The axioms of descriptive geometry, by A.N. Whitehead.

69-72] ANALYTICAL EXPRESSION FOR DISTANCE

71

Then if P1 and P2 are both within the region enclosed by the absolute,
0 is necessarily real.
Hence (cf. ~ 69, equation (1))
{APiA2P2} = =/lA2/X'l2 = e2.
Thus the distance P1P,, written dist (PIP2), can be defined by
dist (P P2) = }Y log {AP1A2PI...................(1).
Therefore
ch dist (P1P2) 1 + c (xx2 + Y.Ic (* + / z,).2)
cosh s- PP- (2).
Y {1 + c (x12 + 2) + C (22 + z222 + Z22)(}2
There will only be one distance between Pi and P2. This must be
associated with the sole segment of the line PiP2 which lies wholly
within the region enclosed by the absolute. This system of metrical
geometry only embraces those points which lie within the region
enclosed by the absolute*. Any point in the region to which the
metrical geometry applies is at an infinite distance from every point
on the absolute.
71. The parabolic formula for the distance, arising when c is indefinitely diminished, can be derived as a limit from either of the other
two cases. Put }2c= + 1, according as c is positive or negative, so
that y increases as c diminishes numerically. Then expanding both
sides of equation (3) of ~ 69, or of equation (2) of ~ 70, and proceeding to the limit, we find
{dist (PiP2)}2 = (i - x2)2 + (1 - y)2 + (Z1 - z2)2......(1).
The parabolic system of metrical geometry embraces all projective
space with the exception of points on the latent plane, which is the
infinite plane in our system of coordinates. This is the ordinary
Euclidean Geometry.
72. Exactly the same procedure can be applied for the measurement of the angle between planes. Let pi and pa be any two planes,
and let t, and t2 be the two real or imaginary planes through the
intersection of pi and p2 and tangential to the absolute. When the
* Metrical Geometry of this Hyperbolic Type was first discovered by Lobatschefskij in 1826, and independently by J. Bolyai in 1832. This discovery is
the origin of the modern period of thought in respect to the foundations of
Geometry.