BISHOP TERROT ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 351 



Cor. If joa=2 7r, cos/j3 + '\/^^ . sinpa=l. 



Hence (cos S + -vATT . sin S), (cos 2 a + \/^3 . sin 2 S) &c. represent the several 

 ^th roots of unity. If, instead of the order 3, 2 a, 33, &c., we arrange the several 

 angles thus in pairs a and p—1 . a, 2 a and p-2 . a, then the several expressions 

 for w minus the several ^th roots of unity, or the several simple factors of the 

 equation a;P—l = 0, taken in pairs corresponding to the above, will be 



(a?— cosa — a/^^ . sina) and (x—co&p—l . a — \/ — 1 . sinjo — 1 . a), 



the latter of which equals (a;-cos .p^-^-V^^ ■ sinjoS-a) 



=2- — cos27r— a — \/— 1 . sin2'7r— a=a;— cosa + \/ — 1 . sina. 



In the same way the next pair must be 



(if— cos2a + \/-l sin2a) and (t— cos 23--/'^. sin2a), and SO on. 



If these several pairs be next multiplied together so as to produce the quadratic 

 factors of .r" -1=0, we obtain the products (a:2_2a;co8a + l), (r'-2x.cos2a + l)&c. 

 And if it be remembered that in every case ^r- 1 =0 is a factor; and that Up be even, 

 x—1 and x + l are simple factors, and consequently .c^ - 1 a quadratic factor ; there- 

 fore ifp be even, 



xP-l = (x^-l) . {x^ -2x coa'^ + 1) . (x' -2 X . cos 2 a + 1) &c. to ^ terms. 



But if /> be odd, 



cci'-l=:(x-V). (x--2a:cos'd + l) . &c. to ^^ terms. 



cy __ 



Where a, it may be observed, equals — . 



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. 

 (J 141.) 



2. That the three angles of a triangle are equal to two right angles. (| 142.j 



3. That the sides are respectively proportional to the sines of the opposite 

 angles. (| 143.) 



4. That cosA=^^i:J^. (f 144.) 



He then asserts, that from these and the preceding propositions, all the for- 

 mulae 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. 



