
BISHOP TERROT ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 349 
ACB. If we call AB the radius or hypothenuse, a, then each of the sides AC CB 
is in length ee and AB (being inclined at an angle of 45° to AD, which we as- 
sume as the original position of the radius) is symbolized by a x 1°°* =a x 1? = 
; = But AC=J75, CB being perpendicular to the original position equals 
phen =i Len ol 
ar: ./—1. (Prop.IV.) Therefore AC +CB=a. [vst 75 |= Fe ae 
2. Let BAC represent a right-angled triangle ae angle at A=60°, then 
AB in length and direction =a. 1° =a. 1'=a. meee g 
/8 

ay oes x? CB = in length 
Bis Ea oe 
Sy Pi Eea AB Ue 
= and therefore in length and direction jointly a. 
a J —3 ay ge) 
AC+CB=5ta. 5) So ee aaa 
3. Let the triangle (Fig. 5.) be equilateral, and AB be taken as the original 
Aus 

position. Let AB=a, AC=a.1*, CB=a.17* 
ne cep =a[ 141" "=«. coe al ae 
=.a al Gare x 2 = Boo 22S =e 
i [ 2 if eee 2 agonal roti 
VI. In the foregoing Propositions and Exam- Fig. 5. 
ples, it has been assumed that we know not only Cc 
the several nth roots of unity, but also their proper 
order, that is, the order in which, as coefficients, ‘ 


they express the radii drawn so as to make angles 
§, 29, 39, &c., with the original radius. But when 
by any analytical process we find the roots of 
—1=0, we procure the symbolical representa- 
tives of these radii in no determinate order. To 
discover this order, we must observe that two roots 
are always of the form a/ —é ; comparing which 
expression with figure 6, it is evident that a is the D 
part symbolical of the cosine, and /—¢ the part symbolical of the sine, because 
it is affected by the coefficient »/—1, and is therefore perpendicular to the original 
radius. It is clear, then, that in the general expression a--/—d, the sign + be- 
longs to those radii which lie in the upper half of the circle, and — to those which 
lie in the lower half; and that the two radii whose symbols differ only in the 
VOL. XVI. PART III. 4 T 
