300, whatever may be the aperture of the telescope.¹

The mention of magnifying powers is calculated to mislead rather than otherwise in comparing the efficient means of different astronomers. The magnifying power used by Galileo did not exceed thirty-two times the diameter. Huygens, who used aerial telescopes of immense length, pushed it to 163 times. Auzout is said to have given a power of 500 to one of these gigantic instruments of 300 feet in length; but it is easily understood that such achievements were little more than nominal, the mechanical difficulty of managing these instruments being excessive. Short professed to carry the magnifying powers of his Gregorian reflectors to 1200, but still they yielded only trifling fruits to astronomy. It required that such instruments should be made and used familiarly, not as objects of luxury of which but one or two were ever brought into actual use. Herschel applied even to his seven-feet Newtonian (his favourite and smallest working size, having 6.3 inches aperture) powers exceeding 2000. To his largest instrument (thirty-nine feet focus) was occasionally applied a power of 6500. The highest powers usefully employed with the gigantic achromatics of Pulkowa and of Cambridge (in America), do not exceed 1822 and 2004 times respectively.

The motion of the earth and the troubled state of

¹ Struve found that with a small telescope, magnifying three times, and having the same aperture with the pupil of the eye (0.2 inch), he could count nearly twice as many stars (accurately 1.83 times) as with the naked eye. This result, interesting on several accounts, deserves farther enquiry.—Études d'Astron. Stellaire, p. 85.

the atmosphere, especially in these climates, impose a speedy limit to the increase of magnifying power, unless in exceptional circumstances. The same objections do not apply to the increased aperture and illuminating power of telescopes. This consideration induced Herschel to construct his forty-foot reflector, the funds being advanced by the king, on the recommendation of the Royal Society. The speculum had a useful surface precisely four feet in diameter, the thickness three and a half inches, the weight when cast above 2000 pounds. The second or plane mirror having been dispensed with, the image was thrown to one side of the object-end of the tube by means of a very slight inclination of the speculum, and was there observed directly by a common eye-piece or single lens, the observer being thus stationed with his back to the object he is viewing. This vast instrument collected a cone of rays from a distant point between six and seven times larger than the twenty-foot instrument commonly employed by Sir W. Herschel, of which the aperture was 18.8 inches. This telescope was not frequently used, partly from the rarity of sufficiently steady weather, and from the difficulty of preserving the figure of the mirror under changes of temperature.¹ Its figure was also affected by flexure under its own enormous weight, and it has been found one of the greatest difficulties by those who have followed Herschel's steps to avoid this source of error. The forty-foot speculum has been religiously preserved by the filial care of Sir John Herschel, in whose possession the writer of these pages had the pleasure of seeing it, now many years since, in a state of high polish. It remains in a deposit hermetically closed, at Slough.²

(182.)
Space-penetrating-power of telescopes.

Herschel made an ingenious determination of what he called the space-penetrating-power of telescopes, by which he discriminated the relative distances at which a given fixed star would become invisible in his several instruments. It was grounded on the assumption that the visibility of the object depends, first, on the density of its light reaching the eye, and secondly, on the number of rays of that density concentrated in the image. The former quantity varies inversely as the square of the given distance, the latter directly as the square of the aperture of the telescope; the limiting visible distance therefore of the radiant body will be simply as the aperture. To this he applied corrections depending on the loss of light by reflection at the specula, and by transmission through one or more eye-glasses. On this scale the space-penetrating-power of the seven-foot Newtonian being twenty, that of the twenty-foot telescope (front view) was seventy-five, and the forty-foot telescope 192. These estimations, as we shall see, form an important step in Herschel's generalizations.

Of new bodies belonging to our system he dis-

covered Uranus and his six satellites, the two satellites of Saturn next to the ring, and several comets. He did not, so far as we know, even suspect the planetary character of Uranus, which he believed to be a comet. The honour of this appears to be due partly to Saron of Paris, partly to Lexell of St Petersburg. Arago supposes that Laplace had also a share, but the evidence on the whole matter is somewhat obscure. Unquestionably it was an exceedingly difficult matter to fix upon a correct orbit for a body moving with such extreme slowness within the limits of the first few weeks after its discovery, and its distance was far beyond what any one calculating a comet's path would readily assume. The original discovery was also a fortunate one, for but eleven days previously the planet was in a position apparently stationary. It has led to results of still higher interest by the prediction of the existence and place of Neptune from the irregularities of Uranus. The satellites move almost perpendicularly to the ecliptic. Strange to say, they remained for, I believe, more than half a century, unseen by any eye but that of their discoverer and his son: and two of them still stand recorded on the single authority of Sir W. Herschel. They are the least specks of light which optical power has ever made visible.

Herschel made many observations on the physical appearance of the sun, moon, and nearer planets, and the times of their rotation. Into these details we cannot now minutely enter. The papers which record them are characterized (like all those of the same author) by the faithful minuteness with which the observations are detailed, and by the mention of every influential circumstance by which the results might be affected. In the case of observations so exceedingly delicate as those of the planets, the astronomer has to keep up an incessant struggle between the anxiety to record all that he sees, and that of recording nothing more than he really and perfectly sees. One or two glimpses are not sufficient. Herschel at one time imagined that the new planet had one or more rings which unquestionably do not exist. His elaborate investigation of the figure, belts, and rings of Saturn, the incredible tenuity of the latter, and the fact of their rotation in 10^h 32^m, were amongst the most interesting of his observations, and were illustrated by drawings made with remarkable care. The edge of the ring was seen in 1789, when it had totally disappeared in all but his forty-foot telescope, and he estimated that its thickness could not exceed 100 miles. How astonishing the magnitude and thinness of this stupendous circular plane, or rather series of flat rings! Some idea of it may be deduced from the fact that if the thickness of the paper on which this is printed be taken to represent that of the rings, their greatest diameter would correspond to nine inches. Observa-

¹ The first object viewed by Sir W. Herschel with his great telescope in 1789 was the nebula in Orion; possibly it was also the last, for he again observed it in 1811 (See Phil. Tr. 1811, p. 322).

² See Weld's History of the Royal Society, II., 193.

tions on the other planets and on the moon we here pass over for want of space.

(185.)
on the Sun:
I have reserved the observations on the nature of the sun to this place, because everything leads us to assimilate the nature of the sun and of the fixed stars. The belief that the luminous disk of the sun is a photosphere or luminous atmosphere of great tenuity surrounding a globe of comparative density and darkness, was long anterior to Herschel, and in fact due to Dr Patrick Wilson of Glasgow, whose admirable paper on this subject was published in the Philosophical Transactions for 1774, in which he explains the phenomena of the solar spots by apertures in the luminous atmosphere, discovering the dark nucleus below, and some shell or shells of intermediate brightness which form the penumbra.¹ These conclusions were most clearly and ably deduced from a careful observation of the changing aspect of the spots, as they move by the solar rotation from the centre to the edge of his disk. It is to be regretted that Herschel does not more pointedly refer to the discoveries of Wilson, which were more than twenty years antecedent to his first paper on the subject, and with which he could hardly fail to have been acquainted. A similar remark applies, in a less degree, to his papers generally, which rarely contain references to the observations and speculations of his predecessors. Herschel adopted Wilson's hypothesis almost literally, and his long series of patient observations on the sun, made with high powers and at an eminent risk to his eyesight, enabled him to classify the singularly varied appearances of that wonderful orb, and to draw some probable conclusions from the excessive rapidity and seeming tumult of the exterior portions of it. That the photosphere is strictly gaseous he rendered very probable, an inference confirmed by the direct observations of Arago as to the unpolarized character of its light. The singular disclosure of faint red prominences extending far beyond the disk, and observed in the total eclipses of 1842 and 1851, shows that there is still much which regards the mysterious nature of the sun within reach of direct observation; and the same may be observed of the direct experiments lately made on the heat and light of different parts of the disk, which diminish to one-half between the centre and the edge, and appear to attain a maximum at the solar equator.

(188.)
Herschel's
sidereal
discoveries.
A convenient, though not a strictly chronological arrangement of Sir W. Herschel's more important sidereal discoveries and speculations may be made under the following heads:—

I. Of double Stars and their mutual connection.

II. Of the Nature of Nebulae, and the so-called Nebular Hypothesis.

III. Of the Grouping of the Stars generally in space, and the significance of the Milky Way.

IV. Of the Motion of our System in space.

I. On Double Stars. Discovery of Binary Systems.—Double stars were noticed as objects of curiosity even before the discovery of the telescope. The group of the Pleiades attracted attention from the earliest times. Amongst the earliest double stars carefully observed were $\backslash xi$ Ursæ Majoris (by Kirch, 1700); $\backslash alpha$ Centauri in 1709; $\backslash gamma$ Virginis and Castor by Bradley (1718 and 1719); Mayer made a considerable catalogue of double stars in 1756. But Lambert first announced in 1761 (in his Lettres Cosmologiques) the probability of the mutual revolution of suns, in these remarkable words (speaking of clusters of stars), "It will perhaps be decided whether there are not fixed stars which make their revolutions in no long periods round their common centres of gravity." Mitchell, in 1767 and 1784, maintained the same views, but supported them by an application of the then young science of probability, hazardous in its principle, and unquestionably wrong in its numerical solution.²

Sir W. Herschel commenced his observations on double stars with the hope of ascertaining the Annual Parallax in the manner previously indicated by Galileo and James Gregory; but, as in many parallel instances, whilst he failed of his main result, he discovered unsought a phenomenon more unexpected and probably more interesting. With the micrometrical means at his disposal, he entirely failed in detecting any semi-annual fluctuation of the interval between the members of the pair of stars, but he found (in some instances) progressive and continually increasing changes both in the relative position and distance of the two. He commenced his observations in 1779, but it was not until 1802 that he thought himself entitled to announce with confidence his discovery of the circulation of one sun round another, or rather of both round their common centre of gravity. Herschel's first list of orbital stars (Phil. Trans. 1803, where this splendid discovery was first published³) includes the chief examples now known; and they have all been confirmed. That of which the revolution is most rapid is $\backslash zeta$ Herculis, which has a period of 31½ years, and consequently has revolved twice round since it was first observed, whilst the slow planet Uranus has not yet returned to the position of its first discovery. Herschel does not appear to have received a medal or other public recognition of this signal success.

Of the subsequent progress of this interesting inquiry we shall speak in the latter part of this chapter. (189.)

¹ Flamsteed appears to have entertained in 1681 a somewhat similar opinion, as we find from a letter published (1855) in Sir David Brewster's Life of Newton, vol. ii., p. 103.

² See art. (85) and note.

³ It was in fact distinctly though incidentally announced in his paper of 1802, though the proofs were given the following year. See Phil. Trans. 1802.

Greater precision has been in some respects attained, but little, if anything, added to the theory of double stars.

II. On Nebulae and the (so-called) Nebular Hypothesis.—The cloudy patches of mild light which are visible in clear but moonless nights in various parts of the sky are called nebulae from their resemblance to light vapours moderately illuminated. Some regions of the sky abound more with them than others. The Milky Way presents to the naked eye a nebulous appearance extending over large tracts, and nebulae more or less dense are frequent within and near its borders. These facts were well known from an early period. Some conspicuous nebulae were described in the sixteenth century; that in Andromeda by Marius and the conspicuous nebula in Orion by Huygens. Halley added several, but the first descriptive catalogue of these objects was published by Messier in 1783 (in the Connaissance des Temps), and it forms the basis of the modern observations.

Sir W. Herschel took up the subject with his customary zeal. In 1786 he published a catalogue of 1000 nebulae, which he had increased by 1802 to 2500. The first step to some appreciation of the nature of these extraordinary appearances was of course to classify them, and this Herschel did with such success that it may be affirmed that his classes remain the best at the present time, as generally representing the phenomena. Without entering into all his details, the following division embraces the main facts:—

A. Clusters of stars.

B. Nebulae proper.

α. Having a certain regularity, as 1. Circular or planetary nebulae; 2. elliptic; 3. annular.

β. Wholly irregular, as the nebula of Orion, the Magellanic clouds, &c.

C. Nebulous stars.

Remarks. A number of faint stars produces a nebulous haze. This is a physiological fact. Parts of the Milky Way which present this appearance to the naked eye are at once shown by a telescope of small power to be composed of stars.¹ Patches which appear nebulous in such a telescope are shown by a better one to be composed of small stars. This is called the Resolution of a nebula. Such resolutions become more and more numerous as the apertures of telescopes are enlarged. Lord Rosse's telescope has resolved several which withstood the highest power of Herschel's. One inference might therefore be that all nebulae are clustered stars, at a greater or less distance. Such an opinion is very general, and very plausible. It may on the other hand be observed that Herschel abandoned this his earlier inference,

whilst, at the same time, he proceeded diligently with the resolution of successive nebulae, by means of his powerful telescopes. He maintained on the contrary to the last that nebulae belong to two classes, resolvable and irresolvable. The latter he believed to be composed of diffuse self-luminous matter, more or less condensed, whilst the milky light of the former is occasioned by the blended gleam of numerous close stars. With this difference, we find his views regarding the changes in the sidereal system and the progress of the condensation and breaking up of nebulae, nearly the same in his earlier and later papers, particularly those of 1785 and 1811.

The opinions of Herschel on these changes were the following:—Having observed that the more regular nebulae (whether with a central star or not) presented almost invariably an appearance of gradually increasing brilliancy towards its centre, he was led to conclude that this is due to the aggregation of the self-luminous parts in virtue of their mutual gravitation; and the conclusion would so far be the same, whether the parts thus coalescing were in fact numerous distinct masses, or were like a vapour diffused through space. This was the process of condensation, which he inferred on the one hand from the law of gravity, on the other from the generally symmetrical arrangement of the luminous matter in nebulae of the first class. In order to detect, by actual observation, the progress of condensation, he invented a simple and highly expressive terminology, indicating the general brightness, the amount of its gradation, and the law of its gradation from the circumference towards the centre. In whatever degree Lord Rosse's late observations may be considered to deprive nebulae of the character of symmetry which Herschel had conferred on them, the general fact of an apparent condensation in a great majority of instances cannot be denied.

The breaking up of nebulae (also described in his paper of 1785, as well as in his later ones) is likewise an inferred change anticipated in large irregular nebulae, which usually exhibit bright spots or centres, with irregular vacant spaces, which sometimes seem like portions of a black and distant profundity seen through the rents of a nebulous veil. Herschel supposes that the luminous centres are also places of greatest attraction, and must gradually draw away the star-like substance from the rarer spaces.

It is curious that these speculations of Herschel (which are substantially common to his papers of 1785 and 1811) had been at least imagined at a very early period. Arago has pointed out that Tycho and Kepler conceived the New Stars of 1572 and 1604 to have resulted from a sudden condensation

¹ The group of the Pleiades presents a peculiar instance. When the eye is turned full upon it, no more than six stars (as is well known) can be counted, but when the eye is turned aside through 10° or 15°, by a peculiarity of indirect vision by means of the lateral parts of the retina, the group appears a mass of nebulous light. This arises from an immense crowd of stars just below visibility, which, probably from the greater sensitiveness of this part of the eye, make a sensible though somewhat indistinct impression. This observation as to the Pleiades, and the inference which explains by it the seeming countlessness of the stars when viewed cursorily, were published by the present writer in Brewster's Journal for 1826.

of diffuse celestial matter, such as the Milky Way exhibits, and Tycho alleged that the new star was surrounded by an obscure space, half as large as the moon's disk. The history of Science scarcely presents a more curious anticipation.

(196.) Herschel's opinions on the distance of nebulae;
One practical difference between the earlier and later views of Herschel on the nature of nebulae, was as to their distance from our system. For if the milky aspect be due to a confusion of small stars, it is assumed that they must be almost incredibly distant. In his memoir of 1785, he estimates this distance at not less than 6000 or 8000 times that of Sirius. If, on the other hand, the nebulous appearance be due to a diffusion of starry matter, the distance may be that of any order of fixed stars; and this opinion derives weight from the evident connection of some nebulae with stars round which they seem to cluster or hang almost like a drapery. Thus in 1811 he supposed that the nebula in Orion need not be more distant than a star of the 7th or 8th magnitude.

(197.) and their changes of form.
It appears from his paper of 1802 (page 500), that his ideas about nebulae were then wavering. Between that time and 1811 he elaborated the nebular hypothesis, so far as it is due to him. He seems to suppose that starry matter was once in a state of indefinite diffusion. That during "an eternity of past duration" (Ph. Tr., 1811, p. 287) it has been "breaking up" by condensation toward centres more or less remote. That the Milky Way—or at least the nebulous parts which it contained—and the dispersed nebulae, are the relics of this former state of things. That where condensation has gone on more energetically, we have nebulae with a gradually or rapidly increasing brightness towards the centre;—if still more energetic, a nucleus, or it may be a planetary nebula;—next a nebulous star, which he supposes our sun to be, and the zodiacal light a relic of its nebula (p. 311);—finally, the completely formed stars may be assumed to be merely consolidated nebulae. (See pp. 284, 285, 299, 310.) This condensation, he believes, must be accompanied by rotation due to the originally irregular distribution of the gravitating particles (p. 312–319).

(198.) Theory of condensation.
The proofs on which the whole of this cosmogony rests are, 1st, the gradation of appearances above described, to be collected from distinct objects in the heavens; and, 2dly, supposed changes observed by him in the nebula of Orion and others, during thirty years, and even during intervals of a few years. As to the last argument, it is admitted, we believe, by the best authorities, to be without weight. The changes of aspect of such curiously faint and graduated objects, even to the same eye using the same telescope, are numberless, and may be due to the slightest atmospheric and physiological influences. This test (as applied hitherto) is therefore generally rejected. Against the first proof much might also be urged, for

which we cannot afford space. Its idea is essentially derived from the Natural History sciences, and I cannot help thinking that Herschel must have derived it from some one more conversant with these than with mechanical physics. This opinion is confirmed by a curious illustration which he uses in the same paper of 1811. "There is perhaps not so much difference between them" (viz., the whole group of nebular and quasi-nebular phenomena), "if I may use the comparison, as there would be in an annual description of the human figure, were it given from the birth of a child until he became a man in his prime" (p. 271).

A theory or hypothesis in some respects similar had been previously expounded by Laplace in the concluding chapter of his Système du Monde. At least it so far resembles it, that he puts forth Tycho's notion of the star of 1572 being a condensation of a widely-spread stellar atmosphere (without, however, naming Tycho), and supposes that our solar atmosphere might also once have extended to the limit of the system, and that the planets were thrown off successively in the form of nebulous rings (subsequently condensed into spheres) from the equatorial parts of this vast revolving mass during its contraction. He ascribes to Saturn's ring a like origin. It required all the authority of a name like that of Laplace to circulate a theory so bold, if not extravagant. He adhered, however (at that time), to the old idea of the constitution of nebulae, which he considered to be composed entirely of stars, and to be at once the most distant and the most massive aggregations of matter in the universe. Herschel could hardly fail of being acquainted in 1811 with Laplace's views, and they probably in some degree influenced his own, particularly as to the cause of the rotation of his condensing suns. But he prudently reserved his hypothesis for sidereal objects, and did not deduce from it a planetary cosmogony. In the later editions of the Système du Monde, Laplace quotes Herschel's observations in confirmation of his views, adopts the notion of a diffuse nebulous matter, and considers that an hypothesis arrived at by a "remarkable coincidence" in opposite directions, is thereby invested with a great probability.

Even Herschel's more limited conclusions have been very dubiously received, and notwithstanding the weighty adhesion of Humboldt and Arago to them, it may be affirmed that natural philosophers generally are content to leave the solution of this cosmical problem to that distant posterity which alone can hope to witness unequivocal evidence of progressive change in those wonderful objects.¹

III. Of the Grouping of the Stars generally in Space, and the Significance of the Milky Way.—The earlier astronomers, and particularly Halley, had the idea that the nebulae—which abound so remarkably in the Milky Way in particular—

¹ The most intelligible account of the nebular hypothesis, in its least objectionable form, will be found in Sir John Herschel's Outlines of Astronomy, Articles 870 to 872.

ably in the Milky Way, and which may in some sense be almost said to compose it—are the parent-matter of stars, whilst stars abound in their neighbourhood, or seem partly to compose them, in a state of incredible aggregation. That the mass of sidereal matter composing the Milky Way (which forms an irregular girdle round the whole heavens¹) has some special reference in its locality to our solar system, was also an early idea, which, towards the middle of the 18th century, had attained to something like a definite speculation in the minds of Kant, Lambert, and other philosophers, chiefly those of Germany. Its origin is perhaps traceable to Thomas Wright of Durham² (1742), who affirmed that the Milky Way is a projection on the sphere of a stratum of stars, in the midst of which our Sun and system are placed—somewhat eccentrically, however, towards Sirius.

I do not know whether Sir W. Herschel derived his first ideas on the subject either from English or German authors. His customary silence (which is to be regretted) on the history of his discoveries, leaves us in doubt of his originality. Be this as it may, he proceeded to test the reasonableness of the speculation, in that definite and thorough manner which was very characteristic of his genius. He proceeded to number the stars after a fashion which, if the early attempt of Hipparchus was pronounced in its day impious, might well have been accounted, even in Herschel's age, impossible. To use his own bold expression, he gauged the heavens, by counting the whole number of stars visible in the field of his 20-feet reflector as many times as possible, and in every region of the sky which was visible from Slough, distinguishing the results of each region. As it was impossible in one lifetime to accomplish the entire survey (for the complete sphere contains 883,000 fields of 15' each in diameter), he took an average for each region, and thus determined the general population of the sky in all directions. The result more than confirmed the suspicions of Wright and Lambert, and demonstrated a remarkable, and, on the whole, a steady law of decrease, from the central zone of the Milky Way in opposite directions, until we reach two poles, one in the Southern, the other in the Northern hemisphere, which are the localities poorest in stars.

This DISCOVERY (for it well deserves the name), which assigns a Law to the distribution of the entire visible bodies of the Universe in space, is surely one of the greatest which man has ever attained. The

simplicity of the means by which it was reached, and the amount of mere mechanical and arithmetical labour which forms its sole basis, and constitutes its essence, is a powerful lesson of how little mankind has a right to scorn the humblest means of knowledge, or to attach the prerogative of genius only to the lofty flights of imagination and speculative philosophy.³ This triumph belongs to Herschel alone; his successors have commented upon and scrutinized his results; his son has "gauged" the Southern Hemisphere, with like results; but the original result is substantially confirmed. We may therefore anticipate so far as to quote the numbers denoting the "plenitude" of stars as we recede from the equator of the Milky Way in either direction, first, as calculated by Struve⁴ and Sir John Herschel, from Sir William Herschel's observations in the Northern hemisphere; secondly, as calculated for the Southern hemisphere by Sir J. Herschel, from his own observations at the Cape of Good Hope.

This table shows the extreme condensation of the stars round the "Galactic" equator; indeed Struve carries the average "plenitude" of the field as high as 122 stars in that plane itself.

According to Herschel's views, the whole visible firmament of stars belongs to the nebula of the Milky Way, of which our own system forms an infinitesimal part. In his paper of 1785, on "the Constitution of the Heavens," he inferred that the general form and comparative limits of this cluster of stars might be found from his "gauges," by supposing that his telescope brought into view every star between the spectator and the surface of the cluster. Supposing the stars to be uniformly distributed within the cluster, it is easy to see that the number counted in a conical space, having an angle of 15' at its vertex, would give the height of the cone, or the depth to which the spectator is immersed in the cluster, in that particular direction; and thus he endeavours to assign the solid form of the cluster and its dimensions, in terms of the distances

¹ Its plane is inclined about 63° to the Celestial Equator: its nodes are in 0^h 47^m and 12^h 47^m of Right Ascension.

² In his Synopsis of the Universe, a rare work. See Professor De Morgan's account of Wright, in the Philos. Magazine for 1848, xxxii., p. 241. Wright believed that the Milky Way was a congeries of stars of a particular form, which it owes to a rotation of the whole round some central point.

³ Another remark, perhaps, deserves to be recorded. Had Hipparchus and his followers been deterred by the threatened charge of impiety from making a catalogue of stars, how great the loss would have been, not to science alone, but to our estimate of the order and magnificence of creation. The telescope only increased our knowledge of the number of worlds, without really enabling us to approach to their limits:—the gauges of Herschel, which he at first supposed to reach to the boundary of the visible firmament, proved to him at last that the wealth of creation was unfathomable by any measure he could apply to it. "Numbers numberless" appeared in those regions where the stratum of the Milky Way is most extended.

⁴ Etudes d'Astronomie Stellaire.

(hypothetically uniform) between any two stars. But in his later memoirs, Herschel seems to have felt the insecurity of his conclusions, and that there is no reason, but the contrary, to believe in a uniform distribution of the stars; that, at least in some directions, the Milky Way is unfathomable, or its limits unattained even by the 40-feet telescope. (See his Memoir of 1817.) He therefore judiciously recurs to photometrical measures as the best criterion of the distance of the stars. The estimate of "the space-penetrating power" of his telescopes, already referred to (182), enabled him to infer the relative distances at which a star would become lost to vision aided by each telescope.

(205.) Calling the distance of a star of the first magnitude 1, and supposing that light takes three years to traverse it,¹—

Yet none of these telescopes appear to touch the boundary of our nebula in the direction of its greatest extension, for fresh optical power still yields fresh harvests of stars.

(206.) In a direction perpendicular to the plane of the Milky Way, not only is the paucity of stars remarkable, but we seem to look out, as Sir John Herschel describes it, into a starless region external to the stratum. Sir W. Herschel concluded, in 1817, that the depth of the stratum in its least dimension is such as to include all stars down to those three times more distant than the naked eye perceives. His son, 20 years later, infers, from the investigation of the southern hemisphere, that the distribution of stars is comparatively independent of direction for all those which are brighter than the 8th magnitude, and that it is by the influence of the hosts of lesser brightness visible in the 20-feet reflector, that the very remarkable law of condensation towards the plane of the

Milky Way takes effect. He is therefore disposed to conclude that "our system is plunged in the sidereal stratum constituting the Galaxy, reckoning from the southern surface to a distance which corresponds to the light of a star of the 9th or 10th magnitude."²

These great and arduous enquiries occupied Sir William Herschel during nearly the whole of his scientific career, extending to almost half a century. His first observations (Struve remarks) dated from 1774, his last observations on this subject which have been published were of 1804, and his latest paper, still on the same topic, was in 1821, when he was almost on the verge of the grave.

Excepting the continuation of his labours by his illustrious son, little has been added to our knowledge of "the constitution of the Heavens" since his death. We shall, therefore, hardly recur to the topic in the latter part of this chapter. Some German astronomers of eminence have indeed tried to deduce from an assumed distribution of the stars in the Milky Way, the necessity of allowing that space is not perfectly transparent, and have even attempted to assign the law of extinction of light through the sidereal intervals; but having examined their reasoning with the best care I have been able to give it, I am bound to say that it appears to me very far short of conclusive.³

IV. Of the Motion of our System in Space.—That the stars are not fixed, but have "proper motions," not explicable by the gyrations of our earth round its centre of gravity (which produce precession and nutation), nor by its revolution round the sun (on which aberration and parallax depend), was a fact first detected by Halley from a comparison of ancient and modern observations. Though small in amount, yet as it continually increases in the same direction, this Proper Motion becomes in the course of ages an important quantity. In 2000 years, Arcturus, $\backslash mu$ Casiopeie and 61 Cygni have moved $2\backslash frac\{1\}\{2\}$, $3\backslash frac\{1\}\{2\}$, and 6 times the moon's diameter.

In 1748, Bradley (in his paper on Nutation) stated the probable explanation of these apparent motions

¹ These startling results appear to be necessarily true, if the following assumptions be admitted:—(1.) That the stars have on an average the same brilliancy in all regions of space. (2.) That the light received from a star is inversely as the square of the distance; or that the laws of radiation and of vis viva are rigorously true, and that there is no absorption of light in space. (3.) That the visibility of a star is exactly as the quantity of light received from it in the telescope. Probably this last assumption is the least sure. We know that magnifying power (independently of illumination) affects the result.

² Cape Observations, p. 383.

³ I ought perhaps to state that in this passage I have referred to the reasonings of M. Struve, contained in the latter part of his Études d'Astronomie Stellaire. Having taken some pains to follow the argument of that work, which professes to be "based entirely upon observation, without any arbitrary hypothesis," I was reluctantly led to the conclusion that M. Struve's assumptions, if tacit, were not the less arbitrary and questionable. Indeed it appears self-evident that no geometrical certainty can be attained as to the relative distances of the stars composing the Milky Way, but from some fundamental hypothesis respecting their magnitudes and distribution. I am persuaded that the popular writers and reviewers who have given additional publicity to the most striking and positive of M. Struve's conclusions, have (very naturally) done so on the strength of the author's well-deserved reputation as an observer, and without attempting to analyse his reasoning, which it must be owned is sometimes obscure. My objections to M. Struve's argument were put in writing several years ago (1850), but not published except in my lectures. It was only whilst correcting the proofs of these sheets (1855) that I saw for the first time a memoir by Professor Encke in the Astronomische Nachrichten, vol. xxvi., No. 622 (published in 1848), maintaining the same view of the invalidity of M. Struve's reasoning, and questioning the hypotheses (of which M. Encke reckons tacitly assumed by him. The Dutch Academy of Sciences (Haarlem) has proposed the points at issue between M. Struve and Encke as the subject of a prize question.

to be the real displacement of our sun and system in space. In 1760 Mayer gave a list of 80 stars whose proper motions he determined by a comparison of Römer's observations. He conceived the possibility of determining the direction of the solar motion by the changing aspect of the constellations, keeping in view, however, that the effect is complicated by real independent movements of each individual star. Lambert and Mitchell entertained similar views, but carried them to no practical result.

Here, then, Herschel appears on the field. He proceeded in 1783 to do what his predecessors had discussed the possibility of doing. His own observations were not available in a problem like this. Taking the best values he could obtain of the Proper Motions of only about 20 stars, he, with great sagacity and success, divined that the solar system is moving somewhere in the direction of $\backslash lambda$ Herculis, a point in the heavens whose right ascension is $257^\backslash circ$, and north declination $25^\backslash circ$. This result he arrived at, almost without calculation, by a mere consideration of the changing perspective of a multitude of objects amongst which the spectator moves. Mayer had compared the change to that seen when we walk slowly through a grove of rather distant trees. Those in front will seem to widen or expand as we approach, those behind will condense towards the point we are quitting. The greatest proper motions will be in the objects on our right and left as we advance. It is not to be supposed that we find in the stars the exact counterpart of this change of place. For each star may by its own absolute movement in space either conspire with or oppose the motion due to perspective. But a combination of the whole results will make apparent the direction of that part of the motion due to a common cause, that is, to the translation of our system in space. A cleverer approximation than Herschel's was never made. He returned to the subject in 1805, but there is reason to think that his first result was the more correct.

As nothing essential has been added to Herschel's discovery of the direction of the solar motion, we shall here refer once for all to the important confirmations which it has since received. Professor Argelander, availing himself in 1835 of the improved state of astronomy since Bradley's observations were made, and since their reduction by Bessel, considered the whole problem in a general and geometrical manner, including every well-determined proper motion¹ by means of appropriate equations of condition, which, being resolved by the method of least squares,

give the direction of solar motion $261^\backslash circ\; 11\text{'}$ of right ascension, and $30^\backslash circ\; 58\text{'}$ of north declination, differing respectively about $4^\backslash circ$ and $6^\backslash circ$ from Herschel's first numbers. Perhaps a still more convincing confirmation was obtained by Mr Galloway (Phil. Trans. 1847), from the proper motions of stars of the southern hemisphere alone, which lead to a result nearly coinciding with the above.

Sir William Herschel's leading discoveries in Side-real Astronomy may therefore be reduced to these—
the discovery of binary systems of stars and the orbits of several revolving stars; the discovery and classification of a prodigious multitude of nebulae; the law of grouping of the entire visible firmament, and its connection with the great nebula of the Milky Way; and lastly, the determination of the fact of the motion of our sun and system in space, and the direction of that motion. Setting aside all that is valuable, ingenious, and noble in his farther speculations, and all that he contributed to the enlargement of our knowledge of the system of sun and planets with which we are more immediately connected, these positive discoveries will ever remain a magnificent trophy of his perseverance and success as a natural philosopher.²

When his increasing years rendered the relaxation of his arduous course of telescopic observation advisable, whilst all his other faculties were in the highest vigour, he began to devote more attention to physical enquiries less directly connected with astronomy. The nature of the emanations of light and heat proceeding from the sun was a matter on which he was forced to speculate in connection with his ideas of the sun's constitution; and even the practical enquiry as to the kind of dark glasses best fitted for defending the eye in observations of that nature, led him, in conformity with his usual habit, to a large series of experiments on solar heat, and its transmissibility through glasses of different colours, and other bodies. In the progress of these he was led to examine the heating power of the different rays of the spectrum, and consequently the refrangibility of heat. He then arrived at the real discovery that the place of greatest heat in a spectrum formed from the sun's rays by a flint-glass prism is considerably beyond the extreme visible red ray. But we shall return to this result in connection with the history of radiant heat. He also made some experiments on

¹ A small star in Ursa Major (1830 of Groombridge's Catalogue) has an annual proper motion of $6\text{'}\text{'}\; 97$; the star 2151 Puppis, is the greatest known, being $7\text{'}\text{'}\; 87$.

² No historian of science, or biographer of Sir W. Herschel, can fail to acknowledge with gratitude the important assistance afforded by MM. Arago and Struve to a compendious acquaintance with his writings and discoveries; the former by his article on Herschel in the Annuaire du Bureau des Longitudes for 1842, the latter in his Etudes d'Astronomie Stellaire. While acknowledging my obligations to these works, I shall not be suspected by any careful reader of having dispensed with a reference to the original memoirs; still less of having adopted without examination the conclusions of these eminent authors, from whom, indeed, on some important points, I differ entirely.

the colours of thin plates, but to these he did not attach much importance.

(215.) Herschel's character; The philosophical character of Herschel is almost included in what precedes; but we will endeavour to sum it up in a few words. He united in a remarkable degree the resolute industry which distinguishes the Germans, with the ardour and constancy of purpose which has been thought characteristic of the Anglo-Saxon. From his native country he brought with him a boldness of speculation which has long distinguished it, and it is probable that he had also a vigorous and even poetic imagination. Yet he was ever impatient until he had brought his conjectures to the test of experiment and observation of the most uncompromising kind. He delighted to give his data a strictly numerical character. Where this was (by their nature) impossible, he confirmed his descriptions by reiterated observation, in different states of weather, with different telescopes, apertures, and magnifying powers; and with praiseworthy fidelity he enabled his readers to form their own judgment of the character of his results by copious and literal transcripts from his journals. On his claims to originality we are unable in all cases to decide, owing to almost his only literary fault, that of rarely al-

luding to the writings of his predecessors or contemporaries, even so far as to acknowledge their existence. The discovery of binary systems is probably that which was most absolutely his own; but even supposing the speculations of Wright, Lambert, and Mitchell were not unknown to him, this would diminish but little the substantive merit of having devised and executed the means of removing them from the regions of almost metaphysical abstraction to that of concrete reality.

His long and tranquil but ever active life corresponded happily to the nature of his pursuits, which (216.) required an absolute devotion of his time, and the means of instituting comparative observations after an interval of many years. There are not many philosophers who could have expected within their lifetime to see at least one pair of suns complete their mutual orbit.

Herschel's career at length drew to a close. He (217.) died peacefully at Slough, near Windsor, where he had resided throughout the greater part of his life, at the age of eighty-three, on the 23d August 1822, one year only after the publication of his latest memoir in the Transactions of the then recently formed Astronomical Society, of which he was the first President.

§ 3. BESSEL—MR AIRY—Modern Observatories. Fixed Star Catalogues—Planetary and Lunar Observations.

(218.) Modern astronomical details since 1810. As we approach the most recent period of the history of astronomy, I feel the increasing difficulty of a due condensation and selection of the interesting matter which claims our notice. Astronomy has been so generally and so zealously cultivated, that it seems almost invidious to select a few names and a few leading discoveries as the topics of discussion. It is, however, necessary to do so, and I shall be guided by the single aim of trying to specify impartially those individuals who have by their labours given most of the impress of the age to the science which they represent. In a subject like this, of almost infinite detail, the reader is not to expect a list of the names of all discoverers and improvers; it is not our business here to dwell upon mere labours of precision, to which so large a part of the most useful industry of astronomers is devoted; but to show the spirit in which these labours must be undertaken, and the general results by which they enrich the knowledge of the passing generation.

(219.) Practical astronomy in England. In the first section of this chapter we have sketched (in connection with the name of Maskelyne) the state of accurate astronomy in fixed observatories, down to about the year 1810. Since that period the multiplication of observatories has been very great. There has been a great improvement in instruments and in the methods of using them. But there has been an incomparably greater advance in the methods of ex-

tracting trustworthy results from observations made with due care. The art of dividing instruments was carried to great perfection in Great Britain by Graham and Bird, but their preference of the quadrant to instruments of a circular form (introduced long before by Römer) retarded the progress of astronomy. Foreign observatories in the early part of this century (as in the last) sought their divided instruments from London; Piazzi and Bessel worked with English circles. Troughton was as an artist the worthy successor of Ramsden; and even to the present time British astronomers, at least, do not admit that the nice contrivances and beautiful workmanship of Repsold and of Merz in Germany, or of Gambey in France, have produced instruments worthy of more confidence than those which have been constructed at home. In the improvement of object-glasses for telescopes, however, Germany and France bear away the palm, although for many years Dollond supplied Europe with his achromatic telescopes. With regard to constancy and fidelity of observation, whether in fixed observatories or in maritime and geographical expeditions, English astronomers have never been backward; and the reputation of Greenwich Observatory in this respect (and also in the punctuality of publication) has, as we have stated elsewhere, earned the approbation and gratitude of all Europe.

But here our praise of British astronomy must (220.)

pause. Until very recent times, with few exceptions, our observers, however industrious, have merely used mechanically the apparatus put into their hands by intelligent and conscientious artists, who yet, never having occasion to apply their own handiwork to the purposes for which it was made, could not be expected to detect deficiencies of construction, nor to possess the mathematical knowledge required to remedy them. To determine in a complete and independent manner the errors, not only of adjustment, but also of construction of the instruments, and to correct them by calculations (deduced when necessary from frequent observations of their amount) is now felt to be the duty of the astronomer himself. This improvement (though, as we shall see, already partially practised even in England) was mainly due to Bessel, whose name we have placed first at the head of this section. But astronomy owes to him much more than this. He first showed that it is part of an intelligent astronomer's duty, not only to observe stars and planets, but to undergo the vastly greater labour of comparing his results with the best theories of his time, and of improving as far as possible those theories, so as to hand over each branch of science to his successor in a more perfect state than that in which he found it. In all these respects practical astronomy commenced a new epoch in Germany, and on this account principally we place the name of Bessel in a conspicuous rank.

(221.)
Bessel—his early labours.

FRIEDRICH WILHELM BESSEL was born at Minden on the 22d July 1784. He was destined for a mercantile career; and it is an interesting fact that it was the expectation of an appointment as supercargo on a commercial voyage which led him to study navigation and astronomy, and finally induced him to devote his entire energies to the latter science, in which he received every encouragement from the amiable Olbers, who fortunately inhabited Bremen, where young Bessel was engaged in business. He commenced his astronomical career by reducing Harriott's observations of the comet of 1607, and was thereafter appointed assistant to Schröter at the observatory of Lilienthal. In 1810 he was removed by the Prussian government to the charge of the observatory about to be erected at Königsberg, where he spent the remainder of his life, and which he very speedily raised by his labours to almost the first rank of European observatories. In fact, he possessed, in a singular degree, the qualifications for directing a great observatory; including a thorough acquaintance with the use of instruments, with the theory of astronomy in all its branches, with the higher mathematics, and also the art and practice of calculation, as well as with many allied branches of natural philosophy.

(222.)
His Fundamenta Astronomice.

The work by which he is perhaps best known, the Fundamenta Astronomice, is not grounded on his own observations, nor even those made in the same century, nor in his own country, but, what may appear singular, upon the observations of Bradley at Green-

wich in 1750 and some following years. In reality, no better observations had been then made. The instruments had undergone no material change during the long and industrious, though not splendid, career of Maskelyne; and Bradley was beyond question the most accomplished astronomer of the 18th century. Let us record it as a fact for the encouragement of conscientious labour in the service of science, that these precious records of the state of the heavens, of which the value seemed unknown to a whole succeeding generation (so that they were not even printed in a crude form until nearly fifty years after they were made), were disinterred, so to speak, by an illustrious foreigner, and in course of time made the basis of what, in respect of precision and method, might be called a new astronomy. Bessel was assisted in this work by a grant from the British Board of Longitude.

The observations especially considered by Bessel are those of the fixed stars. To apprehend their importance, we must recollect that the foundation of an accurate knowledge of the orbits, and of the perturbations of the Solar System, not to mention less apparent but not less remarkable changes of place in the stars themselves, consists in rendering perfectly comparable observations of position at one time with those made at another more or less remote. Now, not only are our transit instruments and quadrants, and circles, and clocks, affected with errors more or less unknown, so that the registered figures require correction; but, in consequence of Refraction and Aberration, we see no body whatever exterior to the earth in its real visual position; and this visual position, when found, is affected by Parallax. Besides all this, the grand points of reference in the sky, the Pole and the line of Equinoxes, are undergoing perpetual, though small, changes of position:—the first star of Aries had already, in the time of Hipparchus, wandered from the point of intersection of the Equator and Ecliptic;—consequently, longitudes and right ascensions have to be reckoned from a directional line in space altogether imaginary. The pole of the heavens wanders amongst the stars, in consequence of Nutation—the obliquity of the Ecliptic is itself changing; and, to crown all these causes of perturbation and seeming unsteadiness, the stars called fixed have (as we have formerly seen) peculiar and individual displacements not accounted for by any of the preceding causes of change. In short, nothing is fixed and comparable, except our measure of time. The length of the sidereal day is subject to no secular variation. The bases of Astronomy,—the Fundamenta Astronomice,—then, evidently consist in the perfect evaluation of all these varying elements, so that the stars, if not really fixed, may become, for us at least, virtually so,—that by their aid the position of the fundamental directional lines of the sphere may at any time be found, and the relative positions of the stars themselves. After this it is easy to

refer the places of wandering bodies to standard points.

(224.)
Results. Bessel's work includes a methodical reduction of the places of above 3000 stars observed by Bradley, and an investigation of the sun's apparent path, together with a full discussion of the principles and application of every correction, instrumental and uranographical, which such observations require, and which are applicable to all others of the same kind. It includes, therefore, dissertations on some of the most delicate points of Astronomy, and "Constants" for the different corrections, which, with slight variation, have been since employed in every observatory. Bessel devised also a remarkably simple mode of finding for each star the varying corrections of its place. The proper motions of the stars were also determined for the first time with approximate exactness, by comparing Bradley's places with more modern observations.

(225.)
Bessel's own observations. In subsequent years (the Fundamenta were published in 1818) Bessel contributed to the science of astronomy numerous and regular observations of the heavenly bodies; the Annals of the Königsberg Observatory were regularly published from 1814, being (I believe) the first foreign establishment which followed the example set by Maskelyne at Greenwich, with the important addition of systematic reduction.

(226.)
His numerous works on astronomy. Of the subsequent labours of Bessel we must speak very shortly. The determination of the parallax of the star called 61 Cygni, perhaps the most original and important of these, we shall refer to another section, where we shall compare it with the results obtained by other astronomers. He made interesting physical observations on Halley's comet at its return in 1835; he prepared the materials for a very extended catalogue of fixed stars, arranged in zones, more recently published by Professor Weisse. In connection with the theory of the pendulum as a measure of gravity, he repeated and extended Newton's important experiments on the uniformity of the gravitating force on all kinds of matter; he applied a new correction to the vibrations of the pendulum, and improved the method of observing them correctly; and he discussed with his habitual mathematical skill and elaborate perseverance the figure of the Earth, from the whole of the then existing observations; whilst he also directed an operation for connecting the Russian triangulations with those of western Europe,—a delicate task, which he performed with consummate skill. It will thus be seen that (without mentioning a host of minor works and memoirs) there was hardly a great department of Astronomy in which Bessel did not take a distinguished part. Even the discovery of Neptune, by calculation from the irregularities of Uranus, was contemplated as a practicable problem by the veteran astronomer in his later years. He died on the 17th March 1846; therefore only a few months previous to the publication of MM. Le-

verrier and Adams' discoveries, and their triumphant consummation by M. Galle.

Even before Bessel's career as an astronomer had properly commenced, Pond, in England, had anticipated, in a good measure, the improvements in practical astronomy to which we have referred in the beginning of this section. Already, in the earliest years of this century, he had made observations at his private residence with a comparatively small circular instrument by Troughton, which led him to detect the gradually increasing errors of Bradley's quadrant still in use at Greenwich, and to recommend the adoption of complete circles (disused since the time of Römer) and the specific examination of errors of division. These principles he carried out at Greenwich, to which establishment he was afterwards appointed. His very indifferent health prevented that incessant activity which the management of a first-rate observatory requires; nevertheless, he is justly regarded as one of the chief reformers of the practical astronomy of those days.

His successor, Mr Airy, has distinguished himself by a more active career. Endowed with very uncommon abilities, and with great physical powers of endurance, he has, from his youth, been ever foremost not only in promoting, in every one of its departments, his favourite science of Astronomy, but also many other allied subjects, particularly Optics, to which we shall have occasion to refer in another section.

Born in the county of Northumberland in 1801, Mr GEORGE BIDDELL AIRY acquired great distinction at Cambridge, where he graduated in 1823, and was appointed to the charge of the Observatory there in 1828, after the death of Professor Woodhouse, a person sincerely attached to astronomy and well skilled in it, yet one who did not succeed in imparting much interest either to its theoretical or practical departments. Mr Airy engaged in the important investigation of a new irregularity in the motion of the Earth and Venus, to which I have referred in the chapter on Physical Astronomy (114), and at the same time he undertook to publish his own Observations in regular annual volumes, including a complete comparison of the results with the best tables, thus presenting at a glance the existing errors of theory. This practice he introduced at Greenwich on succeeding Pond there in 1835; and he has pursued it ever since. But not contented with rendering the annals of the National Observatory a correct reflection of the state of the heavens in his own day, and also of the condition of Astronomical Theories, he did not rest until he had performed for the observations of his predecessors the same service, and thus produced a series of comparisons of the observed and calculated places of the moon and planets, unexampled for extent and accuracy. Beginning with Bradley's Observations in 1750, he investigated and applied all the instrumental and uranographical corrections to each, as Bessel had done for the observations of the sun and

fixed stars, and then calculated the tabular places for the same instants of time. This last, especially in the case of the moon, was a truly formidable task. The whole labour was methodically performed at the expense of government by a staff of no less than sixteen computers, under Mr Airy's vigilant superintendence, and it extended, first and last, over the space of a dozen years. The results, printed in the most compendious form, fill 2200 quarto pages. One consequence of this systematic reduction was, that MM. Leverrier and Adams were separately provided with the places and errors of the Tables of Uranus necessary for their great investigations; and the Lunar Theory has been enriched with Mr Hansen's latest corrections through the same facilities. The final corrections of the co-efficients of the Lunar Tables are concisely stated by Mr Airy in a paper addressed to the Royal Astronomical Society.

Like Bessel, Mr Airy is thoroughly acquainted with the theory and practice of mechanics; and he has applied his knowledge both to the improvement of clocks and of proper astronomical instruments. A great and apparently successful experiment of the latter description is now taking place at Greenwich. A meridian circle (combining in effect a mural circle and transit instrument) has been erected, of great size and optical power (6 feet circle and 8½ inch object-glass). But the novelty of the apparatus is its great weight and strength. Hitherto artists and astronomers have sought to avoid the evils of unsteadiness and flexure principally by the elaborate framing and trussing of light materials (chiefly brass tubes), and (in Germany especially) by an elaborate system of counterpoises. The new Greenwich circular instrument is constructed of as few pieces as possible, of great strength and weight, and in great part of cast iron. The whole was elaborated and prepared for dividing in an engineering workshop, and with the powerful and accurate tools which modern engineers use. A man might stand upon the axis without producing the smallest bad effect. It is understood that the result is perfectly satisfactory as to steadiness and stiffness. Mr Airy has likewise invented and erected a new and most ingenious zenith tube (acting by reflection) for ascertaining the place of $\backslash gamma$ Draconis, a star which, at its superior culmination, crosses the meridian of Greenwich almost in the zenith.

There are few projects for the improvement of astronomy in the last twenty years in which Mr Airy

has not been a counsellor or participant. He has introduced at all times given efficient aid to those engaged in the pursuit of science; and has been himself the first to import from America and to establish on a great scale the method of recording observations by means of a galvanic pressure produced on a sheet of paper surrounding a cylinder; which cylinder is moved in a perfectly uniform manner by means of a connection with a clock producing uniform motion on the principle of the conical pendulum. By means of a simple contrivance the tracer follows a slightly spiral line, so that one barrel may include many hours of continuous observations. He has also established galvanic time signals, and employed them for the determination of longitudes. Besides all this, he has taken part in furthering geodetical measures, in regulating the national standards of length, and in promoting in various ways the collateral sciences. He contributed to the Encyclopædia Metropolitana a most able analysis of the methods for determining the Figure of the Earth, and his conclusions have been very generally adopted. In the same publication he has given a theory of the Tides, already referred to (78, 82). The public and his own university are indebted to him for several very valuable text-books on mathematical physics.

During his incumbency at Greenwich, Magnetical and Meteorological Observations have been added to those of Astronomy, and engage the time of four regular assistants in that department. The astronomical assistants amount to six, besides occasional computers. It may safely be affirmed that there is not a public establishment in a more complete state of efficiency than is Greenwich Observatory at this moment; and that its traditional reputation, expressed in the emphatic eulogium of Delambre, Art. (150), runs no risk of deterioration under the direction of Mr Airy.

Our limits will not allow us to refer to the numerous private observatories in England and public ones abroad which are contributing useful elements to the promotion of astronomy. It is not too much to say that the activity of Greenwich has set a generally good example. The observations at Oxford and Edinburgh have long been reduced on a systematic plan; those at Cambridge had been already so treated. The establishments where sidereal astronomy is particularly cultivated will be noticed in another section.

§ 4. BORDA—KATER—BAILY. The Figure of the Earth from Pendulum Observations—"Reduction to a vacuum;" Mr Stokes. Colonel EVEREST—M. STRUVE. Latest Measures of the Earth. M. Foucault's Pendulum Experiment.

In the first section of this chapter I have mentioned the progress made in determining the figure of the earth by measurements of its surface in the early part of this century. I shall now advert

briefly to the employment of the pendulum for the same purpose.

Clairaut had published in 1743 a very remarkable theorem which connects the force of gravity in any

Connection of gravity and the figure of the earth—
Clairaut's Theorem.

latitude with the compression of the terrestrial ellipsoid.¹ The pendulum is the readiest and most accurate means of determining variations in the force of gravity; the repetition of its oscillations enabling the observer to ascertain their average duration with extreme nicety. The more rapid the oscillation the greater is the force of gravity, because the heavy body is more quickly drawn into its lowest position. Two sets of observations, made in different latitudes, would, rigorously speaking, suffice to determine the polar compression, supposing the earth to be a geometrical ellipsoid, and composed of homogeneous and concentric layers; a larger number would serve to test the accuracy of these assumptions, and would, at all events, be imperatively required to eliminate the errors of observation.

(236.) Early pendulum observations.

It might be supposed that to swing a pendulum, count its vibrations, and measure its length, is a very easy thing. Experience shows that a more difficult practical problem can hardly be proposed. Those persons, therefore, who have in a good measure overcome the difficulties, are to be regarded as promoting materially an enquiry which has always ranked amongst the most interesting connected with astronomy. Some of the observations of the eighteenth century, especially those of Lacaille at the Cape of Good Hope and of Phipps in the Arctic Regions, appear to have been made with much care; but the French astronomer Borda seems to have given the initiative to observations of a higher degree of accuracy, and his methods were ably carried out in connection with the great French meridian arc, by M. Biot, who even extended his stations to the Island of Unst in Shetland.

(237.) Borda improves the method of making them.

JEAN CHARLES BORDA, who was born in 1733 and died in 1799, was devoted to the promotion of the exact sciences, and contributed in an eminent degree to the precision attained in physics and astronomy towards the close of the last century. The name of Borda deserves a record in the history of science, since it has been said of him, by no less an authority than Dr Young, that "he seems to have possessed a considerable share of that natural tact and sagacity which was so remarkable in Newton." His earlier researches were connected with hydrodynamics, and the resisted motion of projectiles; in later years he was chiefly devoted to practical astronomy and geodesy. The portable repeating circle invented by him

has had a very great reputation in France. He was one of the principal designers of the measurement of the French arc, and was employed in its superintendence, especially as regards the measurement of the base. His experiments on the pendulum were originally undertaken (I believe) in 1790, with a view to making it the basis of the national measures, but were afterwards subordinated to the greater scheme of terrestrial measurement. Baily records a fact connected with Borda, not unworthy of mention, as a caution to observers:—the year in which many of his experiments were made is not discoverable from the account of them, though the day, hour, minute, and second, are recorded with praiseworthy fidelity. From this time pendulum experiments assumed an astronomical and geodetical importance. The apparatus of Borda consisted of a slender metallic wire, attached at one end to a knife-edge suspension, and at the other to a small brass cap, nicely fitted by grinding to a sphere of platinum which formed the bob or weight of the pendulum. The oscillations were counted by the method of "coincidences," that is, by placing the experimental pendulum in front of a good clock with a known rate, and observing after how long a time the two pendulums (having started in the same direction) again coincided in their motions by one of them having gained or lost exactly two vibrations; a method which has been used at least since the time of Bouguer. The time of vibration being thus known, the distance from the knife edges to the bottom of the ball was measured by means of a scale; and due corrections for the position of the centre of oscillation were then applied.

By English observers, an invariable pendulum of the form of a flat bar, provided with a bob or weight, has usually been preferred, and on the whole it appears to be more satisfactory. Such a pendulum, after being compared with an astronomical clock at Greenwich, or any other place of reference, is taken to different stations and its rate of vibration determined, and it is then brought back and compared once more at the point of starting. Important series of observations of this kind have been made by Colonel Sabine and Captain Foster, and also by Admiral Du-perrey.

But the two individuals who have most studied the practical determination of the length of the seconds pendulum as a mechanical problem, are Kater and Baily.

Laplace and Mr Stokes on the earth's attraction.

¹ Clairaut (besides giving a theorem which, in the case of an ellipsoid, connects the polar compression with the increase of gravity there) showed that the increase of the force of gravity from the equator will vary as the square of the sine of the latitude. This he proved to be true when the spheroid is homogeneous, or when it is composed of similar concentric layers of varying density. Laplace confirmed the result by a different analysis, and farther showed, that if the earth be composed of concentric strata which are severally homogeneous, and nearly spherical, but otherwise arbitrary in form, and if the surface be that of a fluid in equilibrium (as it practically is when we refer the earth's figure to the sea-level), there exists a necessary connection between gravity and the superficial figure, which, in the case of an ellipsoid of revolution, leads to the same relation of the square of the sine of the latitude. In this case, also, the strata are presumed to be concentric. But in a recent and remarkable paper by Professor Stokes, it is shown that the law connecting the force of gravity and the figure of equilibrium still holds when no hypothesis whatever is made as to the distribution of the matter within the earth, provided always that it be consistent with the observed fact of the ellipsoidal figure of the earth's fluid covering. Thus the confirmation, by means of pendulum experiments, of Clairaut's Theorems cannot be regarded as a proof of the concentric arrangement of the strata, nor (so far) of the primitive fluidity of the earth. See Cambridge Transactions, vol. viii.; and for a simpler proof, Cambridge and Dublin Math. Journal, vol. iv. p. 194.

(238.) Invariable pendulum.

(239.) Relative and absolute measures.

and Baily. As regards the determination of the Earth's figure, the relative length of a pendulum vibrating seconds in different latitudes is all that is required; but the absolute length in a given place is necessary for finding the force of gravity there expressed by the acceleration of falling bodies in vacuo in one second; and as the British standard yard was at that time made to depend on the length of the pendulum vibrating seconds at London, this enquiry was mixed up with the intricate one of National Standard Measures.

CAPTAIN HENRY KATER was a person of remarkable mechanical skill and ingenuity, and his numerous papers in the Philosophical Transactions, between 1813 and 1828, refer chiefly to the accurate construction and use of the Pendulum, the Balance, and certain Astronomical Instruments. He first applied to practical use the beautiful theorem of Huygens respecting the convertibility of the Centres of Oscillation and Suspension of the pendulum. A metallic bar pendulum was provided with two parallel knife-edges facing opposite ways, and upon either of which it might be swung. They were so arranged that when either was used as the point of suspension, the other nearly represented the centre of oscillation, and by means of a small adjustable weight, this condition might be accurately fulfilled; in which case (by Huygens' principle) the distance between the knife-edges corresponded accurately to the length of an equivalent simple pendulum. This elegant application was farther tested by Mr Baily, but it was found liable to serious practical objections. Baily dispensed with the sliding weight, and rendered the pendulum invariable: the distance of the knife-edges was adjusted by grinding.

To Kater we owe the invention of the floating Collimator, for ascertaining the accurate zero or level points of divided astronomical instruments. The optical principle on which it depends is a very beautiful one, and the invention of Kater, with several improvements in point of form, has become the auxiliary of nearly every observatory in the world. It is one of those small but happy improvements which affect materially the progress of Science.

FRANCIS BAILY, who was born 28th April 1774, and who died 30th August 1844, was an instance of a person not endowed with extraordinary talents, yet of singular use in his generation. Those who knew Mr Baily personally, and what he accomplished for Astronomy, are best aware how rare and yet how estimable are such qualifications as his. Born of parents of a middle rank of life, educated at a country school, and at an early age thrown into the vortex of commercial life in the city of London, no probability could have appeared more remote than that he should acquire the respect of even the most accomplished Astronomers by his scientific acquirements and performances. He enjoyed the advantage, in the first place,

of a perfectly sound and uninterrupted state of health until his 70th year. His intellect partook of the equable tone of his bodily powers. Method, and an and character, instead of acting by impulses, seemed inexhaustible and unvaried, were the secrets of his success. Mr Baily never was hurried, and he never was unemployed. He had always leisure to encourage men of like tastes, and he was never unwilling to learn from any one who had knowledge to communicate. To do one thing at a time was his first principle. When thoroughly accomplished, it was completely put aside, and a fresh subject undertaken. Whether an experiment or a calculation was found to require a month or three years, it was pursued to a successful completion, without stint or grudging. No amount of contrariety and failure in such matters was ever known to ruffle his temper, or to make his perseverance falter.

Francis Baily was known, in the first place, as the founder (in 1820) of the Astronomical Society of London, an institution of signal utility, which has ever since been conducted with great judgment and success. His accurate knowledge of past and current astronomical history, and his personal acquaintance with most European astronomers, enabled him to conduct its affairs with great advantage.

Amongst his other acquirements, he studied history and chronology with his customary precision. A very able paper on the date of the celebrated Eclipse of Thales was his earliest scientific publication,¹ and the literature and phenomena of solar eclipses occupied his particular attention ever afterwards. His accounts of the Eclipse of 1820, of the Annular Eclipse of 1836, which he observed at Jodburgh, and the Total Eclipse of 1842, with its marvellous revelation of the rose-coloured protuberances of the solar atmosphere, are among the most interesting and classical of his writings. He promoted greatly the formation and publication of accurate Star Catalogues, and he invented, independently of Bessel, constant numbers for facilitating the calculation of Precession, Nutation, and Aberration, as affecting the places of particular stars. He also published a most useful collection of Astronomical Tables and Formulae, and an elaborate and very curious life of Flamsteed.

I shall now proceed to notice his experiments on the Pendulum, which are more particularly connected with the present section.

Mr Baily's most important observations were on a correction of the motion of a pendulum long misunderstood. The corrections usually admitted were the following:—1. For temperature altering the length of the pendulum. 2. For the height of the station above the level of the sea. Dr Young showed that this correction was commonly overrated, owing to the sensible amount of the attraction of the elevated land on which the experiment is made. 3. For

¹ Phil. Trans. 1811.

the air's buoyancy. The mere resistance of the air to the motion of the pendulum may be shown to produce opposite and compensating effects in the descending and ascending semi-vibrations; but the presence of air, as of any other fluid, diminishes the total effect of gravity on the bob, and that in a degree depending on the density of the air or atmospheric pressure. But besides this obvious "reduction to a vacuum," it appears that the air acts also in a different way. Owing to its inertia and cohesion a portion of air is shoved along with the pendulum, or otherwise put in motion by it, and the force thus spent in moving air is not compensated during the two semi-vibrations. This effect was clearly ascertained and measured by Dubuat about 1786, and described in the second edition of his "Principes d'Hydraulique;" but it appears to have been totally unobserved by his own countrymen and others, who (like Borda) were soon after actively engaged with the same subject. The effect was experimentally re-discovered by Bessel, and made the subject of an admirable series of experiments by Baily, which were published in the Philosophical Transactions for 1832. He vibrated pendulums of different forms in vacuo and in air (as Colonel Sabine had done to a more limited extent shortly before), and he ascertained the corrected "reduction to a vacuum" to be in many instances double of the old correction, and to depend materially on the form and density of the pendulum. It even appeared that when a convertible pendulum is swung in two positions in the manner of Kater, the correction is different in the two cases, owing to the want of symmetry, so that a pendulum convertible in air of a given density is not convertible in vacuo, nor vice versa.

(247.) Bessel and Baily agree in imputing the effect to the clinging of a mass of air to the metallic pendulum, thus rendering it in effect a pendulum of much less density, for which the ordinary correction for buoyancy will have to be increased. This view is probably rather a popular than an exact expression of the fact, and the correction need not be always proportional to that of buoyancy. Poisson has treated the question mathematically, and Professor Stokes has recently resumed the whole subject, both mathematically and practically, and considers that he has arrived at precise mathematical results in several cases. The result may probably be a revival of interest in pendulum observations, which has manifestly very much declined since the existence of this irregularity has been fully established, and the incompetency of existing rules for "reduction to a vacuum" clearly shown.

(248.) Baily deduced from the very elaborate experiments of Captain Foster in both hemispheres the result as to Earth's oblateness. Colonel Sabine had found it $\backslash frac\{1\}\{294\}$ for the earth's compression. Colonel Sabine had found it $\backslash frac\{1\}\{294\}$ from his own observations. The French and Russian observers concur in obtaining a result sensibly greater. Of these experiments, Captain Fos-

ter's is by much the most extensive. Mr Baily's repetition of Cavendish's experiment for determining the earth's density, has been mentioned in a previous section.

In conclusion of the subject of the figure of the earth deduced from observation, I shall now briefly refer to the results of geodetical measures more recent than those noticed in the first section of this chapter, which comprehended the great French arc. In a matter so much of detail, which will be better appreciated from special articles in the Encyclopædia, I shall best fulfil the ends of this discourse, by merely directing attention to the two most conspicuous and important of these measurements of the earth, one in India, the other in central and northern Europe; the first directed by Colonel EVEREST, the second by M. STRUVE, the eminent astronomer of Pulkowa.

The measurement of an arc of the meridian in India by Colonel Lambton in the early part of this century, has been mentioned in a former part of this Dissertation, as well as in Sir John Leslie's. But its value has been prodigiously increased by the extension of the same by Colonel Everest, who was at one time the principal assistant and coadjutor of Colonel Lambton. The arc of Lambton, extending from Punnæ (Lat. $8^\{\backslash circ\}\; 9\text{'}\; 35\text{'}\text{'}$) to Damargida (Lat. $18^\{\backslash circ\}\; 3\text{'}\; 15\text{'}\text{'}$), was measured after the model of the English trigonometrical survey, by means of a 100-feet steel chain for the base, a 3-feet theodolite for the azimuths, and a 5-feet zenith sector for latitudes. Colonel Everest pursued the triangulation in a northerly direction for some years after the death of Lambton in 1823; but losing his health, and being forced to return to Europe, he was provided, by the princely munificence of the East India Company, with an entirely new set of instruments, and with two able assistants, Captains Waugh and Renny.

He resumed operations in 1832, which were continued until 1840, and included an entirely new measurement of an arc extending from Damargida, where Lambton's arc terminated, to Kaliana (Lat. $29^\{\backslash circ\}\; 30\text{'}\; 48\text{'}\text{'}$), a space of 797 miles, covering an arc of $11^\{\backslash circ\}\; 27\text{'}\; 33\text{'}\text{'}$. The latest geodetical improvements were introduced. The bases were measured by General Colby's compensating bars, in which the effect of temperature is self-corrected. The signals from station to station were made during the day by heliotropes reflecting the solar ray parallel to the sides of the triangles to be measured. The azimuths were determined by a 3-feet theodolite of Troughton, in addition to that of Carey used by Lambton; the astronomical observations were made with two circles of large dimensions. The difficulties and annoyances experienced in conducting so vast a work, requiring such excessive accuracy, in so remote a situation, can only be judged of by referring to Colonel Everest's graphic details contained in his elaborate ac-

count of it, published at the East India Company's expense in 1847. The heats and the rains, the dull opaque atmosphere of the Doab, with its boundless and densely-wooded flats, the jungle fever and the wild animals, were natural impediments enough. But Colonel Everest, whose conscientious anxiety reached almost a nervous pitch, seems not to have been satisfied with any one instrument which the mechanics of London could produce; but to have metamorphosed most of them, partly with his own hands, partly by native aid. It says much for his ability in this respect, that the results appear entitled to compete with all the most exquisite of the kind in Europe. His bases, for instance, about $7\backslash frac\{1\}\{2\}$ miles in length, when checked by intermediate triangulation above 400 miles in extent, differed in one instance by 4, in another by 7 inches from the primary measurement. The whole extent of Lampton's and Colonel Everest's operations includes a continuous arc of $21^\{\backslash circ\}\; 21\text{'}$ (1477 miles), by far the greatest at that time executed.

(252.) It is indeed only rivalled in this respect by a vast Russian are by M. Struve. operation very lately executed in Russia and other northern countries of Europe, by which an arc of $25^\{\backslash circ\}\; 20\text{'}$, extending from the banks of the Danube to the shores of the Arctic Sea, near the North Cape, has been measured under the general superintendence and direction of that able astronomer M. Struve, whose meritorious labours in other departments of astronomy will be specified in a succeeding section. The results are still incomplete, though the operations in the field were happily concluded just in time to prevent their total frustration by the unhappy war in which Russia has since engaged.

(253.) The arcs of India and of Russia include a space from Lat. $8^\{\backslash circ\}$ to Lat. $71^\{\backslash circ\}$, with the exception of only about sixteen degrees, and are unquestionably the most important which exist for the determination of the earth's figure. When to them we add the French arc of $12^\{\backslash circ\}\; 22\text{'}$ in a medium latitude, it will scarcely be necessary to take into account any other, at least for the Northern Hemisphere.

(254.) The following brief details of the Russian arc are taken from the provisional report of M. Struve (1852).

(255.) Details of the Russo-Scandinavian Arc. The southern extremity of the Russo-Scandinavian arc is Ismail on the Danube (Lat. $45^\{\backslash circ\}\; 20\text{'}$), the northern extremity is Fuglænes on the island of Qualoe in Finnmarken (Lat. $70^\{\backslash circ\}\; 40\text{'}$). The interval from Tørnæ to Fuglænes ($4^\{\backslash circ\}\; 49\text{'}$) was measured by Swedish and Norwegian engineers; all the remainder by those of Russia, and, in particular, by M. Von Tenner, who, with M. Struve, has since 1816 directed the whole operation. The whole line is remarkably free from considerable inequalities of ground, and from mountain ranges, so that local attractions are probably inconsiderable.¹ On the other hand, extensive forests ex-

tending over dead flats have, in the southern part of the arc (as in India), occasioned great difficulties, and compelled the erection of numerous temporary structures to overlook the country. In the north the extraordinary refractions have, as usual, created some difficulties. Ten different base lines, all at a small height above the sea, form part of the operations. It is a very satisfactory circumstance, that by the care of M. Struve and Mr Airy, the standards of length used in the Indian and Russian arcs have been directly compared.

The calculation of the figure of the earth from the completed Russian arc, in combination with others, has not yet been made, but it is believed that it will indicate an ellipticity somewhat greater than that generally received. The results obtained by Colonel Everest, on the other hand, by comparing his arc with those of Europe, give generally small ellipticities, that is under $\backslash frac\{1\}\{300\}$. The French and Indian arcs, for instance, give $\backslash frac\{1\}\{310\}$. Now Mr Airy had deduced 20 years ago from the best existing observations $\backslash frac\{1\}\{310\}$; Bessel, a few years later, obtained the almost identical result of $\backslash frac\{1\}\{310\}$; and Schmidt of $\backslash frac\{1\}\{310\}$. The determinations by means of the pendulum are somewhat larger. The extensive observations of Colonel Sabine and Captain Foster concur in giving an ellipticity of $\backslash frac\{1\}\{310\}$; but the French experiments by Duperrey and Freycinet lead to a result considerably greater. The discrepancy between the geodetical and pendulum results may of course be a real one depending on local variations of density. The astronomical determination from the lunar inequalities, which might be expected to concur with the results of the pendulum, gives (as we have seen in the chapter on Physical Astronomy, Art. 63) $\backslash frac\{1\}\{310\}$ at a mean.

I regret that my limits do not permit me to speak of the measure of degrees of longitude or arcs of parallel, as another test of the earth's figure. The results, however, cannot compete in point of certainty with those from arcs of the meridian. It may be satisfactory to add, that the British arc of parallel from Greenwich to Valentia (west coast of Ireland) sensibly accords with the earth's figure obtained independently.

In connection with the subject of the pendulum treated in the earlier part of this section, I shall here mention a very remarkable experimental observation communicated to the Paris Academy of Sciences, on the 3d February 1851, by M. Léon Foucault, to whom we also owe some ingenious experiments on the velocity of light. It is not indeed connected with the determination of the earth's figure, but it has a connection with astronomy, inasmuch as it affords the most remarkable and direct proof of the earth's rotation round its axis.

¹ In India, on the contrary, the local attractions of the Himalaya must be very sensible; indeed Mr Pratt has calculated them to be so considerable, as to have given rise to a curious speculation by Mr Airy as to the circumstances which may tend to diminish the attraction of mountain ranges.—(Phil. Trans. 1855.)

(259.)
demonstrating the Earth's rotation.

Suppose a considerable weight suspended by a wire of regular elasticity from a fixed point or stand connected firmly with the ground; and let us first imagine the place of the experiment to be exactly over the North Pole. Let the wire pendulum be swung so as to coincide with the plane of the meridian of London. As the earth rotates, the wire and the ball must evidently rotate too. But the motion of the mass originally impressed parallel to a given plane will continue in that plane, and consequently the plane of motion will coincide in the course of 12 hours with every meridian in succession, and the apparent rotation will be entirely completed in a uniform manner in 24 hours. In any other latitude than $90^\backslash circ$, but greater than $0^\backslash circ$, a continuous and regular apparent change of motion must also occur, since a meridian of the globe does not preserve its parallelism during diurnal motion, except only at the equator; consequently, at all points of the earth's surface, except at the equator, the plane of motion of the pendulum will vary uniformly in azimuth—quicker, however, in high than in low latitudes. To find this velocity, it is only necessary to decompose the rotation of the earth round its axis into two, one of which (which is alone effective) is round the vertical of the place of observation. The apparent angular motion is thus proportional to the sine of the angle of latitude. Thus, in an hour, it is $15^\backslash circ\; \backslash times\; \backslash sin\; \backslash text\{\; lat\}$.

M. Foucault's experiment was made, in the first instance, with a pendulum of steel wire from .03 to .05 inch in diameter, bearing a ball of 12 pounds weight. It is desirable to make the pendulum vibrate in small arcs, in consequence of the tendency to ellipticity in the vibrations, which is necessarily accompanied by a rotation of the major axis of the ellipse, which might easily be mistaken for the influence of the earth's motion. To take account of this disturbing force, we have only to measure accurately the greater and less axes of the ellipse described. Then a revolution of the apsides (from this cause only) will be performed in a time which will be found by multiplying the time of a double vibration of the pendulum by 8 times the square of the length of the pendulum, divided by 3 times the product of the two axes of the ellipse. This formula is due to Mr Airy. The theory of M. Foucault's beautiful experiment has been verified by numerous experiments in different latitudes.

(260.)
Corrections to be attended to.

Very recently the theory of the motion of a pendulum suspended by a thread or wire has been considered in the most general manner by M. Hansen, the physical astronomer, with reference to the earth's motion.¹ It is an intricate problem of analytical mechanics. But the results show, as might have been anticipated, that all the sensible results are those which the geometrical treatment of the question indicate.

§ 5. M. ENCKE. Cometary Astronomy—Periodic Comets of Halley and Encke. GAMBART'S and Biela's Comet—Comets of 1811 and 1843. Mr HIND—New Planets or Asteroids. Mr LASSELL—Newly discovered Satellites. Mr Bond.

(262.)
Professor Encke. Periodic Comets.

PROFESSOR JOHANN FRANZ ENCKE, Director of the Observatory of Berlin, is one of the most eminent physical and practical astronomers of the present day. The author of many valuable observations and important memoirs, he is best known by those which are connected with the motions and theory of Comets. I shall therefore devote this section to an abstract of the progress of this interesting subject, and more particularly of M. Encke's discoveries and speculations. I must premise, however, that, easy and agreeable as it would be to introduce here a detailed essay on Cometary Astronomy, the design and extent of this discourse alike forbid it, and at the cost of some self-denial, I will endeavour to confine myself entirely to what is most new and characteristic in the Cometary history of later years.

(263.)
Halley's Comet—period 76 years.

Halley's Comet.—Before proceeding to describe the remarkable comet especially connected with the name of ENCKE, it will be proper to resume the history of Halley's Comet where it has been left off by Sir John Leslie in his Dissertation. That Comet—the comet of 1682—must ever remain memorable, perhaps the most so of its class, as being the first

whose return was confidently predicted, in firm reliance on the Newtonian Theory of Gravity. Halley's announcement—grounded not on vague analogies, but on laborious computations—that it would reappear early in 1759, was realized almost to the letter; its return and Clairaut, whose surprising analytical ability often left but little to his successors to accomplish, calculated the perturbations with an accuracy which even the present state of physical astronomy has hardly exceeded. Indeed no general method for calculating perturbations in highly elliptic orbits is as yet in use, and though the methods of Clairaut have been superseded by those of Lagrange and Bessel and Leverrier, the summation of the effects by the method of "quadratures" is always used,—the periodic time of the comet being divided into short intervals throughout which the elements are considered invariable, and during which the configurations and perturbing effects of the principal planets are computed with much labour.

The chief improvement in the calculation of perturbations is the introduction by Lagrange of the celebrated method of the Variation of the elements (44.)

¹ Theorie der Pendelbewegung. Danzig, 1853.

Its pertur-
bations
calculated
by various
astrono-
mers.

His successors have distinguished themselves chiefly by the praiseworthy minuteness and extent of their calculations, which are amongst the most laborious, if not indeed the most laborious, which occur in Physical Astronomy. The computers who calculated the return of Halley's Comet in 1835 were four in number, MM. Danoiseau, Pontecoulant, Lehmann, and Rosenberger. Their memoirs are all considered by competent judges to be excellent, but especially that of Rosenberger, who calculated more fully than the others the perturbations from 1682 to 1759, and who has introduced a theoretical correction of some importance. Some idea of the extent of these calculations may be formed from the fact, that in some parts of the orbit the Elements were made to vary for intervals of only two days. The Comet of Halley was rediscovered at Rome on the 6th August 1835, in the Jesuits' College. The error of Rosenberger's Ephemeris was only seven minutes of arc, and the perihelion passage took place on the 16th November (civil reckoning), five days after the predicted time. Bessel states the remarkable fact, that the coincidence of the comet's path with the results of previous calculation is as close as the use of five-place logarithms in computing the perturbations would permit. All this is very creditable to the state of Astronomy; still it is infinitely less remarkable than Clairaut's approximation, only a little less close, made 77 years before. It is a matter of regret that neither in the matter of this comet, nor in any other point of theory connected with Cometary movements, have our countrymen made any considerable advance since the time of Halley.

(265.)
Its return
in 1835.

At its return in 1835, this Comet was watched until May 1836, and in the course of its long visibility was made the subject of minute and admirable telescopic researches, by Bessel in Europe, and by Sir John Herschel at the Cape of Good Hope. Their observations strictly confirmed a remark of M. Valz, that the Tails of Comets, though evidently generated under the influence of the neighbourhood of the Sun, yet commonly disappear at the period of closest approach. This, at least, was the case with Halley's Comet. The tail began to grow on the 2d October, two months after its discovery, and diminished after the 15th (a month before perihelion). The comet reappeared in the end of January without any vestige of a tail, and then dilated in absolute bulk with incredible quickness. But before its final disappearance, on the 5th May, the tail is supposed to have been reabsorbed into the nucleus, which presented then a uniform circular outline. The evolution of the highly-expansible luminous matter of the tail took place from the nucleus in luminous fan-like jets, directed on the whole towards the Sun, and issuing on the side of the Comet exposed to that body. The jets had a vibratory motion. The tail

Its peculiar
features.

(directed from the Sun) was formed by the abrupt inflexion of these effluent jets, in a manner resembling electrical repulsion; and Sir John Herschel has not hesitated to ascribe the phenomenon to the combination of a repulsive force as respects the Sun, ("of an energy very far exceeding the gravitating force towards that luminary") with "a peculiar and highly energetic attraction to the nucleus, differing from, and exceptional to, the ordinary power of gravitation."¹ The opinion that polar forces resembling magnetism or electricity are necessary to explain the phenomena of Comets, has been for some time current in Germany, and received, I believe, the countenance of Bessel. That there exist in the movements of these bodies anomalies so great as to render the sufficiency and completeness of the Theory of Gravity suspected by such high authorities, makes the statement of them very important, though I confess a great reluctance to share in the conclusion.

I shall terminate this notice of Halley's Comet with a table of the dates of its probable perihelion passages, according to M. Laugier and Mr Hind.² The earlier appearances are deduced from the Chinese Annals:—

The differences in the above periods are attributed to the effects of perturbation. The extreme distance of the Comet's path from the Sun is 35½ times the radius of the Earth's orbit, which is only one-sixth part greater than the distance of Neptune, and therefore nearly within the recognized limits of the planetary system.

Comet of Encke.—Next in interest to the Comet of Halley is that discovered by Pons of Marseilles in November 1819, which was first suspected, then proved by Professor Encke, to revolve in an elliptic orbit of short period (considerably interior, in fact,

¹ See Sir John Herschel's Outlines of Astronomy, and Results of Observations at the Cape of Good Hope.

² Taken from Mr Hind's small work on Comets.

to the orbit of Jupiter), and made the subject by him of a series of investigations altogether peculiar. On this account it bears the name of Encke, instead of that of its discoverer Pons, although M. Encke himself, with unaffected modesty, always describes it as the Comet of Pons.

(268.)
Cometary orbits derived from observation.

Newton gave, in the Principia, his celebrated solution of the problem of determining a Comet's orbit assumed to be parabolic, from three geocentric places. This solution has been simplified and improved by Lagrange and Boscovich, and also by Olbers. Laplace gave a method for an elliptic orbit, which may represent any number of observations. Gauss, in his Theoria Motus Corporum Celestium, treated the subject with great skill and generality. I am unable to state who first attempted to discriminate an elliptic from a parabolic cometary orbit, or to determine the period in the former from observations at one apparition only. It is evident that such outstanding differences as are irreconcilable with a parabolic orbit, will be most perceptible in the case of comets whose orbits have a tolerably short major axis, or whose period is not very great, and will be materially increased by watching a comet through a considerable part of its orbit, which the assiduous application of telescopes to every part of the heavens has of late years rendered much more frequent than formerly. Amongst others, Bessel, who has signalized himself by a capital performance in this, as in every other department of Astronomy, applied rigorous methods to determine the orbits of the comets of 1807 and 1815; the latter of which will very probably return to its perihelion in 1887. It is undeniable, however, that expert calculators have often been deceived in assigning orbits, even when believed to be of short period, founded upon a single apparition.

(269.)
M. Encke's researches on the comet of 1819—period 3½ years.

M. Encke, however, was more fortunate in the case of the first comet of 1819. Using the methods of Gauss, he showed that an Elliptic Orbit of about 3½ years must be admitted, and that the comet had probably been already observed in 1786, by Méchain, in 1795 by Miss Herschel,¹ and in 1805 by Pons. He investigated with great labour the effects of the planetary perturbations on this body, which, in the case of Jupiter, are occasionally very large, if that planet be in the part of its orbit near the aphelion position of the comet, when it approaches the orbit of Jupiter. The careful calculations of M. Encke for the next return in 1822 were verified by the observations of Sir Thomas Brisbane, at that time fortunately governor of New South Wales, where he had, with characteristic liberality, founded an Observatory. Since then, this body, insignificant in its physical appearance (being to all appearance a small cloud of vapour without a solid nucleus), has been detected in one or

another part of the world at every revolution:—namely, in 1825, 1828, 1832, 1835, 1838, 1842, 1845, 1848, and 1852; so that the Comet has been observed at fourteen (not all consecutive) returns. Its apparitions.

(270.)
Its accelerated motion.

The complete establishment of the existence and periodicity of a comet, quite in the interior of the planetary system (its greatest distance from the Sun being four times the Earth's distance, and its least distance but one-third of the Earth's), was a discovery in itself highly interesting. But something yet remained behind. Professor Encke, in comparing the earlier with the later apparitions of the Comet, detected a gradual acceleration of its movement, which amounted between 1786 and 1838, to 1.8 days, on a period of about 1211 days; being about 2½ hours per revolution. Whatever may be the cause of this, the fact is undisputed, even by Bessel, who was indisposed to accept M. Encke's explanation. This fact, it will be observed, is unique in Astronomy. The major axis and periodic times of the planets and satellites have, as we have seen in the chapter on Physical Astronomy, no secular variation. The moon's apparent acceleration has been otherwise accounted for. M. Encke at once, attributed and at an early period, attributed the acceleration of the Comet's mean motion to the effect of a slightly resisting medium, insensible in the case of the planets, partly owing to their incomparably greater density (for this Comet appears to be one of the most loosely aggregated bodies known, being transparent to its very centre); and also, to the circumstance, that the density of the ether or resisting medium is assumed to diminish rapidly at a distance from the Sun. M. Encke supposes it to decrease in density with the square of the distance, and only to affect the Comet sensibly within 25 days preceding or following its perihelion. That the effect of resistance is to accelerate the return of the Comet is evident, by considering that the projectile force becoming gradually extinguished, the Sun's attraction must be more available to pull the body inwards at each revolution, thus shortening the major axis of the ellipse, and diminishing the time.

(271.)
Perturbations of Encke's comet applied to rectify the masses of Mercury and Jupiter.

In order satisfactorily to arrive at any such conclusion, it was of course necessary to estimate with great accuracy the perturbing effects of the planets on the Comet's motion; and it is not a little curious and satisfactory, that the movements of this insignificant erratic body should have occasioned a material rectification of the masses of two of the Planets. M. Encke very early suspected that the received mass of Jupiter was too small, a fact clearly established afterwards by Mr Airy; and in 1838 M. Encke showed that the mass of Mercury (which, not having a satellite, was little more than guessed at previously) had been assumed nearly three times too great by Lagrange. The perihelion of the Comet approaches much

Miss Caroline Herschel.

¹ Caroline Lucretia Herschel, sister of Sir William and aunt of Sir John Herschel, deserves a passing notice, not only as the independent discoverer of eight comets (of which five were first seen by her), but as the indefatigable and intelligent assistant of Sir William Herschel during the busiest years of his life. For this service King George III., carrying out his judicious liberality to her brother, granted her a small pension. She died at Hanover 9th January 1848, aged 97.

more nearly to the orbit of Mercury than the aphelion does to that of Jupiter; consequently at times the perturbations due to the former planet may be very great, and though the gravitating mass of the Comet is utterly unknown, yet since the momentary direction of its motion depends solely on the ratio of the attractive force of the Sun and Mercury, its observed course gives the means of estimating that ratio.¹

(272.) Theory of a resisting medium not altogether favourably received.

The theory of a resisting medium was, on the whole, well received, especially in England, where some of our first authorities gave it their adhesion. The then recent establishment of the Undulatory Theory of Light, was thought by many to receive a confirmation from this evidence of something material filling the planetary spaces. In Germany the hypothesis of resistance received the complete opposition of Bessel's high authority; who declared that "a hundred other reasons" might be found for the fact of the acceleration, which he admits to be true. Encke, in reply, reduces these 100 possible hypotheses to four, of which we shall mention only one, as seemingly important, namely, the forces exerted with so much intensity within the body of the Comet itself, as indicated by the projection of the tail. But he observes, with great sagacity, that these forces, being apparently usually excited in the line of the radius-vector joining the Comet and the Sun, can hardly be supposed to affect the periodic time. It having also been objected that Halley's Comet shows no trace of acceleration, but, if anything, of the reverse, M. Encke truly says, that its perihelion distance does not lie within the assumed limits of the denser ether.

(273.) and still questioned.

Nevertheless, the theory of a resisting medium in space is not perhaps very popular, except in England. Although M. de Humboldt appears to favour it, I understand that the German astronomers in general scarcely regard it as in any degree proved.

(274.)

Yet, if not true, the cardinal fact remains unexplained. The anomalous phenomena of the Tails of Comets, considered by Herschel to be altogether inexplicable by the law of gravity, demand the closest scrutiny; and one can hardly help supposing that the two difficulties may be in close connection. As the Newtonian law is now considered (since the discovery of Neptune, and the latest corrections of the Lunar Tables) to be absolutely sufficient to account for everything connected with planetary motion, the Astronomy of Comets will be looked to with increasing interest, as likely to reveal some laws of nature not otherwise to be detected. In this respect, Professor Encke's labours are likely to be more and more important in their results.

(275.) M. Encke on perturbations.

With reference to this very eminent astronomer, we have only to add, that he has for a great many years been at the head of the Observatory at Berlin, and in that capacity has published an Astronomical Ephemeris of first-rate excellence. It is as a phy-

sical astronomer, however, that he will be principally remembered. Besides his admirable investigations connected with the Comet, he improved the theory of Vesta, and has very lately published a new Method of Computing Perturbations, especially for orbits considerably elliptical. Neptune was discovered at his Observatory, by the assistant astronomer, M. Gallo.

(276.) Gambart's and Biela's Comet.—JEAN GAMBART, one of the most promising astronomers in France, died of consumption at a comparatively early age, I believe in 1836. He was director of the Observatory at Marseilles, which, notwithstanding its very unfavourable position in the midst of the town, has acquired considerable celebrity as regards the discovery and observation of Comets. Pons, by whom Encke's Comet was found, both in 1805 and 1818, conducted the Observatory; but its mounting was as bad as its situation, and Pons used despairingly to describe his telescope as rather paralytic than parallactic. To this crippled establishment M. Gambart succeeded, and by his skill in managing his defective instruments, and by his patience in sweeping for Comets, he discovered and subsequently computed the orbits of a number of these bodies between 1822 and the period of his death. Gambart was highly esteemed, both by French and foreign Astronomers. Pons also deserves great credit for his extraordinary diligence in the discovery of Comets, and M. Valz, who still directs the Observatory of Marseilles has cultivated this and other branches of the science with success.

(277.) Gambart's most remarkable discovery was the periodicity of the first Comet of 1826, having detected that body independently at Marseilles, though it had been observed some days previously in Bohemia, by Biela, an officer in the Austrian service. It is most usually called Biela's Comet, though it might with equal right be termed Gambart's, who assigned its path and predicted its return. Clausen, about the same time with Gambart, assigned it a period of about 7 years; and it was identified with former appearances in 1772 and 1805-6. Its period thus appeared to be 2460 days, or 6$\backslash frac\{1\}\{2\}$ years; its aphelion is a little exterior to Jupiter's orbit, and its perihelion is not much within the Earth's. This Comet's orbit very nearly intersects in one place the orbit of the Earth, so that had the earth been one month forwarder in its annual course in 1832, a collision would have taken place, or at least the Earth would have been enveloped in a cometary haze; for it is difficult to imagine a collision with a body whose tenuity is so excessive, that Sir John Herschel perceived through its entire thickness (estimated at 50,000 miles) stars of the most excessive minuteness (16th or 17th magnitude) as seen by his 20-feet reflector. It is an interesting circumstance, that the first predicted perihelion passage, in 1832, took place within some hours of the time fixed by MM. Santini and Damoiseau,

¹ On the Masses and Densities of the Planets, see Encke in Astron. Nachrichten, vol. xix., col. 187.

though the perturbations of Jupiter were, as usual, large.

(278.) Biela's Comet has been since recognised in 1846 and 1852, but it was not seen in 1839. It does not appear to be admitted that it shows any acceleration due to a resisting medium. Its perihelion distance is, however, considerable.

(279.) Division of the body of the comet in 1846. At the apparition of 1846 an extraordinary circumstance occurred. When discovered in the end of November 1845 it appeared round and single. On the 19th December it was observed by Mr Hind to be elongated, and ten days later was seen in America (and soon after at Cambridge and elsewhere) to have divided into two seemingly distinct nebulous parts.¹ These continued to subsist and move independently throughout the remainder of the apparition; the real distance of the centres being about 150,000 English miles. In 1852 the comet was rediscovered at Rome; the division into two still subsisting, but the interval of separation being increased about eight-fold.

(280.) Other periodical comets. Besides the comets of Encke and Biela, there are several others which are suspected on good grounds to have periods of from $5\backslash frac\{1\}\{2\}$ to $7\backslash frac\{1\}\{2\}$ years, their aphelia all lying in tolerable proximity to the orbit of Jupiter. But among these the return of only one has yet been verified by observation; namely, the comet of Faye, which, after passing its perihelion 17th October 1843, returned to it 3d April 1851, within an hour of the time predicted by M. Leverrier. The motion of comets of short period seems to be invariably direct or conformable to that of the planets. The inclination of their orbits to the Ecliptic is usually moderate.

(281.) Great comets of 1811 and 1843. Great comets of 1811 and 1843. The finest comets of the last hundred years were those of 1811 and 1843. The former was observed for a length of time altogether unusual, having been visible from March 1811 to August 1812. There is pretty good reason to think that its period is not much less than 3000 years. The comet of 1843 was even more splendid, but its flight was more rapid, and it was not favourably seen in northern latitudes. It was visible at many places in broad daylight when less than $4^\{\backslash circ\}$ from the Sun, and at one time a tail $65^\{\backslash circ\}$ in length could be traced. The circumstance which distinguishes this comet from all others which have been computed is the smallness of its perihelion distance, which was only $\backslash frac\{1\}\{3\}\backslash frac\{1\}\{2\}$ of the radius of the Earth's orbit, or the comet approached the Sun's body within one-seventh of his radius. The solar disk then subtended an angle at the comet of $121\backslash frac\{1\}\{2\}^\{\backslash circ\}$, or the glare was equal to that of 47,000 suns as seen by us! The heat to which the comet was exposed is supposed

to have exceeded 24 times that concentrated by our most powerful burning-glasses by which even rock crystal has been fused.²

Mr HIND.—Discovery of New Planets.—We have spoken in a former section (161), of the discovery of four small planets or asteroids between the orbits of Mars and Jupiter. They were found between the years 1801 and 1807. An interval of nearly forty years elapsed without any addition to the members of our system. In 1845 a new asteroid, Astræa, was found by M. Hencke; the following year was distinguished by the discovery of Neptune under unparalleled circumstances; and since 1847 every year, down to the present time (1855), has added to our knowledge of the group of asteroids.

Among the discoverers of these planetary bodies Mr Hind has been distinguished by frequent success, under circumstances which appeared by no means peculiarly advantageous. This indefatigable observer and computer commenced (I believe) his astronomical career as one of the assistants at Greenwich, and afterwards had the sole charge of the private observatory of Mr Bishop, a wealthy citizen of London, together with the use of a fine refractor equatorially mounted. It is within the Regent's Park, close to the smoke of the metropolis, that Mr Hind has discovered a larger number of planetary bodies than any other person living. Next to him M. de Gasparis of Naples has been most successful. Unquestionably the impulse towards these new discoveries has been given by the indefatigable industry of astronomers (principally those of Germany), in constructing minutely accurate star-maps. Mr Hind is also advantageously known by the discovery of several comets, and by his ingenious observations in sidereal astronomy, especially on variable stars. I shall here give a table of the asteroids in the order of discovery as at present known (July 1855).

¹ A similar phenomenon is related by Seneca. See Grant's History of Astronomy, p. 302.

² See many other interesting particulars of this comet in Sir J. Herschel's Outlines of Astronomy, arts. 589, &c. See also interesting details on the subject of comets generally in Mr Hind's and Mr Milne's works on comets, and in Mr Grant's excellent History of Physical Astronomy.

MR LASSELL.—New Secondary Planets.—Mr Lassell of Liverpool deserves a more lengthened notice than our limits will permit, not only as a distinguished discoverer, but as one whose success cannot be too widely made known as an encouragement to others. This gentleman, engaged in mercantile pursuits in an eminently commercial town, possessing little leisure and no enormous fortune, has contrived, in the intervals of business, to construct with his own hands telescopes which in accuracy of definition appear to rival any which art stimulated by national liberality has yet constructed elsewhere, and to use them with a degree of skill and success which has not been exceeded (nor in some respects equalled) by any astronomer whether professional or otherwise. I speak, let it be observed, of accuracy of definition, such as is necessary to display minute points of light, like the satellites of Uranus. In respect of the amount of illumination requisite for the display of many diffuse faint objects among the nebulae, the gigantic telescopes of Herschel and Lord Rosse are of course superior.

Mr Lassell's observatory near Liverpool was erected in 1840. The principal instrument is a reflecting telescope of 24 inches aperture (completed, however, only some years later), mounted equatorially, an arrangement requiring great mechanical skill, but, as the results show, most effectually accomplished. The speculum was worked and polished by machinery constructed by Mr Nasmyth, but principally devised by Mr Lassell, after he had examined and tried Lord Rosse's method. I should think it must be admitted to be the most perfect optical work of its kind ever made: for I believe there is no test object in existence which Mr Lassell has not seen with it; in fact he has discovered the most delicate tests himself,—the 6th star of the group $\backslash delta$ Orionis (though not first seen

by him), a satellite of Neptune, an eighth satellite of Saturn, and several satellites of Uranus.

Not many days had elapsed after the discovery of Neptune, before Mr Lassell, directing his telescope to it, perceived a satellite (as he believed) on Oct. 10, 1846. The discovery was fully made out in the following year, and was soon after verified by the great refractors of Pulkowa and Western Cambridge (U.S.) Its period is about 5.879 days,¹ and its discovery was of singular importance as leading to a knowledge of the mass of the planet.

Till 1848 only seven satellites of Saturn were admitted. The two closest to the planet were detected by Sir William Herschel in 1789, and have been seen by very few astronomers since. During five years' residence at the Cape, Sir John Herschel never but once obtained even a doubtful glimpse of the closest with an 18-inch mirror. The third, fourth, and fifth were discovered by Cassini in 1684; the sixth and most conspicuous by Huygens in 1654: the outermost by Cassini in 1671. To these an eighth satellite, intermediate in position between the two last, was added by Mr Lassell on the 19th September 1848. By a singular coincidence, it was recognised as a satellite the very same evening by Mr Bond of Cambridge (in America) with the great Munich refractor. The new body was called Hyperion, in conformity with Sir John Herschel's suggestion of distinguishing the satellites as well as the planets by mythological names. On the 22d November 1850 Mr Lassell saw at once Saturn with his whole train of eight satellites—a glorious spectacle probably enjoyed by no other astronomer. In the same month of November Mr Bond discovered a faint or dusky ring of Saturn interior to the two long known. It is probably nebulous, for by Mr Lassell's observations and Mr Jacob's it appears to be transparent.

Sir William Herschel thought that he recognised six satellites of Uranus. The second and fourth of his table have been observed by several astronomers, particularly Sir John Herschel and M. Lamont. Their periods are 8^d 17^h and 13^d 11^h. To these Mr Lassell, aided by the fine climate of Malta (to which for two seasons he removed his telescope), has conclusively added two more: one, appearing to coincide with the closest of Herschel's, of which the period is 4^d 3^h 28^m, and one still nearer the planet revolving in 2^d 12^m 29^s, the shortest orbital revolution in the solar system. Mr Lassell doubts seriously the existence of any other satellite.²

¹ From Mr Lassell's observations at Malta, 1852-3.

² See Astron. Society's Notices, xiii. 151; and xiv. 133.

§ 6. Sidereal Astronomy since 1820.—M. STRUVE—Double Stars. Observatories of Dorpat and Pulkowa. SIR JOHN HERSCHEL—Orbits of Double Stars. Magnitudes of Stars. Variable Stars. EARL OF ROSSE—His Telescopes. Nebulæ. HENDERSON and BESSEL—Parallax of Stars.

As it is absolutely necessary to bring this chapter to a speedy close, and as I have already anticipated, in the account of Sir William Herschel's discoveries, part of what relates to the recent history of the Astronomy of the Fixed Stars, particularly the "Constitution of the Heavens," and the motion of our system in space, I shall condense within brief compass a few leading facts connected, in the first place, with the Orbits of Double Stars, the Brightness of Stars, and the constitution of Nebulæ, and these subjects I shall connect with the names of the elder M. Struve, Sir John Herschel, and Lord Rosse; the second topic shall be the Parallax and distance of the fixed stars, as ascertained more particularly by the late professors Henderson and Bessel.

FRIEDRICH GEORG WILHELM STRUVE has been the most assiduous observer of double stars since the time of Sir William Herschel. No discovery in this department can for a moment compete with the great one of the orbital revolution of one star round another. But M. Struve, by devoting his chief energies during the most active years of his life, since 1813, to the assiduous continuation of Herschel's observations, has added immensely to our knowledge of these systems, and has earned the reputation of one of the most skilful of modern practical astronomers. His most elaborate observations were made at the Russian Observatory at Dorpat, with a noble refractor by Fraunhofer, nearly 10 inches aperture, and 13½ feet in focal length. He has published three works on the subject of double stars, one in 1824, one in 1837, and one in 1852, besides minor papers. The second of these works contains the particulars of about 3000 double stars, deduced from a survey of the heavens, in which at least 120,000 stars were examined.

M. Struve's papers are distinguished by the elaboration of the reductions, and of the statistical results deduced from them. In his last publication he has made an interesting estimate of the number of true double stars in the heavens, which, it appears, is much greater in proportion to the whole number than is usually believed. But it is first necessary to distinguish those which are physically double from those which are merely apparently or optically so. The criteria on which he principally depends are—(1.) the fact of observed orbital revolution; but as this is established in comparatively few instances, he very reasonably admits (2.) a common proper motion of the two components as a proof of their connection. He thus finds the evidence for physical duplicity to be much stronger for the closer double stars, and also for brighter or nearer stars, as compared with those of less magnitude. On the whole he concludes, that of

1973 double stars, 1702, or six-sevenths of the whole, are physically connected, and that, amongst at least the brighter stars, the number of compound systems is to that of isolated stars in a ratio not less than 1 to 3, perhaps even 1 to 2.

M. Struve has also determined with extreme care (292.) the places and proper motions of the double stars; and he has formed a decided opinion, that the proper motions diminish on the whole regularly with the order of magnitude,—thus confirming the criterion of increased distance from diminished brilliancy, by that of the apparent displacement of the stars by the motion of our system.

Besides these important works on Sidereal Astronomy, M. Struve is well known as the head and director of perhaps the best organized observatory in the world, that of Pulkowa near St Petersburg, of which he has published a very interesting description. Besides other noble instruments, it contains the finest refractor in Europe, that by Merz, 15 inches in diameter, and 22 feet focal length. The observations with this noble telescope are chiefly made by his son M. Otto Struve, the author of many good papers. We have seen in a former section (252), that we owe in great part to M. Struve the conduct of the most extensive trigonometrical operation ever undertaken.

SIR JOHN FREDERICK WILLIAM HERSCHEL, son of (294.) Sir William Herschel, whilst conversant with almost every branch of science, has devoted himself with remarkable success to the cultivation of Sidereal Astronomy. "Bearing a name honoured and revered by all, his career at Cambridge reflected upon it fresh lustre; the variety and extent of his acquirements gave him a reputation amongst his college contemporaries, afterwards fully confirmed by the not more impartial voice of mankind at large." He was senior wrangler in 1813. "Since that time he has been indefatigable as an author:—first, in systematizing the higher mathematics, and in forwarding their study in his own university;—afterwards by treatises contributed to the Encyclopædia Metropolitana on Sound, Light, and Physical Astronomy, which still rank among the clearest, completest, and most philosophical in our language. About the same time he wrote experimental essays on different branches of Chemistry, Magnetism, and Optics, and commenced his purely astronomical investigations, chiefly on nebulæ and double stars, partly in conjunction with Sir James South, of which the details are given in different volumes of the Astronomical, and of the Royal Society's Transactions. These memoirs collectively include a complete revision of the objects of the same description catalogued and classified by Sir

William Herschel."¹ But the most considerable monument to Sir John Herschel's love of science is the record of his four years' labours for the advancement of Sidereal Astronomy at the Cape of Good Hope, where he applied his father's methods of observation to the southern hemisphere. His Results of Astronomical Observations, which fill a large quarto volume, and which include "the completion of a Telescopic Survey of the visible Heavens, commenced in 1825," form one of the most considerable and most interesting works of our time. The instruments employed were a 20-feet reflector, of 18$\backslash frac\{1\}\{2\}$ inches aperture, and a 7-feet achromatic, with 5 inches of aperture. With these the nebulae and double stars of southern skies were examined and measured, and that wonderful "Gauging of the Heavens" completed, of which I have spoken in the account of Sir William Herschel (201). There is an admirable chapter on the apparent magnitude of the stars, to which I shall refer presently, and one on Halley's comet, besides other matters of interest.

(295.) His high character amongst his contemporaries.
Since his return to England in 1838, Sir John Herschel has withdrawn from the labours of practical astronomy, but he continues to advance different branches of science, and to expound them by his able and lucid writings in a way which has made his authority equally respected by philosophers and by men of the world. The career of Sir John Herschel has been marked by an almost total absence of the element of ambition, so often a powerful excitement in the pursuit of discovery. Had he sought notoriety and posthumous fame, he would have confined his efforts within a more circumscribed range. But his versatile talents sought their appropriate exercise in all departments of exact science, and even (it is believed) in pursuits widely distinct from these, in natural history, belles lettres, and the fine arts. In all this he no doubt considered simply the useful and pleasurable employment of his mental activities. Truth seemed to him as desirable whether attained by the labours of others or by his own; and in his numerous writings he has expounded these with a zest which a less generous spirit might have reserved for his peculiar achievements. What he may have lost in future fame by this enlargement of his sympathies and interests, he has gained in the respect and good-will of all his contemporaries. Sir John Herschel recently filled the post of Master of the Mint, to which, like his illustrious predecessor Newton, he devoted a considerable share of his time. His general eminence as a man of science has been acknowledged by his nomination in 1855 to the distinguished honorary position of one of the eight foreign Associates of the French Academy of Sciences.

(296.) On the orbits of double stars.
Orbits of Double Stars.—Though not absolutely the first to apply calculation to the orbits of double stars, this step in their theory may not unfitly be con-

nected with the name of Sir John Herschel, from the ardour of his researches and the neatness of his methods. To Savary of Paris is due the merit of ascertaining the form and position of the orbit of $\backslash xi$ Ursæ Majoris in 1827, which was followed by a more purely analytical method by M. Encke, and one chiefly graphical by Sir J. Herschel,² in which angles of position of the component stars are used nearly to the exclusion of the more doubtful measures of distance. On the whole, these investigations not only confirm Sir William Herschel's anticipations, but render it highly probable that the relative orbits are really ellipses, and consequently that the law of force is that of the inverse square of the distance. The reader will find in Sir J. Herschel's Cape Observations a very curious discussion of the orbit of $\backslash gamma$ Virginis, a remarkable double star, whose interval was in 1783 five seconds and two-thirds, which diminished till 1836, when the two stars appeared united in one, as seen even in the best telescopes. This was the perihelion passage of these two suns, and the angle of position must then have varied (could it have been measured) at the rate of 70° per annum, or 1° in 5 days. The following are some of the best ascertained periods of sidereal revolutions in years:—$\backslash zeta$ Herculis 36$\backslash frac\{1\}\{2\}$; $\backslash xi$ Ursæ Majoris 61$\backslash frac\{1\}\{2\}$; $\backslash alpha$ Centauri 77$\backslash frac\{1\}\{2\}$; $\backslash rho$ Ophiuchi 80 or 90$\backslash frac\{1\}\{2\}$; $\backslash epsilon$ Coronæ Borealis 600 or 700. M. Mädler, Admiral Smyth, and Mr Hind, have added much to our knowledge of this interesting subject.

(297.) Brightness of Stars, and Variable Stars.—Sir John Herschel has attempted by an elaborate system of inter-comparison to assign the correct relative brightness of the stars, and to give precision to the ordinary terminology of Magnitudes. His "Method of Sequences" described in his Cape Observations, appears to be one of the happiest specimens of generalization which experimental science affords. Whilst regretting the impossibility of here giving even the slightest sketch of it, I cannot but recommend it to the student of natural philosophy as a model of research. Having ascertained, in a way independent of every sort of hypothesis, the relative brightness of the stars upon the scale of Magnitudes usually adopted, but which is wholly arbitrary, Sir J. Herschel proceeds, by properly photometric methods, to give a scientific precision to this notation; and he arrives at this singular and fortunate conclusion, that by adding a small and constant correction to the received scale of Magnitudes, the numbers will represent the distances of the respective stars from our system on the supposition of an intrinsic equality in the brightness of the stars themselves.

(298.) Variable Stars.
This subject naturally includes that of Variable Stars, which may be divided into those which undergo periodic or irregular fluctuations, and the latter class may embrace new stars, and stars which have

disappeared. The last two hundred years have not presented any such astonishing phenomena as the new stars recorded in the sixteenth and seventeenth centuries; but singular variations in the brightness of some of the most conspicuous stars, as $\backslash alpha$ Orionis and $\backslash eta$ Argus, have been discovered by Sir John Herschel. Several stars of short and irregular periods of varying brightness were recorded towards the close of last century. But upon this very interesting subject I must content myself with referring to the details given by Professor Argelander in the third volume of Humboldt's Cosmos.

(259.)
Lord Rosse
—his re-
fecting
telescope.

EARL OF ROSSE. Latest observations on Nebulae.—It is a remarkable circumstance, that as the reflecting telescope was a British invention, so the more important improvements and applications of it have been almost confined to the United Kingdom. It is also worthy of notice that the manufacture has prospered more in the hands of amateurs than of regular opticians. Sir William Herschel appeared at one time to have brought the invention to its highest perfection, but the Earl of Rosse has made an important step farther; not only by constructing a larger telescope than had been made before, but by adapting machinery driven by steam power, to the grinding and polishing of the mirror; so that the largest speculum may be finished with nearly the same accuracy and expedition as the smallest. The chef d'œuvre of Lord Rosse is a telescope of 6 feet aperture, and 53 or 54 feet of focal length. It was completed in the latter part of 1844, and erected at Parsonstown in Ireland.

(300.)
Difficulties
encoun-
tered.

Let me here record the important fact, that neither rank nor wealth could absolve Lord Rosse from those toils and disappointments which attend all new and original efforts. There is no royal road to such triumphs. The Irish nobleman owes his success entirely to his unwearying perseverance and mechanical skill. Even his assistants were countrymen instructed by himself in his own workshops, where the very steam-engine which drives the polisher was fabricated. His labours to improve the telescope date from 1828 (when he was Lord Oxmantown), or even earlier, and they appear to have been unremitting until 1844; indeed I might say until the present time. Commencing with a variety of ingenious attempts to correct spherical aberration, and to overcome the extreme difficulty of procuring large castings of so excessively brittle a material as speculum-metal, they terminated in the rejection of all minor helps and expedients, and in the fortunate completion of immense mirrors at a single casting, and of correctly parabolic figure when ground and polished. The speculum of his largest telescope weighs four tons. It was polished in six hours, and its surface is con-

siderably more than twice as large as that of Sir William Herschel's forty-feet instrument.

We cannot enter into the details of the methods, (301.)
which evince no small mechanical skill and scientific ingenuity, together with a perseverance admirable in itself.¹ I will only mention how the upper and lower surfaces of the casting were made to cool nearly equally fast. To effect this the lower surface of the mould (which naturally retains the heat more than the upper) was made of iron, a good conductor, whilst the upper surface was made of sand. This effected the purpose; but it being found that air bubbles entangled in the fluid metal could not escape beneath, and injured the casting, the iron bed was constructed of hoops set on edge and closely packed, the crevices allowing the escape of air, whilst the cooling proceeded as before; and this ingenious contrivance was perfectly successful.

Many difficulties in detail have been found in the (302.)
mounting and use of so gigantic a mass, particularly
on account of the distortion of the mirror by flexure.
But these have gradually been surmounted by Lord
Rosse. His published observations (Philosophical
Transactions, 1850) relate almost entirely to objects
of the class of nebulae; and as I cannot enter into
details, I may state the general results in two or three
sentences. (1.) As might have been expected, many
nebulae which resisted the power of former telescopes
(for, except in rare instances, nothing greater than
eighteen-inch apertures have been directed to them)
have been "resolved" into stars by the six-feet specu-
lum. (2.) The aspect of a great number of nebulae
described by the two Herschels is materially modified
by the power of the telescope to embrace the fainter
prolongations of these singular objects. In general,
the symmetrical forms are very much cut up and
confused, and in many cases vanish altogether.
(3.) Instead of these, a certain species of symmetry,
of a vague yet very remarkable description, has been
detected by Lord Rosse, probably for the first time.
It is a spiral arrangement of the nebulous coils
round a centre, resembling somewhat the spiral
emanations of revolving fireworks. The well-known
nebula No. 51 of Messier's Catalogue shows this
in a remarkable manner.

Some observations have been made upon the moon. (303.)
It is much to be desired that these were continued,
and that the planets could also be observed; but I
believe that the climate of Parsonstown affords but
few nights favourable to observation.

HENDERSON AND BESSEL. Parallax and distance (304.)
of the Fixed Stars.—THOMAS HENDERSON, at one
time government astronomer at the Cape of Good
Hope, and subsequently professor of practical as-
teronomy in the university of Edinburgh, and his

¹ For a comparison of Lord Rosse's and Messrs. Nayamith and Lassell's (subsequent) methods of mechanical grinding and polishing, see Astron. Soc. Notices, vol. ix, p. 110. In Lord Rosse's apparatus every stroke of the polisher is almost a straight line; in Mr. Lassell's it never is.

Majesty's astronomer for Scotland, was born at Dundee, in Scotland, 28th December 1798. He died at Edinburgh 23d November 1844. His career was an instance of the conquest of disadvantages of many kinds by a patient, devoted, and conscientious spirit, and of the attainment of a station of great eminence in the world of science by singleness of purpose, and an ardent love of knowledge.

He was fortunate in having Mr Duncan (now of St Andrews) as his instructor in mathematics at Dundee, and it was "while employed as an attorney's clerk in a provincial town that he laid the foundations of that extensive acquaintance with astronomy for which he was afterwards distinguished." It was his good fortune to attract in Edinburgh the discerning notice of Sir James Gibson-Craig and his family; through whose influence, probably, he obtained professional employments of a kind which permitted him considerable leisure, and even gave him an opportunity of forming scientific acquaintances in London. His early tastes were towards the practical calculations of astronomy, such as occultations and ephemerides; and from his merits alone he was commended and patronized by Dr Thomas Young. Had that great man lived, his promotion to a scientific post would have probably been earlier. As it was, he obtained in 1831 an honourable, and, for him, lucrative appointment as astronomer at the Cape; at the sacrifice, however, of quitting his native country. In the thirteen months during which he held that situation, he performed an amount of first-rate work in practical astronomy which may bear a comparison with any similar effort. Charged with an instrument notoriously defective (Jones's circle), he had to examine its errors and their compensations; and it is no small credit to him to say that with such an indifferent tool he carried off a prize which had been the aspiration of so many astronomers before him—the determination of the parallax of a fixed star.

We must not go back to the history of this problem previous to the nineteenth century, when alone instruments were so far perfected as to give results by which a parallax, or displacement by perspective of a star in consequence of the earth's motion had a chance of being discovered, since it certainly does not amount to above a second or two, probably never attains even the former amount.

(307.)
had been investigated by Pond and Brinkley.

The earlier part of this century witnessed a memorable contest on this subject between Mr Pond, the Astronomer-Royal, and Dr Brinkley, afterwards Bishop of Cloyne; the former founded on observations

at Greenwich, the latter at Dublin, both with instruments of great power, being meridian circles of the largest size. It is sufficient here to note that Dr Brinkley attributed to some of the brighter stars, such as $\backslash alpha$ Lyre, a parallax of 2.5 (which, however, he afterwards reduced to little more than 1"), whilst Mr Pond could arrive at no such result. There is no doubt that Mr Pond was correct.¹

The first case of parallax which was determined with some certainty by the use of ordinary meridional instruments was that of $\backslash alpha$ Centauri, a bright star of the southern hemisphere, which was deduced by Henderson from his observations at the Cape long after they had been made, and what is perhaps still more satisfactory, made without reference to this particular question. The result, which gave to this star an annual parallax of 0.91 of a second, is believed to be correct, because it has been confirmed by Mr Maclear, Henderson's successor at the Cape. It may, however, be not unreasonable to desire that the observations should be repeated in another locality, and with a different instrument; for it has not been unusual (as in the case of Brinkley) to obtain under the same circumstances perfectly consistent, yet erroneous, and therefore inexplicable results.

The very considerable amount of parallax in this instance, corresponding to a large proper motion (3.6), gives a strong independent probability that it is not materially erroneous; and in this respect the most competent and impartial historians of science have given full credit to Mr Henderson for having, by superior skill in the use of his instruments, and the happy choice of an object, made a discovery which so many eminent observers had long sought for in vain.

Subsequent to the date of Henderson's observations, but before their publication, Bessel (whose biography we have given in Section 3 of this chapter) determined the parallax of the star 61 Cygni, in a different way. It was a happy suggestion of Galileo, that if two stars be selected apparently near one another, but really disconnected, and having very unequal magnitudes (therefore probably at very different distances from our system),—and if the apparent angular distance between the two stars be measured from opposite parts of the earth's orbit, it must sensibly vary by an obvious effect of perspective. It was in pursuit of this happy idea that Sir W. Herschel discovered the fact that the connection of such pairs of stars is often real, not apparent—but it was Bessel who, guided in his choice of an object by the critical character of the

Notice of Brinkley, Bishop of Cloyne.

¹ John Brinkley, Bishop of Cloyne, was born in England, and educated at Cambridge, where he was senior wrangler in 1788. He acted for a short time as Maskelyne's assistant at Greenwich, and was subsequently appointed Professor of Astronomy at Dublin, where he made many excellent observations, especially those on Nutation and Aberration. Brinkley ought to have been mentioned in Art. (30) of this Dissertation, as having contributed materially to the progress of the study of the Continental Mathematics in the United Kingdom.

considerable proper motion of an otherwise un-
conspicuous star, first reduced Galileo's theory to
successful practice. The details of this elaborate in-
vestigation have been considered by competent judges
as among the happiest specimens of astronomical in-
duction; and like Professor Henderson's, they have
also had the advantage of subsequent and still more
independent corroboration. Professor Johnson of
Oxford, using, like Bessel, a divided object-glass
micrometer (or heliometer, as it is not very appro-
priately called, never being applied to the sun), has
obtained results for the parallax of $\backslash alpha$ Cygni almost
identical with Bessel's, which was 0.35 of a second.
I am sorry that my limits will not allow me to
explain more particularly the details of Bessel's

method, which, however, may be found in several
modern works on astronomy.

M. Peters of the Observatory of Pulkowa has an-
nounced several additional parallaxes, all determined ^(311.)
with the aid of a meridian instrument. He has even
attempted to infer the mean parallax of stars of the
1st, 2d, &c. magnitudes, which M. Struve has com-
pared with his estimate of relative distance derived
from other considerations; but the numbers of as-
certained parallaxes, as well as their amount, are too
small, and the anomalies still too apparent (as in
the case of the star No. 1830 of Groombridge's
Catalogue), and the parallactic distances too often
inverted on the scale of brightness, to allow us to
attach much importance to these generalizations.

CHAPTER IV.

MECHANICS OF SOLID AND FLUID BODIES, CIVIL ENGINEERING, AND ACOUSTICS.

§ 1. WATT.—Condition of Practical Mechanics previous to the time of Watt. His genius for the application of Science to Practice. His successive Improvements on the Steam-Engine. Steam Navigation.

JAMES WATT may be considered as the most dis-
tinguished practical man of science of the last cen-
tury, or even for a much longer period. But this is
not all. Few men achieve such a reputation as his
without having done more than originate a great in-
vention for the use and benefit of mankind in all ages:
—He also taught men to raise the useful arts to a new
dignity, to marry them to genuine, unpretending,
and inductive science, to disparage ignorance and
empiricism, and to render the labours of the work-
shop subservient to intellectual progress.

I have attempted, in the first chapter of this Dis-
sertation, to place the scientific part of engineering in
its due relation to pure physics, and I have compared
the relation between them to that subsisting between
Mathematics and Physics—the one as an instrument,
the other as an end. Now it was this, in particular,
which made Watt the important character he really
was. He brought out the dependence of the former sub-
jects, as Newton and others had successfully taught
and demonstrated that of the latter.

To appreciate Watt's merit, we must compare the
purely mechanical contrivances of the period preced-
ing the date of his improvements on the steam-engine
with those of a similar space of time succeeding it.
In the former we find multitudes of contrivances on
paper—ingenious, indeed, but many of which could
not be executed; the greater part of the remainder
could not be carried into effect through want of know-
ledge in the inventor. We have large promises of

"semi-omnipotent engines," perpetual motions,
"quintessences of motion," and the like, mingled
with trivial mechanical toys; or we have elaborate
diagrams of mill-gearing, lathes, fountains, and saw-
ing machines without end, illustrated with showy and
expensive plates, but destitute, for the most part, of
the slightest novelty of principle, or truly mechani-
cal skill in application. Here and there, no doubt,
elegant and appropriate contrivances for communi-
cating or sustaining motion occur; but nearly all
the best forms of elementary machinery were of re-
mote antiquity, excepting those connected with clock-
work, which, including the great and truly scientific
application of the principle of Isochronism, formed
the only considerable step in philosophical mechanics
for very many years previous to the conception of
the steam-engine. The statical part of mechanics
had made more progress. Masonry and Carpentry
had attained a degree of perfection in many respects
admirable, under the Italian and Norman architects;
but the really difficult theory of machinery in motion
was little understood before Watt's day, and the
knowledge which then existed was unassociated with
practical skill or commercial enterprize. It could
be found only in profound treatises of theoretical
mechanics, and in experimental courses of natural
philosophy. The sources of power were almost ex-
clusively those derived from simple gravity and the
impact of fluids. It was not, indeed, the good for-
tune of Watt to be the first to employ the admirable

properties of steam as a source of power; but he was almost the first to study them as a philosopher, and to apply, with exemplary patience and skill, principles sought in the laboratory, to make available the most convenient of all motive forces. This was done by means of a series of contrivances so ingenious—so strictly connected by scientific relations—as to be a model of experimental research, quite as much as a triumph of mechanical art.

(315.) Philosophical spirit infused into practical mechanics by Watt.

Such combinations of theory and practice have now become far from rare. They have followed the march of physical learning, they have borrowed from it, and they have contributed to it. But it was Watt who chiefly gave the happy example. Himself by education and habit strictly a mechanic, he had the peculiar merit of apprehending the value of theory, and of acquiring a kind of knowledge then altogether uncommon amongst persons of his profession. He was no doubt a successful speculator, and a shrewd ingenious man besides; and this, his ostensible character, constituted possibly in the eyes of many his world-wide celebrity. But such considerations were little likely to influence the opinions of contemporary scientific men well qualified to judge, and least of all of eminent foreigners, who generally regard with little partiality the presumed commercial character of their insular neighbours. Dr Black, who was by no means prodigal of praise, termed the steam-engine, as improved by Watt, "an invention which is in its present state the master-piece of human skill," not "the production of a chance observation, but the result of deep thought and reflection, and really a present by philosophy to the arts."¹ Professor Robison, who knew Mr Watt intimately, was even more enthusiastic in his appreciation of his genius; and Sir Humphry Davy, in a speech manifesting a just estimate of his peculiar merits, did not hesitate to place him on a level with Archimedes.² But a testimony more authoritative and unbiased than any of these, is the fact that Watt was elected first a corresponding member of the French Institute, and finally one of the eight foreign Associates of the Academy of Sciences. This honour, to which so few can attain, which Newton once owned, and which now graces or lately graced the names of Young, Humboldt, Oersted, Brewster, and Robert Brown, is a sure passport to scientific immortality. Here, at least, no utilitarian pride, nor even the laudably patriotic emotion of gratitude to one who had proved, in more ways than one, his country's benefactor, can be supposed to have influenced in the remotest degree his election.

(316.) Having said thus much on the position to which Watt's inventions entitle him in the narrative of the history of science, we may refer with brevity to the generally well-known improvements of the steam-

engine, in which they mainly consist. In Sir John Leslie's Dissertation, a space of but a few lines has been devoted to them, which seems inadequate to their importance. For details, however, we must refer to the articles in the Encyclopedia where they are explained at full length.

No doubt we cannot in strictness call Watt the inventor of the steam-engine. The grand principle of rendering the heat contained in steam available as an economical source of moving power may be traced so far back that we lose the clue altogether in the obscure, or impracticable, or simply puerile shapes in which the idea was contained. Even in the time of Worcester (1663) we must be allowed to doubt whether the history of the steam-engine had outgrown the mythical stage; Papin, indeed, proposed a piston and cylinder in which the vacuum was produced by steam instead of by the air-pump (as already suggested or practised by Guericke); but Savery (1698) was the first who constructed a steam-engine, and applied it to the drainage of mines. His invention included the two capital properties of steam, its power of producing a vacuum by condensation, and its elastic force at high temperatures. A few years later the piston-form was introduced or re-invented by Newcomen and Cawley, as well as the valuable expedient of producing condensation by a squirt of cold water injected into the cylinder; and in this condition the Atmospheric Engine remained with slight improvement for above half a century, doing the work for which it was invented,—the pumping of water out of shafts (the pump being moved by a chain attached to the end of a horizontal oscillating beam),—wherever economy of fuel was unimportant. Such was the case at coal pits, but in other mines, usually situated remote from coal, it was of comparatively little use, on account of the enormous consumption of fuel.

(318.) JAMES WATT was born at Greenock in 1736, and, owing to feeble health, seems to have enjoyed little advantage from regular tuition of any kind; which was, however, to a great extent made up for by the intelligent spirit with which he acquired knowledge without assistance on a great variety of subjects. At the age of nineteen he proceeded to London, where he learned mathematical-instrument making, but he soon returned to Scotland, intending to pursue that business in Glasgow. Here he met with obstacles, but finally, by the patronage of Dr Adam Smith, Dr Black, and other professors, he was established as instrument-maker to the university within the college buildings. I mention these details because they show that Watt, as early at least as 1757, had been favourably noticed by the most celebrated professors then in Glasgow, and had received a special pledge of their good will.³ It is evident that the professors of Cho-

¹ Lectures, i., 181.

² Speech at Freemasons' Hall, preserved in Arago's Eloge of Watt, and in Davy's Works, vol. vii.

³ It appears from Mr Muirhead's work on The Origin and Progress of the Mechanical Inventions of James Watt (published since the greater part of the text of this section was written), that Watt's introduction to the college took place "through the instru-

and connection with Dr Black. mistry and Natural Philosophy (Dr Black and Dr Dick), who had obtained for him his appointment, must have been also his chief academical employers; and we may safely conclude that Watt constructed much of Dr Black's apparatus, as he was certainly admitted to intimate and confidential intercourse with that great and amiable man. Dr Black removed to Edinburgh in 1766, but the friendship which he had extended to the young mechanic remained unaltered during their joint lives, although their personal intercourse must have been extremely slight after that time. Now Watt's first thought of improving the steam-engine dates (as we shall see) from the session 1763-4, the last but one of Black's stay in Glasgow; their intimacy was therefore fully established by that time; and we find Watt acknowledging his "obligations to him for the information received from his conversation, and particularly for the knowledge of the doctrine of latent heat;"¹ and we also find that during Watt's experiments on steam subsequent to the last-mentioned date, Black was cognizant of them, and assisted in their contrivance.² Moreover, Watt admits Dr Cullen's well-known experiment of reducing the temperature of ebullition under the air-pump to have been one of his starting points in the improvement of the steam-engine.³ Yet this experiment was not published until 1770,⁴ and Watt must have heard of it from Black himself, or from some one attending his lectures, where the fact was almost certainly mentioned. Whether formally a student of Dr Black's chemistry class or not, it is therefore evident that Watt enjoyed advantages in the prosecution of experimental physics which nineteen-twentieths of enrolled students never attain. He had the privilege of unreserved personal intercourse, amounting at last to intimate friendship, with the first authority of the last century on the subject of Heat, and one of the

most cautious and accomplished of inductive philosophers.⁵ Watt's eminent merits, and doubtless his merits alone, gained him this happy position; but had he remained either at Greenock, or with the optician in Cornhill, he might have failed to combine so admirably as he did the character of the practical man and the philosopher. His place in science was His position in Glasgow College. well typified by his position in Glasgow. His was "the workshop within the College." Whilst the laboratories of the classes of Chemistry and Natural Philosophy must have been his familiar resort, his own rooms were frequented by the most intelligent students, including his contemporary Dr Robison, where subjects of science particularly connected with mechanics were diligently canvassed. The extent of his knowledge and the variety of his resources were fully tested, and the result, as stated by the generous pen of Robison, was a conviction of the superiority of Watt in these respects to any of his contemporaries.

It is a fact worthy of note that the immediate occasion of Watt's improvements was the commercial consideration of economy. In 1763 or 1764, being called on to repair a model of the Atmospheric Engine in the Natural Philosophy class at Glasgow (which model is still preserved), he found the amount of steam expended in heating the cylinder at each stroke to be so great that the boiler was insufficient to supply it properly. He then commenced experiments on the amount of steam thus consumed, and on the means of diminishing it. Though the primary object was the repair of a model, it is not to be doubted that Mr Watt had in view the practical improvement of the engine on the great scale, with the use of which in the coal-fields of the west of Scotland he was probably familiar, having the intention of becoming himself a civil engineer, and being already acquainted with the writings of Desaguliers and Belidor.⁶

mentality of his mother's kinsman Mr George Muirhead, who had then (1754) just exchanged the professorship of oriental languages for that of Latin." It will be seen that this refers to a period antecedent to his journey to London.

¹ Letter to Dr Brewster in Robison's Mechanical Philosophy, ii., p. v.

² Robison's Mechanical Philosophy, ii., 115, note (by Mr Watt).

³ Ibid., p. vii., and 114, note.

⁴ In the Edinburgh Physical and Literary Essays, vol. ii., Dr Black (who was intimate with Cullen) knew of it at least in 1767. Lectures, i., 525.

⁵ It is perhaps unfortunate that Mr Watt, when on the verge of four-score, contrary to his own intentions, but yielding to "the representation of friends," recorded a complaint that his best friends, Black and Robison, had refused him his due share of merit in the improvement of the steam-engine, and disclaimed, as injurious, the appellation of "a pupil of Dr Black" (letter to Dr Brewster in Robison's Mechanical Philosophy, vol. ii., p. v.) Admitting (as he does on the same page) that the doctrine of latent heat was due to Dr Black, and that he first learnt it from him, he adds, "this theory did not lead to the improvements I afterwards made in the engine," p. viii. No one ever ascribed to Black the beautiful invention of a separate condenser, but even Watt's most ardent eulogists (Lord Brougham and M. Arago) admit that the theory of latent heat "forms the key to the economical appreciation of the steam-engine." So far was Watt from being independent of Dr Black's assistance in this matter, that he himself tells us twice over, in substantially the same language, that when in the course of his experiments on the model engine in Glasgow College in 1764 he was "at a loss to understand how much cold water could be heated by so small a quantity in the form of steam," he "applied to Dr Black, and then first understood what was called latent heat" (Robison, ii., p. v.; also p. 116, note). Mr Watt might have been the discoverer of latent heat, and the solver of his own dilemma, had not Dr Black been at hand. It is the highest compliment we can pay him to say that such an achievement was not too much to expect from him.

[These conclusions are fully substantiated by the interesting documents lately published by Mr Muirhead in the Correspondence of Watt on the Composition of Water, p. 6, and in his Mechanical Inventions of James Watt, vol. i., p. lxxxii.; vol. ii., pp. 116, 118, 119, 275. In the last-cited passage Watt himself fixes the date and manner of his receiving from Dr Black his knowledge of the doctrine of latent heat. He says, "I myself never attended his (Dr Black's) lectures; but the doctor explained his doctrines to me about the year 1763." Note added during printing.

⁶ See Mr Watt's own narrative in Robison's Mechanical Philosophy, ii., 113, note; and in Mr Muirhead's Mechanical Inventions of James Watt, vol. i.

When Watt discovered by means of his model that the condensation and loss of steam in the cylinder at every stroke of the piston exceeded that which is usefully employed in producing the vacuum, he proceeded very methodically to ascertain the chief numerical data or constants upon which the working of steam-engines depend; as, the bulk of steam of given elasticity compared with that of the water producing it, the elasticities of steam at different temperatures, especially above the boiling point, the evaporating power of a pound of coals, and finally, the expenditure of steam and of injection-water for a single stroke. The last, the amount of cold water required effectually to condense the cylinderful of steam, appeared so very large that it at once betrayed the source of the wasteful expenditure of fuel. All the heat which was abstracted by the injection-water had to be supplied afresh at a vast expense of steam, and consequently of coal, before a fresh stroke could be made. The mere amount of steam necessary to fill the cylinder and elevate the piston was a trifle in comparison. He naturally ascribed it to the absorption of heat by the material of the cylinder, which no doubt was an important element; and he tried cylinders of different materials, and ascertained (in most instances for the first time) the specific heats of the substances used. But still the amount of injection-water remained unaccounted for, and the regulation of it was the great difficulty of the old engine. If a tolerable vacuum was desired, the cylinder must be flooded with cold water, and all the heat abstracted must be restored before the next stroke was produced. If, on the other hand, the injection was imperfect, so that the condensed steam and water together retained a high temperature, an atmosphere of vapour so elastic spoiled the vacuum, and in many cases the working efficiency of the engine was reduced to about half of the atmospheric pressure.

The theory of latent heat, showing that steam contains as much heat as if the water yielding it had been raised through 1000° of temperature, gave a clear explanation of the enormous amount of injection-water necessary. How to escape from the difficulty was another question, which was solved solely by Mr Watt, who, early in 1765, first conceived the idea of retaining the cylinder always at the boiling heat, and effecting condensation in a separate vessel kept constantly as cold as possible, in which injection should also take place, and into which the steam of the cylinder would spontaneously rush, on a communication being opened at the proper instant. The success of this device was complete, and it was speedily followed by a series of admirable and mutually de-

pendent improvements which Watt introduced. To keep the cylinder as warm as possible, it was first provided with a cover, through which the piston-rod passed by means of a stuffing-box. Then steam itself was employed to press down the piston, instead of atmospheric air which cooled the cylinder injuriously.

Having thus steam on each side of the piston, the step was easy to make the engine a double-acting (322.) Double-setting engine. so that the volume of steam which by its elasticity impelled the piston during its descent, being thrown into the condenser, produced the vacuum necessary to make the ascending stroke equally advantageous.¹ But in pumping-engines, which had hitherto been the main employment of steam, or fire engines (as they were then called), it was sufficient that the power was produced in one direction, that is, to lift the pump-rods;² but the double action of a pushing and pulling force was evidently applicable to every sort of machinery by the use of a crank and fly-wheel. A difficulty, however, was experienced in applying this push-and-pull movement to the oscillating beam which had so long formed a part of the atmospheric engine, and which it was so convenient to use. Mr Watt obviated it with his customary sagacity, and succeeded in inventing a connection of rigid (323.) Parallel motion. jointed links which effectually united the piston-rod with the end of the beam, the former moving in a vertical straight line, the latter in a circular arc, so that pressure was transmitted in the manner of either push or pull without the slightest practical inconvenience. No doubt the locus of the motion of the piston-rod head is not rigorously straight. It is, when fully developed, a curve of the fourth order, resembling a very elongated figure 8; but the portion of the curve in use is practically straight. This happy solution, the most elegant perhaps in the science of pure machinery, or kinematics, was not the result of a formal mathematical research, but presented itself to the mind of Watt, as he tells us, almost intuitively. It became from henceforth an important element of mechanical combinations.

Other methods of producing direct rotation occurred to the ingenious mind of Watt, and many of them were shown to be practicable by being actually executed in model size. But the next great practical improvement introduced by him was the principle of expansive working, whereby a saving of steam is produced (particularly in single-acting engines), second in importance only to the result of the separate condensation. This consists in allowing the steam to flow in upon the piston during only the first part of the stroke; the rest being completed partly

¹ This happy idea seems to have been suggested by Dr Small of Birmingham in a letter to Watt, of date 5th November 1769. "I see no reason . . . why you should not condense at both ends of your cylinder, and drive all before you, back stroke and fore stroke."—Mechanical Inventions of James Watt, i. p. 81. The uncommon sagacity of Dr Small, and the important services which he rendered in many ways to Watt, come out clearly for the first time in this work.

² So associated were these engines with the operation of raising water, that we find they were at one time used to supply common water-wheels with a constant stream, and thus to produce continued motion for mill-work, &c.

by the inertia of the moving machinery, but chiefly by the continuous though constantly diminishing elasticity of the steam as it expands and fills the entire cylinder.¹ It is evident that, in the extreme case, the work required (in pumping) will be accomplished if the piston reach the end of the stroke just as its momentum is exhausted, and that, on the other hand, if the pressure of the steam remain constant throughout, the velocity will be a maximum just when the work being done by the machine is suddenly completed, and so much moving power will be wasted. A medium distribution produces the most favourable result in practice, and it depends entirely upon the ratio between the resistance and the effective pressure of the steam what this most favourable proportion will be. In the single-pumping engines of Cornwall, in which the economy of fuel is carried to the highest perfection, it frequently happens that not more than one-eighth of the stroke is performed under the full pressure of the steam, which acts expansively during the remainder of the stroke; but these engines, though condensing, are moved by steam of four times the elasticity of the atmosphere. The Cornish engines have been very elaborately perfected, and they have been more carefully tested in respect to performance than any others. It is perhaps not too much to say that they do not contain a single contrivance of any importance (beyond of course what they have in common with the atmospheric engine), which was not the unaided invention of Mr Watt.

In the Cornish engine we see the Energy of Heat rendered available to an extent which the inventor himself would at one time have thought scarcely credible. The combustion of a bushel of coal which in a Newcomen's engine improved by Smeaton was capable of raising 3,000,000 pounds through one foot, in Watt's improved pumping engine raised 20,000,000 pounds the same height. But by the indefatigable skill and perseverance of engineers the Cornish pumps now yield at least five times the last amount. This, however, is not the place to enter upon these details, nor can we stop to particularize the other and various mechanical inventions introduced by Watt in the form of valves, governors, and steam-indicators. Still less can we enlarge upon the endless and still multiplying applications of this admirable moving power, which is as capable of superseding the greatest natural forces hitherto applied by man to the useful arts, as it is adapted by its easy regulation to replace human industry in the most delicate operations; "the trunk of an elephant, which can pick up a pin or rend an oak, is as nothing to it."²

Of the applications of the steam-engine with which (325.) Mr Watt was less immediately connected, its adaptation to locomotion in the case of ships in the last century, and in that of railway trains in the present, have been the most striking, and fraught with consequences the most important to mankind. Of the latter we shall have occasion to speak in a future section; of the origin of steam navigation we may here say a very few words. Passing over projects which never were realized, of moving barges by steam, or other inanimate power, against wind and tide,—such as those of Worcester, Papin, and Hulls,—we find that the first experiment entitled to be called successful was made by Mr Miller of Dalswinton in Miller—Scotland, conjointly with Mr James Taylor, tutor in Taylor—his family, who together formed the project of moving vessels by means of paddle-wheels driven by a steam-engine, and realized it with the aid of Symington, a practical engineer. As we shall also find in the case of steam-carriages, the idea of the application of the steam-engine to move ships was already a familiar one to the minds of many persons about the middle of the last century. To put it in practice with advantage was the step required. Mr Miller's first boat was launched on Dalswinton Loch in Dumfrieshire, in October 1788, and attained a speed of five miles an hour. The subject was pursued by Symington and others. In 1789 a larger vessel was propelled on the Forth and Clyde Canal. Subsequently, however, the invention languished. The want of co-operation, of capital, and ingenuity, naturally extinguishes many valuable inventions. Watt himself was only rescued from the same difficulty by the unusual intelligence of Boulton and Small, his coadjutors at Birmingham. Symington was less fortunate, as well as probably less meritorious; and though it is well established that Fulton, who passes on the other side of the Atlantic for the inventor of steam-ships, had seen the reliefs of Symington's second experiment, we must do the Americans the justice to say that the application of steam to navigation first flourished in the United States. In 1807 Fulton started a river boat with an engine of Boulton and Watt. In 1813 the example was tardily imitated on the Firth of Clyde. The subsequent improvements need not here be specified. They have been very great and striking, but with the exception of the recent substitution of the screw-propeller for paddle-wheels, they scarcely involve any new principles.³

We may briefly close what we have to say of Mr (326.) Watt personally. His health was feeble from childhood, but being blessed with much calmness of temper he prolonged his life to a great age, and passed through its struggles, though they were to him con-

¹ The double-acting engine was planned in 1774 or 1775. The expansion principle was first used in 1776; the parallel motion was patented in 1784.—Watt's Notes on Dr Robison's Article. Robison's Mech. Phil., vol. ii.

² Lord Jeffrey.
³ For further details on the history of Steam Navigation, see that article in the Encyclopædia; and Mr Bennet Woodcroft's work on the subject, London, 1848.

siderable, without serious injury. Vexations lawsuits connected with his valuable patent rights afforded him but too much occupation and anxiety. He wrote little, and he did not covet fame. He was fond of chemistry as well as mechanics, and was well acquainted with the theory and practice of that science as it then existed. He put forth opinions on the chemical constitution of water which, in the judgment of some, entitle him to contest priority in the discovery with Cavendish. He must have been versed in mathematics in his early years, as we learn from his friend Dr Robison, but there is no evidence that he ever attempted a strict theory of his own engine. Nor were his successors in this respect more fortunate. The best practical writers on the steam-engine, up even to a late period, gave most rude and inaccurate rules for computing its effects; and it is to a Frenchman, M. de Pambour, that we owe the first philosophical, and, at the same time, elementary analysis of this noble machine.

(327.)
Personal character of Watt, continued.

Much light has been thrown upon the character of Watt by the recent publication of his correspondence by Mr Muirhead.¹ We there see the pressure of physical infirmity, and mental despondency and indifference, under which he laboured from boyhood. We learn the accumulated difficulties, arising from the backward state of the mechanical arts in his time, which delayed for years the successful prosecution of his happiest, and what would appear in our day most easily realized, conceptions. We see him at times ready to abandon fame and profit for the enjoyment of the humblest competence with tranquill-

ity. We find him, mechanic though he was, shrinking from possible collision with the opinions or interests of others, and in his early as in his latest days, solicits to avoid responsibility. He had, in a word, throughout, the finely strung susceptibility of a man of genius, singularly at variance with the necessity he was under of pushing his way in the world, and of turning his inventions to the best commercial account. Providentially he was thrown in the way of friends to whom, by his private character, he was greatly endeared, and who supplied the elements necessary to the successful prosecution of his schemes. The sanguine zeal of Roebuck, the commercial sagacity of the capitalist Boulton, and not least, the sympathizing friendship of Dr Small, who was well fitted by his character and attainments to mediate between Watt and the other two, were all essential to the realization of the improved steam-engine.

When Mr Watt was finally relieved of the oppression and chicaneries of his opponents in the courts of law, he was settling down into a peaceful old age. (328.)
Close of his life.

He probably hoped to live over again some of the scientific passages of his youth in sympathy with his second son, Gregory, who possessed a decided taste for science, but was unfortunately early cut off. Respected and beloved by a large group of friends, many of whom survived him, and admired by a far wider circle, he died at Heathfield, near Birmingham, 25th August 1819. Statesmen, philosophers, and men of the world, united in extolling the worth of his character and the greatness of his genius.

§ 2. ROBISON.—Application of Statical Principles to Engineering—especially to Practical Masonry. COULOMB.—Friction—Force of Torsion.

(329.)
Robison a practical philosopher.

The name of Robison may perhaps not appear to be sufficiently identified with any great discovery to merit a place in this condensed sketch of the progress of science biographically illustrated. Were there no other claim, I should consider it a sufficient one to entitle him to at least a brief notice, that he was by far the most important, and, as M. Arago has justly called him, "most illustrious contributor" to the earlier editions of the Encyclopaedia Britannica. But he was also a philosopher in a high sense of the word. His knowledge was multifarious in no ordinary degree. He had little of pretension to originality, yet he brought to bear upon matters of science an unfailing amount of excellent common sense, and his personal acquaintance with the Arts which may be called Philosophical far exceeded, I imagine, that of any man of his time. Though not without his prepossessions, he was generous in the highest degree in his estimation of others, who in

some sense might have been considered his rivals; he was eminently patient in his study of the works of his contemporaries, and in his published writings he laboured to render the results generally accessible to ordinary readers, by means of laborious abstracts, intermingled often with highly original views; and he explained them with conscientious energy in his lectures to the students of Natural Philosophy, whom for thirty years of his life it was his pride and pleasure to instruct. Amidst these congenial labours he found little time for making prolonged original trains of experiment, though the specimens which he has almost incidentally left us give the fullest proof of his ability in this respect; and the explanation is, I have no doubt, to be found in the peculiar circumstances of his early life. Until his settlement in Edinburgh, he passed his time in a series of active pursuits, having much more of the character of stirring practical life than of literary ro-

¹ The Origin and Progress of the Mechanical Inventions of James Watt. 3 vols. 1856.

pose or scientific contemplation. He was thirty-five years old when he began to lecture in Edinburgh; and though an excellent working age, it is rare indeed to find that the talent of discovering new truths has been formed and developed so late in life. A major part of the great steps in science have been taken at even a much earlier period.

(330.) His eminent services as an expositor of science.

That Robison possessed all the elements of an original thinker we shall presently endeavour to show; but had his excellences consisted alone in those we have specified, he would have been a person eminently useful in forwarding the march of science. In fact, a few more such authors in every generation would be cheaply purchased by the postponement of some second-rate discoveries. Men—and we include men of science—are in too great a hurry to push on (actuated often by a morbid love of praise) to acquire something they may call their own, whilst they are little acquainted with the important contributions of their rivals in the race of fame, and of the predecessors to whom they really owe so much of what they may choose to consider their peculiar property. To methodize knowledge from time to time—to present discoveries in a form different from that in which they were first published, and thus connect them with what is already known and what remains to be clearly proved—these are real services to science which contribute in a very essential manner to its progress. Robison was a teacher, not only to the youth of his native country, but to the men of science and of practice of all countries, and of many succeeding years.

(331.) His early life and friendship with Watt.

JOHN ROBISON was born in 1739, entered the University of Glasgow at the early age of eleven, and graduated at seventeen. His early tastes were directed towards Natural Philosophy. He studied Mathematics (as in after life) chiefly with reference to its applications. Whilst at Glasgow he formed the intimate acquaintance of James Watt, then a practical instrument maker. In a passage of a private paper, published first in Arago's Eloge of Watt,¹ Robison, with his characteristic generosity, describes his mortification when "yet a young student" at finding him much his superior in mathematical and mechanical knowledge; but M. Arago has omitted to state that Watt was, at least, three years the senior, which at such an age, and on such subjects, might make all the difference between a beginner and a proficient. But the truth is, that with a rarely generous rivalry in excellence, each esteemed the other most; for we find Watt bearing a similar testimony to the supe-

riority of his friend and junior: "I was happy to find in him [Robison] a person who was so much better informed on mathematical and philosophical subjects than I was."² It was by Robison that Watt had his attention first directed to the steam-engine; their correspondence appears to have been frequent during their joint lives; and near the close of his life Watt acknowledged "his obligations to him for very much information and occasional assistance in his pursuits;" and in the Edinburgh professor he found not only the zealous defender before a court of law of his rights as an inventor, but also the first who expounded methodically the principles and details of the steam-engine in a manner which, at least until lately, had not been superseded.

(332.) His adventures.

In 1758 he left Glasgow, and the following year he went to sea as tutor to a son of Admiral Knowles. His life in a man-of-war, during which he saw some active service in Canada, was favourable to the development of his practical turn of mind, and doubtless gave him an interest in seamanship, naval architecture, and other subjects, which he afterwards turned to good account; and a subsequent expedition to Jamaica, for the trial of Harrison's Time-keeper, exercised him in some of the practical parts of astronomy. He returned, however, to Glasgow in 1761, and attached himself with such success to the study of Chemistry under Dr Black that he taught the Chemical Class in the university for several sessions.³ But his active life was not at an end. In 1770 he accompanied his first patron, Admiral Knowles, to Russia, and for some years was employed, first as his secretary, superintending improvements in the marine establishment, and afterwards as professor of Mathematics in the naval school of Cronstadt. He spoke and wrote the Russian language with facility,⁴ and performed his duties to the satisfaction of all. But in 1774 he and his appointment could not resist the honourable invitation which he received to fill the chair of Natural Philosophy in the University of Edinburgh, where he spent the remainder of his life, which terminated in January 1805, amidst incessant literary occupation, even when repeated attacks of a painful disorder had prevented him from personally continuing his lectures.

(333.) Character of his articles in the Encyclopædia Britannica.

This brief sketch⁵ of a career rather unusual for a man of science throws light upon Robison's peculiar merits. He had extensive, and then uncommon, opportunities of acquiring information, of seeing various countries, and of noticing their physical peculiarities; of being introduced to their society and literature;

¹ And since, in extenso, in Muirhead's Mechanical Inventions of James Watt, vol. 1., p. xli.

² Ibid., vol. ii., p. 293.

³ The MS. of these lectures, written with curious care, but with the use of continual abbreviations of the larger words for the sake of compression, is now in my possession, having been given to me by his son, the late Sir John Robison. I have also many others of his MSS., which for the most part seem to have been printed in some form or other. Dr Robison was an indefatigable penman, and wrote and re-wrote his lectures with great labour. He also made elaborate analyses of his reading, but the bad habit of contracting words remained with him through life.

⁴ The private marks on his MSS. are often in the Russian character.

⁵ A much fuller one will be found in Playfair's Works, vol. iv.

of taking part in the actual execution of scientific designs; of studying foreign languages; and of methodizing his knowledge for the purposes of instruction. Every one of these influences may be clearly traced in his writings, which will challenge comparison with those of any English writer, except perhaps Dr Thomas Young, for variety and fulness of information, and for the general soundness and strongly practical character of the mechanical knowledge with which they abound. They would also doubtless have appeared to far greater advantage had they been the product of his most vigorous days; but he was little known as a writer until the year 1793, when he commenced a series of important contributions to the Encyclopaedia Britannica, embracing at least forty-six articles, all scientific, some of great length and elaboration, which were published during the succeeding eight years, a period when he was seriously afflicted with chronic disease. They embraced disquisitions on general philosophy, as in the articles Philosophy and Physics; of strict science, as in Astronomy, Dynamics, Projectiles, Pneumatics; of the more experimental sciences, as in Magnetism, Electricity, Sound; of the art of music (with which he was practically conversant), in Temperament, Piano, &c.;—but his great strength lay in the articles in which just mechanical principles were applied successfully, and often with marked originality, to practice, as in Arch, Roof, Carpentry, and Strength of Materials; Resistance, Rivers, and Waterworks; Seamanship, Steam and Steam-Engine; Machinery, Telescope, and Watchwork.

(334.) His Mechanical Philosophy.
The last-named articles (most of which have been collected in a compilation, edited by Sir David Brewster, in four thick volumes, entitled Robison's Mechanical Philosophy) constitute a body of knowledge in civil engineering which has not yet been surpassed in clear exposition of physical principles and their application—in the extensive acquaintance it shows with the details of practice—and in proofs of elaborate and impartial study of authors, both British and foreign, on the subjects of which it treats. Several of these articles have been resorted to by the most accomplished engineers of our own time as stores of sound experience, and they have been lavishly borrowed from by some writers on similar subjects, occasionally without acknowledgment. The original matter which they contain is not always easily separated from that which is compiled. In every instance Dr Robison lays claim to less than his own share, and is so scrupulous in quoting the names of other authors, that he has unquestionably received less reputation from these Essays than he deserves. He writes like a man who has himself used the saw and hammer, who has had the responsibility of success or failure,

who might have been called upon for a scientific opinion (of far more weight than what generally goes by the name) on almost any practical subject upon which he might have been consulted; and that not merely in its general outlines, but in its strictest details; not only the design for a roof or a centre, but the scantling of the timber to be used; the minutiae of a pump or hot-air stove; the curvatures of an achromatic object-glass; the temperament of a piano; or the angle for the pallets of an escapement. In such matters nothing important either in the theory or the practice of his times escaped him. His opinions were very generally formed on original considerations, supported by experiments equally well devised and carried out. His method of finding mechanically the relation between the intrados and extrados of a properly balanced arch by means of suspended pieces of chain is as ingenious as it is elementary; but his observations on the manner in which stone arches, when overloaded, break up, established on direct observation and experiment, unquestionably gave a just foundation to the theory of masonry, till then so generally and erroneously treated of by mathematicians with the preposterous abstraction of the forces of friction and cohesion, the action of which in many instances vastly exceeds the direct effect of gravity.

But on this we must not dwell. Robison's articles (335.) on Electricity and Magnetism are deserving of nearly equal praise with reference to the state of knowledge of the time. Probably there was not one author of merit on these subjects, whatever his country or language, whose works he had not laboriously consulted and analysed the conclusions; whilst a multitude of ingenious experiments give evidence of the skill and patience of the writer. To Dr Robison we are, indeed, indebted for the approximate knowledge of the primary law of electric attractions and repulsions; for a careful consideration of those beautiful curves formed by iron filings round magnets,¹ to which such an enlarged importance has since been given by the beautiful generalizations of Mr Faraday; and I believe likewise for the first suggestion of combining the voltaic elements in a pile or column.² Dr Robison, though for many years secretary of the Royal Society of Edinburgh, published little in its Transactions, and almost nothing, so far as I am aware, in any periodical work. Hence his admirable adaptations of experiments were little known beyond his own class and friendly circle.

Of his more abstract mathematical writings we (336.) need say little. He was thoroughly acquainted with the works and methods of Newton, and with nearly all those of the same school, particularly of Boss-covich. He laboured incessantly to reduce the demonstrations of the higher mechanics and astronomy.

¹ Indirectly we owe to him also the first exact determination of the mathematical properties of those curves, made at his request by Mr Playfair.—Robison's Mech. Phil., iv. 350.

² See the Fifth Dissertation (by Sir John Leslie), p. 739.

my to their simplest and most geometrical elements. But the result (as might have been foreseen) was in many cases a wearisome prolixity; and originality could not be expected in these departments without applying the continental improvements in analysis: not that Dr Robison overlooked these; but he took little pleasure in them; and as regards Physical Astronomy he adhered to the older methods. Having been so long at St Petersburg, the writings of Euler must have been familiar to him, as indeed they evidently were as far as regards all subjects connected with mechanics: he justly, however, considered Euler as a superficial natural philosopher, though an incomparable mathematician.

(337.)
Robison as a lecturer.

As a lecturer in his own department, Robison was the most eminent of his time, at least in Britain. That his courses were not considered popular will easily be understood from a slight inspection of his writings. The demonstrations were long and copious, but too rapidly delivered. "The singular felicity of his own apprehension," says Mr Playfair, "made him judge too favourably of the same power in others." The lectures must have abounded in practical details, which ordinary students rarely appreciate; and they were deficient in experiments, which unquestionably arose from no want of the ingenuity necessary either to invent or execute them. On the other hand, the effect of his discourses was greatly enhanced by his striking and energetic delivery, and by the stores of his memory, which often recalled the incidents of the stirring life in which he had once been engaged; and to the more thoughtful and philosophic they were rendered at once attractive and elevating in no ordinary degree by the strain of fervent thought by which they were accompanied, and the impress which they frequently bore of the pure morality and exalted piety of their author.¹

(338.)
His philosophical character.

Apart from his local usefulness as a professor, we regard Dr Robison's place in science as eminent chiefly on account of the sagacity with which he applied knowledge to practice, and analysed complicated effects of force as manifested in engineering constructions. This he did so ably as to guide future practice, and to reflect much light on the theory of solids more or less elastic and tenacious, and subject to the intricate strains which gravity produces. The difficulty and merit of these investigations will best be gathered from the slow progress of his successors in the same field. The criticism with which his writings abound on the theories of even his more celebrated contemporaries and predecessors show remarkable acuteness, patience, and independence of thought. We may perhaps sometimes think him prejudiced, but his decisions are never uttered without an

elaborate statement of reasons, nor ever sullied by the suggestions of jealousy or self-conceit. Had he been an accomplished analyst he must have been less distinguished in the equally important walk in which he stood pre-eminent. The limits of human life and faculties prevents universal attainment, but he was surely no mean philosopher of whom the sexagenarian James Watt could say, "He was a man of the clearest head and the most science of anybody I have known."²

Amongst the contemporaries of Robison was one (339.)
Coulomb on friction and passive strength.

whose requirements were in many respects very similar to his own, and who contributed, in a degree second to no other philosopher of his day, to promote sound views in the very same branches of science. CHARLES-AUGUSTIN COULOMB (born 1736, died 1806) was, like Robison, addicted through life to practical enquiries, and was intimately acquainted with all the details of the civil engineering of his day. To him we owe a correct knowledge of the laws of friction in most ordinary cases, and the right application of them to the theory of machines, and to that of the stability of structures. In a very remarkable paper, published in 1776, he analysed, from the basis both of theory and experiment, the manner in which columns of masonry give way under longitudinal pressure; not, as had previously been supposed, by flexure under the imposed weight, like a steel spring or a rod of deal, but by the sliding of one portion of the column over another, at an angle determined by the cohesion of the stone, and capable of being reduced to a problem of maxima and minima. This important principle is now known to apply to wrought and cast iron, and many other substances.

One of Coulomb's happiest investigations was on (340.)
On torsion.

the force of torsion, or the resistance of wires to twisting; which he showed to vary directly with the angle of torsion, inversely as the length of the wire, and directly as the square of its section. These mechanical principles he applied with address to determine the viscosity of fluids, and with still more consummate skill and success to the measurement of electrical and magnetic forces by the construction of a Torsion Balance, similar in principle but anterior to that employed by Michell and Cavendish for estimating the force of gravity.

Coulomb's researches on electricity (which have (341.)
His researches resembled Robison's.

been only partially published) will be made the subject of discussion in another chapter of this Dissertation; they were the most sustained and elaborate of his investigations, and display very considerable mathematical resources. In this respect he had a superiority over Robison, who, as we have seen, cul-

¹ See Dr Chalmers' Life, vol. I. That great man had a peculiar veneration for Dr Robison, and is understood to have received impressions from attending his lectures which materially influenced his future life.

² Mechanical Inventions of James Watt, ii., p. 290.

tivated precisely the same branches of science which gave distinction to the career of Coulomb. When we compare the two philosophers, we find that the former was the more discursive reasoner and experimenter, and was diverted, perhaps by the copiousness of his erudition, and the attention with which he studied the works of others, from doing full justice to his own original powers. The latter excelled as a ma-

thematician; he concentrated his efforts more methodically, and displayed the results to the world (so far as they were published) in a more consecutive and lucid form. In uprightness of character and high morality, the two philosophers bore a marked resemblance. They were both sufferers from bad health, and they died within about a year of each other, at nearly the same age.

§ 3. THOMAS YOUNG—Strength of Materials, and Art of Construction (continued).—TELFORD—Introduction of Iron into permanent Structures. Suspension Bridges.—Tredgold; Mr Hodgkinson; M. Navier.—Mr ROBERT STEPHENSON—Tubular Bridges.

(342.) Difficulty and importance of the enquiry into the mechanical properties of solids.

It is a circumstance not unproductive as to the progress and achievements of science, that the greatest modern philosopher who preceded Newton—Galileo—and one of the most eminent, if not the most eminent, of his successors—Young—should have laboured with minute and practical care, and with corresponding success, on a subject apparently so humble and mechanical as the Strength of Materials, and the Resistance of Beams to fracture. Newton himself condescended to swing pendulums, and to observe the collisions of elastic worsted balls. It is sufficient here to advert to the exceeding interest of enquiries which throw so much light upon the internal constitution of bodies, and in some instances intimately connect them with the laws of vibration of elastic media, to which so much of Modern Physics is intimately allied.

(343.) Progress during the eighteenth century.

The eighteenth century was in this, as in so many other departments of science, sluggish and mechanical, or else abstract and ultra-geometrical. The learned labours of Euler and the Bernoullis on elastic curves, and the strength of pillars, were for the most part elegant mathematical amusements, and with the exception of the experiments of Muschenbroek in the earlier half of the century, and the skilful but more limited researches of Coulomb at its close, little valuable in the way of precise theory or of accurate data derived from practice had been added to this important branch of mechanical engineering.

(344.) Thomas Young.

Robison, indeed, with the peculiar tact and skill which I have already ascribed to him, wrote several papers (contributed to an early edition of the Encyclopædia Britannica, and printed in his collected works) full of acute observation and reasoning, adapted to the imperfect experiments of his time, and connected by sound scientific deductions, which are still well worthy of careful perusal; but it was to the penetration of Dr THOMAS YOUNG,¹ who partook strongly of Robison's mechanical tastes, whilst he surpassed him in facility of mathematical resource, that we owe a great revision of the doctrine of the strength of materials. In the "Syllabus of Lec-

tures" (1802), into which he condensed, in a manner peculiar to himself, an incredible amount of positive knowledge; in the Lectures themselves (1806), with the admirable "Catalogue of References;" and in the articles on "Bridges," and the supplementary propositions on "Carpentry," which he contributed to this Encyclopædia—we find (stated, as usual, not without some obscurity) a multitude of theorems and problems embracing the whole principles of construction, and based upon mechanical laws and the most probable interpretation of experiments.

The forces tending to alter the figure or dimensions of substances usually called solid may be thus classified: (1.) Extending forces, or such as produce elongation in a body when applied in a direct manner. (2.) Compressive forces. (3.) Force producing detraction, or the slipping of one portion of the substance over another. (4.) Force producing flexure. (5.) Torsion or twisting force. The resistance of bodies to extension was examined by Hooke and Gravesande, and is held to be directly as the area of section of the body, and to increase directly as the amount of elongation produced, at least within certain limits. The measure of this resistance Young termed (not very happily) Modulus of Elasticity, expressing the force required to produce unit of elongation (or to double the length) of a prism of the substance under experiment. This quantity may be measured either by the length of a depending prism of the substance which would produce the requisite strain, or more simply by the strain expressed in pounds or tons, which, supposing the elongations to increase without limit as the extending forces, would double the length of the prism under experiment. Thus, in round numbers, a bar of wrought iron an inch square will be extended $\backslash frac\{1\}\{10000\}$ part by a pressure of one ton—hence the modulus of elasticity is about 10,000 tons. The elasticity of wrought iron remains perfect to about half the breaking weight, after which the elongations appear to double for each addition of about $\backslash frac\{1\}\{10\}$ or $\backslash frac\{1\}\{20\}$ of the breaking weight. Thus, in a recent experiment by Mr Edwin Clark, a bar of

¹ I reserve to the chapter on Optics a fuller account of Young and his writings.

wrought iron, one inch square and ten feet long, extended $\backslash frac\{1\}\{10000\}$ of its length for every ton of weight up to 12 tons, from which point the extensions nearly doubled successively for every two tons of load, and the bar was finally torn asunder by 23 tons.

(346.) Compression. The compression of bodies proceeds (like the extension) at first uniformly with the load. Some bodies resist compression more than extension (as cast iron); some the reverse (as wrought iron). Substances give way under compression after different fashions. Hard bodies divide into prisms parallel to the compressing force; slender elastic bodies bend laterally; soft bodies bulge horizontally; bodies of a medium hardness divide into wedges, and the surfaces slide along the plane of spontaneous fissure.

Detraction. Detraction marks more particularly the mode of giving way by the sliding of surfaces in the interior of solids. Though seldom due to force directly applied, it is an important element in most cases of the rupture of semiductile solids.

(347.) Flexure of beams. The force of flexure is that by which the resistance of the greater number of solids is most easily overcome, but which it is of most importance to resist; as when a beam is fastened by one end into a wall, and loaded at the other, or when it spans a horizontal space. It had not escaped the notice of James Bernoulli, Duhamel, and other writers of the earlier part of the 17th century, that the fibres on the concave side of a loaded beam are in a state of compression and not of extension, and that there is therefore a point, or rather a line, in every beam, in which the fibres are neither extended nor compressed. But the clear modification of the theory prevalent in the time of Leibnitz and Marriotte which this consideration introduced, was probably first developed by Coulomb, Robison, and Young, who in their respective publications insisted upon it with great judgment; and it is difficult to overrate its importance in mechanical engineering, although the first great canon of Galileo remains still true, that the ultimate strength of a solid rectangular beam varies as the breadth and as the square of the depth. The writings of Young and Robison did not immediately attract the attention of practical men, and Coulomb, who was by far the ablest French experimenter on subjects of mixed mechanics, seems to have done less on the theory of strains producing flexure than in the case of torsion, which he studied with so much success, and applied to such excellent purpose. Nevertheless in his memoir on the Resistance of Masonry, in the 7th vol. of the "Mémoires Présentés" (1776), he had already laid down very clearly the effect of compression on a beam.¹

It is, however, to Young that we owe the application of these principles in unfolding their legitimate consequences. In a series of remarkable propositions contained in the writings I have quoted (344), he assigns numerical relations between the flexure of a beam under almost every supposable circumstance, and the resistance of the material to direct strains. These results have been extensively used by all subsequent writers. They are not equally verified in all classes of substances. This, however, is not wonderful; flexure is not due to direct compressive and extending strains alone; deformation may take place in a solid without appreciable change of density, thus giving rise to some of the nicest questions in molecular physics.

The laws of Torsion, as laid down by Coulomb, (348.) have been mentioned in the last section (340). Torsion.

The mathematical investigations of Young on (349.) mechanical problems were conducted with bold directness and in defiance of the generalizing methods and symmetrical notation of foreign writers on such subjects. But his pre-eminent sagacity in laying hold on the salient points of the questions he discussed, and in conducting his argument to a practical conclusion, was unequalled, and deserves imitation.²

A great revival in the study of the properties of (350.) elastic matter, as regards strength, took place about the year 1820, probably in consequence of the introduction of wrought iron into the construction of suspension bridges, which has been attended with important results. General use of wrought iron.

THOMAS TELFORD, though neither the contriver of (351.) suspension bridges, nor the introducer of them into Britain,³ deserves notice from the superior boldness and solidity of the noblest work of the kind which has yet been executed—the Menai Bridge. Telford (and the same may be said of his contemporary Rennie) was more distinguished as a man of judgment, integrity, and experience, than as eminently original or philosophical. In this respect both yield to Smeaton, who, with Watt, was the founder (each in his own department) of modern engineering. But the beautiful and truly workmanlike structure of the Menai Bridge inaugurated the era of the extensive introduction of that admirable material, WROUGHT IRON, into great permanent structures exposed to heavy strains. Cast iron had been used much earlier, as in the bridge erected at Colebrookdale in 1777 by Mr Derby, and in the very beautiful arch at Sunderland, which dates from 1796. The span of the Menai Bridge is 580 feet, the whole quantity of iron used was 2186 tons, the transverse section of the suspending chains or bars was 260 square inches, supporting a strain of 1094½ tons. This

¹ Reprinted in the Théorie des Machines Simples, Paris, 1821.

² A copious selection from Young's mechanical writings may be found in his Miscell. Works edited by Dr Peacock, vol. ii.

³ Captain Samuel Brown erected the first considerable chain bridge in this country across the Tweed in 1819.

was a work quite unexampled at the time of its erection (1826), and showed a sagacious confidence in the employment of a material then comparatively little trusted.

(352.) Telford never made extensive experiments on the Data of resistance of solids. Some special ones were indeed made under his direction on wrought iron in particular, but in general he seems to have relied upon the old ones of Musschenbroek and Buffon. The two persons who first in recent times vigorously applied themselves to the practical determination of the data of resistance so long deficient, were Tredgold, a private engineer, and Professor Barlow of Woolwich. The data they obtained have since been generally used, not only in this, but in other countries. Tredgold's works (on Carpentry, Strength of Timber, &c.) show a very great aptitude in applying the results of science to practice, and an acquaintance with both which is rarely attained. Mr Eaton Hodgkinson has made many valuable additions to Tredgold's work, and has contributed an excellent paper on the strength of pillars to the Philosophical Transactions (1840.)

(353.) To Mr Hodgkinson we are also indebted for a useful investigation (in the Manchester Transactions) into the figure assumed by the chains of suspension bridges. The elegant properties of the simple or geometrical catenary were fully investigated a century and a half since by the Bernoullis and by David Gregory, but the application of suspended structures of immense weight to purposes of utility suggested new problems. Amongst these, perhaps the most interesting was the catenary of uniform strength, in which the section of the suspending chains is made everywhere proportional to the strain which they have to resist at that particular point. Its equation was investigated by Mr Davies Gilbert in 1826. An elegant and valuable contribution to the geometry of catenary curves was made by the late Professor Wallace of Edinburgh,¹ with particular application to curves of equilibration for bridges of masonry after the ingenious manner of Robison mentioned in Art. (334).

(354.) In connection with suspension bridges, and also M. Navier. with researches on the yielding of elastic materials, we must record the name of M. Navier, a very eminent French engineer and writer on practical and theoretical mechanics. His work on suspension bridges (1823) is one of the earliest and best. He is also well known for his physico-mathematical researches on the yielding of elastic solids to pressure

under given circumstances, in the course of which he came into collision with Poisson, who gave a somewhat different theory. The subject is one of extreme difficulty, owing to our ignorance of the molecular constitution of bodies; and it is believed that all these investigations were so far erroneous that they were based upon the assumption of a single constant to represent the resistance of bodies to change of form and dimension. These (form and dimension) are two very different things, and require distinct treatment.² British mathematicians have lately paid much attention to these enquiries, with the prospect of a solid improvement in engineering theories.³

The art of bridging over great spaces has been pushed, by the requirements of the railway system, to an astonishing extent, and under circumstances of peculiar difficulty. I shall connect these improvements with the name of Mr ROBERT STEPHENSON, the inventor of the Tubular Bridge, a work which, in its very simplicity, is a triumph of art, and being nothing more than a hollow beam of somewhat peculiar construction, supported at the ends, it is an admirable instance of a structure of which the stability may be easily reduced to calculation.

The wooden bridges of Switzerland were for a long time unequalled as skilful works of carpentry. During the last century the Rhine at Schaffhausen was crossed by two spans of 171 and 193 feet. At Trenton, in America, the river Delaware is crossed by a wooden bridge, of which one arch is 200 feet in span. It is on the bow principle, an elastic wooden arch, convex upwards, being skilfully braced and united to a level roadway passing through the spring of the arches. The American lattice bridge, very simply and skilfully contrived, has great firmness, owing to the depth of the framing, and exercises no horizontal thrust on the piers. The widest spanned wooden bridge in the world, 340 feet, across the Schuylkill, at Philadelphia, designed by Wernwag, combines the bow and lattice principle.

In these we might see foreshadowed in some faint degree the principle of the TUBULAR BRIDGE, the greatest discovery in construction of our day. But in reality the idea of it arose from a different consideration.

During the first ten or fifteen years of railway experience, engineers had gradually acquired a correct perception of the manner in which cast and wrought iron may most effectually and economically be formed

¹ Edinburgh Transactions, vol. xiv.

² About twenty years ago, the present writer showed that India rubber, which possesses to such a remarkable extent the quality which may be termed subcylindrical flexibility, is yet scarcely at all compressible—in fact, just as much as water, and no more. Though not otherwise published, he has been in the habit of demonstrating this in his annual course of lectures.

³ Professor Stokes in Cambridge Transactions, vol. viii.; Mr Clerk Maxwell in Edinburgh Transactions, vol. xx. Mr M. Rankine in Cambridge and Dublin Math. Journal for 1851 and 1852. Experimental data are still deficient; but M. Wertheim has lately published some valuable ones (which are still in progress) in the Annales de Chimie.

into girders or beams supported at the ends, and adapted for sustaining enormous loads. Such beams were constructed, often of single castings, so as to include three portions; an upper flange, a lower flange, and a web, or thinner vertical plate connecting the two. The relative section of the upper and lower flange was made to vary with the material. In cast iron, which yields far more easily to tensile than to compressive strains, the lower flange should be almost incomparably greater than the upper; in wrought iron a slight predominance should be given to the upper flange for the converse reason.

(359.) The idea of the tubular bridge derived from these. Hence it will be easily understood how, when Mr Robert Stephenson was desired to construct a railway bridge across the Menai Strait,—subject to the onerous condition imposed by the Admiralty, that it should (even at the abutments) be without lateral struts or diagonal pieces below the roadway,—he should have entertained the idea of a gigantic girder with a top and bottom flange of proportionate extent, with a deep web uniting them, or rather of two such girders placed side by side, thus forming square tubes, of which the lower flanges should constitute the bottom, the upper flanges the top, and the two webs the sides.

(360.) Progress of the invention. It is not for me in this place to explain how, step by step, the idea of a tubular bridge of wrought iron assumed the practical shape, now to be seen at Conway, and near Bangor, in North Wales. It is unfortunately notorious that there has existed an unhappy rivalry as to the share of merit due to the several persons who of necessity were jointly concerned in the completion even of the design of these astonishing works. Unfortunately for Mr Stephenson's tranquillity, the tremendous responsibility of this novel, gigantic, and costly experiment, was thrown upon him during the very height of the commercial and engineering excitement (not unjustly called mania) which prevailed in 1845 and 1846, on the subject of railway projects. Instead of the uninterrupted leisure which he required to superintend his preliminary experiments, to consider his plans, and perform his calculations, Mr Stephenson, as well as every other engineer of eminence was at that time engaged all day and a great part of the night in the unparalleled worry of Parliamentary contests. As a matter of course, much was trusted to able, confidential, and highly paid assistants. The experiments on models of different forms, which alone cost many thousand pounds, could not all be conducted in the presence of the chief engineer. Yet he alone was responsible for the failure or success of the plan.

(361.) The cellular top. The comparatively great strength of tubes was a fact known from the time of Galileo. Their defect was a liability to crumple or pucker. Round, oval, and rectangular tubes were tried, and the last (Mr Stephenson's original conception) were, as might be supposed, found to be stronger than the other two. When supported at the ends, and loaded in

the middle, model tubes of this form invariably gave way at the top. How to strengthen the top against compressive strains was the question. The excessive stiffness of corrugated iron and zinc plates (long previously used for roofs) came to the engineer's assistance. A combination of two longitudinally corrugated or goffered wrought-iron plates running along the top of the model, and forming long and nearly cylindrical cells, was found to give the required stiffness with the least increase of weight. Ultimately a square arrangement of cells was adopted, principally to give facility for painting and repair. The tubes were hindered from racking by means of numerous wrought-iron frames employed to stiffen them, whose section resembled the letter T, and which were called T irons. Suitable diaphragms were also inserted at short distances along the tubes. The widest spaces to be spanned at the Menai Strait were Dimensions 460 feet, there being two intervals of this width, and two of 230 feet. The tubes are 30 feet high and 14 strength of the bridge. broad, containing a transverse section of about 1500 square inches of wrought iron. The weight of one principal tube is about 1450 tons, and its strength, measured by the breaking load at the centre, above 2300 tons. This last number is calculated from the experiments on the breaking weight of models, one of which was on no less than a sixth of the true scale. The boiler plates, of which the tubes are composed, are united by mechanical pressure by means of hot rivets; and it may safely be affirmed that without this ingenious and perfect method of combination (which is due to Mr Fairbairn), the structure would have been impossible.

The success of this astonishing piece of engineering has been complete; the stiffness of the tubes, whether under constant pressure or during the rapid transit of trains, is almost incredibly great. (362.) Its complete success.

To Mr Robert Stephenson is clearly due the credit (363.) of undertaking, on his sole responsibility, a project of equal boldness and novelty, and of contriving, not perhaps in every detail, but in its totality, the means by which so signal a triumph of art and of science was carried into effect, an honour to his own age, and a lesson to posterity. To Mr Fairbairn and Mr Hodgkinson, his assistants, selected by himself, much praise is also due for the manner in which the experiments were managed, and the principles established by these men educated. Mr Fairbairn, a practical engineer of Manchester, well known for his experience and sagacity, gave to Mr Stephenson, as a matter of honour, the full benefit of both; and his confidence in the result helped no doubt to sustain the manly courage of his principal amidst a storm of opposition. Mr Hodgkinson, well known for his able enquiries into the strength of pillars and girders of different forms, conducted the mathematical enquiries, and determined the relative strengths of the models. His confidence in the result was less encouraging than that of his coadjutor, which serves to show the greatness

of the responsibility of the engineer-in-chief. As neither Mr Fairbairn nor Mr Hodgkinson could have incurred any just blame had the vast structure when on the eve of completion doubled up under its own weight, and blocked up, perhaps for ever, the navigation of half the Menai Strait, so neither can they possibly claim more than a subordinate share in the success of the undertaking.¹

(364.) Commission on railway bridges.
I shall here only refer to the work of Mr Edwin Clark, the resident engineer of the Britannia Bridge, for further details of its principles and construction, and to the report of a royal commission (published in 1849) on the application of iron to railway structures, for many curious researches connected with the subject of this section. In particular, we find a theoretical and practical solution of the very delicate problem of the influence of the speed of passing loads on the deflection of bridges, to which Professors Willis and Stokes, and Colonel James, R.E., are contributors.

(365.) Mr Robert Stephenson is the son of Mr George Stephenson, who will be mentioned in a succeeding

section. He was born in 1803; educated (in part) at the University of Edinburgh, under Leslie, Hope, and Jameson; he long occupied the chief position in the locomotive factory established by his father at Newcastle, having in the first place constructed under his direction the celebrated "Rocket" engine which gained the prize at the opening of the Liverpool and Manchester railway. To his own exertions, both before and after that period, the locomotive owes much of its present perfection. He surveyed and principally carried through the London and Birmingham railway, the second great line in the kingdom; and he has been engaged in a large proportion of the most remarkable engineering works connected with railways, both in this country and abroad. He has personally superintended the construction of railways amidst the blowing sands of Egypt, and in Norway with its heavy winter snows and deeply frozen soil. His high personal character, both for skill and integrity, has everywhere procured him the respect and confidence of his profession and of the public.

§ 4. BRUNEL.—Self-acting Machinery.—The Thames Tunnel.—Mr BABBAGE'S Calculating Engines.

(366.) Marc Isambart Brunel.
Sir MARC ISAMBART BRUNEL, born at Haqueville in Normandy on the 25th April 1769, was one of the most inventive mechanicians and engineers of his day. As his genius gave a strong impression to contemporary art, we associate his name with the progress of civil engineering in the earlier part of the present century, particularly in connection with mechanism. Like most of his eminent coevals in the same profession, he had not the benefit of a scientific education; but he more than most of them supplied its defects by a singular capacity for correct induction and by great mechanical ingenuity. Though a native of France, it was in Great Britain that his talents were to find their full scope, and it became his thoroughly adopted country.

(367.) His early history.
Disgusted by the horrors of the first revolution, he quitted France in 1793 in the capacity of a common sailor, a position far below that which either his birth or his intellect entitled him to hold, yet in which he made himself remarked by his excellent disposition and mental superiority. His destination was New York, where in 1794 he commenced his

career as a civil engineer, his boyish tastes having already indicated this as his natural calling. He executed some considerable works, and planned many more; it is stated that he there devised the essential parts of his block machinery. About 1799 he decided on settling in England.

(368.) Employed by the English government.
It is probable that his talents and ingenuity alone recommended him to a government employment at a time when the mere fact of his being a Frenchman must have acted as a powerful obstacle to his success. Those who recollect the vivacity and bright intelligence of even his later years, will understand that in his more active days it must have been difficult to refuse Brunel at least a hearing. And it is to the credit of Lord Spencer, then one of the Lords of the Admiralty, and of General Sir Samuel Bentham, inspector of naval works, that Brunel was engaged in 1802 to superintend the erection of his celebrated block machinery.

(369.) Self-acting machinery.
The invention of self-acting machinery to supercede the work of artisans was of course not new. The saw-mill and the spinning-jenny were already in

¹ It has been alleged that Mr Stephenson's original proposal to allow the suspension chains (which were primarily intended to be used in putting together the tubes in their final positions) to remain in aid of the rigidity of the structure, manifested a want of confidence in his own great idea. But a dispassionate consideration of his evidence before the committee of the House of Commons would alone clearly show (independent of Mr Stephenson's declarations on the subject) that he was forced into the admission that the chains might give an ulterior guarantee against miscarriage of the whole plan, simply to save the bill from being thrown out by the not unnatural incredulity of those to whom a proposal so new, so gigantic, and affecting the lives of so many persons, as well as so great pecuniary and other interests, was for the first time and suddenly proposed. Besides this, even his own coadjutors did not all entirely support him. Mr Hodgkinson, whose character for scientific knowledge carried great weight with the committee, recommended in his report the ultimate additional security of chains.

use, but the invention of Brunel was not less important as creating an epoch in art. Not only is it possible to execute in a comparatively short time, and with a prodigious economy, objects such as blocks and pulleys, which are required in vast numbers and precisely alike, but the nicety and accuracy of the manufacture is thereby increased, and owing to the facility with which inanimate force may be concentrated on machinery, works which transcend the power of unaided muscular labour are as surely and exactly executed as those of smaller dimensions. For example, by no enlargement of the common turning-lathe would it be possible to construct an accurately turned iron steam-cylinder 8 feet or more in diameter, which is yet readily executed under the direction of a very ordinary workman by means of steam power and self-acting machinery.

(370.) The block-machinery at Portsmouth consists of a series of engines impelled by steam, and by means of the materials of wood and metal employed in the construction of ships' blocks are reduced to exact forms in graduated sizes, and are finally put together with very little manual labour. These machines, with the exception of the turning-lathe and circular saw, were wholly new, and, it is stated, were devised in part by General Bentham, who gave to Brunel at least the benefit of his advice and previous experiments. In some of them we have the first germ of implements now used by every machine-maker in the kingdom; and the ingenuity of the movements, and the variety of effects produced, earned for this great invention a just celebrity. Such a beginning could not have been made without the aid of government. To construct the tools was an expensive and troublesome business, and to start the manufactory cost £53,000, which was speedily saved by the economy of the process. In the course of a year 140,000 blocks of no less than 200 different patterns were produced, and the number of workmen was diminished in the proportion of about 11 to 1. As a reward, Mr Brunel received £16,000, being two-thirds of the first year's saving, itself a sufficient proof that he was the bona fide inventor of this admirable apparatus, whatever hints he may have received from his immediate superior.

(371.) So successful an experiment produced ultimately, though with characteristic slowness, its effect on the mercantile world; nearly twenty years elapsed before such a splendid example of ingenious economy and artistic precision was at all generally imitated. Yet before his death, Sir Mare Brunel saw the fruit of his ingenuity almost indefinitely multiplied in the workshops of London, Manchester, Glasgow, Newcastle, and Birmingham, and highly appreciated if less extensively imitated abroad.

(372.) The more we reflect on the comparative state of the arts now and a century ago, the more we shall find reason to estimate highly the introduction of correct and scientific ideas of machinery and of tools for con-

structing other machines and structures. It was, in fact, the necessary complement of the invention of the steam-engine. Watt contrived the mighty Heart which was to give a new impulse to social life, Brunel and others of the same stamp added limbs and muscles, whereby its energies were rendered thoroughly practical. The sixteenth, seventeenth, and part of the eighteenth centuries, had given to the world designs for countless mechanical contrivances, often highly original and ingenious. But many were grounded on fallacies, and others belong to the class of elaborate trifles. At that period the slide-rest of the turning-lathe, the planing machine, and the circular saw, were practically unknown; at least the two former, which are incomparably the most important inventions of their class, and which belong to no certain author, having almost imperceptibly come into use—the slide-rest about the end of the last century, the planing machine as lately as about 1820. The former of these readily forms surfaces of revolution with geometrical accuracy, the latter plane surfaces, and in either case the application to metals is most important. The circular saw and slide-rest form part of Brunel's series of machines, and he afterwards constructed the former on a very great scale for the manufacture of wooden veneers. To them he added the mortising machine, and these, it will be seen (together with the planing engine), form the staple of the magnificent and varied apparatus with which, driven by the gigantic power of steam, our mechanical factories are now so generally provided. We again repeat that the triumphs of art in which our generation glories, our railroads, our locomotives, our crystal palaces, and our steam navies, would have been impossible feats but for the improvement of tools and the substitution of steam for muscular power.

(373.) Every one is, however, aware that Brunel owed his reputation to other achievements as well as his improvements of mechanical tools. The Thames Tunnel will ever be considered as his most arduous triumph. It is a structure of exquisite firmness laid in a quicksand. It will endure like the cloacæ of regal Rome, when the palace and the cathedral have crumbled to dust. Yet here also we perceive that it was Brunel's exquisite mechanical tact and ingenuity which enabled him to succeed. The problem of the tunnel is not one of balancing vaults; the statical conditions of stability are simple enough, and it was not in the solution of such that Brunel peculiarly excelled. The practical problem was to introduce a rigid tube of brick horizontally into the middle of a quaking mass of mud; and the solution was the invention of a tool which should enable men to make the excavation and to proceed with the building in safety. It was the shield which carried the tunnel under the Thames,—a moveable vertical frame of cast iron, provided with thirty-six cells, in each of which a man was placed with a pick to ex-

cavate the area required for the construction of the tunnel. By a simple but most ingenious contrivance, every part of the face of unstable clay was firmly supported by boards which leaned upon the frame or shield, which, in its turn, pressed against the part of the brickwork of the tunnel already completed. Each workman could remove one or more of these small boards at pleasure, and excavate a short way into the yielding mass before him, then advance the boards and sustain the slippery face. When the whole face had thus undergone piecemeal excavation, the frame or shield was moved bodily forwards by powerful screws, and the bricklayers brought up the masonry behind, which was then beyond the reach of injury.

(374.) Completion of the tunnel.
The idea of the shield was derived, it is stated, from a specimen in the arsenal at Chatham, showing the operations of a testaceous worm which bores under water, and which nature has provided with a protective covering. But the analogy is certainly indirect, since water could hardly retard the operations of such an animal. Repeated irruptions of the Thames several times drowned the work, which was as often abandoned and renewed, but every difficulty was met by fresh resources on the part of the engineer. The failure of funds was a far more serious obstacle, and government at last came to the aid of an undertaking of such consummate ingenuity that its completion was deemed due to the honour of the nation. The tunnel was commenced on the 2d March 1825, and finished 25th March 1843. Brunel survived the completion of his great work above six years, dying on the 12th December 1849, aged 81.

(375.) Variety of his works.
We have not in this brief sketch glanced at one half of his ingenious projects and successful enterprises. Scarcely any branch of his multiform profession but received some improvement at his hand. The discovery of the condensation of several gases in 1823, by Mr Faraday, suggested to Brunel their application as a moving power; and his want of success did not arise from any deficiency on his part of skill or forethought. He was one of the first to construct a roof of extreme lightness, somewhat resembling those now in use for railway stations. He erected a suspension bridge in the Isle of Bourbon on an original plan; and he pointed out with characteristic shrewdness how much of the stability of arches depends upon the cohesion of the parts, so that the vault may in some cases be entirely dispensed with.

(376.) It will be understood that we have selected Sir Marc Brunel as the representative of a class, the eminently mechanical engineers, a class now extensively multiplied, and amongst whom his son, Mr Brunel, occupies an eminent position.

(377.) Calculating machines of Mr Babbage.
It cannot be expected in an essay like the present that I should enter into the details of the variety of mechanical inventions which have now become so numerous, and which have been marked by every

gradation of originality and resource. But as illustrating a class of contrivances altogether different from those of Brunel, though like them tending to produce a great influence on the improvement of the mechanical arts, I will briefly refer to the Calculating Machines of Mr Babbage, which have at different times excited the interest of the public and of scientific men.

(378.) Mr Babbage was a fellow student at Cambridge with Sir John Herschel and Dean Peacock, and along with them he contributed by his writings and personal efforts to introduce into that university the improved Continental mathematics. A few years after leaving college he originated the plan of a machine for calculating tables by means of successive orders of differences, and having received for it in 1822 and the following year the support of the Astronomical and Royal Societies, and a grant of money from government, he proceeded to its execution. It is believed that Mr Babbage was the first who thought of employing mechanism for computing tables by means of differences; the machine was subsequently termed the difference engine. In the course of his proceedings Mr Babbage invented a mechanical notation (described in the Philosophical Transactions for 1826), intended to show the exact mutual relations of all the parts of any connected machine, however complex, at a given instant of time. He also made himself acquainted with the various machines used in the arts, with the tools used in constructing them, and with the details of the most improved workshops. Employing Mr Clements, a skilful mechanist, a portion of the calculating machine, very beautifully constructed, was brought into working order, and its success so far answered the expectations of its projector. But, notwithstanding several additional grants from government, the outlay on this most expensive kind of work soon exceeded them. The part actually constructed is now placed in the Museum of King's College, London; it employs numbers of nineteen digits, and effects summations by means of three orders of differences. Though only constituting a small part of the intended engine, it involves the principles of the whole. The inventor proposed to connect with it a printing apparatus, so that the engine should not only tabulate the numbers, but also print them beyond almost the possibility of error.

(379.) At this stage (1834) Mr Babbage contrived a machine of a far more comprehensive character, which he calls the Analytical Engine, extending the plan so as to develop algebraic quantities, and to tabulate the numerical value of complicated functions when one or more of the variables which they contain are made to alter their values. Had this engine been constructed, it would necessarily have superseded what had already been done. Government were not unnaturally startled by this new proposal, and as about the same time Mr Babbage's relations to

Mr Clements were broken off, the difficulties of the affair became insurmountable, and the construction of either engine has for some years been in abeyance. The opinions of men of science are not unanimous as to the great practical importance of calculating tables by machinery, but the improvements of mechanical contrivance which the joint skill of Mr Babbage and Mr Clements introduced into engineering workshops are unquestionably of great importance to the arts. Though the details of Mr Babbage's plans have not been published, there can be no doubt that, whether economical or not as substitutions of machinery for human labour, they were devised with remarkable skill and ingenuity, and even on this account merit preservation.¹

(380.) Recently (1855) attention has been directed in London to a simple and effective Difference Engine constructed and patented by M. Scheutz, confessedly on the principles of Mr Babbage, though without an acquaintance with his mechanical contrivances. The result is stated to be satisfactory. The engine deals with fifteen digits or figures, and with four orders of differences. Only eight figures are preserved in the result, the others being reserved to prevent errors

arising from the accumulation of still lower digits omitted. The engine not only computes with facility and accuracy, but, by means of steel punches impressing lead, provides for the perpetuation of the numbers in the form of stereotyped plates. The workmanship of the whole requires no particular nicety of execution, is not liable to derangement, and can by scarcely any contingency produce inaccurate results.

Before closing this section, we may advert to improvements in the theory of machines by those who have regarded it rather from the geometrical side than from that of routine practice. Our French neighbours have been distinguished in this respect. Carnot and De Prony, MM. Hachette, Poncelot, and Morin, have been or are accomplished mechanists in this respect; and in the French repertories we must look for some of the earliest good scientific descriptions of machinery, even when of English invention.

In England, besides Mr Babbage, Professor Willis of Cambridge has shown a peculiar aptitude in this department, and has published a very valuable work on machinery, regarded in a strictly geometrical sense.² To Mr Moseley we are likewise indebted for some valuable contributions to the theory of engineering.

§ 5. TREVITHICK.—GEORGE STEPHENSON.—The Locomotive Steam-Engine.—Rise and Progress of Railways.—M. de Pambour on Locomotives.

(383.) Of all the inventions which have powerfully affected the interests of mankind, none have been more slowly perfected, or can be less certainly traced to a single individual as the inventor, than those of the Locomotive engine and the Railway. These two great and essentially connected portions of the greatest mechanical and commercial effort of any age or country had their origin in obscurity. Each appeared several times to be rising into the importance it deserved, but failing the concurrence of the fortunate circumstances which are necessary to give permanence to invention, was once more forgotten and was left for re-discovery at a happier epoch.

(384.) With regard to Steam-Carriages, passing over still earlier speculations, we find that Dr John Robison, at the age of twenty-one, published a design for a steam-carriage in the Universal Magazine for November 1757, and that he also directed Mr Watt's attention to the steam-engine in the same year, with a view to this very application. The cylinder of the proposed machine was an inverted one, and Watt actually made a rude model on Robison's suggestion.³ From this time the steam-carriage seems never to have

been long lost sight of by mechanical speculators. It was included in a patent by one Moore, a linen draper, in 1769.⁴ In the same year it is stated that Cugnot, a native of Lorraine, actually constructed a steam-carriage, which, like the nearly contemporary but unsuccessful efforts of his countrymen to effect steam navigation, fell speedily into oblivion. About 1773 Edgeworth of Edgeworthstown urged the construction of steam-carriages, and at a later period expressed, in terms of unequivocal anticipation, the triumph arising from their connection with railways. "I have always thought," he wrote in 1813, "that steam would become the universal lord, and that we should in time scorn post-horses. An iron railroad would be a cheaper thing than a road on the common construction." At Soho the movement of carriages as well as of boats by steam never was or could be forgotten. In Watt's patent of 1784 the steam-carriage forms the seventh article, and in the same year Mr William Murdoch, a member of Boulton and Watt's establishment, made a model, acting by high-pressure steam, which drove a small waggon round the room. Hence it required no prophetic power in Darwin, the intimate friend of Watt, to

¹ For historical details connected with Mr Babbage's engine, see Weld's History of the Royal Society, vol. ii. An account of the principles and action of the Difference Engine may be found in the Edinburgh Review for July 1834; and those of the Analytical Engine in Taylor's Scientific Memoirs, vol. iii.

² Principles of Mechanism. Camb., 1841.

³ Mechanical Inventions of James Watt, l. 52.

⁴ Mechanical Inventions of James Watt, ll. 294.

⁵ Translation of Arago's Elogium of Watt, p. 120, note.

write those often quoted lines in the Botanic Garden (canto i. line 290):—

"Soon shall thine arm, unconquered Steam, afar
Drag the slow barge, or drive the rapid car."

(385.) A somewhat longer pause now occurs. But in 1802 we find Richard Trevithick, a Cornish "captain" of a mine, taking out a patent along with Vivian for the high-pressure steam-engine, and applying it specifically and practically to the movement of carriages or waggons along a railway at Merthyr Tydvil in South Wales. Mr Muirhead informs us that Trevithick saw Murdoch's model at Redruth in Cornwall. But admitting this, it is plain that the idea was much older still, and also that many years elapsed without its ever being brought practically to bear until the year 1804, when Trevithick's locomotive was actually used.

(386.) RICHARD TREVITHICK appears to have been one of the most ingenious men of his time; but (from the scanty notices which I have been able to collect¹) to have been also of an inconstant speculative disposition, which prevented him from bringing any of his numerous inventions to perfection. Yet he had the good fortune, which so many inventors have missed, of meeting with partners able and willing to assist him in carrying out his designs. Amongst these was Andrew Vivian, with whom in 1802 he took out the patent already mentioned for the construction of high-pressure engines, and their application to the movement of carriages along rails or common roads. As the first practical employer of high-pressure steam of 60 to 80 lb. pressure on the square inch, Trevithick deserves especial notice. He borrowed the notion of it, as well as the ingenious invention of the four-way cock, from an old scheme of Leupold's, but he overcame for the time the prejudice which had always existed even in the mind of Watt against its adoption. His earliest engine is stated to have been connected with a common stage coach which ran on the streets of London; but his more successful and important effort was made in dragging waggons along the Merthyr Tydvil railway in South Wales, which was successfully tried on the 21st February 1804, when the engine drew carriages containing ten tons of bar iron for a distance of nine miles at the rate of five miles an hour. This was unquestionably the first successful example of this modern species of locomotion.

If we look more closely at the means by which it was accomplished, we find still more reason to commend the sagacity of the inventor, and to wonder at the interval of nearly thirty years which elapsed before the general adoption of his plan. "A square iron case containing the boiler and cylinder was placed behind the large or hinder wheels of the carriage, and was attached to a frame supported from the axles of

those wheels. The cylinder was in a horizontal position, and the piston-rod was projected backwards and forwards in the line of the road towards the front of the carriage. Across the square frame, supported by the wheels of the carriage, an axle was extended reaching a little beyond the frame on each side; this axle was cranked in the middle, in a line with the centre of the cylinder, and a connecting-rod passing from the end of the piston turned this axle round, and produced a continued rotatory motion of it when the piston was moved backwards and forwards in the cylinder; upon both ends of this axle cog-wheels were fixed, which worked into similar cog-wheels upon the axles of the wheels of the carriages, so that when a rotatory motion was produced in the cranked axle by the piston-rod it was communicated to the axle of the larger or hinder wheels of the carriages, and these wheels being fixed upon and turning round with the axle, gave a progressive motion to the carriage. Upon one end of the axle was fixed a fly-wheel to secure a rotatory motion in the axle at the termination of each stroke."²

We here find the cranked axle and the horizontal cylinder of modern locomotives, both of which were departed from by Trevithick himself, probably in consequence of difficulties of execution. When we add to this plain description, that the fly-wheel was furnished with a break, that the boiler had a safety-valve or a fusible plug beyond the reach of the engineer, and that the patent includes the production of "a more equable rotatory motion, . . . by causing the piston-rods of two cylinders to work on the said axis by means of cranks at a quarter of a turn asunder," it is scarcely too much to say that nothing material was added to the design of the Locomotive until the invention of the tubular boiler in 1829.

The Merthyr locomotive blew up, and the prejudice against high-pressure steam revived. The inventor, in the meantime, diverted his attention to other schemes, and continued his profession as a Cornish mining engineer.

A singular circumstance opened for Trevithick a new sphere. A Spanish-American gentleman named Uvillé, who was engaged in working the silver mines of Peru, came to England in 1811, with a view to discover an engine fit for draining those mines whose high elevation rendered condensing steam-engines working under atmospheric pressure comparatively inert. In London he met accidentally with a model of Trevithick's engine, and having carried it to the heights of Pasco in Peru, and being satisfied with its work, he did not rest until he had returned to England and transported nine high-pressure engines in 1814 to the scene of operations. In 1816 Trevithick himself followed, with coining engines and purifying furnaces of his own contrivance. Had he

¹ It is stated that the Society of Civil Engineers have in vain proposed a medal for a biography of Trevithick.

² Wood on Railways. This description corresponds with Trevithick's specification and drawings.—Repertory of Arts, &c., Second Series, vol. IV.

been a prudent man his fortune was now made; but it is stated that after a year or two he returned to this country impoverished and disappointed. I am unacquainted with his further history.

(391.) In the meantime the locomotive engine, which Trevithick had long abandoned to its fate, was becoming known in the hands of a man perhaps of less genius but of greater sagacity and perseverance.

(392.) GEORGE STEPHENSON, civil engineer, was born in 1780 near Newcastle, of respectable persons in the humblest rank of life. His father was either a common pitman or otherwise employed about the collieries of the district, and young Stephenson, without any advantages of education, began to labour for his bread at an early age. His work appears to have been always connected with the machinery of the pits above ground, and not with their excavation. Thus he rose gradually to be an engine-man at the wages of twelve shillings a week. This was at Killingworth near Newcastle, where he showed considerable mechanical ingenuity, and gradually gained the confidence of his employers. Having married in 1802, he had a son born the following year, the present Mr Robert Stephenson, M.P., whom he brought up with the tenderest care, and whom he ever and justly regarded with a father's pride. In order to bestow upon him the advantage of that education of which he had himself felt the want, it is stated that he made money at extra hours by mending his neighbours' clocks and watches, and finally, in more prosperous days, sent his son to complete his education at the University of Edinburgh. George Stephenson never acquired much book-learning himself, but by natural sagacity and observation he attained to a sound knowledge of mechanical principles. We do not claim for him, however, the character of great inventiveness. His skill rather lay in perceiving how far methods and contrivances already known might be pushed to an advantageous result. He possessed that shrewd decision which ingenious persons often want, enabling him to detect what is truly valuable in the numerous mechanical schemes which at any time are afloat, and to devise the means of realizing them. He also possessed that confidence in his own judgment which is necessary to carry out principles to their legitimate extent, but from which feeble or less practical minds usually shrink.

His early difficulties, and character.

(393.) He contrives a safety lamp;

the end have led him to a construction analogous to that of the safety lamp. As matters stood, it is not surprising that his efforts, though highly meritorious, led him slowly and uncertainly towards the goal which Davy, having once sighted, arrived at with that rapid instinct in which he has never been surpassed. Stephenson was left behind, but was rewarded by a handsome gift offered by his local admirers, who, in doing so, naturally rather considered the difficulties overcome by their humble neighbour than the strictly comparative merit of the two inventions.

But it is of the locomotive and of the railway that we have here to speak. (394.)

The former, we have seen, had been brought to considerable perfection by Trevithick. An engine on his plan had been constructed and used by Mr Blackett of Wylam in Northumberland, near the place where Stephenson resided, and was the basis of his improvements. Blackett's engine had two cylinders, an addition often ascribed to Stephenson, but which, as we have said, was included in Trevithick's patent. What he saw of the performance of this machine appears to have convinced Stephenson, once for all, of the groundlessness of an opinion which then and for long after haunted the minds of railway engineers. This opinion was, that the adhesion between the wheels of a locomotive engine and the smooth iron surfaces of the rails must be insufficient to allow the impulsion of the train, at least with any degree of velocity, or up the smallest inclination. Trevithick had a scheme for increasing the adhesion, and this ideal improvement was the subject of repeated patents, some of a singular nature, between 1802 and 1824, one of which, Blenkinsop's, provided a cog-wheel in the engine working into a rack on the rail, which was actually in use down at least to 1830. It is rather a singular thing that men spurning theories, as was the fashion of the engineers of that day, and especially those of Smeaton's school, should have thought as little of an appeal to experiment on so simple a matter as did the followers of Aristotle in the seventeenth century, when Galileo offered to convince them that light and heavy bodies fall equally fast. Forgetting that direct friction is always large, and that it varies in proportion to the pressure, these practical men could not get over their first impression, that iron must slide on iron long before a heavy train could be set in motion. It was characteristic of Stephenson's decision of character, that he dismissed all doubts on the subject so soon as his observations seemed distinct, and that he did not hesitate to carry out his belief to its consequences, and to maintain his confidence in the locomotive engine against all antagonists.

I shall not stop to particularize Stephenson's first improvements on the locomotive, which were rather in detail than in principle. He saw clearly all along that if it was to work at high speeds he must in every (396.) G. Stephenson's improvements.

possible way diminish the vibrations and strains to which it was subject, and which would otherwise rapidly wear out the machinery. For this purpose he proposed to connect the engine with its carriage by means of steam acting on six pistons in lieu of springs. But perhaps his most material improvement consisted in the very simple one of throwing the waste high-pressure steam as a blast into the chimney, which was found to increase enormously the force of the fire, and the evaporating power of the boiler. Engines having this improvement, and with two vertical cylinders, as constructed in 1818, are, or were lately (1854), still at work on the Killingworth railway dragging coals at the rate of five or six miles an hour.

One of Stephenson's clear practical opinions was this,—that the locomotive and the railway are part of one mechanism, and must be adapted to one another. He was not a friend to steam-carriages on common roads, and the event proved his sagacity.

If the idea of a locomotive belongs to no one man, still less does that of a railway, which being one of the most elementary of mechanical contrivances, may be traced, under some modifications, almost indefinitely backwards, as a means of conveying heavy loads with facility. Hence it was at first confined chiefly to quarries and collieries, especially in underground passages or drifts. The gauge of these subterranean railways, or tram ways, was only about 18 inches. The material of the rails was first wood, then cast iron, finally wrought iron, as being less liable to wear and to accident. The wrought iron rail, though not absolutely new, was first generally introduced in 1820. About the same time, or rather sooner, the rails began to be made plain, that is, without any vertical guide or flange to prevent the wheels of the carriages from leaving the rail, and the flange was transferred to the wheels of the locomotive. Even this was not new, for it had been used by Jessop in 1789. The weight of the rails has been constantly on the increase. The original cast-iron rails weighed only 15 lb. a yard; the malleable-iron rail in 1821 weighed about 28 lb., then 35, afterwards 64, and now rails of 80 lb. a yard are generally used.

One of Stephenson's first cares was to make his railways solid and level, and to prevent jerks at the junction of the rails. The gauge he adopted, or the interval between the rails (now generally used, except on the Great Western Railway and its branches), was 4 ft. 8½ inches, and was derived from the accidental width of the parent railways in Northumberland. Like Watt and all other innovators, his great difficulty was to get the machinery of his locomotives properly made, and the great railway movement of 1825 was anticipated by the establishment in 1820 of an engine factory at Newcastle, which, till after the

opening of the Liverpool and Manchester Railway in 1831, remained the only one, and for long afterwards the best of its class. The cranked axle contrived by Trevithick, and abandoned because it could not be properly welded, was now restored; the heavy locomotive was placed on strong but easy steel springs, wrought iron was skilfully introduced into the wheels of the carriages, and the whole machinery was made to work with precision, and to combine a degree of resistance never before anticipated with comparative lightness. The factory was established in 1821, and the first passenger locomotive was started on the Darlington and Stockton Railway in 1825.

I ought, perhaps, to apologize for these details, but they illustrate so well the exceedingly gradual progress of mechanical invention, that I have thought them worthy of mention here.¹ The subsequent history of the locomotive and the railway is more generally known. From the date of 1825, both grew and flourished; the railway first and most steadily; the locomotive was introduced more cautiously, and met with much opposition; its triumph was almost entirely due to the steadiness of George Stephenson.

The year 1825, so fertile in speculation, produced a series of projects for railways to an extent not commonly known, since few of them came into existence or were even commenced for many years later. The projected capital of these companies amounted to not less than £30,000,000 or £40,000,000. But the only considerable undertaking which was at that time seriously supported was the railway from Liverpool to Manchester, and on that battle-field were fought the great questions of the superiority of railways to common roads,—of high to low velocities of transport,—and of locomotives to fixed engines.

On these three important points, George Stephenson was in advance both of the science and of the practice of his age; and, accordingly, backed chiefly by commercial men, who had entire confidence in his sagacity, he had to maintain the conflict almost single-handed against general and professional prejudice. With respect to the Railway, he had long decided in his own mind against the use of steam-carriages on common roads. This conclusion was scientifically based on his own experiments on the friction of wagons on railways made in conjunction with Mr Nicholas Wood, civil engineer at Newcastle, as far back as the years 1815 and 1816. A simple dynamometer of Stephenson's invention was used, and by means of it the two fundamental propositions were established, that the friction is directly as the pressure, and that it is quite independent of velocity (at least when the speed was moderate). It may be said that these propositions were already known; but, besides that probably Stephenson and Wood were equally unacquainted with the writings of Coulomb, they could not have

¹ I have found many curious details of the early history of railways in a series of articles on the life of George Stephenson in the Civil Engineer Journal for 1848 and 1849. I am indebted to Mr Robert Stephenson, M.P., for many interesting particulars respecting his father's inventions.

dispensed with verifying his results under circumstances so peculiar as those of a railway and a train of carriages. The necessity of doing so was manifested by the opposition, and even ridicule, with which the idea of friction being in this case independent of velocity was received, showing, as has been correctly observed, "how small was the amount of science at that time blended with engineering practice." The friction of even the indifferent railways of those days amounted to only 10 lb. per ton of load; consequently an incline of only 1 foot in 100 would increase by one-half the resistance to the motion of a carriage on a railway. Hence Stephenson determined to construct railways having only the smallest inclinations, and to use fixed engines for higher slopes. With respect to common roads, he showed by powdering even a level railway with sand, that the most powerful locomotives then in use speedily came to rest; this, with the previous objection, overruled in his mind the possibility of advantage in that case.

(403.) Practicability of high velocities on railways.

With low gradients and small resistances, together with the proved invariability of friction with speed, there necessarily came into Stephenson's mind the practicability of using high velocities. At a very early period (1816) he spoke in one of his patents of conveying goods "at nearly double the rate at which they were then usually carried along railways," in other words, at 10 or 12 miles an hour, and this he states "with no hesitation, speaking from experiments already made," referring, no doubt, to those on friction made along with Wood about this time. Yet after nine years of farther experience, his old coadjutor Wood deserted him on this grand point, and in the first edition of his book on Railways (1825, p. 290) he disclaims the "ridiculous expectation" that locomotives will be seen to travel at "12, 16, 18, or 20 miles an hour," and scorns "the promulgation of such nonsense." Even in 1829 he reported to Messrs Walker and Rastick, who were referees on the Liverpool and Manchester Railway, "that no locomotive engine should travel more than 8 miles an hour." If at this comparatively late period perhaps the most practised railway engineer in England held these opinions, and that with the full knowledge of his friend Stephenson's matured convictions (which it is difficult to believe were not pointed at in this paragraph), we may imagine the opposition which the plans of the latter were likely to meet with from interested or even indifferent persons.

(404.) The Manchester and Liverpool Railway.

At last a company was formed, and funds provided to construct the Liverpool and Manchester Railway. It is unnecessary to state how successfully Stephenson conquered the engineering difficulties of the line, and refuted the predicted impossibility of crossing the Chat Moss. In every respect this railway became a model for those which succeeded, and

in essentials very little has been added by 25 years' experience on lines of the same gauge. But now came the struggle as to how this beautiful road was to be worked;—with horses,—by means of fixed engines,—or by locomotives. It was not without a struggle that Stephenson gained his point. Even in 1829 the prejudices of the engineering profession were still strong against the locomotive. And it is curious to read in the contemporary documents with what distrust they were regarded. The clumsy expedient of a series of stationary engines $1\backslash frac\{1\}\{2\}$ miles apart, dragging the trains by ropes, would probably have been adopted to the disgrace of the age, but for the energy of Stephenson and his commercial friends. A competition of locomotives was at last agreed to, which took place on October 6, 1829, on a level piece of railway at Rainhill near Liverpool. Though the makers of engines had their energies hampered by various needless conditions (particularly as regards the weight of the engines, under the mistaken notion, that velocity could only be combined with lightness), several excellent engines appeared; but the "Rocket" made at Stephenson's factory at Newcastle, not only gained the prize, but far exceeded in its performances the limits assigned in the programme. It weighed 44 tons, and dragged a gross load of 17 tons, at the rate of 15 miles an hour, but moved itself with a velocity of 35 miles an hour. The "Novelty" of Messrs Braithwaite and Ericson was also very successful. The prize was awarded to Stephenson, and this success was mainly due to the admirable invention of the multi-tubular boiler, imagined by Mr Booth, and carried out by Stephenson. To distribute the water of the boiler in tubes, and allow the heat of the furnace to act around them, was an idea as old as the time of Watt, but it did not succeed. To carry the hot air of the furnace through tubes surrounded by water, was the more successful arrangement of Booth and Stephenson, to the right working of which the draught occasioned by the steam-blast in the chimney was essential. The idea, it is said, had occurred both in France and America, but it certainly remained practically inefficient, perhaps on account of the want of draught.

(405.) Stephenson's success.

This invention, by increasing almost without limit the evaporating power of the boiler, which is the key to the efficiency of a locomotive, completed for the time the skilful improvements on locomotives and railways, which, as has been seen, we owe mainly to Stephenson. The comparatively trifling ameliorations which have occurred in either, and the stereotyped character of even the minor arrangements, such as those of stations and of passenger carriages, show how much the sagacity of the engineer had anticipated the accommodation of the public.¹

(406.) and its consequences.

I here close my account of Mr Stephenson and of

¹ I do not overlook of course the modifications introduced in the broad-gauge system of the Great Western Railway. Mr Brunel indeed tried to show how far he could deviate without positive injury from Stephenson's plans; in some points, perhaps, he did so with advantage, yet, on the whole, the results do not shake Stephenson's position as the commanding engineer of his time.

the railway system. The locomotive engine will ever remain as an invention entirely distinct from those created by the genius of Watt. It is in reality independent of his great principle of separate condensation. It has a power of adaptation and a perfection of performance more astonishing than any contrivance of our time, except perhaps the electric telegraph. In 1552, Jerome Cardan took twenty-three days to travel from London to Edinburgh; that the same should be done 300 years later in eleven hours, and every day, is a fact as striking as any which the progress of science presents. Whilst to Trevithick and George Stephenson we are mainly indebted for these results, the theory of the steam-engine is still in ar-

rear. To M. de Pambour we are indebted¹ for by M. de Pambour the best account of the elements of force contained in the locomotive, and of the resistances which they have to overcome; but it is not to be doubted that much yet remains to be done in this direction.

George Stephenson died 12th August 1848, at the age of sixty-eight, and generally respected for his private character as well as for his talents. His son had the honour of completing the second great railway work in Britain, from London to Birmingham, and by the invention of the tubular bridge, described in a previous section, he has added the most important, as well as the most scientific auxiliary to the extension of railways since 1830.

§ 6. Hydrodynamics, DUBUAT, VENTURI, Professor STOKES.—Friction and Resistance of Fluids. MM. WEBER, Mr SCOTT RUSSELL.—Propagation of Waves. Influence on Canal Navigation. MM. FOURNEYRON and PONCELET.—Improved Hydraulic Machines; Turbine. Reference to the subject of Capillary Attraction.

(408.) There are few subjects less adapted for the kind of discussion which I have adopted in this Dissertation than Hydrodynamics, whether in its abstract or practical form. Not only is it difficult to fix upon individuals who, in the more recent progress of the subject, have attached themselves to it in so especial a manner as to have impressed the science with the individual character of their own minds, but the progress made in either branch has been of a remarkably fragmentary kind, usually bearing upon the solution of individual problems, and tending to the improvement of particular machines. It must be owned, too, that however important in practice may be the efficiency of a water-wheel or the discharge of a pipe, the solution of such problems does not present the attractive interest which attaches to many less difficult ones in Natural Philosophy. As regards more general problems of which a direct solution from first principles might appear possible, it has been found that these are as yet so limited as to give a character (which is generally admitted) of great barrenness to the theoretical investigations.

(409.) While, then, I shall endeavour, in conformity with my general plan, to preserve something of the biographical character in the history, suppressing details and many minor and even some important steps, this section will necessarily have somewhat of a fragmentary character, and its deficiencies must be supplied by a reference to the numerous articles of the Encyclopaedia which treat more or less fully of these subjects and their history.

(410.) 1. Friction and Resistance of Fluids.—The first name I shall mention is that of the Chevalier DUBUAT, who has the signal merit of having attributed due importance to, and of having considered

with much sagacity, the effects of the friction of fluids against their own particles, and against the sides of the solid bodies used to confine them. He had the full advantage of an acquaintance with the Abbe Bossut's excellent experiments and judicious writings, but he was, I believe, the first who succeeded in ascribing to the different forces which act on fluids in a state of uniform motion their effective share in determining their velocity. I refer particularly to the discharge of pipes and of rivers. Dubuat showed that when the motion of water in such circumstances becomes uniform, the accelerating force which acts is the measure of the total resistances to the fluid motion, whether arising from the inequalities of the bed or the viscosity of the fluid. These resistances are assumed to be proportional to the square of the velocity. Since they are exactly in proportion to the length of the pipe or channel, and since the moving pressure increases in the same proportion, the velocity is independent of the length of the pipe, whilst the inclination remains the same—a simple result not previously noticed. In the case of a pipe the head or superincumbent pressure may be divided into two parts; one, requisite to force the water into the tube with the requisite velocity, which is independent of the distance to be travelled; the other, which balances the resistance due to the length of the pipe, which for a given diameter varies as the length, or if the slope be constant is independent of the length.

The relation of the area of the stream to the perimeter or rubbing surface of the channel is then taken into account. This ratio is called by some writers the mean hydraulic depth. The manner in which Dubuat derives his formula (which I shall not here set down) from direct experiment, guided by a few general notions of theory, is a very good specimen of this kind

¹ The Theory of the Steam-Engine (1839), and A Practical Treatise on Locomotive Engines (1840), by the Comte F. M. G. de Pambour.

of induction, without any pretence whatever of proceeding upon the mathematical theory of fluids in motion. The same views were ably expounded and enlarged by Dr Robison in his excellent article on Rivers and Resistance in this Encyclopædia, through which principally the theory of Dubuat became known in this country. Indeed, viewed apart from the technical value of the enquiry, the research as to the natural economy of rivers is a wonderful and striking branch of terrestrial physics, and one which had long afforded a subject of anxious and perplexing enquiries to Italian engineers from the time of Leonardo da Vinci, though comparatively little studied elsewhere. The collected treatises of Italian authors form an important body of hydraulic information.¹ The rivers of the north of Italy, like those of Holland, conveying vast masses of water charged with mud under very feeble slopes to the sea, present a formidable difficulty which compels attention,—vast territories being increasingly subject to inundation, as the beds of the rivers are raised by deposition above the general level of the soil.

Dubuât's formulas have been modified, and made to represent experiments better, by subsequent writers, particularly by De Prony, Langsdorf, Eyelwein, and Thomas Young; and the discharge from mere orifices has been discussed by numerous authors who have given empirical co-efficients to represent the phenomena. But little has been added to a philosophical view of the real laws which govern the fluid motion in this case. Nevertheless Savart and Magnus have made some ingenious observations on the constituent parts of effluent streams.

The viscosity or imperfect fluidity of water is the property most difficult to be taken into account in these and other hydraulic problems. It is that which causes the velocity of a stream to diminish from the surface to the bottom, and from the centre to the sides; these proportions were also sought by Dubuat. Each layer of water in motion exerts a dragging or "tangential force" upon other layers, which from any cause are comparatively quiescent; and the experiments of VENTURI about the end of the last century, showing that a stream in motion draws towards it the particles of still water with which it may be in contact, with a force sufficient to overcome considerable hydrostatic pressure, attracted much attention. About the same time Coulomb, with his usual address, made experiments on the friction of fluids by observing the rapidity with which cylinders oscillating by means of torsion in different fluids had their original motion destroyed.

Intimately connected with the friction, and consequent mutual action of fluids, is the resistance which they offer to the passage of solid bodies through them,

and this favourite problem of mathematicians was treated by Dubuat, in his inductive way, in an able and practical manner. Three cases may be specified on account of their theoretical or practical importance:—1st, In the case of a body oscillating like a pendulum, with small velocities, the body being immersed in a resisting medium; 2dly, The resistance to vessels floating on and propelled through water; 3dly, The resistance of the air to projectiles whose velocity is very great.

Under the first head Dubuat made those ingenious experiments long overlooked, but lately brought into notice, which I have mentioned in the section on the Pendulum in the chapter on Astronomy, Art. (246). The cause of the neglect of these striking and original observations has probably been correctly stated, by saying that in them the pendulum was made subservient to hydraulic experiments, and not the theory of fluids to the improved use of the pendulum; and they were therefore overlooked by those whose studies were connected with the pendulum and its applications. The fact observed by Dubuat was, that a large mass of air (or of water in the corresponding case) is carried along with a pendulum in motion, and affects in a sensible manner the time of vibration, quite independent of the diminution of gravity due to the buoyancy of the pendulum. The moved mass of air was proved by hanging a film of worsted from an arm a foot long in advance of the moving sphere, when it was found to be but slightly driven by the inertia of the air through which the pendulum moved. Dubuat significantly calls the mass of moved air "the prow" of the moving body, and it is easy to anticipate the sort of effect which such a graduated condition of the surrounding air from motion to absolute rest would produce.

But the most surprising thing is, that mathematicians should have attempted to compute the effect, Professor or should have been in any degree successful in doing so; yet after the preliminary efforts of Poisson and Green, Professor STOKES has introduced for the first time a correct definition of the "index of friction" of a fluid, and after great labour has succeeded in finding exact expressions for the motions of a solid sphere and cylinder. This investigation may be found in a very elaborate paper in the Cambridge Transactions,² in which he solves the equations found by him in a previous paper,³ in the cases of pendulums having the forms just mentioned. Another interesting result of his investigation is the immense effect of fluid friction in retarding the fall of minute rain drops, which he states to be such as to explain satisfactorily the suspension of clouds. In the second part of the paper I have first cited Mr Stokes proceeds to compare his theory with the observations on the pen-

¹ Raccolta di Autori che trattano del moto dell' acque, 10 vols., 1822-26; and Nuova Raccolta, 7 vols., 1823-45. See also the admirable methodized catalogue of writers on Hydraulics in the second volume of Young's Lectures on Natural Philosophy.

² Vol. ix. part ii.—On the effect of the internal friction of Fluids on the motion of Pendulums.

³ On the friction of Fluids in motion. Camb. Trans., vol. viii., part iii.

dulum mentioned in Art. (246), and the agreement appears satisfactory. With reference to Coulomb's experiments on oscillating disks there remains some doubt whether the theory applies satisfactorily to viscid fluids such as oil.

(417.) Naval architecture. On the very difficult and still empirical subject of naval architecture as regards forms of least resistance, I shall not here speak; but one very interesting case of the resistance of fluids in canals will be noticed in the course of this section—Arts. (422), &c.

(418.) Resisted motion of projectiles. The theory of military projectiles has perhaps received no improvement so considerable during the last 100 years as by the previous experiments of Robins, and the invention of the Ballistic Pendulum referred to in the preceding Dissertation; though the experiments of Borda in France, and of Hutton in this country, have of course increased the technical precision of artillery. Poisson has considered, in a mathematical way, some of the simpler cases of projectiles moving with a moderate velocity.

(419.) Experiments on waves in water. II. Experiments on Waves.—MM. WEBER—MR. RUSSELL.—Whilst the theory of the dilated and compressed waves which constitute sonorous vibrations in elastic fluids was being successfully investigated by Lagrange and Laplace, the case of waves in water, due to a disturbance of hydrostatic pressure only, was attacked by the same mathematicians with far less success. It is generally allowed that the more recent and abstruse researches of Poisson and M. Cauchy, though very valuable as improvements in pure mathematics, have been also singularly barren of valuable results admitting of the very desirable confirmation of direct experiment. The greatest of all hydrodynamical problems, that of the Tides, is, as has been seen in the second section of the chapter on Physical Astronomy, Art. (69) &c., so excessively complex as to be the last, instead of the first, which analysis might have been expected to resolve. When, on the other hand, we see the light which has been thrown even upon it by experiments on a comparatively small scale, we learn to value them in proportion to their rarity.

(420.) Observations of MM. Weber on the form of waves. The brothers ERNST HEINRICH WEBER and WILHELM WEBER, known for many ingenious experiments in physics, and particularly in magnetism, are the authors of a useful book on waves.¹ They made experiments on the velocity of waves in glass troughs, by means of which they determined in several cases the velocity of the wave with different depths of fluid; they also ascertained mechanically the form of the wave, which they found to be that of the curve of sines. But the most important of the Webers' experiments is probably the determination of the motion of individual particles of the water, which they ascertained by watching from the exterior of the glass trough the

curve described by minute floating particles as the wave passed over them. The usual form of the trajectory is an ellipse having the greater axis horizontal, the whole ellipse being described when the wave includes a ridge followed by a depression of the surface; but if there be only an elevated wave propagated, then a semi-ellipse is described in the direction in which the wave moves, and the particle returns to rest in a new position; if the wave be a hollow one the converse takes place. The limits of oscillation diminish with the depth of the fluid, particularly the vertical limits; consequently at some depth the motion of the particles is nearly in a short straight line. The rapidity of the degradation of the individual motions depends on the relation between the length of a wave compared to the depth of the fluid: where the wave is very short compared to the depth of fluid, the ellipses near the surface become circles, and the motion rapidly disappears beneath; when the length of the wave is great compared to the depth of fluid, the motion of the particles is nearly the same from top to bottom. The conclusions of theory are on the whole confirmed by these experiments.

The experiments of Mr JOHN SCOTT RUSSELL, so far as we shall notice them, refer to two closely connected subjects:—the transmission of waves, and the resistance of water to vessels propelled through it as affected by waves. They are principally to be found in the Edinburgh Transactions, vol. xiv., and the British Association Report for 1837. The velocity of waves in troughs of different depths and sections was ascertained by allowing the wave to be reflected at either end of the trough, which does not in any way affect the time of its propagation, and gives great facility for accurate observation, as the distance travelled over may thus become large, at the same time that all casual and interfering waves are gradually eliminated. The instant of the passage of the wave was ascertained by reflecting the light of a candle vertically upwards from its horizontal summit. The length of the trough was 20 feet, but since the wave was made to traverse it as often as 60 times, a distance of 1200 feet was really observed. The waves were single or "solitary" waves, either "positive" produced by a gush of water from behind a sluice, or "negative" by withdrawing water suddenly. Mathematicians seem not agreed as to whether or not the "solitary" waves of Mr Russell are to be treated as a species apart.² Unquestionably it has long been known that a partial wave either positive or negative may be propagated without any farther disturbance of the fluid. Theory also shows that the velocity of a wave long in proportion to the depth of the water is nearly as the square root of the depth; but it does not clearly ap-

¹ Wellenlehre auf experimente gegründet, oder über die Wellen tropfbarer Flüssigkeiten mit anwendung auf die Schall und Lichtwellen. Leipzig, 1825.

² Compare Mr Airy in the article "Tides and Waves," Encycl. Metropolitana; Prof. Stokes in British Assoc. Report for 1846, and Mr Earnshaw in Cambridge Trans., vol. viii.

pear whether the disturbed or undisturbed depth is to be taken. The analysis of Mr Airy seems to show that a depth somewhat greater than that due to the utmost effect of disturbance is to be preferred. Mr Russell also made experiments on the propagation of waves in channels of different forms of section. Professor Kelland has given a very simple expression for the velocity of the wave in this case,¹ which on the whole agrees with experiment.

(422.) Peculiar effect of wave transmission on canal navigation.

Sometime about the year 1830, attention was drawn to a singular fact connected with the resistance of water in the case of canal navigation. It was first noticed I believe in Scotland, probably on the Forth and Clyde Canal. It amounted to this, that whereas at moderate or rather slow velocities, the resistance to a boat increases with the square of the velocity,—after a certain point, not differing very much from 7 miles an hour, the resistances not only cease to increase according to the same rapid law, but actually diminish to some extent when the speed is greater. Different experiments were made by the canal proprietors with a view to meet in some degree the active competition by railways, then commencing. Mr Russell was employed by the directors of the Forth and Clyde Canal, and to his experiments we now refer. It appears from his tables that the resistances increased on the whole faster than the squares of the velocities up to 7½ miles an hour, when they suddenly diminished between 7½ and 8½ miles an hour by one-fifth part in one experiment, and by no less than one-third in another.² It was not until about 12 miles an hour, that the resistance reached the same amount as at 7½ miles.

The occurrence of this singular transition was attended with a phenomenon easily noticeable. In every ordinary case a boat in a canal drives a wave before it, which is in fact a heap of water resisting the boat by increasing the pressure against its bows, which wave may be called a forced wave, having this peculiarity that it travels with the speed of the boat and never quits it; whilst a free wave, by whatever cause excited, is propagated at a rate depending only on the dimensions of the canal, particularly its depth. Now the diminished resistance takes place when the boat is by the force of traction partly drawn out of the water, and lifted up upon the wave to which its own motion gives rise. It is said to ride upon the wave, and the head of water pressing against its bows is visibly diminished. The most advantageous rate of transport was found to be about one-third greater than that required merely to mount the wave, which last depends principally on the depth of the canal. Thus on three different canals, 3½, 5½, and 9 feet deep, the most advantageous velocities were 8, 11, and 15½ miles an hour. The actual velocities of the free wave were ascertained by Mr Russell in an in-

genious and satisfactory manner. When the boat is dragged to most advantage, the draught of water is less at the stem and stern than in the centre. All these circumstances have been very ingeniously and satisfactorily explained by Mr Airy in his paper on Tides and Waves, articles (404–411).

Many persons (amongst whom are Colonel Henry Beaufoy, Mr Scott Russell, and the American ship-builders) have bestowed much attention on the forms of vessels for ensuring speed, especially by the avoidance of waves of various kinds generated by steam-vessels in motion. Every one who can compare the performance of such vessels during the last twenty years, and the still surface which waters navigated by steam vessels now present, as if they were merely cut open and closed again before and after the passage of the ship, instead of being tossed into dangerous billows consuming uselessly the propelling force, will readily admit that, however imperfect the theory, practical art has made real progress in this direction.

III. Improved Hydraulic Machines—Turbine.— (425.) Before concluding this section, I will refer to the most considerable improvement made of late years in the application of hydraulic pressure to motive purposes, and I shall couple it with the names of two French engineers, MM. FOURNEYRON and PONCELET, the former the inventor of the Turbine (the machine referred to), at least in its improved practical form; the latter an important contributor to the useful application of hydraulics, an accomplished mathematician, and the author of several standard works connected with industrial mechanics.

The defects of common vertical water-wheels, whether overshot or undershot, are so great and so notorious, that only their simplicity, and the fact that in very many cases water-power costs next to nothing, and may be squandered with impunity, could justify their use. The advantage of using the simple pressure of a fluid as a moving power had been foreseen in that application of reaction called Barker's Mill; which, though well known in models, was seldom if ever applied in practice. Mathematicians were, however, aware that it offered important advantages. Of late years a patent has been taken out in Scotland for a modification of it, which is found, I believe, to work well. But the Turbine or horizontal water-wheel imagined by Burdin and Fourneyron, and brought to a high state of perfection by the latter about the year 1833, appears to exhaust all that is valuable in this mode of applying water.

Referring to other parts of the Encyclopædia for the details, I may here explain generally that the Turbine consists of two parts, one a fixed cylinder or drum of small height compared to its diameter; the other a portion of a cylinder exterior to the former, and moveable round it, so that

¹ Velocity = $\backslash sqrt\{\backslash frac\{gA\}\{b\}\}$; where A is the area of section of the canal, b the breadth of the water at the surface, and g the accelerating effect of gravity. Edinburgh Trans. vol. xiv.

² Edin. Trans., vol. xiv., p. 48.

the inner surface of the moveable cylinder is all but in contact with the circumference of the fixed cylinder. Further, water under a greater or less hydrostatic pressure, and moving with the consequent velocity, is introduced at the centre of the fixed cylinder, and conveyed to its circumference by channels of a peculiar form constructed by means of vertical partitions or guides of continuous curvature, from which the water is discharged against float boards within the external cylinder, which are also curved, but the curvature is turned the other way so that the partitions in the first cylinder are where they terminate nearly perpendicular to the internal surface of the moveable or second cylinder where they commence. These latter partitions or float-boards terminate exteriorly in a direction nearly tangential to the outer cylindrical surface where the water emerges.

The hydraulic principle of greatest possible advantage to which these Turbines are made as nearly as possible to approximate is this, that the water shall enter the moveable apparatus without shock, and shall quit it without velocity, being simply left behind by the wheel as it escapes from it. Since both these conditions cannot absolutely be fulfilled, the first part of the condition is left imperfect. Some secrecy is, I believe, still maintained as to the forms and dimensions of these machines; but their actual efficiency has been most thoroughly tested by means of De Prony's Friction Dynamometer¹ by Colonel Morin, a most competent authority. His experiments leave no doubt of the admirable qualities of these machines. In particular, the useful effect compared to the theoretical effect represented by the fall of water expended rises higher than probably in any other hydraulic machine, being under favourable circumstances about eighty per cent. Now water-wheels moved principally by the shock of the fall seldom, in the most advantageous conditions, realize thirty-five per cent., often not seven per cent. This superiority of action of the Turbine is due entirely to the approximation which it gives to the theoretic condition above mentioned of perfect efficiency.

But what is not less striking in the performance of this machine, is the variety of circumstances under which it acts advantageously; however great may be the variation in the size and velocity of the wheel, the height of fall, and the power disposable. Turbines have been made of as small diameter as 2 feet with the enormous fall of 350 feet, making 2300 revolutions per minute. They work with nearly equal advantage (relatively to the power expended) whether the supply of water be small or great. They may be completely

drowned or buried under water to a considerable depth without any sensible variation in their efficiency, thus preventing any inconvenience from floods.

It is remarkable that, with these manifest advantages, the Turbine has been so sparingly introduced at least in this country. No doubt the first establishment of it is attended with considerable expense. (430.)

M. PONCELET, an active and intelligent officer of Génie, and member of the Institut, is favourably known by his hydraulic observations and inventions, as well as by his skilful investigation of the effects of machines, and his excellent works and memoirs on several subjects. He has investigated with much patience and geometrical nicety the form and discharges of spouting fluids, and was one of the first to improve materially the ordinary water-wheels, by introducing a kind of breast-wheel (which thirty-five years ago was scarcely known in France) in which the water is conveyed without shock into compartments on the descending side, from which again it was allowed to escape with all its acquired velocity spent, or nearly so. The efficiency of these wheels is equal to about two-thirds of the power expended. Before the Turbine had been finally improved by M. Fourneyron, M. Poncelet had invented an engine on the same principle, in which the water enters at the circumference of the horizontal wheel, and is delivered at the centre. (431.)

I am aware how imperfect this section remains as a history of Hydrodynamics. But I must again refer to special articles on the subject, the plan of the Dissertation not admitting of farther detail. As nothing material has been added to the doctrine of Capillary Attraction since the publications alluded to in Sir John Leslie's Dissertation (although M. Poisson has written a treatise on the subject), I will for the sake of compression not enlarge upon it. I do so with the less regret, because I cannot regard the excessive mathematical illustration which it has received as altogether justified by the certainty and due appreciation of the physical principles involved, such as can alone give to applied mathematics their distinctive value. The theory of Laplace, so far as it was based on novel grounds, was anticipated by Dr Young, and gave rise to several controversial articles by that most eminent philosopher, of which an account will be found in Dr Peacock's Life of Young, pages 199-210, as well as a most excellent review of the subject of Capillary Attraction, which, indeed, by its candour and completeness, supersedes anything which I should have felt disposed to say on the subject. (432.)

¹ Gaspard de Prony, born in 1755, was an eminent engineer, especially in the department of hydraulics, and the author of a voluminous work entitled Nouvelle Architecture Hydraulique; but his originality was not great enough to authorize his being placed among the leaders of his age. His simple invention of the Frein Dynamométrique, or friction dynamometer, is the one by which perhaps he will be longest remembered. It consists of an iron collar with tightening screw, which may be clasped on a horizontal wooden arbor connected with uniformly revolving machinery. A lever, to which a weight may be applied, is attached so as to form part of the collar. The clamping screw being moderately tightened, the collar and lever are of course carried round by the machinery until a weight is applied sufficient to check the velocity, and to generate an amount of friction which is in fact the useful effect of the machine for that velocity, and which is determined by the momentum of the weight overcome in one second. De Prony was a most amiable man, and died in 1839, in the 84th year of his age.

§ 7. Progress of Acoustics. CHLADNI—SAVART. Laplace's Correction of the Theory of Sound. Vibrating Plates and Acoustic Figures. Cagniard de la Tour's "Sirene."

(433.) The mathematical theory of the propagation of sound, considered as a branch of analytical mechanics, made far greater progress during the eighteenth century, in harmony with the general character of the science of that period, than the inductive doctrines of acoustics. Newton here, as in other departments, overstepping the limits of knowledge of his day, left a legacy of toil to his immediate successors. Lagrange had the most distinguished good fortune in reducing the theory of aerial tremors under their most general conditions to the laws of mechanics by the calculus of partial differentials; and Laplace supplied the link which was wanting to reconcile the result with the known mechanical properties of air. As the former of these matters belongs more properly to the period of the previous Dissertation, and as the beautiful discovery of Laplace has been more especially touched upon by Sir John Leslie, it will be sufficient here to recall the fact that the spring of air, or the effort by which it tends to re-expand under sudden compression or to contract to its former bulk when suddenly dilated, is increased by the heat extricated in the former case, as well as by its absorption in the latter. And as sonorous pulsations are held to consist of a series of compressed and rarified waves whose velocity is affected by the recoil of air, it appears certain that the velocity must be increased by this circumstance, though it is difficult to determine experimentally the exact amount.

(434.) The revival of experimental acoustics is due to ERNST CHLADNI a native of Saxony, but of Hungarian extraction, born in 1756. Little had been done in this department since the time of Sauveur, who ascertained the nature of the harmonic vibrations of strings, and that of Daniel Bernoulli, who considered the analogous case of organ pipes. We are indebted to Chladni for two classes of original experiments—his measure of the velocity of sound in a variety of bodies by peculiar and ingenious methods; and his observations on the vibrations of plates by means of the ingenious expedient of strewing them with sand and other powders.

(435.) Velocity of sound in different media; I. Chladni observed the velocity of sound in air of different densities, and in different gases, by using a flute of metal which was sounded by means of the elastic fluid required, and the resulting note enabled him to determine by an easy calculation the speed of propagation of the tremor. This method (using an organ pipe) has been more lately resorted to by Dulong for the purpose of deducing the properties

of different gases with respect to heat, by ascertaining from experiment the co-efficient in Laplace's correction for the velocity of sound. Chladni was also probably the first to notice the longitudinal oscillations of strings and rods which always yield a note immensely sharper than the lateral vibrations. By means of the former he ascertained the velocity of sound in a variety of woods and metals, in some of which it is no less than seventeen times greater than in air. These observations are not only interesting in themselves, but as throwing light on the constitution of solid bodies. To Chladni we likewise owe a knowledge of the twisting vibrations of rods, which exhausts the modes of vibration of such bodies. To connect theoretically the periods of these different kinds of movement, has been a favourite problem with recent mathematicians, but has not even yet been quite successfully performed.

The determination of the velocity of sound in water, an experiment by no means difficult, was reserved for MM. Colladon and Sturm, who observed it on the Lake of Geneva, and found it to be 4708 English feet per second, a result closely conforming to the theoretical amount deduced from Oersted's observation on the compressibility of water.

(436.) II. But Chladni's experiments on the vibrations of plates are of still greater interest and originality. Though it has been affirmed that Galileo strewed sand or light substances upon the sounding boards of musical instruments,¹ Chladni deserves the entire credit of rendering this an exact method of ascertaining the nodal lines or points of rest in bodies vibrating in a stable or permanent manner. He first applied it to plates round, square, or of different figures, supported horizontally and caused to vibrate by applying a violin bow to their edges. Dust or fine sand strewed or sifted uniformly over such a plate, arranges itself in a variety of beautiful figures, being tilted from the greater part of the surface, and heaped up on those parts which are at rest in consequence of the vibratory motion of adjacent parts taking place simultaneously in opposite directions; just as the nodal points of a string vibrating harmonically are without motion on the same account. The number and variety of figures thus producible in the same plate is very great, and corresponds, as Chladni clearly saw, to different simple harmonical vibrations of the elastic plate, being accompanied by their appropriate notes; or by the superposition of several such modes of vibration, and of the corresponding sounds. The tract published by him in 1787 entitled, Entdeckungen über die Theorie des Klanges, contains numerous figures of these appearances, which,

¹ This, however, is very doubtful. See Dove, Reptoricon, iii. 106.

more recently, Savart found the means of preserving by transferring them to sheets of gummed paper. There are few experiments in physics of a more striking character, or which so plainly reveal minute and complex motions so small and so rapid as to be difficultly appreciated otherwise. Mr Faraday and Mr Wheatstone have pursued the same enquiry experimentally, and the latter has satisfactorily deduced the figures of Chladni's square plates from the mechanical superposition of simple modes of vibration which are symmetrical and isochronous. (Phil. Trans. 1833.) By so doing he has succeeded better than the mathematicians, whose results on this subject have been very little practical.

(438.) In bodies possessing unequal elasticity. Chladni was the first to make experiments on the vibrations of bodies whose elasticity varies in different directions. Thus he cut plates out of different kinds of wood, and found the nodal curves unsymmetrical in different directions depending on the course of the fibres. The experiments were naturally afterwards employed to illustrate the theory of ellipsoidal waves on the undulatory hypothesis of Optics.

(439.) Chladni on meteoric bodies. The experiments of Chladni procured for him the especial notice of Napoleon, by whose wish one of his works was translated into French. He died in 1827, and besides his acoustical discoveries, he will be remembered by the sagacity and boldness with which he maintained the popular opinion of the fall of heavy meteors from the sky, contrary to the prevalent scepticism of philosophers. Chladni's success in establishing this important fact in natural history is due, like his other physical inductions, to the constancy and simple-mindedness with which he attached himself to a strictly definite enquiry.

(440.) Young—Robison. We must not here enlarge upon the ingenious and important investigations of Dr Thomas Young connected with acoustics. Being chiefly connected with his admirable Theory of Light, they will be noticed in the chapter on Optics. The peculiarly practical and sagacious views of Dr Robison connected with the Theory of Musical Instruments and Acoustics generally, must also be passed over.

(441.) Savart. In FELIX SAVART we find a man like Chladni who was especially devoted to a single and circumscribed branch of science—acoustics. His published researches are almost all detached notices in the Annales de Chimie, with a few memoirs in the publications of the Institute; and whilst they are very interesting, exact, and instructive, I doubt whether it would be possible to place the results in a connected and comprehensive view before the reader. They are therefore rather to be sought in the special articles of the Encyclopædia devoted to the subject. In general they may be stated to refer to the following topics:—(1.) To the manner of propagation of sound through the air and through liquids, with an attempt to explore the manner in which sounds spread in apartments of different forms; an enquiry as difficult as it is important; (2.) To the generation and audibility of

musical notes. Previous authors (for example, On the production of musical notes. Chladni, Biot, and Wollaston) having differed materially as to the range of audibility of repeated equidistant impressions which affect the human ear as musical notes, Savart used a simple method, no doubt original to him, but anticipated I believe by Robison, in which a card is held near or touching a revolving wheel, and the number of impulses (each double) given to the air by every tooth as it passes the card, is readily measured. He thus found that a note occasioned by 24,000 double vibrations in a second is perfectly audible; and, at the other limit of the musical scale, from seven to eight equidistant beats constitutes a sound having a distinct pitch. According to Savart, two consecutive double impulses of whatever duration are sufficient to convey to the ear the sensation of pitch. But a more elegant and accurate instrument for the numeration of sonorous pulsations is the Sirène of M. CAGNIARD DE LA TOUR, unquestionably one of the most exact and satisfactory additions lately made to our experimental apparatus. In it a current of air is repeatedly interrupted and renewed, giving rise to a series of impulses similar to those of the toothed wheel; and this apparatus is ingeniously contrived, so as to maintain its own motion, and record its indications. It is by far the most accurate known method of ascertaining the pitch of a given note. It may also be worked with water. Robison had also the merit of the primary idea of the Sirène, by making a stopcock revolve rapidly whilst applied to a tube emitting a blast of air. (3.)

Savart extended the researches of Chladni by means of sand to many new cases, and with interesting results; in particular he exhibited the effects of the unequal mechanical elasticity of crystals cut in different directions. He has also examined with great care and ingenuity, the nature of the vibrations which occasion the accumulations of sand on the nodal lines of plates, and he comes to the conclusion that they are determined by simultaneous transverse and longitudinal movements (the latter of which are parallel to the surface of the plate). In proof of this he shows that in long rods or hollow cylinders, the position of the nodes is intermediate and opposed upon the contrary sides of the rod or hollow cylinder. Savart made many experiments on the communication of vibrations from one body to another; showing that the molecular movements generally preserve their parallelism, so that a longitudinal vibration of one body may give rise to transversal movements in another; and he applies this to the theory of musical instruments.

Savart was born at Mezières in the department of Ardennes, on the 30th June 1791; and died somewhat prematurely on the 16th March 1840. He had some peculiarities of temper, amongst which was his unconquerable prejudice to everything English. He did not even acknowledge the intimation

of his election as a foreign member of the Royal Society of London. It is to be regretted that Savart never published a connected account of his observations. He had caused to be collected at the Collège de France, where he was professor, an unequalled collection of acoustical apparatus, a great deal of which was contrived by himself, and where he delivered extensive courses of lectures on this subject alone.

Several English philosophers, in particular Professor Willis, Mr Hopkins, and Mr Wheatstone, have written several important detached memoirs on particular practical points, for which we must refer to special treatises.¹ The first and last of these gentlemen, together with some Germans, have approximated in some degree to the formation (on empirical principles) of a speaking machine.

§ 1. THOMAS YOUNG.—The Undulatory Theory of Light.—Its history from the time of Hooke and Huygens.—The Law of Interference.—Its application to Diffraction—to the Rainbow—and to other subjects.—The Theory of Polarization referred to another section.

(444.)
Small progress of optics in the 18th century.

THE history of Optics in the eighteenth century is one of the blankest pages of scientific story; at least if we allow Bradley's discovery of aberration to be (as it really is) rather an astronomical than an optical discovery. The most notable advance was unquestionably the invention of the achromatic telescope as narrated in the Fifth Dissertation,² founded on the proof of Newton's oversight in the matter of dispersion. The construction of refracting telescopes made rapid advancement in the workshop of Dollond, whilst reflecting telescopes, in the hands first of Short, but far more conspicuously, of Sir William Herschel, were shown to be capable of making unimaginable discoveries. The geometrical theory of optical instruments was also greatly improved; but all this led to little increased knowledge concerning Light itself. If we except the valuable though imperfect treatises of Bouguer and Lambert on the subject of photometry, and a paper by Mr (now Lord) Brougham in the last years of the century, recalling attention to the inflexion of light, the history of Physical Optics (as that part of the science touching more immediately the nature and qualities of light is now usually termed) is almost a blank from the publication of the Optics of Newton in 1704 to that of Young's papers almost one hundred years later.

(445.)
Thomas Young.

It is not therefore from overlooking Young's predecessors that we open our review of the recent progress of optics with his discoveries. We here meet

with a man altogether beyond the common standard, one in whom natural endowment and sedulous cultivation rivalled each other in the production of a true philosopher; nor do we hesitate to state our belief that since Newton, THOMAS YOUNG stands unrivalled in the annals of British science.

He was born at Milverton in Somersetshire on the 13th June 1773, and his biographers dwell with complacency on the prodigies of his youth, uncertain as such attainments confessedly are in stamping the greatness of the future character. At the age of fourteen he had learned (principally for amusement) seven languages besides his own, and besides had made a point of mastering every subject, whether in science or miscellaneous knowledge, which he had once determined upon prosecuting. Thus, whilst studying botany he resolved to learn how to make a microscope, but finding in Martin's Optics the notation of fluxions, he became his own preceptor in that branch of analysis. "He acquired a great facility in writing Latin. He composed Greek verses which stood the test of the criticism of the first scholars of the day, and read a good deal of the higher mathematics. His amusements were the studies of botany and zoology, and to entomology, in particular, he at that time paid great attention."³ Dr Young's education was almost completely private. Having been brought up according to the tenets of the Society of Friends, he had not thought of going to Cambridge

¹ See also Herschel on Sound (Encyc. Metrop.); and Whewell's History of the Inductive Sciences, vol. ii.

² It may be mentioned, however, that the credit usually ascribed to Dollond must be divided, at least, with Mr Hall, a private gentleman of Worcestershire, who not only imagined but constructed achromatic telescopes as early as 1733 (Gentleman's Magazine, 1790, and Phil. Mag., vol. ii.) The improvement by Dr Blair of Edinburgh has been alluded to in Sir John Leslie's Dissertation. It consisted in enclosing fields in the object glass, of such composition as to disperse the several rays of the spectrum in the same proportion to one another (though not to the same absolute amount) as the glass with which it was combined; thus rendering the achromatism more perfect.

³ Memoir of the Life of Thomas Young, M.D. Svo, 1831. The present section of the Dissertation was written shortly before the publication of the Life and Miscellaneous Works of Dr Thomas Young, for which the public is indebted to Dr Peacock, Dean of Ely, the possession of which would have materially facilitated my task. Wherever Dr Peacock's information has

(which would have been his natural destination), and entered the university of Edinburgh as a medical student at the age of twenty-one. He had already declined the overtures of such distinguished patrons as Windham and Burke, resolving to devote himself to the pursuit of science, for which a medical education seemed to him a fit entrance;—his studies being made under the more immediate advice of his uncle, Dr Brocklesby. He attended Black's lectures in Edinburgh; whether he was known to Robison I am not aware, though I should be inclined to infer that he was from the terms in which Robison speaks of Young when criticising his strictures upon Smith's Harmonics. Robison disagreed with him on this point, and also about the nature of light, yet he speaks of Young and of his paper on Sound with very marked respect. More than a year before his enrolment at Edinburgh (which took place in autumn 1794), he communicated to the Royal Society of London a paper on vision, of which we shall presently give some further account; and was elected a fellow of the Society when just of age.

From Edinburgh he proceeded to Göttingen where he graduated; acquiring the German language, and leaving a vivid impression of his astonishing versatility of talent and powers of memory. Early in 1797 he returned to England, and soon after entered himself at Emmanuel College, Cambridge, in order to comply with the requisitions of the London College of Physicians, and thus to obtain a license to practice. For the next few years his time was divided much between Cambridge and London. He was now twenty-three years of age, and his mental habits too much formed to bend to the rules of Cambridge study. When the master of his college introduced him to the fellows, he is reported to have said, "I have brought you a pupil qualified to read lectures to his tutors," and such, no doubt, was the fact. We hear little of his occupations at Cambridge, but we can hardly doubt that his private studies then ranged over the vast fields of erudition which he afterwards proved that he had made so completely his own; and we cannot doubt that he was then preparing the groundwork of his theory of optics, although his discovery of interference was certainly not made at Cambridge, and probably in London after his settlement there in 1800.¹ His first paper on sound and light is dated from Cambridge in July 1799.

I have entered into these details because they throw light on the peculiarities of Young's character and attainments. He was to a great degree self-educated; and his studies in consequence may be called desultory, though none would dare to call

them superficial. Mathematicians may consider his acquaintance with their science as not technically complete, yet one of them admits that "he could make a small amount of mathematics go farther than any one else." Had he been a consummate analyst it is unlikely that we should have had in him the author of the undulatory theory, the difficulties of which in its earlier stages made it unpalatable to Laplace, Poisson, and the most considerable French mathematicians. Having thought out for himself every one of the multifarious subjects with which he grappled, his writings have a striking force and originality, and his reports of the labours of others are almost invariably drawn from a study of their original works. His earliest principle was, that what one man has done another may accomplish; and one of the many respects in which he resembled his great predecessor Newton, was unbounded confidence in the powers of "patient thought." Not that he confined the desire to excel to purely intellectual matters. What he found it worth while to do at all, he thought it worth doing well. He chose to be first-rate in dancing and in equitation;—his penmanship was (in his early days) as scrupulously elegant as his scholarship.

In 1801 Young was appointed Professor of Natural Philosophy at the Royal Institution, where he was the colleague of Davy. Young had not the qualifications of a popular lecturer, and the most important result of his short connection of two years with the Royal Institution was the publication in 1807 of his Lectures on Natural Philosophy, in two large quarto volumes. It is a work peculiarly characteristic of the author; and is rather adapted for reference by the scholar, than to be studied as an elementary treatise. Its condensation is such as to render it in many places obscure; though when read by one conversant with the subject, its comprehensiveness and precision are surprising. It is a truly admirable monument of labour and genius combined. Embracing the arts as well as the whole of natural philosophy, it seems to include the mention of everything connected with his vast subject from the simplest tool of the artisan to the highest speculations of Newton and Lagrange; and yet it is evident, by the masterly manner in which he handles it, that the author had made all this mass of knowledge completely his own. The catalogue of references with which it closes indicates an extent of bibliographical research which would have done honour to any one who had made that an exclusive object of study. Even the plates are drawn with a studious care, betokening well his own mechanical talent.

enabled me to improve the text I have not hesitated to use it. The facility of consultation afforded by the collection of Dr Young's widely scattered writings is a most important aid to all future students of science, and one which cannot fail to raise still higher the great reputation of their author.

¹ See Sect. vii. of the article POLARIZATION in this Encyclopædia, where, in a bracketed interpolation by the translator, Dr Young, he speaks of this fundamental experiment being made "in the room and at the table on which he is now writing." This must have been in Welbeck Street, London.

Many of the figures, though of hackneyed subjects, are represented in a novel manner; there seems not a line drawn at random. Such figures often illustrate better than pages of description, the clearness with which an author has conceived to himself the necessary results of his own principles. An example of this may be found in Plate XXX., fig. 442 of the first volume, which, as Arago relates, served to demonstrate to him, when he visited Dr Young in 1816, that he and Fresnel had been anticipated on one point more than they believed.¹

(450.) The absence of algebraic formulae from this work was as characteristic of Dr Young as their copious introduction into articles which he subsequently contributed to the Quarterly Review. He had decided upon writing a book without symbols, and he wrote it, though it gave additional trouble both to himself and the reader.

(451.) We shall now proceed to trace the progress of the undulatory theory of light, the greatest physico-mathematical discovery of our time, in the establishment of which Young acted the leading part.

(452.) Undulations of Light—Hooke and Huygens.—
The undulatory theory of light—Hooke.
The idea of accounting for the effects and modifications of luminous impressions by disturbances propagated through a very elastic medium was by no means new at the commencement of the present century. We do not, indeed, attach much importance to the so-called anticipations of Grimaldi and Hooke in the seventeenth century. The former, amongst his valuable experiments on the deflection of light and fringes of shadows, had used an expression as to illumination being diminished by the addition of light, which is true in fact, and is a correct deduction from the law of interference as we now understand it. Hooke, in his Micrographia, asserts light to consist in "quick, short, vibrating motion;" but his explanation of refraction by it is altogether erroneous; and his application of it to the doctrine of the colours of thin plates, though admitted by the candid Young to be an anticipation (unknown to him at the time) of his own, has in it no more than a germ of truth (like so many of Hooke's ingenious hints, afterwards claimed by him as discoveries), which yet only explains the fact on which it is founded by means of an additional and gratuitous assumption. The germ of truth in Hooke's writings is this, that the colours in question depend upon a mixture of the light reflected at the first surface of the thin plate with "a kind of fainter ray" propagated from the second surface backwards: the gratuitous assumption is, that "this compound or duplicated pulse does produce on the retina the sensation of a yellow;" why it does so

is not explained. This was in 1664, before even Newton was acquainted with the analysis of white light. Hooke's idea that yellow, or any other colour, was the result of the conflict of pulses simultaneously reaching the eye, was an assertion, admissible, perhaps, at that time, as expressing a fact; but surely not a proof of interference producing reinforcement or annihilation of light, as taught by Young. I am not aware that Hooke ever even reiterated his opinions on this subject after Newton had analysed the phenomena experimentally, and shown that the colours of thin plates result from the superposition of bright and dark rings of different prismatic hues, each with its appropriate diameter. It was then apparent that colour was only an indirect effect of interference.

But whatever may be thought of the theories of Hooke, those of Huygens deserve a far more eminent place in history. Having already been succinctly adverted to in Professor Playfair's dissertation, we will only observe that the Traité de la Lumière (1690) is an admirably composed and reasoned treatise on the phenomena of light on the undulatory hypothesis. The uniformity of its propagation through the celestial spaces, its rectilinear course in ordinary circumstances, the laws of its reflection and refraction, are there explained with a degree of elegance and precision which ought to have excited (we must think) general attention and assent, but for the ascendancy of Newton's authority, and the astonishing and beautiful nature of the experiments on which his theories were based; whereas Huygens referred to few experiments except those of the simplest kind, and the phenomena of colour were (for good reasons) left chiefly out of view.² Such being the case, Huygens may fairly be considered as the author of the undulatory theory, which he supported by such convincing proofs.

(453.) The fundamental principle of the Huygenian doctrine was the same as that which Hooke admitted, which was probably far older than his time, namely, that all space, including the interior of transparent bodies, contains an ether whose pulsations communicate the sense of light to the eye, as waves in air convey to the ear impressions of sound. To this he added the assumption, that in refracting media, such as glass, the pulsations are retarded; whereas in Newton's theory, as is well known, the propagation of light is assumed to be fastest in dense media. The "law of the sines" in refraction, is deduced as a consequence; and one of the prettiest applications made of it is to the phenomena of atmospheric refraction. But the most important features of the whole investigation are these two—(1), the

¹ The reference of Arago, in his original éloge of Young, is to p. 357 of the Lectures, obviously by mistake. In the first volume of his (Arago's) collected works it is corrected into p. 787, which is as certainly correct. I have supposed fig. 442 to be more probably the one referred to than 445, to which Dr Peacock refers (Life of Young, p. 389).

² Nevertheless the Huygenian Theory of Light was propounded as the subject of a thesis at St Andrews by David Gregory, whilst professor of mathematics there, within about a year of its publication (including also the Newtonian doctrine of gravity),—a pleasing proof of the activity which then reigned in that university. Principal Lee possesses the original programme.

principle that the impression of a wave of light on a screen is to be found by considering the simultaneous effect of the wavelets propagated from all the points disturbed at any previous moment (which points form what is called the front of a wave). It is only when these partial impressions concur that they are strong enough to affect the sense of vision;—(2), the representation of the phenomenon of double refraction by a wave of two sheets, or the simultaneous propagation of spherical and spheroidal waves in one and the same medium. Of the last of these doctrines I shall speak in the next section in connection with the discoveries of Malus. With respect to the former I may here observe that it gives the only satisfactory explanation of the primary difficulty of the undulatory hypothesis,—namely, that a beam of light admitted by a hole in a screen pursues a rectilinear course afterwards, instead of spreading sideways, as do waves in water, and waves of sound. Huygens shows, on elementary and convincing principles, that the lateral impressions of the wave are rapidly extinguished by the want of concurrence of the impulses which they communicate to the ether. This is necessarily true when the breadth of the aperture is such as to exceed vastly the length of a wave; and such is always the case with light, but rarely in any other instance. It is, in short, only immediately in front of the aperture that the disturbances originating in every part of the front of the original wave embraced within the aperture, concur in producing an accordant movement on the ether.

This principle, more fully stated, by which every luminiferous disturbance of the ether is considered as the resultant of all the pre-existing disturbances to which it is due, constitutes what is sometimes called the principle of Huygens, of which I shall have more to say hereafter.

Neither at the time of its publication, nor for more than a century afterwards, was the value of these reasonings understood. It would be beside our present object to discuss Newton's opinions; but it is too certain that he did not allow Huygens' arguments on the undulatory nature of light to have any weight with him. Not that he was averse (as is often supposed) to the presence of Ether as modifying the corpuscular theory of light; on the contrary, in many of his minor writings he speaks of its existence as all but certain, and as a requisite adjunct to the corpuscular hypothesis to which he had been led by the facts of reflection and refraction.¹ But he never adjusted the terms of a compro-

mise, and must be held, we think, to have left behind him no substantive theory of light worthy of the name. The question never perhaps very seriously engaged his attention after the publication of Huygens' book;² and we know that about that time his intellectual energies received a shock which left him indisposed for the fatigue of constructing new theories, and still more disinclined to publish them.

Euler, though he professed to defend an undulatory theory of light, treated the subject in a point of view chiefly mathematical, and betrayed that unconcern about physical theories which characterized a mind steeped in geometric abstractions. He even retrograded; for he did not maintain the Huygenian explanation of the cause of definite shadows.

Young on the Undulatory Theory—Diffraction. (458.)
—One hundred and ten years after the publication of the Traité de la Lumière of Huygens, Dr Thomas Young re-opened the theory of light or "Physical Optics," as he termed it. His experiments and reasonings will be found in a series of papers in the Philosophical Transactions for 1800, 1801, 1802, and 1803.³ These memoirs are of no great length, and deserve the most careful study. They are perhaps among the clearest and plainest of Young's writings, although blamed at the time for defects precisely the reverse. They are eminently marked by penetration, profound induction, and candour of argument. Starting from his studies in acoustics, the transition to optical questions is extremely gradual. Young was, characteristically, a good musician in practice, as well as a profound one in theory, and his paper of 1799 is principally acoustical. In it he attaches consequence to showing that the divergence of sound from the direction of its emission is slower and less complete than it is commonly believed to be, and he applies the analogy to the existence of rays of light and definite shadows. In one short section he sums up the chief points of optical doctrine which lead him to prefer the theory of Huygens to that of Newton. Amongst the facts better explained by waves than corpuscles, we find reckoned Inflection and the Colours of thin plates. But all this is stated in a very general way, evidently rather as a conclusion towards which his mind had for some time been tending, than as the result of demonstrative proofs. In his paper of 1801 the undulatory doctrine is methodically expounded in a series of propositions, accompanied by proofs. The accurate definition of shadows is shown to be possible and natural on that theory, as well as the usual phenomena of reflection, refraction, and total reflection.

¹ Thus Newton writes in 1675:—"Were I to assume an hypothesis, it should be this, if propounded more generally, so as not to determine what light is farther than that it is something or other capable of exciting vibrations in the ether; for thus it will become so general and comprehensive of other hypotheses as to leave little room for new ones to be invented." And again,—"Do not the most refrangible rays excite the shortest vibrations [of the retina], the least refrangible the largest?"—Birch's Hist. of the Royal Society, quoted in Young's Lectures, ii. 615, 617. Sir D. Brewster (Life of Newton, 1855, vol. i. p. 148) considers that some passages in the later editions of Newton's Optics show that he had departed from any theory of undulations.

² The Optics, though published in 1704, had been written principally in 1675 and 1687 (see Preface).

³ They may be more conveniently consulted as reprinted in the second volume of his Lectures on Natural Philosophy, and in his Miscellaneous Works, vol. i.

The partial reflection which always accompanies refraction is strongly and justly insisted on as an obvious consequence of the theory, while it requires a separate hypothesis on Newton's. But the chief weight is claimed for the evidence from the colours of heated surfaces, of thin plates, and of diffracted shadows, all of which the author explains by the mixture of two portions of light conveying to the same particle of ether at the same time either accordant or opposing motions, thus redoubling or destroying the light. Of these the splendid iridescent colours reflected by surfaces having fine equidistant lines drawn upon them, admit of the most elementary and striking explanation. The reflected image of a luminous point viewed in a mirror thus cut up by parallel lines, consists of one common reflection and numerous lateral images which are coloured, and in which the angles of incidence and reflection are not equal, thus contradicting one of the axioms of common optics. Young showed that the scattered waves of light recover the faculty of appearing when the surface of the plate is seen under such an angle that foreshortened intervals between the scratches amount severally to the length of one undulation or a multiple of it; for then the waves of light scattered by the reflecting surface will not come entire to the eye, but each will have a part systematically suppressed by the non-reflecting space of the groove, so that the remainders being nearly in one phase, concur in making a general impression. This experiment, therefore, literally presents us with the paradox that by suppressing half the light, the remainder is not suffered to be extinguished by it. The different colours appear reflected at different angles, because the obliquity must vary in order to be accommodated to their several wave-lengths, and each colour undergoes several repetitions corresponding to breadths representing the successive multiples of a wave-length.

Precisely similar in their origin are the coloured rays scattered by fibrous substances when held between the eye and a small point of light. If they be numerous and all of the same diameter, such fibres will suppress symmetrically portions of waves, and suffer the oblique effect to be perceptible. Dr Young most ingeniously applied this principle to construct an Eriometer or measurer of the fineness of fibres. The diffracted light of any order and colour from a distant flame will be seen at an angle with the principal or white image about four and a half times greater when viewed through the down of the beaver, than in the case of Southdown wool; being the inverse proportion of the diameters of the fibres which compose them.

Fundamental Law of Interference.—But the critical and characteristic experiment of interference in its simplest shape was published two years later, in

the Bakerian Lecture for 1803.¹ A small hole being made with a needle point in a piece of paper applied to a window-shutter, and a sunbeam being directed upon it by means of a mirror from without, a cone of light is thrown into a darkened room. A slip of card one-thirtieth of an inch wide being held in the sunbeam, its shadow was observed on the opposite wall or on a moveable screen. There were seen fringes of colour exterior to the shadow on each side, such as Newton had described, and on which Mr Brougham and others had made experiments. But besides these, the narrower and less conspicuous fringes seen in the interior of the shadow, and first described by Grimaldi, were found by Young to have this remarkable property, that they disappeared the moment that the light passing either edge of the card was intercepted, whilst the exterior fringe was not at all affected by that circumstance excepting on the side where the light was stopped.

Young at once perceived the significance of this admirable fact. The existence of light within the shadow at all, was evidently due to the bending of the wave round the opaque edge; but the alternation of light and dark spaces required the union of the two lights from opposite edges, which, immediately behind the centre of the obstacle, must have described exactly equal paths, and therefore united in the same phase; but a little way either to the right or left of the centre the phases were discordant, and complete and effectual annihilation of the light resulted. In fact, when the experiment is performed under favourable circumstances, the result of the union of the light is perfect blackness in these places, but if half the light is stopped the dark spaces become luminous!

This splendid paradox may also be demonstrated without any bending round the edges of bodies, and consequently without any inflexion in Newton's sense of the word; and this simplifies the conditions materially. In order to effect this, Fresnel (many years after) produced interference bands by allowing light emitted from a very small luminous point (an image of the sun formed by a lens of short focus) to fall upon two mirrors touching at the edge, and inclined to one another at an angle very little less than 180°. By the common principles of reflection there will be a space beyond the mirrors where the light reflected from the respective mirrors overlaps, and except in a single line within that space, the paths of the two rays meeting in any point will be different. When this difference amounts to a whole number of undulations, an exalted brightness results; when the undulations arrive in opposite phases or the centres of one set of waves concur with the ends of the other, blackness results. The experiment may even be made with a single mirror which a ray of light just grazes, and after reflection mixes with the direct light

which had in the first instance passed clear of the mirror.¹ Here evidently we have the required condition of double illumination with difference of paths. The same effect is obtained without either Inflexion or Reflection by refraction through an excessively flat prism.

(463.) Length of the waves of coloured light. It is an easy matter (comparatively) to assign the lengths of a wave of light, from the intervals of the interference lines, or still better from the elongations of the coloured images produced by striated surfaces, the intervals of the striæ being given. Newton's measures of the intervals between the lenses producing coloured rays, gave Dr Young the following for the number of undulations contained in an inch producing each colour:

Now the velocity of light is known, that is, the rate of propagation of a disturbance in ether; but the duration of an impulse, or rather the interval between two successive impulses striking the eye and producing the effect of colour, is the time that an impulse takes to travel over the length of a wave. It is easy to see how almost infinitely short this must be: 460 millions of millions of such impressions in a second of time go to make up the sensation of redness, 735 millions of millions that of violet light.

(464.) Opposition to Young's views. It might be supposed that Young's discovery and its application excited the notice and applause of all persons interested in optics. This was very far from being the case. Though he brought it several times in succession and in different forms before the Royal Society of London, there is no evidence, so far as I at present know, of his having then obtained a single adherent. Davy was no optician; Wollaston was too cautious to commit himself, though probably giving a tacit assent; Cavendish was aged, and besides had attended less to this subject than to most others; Sir William Herschel had only lately taken up physical optics, and that with reference to the qualities of the spectrum least connected with Young's observations. At the Royal Institution Young vainly attempted, in the elaborate course of lectures which he there delivered for two years (1801-3) on natural philosophy and the arts, to arouse a popular interest in the unveiling of these mysteries. The abstruseness of his discourses scared that mixed audience, and his colleague Davy, in a letter, incidentally observes that Young would be satisfied if any one would even offer criticisms on his opinions. Criticism of a certain kind, however, he soon got in abundance. The Edinburgh Review, in its second number, under-

took to crush at once the theory of Young and his reputation as a philosopher, and this (in singular contrast to its avowed principles), not by argument, but by an appeal to the weight of prescriptive authority in favour of the Newtonian hypothesis, concluding with an admonition to the Royal Society to cling to its old standards and old celebrities, and to save its Transactions from degenerating into volumes of miscellanies. This attack, paltry as it was, seriously prejudiced the reception, or even the dispassionate consideration of Young's views. His anxious vindication put forth in a separate pamphlet was unread, and the doctrine of interference was first understood and relished in France ten years later.

Theory of the Rainbow.—It is a matter of interest (465.) in several points of view that the phenomenon of the rainbow, which gave the first suspicion of the varying refrangibility of light, and which, when explained and reduced to calculation by Newton, so convincingly proved the truth of the doctrine of the composite nature of white light, was destined in the hands of Young and of his successors to yield one of the most refined evidences of the extensive application of the doctrine of interference. The general fact of the arrangement of colour in the primary and secondary bows Newton accounted for. But the spurious or supernumerary bows occasionally seen within the primary, and far more rarely beyond the secondary, consisting of reddish and greenish bands, remained unexplained. The brilliancy of any given portion of the rainbow depends upon the deviation of the sun's rays by two refractions and one reflection, approaching to a limit which it cannot overpass. But except at this precise limit an amount of scattered light will reach the eye, which, though not reflected under the most favourable circumstances, yet is still sufficiently intense to be visible. This light must be composed, as Young showed, of two portions, entering the eye in the same direction, but which have pursued different paths within the drop, and which never coincide except at the extreme geometrical limit before mentioned. When one of these paths differs from the other by the length of half a wave of the particular kind of light considered, darkness will result, but a feeble maximum will be again attained when the interval rises to a whole wave-length, or to two or more. Hence these consequences follow—first, that the bright part of each colour is limited by its self-destruction to a narrow band, and thus the purity of prismatic colour so striking in a well-formed rainbow is preserved; secondly, that each colour may attain (by interference) a second and a third maximum, corresponding in fact to the position of the spurious bows; thirdly, that these phenomena of perfect definition of the primary and secondary bows, and of repeated maxima in the supernumerary bows, depend essentially on this condition, that the drops of falling rain shall approach to

¹ This experiment is usually (and justly) ascribed to an eminent and amiable British philosopher. But it had already been performed by Fresnel with a special object. See Ann. de Chimie et de Physique, second series, xv. 382.

a general equality of size. For the effects of interference depend on the precise diameter of the drop; if these be very various the resulting positions of maxima and minima will be altogether confused. Dr Young pointed out that to accord with the phenomena the falling drops must be about $\backslash frac\{1\}\{2\}$ of an inch in diameter.

(466.) Delicacy of the phenomenon of the rainbow as a test of Theory.

I have merely indicated the nature of an argument of extreme interest and beauty. It would be difficult to cite (except perhaps in the science of physical astronomy) a more complete specimen of gradual inductive research. Here is a phenomenon—the rainbow—as familiar as it is beautiful. Even a partial insight into its cause confers a certain reputation upon one individual (De Dominis), its farther explication gave Newton one of his most popular triumphs. It is then found that the rainbow is not so simple a fact as was supposed, and that Newton's theory accounts for only its broader features. Then, as in the theory of gravity, a long period of uncertainty ensues; but observations are continued. A perfect rainbow is found to be one of the rarest of natural phenomena, instead of the commonest. Not above two or three individuals have ever seen, or at least described one. Then comes Dr Young, with his theory of interference and diffraction. This theory not only accounts for the spurious bows, but for the precise appearance of the principal ones, which, but for it, would have been different from what Newton supposed. Finally, after being canvassed for more than two centuries, the theory of Young is carried out into its rigorous consequences by Mr Airy¹ and Professor Stokes² (who must first invent a new mathematical method for the purpose) and illustrated by the ingenious experiments of M. Babinet and Professor Miller;³ until at last we begin to believe that we understand this matter completely.

(467.) Exterior fringes of shadows;

Exterior Fringes of Shadows.—I have mentioned only generally Young's application of his theory to the coloured fringes observed by Grimaldi and Newton to surround the outline of bodies, as thrown in shadow by a luminous point upon a distant screen. I have done so because Young's explanation was imperfect, not to say incorrect. But as it would be inconvenient to discuss the subject here, I shall briefly indicate its history and result. Dr Young expresses himself more obscurely in his paper of 1801 on this point than on any other, indicating three possible explanations. In 1803, however, he distinctly adopts the opinion that the periodical colours in question are due to the interference of direct light passing near the opaque edge with a portion of light very obliquely reflected from that edge; and he enters into calculations to show that such a theory represents sufficiently well Newton's measures. But it is unaccountable that Young should have been satisfied with the belief that the screens employed

should in every case have reflected an appreciable quantity of light (or indeed any light at all) in the required direction. It might be conceivable in the case of a cylindrical wire or a cylindrical hair; but how could a film of gold-leaf or a slip of paper receiving the light on its broad side furnish such a degree of oblique illumination? It is wonderful that Young's intuitive sense did not perceive that the portion of a luminiferous wave passing near an opaque edge, is deficient on one side of the interfering wavelets which are necessary to make the boundary of the shadow definite, and to extinguish the laterally-spreading light. In short, he did not allow to Huygens' principle (see art. 455) the full breadth of its application—a discovery made some years later by Fresnel, who has the credit of first explaining these exterior fringes.

(468.) That great philosopher (the worthy rival of Young in this career of discovery) found the means of computing, on strict geometrical principles, the sum total of the disturbance produced at any point of a screen by the whole effective portion of a luminiferous wave partially stopped by an obstacle of a given form. The principle of the calculation is simple enough. The origin of the light being distant, the front of the wave is considered as flat when it breaks against the opaque body. Its front is then divided (in thought) into small elementary portions, each of which is considered as the source of a disturbance propagated as from a new origin. The effect of each wavelet is calculated in terms of the co-ordinates of its origin, and of the point where its effect is to be considered. The sum of all these simultaneous effects is collected by integration, a process which unfortunately is only rigorously possible in a limited number of cases. Some of these cases were solved with great ingenuity by Fresnel, and compared with observation. The result was extremely satisfactory. Yet it is curious to observe that Young's explanation, if it had had a sufficient physical basis, leads to nearly similar results. In the case of an indefinite opaque body with a straight edge, the illumination precisely at the boundary of the "geometrical" shadow is, on Fresnel's theory, one-fourth of what it would have been were the body removed. Within this line the light dies away gradually, having no maxima or minima. Without it, a series of dark and light bands occur, which rapidly blend into a uniform illumination. The same theory leads to results as to the position of the interior bands which are also somewhat different from the simpler calculation of Young, and still more conformable to experience. Amongst the most singular of these results is this (which is perfectly confirmed by observation), that the shadow of a small round opaque body (as a spot of tin foil) is illuminated by a speck of diffracted light at its centre precisely as bright as if the disk were removed! How, after

this, can we talk of light as moving in straight lines only?

We must now advert to the peculiar and disadvantageous manner in which Dr Young laid his farther researches before the world. The optical papers in the Philosophical Transactions, ending with 1803, were the last which he published connected with his theory in that work, although he continued to be Foreign Secretary until his death twenty-five years afterwards, and although during all that time he never ceased to extend and perfect his views on the subject of his predilection. The explanation of this paradox is to be found principally in the strictness with which he interpreted the allegiance which he owed to the medical profession. He had determined to be a practical physician; his early principles of action prevented him from doing anything by halves; and all experience affirmed, that to gain confidence as a physician in the metropolis he must cultivate sparingly, and as it were by stealth, the studies of abstract science and of philology in which he delighted. Unquestionably he was also disgusted by the absence of one single supporter amongst the members of the great society referred to,—by the injurious petulance of the then popular critical journal,—and by the impossibility under which he laboured of communicating orally his knowledge to a general audience in an interesting and acceptable manner. The result of all this was the suppression of many of his opinions, and the publication of others in so concealed and uninviting a form that they remained for years nearly buried and unknown to men of science. He contributed a series of articles on sub-

jects connected with light to the Quarterly Review; and we may well smile at the abstruse and really obscure dissertations on detached points of science—often unmercifully loaded with algebra—thus interspersed with articles of popular criticism for the entertainment of the reading public. From some of these papers we may readily gather the soreness which he felt at the cold reception of his discoveries. Farther and still more important original speculations were contained in a series of anonymous papers (sixty-three in number) on a vast variety of subjects, both in science and philology, contributed to the Encyclopædia Britannica. It is not in a work such as this that we usually look for the first publication of great and original views: the articles being anonymous could only very gradually attract notice by their intrinsic merit; and the obscurity of some of those written by Young rendered this difficult enough. But it is most fortunate that he was induced thus to write: many of his most original thoughts must have been lost but for these concealed repositories. In the articles in the Quarterly Review, for example, we watch with interest the impression which contemporary discoveries made upon his mind. The spheroidal wave of extraordinary refraction is explained by unequal elasticity of the

crystal in different directions;¹ the discovery of polarization by reflection is received with characteristic candour, as giving a temporary blow to the undulatory theory;² whilst in a later paper the cause of chromatic polarization is convincingly deduced from the principle of interferences, and in the space of two lines the peculiar coloured laminae occurring in Iceland spar, which had been noticed by several experimenters, are accounted for.³

From about 1815 the optical discoveries of Young (470.) were so intimately connected with those of his younger friend and rival Fresnel, that it seems best to defer our account of them until we consider (in § 3 of this chapter) the peculiar researches of Fresnel, which ultimately rendered the phenomena of polarization the most impregnable position of the partizans of the undulatory theory. The first great step was the conception of transverse vibrations of ether, as constituting polarization. This, as we shall see, was first published by Young. It is to be regretted that the tardy and imperfect publication of Fresnel's memoirs on the one hand, and the resolution of Young to adhere to an anonymous and indirect mode of announcing his discoveries, on the other, render the history of the subject sometimes obscure. The correspondence between them, first fully published by Dr Peacock, throws some light upon it; but several important letters have not been recovered.

I had intended devoting a portion of this section to Dr Young's important and ingenious researches on the physiology of vision. But the length to which it has already extended obliges me reluctantly to omit it. I also refer to the chapter on mechanics (Art. 344, &c.) for some notice of his masterly reasonings on the principles of carpentry and the flexure of elastic substances. They are characterized by directness of purpose and a consummate command of ordinary mathematics, unaccompanied by any pretension to symbolical display;—it might be added too, by the obscure conciseness of Dr Young's habitual style. His researches (preceding and anticipating those of Laplace) on capillary attraction have also been referred to (432), as well as his masterly investigation of the tides (80, 81). It does not belong to this treatise to speak of his discovery of the interpretation of hieroglyphics in certain cases which gave the first real impulse to this obscure but interesting subject. The successes of Champollion, Rawlinson, and others, in similar undertakings, must logically be connected with the first great step of deciphering the polyglot stone of Rosetta. It may safely be affirmed that no philologist ever before made such a discovery in science as the law of interference, and that no natural philosopher ever made such a step in the interpretation of a lost tongue as the formation (up to a certain point) of an Egyptian alphabet.

We cannot close this imperfect sketch of one of the greatest ornaments of our age and nation, without (472.)

¹ Quart. Rev., vol. ii.

² Ibid., vol. iii.

³ Ibid., vol. xi.

adding, that in private life Dr Young was exemplary; endowed with warm affections, philosophic moderation, and high moral and religious principles. His office as secretary to the Board of Longitude (the only public promotion he received), was attended not only with immense labour in editing the Nautical Almanac, but with vexatious contentions, which in his, as in so many other cases, tended to diminish his usefulness and even shorten his life. To the petty persecutions with which he was assailed, it was owing that the health which the unbroken study of fifty years had not impaired, at length gave way, and he died yet in the prime of intellect, the 10th May 1829, within a few months of his honoured associates and friends, Wollaston and Davy. He had been elected two years previously one of the eight foreign associates of the Academy of Sciences of Paris.

Dr Young's philosophical character approached in many important particulars to that of Newton. With much of the inventive fire of Davy, and of the reasoning sagacity of Wollaston, he combined an amount of acquired learning, and a versatility in its application, far superior to both. We do not ascribe to him an intuitive insight so rapid and almost divine as that which distinguished the author of the Principia above all other men, nor had Young the same strictly mathematical ability; but like Newton, whatever he did was practical and sound; nothing was done for show, nothing omitted through haste. "The power of patient thought" was the lever with which he moved the world. His self-confidence was great but unobtrusive. He attained, as he himself said, all the main objects to which he had looked forward in life, "such fame as he valued, and such acquirements as he might think to deserve it."

§ 2. MALUS.—Discovery of the Polarization of Light by Reflection.—Early History of Double Refraction and Polarization.

ETIENNE LOUIS MALUS was born at Paris on the 23d July 1775, and died on the 24th February 1812, after a too brief but brilliant career. His principal discovery, that of the polarization of light by reflection, is so intimately connected, both historically and by the nature of the case, with double refraction, that I shall briefly sum up the scanty progress of that singular subject previous to his time.

It was known to Bartholin of Copenhagen, about 1669, that Iceland or calcareous-spar has the property of dividing a ray of light, which falls upon it in almost any direction, into two; one of which is refracted according to the usual law, but the other in an extraordinary manner, which was first analyzed by Huygens—a problem of great difficulty, in which Newton not only failed, but he also erred in continuing to pronounce Huygens' solution false. The solution was this, that there is one direction in the crystal parallel to which both the rays (called the ordinary and the extraordinary) move in a similar and uniform manner. In other directions their propagation may be expressed by considering the ordinary ray within the crystal to be due to a spherical wave (the centre of which coincides with the point of incidence), whilst the extraordinary ray corresponds to a flattened spheroidal wave concentric with the former, and having its axis coincident with a diameter of the sphere, and parallel to the mineralogical axis of the crystal. Both rays, on the Huygenian hypothesis, move slower than in air, but the extraordinary ray everywhere faster than the ordinary ray, excepting only in the axial direction. A perfectly plain though necessarily complex construction was given by Huygens for the purpose of tracing both rays in the course of their refraction, founded on this idea.

Newton's opposition to Huygens' law as a statement of fact left it for more than a century under partial doubt. Håuy is stated to have verified it, or at least to have shown that it approached nearer to the truth than Newton's; but Dr Wollaston first re-established it in 1802 by conclusive experiments, which, however, he found it impossible to connect by a law until the previous generalization of Huygens had been pointed out to him,—most probably by Dr Young.

It was several years later that Malus directed his attention to the subject, unaware of what had been accomplished by Wollaston. He had returned in 1801 from the unfortunate French expedition to Egypt, where he was engaged as an officer of engineers, and had ruined his health through fatigue and the insalubrity of the climate. He was an accomplished mathematician, having acted as professor both at the Polytechnic School and that of Metz, and was of course a member of the Institute of Cairo. On his return to France, during the intervals of his military duties, he occupied himself in the composition of an elaborate analytical treatise on optics, which had already occupied his attention in Egypt. This led him to the subject of double refraction, and he verified by numerous experiments the accuracy of Huygens' law. La Place wrote a paper on the mathematical law of the velocity of the extraordinary ray, in which he introduced the idea of a repulsive force emanating from the axis of the crystal; but it may be truly affirmed that the notion of a spheroidal undulation so happily introduced by Huygens is the only one which really fits the case; and by the very impossibility of expressing the facts intelligibly without it, gives an undisputed advantage to that theory. A prize having been proposed by the Academy of

Sciences for the theory of double refraction, Malus wrote a second paper, which was crowned, but which, however admirable as a specimen of mathematical address, added little to what was previously known.

Malus had already, in the end of 1808, announced a property of light which, if not absolutely new, was entirely so with reference to the circumstances in which it was produced. The polarization of light was in reality discovered by Huygens previous to 1680. He had observed that the two rays into which common light is divided in passing through Iceland spar have a singular diversity of character, which Newton afterwards described as an opposite polarity. Huygens showed that if two rhombs of doubly refracting spar are laid symmetrically one upon the other, the "extraordinary" ray yielded by the first is extraordinarily refracted by the second, and the "ordinary" ray from the first is ordinarily refracted by the second. But when we revolve the upper rhomb as it lies upon the lower one through a right angle, a remarkable change appears. The extraordinary ray escaping from the first is now ordinarily refracted by the second, and vice versa, so that the qualities of the two rays differ but only so far as this, that either may be assimilated to the other by making it (or the crystal from which it derives its properties) revolve round ninety degrees. A definite notion of such a distinction may be formed by imagining a musical string vibrating at one time in a vertical, at another in a horizontal, plane. If we could possibly imagine light to consist of vibrations of this description, the two rays of Iceland spar might be conceived the one to vibrate in a plane passing through the axis of the crystal, the other in a plane perpendicular to that. Such light might truly be said to have acquired the property of having sides. In the language of Newton, it is polarized.

The discovery of Malus consisted in showing that light may acquire properties identical with those of either ray yielded by refraction through Iceland spar, by the very simple process of simple reflection at a particular angle from any transparent body. Thus for a surface of water he found this angle to be $52^\{\backslash circ\}\; 45\text{'}$ with the perpendicular, and for glass $54^\{\backslash circ\}\; 35\text{'}$. The reflected light in either case has exactly the property of the ordinary ray transmitted by a crystal whose principal section (that is, a section passing through the axis of the crystal) is parallel to the plane of reflection. Consequently this light will be acted on by a doubly refracting crystal placed in its way precisely as if it had emerged from a similar crystal; and, on the other hand, if the two rays emerging from a crystal be incident on water or glass as above mentioned, the one will be copiously reflected from the surface, whilst the other will not be reflected at all, but pass entirely into the transparent substance. Farther, as might be expected, light thus polarized by reflection, when

it falls on a second similar reflecting surface at the same angle as before, will be copiously reflected if the planes of reflection coincide, but will refuse to be reflected in an appreciable degree when the planes of reflection are perpendicular.

This phenomenon was detected by Malus by casually observing that a ray from the setting sun reflected from a distant window, and viewed through a piece of Iceland spar, afforded but a single image in two positions of the latter. The experiment attracted universal attention, and became the germ of a series of optical discoveries almost unprecedented for their beauty and variety; yet most of the experiments may be made quite as well when light is polarized by the method of Huygens as by that of Malus. Nevertheless the former had remained a sterile fact for 130 years. Upon such trifling circumstances does the progress of knowledge often depend.

Malus survived his discovery only four years, and saw but the borders of that land of promise which he had pointed out to others. A few results he however obtained, which are worthy of notice. Thus he found that in every instance where light is polarized in any plane there is also produced a certain proportional amount of light polarized in the perpendicular plane. Arago afterwards proved the very important fact, that these two portions are universally equal. Huygens had shown in the experiment of the two rhombs, that when their positions are neither symmetrical nor perpendicular, each ray emerging from the first is duplicated by the second. When the principal sections of the rhombs are inclined $45^\{\backslash circ\}$, the duplicated rays are equally bright; as they approach parallelism or perpendicularity, one pair of the rays brighten and the other pair is enfeebled. Malus ascertained the law of change of brightness, which is the same for rhombs of spar or for plates of glass whose planes of reflection vary whilst the angle of reflection remains constant. In either case the intensity of the light varies as the square of the cosine of the angle formed by the principal sections of the crystals or the planes of reflection of the plates. This important law, the best established in photometry, has been applied by Arago to the measurement of light in many instances, but the details were unfortunately not made public before his decease.

To Malus is also due the discovery of the polarization of light by common refraction. When light is incident on glass or water, the refracted beam contains precisely as much polarized light as the reflected beam, but oppositely polarized. The metals were at first believed by Malus to polarize no appreciable quantity of light. He afterwards found that at great incidences the reflected light is partly polarized.¹ He likewise ascertained the fact of the "depolarization" (as it was termed) of light by many crystals, and also by organized substances, such as hair, horn, and whale-

¹ See an interesting letter from Malus to Dr Young in Thomson's Annals, vol. iii.

bone. He does not appear to have noticed the phenomena of colour accompanying such depolarization, though he arrived at it so nearly that Arago, in all probability, anticipated him by presenting a memoir on the subject to the Institute just one week previous to Malus's announcement of what he had observed (19th August 1811). In less than six months later Malus was no more.

(484.) The writings and discoveries of Malus present evi-

dence of great talent, but of far less fertility of combination than those of Fresnel, presently to be noticed. He maintained the Newtonian theory of light. His reputation amongst his intimates was extremely high, and it was generally believed, that had he survived, his discoveries would have extended much farther. To him was applied Newton's saying on the death of Cotes,—"If Cotes had lived we should have learned something."

§ 3. FRESNEL.—The Undulatory Theory of Light continued.—Diffraction.—Transverse Vibrations; Young.—Polarization and Double Refraction explained.—Lighthouse Illumination.

(485.) AUGUSTIN FRESNEL was born at Broglie, in France, 10th May 1788, of a feeble constitution, and he continued throughout his too short life a prey to attacks of bad health. As a boy, his slow apprehension and uncertain memory gave no indication of the maturity of his judgment. He entered first the Polytechnic School, then that of Ponts et Chaussées. His fidelity to the Bourbon cause occasioned his being harshly treated by Napoleon, and he retired to Normandy in the beginning of 1815, to pursue the scientific studies which he had always loved.

(486.) Diffraction of light. Diffraction.—The theory of light in particular attracted his attention, and he had a steady belief that the Newtonian doctrine was erroneous, though in ignorance, as it appears, of the undulatory doctrines of Hooke, Huygens, and Young. The phenomena of diffraction, or the coloured fringes which are seen in the interior of the shadows of opaque bodies when illuminated by a minute source of light, attracted his attention as most proper for deciding the delicate question of the molecular or undulatory character of light. The results of his experiments were detailed in a memoir confided in the first place to his friend Arago, and by him communicated to the Institute of France (October 1815). This remarkable paper contained much which Dr Young had already discovered, and the explanations of the experiments which it described, both new and old, by the theory of undulations, were common to both. Dr Young having anticipated the publication by at least a dozen years, there could be no question of priority; but it is equally certain that Fresnel was unaware of what Young had done until it was pointed out to him by Arago. His memoir, which was published in great part in the Annales de Chimie for 1816, contains much which is interesting. The mode of observing the diffraction bands directly by means of a lens, without the intervention of a screen, was equally new and important. The observation that the interior fringes of the shadow of a narrow body, such as a wire, disappear when the light is intercepted on either side of the wire, leading to the conclusion that the union of the light from both sides is neces-

sary for their occurrence, was (as we have seen) one of Young's capital experiments. The explanation of Newton's rings, by the interference of the light reflected from two adjacent surfaces, though partly anticipated by Hooke, was equally important. Numerous measures of the distances of the exterior diffraction bands from the geometrical shadow, as formed by homogeneous red light, are then given and compared with theory. Here Fresnel was on original ground. These accurate numerical comparisons, afterwards pursued to a greater extent, constituted one of the most important bases of the new theory. In obtaining them he was materially aided by Arago, who, though considerably his senior, generously assisted him in every respect, and gave him the full advantage of his station as a member of the Institute, and of his experience.

Fresnel's first memoir on diffraction justly excited (487.) so much notice that the subject was proposed by the Academy of Sciences in 1817 for one of their prizes. The new essay which Fresnel then wrote was, as probably had been anticipated, the successful one. In this memoir he made an important step, by showing that the exterior fringes in diffraction shadows do not depend (as Young had supposed) upon the interference of the direct light with that reflected at a great obliquity from the edge of the diffracting body, but from the interference of the different elementary undulations which proceed from the disturbed surface forming the front of the grand wave. Decomposing the front of the wave into small portions after the manner of Huygens, he computed the disturbance produced by the integral effect of the whole at a given point of the screen where the picture of the shadow fell and was submitted to examination, and he found that such integral effects have a periodic character, presenting points of maximum and minimum disturbance, or of greatest and least illumination as we recede from the geometrical shadow. These distances being measured in homogeneous red light were found to agree with the results of an arduous computation, requiring, as will easily be seen, an intimate acquaintance with the integral calculus and much skill in

using it. Such Fresnel possessed, though he always refers with great modesty to his limited facility of employing the higher mathematics.

Transverse Vibrations—Young and Fresnel.—Considerable obscurity hangs over the first publication of this important discovery. A clear and impartial abstract of the facts will be found in the second volume of Dr Whewell's History of the Inductive Sciences, and some further documentary evidence, including interesting letters which passed between Young and Fresnel, have more recently been published in the Life and Works of the former, edited by Dr Peacock. The difficulty of apportioning the credit between Young and Fresnel partly arises from the unfortunate system of imperfect publication, or non-publication, adopted on professional grounds by the former, and partly from the grievous delays imposed upon the latter by the opposition with which his opinions and experiments were received at the French Academy of Sciences. This continued to the very close of Fresnel's career. His greatest work was not published in the Memoirs of the Institute until six years after date; another was mislaid for above twenty years, and even the hardy friendship of Arago sometimes almost recoiled before the storm of opposition which the novelties of his associate were sure to excite in the minds of the dominant mathematical section. It is quite impossible to say precisely at what period Young first imagined that the differences of oppositely polarized rays of light might be explained by perpendicularity in the directions of vibration of the ethereal molecules, which he compared to the vibrations of a cord in which the elementary movements are at right angles to the direction of wave-propagation. It seems evident that Young was not possessed of this idea in 1814, when he partly explained depolarization in a few pages of an article in the Quarterly Review. It is equally certain that he announced it to Arago (with whom he became personally acquainted in 1816) in January, 1817; and that he then speaks of it as an idea which apparently had recently occurred to him, most probably since their interview.¹ Arago and Fresnel had already, in 1816, made experiments demonstrating that rays oppositely polarized do not produce dark bands by their interference, a memorable discovery, requiring very great nicety for its satisfactory proof, which, however, was completely attained. It was this observation which (naturally) gave Fresnel the first idea of transverse vibrations, and it is much more than probable that Young worked out a similar solution of the great problem, in consequence of the account of these experiments which he received from Arago in the summer of 1816.² Be this as it may, Young

and Fresnel unquestionably imagined the theory separately, but Young first announced it, Fresnel being discouraged by the doubts of Arago, and by his awe of the Institute. As clearly, the experiment of non-interference was the first which gave a colour to so bold an assumption, and in the details of its application to double refraction Fresnel had the undivided merit. What is not least worthy of notice in the affair, is that neither of the amiable rivals (Young and Fresnel) ever published a word in disparagement of the other, nor a single unfriendly reclamation of priority.

The doctrine of transverse vibrations being allowed, (489.)
its applications and severest tests were twofold, 1st, To the phenomena of ordinary reflection and refraction including the polarization produced by these operations; and 2dly, to double refraction and the universally concomitant polarization. In these bold speculations and laborious inductions, Fresnel was nearly alone. Young did not appear as a competitor; even his friend Arago, though sympathizing with and proud of his success, was not associated with him.

Laws of Reflection and Refraction.—With regard to the reflection and refraction of light, its intensity has to be defined, and also its condition as to polarization. The fundamental laws of the direction of the rays are not affected by this theory. Rigorous mathematicians who then doubted the possibility of transverse vibrations having more than a transitory existence, if they existed at all, could not be expected to supply the theory of their reflection and refraction at the bounding surfaces of different media. Fresnel, however, guided by probable mechanical analogies, with an intuitive insight worthy of Newton himself, gave a formula for the intensity of reflected transverse vibrations, both when the plane of vibration of the molecules is in the plane of reflection, and when it is perpendicular to it; and he conceived common light to act as if equally composed of both sets of vibrations. His formulae embrace the non-reflection of polarized light at the critical angle, under the circumstances explained in the last section. It is a most remarkable fact that these inferences by Fresnel as to the numerical relations of the intensity of the reflected to the incident light through all angles of incidence, anticipated almost every trustworthy photometrical measure; and from their singular though indirect accordance with many phenomena, they have been generally accepted as an expression of a natural law of great complexity, even by those who were not favourable to the theoretical views on which they are based.

The modifications of the state of polarization of light which takes place by reflection, was equally (491.)

¹ "I have also been reflecting on the possibility of giving an imperfect explanation of the affection of light which constitutes polarization, without departing from the genuine doctrine of undulation." He then refers to "a transverse vibration propagated in the direction of the radius, the motion of the particles being in a certain constant direction with respect to that radius; and this," he adds, "is a polarization."—Young's Miscell. Works, vol. i., p. 383.

² Peacock's Life of Young, p. 390.

embraced in Fresnel's theory, and equally (though unknowingly) confirmed by Sir D. Brewster's labours. But on one point Fresnel himself obtained a signal triumph. Having deduced expressions for the intensity of refracted light, on pushing them to the limit where refraction out of a denser into a rarer medium becomes impracticable because the light undergoes total internal reflection, the formulae became affected by the multiplier $\backslash sqrt\{-1\}$, and were unsusceptible of arithmetical evaluation. In endeavouring to attach a meaning to these expressions, it occurred to him that as the intensity of the totally reflected ray undergoes no change with the angle of incidence, the expression in question might in some way determine the alteration of the phase of the wave (the position and direction of motion of the molecules under consideration) which took place at the instant of reflection. Now, admitting this as likely, it appeared that the phase would vary not only with the angle of internal reflection, but with the plane of polarization of the ray. It had previously been shown by Arago and by himself, that when two oppositely polarized rays meet or interfere, though there is then no destruction of the light, there is usually a remarkable change in its character. There is one position of the interfering wave relative to the primary one in which the combination produces light polarized in a plane exactly intermediate between the planes of previous polarization. If either ray be now accelerated by half a wave-length on the other, the new plane of polarization becomes perpendicular to the former; but if the shift of either of the primary rays amounts to only one quarter of a wave-length, the motion of the molecules takes place in a circle, and the undulation has a helical form. Now, Fresnel tested his hypothesis concerning totally reflected light by calculating the circumstances of incidence which should produce an effect equivalent to this; and the result completely verified his bold conjecture. The apparatus employed is called Fresnel's Rhomb, which transforms plane-polarized light into light equally reflexible in all azimuths, yet not common light, because it possesses properties which common light does not (such as displaying the rings in crystals); this is termed circularly polarized light.

(492.) Theory of Double Refraction.—The difficulty of accounting for double refraction did not consist in showing how a spheroidal wave might be propagated. Young had already shown, in 1809, that it would result from supposing a lamellar arrangement of the crystalline molecules so that the ether was differently elastic in a direction parallel to the axis than in a plane or planes perpendicular to that line.¹ Huygens had shown something similar in accounting for terrestrial atmospheric refraction. The difficulty was, to account for two waves travel-

ling at the same time through the same portion of matter with unequal velocities. The moment that the idea of molecular movement transverse to the line of propagation was admitted, it was easy to see that no contradiction was involved in the idea. Two waves might simultaneously travel in the same direction, and through the same medium, provided the molecular displacements were in different planes. So happy a solution could hardly fail to strike such minds as those of Young and Fresnel with the impress of conviction. A closer analysis confirmed the probability. Iceland spar (or rather the ether imprisoned within it) is conceived of as a medium of uniform elasticity in all planes perpendicular to the axis, but of a different and greater elasticity in any direction parallel to the axis. It is shown to result from this, that in the direction of the axis alone is the motion of light independent of the plane of the vibrations of which it is composed, and consequently no separation of rays occurs. When a ray moves parallel to what may by an analogy be called the equatorial plane of the crystal, its undulation will, generally speaking, be resolved into two whose vibrations are parallel and perpendicular to that plane, and which travel with different velocities though in the same direction. If the ray take any other direction through the crystal, both the direction and velocity of the divided rays differ. The form of the extraordinary wave is exactly the spheroid of Huygens.

But what are we to conclude concerning those crystals (of the discovery of which we shall speak in § 5) presenting two axes of double refraction? Fresnel at once assumed that the elasticities must in that case vary in three rectangular directions, and he proceeded to calculate the manner of propagation of a wave through a medium thus constituted. I had proposed to attempt some explanation of the steps of his most ingenious and profound argument, but I find it incompatible with the space at my disposal, and at any rate hardly to be apprehended without the use of symbols or figures. For these reasons I shall merely state the results. When the medium presents unequal elasticities in three rectangular directions, the surface of the wave consists of two sheets each travelling with its peculiar velocity. But neither of these being spherical, the result cannot be expressed by the ordinary law of refraction. In two directions within the crystal, the wave surfaces coincide, or the two rays coalesce. These directions are evidently the optic axes, and the wave surface in their neighbourhood has very interesting geometrical and physical properties which have been elucidated by British philosophers, as will be noticed in another section. The true optical axes cannot exceed two, and when two of the three elasticities become equal, they merge into one. This is the

¹ Young's reasoning (Quarterly Review 1809, and Works, vol. I. p. 228) is based on an experiment by Chladni on the differing velocity with which sound is propagated in wood, depending on the direction of the fibres.

case in Iceland spar and similar crystals. At the same time the wave surface degenerates into the united sphere and spheroid. The equation to the wave surface was deduced by Fresnel in an indirect and somewhat tentative manner. It was demonstrated by Ampère directly, but inelegantly. M. Cauchy, Mr Archibald Smith, and Professors Sir W. R. Hamilton and Macculloch gave more complete and elegant solutions.

Fresnel submitted his theory (as usual) to experiment. He found that in topaz, which is a biaxial crystal, neither ray follows the law of common refraction. The plane of polarization (which is always perpendicular in the two rays) follows very nearly indeed, by theory, the law which M. Biot had assigned by experiment. Fresnel thus stated the ground of his conviction of the truth of his theory, and it would be difficult to express more appropriately the characteristics of a just hypothesis:—"The theory which we have adopted, and the simple construction which we have deduced from it, present this remarkable character, that all the unknown quantities are at once determined by the solution of the problem;—the velocity of the ordinary and extraordinary rays, and their respective planes of polarization. Physicists who have studied with attention the laws of nature, will admit that this simplicity and these intimate relations between different parts of the same phenomenon present a great probability in favour of the theory by which they are established."

The memoir on double refraction was received with much incredulity and partial applause. It was not to be supposed that a theory in opposition to that imagined by Newton, and received with almost general assent for more than a century and a half, would not meet with many opponents; but in the case of double refraction and polarization it was also essentially coupled with the idea of transverse vibrations, whose exact mechanism was admitted on all hands to be extremely obscure. Laplace, now more than seventy years of age, opposed the new opinion to the last. His reason for doing so was eminently characteristic of the great geometer—"it was one to which analysis could not be applied without much difficulty;" to which Fresnel replied, "that it was still harder to believe that the laws of nature were arrested by such obstacles." Poisson, as might have been expected, was equally opposed to the undulatory doctrine, for he was still less of a physicist than Laplace. His standing argument against it was its inability to explain dispersion. M. Biot, also a keen supporter of Laplace, was still more strongly compromised to the theory of emission. The inertia of such authorities at the Institute retarded of course the growth of Fresnel's reputation at home, notwithstanding the great weight of his friend Arago's opinion. It was in fact in England that the merits of

Fresnel were first most generally and liberally acknowledged; as, singularly enough, Young had received almost the first expression of sympathy in his optical discoveries from France. In 1825 Fresnel received the distinguished honour of being elected a foreign member of the Royal Society of London, only two years subsequent to his election into the Institute, and whilst his greatest paper was as yet known only by an abstract. In 1827 he received the Rumford medal from the same body. This recognition of his merits was due, as we learn on the authority of Dr Young (who was then Foreign Secretary of the Royal Society) to the influence of Sir John Herschel, at that time and afterwards a zealous supporter of the undulatory theory of light, and by whom it became first generally known in England through the medium of his admirable Essay on Light. Dr Young, though present, was silent; "from being," as he himself tells us, "too much interested in the subject" on account of his personal share in the matter. In announcing this distinction officially to Fresnel (then in the last stage of consumption), Young characteristically observed, "I too should claim some right to participate in the compliment which is tacitly paid to myself in common with you by this adjudication; but considering that more than a quarter of a century is past since my principal experiments were made, I can only feel it a sort of anticipation of posthumous fame which I have never particularly coveted."²

I have stated in the opening of the section that Fresnel, who was attached to the Bourbon cause, had retired to Normandy near the close of Napoleon's career. On the re-establishment of the monarchy in 1815 he was recalled from his retreat and appointed to an office in the departments connected with his profession as an engineer; but in 1817 he was brought to Paris with the express view of giving him more facility in his researches. In 1819 he was placed on the Commission for the Management of the Lighthouses of France (of which he afterwards became Secretary), and he entered with ardour on the application of his favourite science of optics to the duties of his profession and the benefit of mankind.

The use of lenses in place of reflectors for preventing the indefinite dispersion of the light employed, and the effectual concentration of it in the direction where it will be most useful, was not altogether new. The construction of immense lenses of glass of no great thickness, formed by grinding out a series of concentric refracting surfaces having a common focus, had also been proposed by Buffon, and the idea of constructing these rings or échelons separately and then uniting them had been suggested by Condorcet in his Éloge of Buffon, as well as at a later period

² Peacock's Life of Young, p. 401.

by Sir D. Brewster. These proposals were all alike unknown to Fresnel, who had the grand merit, in a case of this kind, first, of carrying his happy idea into effectual execution, and secondly, of giving it a wonderful extension by the invention of a multitude of other forms of refracting and totally reflecting apparatus, till then unimaginable as well as unexecuted.¹ In 1823 the lighthouse of Corduan, at the mouth of the Garonne, was furnished with the new lenticular system, which was very skilfully executed by Soleil of Paris. The illumination was provided by means of a beautiful and powerful lamp with several concentric wicks, the joint invention of Fresnel and Arago, which gave twenty-five times the light of the best Argand then in use. The system was found to work so well that it was speedily extended in France, then to Holland, and in the third place to Scotland, principally through the energy of the late Robert Stevenson and the present Mr Alan Stevenson, his son, to the latter of whom we owe the best and most compendious treatise on the subject of lighthouses,² as well as the noblest exemplification of it in the Skerryvore Lighthouse, erected by him in 1843. The same small work contains the details of Fresnel's admirably ingenious applications of the principle of refraction to the distribution of light under almost every circumstance, which were not, however, published by their inventor.

(498.)
His premature death.

In 1824, consequent upon his exertions as examiner at the Polytechnic School, Fresnel had the first seizure of the malady which brought him to the grave at the premature age of 39, on the 14th July 1827. Eight days previously to his death he had received at the hands of Arago the Rumford medal before referred to, which his distinguished friend

Malus had obtained under like melancholy circumstances 16 years before.

From what has been stated it will appear that (499.)
Fresnel eminently possessed the qualities requisite Character of Fresnel. for original investigation. So finely balanced a combination of mathematical skill and attainment with profound inductive sagacity has rarely been witnessed. Had Young not happened to precede him there can be no question but that he would have made the undulatory theory entirely his own. Fresnel was superior to Young in the talent for devising and executing critical experiments, in which indeed he showed a degree of skill equally rare and admirable. It is hoped that his surviving brother, M. Léonor Fresnel, who is well qualified for the task, will collect his scattered papers and edit them, with a suitable biography. The éloge of Fresnel, written by the man most competent to render him justice—Arago—remained more than twenty years among the unedited papers of that philosopher, and has only appeared since his death in 1853. The cause of this suppression was one of those partly political and partly personal disputes which seem almost inseparable from the proceedings of the Institute. The éloge was announced to be read just two days before the Revolution of 1830 burst forth; Arago could not persuade himself at such a moment to discuss the merits of the Theory of Double Refraction without committing himself also on the politics of the day. Disputes arose, friendships cooled, and the unlucky work was returned to the author's desk. Hence no biography of Fresnel appears in the publications of the Institute, but his reputation will be treasured in France and elsewhere, when the more conspicuous laurels of many of his compeers are withered and half-forgotten.

§ 4. ARAGO.³—Short Account of his Scientific Career—He discovers the Colours of Polarized Light—Laws and Theory of Depolarization; M. Biot; Young; Fresnel.—Non-interference of oppositely Polarized Rays—Rotatory Action of Quartz.—M. Foucault's Experiment on the Velocity of Light.

(500.)
Arago:—

DOMINIQUE FRANÇOIS JEAN ARAGO, one of the most generally known of the philosophers of the half century just elapsed, though the author of a large number of miscellaneous writings which since his death have been edited in a collected form, has not left an amount of positive contribution to any one of the sciences at all in proportion to the reputation for ability which he very justly enjoyed.

(501.)
His early life;

He was born at Estagel near Perpignan, on the 26th February 1786, and the ardent temperament of a native of the south was one of his chief character-

istics. From a fragment of his early history which he left behind him, it appears that he educated himself almost without assistance, and that when he was admitted to the Polytechnic School in 1803 (consequently at the age of 17), he was intimately acquainted with the chief writings of Lagrange, had studied the Mécanique Céleste, and had consequently in his possession far more mathematical knowledge than would have been required of him on leaving that celebrated institution. From the Polytechnic School he passed into the position of Secre-

¹ One of Fresnel's lenses was used in 1821 for the geodetic operations connecting France and England, and the light was observed at a distance of fifteen marine leagues one hour after sunset.

² Rudimentary Treatise on Lighthouses, by Alan Stevenson. Weale, 1850. See also the Account of the Skerryvore Lighthouse, with numerous plates, in 4to, 1848.

³ I may perhaps be thought to give Arago too prominent a place in the history of Optics. If so, it has arisen in part from

tary to the Paris Observatory. He pursued with M. Biot experiments on the refraction of the gases, and in 1806 the two young philosophers were despatched to the south of France and to Spain to continue the triangulation interrupted by the death of Méchain.¹ The next three years were spent by Arago in a series of voluntary and involuntary journeys, perils by land and sea, from robbers, and from the Spanish government and populace, such as have been rarely equalled, — perhaps never in the pursuit of science. The declaration of war against France rendered his stay either in Valencia or in the Balearic Isles impossible, and he was conveyed in disguise from Majorca to Algiers, whence he twice essayed to reach Marseilles, but was once driven back to Africa by a storm, once made prisoner by a Spanish corsair. After great suffering, he at length reached France in July 1809, carrying with him the precious record of his geodetical operations. From this time his promotion was assured, and his life became tranquil and inactive, although the deep attachment which he formed with Baron Humboldt immediately on his return to France would probably have induced him to accompany that enterprising traveller to Central Asia, had that journey ever been accomplished. At the early age of 23 Arago attained the position of Member of the Institute, and was again attached to the Paris Observatory, of which at a later period he became director. He took a very active share in the proceedings of the Academy of Sciences, and became one of its secretaries in 1830.

From the period of his election to the Institute, Arago's career was destitute of stirring incidents, but was, from first to last, devoted chiefly to science. It is certain, however, that he was deficient in that power of continuous application, to which alone great discoveries are commonly due. Full of ingenious, original, and even profound conceptions, he shunned the labour of realizing them. His appointment to the charge of the Observatory of Paris was perhaps unfortunate. Well versed in the theory of astronomy, the minute drudgery of observation and the control of numerous assistants, was altogether uncongenial to him. It was a duty imperfectly fulfilled, to say the least, for 40 years: and it is needless to add how much the consciousness of habitual neglect of a duty deadens the faculty of useful application to anything else.

If the science of astronomy then owes little to Arago, beyond the part which he took in geodetical

observation, what are we to consider his chief claims His optical to a place in this history? I have no hesitation in saying that they are to be found in connection with the discoveries and labours of his attached friends, Malus and Fresnel, and therefore we group them together in this chapter. Arago not only himself made some important optical discoveries in 1811 and the following years, but he was instrumental, as we have seen, in a very important degree, in calling forth the genius of Fresnel, and in obtaining a public recognition of the labours of Young; a service not the less worthy of note because of its eminently disinterested character. The undulatory theory of light, one of the greatest triumphs, if not the greatest, of our age, stands where it does in no slight degree through the instrumentality of Arago.

One of his most considerable discoveries was that of the colours which crystallized bodies develop in white light polarized before incidence on the crystal, and afterwards transmitted through a rhomb of calc-spar. These colours, by their order, the singular manner of their occurrence and disappearance, and in certain cases by their extraordinary and beautiful forms, offered a problem at once the most attractive, the most definitely marked, and the most seemingly inexplicable which had been met with in optics for much more than a century. They exemplified a new mode of analyzing light, evidently connected with the molecular forces concerned in crystallization; and for their display it was necessary that light should be in that peculiar and yet mysterious state called polarized.

The substances he employed were principally selenite, rock-crystal, and mica. When plates of the first and last of these minerals, formed by their natural cleavage, are placed in a beam of polarized light, and the light transmitted by them is then analyzed, by being passed through a doubly-refracting prism or thrown on a screen after reflection at the critical angle from glass, splendid colours are the result. These colours vary with the thickness of the plate, with its inclination to the incident light, and, what is most remarkable, they vary in intensity by merely turning the plate of selenite round in its own plane. When only this last motion is made, there are two positions of the plate where no colour results, the light passing through unchanged; and these positions are at right angles to one another. At all intermediate angles colours appear, the light is said to be depolarized, and this depolarization is most complete when the plate is moved $45^\backslash circ$ from

the difficulties inseparable from the biographical system which I have adopted. My intention was to have thrown together the labours of Malus, Fresnel, and Arago into one section. But having written the different portions separately, there seemed so much precision and facility of explanation to be derived from treating of them consecutively, that I sacrificed, in some degree, the biographical principle to that of systematic classification; placing under the name of Malus what referred to the empirical laws of double refraction; under that of Fresnel the doctrine of transverse vibration (though mainly due to Young); and under Arago the discoveries of Young, Fresnel, Biot, and others, relative to the great subject of chromatic polarization, to which he gave the first impulse. The establishment of the undulatory theory, principally due to Young, Fresnel, and Arago, I have considered as deserving of a more detailed and systematic treatment than almost any other of the numerous discoveries of which I have to speak in this Dissertation. I may add, that the biography of Arago not appearing in its alphabetical place in the Encyclopædia, M. Arago being still alive at the date of the publication of that part of the work, it has been incumbent upon me to enter into more details than I should otherwise have done.

¹ See Chapter III., Art. (168) of this Dissertation.

the positions in which the light passed through unchanged.

(506.) Observation of Malus, Arago, Sir D. Brewster, and M. Biot.

The theory of this simple yet admirable experiment is one of the happiest examples of Fresnel's mechanical explanation of double refraction. But it was not attained by a single step, nor effected by a single hand. Malus had observed the fact of depolarization, and the existence of perpendicular neutral axes in the crystalline plate. Arago added to this the knowledge of the phenomena of colour, which Sir David Brewster also observed independently somewhat later. He also invented a particular theory to explain them by what he termed moveable polarization. But it was not a successful effort. M. Biot assiduously studied the empirical laws of these periodic colours, and traced their dependence on the thickness of the interposed plate, according to a law similar to that of Newton's rings. Young made an important step farther. He

attributed to interference by Young.

attributed the colour to the interference of the ordinary and extraordinary rays into which the incident light was divided on entering the crystalline plate; and he showed by accurate calculation that the retardation of the slower-moving of the two rays during their passage through the plate, did in fact produce the difference of phase necessary for developing the tints observed by their reunion on leaving the plate, according to the usual laws of interference; and he showed that this theory coincided with M. Biot's rules.¹ But even this was not enough to explain the facts. It was not clear why the colours due to doubly-refracting plates should not be seen without reaching the eye through an analyzer, or calc-spar prism. To Arago and Fresnel jointly we owe the important reply to this difficulty, which in fact forced upon the latter the idea of transversal vibrations. Their joint experiments (mentioned in the last section, art. 488), had shown that oppositely polarized rays cannot interfere unless they have, first, a common origin and a common plane of polarization; and, secondly, unless they be reduced to a common plane of polarization (that is, analyzed) before falling on the eye or the screen: so that the theory of the colours of crystallized plates is, briefly, as follows:—

(507.) Depolarization explained on the undulatory theory.

Polarized light is represented by transverse vibrations of ether, the particles moving all in one plane. The crystallized plate has, we will suppose (in order to take the simplest case) one axis of double refraction, and the direction of that axis is in the plane of the lamina. When the vibrations of the light falling on the crystallized plate are either wholly parallel or wholly perpendicular to this axis, the light is transmitted without alteration, either as an extraordinary or as an ordinary ray, and it is then reflected or not by the analyzer, as would have been the case had it not been transmitted at all through the crystal of

selenite. But if the axis of the crystal be inclined, suppose 45°, to the plane of vibration of the incident polarized ray, the vibration is mechanically resolved into two, which are oppositely polarized. After traversing the thickness of the crystallized plate with the different velocities due to the motion of the ordinary and extraordinary rays, they are reunited at emergence, but in altered relative phases, so that by their union they form a beam no longer polarized (unless by an exception) in the same plane as at first, nor indeed plane-polarized at all, but most likely performing elliptical or circular vibrations, which, again, falling on the analyzer, are reflected or transmitted, or partly both, in a manner quite different from what would have happened to the light unchanged by crystalline transmission.

White light becomes coloured because the state of polarization of the emergent ray depends on the difference of length of path for the two rays which underwent crystalline separation within the plate, and also on the length of a wave of light (for this determines the phase of polarization at emergence). But the wave of light varies in length for each colour, consequently every colour has its maximum under different circumstances, and if the incident light be white, the light falling after reflection on a screen will present mixed tints similar to those of Newton's rings.

This extraordinary property of a crystallized plate (508.) (which, in common light, appears equally transparent and homogeneous in every direction), of modifying light, or dyeing it with the most gorgeous colours, when the plate is merely turned in its own plane, is one of the nicest tests of the polarization of the light, and has been used to detect the analogous polarization of radiant heat, and the concomitant phenomenon of double refraction, which, except in the case of the heat accompanying the solar rays, has not yet been independently recognized.²

Arago applied his discovery to the construction of a polariscope, for estimating the feeblest amount of polarization; and he used this instrument for some very interesting experiments on the polarization of the light of the sky (which is sunlight polarized by reflection from the atmosphere), and on that of different incandescent and reflecting surfaces. He also found that the moon and the tails of comets send light to the eye which is slightly polarized, thus betraying its borrowed origin. But that of the sun, being absolutely neutral, is only comparable (according to Arago) to the light arising from incandescent vapours, thus distinguishing the sun from a solid or liquid globe.

We cannot do more than allude to Arago's other optical papers and experiments. He was, probably, the only Frenchman of his time who was well acquainted with Young's discoveries. The explanations and

¹ Quarterly Review, vol. xi.; Miscell. Works, vol. i., p. 286.

² See Researches on Heat by the present writer. Edinburgh Transactions, vols. xiii. and xiv.

tion by the doctrine of interference of the colours of Newton's rings received an important confirmation from an experiment of Arago's, which proved them to arise from the mixture of the pencils of light reflected at the two neighbouring surfaces. He pressed a lens of glass against a plate of metal, in which case the central spot is white or black when light polarized perpendicularly to the plane of incidence is reflected at an angle greater or less than the polarizing angle for glass; and the rings vanish altogether at the polarizing angle;—results which have been found conformable to the undulatory theory.¹ He also discovered the peculiarity of the rays transmitted along the axis of a crystal of quartz. These depolarize light, or produce colours similar to those of crystallized plates, varying according to a well-marked law with the thickness of the plate. The most singular fact is, that by turning round the analyzing plate, no position of neutrality is found, but a series of colours similar to those of Newton's scale succeed one another. Arago showed that this effect is due to a rotatory motion of the plane of polarization within the crystal. The rotation is greater for the violet than the red ray; this was shown by M. Biot, who also discovered that in some specimens the rotation takes place from right to left, in others from left to right—a peculiarity connected with certain crystallographic modifications, as was first shown by Sir John Herschel.

MM. Seebeck and Biot discovered an analogous property in oil of turpentine, and in various saccharine fluids, an observation which, in many cases, allows the substitution of an instantaneous optical, for an operose chemical test. Fresnel has shown that the phenomena of quartz may be represented on the supposition that the rays traversing the axis consist of two rays circularly polarized in opposite directions, and travelling with different velocities; and Mr Airy succeeded in calculating, by the aid of this fundamental hypothesis,² a number of most beautiful and complicated phenomena, such, for example, as those which occur when plates of right and left handed quartz are superposed. Macullagh has shown how the generation of elliptical or circular vibrations may be deduced from the general equations of motion, but he has not invented a mechanical theory to explain them. This is a point of the very highest interest, inasmuch as Dr Faraday has succeeded, for the first time, in inducing artificially in a substance the power of rotating the plane of polarization by the presentation of it to the poles of a most powerful magnet.³

Arago's experiments on the non-interference of rays of light oppositely polarized, being undertaken in conjunction with Fresnel, have been already re-

ferred to, arts. 488 and 506; but to Arago alone is in dense due the ingenious idea of interposing a thin slip of mica or blown glass in the path of one of the interfering pencils, and observing the displacement of the interference-bands, which is always towards the side of the interposed slip, showing that the movement of the wave has been slower within the denser medium.

Arago continued to attach great importance to the obtaining of a still more direct proof of this fact, which he considered as a crucial one between the rival hypotheses of Newton and Huygens. In his last years he had the satisfaction of witnessing the accomplishment of it, with the result he anticipated, and by a method which he had himself indicated. In 1838, he had already indicated the application of Mr Wheatstone's beautiful invention of the revolving mirror,⁴ as a means of measuring intervals of time incredibly short, in order to compare the velocity of light in air, and in a corresponding length of water. He even caused an apparatus to be partly prepared, but we have seen that Arago's forte was rather in suggesting than in completing researches. After his increasing failure of sight rendered it physically impossible that he should ever realize his own idea, it was skilfully adopted by M. Foucault (the author of the admirable experiment with the pendulum, demonstrating the earth's motion⁵), who, by an ingenious combination of fixed and revolving mirrors, succeeded in 1850 in demonstrating the retardation of light in a tube of water only $6\backslash frac\{1\}\{2\}$ feet long, and with a velocity of rotation of the moveable mirror not exceeding 200 turns in a second (a rapidity four times less than had already been obtained by Mr Wheatstone). The rotation was produced by means of the Sirène of M. Cagniard de la Tour, acting by steam. The velocity was thus raised to 1000 revolutions. It was afterwards, however, carried by MM. Fizeau and Breguet to 2000 revolutions. It will be understood, from the account of the method, as applied to the measurement of the velocity of electricity, in another chapter, that the retardation is shown by the displacement of the image of a minute object seen through the water, relatively to the image of the same object seen in air. If light moves faster in water (as Newton imagined), the displacement of the water-image will be (let us say) to the right; but if slower (as Huygens and Young believed), it will be to the left. The calculated displacement, with 800 revolutions in a second, was .004 inch on the first supposition, and .003 in the opposite direction in the second, quantities easily visible with a high magnifying power. The result, as has been stated, confirmed Arago's original experiment of 1815 on the displacement of the interference fringes.

¹ Sir W. Herschel first formed Newton's rings between glass and metal. Arago's experiment was reproduced (unknowingly) by Mr Airy in 1831, who first explained it fully in the undulatory sense. Cas. Trans., vol. iii.

² With this addition, that rays inclined to the axis are elliptically polarized, and that with a greater ellipticity as the inclination increases.

³ See the chapter on Electricity, § 5.

⁴ See Electricity, § 6.

⁵ See the chapter on Astronomy, Art. (258).

(515.) Arago's electro-magnetic observations.
To complete our biographical notice, we will here just allude to M. Arago's principal researches unconnected with optics. One was on the magnetization of iron filings, and the formation of a temporary magnet, by means of a helical conductor of electricity;¹ the other was the important observation of the seeming magnetism of copper, and other non-magnetic metals, when put in rapid rotation near a permanent magnet. Neither of these happy experiments was carried out by their author. The former was left in the hands of Ampère, Sturgeon, and Henry; the latter was only rightly understood and valued when it was engrafted by Mr Faraday on his splendid series of researches on Magneto-Electric Induction.²

(516.) Essays on meteorology.
Arago is fairly entitled to be regarded as having proved the long-suspected connection between the aurora borealis and the freely suspended magnet; and this in the face of urgent contestation. His contributions to Meteorology (founded rather upon the observations of others than upon his own) were of considerable importance, and several of his popular papers, appended to the smaller Almanac (Annuaire) of the Board of Longitude, contain a great deal of well-digested and curious information.

(517.) As Secretary of the Academy of Sciences in the

Mathematical Department (in which office he succeeded Fourier, in 1830), the duty devolved on him of writing the biographies of eminent deceased members of the Academy. He bestowed extraordinary pains on these compositions, and strove to render them popular without sacrificing their scientific character. In this difficult attempt he was not always successful. Abrupt transitions, piquant anecdotes, paradoxical arguments, and political allusions, appear, at least to the English reader, to be unacademical adjuncts to the history of contemporary discovery. The special pleader is too often visible, and even the occasional sacrifices of the strong spirit of nationality by which he was commonly actuated, to some chivalrous adjustment of the rights of discovery, do not always carry conviction to the mind of the reader. The Eloge of Watt, probably the most popular, appears to us far from being the best of these biographies. Those of Sir W. Herschel and of Dr Young are ably executed, and display much research and candour.

After having been for three years almost withdrawn from science by lingering disease, and nearly (518.) His death. complete blindness, Arago expired at the Observatory of Paris, on the 2d October 1853, aged 67.³

§ 5. SIR DAVID BREWSTER—Progress of Experimental Optics—Laws of Polarization—Double Refraction produced by Heat and Compression—Discovery of Biaxial Crystals—Laws of Metallic Reflection—Absorption of Light; and Lines of the Solar Spectrum; FRAUNHOFER.—Seebeck; M. BIOT.

(519.) Sir David Brewster.
We have pleasure in ranking amongst the foremost promoters of the science of optics in its surprising revival in the earlier part of this century, a philosopher who still lives amongst us and pursues with ardour the investigations of his youth.

(520.) His early studies.
Sir DAVID BREWSTER was born at Jedburgh, in Scotland, on the 11th December 1784. He was educated for the Scottish Church, and having entered the University of Edinburgh at a very early age, pursued his studies under Robison, Playfair, and Stewart, and formed the friendship of those eminent men. Amongst fellow-students of no common distinction who at that time frequented the college lectures, and of whom not a few were destined to signalize themselves in literature, science, and the career of politics, he formed the particular acquaintance of Mr, now Lord Brougham, and through him was led to study the inflection of light, and to repeat Newton's experiments. This was in 1799; nor did he afterwards lose sight of a science which he was so signally to improve. The distraction of other occupations, the calls of his profession, and his indifferent health prevented, however, any very constant application to optics; and the part of

the subject he then principally studied was rather connected with the use and theory of instruments than with physical optics in the sense in which we have explained it. This is evident from his first separate publication in 1813, "on new Philosophical Instruments," which, though containing many ingenious and valuable suggestions, fell short of the importance of his subsequent publications.

Sir David Brewster's genius was first called forth (521.) by the announcement of Malus's great discovery in 1808 of the polarization of light by reflection. But for the unfortunate political relations of France and England at the time, which prevented, to a degree which now appears almost incredible, the transmission of even the most interesting scientific facts from one country to the other, our countrymen would have borne a larger share in the discoveries which immediately followed; and it would have been an easier task to apportion with historical accuracy what was due to each. As the French philosophers remained long in ignorance of the discoveries of Davy, and were anticipated in every important step in voltaic science, so Malus and Arago pursued and published researches and bril-

¹ See Electricity, § 4.

² Electricity, § 5.

³ The article POLARIZATION OF LIGHT in this Encyclopædia, the production of Arago, contains an excellent review of many of the topics of this Section.

liant discoveries which, literally for years, remained unknown in England to those most interested and solicitous to learn them. Thus Sir D. Brewster learned first in February 1814 that Malus had in March 1811 published the discovery of the polarization of light by refraction, which he also had made; whilst Arago's experiments on coloured polarization were likewise unknown to him through the same want of international communication.

(522.) Treatise on new philosophical instruments. The work on Philosophical Instruments, mentioned above, contains, besides what its name more particularly imports, numerous observations on refractive and dispersive powers, including the discovery of substances more refractive than diamond, and less so than water. It also describes the property of some agates to transmit light polarized in only one plane. The imperfect polarization of light by metals and by a serene sky had been anticipated by Malus and Arago.

(523.) His researches on optical subjects. From this time (1813) Sir David Brewster became a regular contributor to the London Philosophical Transactions, which, as well as those of Edinburgh, contain a series of elaborate experimental investigations due to him, which have hardly been surpassed. It is difficult to overrate the importance of these researches, whether for the intrinsic interest of the phenomena they reveal, or for the significance of the empirical laws by which their author, with a rare sagacity, succeeded in classifying facts, and afforded a sure basis for farther generalization. The number and variety of these researches is so exceedingly great, and in many cases so impossible to explain without entering into minute detail, that I shall, in accordance with the plan of this essay, merely indicate some of the most generally important by arranging them in groups. Such are—

(524.) Enumerations of the most important. I. The laws of polarization by reflection and refraction, and other quantitative laws of phenomena. II. The discovery of the polarizing structure induced by heat and pressure.

III. The discovery of crystals with two axes of double refraction, and many of the laws of their phenomena, including the connection of optical structure and crystalline forms.

IV. The laws of metallic reflection.

V. Experiments on the absorption of light.

(525.) Law of polarization by reflection. I. Malus had failed to discover a connection between the angle at which light is completely polarized by reflection, and the other known optical properties of bodies. In 1814, Sir D. Brewster discovered the beautiful and simple law, "that the index of refraction is equal to the tangent of the angle of polarization." He had suspected it much

sooner, but he had been baffled by the irregular results obtained by reflection from glass, whose surface he found to undergo an almost imperceptible chemical change. He further observed that it is only in bodies of low refractive power that the polarization is sensibly complete, a result of great importance, which has been too much overlooked until the recent and valuable paper of Jamin on the same subject. He deduced as a corollary, that at the maximum polarizing angle the incident and refracted rays are at right angles to one another, and also Malus's experimental result that the rays reflected from the first and second surfaces of plates are simultaneously polarized. He further discovered the fact that light may be completely polarized (as to sense) by a sufficient number of reflections at any angle, and drew the conclusion that the whole light undergoes some change at each reflection, in opposition to the view of Malus, who maintained that, except at the polarizing angle, a portion of the light is polarized, and the rest is unchanged.

Sir David Brewster independently observed the polarization of light transmitted obliquely through glass, and he calculated the number of plates necessary to polarize it with sensible completeness. All these researches he resumed some years after (Phil. Trans., 1830), endeavouring to give a photometric estimate of the effects of reflection and refraction under all circumstances. The results as regards partially polarized light may still be considered as subject to doubt. His skill in obtaining a mathematical representation of the phenomena was again displayed in a number of laws connecting the experimental results.¹

II. Malus had observed that a vast number of substances depolarized light more or less completely; and Arago found feeble traces of chromatic polarization in some specimens of glass. But the more definite characters of the beautiful phenomena of glass not perfectly annealed (which proved to be of unexpected importance) were noticed, independently, by Sir D. Brewster and Dr Seebeck of Nurnberg. The latter had priority in publication,² but the former correctly referred them to their immediate cause—the constraint produced by rapid cooling. Sir D. Brewster noticed that the unannealed glass which forms what are called Prince Rupert's Drops, had a remarkable power of depolarization; and he also observed subsequently that the plates of glass between which he was in the habit of squeezing heated wax and resins, for the purpose of optical examination, transiently communicated tints to polarized light. These observations, duly developed, proved on the one hand that glass (and generally refracting

¹ Thus he found that the effect of refraction on the plane of polarization of the incident light may be expressed by this simple formula—$\backslash cotan\; \backslash alpha\text{'}\; =\; \backslash cotan\; \backslash alpha\; \backslash cos\; (i\; -\; i\text{'})$, where $\backslash alpha$ and $\backslash alpha\text{'}$ are the azimuths of the planes of polarization of the incident and refracted rays measured from the plane of reflection, and $i$ and $i\text{'}$ the angles of incidence and refraction. This result, admirably verified by experiment, is also conformable to Fresnel's theory.

² In Schweigger's Journal for 1813, vol. vii. I have not been able to find in this paper (which contains the first account of the beautiful symmetric coloured figures displayed in cubes and cylinders of glass) the smallest trace of the true cause of the phenomenon, viz., the sudden or partial cooling of the glass.

(526.) Imperfect polarization.

(527.) Polarizing structure in heated glass;

solids not usually possessed of any polarizing action) acquired such properties by being suddenly and unequally cooled; and on the other, that such substances undergoing partial changes of temperature, possess the same property. These phenomena are exceedingly beautiful, and easily displayed. It is sufficient to place a rectangular piece of glass, somewhat thick, with one of its longer edges in contact with a hot iron, and placing it between a polarizing and analyzing plate, to incline the heated edge $45^\backslash circ$ to the plane of polarization. Bands of light and shade are seen to traverse the glass parallel to the same edge, and simultaneously appear also on the side farthest from the heated metal. They pass gradually into rich coloured tints diffused with geometric regularity. In the case of cubes or cylinders of glass, suddenly, and therefore unequally cooled, the phenomena are permanent, and the colours splendid, being arranged in patterns which may be made to resemble those of natural crystals.

(528.) Sir David Brewster at first ascribed these effects to the direct effect of the heat in the glass upon light, and compared its simultaneous influence over a whole plate to a polar influence. Another curious discovery, also due to him, leads to the simpler conclusion, that in every case the development of a polarizing structure is due to the varying tension (transient or permanent) into which the particles of the glass are thrown by local expansion or by irregular cooling. This discovery was, that similar effects may be produced in jellies and soft transparent substances by the effect of pressure, and that even glass itself, when strained in any way by mechanical force, shows depolarizing bands; and crystals may have their peculiar optical phenomena altered by pressure. Fresnel, not satisfied with inferring that the chromatic display is due to a doubly-refracting structure communicated to the glass, contrived, by an ingenious arrangement of prisms, actually to exhibit the separation of the images under the action of powerful pressure.

(529.) Discovery of biaxial crystals.

III. Possibly the most remarkable of Sir David Brewster's discoveries,—at all events, that which probably cost him most labour to develop,—was that there are crystals possessing two axes of double refraction, and showing many remarkable phenomena, which indicate a connection between optical structure and crystalline form. No one before him suspected the existence of a doubly-refracting structure differing from that so ably investigated by Huygens and Malus in Iceland spar. In 1813 Sir David Brewster had discovered coloured rings in topaz when viewed by polarized light. Though intimately connected with M. Arago's observation of colours in crystallized plates, these interesting phenomena had

a still more extraordinary and geometrical character. When a plate of topaz, split by natural cleavage, was presented between the polarizing and analyzing plate (or rhomb of calc-spar), and at the same time inclined in a certain manner, the colours were no longer in broad sheets, but, if viewed closely, they arranged themselves in oval rings of great beauty, presenting orders of mixed colour analogous to those described by Newton when formed between convex glasses, and they were traversed by dark or white brushes as the analyzing plate was held in the dark or bright position. A second such system was observed inclined at an (apparent) angle of $65^\backslash circ$ in the same plate.

Dr Wollaston afterwards (1814) discovered a phenomenon equally beautiful in calcareous spar, of which Sir D. Brewster had already perceived traces in some other crystals. Concentric with the position of the axis of double refraction (or optic axis), in a crystal of that mineral cut with two parallel faces perpendicular thereto, a series of perfectly symmetric and exquisitely coloured rings are seen in polarized light, having a white or black cross traversing them, according to the position of the analyzing plate. This magnificent phenomenon, which (excepting, of course, the rings of biaxial crystals mentioned above) has perhaps no parallel in optical science, is seen in the most perfect manner possible in an apparatus constructed solely of Iceland spar, cemented by Canada balsam. A Nicol's single-image prism¹ is used to polarize, another to analyze the light, and between them is a plate of calcareous spar properly cut. It is a truly astonishing paradox to see the union of three perfectly transparent and colourless crystals display by their union such an exquisite combination of form and colour. The pole or centre of the rings in calc-spar coinciding with the axis of double refraction, of necessity suggested the idea that topaz, which shows two systems of rings arranged round two poles, must possess two axes of double refraction; in other words, that there must exist within the crystal two directions (not mere lines), parallel to which a transmitted ray emerges without subdivision into two pencils.

This probable conjecture was verified by careful observation, not only in topaz, but also in a vast variety of other crystals which were found by Sir David Brewster much more commonly to possess two than one system of rings. Amongst the earliest examples which he observed were nitre, mica, acetate of lead, and Rochelle salt. Of these the first is exceedingly remarkable for the small inclination of the axes (about $5^\backslash circ$) which permits both systems of rings to be readily observed at once. It was not, however, for some years (1817) that he reduced these most

¹ The invention of a most ingenious person, the late Mr William Nicol of Edinburgh. One of the doubly-refracted rays is thrown completely out of the field by undergoing total reflection at the surface of a film of Canada balsam in contact with the spar, whilst the other less refrangible ray proceeds quite isolated, and with scarcely any loss of brilliancy. It is of almost universal application in this branch of Optics.

curious phenomena to anything like a law. It was evident almost from the first, that the axes in question (which he termed axes of no polarization) are only the resultants of remoter fundamental actions of the crystalline constitution. For example, these axes vary in position according to the colour of the light used to display them; their position within the crystal varies (as was shown by Mitscherlich) with the temperature of the body, nor is it obviously related to any of the geometrical lines of crystallization.

Sir David Brewster succeeded in finding the law of the tints expressed upon the surface of a sphere of which the directions of the two axes form diameters. M. Biot expressed the law more elegantly by saying, that the tint developed by a biaxial crystal in any ray, is proportional to the product of the sines of the angles which the ray in question makes with the two optic axes. The tints are consequently arranged round the two poles of the axes, in a series of curves resembling the figure 8, having each this property, that the product of the sines of the angular distances of each point of one curve from the two poles is equal to a constant quantity. Such curves are called lemniscates, and are beautifully seen in nitre, especially when viewed by homogeneous light.

A series of researches of the most elaborate description led Sir David Brewster to this additional and admirable discovery, viz., that the optical characters of single refraction, double refraction with one axis, and double refraction with two axes, have reference invariably to the primitive crystalline form of the mineral, and that the complexity of the optical character is as invariably related in degree to the complexity of the crystalline figure. Cubical and regularly octahedral crystals (as rock salt and fluor spar) being possessed of perfect symmetry in three principal directions, possess also simple refraction. Crystals with one predominant line or axis of symmetry—as rhombohedrons, octahedrons with square bases, right prisms with square or hexagonal bases—have a single axis of double refraction. Such, for instance, are Iceland spar, zircon, ice, beryl. Finally, all crystals unsymmetrical in the three principal directions, including prisms and octahedrons whose bases are not square, and those which are oblique, have two axes of double refraction. The rarity and minuteness of many crystals, the difficulty of cutting them, and when cut, of detecting their optic axes, evidently made the research one of extreme labour, yet highly remunerative, not only through the discovery of the general principle, but by the vast amount of beautiful and varied optic displays witnessed in the course of it. Sir David Brewster was nearly, if not quite, alone in this research, and after a short resistance on the part of some mineralogists, his principle of discrimination of primitive forms of crystallization by optical characters has been perfectly established.

IV. The action of metals on the light which they

reflect is very peculiar. Malus, who at first believed that they were incapable of polarizing it in any degree, afterwards changed his opinion, and ingeniously suggested that whilst transparent bodies reflect, at the polarizing angle, light polarized only in one plane, metals reflect rays oppositely polarized and then mixed. Sir David Brewster has the merit of having, after several unsuccessful attempts, deduced the leading empirical laws of metallic polarization, having been partly guided (as he states in his paper in the Philosophical Transactions for 1830) by Fresnel's remarkable experiments on circular polarization produced by total reflection in glass (see Art. 491). Having found qualities somewhat analogous in light reflected one or several times from metallic plates at various angles depending on the nature of the substance, he gave to the light so reflected the name of elliptically polarized light. It was afterwards satisfactorily proved by Mr Airy that the light so named by Sir D. Brewster is in fact identical in its qualities with the elliptically polarized light of Fresnel.

The subject of metallic polarization is rather too abstruse to be explained in a popular way; and the phenomena produced with depolarizing plates of different metals are not so well known as, according to their discoverer, they deserve to be. My limits only permit me at present to state that the care and accuracy of Sir D. Brewster's results are unquestionable; that they have formed almost the sole data upon which M. Cauchy and other mathematicians have based their theories of metallic reflection; and that, by generalizing the more limited views entertained by Fresnel as to the constitution of media and the nature of reflected light, they have been mainly instrumental in fixing the later views of optical writers as to the precise phenomena of polarization as produced, not only by metals, but by other substances. To these views I shall briefly advert in the next section.

The laws of the reflection of light at crystallized surfaces have also been studied by Sir D. Brewster. In this case observation is still in advance of theory.

V. Of Sir David Brewster's experiments on the absorption of light we must speak much more briefly. White light is coloured or analyzed by refraction (as in a prism); by simple interference, as in Newton's rings; by double refraction combined with polarization. But it is also decomposed in a way which, primarily at least, seems different from all these,—by passing through coloured, or rather, colouring media, whether solids, liquids, or gases, as red glass, ink, chlorine. This, the most familiar mode of coloration, is the most difficult to account for, and has been (on account of its obscurity) less studied than the others. In some instances, the complementary colour (that which, added to the transmitted tint, makes up white light) is entirely absorbed or lost; in other cases it is reflected at or near the first surface of the medium. Sometimes

the transmitted light is made up of different portions of the spectrum curiously blended, whilst rays intermediate in the order of refrangibility are wholly stifled. Many crystals have the curious property of dichroism, that is, of transmitting light of different colours in different directions. All these facts have been very carefully studied by Sir David Brewster.

But the most remarkable phenomenon to be noticed under this head is the wonderful action of nitrous acid gas upon light.¹ When a beam, either of sunlight, or the light of a lamp, is passed through a bottle containing a small quantity of fuming nitrous acid, the light emerges of a tawny orange colour, which may be deepened indefinitely by heating the acid. If this light be then analyzed by a common prism, a wonderful spectacle is seen. The spectrum appears traversed by countless bands or dark spaces, whilst the blue and violet colours are nearly absorbed. The effect of the gas, then, is this,—to stifle or absorb countless minute portions of light seemingly selected at random from every part of every colour in the whole spectrum. Some of these deficient rays are broad and palpable, but most of them are so fine as to be visible only with the telescope. To understand the full import of this discovery, it is necessary to describe first the lines of Fraunhofer.

JOSEPH FRAUNHOFER, born in Bavaria, of humble parents, in 1787, raised himself by his unassisted efforts to be the first practical optician of the day. He had also the merit of devoting his leisure and the fine apparatus at his command to the observation and discovery of many optical phenomena, particularly those diffractive colours produced by fine gratings, which are known under the name of Fraunhofer's spectra.² His principal discovery, however, was (in 1814) that of countless deficient or dark lines in the solar spectrum, resembling those which, as we have mentioned, were afterwards observed by Sir D. Brewster, to be produced in any kind of light by the action of nitrous gas. The deficient rays of solar light had, indeed, been observed still earlier (in 1802) by Dr Wollaston, but he counted only a very few of the more conspicuous ones; he described them merely incidentally, and (unusually with him) seems not to have perceived the great value of the discovery both in a theoretical and practical point of view.

Fraunhofer's beautiful map of the spectrum, traversed by lines of every grade of darkness, and clustered with every conceivable variety of distribution, was published in the Munich Transactions. He counted 590 lines, but Sir D. Brewster states that he has carried the number to 2000. Like the stars, they are probably countless. These lines characterize solar light. The light of the fixed stars and that of the

electric spark have their peculiar deficiencies different from those of our sun. These were discovered by Fraunhofer, as well as their occurrence in certain coloured flames. The order and number of the lines is, in each case, independent of the kind of prism used; but the angular distribution of the deficient rays varies with the material. Thus an oil of cassia prism expands most in proportion the less refrangible end of the spectrum; while water and sulphuric acid act with disproportionate dispersive energy on blue and violet light. This property of substances had been already studied by Sir D. Brewster with his usual diligence; but the importance of Fraunhofer's discovery was this, that the lines (the larger of which he distinguished by letters of the alphabet) furnish landmarks which define special rays of light invariably recognisable under all circumstances, which the vague description of their tints is quite incompetent to do. This enabled, on the one hand, the practical optician to discover the kinds of glass most fit for achromatic combination; and, on the other, it afforded precise numerical measures of the quality of dispersiveness in bodies which have been partly already, and will yet much more become, tests of some of the more obscure and difficult portions of the theory of light,—those, namely, which are connected with dispersion and absorption. Fraunhofer was, after Dollond, the most eminent and scientific manufacturer of achromatic telescopes, of which he vastly increased the aperture. He died at Munich in 1826.

Returning to Sir David Brewster's discovery of the artificial production of analogous lines or deficient rays in light from any source, its importance is easily perceived:—for, in the first place, it so far accounts for the strange phenomenon of the deficient rays of the sun's light, by showing that it may be caused by a gas resembling nitrous acid gas in its properties existing in the solar atmosphere; and, farther, if so astonishing a result of absorption is ever to be explained by theory, the first step is to be able to produce the phenomenon at pleasure, and to examine the qualities of the bodies producing it. The phenomena of coloured flames which possess standard deficient rays, present perhaps a closer analogy to the sidereal spectra. Sir David Brewster has farther found that the absorptive action of the earth's atmosphere (detected by the varying character of the spectrum for different angular altitudes of the sun) increases the number and also the breadth of these lines.

Intimately connected with, and nearly of the same date as these experiments, was an observation of Sir D. Brewster's, which has received less general assent than any other of the numerous and important ones

¹ Edinburgh Transactions, vol. xii. (1833).

² The peculiarity of these spectra is this, that they consist of pure colours, whilst almost all interference-colours are, like those of Newton's rings, mixed and impure. One result is very remarkable. Fraunhofer obtained his spectra of such brilliancy as to be able to measure the position of the dark lines (an evidence of their exceeding purity), thus obtaining a standard spectrum in which the material of the prism has no influence whatever in varying the ratio of the dispersion of the various colours.

which we owe to his genius. It is an analysis of the coloured light of the (so-called) homogeneous rays of the pure spectrum by the specific action of absorbing substances. Sir D. Brewster believes that he has separated the homogeneous orange of Newton into red and yellow, the green into yellow and blue; and that, in fact, each of the three primary colours exists at every point of the spectrum. But as grave doubts have been thrown on the results, especially by the recent careful experiments of Helmholtz, I shall not further insist upon them here.¹ Still less can I take notice of a multitude of microscopic researches on a variety of objects in the animal, vegetable, and mineral kingdom, and on the physiology of vision, with which Sir D. Brewster has filled a multitude of memoirs, each bearing testimony to the zeal and acuteness by which his researches are directed.

(542.)
His other scientific papers.

Sir David Brewster received, in 1816, jointly with Seebeck, one of the great prizes of the Institute; he also received, in succession, all the medals in the gift of the Royal Societies of London and Edinburgh, and he is an honorary member of the principal academies of Europe. In particular, he is one of the eight associate-members of the French Academy of Sciences. To meteorology he has been a valuable contributor, having discussed in an able paper the law of the distribution of temperature over the globe, and pointed out the near coincidence of two regions or centres of greatest cold in the northern hemisphere, with the magnetic poles. His papers are so numerous, and their variety is so great, as to render an enumeration, even of those containing what may reasonably be termed discoveries, impossible within our limits. Few persons have made with their own eyes so vast a number of independent observations; few have ever observed better, or recorded their observations more faithfully. He has discovered (as we have partly seen) a multitude of laws of phenomena of the greatest importance in the construction of a theory, but he has not been forward in proposing such a theory. Neither the moveable polarization of Biot, nor the transverse undulations of Young and Fresnel, received his cordial assent. Generally speaking, he has been favourable to a corpuscular theory of light, without, however, attempting to render the Newtonian view mechanically consistent with the astonishing variety of complex phenomena which he aided in discovering, and which would evidently require it (to say the least) to be completely remodelled. His scientific glory is different in kind from that of Young and Fresnel; but the discoverer of the law of polarization, of bixial crystals, of optical mineralogy, and of double refraction by compression, will always occupy a foremost rank in the intellectual history of the age.

(543.) Before closing this section I shall add a few words respecting the discoveries of MM. Seebeck and Biot,

which have a very close relation to those of Sir David Brewster.

THOMAS SEEBECK was born in 1770. We have (544.) seen that he was one of the discoverers of the depolarizing structure of heated and compressed glass. Seebeck. In 1816 he observed, independently of M. Biot, the property of oil of turpentine and other fluids to rotate the plane of polarization of light transmitted through them, thus acting similarly to a crystal of quartz on a ray passing along its axis (512). Previously to these discoveries he had repeated Sir William Herschel's experiment on the position of maximum heat in the spectrum, and found it to vary with the material of the prism. When the science of electro-magnetism was created by Oersted in 1819, his attention became chiefly directed to that class of phenomena, and in 1823 he was fortunate enough to discover thermo-electricity. He also wrote many papers on allied subjects. He was a skilful observer, but deficient in the power of physical analysis. He died in 1831.

M. BIOT, at the time I write, the oldest member (545.) (I believe) of the Academy of Sciences, and one of the veterans of European science, was born at Paris in 1774, and has lived to the age of 80, a life of almost unintermitted intellectual labour. It is impossible not to be touched by the evidence of such unconquerable love of knowledge. He was, if I mistake not, one of the original pupils of the Polytechnic School; and his talents being first developed in an almost purely mathematical direction, he attracted the notice of Laplace, who introduced him to the Institute, and by whom he was always befriended. In 1802 he published a work on curves and surfaces of the second degree, and was the first after Lambert who thought of applying mathematics to the theory of conducted heat. From this time his attention was almost exclusively directed to the applied sciences, and the number and variety of his experiments and writings almost baffles enumeration. Descriptive and practical astronomy, the theories of sound, of light, of the voltaic pile, of terrestrial magnetism, of electro-magnetism, of heat, radiant and combined, have been the subjects of his studies and writings. We find him in the earlier part of his career associated with Gay Lussac in his first aeronautic expedition, and with Arago in the geodetical and astronomical operations of the great arc of the meridian. He afterwards carried the pendulum to the Island of Unst, the northmost land in Shetland; and he made original experiments on the propagation of heat and of sound. He wrote a voluminous treatise on descriptive and practical astronomy, one still more elaborate on general physics, and a vast number of miscellaneous papers in the Journal des Savans and the Biographie Universelle. His original memoirs in the Transactions of the Academy are usually very long and elaborate, his calculations and empirical formulae

¹ See Sir D. Brewster's statement and defence of his opinions in his Life of Newton, vol. I., p. 117, &c.

laboriously accurate. One of his papers fills an entire volume of the Academy's Memoirs. Even the astronomical hieroglyphics of the Egyptians, and the chronology of Chinese eclipses, have drawn from his pen learned treatises; and he has expounded the labours and discoveries of his countrymen and others with almost as much care and effort as if they had been proper to himself. But his subject by predilection was optics, and here he made his most considerable discovery, and that which he has followed out with most minute industry, namely, the rotatory action of fluids, in which he had Seebeck for a co-discoverer. (See Art. 512.) He studied the colours of crystallized plates with exemplary patience, and as we have seen in the preceding section, by his accurate observations on the law of the tints, prepared the way for the theory of transverse vibrations; but his own doctrine of moveable polarization, which he imagined to explain them, made no impression on the progress of science. He was the first who divided doubly-refracting crystals into positive (as

quartz), and negative (as calcareous spar). In the former the extraordinary wave is a prolate spheroid, and inclosed within the ordinary spherical wave; in the latter the spheroid is oblate, and exterior to the sphere. He also discovered (very approximately) the law regulating the plane of polarization of the rays in biaxial crystals.

M. Biot has for about half a century been an active professor and member of the Institute. His researches, always marked by precision, are perhaps deficient in bold conjecture and happy generalization. They are conducted with a mathematical stiffness which allows little play to the fancy, and in hypothetical reasoning he rarely indulges. His style is formal yet diffuse, and consequently somewhat repulsive to the student. His works are consequently not easily read, and have contributed less to the progress of knowledge than the scrupulous care often evinced in their compilation might seem to warrant. Yet the name of Biot will be ever associated with devotion to science, and especially with the progress of optics in our own day. (546.)

§ 6. Mr AIRY, Sir WILLIAM R. HAMILTON, and Professors LLOYD and MACCULLAGH.—Confirmation of Fresnel's Theory—Investigation of the Wave Surface completed; Conical Refraction.—M. CAUCHY. Mechanical Theory of Elastic Media, and of Ordinary and Metallic Refraction; M. Jamin.—Theory of Dispersion; Professor Powell.

(547.) It would not be possible, in one short section, to do justice to the various improvements and additions which the undulatory theory of light—the joint creation of Huygens, Young, and Fresnel—has received since the nearly simultaneous decease of the two last-named philosophers. But while a vast amount of labour and of mathematical and experimental skill has been thus expended, of which it would be in vain to attempt within our limits to give an account, we may pause upon two or three of the more conspicuous results of these researches, which, in conformity with the plan of this dissertation, may give a tolerable idea of their general tendency.

(548.) Looking at the history generally, we find one curious peculiarity in the progress of this remarkable theory. Its origin in the seventeenth century was unattended with sympathy or success. It received little support, and was well nigh forgotten for more than an hundred years; it was then resumed (we might almost say re-invented) in England, but it remained unpopular and almost unknown until re-echoed from a foreign land; while in France itself the views of Fresnel were (with one or two exceptions) as little appreciated as those of Young had been in England. From this period England became the place of its chief development; and with the exception of one eminent philosopher, M. Cauchy, its supporters and extenders, whether by analysis or experiment, have

belonged to Great Britain,¹ a few of the most conspicuous of whom are named at the head of this section.

The attention of the British public was forcibly called to the theory of Young and Fresnel, by an able treatise on Light, contributed by Sir John Herschel in 1827 to the Encyclopædia Metropolitana. The excellent method, lucid explanations, and intelligent zeal which marked this essay compelled the notice of men of science, too long deterred from the study of the fragmentary and abstruse writings of Young. It was followed four years later by a most able and precise mathematical exposition of the theory, and its application to optical problems, by Mr Airy (now Astronomer Royal), who was then Plumian Professor at Cambridge, and who introduced this part of optics as a branch of study in that university. Whilst the excellent tract on the undulatory theory (published in 1831 in his Mathematical Tracts) opened up the subject in a most accessible form to British mathematicians, his original papers in the Cambridge Transactions confirmed the doctrines of Fresnel by a number of new and admirably contrived experiments, some in connection with interference, some with polarization, and all were confronted with the rigorous results of the mathematical theory. The paper on Quartz, and that on the Rainbow, have been already referred to (art. 466, 512). The writings of Mr Airy and of Sir John Herschel have continued to be the

¹ M. Moigno (a Frenchman), writing in 1847, laments that France was then perhaps the only country in which the experiment of "conical refraction" (the triumph of Fresnel's theory to be presently mentioned) had never been repeated.

main sources of information on this subject, and on physical optics generally, not only in this country but on the Continent. It is remarkable that in France, which possesses so many admirable scientific books, there should not exist a single good treatise on optics. Had not Mr Airy's attention been necessarily withdrawn from optics to astronomy, it is very evident that the theory of light would have received from him many further important additions.

(550.)
Sir Wm.
R. Hamilton and Dr
Lloyd.

Whilst an impulse was thus given to the mathematical theory of light in the University of Cambridge, a similar progress was being made in the sister University of Dublin, where three of her most eminent professors, Sir WILLIAM ROWAN HAMILTON and Professors LLOYD and MACCULLAGH devoted themselves energetically to its improvement and verification.

(551.)
Conical refraction in bisaxal crystals.

To the two former of these we owe the prediction and ocular demonstration of the most singular and critical of all the results of Fresnel's theory. Sir William Hamilton, a geometer of the first order, having undertaken the more complete discussion of the wave surface of Fresnel (see Section Third of this Chapter), to the equation of which he gave a more elegant form than heretofore,¹ ascertained the exact nature of that surface, and consequently the exact direction of refracted rays in the neighbourhood of the "optic axes." It had been shown by Fresnel that, in the case of crystals with two axes, a plane section in a certain direction cuts the two sheets of the wave surface in a circle and in an ellipse, which necessarily intersect each other in four places. (See the annexed figure.) In the lines joining these four points with the centre of the figure the velocity of the two rays is equal. Now the cusps or sharp inflections of the wave surface in these particular directions, occur not only in the particular plane of section which we have considered, but in any section of the wave surface passing through these lines of equal ray-velocity. In the figure, therefore, of the compound sheet there is not a furrow, as Fresnel had supposed, but a pit or dimple, with arched sides something like the flower of a convolvulus, and the surfaces meet at the bottom of the pit at a definite angle. Let the circle and ellipse, in the annexed figure, represent the section of the wave surface we have described; then O P is the line of uniform propagation, and P is the bottom of the

conoidal pit M P N. Now suppose a slender ray of light to move through the crystal in the line O P, and to emerge into air at a surface of the crystal cut perpendicular or nearly so to the direction of single ray-velocity O P. If we confine our attention at first to the plane of the figure only, that ray having intersecting

tangents both proper to the wave surface, would give rise, on Huygens' construction (art. 475), to two emergent rays inclined at an angle. But since this is the case, not only in the plane of the figure, but (as has been stated) in any plane passing through the ray in question, the emergent light must form a conical luminous sheet, the angle of the cone being determined by the refractive properties of the crystal. This beautiful and unexpected result was verified with great skill and address by Dr Lloyd in the case of Arragonite, which is a bisaxal crystal, and he found the position, dimensions, and conditions of polarization of the emerging cone of light to be exactly such as theory assigns. When all the necessary corrections are attended to, the angle of the cone of light is about 3°. There is another case of conical (or it might be called cylindrical) refraction, which occurs nearly in the same portion of a crystal, which was predicted and discovered in like manner, but which we will not stop to particularize.² The observations of Dr Lloyd have been extended by M. Haidinger to the case of Diopside, a crystal also having two optic axes.

Every one capable of appreciating such evidence, will feel the irresistible impression which so curious an anticipation, so accurately fulfilled, gives us of the positive truth of a theory admitting of such verifications. The names of Sir W. Hamilton and Dr Lloyd will be handed down to posterity in connection with this admirable discovery. But they have also other claims to our respect, to which we can here only refer in the most general terms. The former has generalized the most complicated cases of common geometrical optics by a peculiar analysis developed in his essays on "Systems of Rays" (Irish Academy Transactions, vols. xv.-xvii.).³ To Dr Lloyd

¹ The equation is— $$\backslash frac\{\backslash alpha^2\; x^2\}\{x^2\; +\; y^2\; +\; z^2\; -\; \backslash alpha^2\}\; +\; \backslash frac\{\backslash beta^2\; y^2\}\{x^2\; +\; y^2\; +\; z^2\; -\; \backslash beta^2\}\; +\; \backslash frac\{c^2\; z^2\}\{x^2\; +\; y^2\; +\; z^2\; -\; c^2\}\; =\; 0.$$

² It was shown by Sir W. Hamilton that the tangent plane M N touches the wave surface, not in two points merely, but in a circle of contact; consequently, the perpendicular to this tangent plane, O M, is the direction of one of the optic axes (or the velocity is the same for both portions of the compound wave). Hence a ray incident externally so as to be refracted along this perpendicular, will at entrance spread into a hollow cone interior to the crystal, and on emergence at a parallel face each portion of the ray recovers a direction parallel to its primitive direction, and a luminous hollow cylinder is the result. See Dr Lloyd, in the Irish Academy Transactions, vol. xvii., and Sir W. R. Hamilton's third supplement to his "Systems of Rays" in the same vol.

³ Sir W. R. Hamilton is also a discoverer in pure analysis and its connection with geometry. Following up the ideas of Mr Warren on the geometrical significance of the symbol $\backslash sqrt\{-1\}$, as indicative of direction, Sir W. Hamilton has developed the theory of a new class of imaginary quantities, which he terms quaternions, by means of which he contrives to express simultaneously the direction in space and magnitude of a line or form; and this calculus he has applied to the solutions of problems of geometry and physical astronomy. The quaternion appears to express something even beyond this; and this redundancy has been considered as a difficulty by some mathematicians. The superfluous number is considered by Sir W. Hamilton as representing time in mechanical problems.

we are indebted for several interesting experimental papers on optics, for an able and impartial review of the progress of the science,¹ and for an excellent elementary treatise on the Wave Theory, which forms by far the best popular introduction to the subject.

(553.) Closely associated in his pursuits, as in personal friendship, with Sir W. R. Hamilton and Dr Lloyd, was JAMES MACCULLAGH, a native of Tyrone, born in 1809, and who died prematurely and unhappily October 24, 1847.² His first paper was communicated to the Royal Irish Academy at the age of 21. It was one of the earliest original contributions of this country to the development of the theory of Fresnel. The construction of the wave surface in biaxial crystals was simplified and improved; and in 1835 a second paper appeared, in which geometrical constructions of great elegance were employed for the farther investigation of the subject. Mr Macculagh next attacked the theory of the undulations of ether in quartz crystals, to which he gave a mathematical expression (see art. 512). In 1838 he published a paper on the laws of crystalline reflection, in which he adopted certain hypotheses, such as that the vibrations of the ethereal particles, in the case of polarized light, are parallel to the plane of polarization (contrary to Fresnel's opinion), and that the density of ether is the same in all media. In a subsequent memoir on the dynamical theory of reflection and refraction, he arrived at similar results with fewer physical assumptions, and by a more purely mathematical treatment of the subject. Subsequent researches, presently to be mentioned, have diminished the value of these theoretical investigations.

(554.) Numerous mathematicians of eminence at home and abroad entered upon the same arduous enquiry. To attempt to deduce from the hypothetical constitution of a very rare highly-elastic medium, together with the known dynamic laws, the various complicated facts of optics, was a problem whose difficulty was only equalled by its indefinite character. For how little do we know of the molecular constitution of such fluids as air and water? How much less then of a fluid (if we may so term it) almost infinitely rarer, and incapable of being inclosed, measured, or weighed? The bare possibility of transversal undulations was long contested by very able mathematicians; and conceding it, the mutual influence of such an ether and the particles of gross matter (as shown by reflection and refraction) must, it would seem, for ever remain problematical. Yet, however gratuitous or even erroneous our reasonings about such ultimate questions may be, there is no doubt a real benefit in obtaining

from them at least a mathematical congruity with observed facts. The progress of science shows that there is a practical usefulness in this step. On the other hand, we must not be discouraged to find that there are so many handles to the matter, that even the most profound thinkers may conceive that they have reached the proposed end by different and incongruous routes. Besides the mathematicians whom I have mentioned, many others were in the field; for the result of tremors propagated through elastic media of different kinds is an enquiry of excessive generality, and forms a part of many branches of science besides optics. MM. Cauchy, Navier, Poisson, Coriolis, Green, Kelland, and Neumann are amongst those who attacked the problem. For a series of years memoirs rose fast and thick on this favourite battle-field; and even a skilful mathematician might find it no small penance to discuss the merits of the various hypotheses and the solidity of the respective deductions which were proposed.

(555.) It must be owned that a great part of this vast mathematical toil has been without immediate result. Their bearing on optics. It is by comparing the conclusions arrived at by authorities of seemingly equal weight, that we learn the difference between a stable physical induction and a clever mathematical hypothesis. Macculagh, for instance, maintains the vibrations of ether to be parallel to the plane of polarization; Fresnel and M. Cauchy³ that they are perpendicular. The first mathematician considered that the vibrations are wholly transversal, the last believes that the normal vibrations have also their share in affecting the phenomena, whilst Poisson denies that transversal vibrations can be propagated to a sensible distance. Macculagh finds that, to account for metallic reflection, the indices of refraction of mercury and silver are 15.0 and 35.0, whilst Cauchy makes them but 1.77 and 0.34. One theorist assumes that the density of ether is the same in all bodies, another that it is greatest in a vacuum, and others precisely the reverse; one that vibration is not accompanied by change of density, another that it is;⁴ and so on in almost endless variety. A matured opinion can only be formed after the results of the various assumptions, and their congruity with facts, have been more thoroughly worked out. Already two of Macculagh's essential postulates have lost much of their plausibility; that respecting the direction of the vibrations in polarized light has been probably decided by Professor Stokes in favour of Fresnel's opposite view, by an admirably devised experiment on the effect of diffraction on polarized

¹ British Association Reports for 1834.

² It is unnecessary to suppress the fact that Mr Macculagh died by his own hand, under the pressure of a fit of despondency, brought on (it is believed) by over-work; a fate happily extremely rare amongst students of exact science. I say it is needless to suppress the fact, because it infers no blame. Mr Macculagh was an amiable, pious, and exemplary man. Irresponsible insanity was, of course, the cause.

³ This is M. Cauchy's last view; for some years he adopted the opposite one, which he then surrendered with great frankness.

⁴ If, however, the vibrations were wholly transversal, it seems to be admitted that they would not affect the density of the medium.

rays¹; the existence of normal vibrations seems to be proved by the ingenious experiments of M. Jamin, showing that at no angle is light perfectly polarized by reflection. The more we see of these diversities arising in the progress of science, the less are we disposed to found on merely mathematical conclusions from an assumed constitution of elastic bodies; the more, on the other hand, do we admire the admirable sagacity of Fresnel—the real Newton of modern optics—few of even the least of whose suggestive anticipations have fallen to the ground.

(556.) M. AUGUSTIN LOUIS CAUCHY has been long known as one of the ablest and most prolific mathematical writers of this century. Besides numerous and important memoirs, on nearly every branch of pure and applied mathematics, published in the Journal de l'Ecole Polytechnique, and the Memoirs of the Institute, he has published, in a separate form, Exercices des Mathématiques in two series of volumes; and for many years scarcely a weekly meeting of the Academy of Sciences occurred without a mathematical memoir of this prolific author being laid on the table, and subsequently printed in the Comptes Rendus. The integral calculus and other parts of analysis form the subjects of a large part of these writings; but the theory of hydrodynamics in the earlier, and of optics in the later part of his career, are largely represented. So diffuse and desultory a mode of publication has been little favourable to those who wish to make themselves acquainted with what has been accomplished by M. Cauchy. The scientific world is indebted to Abbé Moigno² in France, M. Radicke³ in Germany, and Professor Powell⁴ in England, for analyzing in part his optical labours. As the present brief notice is evidently inadequate to include even the most superficial view of the whole, I shall say a few words upon two of his theoretical researches on light, which have attracted most general attention. The first is upon the theory of Reflection and Refraction, framed so as to include the phenomena of metallic reflection; the second is upon the Dispersion of Light.

(557.) I shall first mention some seemingly exceptional

facts to the ordinary laws of polarization by reflection. As early as 1814 (Philosophical Transactions, 1814, p. 230), Sir David Brewster had remarked that such highly refractive substances as Realgar, Diamond, and Chromate of Lead, do not polarize, at any angle, the whole of the reflected light. Mr Airy afterwards showed that light reflected from diamond near the maximum polarizing angle, possesses qualities resembling those of light reflected from metals. The same view was more generally stated by Mr Dale;⁵ and last of all, M. Jamin showed that all transparent substances polarize elliptically the light which they reflect,—the difference of "phase" of the two component vibrations increasing from 180° at a perpendicular incidence, to 360° at an incidence of 90°; and that the laws of reflection at transparent surfaces, as also in the case of metals, depend upon two constants—the index of refraction and the coefficient of ellipticity. And he has determined in numerous cases the values of these constants.

Thus Fresnel's theory of reflection requires undoubted modification. It only holds true for substances whose index of refraction is nearly 1.46, that of the glass which he examined. The complication is held to arise from the existence of vibrations (called normal) in the direction of transmission of the luminiferous wave, such as those which produce the effects of Sound in air, and which produce certain effects on Light at the bounding surfaces of two media. In the theory of Fresnel, as also in those of Maccullagh and Neumann, this influence is neglected. To Green and to M. Cauchy belongs the merit of laying down a more comprehensive theory. Mr Green's theory, published in 1837⁶ (not long before his death), is so far incomplete that it involves only one constant. M. Cauchy's investigations, published two years later,⁷ embrace the phenomena of metallic reflection by the introduction of the two constants mentioned above, thus completing the theory of reflection and refraction both for transparent and metallic surfaces.

The fact of the unequal refrangibility (dispersion) of light has ever been felt to be one of the most real as well as prominent difficulties in admitting

¹ Cambridge Transactions, vol. ix.

² Handbuch der Optik, Band I, 1839.

³ See Moigno Rep. d'Optique, p. 1395.

⁴ Cambridge Transactions, vol. vii.

The following extract from this able paper shows the independence of physical assumptions which characterizes these ultra-mathematical optical theories:—"We are so perfectly ignorant of the mode of action of the elements of the luminiferous ether on each other, that it would seem a safe method to take some general physical principle as the basis of our reasoning. . . . The principle selected as the basis of the reasoning contained in the following paper is this: In whatever way the elements of any material system act upon each other, if all the internal forces exerted be multiplied by the elements of their respective directions, the total sum for any assigned portion of the mass will always be the exact differential of some function. But this function being known, we can immediately apply the general method given in the Mécanique Analytique, and which appears to be more especially applicable to problems that relate to the motions of systems of an immense number of particles mutually acting on each other. One of the advantages of this method, of great importance, is, that we are necessarily led by the mere process of the calculation, and, with little care on our part, to all the equations and conditions which are requisite and sufficient for the complete solution of any problem to which it may be applied."

A consideration of the candid admissions of the preceding paragraphs (especially the last sentence) will lead the reader to see how short a way a theory of so general a kind—the chief characteristic of which consists in eluding every troublesome physical enquiry—can go towards explaining the relations of Light to Matter; yet it may be of use by indicating the kind of solutions which more restricted hypothesis may be expected to give of the laws of phenomena.

⁷ Comptes Rendus de l'Acad. des Sciences.

the undulatory theory. That theory, in its simple form, enables us indeed to explain clearly enough the refraction of light owing to a change of velocity in the wave as it passes from one medium to another; but it assigns no reason why that change of velocity should be different for light of various colours, in other words, of different wave-lengths. The corpuscular theory, on the other hand, furnishes at least a plausible explanation, by assuming a variable attraction between refracting media and the molecules composing the different rays.

When, however, undulationists were pressed on the subject, it was easy to see the direction in which at least a plausible explanation might be sought. In the usual form of equation for vibrations in air, given by Lagrange, the integration is effected by assuming the intervals between the particles evanescent compared with the length of a wave. This is perfectly true in the case of sound, and all sounds appear in consequence to travel uniformly. But should it fail in the case of light, that is, should the intervals of the ethereal particles bear some sensible ratio to that very small quantity, the length of a wave, what would be the result? M. Cauchy has made out, by a very complex analysis, that in this case the longer waves will travel most rapidly, and consequently be least refracted. Several other writers, especially Professor Kelland, obtained similar results; and Mr Airy, by very simple, though only approximate, considerations, showed the dependence of refraction on the length of a wave.¹ M. Cauchy's memoir appeared in 1835 at Prague, in a bulky and abstruse form: the mathematical investigations are very long and complex, the numerical verifications

scarcely less so.² The indices of refraction, observed by Fraunhofer, for different lines of the spectrum, in different kinds of glass, together with the corresponding wave-lengths for these rays, formed the principal data for comparison. But others have since been obtained by Rudberg and Professor Powell, and carefully compared with M. Cauchy's theory by the latter, and the coincidence appears satisfactory.³ But in estimating the value of this coincidence, it is to be observed that of 7 indices of refraction observed for each substance, 3 must be used to ascertain the constants in the formula, and only 4 remain to be calculated.

One difficulty, however, remains. If the rays of light travel through space with variable velocities, the images of the stars would present tails of colour inconsistent with observation. M. Cauchy eludes this difficulty by the following hypothesis respecting ethereal media:—The existence of transversal vibrations (according to him) requires that the law of force between particle and particle of ether must not be intermediate between the inverse 2d and inverse 4th power of their distance. In the former case the force is an attractive, in the second a repulsive one. In the first, the velocity of propagation depends on the length of a wave, in the second it is independent of it. [The first case, too, alone will be consistent with permanent longitudinal vibrations.] Consequently if we suppose that in free space the particles of ether are arranged in close order, and exert a repulsive force on each other, no dispersion results; but in refracting media, supposing the distance of the molecules increased, and the mutual action attractive, then dispersion occurs.⁴

§ 7. RITTER.—Chemical Rays of the Spectrum.—NIEPCE; DAGUERRE; Mr TALBOT. Art of Heliography or Photography—Daguerreotype—Calotype.—Professor STOKES. Chemical Rays rendered visible—Fluorescence.

A very curious chapter of the history of Light remains to be written, respecting the chemical energies which it is capable of exerting, or which at least are found in those parts of the solar rays which are dispersed by a prism. These are in part luminous and partly invisible under ordinary circumstances, the latter possessing these chemical qualities in a still higher degree than the others. Though not perhaps very closely associated with the optical discussions of the previous sections, it seems impossible to separate this part of the subject from the rest, since the rays called Chemical may, as we have reason to think, be reflected, refracted, polarized, absorbed, and made to interfere like visible light; and farther, because to the extreme

limit of their sensible action they may, by certain treatment, be made visible to the eye. The art of photography, though belonging quite as much to chemistry as optics, being a means of inquiry into the qualities of the solar radiations invaluable to the natural philosopher, cannot by any means be excluded from a sketch, however general, of the progress of physical science.

J. W. RITTER, Professor of Chemistry at Jena, and well-known for his numerous contributions to the earlier progress of voltaic electricity, has the merit of having first clearly pointed out in 1801 the separate existence of chemical rays in the spectrum which extend beyond the most refrangible or

¹ In all these cases the expression for refrangibility depends on the ratio of the sine of an arc to the arc itself; which again includes the ratio of $\backslash Delta\; x$, the distance of the molecules to $\backslash lambda$, the length of a wave. When $\backslash Delta\; x$ becomes very small, the first ratio becomes unity.

² A portion of these researches had, however, been printed in Paris (privately, I believe) in 1830.

³ Powell on the Undulatory Theory. 1841.

⁴ Moigno, Repertoire d'Optique, p. 128.

violet rays (when the sun's light is decomposed by a prism), in the same manner as the invisible rays of heat were found by Sir Wm. Herschel to extend beyond the visible red. Scheele had indeed previously noticed that the power of the sun's light to decompose and blacken salts of silver increased rapidly from the red towards the violet ray; and there is little doubt that Herschel's discovery suggested that of Ritter, of the independent or non-luminous rays of the spectrum.

Ritter attributed to these rays a deoxidizing quality. Dr Wollaston, who also made experiments on the subject, and discovered the specific action of the different rays on gum-guaiacum, prudently suggested the more comprehensive term of chemical rays. Various other denominations have been proposed, which need not be here dwelt upon; all that can be said is, that deoxidation does not represent the solar action completely. Whether it be really an independent principle in the sun's rays which causes these effects, or merely light and its modifications, is, as in the corresponding case of heat, yet undecided. But it is remarkable that, as shown by M. E. Becquerel, the discontinuity of the luminous spectrum producing "Fraunhofer's Lines" exists equally for the chemical rays.

Soon after Ritter's first experiments, Dr Young proved the interference of the obscure chemical rays (Phil. Trans., 1803), a conclusion successively claimed since by different physicists. Berard in 1812 showed that these rays are polarized by reflection. Seebeck observed that the different rays impressed different colours upon salts of silver; and M. Edmond Becquerel long afterwards showed that the red and yellow rays, though incapable of commencing chemical action, in some instances have the power of continuing it when once excited by the more refrangible rays.

In 1835 Mrs Somerville made some interesting experiments on the permeability of different bodies to the chemical rays, similar to those of Melloni on the heating rays (see chap. VI., § 8), and she found great and seemingly capricious variations in this respect.¹ The account of them was addressed to Arago, and published in the proceedings of the French Academy. She found that green glass, coloured by copper, intercepted entirely the chemical rays, yet this was not due so much to its colour as to its other qualities (which are also peculiar as regards radiant heat), for

the emerald transmits the same rays. Red glass stops most of the chemical rays, whilst the garnet transmits them. Though white glass has generally been considered very transparent for these rays, Mr Stokes has shown that it entirely stops those of the very highest refrangibility, which are readily transmitted by quartz. Sir John Herschel and Mr Hunt have made many interesting experiments with coloured media on the particular parts of the chemical spectrum absorbed, but a great deal remains to be done, especially as regards the nature of the substances employed.

But the great impulse given to this subject was derived from the invention or discovery of the beautiful art of PHOTOGRAPHY.

In 1802 Mr Thomas Wedgwood and Sir Humphrey Davy succeeded in forming pictures of objects laid on paper prepared with nitrate of silver, and in taking profiles (silhouettes) by means of shadows. They proposed to obtain similar effects by means of the camera obscura, but their paper was not sufficiently sensitive. The effectual bar to their proceedings was, however, this: that they could discover no means of fixing the shadows which they had obtained, or preventing the whole surface of the paper from being gradually blackened by exposure to light.

In 1814 J. NICEPHORE NIEPCE, a retired proprietor at Chalons sur Saone,² entered into a similar enquiry, but by methods quite different. He employed the solar effect upon resinous bodies, and some at least of his pictures were executed on plates of pewter or of rolled silver. They were mostly copies of engravings, and the light parts corresponded to the lights of the originals. He, however, at length succeeded in fixing impressions of views in the camera obscura, though in an imperfect manner, and after very long exposure. The pictures thus obtained had this in common with more perfect processes, that the luminous impression was first brought into view by a chemical process subsequent to exposure in the camera. In 1825 Nicephore Niepce became associated with Daguerre, who had previously been engaged in the same research; they agreed to communicate the results of their several experiments. The result, as is well known, was the invention of the DAGUERREOTYPE, not improperly called after Daguerre, who seems really to have worked it out almost entirely for himself, after the death of Niepce in 1833; whilst so patient and determined was Daguerre in

Mrs Mary Somerville Greig, afterwards her relative Dr Somerville. She is known in British science not only as the able commentator of Laplace's Mécanique Céleste, but as the author of some ingenious and apparently convincing experiments on the magnetizing power of the violet ray. Some anomaly, however, remains to be explained on this subject, as the result cannot always be obtained. Several years of light [i. before Dr Faraday made his discovery of what he terms the "magnetization of light" (see chapter vii., § 5), the writer of these pages supposed that the reaction of light and magnetism, observed by Morichini and Mrs Somerville, might be due to a latent and casual polarization of the light which was not present in all the experiments; and, in particular, he suspected that circularly polarized light might have a magnetic influence; but his experiments to this effect, in May 1836, were not successful, though he thinks them worth repeating.

² Probably one of the MM. Niepce who, in the early part of the century, are said to have propelled a boat on the Saone by a peculiar kind of air-engine, called pyreolophore. Delambre, Rapport Historique, 1810, p. 242.

keeping his secret until brought to perfection, that he did not even show his results until early in 1839, when the numerous specimens he had to exhibit rivalled in delicacy anything that the art has since produced.¹

The daguerreotype is depicted in the camera on a plate of silver, coated with an evanescent film of iodide, by exposure for a short time to the vapour of iodine. After the light has acted, and the plate has been withdrawn from the camera, no trace of a picture is visible until it has been exposed to the vapour of mercury, which, by its peculiar action on the places where light has acted, produces a correct picture of the object (or positive image, as it is called, light answering to light). It is then fixed, as it is called, by a bath of dissolved hypo-sulphite of soda, which has the property of removing the iodide of silver wherever the light has not acted. This singular and elaborate process has since been but slightly modified by various plans for rendering the iodide more exquisitely sensitive. To M. Niepce belongs the credit (1) of having fixed an impression of light, (2) of using metal plates, (3) of forming a picture by means of a camera obscura; to Daguerre, on the other hand, the novel and ingenious use of vapours instead of washes, and the whole succession of operations in the daguerreotype. When the French government acquired for the public (the French public, however, only) a right of property in the invention, they marked their sense of the share of merit of the inventors, by awarding to Daguerre 6000, to M. I. Niepce 4000 francs per annum.

Mr W. H. FOX TALBOT, a Wiltshire gentleman of great ingenuity and perseverance, and well acquainted with mathematics and physics, applied himself in 1834 to the problem of fixing shadows, in entire ignorance of what Wedgwood and Davy had attempted. He used paper washed with the nitrate of silver, and soon succeeded in obtaining impressions of lace and leaves of plants, but without fixing the shadows. This great step he, however, made a year or two later: a wash of iodide of potassium, or of common brine, was found to effect it. The announcement of his success and of his methods was called forth early in 1839 by the first reports of Daguerre's discovery. His method then consisted in dipping writing paper alternately in nitrate of silver and common salt, drying between the operations, and afterwards fixing the image. An unnatural or negative picture was thus obtained, the lights of na-

ture being darkened on the paper, and vice versa; but the truth was restored by pressing the drawing thus obtained against a second prepared sheet of paper, and exposing it to light, when the natural lights and shades were of course obtained. This derived impression is called a positive. This process has evidently several advantages over Daguerre's;—such as, that paper is used instead of metal plates; and that from a single impression (negative from nature) copies may be at leisure indefinitely multiplied. Pictures were thus obtained by Mr Talbot with the camera obscura.

Mr Talbot's chief improvement on his first method he called the Calotype (1841), and consisted in washing the paper successively with nitrate of silver, iodide of potassium,² and gallo-nitrate of silver. It is then exposed in the camera, but no impression appears until again washed with the gallo-nitrate of silver. It is then fixed with bromide of potassium, or with hypo-sulphite of soda, as in Daguerre's process.

By a subsequent invention, Mr Talbot has obtained what he justly calls an instantaneous process.³ An image was formed in a camera of a revolving wheel, to which was affixed a printed bill; the room being darkened, and the wheel made to revolve with the speed of 200 revolutions in a second, and being then illuminated by an electric spark, a legible impression of the printing was obtained. We doubt if, in the whole history of physics, a more astonishing result is recorded. Thus Mr Fox Talbot, by his rare energy, brought his inventions almost to perfection. Numerous competitors, of course, appeared on the field, and obtained many interesting results. The only one of much importance to the art of photography is the substitution of a film of iodized collodion on a glass plate, for the prepared paper in the first or negative process.

The daguerreotype and calotype processes, though seemingly so different, have much in common:—(1) a sensitive surface has to be prepared (iodide of silver is the basis in both); (2) it is exposed in the camera; (3) the picture (still invisible) is developed; (4) it is fixed. In the calotype the printing process for obtaining positives must be added to these.

The chemical theory is very far behind the art of photography. The most important steps of these curious and complicated processes have been attained by a kind of divination after a multitude of failures. The salts of silver are of a highly decomposable nature, the iodides and bromides pecu-

¹ The present writer had the benefit of seeing Daguerre's marvellous productions, and making his acquaintance at Paris, through the courtesy of M. Arago, while the secret was still preserved, and the public interest was excited to the highest pitch. About the same time he saw M. Isidore Niepce, son of the first photographer, and the specimens in his hands, as well as those in the possession of Mr Bauer of Kew, with whom they had been left by M. Nicéphore Niepce, when he visited England in 1827 and exhibited them at the Royal Society. One of the latter was engraved on a plate resembling pewter.

² Communicated to the Royal Society of London, 31st January, and printed in the Philosophical Magazine for March.

³ Mr Talbot says (Phil. Mag., March 1839, p. 203) that Sir H. Davy had recommended iodide of silver as a sensitive substance; but in his earlier experiments he had found it the contrary. But now, like Daguerre, he requires no immediate visible action of light, but develops it by a subsequent reaction.

⁴ The process is described in Hunt's Researches on Light, 2d edit., p. 140.

liarly so. The action of light appears to be to reduce the metallic silver, at least partially, an operation which is completed in the daguerreotype process apparently by the affinity of the mercurial vapour for iodine and for oxygen.¹

(576.) If anything were wanting to show the impossibility of separating the scientific arts from the history of science itself, it would be the case before us. The art of photography is far in advance of the theory, yet it constitutes in itself the greater part of what we know in a highly interesting branch of Natural Philosophy—the chemical agency of the spectrum. The surfaces of Talbot and Daguerre are the philosophical tools by which farther discoveries can alone be made.

(577.) So intimately connected with the discovery of the Chemical Rays of the spectrum, is that of revealing their existence (at least the existence of rays of the spectrum equally refrangible with those possessing chemical properties) to the sense of sight, that we may conveniently include in this section a very brief notice of Professor Stokes' remarkable experiments.

(578.) Mr GEORGE GABRIEL STOKES, a fellow of Pembroke College, Cambridge, and senior wrangler in 1841, at present holds the distinguished position of Lucasian Professor of Mathematics. It is almost needless, therefore, to state that his mathematical talents are generally acknowledged, and that he has displayed them by a ready application to many difficult problems in optics and mechanics which had not previously been accomplished. I may refer, however, to the integration of complex differential equations occurring in the theory of the flexure of solids (see art. 364), and in that of the rainbow (466), and in his elaborate investigation of certain cases of the friction of fluids (416). But it is more to our present purpose to observe that he combines this profound and technical command of analysis with singular skill in the experimental department of optics;—not merely in investigations closely connected with the wave theory, and expressible by mathematical formulae,² but in those which depend on the judicious questioning of nature by critical experiments not necessarily quantitative;—such, in short, as Newton discusses in his Optics; and indeed since Newton himself occupied the Lucasian chair, there have been perhaps few philosophers who have shown so remarkable an aptitude for both kinds of research.

(579.) In 1852 Professor Stokes announced that the refrangibility of light, which has hitherto been considered its most inherent and invariable quality, may in some circumstances be altered. The fundamental experiment is this:—A beam of solar light is caused

to form a pure and highly dispersed spectrum by passing through two or three prisms in succession, associated with a lens. The spectrum is made to fall on a glass vessel containing a solution of sulphate of quinine (a colourless fluid). Whilst the red, orange, and indeed the greater part of the luminous spectrum, pass through as if the fluid had been merely water, from about the middle of the violet "the path of the rays is marked within the fluid by a sky-blue light, which emanates in all directions from the fluid, as if it (the medium) had been self-luminous." This appearance extends far beyond the visible violet, the presence of the invisible rays (called chemical) becoming disclosed by the reaction of the quinine; and the dark bands, both of the visible and the (usually) invisible spectrum, are marked by obscure planes traversing the mass of diffuse light. When the light thus emanating from the interior of the quinine is examined by a prism, it is found to consist of rays of various colours and refrangibilities corresponding to those of the ordinary spectrum; from which Mr Stokes concludes that, since only violet rays and the invisible rays beyond them could have furnished or excited this emanation, the rays of light themselves have been transformed into others, with a lower degree of refrangibility, and possessing the corresponding optical properties. In conformity with this explanation, Professor Stokes has considered and accounted for a number of curious phenomena described by Sir David Brewster and by Sir John Herschel; called by the former "internal dispersion," by the latter "epipolic dispersion," all of which, so far as they involve anything peculiar, may be reduced to the general principle that certain bodies by their action on light have the power of lowering the refrangibility of the rays incident upon them, whether belonging to the visible or invisible spectrum—that is, of emitting rays of a lower, while under the excitement of rays of a higher refrangibility. Hence the phenomenon has been termed the "degradation" of light. Mr Stokes has also called it "fluorescence," a word involving no hypothesis, being derived from the characteristic action of a certain kind of fluor-spar described by Sir David Brewster.

Many substances, solid and fluid, are found to possess these qualities.³ Amongst others, and in the highest degree, glass coloured yellow by oxide of uranium, called in commerce canary glass, and solutions of horse-chestnut bark.

By using sources of light richer than sunlight in the highly refrangible rays; by also using quartz for the prisms instead of glass (which has the power of absorbing these to a considerable extent)—Pro-electric light.

¹ I may refer to a short but clear paper on the theory and practice of photography by the Rev. W. T. Kingsley, in the Journal of the Photographic Society, vol. 1., August 1853. Mr Kingsley is one of the most scientific and practical improvers of the art.

² This he has also done in a beautiful paper on the effect of polarization in modifying the phenomena of diffraction. Camb. Trans., vol. ix.

³ The greater part of these had been detected by Sir D. Brewster, with his usual acuteness and perseverance, in the course of his researches on "internal dispersion." The colouring matter of green leaves is one of the most remarkable.

fessor Stokes has obtained a result truly astonishing. In the case of the light derived from a voltaic arc, produced by the battery of the Royal Institution with metallic poles, the visible spectrum formed upon uranium-glass extended no less than six or eight times the length of the ordinary spectrum. If, by way of contrast, a porcelain tablet be used as a screen, the spectrum terminates at the usual point.¹

naming Mr Wheatstone's beautiful invention of the Stereoscope, as by far the most interesting contribution recently made to the theory of vision, regarded in a point of view not strictly anatomical. Although Mr Wheatstone's paper was published in the Philosophical Transactions for 1838, and the Stereoscope became at that time known to men of science, it by no means attracted, for a good many years, the attention which it deserves. It is only since it received a convenient alteration of form (due, I believe, to Sir David Brewster), by the substitution of lenses for mirrors, that it has become the popular instrument which we now see it, but it is not more suggestive than it always was of the wonderful adaptations of the sense of sight.]

(581.) [I have already (art. 471) expressed my regret that the limits of this Essay prevent me from devoting a section to the physiological part of Optics; but I cannot close the chapter without at least

CHAPTER VI.

HEAT, INCLUDING SOME TOPICS OF CHEMICAL PHILOSOPHY.

§ 1. BLACK.—Latent and Specific Heat.—Irvine.—Hutton.—Doctrines of Heat applied to some Natural Phenomena.

(582.) Heat considered in the 18th century as a branch of chemistry.

Down to the close of the 18th century, the science of Heat was studied and advanced mainly by chemists, and it was in all respects treated as a branch of Chemistry; a position of which we still find traces in the introduction of the doctrines of heat (even of radiant heat) into most of our approved treatises on Chemistry. This circumstance brings us, in this chapter, into close contact with the most illustrious chemical names of the second half of the last century and of the first years of the present. Such were Black, Cavendish, Lavoisier, and Dalton. Of the last and two first it may be doubted whether they were not as prominent discoverers in Physics as in Chemistry. Davy occupies a similar position. It was not, indeed, until the 19th century had made some progress that Chemistry assumed a strongly distinctive position of its own, and began to attain that large development and complex character of detail which render it a science now hardly accessible to those who do not devote to it their almost undivided attention. In the days of Black and Cavendish it was otherwise; and in the first section of the present chapter I shall attempt to give an outline of the characters of the very remarkable men who then advanced simultaneously the doctrines of Physics and Chemistry; referring, of course, chiefly to the former, but not entirely to the exclusion of the latter portion of their researches, particularly with respect to the atomic and gaseous theories of Dalton, which have a strongly physical aspect. I have

elsewhere noticed the barrenness of the greater part of the 18th century in contributions to the experimental sciences; the temptation is therefore the greater to dwell a little even on the personal history of men so celebrated and influential as Black, Cavendish, and Dalton.

JOSEPH BLACK was born at or near Bordeaux in France, in 1728. His biography, little eventful and almost exclusively academic, has been recorded in some detail by his companions and friends Adam Ferguson and John Robison (the former of whom was a relation), in the preface to the posthumous publication of his Lectures on Chemistry. It is sufficient for me to state that he entered the University of Glasgow as a student in 1746. Being destined for the medical profession, he removed in 1750 or 1751 to Edinburgh, where he benefited especially by the lectures of Cullen, a most eminent physician, and the author of a beautiful experiment on the cold produced during evaporation. Before Black graduated (in 1754) he had entered upon a course of chemical experiments connected with the causticity of many earthy bodies, which ended in his first (and perhaps most famous) discovery of the existence of fixed air or carbonic acid gas as an essential constituent of marble and other solids, together with a train of important consequences. Few inaugural dissertations have been so interesting to science as that on Magnesia, printed at Edinburgh in 1754, which contained these results. But on this purely chemical question we will not enlarge.

¹ On Mr Stokes's experiments, see Phil. Trans., 1852-53; and Proceedings of the Royal Institution.

The discoveries of Black with which we are chiefly concerned are those of latent and specific heat. The former, at least, is Black's sole and exclusive property. When we look back to the state of the science of heat in the first half of the 18th century, we are surprised at the exceeding slowness of its progress. The thermometer had been improved by the use of the freezing and boiling temperatures of water for its fixed points, and the adoption of mercury in its construction by Fahrenheit; the correspondence of its scale with true increments of heat had been tested, though imperfectly, by Brooke Taylor; and Dr Martine of St Andrews had published an ingenious work (the best of its period) on the expansion of different liquids, and on some kindred subjects; but in general no great step was made until Black, in 1757, or previously,¹ concluded that during the melting of snow or ice, a great quantity of heat enters into the body without affecting the thermometer in an appreciable degree. The heat therefore spends itself or is absorbed in effecting liquefaction. Black called it latent (as opposed to sensible) heat. He was led to this discovery by the very simple observation of the extreme slowness with which ice is melted by the application of an amount of heat which would have raised the temperature of water to an enormous extent; the thermometer plunged in the thawing ice remaining stationary until it is entirely reduced to water. When that occurs, heating immediately commences according to the usual laws. One of Black's original experiments clearly illustrates his mode of procedure. He suspended equal weights of ice at 32°, and of water as near as might be at the same temperature, in two thin glass vessels 18 inches apart, in a spacious room having a temperature of 47°. The vessel containing water rose 7° in temperature in half an hour, but the equal weight of ice had not wholly melted, nor had its temperature even slightly increased until 21 half hours had elapsed. Whence Black concluded (approximately) that as much heat is requisite merely to thaw ice as would raise an equal weight of water through $7\; \backslash times\; 21$ or its amount. 147°; a result almost correct, although the experiment in this form is manifestly not unexceptionable. The converse process of freezing shows how prolonged must be the application of cold to discharge water at 32° of its latent heat, or heat of liquefaction, and to convert it into ice.

Nothing is more admirable in these results than the light they throw upon certain natural processes. Were instant liquefaction the result of the smallest application of heat to snow at 32°, we should many times a year be the victims of uncontrollable floods; and did water instantly become ice on its temperature reaching the freezing point, our lakes and rivers would be rapidly consolidated to the very bottom on

occasion of every frost, and animal life would be impossible.

The analogy of the gradual formation of steam in boiling, to the gradual liquefaction of ice, was so latent as to lead Black to conclude, without any special experiment, that a great deal of heat becomes latent during the conversion of liquids into vapour. It appears to have been in 1762 that he attempted to determine roughly the amount, by a method similar to that just described for the heat of liquefaction. He compared the time required under the action of a uniform source of heat to raise the temperature of a certain quantity of water from 50° to the boiling point, with that required to boil it away; and inferring that heat was continually combining with the water at the same rate, he estimated the latent heat, or heat of vapour, to be as great as would have raised the temperature of the water had that been possible by 810°. This was in October 1762;² about two years later his pupil, Irvine, made experiments under his direction with a still and refrigerator, by which he estimated the heat given out by steam during its reconversion into water. Mr Watt prosecuted the same inquiry soon after.³ How happily Mr Watt applied the doctrine of latent heat thus brought experimentally under his notice, to the improvement of the steam-engine, has been already recorded.

One thing strikes us very much on a review of these discoveries of Black; it is the great interval which separates the clear perception of a fact from the explanation of it; or, in other words, from the clear expression of the more general fact which embraces it, although when once given, the explanation may seem to be almost expressed in the very enunciation of the fact. Nearly a century before Black, the lynx-eyed Hooke had noticed that water during the process of congelation remained unaltered in temperature, and that the same takes place when it boils, and this observation had been numberless times verified in ascertaining the fixed points of thermometers; yet no one before Dr Black had even guessed that the heat which enters into ice during liquefaction, and into water during vaporization, remained as it were a constituent of the water and the steam thus formed, and could at any time be recovered by a converse process. Numberless instances occur in the history of science in which it requires the utmost attention to appreciate the importance and difficulty of drawing such seemingly obvious conclusions.

The other great discovery of Black, Specific Heat, was in like manner a correct interpretation of a fact already known. Boerhaave and Fahrenheit had found that when quicksilver and water were mixed together, their temperatures being different, the heat of the mixture was not, as might have been expected, the average of that of its ingredients, whether

the volumes or the weights of those ingredients were equal. "Quicksilver, whether it were applied hot to cold water or cold to hot water, never produced more effect in heating or cooling an equal measure of the water than would have been produced by water equally hot or cold with the quicksilver, and only two-thirds of its bulk." Of course, when equal weights were used, the inequality of effect was still more striking, for, from the great density of quicksilver, it required no less than 30 times the weight of the water mixed with it in order that it should contribute in an equal degree to the production of a mean temperature. Strange to say, the interpretation of this important experiment remained to be made by Dr Black after half a century. He taught that temperature is an effect of heat, which is neither the same in all bodies nor in the same body under differing circumstances; that the superior effect of the water to the mercury in determining the temperature of the mixture was caused by the fact, that it is the nature of quicksilver to require a smaller amount of heat to raise its temperature through one degree than an equal volume or weight of water would require under like conditions. Black made many experiments to satisfy himself of the constancy of this property in bodies; and with the assistance of Irvine,¹ probably ascertained its numerical value (the amount of heat necessary to raise an unit of mass of water through one degree of temperature being the standard) in different cases; but he left the subject chiefly in the hands of that observer and of Watt. The former gave the name of capacity for heat to this property, which was afterwards more happily termed specific heat by Gadolin, who made many experiments on the subject, as did also Cavendish and Adair Crawford whose theory of animal heat, published in 1778, turns entirely on this property of bodies.

(589.) Irvine had the merit, such as it was, of proposing a theory on which his term capacity for heat was principally founded, which occasioned for many years after, although now comparatively forgotten, much discussion. He assumed that the changes in the temperature of bodies, whether by alteration of their mechanical condition or by chemical combination, were due to a change in the capacity for heat of the substance or the mixture. He also assumed that the total amount of heat contained in a body is proportional to the amount of heat necessary to raise its temperature through one degree (for example, a pound of mercury contains altogether one-thirtieth part of the entire heat contained in a pound of water). From these principles he deduced the temperature of absolute zero or privation of heat as follows:—When sulphuric acid and water are mixed together, the temperature rises. This rise, according to him, is caused by the capacity for heat of the mixture being less

than the average due to its ingredients. He has, therefore, got the ratio of the whole heat in the bodies before mixture and after mixture. He has also got the number of degrees of temperature corresponding to this difference. Having the ratio of these quantities, and also their difference, the quantities themselves, or the whole amount of heat expressed in degrees of temperature before and after mixture, become known, and the temperature of absolute privation of heat is also known. He applied the same reasoning, with great ingenuity, to explain the latent heat of liquids and vapours, which he ascribed to their increasing capacities. He thence deduced other values for the absolute zero; but whereas all these determinations ought to have agreed, at least approximately, they were found by later experimenters, especially by Lavoisier and Dalton, to differ so widely—even by several thousand degrees—that, since the time of the latter, this ingenious theory has been nearly abandoned, at least as far as the search after the zero is concerned; although undoubtedly change in the specific heat of bodies is often an important element in determining their temperature.

To return to Dr Black. From 1756 to 1766 he filled the chair of Chemistry and Medicine at Glasgow, where he also practised as a physician. In 1766 he left Glasgow for Edinburgh, to succeed Dr Cullen as Professor of Chemistry—a position which he held, with great credit to himself and with benefit to the University, till his death in 1799. His health, during the greater part of that time, was feeble, owing to a pulmonary affection, which often interrupted his lectures, and, it is stated, prevented him from engaging in severe study without immediate injury. Though he published one or two papers during these thirty years, they were of comparatively trifling importance. His influence on science was chiefly exerted through the medium of his pupils and of his intercourse with general society. His lectures are described by those who had the good fortune to hear them as inimitable of their kind—grave, dignified, and so interesting as to rivet the attention. "Perfect elegance as well as repose was the phrase by which every hearer and spectator naturally, and as if by common consent, described the whole delivery."² It is probable, also, that in his private intercourse with his pupils, he inspired them with that love of research which distinguished his own early days, and his taste for neat and accurate experiment could hardly fail of being imparted to a certain extent. Yet we may be permitted to regret that his constitutional indolence, almost apathy, had perhaps as great a share as bad health in the interruption of his career of discovery. Even when quite a young man, his most interesting conclusions were so gradually evolved, that he himself had difficulty afterwards in fixing their date; and some of them were delayed even for years, until he found time to

¹ Sir J. Leslie has, I think inadvertently, given the credit of the discovery of specific heat to Dr Irvine.

² Lord Brougham.

make the requisite experiments. So cool a temperament was not likely to grow warmer as age advanced. Almost as indifferent to the honours of discovery as his stoical contemporary Cavendish, unlike him he enjoyed in a high degree intercourse with the congenial society which Edinburgh at that time afforded. He loved to converse infinitely better than to write; especially when he could converse intimately with such men as Adam Smith, David Hume, Adam Ferguson, Principal Robertson, Dr John Robison, John Home, Clerk of Eldin, and Dr James Hutton.

HUTTON was perhaps Black's dearest friend, and may be mentioned here (episodically) as no ordinary thinker in Natural Philosophy, as well as in Geology and Metaphysics. Besides the "theory of the Earth," which will ever bear his name, and which, after various transmutations in name and form, is now by far the most widely prevalent, his theory of Rain was an ingenious and important speculation. Other branches of Meteorology also claimed his attention, particularly, as might have been expected, those which are connected with the temperature of the earth. He was one of the first who drew conclusions from the temperature of springs, with regard to change of climate due either to increased latitude or to increased height above the sea. His hygrometer, in which the dampness of the air was estimated by the coolness due to evaporation, was unquestionably the first suggestion of a method now in general use. His ideas on the constitution of matter were bold and ingenious, though not on all points tenable. They resembled those of Boscovich, though independent of them. He published a voluminous treatise on several subjects in Natural Philosophy, and a still more formidable one on the Principles of Knowledge, neither of which attracted much attention at the time, and have been long forgotten; yet it is not unlikely that some of his speculations in metaphysics might be worth the labour of re-

examination. His friend and commentator, Playfair, (whose style was as remarkable for perspicuity as Dr Hutton's was the contrary) has drawn the following lively contrast between the characters of Hutton and Black, which may properly conclude this notice:—
"Ardour and even enthusiasm in the pursuit of science, great rapidity of thought, and much animation distinguished Dr Hutton on all occasions. Great caution in his reasonings, and a coolness of head which even approached to indifference, were characteristic of Dr Black. On attending to their conversation, and the way in which they treated any question of science and philosophy, one would say that Dr Black dreaded nothing so much as error, and that Dr Hutton dreaded nothing so much as ignorance; that the one was always afraid of going beyond the truth, and the other of not reaching it. The curiosity of the latter was by much the more easily awakened, and its impulse most powerful and imperious. With the former, it was a desire which he could suspend and lay asleep for a time; with the other, it was an appetite that might be satisfied for a moment, but was soon to be quickly renewed. . . . Each had something to give which the other was in want of. Dr Black derived great amusement from the vivacity of his friend, the sallies of his wit, the glow and original turn of his expression; and that calmness and serenity of mind which, even in a man of genius, may border on languor and monotony, received a pleasing impulse by sympathy with more powerful emotions."¹

Black died on the 6th December 1799.² His death, as recorded by his kinsman, Adam Ferguson,³ was one of the most touching on record. It succeeded his customary state of health by an interval inappreciably short, and, as appeared by the accompanying circumstances, without the slightest physical emotion. The philosophic composure of his whole life was mirrored in the serenity of its close.

§ 2. CAVENDISH.⁴—His Singular Character and Attainments—Eminent Chemical Discoveries—Observations on Heat and on other Branches of Physics—LAVOISIER—The Calorimeter—Theory of Combustion and of Oxidation.

Cuvier has justly remarked, in his biography of Cavendish, that he had to struggle in his scientific career against obstacles much more rarely encountered, and perhaps less easily overcome, than those

which beset the progress of genius cramped by poverty and neglect. Cavendish was the descendant of one of England's noblest families, and he was likewise the possessor of enormous wealth; yet neither of

¹ Playfair's Biographical Account of Dr James Hutton, Works, vol. iv.

² Muirhead's Correspondence of James Watt. Intro., p. xxii.

³ Preface to Black's Lectures, by Robison, p. lxxiv.

⁴ I find an apology almost necessary for introducing at some length the biography of Cavendish into a chapter professedly on Heat, his positive discoveries connected with which were less notable than in some other departments. But besides that his position in the first rank of chemists naturally indicates his place to be between Black and Dalton, I felt a wish to bring out the relief of the striking intellectual characteristics of those three remarkable men, by placing them in juxtaposition. I may add that these three sections were the earliest written of this Dissertation, at a time when I had hoped to interweave into its composition more of the purely biographical character of each period of scientific history than I found it afterwards practicable in all instances to carry out. I trust, however, that it may be found a not unwelcome variety amidst the abstruser details of science. In the case of Cavendish, too, so various are his claims on our notice, that it was inevitable to recur to them in different chapters, especially in those on Astronomy and Electricity. It was not, therefore, really material under which head the more strictly personal details were given to which I have alluded.

these powerful temptations could withdraw him even for an hour from the course of study which he had marked out; and which constituted for him at once labour and relaxation, the end of living, and almost life itself.

(594.) His personal history.
The Honourable Henry Cavendish, son of Lord Charles and Lady Anne Cavendish, was born at Nice, October 10, 1731.¹ He entered St Peter's College, Cambridge, 24th November 1749, where he resided during the usual terms for above three years, when he ought naturally to have graduated, which, however, he never did. That he had pursued at least his mathematical studies with ability and success is, however, nearly certain from the firm hold which he ever after retained of them. He joined the Royal Society of London in 1760, and published his first paper in their Transactions in 1766. From the time of his leaving Cambridge for some years his personal history is not known, though he probably resided in London.

(595.) His wonderful range of scientific knowledge.
The subsequent history of Cavendish is the history of his studies and his discoveries. The latter were published in concise memoirs, written with scrupulous precision, and all printed in the Philosophical Transactions. If collected (which they have not been) they would fill but an insignificant volume, yet include all the requisites to establish a first-rate reputation. His studies no doubt were enormous, for they occupied every disposable moment of a life prolonged almost to fourscore. They may be guessed at (in addition to the published results) from his manuscript remains, a few of which have been edited recently, but the larger part remains in manuscript in the possession of his heirs. Few if any branches of exact science were unfamiliar to him; and his published papers include astronomy, mechanics, electricity, heat, chemistry, and meteorology, besides which he cultivated mathematics and geology. His reputation is one of those which may be called in a peculiar sense European or universal, because it marked a great epoch in science to which the publication of his writings materially contributed. That epoch was when chemistry became a science of weight and measure. Cavendish was a weigher and measurer almost by nature, and entirely so by habit. It appears from his note-books that he took the most scrupulous pains to ascertain and record the quantities of the ingredients employed in every experiment, even though they might be immaterial to the result; and his whole life was methodical in the same proportion. The immense importance of Cavendish's labours to the progress of science in his day is found in the unanimous testimony of his contemporaries; and notwithstanding the extreme retirement in which he lived, and the rarity of his appearances as an author, he was generally regarded as perhaps the leading man of science in England of

his day; and his good opinion was considered by contemporary philosophers as their highest praise. Sir Humphry Davy's opinion of him (recovered and published by Dr Davy from a manuscript lecture) represents that of such of his countrymen as were qualified to form one, and the writer of it was usually fastidious in his judgments of others:—"It may be said of Mr Cavendish, what can perhaps hardly be said of any other person, that whatever he has done has been perfect at the moment of its production. His processes were all of a finished nature. Executed by the hand of a master, they required no correction; and though many of them were performed in the very infancy of chemical philosophy, yet their accuracy and their beauty have remained amidst the progress of discovery, and their merits have been illustrated by discussion, and exalted by time."

(596.) Chemical discoveries of Cavendish.
After this eulogium of so competent a judge, I shall be satisfied by a simple enumeration of those chemical discoveries which were the most splendid results of his career, but which hardly come within the immediate scope of this dissertation. To him we are mainly or entirely indebted for the knowledge of hydrogen as a distinct elastic fluid or gas; of the exact constitution of the atmosphere, and the wonderful constancy of its ingredients; of the composition of nitric acid; and finally, according to the opinion of most persons (at least till lately), of the non-elementary nature of water, and of its precise constituents. This last and crowning discovery has indeed been contested, and made the subject of a prolonged and bitter scientific controversy, which hardly could be said to exist until the contemporary generation who witnessed the facts, and also the succeeding one, had passed away. For Cavendish received until his death, and for nearly thirty years after, the unquestioned tribute of at least the primary merit in so great a step in science. The topics more peculiar to the present essay (sufficiently extensive besides) enable us to dispense with the unwelcome task of once more analyzing a controversy purely personal, and which has almost filled volumes. I shall content myself with stating two considerations, which must have great weight in turning the balance of the evidence (supposing it balanced) in Cavendish's favour. The first is founded on the behaviour of Watt himself; the second on the known character of Cavendish.

(597.) As to the former consideration, I would remark that Watt withdrew the letter to Priestley which he had submitted to the Royal Society and which contained controversy on discovery of the composition of water.² The position of causes of this suppression are candidly stated by Mr Muirhead,³ in which he states that he "thought

¹ Life of Cavendish, by Dr George Wilson; a valuable biography, which has been printed in the series of publications of the "Cavendish Society," and thus unfortunately has had but a limited circulation.

² Correspondence on the Composition of Water, p. 30.

³ Ibid. p. 52.

it prudent to delay the publication until he should have considered it [the theory of the composition of water] more maturely, and have made some experiments to determine the proof or falsehood of it.¹ This, it must be owned, was not the language of a man who had acquired that amount of conviction which is needful,—not to broach a theory,—but to hold fast by it against all opposition. With the caution which formed part of his character, Watt was unwilling to hazard his reputation by adhering to a doctrine which appears at the time to have received no support from his scientific friends, especially Dr Black. Having read the correspondence published by Mr Muirhead, I cannot doubt that Watt, whatever his private opinions might continue to be, would never have urged his views on an unwilling public, but would have finally suppressed the letter to Priestley, had not the experiments and claims of Cavendish at home, and of Lavoisier in France, reanimated all his zeal for the assertion of his opinion. This indeed Watt ingeniously admits in the letter to Sir Joseph Banks just cited, where he states that the fact of similar theories having since been supported by philosophers of first-rate abilities, removed his second objection to publication. As the suppression of his paper would have relieved Watt of all the responsibility of error, it seems impossible to allow him the advantage—of which that suppression deprived him—of anticipating the date of his matured conviction; and to this conviction we have his own evidence that Cavendish's publication as well as certain additional experiments of his own influentially contributed. Watt, in after life, may be said to have tacitly relinquished to Cavendish the honour which, in the first irritation of the conflict of their claims, he showed no disposition to do; it is, therefore, reasonable to infer that, on reflection, he saw good reasons for doing so. By this I mean that he suffered judgment to be passed in favour of Cavendish's claim in the writings of many of his eminent contemporaries, without attempting publicly to correct the all but universal impression which they made. In one instance, he almost homologated this adverse judgment:—In the article on Steam, written by Robison, and revised by Watt in his last years and after Cavendish's death, this passage appears,—“This is fully evinced by the great discovery of Mr Cavendish of the composition of water:”² from which it must be concluded, first, that Robison, the intimate friend of Watt, and the almost chivalrous defender of his fame, believed Cavendish to be the true discoverer;² secondly, that Watt, in commenting on this article in 1814, permitted the fact to be thus transmitted to posterity. For, in

his numerous animadversions on other parts of the same papers, he gives (as I have pointed out in the note to art. 318) free expression to the sensitiveness which he felt lest Dr Black should derive any credit to which he was not entitled in connection with the steam-engine, but he suffers the passage just quoted to pass without remark. Such being the case, and waiving all purely chemical discussion, I am of opinion that Watt's friends should have left the matter as he was content to leave it.

With reference to the argument from Cavendish's character, I would remark that whilst the claim of Watt cannot be maintained without impeaching the honour and integrity of his rival, and showing that he stooped to subterfuge in order to appropriate to himself a discovery due to another—it would yet be difficult to find in the whole range of scientific history (without excepting the venerable name of Newton), an individual so devoted to knowledge for its own sake, so indifferent to the rewards of discovery, so averse to the publication of what he felt to be important, and knew to be original, so insensible to the voice of praise when applied to himself, so ardent in acquainting himself with the labours of others, and so liberal in assisting them. Such a man was Cavendish; and that he should stoop even to the common artifices of little minds for exalting his own reputation at the expense of others, would itself be incredible; how much more the insinuation (grounded solely on alleged circumstantial evidence) that in doing so he disregarded the plainest dictates of honour and justice.

Black and Cavendish were nearly contemporaries, the former having been born only three years earlier. But I doubt whether these two men, so congenial in their studies, so different in almost every circumstance, whether of fortune or temperament, ever met. They had this, however, in common, that they published their discoveries and observations with reluctance, and were fastidious in the manner of doing so. Black's experiments, both in chemistry and on heat, preceded those of Cavendish, and with regard to the latter subject (heat) it is probable that Cavendish pursued investigations and made original experiments, in which he had been unknowingly anticipated by the Scotch professor. His researches on latent and specific heat (though he did not employ these terms) appear to have been subsequent to Black's, but to have preceded those of Crawford and Wilecke; and it is not impossible that his was the earliest, or one of the earliest, determinations of the amount of heat absorbed in the conversion of water into steam, for which he obtained various results between 923° and 982°, a mean of which would be very near the

¹ Robison's Mechanical Philosophy, ii., 21.

² The article WATER in the early editions of the Encyclopædia Britannica has been quoted as the production of Robison, of which there does not appear to exist any proof, whilst the probability, as shown in the text, lies all the other way. I learn upon the best authority that the proprietors of this Encyclopædia have no clue to the authorship of that article, and that it is not included in the lists of Robison's known contributions. The part relating to the present question was expunged in the Fourth Edition, and a reference made to the article Chemistry where Cavendish received the credit of the discovery.

truth. But it is needless to dwell upon these observations, however original, because they were voluntarily suppressed by the author, and have only recently been brought to light from his manuscripts.¹ What he did publish in connection with this subject was a paper on the construction and graduation of meteorological instruments, especially thermometers, and others on the temperature at which mercury freezes, and on freezing mixtures. The former of these papers was, as might be expected, far in advance of its age in the degree of exactness which was shown to be attainable in the construction of thermometers, and scarcely even now can it be considered as obsolete. The papers on freezing mercury finally corrected the exaggerated notion at first entertained of the extreme cold at which that metal becomes solid, and also contain valuable views on the subject of congelation, and fixed the latent heat of water at 150°. He calls this, "generation of heat" during liquefaction, objecting to Black's term as relating "to an hypothesis, depending on the supposition that the heat of bodies is owing to their containing more or less of a substance, called the matter of heat; and as I think Sir Isaac Newton's opinion that heat consists in the internal motion of the particles of bodies much the most probable, I chose to use the expression, heat is generated."²

Two of Cavendish's most important researches refer to the attraction and density of the earth, and to the mathematical theory of electricity. The former (which, in principle, was derived from the Rev. John Michell) has been already analyzed in the chapter on Astronomy (art. 156). The latter will be more conveniently referred to in our chapter on Electricity.

Cavendish's publications extended over the greater part of his active life, but those on chemistry and electricity, on which his fame principally depends, do not extend beyond the year 1775; the date of his paper on the density of the earth is 1798. He died 24th February 1810, at the age of 79. He appears to have exercised scarcely less influence by his general devotion to science, than by his specific discoveries, great and original as they were. In 1782, when Playfair met him incidentally in London, he described him as being generally looked up to as one possessed of talents confessedly superior, and as the only member of the Royal Society who then united the knowledge of mathematics, chemistry, and experimental philosophy. The absolute devotion of his life to inquiries the most abstractly scientific, whilst he showed an entire indifference to the luxuries which his wealth might have commanded, and the social station to which his birth entitled him, could not fail to inspire respect for his character, as well as to obtain the homage of mankind for pursuits so dignified and so generally disregarded. Many

curious anecdotes are related of the annoyance which the inevitable accumulation of his unspent income occasioned. He no doubt would have distributed more liberally what he so little valued, but for the amount of time and inquiry which such a course must have compelled him to withdraw from his beloved pursuits. Some instances of his generosity are on record, and others, no doubt, will never come to light. M. Biot's epigrammatic description of him will probably long remain applicable,—"Il était le plus riche de tous les savans, et probablement aussi le plus savant de tous les riches."

This isolation of interest was doubtless due, quite as much to a constitutionally morbid temperament, as to a real misanthropy. He avoided even the most casual intercourse with his fellow men, excepting only when it was likely to bear the immediate fruit of scientific information. He almost never visited his relatives, and his heir paid him a visit of a few minutes once a year; but he frequented regularly the social meetings of the Royal Society Club, and the evening reunions of Sir Joseph Banks. But he came, not to participate, but to increase his stores; if he spoke, it seemed to be by inadvertence, and he was silenced by a question, or even by a look. "A sense of isolation from his brethren made him shrink from their society, and avoid their presence; but he did so as one conscious of an infirmity, not boasting of an excellence. He was like a deaf-mute sitting apart from a circle, whose looks and gestures show that they are uttering and listening to music and eloquence in producing or welcoming which he can be no sharer. . . . He was one of the unthanked benefactors of his race who was patiently teaching and serving mankind, whilst they were shrinking from his coldness, or mocking his peculiarities. . . . Such was he in life, a wonderful piece of intellectual clock-work, and as he lived by rule he died by it, predicting his death as if it had been the eclipse of some great luminary, . . . and counting the very moment when the shadow of the unseen world should enshroud him in its darkness."³

I shall only add, that Cavendish was elected one of the eight Associates of the French Institute in 1803. This is a distinction perhaps the highest, of a formal kind, to which a scientific man can aspire, and was given at a time when, as an Englishman, he must have felt it to be peculiarly honourable.

The philosophical character of Cavendish resembled in many respects that of Newton; and with but a slight modification of its secondary ingredients, he might have been, perhaps, another Newton in experimental physics. His singular incommunicativeness, and the absence of a laudable ambition to perpetuate his name by the establishment of great theories, are perhaps the main reasons why his reputation, except in chemistry, did not stand yet higher than we find it.

¹ By the Rev. W. Vernon Harcourt in British Association Report for 1839. Compare Wilson's Life of Cavendish, p. 446.

² Phil. Trans. 1783.

³ Wilson's Life of Cavendish.

A man indifferent to external relations pursues even his studies at a disadvantage; and the patient labours of so long a life devoted to a single object, would perhaps have told to greater effect had he published the results more frequently, and had he communicated more freely with those qualified to interchange their views with his. His excellent mathematical education, and his unusual skill in experiment, combined with a habit independent of either, but not less valuable, of patiently drawing inferences, might have placed him in the first rank as a discoverer in Heat, in Electricity, or in Optics,¹ each of which sciences was so soon to take a surprising step in advance. In several of his important researches, he was more or less anticipated; a circumstance which his cold nature would perhaps scarcely have allowed him to make an effort to prevent. Black preceded him in most of his excellent experiments on heat; Epinus in his theory of Electricity; Watt was at least close on his traces in suspecting that water consists of oxygen and hydrogen; and the admirable experiment on gravitation had been devised by his friend Michell, and was not improbably recalled to his recollection, by the happy use of the torsion balance by Coulomb. It is given but to a few to achieve great discoveries, nor is the longest life always the most productive. Cavendish had his share, and some of the most considerable of these were even made later in life than is usual amongst experimentalists.² It is no mean eulogy of him to say, that the purity and ardour of his pursuit of truth were never exceeded, and that had he been more ambitious of praise, he might have stood as pre-eminent in mathematical physics as he did in chemistry.

(605.)
Lavoisier. ANTOINE LAURENT LAVOISIER stands in intimate connection with Cavendish, as well by the nature of his pursuits generally, as by the brilliancy and importance of his chemical discoveries, which were nearly contemporaneous with those of Cavendish. He was born in 1745, and suffered by the guillotine on the 8th May 1794, without even the shadow of a misdemeanour. He was attached to the sciences of Heat and Chemistry, which he prosecuted with admirable success. He happily availed himself of the discoveries of Black, Priestley, and Cavendish, as well as his own, to establish the important chemical theory which has immortalized him. It is to be regretted that he was not always just in citing the English writers from whom he so freely borrowed. Such looseness was, however, common at that period, and (unfortunately) has continued to be so in France even to our own time.

(606.)
His contributions to heat and chemistry. Lavoisier's more important papers may be classed under two heads; those referring more immediately to the subject of Heat as a branch of physics; and those of a more strictly chemical character, principally in support of the "Oxygen-Theory," and con-

sequently also intimately connected with the doctrine of heat considered in reference to its most ordinary source, Combustion.

He published in 1772 a paper on the Latent Heat (607.) of Water, and some years afterwards one on the Latent Heat of Steam, which, in general, merely reproduce the views of Black. In the memoirs of the Academy of Sciences for 1780 (published 1784), we find an important paper on Heat written in conjunction with Laplace, in which the calorimeter is described, though not under that appellation, together with its applications and their results. The principle of the calorimeter is too well known to require to be detailed here; but the authors of the joint memoir refer, with commendable precision, to the previous labours of Wilcke of Sweden, who first employed the melting of snow to measure the quantities of heat given off by bodies in cooling. To Laplace and Lavoisier, is, however, due an important addition which could alone impart any value to the results,—that of the external chamber of ice which prevents fusion taking place by the contact of the external air and by radiation. The French philosophers were not so successful in eliminating the other source of inaccuracy specified by Wilcke, which arises from the difficulty of drawing off the whole of the melted water. Sir John Herschel has of late years proposed what seems to be an important improvement on the calorimeter, by filling the interstices of the snow or ice with water, and estimating the quantity of the former melted by the contraction of volume of the compound mass. I am not aware that it has been as yet practised. I have stated in Art. (88) one ground on which the idea of the calorimeter (so far as not anticipated by Wilcke), may be probably ascribed to Laplace. Another is to be found in the fact that, in the opening of the description of the method in the paper which we are considering, Laplace writes of himself in the first person.

Of the memorable revolution which Lavoisier introduced into chemistry, more immediately in connection with the subject of combustion, I cannot be expected to speak here at length. It is well known that the early chemists entertained more correct views as to the calcination of metals than those prevalent during the greater part of the eighteenth century under the influence of Stahl's theory of Phlogiston; and that Lavoisier, in the first instance, only led chemists back into the right road by insisting that the increase of weight observed when metals are calcined in air, must be due to some ponderable substance associated with the metal and derived from the air, and not to the escape of an imaginary spirituous substance, endowed with positive levity, and termed Phlogiston. But it required the progress which had already been made in pneumatic chemistry by Black and Priestley, and especially the discovery of oxygen

¹ This last subject seems to have been comparatively indifferent to him.

² His discovery of the composition of water was made when he was about fifty years of age.

by the latter (a fact sparingly alluded to by Lavoisier), to give to the true theory a stable foundation. That metals calcine, and that flames burn by the aid of the vivifying principle of the atmosphere abstracted from it, and which adds its weight to the compounds produced, was the primary step made by Lavoisier; to which, by cautious, yet rapid inductions, he added the knowledge of oxygen as the usual (he believed the sole) acidifying principle, and demonstrated the true nature of carbonic acid. To these various phenomena thus happily reconciled, he added the theory of respiration, confirming it by the effects observed in air which has undergone that process. These, and many other consequences of the "oxygen theory," were developed in a numerous series of admirable memoirs. The reception of it was anything but in-

stantaneous; and the hesitation and delay which occurred, enable us, as Dr Whewell has well remarked, to estimate the force of mind which was required to promulgate the theory,—as the subsequent course of discovery, and its infinite applications in practice, best attest its importance.

Lavoisier was still occupied in extending the conclusions of his chemical doctrines, when he was overtaken by the unprovoked sentence to a violent death. Like Archimedes, he begged a short respite for the completion of experiments in which he was immediately engaged, but he was silenced by the brutal reply that "the Republic had no need of philosophers." He left a name equal to any in the science of his time, and adorned by the memory of public and private virtues.

§ 3. DALTON.—Theory of Gases and Vapours.—Law of Expansion by Heat.—Atomic Theory of Chemistry.—GUY-LUSSAC.

(610.)
Dalton—
his early
circum-
stances;

JOHN DALTON, the chief author of the theory of chemical equivalents or the Atomic Theory (as he preferred to call it), and of many important researches on the constitution of elastic fluids, was born at Eaglesfield, near Cockermouth in Cumberland, on the 5th of September 1766.¹ His parentage was humble, and his family belonged to the sect of Quakers, whose tenets he, to the close of his life, professed. Had we wished to invent a striking antithesis in the personal histories of the cultivators of science, or to illustrate merely the various soils on which rich crops of discovery may be reared, we could scarcely have imagined more striking contrasts than in the social positions and advantages of Black, Cavendish, and Dalton—three of the names which we have selected in illustration of the history of physics of their age. Cavendish we have seen connected with one of the noblest families in England, and wealthy almost beyond the dreams of the covetous; spending his life in or near London, and enjoying every facility of direct communication with the first scientific celebrities at home and abroad;—Black, almost the beau idéal of an Academic, not wealthy indeed, but surrounded by all the opportunities of study, of information, and of social intercourse which he desired; passionless almost to a fault; admired by his pupils and friends; enjoying, in short, all the advantages which education and a literary position can afford for the prosecution of a favourite study;—Dalton, on the other hand, poor and hardly winning a well-earned subsistence by private tuition, from the time he was himself a child until near the close of his long career,—with few friends, a scanty education, and a scantier library,—attaining, through his unaided and long almost unheeded efforts, and by means of an

apparatus constructed entirely by himself, a position in the world of science unquestionably not second to that of either of his more highly-favoured contemporaries.

At the age of thirteen he had commenced the arduous office of an instructor; and from 1781 to 1792 pursued the same occupation in a humble sphere at Kendal, where he fortunately became acquainted with Mr Gough, a blind gentleman of some fortune, who devoted his time to the prosecution of science in nearly all its branches, and particularly of mathematical subjects, of which he has left an enduring record in many ingenious papers, published chiefly in the Manchester Transactions and in Nicholson's Journal. He patronized young Dalton, giving him free access to his library and apparatus, and receiving from him in return the benefit of his assistance in prosecuting his experiments. Dalton always recognised (as he had unquestionably good reason to do) the merits of his patron, and the importance of the advantages which he had derived from his advice and example. Indeed, without some such fortunate concurrence of circumstances (and something similar may be noted in the history of nearly all self-educated men), it could hardly have been hoped that Dalton would have been so well grounded in the mathematical principles of at least some branches of Natural Philosophy as he probably was. For, though his discoveries bear mainly on the science of chemistry in the wide sense in which it was then understood, yet geometrical precision is after all their fundamental characteristic; and whether in treating of the constitution of a gas, or of the scale of a thermometer, or of the composition of a salt, it is evident that numbers and ratios were the ideas predominating in

¹ Life of Dalton, by Dr Henry, in the publications of the Cavendish Society. The present section was originally written before the appearance of this biography.

his mind. In this he resembled Cavendish; but he did not resemble him, and in one sense may be said to have surpassed him, in the boldness, we might call it audacity, with which on a frequently too slender foundation of facts Dalton established, to his own satisfaction, by reasoning almost as much a priori as that of the mathematician, comprehensive physical theorems expressing the laws of phenomena.

(612.) Dalton's writings. The more important of these refer to the constitution of gases and vapours, together with their relations to heat, and to the combinations of bodies by chemical affinity. We shall endeavour to give some account of these researches separately. They are to be found detailed in the Memoirs of the Literary and Philosophical Society of Manchester (particularly the fifth volume), and in Dalton's New System of Chemical Philosophy, of which three successive parts appeared in 1808, 1810, and 1827, of which the first is the most original and important.

(613.) Researches on gases and vapours. I. We shall first speak of his researches connected with gases and vapours. It had been noticed by Priestley and others that when gases exercising apparently no chemical action upon one another, are mixed in a confined space, they become, after the lapse of a longer or shorter time, completely intermingled, and that without any regard to even great differences in their density, the particles of the lighter gas diffusing themselves contrary to gravity through those of the denser, and causing them in their turn to ascend. The uniform composition of our atmosphere, in all circumstances and at all heights, is a striking example of this property. In explanation of it, Dalton had in 1801 arrived at the conclusion, "that the particles of one gas are not elastic or repulsive in regard to the particles of another gas, but only to the particles of their own kind." The fundamental experiment on which this singular conclusion was based was the following:—That the quantity of vapour of water (which, when purely elastic, may be considered as a gas) which can exist uncondensed in a given space, depends solely upon the temperature, and is independent of the presence of air and of its own density. Thus water, or any other evaporable liquid, being introduced into a space containing air, or any other gas, of any density, but subject to a constant external pressure, the space in question becomes damp by the evaporation of the liquid; now the amount of the latter converted into vapour depends, according to Dalton, upon the single circumstance that the vapour yielded by it must have the precise elasticity due to its temperature.

Its elastic force being added to that of the dry air, the whole will expand until equilibrium is restored with the constant pressure without, and this will occur as soon as the elasticity of the dry air alone (proper to its increased volume), added to the elasticity of the vapour alone (depending solely on its temperature), are together equal to the pressure which they have to support. In short, to use the precise enunciation of Dalton himself, "in all cases the vapour rises to a certain force, according to temperature, and the air adjusts the equilibrium by expanding or contracting as may be required."¹

The importance of this law (easily verified in the particular case) is readily perceived. Not only did it affect the results of almost every experiment in pneumatic chemistry, but it rendered a new theory of hygrometry indispensable. The older theory of Halley, Leroy, and Franklin was, that the direct affinity between air and water drew up a portion of the latter into the former with the aid of heat, whilst De Saussure (Essai sur l'Hygrométrie, 1783) believed that the conversion of water into vapour took place first and independently by the action of heat, and that it was then drawn up into the atmosphere by the attraction of the gases which compose it. Dalton's experiments show that the air and the vapour mix without the slightest mutual interference or reaction. He founded thereupon an excellent practical method of determining the amount of vapour in the atmosphere. He first formed a table of the elasticities of watery vapour or steam for all temperatures between 32° and 212°; and so simple and accurate were his methods, at least for the lower degrees of the thermometer, that his numbers are still received as amongst the best we have. He operated merely with a carefully constructed barometer, into the vacuum of which he introduced a few drops of water, and raising the temperature by means of a tube embracing it, and which could be filled with water at pleasure, he observed carefully the depression of the mercurial column. The elasticities thus determined commenced with 0.2 inches of mercury at 32° up to 30 inches at 212°. These numbers, consequently, represent the utmost elasticities of vapour which can exist either in air or without air at the corresponding temperature. If we attempt to add more vapour, or to lower the temperature, in either case moisture will be deposited. Hence to find the quantity of vapour in the atmosphere when not absolutely damp, it is sufficient to ascertain the temperature at which it becomes so. This Dalton did by filling a

¹ The exact expression of the effect is this, $v\; =\; \backslash frac\{p\}\{p-f\}$, where $p$ represents the pressure expressed in inches of mercury upon a given volume (equal to unity) of dry air; $f$ the force of the vapour in vacuo at the temperature of experiment, also in inches of mercury; and $v$ the volume which the mixture of air and vapour occupies under the given pressure $p$ after saturation. It is evident that $p-f$ being the pressure due to the elasticity of the dry air apart from the vapour, when we affirm that the volume (1) becomes $\backslash frac\{p\}\{p-f\}$, we in effect affirm that Mariotte's (or Boyle's) law connecting volume and pressure holds true for air which is mixed with vapour, just as though vapour were absent or its space void.

thin glass with cold spring water, or by adding, if necessary, the solid nitrate of ammonia, until the temperature fell so far that dew began to be deposited on the surface. This excellent method constitutes in fact the dew-point hygrometer. It has received various forms from Mr Daniell and others, but the original one is probably the best.

(615.)
Laws of mixed gases.