588 
A SIXTH MEMOIR UPON QUANTICS. 
[158 
218. And we may from the first formula of either set, deduce for the distance 
of the point (x, y, z) and the line (£', y, £"), the expression 
sin“ 1 V K (£x + yy + %'z) 
V(a, y, zf V(&, 77', O 2 ’ 
as may be easily seen by writing + Ji%y' + for x', y\ z, or ax + hy + gz,... for 
77, £, and putting sin -1 for cos -1 . 
219. It may be noticed that there are certain lines, viz. the tangents of the 
Absolute, in regard to which, considered as a space of one dimension, the Absolute is 
a pair of coincident points; and in like manner certain points, viz. the ineunts of 
the Absolute, in regard to which, considered as a space of one dimension, the Absolute 
is a pair of coincident lines. 
220. We may, in particular, suppose that the Absolute, instead of being a proper 
conic, is a pair of points. The line through the two points may be called the Absolute 
line; such line is to be considered as a pair of coincident lines. Any point what 
ever determines with the Absolute, two lines, viz. the lines joining the point with 
the two points of the Absolute; this line-pair is the Absolute for the point con 
sidered as a space of one dimension or locus in quo of a pencil of lines, and the 
theory of the distances of lines through a point is therefore precisely the same as in 
the general case. But any line whatever determines with the Absolute (meets the 
Absolute line in) a pair of coincident points, which pair of coincident points is the 
Absolute in regard to such line considered as a space of one dimension or locus in 
quo of a range of points, and the theory of the distance of points on a line is 
therefore the theory before explained for this special case. But we cannot, in the 
same way as before, compare the distances of points upon different lines, since we have 
not in the present case the quadrant as a unit of distance. The comparison must 
be made by means of the circle, viz. in the present case any conic passing through 
the two points of the Absolute is termed a circle, and the point of intersection of 
the tangents to the circle at the two points of the Absolute (or what is the same 
thing, the pole of the Absolute line in respect to the circle) is the centre of the 
circle. The Absolute line itself may, if it is necessary to do so, be considered as the 
axis of the circle. It is assumed that the points of the circle are all of them 
equidistant from the centre, and by this assumption we are enabled to compare 
distances upon different lines. In fact we may, by a construction precisely similar to 
that of Euclid, Book I. Prop. II., from a given point A draw a finite line equal to 
a given finite line BC, and thence also upon a given line through A, determine the 
finite line AD equal to the given finite line BG. Since the unit of distance for 
points on a line is arbitrary, we cannot of course compare the distances of points 
with the distances of lines. The distance of a point from a line does, however, admit 
of comparison with the distance of two points; we have only to assume as a definition 
that the distance of a point from a line is the distance of the point from the point 
of intersection of the line with the quadrantal line through the point.