350 BISHOP TEEROT 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 



radii in the upper half of the circle, that which 



has a, representing the cosine greatest, is the 



nearest to the original radii. Thus the roots of 



/('■_1 = 0, in the order given by Dr Peacock, 



.... ,.,„ , _i+-v/33 -i-V^s A|. 



Alg. a., p, 128, are l, 



l_^/33 i + V^S 



•1, 



2 ' 2 



To arrange these in 



2 ' 2 



their proper order, if + 1 be placed first, then 

 - 1, as having no sinal part, and being therefore, 

 neither in the upper nor lower half, must stand in the middle of the remaining 



AorA 



roots. 



Next these are two roots, "*" „ — and "^ 



3 



', each having the sinal 



2 2 



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 



-l-\/-3 l-V-3 



and 



2 



i+V- 



', because they are thus equidistant from unity with 



■ 1 + 1 



Hence the roots in their proper sequence are 

 1+^/33 -i + ^^Ts _i_/v/T:3 i_-v/z:3 



"2 ' 2 ' ' 2 2' 



symbohzing severally the radii drawn to the extremities of the arcs 

 or 360°, 60", 120°, 180°, 240°, 300°. 

 VI. It appears from Props. IV., V.. that the radius drawn to the extremity 



_9_ 



of an arc 3, is properly expressed by I"''', and this again by ad='\/^^, where a is 

 what is called in trigonometry the cosine of a, and Vb the sine. 



Now let CAi (Fig. 6.) make with CA an angle a, CAa an angle 2 a . 

 angle p a. 



Then CA, = CD + .\/^ . DAi^cosa + V^ . sba 



CAy=cospa + \/3l . sin^a. 



But by Prop. II. CA,,="CA;|p=(cosa + -/-T. sina)'' 



(cosa + 's/^. sina)'' = cospa + \/ + l . sinjoa, 

 which is Demoivre's Theorem. 



CAp an 



