8 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
Fig. 3 
Let AB, CD, be perpendiculars 
on AC (Fig-. 3); take arbitrarily 
AB = CD. Assuming- S < tt, the 
angles ABD, CDB, are acute; the 
perpendiculars BB", DD", on BD 
are situated in the interior of the 
fig-ure B'BDD', without ever meeting-, and make with BB' and 
DD' the angles B'BB", D'DD", whose infinite surfaces are smaller 
than the surface B'ACD', contrary to what Bertrand wished to 
maintain of all angles without exception. 
Bertrand gives another form of his demonstration considering- 
infinite surfaces in angles only. 
In the triangle ABC (Fig- 4), 
call A, B, C the angles opposite 
the sides #, b, c, which prolong-: 
AC throug-h A to A", AB 
throug-h B to B M/ , BC throug-h 
C to C’. The surfaces X+a, 
Y+y, Z, in the opening's of the 
exterior angles tt— A, tt— B, 
7T — C, extended indefinitely, 
form about the point C toward 
all sides an unlimited surface, 
where is omitted the surface 
ABC, which because of its smallness may be neglected, 
sig-nifies that tt— A+tt — B+tt — C= 2~, whence A+B J r C=7r. 
Verify now this reasoning-, after limiting- the surfaces by arcs 
of circles described from the points A, B, C as centers with radii 
equal to CC ' =r. 
The circle about center C, meeting- the sides of the angle tt— A 
at A’ and B’ divides the surface in the opening- of tt — A into two 
parts; the one X within, the other oc without the entire circle with 
center C. 
The same circle, meeting- at B', C' the sides of the angle 7r — B, 
divides the surface in the opening- of tt — B into two parts: Y within, 
y without the circle; but besides, the piece & of the entire circle 
does not appertain to the angle 7r — B. 
That gives all possible cases which our fig-ure can present, pre- 
supposing- r>c, a>c. 
Designating- now by A the area of the triangle ABC, by R the 
area of the circle of radius r we find 
Fig. 4. 
This 
