352 BISHOP TERROT ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 



IX. sin (A + B)=A X cos B + cos Ax sinB 



cos (A + B) = cos A X cos B — sin A . sin B. 

 Let arc AB (Fig. 7.) =A, BD„ and ADj each = B. 



Then by Prop. III., CB=r . 1^% 



B A + B 



CD, = ;-.l-% CD, = r.l ^' 



But Prop. VII., 



A 



1~'^= COS A +\/ — 1 . sin A 



B 



1'"= cosB+ V — 1 . sin B 



A + B 



1'" =:cos A X cosB— sin A . sinB + '\/ — 1 • (sin A . cosB + cos A . sinB 



A + B 



but 1 ^ "■ = cos A + B + V-^ . sin A + B 



Equating, then, the possible and impossible, or, more properly, the sinal and 

 cosinal, parts of these equal forms 



cos A X cos B— sin A . sin B = cos A + B 

 and sin A X cos B + cos A . sin B= sin A + B. 



This demonstration is the same in princii^le, and nearly the same in detail, 

 as that given by Dr Peacock, in his Algebra, vol. i., p. 392. In his 2d volume, 

 Dr Peacock goes more fully into the consideration of the roots of imity as coeffi- 

 cients of direction. Yet there he proves these propositions, not upon that consi- 

 deration, but by the ordinary geometrical method. 



Def. It should be observed that in the following propositions, a line ex- 

 pressed by letters simply as AB, must be understood as considered in respect 

 both of length and direction ; while by the same letters in brackets, thus (AB), 

 is understood the same line in regard to its length only. Thus, if a be the angle 



which AB makes with unity, (AB) . l^'^^AB. 



X. In any right-angled triangle, the sum of the squares of the sides is equal 

 to the square of the hypothenuse. 



• Let CA (Fig. 6)=r, then CA, = /- . l^', and CA„_i=:r . l'^-- 



CAj X CA„_i = r2 X 1^ X ^=/' 

 1^ 



