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 Zuylchen, 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 Physick 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 commendatory 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 emeritus professor in 1761, and died on the 1st of October 1763.

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 farther 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 $3\backslash frac\{1\}\{7\}$ 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 1595 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-

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 formulæ 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 disproportioned 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 Loc 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 angle, 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 Beauve, 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 Bossut 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 28th 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 Bachelet 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, Poincaré, 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, 23, 496, 8128, 33550336, 3359869056, 137438691328, and 2305843003130952128. 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 Iovis, 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 completest work on this curious subject was compiled by the late Professor Mollweide of Leipzig, the ingenious editor of Klügel'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 formulæ. 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 explication 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. Ferreo, 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 $\backslash sqrt\{\backslash phantom\{x\}\}$ (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 complete 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 expence, 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 Hugonot, 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 1770. 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 Vitry 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 toiling 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 Dr 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 Burekhardt, 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 Éléges 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 Fluxents 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 $\backslash mu\; x^\{n-1\}$ or $x$. 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 Edmonton in Middlesex in 1685; 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 formulae.

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 differentiali 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 Peakirk 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 $x$ to that of $x^n$. Assume $v$ a proximate value or state of the variable $x$; then by division $\backslash frac\{x^n\; -\; v^n\}\{x\; -\; v\}\; =\; x^\{n-1\}\; +\; x^\{n-2\}v\; +\; \backslash dots\; +\; x^2v^\{n-2\}\; +\; v^\{n-1\}$, 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 Analysen 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'Hopital, 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 fame 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 Theoræ Turbinum, wherein he showed that every body having a determinate figure, which after combined impulses 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 Philæthon 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, "Eodem mutata recurgo."

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 1705. 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 endued 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 Opuscula 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, exa-

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 Croisic 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 Graduation of Light appeared in 1729, and established his high reputation. His celebrity always increasing, he was appointed in 1735 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 concourse, 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 1758. 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 bellow 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 Encyclopædia.

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 $\backslash frac\{4\}\{3\}$ is $\backslash frac\{1\}\{2\}$, and between $\backslash frac\{4\}\{3\}$ and $\backslash frac\{5\}\{4\}$ it is $\backslash frac\{1\}\{6\}$; which fractions indicate a tone and a semi-tone, the latter differing from unit only by about half the difference of the former. But $\backslash frac\{5\}\{4\}$ may be split into $\backslash frac\{5\}\{6\}$, $\backslash frac\{6\}\{5\}$, and $\backslash frac\{1\}\{6\}$; and hence $\backslash frac\{5\}\{6\}$ and $\backslash frac\{6\}\{5\}$, though not quite equal, are both deemed tones, the former being called the Major and the latter the Minor. The fraction $\backslash frac\{7\}\{6\}$ is therefore composed of $\backslash frac\{5\}\{6\}$, $\backslash frac\{6\}\{5\}$, $\backslash frac\{1\}\{6\}$, 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 $\backslash frac\{7\}\{6\}$, 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 Thaumatrope.

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 Agguinto, 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 princi-

ples, 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 formula $\backslash frac\{1\}\{2\}d$ and $\backslash frac\{1\}\{2\}d\text{'}$, 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. 329.

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. Nollet,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 Nollet had in 1746 drawn a parallel, in which he compared the con-

¹ John Anthony Nollet, 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 conge-lation, 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 conglobating 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 succussions, 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 Oersted 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 Epinus 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 Dunciad:
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 Cabeus, 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 $7\text{'}$, but it mounted to $13\backslash frac\{1\}\{2\}\text{'}$ 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 feruginous 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 $3\text{'}$. 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 $75^\{\backslash circ\}\; 7\text{'}$, and the South Pole in the latitude of $60^\{\backslash circ\}\; 37\text{'}$; assigning to the former a western longitude of $91^\{\backslash circ\}\; 54\text{'}$, and to the latter only $38^\{\backslash circ\}\; 36\text{'}$.

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

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 Burckhardt, 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 $24^\{\backslash circ\}\; 26\text{'}$, 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° or 80°.

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 feruginous 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 retained, 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 Klingenstierna, 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. Klingenstierna 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 esset, ut excogitaretur Photometrum thermometro analogum, quod lumini expositum ejus intensitatem atque claritatem indicaret. He subjoins, Eiusmodi ipse oculus ejus sinit exemplar, quippe pupilla aperta 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 Umbre, 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." (Optics, 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 Alcmair 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 excluded 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 Carthagen, 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 concave surface, 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 corpule 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 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 $98^\{\backslash circ\}$ of Celsius, poured upon an equal weight of snow, had its temperature reduced to $13^\{\backslash circ\}$, and that three parts of water at $40^\{\backslash circ\}$, joined to one of snow, produced the common temperature of $12^\{\backslash circ\}$. These trials gave both the same result, or $73^\{\backslash circ\}$ of Celsius, corresponding to $129.6^\{\backslash circ\}$ 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 $72^\{\backslash circ\}$, 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 $1318^\{\backslash circ\}$ 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 $169^\{\backslash circ\}$, or from $32^\{\backslash circ\}$ to $201^\{\backslash circ\}$, the whole extension below the point of congelation would be $8^\{\backslash circ\}\; \backslash times\; 169^\{\backslash circ\}$, or $1352^\{\backslash circ\}$. 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 $98^\{\backslash circ\}\; -\; 13^\{\backslash circ\}$, or $85^\{\backslash circ\}$, while the snow water gained only $13^\{\backslash circ\}$, or had absorbed $72^\{\backslash circ\}$; in the second experiment the hot water had transferred $3(40\; -\; 12)$ or $84^\{\backslash circ\}$, and consequently $84^\{\backslash circ\}\; -\; 12^\{\backslash circ\}$, or $72^\{\backslash circ\}$, was spent in liquefying the snow. If every precaution were used, the measure of heat absorbed would amount to $75^\{\backslash circ\}$ of Celsius, or $135^\{\backslash circ\}$ 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 Saussure and Pictet, 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

¹ This 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 tin foil 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 inclined 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 LIQUID 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 $\backslash frac\{3\}\{4\}$; the naked ball emitting now $6\backslash frac\{1\}\{2\}$ parts of heat, and the gilt one only $2\backslash frac\{1\}\{4\}$. 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 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 Hygrometer 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 Hygrometer 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 æra 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 Eolipile, 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 collocavi 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 formulae 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 1764 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 20th 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-

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 36, 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 Calandrini, 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 apsides, 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 apsides 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 formulæ 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 formulæ, 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 apercevoir 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 connoissance des mouvemens de la lune amènera 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 excrescence, 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 Fellows' 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 $25\text{'}\text{'}$, 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 $9\backslash frac\{1\}\{2\}$ 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 dissevered 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 depart 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.

... of the smaller stars are found to have
proper motions some of them in an extent of
a few degrees. They appeared to show the
shifting stars according to their places in the
heaven and reminded that while these stars
of two opposite quarters appeared nearly at
the same time and had a similar position, the
other half of the sky appeared to be at a
different time of its appearance and position
in the sky. This is the reason why the
stars in the sky appear to move in a
circle. All these stars are found to be
extremely remote, so that we can only
see their light, and not their
forms. It is distinguished only by the position of
the most powerful telescope. It is found to
be situated in the sky, and is seen
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found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the
other stars, and is seen in the same
direction. It is found to be in the same
direction as the other stars, and is seen
in the same direction. It is found to be
in the same direction as the other stars,
and is seen in the same direction. It is
found to be in the same direction as the

INDEX

TO

THE DISSERTATIONS.