4 
TRANSACTION'S OF THE TEXAS ACADEMY OF SCIENCE. 
the proposition that the sum of the angles of a triangle can not be 
greater than tt, than two right angles. 
There also he wished to prove that this sum could not be < tt. 
But here he has not noticed that just then when the value of the 
sum deduced in another way shows some contradiction, the lines 
possibly no longer make a triangle. 
I do not think it necessary to insist here in more detail on this 
error, which Legendre himself conceded later, when he declared, 
although the principle used as foundation was subject to no doubt, 
yet he found difficulties which he could not overcome.* 
In the Memoires of the Academy of Science of Paris for 1833, he 
has also published the theorem, that the sum of the angles in every 
rectilineal triangle must equal tt, if in any one it has this value. 
I had to prove the same proposition in the theory I wrote in 1826. 
I even find that Legendre often came upon the way which I have 
chosen with such success. But probably his prejudice in favor of 
the proposition assumed by all caused him, at each attempt, either 
to be hasty in the deduction or to fill it out with things which un- 
der the new theory are no longer admissible. 
Let us analyse all he has printed on this subject in the Memoires 
of the French Academie for the year 1833. 
In the triangle ABC (Fig. 1) draw from 
A through the midpoint I of the side BC 
the straight [sect] AC'=AB. Prolong 
AB to make AB' =2AI. Thus we get a 
triangle AB'C' in which B'C' =AC, and 
the sum of the angles S is the same as in the first triangle ABC, of 
which the angle CAB passes into the A AB'C', dividing into the 
two at the points A and B' . 
Suppose now AB the* largest side of the triangle ABC or anyhow 
not less than the others, and also BC<AC. Then is AC ' >C ' B', 
and the angle opposite the side B'C' in A AB ' C' is at most half 
as great as the angle CAB. Continuing thus, we will arrive at a 
triangle where two of the angles will be as small as we choose, 
while the sum of all three has the same value as in the first tri- 
angle ABC. 
Legendre would from this conclude that in diminishing the two 
angles the approach of the opposite sides toward the third side 
would finally convert the remaining angle into two right angles, 
*Ivegendre’s own words are: “Nous devons avouer que cette seconde propo- 
sition, quoique le principe de la demonstration fut bien connu, nous a present^ 
des difficulty que nous n’ avons pu entierement resoudre. ” (Memoires de 
l’Acad. des Sc. de l’lnst. de France, Tome XII. 1833, p. 371.) 
Fig. 1. 
