300 BISHOP TERROT ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 
sign of ./—, are at equal angles to the original radius, in different directions, 
that is, on different sides of it. 
Again, of those roots which symbolize the wi 
radii in the upper half of the circle, that which A? 
has a, representing the cosine greatest, is the 
nearest to the original radii. Thus the roots of 
n6—1=0, in the order given by Dr PEacock, 
—144/—B Ten 
2 2 
? 


tala 
Ales i, p,%128, “are “1, 
; 1 Wes, ay a 
q 2 ‘ 2 p 
their proper order, if +1 be placed first, then ~ n-1 
—1, as having no sinal part, and being therefore, 
neither in the upper nor lower half, must stand in the middle of the remaining 
ee aml aie 
To arrange these in 
roots. Next these are two roots, , each having the sinal 
part +, which must be arranged in this order, because the sign of 1 in the former 
indicates that the cosine is in CA, and in the latter in CA,. Finally, considering 
those roots of which the sinal part is minus; we must place them in the order 
plist 9) ene because they are thus equidistant from unity with =i 
2 
1+/—3 3 
and jo Hence the roots in their proper sequence are 
1 1t+V¥—-38 -1l4+V—-3 _, -1-V-3 1-V-3 
“apelin: RETO EOS es mer 
symbolizing severally the radii drawn to the extremities of the arcs 
0 or 360°, 60°, 120°, 180°, 240°, 300°. 
VI. It appears from Props. IV., V.. that the radius drawn to the extremity 
3 


of an arc 9, is properly expressed by 1°"*, and this again by a+/—é, where a is 
what is called in trigonometry the cosine of 3, and »/é the sine. 
Now let CA, (Fig. 6.) make with CA an angle 3, CA, an angle 23 .. . CA, an 
angle ps. 
Then CA,=CD+/7-1. DA, =cos3+V7—1. sind 
CA, =cos p3+/—] . sinps. 
But by Prop. Il. CA,=CA,|?=(cos3+/7—T. sin3)? 
(cos3+/—1 Z sin3)?=cospS+V7 41 . sin p 8, 
which is DemMorvre’s Theorem. 
