162 On the Application of algebraic Functions 



another are accurately defined, the conditions of their not in- 

 tersecting, or of their being parallel, are immediately inferred. 

 And again, the properties of parallel lines being known, we 

 arrive at the famous theorem, the 32d of book 1st; namely, 

 that the sum of the three angles of every triangle is equal to 

 two right angles. Now, if we could prove this last proposi- 

 tion independently of parallel lines, and without referring to 

 the undemonstrated postulate, we should, by inverting the 

 order of investigation, be able to lay down the elements with 

 the strictest accuracy, and to remove the blemish which has 

 always been viewed by geometers with so much regret. There 

 is no doubt that, in this manner, geometry may be treated with 

 all the rigour of ancient demonstration, without any peculiar 

 axiom relating to parallel lines ; and it is some satisfaction to 

 know that this may be done, although the process of reason- 

 ing may be too long and cumbersome to be introduced in an 

 elementary treatise. It is by pursuing the same rout that the 

 analyst professes to demonstrate the properties of parallel 

 lines by means of the theory of functions. 



Legendre's demonstrations respecting parallel lines are 

 founded on what is called the law of homogeneity in algebraic 

 functions. In the problems of plane geometry, where lines 

 and angles are combined in the same equations, the quantities 

 depending on the angles invariably contain in their expression 

 nothing else but ratios, or the quotients of homogeneous mag- 

 nitudes ; which renders the equations independent of the man- 

 ner in which the angles are themselves compared or measured. 

 It is not the same with regard to lines; for the algebraic sym- 

 bols of these always involve an arbitrary unit. Hence we 

 may lay down this general rule, that the expression of an an- 

 gular quantity, deduced from an equation, can involve the 

 ratios only of the lines concerned ; for otherwise it would be 

 dependent on the arbitrary measures of the lines, and would 

 have no determinate value. But in the expression of a line 

 no conditions are lequired with respect to angular quantities, 

 their values being in no respect arbitrary. As the principle 

 of homogeneity is clear, and liable to no difficulty or objec- 

 tion, we shall not extend further our remarks upon it. 



Two functional equations are employed by Legendre to 

 prove that the third angle of a triangle depends wholly upon 

 the other two. Put c for the base of a triangle, C for the op- 

 posite angle, and A, B for the other two angles; then the first 

 of the equations is thus investigated. 



It is proved by superposition, that two triangles are identi- 

 cal when their ba ;es, and the angles at their bases, are re- 

 spectively equal. It follows that the vertical angle of a tri- 

 angle 



