1 96 On the Use uf Analysis in investigating 



merely changes the forms of the conceptions on which it ope- 

 rates, or combines a number into one general expression. But 

 the question is, with respect to the applicability of such re- 

 marks to the subject on which Mr. Leslie had just been com- 

 menting. Does Legendre proceed a priori, or attempt to de- 

 duce the primary properties of extension 'without appealing in 

 the Jirst instance to external observation ? I certainly think 

 not; and it will not be difficult to show cause for my opinion. 

 — Whence, I would ask, is his fundamental equation de- 

 rived, if not from superposition or experiment ? and what more 

 is necessary than this single appeal, to give the object he con- 

 templates its complete definition? The thirty-second Prop, of 

 Euclid is immediately traceable to the fourth and to the twelfth 

 axiom, Legendre involves the axiom, and derives his equa- 

 tion from an equivalent to the fourth ; and there is thus no 

 constitutio)ial difference betwixt the processes of the geometer 

 and the analyst. But it will be further evident if we view 

 the subject in its most elevated light. 



A triangle analytically considered, is merely one of a class 

 of functional equations which repi'esent a relation betwixt six 

 quantities, three of which are heterogeneous to the other three; 

 or it is a particular case of the equation 



a = ^{b, c, A, B, C, y) 



Now there will always be a limit to the number of possible 

 ways in which such a set of quantities can be connected to- 

 gethei*. This limit is fixed in geometry, for each particular 

 case, by what Euclid terms definitions. It is fixed by the vi- 

 sible Jigiire, whose existence the definition declares jiossible ; 

 and the continued use of this figure prevents the reasoner from 

 falling into the supposition of impossible combinations. The 

 same limit is fixed in analysis, with the utmost generality, by the 

 rationalform which the foregoing equation necessarily assumes. 



The use of visible figure, then, and the operation of the law 

 of homogeneity are identical in this respect, and we shall quickly 

 recognize the entire sameness of the other parts of the pro- 

 cesses. The analyst and the geometer are at this point in 

 precisely similar states. The geometer, in assenting to his de- 

 finition, has assented to the possibility of the existence of re- 

 lations betwixt certain quantities ; and the analyst, in writing 

 the foregoing equation, has done the same thing. But the 

 great inquiry into the nature of these relations, in each case, is 

 now before them. The geometer has to translate into the 

 language of intellect the visible figure under his contempla- 

 tion; 



