sjict. i j DISSERTATION SECOND. 17 



Proceeding one step farther, he sought for the sura of 

 the cubes of the same lines, and found it to be one fourth 

 of the greatest, taken as often as there are terras ; and, 

 continuing this investigalion, he was able to assign (he sura 

 of the nth powers of a series in arithmetical progression, 

 supposing always the difference of the terras to be infinite- 

 ly small, and their number to be infinitely great. The 

 number of curious results obtained from these investiga- 

 tions may be easily conceived. It gave, over geometrical 

 problems of the higher class, the same power which the 

 integral calculus, or the inverse method of fluxions does, 

 in the case when the exponent of the variable quantity is 

 an integer. The method of indivisibles, however, was not 

 without difficulties, and could not but be liable to objection, 

 with those accustomed to the rigorous exactness of the an- 

 cient geometry. In strictness, lines, however multiplied, 

 can never make an area, or any thing but a line ; nor can 

 areas, however they may be added together, compose a 

 solid, or any thing but an area. This is certainly true, 

 and yet the conclusions of Cavalleri, deduced on a con- 

 trary supposition, are true also. This happened, because, 

 though the suppositions that a certain series of lines, in- 

 finite in number, and contiguous to one another, may com- 

 pose a certain area, and that another series may compose 

 another area, are neither of them true ; yet is it strictly 

 true, that the one of these areas must have to the other 

 the same ratio, which the sum of the one series of lines has 

 to the sum of the other series. Thus, it is the ratios of 

 the areas, and not the areas absolutely considered, which 

 are determined by the reasonings of Cavalleri ; and that 

 this determination of their ratios is quite accurate, can 

 very readily be demonstrated by the method of exhausv 

 tions. 



