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

POSTULATES 47
its sides; its interior is the class of points such that, if
A and B be any two of them, segment AB contains no
point of h or k; its exterior is composed of all other points
of the plane except O and the points of h and k.
(I7) Given an angle (h, k); a line a' in a plane a; a
point O of a'; and in a a half-ray h' emanating from O;
then in a and emanating from O there is one and but one
half-ray k' such that the angle (h', k') is congruent to (h, k)
and that the interior of (h', k') is on a given side of a'.
(I8) If the angle (h, k) is congruent to (h', k') and to
(h", k"), then (h', k') is congruent to (h", k").
(I9) If, in the triangles ABC and A'B'C', AB, AC and
angle BAC are respectively congruent to A'B', A'C' and
angle B'A'C', then the angles ABC and ACB are respectively congruent to the angle A'B'C' and A'C'B'.
The Postulate of Continuity (or of Archimides)
(20) Let the point A1 be between any two given
points A and B of a straight line a. Let the points
A2, A3, A4,.... of a be such that A1 is between A and
A2, A2 is between A1 and A3, and so on, and that the
segments AAi1, A1A, AA3,... are mutually congruent.
Then in the point series there is a point An such that B
is between A and AI.
Postulate of Completeness
(21) To a system of points, lines and planes it is not
possible to add other elements such that the system thus
generalized shall form a new geometry in which al the
postulates of the foregoing five sets are valid.
Such is a list of the postulates devised by Hilbert to
serve as a foundation of Euclidean geometry. I regret