which teaches how to find the area of a triangle by means of its three sides, without the aid of a perpendicular. Maurolicus was a respectable geometer, and wrote on various subjects; his treatise on the Conic Sections is remarkable for its perspicuity and elegance. Aurispa, Batecombe, Butes, Ramus, Xylander, Fortius, Cardan, Fregius, Bombelli, Ficinus, Durer, Zeigler, Fernel, Ubaldi, Clavius, Barbaro, Byrgius, Commandine, Pelletier, Dryander, Nonius, Linacre, Sturmius, Saville, Ghetaldus, R. Snellius, and many others who flourished at this period, were cultivators of Geometry; and if they made few discoveries, still their labours as translators, commentators, or teachers, were beneficial in diffusing knowledge, and merit our grateful acknowledgments.

Various approximations to the ratio of the circumference of a circle to its diameter, were given about the beginning of the 17th century, approaching much nearer the truth than any that had hitherto appeared; viz. by Adrian Romanus, Willebrord Snellius, Peter Metius, and Ludolph Van Ceulen; according to the conclusion of Metius, if the diameter be 113, the circumference will be 355, which is very near the truth, and has the advantage of being expressed by small numbers. By continual bisection of the circumference, Van Ceulen found, that if the diameter be 1, the circumference will be 3,14159, &c. to 36 places of decimals; which discovery was thought so curious, that the numbers were engraved on his tomb in St. Peter's Church-yard, at Leyden ".

The simplest (and consequently least accurate) ratio of the diameter to the circumference is as 1 to 3 ; a ratio somewhat nearer than this, is as 6 to 19. We have noticed before that Archimedes determined the ratio to be as 7 to 22 nearly, which is nearer than the above.

Geometrical problems had long before this period been solved algebraically, by Cardan, Tartalea, Regiomontanus, and Bombelli; but a regular and general method of applying Algebra to Geometry, was first given by Vieta, about the year 1580; as also the elements of angular sections. Des Cartes improved the discovery of Vieta, by introducing a general method of representing the nature and circumstances of curve lines by algebraic equations, distributing curves into classes, corresponding to the different orders of equations by which they are expressed; A.D. 1637. A method of tangents, and a method de maximis et minimis, much resembling that of fluxions or increments, owe their origin to Fermat, a learned countryman and competitor of Des Cartes, with whom he disputed the honour of first applying Algebra to curve lines, and to the geometrical con⚫struction of equations, secrets of which he was in possession before Des Cartes' Geometry appeared. About this time, or a little earlier, Galileo invented the cycloid; its properties were afterwards demonstrated by Torricellius.

The improvement of Des Cartes, now called the new Geometry, was cultivated with ardour and success by mathematicians in almost every part of Europe; his work was translated out of French into Latin, and published by Francis Schooten, with a commentary by Schooten, and notes by M. de Beaune, 1649. The Indivisibles of Cavalerius, published in 1635, was a new and useful invention, applied to

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Van Ceulen's numbers, as mentioned above, were extended to 72 places of figures by Mr. Abraham Sharp, about 1706; Mr. Machin afterwards extended the same to 100 places, and M. De Lagni has carried them to the amazing length of 128 places: thus, if the diameter be 1000, &c. (to 128 places) the circumference will be 31415, 92653, 58979, 32384, 62643, 38327, 95028, 84197, 16939, 93751, 05820, 97494, 45923, 07816, 40628, 62089, 98628, 03482, 53421, 17067, 98214, 80865, 13272, 30664, 70938, 446+, or 7-. This number (which includes those of Van Ceulen, Sharp, and Machin) is sufficiently near the truth for any purpose, so that except the ratio could be completely found, we need not wish for a greater degree of accuracy.

determine the areas of curves, the solidities of bodies generated by their revolution about a fixed line, &c. Roberval, aş early as 1634, had employed a similar method, which he applied to the cycloid, a curve at that time much celebrated for its numerous and singular properties; he likewise invented a general method for tangents, applicable alike to geometrical and mechanical curves. The inverse method of tangents derived its origin from a problem, which De Beaune proposed to his friend Des Cartes, in 1647. In 1655 the learned Dr. Wallis published his Arithmetica Infinitorum; being either a new method of reasoning on quantities, or else a great improvement on the Indivisibles of Cavalerius above mentioned; speculations which led the way to infinite series, the binomial theorem, and the method of fluxions: this work treats of the quadrature of curves and many other problems, and gives the first expression known for the area of a circle by an infinite series.

One of the greatest discoveries in modern Geometry was the theory of evolutes, the author of which was Christian Huygens, an ingenious Dutch mathematician, who published it at the Hague in 1658, in a work entitled, Horologium Oscillatorium, sive de Motu Pendulorum, &c.

In 1669 were published Dr. Barrow's Optical and Geometrical Lectures, containing many very ingenious and profound researches on the dimensions and properties of curves, and especially a method of tangents, by a mode of calculation differing from that of fluxions or increments in scarcely any particular, except the notation. About this time the use of geometrical loci for the solution of equations, was carried to a great degree of perfection by Slusius, a canon of Liege, in his Mesolabium et Problemata Solida: he likewise inserted in the Philosophical Transactions, a short and easy method of drawing tangents to all geometrical curves, with a demonstration of the same; and likewise a tract on the

Optic Angle of Alhazen. Besides those we have mentioned, many others of this period devoted their attention to the rectification and quadrature of curves, &c. of whom Van Heuraet, Rolle, Pascal, Briggs, Halley, Lallouère, Torricellius, Herigon, Niell, Sir Christopher Wren, Faber, Lord Brouncker, Nicholas Baker, G. St. Vincent, Mercator, Gregory, and Leibnitz, were the principal.

The seventeenth century is famed for giving birth to two noble discoveries; namely, that of logarithms in 1614 by Lord Napier, whereby the practical applications of Geometry are greatly facilitated; and that of fluxions, to which problems relating to infinite series, the quadrature and properties of curves, and other geometrical subjects connected with Astronomy, Physics, &c. and which were formerly considered as beyond the reach of human sagacity, readily submit. For this sublime discovery, the learned are indebted either to the profound and penetrating genius of Sir Isaac Newton', or

Sir Isaac Newton, one of the greatest mathematicians and philosophers that ever lived, was born in Lincolnshire, in 1642. Having made some proficiency in the classics, &c. at the grammar school at Grantham, he (being an only child) was taken home by his mother (who was a widow) to be her companion, and to learn the management of his paternal estate: but the love of books and study occasioned his farming concerns to be neglected. In 1660 he was sent to Trinity College, Cambridge: here he began with the study of Euclid, but the propositions of that book being too easy to arrest his attention long, he passed rapidly on to the Analysis of Des Cartes, Kepler's Optics, &c. making occasional improvements on his author, and entering his observations, &c. on the margin. His genius and attention soon attracted the favourable notice of Dr. Barrow, at that time one of the most eminent mathematicians in England, who soon became his steady patron and friend. In 1664 he took his degree of B. A. and employed himself in speculations and experiments on the nature of light and colours, grinding and polishing optic glasses, and opening the way for his new method of fluxions and infinite series. The next year, the plague which raged at Cambridge obliged him to retire into the country; here he laid the foundation of his universal system of gravitation, the first hint of which he received from seeing an apple fall from a tree; and subsequent reasoning induced him to conclude, that the same force which brought down the apple might possibly extend to the moon, and retain her in her orbit: he afterwards extended the doctrine to all the bodies which compose the solar system, and..

to that of Leibnitz, or to both, for both laid claim to the invention. No sooner was the method made public, than a

demonstrated the same in the most evident manner, confirming the laws which Kepler had discovered, by a laborious train of observation and reasoning; namely, that "the planets move in elliptical orbits;" that "they describe equal areas in equal times;" and that "the squares of their periodic times are as the cubes of their distances." Every part of natural philosophy not only received improvement by his inimitable touch, but became a new science under his hands: his system of gravitation, as we have observed, confirmed the discoveries of Kepler, explained the immutable laws of nature, changed the system of Copernicus from a probable hypothesis to a plain and demonstrated truth, and effectually overturned the vortices and other imaginary machinery of Des Cartes, with all the improbable epicycles, deferents, and clumsy apparatus, with which the ancients and some of the moderns had encumbered the universe. In fact, his Philosophiæ Naturalis Principia Mathematica contains an entirely new system of philosophy, built on the solid basis of experiment and observation, and demonstrated by the most sublime Geometry; and his treatises and papers on optics supply a new theory of light and colours. The invention of the reflecting telescope, which is due to Mr. James Gregory, would in all probability have been lost, had not Newton interposed, and by his great improvements brought it forward into public notice.

In 1667 Newton was chosen fellow of his College, and took his degree of M.A. Two years after, his friend Dr. Barrow resigned to him the mathematical chair; he became a Member of Parliament in 1688, and through the interest of Mr. Montagu, Chancellor of the Exchequer, who had been educated with him at Trinity College, our author obtained in 1696 the appointment of Warden, and three years after that of Master, of the Mint: he was elected in 1699 member of the Royal Academy of Sciences at Paris; and in 1703 President of the Royal Society, a situation which he filled during the remainder of his life, with no less honour to himself than benefit to the interests of science.

In 1705, in consideration of his superior merit, Queen Anne conferred on him the honour of knighthood: he died on March 20th, 1727, in the 85th year of his age. Virtue is the brightest ornament of science: Newton is indebted to this for the best part of his fame; he was a great man, and good as he was great to the most exemplary candour, moderation, and affability, he added every virtue necessary to constitute a truly moral character; above all, he felt a firm conviction of the truth of Revelation, and studied the Bible with

the greatest application and diligence. But such is the folly of man,

that the tribute, which is due to the GREAT FIRST CAUSE ALONE, we transfer to the instrument; Newton, Marlborough, Nelson, Wellington, &c. have all our praise, while the GREAT SOURCE of knowledge, strength, victory, and every benefit we enjoy, is forgotten. How would the modest Newton have reddened with shame and indignation, could he have heard all the extravagant encomiums, little short of adoration, which have with foolish and