2iJ$ ON A NEW PRINCIPLE OF LEGENDRE*S. 



Proof of this. The principle itself, as well as the small reliance that cas 

 be placed on it, may be understood from the following 



Theorem*. 



If two sides of one triangle are equal respectively to two 

 sides of another, the third sides also are equal. 



For let A and B be two sides of a triangle, p the included 

 angle, C the opposite side. If A, B, and p be given, C 

 will evidently be completely determined. C therefore is a 

 function of A, B, andp. But it is plain, that/? cannot 

 enter into this function: for let some line, as D, be repre-. 

 sented by unity : then A, B, and C are numbers, and if 

 there could be an equation between A, B, C, and p, we 

 might find p in terms of A, B, C ; whence p would be a 

 number, which is absurd. It follows from this, that C is 

 a function of A and B only ; whence the truth of the pro- 

 position is manifest. Q. E. D. 



It is needless to say, that the principle must be erroneous, 

 which leads to such a conclusion. 

 Thinesthat "^ writer in the Edinburgh Reviewf asserts, that this rea- 



cannot be soning " takes for granted nothing, but that an angle and a 

 therefore inde- " bneare magnitudes, which admit of no comparison." It 

 pendent of is a sufficient answer to this ; that the quantity of grain in onr 

 bams, and the weather which preceded the collecting it there, 

 are quantities which admit of no comparison ; and yet the 

 former has a pretty evident dependance on the latter. 

 Use of the It may be observed in the above proposition, thnt th€ 



word function, jg^j^ J'unction is used in a very restricted sense: merely 

 denoting numerical equality, or at most equality of homo- 

 geneous quantities; whereas every one knovvs, that a quan- 

 tity may be a function of (or dependant on) another, with- 

 out any such abfoluie equal \ty as is there supposed. 

 fallncTof It is of no use therefore, to have shown, tliat there can 



Legeiidre's be no equation, properly so called, as C— 9 (A, B, p) 

 e ,oa! .fe between A, B, C, p, unless it could farther be proved, 



that there can be no analogy as C oc (p (A, B, p) between the 

 same quantities. This maybe very simply exemplified; 



* This diiTers from the first of Mr, Legendre's only in this j that I 

 have change'] anj^les into side^, and the side into an angle. 

 t tJa 29, p 4. 



In 



