to prove the Properties of parallel Lines. 1 65 



12th axiom. But we may also object to the soundness of his 

 reasoning. 



Since the symbols A, B, C denote ratios, by the law of 

 homogeneity, there cannot be an equation between them and 

 the single line c; for, on account of the arbitrary measure of 

 c, such equation would have no fixed or determinate meaning. 

 Therefore the equation 



C = $ (c, A, B) 

 is absurd. But from an absurd equation, or rather from no 

 equation at all, you cannot, in a legitimate manner, infer that 

 another equation is true, viz. 



_ C = $ (A, B). 



A process of reasoning ought to be stopped whenever the re- 

 lation between the magnitudes concerned implies contradiction, 

 or can no longer be clearly understood. It seems to be an 

 odd argument to infer that there must be an equation in one 

 form, merely because there cannot possibly be an equation in 

 another form. The proper conclusion seems to be, that, in 

 analysis as well as in geometry, there is, in the present in- 

 stance, a peculiar difficulty, or a failing and breaking off in 

 the usual train of investigation. 



Professor Leslie has attacked Legendre's demonstrations 

 on other grounds. He objects to the law of homogeneity : 

 but, in a case so very clear and undeniable, his arguments 

 have no weight. He contends likewise that the measure of 

 an angle estimated by its proportion to a right angle is just 

 as arbitrary as the measure of a line in yards. This is more 

 specious. It is urged with great confidence, and clothed, as 

 usual, with the imposing garb of philosophical accuracy. 

 Perhaps Legendre and his supporters have not furnished the 

 proper answer to it. But there are few mathematicians who 

 would oppose the opinion of that eminent geometer on a point 

 of the modern analysis, without great diffidence; and some 

 reflection will show that the functional equations may be vin- 

 dicated from the charge brought against them by the learned 

 Professor. 



Conceive that A, B, C, instead of denoting angles simply, 

 are the arcs subtending the angles in a circle of which the 

 radius is r; and let Q be the quadrant of the same circle; 



then — , — r-, — , which are no other than Legendre's an- 

 gles, are ratios, or quantities of no dimensions, that remain 

 the same in all circles, and however the arcs are measured 

 or compared. Now we can obtain equations between the 



sides of the triangle and the trigonometrical quantities ' 



