GEORGE BRUCE HALSTED — NEW ELEMENTS OF GEOMETRY. 
7 
a ratio which with increase of the surfaces Y and X, for n— go be- 
comes nul, as Bertrand also has assumed. 
If, however, instead of assuming- AB=na, we make AB=n 2 a, 
but CD is left as before, then this time we find the ratio 
Y _ _4 
X “ 7T , 
which is constant for every n and consequently also for n = oo, 
where both surfaces become infinite. 
Thus the ratio Y/X turns out each time different, according- as 
we limit the surfaces in the beginning-, and according- as they sub- 
sequently increase into the infinite. 
Now limit both surfaces X and Y by an arc FDE described from 
C as center with radius CD —na. 
Under the assumption, that in every triangle the angle-sum 
S > 7T or S = 7T, it is easy to see that the ratio Y : X becomes nul 
for n— oo. 
This means that in both cases Y becomes an infinite of the first 
order, but X an infinite of the second order, as also Bertrand 
thoug-ht. 
On the other hand, with the assumption S< w, we find the ratio* 
Y 
X 
2 ( e na -\-e~ na ) arc sin 
e 2a -l 
2 na 
+ 1 
2 a 
+ 1 e 2na —l 
— 4 arc sin 
e a — e 
7T ( e na + e~ na — 2) 
where the number e>l is independent of n; and consequently for 
n — oo 
Y 2 
= arc sin 
X 
f e 2a — l'] 
e 2a +l ’ 
V 1 y 
which is not null so long- as a > 0, and which can be neglected only 
when a is infinitesimal. 
From another point of view it is easy to see that we can not pre- 
sume the ratio Y :X for X= co to become null in case S< tu 
* Designate by e the base of the Naperian logarithms, and leaving indeter- 
minate the unit for length, put 
2 . , 
sin <P =■ tan r cot x , 
tan r' — 
tan x' — 
Then we find as the expression for the piece of the circle comprised between 
two perpendiculars to the radius r, of which one passes through the center 
and the other is at distance from it (Imaginary Geometry). 
— arc cot (sin r' cot (}> )- <p . 
sin r' 
