that is, 'amounts' belonging to existing Triangles: then any 'amount,' intermediate to and , is also 'possible'. 8 Corollary 1. Among angular magnitudes there is one, and only one, 'possible region'. 9 Corollary 2. 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'. " V. Theorem. The angles of any Triangle are together less than three right angles. 10 VI. Theorem. There is a Triangle whose angles are together not-greater than two right angles. " Corollary. The 'possible region' does not lie wholly above two right angles. 13[Pg xxvii] Book II. Certain universally true Propositions, not provable from genuine Axioms, but provable if the following Axiom be accepted. Axiom 1. In any Circle, the inscribed equilateral Tetragon is greater than any one of the Segments which lie outside it. 14 Propositions:— " I. Theorem. An isosceles Triangle, whose vertical angle is one-eighth of a right angle, has its base less than either of its sides. 15 Corollary. 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. 17 II. Theorem. The angles of any Triangle are together not-less than one-eighth of a right angle. 18 Corollary. The 'possible region' does not extend below one-eighth of a right angle. " III. Theorem. There is a Triangle whose angles are together not-less than two right angles. 19 Corollary. The 'possible region' does not lie wholly below two right angles. 21 IV. Theorem. There is a Triangle whose angles are together equal to two right angles. 22 V. Theorem. There is a quadrilateral Figure which is 'rectangular,' that is, which has all its angles right angles. "[Pg xxviii] Definition. 23 Propositions (continued):— VI. Theorem. The opposite sides of a Rectangle are equal. 24 VII. Theorem. There is a Pair of Lines, each of which is 'equidistant' from the other, that is, is such that all Points on it are equally distant from the other Line. " Corollary 1. If a Pair of Lines have a common perpendicular: each of them is equidistant from the other. 25 Corollary 2. It is possible to form a Rectangle of any given width and height. 26 VIII. Theorem. The angles of any Triangle are together equal to two right angles. " IX. Theorem. A Pair of Lines, which are equally inclined to a certain transversal, are so to any transversal. 27 Axiom 2. If two homogeneous magnitudes be both of them finite: the lesser may be so multiplied, by a finite number, as to exceed the greater. " Propositions (continued):— X. Theorem. If a Pair of Lines make, with a certain