Because is not-less than ; Triangle is not-less than Triangle ; [I. Prop. 2. , a fortiori, Segment is greater than Triangle . Hence if, in any Sector &c. Q.E.D. PROP. B. Theorem. [Pg 34] 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. Let be a Sector such that each of the equal sides of its inscribed isosceles Triangle is not-less than its Radius , and such that its outer Segment is greater than times its central Triangle . Make angle equal to angle ; bisect angles , by Lines , ; and join , , , . Then vertical angle of Sector is twice as great as that of Sector . It shall be proved that its outer Segment is greater than times its central Triangle . Because angle is greater than angle , and that angle is less than angle ; angle is greater than angle ; is greater than ; [Euc. I. 25. but is not-less than ; is greater than ; Triangle is greater than Triangle ; [I. Prop. 2. to each of these add Triangle ; Figure is greater than twice Triangle . Again, Segment is given to be greater than times Triangle ; Segments , are together greater than times Figure ; i. e. are together greater than times Triangle ; , a fortiori, Segment is greater than times Triangle . Hence if, in any Sector &c. Q.E.D. Axiom. [Pg 35] [An alternative Axiom, to be substituted for Axiom 1, at p. 14, if the Reader feel any difficulty in granting that Axiom. In this case, certain portions of the foregoing Propositions will also need to be replaced by new matter, which is hereto appended.] 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. Note.—The reader can assign to '' any finite value which he finds large enough to induce him to accept this Axiom. For example, if he be willing to grant that 1024 times the Tetragon is greater than the Segment, he can assign to it the value '10.' PROP. C. Theorem. [Pg 36] [To be substituted for Prop. I, at p. 15.] An isosceles Triangle, whose vertical angle is of a right angle, has its base less than either of its sides. Let of a right angle be represented by '.' Let be an isosceles Triangle, whose vertical angle at is . It shall be proved that is less than . Draw at right angles to , and, with centre , and distance , describe Quadrant. (See Note.) And join . Then Triangle is one-fourth of an equilateral Tetragon inscribed in the Circle. Hence times this Triangle is greater than Segment . [Alternative Axiom. Now, if we deny that is less than , we must assert that is not-less than . Let this be our Hypothesis.