16 DISSERTATION SECOND. [part i. 



no less than eighty-four such solids, which he proposed 

 for the consideration of mathematicians. He was, how- 

 ever, himself unequal to the task of resolving any but a 

 small number of the simplest of these probleuis. In these 

 solutions, he was bold enough to introduce into geometry, 

 for the first time, the idea of infinitely great and infinitely 

 small quantities, and by this apparent departure from the 

 rigour of the science, he rendered it in fact a most essen- 

 tial service. Kepler conceived a circle to be composed 

 of an infinite number of triangles, having their common 

 vertex in the centre of the circle, and their infinitely small 

 bases in the circumference. It is to be remarked, that 

 Galileo had also introduced the notion of infinitely small 

 quantities, in his first dialogue, De Mechanica, where he 

 treats of a cylinder cut out of a hemisphere ; and he has 

 done the same in treating of the acceleration of falling 

 bodies. Cavalleri was the friend and disciple of Galileo, 

 but much more profound in the mathematicks. In his 

 hands the idea took a more regular and systematick form, 

 and was explained in his work on indivisibles, published 

 in 1635. 



The rule for summing an infinite series of terms in arith- 

 metical progression had been long known, and the appli- 

 cation of it to find the area of a triangle, according to the 

 method of indivisibles, was a matter of no difficulty. The 

 next step was, supposing a series of lines in arithmetical 

 progression, and squares to be described on each of them, 

 to find what ratio the sum of all these squares bears to the 

 greatest square, taken as often as there are terms in the 

 progression. Cavalleri showed, that when the number of 

 terms is infinitely great, the first of these sums is just one 

 third of the second. This evidently led to the cubaturc 

 of many solids. 



