(7) [contranominal of (6)] A Pair of Lines, such that two Points on one of them are non-equidistant from the other, 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 the other). *(8) A Line cannot recede from and then approach another; nor can it approach and then recede from another while on the same side of it. *(9) In any Circle, the inscribed equilateral Tetragon is greater than any one of the Segments which lie outside it. If any one of these 9 Theorems be granted as an Axiom, the rest can be proved from it. But only 3 of them, so far as I know, have been used as Axioms—No. 5 by Clavius, No. 8 by Dr. R. Simson, and No. 9 by myself. Clavius' Axiom requires us to assure ourselves that it will continue true when the Lines are produced without limit; and the strain on the imagination, caused by the effort of following them into Infinite Space, is one to be, if possible, avoided. Dr. R. Simson's Axiom gains a certain plausibility from the fact that, for intersecting Lines, it admits of actual proof (see § 1. (4)): where, however, the Lines are not known to intersect, it is an appeal to the eye, of very doubtful force. § 3. [Pg 65] Certain universally-true Theorems, not provable from genuine Axioms, but provable if any one of the 'Nine Quasi-Axioms,' given in § 2, be accepted. (1) There is a finite angular magnitude such that the angles of any Triangle are together not-less than it. (2) There is a Triangle whose angles are together not-less than two right angles. (3) There is a Triangle whose angles are together equal to two right angles. (4) The angles of any Triangle are together equal to two right angles. [Euc. I. 32.] § 4. Certain partially-true Theorems, not provable from any universally-true Axioms, whether genuine or 'quasi,' but provable if any one of themselves be accepted as an Axiom. [N.B. By 'partially-true' is meant 'true for finite magnitudes.' They become universally-true, if 'magnitude' be taken to mean 'finite magnitude'; 'equal' to mean 'not differing by a finite difference'; 'unequal' to mean 'differing by a finite difference'; 'multiplied' to mean 'multiplied by a finite number'; and 'intersectional' to mean 'intersectional at a finite angle.' These Theorems will be hereafter referred to as the 'Nine Pseudo-Axioms,' the name being chosen to indicate that they are not even universally true—far less self-evident.] [Pg 66] (1) The lesser of two homogeneous Magnitudes may be so multiplied as to exceed the greater. (2) A Pair of Lines, which are unequally inclined to a certain transversal, are intersectional. [Euclid's Axiom.] (3) [contranominal of (2)] A Pair of non-intersectional Lines are equally inclined to any transversal. [Euc. I. 29.] (4) A Pair of Lines, such that two Points on one of them are non-equidistant from the other, are intersectional.