22 DISSERTATION SECOND. [rART n. 



chimedes. These geometers observed, and, for what we 

 know, were the first to observe, that the approach which 

 a rectilineal figure may make to one that is curvilineal, 

 by the increase of the number of its sides, the diminution 

 of their magnitude, and a certain enlargement of the an- 

 gles they contain, may be such that the properties of the 

 former shall coincide so nearly with those of the latter, 

 that no real difference can be supposed between them 

 without involving a contradiction ; and it was in ascertaining 

 the conditions of this approach, and in showing the contra- 

 diction to be unavoidable, that the method of Exhaustions 

 consisted. The demonstrations were strictly geometrical, 

 but they were often complicated, always indirect, and of 

 course synthetical, so that they did not explain the means 

 by which they had been discovered. 



At the distance of more than two thousand years, Cava- 

 Ueri advanced a step farther, and, by the sacrifice of some 

 apparent, though of no real accuracy, explained, in the me- 

 thod of indivisibles, a principle which could easily be made 

 to assume the more rigid form of Exhaustions. This was a 

 very important discovery ; — though the process was not 

 analytical, the demonstrations were direct, and, when applied 

 to the same subjects, led to the same conclusions which the 

 ancient geometers had deduced ; by an indirect proof also, 

 such as those geometers had adopted, it could always be 

 shown that an absurdity followed from supposing the results 

 deduced from the method of indivisibles to be other than 

 rigorously true. 



The method of Cavalieri was improved and extended by 

 a number of geometers of great genius who followed him ; 

 Torricelli, Roberval, Fermat, Huygens, Barrow, who all ob- 

 served the great advantage that arose from applying the 

 general theorems concerning variable quantity, to the cases 

 where the quantities approached to one another infinitely 



