16 REPORT—1883. 
course been introduced only for facility of expression; it may be proper 
to add that in speaking of the two figures I have been following Briot 
and Bouquet rather than Riemann, whose representation of the function 
of an imaginary variable is a different one. 
I have been speaking of an imaginary variable (# + iy), and of a 
function ¢(« + iy) = X + 7Y of that variable, but the theory may equally 
well be stated in regard to a plane curve: in fact, the « + zy and the 
X + <Y are two imaginary variables connected by an equation; say their 
values are u and v, connected by an equation F(u, v) = 0; then, regard- 
ing u, v as the coordinates of a. point im plano, this will be a point on the 
curve represented by the equation. The curve, in the widest sense of the 
expression, is the whole series of points, real or imaginary, the coordinates 
of which satisfy the equation, and these are exhibited by the foregoing 
corresponding figures in two planes; but in the ordinary sense the curve 
is the series of real points, with coordinates wu, v, which satisfy the 
equation. 
In geometry it is the curve, whether defined by means of its equa- 
tion, or in any other manner, which is the subject for contemplation and 
study. But we also use the curve as a representation of its equation— 
that is, of the relation existing between two magnitudes 2, 7, which are 
taken as the coordinates of a point on the curve. Such employment of 
a curve for all sorts of purposes—the fluctuations of the barometer, the 
Cambridge boat races, or the Funds—is familiar to most of you. It is in 
_ like manner convenient in analysis, for exhibiting the relations between 
any three magnitudes a, y, z, to regard them as the coordinates of a 
point in space; and, on the like ground, we should at least wish to regard 
any four or more magnitudes as the coordinates of a point in space of a 
corresponding number of dimensions. Starting with the hypothesis of 
such a space, and of points therein each determined by means of its 
coordinates, it is found possible to establish a system of n-dimensional 
geometry analogous in every respect to our two- and three-dimensional 
geometries, and to a very considerable extent serving to exhibit the 
relations of the variables. To quote from my memoir ‘On Abstract 
Geometry’ (1869) : ‘ The science presents itself in two ways: as a legiti- 
mate extension of the ordinary two- and three-dimensional geometries, 
and as a need in these geometries and in analysis generally. In fact, 
whenever we are concerned with quantities connected in any manner, 
and which are considered as variable or determinable, then the nature of 
the connection between the quantities is frequently rendered more intel- 
ligible by regarding them (if two or three in number) as the coordinates 
of a point in a plane or in space. For more than three quantities there 
is, from the greater complexity of the case, the greater need of such a 
representation ; but this can only be obtained by means of the notion of 
a space of the proper dimensionality ; and to use such representation we 
require @ corresponding geometry. An important instance in plane 
