GEORGE BRUCE HALSTED — NEW ELEMENTS OE GEOMETRY. 
5 
and that therefore in the original and consequently in every tri- 
angle S = 7 r. (Reflexions sur la theorie des paralleles. Memoires 
de l’Acad. des Sc. Tome XII. 1833, p. 390.) 
However, this reasoning- is not correct, because the sides of the 
triangle here increase without limit and therefore also we can as- 
sume the limit of approach so that the angle AC / B' ==S< w. 
Let A, B, C be the angles of A ABC at the points marked by 
these letters; let ABC' be the angles of A AB'C' at the points 
A, B', C'; finally let li be the perpendicular let fall from C' on 
ABC 
With help of the Imaginary Geometry, supposing- S < tt and 
desig-nating- by e the base of Naperian log-arithms, we find 
cot A' = cot A + 
cot B' = cot A -j- 
sin C 
sin A sin B 
sin B 
sin A sin C 
e 
h 
— e~ 
cos -J-S cos (J S— A) cos (£S — B) cos J (S— C) 
sin 2 B + sin 2 C + 2 sin B sin C cos A 
The first two equations show that A' and B' are always real, 
and with the transformation of the triangle decrease toward limit 
zero. The last equation gives always the heig-ht h and determines 
the limit of approach 
h — log- cot \ S, 
where the log-arithm is Naperian. 
Althoug-h Leg-endre designates his demonstration as completely 
rig-orous, he without doubt thoug-ht otherwise, for he adds the pro- 
viso, that a difficulty which one would perhaps still find can always 
be removed. For this, he has recourse to calculations founded on 
the first formulas of plane trig-onometry, which it would be neces- 
sary first to establish, and which just in this case are useless and 
lead to no result. 
To omit no argmment in favor of his theorem, Leg-endre re- 
marks, that congruent triangles placed tog-ether with, throug-hout, 
different angles by threes at a point, make a ribbon which can be 
prolong-ed indefinitely and which then is bounded by two broken 
lines concave toward one another for S < tt, for S > tt convex one 
toward the other. The proved impossibility of the latter case in- 
duces also to reject the first, where the lines, turned toward each 
other like two circle-arcs, would seem necessarily to meet. 
• It seems superfluous to analyse and judg-e such an arg-ument, 
where there is not even the shadow of a rig-orous demonstration. 
I will only add, that lines concave toward each other only approach 
each other because of the notion assumed in the ordinary g-eom- 
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