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 ix)f(^) +fix) F (y) 



and (2) af(x + i/)=f{x)f{^)-cF(x)F(^). 



These are laws suggested by the known relation between certain functions 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 ftmctions of angles or circular 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. 



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

 MoiVRE, that 



(cos X + ( — )* sin a;)" r= cos nx + ( — )* sin n x. 



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 line, and 

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



a line bearing another relation in magnitude to a, then a (cos x -t- ( — )* sin x\ 



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

 we are to measure another line a ^m x; but in consequence of the sign of opera- 

 tion ( — )*, 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 turn it round to the position BC ; and thus, ^^^ 



geometrically, we arrive at the point C. Also, ^ 



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



that the line AC will be equal to a, and thus the expression a (cos x -v ( — y sin x?^ 

 is an operation expressing that the line whose length is a, is turned through an 



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



1 

 that indicated by (+)'», the difference being, that, in the former, we refer to rec- 

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

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

 ployed one which, though disguised under an inconvenient and arbitrary notation, 



