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

EUCLID'S ASSUMPTIONS

3

refrained from employing it in other places, where it would
have shortened the demonstration.
Again, Postulate I., which asserts the possibility of drawing
a straight line from any one point to any other, must be held
to declare that the straight line so drawn is unique, and that
two straight lines cannot enclose a space. And Postulate II.,
which asserts the possibility of producing a finite straight line
continuously in a straight line, must also be held to assert that
the produced part in either direction is unique; in other words,
that two straight lines cannot have a common segment.
But the following more fundamental and distinct assumptions are made by Euclid, without including them among the
axioms or postulates:
(i) That a straight line is infinite.
This property of the straight line is required in the proof of
I. 16. The theorem that the exterior angle is greater than
either of the interior and opposite angles does not hold in the
Non-Euclidean Geometry in which the straight line is regarded
as endless, returning upon itself, but not infinite.
(ii) Let A, B, C be three points, not lying in a straight line, and
let a be a straight line lying in the plane ABC, and not passing
through any of the points A, B, or C. Then, if a passes through
a point of the segment AB, it must also pass through a point of the
segment BC, or of the segment AC (Pasch's Axiom).
From this axiom it can be deduced that a ray passing through
an angular point, say A, of the triangle ABC, and lying in the
region bounded by AB and AC, must cut the segment BC.
(iii) Further, in the very first proposition of the First Book
of the Elements the vertex of the required equilateral triangle
is determined by the intersection of two circles. It is assumed
that these circles intersect. A similar assumption is made in
I. 22 in the construction of a triangle when the sides are given.
The first proposition is used in the fundamental constructions
of I. 2 and I. 9-11.
Again, in I. 12, in order to be sure that the circle with a
given centre will intersect the given straight line, Euclid makes
the circle pass through a point on the side opposite to that in
which the centre lies. And in some of the propositions of
Book III. assumptions are made with regard to the intersection of the circles employed in the demonstration. Indeed