38 
SECTION A. 
This is a difficulty which a theory of mere direction cannot get over. 
Led by his theory of perpendicularity, Bue6 considers the question : What 
does a conic section become, when its ordinates become imaginary ? Con 
sider a circle; when x has any value between — a and -f- «> then 
y = dri/ a 2 — x 2 
But when x is greater than a, or less than —a, let it be denoted by x', and 
the analogue of y by y', then 
y' = |/x' 2 — a 2 . 
Buee advances the view that the circle in the plane of the paper changes into 
an equilateral hyperbola in the plane perpendicular to the plane of the paper; 
but he does not prove the suggestion, or test it by application to calculation. 
A similar view has been developed by Phillips and Beebe in their “Graphic 
Algebra.” It appears to me that here we have a fundamental question in 
the theory of \/—1. The expression 1/a 2 —x 2 denotes the ordinate of the 
circle, what is represented by |/ —1 \/x' 2 —a 2 , x' being greater than a? 
The former is constructed by drawing from the extremity of x a straight 
line at right angles to it in 
the given plane, and de 
scribing with centre 0 a 
circle of radius a the point 
of intersection P determin 
ing the length of the ordi 
nate, and—\/ a?—x 2 is equal 
and opposite. Now (fig. 4) 
~\/x' 2 —a 2 is equal in length 
to the tangent from the ex 
tremity of x' to the circle, 
and p 7 "—1 appears to indi 
cate the direction of the 
tangent, which varies in inclination to the axis of x, but is determined by 
always being perpendicular to the radius at the point of contact. Hence 
if x 1 be considered a directed magnitude, the expression 
x' +}/—1 \/x' 2 —a 2 
denotes the radius from O to the one point of contact T, while 
x'—\/—1 ]/x 2 —a 2 
denotes the radius to the other point of contact T'. This construction 
does not necessitate going out of the given plane; and if space be consid 
ered we have a whole complex of ordinates to the sphere, as well as a 
complex of tangents to the sphere. The ordinary theory of minus gives 
no explanation of the double sign in the case of the tangent. It is true in 
the case of the two ordinates, that the one is opposite to the other in direc 
tion, but it is not true of the two tangents. In the case of the sphere the 
ordinate may have any direction in a plane perpendicular to x, while the 
tangent may have any direction in a cone of which x is the axis This 
other and hitherto unnoticed meaning of \/—1 will be developed more 
fully in the investigation which follows (p. 52).