My own belief is that No. (3) is, on the whole, the property best adapted for practical use as a Definition. APPENDIX IV. How the Question stands to-day. I will first enumerate certain Theorems connected with Pairs of Lines. I will then notice three other methods which have been suggested for superseding Euclid's Axiom. And, in conclusion, I will indicate what seem to me the most hopeful directions for future efforts at exploring this fascinating, but very obscure, region of mathematical research. § 1. [Pg 62] Certain universally-true Theorems, provable from genuine Axioms (i. e. from Axioms whose self-evident character is indisputable). (1) A Pair of Lines, which are equally inclined to a certain transversal, are non-intersectional (i. e. never intersect, however far produced). [Euc. I. 27, 28.] (2) [contranominal of (1)] A Pair of intersectional Lines (i. e. which either intersect or would do so if produced) are unequally inclined to any transversal. [Euc. I. 16, 17.] (3) A Pair of Lines, such that two Points on one of them are on the same side of, and equidistant from, the other, are non-intersectional. (4) [contranominal of (3)] A Pair of intersectional Lines are non-equidistantial (i. e. are such that any two Points on either of them, which are on the same side of the other, are non-equidistant from it, that which is further from the Point of intersection being also further from the other Line). (5) If there be a Triangle whose angles are together equal to two right angles: the angles of any Triangle are together equal to two right angles. (6) There is a Triangle whose angles are together not-greater than two right angles. § 2. [Pg 63] Certain universally-true Theorems, not provable from genuine Axioms, but provable if any one of them be accepted as an Axiom. [N.B. These will be hereafter referred to as the 'Nine Quasi-Axioms,' their self-evident character being disputable.] (1) Through a given Point, outside a given Line, a Line may be drawn, such that the Pair shall be equally inclined to any transversal. (2) A Pair of Lines, which are equally inclined to a certain transversal, are so to any transversal. [Deducible from Euc. I. 27, 28, 29.] (3) [contranominal of (2)] A pair of Lines, which are unequally inclined to a certain transversal, are so to any transversal. [Pg 64] (4) If a Point move so as to be at a constant distance from a given Line, its path shall be a straight Line. *(5) Through a given Point, outside a given Line, a Line may be drawn equidistantial from the given Line (i. e. such that any two Points on it shall be equidistant from the given Line). (6) A Pair of Lines, such that two Points on one of them are on the same side of, and equidistant from, the other, are equidistantial (i. e. are such that any two Points on either of them are equidistant from the other).