to prove the Properties of parallel Lines. 163 



angle will be of the same magnitude in all cases when the base 

 and the other two angles have the same values. Therefore, 

 in every triangle, the first of these quantities must be the same 

 function of the other three; which may be thus expressed 

 analytically, viz. 



C = $( C ,A,B). 



It the foregoing process seem clear when we consider it as 

 an algebraic calculation, it becomes clouded with obscurity 

 when we apply it to geometry. It is so shortly expressed, 

 that the geometrical grounds of the demonstration cannot be 

 brought into view, without expanding the steps of the rea- 

 soning. For this purpose, conceive another triangle of which 

 the vertical angle is C, the base c, and the other two angles 

 the same as in the first triangle ; and suppose that, in this in- 

 stance, we have this equation, viz. 



> C'=^( C ',A,B), 

 the form of the function being different from what it was in 

 the former case. But when the two bases c and c' become 

 equal, the triangles will be identical, and the angle C will be 

 equal to C: wherefore by puting c=c and C = C, we have 

 C = $ (c, A, B) 

 C=*(c,A,B); 

 and this proves that the function must be the same for all tri- 

 angles. 



The real ground of the demonstration is now apparent. 

 The force of the reasoning turns upon this assumption ; that 

 the base of a triangle may vary from c to c, while the angles 

 at the base remain the same. This is assumed, not proved. 

 If any one deny it, there is no argument to compel his assent. 

 Now if we follow the laudable example of the ancient geo- 

 meters and state separately the principles of our reasoning, we 

 shall be led to this postulate : Let it be granted that upon 

 any proposed base a triangle can be constructed having the 

 angles at the base equal to two angles of a given triangle. If 

 this postulate be admitted, Legendre's demonstration will be 

 legitimate and conclusive ; otherwise, it will lose all its force. 



But the postulate is deducible from, and therefore subor- 

 dinate to, Euclid's 12th axiom. What then does geometry 

 gain by the functional process ? 



We shall next give the investigation of the second functional 

 equation. It has already been proved that 



C = <f> (c, A, B), 

 the form of the function being the same in all triangles. But 

 the angle C is a determinate ratio, or a value independent of 

 all arbitrary measurement ; and, since the measure of the line 



X 2 cis 



