, fall outside the Triangle ; Triangle is greater than Triangle . Therefore, if two isosceles Triangles &c. Q.E.D. [Pg 6] PROP. III. Problem. Given a certain angle; and given that any isosceles Triangle, whose vertical angle is not-greater than the given angle, has its base not-greater than either of its sides: to describe, on a given base, an isosceles Triangle having each base-angle equal to the given angle. Let be given base. At , in Line , make angle equal to given angle, making ; and join . Then, by hypothesis, is not-greater than : i. e. it is either equal to it, or less than it. First let be equal to . (Fig. 1.) Then Triangle is equilateral: i. e. it is an isosceles Triangle, on given base , and having each base-angle equal to given angle. Q.E.F. Secondly, let be less than . (Fig. 2.) Then angle is less than angle . [Euc. I. 18. At make angle equal to angle . [Euc. I. 23. Then Triangle is isosceles, and is on given base , and has each base-angle equal to given angle. Q.E.F. [Pg 7] Corollary. The isosceles Triangle, so described, has its vertical angle not-less than either of its base-angles. For, in Fig. 1, ; . Q.E.D. Again, in Fig. 2, ; . But angle is greater than angle ; [Euc. I. 16. and angle is less than angle ; . Q.E.D. [Pg 8] PROP. IV. Theorem. Either all Triangles have the same 'amount'; or else, if , be two 'possible amounts,' that is, 'amounts' belonging to existing Triangles, then any 'amount,' intermediate to and , is also 'possible.' If all Triangles have the same amount, the Proposition is true. If not, let , be two Triangles whose amounts are different. Call their 'amounts' ', .' And let the two Triangles be placed so as to have a common vertex at , and their bases in the same straight Line. Now Triangle may be converted into Triangle by making the point, where intersects , move from to , remaining stationary. And, during this process, the angle at will vary continuously, [Ax. 4. and the angle, at the point where the revolving Line intersects , will vary continuously; [Ax. 5. the sum of these angles will, if it vary at all, vary continuously. [Ax. 3. Similarly, Triangle may be converted, first, into Triangle , by making the point, where intersects , move from to , remaining stationary, and then into Triangle , by making the point, where intersects , move from to , remaining stationary. And, during the whole process, the 'amount' of the changing Triangle will, if it vary at all, vary continuously. Hence, in changing from the value to the value , it must pass through all intermediate 'amounts': i. e. all intermediate 'amounts' are 'possible.' [Ax. 2. Therefore either all Triangles have &c. Q.E.D. Corollaries. [Pg 9] 1. Among angular magnitudes there is one, and only one, 'possible region.' 2.