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

ESSENTIAL DISCRIMINATIONS 113
tries (actual or potential) initiated by Julius Plfcker's
great creation of Line Geometry, it employs some spatial
entity or entities other than the point, line and plane
(of Euclid's Elements) for primary element or elements, or
subject-matter. For all such distinctions are sufficiently
important. On the other hand, as I need hardly say, a
merely idiomatic or expressional difference,-such, for
example, as the Greek's saying, " a straight line can be
drawn from every point to every point " whereas we say
"from any point to any point,"-is no warrant for calling the latter non-Euclidean. So much for the generic
meanings of the adjectives-Euclidean and non-Euclidean
as applied in geometry. And now let us be very clear
as to what the specific and more common meaning of each
term is. One of Euclid's postulates-his postulate 5 -had the fortune to be an epoch-making statementperhaps the most famous single utterance in the history
of science. It is this:
If a straight lne falling on two straight lnes
make the interior angles on the same side less than two
right angles, the two straight lines, if produced indefinitely,
meet on that side on which are the angles less than two
right angles.
Apparently convinced that this proposition could not
be deduced as a theorem from his other postulates and
axioms, or common notions, Euclid assumed it. It was
for him an assumption, an hypothesis, a primitive proposition, a postulate-a basal proposition of the Elements.
It is commonly known as Euclid's parallel-postulate
because it is equivalent to the postulate that, if P be a
point and L a line, there is but one line through P parallel
to L. Unlike Euclid, his successors for two thousand
years, like his predecessors, were not convinced that the