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

80 MATHEMATICAL PHILOSOPHY
Let me suggest that, ag an exercise, you make a figure
illustrating that the theorem (corresponding to Hilbert's
theorem 5) dealt with in the preceding lecture, is verified
in the present interpretation.
Let us turn to the parallel postulate (I3). That it is
satisfied is clear in the light of Fig. I7. The given
pathocircle is a; A is a point not on a; through A there
is evidently one and but one pathocircle b having no
point in common with a; a and b are, of course, parallel to
each other. This postulate, as you know, is the Euclidean
FIG. 17.
postulate par excellence-the one that mainly distinguishes Euclidean geometry from the famous non-Euclidean geometries of Lobachevski and Riemann. And so
you see, in passing, that all interpretations of HzF or
HAF' yield doctrines of Euclidean type-in the sense that
in them the foregoing postulate of parallels is satisfied:
they all of them contain some theorems whose proofs
depend upon that postulate.
That all of the postulates of HAF are verified by the
meanings we have assigned to their variables may be
quickly made evident by help of the inversion transforma