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BRITISH ASSOCIATION, SEPTEMBER 1883. 
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regard to a plane curve: in fact, the cc + iy and the X + iY are two imaginary 
variables connected by an equation; say their values are u and v, connected by an 
equation F (u, v) = 0; then, regarding u, v as the coordinates of a point in piano, 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 u, v, which satisfy the equation. 
In geometry it is the curve, whether defined by means of its equation, 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 x, y, 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 x, 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), [413]: “The science presents itself in two ways: as a legitimate 
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 connexion between the quantities is frequently rendered more intelligible 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 a corresponding geometry. An important instance in plane 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-dimen 
sional 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 x, y, z to be, not equal, 
c. xi. 56