OF NEGATIVE QUANTITIES. 243 



(2) Hence, if a be a positive quantity, a line equal in length to a and drawn 

 at right angles to + 1 will be equal to a ^—1. 



(3) Let it be required to find the length and direction of t/— 1 ; 

 f/iZY — .i/^^Tij .-. 4/~l i^s a mean proportional between + 1 and ^— 1, 



.•. ^— 1 is equal in length to + 1 ; 



Also it has been proved that ^— I is inclined to + 1 at 90°, 



/. ^— 1 is inclined to + I at 45° ; 



/. 4/— T is a line equal in length to + 1, and inclined to + 1 at 45°. 



(4) As ^—l is a line drawn in an oblique direction, let it be required to 

 find an expression for it, considered as the sum of two quantities, the one either 

 positive or negative, and the other the square root of a negative quantity. 



Since ^— 1 is equal in length to + 1 and is inclined to + 1 at 45°, and ad- 

 dition is performed in the same manner as composition of motion, 

 y^ = cos 45° + sin 45° J^ 



~ 71 + 71 • ^/~^' 



(5) To show that — ^ + ~T| -\/— 1 is a true value of ^Z— 1 according to 

 common algebra ; 



Let ^ — 1 = ir, 

 then X +1=0, 

 an equation, one of whose roots is — ;= + —r^ ^/— I, 



••• -j^ + -j~ ^/^ is a true value of ^—\. 



In like manner other examples might be given, but these will suffice to 

 illustrate the definition ; and for further information I must refer to a treatise 

 which I published on this subject in April 1828. 



Since the publication of that work, several objections have been made to 

 the geometrical representation of the square roots of negative quantities. 

 First, that impossible roots are merely signs of impossibility ; that, if, in the 

 solution of any question, we arrive at an equation, all whose roots are impos- 

 sible, the only conclusion to be drawn, is, that the question involves an impos- 



2 i2 



