The method of indivisibles, from the great facility with which it could be managed, furnished a most ready method of ascertaining the ratios of areas and solids to one ano ther, and, therefore, scarcely seems to deserve the epithet which Newton himself be stows upon it, of involving in its conceptions something harsh; (durusn,) and not easy to be admitted. It was the doctrine of infinitely small quantities carried to the extreme, and gave at once the result of an infinite series of successive approximations. Nothing, perhaps, more ingenious, and certainly nothing more happy, ever was contrived, than to arrive at the conclusion of all these approximations, without going through the approxi mations themselves. This is the purpose served by introducing into mathematics the con sideration of quantities infinitely small in size, and infinitely great in number ; ideas which, however inaccurate they may seem, yet, when carefully and analogically reasoned upon, have never led into error.
Geometry owes4o Cavalleri, not only the general method just described, but many par ticular theorems, which that method was the instrument of discovering.. Among these is the very remarkable proposition, that as four right angles, to the excess of the three angles of any spherical triangle, above two right angles, so is the superfries of the hemisphere to the area of the triangle. At that time, however, science was advancing so fast, and the human mind was everywhere expanding itself with so much energy, that the same dis covery was likely to be made by more individuals than one at the same time. It was not known in Italy in 1632, when this determination of the area of a spherical triangle was given by Cavalleri, that it had been published three years before by Albert Girard, a mathematician of the Low Countries, of whose inventive powers we shall soon have more occasion to speak.
The Cycloid afforded a number of problems, well calculated to exercise the proficients in the geometry of indivisibles, or of infinites. It is the curve described by a point in the circumference of a circle, while the circle itself rolls in a straight line along a plane. It is not quite certain when this curve, so remarkable for its curious properties, and for the place which it occupies in the history of geometry, first drew the attention of mathemati cians. In the year 1639, Galileo informed his friend Torricelli, that, forty years before that time, he had thought of this curve, on account of its shape, and the graceful form it would give to arches in. architecture. The same philosopher had endeavoured to find the
area of the cycloid ; but though he was one of those who first introduced the consideration of infinites into geometry, he was not expert enough in the use of that doctrine, to be • able to resolve this problem. It is still more extraordinary, that the same problem proved too difficult for Cavalleri, though he certainly was in complete possession of the principles . by which it was to be resolved. It is, however, not easy to determine whether it be to Torricelli, the scholar of Cavalleri, and his successor in genius and talents, or to Roberval, a French mathematician of the same period, and a man also of great originality and inven tion, that science is indebted for the first quadrature of the cycloid, or the proof that its area is three times that of its generating circle. Both these mathematicians laid claim to it. The French and Italians each took the part of their own countryman ; and in their zeal have so perplexed the question, that it is hard to say on which side the truth is to be found. Torricelli, however, was a man of a mild, amiable, and candid disposition ; Ro berval of a temper irritable, violent, and envious ; so that, in as far as the testimony of the individuals themselves is concerned, there is no doubt which ought to preponde rate. They had both the skill and talent which fitted them for this, or even for more difficult researches.
The other properties of this curve, those that respect its tangents, its length, its curva ture, &c. exercised the ingenuity, not only of the geometers just mentioned, but of Wien, 1 Wallis, Huygens, and, even after the invention of the integral calculus, of Newton, Leib nitz, and Bernoulli.
Roberval also improved the method of quadratures invented by Cavalleri, and extended his solutions to the case, when the powers of the terms in the arithmetical progression of which the sum was to be found were fractional ; and Wallis added the case when they were negative. Fermat, who, in his inventive resources, as well as in the correctness of his mathematical taste, ' yielded to none of his contemporaries, applied the consideration of infinitely small quantities to determine the maxima and minima of the ordinates of curves, as also their tangents. Barrow, somewhat later, did the same in England. After wards the geometry of infinites fell into the hands of Leibnitz and Newton, and acquired that new character which marks so distinguished an era in the mathematical sciences.