ME. A. CAYLEY’S SIXTH MEMOIE UPON QUANTICS. 
89 
for that of the lines (|, ri, 1), (!', I'), 
_i W + . 
* 
and for that of the point (x, y, z) and the line (!', rj, t), 
• -1 ^x + ^y + ^z 
V + -\- V 
226. Suppose {x, y, z) are ordinary rectangular coordinates in space satisfying the 
condition 
the point having {x, y, z) for its coordinates will be a point on the surface of the sphere, 
and (the last-mentioned equation always subsisting) the equation lx-\-ny-\-lz~^ will be 
a great circle of the sphere ; and since we are only concerned with the ratios of t-, 
we may also assume ^=1. We may of course retain in theformulse the expres- 
sions a^^y^-^z^ and without substituting for these the values unity, and it is 
in fact convenient thus to preserve all the formulae in their original forms. We have 
thus a system of spherical geometry ; and it appears that the Absolute in such system is 
the (spherical) conic, which is the intersection of the sphere with the concentric cone 
or evanescent sphere The circumstance that the Absolute is a proper 
conic, and not a mere point-pair, is the real ground of the distinction between spherical 
geometry and ordinary plane geometry, and the cause of the complete duality of the 
theorems of spherical geometry. 
227. I have, in all that has preceded, given the analytical theory of distance along 
with the geometrical theory, as well for the purpose of illustration, as because it is 
important to have the analytical expression of a distance in terms of the coordinates; 
but I consider the geometrical theory as perfectly complete in itself: the general result 
is as follows, viz. assuming in the plane (or space of geometry of two dimensions) a conic 
termed the Absolute, we may by means of this conic, by descriptive constructions, divide 
any line or range of points whatever, and any point or pencil of lines whatever, into an 
infinite series of infinitesimal elements, which are (as a definition of distance) assumed 
to be equal ; the number of elements between two points of the range or two lines of 
the pencil, measures the distance between the two points or lines ; and by means of the 
quadrant, as a distance which exists as well with respect to lines as points, we are 
enabled to compare the distance of two lines with that of two points ; and the distance 
of a point and a hne may be represented indifferently as the distance of two points, or 
as the distance of two lines. 
228. In ordinary spherical geometry, the general theory undergoes no modification 
whatever ; the Absolute is an actual conic, the intersection of the sphere with the con- 
centric evanescent sphere. 
229. In ordinary plane geometry, the Absolute degenerates into a pair of points, viz. 
the points of intersection of the line infinity with any evanescent circle, or what is the 
