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HISTORY OF THE INFINITESIMAL CALCULUS — ARCHIMEDES — PAPPUS — NAPIER — EDWARD WRIGHT — KEPLER'S TREATISE ON STEREOMETRY — CAVALIERI'S GEOMETRIA INDIVISIBILIUM — ROBERVAL — TORICELLI — FERMAT — WALLI'S ARITHMETICA INFINITORUM — HUDDE — GREGORY — SLUSIUS — NEWTON'S DISCOVERY OF FLUXIONS IN 1655 — GENERAL ACCOUNT OF THE METHOD, AND OF ITS APPLICATIONS — HIS ANALYSIS PER EQUATIONES, ETC. — HIS DISCOVERIES COMMUNICATED TO ENGLISH AND FOREIGN MATHEMATICIANS — THE METHOD OF FLUXIONS AND QUADRATURES — ACCOUNT OF HIS OTHER MATHEMATICAL WRITINGS — HE SOLVES THE PROBLEMS PROPOSED BY BERNOULLI AND LEIBNITZ — LEIBNITZ VISITS LONDON, AND CORRESPONDS WITH THE ENGLISH MATHEMATICIANS, AND WITH NEWTON THROUGH OLDENBURG — HE DISCOVERS THE DIFFERENTIAL CALCULUS, AND COMMUNICATES IT TO NEWTON — NOTICE OF OLDENBURG — CELEBRATED SCHOLIUM RESPECTING FLUXIONS IN THE PRINCIPIA — ACCOUNT OF THE CHANGES UPON IT — LEIBNITZ'S MANUSCRIPTS IN HANOVER.

IN the history of Newton's optical and astronomical discoveries, which we have given in the preceding chapters, we have seen him involved in disputes with his own countrymen as well as with foreigners, in reference to the value and the priority of his labours. Such extreme sensitiveness as that with which he felt the criticisms and discussed the claims of his opponents, has been seldom

In the history of Newton's mathematical discoveries, which the same dread of controversy had induced him to withhold from the world, we shall find him involved in more exciting discussions, — in what may even be called quarrels, in which both the temper and the character of the disputants were severely tried. In the controversy respecting the discovery of fluxions, or of the differential calculus, Newton took up arms in his own cause, and though he never placed himself in the front rank of danger, he yet combated with all the ardour of his comrades. Hitherto it had been his lot to contend with

Painful as the sight must always be when superior minds are brought into collision, society gains from the contest more than the parties lose. We are too apt to regard great men, of the order of Newton and Leibnitz, as exempt from the common infirmities of our nature, and to worship them as demigods more than to admire them as sages. In the history upon which we are about to enter we shall see distinguished philosophers upon the stage, superior, doubtless, to their fellows, but partaking in all the frailties of temper, and exposed to all the suspicions of injustice, which embitter the controversies of ordinary life.

Although the honour of having invented the calculus of fluxions, or the differential calculus, has been conferred upon Newton and Leibnitz, yet, as in every other great invention, they were but the individuals who combined the scattered lights of their predecessors, and gave a method, a notation, and a name, to the doctrine of quantities infinitely small.

By an ingenious attempt to determine the area of curves the ancients made the first step in this interesting inquiry. Their principles were sound, but their want of an organized method of operation prevented them from

Pappus of Alexandria, who flourished about the close of the fourth century, followed Archimedes in the same inquiries, and his celebrated theorems on the centre of gravity

Guldinus gave this theorem in 1635, and seeing that he was acquainted with Pappus, Montucla and others were disposed to regard him as a plagiarist. Had they studied Pappus in Condamine's Latin, in place of that of Halley, they never would have known the theorem but from Guldinus.

is the only fruit which sprung from the seed sown by the Greek geometer till we reach the commencement of the seventeenth century. We search in vain the writings of Cardan, Tartaglia, Vieta, and Stevinus, for any proof of their power to employ the infinitesimal principle.Our countryman, John Napier of Merchiston, and his contemporary, Edward Wright, were not only the first to revive the use of the infinitesimal principle, but the first who applied it in an arithmetical form. They

In his treatise on Stereometry, published in 1615, Kepler made some advances in the doctrine of infinitesimals. In consequence of a dispute with a wine-merchant he studied the mensuration of round solids, or those which are formed by the revolution of the conic sections round any line whatever within or without the section. He considered the circle as consisting of an infinite number of triangles, having their vertices in the centre, and their infinitely small bases in the circumference. In like manner, he considered the cone as composed of an infinite number of pyramids, and the cylinder of an infinite number of prisms, and by thus rendering familiar the idea of quantities infinitely great and infinitely small, he gave an impulse to this branch of mathematics.

The failure of Kepler in solving some of the more difficult problems which he himself proposed, drew the attention of geometers to the subject of infinitely small Geometria Indivisibilium did not appear till 1635, nor his Exercitationes, containing his most remarkable results, till 1647. He considers a line as composed of an infinite number of points, a surface of an infinite number of lines, and a solid of an infinite number of surfaces, and he assumes as an axiom, that the infinite sums of such lines and surfaces have the same ratio, existing in equal numbers in different surfaces or solids, as the surfaces or solids to be determined. As it is not true that an infinite number of infinitely small points can make a line, nor an infinite number of infinitely small lines a surface, Pascal proposed to return to the idea of Kepler by considering a line as composed of an infinite number of infinitely short lines, — a surface as composed of an infinite number of infinitely narrow parallelograms, and a solid of an infinite number of infinitely thin solids. If Cavalieri had been more advanced in algebra he might, perhaps, have gone farther; but he was undoubtedly the first who applied the algebraical process to the quadrature of parabolas of an integer order; and his chief instrument, as it was afterwards that of Wallis, was the theorem, that