[951 
951] 
NON-EUCLIDIAN GEOMETRY. 
483 
tersections 
: the two 
mt planes 
me-pencil : 
e tangents 
common 
dproeal or 
3S i to a 
reciprocal 
ses always 
point of 
reciprocal 
Similarly 
in regard 
)lute ; the 
rocal line, 
; may be 
rsely, that 
if a line 
! point of 
in regard 
; are said 
nt of the 
. plane of 
x to the 
rough the 
B in the 
B of the 
BG, CA. 
point, say 
six parts 
lines, and 
g to the 
s that of 
are each 
= \tt, then the opposite angles are each =\nr, and the included angle and the 
opposite side have a common value; and so also if two angles are each =£7r, then 
the opposite sides are each = ^7r, and the included side and the opposite angle 
have a common value. 
5. Let A, G be points on a line, and B, D points on the reciprocal line; by 
what precedes, each of the lines AB, AD, GB, CD is =\tt: also each of the angles 
ACD, ACB, CAB, GAD is =\tt. The line AG is x to the plane BCD and to the 
lines BG, CD, in that plane; it is also x to the plane BAD and to the lines BA, 
AD in that plane; and similarly for the line BD. From the trihedral of the planes 
which meet in C, distance of planes ACB, ACD = distance of lines BG, CD, viz. the 
dihedral distance of two planes through the line AG is equal to the angular distance 
of their intersections with the x plane BCD; and it is therefore equal also to the 
Fig. 2. 
A 
I) 
linear distance of their intersections with the other x plane BAD: and so from the 
triangle BCD, where BG, CD are each — l tt, the angular distance BCD is equal to 
the linear distance BD; that is, the distance of the planes ACB, ACD, that of the 
lines BG, CD, that of the lines BA, AD, and that of the points B, D, are all of 
them equal; say the value of each of them is = 6. And in like manner the 
distance of the planes ABD, CBD, that of the lines AB, BG, that of the lines AD, 
DC, and that of the points A, G, are all of them equal: say the value of each of 
them is = 8. 
The theorem may be stated as follows : all the planes x to a given line intersect 
in the reciprocal line: and if we have through the given line any two planes, the 
distance of these two planes, the distance between their lines of intersection with 
any one of the x planes, and the distance between their points of intersection with 
the reciprocal line, are all of them equal. 
And it thus appears also that a distance may be represented indifferently as a 
linear distance, an angular distance, or a dihedral distance. 
6. Consider a point and a plane: we may through the point draw a line x to 
the plane, and intersecting it in a point called the “ foot ”: the distance of the point 
and plane is then (as a definition) taken to be equal to that of the point and foot. 
It may be added that the x line is, in fact, the line joining the point with the 
61—2 
x