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 -y=, and AB (being inclined at an angle of 45^ to AD, which we as- 

 sume as the original position of the radius) is symboUzed by a x 1'*'°= = a x 1' = 

 a ■ — -T= — . But AC =-75, CB being perpendicular to the original position equals 



^V:rT. (Prop. IV.) Therefore AC + CB=«.[;^ + ^^]=a.^^t^=AB. 

 2. Let BAC represent a right-angled triangle whose angle at A = 60°, then 



l-±~^, AC=-|-, CB = in length 



AB in length and direction 

 V3 



a. l'^"=a.r 



= a 



, and therefore in length and direction jointly a . 



Vs 



-1 



V-3 



AC-l-CB=2 + « • ~2~~''' 2 



=AB. 



3. Let the triangle (Fig. 5.) be equilateral, and AB be taken as the original 

 position. Let AB=a, AC=o . F, CB=a . 1~^ 



r-l + V-3 ^-1 2 rl + v-a -i -1 



-'' I — 2 — +iJ><rT7ri=i— 2— ^r77TiJ= 



a = AB. 



Pig. 5. 



VI. In the foregoing Propositions and Exam- 

 ples, it has been assumed that we know not only 

 the several wth roots of unity, but also their proper 

 order, that is, the order in which, as coeflficients, 

 they express the radii drawn so as to make angles 

 a, 2 a, 3 a, &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 ± -/^^ ; comparing which 

 expression with figure 6, it is evident that a is the 

 part symbolical of the cosine, and V^^ the part symbolical of the sine, because 

 it is affected by the coefficient V^, and is therefore perpendicular to the original 

 radius. It is clear, then, that in the general expression a =fc V^^, 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 



