Figure is greater than twice Figure , i.e. greater than 4 times Triangle . Again, is not-less than ; Triangle is not-less than Triangle ; [I. Prop. 2. Triangles , , , are together not-less than Figure ; i.e. they are together greater than 4 times Triangle ; , a fortiori, Segment is greater than 4 times Triangle . But this is absurd, since it has been already proved less than 4 times this Triangle. Hence our Hypothesis, that is not-less than , is false; i.e. is less than . Therefore an isosceles Triangle &c. Q.E.D. Corollary. [Pg 17] Hence, by Book I, Prop. III, it is possible to describe, on a given base, an isosceles Triangle having each base-angle equal to one-eighth of a right angle. PROP. II. Theorem. [Pg 18] The angles of any Triangle are together not-less than one-eighth of a right angle. Let one-eighth of a right angle be represented by '.' Let be a Triangle; it shall be proved that its 'amount' is not-less than . If we deny this, we must assert that its 'amount' is less than . Let this be our Hypothesis. Hence each of its angles is less than . On describe an isosceles Triangle having each base-angle equal to ; [II. Prop. 1. Cor. hence , , must lie within this Triangle; i.e. Triangle must lie within it; angle is less than angle ; [Euc. I. 21. i.e. less than ; but angle is not-less than angle ; [I. Prop. 3. Cor. i.e. not-less than ; which is absurd. Hence our Hypothesis, that 'amount' of Triangle is less than , is false; i.e. it is not less than . Therefore the angles of any Triangle &c. Q.E.D. Corollary. The 'possible region' does not extend below one-eighth of a right angle. PROP. III. Theorem. [Pg 19] There is a Triangle whose angles are together not-less than two right angles. If we deny this, we must assert that any 'amount' is less than . Let this be our First Hypothesis. It shall be proved that, if we assert this, we must also assert that any 'amount' is not-less-than , where '' represents one-eighth of a right angle. If we deny this, we must assert that there is an 'amount' less than . Let this be our Second Hypothesis. Now we know that the 'possible region' does not extend below ; [II. Prop. 2. Cor. i.e. it has an 'inferior limit.' Let a 'possible amount' be selected, more than half-way from to this 'inferior limit,' and call it '.' Then it is evident that will lie below the 'inferior limit,' and will therefore be an 'impossible amount.' Let a Triangle be taken, whose 'amount' is ; any one of its angles, which is not-greater than either of the others, is not-greater than ; i.e. is less than . Call this angle '.' Now one, at least, of the remaining angles must be acute. [Euc. I. 17. Call this ''; and call the third angle . [Pg 20] Let 2 such Triangles, and , be taken; and let them be so placed that their -vertices coincide and their -sides lie in one straight line; and join . On describe an isosceles Triangle , having each base-angle equal to .