1 14 Treatment of Geometry as a branch of Analysis. [No. 134. 



figure has its peculiar but invariable formula of calculation. The 

 geometrical definition prescribes an invariability of form as regards 

 figure : when we transfer the question into the domains of analysis, we 

 introduce a consideration equivalent to this, it is the invariability of 

 form as regards calculation. 



Legendre's own attempt to clear up this point is not even specious, 

 although while his impregnable positions were hotly attacked, the 

 weakest escaped all but the practised eye of Sir James Ivory. He had 

 to prove that the formula by which the third angle is calculated from 

 the base and base-angles applies to all triangles. He imagines two 

 triangles, one constructed with the data a, B, C, and another with a', 

 B, C, having if possible different formula, the first say (f>, the second 

 $'. Then considering a' to vary to a, he obtains a third triangle. But 

 this third triangle has the same data as the first, and its third angle is 

 therefore equal to that of the first. Hence it must be calculated by the 

 same formula. But the formula of the third is that of the second, that 

 is <p', hence <f> & <j> are the same formula. The words in italics beg the 

 question glaringly : if the variation of an element can make a formula 

 vary (which is to be disproved,) then the change of a' into a gives the 

 third triangle some new formula more or less different from <j>' : the 

 principle of superposition shews that it is identical with (j>, hence (j> 

 differs from </>', and there is no absurdity forced upon the adversary. 



5. The geometrical weight of this flaw is of importance and great 

 interest. It was pointed out by Sir J. Ivory, that to assume a' to 

 change to a while the base angles remain B and C as before, is equi- 

 valent to drawing from the ends of a base «, lines making with it 

 angles equal to those of a given triangle A' B C. To assume further 

 that the formula of A' B C will apply to the new figure is to assume, 

 that the new lines will form a triangle with the new base a. The 

 double assumption amounts therefore to stating, that two lines making 

 given angles with a third will always meet, the only thing known re- 

 garding those given angles being that they are less than two right 

 angles ; since they are angles of a given triangle. This is nothing more 

 nor less than Euclid's axiom, and therefore Legendre's process involves 

 the assumption of that axiom. The analytical investigation therefore 

 rests on an assumption, that of the invariability of formula as dis- 

 tinguishing a defined geometrical figure, which no skill can do away 



