xii. If two right lines (AB, CD) meet a third line (AC), so as to make the sum of the two interior angles (BAC, ACD) on the same side less than two right angles, these lines being produced shall meet at some finite distance. This axiom is the converse of Prop. xvii., Book I. Explanation of Terms. Axioms.—“Elements of human reason,” according to Dugald Stewart, are certain general propositions, the truths of which are self-evident, and which are so fundamental, that they cannot be inferred from any propositions which are more elementary; in other words, they are incapable of demonstration. “That two sides of a triangle are greater than the third” is, perhaps, self-evident; but it is not an axiom, inasmuch as it can be inferred by demonstration from other propositions; but we can give no proof of the proposition that “things which are equal to the same are equal to one another,” and, being self-evident, it is an axiom. Propositions which are not axioms are properties of figures obtained by processes of reasoning. They are divided into theorems and problems. A Theorem is the formal statement of a property that may be demonstrated from known propositions. These propositions may themselves be theorems or axioms. A theorem consists of two parts, the hypothesis, or that which is assumed, and the conclusion, or that which is asserted to follow therefrom. Thus, in the typical theorem, the hypothesis is that X is Y , and the conclusion is that Z is W. Converse Theorems.—Two theorems are said to be converse, each of the other, when the hypothesis of either is the conclusion of the other. Thus the converse of the theorem (i.) is— From the two theorems (i.) and (ii.) we may infer two others, called their contrapositives. Thus the contrapositive of (i.) is, If Z is not W, then X is not Y ; (iii.) of (ii.) is, If X is not Y , then Z is not W. (iv.) The theorem (iv.) is called the obverse of (i.), and (iii.) the obverse of (ii.). A Problem is a proposition in which something is proposed to be done, such as a line to be drawn, or a figure to be constructed, under some given conditions. The Solution of a problem is the method of construction which accomplishes the required end. The Demonstration is the proof, in the case of a theorem, that the conclusion follows from the hypothesis; and in the case of a problem, that the construction accomplishes the object proposed. The Enunciation of a problem consists of two parts, namely, the data, or things supposed to be given, and the quaesita, or things required to be done. Postulates are the elements of geometrical construction, and occupy the same relation with respect to problems as axioms do to theorems. A Corollary is an inference or deduction from a proposition.