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IV. A Memoir on Abstract Geometry. By Professor Cayley, F.R.S. 
.Received October 14, — Read December 16, 1869. 
I submit to the Society the present exposition of some of the elementary principles of an 
Abstract m-dimensional Geometry. The science presents itself in two ways, — as a legi- 
timate 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 together in any manner, and which are, or are considered as variable 
or determinable, then the nature of the relation between the quantities is frequently 
rendered more intelligible by regarding them (if only 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 the geometry of such space. An important 
instance in plane geometry has actually presented itself in the question of the determi- 
nation of the number of the curves which satisfy given conditions : the conditions imply 
relations between the coefficients in the equation of the curve ; and for the better under- 
standing of these relations it was expedient to consider the coefficients as the coordinates 
of a point in a space of the proper dimensionality. 
A fundamental notion in the general theory presents itself, slightly in plane geometry, 
but already very prominently in solid geometry ; viz. we have here the difficulty as to 
the form of the equations of a curve in space, or (to speak more accurately) as to the 
expression by means of equations of the twofold relation between the coordinates of a 
point of such curve. The notion in question is that of a /r-fold relation, — as distinguished 
from any system of equations (or onefold relations) serving for the expression of it, and 
as giving rise to the problem how to express such relation by means of a system of equa- 
tions (or onefold relations). Applying to the case of solid geometry my conclusion in 
the general theory, it may be mentioned that I regard the twofold relation of a curve 
in space as being completely and precisely expressed by means of a system of equations 
(P=0, Q— 0, . . . T=0), when no one of the functions P, Q, . . . T is a linear function, 
with constant or variable integral coefficients, of the others of them, and when every 
surface whatever which passes through the curve has its equation expressible in the form 
U=AP-j-BQ . . .+KT, with constant or variable integral coefficients, A, B, . . . K. It 
is hardly necessary to remark that all the functions and coefficients are taken to be 
rational functions of the coordinates, and that the word integral has reference to the 
coordinates. 
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