352 BISHOP TERROT ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 
IX. sin (A+B)=A x cos B+ cos A x sin B 
cos (A+ B)=cos A x cos B—sin A. sin B. 
Let arc AB (Fig. 7.) =A, BD, and AD, each=B. 



A Fig. 7. 
Then by Prop. Ill., CB=r . 17”, Dgsysn igs 
By A+B 
CD,=r.1°7, CD,=r.17” « 
ea gs 
OD,=rx1°7x 1°” 
But Prop. VIL, = 
ee 
1°*= csp A +V—1.sinA 
B. 
177= cosB+ /—1.sinB 
A+B 
1°” =cos A xcosB—sinA.sinB+/—1. (sin A . cos B+cos A. sin B 
A+B 
but 177 =cosA+B+V7—1.sinA+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 principle, 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 unity as coeffi- 
cients of direction. Yet there he proves these propositions, not upon that consi- 
deration, but by the ordinary geometrical method. 
Der. 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 3 be the angle 
3 
which AB makes with unity, (AB) . 127=AB. 
X. In any right-angled triangle, the sum of the squares of the sides is equal 
to the square of the hypothenuse. 
9 9 
. Let CA (Fig. 6)=r, then CA,=r . 127, and CA,_;=r.1 27 
9 
CPCUA, 17 ox 127 x 
flows 
ay 
127 

