1843.] Treatment of Geometry as a branch of Analysis. 117 



Calling the functions <j> and \p by the names sin and cos, we have 



a — c cos B = c sin A, and b = c cos A = c sin B. 

 These considerations premised, it is easy to determine the general 

 form of the functional equation for any triangle ABC. Drop a per- 

 pendicular from A on a, then a will be divided into two parts, the one 

 adjacent to the angle B must as above be equal to c cos B, and the 

 other, adjacent to the angle C must also be b cos C. Hence 

 a = b cos C -}- c cos B. 

 Besides, the perpendicular in the one triangle equals b sin C, in the 

 other it is c sin B ; these are therefore equal or 

 b sin C — c sin B = 

 The conditions of symmetry give us two other pairs of equations, 

 « = b cos C -f- c cos B 1 b sin C — c sin B = 01 



b = a cos C + c cos A > ..(a) a sin C — c sin A = > ...(|3) 

 c = a cos B +~b cos A J a sin B — b sin A = OJ 



1 1 . We must remember, however, that the functions sin and cos are 

 only intelligible with regard to acute angles, since from the consideration 

 of such only they were derived in (10). The formulae above apply 

 therefore only to acute angled triangles, unless we are able to put such 

 an interpretation on sin and cos in the case of right and obtuse angles, as 

 will permit us to consider (a) and (/3) universal forms. 



If (a) and (p) are to apply to all triangles, then if C were a right 

 angle we should have 



b = a cos p )+ c cos A and a sin 5) e sin A = 



But examining a triangle right angled at C, we perceive as in (10), 



b = c cos A and a — c sin A = 0. 

 Hence to admit the generality of (a) and (|3) we must interpret 



cos M as and sin F ) as 1 . 



If the triangle again were obtuse at C, the perpendicular from A 

 would fall on a produced, hence a would be the difference of c cos B 

 and b cos \ir — C) or a = c cos B — b cos \ir — C). The perpendicular 

 is also in one case c sin B, in another b sin (7r — C) ; or 

 c sin B — b sin {it — C) = 

 Compare these with (a) and (fi) supposed to be universal, and it 

 must follow that 



cos C = — cos (w — C) and sin C = sin {tt — C) 

 are the only interpretations that can be put on the sin and cos of the 



