(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 began 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.

*Principia, lib. i. prop. i.

From

Ib. lib. i. prop. xi.

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 Binomial Theorem-and the Doctrine of Fluxions in 1666-His Manuscript Work containing this Doctrine communicated to his Friends-His Treatise on Fluxions-His Mathematicul 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 are of the curve, and in the case of the parabola it we 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

P

number of infinitely small points can make a line, or an infinite number of infinitely small lines a surface, Pascal removed this verbal difficulty 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. But, independent of this correction, the conclusions deduced by Cavaleri are rigorously true, and his method of ascertaining the ratios of areas and solids to one another, and the theorems which he deduced from it may be considered as forming an era in mathematics.

By the application of this method, Roberval and Toricelli showed that the area of the cycloid is three times that of its generating circle, and the former extended the method of Cavaleri to the case where the powers of the terms of the arithmetical progression to be summed were fractional.

In applying the doctrine of infinitely small quantities to determine the tangents of curves, and the maxima and minima of their ordinates, both Roberval and Fermat made a near approach to the invention of fluxions--so near indeed that both Lagrange and Laplace* have pronounced the latter to be the true inventer of the differential calculus. Roberval supposed the point which describes a curve to be actuated by two motions, by the composition of which it moves in the direction of a tangent; and had he possessed the method of fluxions, he could, in every case, have determined the rela tive velocities of these motions, which depend on the nature of the curve, and consequently the direction of the tangent which he assumed to be in the diagonal of a parallelogram whose sides had the

* "On peut regarder Fermat," says Lagrange, "comme le premier inventeur des nouveaux calculs ;" and Laplace observes, "Il paraitque Fermat, le veritable inventeur du calcul differentiel, l'ait envisagé comme un cas particulier de celui des differences," &c.

same ratio as the velocities. But as he was able to determine these velocities only in the conic sections, &c. his ingenious method had but few applications.

The labours of Peter Fermat, a counsellor of the parliament of Toulouse, approached still nearer to the fluxionary calculus. In his method of determining the maxima and minima of the ordinates of curves, he substitutes +e for the independent variable in the function which is to become a maximum, and as these two expressions should be equal when e becomes infinitely small or 0, he frees this equation from surds and radicals, and after dividing the whole by e, e is made=0, and the equation for the maximum is thus obtained. Upon a similar principle he founded his method of drawing tangents to curves. But though the methods thus used by Fermat are in principle the same with those which connect the theory of tangents and of maxima and minima with the analytical method of exhibiting the differential calculus, yet it is a singular example of national partiality to consider the inventer of these methods as the inventer of the method of fluxions.

"One might be led," ays Mr. Herschel," to suppose by Laplace's expression that the calculus of finite differences had then already assumed a systematic form, and that Fermat had actually observed the relation between the two calculi, and derived the one from the other. The latter conclusion would scarcely be less correct than the former. No method can justly be regarded as bearing any analogy to the differential calculus which does not lay down a system of rules (no matter on what considerations founded, by what names called, or by what extraneous matter enveloped) by means of which the second term of the development of any function of ate in powers of e, can be correctly calculated, 'quæ extendet se,' to use Newton's expression,