158] A SIXTH MEMOIR UPON QU ANTICS, 
591 
for that of the lines (£, ?7, £), (£', y, £'), 
cos - 
ir+w'+sr 
Vf+ i? 2 +r Vf 2 + V 2 + f 2 ’ 
and for that of the point (x, y, z) and the line (£', 77', £'), 
fa? + Vy + K'z 
V« 2 + 2/ a + s 2 V f 2 + r/ 2 + f 8 ’ 
sin" 
226. Suppose (a;, y, z) are ordinary rectangular coordinates in space satisfying the 
condition 
x- + y 2 -M 2 = 1, 
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 %x + yy + £z = 0 
will be a great circle of the sphere; and since we are only concerned with the ratios 
of f y, we may also assume f + rf + = 1. We may of course retain in the formulae 
the expressions x 2 4- y 2 + z 2 and f + ?7 2 + £ 2 , 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 x 2 + y 2 + z 2 = 0. 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 line may be represented indifferently 
as the distance of two points, or as the distance of two lines. 
228. I11 ordinary spherical geometry, the general theory undergoes no modification 
whatever; the Absolute is an actual conic, the intersection of the sphere with the 
concentric evanescent sphere.