Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

NON-EUCLIDEAN GEOMETRIES 353
arise and what they are. And I will begin with that of
the Russian.
The point of departure is Euclid's famous postulate,
-his postulate F,-which I stated in a previous Lecture
(VII); this postulate,-or assumption, for that is what
it is,-is pretty long; it is known, however, to be equivalent to the briefer assumption:
Through any point there is one and but one line
parallel to a given line.
For many centuries geometricians, great and small,
tried to deduce this assumption,-which may be called
the one-parallel assumption,-as a theorem from Euclid's
other assumptions (conscious and unconscious). They
failed, and today we know why-the assumption is not
implied in the other ones and so is not deducible from
them, not even by demons or archangels or gods. Now,
what the adventurous spirit of Lobachevski led him to
do is simply this: retaining all of the Euclidean assumptions save the one respecting parallels, he replaced the
latter by an assumption contradicting it, and then proceeded to deduce the consequences of the set of assumptions he had thus adopted as postulates. What is the
assumption with which he replaced Euclid's postulate of
the single parallel? It may be stated as follows by help
of Fig. 30: If line PF rotate (in the plane of the figure)
about P, say counterclockwise, it will come to a position,
call it PK, where it first fails to cut line L, and then, without cutting L, it will rotate through a finite angle KPH
into a position PH such that, if it rotate further, it will
cut L to the left of F, the angles FPK and FPK' being
equal.
According to this assumption the lines of the Pencil