Page:Carroll - Euclid and His Modern Rivals.djvu/75

Table III.

Containing five Propositions, taken from Table II, which have been proposed as Axioms.

Euclid's Axiom.

A Pair of Lines, which have a separate point and make, with a certain transversal, two interior angles on one side of it together less than two right angles, are intersectional on that side.

[This is one case of II. 2, with an additional statement as to the side of the transversal on which the Lines will meet.]

T. Simpson's Axiom.

A Pair of Lines, which have a separate point and of which one has two points on the same side of, and not equidistant from, the other, are intersectional.

[This is II. 7.]

Clavius' Axiom.

Through a given Point, without a given Line, a Line may be drawn equidistantial from the given Line.

[This is part of II, 8.]

Playfair's Axiom.

A pair of intersectional Lines cannot both be separational from the same Line.

[This is II. 16 (a).]