(5) [contranominal of (4)] A Pair of non-intersectional Lines are equidistantial. (6) On either of two intersectional Lines a Point may be found, whose distance from the other Line shall exceed any assigned length. (7) On either of two intersectional Lines a Point may be found, such that the distance, from its projection on the other Line to the point of intersection of the two Lines, shall exceed any assigned length. (8) A Pair of intersectional Lines cannot be, both of them, non-intersectional with a third Line. (9) [contranominal of (8)] A Pair of Lines, which are, both of them, non-intersectional with a third Line, are non-intersectional. Six of these have been used as Axioms, viz. (2), which is Euclid's celebrated Axiom; (4), by T. Simpson; (6), by Proclus; (7), by Franceschini; (8), by Ludlam, Playfair, &c.; (1), by Legendre, in the 12th Volume of the 'Memoirs of the Institute,' being his "latest attempt" (I quote from De Morgan's article on Parallels in Knight's Cyclopædia) "at the solution of the problem." He only succeeds, however, in proving Prop. 6 at p. 10 of this book, and in proving that Prop. 8, at p. 26, follows logically from Prop. 4, at p. 22. In order to prove Euc. I. 32, he introduces the principle of Limits and Vanishing Quantities, which takes us at once into the region of Infinities and Infinitesimals. But a greater success than this has, I understand, rewarded some recent investigations made by Professor J. Cook Wilson, of Oriel College, Oxford, who has deduced Euclid's 12th Axiom from No. (1) of this Section. In all these systems, however, including Euclid's, the deductions, from the proposed Axiom, labour under the same defect as the Axiom itself, that is, they are only partially, and not universally, true. § 5. [Pg 67] Other methods of treatment. Three other methods of treating this subject call for notice. Playfair and (more recently) Wilson have tried to deduce the properties of Parallels (and thence Euc. I. 32) from the idea of sameness of direction, as predicated of two non-coincidental Lines (i. e. which do not lie in one and the same straight Line). The foundation-stones of this Theory—without which it has no raison d'être whatever—are the two Axioms, which must necessarily be somewhere assumed, whether expressly or tacitly, first, that it is possible for non-coincident Lines to have 'the same direction'; and secondly, that Lines, which have the same direction, make equal angles with all transversals.