A MEMOIR ON ABSTRACT GEOMETRY. 
[From the Philosophical Transactions of the Royal Society of London, vol. clx. (for 
the year 1870), pp. 51—63. 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 
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 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 repre 
sentation ; 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 determination of the number of the curves which satisfy given con 
ditions : 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. 
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 &-fold relation,—as dis 
tinguished 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 equations (or onefold relations). Applying to the case of solid geometry