The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

2, 3] THE PRINCIPLE OF CONTINUITY

5

Postulate of Archimedes, which will be frequently referred to in
the following pages:
If two segments are given, there is always some multiple of
the one which is greater than the other.*
~ 3. An interesting discovery, arising out of the recent study
of the Foundations of Geometry, is that a great part of Elementary Geometry can be built up without the Principle of
Continuity. In place of the construction of Euclid I. 2, the
proof of which depends upon this Principle, the following
Postulate t is made:
If A, B are two points on a straight line a, and if A' is a point
upon the same or another straight line a', then we can always find
on the straight line a', on a given ray from A', one and only one
point B', such that the segment AB is congruent to the segment
A'B'.
In other words, we assume that we can always set off a
given length on a given line, from a given point upon it,
towards a given side. By the term ray is meant the half-line
starting from a given point.
With this assumption, for Euclid's constructions for the
bisector of a given angle (I. 9), for the middle point of a given
straight line (I. 10), for the perpendicular to a given straight
line from a point upon it (I. 11), and outside it (I. 12), and,
finally, for an angle equal to a given angle (I. 23)-all of which,
in the Elements, depend upon the Principle of Continuitywe may substitute the following constructions, which are
independent, both of that Principle and of the Parallel
Postulate.
*For the proof of the Postulate of A rchimedes on the assumption of
Dedekind's Postulate, see Vitali's article named above, ~ 3. Another
treatment of this question will be found in Hilbert's Grundlagen der
Geometrie, 3rd ed. ~ 8. An English translation of the first edition was
made by Townsend (Chicago, 1902). The Postulate of Archimedes
stated above for linear segments is adopted also for angles, areas, and
volumes.
t Cf. Hilbert, loc. cit. 3rd ed. ~ 5, Axioms of Congruence.
t The constructions in Problems 1, 2, 3 and 5 are given by Halsted
in his book, Rational Geometry (2nd ed. 1907). Those for Problems 4
and 6 in the text are independent of the Parallel Postulate, and
replace those given by Halsted, in which the Euclidean Hypothesis is
assumed.