562 
A SIXTH MEMOIR UPON QUANTICS. 
[158 
word line mean point. It is, I say, necessary to do this, for in such geometry of 
two dimensions we have systems of points in a line and of lines through a point, 
and each of these systems is in fact a system belonging to, and which can by such 
extended signification of the terms be included in, the geometry of one dimension. 
And precisely because we can by such extension comprise the correlative theorems 
under a common enunciation, it is not in the geometry of one dimension necessary 
to enunciate them separately; it may be and very frequently is necessary and proper 
in the geometry of two dimensions, where we are concerned with systems of each 
kind, to enunciate such correlative theorems separately. It may, by way of further 
illustration, be remarked, that in using the geometry of one dimension in reference 
to geometry of three dimensions considered as a geometry of points, lines, and planes 
in space, it would be necessary to consider,—1°, that the words point and line may 
mean respectively point and line; 2°, that the word line may mean point in a plane 1 , 
and the word point mean line, viz. the expression points in a line mean lines through 
a point and in a plane; 3rd, that the word line may mean line and the word point 
mean plane, viz. the expression points in a line mean planes through a line. And 
so in using the geometry of two dimensions in reference to geometry of three dimen 
sions considered as a geometry of points, lines, and planes in space, it would be 
necessary to consider,—1°, that the words point, line, and plane may mean respectively 
point, line, and plane; 2°, that the words point, line, and plane may mean respectively 
plane, line, and point. But I am not in the present memoir concerned with geometry 
of three dimensions. The thing to be attended to is, that in virtue of the extension 
of the signification of the terms, in treating the geometry of one dimension as a 
geometry of points in a line, and the geometry of two dimensions as a geometry of 
points and lines in a plane, we do in reality treat these geometries respectively in 
an absolutely general manner. In particular—and I notice the case because I shall 
have occasion again to refer to it—we do in the geometry of two dimensions include 
spherical geometry; the words plane, point, and line, meaning for this purpose, spherical 
surface, arc (of a great circle) and point (that is, pair of opposite points) of the 
spherical surface. And in like manner the geometry of one dimension includes the 
cases of points on an arc, and of arcs through a point. 
148. I repeat also a remark which is in effect made in the same No. 4; the 
coordinates x, y of the geometry of one dimension, and the coordinates x, y, z and 
f, V> K °f the geometry of two dimensions are only determinate to a common factor 
pres (that is, it is the ratios only of the coordinates, and not their absolute magni 
tudes, which are determinate); hence in saying that the coordinates x, y are equal 
to a, h, or in writing x, y = a, b, we mean only that x : y = a : b, and we never as 
a result obtain x, y = a, b, but only x : y = a : b. And the like with respect to the 
coordinates x, y, z and f, rj, £ (In the geometry of two dimensions, x, y = a, b, is 
for this reason considered and spoken of as a single equation.) But when this is 
once understood, there is no objection to treating the coordinates as if they were 
completely determinate. 
1 It would be more accurate to say that the word line may mean point-in-and-with-a plane, viz. the 
locus in quo of lines through the point and in the plane. Added, June 16, 1859.—A. C.