
BISHOP TERROT ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 351 
Cor. Ifp3=27, cosp34+V7—1. sinp3=1. 
Hence (cos 3 + /—1. sin3), (cos 29+/—1. sin29) &e. represent the several 
pth roots of unity. If, instead of the order 3, 23, 39, &., we arrange the several 
angles thus in pairs 3 and p—1.3, 23 and p—2.3, then the several expressions 
for z minus the several pth roots of unity, or the several simple factors of the 
equation 2?—1=0, taken in pairs corresponding to the above, will be 
(a—cos3—V/ —1 . sin$) and (z—cosp—1 .3—»/—1 . sin p—1 .9), 
the latter of which equals (x—cos . p3—93—W/—1. sinp3—9) 
=r—cos27—3—V7 —1 . sin2 T—S=x—cos$+W/—1. sind. 
In the same way the next pair must be 
(a—cos29+4+/—1 sin23) and (x—cos29—s/—1. sin23), and so on. 
If these several pairs be next multiplied together so as to produce the quadratic 
factors of 2”—1=0, we obtain the products (#?—2%cos3+1), (22-22. cos23+1) ke. 
And if it be remembered that in every case z—1=0 isa factor; and that if p be even, 
z—1and +1 are simple factors, and consequently x?—1 a quadratic factor ; there- 
fore if p be even, 
a? —1=(a?—1) . (a? -—2xcos3+1).(#?-—22%. cos23+1) &e. tof terms. 
But if p be odd, 
x? —1=(4—1). (@ —2a@c0s$+1). &e. vee terms. 
Where 3, it may be observed, equals “S. 
VIII. From these fundamental propositions, Mr WarREN, in his Treatise on 
Negative Roots, has deduced— 
1. The value of each side of a triangle in terms of the other sides and angles. 
(§ 141.) 
2. That the three angles of a triangle are equal to two right angles. (§ 142.) 
3. That the sides are respectively proportional to the sines of the opposite 
angles. (( 143.) 
4. That cosA="*°—". (9 144) 
He then asserts, that from these and the preceding propositions, all the for- 
mule of plane trigonometry may easily be deduced. In the following proposi- 
tions, I have applied his principles to the solution of some of the most simple, 
and to some of comparatively the more difficult problems usually given in ele- 
mentary books of trigonometry. 
