248 MR. WARREN ON THE SQUARE ROOTS 



.-. There is a connexion between what are called impossible roots and the 

 circumference of the circle. 



But by examining the series more accurately, we may find a greater con- 

 nexion between the series and the properties of the circle 



(y X^ C X 



Fo'- 1 - r:2 + 1727174 - ^"- = ^^s ^^' 



and cw - ^-^ + 1,2.3.4.5 - ^'- = '^" "^' 



.-. r = cos ex + sin ex . »y — I. 



Now lest there should be any error in the method of expanding I'' as given 

 above, let us try whether we can verify the last equation by some independent 

 process ; 



Let 07 = — , where p and q are whole numbers, 



and let 1 = ?/ then ?/' = 1, 

 an equation, one of whose roots is cos — + sin — \/ — i , where c is the cir- 

 cumference of a circle, whose radius is unity, 

 1 



.-. 1 = cos — + sm — • -/ — 1, 



q I C . C \P P ■ P 



.-. 1 = (cos — + sm — . -/ - 1 ) = cos^c + sm^c. ^ -I, 



^' or r = cos c 0? + sin c a- . ^ — 1, 



the same equation as that obtained above by the expansion of T. 



From what has just been proved, it appears that there is a connexion 

 between the properties of the circle and the quantities commonly called impos- 

 sible roots, that is between geometry and algebra ; therefore it is so far from 

 being improper to introduce geometric considerations into questions purely 

 algebraic, that it is to geometry we must look (and to geometry alone as far 

 as we know at present), if we expect to arrive at a true theory respecting the 

 square roots of negative quantities. 



It may be proper to observe here, that the object of the above investigation 

 is not to expand all the values of 1" in a series, but merely to show that one 



