246 
MR. A. CAYLEY’S INTRODUCTORY MEMOIR UPON QUANTICS. 
4. I have spoken of the coordinates of a point in space. I consider that there is an 
ideal space of any number of dimensions, but of course, in the ordinary acceptation 
of the word, space is of three dimensions ; however, the plane (the space of ordinary 
plane geometry) is a space of two dimensions, and we may consider the line as a 
space of one dimension. I do not, it should be observed, say that the only idea 
which can be formed of a space of two dimensions is the plane, or the only idea 
which can be formed of space of one dimension is the line ; this is not the case. To 
avoid complexity, I will take the case of plane geometry rather than geometry of 
three dimensions ; it will be unnecessary to speak of space, or of the number of 
its dimensions or of the plane, since we are only concerned with space of two dimen- 
sions, viz. the plane; I say, therefore, simply that x, y, z are the coordinates of a 
point (strictly speaking, it is the ratios of these quantities which are the coordinates, 
and the quantities x, y, z themselves are indeterminate, i. e. they are only determi- 
nate to a common factor pres, so that in assuming that the coordinates of a point 
are go, (3, y, we mean only that x :y : z=a : \ 3 : y, and we never as a result obtain 
x, y, z = u, (3, y, but only x \y : z=u : (3 : y; but this being once understood, there is 
no objection to speaking of x,y, z as coordinates). Now the notions of coordinates 
and of a point are merely relative; we may, if we please, consider x:y:z as the 
parameters of a curve containing two variable parameters ; such curve becomes 
of course determinate when we assume x:y : z=a : (3 : y, and this very curve is 
nothing else than the point whose coordinates are a, (3, y, or as we may for short- 
ness call it, the point (a, {3, y). And if the coordinates (x, y, z) are connected by an 
equation, then giving to these coordinates the entire system of values which satisfy 
the equation, the locus of the points corresponding to these values is the locus repre- 
senting or represented by the equation ; this of course fixes the notion of a curve of 
any order, and in particular the notion of a line as the curve of the first order. 
The theory includes as a very particular case, the ordinary theory of reciprocity in 
plane geometry ; we have only to say that the word c point’ shall mean 4 line,’ and the 
word ‘line’ shall mean ‘point,’ and that expressions properly or primarily applicable 
to a point and a line respectively shall be construed to apply to a line and a point 
respectively, and any theorem (assumed of course to be a purely descriptive one) 
relating to points and lines will become a corresponding theorem relating to lines 
and points ; and similarly with regard to curves of a higher order when the ideas of 
reciprocity applicable to these curves are properly developed. 
5. A quantic of the degrees m, m ! .. in the sets (x,y..), (x, y' ..) &c. will for the 
most part be represented by a notation such as 
m m' 
{*Jx,y..Jpd, y. .)..), 
where the mark * may be considered as indicative of tne absolute generality of the 
quantic; any such quantic may of course be considered as the sum of a series of 
terms x a y p ..x la 'y w ' &c. of the proper degrees in the different sets respectively, each 
