the parallelogram, each of the lines is n and their number is n ; hence the total sum of their squares is n 3 . The ratio between the two sums is therefore n ( n + 1 ) ( 2 n + 1 ) ÷ 6 n 3 = 1 3 , since n is infinite. From this he concludes that the pyramid or cone is respectively 1 3 of a prism or cylinder of equal base and altitude, since the polygons or circles composing the former decrease from the base to the apex in the same way as the squares of the lines parallel to the base in a triangle decrease from base to apex. By the Method of Indivisibles, Cavalieri
solved the majority of the problems proposed by Kepler.
Though expeditious and yielding correct results, Cavalieri's method lacks a scientific foundation. If a line has absolutely no width, then no number, however great, of lines can ever make up an area; if a plane has no thickness whatever, then even an infinite number of planes cannot form a solid. The reason why this method led to correct conclusions is that one area is to another area in the same ratio as the sum of the series of lines in the one is to the sum of the series of lines in the other. Though unscientific, Cavalieri's method was used for fifty years as a sort of integral calculus. It yielded solutions
to some difficult problems. Guldin made a severe attack
on Cavalieri and his method. The latter published in 1647, after the death of Guldin, a treatise entitled Exercitationes geometricæ sex, in which he replied to the objections of his opponent and attempted to give a clearer explanation of his method. Guldin had never been able to demonstrate the theorem named after him, except by metaphysical reasoning, but Cavalieri proved it by the method of indivisibles. A revised edition of the Geometry of Indivisibles appeared in 1653.
There is an important curve, not known to the ancients, which now began to be studied with great zeal. Roberval gave it the name of "trochoid," Pascal the name of "roulette,"
Galileo the name of "cycloid." The invention of this curve
seems to be due to Galileo, who valued it for the graceful form it would give to arches in architecture. He ascertained its area by weighing paper figures of the cycloid against that of the generating circle, and found thereby the first area to be nearly but not exactly thrice the latter. A mathematical determination was made by his pupil, Evangelista Torricelli
(1608–1647), who is more widely known as a physicist than as a mathematician.
By the Method of Indivisibles he demonstrated its area to be triple that of the revolving circle, and published his solution. This same quadrature had been effected a few years earlier by Roberval in France, but his solution was not known
to the Italians. Roberval, being a man of irritable and violent disposition, unjustly accused the mild and amiable Torricelli of stealing the proof. This accusation of plagiarism created so much chagrin with Torricelli that it is considered to have been the cause of his early death. Vincenzo Viviani,
another prominent pupil of Galileo, determined the tangent to the cycloid. This was accomplished in France by Descartes and Fermat.