Hence our Third Hypothesis is false; i.e. no right-angled Triangle can have the amount . Thirdly, let it be acute-angled. Bisect at ; join , and produce it to , making ; and join . Now angles , , are equal, being vertical; [Euc. I. 15. Triangles , , are equal in all respects; [Euc. I. 4. angle = angle ; and angle = angle ; 'amount' of Triangle = that of Triangle . Again, angle = angle ; angles , together = angle ; they are together less than ; angle is, by our First Hypothesis, greater than ; i. e. Triangle is obtuse-angled; its 'amount' cannot be ; 'amount' of Triangle cannot be . Hence our Fourth Hypothesis is false; i.e. no acute-angled Triangle can have the 'amount' . Hence, no Triangle can have this amount. Hence our First Hypothesis, that any 'amount' is greater than , is false. Therefore there is a Triangle &c. Q.E.D. Corollary. [Pg 13] The 'possible region' does not lie wholly above two right angles. Book II. Certain universally-true Propositions, not provable from genuine Axioms, but provable if the following Axiom be accepted. [N.B. This Axiom cannot claim to be more than a 'Quasi-Axiom,' i. e. one whose self-evident character is disputable.] Axioms. [Pg 14] 1. In any Circle, the inscribed equilateral Tetragon is greater than any one of the Segments which lie outside it. Note.—If, in any Circle, 2 Diameters be drawn at right angles to each other, and their extremities joined, the joining Lines will, by Euc. I. 4 be equal to each other. Hence the Figure, thus formed, will be an inscribed equilateral Tetragon. (It will also be equiangular; but that is of no importance for our present purpose.) PROP. I. Theorem. [Pg 15] An isosceles Triangle, whose vertical angle is one-eighth of a right angle, has its base less than either of its sides. Let be an isosceles Triangle, whose vertical angle at is one-eighth of a right angle. It shall be proved that is less than . If we deny this, we must assert that is not-less than . Let this be our Hypothesis. Construct 7 more Triangles , &c., equal to . With centre , and distance , describe quadrant passing through , , &c. (See Note.) And join , , , , , , . Note.—The Reader is requested to imagine chords drawn to the arcs , , &c. [Pg 16] Then Triangle is one-fourth of an equilateral Tetragon inscribed in the Circle. Hence, 4 times this Triangle is greater than Segment . [II. Ax. 1. Because angle is greater than angle , and that angle is less than angle ; angle is greater than angle ; is greater than . [Euc. I. 19. Similarly, is greater than . Hence, on our Hypothesis, and are both of them greater than . Also, is greater than ; Triangle is greater than Triangle ; [I. Prop. 2. to each of these add Triangle ; Figure is greater than twice Triangle . Again, is greater than ; Triangle is greater than Triangle ; [I. Prop. 2. Triangles , are together greater than Figure ; to each of these add Figure ;