Page:Carroll - Euclid and His Modern Rivals.djvu/71

. II. § 4.] with only three Propositions. There are, however, many other Propositions concerning them, which are fully admitted to be true, though no one of them has yet been proved from undisputed Axioms: and we shall find that they are so related to one another that, if any one be granted as an Axiom, all the rest may be proved; but, unless some one be so granted, none can be proved. Two thousand years of controversy have not yet settled the knotty question which of them, if any, can be taken as axiomatic.

If we are told that a certain Pair of Lines fulfil some one of the following conditions:—
 * (1) they are separational;
 * (2) they have a separate point and are equally inclined to a certain transversal;
 * (3) they have a separate point, and one of them has two points on the same side of and equidistant from the other;

we may prove (though not without the help of some disputed Axiom) that they fulfil both the following conditions:—
 * (1) they are equally inclined to any transversal;
 * (2) they are equidistantial from each other.

These Propositions, with the addition of my own I. 30, I. 32, and certain others, I will now arrange in a tabular form, placing Contranominals in the same section.