246 Further Remarks on the Theory of parallel Lines. 



a vessel in which such a pile is placed, after each draught, 

 we may have a competent substitute for a chalybeate spring. 



Clean copper plates alternating with iron * would answer, 

 or a clean copper wire entwined on an iron rod ; but as the 

 copper when oxidated yields an oxide, it is safer to employ 

 silver. 



XL. Further Remarks on the Theory of parallel Lines. 

 [Seep. 161.] 

 1VTO apology can be required for resuming the discussion 

 ■*- of parallel lines with the view of extending and com- 

 pleting the observations already made in the last Number of 

 this Journal. A subject liable to many subtile distinctions 

 must be placed in a variety of aspects, before the reader can 

 form a decided opinion upon it. Besides, the validity of Le- 

 gendre's demonstrations by algebraic functions has been so 

 keenly contested by men of great eminence, that the full elu- 

 cidation of this point must be not only very curious and in- 

 teresting, but is even of some importance in geometry. 



1. If we have a number of algebraic equations, viz. 

 C = $ (c , A, B), 

 C'=$V,A,B), 

 C"= f (c\ A, B), 

 &c. 

 in which the letters p, <p', p", &c. are the marks of functions 

 of unknown forms ; and if the numbers that vary from one 

 equation to another, be so related that when c= c'= c", we 

 must likewise have C = C / = C' / ; it will be evident that the 

 functions cannot be of different forms, and that all the equa- 

 tions will be represented generally by the expression 

 C=<p(c, A,B). 



Now if, with Legendre, we apply the foregoing reasoning 

 to triangles, we must conceive a separate figure answering to 

 every equation ; the bases being, c, c', c" ; the vertical angles, 

 C, C, C"; and the other angles common to all the triangles 

 and equal to A, B : then because, by the principle of super- 

 position, we know that, when the bases c, c', c" are equal, 

 the vertical angles C, C, C, will likewise be equal; it will 

 follow that, in all the triangles, the vertical angle is the same 

 function of the base and the other two angles ; or, that the 

 equation C = <p (r, A, B) 



comprehends every case. 



That we have here faithfully explained Legendre's process 

 of reasoning is manifest from his own words: Car, si plusieurs 

 angles C pouvaient coirespondre aux trois donnets c, A, B, il y 



aurait 



