and on Algebra as the Science of Pure Time. 417 
primary unit (1,0). The mth root of any couple has therefore m distinct values, 
and no more, because the mth root of the primary unit (1, 0) has m distinct values, 
and no more, since it may be thus expressed, by (147.) and (131.), 
Z 2ur . 207 
a, O)m= ( cos. —-— 5 sin. a) ' (149.) 
so that, by the law of periodicity (133.), for any different whole number w, 
(1, 0) m = 0)", (150.) 
and therefore generally, - 
(b, b) = h, ae : (151.) 
if ; é : 
w —w+im, (152.) 
but not otherwise. For example, the cube-root of the primary unit (1, 0) has three 
distinct values, and no more, namely 
a1 ENS} 
a,OF = 4,0); a,07=(—3, 2);a,or=("5,-); a3, 
27 2 
60 that each of these three couples, but no other, has its cube =(1, 0). Again the 
couple (—1, 0) has two distinct square-reots, and no more, namely 
1 1 
(—1, 0) =" (O05 (-1, 0)? =(0, —1). (154.) 
In general we may agree to denote the principal square-root of a couple (b,, b:) by 
the symbol 
Jb, &) = @, %)7; (155.) 
and then we shall have the particular equation 
V(—1, 0)=©, 1); (156.) 
which may, by the principle (61.), be concisely denoted as follows, 
¥—1=(0, 1). (157.) 
In the THEORY OF sINGLE NUMBERS, the symbol /—1 is absurd, and denotes an 
IMPOSSIBLE EXTRACTION, or a merely IMAGINARY NUMBER; but in the THEORY OF 
couptes, the same symbol ¥ —1 is significant, and denotes a POSSIBLE EXTRACTION, 
