remarks (Part 11. Ep. 72), "In searching out the solution of geometrical questions, I always make use of lines parallel and perpendicular as much as possible: and I consider no other theorems than the two following; the sides of similar triangles are proportionals; and in right-angled triangles the square of the hypothenuse is equal to the squares of the two sides. And I am not afraid to suppose several unknown quantities, that I may reduce the proposed equation to such terms as that it may depend on no other theorems than these two."

Buonaventura Cavalieri, better known by the Latinized appellation, Cavalerius, was born at Milan in 1598. He was a pupil of the celebrated Galileo, and became Professor of Mathematics at Bologna; he died in 1647. He was the author of several works on the mathematical sciences, the most important of which is a treatise on Indivisibles in seven books, which he put forth in 1635: it is important as being one of the first attempts to extend the powers of the ancient Geometry. He conceived a line to be made up of an infinite number of points; a surface to be formed of an infinite number of such lines; and a solid to be composed of an infinite number of such surfaces. These elements of geometrical magnitudes he named Indivisibles: and the principle he assumed in the application of these assumptions was, that the ratio of the infinite sums of lines, or of planes as compared with the unit of surface or volume, was the same as that of the surface or volume of which they were the measures. He shewed that his new principle was, in effect, the same as the method of Exhaustions, but a more convenient mode of reasoning, being less tedious and more direct. In the first six books he explains and applies his theory of Indivisibles, and in the seventh book he proves the same results by methods independent of Indivisibles; with the view of shewing the agreement of the results, and the consequent truth of his new principle. Guldin controverted and wrote against the doctrine of Indivisibles, and was answered by Cavalerius in the third of his "Exercitationes Geometrica sex," which were published in 1647. This work consists of exercises in the Method of Indivisibles, with answers to Guldin's objections. The method of Cavalerius is not free from error, as he applies the process of simplification at too early a stage of the investigation, by which means the strict logic of the reasoning is violated, while the correctness of the result is not affected.

Roberval was born in 1602, and adopted the theory of Cavalerius, but improved his modes of expression. His Traité des Indivisibles was published in 1693, after his death, and contains his application of the new principles. His method of drawing tangents was an approximation to that applied afterwards by the principles of Fluxions and the Differential Calculus.

Albert Girard was a Fleming who displayed great genius in the Mathematics. He was the first who announced the restoration of the three books of the Porisms of Euclid, in a work on Trigonometry, which was printed at the Hague in 1629. To what extent he succeeded is not known, as the results of his labours have never been published. To him are due some general theorems for the measuring of solid angles.

Marinus Ghetaldus was a distinguished geometer at the beginning of the seventeenth century: he was the author of several works on the ancient Geometry: he attempted, from what could be gathered from

Pappus, a restoration of the lost book of Apollonius on Inclinations, and published it in 1607, with the title of Apollonius Redivivus. In the same year also he put forth a supplement to the Apollonius Gallus of Vieta, and a collection of problems. Among his other writings may be mentioned his Archimedes Promotus, which was first published

in 1603.

Blaise Pascal was born at Clermont in 1623, and at an early age gave proofs of extraordinary ability. He was of an enquiring mind, desirous of knowing the reasons of every thing. It is reported of him that whenever he could not obtain from others sound reasons, he used to seek them out for himself; never giving his assent except on conviction. Pascal challenged the mathematicians of his day to prove some properties of the cycloid; but as the answers he received from Wallis and other eminent men were unsatisfactory, he himself gave the complete proofs of all the properties mentioned in his challenge. He invented the arithmetical machine which bears his name, and was the author of some pieces on other mathematical subjects. His intense application to study injured his health. He became a Jansenist, and retired to the Abbey of Port Royal, near Paris, where he composed his Provincial Letters, and wrote his thoughts on Religion and other subjects. These were published after his death, which happened when he was only thirty-nine years of age. His works were collected and published at Paris in 1779.

Schooten was Professor of Mathematics at Leyden. In 1649 he published an edition of Descartes' Geometry, and in 1657 an original work entitled Exercitationes Mathematicæ. He attempted the restoration of the Loci Plani of Apollonius. He has given the synthesis only, omitting the analysis, except in a few instances. He has also omitted the determinations and the distinction of cases; and in the preface he acknowledges that his attempted restoration was designed to be an illustration of the Geometry of Descartes, by furnishing appropriate examples to his method.

Christian Huygens was born at the Hague in 1629, where he died in 1696. He was one of the most ingenious mathematicians of his age. He was the author of two treatises, one entitled "Theoremata de Quadratura Hyperbolæ, Ellipsis et Circuli, ex dato portionum gravitatis centro;" and the other, "De Circuli magnitudine inventa;" which, at the time of their publication, were highly esteemed. He was also the author of some other pieces on Geometry, which were published at Paris in 1693, and he discovered the theory of Evolutes. His studies were not confined merely to the speculative portion of the mathematical sciences, but were extended to questions of practical utility. Among them may be named his improvements in telescopes, and his method of rendering the oscillations of pendulums isochronous. He was the discoverer of Saturn's ring, and a third satellite of that planet. When nearly sixty years of age he read and admitted the theory of centripetal forces, and the gravitation of the planets to the sun, which had been proved in Newton's Principia. His writings are

numerous.

Dr Isaac Barrow was a distinguished scholar and geometer, and the tutor of Isaac Newton when an undergraduate at Trinity College. Dr Barrow was appointed Greek Professor at Cambridge in 1660, and Gresham Professor of Geometry in 1662: the latter office he resigned

on his appointment to the new Professorship, founded at Cambridge by Mr Lucas, in 1663. This also he resigned, in favour of Mr Newton, in 1669. He became Master of Trinity College in 1672, and died in 1677 at the age of 47 years. In 1655 he edited, in Latin, the thirteen books of Euclid's Elements of Geometry, which were translated into English and published in 1660. He also put forth Euclid's Data in 1657. His Lectiones Geometrica were published in 1670, and translated into English, by Stone, in 1735. They contain his method of drawing tangents to curves, which is similar to that by the method of fluxions or the differential calculus: the difference being only in the notation. His Lectiones Optica were published in 1669. He also edited Archimedes, Apollonius and Theodosius. The Lectures which he delivered, as Lucasian Professor, were published after his death in 1683, with the title of Lectiones Mathematicæ, and were translated into English by Kirkby, in 1734: they are confined to Euclid's Elements of Geometry. He also applied the method of indivisibles to the propositions of Archimedes on the Sphere and Cylinder. His treatise was printed in 1678.

Sir Isaac Newton was one of the greatest philosophers that ever lived. The inscription on the pediment of the statue of Newton in the chapel of Trinity College, "Qui genus humanum ingenio superavit," records the unquestioned judgment of posterity. He was the original inventor of the method of fluxions and fluents; in 1665, his attention was first directed to the subject, and his method was completed before 1669. The merit of the invention was claimed for Leibnitz, who had put forth, in 1684, his view of the principles of infinitesimals or differentials. The terms employed and the notations adopted by these two great men were different; they agreed in the substance and object of the theory, but there was a considerable difference in their conception of the principles. An angry controversy arose respecting these claims, and an appeal was at length made to the Royal Society, by which a committee of enquiry was appointed. The result of their enquiries and deliberations was a decision in favour of Newton. The papers relating to this enquiry were printed in 1712, with the title of Commercium Epistolicum de Analysi promota. His discoveries in optics, and his new theory of light and colours for the explanation of optical phenomena, he did not publish till 1704: nearly thirty years after his chief discovery in that science.

His greatest discovery, however, was that of universal gravitation. His thoughts were first led to the subject in 1666, just after he had left Cambridge for the country, on account of the Plague. As he was sitting alone in a garden, some apples fell from a tree, and his thoughts were led to the subject of gravity. He considered that as this power is not found to be sensibly diminished at the remotest distance from the surface of the earth to which we can rise, it seemed reasonable to conclude that it must extend much further than is commonly believed. He enquired: "Why not as high as the moon? and if so, her motion must be influenced by the force of gravity: perhaps she is retained in her orbit by it. Though the power of gravity is not sensibly lessened, in the little change of distance at which we can place ourselves from the surface of the earth, yet it is possible that at the height of the moon this power may differ much in degree from what it is here.' To make an estimate of the amount of this diminution, he considered,

d

that if the moon were retained in her orbit by the force of gravity, no doubt the primary planets are carried round the sun by the like power: and by comparing the periods of the several planets with their mean distances from the sun, he found, that if any power like gravity held them in their courses, its intensity must decrease inversely as the square of their distances from the sun. He arrived at this conclusion, from considering them to move in circles concentric with the sun; from which form the orbits of the greater part of them do not greatly differ. By supposing therefore the force of gravity, when extended to the moon, to decrease in the same manner, he computed whether that force would be sufficient to keep the moon in her orbit. In this computation, taking sixty miles to a degree, the result at which he arrived did not shew the power of gravity to decrease as the inverse square of the moon's distance from the earth: whence he concluded that some other cause must, at least, combine with the action of gravity on the moon. Being unable to satisfy himself respecting this, he laid aside, for that time, all thought upon the subject. In the winter of 1676, he discovered the two grand propositions, that by a centripetal force varying inversely as the square of the distance, a planet must revolve in an ellipse about a centre of force, placed in one of the foci; and by a radius vector drawn to that focus, describe areas proportional to the times. These are proved in the second and third sections of the first book of the Principia. After the year 1679 his thoughts were again turned to the moon; and, by using in his computation the more accurate length of a degree, he arrived at the conclusion, that that planet appeared, agreeably to his former conjecture, to be retained in her orbit by the force of gravity, varying as the inverse square of the distance. On this principle he proved that the primary planets really moved in such orbits as Kepler had supposed. He afterwards drew up about twelve propositions relating to the motion of the primary planets round the sun, and sent them, at the end of the year 1683, to the Royal Society. Soon after this communication, Dr Halley became known to Newton, and having learned from him that the proof of the propositions respecting the primary planets was completed, earnestly solicited him to finish the work. Accordingly the first edition was printed under the care of Dr Halley, with the title of "Philosophiæ Naturalis Principia Mathematica," and published in 1687.

The reader of the Principia cannot fail of perceiving that Sir Isaac Newton was a most profound geometer. Newton and his cotemporary Maclaurin were the first who applied the consideration of the degrees of equations to the discovery of the general properties and characteristics of curved lines, and to them and Cotes are due the discovery of their most important general properties. Newton has given the results of his investigations on this subject, in his "Enumeratio Linearum tertii ordinis," which, with his tract on the quadrature of curves by the method of fluxions and fluents, were first printed at the end of the first edition of his Optics, in 1704. His principal object was the enumeration of lines contained in an equation of the third degree between two variables. He discovered seventy-two different species, and four more were added by Stirling. In his "Arithmetica Universalis," he has applied the method of Descartes to the solution of geometrical problems and the construction of the roots of equations. The whole works of Newton were edited by Dr Horsley in five quarto volumes, in 1779.

Edmund Halley was one of the most eminent geometers and astronomers of his age. In 1703 he succeeded Dr Wallis as Savilian Professor of Geometry at Oxford. Soon after his appointment, he commenced a translation from the Arabic, of the treatise of Apollonius De Sectione Rationis, and attempted the restoration of his two books De Sectione Spatii, from the account given by Pappus in the seventh book of his Mathematical Collections: the whole of these he published in 1706. The tract de Sectione Rationis, recovered from the Arabic, is a complete specimen of the ancient method of analysis. He also published at Oxford in 1710, an amended translation of seven books of the Conics of Apollonius, and attempted a restoration of the eighth, which was lost. This magnificent folio edition contains the first four books in the original Greek, together with a Latin translation; the next three in Latin from the Arabic version; and the last book in Latin as restored by Dr Halley himself, together with the Lemmas of Pappus and the Commentaries of Eutocius. Halley united a profound knowledge of the ancient geometry with the new geometry of Descartes, and applied the latter to the construction of equations of the third and fourth degrees by means of a parabola and a circle. His memoir on this subject was published in the Philosophical Transactions for 1687. At 63 years of age, he succeeded Flamsteed as Astronomer Royal at Greenwich; and for eighteen years discharged the duties of that office without an assistant. He died at the age of 86, in the year 1742.

James Gregory was an eminent mathematician, the cotemporary and correspondent of Newton, Huygens, Wallis, and others of that time. He was the inventor of a reflecting telescope which bears his name. He discovered a method of drawing tangents to curves, geometrically. His "Geometria Pars Universalis" was first published at Padua, in 1668, and his "Exercitationes Geometrica" in the same year.

Dr David Gregory, the nephew of James, was chosen Savilian Professor of Astronomy, in 1691, in preference to Dr Halley. In 1702 he published his chief work, entitled " Astronomia Physicæ et Geometrica Elementa," founded on the Newtonian hypothesis; and in the following year, the works of Euclid, in Greek and Latin.

Abraham Sharp, a skilful mathematician and expert mechanic, became the amanuensis of Flamsteed, in 1688, and assisted him in his "Historia Cælestis." He had the chief hand in constructing the mural arc at the Greenwich Observatory. He published in 1717, "Geometry Improved, by A. S. Philomath ;" an elaborate treatise, and containing solutions of many difficult problems: he died in 1742, at the age of 91.

Alexis Claude Clairaut was born in 1713, and died in 1765. To Clairaut is due the merit of having exhibited methodically the doctrine of three co-ordinates in space, applied to curve surfaces and the lines which originate in their intersections. His celebrated treatise on this subject was published in 1731, entitled "Traité des Courbes à double Courbure." Before Clairaut, however, M. Pacent, in a memoir read before the French Academy of Sciences, in 1700, had illustrated the extension of the principle of Descartes by a curve surface, expressed by an equation between three variables. Also John Bernouilli had expressed surfaces by an equation, involving three co-ordinates in his solution of the problem, "To find the shortest line which can be drawn on a surface between two given points."