, 
a 
BISHOP TERROT ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 347 
| that », which is the index of the last term, is also the coefficient of the angle which 

it makes with that line whose coefficient we assume to be unity, that is, with CA. 
But »3=2r7, or an integer number of complete circumferences. Hence the radius 
symbolized by a” coincides in length and position with the original AC, or a”=1, 
Iai i9 
therefore a’ =1" =12"". 
Now we know, on ordinary algebraic principles, that the several nth roots 
of unity are properly represented by the several terms of the geometric series 
a, a?,a?....a”, or 1. Since, then, the two series, first that of the successive radii 
of a circle making equal angles with one another, and secondly, that of the several 
nth roots of unity are in symbolism the same, it follows, that, dropping this com- 
mon symbolism, we may take the several roots of unity to represent the succes- 
Sive radii, and conversely. 
If, as before, we take not unity but R for the numerical length of the radius, 
$ 
then R . 1””” is the expression for that radius which is inclined to that symbol- 
ized by Rx 1. atan angle 3. And as the direction of the radius, or its angularity 
> 
to the original position is noted by the numerator of the index, we call 17" * the 
coeficient of direction. We have thus found a function of the angle of inclination 
which, being affixed as a coefficient or multiplier to the arithmetical expression 
for the length of the radius, represents the radius so inclined, both in length and 
position; and which may be employed according to the ordinary rules of alge- 
braic calculation, to find the length and position of other lines under conditions of 
relation to it. 
These coefficients of direction, however, it must be observed, have no quanti- 
tative or arithmetical value. Thus a. cles) expresses a line whose length is 
affecting not the length, but only the direction 
simply a; the coefficient Lives 
of the line. 
IV. As illustrative of this reciprocal symbolism, let us suppose that the suc- 
cessive radii are two in number, or, in other words, that a radius revolving round 
C takes only one fixed position, and makes only two equal angles before it returns 
to its original position (Fig. 2). Then the circumference is divided into two equal 
parts, AB is the diameter, and if CA=1, CB=—1. In this case n=2, therefore 
@?=1 or a?-1=0 .. a=+1. But the radii being a, a?, a must evidently be 
—1, and a?=+1., 
Next let the circumference (Fig. 2.) be divided into four equal parts, then 
CA, CD, CB, CE are the four roots of the equation at_1=0. But these roots are 
+1 and +/ =i. 
