6 
TRANSACTION S OF THE TEXAS ACADEMY OF SCIENCE. 
etry, while on the assumption S < n nothing- prevents supposing 
their prolongation while remaining- equidistant. 
Bertrand [of Geneva] and, after his example, Legendre, have 
wished to compare infinite surfaces in angles and between perpen- 
diculars. 
Demonstrations of this kind should be preceded by a determi- 
nation of the mag-nitude idea, which in g-eometry one can only un- 
derstand in connection with measurement, if besides it is convened 
beforehand by what characteristic to discriminate greater from 
lesser. 
For example, a piece of a plane enclosed by a curve is considered 
greater than a polyg-on comprised entirely within it; on the con- 
trary, smaller, if inversely the curve is entirely within the polyg-on; 
and this is so, even if no means be known of measuring- these sur- 
faces. 
But as to infinite surfaces, it is necessary here, as everywhere 
else in mathematics, to understand as the ratio of two infinites its 
limit when its two terms increase indefinitely. 
Besides, it is necessary to understand here by g-eometric mag-ni- 
tude at least such a one as we can approximately determine, esti- 
mating- by the characteristics of inequality. 
In this reg-ard, the demonstration of Bertrand, and all the analo- 
gous proofs are far from satisfactory since in them we see no pro- 
cedure for measuring- the surfaces; not to mention, that the sur- 
faces must first be bounded, to increase into the infinite through 
widening of their boundaries. 
Suppose we wish to compare the surface 
X (Fig. 2), in the opening of the angle DCE, 
with the surface Y, between the two per- 
pendiculars AB, CD to AC, for AC —a. 
The ratio of the two surfaces X and Y, 
even when they increase to the infinite, will 
turn out different according to the way we 
convene in the start to limit them. 
Suppose for example that in every triangle the angle-sum S = n. 
Make AB = CD =na, where n is a whole number. 
Then draw the straight DB. On the other hand, limit the sur- 
face in the opening of the angle DCE by a circle-arc described from 
C as center with radius CD = na. 
We find 
Y —na 2 , ~X.—\nn 2 a 2 \ 
Whence Y _JL 
X * n 
B D 
