16 1 On the Application of algebraic Functions 



c is arbitrary, it follows, from the law of homogeneity, 

 that the line cannot enter into the expression of the angle. 

 Wherefore the equation must simply be, 



C = $(A,B). 

 And this proves that the angle C is determined by the other 

 two angles independently of the sides of the triangle. 



In this investigation triangles that have the same angles at 

 their bases are compared together ; and the existence of such 

 triangles is therefore assumed. 



It is not difficult, it may be argued, to admit that the base 

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

 the same; and when this is granted, the functional demon- 

 strations are not liable to objection. But the present question 

 relates solely to what is consonant to the principles of geome- 

 try ; and that science does not permit the gratuitous assump- 

 tion of related figures. You may draw one triangle, for in- 

 stance that upon the base c ; and you may assume as many 

 bases c', c", &c. as you please ; but you cannot be allowed 

 to suppose that, upon these bases, triangles exist which 

 have two angles in common with that already drawn. In this 

 manner we might be led to reason about figures that have no 

 existence ; an absurdity which is guarded against in geometry 

 by the postulates *. You have all the data necessary for con- 

 structing the figures about which you are to reason; and you 

 must construct them by the admitted postulates, without the 

 help of Euclid's 12th axiom, which you propose to prove. 

 But the triangles cannot be so constructed; and therefore the 

 functional demonstrations cannot be held good in geometry. 



The difficulty about parallel lines lies in the strictness of 

 geometrical principles. The least relaxation of these would 

 open a door to innumerable solutions. If the analyst, by a 

 latent assumption, evade any of the rules imposed on the geo- 

 meter, the different results of the two modes of investigation 

 can occasion little surprise. 



Without objecting to Legendre's mode of reasoning, we 

 have endeavoured to show that his analytical demonstrations 

 cannot be applied in geometry, unless it be assumed, that the 

 base of a triangle may vary while the angles at the base re- 

 main the same ; a proposition which is deducible from Euclid's 



* In the Greek geometry the postulates are not to be considered as 

 having a reference to practice. They are theoretical principles precluding 

 the geometer from introducing arbitrary figures and constructions in the 

 course of his reasoning. Legendre does not enumerate the postulates. 

 And even Professor Leslie has degraded from its place, and thrown into the 

 6hadc, this important and essential part of the geometry of Euclid. 



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