18 DISSERTATION SECOND. [pa«t i. 



The method of indivisibles, from the great facility with 

 which it coidd be managed, furnished a most ready method 

 of ascertaining the ratios of areas and solids to one ano- 

 ther, and, therefore, scarcely seems to deserve the epithet 

 which Newton himself bestows upon it, of involving in its 

 conceptions something harsh, (durum,) and not easy to be 

 admitted. It was the doctrine of infinitely small quanti- 

 ties carried to the extreme, and gave at once the result of 

 an infinite series of successive approximations. Nothing, 

 perhaps, more ingenious, and certainly nothing more hap- 

 py, ever was contrived, than to arrive at the conclusion of 

 ail these approximations, without going through the ap- 

 proximations themselves. This is the purpose served by 

 introducing into malheuiaticks the consideration of quanti- 

 ties infinitely small in size, and infinitely great in number ; 

 ideas which, however inaccurate they may seem, yet, when 

 carefully and analogically reasoned upon, have never led 

 into errour. 



Geometry owes to Cavalleri, not only the general method 

 just described, but many particular theorems, which that 

 method was the instrument of discovering. Among these 

 is the very remarkable proposition, that as four right an- 

 gles, to the excess of the three angles of any spherical 

 triangle, above two right angles, so is the superficies of 

 the hemisphere to the area of the triangle. At that time, 

 however, science was advancing so fast, and the human 

 mind was every where expanding itself with so much ener- 

 gy, that the same discovery was likely to be made by 

 more individuals than one at the same time. It was not 

 known in Italy in 1632, when this determination of the area 

 of a spherical triangle was given by Cavalleri, that it had 

 been published three years before by Albert Girard, a 

 mathematician of the Low Countries, of whose inventive 

 powers we shall soon have more occasion to speak. 



