The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

94

NON-EUCLIDEAN GEOMETRY

[CH. IV.

On the ray Ai2 cut off the equal segments AA1, A1A2, A2A3....
Let the Concentric Limiting-Curves through Al, A2, A3,...,
cut the ray B82 in B1, B2, Bs,....
Then we have, by ~ 54 (1),
AA1 = BB1= B1B2 = B2B3= etc.
Also, from ~ 48 and ~ 54 (3),
ar AB: arc ABi =arc A.B,: arc AB2 =arc A2B2: arc A3B3 =etc.
This ratio is greater than unity, and depends only on the
length of AA1.
We may choose the unit segment so that the ratio is equal to
e, when AA = AA = AA3 =...= the unit segment.
Let the arcs AB, A1B1, A2B2, etc., be denoted by s, sl, s2,
etc., when the segment AA1 is the unit of length.
Then we have
S: S1-S1: S2= S2: 83. =e.
Thus s,= se-, when n is a positive integer.
A
5 S* n
B
FIa. 66.
It is easy to deduce from this that when the segment AP
(Fig. 66) is x units, x being any rational number, and the arc
PQ is denoted by sx, then we have
sx = se -.
We obtain the same result for an irrational number x by
proceeding to the limit.