This 'possible region' either consists of one single angular magnitude, such that it, and it alone, is a 'possible amount'; or it consists of a continuous series of angular magnitudes, lying between 2 'limits,' which 2 limits are such that any magnitude, lying between them, is a 'possible amount,' and any magnitude, lying outside them, is an 'impossible amount.' PROP. V. Theorem. [Pg 10] The angles of any Triangle are together less than three right angles. Let a right angle be represented by '.' Now any 2 of the angles of a Triangle are together less than ; [Euc. I. 17. , adding together the 3 pairs which may be taken, the 3 angles of a Triangle, taken twice over, are together less than ; , taken once only, they are together less than . Therefore the angles of any Triangle &c. Q.E.D. PROP. VI. Theorem. There is a Triangle whose angles are together not-greater than two right angles. If we deny this, we must assert that any 'amount' is greater than . Let this be our First Hypothesis. Now any 'amount' is less than . [Prop. 5. Hence the 'possible region' lies below ; i. e. it has a 'superior limit.' Now let a 'possible amount' be selected, more than half-way from to the 'superior limit' of this region; and call it '.' Then it is evident that will lie above this limit, and will therefore be an 'impossible amount.' [Pg 11] Now any Triangle, whose 'amount' is , must be either obtuse-angled, or right-angled, or acute-angled. Hence we must assert that there is either an obtuse-angled Triangle, or a right-angled Triangle, or an acute-angled Triangle, whose 'amount' is . Let these be our Second, our Third, and our Fourth Hypotheses. Call the Triangle '.' First, let it be obtuse-angled; and let be the obtuse angle. At Point , in Line , make angle equal to angle , making equal to ; and join and . Then Triangle is equal, in all respects, to Triangle ; [Euc. I. 4. its 'amount' = ; also 'amount' of Triangle is, by our First Hypothesis, greater than ; 'amounts' of the 3 Triangles are together greater than . But these make up 'amount' of Triangle , plus angles about , which = ; [Euc. I. 13. Cor. 'amount' of Triangle , plus , is greater than ; this 'amount,' alone, is greater than ; which is absurd, since the latter lies above the 'superior limit,' and is therefore an 'impossible amount.' Hence our Second Hypothesis is false; i.e. no obtuse-angled Triangle can have the amount . [Pg 12] Secondly, let it be right-angled; and let be the right angle. At Point , in Line , make angle equal to angle , i. e. equal to ; and make ; and join . Then , , are in one straight Line. [Euc. I. 14. Also Triangle is equal, in all respects, to Triangle ; [Euc. I. 4. its 'amount' = ; 'amounts' of the 2 Triangles together = . But these make up 'amount' of Triangle , plus angles at , which = ; 'amount' of Triangle , plus ; this 'amount,' alone, = ; which is absurd.