442 
PRESIDENTIAL ADDRESS TO THE 
[784 
but only proportional to the distances of a point from three given lines ; the ratios of 
the coordinates (x, y, z) determine the point ; and so in one-dimensional geometry, we 
may have a point determined by the ratio of its two coordinates x, y, these coordinates 
being proportional to the distances of the point from two fixed points ; and generally in 
?i-dimensional geometry a point will be determined by the ratios of the (w+l) coordinates 
(x, 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 variables, 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 determining 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 corresponding 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 the plane of the picture of the line joining the two positions of the 
eye) ; the figure in space is thus represented on the plane by two figures, which are 
such that the lines joining corresponding points of the two figures pass always through 
the fixed point. And such two figures completely replace the figure in space; we can by 
means of them perform on the plane any constructions which could be performed on the 
figure in space, and employ them in the demonstration of properties relating to such 
figure. A curious extension has recently been made : two figures in space such that the 
lines joining corresponding points pass through a fixed point have been regarded by the 
Italian geometer Veronese as representations of a figure in four-dimensional space, and 
have been used for the demonstration of properties of such figure. 
I referred to the connexion of Mathematics with the notions of space and time, but 
I have hardly spoken of time. It is, I believe, usually considered that the notion of 
number is derived from that of time ; thus Whewell in the work referred to, p. xx, says 
number is a modification of the conception of repetition, which belongs to that of time. 
I cannot recognise that this is so : it seems to me that we have (independently, I 
should say, of space or time, and in any case not more depending on time than on space) 
the notion of plurality ; we think of, say, the letters a, b, c, &c., and thence in the case