242 
MR. WARREN ON THE SQUARE ROOTS 
examine into the nature of algebraic operations, with a view, if possible, of 
arriving at these general definitions and fundamental principles : and I found, 
that, by considering algebra merely as applied to geometry, such principles 
and definitions might be obtained. The fundamental principles and definitions 
which I arrived at were these : that all straight lines drawn in a given plane 
from a given point, in any direction whatever, are capable of being algebra- 
ically represented, both in length and direction ; that the addition of such lines 
(when estimated both in length and direction) must be performed in the same 
manner as composition of motion in dynamics ; and that four such lines are 
proportionals, both in length and direction, when they are proportionals in 
length, and the fourth is inclined to the third at the same angle that the second 
is to the first. From these principles I deduced, that, if a line drawn in any 
given direction be assumed as a positive quantity, and consequently its oppo- 
site, a negative quantity, a line drawn at right angles to the positive or nega- 
tive direction will be the square root of a negative quantity, and a line drawn 
in an oblique direction will be the sum of two quantities, the one either posi- 
tive or negative, and the other, the square root of a negative quantity. 
This may be illustrated by the following examples : 
(1) Let it be required to find the length and direction of — 1 ; 
First to find the direction of *y~ 1, 
— 1 is evidently a mean proportional between + 1, and — 1 ; 
Now by the definition of proportion, if 4 lines be proportionals, the fourth is 
inclined to the third at the same angle that the second is to the first, 
.•. if three lines be proportionals, the third is inclined to the second at the 
same angle that the second is to the first, 
.-. a mean proportional between any two lines must lie in such a direction 
as to bisect the angle at which those lines are inclined to each other, 
.*. J-\ bisects the angle at which — 1 is inclined to + 1 ; 
But — 1 is inclined to + 1 at 180°, 
.-. ^/ — 1 is inclined to -f- 1 at 90°; 
Next to find the length of — 1 ; 
Since — 1 is a mean proportional between + 1 and — 1, and + 1 and — 1 
are equal in length, — 1 is equal in length either to + 1 or — 1 ; 
.-. N / — 1 is a line equal in length to + 1, and drawn at right angles to + 1. 
