The sum of the angles of a Triangle is called its 'amount.' 2. Any angular magnitude is called a 'possible amount,' if there be a Triangle whose 'amount' is equal to it: but, if there be no such Triangle, it is called an 'impossible amount.' 3. If any such angular magnitude vary continuously: whenever it changes from a 'possible amount' to an 'impossible amount,' it is said to pass from a 'possible region,' to an 'impossible region': and vice versâ. [Pg 2] 4. If there be two fixed angular magnitudes such that the varying magnitude, while it continues between them, is always a 'possible amount,' but becomes an 'impossible amount' when it passes beyond them: they are called the 'superior limit,' and the 'inferior limit,' of the 'possible region' which lies between them. Axioms. [Pg 3] 1. If a Magnitude change from one value to another; and if, in doing so, it vary continuously; and if a certain value, intermediate to its first and last values, be selected: the Magnitude must, at some moment during the process of change, have that selected value. 2. If two or more Magnitudes be such that, whenever any one of them varies, it varies continuously: then, whenever their sum varies, it varies continuously. 3. If , , be two Lines intersecting at ; and if be turned about , remaining stationary: each of the angles at varies continuously. 4. If , , be two intersecting Lines; and if be turned about , remaining stationary: then, so long as the Lines continue to intersect, each of the 4 angles at the Point of intersection varies continuously. Propositions. [Pg 4] PROP. I. Theorem. If a Pair of Lines make, with a certain transversal, either (1) a pair of alternate angles equal, or (2) an exterior angle equal to its interior opposite angle on the same side of the transversal, or (3) a pair of interior angles on the same side of the transversal supplementary: they will make, with that transversal, (4), each pair of alternate angles equal, and (5) each of the four exterior angles equal to its interior opposite angle on the same side of the transversal, and (6) each pair of interior angles on the same side of the transversal supplementary. This Proposition is easily deduced from Euc. I. 13, 15. Definitions (continued). 5. Such a Pair of Lines may be said to be 'equally inclined' to that transversal. PROP. II. Theorem. [Pg 5] If two isosceles Triangles have equal bases but unequal sides: that Triangle, which has the greater sides, has the greater area. Let the Triangles be set on the same base, and call them , . And let , , be respectively greater than , . Now cannot fall within the Triangle , or upon either of its sides; for then , would be less than , ; [Euc. I. 21. neither can it fall on ; for then , would be equal to , ; neither can intersect , nor intersect ; for, in either case, if were joined, there would be two Triangles, on the same base , having their coterminous sides equal; which is impossible; [Euc. I. 7.