ADDRESS. 17 
geometry has already presented itself in the question of the number of 
curves which satisfy given conditions; the conditions imply relations 
between the coefficients in the equation of the curve; and for the better 
understanding of these relations it was expedient to consider the 
coefficients as the coordinates of a point in a space of the proper 
dimensionality.’ 
It is to be borne in mind that the space, whatever its dimensionality 
may be, must always be regarded as an imaginary or complex space 
such as the two- or three-dimensional space of ordinary geometry ; the 
advantages of the representation would otherwise altogether fail to be 
obtained. 
I have spoken throughout of Cartesian coordinates ; instead of these 
it is in plane geometry not unusual to employ trilinear coordinates, and 
these may be regarded as absolutely undetermined in their magnitude— 
viz. we may take 2, y, z to be, not equal, but only proportional to the 
distances of a point from three given lines; the ratios of the coordinates 
(a, y, 2) determine the point; and so in one-dimensional geometry, we 
-may have a point determined by the ratio of its two coordinates a, y, these 
coordinates being proportional to the distances of the point from two 
fixed points ; and generally in n-dimensional geometry a point will be de- 
termined by the ratios of the (n+1) coordinates (a, y, z...). The 
corresponding analytical change is in the expression of the original 
magnitudes as fractions with a common denominator; we thus, in place 
of rational and integral non-homogeneous functions of the original vari- 
ables, introduce rational and integral homogeneous functions (quantics) 
of the next succeeding number of variables—viz. we have binary quantics 
corresponding to one-dimensional geometry, ternary to two-dimensional 
geometry, and so on. 
It is a digression, but I wish to speak of the representation of points 
or figures in space upon a plane. In perspective we represent a point in 
space by means of the intersection with the plane of the picture (suppose 
a pane of glass) of the line drawn from the point to the eye, and doing 
this for each point of the object we obtain a representation or picture of 
‘the object. But such representation is an imperfect one, as not deter- 
mining the object: we cannot by means of the picture alone find out the 
form of the object; in fact, for a given point of the picture the corre- 
sponding point of the object is not a determinate point, but it is a point 
anywhere in the line joining the eye with the point of the picture. To 
determine the object we need two pictures, such as we have in a plan and 
elevation, or, what is the same thing, in a representation on the system of 
Monge’s descriptive geometry. But it is theoretically more simple to 
consider two projections on the same plane, with different positions of the 
eye: the point in space is here represented on the plane by means of two 
points which are such that the line joining them passes through a 
fixed point of the plane (this point is in fact the intersection with 
c 
