Mathematics are divided into two kinds, pure and mixed. In pure mathematics magnitude is considered in the abstract; and as they are founded on the simplest notions of quantity, the conclusions to which they lead have the same evidence and certainty as the elementary principles from which these conclusions are deduced. This branch of mathematics comprehends:

1. Arithmetic, which treats of the properties of numbers; 2. Geometry, which treats of extension as endowed with three dimensions, length, breadth, and thickness, without considering the physical qualities inseparable from bodies in their natural state; 3. Algebra, sometimes called universal arithmetic, which compares together all kinds of quantities, whatever be their value; and 4. The direct and inverse method of fractions, or the differential and integral calculus, which consider magnitudes as divided into two kinds, constant and variable, the variable magnitudes being generated by motion, and which determines the value of quantities from the velocities of the motions with which they are generated.

Mixed mathematics are the application of pure mathematics to certain established physical principles, and comprehend all the physico-mathematical sciences, as, 1. Mechanics; 2. Hydrodynamics; 3. Optics; 4. Astronomy; 5. Acoustics; 6. Electricity; and 7. Magnetism.

In attempting to discover the origin of arithmetic and geometry, it would be a fruitless task to conduct the reader into those ages of fable which preceded the records of authentic history. Our means of information upon this subject are extremely limited and imperfect; and it would but ill accord with the dignity of a science the principles and conclusions of which are alike irresistible, to place its history upon a conjectural or fabulous basis. But notwithstanding the obscurity in which the early history of the sciences is enveloped, one thing appears certain, that arithmetic and geometry, and some of the physical sciences, had made considerable progress in Egypt, when the mysteries and theology of that favoured kingdom were transplanted into Greece. It is highly probable that much natural and moral knowledge was taught in the Eleusinian and Dionysian mysteries, which the Greeks borrowed from the Egyptians; and that several of the Grecian philosophers were induced by this circumstance to travel into Egypt in search of those higher degrees of knowledge which an acquaintance with the Egyptian mysteries had taught them to look for in that country.

We accordingly find that Thales and Pythagoras were successively under the tuition of the Egyptian priests, and returned into Greece loaded with the intellectual treasures of Egypt. By the establishment of the Ionian school at Miletus, Thales instructed his countrymen in the knowledge which he had received, and gave birth to that spirit of investigation and discovery with which his followers were inspired. He taught them the method of ascertaining the height of the pyramids of Memphis by the length of their shadows; and there is reason to believe that he was the first who employed the circumference of a circle for the mensuration of angles. That he was the author of greater discoveries, which have been either lost or ascribed to others, there can be little doubt; but these are the only facts in the history of Thales of which we have any certain information.

The science of arithmetic was one of the chief branches of the Pythagorean discipline. Pythagoras attached several mysterious virtues to certain combinations of numbers. He swore by four, which he regarded as the chief of numbers. In the number three he supposed that many wonderful properties existed; and he regarded a knowledge of arithmetic as the chief good. But of all Pythagoras's discoveries in arithmetic, none has reached our times except his multiplication table. In geometry, however, the philosopher of Samos seems to have been more successful. The discovery of the celebrated proposition which forms the forty-seventh of the first book of Euclid's Elements, that in every right-angled triangle the square of the side subtending the right angle is equal to the sum of the squares of the two sides containing it, has immortalized his name; and whether we consider the inherent beauty of the proposition, or the extent of its application in the mathematical sciences, we cannot fail to class it amongst the most important truths in geometry. From the same proposition its author concluded that the diagonal of a square is incommensurable with its side; and this gave occasion to the discovery of several general properties of other incommensurable lines and numbers.

During the time which elapsed between the birth of Pythagoras and the destruction of the Alexandrian school, the mathematical sciences were cultivated with great ardour and success. Many of the elementary propositions of geometry were discovered during this period; but history does not enable us to refer each discovery to its proper author. The method of letting fall a perpendicular upon a right line from a given point (Euclid, b. i. prop. 11), of dividing an angle into two equal parts (Euclid, b. i. prop. 9), and of making an angle equal to a given angle (Euclid, b. i. prop. 23), were invented by Cnepidus of Chios. About the same time Zenodorus, some of whose writings have been preserved by Theon in his commentary on Ptolemy, demonstrated, in opposition to the opinion then entertained, that isoperimetric figures have equal areas. Coeval with this discovery was that of the theory of regular bodies, for which we are indebted to the Pythagorean school.

About this time the celebrated problem of the duplication of the cube began to occupy the attention of the Greek geometers. In this problem it was required to construct a cube whose solid content should be double that of a given cube; and the assistance of no other instrument but the rule and compasses was to be employed. The origin of this problem has been ascribed by tradition to a demand of one of the Grecian deities. The Athenians having offered some affront to Apollo, were afflicted with a dreadful pestilence; and upon consulting the oracle at Delos, received for answer, "Double the altar of Apollo." The altar alluded to happened to be cubical; and the problem, supposed to be of divine origin, was investigated with ardour by the Greek geometers, though it afterwards baffled all their acuteness. The solution of this difficulty was attempted by Hippocrates of Chios. He discovered, that if two mean proportionals could be found between the side of the given cube and the double of that side, the first of these proportionals would be the side of the cube sought. In order to effect this, Plato invented an instrument composed of two rules, one of which moved in grooves cut in two arms at right angles to the other, so as always to continue parallel with it; but as this method was mechanical, and likewise supposed the description of a curve of the third order, it did not satisfy the pure taste of the ancient geometers. The doctrine of conic sections, which was at this time introduced into geometry by Plato, and which was so widely extended as to receive the name of the "higher geometry," was successfully employed in the problem of doubling the cube. Menechmus found that the two mean proportionals mentioned by Hippocrates might be considered as the ordinates of two conic sections, which being constructed according to the conditions of the problem, would intersect one another in two points proper for the solution of the problem. The question having assumed this form, gave rise to the theory of geometrical loci, of which so many important applications have been made. In doubling the cube, therefore, we have only to employ the instruments which have been invented for describing the conic sections by one continued motion. It was afterwards found, that instead of employing two conic sections, the problem could be solved by the intersection of the circle of the parabola. Succeeding geometers employed other curves for this purpose, such as the conchoid of Nicomedes and the cissoid of Diocles. An ingenious method of finding the two mean proportionals, without the aid of the conic sections, was afterwards given by Pappus in his mathematical collections.

Another celebrated problem, the trisection of an angle, was agitated in the school of Plato. It was found that this division of the angle depended upon principles analogous to those of the duplication of the cube, and that it could be constructed either by the intersection of two conic sections, or by the intersection of a circle with a parabola. Without the aid of the conic sections, it was reduced to this simple proposition: To draw a line to a semicircle from a given point, which line shall cut its circumference, and the prolongation of the diameter that forms its base, so that the part of the line comprehended between the two points of intersection shall be equal to the radius. From this proposition several easy constructions may be derived. Dinostratus of the Platonic school, and the contemporary of Menechmus, invented a curve by which the preceding problem might be solved. It had the advantage also of giving the multiplication of an angle, and the quadrature of a circle, from which it derived the name of quadratrix.

Whilst Hippocrates of Chios was paving the way for the Hipparchian method of doubling the cube, which was afterwards given by Pappus, he distinguished himself by the quadrature of the lunule of the circle; and had from this circumstance the honour of being the first who found a curvilinear area equal to a space bounded by right lines. He was likewise the author of Elements of Geometry, a work which, though highly approved of by his contemporaries, has shared the same fate with some of the most valuable productions of antiquity.

After the conic sections had been introduced into geometry by Plato, they received many important additions from Eudoxus, Menechmus, and Aristaeus. The latter of these philosophers wrote five books on conic sections, which, unfortunately for science, have not reached our times.

About this time appeared Euclid's Elements of Geometry, a work which has been employed for two thousand years in teaching the principles of mathematics, and which is still reckoned the most complete work upon the subject. Peter Ramus has ascribed to Theon both the propositions and the demonstrations in Euclid. But it is the opinion of some that the propositions belong to Euclid and the demonstrations to Theon, whilst others have ascribed to Euclid the honour of both. It seems most probable, however, that Euclid merely collected and arranged the geometrical knowledge of the ancients; and that he supplied many new propositions in order to form that chain of reasoning which runs through his Elements. This great work of the Greek geometer consists of fifteen books, eleven of which contain the elements of pure geometry, and the rest embrace the general theory of ratios, together with the leading properties of commensurable and incommensurable numbers.

Archimedes, the greatest geometrician amongst the ancients, flourished about half a century after Euclid. He was the first who found the ratio between the diameter of a circle and its circumference; and, by a method of approximation, he determined this ratio to be as 7 to 22. This result was obtained by taking an arithmetical mean between the perimeters of the inscribed and circumscribed polygon, and is sufficiently accurate for every practical purpose. Many attempts have since been made to assign the precise ratio of the circumference of a circle to its diameter; but, in the present state of geometry, this problem does not seem to admit of a solution. The limits of the present article will not permit us to enlarge upon the discoveries of the philosopher of Syracuse. We can only state, that he discovered that the superficies of a sphere is equal to the convex surface of the circumscribed cylinder, or to the area of four of its great circles, and that the solidity of the sphere is to that of the cylinder as 3 to 2. He discovered that the solidity of the paraboloid is one half that of the circumscribed cylinder, and that the area of the parabola is two thirds that of the circumscribed rectangle; and he was the first who pointed out the method of drawing tangents and forming spirals. These discoveries are contained in his works on the dimension of the circle, on the sphere and cylinder, on conoids and spheroids, and on spiral lines. Archimedes was so fond of his discovery of the proportion between the solidity of the sphere and that of the cylinder, that he ordered to be placed upon his tomb a sphere inscribed in a cylinder, and likewise the numbers which express the ratio of these solids.

Whilst geometry was thus advancing with rapid strides, Apollonius Pergaeus, who was so called from being born at Perga in Pamphylia, followed in the steps of Archimedes, and widely extended the boundaries of the science. In addition to several mathematical works, which are now lost, Apollonius wrote a treatise on the theory of the conic sections, which contains all their properties with relation to their axes, their diameters, and their tangents. He demonstrated the celebrated theorem, that the parallelogram described about the two conjugate diameters of an ellipse or hyperbola is equal to the rectangle described round the two axes, and that the sum or difference of the squares of the two conjugate diameters is equal to the sum or difference of the squares of the two axes. In his fifth book he determines the greatest and the least lines that can be drawn to the circumferences of the conic sections from a given point, whether this point be situated in or out of the axis. This work, which contains everywhere the deepest marks of an inventive genius, procured for its author the appellation of the Great Geometer.

There is some reason to believe that the Egyptians were a little acquainted with plane trigonometry; and there can be no doubt that it was known to the Greeks. Spherical trigonometry, which is a more difficult part of geometry, does not seem to have made any progress till the time of Menelaus, an excellent geometrician and astronomer. In his work on spherical triangles, he gives the method of constructing them, and of resolving most of the cases which were necessary in the ancient astronomy. An introduction to spherical trigonometry had already been given to the world by Theodosius in his Treatise on Spheres, where he examined the relative properties of different circles formed by cutting a sphere in all directions.

Though the Greeks had made great progress in the science of geometry, they do not seem to have hitherto considered quantity in general or in the abstract. In the writings of Plato we can discover something like traces of geometrical analysis; and in the seventh proposition of Archimedes's work on the sphere and the cylinder, these traces are more distinctly marked. He reasons about unknown magnitudes as if they were known, and he finally arrives at an analogy, which, when transferred into the language of algebra, gives an equation of the third degree, which leads to the solution of the problem.

It was, however, reserved for Diophantus to lay the foundation of the modern analysis, by his invention of the Diophantine analysis of indeterminate problems; for the method which he employed in the resolution of these problems has a striking analogy to the present mode of resolving equations of the first and second degrees. He was likewise the author of thirteen books on arithmetic, several of which are now lost. The works of Diophantus were honoured with a commentary by the beautiful and learned Hypatia, the daughter of Theon. But the same fanaticism which led to the murder of this accomplished female was probably the cause that her works have not descended to posterity.

Near to the end of the fourth century of the Christian Pappus era, Pappus of Alexandria published his mathematical collections, a work which, besides many new propositions of his own, contains the most valuable productions of ancient geometry. Out of the eight books of which this work consisted, two have been lost; and the rest are occupied with questions in geometry, astronomy, and mechanics.

Diocles, whom we have already had occasion to mention as the inventor of the cissoid, discovered the solution of a problem proposed by Archimedes, namely, to cut a sphere by a plane in a given ratio. The solution of Diocles has been conveyed to us by Eutocius, who wrote commentaries on some of the works of Archimedes and Apollonius, in the year 520. About the time of Diocles flourished Serenus, who wrote two books on the cylinder and cone, which have been published at the end of Halley's edition of Apollonius.

Geometry was likewise indebted to Proclus, the head of the Platonic school at Athens, not only for his patronage of men of science, but also for his commentary on the first book of Euclid. Mathematics were likewise cultivated by Marinus, the author of the Introduction to Euclid's Data; by Isidorus of Miletus, who was a disciple of Proclus; and by Hero the younger, whose work, entitled Geodesia, contains the method of determining the area of a triangle from its three sides.

Whilst the mathematical sciences were thus flourishing in Greece, and were so successfully cultivated by the philosophers of the Alexandrian school, their very existence was threatened by one of those great revolutions with which the world has been convulsed. The dreadful ravages which were committed by the successors of Mahommmed in Egypt, in Persia, and in Syria, the destruction of the Alexandrian library by the caliph Omar, and the dispersion of a number of those illustrious men who had flocked to Alexandria as the cultivators of science, gave a deadly blow to the progress of geometry. When the fanaticism of the Mahommmedan religion, however, had subsided, and the termination of war had turned the minds of the Arabs to the pursuits of peace, the arts and sciences engaged their attention, and they began to rekindle those very intellectual lights which they had so assiduously endeavoured to extinguish. The works of the Greek geometers were studied with care; and the arts and sciences, reviving under the auspices of the Arabs, were communicated in a more advanced condition to the other nations of the world.

The system of arithmetical notation at present adopted Arabian in every civilized country had its origin amongst the Arabs. Their system of arithmetic was made known to Europe by the famous Gerbert, afterwards Pope Sylvester II, who travelled into Spain when it was under the dominion of that nation.

The invention of algebra has been ascribed to the Arabs by Cardan and Wallis, from the circumstance of their using the words square, cube, quadrato-quadratum, &c., in stead of the second, third, fourth, &c. powers, as employed by Diophantus. But whatever truth there may be in this supposition, it appears that they were able to resolve cubic, and even biquadratic equations, as there is in the Leyden library an Arabic manuscript, entitled the Algebra of Cubic Equations, or the Solution of Solid Problems.

The various works of the Greek geometers were translated by the Arabs, and it is through the medium of an Arabic version that the fifth and sixth books of Apollonius have descended to our times. Mahammed Ben Musa, the author of a work on Plane and Spherical Figures, and Geber Ben Apha, who wrote a commentary on Plato, gave a new form to the plane and spherical trigonometry of the ancients. By reducing the theory of triangles to a few propositions, and by substituting, instead of the chords of double arcs, the sines of the arcs themselves, they simplified this important branch of geometry, and contributed greatly to the abridgment of astronomical calculation. A treatise on the art of surveying was likewise written by Mahammed of Bagdad.

After the destruction of the Alexandrian school founded by Ptolemy Lagus, one of the successors of Alexander, the dispersed Greeks continued for a while to cultivate their favourite sciences, and exhibited some marks of that genius which had inspired their forefathers. The magic squares were invented by Moschopolus, a discovery more remarkable for its ingenuity than for its practical use. The same subject was afterwards treated by Cornelius Agrippa in his work on occult philosophy; by Bachet de Meziria, a learned algebraist, about the beginning of the seventeenth century; and in later times by Frencle de Bessi, M. Poignard of Brussels, De la Hire, and Sauveur.

The science of the pure mathematics advanced with a doubtful pace during the thirteenth, fourteenth, and fifteenth centuries. The algebra of the Arabians was introduced into Italy by Leonardo di Pisa, who, in the course of his commercial speculations in the East, had considerable intercourse with the Arabs. A work on the Planisphere, and ten books on arithmetic, were written by Jordanus Nemorarius. The Elements of Euclid were translated by Campanus of Novara. A work on algebra, entitled Summa de Arithmetica, Geometria, Proportione et Proportionatilitate, was published by Lucas Pacioli; and about the same time appeared Regiomontanus's treatise on trigonometry, which contains the method of resolving spherical triangles in general, when the three angles or three sides are known.

During the sixteenth century, algebra and geometry advanced with rapidity, and received many new discoveries from the Italian philosophers. The formula for the solution of equations of the third degree was discovered by Scipio Ferri, professor of mathematics at Bologna, and perhaps by Nicholas Tartalea of Brescia; and equations of the fourth order were resolved by Louis Ferrari, the disciple of Hieronymus Cardan of Bononia. This last mathematician published nine books of arithmetic in 1539; and in 1545 he added a tenth, containing the doctrine of cubic equations, which he had received in secrecy from Tartalea, but which he had so improved as to render them in some measure his own. The common rule for solving cubic equations still goes by the name of Cardan's Rule.

The irreducible case in cubic equations was successfully illustrated by Raphael Bombelli of Bologna. He has shown in his algebra, what was then considered as a paradox, that the parts of the formula which represents each root in the irreducible case, form, when taken together, a real result; but the paradox vanished when it was seen from the demonstration of Bombelli that the imaginary quantities contained in the two numbers of the formula necessarily destroyed each other by their opposite signs. About this time Maurolycus, a Sicilian mathematician, discovered the method of summing up several serieses of numbers, such as the series 1, 2, 3, 4, &c.; 1, 4, 9, 16, &c.; and the series of triangular numbers, 1, 3, 6, 10, 15, 21, &c.

The science of analysis is under great obligations to Vieta, Francis Vieta, a native of France. He introduced the present mode of notation, called literal, by employing the letters of the alphabet to represent indefinite given quantities; and we are also indebted to him for the method of transforming one equation into another, whose roots are greater or less than those of the original equation by a given quantity; for the method of multiplying or dividing their roots by any given number, of depriving equations of the second term, and of freeing them from fractional coefficients. The method which he has given for resolving equations of the third and fourth degree is also new and ingenious, and his mode of obtaining an approximate solution of equations of every order is entitled to still higher praise. We are also indebted to Vieta for the theory of angular sections, the object of which is to find the general expressions of the chords or sines for a series of arcs that are multiples of each other.

Whilst analysis was making such progress on the continent, Napier of Merchiston in Scotland was bringing to perfection his celebrated discovery of the logarithms, a set of artificial numbers, by which the most tedious operations in multiplication and division may be performed merely by addition and subtraction. This discovery was published at Edinburgh in 1614, in his work entitled Logarithmorum Canonis Descriptio, seu Arithmeticae Supputationum Mirabilis Abbreviatio. It is well known that there is such a correspondence between every arithmetical and geometrical progression, viz. \{0, 1, 2, 3, 4, 5, 6,\} that any terms of the geometrical progression may be multiplied or divided by merely adding or subtracting the corresponding terms of the arithmetical progression; thus the product of four and eight may be found by taking the sum of the corresponding terms in the arithmetical progression, viz. 2 and 3, for their sum 5 points out 32 as the product of 4 and 8. The numbers 0, 1, 2, 3, &c. are therefore the logarithms of 1, 2, 4, 8, &c. The choice of the two progressions being altogether arbitrary, Napier took the arithmetical progression which we have given above, and made the term 0 correspond with the unit of the geometrical progression, which he regulated in such a manner that when its terms are represented by the abscissae of an equilateral hyperbola in which the first abscissa and the first ordinate are equal to 1, the logarithms are represented by the hyperbolic spaces. In consequence, however, of the inconvenience of this geometrical progression, Napier, after consulting upon the subject with Henry Briggs of Gresham College, substituted the decuple progression, 1, 10, 100, 1000, of which 0, 1, 2, 3, 4, &c. are the logarithms. Nothing now remained but to construct tables of logarithms, by finding the logarithms of the intermediate numbers between the terms of the decuple progression. Napier, however, died before he was able to calculate these tables; but his loss was in some measure supplied by Mr Briggs, who applied himself with zeal to this arduous task, and published in 1618 a table of the logarithms of all numbers from 1 to 1000. In 1624 he published another table containing the logarithms from 1000 to 20,000, and from 90,000 to 100,000. The defects in Briggs's tables were filled up by his friends Gellibrand and Hadrian Vlacq, who also published new tables containing the logarithms of sines, tangents, &c. for 90 degrees.

During the time when Napier and Briggs were doing honour to their country by completing the system of logarithms, algebra was making great progress in the hands of our countryman Harriot. His Artis Analyticae Praxis, which appeared in 1620, contains, along with the discove- ties of its author, a complete view of the state of algebra. He simplified the notation, by substituting small letters instead of the capitals introduced by Vieta; and he was the first who showed that every equation beyond the first degree may be considered as produced by the multiplication of as many simple equations as there are units in the exponent of the highest power of the unknown quantity. From this he deduced the relation which exists between the roots of an equation and the co-efficients of the terms of which it consists.

About the same time, a foreign author named Fernel, who was physician to King Henry II. of France, had the merit of being the first who gave the measure of the earth. By reckoning the number of turns made by a coach-wheel from Amiens to Paris, till the altitude of the pole star was increased one degree, he estimated the length of a degree of the meridian to be 56,746 toises, which is wonderfully near the truth. He also wrote a work on mathematics, entitled De Proportionibus. About this time it was shown by Peter Metius, a German mathematician, that if the diameter of a circle be 113, its circumference will be 355. This result, so very near the truth, and expressed in so few figures, has preserved the name of its author.

The next author whose labours here claim our attention is the illustrious Descartes. We do not allude to those wild and ingenious speculations by which this philosopher endeavoured to explain the celestial phenomena, but to those great discoveries with which he enriched the kindred sciences of algebra and geometry. He introduced the present method of marking the powers of any quantity by numerical exponents. He first explained the use of negative roots in equations, and showed that they are as real and useful as positive roots; the only difference between them being founded on the different manner in which the corresponding quantities are considered. He pointed out the method of finding the number of positive and negative roots in any equation where the roots are real; and developed the method of indeterminates, which Vieta had obscurely hinted at.

Though Regiomontanus, Tartalea, and Bombelli, had resolved several geometrical problems by means of algebra, yet the general method of applying geometry to algebra was first given by Vieta. It is to Descartes, however, that we are indebted for the beautiful and extensive use which he made of his discovery. His method of representing the nature of curve lines by equations, and of arranging them in different orders according to the equations which distinguished them, opened a vast field of inquiry to subsequent mathematicians; and his methods of constructing curves of double curvature, and of drawing tangents to curve lines, have contributed much to the progress of geometry. The inverse method of tangents, which it was reserved for the fluxionary calculus to bring to perfection, originated at this time in a problem which Florimonde de Beaune proposed to Descartes. It was required to construct a curve in which the ratio of the ordinate and sub-tangent should be the same as that of a given line to the portion of the ordinate included between the curve and a line inclined at a given angle. The curve was constructed by Descartes, and several of its properties detected; but he was unable to accomplish the complete solution of the problem. These discoveries of Descartes were studied and improved by his successors, amongst whom we may number the celebrated Hudd, who published, in Schooten's commentary on the geometry of Descartes, an excellent method of determining whether an equation of any order contains several equal roots, and of discovering the roots which contains.

The celebrated Pascal, who was equally distinguished by his literary and his scientific acquirements, extended the boundaries of analysis by the invention of his arithmetical triangle. By means of arbitrary numbers placed at the vertex of the triangle, he formed all the figurate numbers in succession, and determined the ratio between the numbers of any two cases, and the various sums resulting from the addition of all the numbers of one rank taken in any possible direction. This ingenious invention gave rise to the calculation of probabilities in the theory of games of chance, and formed the foundation of an excellent treatise of Huygens, entitled De Ratiocinis in Ludo Aleae, published in 1657.

Several curious properties of numbers were at the same time discovered by Fermat at Toulouse. In the theory of prime numbers, particularly, which had first been considered by Eratosthenes, Fermat made great discoveries; and in the doctrine of indeterminate problems he seems to have been deeply versed, having republished the arithmetic of Diophantus, and enriched it with many valuable notes of his own. He invented the method of discovering the maxima and minima of variable quantities, which serves to determine the tangents of geometrical curves, and paved the way for the invention of the fluxionary calculus.

Another step towards the discovery of fluxions was at this time made by Cavalieri in his geometry of indivisibles. In this work, which was published in 1635, its author supposes every plane surface to consist of an infinite number of planes; and he lays it down as an axiom, that these infinite sums of lines and surfaces have the same ratio when compared with the unit in each case as the superficies and solids to be measured. This ingenious method was employed by Cavalieri in the quadrature of the conic sections, and in the curvature of solids generated by revolution; and in order to prove the accuracy of his theory, he deduced the same results from different principles.

Problems of a similar kind, which had been solved by Roberval, Fermat and Descartes, now occupied the attention of Roberval. The latter of these mathematicians began his investigation of this subject about a year before the publication of Cavalieri's work, and the methods which both of them employed were so far the same as to be founded on the principles of indivisibles. In the mode, however, which Roberval adopted, planes and solids were considered as composed of an infinite number of rectangles, whose altitudes and the thickness of their sections were infinitely small. By means of this method, Roberval determined the area of the cycloid, the centre of gravity of this area, and the solids formed by its revolution on its axis and base. He also invented a general method for tangents, similar in metaphysical principles to that of fluxions, and applicable both to mechanical and geometrical curves. By means of this, he determined the tangents of the cycloid; but there were some curves which resisted its application. Considering every curve as generated by the motion of a point, Roberval regarded this point as acted upon at every instant by two velocities which were ascertained from the nature of the curve. He constructed a parallelogram having its sides in the same ratio as the two velocities; and he assumed as a principle, that the direction of the tangent must fall on the diagonal, the position of which being ascertained, gave the position of the tangent.

In 1644, solutions of the cycloidal problems formerly resolved by Roberval were published by Torricelli as invented by himself. The demonstrations of Roberval had been transmitted to Galileo, the preceptor of Torricelli, and had also been published, in 1637, in Mersenne's Universal Harmony. The Italian philosopher was consequently accused of plagiarism by Roberval, and the charge so deeply affected his mind as to bring him prematurely to the grave. It is obvious, however, from the demonstrations of Torricelli, that he had never seen those of Roberval, and that he was far from meriting that cruel accusa- The cycloid having attracted the notice of geometers, from the number and singularity of its properties, the celebrated Pascal proposed to them a variety of new problems relative to this curve, and offered prizes for their solution. These problems required the area of any cycloidal segment, the centre of gravity of that segment, the solids, and the centres of gravity of the solids, which are generated either by a whole revolution, a half, or a quarter of a revolution, of this segment round an abscissa or an ordinate. The resolution of these problems was attempted by Huygens, Sluze, Sir Christopher Wren, Fermat, and Roberval. Sluze discovered an ingenious method of finding the area of the curve. Huygens squared the segment comprised between the vertex, and as far as a fourth of the diameter of the generating circle; and Sir Christopher Wren ascertained the length of the cycloidal arc included between the vertex and the ordinate, the centre of gravity of this arc, and the surfaces of the solids generated during its revolution. These attempts were not considered by their authors as solutions of Pascal's problems, and therefore they did not lay claim to his prize. Our countryman Wallis, however, and Lallouère a Jesuit, gave in a solution of all the problems, and thought themselves entitled to the proffered reward. In the methods employed by these mathematicians, Pascal detected several sources of error; and it was reserved for that great genius to furnish a complete solution of his own problems. Extending his investigations to curtate and prolate cycloids, he proved that the length of these curves depends on the rectification of the ellipse, and assigned in each case the axis of the ellipse. From this method he deduced this curious theorem, That if two cycloids, the one curtate and the other prolate, be such that the base of the one is equal to the circumference of the circle by which the other is generated, the length of these two cycloids will be equal.

Whilst these discoveries were making on the continent, the friends of science in Britain were actively employed in promoting its advancement. In 1653, Wallis published his *Arithmetica Infinitorum*, a work of great genius. He attempted to determine, by the summation of infinite series, the quadrature of curves, and the curvature of solids; subjects which were afterwards investigated in a different manner by Ishmael Bullialdus. By Wallis's method, curves were squared when their ordinates were expressed by one term; and when their ordinates were complex quantities raised to entire and positive powers, these ordinates were resolved into series, of which each term was a monomial. Wallis attempted to extend his theory to curves the ordinates of which were complex and radical, by attempting to interpolate the series of the former kind with a new series; but he was unsuccessful.

It was reserved for Newton to remove this difficulty. He solved the problem in a more direct and simple manner by the aid of his new formula for expanding into an infinite series any power of a binomial, whether its exponent was positive or negative, an integer or a fraction. Algebra was also indebted to this illustrious mathematician for a simple and extensive method of resolving an equation into commensurable factors; for a method of summing up the powers of the roots of an equation, of extracting the roots of quantities partly commensurable and partly incommensurable, and of finding by approximation the roots of literal and numerical equations of all orders.

About this time, William Lord Brouncker, in attempting to demonstrate an expression of Wallis on the magnitude of the circle, discovered the theory of continued fractions. When an irreducible fraction is expressed by numbers too great and complicated to be easily employed by the analyst, the method of Lord Brouncker enables us to substitute an expression much more simple, and nearly equivalent. This theory, which enables us to find a very accurate relation between the diameter and circumference of the circle, was employed by Huygens in the calculation of his planetary automaton for representing the motions of the solar system, and was enlarged and improved by other celebrated geometers. Lord Brouncker had likewise the merit of discovering an infinite series to represent the area of the hyperbola. The same discovery was made by Nicholas Mercator, who published it in his *Logarithmotechnia* in 1668.

The subject of infinite series received considerable addition from Mr James Gregory. He was the first who gave the tangent and secant in terms of the arc, and inversely, the arc in terms of the tangent and secant. He constructed series for finding directly the logarithm of the tangent and secant from the value of the arc, and the logarithm of the arc from that of the tangent and secant; and he applied this theory of infinite series to the rectification of the ellipsis and hyperbola.

The differential triangle invented by the learned Dr Barrow, for drawing tangents to curves, may be regarded as another contribution towards the invention of fluxions. This triangle has for its sides the element of the curve and those of the abscissa and ordinate, and those sides are treated as quantities infinitely small.

The doctrine of evolutes had been slightly touched upon by Apollonius. It remained, however, for the illustrious Huygens to bring it to perfection. His theory of evolutes is contained in his *Horologium Oscillatorium*, published in 1673, and may be regarded as one of the finest discoveries in geometry. When any curve is given, Huygens has pointed out the method of constructing a second curve, by drawing a series of perpendiculars to the first, which are tangents to the second; and of finding the first curve from the second. From this principle he deduces several theorems on the rectification of curves, and that remarkable property of the cycloid, in which an equal and similar cycloid is produced by evolution.

In contemplating the progress of analysis from the beginning of the seventeenth century to the invention of the fluxions, we cannot fail to perceive the principles of that calculus gradually unfolding themselves to view. The human mind seemed to advance with rapidity towards that great discovery, and it is by no means unlikely that it would soon have arrived at the doctrine of fluxions, even if the superior genius of Newton had not accelerated its progress. In Cavalieri's *Geometria Indivisibilibus* we perceive the germ of the infinitesimal calculus; and the method of Roberval for finding the tangents of curves bears a striking analogy to the metaphysics of the fluxionary calculus. It was the glory of Newton, however, to invent and illustrate the method of fluxions; and the obscure hints which he received from preceding mathematicians do not in the least detract from the merit of our illustrious countryman.

On the claims of Leibnitz as a second inventor of fluxions we shall not here enter at any considerable length; but We shall merely give a brief view of the facts which relate to the discovery of the higher calculus, and make a few observations on the conclusions to which they lead. (See Professor Playfair's Dissertation, prefixed to this work.)

In the year 1669, a paper of Sir Isaac Newton's, entitled *De Analysis per Equationes numero terminorum infinitas*, was communicated by Dr Barrow to Mr Collins, one of the secretaries of the Royal Society. In this paper the author points out a new method of squaring curves, both when the expression of the ordinate is a rational quantity, and when it contains complex radicals, by evolving the expression of the ordinate into an infinite number of simple terms, by means of the binomial theorem. In a letter from Newton to Collins, dated 10th December 1672, there is contained a method of drawing tangents to curve lines, without being obstructed by radicals; and in both these works, an account of which was circulated on the continent by the secretaries of the Royal Society, the principles of the fluxional calculus are plainly exhibited; and it is the opinion of all the disputants, that those works prove that Newton must have been acquainted with the method of fluxions when he composed them.

Leibnitz came to London in the year 1673; and though there is no direct evidence that he saw Newton's paper De Analyti per Equationes, yet it is certain that he had seen Sir Isaac's letter to Collins of 1672; and it is highly improbable that such a man as Leibnitz should have been ignorant of a paper of Newton's, which had been four years in the possession of the public, and which contained discussions at that time interesting to every mathematician.

A letter from Newton to Oldenburg, one of the secretaries of the Royal Society, dated 24th October 1676, was communicated to Leibnitz. This letter contains several theorems without the demonstrations, which are founded on the method of fluxions, and merely states that they result from the solution of a general problem. The enunciation of this problem he expresses in a cipher, the meaning of which was, an equation in which any number of flowing quantities being given, it was required to find the fluxions, and inversely. In reply to this communication, Leibnitz transmitted a letter to Oldenburg, dated 21st June 1677, where he explains the nature of the differential calculus, and affirms that he had long employed it for drawing tangents to curve lines.

The correspondence between Leibnitz and Oldenburg having been broken off by the death of the latter, Leibnitz published, in the Acta Eruditorum Lips., for October 1684, the principles of the new analysis, under the title of Nova Methodus pro maximis et minimis, itemque tangentibus, quae nec fractas, nec irrationales quantitates moratorium, et singularis pro illis calculis. This paper contains the method of differencing simple, fractional, and radical quantities, and the application of the calculus to the solution of some physical and geometrical problems. In 1685 he likewise published two small pamphlets on the quadrature of curves, containing the principles of the Calculus Summatorius, or the Integral Method of Fluxions; and in 1686 there appeared another tract by the same author, On the Recendite Geometry, and the Analysis of Indivisibles and Infinites, containing the fundamental rule of the integral calculus.

Towards the close of the year 1686, Sir Isaac Newton gave to the world his immortal work entitled Philosophiae Naturalis Principia Mathematica. Some of the most difficult problems in this work are founded on the fluxional calculus; and it is allowed by Bossut, one of the defenders of Leibnitz, "that mathematicians did Newton the justice to acknowledge, that at the period when his Principia was published, he was master of the method of fluxions to a high degree, at least with respect to that part which concerns the quadrature of curves." The claim of Leibnitz as a separate inventor of the differential calculus is evidently allowed by Newton himself, when he observes, that Leibnitz had communicated to him a method similar to his own for drawing tangents, &c., and differing from it only in the enunciation and notation.

About this time it became fashionable amongst geometers to perplex each other by the proposal of new and difficult problems; a practice which powerfully contributed to the progress of mathematics. The dispute in which Leibnitz was engaged with the Cartesians respecting the measure of active forces, which the former supposed to be as the simple velocity, whilst the latter asserted that they were as the square of the velocity, led him to propose the problem of the isochronous curve, or "to find the curve which a heavy body must describe equally, in order to approach or recede from a horizontal plane in equal times." This curve was found by Huygens to be the second cubic parabola; but he gave only its properties and construction, without the demonstration. In 1689, the same solution, along with the demonstration, was given by Leibnitz, who, at the same time, proposed to geometers to find the paracentric isochronal curve, or the curve in which a body would equally approach to or recede from a given point in equal times.

It was at this time that the two brothers, James and John Bernoulli, began to display those talents from which the physical and mathematical sciences received such immense improvements. James was born in 1654, and died in 1705; and John, who was his pupil, was born in 1667, and lived to the advanced age of sixty-eight years. In 1690, James Bernoulli gave the same solution of the isochronous curve which had been given by Huygens and Leibnitz, and proposed the celebrated problem of the catenary curve, which had formerly perplexed the ingenuity of Galileo. In two memoirs, published in 1691, he determined, by means of the inverse method of fluxions, the tangents of the parabolic spiral, the logarithmic spiral, and the loxodromic curve, and likewise the quadratures of their respective areas.

The problem of the catenary curve having occupied the attention of geometers, was resolved by Huygens, Leibnitz, and John Bernoulli. In these solutions, however, the gravity of the catenary curve was supposed to be uniform; but James Bernoulli extended the solution to cases where the weight of the curve varies from one point to another, according to a given law. From this problem he was also conducted to the determination of the curvature of a bendèd bow, and that of an elastic bar fixed at one extremity, and loaded at the other with a given weight. In the hopes of contributing to the progress of navigation, the same mathematician considered the form of a sail swollen with the wind. When the wind, after striking the sail, is not prevented from escaping, the curvature of the sail is that of the common catenary curve; but when the sail is supposed to be perfectly flexible, and filled with a fluid pressing downwards on itself, as water presses on the sides of a vessel, the curve which it forms is one of those denominated linearia, which is expressed by the same equation as the common elastic curve, where the extensions are reckoned proportional to the forces applied at each point. The same problem was solved by John Bernoulli, in the Journal des Sciences for 1692; but there is satisfactory evidence that it was chiefly borrowed from his brother James.

The attention of James Bernoulli was now directed to the theory of curves produced by the revolution of one Bernoulli curve upon another. He considered one curve rolling upon another, a given curve equal to the first, and immovable. He determined the evolute and the caustic of the epicycloid, described by a point of the moving circle; and he deduced from it other two curves, denominated the anti-evolute and pericaustic. He found also that the logarithmic spiral was its own evolute, caustic, anti-evolute, and pericaustic; and that an analogous property belonged to the cycloid.

About this time Signor Viviani, an Italian geometer, distinguished as the restorer of Aristote's conic sections, required the solution of the following problem, viz. That there existed a temple of a hemispherical form, pierced with four equal windows, with such skill that the remainder of the hemisphere might be perfectly squared. With the aid of the new analysis, Leibnitz and James Bernoulli immediately discovered a solution, whilst that of Viviani was founded on the ancient geometry. He proved that the problem might be solved by placing parallel to the base of the hemisphere, two right cylinders, the axes of which should pass through the centres of two radii, forming a diameter of the circle of the base, and piercing the dome each way.

Prior to some of these discussions, the curves called caustic, and sometimes Tschirnhausenian, were discovered by Tschirnhausen. These curves are formed by the crossing of the rays of light, when reflected from a curved surface, or refracted through a lens so as not to meet in a single point. With the assistance of the common geometry, Tschirnhausen discovered that they are equal to straight lines when they are formed by geometrical curves; and he also found out several other curious properties. By the aid of the higher calculus, James Bernoulli extended these researches, and added greatly to the theory of caustics produced by refraction.

The problem of the paracentric isochronal curve, proposed by Leibnitz in 1689, was solved by James Bernoulli, who took for ordinates parallel straight lines, and for abscissas the chords of an infinite number of concentric circles described about the given point. In this way he obtained a separate equation, constructed at first by the rectification of the elastic curve, and afterwards by the rectification of an algebraic curve. The same problem was solved by John Bernoulli and by Leibnitz.

In 1694, a branch of the new analysis, called the exponential calculus, was invented separately by John Bernoulli and by Leibnitz. It consists in differentiating and integrating exponential quantities or powers with variable exponents. To Leibnitz the priority in point of invention certainly belongs, but John Bernoulli was the first who published the rules and uses of the calculus.

The Marquis de l'Hospital, who, in 1695, had solved the problem about the curve of equilibration in draw-bridges, and shown it to be an epicycloid, published in the following year his Analysis of Infinites for the understanding of Curve Lines. In this celebrated work, the differential calculus, or the direct method of fluxions, was fully explained and illustrated; and as the knowledge of the higher geometry had been hitherto confined to a few, it was now destined to enlighten the different nations of Europe.

The methods which were employed by Descartes, Fermat, and others, for finding the maxima and minima of quantities, yielded in point of simplicity and generality to that which was derived from the doctrine of fluxions. Another class of problems, however, of the same kind, but more complicated, from their requiring the inverse method of fluxions, began now to exercise the ingenuity of mathematicians. A problem of this class, for finding the solid of least resistance, was solved by Newton in the thirty-fourth proposition of the second book of his Principia. After having determined the truncated right cone, which being moved in a fluid by the smallest base (which is unknown), experiences the least resistance, he gave, without any demonstration, the ratio from which might be derived the differential equation of the curve that generates by a revolution of its axis the solid of least resistance. A general solution, however, was still wanting, until the attention of geometers was directed to the subject by John Bernoulli, who proposed, in 1697, the celebrated problem of the Brachystochronon, or the curve along the concave side of which if a heavy body descend, it will pass in the least time possible from one point to another, the two points not being in the same vertical line. This problem was resolved by Leibnitz, Newton, the Marquis de l'Hospital, and James Bernoulli, who demonstrated that the curve of quickest descent is a cycloid reversed. This result will appear at first surprising, when we consider a line as the shortest distance between two points; but the surprise will cease, when we reflect, that in a concave curve lying between the two given points, the moving body descends at first in a more vertical direction, and therefore acquires a greater velocity than when it rolls down an inclined plane. It follows that this addition to its velocity, at the commencement of its path, may balance the increase of space through which it has to move.

At the close of this discussion commenced, between Sir James and John Bernoulli, that celebrated dispute about isoperimetrical problems, in which the qualities of the two heads were much more conspicuous than those of the heart. These illustrious characters, connected by the strongest ties of affinity, were, at the commencement of their distinguished career, united by the warmest affection. John was initiated by his elder brother into the mathematical sciences; and a generous emulation, softened by friendship in the one, and gratitude in the other, continued for some years to direct their studies, and accelerate their progress. There are few men, however, who can support at the same time the character of a rival and a friend. The success of one party is apt to awaken the envy of the other, and success itself is often the parent of presumption. A foundation is thus laid for future dissension; and it is a melancholy fact in the history of learning, that the most ardent friendships have been sacrificed on the altar of literary ambition. Such was the case between the two Bernoullis. As soon as John was settled as professor of mathematics at Groningen, all friendly intercourse between the two brothers came to an end. Regarding John as the aggressor, and provoked at the ingratitude which he exhibited, his brother James challenged him by name to solve the following problems:—1. To find, amongst all the isoperimetrical curves between given limits, such a curve that, constructing a second curve, the ordinates of which shall be the functions of the ordinates or arcs of the former, the area of the second curve shall be a maximum or a minimum. 2. To find amongst all the cycloids which a heavy body may describe in its descent from a point to a line, the position of which is given, that cycloid which is described in the least possible time. A prize of fifty florins was promised to John Bernoulli, if, within three months, he engaged to solve these problems, and publish within a year legitimate solutions of them.

In a short time John Bernoulli produced his solution, and demanded the prize. He succeeded in constructing the problem of swiftest descent; but his solution of the other problem was radically defective. This failure mortified the vanity with which he gloried in his apparent success. He acknowledged the mistake in his solution, and, with the same imperious tone, transmitted a new result, and re-demanded the prize. This new solution, which was still defective, drew down the wit and ridicule of James Bernoulli, which his brother attempted to repel by a torrent of coarse invective.

Leibnitz, Newton, and the Marquis de l'Hospital, being appointed arbiters in this dispute, James Bernoulli published, in 1700, the formulae of the isoperimetrical problem, without any demonstration; and John transmitted his solution to the French Academy in February 1701, on condition that it should not be opened till his brother's demonstrations were published. In consequence of this, James Bernoulli published his solution in May 1701, in the Acta Eruditorum, under the following title, Analysis magni Problematis Isoperimetrici, and gained great honour from the skill which it displayed. For five years John Bernoulli was silent upon the subject; but his brother dying in 1705, he published his solution in the Memoirs of the Academy for 1706. About thirteen years afterwards, John Bernoulli having perceived the source of his error, confessed his mistake, and published a new solution, not very different from that of his brother, in the Memoirs of the Academy for 1718.

In the problem relative to the cycloid of swiftest descent, John Bernoulli obtained a result similar to that of his brother, by a very ingenious method, which extended the bounds of the new analysis. In his investigations he employed the synchronous curve, or that which cuts a series of similar curves placed in similar positions; so that the arcs of the latter included between a given point and the synchronous curve shall be described by a heavy body in equal times. He demonstrated, that of all the cycloids thus intersected, that which is cut perpendicularly is described in less time than any other terminating equally at the synchronous curve. But being unable to give a general solution of the problem, he applied to Leibnitz, who easily resolved it, and at that time invented the method of differencing de curva in curvam.

About a month after the death of the Marquis de l'Hospital, John Bernoulli declared himself the author of a rule, given by the marquis in his Analysis of Infinites, for finding the value of a fraction the numerator and denominator of which should vanish at the same instant, when the variable quantity that enters into it has a certain given value. The defence made by the marquis's friends only induced John Bernoulli to make greater demands, till he claimed as his own the most important parts of the Analysis of Infinites. But it does not appear, from an examination of the subject, that there is any foundation for his claims.

Towards the close of 1704, Sir Isaac Newton published, at the end of his Optics, his Enumeratio Linearum Tertiae Ordinis, and his treatise De Quadratura Curvarum. The first of these papers displays great ability, but is founded only on the common algebra, and the doctrine of series which Newton had brought to such perfection. His treatise De Quadratura Curvarum contains the resolution of fluxional formulae, with one variable quantity which leads to the quadrature of curves. By means of certain series he obtains the resolution of several complicated formulae, by referring them to such as are more simple; and these series being interrupted in particular cases, give the fluents in finite terms. From this several interesting propositions are deduced, amongst which is the method of resolving rational fractions. In 1711 Newton published his Method of Fluxions. The object of this work is to determine, by simple algebra, the linear co-efficients of an equation that satisfies as many conditions as there are co-efficients, and to construct a curve of the parabolic kind passing through any number of given points. Hence arises a simple method of finding the approximate quadrature of curves, in which a certain number of ordinates are determinable. It has been the opinion of some able mathematicians that this treatise contains the first principles of the integral calculus with finite differences, afterwards invented by Dr Taylor. A posthumous work of Newton's, entitled the Method of Fluxions and of Infinite Series, was published by Dr Pemberton about nine years after the death of its author; but it does not contain any investigations which may be considered as having accelerated the progress of the new analysis.

The mathematical sciences were at this time indebted to the labours of Manfredi, Parent, and Saurin. The former of these geometers published a very able work, De Constructione Equationum differentialium primi gradus. To Parent we are indebted for the problem by which we obtain the ratio between the velocity of the power and the weight, for finding the maximum effect of machines; but his reputation was much injured by the obscurity of his writings. Saurin was celebrated for his theoretical and practical knowledge of watchmaking, and he was the first who elucidated the theory of tangents to the multiple points of curves.

Whilst the science of analysis was thus advancing with rapidity, the dispute between Newton and Leibnitz began to be agitated amongst the mathematicians of Europe. These illustrious rivals seemed to have hitherto contented with sharing the honour of having invented the fluxionary calculus. But as soon as the priority of invention was attributed to Newton, the friends of Leibnitz came forward with eagerness to support the claims of their master.

In a small work on the curve of swiftest descent and the solid of least resistance, published in 1699, Nicholas Facio de Duillier, an eminent Genoese, attributed to Newton the first invention of fluxions, and hinted that Leibnitz, as the second inventor, had borrowed from the English philosopher. Exasperated at this improper insinuation, Leibnitz came forward in his own defence, and appealed to the admission of Newton in his Principia, that neither had borrowed from the other. He expressed his conviction that Facio de Duillier was not authorized by Sir Isaac to prefer such a charge, and he threw himself upon the testimony and candour of the English geometer.

The discussion rested in this situation for several years, until our celebrated countryman Dr Keill, instigated by an attack upon Newton in the Leipsic Journal, repeated the same charge against Leibnitz. The German philosopher made the same reply as he did to his former opponent, and treated Dr Keill as a young man incapable of judging upon the subject. In 1711 Dr Keill addressed a letter to Sir Hans Sloane, secretary to the Royal Society, and accused Leibnitz of having adopted the differential notation, in order to have it believed that he did not borrow his calculus from the writings of Newton.

Leibnitz was with reason irritated at this accusation, and called upon the Royal Society to interfere in his behalf. A committee of that learned body was accordingly appointed to investigate the subject, and their report was published in 1712, under the title of Commercium Epistolicum de Analyse promota. In this report the committee maintain that Leibnitz was not the first inventor, and absolve Dr Keill from all blame in giving the priority of invention to Newton. They were cautious, however, in stating their opinion upon that part of the charge in which Leibnitz was accused of plagiarism.

In answer to the arguments advanced in the Commercium Epistolicum, John Bernoulli, the particular friend of Leibnitz, published a letter, in which he has the assurance to state that the method of fluxions did not precede the differential calculus, but that it might have taken its rise from it. The reason which he assigns for this strange assertion is, that the differential calculus was published before Newton had introduced an uniform algorithm into the method of fluxions. But it may as well be maintained that Newton did not discover the theory of universal gravitation, because the attractive force of mountains and of smaller portions of matter was not ascertained until the time of Maskelyne and of Cavendish. The principles of fluxions are allowed to have been discovered before those of the differential calculus; and yet the former originated from the latter, because the fluxionary notation was not given at the same time.

Notwithstanding the ridiculous assertion of John Bernoulli, it has been admitted by all the foreign mathematicians that Newton was the first inventor of the method of fluxions. The point at issue, therefore, is merely this—Did Leibnitz see any of the writings of Newton which contained the principles of fluxions, before he published, in 1684, his Nova Methodus pro maximis et minimis? The friends of Leibnitz have produced some presumptive proofs that he had never seen the treatise of Newton De Analysi, nor the letter to Collins, in both of which the principles of the new calculus were to be found; and in order to strengthen their argument, they have not scrupled to assert that the writings already mentioned contained but a vague and obscure indication of the method of fluxions, and that Leibnitz might have perused them without having discovered it. This subsidiary argument, however, Mathematics.

rests upon the opinion of individuals; and the only way of repelling it is to give the opinion of an impartial judge. Montucla, the celebrated historian of the mathematics, who, being a Frenchman, cannot be suspected of partiality to the English, has admitted that Newton, in his treatise De Analysis, has disclosed, in a very concise and obscure manner, his principles of fluxions, and "that the suspicion of Leibnitz having seen this work is not destitute of probability;" for Leibnitz admitted, that in his interview with Collins, he had seen a part of the epistolary correspondence between Newton and that gentleman." It is evident, therefore, that Leibnitz had opportunities of being acquainted with the doctrine of fluxions before he had thought of the differential calculus; and as he was in London, where Newton's treatise was published, and in company with the very men to whom the new analysis had been communicated, it is very likely that he then acquired some knowledge of the subject. In favour of Leibnitz, however, it is but justice to say, that the transition from the method of tangents by Dr Barrow to the differential calculus is so simple, that Leibnitz might very easily have perceived it; and that the notation of his analysis, the numerous applications which he made of it, and the perfection to which he carried the integral calculus, are considerable proofs that he was innocent of the charge which the English have attempted to fix upon his memory.

In 1708, Remond de Montmort published a curious work, entitled the Analysis of Games of Chance, in which the common algebra was applied to the computation of probabilities, and the estimation of chances. Though this work did not contain any great discovery, yet it gave extent to the theory of series, and admirably illustrated the doctrine of combinations. The same subject was afterwards discussed by M. Demoivre, a French Protestant residing in England, in a small treatise entitled Mensura Sortis, in which are given the elements of the theory of recurrent series, and some very ingenious applications of it. Another edition was published in English in 1738, under the title of the Doctrine of Chances.

A short time before his death, Leibnitz proposed to the English geometers the celebrated problem of orthogonal trajectories, which was to find the curve that cuts a series of given curves at a constant angle, or at an angle varying according to a given law. This problem was put into the hands of Sir Isaac Newton when he returned to dinner greatly fatigued, and he brought it to an equation before he went to rest. Leibnitz being recently dead, John Bernoulli assumed his place, and maintained that nothing was easier than to bring the problem to an equation, and that the solution of the problem was not complete until the differential equation of the trajectory was resolved. Nicholas Bernoulli, the son of John, resolved the particular case in which the intersected curves are hyperbolas with the same centre and the same vertex. James Hermann, and Nicholas Bernoulli, the nephew of John, treated the subject by more general methods, which applied to the cases in which the intersected curves were geometrical. The most complete solution, however, was given by Dr Taylor in the Philosophical Transactions for 1717, though it was not sufficiently general, and could not apply to some cases capable of resolution. This defect was supplied by John Bernoulli, who, in the Leipzig Transactions for 1718, published a very simple solution, embracing all the geometrical curves, and a great number of the mechanical ones.

During these discussions, several difficult problems on the integration of rational fractions were proposed by Dr Taylor, and solved by John Bernoulli. This subject, however, had been first discussed by Roger Cotes, professor of mathematics at Cambridge, who died in 1710. In his posthumous work, entitled Harmonia Mensurarum, published in 1716, he gave general and convenient formulae for the integration of rational fractions; and we are indebted to this young geometer for his method of estimating errors in mixed mathematics, for his remarks on the differential method of Newton, and for his celebrated theorem for resolving certain equations.

In 1715, Dr Taylor published his learned work, entitled Methodus Incrementorum directa et inversa. In this work the doctor gives the name of increments or decrements of variable quantities to the differences, whether finite or infinitely small, of two consecutive terms, in a series formed according to a given law. When the differences are infinitely small, their calculus belongs to fluxions; but when they are finite, the method of finding their relation to the quantities by which they are produced forms a new calculus, called the integral calculus of finite differences. In consequence of this work, Dr Taylor was attacked anonymously by John Bernoulli, who lavished upon the English geometer all that dull abuse and angry ridicule which he had formerly heaped upon his brother.

The problem of reciprocal trajectories was at this time proposed by the Bernoullis. This problem required the curves which, being constructed in two opposite directions in one axis, given in position, and then moving parallel to one another with unequal velocities, should perpetually intersect each other at a given angle. It was long discussed between John Bernoulli and an anonymous writer, who proved to be Dr Pemberton. By an elegant solution of this problem the celebrated Euler first began to be distinguished amongst mathematicians. He was a pupil of John Bernoulli, and continued throughout the whole of his life the friend and rival of his son Daniel. The great object of his labours was to extend the boundaries of analysis; and before he had reached his twenty-first year he published a new and general method of resolving differential equations of the second order, subject to certain conditions.

The common algebra had been applied by Leibnitz and John Bernoulli to determine arcs of the parabola, the difference of which is an algebraic quantity; imagining that such problems, in the case of the ellipse and hyperbola, resisted the application of the new analysis. The Count de Fagnani, however, applied the integral calculus to the arcs of the ellipsis and hyperbola, and had the honour of explaining this new branch of geometry.

In the various problems depending upon the analysis of infinites, the great difficulty is to resolve the differential equation to which the problems may be reduced. Count James Riccati having been puzzled with a differential equation of the first order, with two variable quantities, proposed it to mathematicians in the Leipzig Acts for 1723. This question baffled the skill of the most celebrated analysts, who were merely able to point out a number of cases in which the indeterminate can be separated, and the equation resolved by the quadrature of curves.

Another problem, suggested by that of Viviani, was proposed in 1718 by Ernest Von Offenburg. It was required to pierce a hemispherical vault with any number of elliptical windows, so that their circumferences should be expressed by algebraic quantities; or, in other words, to determine on the surface of a sphere, curves algebraically rectifiable. In a paper on the rectification of spherical epicycloids, Hermann imagined that these curves were algebraically rectifiable, and therefore satisfied the question of Offenburg; but John Bernoulli (Mém. Acad. Par. 1732) demonstrated, that as the rectification of these curves depended upon the quadrature of the hyperbola, they were only rectifiable in certain cases, and gave the general method of determining the curves which are algebraically rectifiable on the surface of a sphere.

The same subject was also discussed by Nicole and Clairaut (Mém. Acad. 1734). The latter of these mathe- Mathematicians had already acquired fame by his *Recherches sur les Courbes à double Courbure*, published in 1730, before he was twenty-one years of age; but his reputation was extended by a method of finding curves whose property consists in a certain relation between these branches expressed by a given equation. In this research Clairaut pointed out a species of paradox in the integral calculus, which led to the celebrated theory of particular integrals, which was afterwards fully illustrated by Euler and other geometers.

The celebrated problem of isochronous curves began at this time to be re-agitated amongst mathematicians. The object of this problem is to find such a curve that a heavy body descending along its concavity shall always reach the lowest point in the same time, from whatever point of the curve it begins to descend. Huygens had already shown that the cycloid was the isochronous curve *in vacuo*. Newton had demonstrated that the same curve was isochronous when the descending body experiences from the air a resistance proportional to its velocity; and Euler and John Bernoulli had separately found the isochronous curve when the resistance was as the square of the velocity. These three cases, and even a fourth in which the resistance was as the square of the velocity added to the product of the velocity by a constant co-efficient, were all resolved by Fontaine, by means of an ingenious and original method; and it is very remarkable that the isochronous curve is the same in the third and fourth cases. The method of Fontaine was illustrated by Euler, who solved a fifth case, including all the other four, when the resistance is composed of three terms, the square of the velocity, the product of the velocity by a given co-efficient, and a constant quantity. He found also an expression of the time which the body employs to descend through any arc of the curve.

The application of analytical formulae to the physico-mathematical sciences was much facilitated by the algebra of sines and cosines with which Frederick Christian Mayer and Euler enriched geometry. By the combination of arcs, sines, and cosines, formulae are obtained which frequently yield to the method of resolution, and enable us to solve a number of problems which the ordinary use of arcs, sines, and cosines would render tedious and complicated.

About this time a great discovery in the theory of differential equations of the first order was made separately by Euler, Fontaine, and Clairaut. Hitherto geometers had no direct method of ascertaining if any differential equation were resolvable in the state in which it was presented, or if it required some preparation prior to its resolution. For every differential equation a particular method was employed, and their resolution was often effected by a kind of tentative process, which displayed the ingenuity of its author, without being applicable to other equations. The conditions under which differential equations of the first order are resolvable were discovered by the three mathematicians whom we have mentioned. Euler made the discovery in 1736, but did not publish it till 1740. Fontaine and Clairaut lighted upon it in 1739, and Euler afterwards extended the discovery to equations of higher orders.

The first traces of the integral calculus with partial differences appeared in a paper of Euler's in the Petersberg Transactions for 1734; but D'Alembert, in his work *Sur les Vents*, has given clearer notions of it, and he was the first who employed it in solution of the problem of vibrating chords proposed by Dr Taylor, and investigated by Euler and Daniel Bernoulli. The object of this calculus is to find a function of several variable quantities when we have the relation of the co-efficients which affect the differentials of the variable quantities of which this function is composed. Euler exhibited it in various points of view, and showed its application to a number of physical problems; and he afterwards, in his paper entitled *Investigatio Functionum ex data Differentialium conditione*, completely explained the nature and gave the algorithm of the calculus.

Whilst the analysis of infinites was making such rapid progress on the continent, it was attacked in England by Dr Berkeley, bishop of Cloyne, in a work called the Analyst, or a Discourse addressed to an Infidel Mathematician, wherein it is examined whether the object, principles, and inferences of the Modern Analysis are more distinctly conceived than Religious Mysteries and Points of Faith. In this work the doctor admits the truth of the conclusions, but maintains that the principles of fluxions are not founded upon reasoning strictly logical and conclusive. This attack called forth Robins and MacLaurin. The former proved that the principles of fluxions were consistent with the strictest reasoning; whilst MacLaurin, in his Treatise of Fluxions, gave a synthetical demonstration of the principles of the calculus after the manner of the ancient geometricians, and established it with such clearness and satisfaction that no intelligent man could refuse his assent. The differential calculus had been attacked at an earlier period by Nieuwentiet and Rolle, but the weapons wielded by these adversaries were contemptible when compared with the ingenuity of Dr Berkeley.

Notwithstanding this attack upon the principles of the new analysis, the science of geometry made rapid advances in England in the hands of Thomas Simpson, Landen, and Waring. In 1740, Mr Simpson published his Treatise on Fluxions, which, besides many original researches, contains a convenient method of resolving differential equations by approximation, and various means of hastening the convergence of slowly converging series. We are indebted to the same geometer for several general theorems for summing different series, whether they are susceptible of an absolute or an approximate summation. His Mathematical Dissertations, published in 1743; his Essays on several Subjects in Mathematics, published in 1740; and his Select Exercises for Young Proficients in the Mathematics, published in 1752, contain ingenious and original researches, which contributed to the progress of geometry.

In his Mathematical Lucubrations, published in 1755, Landen; Mr Landen has given several ingenious theorems for the summation of series; and the Philosophical Transactions for 1777 contain his curious discovery of the rectification of an hyperbolic arc, by means of two arcs of an ellipse, which was afterwards more simply demonstrated by Legendre. His invention of a new calculus, called the residual analysis, and in some respects subsidiary to the method of fluxions, has immortalized his name. It was announced and explained in a small pamphlet published in 1715, entitled a Discourse concerning the Residual Analysis.

The progress of geometry in England was accelerated by Waring, the labours of Mr Edward Waring, professor of mathematics at Cambridge. His two works entitled *Meditationes Analytice*, published in 1769, and *Meditationes Algebraeae*, and his papers in the Philosophical Transactions on the summation of forces, are filled with original and profound researches into various branches of the common algebra and the higher analysis.

It was, however, from the genius of Lagrange that the Discoveries of the higher calculus received the most brilliant improvements, i.e., of LaGrange. This great man was born in Piedmont. He afterwards removed to Berlin, and from thence to Paris. In addition to many improvements upon the integral analysis, he enriched geometry with a new calculus called the Method of Variations. The object of this calculus is, when there