igoy \ Henderson — Foundations oe Geometry. 
5 
making with a transversal equal alternate angles are 
parallel, is easily demonstrated. But in order to prove its 
inverse: that parallels cut by a transversal make equal 
alternate angles, he is forced to resort to the following pos- 
tulate axiomatically stated (Williamson’s translation, Ox- 
ford, 1781): 
11. And if a straight line meeting two straight lines makes 
those angles which are inward and upon the. same side of 
it less than two right angles, the two straight lines being 
produced indefinitely will meet each other on that side 
upon which the angles are less than two right angles ( Fig . 
i. Angle A + Angle B less than i8o ° ). 
The points to be observed in connection with this postulate are 
two in number. First, “no one had a doubt of the external 
reality and exact applicability of the postulate. The Euclid- 
ian geometry was supposed to be the only possible form of 
space-science, that is, the space analyzed in Euclid’s axioms 
and postulates was supposed to be the only non-contradictory 
sort of space.” Second, the postulate was neither so axiom- 
atic nor so simple as the proposition it was used to prove; 
and hence the world of mathematicians concluded, with 
Proklos, that this postulate could be deduced as a theorem 
from the other assumptions and the twenty-eight preceding 
theorems. And so, for hundreds and hundreds of years, the 
