ipo?] Henderson — Foundations of Geometry. 
7 
pose a substitute for the parallel-postulate, such as “Two 
straight lines which intersect cannot both be parallel to the 
same straight line” (Ludlam), and “Any three points are col- 
linear or concyclic” (Bolyai). And the celebrated Hilbert, in 
his Vorlesung ueber Enklidische Geometric , (winter semester, 
1898-9) cites the following theorems: 
1. The sum of the angles of a triangle is always equal to 
two right angles. 
2. If two parallels are cut by a third straight line, then 
the opposite (corresponding) angles are equal. 
3. Two straight lines, which are parallel to a third, are 
parallel to each other. 
4. Through every point within an angle less than a 
straight angle, one can always draw straight lines 
which cut both sides (not perhaps their prolonga- 
tions). 
5. All points of a straight line have from a parallel the 
same distance. 
His comment is, “Finally we remark, that it seems as if 
each of these five theorems could serve precisely as the equiv- 
alent of the Parallel Axiom .” 
The third class of investigators consisted of those geom- 
eters who foundered upon the rock of the attempt to deduce 
Euclid’s parallel-postulate from reasonings about the nature 
of the straight line and the plane angle, helped out by 
Enclid’s other assumptions and his first twenty-eight theo- 
rems. Euclid took pains to prove things which were more 
axiomatic by far — for instance, that the sum of two sides of 
a triangle is greater than the third side — a thing which any 
ass knows. To give one illustration of the many so-called 
proofs, take the most plausible one, exposed by Charles L. 
Dodgson, in his Cunosa Mathematical Part I. pp. 70-71, 3rd 
edition, 1890: 
“Yet another process has been invented —quite fascinating 
in its brevity and its elegance— which, though involving the 
