transversal, two interior angles, on the same side of it, which are together less than two right angles, the defect being a finite angle: these Lines are intersectional on that side of the transversal. 28[Pg xxix] Appendix I. Containing an alternative Axiom, which may be substituted for Axiom 1 at p. 14. Definition. 32 Propositions:— (A) Theorem. If, in any Sector of a Circle, its Chord be not-less than its Radius: then, in a Sector whose vertical angle is twice as great, its outer Segment is greater than its central Triangle. 33 (B) Theorem. If, in any Sector of a Circle, each of the equal Sides of its inscribed isosceles Triangle be not-less than its Radius; and if its outer Segment be greater than a certain multiple of its central Triangle: then, in a Sector, whose vertical angle is twice as great, its outer Segment is greater than twice that multiple of its central Triangle. 34 Axiom. In every Circle, the inscribed equilateral Tetragon, multiplied by ('' being a certain selected finite number), is greater than any one of the Segments which lie outside it. 35 Propositions (continued):— (C) Theorem. An isosceles Triangle, whose vertical angle is of a right angle, has its base less than either of its sides. 36 Corollary. Hence, by Book I, Prop. III, it is possible to describe, on a given base, an isosceles Triangle having each base-angle equal to of a right angle. 38[Pg xxx] (D) Theorem. The angles of any Triangle are together not-less than of a right angle. 38 (E) Theorem. There is a Triangle whose angles are together not-less than two right angles. 39 Appendix II. Is Euclid's Axiom True? § 1. Infinite and Finite Magnitudes. 40 § 2. Infinitesimal Lines and Strips. 43 § 3. Infinitesimal Angles and Sectors. 48 § 4. Pairs of Lines. 49 Appendix III. How should Parallels be defined? 59 Appendix IV. How the Question stands to-day. § 1. Certain universally-true Theorems, provable from genuine Axioms (i.e. from Axioms whose self-evident character is indisputable). 62 § 2. Certain universally-true Theorems, not provable from genuine Axioms, but provable if any one of them be accepted as an Axiom. 63[Pg xxxi] § 3. Certain universally-true Theorems, not provable from genuine Axioms, but provable if any one of the 'Nine Quasi-Axioms' be accepted. 65 § 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. " § 5. Other methods of treatment. 67 Playfair's theory of 'direction'. " Fallacious proof for Euc. I. 32. 70 Bertrand's Theorem. 71 W. Hanna's fallacy. 72 J. Walmsley's fallacy. " § 6. The Outlook. 73 Advice to future explorers. " Unproved Theorems which would suffice for our purpose. 75 New Definition needed for Right Line. " [Pg 1] A NEW THEORY OF PARALLELS. Book I. Certain universally-true Propositions, provable from genuine Axioms. Definitions. 1.