OF ALGEBRAIC QUANTITIES. 
359 
61. With respect to quantities such as sin (a + b — 1), cos (a b ^ — 1), 
&c., quantities strictly speaking not algebraic, it may be observed, that if it be 
found useful to introduce these quantities into algebra, they may without im 
propriety be introduced, by giving to sines, cosines, &c., algebraic definitions ; 
thus sin A may be defined to signify 
a - a 
cr-(o T jf) 
A V - 1 
£\ — A V - 1 
(?) 
2 - J 
2 a/~ 
'E \ a J ~ 1 — * 
and we shall have 
sin(a + fe N /— 1) = 
(?r -(?) 
E\ _ a J - 1 + b 
(?)~-(0 T - (!)••(!) 
2 J 
2 v' — 1 2 */ — 1 
therefore, considering sines, cosines, &c., merely as algebraic quantities, we 
may make use of the expressions sin (a + b — 1 ), cos ( a + b y/ — 1 ), &c., 
if such expressions be found convenient in algebraic operations. 
Jesus College, Cambridge, 
April 22, 1829. 
J. W. 
