BISHOP TERROT ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 347 



that n, which is the index of the last term, is also the coefficient of the angle which 

 it makes with that line whose coefiBcient we assume to be unity, that is, with CA. 

 But M a=2 /■ TT, or an integer number of complete circumferences. Hence the radius 

 symbolized by a" coincides in length and position with the original AC, or a"=l, 



i ^ 



therefore a^=r'=:l-'"^ 



Now we know, on ordinary algebraic principles, that the several rath 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 

 «th 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, 



_9_ 



then E . 1' '" is the expression for that radius which is inclined to that symbol- 

 ized by E, X 1 . at an angle a. And as the direction of the radius, or its angularity 



_9_ 



to the original position is noted by the numerator of the index, we call 1' '' " the 

 coefficient of direction. We have thus found a function of the angle of inclination 

 which, being aflQxed a,s 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 . "^ ~ , expresses a line whose length is 



simply a ; the coefficient "^ ~ aifecting not the length, but only the direction 



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 ?s=2, therefore 

 a^=l or a2-l=o .-. a==fcl. But the radii being a, a^, a must evidently be 

 -1, and a2=+l. 



Next let the circumference (Fig. 2.) be divided into four equal parts, then 

 CA, CD, CB, CE are the four roots of the equation a^-l=0. But these roots are 

 ±1 and ±^/3l. 



