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

46 MATHEMATICAL PHILOSOPHY
DEFINITIONS.-A pair of points, A and B, on a lne,
is a segment AB or BA; A and B are the segment's ends;
the points between A and B are the segment's points.
(I1) Let A, B, C be three non-collinear points and
let a be a line of their plane but not containing any of
them. If a contains a point of segment AB, it contains
a point of segment BC or of segment AC.
Postulate of Parallels
(I3) If a straight line a and a point A, not in a, be in
a plane a, there is in a one and only one straight line
containing A but no point of a.
Postulates of Congruence
(I4) If A and B are two points on a straight line a,
and if a point A' be on a straight line a', then on either
side of A' there is one and but one point B' such that
the segment AB is congruent to the segment A'B'.
Every segment is congruent to itself.
(I5) If a segment AB is congruent to a segment
A'B' and to a segment A"B", then A'B' is congruent
to A"B".
(I6) If segments AB and BC of a straight line a have
no common point but B, and if segments A'B' and B'C'
of a straight line a' have no common point but B', then,
if AB and BC are respectively congruent to A'B' and
B'C', AC is congruent to A"C".
DEFINITIONS.-If O be a point of a straight line a,
the points of a on a same side of O constitute a half-ray
emanatingfrom O; a pair of half-rays, h and k, emanating
from a point O and not being parts of a same straight
line is an angle (h, k); O is the angle's vertex, and h and k