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public exercises or acts founded on every part of the Newtonian system were very common about 1707, and so general were such studies in the university, that the Principia rose to four times its original price.* One of the most ardent votaries of the Newtonian philosophy was Dr. Laughton, who had been tutor in Clare Hall from 1694, and it is probable that during the whole, or at least a greater part, of his tutorship he had inculcated the same doctrines. In 1709-10, when he was proctor of that college, instead of appointing a moderator, he discharged the office himself, and devoted his most active exertions to the promotion of mathematical knowledge. Previous to this, he had even published a paper of questions on the Newtonian philosophy, which appear to have been used as theses for disputations; and such was his ardour and learning that they powerfully contributed to the popularity of his college. Between 1706 and 1716, the year of his death, the celebrated Roger Cotes, the friend and disciple of Newton, filled the Plumian chair of astronomy and experimental philosophy at Cambridge. During this period he edited the second edition of the Principia, which he enriched with an admirable preface, and thus contributed, by his writings as well as by his lectures, to advance the philosophy of his master. About the same time, the learned Dr. Bentley, who first made known the philosophy of his friend to the readers of general literature, filled the high office of master of Trinity College, and could not fail to have exerted his utmost influence in propagating doctrines which he so greatly admired. Had any opposition been offered to the introduction of the true system of the universe, the talents and influence of these individuals would have immediately suppressed it; but no such opposition seems to have been made;

*Nichols's Literary Anecdotes, vol. ii. p. 322. Cotes states in his preface to the second edition of the Principia, that copies of the first edition could only be obtained at an immense price.

and though there may have been individuals at Cambridge ignorant of mathematical science, who adhered to the system of Descartes, and patronised the study of the Physics of Rohault, yet it is probable that similar persons existed in the universities of Edinburgh and St. Andrew's; and we cannot regard their adherence to error as disproving the general fact, that the philosophy of Newton was quickly introduced into all the universities of Great Britain.

But while the mathematical principles of the Newtonian system were ably expounded in our seats of learning, its physical truths were generally studied, and were explained and communicated to the public by various lecturers on experimental philosophy. The celebrated Locke, who was incapable of understanding the Principia from his want of mathematical knowledge, inquired of Huygens if all the mathematical propositions in that work were true. When he was assured that he might depend upon their certainty, he took them for granted, and carefully examined the reasonings and corollaries deduced from them. In this manner he acquired a knowledge of the physical truths in the Principia, and became a firm believer in the discoveries which it contained. In the same manner he studied the treatise on Optics, and made himself master of every part of it which was not mathematical.* From a manuscript of Sir Isaac Newton's, entitled "A demonstration that the planets, by their gravity towards the sun, may move in ellipses,† found among the papers of Mr. Locke, and published by Lord King," it would appear that he himself had been at considerable trouble in explaining to his friend that interesting doctrine. This manuscript is endorsed, “Mr. Newton, March, 1689." It begins with three hypotheses

Preface to Desaguliers's Experimental Philosophy. Dr. Desaguliers states that he was told this anecdote several times by Sir Isaac Newton himself. †The Life of John Locke, p. 209-215, Lond. 1829.

(the first two being the two laws of motion, and the third the parallelogram of motion), which introduce the proposition of the proportionality of the areas to the times in motions round an immoveable centre of attraction.* Three lemmas, containing properties of the ellipse, then prepare the reader for the celebrated proposition, that when a body moves in an ellipse,f the attraction is reciprocally as the square of the distance of the body from the focus to which it is attracted. These propositions are demonstrated in a more popular manner than in the Principia, but there can be no doubt that, even in their present modified form, they were beyond the capacity of Mr. Locke.

Dr. John Keill was the first person who publicly taught natural philosophy by experiments. Desaguliers informs us that this author" laid down very simple propositions,which he proved by experiments, and from these he deduced others more compound, which he still confirmed by experiments, till he had instructed his auditors in the laws of motion, the principles of hydrostatics and optics, and some of the chief propositions of Sir Isaac Newton concerning light and colours. He began these courses in Oxford about the year 1704 or 1705, and in that way introduced the love of the Newtonian philosophy." When Dr. Keill left the university, Desaguliers hegan to teach the Newtonian philosophy by experiments. He commenced his lectures at Harthall in Oxford, in 1710, and delivered more than a hundred and twenty courses; and when he went to settle in London in 1713, he informs us that he found "the Newtonian philosophy generally received among persons of all ranks and professions, and even among the ladies by the help of experiments." Such were the steps by which the Newtonian philosophy was established in Great Britain. From † Ib. lib. i. prop. xi.

* Principia, lib. i. prop. 1.

the time of the publication of the Principia, its mathematical doctrines formed a regular part of academical education; and before twenty years had elapsed, its physical truths were communicated to the public in popular lectures illustrated by experiments, and accommodated to the capacities of those who were not versed in mathematical knowledge. The Cartesian system, though it may have lingered for a while in the recesses of our universities, was soon overturned; and long before his death, Newton enjoyed the high satisfaction of seeing his philosophy triumphant in his native land.

CHAPTER XII.

Doctrine of Infinite Quantities-Labours of Pappus-Kepler-Cavaleri -Roberval-Fermat-Wallis-Newton discovers the Kinomial Theorem-and the Doctrine of Fluxions in 1666-His Manuscript Work containing this Doctrine communicated to his Friends-His Treatise on Fluxions-His Mathematical Tracts-His Universal ArithmeticHis Methodus Differentialis-His Geometria Analytica-His Solution of the Problems proposed by Bernouilli aud Leibnitz-Account of the celebrated Dispute respecting the Invention of Fluxions-Commercium Epistolicum-Report of the Royal Society-General View of the Controversy.

PREVIOUS to the time of Newton, the doctrine of infinite quantities had been the subject of profound study. The ancients made the first step in this curious inquiry by a rude though ingenious attempt to determine the area of curves. The method of exhaustions which was used for this purpose consisted in finding a given rectilineal area to which the inscribed and circumscribed polygonal figures continually approached by increasing the number of their sides. This area was obviously the area of the curve, and in the case of the parabola it was found by Archimedes to be two-thirds of the area

formed by multiplying the ordinate by the abscissa. Although the synthetical demonstration of the results was perfectly conclusive, yet the method itself was limited and imperfect.

The celebrated Pappus of Alexandria followed Archimedes in the same inquiries; and in his demonstration of the property of the centre of gravity of a plane figure, by which we may determine the solid formed by its revolution, he has shadowed forth the discoveries of later times.

In his curious tract on Stereometry, published in 1615, Kepler made some advances in the doctrine of infinitesimals. Prompted to the task by a dispute with the seller of some casks of wine, he studied the measurement of solids formed by the revolution of a curve round any line whatever. In solving some of the simplest of these problems, he conceived a circle to be formed of an infinite number of triangles having all their vertices in the centre, and their infinitely small bases in the circumference of the circle, 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, too, in solving some of the more difficult of the problems which he himself proposed roused the attention of geometers, and seems particularly to have attracted the notice of Cavaleri.

This ingenious mathematician was born at Milan in 1598, and was Professor of Geometry at Bologna. In his method of Indivisibles, which was published in 1635, he considered 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 lays it down as an axiom that the infinite sums of such lines and surfaces have the same ratio when compared with the linear or superficial unit, as the surfaces and solids which are to be determined. As it is not true that an infinite

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