62 
ME. A. CAYLEY’S SIXTH IMEMOIE LTOX QUAXTICS. 
and planes in space, it would be necessary to consider,-!", that the words point and 
line may mean respectively i,mnt and line-, 2”, that the word Ime may mean y-onif m 
a plane*, and the word point mean line, viz. the expression pomts in a hue mean i,m 
through a ]>mnt and in a plane; 3rd, that the word hue may mean and the word 
point mean plane, viz. the expression points in a line mem planes though a line. - 
so in using the geometry of two dimensions in reference to geometry of three dimensions 
considered as a geometry of points, lines, and planes in space, it would be neces^ to 
consider,— 1", that the words point, line, and plane may mean respectively jwmt, line. 
and plane; 2”, that the words point, line, and plane may mean respectively 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 si^- 
iication of the terms, in treating the geometry of one dimension as a geometiy of pomts 
in a line, and the geometry of two dimensions as a geometiy of points and Imes m a 
plane, we do in reality treat these geometries respectively in an absolutely genera 
manner. In particular— and I notice the case because I shall have occasion agam to 
refer to it— we do in the geometry of two dimensions include sphencal geometry; t e 
words plane, point, and line, meaning for this pm'pose, spherical surface, arc (of a gl'eat 
circle) and point (that is, pair of opposite points) of the spherical sm-face. And m like 
manner the geometry of one dimension includes the cases of pomts on an arc, and o 
arcs through, a point. j- 
148. I repeat also a remark which is in eifect made in the same No. 4 ; e cooi ina ^ 
^ i/ of the geometry of one dimension, and the coordmates a?, z and ^ o t e 
geometry of two dimensions are only determinate to a common factor^^?-es (that is, it i^ 
the ratios only of the coordinates, and not them absolute magnitudes, wiic aie 
determinate) ; hence in saying that the coordinates .r, y are equal to a, h, or in imtmg 
b, we mean only that x:y=a: b, and we never as a result obtain 
oiily x-y~a-b. And the like with respect to the coordinates y, z and s, n, (Hi 
the geometry of two dimensions, (T, y=^a, b, is for this reason considered and spoken ot 
as a single equation.) But when this is once understood, there is no objection to tiea 
ing the coordinates as if they were completely determinate. 
On Geometry of One Dimension, Nos. 149 to 168. 
149 In geometry of one dimension we have the line as a space or locus in quo, which 
is considered as made up of points. The several points of the line aie determnmd b> 
the coordinates (*, y), viz. attributing to these any specific values, or writmg x, y a, , 
we have a particular point of the line. And we may say also that the Ime is the locti 
in quo of the coordinates {x, y). 
. It would be more accurate to say that the word line may mean toint-inn,nd-witl-u pUne, viz. the tea.- 
in quo of lines through the point and in the plane. Added, June 16, 1889. A. 0, 
