Further Remarks on the Theory of parallel Lines. 247 



aurait autant de triangles different s qui auraient un cote egal 

 adjacent a deux angles egaux, ce qui est impossible*. For this 

 means that, if the functions <p, <p\ <p" were of different forms 

 in the several triangles, the vertical angles C, C, C" would be 

 unequal, when the bases c, c', c" are supposed to be equal, 

 which is contrary to what is proved by superposition. 



It is manifest therefore that the reasoning of Legendre ne- 

 cessarily supposes the existence of triangles that have differ- 

 ent bases, viz. c, c\ c", and the same angles at their bases, viz. 

 A, B ; or, which is the same thing, it supposes that the base 

 of a triangle may vary while the angles at the base remain 

 the same. We may therefore inquire what authority there 

 is for this assumption. 



If the base of a triangle vary, we may adopt two supposi- 

 tions with respect to the angles at the base : either they may 

 remain the same when the base varies, or they will neces- 

 sarily undergo concomitant changes. Since it is admitted 

 that the vertical angle may vary with the base, there can be 

 no good reason for exempting the other two angles from the 

 possibility of a like variation. The two cases we have men- 

 tioned are a complete enumeration ; and it never can be main- 

 tained that one is true and demonstrated, until the other be 

 excluded. Thus the assumption, on which Legendre's de- 

 monstrations are founded, is one of two hypotheses that seem 

 equally possible. It is therefore certain, as we have already 

 shown in the last Number, that the functional investigation 

 respecting parallel lines, like the geometrical process of 

 Euclid, rests upon a peculiar postulate, in this respect per- 

 fectly resembling almost every other method that has been 

 proposed for overcoming the same difficulty. 



2. Admitting, with Legendre, that the angles A, B, C de- 

 note ratios, or numbers independent on arbitrary measure- 

 ment, it follows, from the principle of homogeneity, that there 

 cannot be an equation between A, B, C and the side c which 

 is measured by an arbitrary unit. Thus the two equations, 



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



C=<p( c ,A,B) 

 are equally impossible and absurd: the first, because a mag- 

 nitude that may be converted into a number by any assumed 

 measure, cannot be expressed by three determinate numbers; 

 and the second, because a determinate ratio cannot involve in 

 its expression, a number that may be varied in an arbitrary 

 manner. But if the equation, 



C = ? (c, A, B) 



• Elcm. de Geom. edit. lOme, p. 280. 



be 



