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

4

NON-EUCLIDEAN GEOMETRY

[COH. I.

right through the Elements constructions are effected by means
of straight lines and circles drawn in accordance with Postulates
I.-III. Such straight lines and circles determine by their intersection other points in addition to those given; and these
points are used to determine new lines, and so on. The existence of these points of intersection must be postulated or
proved, in the same way as the existence of the other straight
lines and circles in the construction has been postulated or
proved.
The Principle of Continuity, as it is called, is introduced to
fill this gap. It can be stated in different ways, but probably
the simplest is that which Dedekind originally adopted in discussing the idea of the irrational number. His treatment of
the irrational number depends upon the following geometrical
axiom:
If all the points of a straight line can be separated into two
classes, such that every point of the first class is to the left of every
point of the other class, then there exists one, and only one, point
which brings about this division of all the points into two classes,
this section of the line into two parts.*
This statement does not admit of proof. The assumption of
this property is nothing less than an axiom by which we assign
its continuity to the straight line.
The Postulate of Dedekind, stated for the linear segment, can
be readily applied to any angle, (the elements in this case
being the rays from the vertex), and to a circular arc. By
this means demonstrations can be obtained of the theorems
as to the intersection of a straight line and a circle, and of
a circle with another circle, assumed by Euclid in the propositions above mentioned.t The idea of continuity was
adopted by Euclid without remark. What was involved in the
assumption and the nature of the irrational number were
unknown to the mathematicians of his time.
This Postulate of Dedekind also carries with it the important
* Dedekind, Stetigkeit und irrationale Zahlen, p. 11 (2nd ed., Braunschweig, 1892); English translation by Beman (Chicago, 1901).
f This question is treated fully in the article by Vitali in Enriques'
volume, Questioni riguardanti la geometria elementare (Bologna, 1900);
German translation under the title, Fragen der Elementargeometrie,
vol. i. p. 129 (Leipzig, 1911). See also Heath's Euclid, vol. i. p. 234.