gHcr. i.] DISSERTATION SECOND. 15 



making more discoveries of (be same kind. It was a de- 

 monstration purely synthetical, and required, as all indi- 

 rect reasoning must do, that the conclusion should be known 

 before the reasoning is begun. A more compendious, and 

 a more analytical method, was therefore much to be wished 

 for, and was an improvement, which, at a moment when the 

 field of mathematical science was enlarging so fast, seemed 

 particularly to be required. 



Cavalleri, born at Milan in the year 1598, is the person 

 by whom this great improvement was made. The princi- 

 ple on which he proceeded was, that areas may be consi- 

 dered as made up of an infinite number of parallel lines ; 

 solids of an infinite number of parallel planes ; and even 

 lines themselves, whether curve or straight, of an infinite 

 number of points. The cubature of a solid being thus re- 

 duced to the summation of a series of planes, and the quad- 

 rature of a curve to the summation of a series of ordinales, 

 each of the investigations was reduced to something more 

 simple. It added to this simplicity not a little, that the 

 sums of series are often more easily found, when the num- 

 ber of terms is infinitely great, than when it is finite, and 

 actually assigned. 



It appears that a tract on stereometry, written by Kep- 

 ler, whose name will hereafter be often mentioned, first 

 led Cavalleri to take this view of geometrical magnitudes. 

 In that tract, which was published in 1615, the measure- 

 ment of many solids was proposed, which had not before 

 fallen under the consideration of mathematicians. Such, 

 for example, was that of the solids generated by the re- 

 volution of a curve, not about its axis, but about any line 

 whatsoever. Solids of that kind, on account of their af- 

 finity with the figure of casks, and vessels actually employ- 

 ed for containing liquids, appeared to Kepler to offer both 

 curious and useful subjects of investigation. There were 



