MR GREGORY ON THE REAL NATURE OF SYMBOLICAL ALGEBRA. 215 



V. The last class of operations I shall consider is that involving two opera- 

 tions connected by the conditions 



(1) aF{x+2,) = F{x)f(j/)+f{x) F(y) 



and (2) af(x+y)=f (x) f{y) - c F (x) F (y) . 



These are laws suggested by the known relation between certain ftinctions of 

 elliptic sectors ; and when a and c both become unity, they are the laws of the 

 combinations of ordinary sines and cosines, which may be considered in geometry 

 as certain functions of angles or cu-cular sectors, but in algebra we only know of 

 them as abbreviated expressions for certain complicated relations between the 

 first three classes of operations we have considered. These relations are. 



S^x = x-—^ + ,^^^-!.c. 

 Co.x=\-^+^^&c. 



The most important theorem proved of this class of functions is that of De- 

 MoivRE, that 



(cos X + ( — )' sin xy = cos nx-\- ( — )' sin nx. 



It is easy to see that, in arithmetical algebra, the expression cos x + ( — )^ sin x 

 can receive no interpretation, as it involves the operation ( — )i In geometry, 

 on the contrary, it has a very distinct meaning. For if a represent a hne, and 

 a cos X represent a line bearing a certain relation in magnitude to a, and a sin oc 

 a line bearing another relation in magnitude to a, then a (cos x + ( — )- sin r) 

 wiU imply, that we have to measure a line a cos x, and from the extremity of it 

 we are to measure another line a sm x\ but in consequence of the sign of opera- 

 tion ( — )i, this new line is to be measured, not in the same direction as a cos x, 

 but turned through a right angle. As, for in- 

 stance, if AB = a cos x, and BC = a sin x, we 

 must not measure it in the prolongation of AB, 

 but tvu-n it round to the position BC ; and thus, 

 geometrically, we arrive at the poiut C. Also, 

 from the relation between sin x and cos x, we Iniow 



that the Una AC wiU be equal to a, and thus the expression a (cos x + ( — )* sin x^ 

 is an operation expressing that the line whose length is a, is tm-ned through an 



angle x. Hence, the operation indicated by cos — -i- ( — ) sin — is the same as 



that indicated by (4-)", the difference being, that, in the foimer, we refer to rec- 

 tangular, in the latter to polar co-ordinates. Mr Peacock has made use of the 

 expression cos x + ( — )^ sin x to represent direction, Avhile Mr Warren has em- 

 ployed one which, though disguised under an inconvenient and ai-bitrary notation. 



