The problems proposed by Kepler probably led to the invention of the methods of Guldin and Cavallerius. The principal discovery of Guldin consisted in an application which he made of a property of the centre of gravity, to the measure of solids produced by revolution. " Every figure," says he, " formed by the rotation of a line, or a surface about an immoveable axis, is the product of the generating quantity by the line described by its centre of gravity. This- principle, certainly one of the most beauti ful discoveries in geometry, was however known in the days of Pappus ; for it is distinctly stated at the end of the preface to his seventh book ; yet Guldin takes no notice of the circumstance.
To Cavallerius we are indebted for the doctrine of indi visibles, which he published in 1635. In this, he consider ed a line as made up of an infinite number of points, a sur face, of an infinite number of lines, and a solid, an in finite number of surfaces : these elements of magnitudes he called Indivisibles. The introduction of so bold a pos tulate into geometry, was opposed by some of his contem poraries; but in answer, the Italian geometer explained that this hypothesis was by no means an essential part of his theory, which, in fact, was the same as the ancient method of exhaustions, but free from its tedious and indi rect modes of reasoning.
In the first place, he considered such figures as had their increasing or decreasing elements at equal heights above the base, always in a given ratio ; and he shewed that the figures themselves were to each other in the same given ratio. Next, he compared figures composed of an increasing or decreasing series of elements, with others in which the elements were all equal: for example, a cone, which he considered as composed of an infinite number of circles, increasing from the base to the vertex, with a cy linder, which is composed of an infinite number of circles) all of the same size ; and to determine the ratio of the con tents of the two solids, he found the ratio of the sum of the decreasing circles in the cone, to the sum of the circles which were equal to one another in the cylinder. In the cone, these circles decrease from the base to the vertex as the squares of the terms of an arithmetical progression. In other solids, they form other progressions: for exam ple, in the parabolic conoid, it is simply that of an arith metical progression. The general object of the method is to assign the ratio of this sum of an increasing or decreas ing series of terms, with that of the equal terms which form an uniform and known figure of the same base and altitude. The method of indivisibles is now superseded by the more extensive doctrine of fluxions ; yet it was of im mense importance at the time it was invented, and in fact was one step towards that grand discovery.
The French geometers pursued the same career of dis covery, and almost at the same time. as Cavallerins ; they even resolved more difficult problems. In 1636, Fermat had found the area of a spiral, of a different nature from that which Archimedes had handled; and soon afterwards, he proposed to Roberval to determine the areas of parabo lic curves of the higher orders, (See FERMAT.) Roberval quickly resolved the problem ; and he also determined their tangents. Fermat, again, on his part, found their centres of gravity. Roberval claimed the merit of having invented for himself a theory altogether similar to that of Cavallerius, before the latter had made his known ; but as his selfish views led him to conceal it, that he might tri umph over his contemporaries, he has hut little claim on the gratitude of posterity as a discoverer, although he de serves credit for his skill as a geometer. Roberval did not venture to deviate so much from the common language of geometry as Cavallerius; he conceived his surfaces and solids to be made up of an indefinite number of very nar now rectangles and thin prisms, which decreased according to a certain law.
The celebrated Descartes contributed in no small de gree to the developement of these new and brilliant disco veries in geometry. When Mersennus had sent him an account of Fermat's method of finding the centre of gra vity of conoids, Descartes quickly sent him the determina tion of the centres of gravity of all parabolas, also their general quadrature, their tangents, and the ratios of their conoids.
It was in this period that the logarithmic spiral and cy cloid were brought into discussion ; the former was sug gested by Descartes, the latter was first noticed by Galileo.
See EPICYCLOID.
Passing over many geometers of ordinary merit, we must notice Pascal, who, at the age of twelve, had such a turn for geometry, that he undertook to construct a system for himself, guided by the recollection of the conversations which he had heard among the mathematicians that visited his father, who was himself a mathematician. He had gone as far as to discover that the three angles of any triangle were equal to two right angles, when he was •observed by his father. At the age of 16, he is said to have composed a treatise on conic sections, in which all that Apollonitts had demonstrated was elegantly deduced from a single proposition : this was shewn to Descartes, but the philoso pher could not believe it to be the work of so young a geo meter. The hopes lie had so early excited, and the ele gance of his disquisitions on the cycloid, gave geometers reason to regret that a larger portion of his short life was not dedicated to the science. He died in 1662, aged 39.