I primi sei libri degli Elementi di Euclide

Le prime righe

If X is Y, then Z is W, 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— If Z is W, then X is Y. 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. A Lemma is an auxiliary proposition required in the demonstration of a principal proposition. A Secant or Transversal is a line which cuts a system of lines, a circle, or any other geometrical figure. Congruent figures are those that can be made to coincide by superposition. They agree in shape and size, but differ in position. Hence it follows, by Axiom viii., that corresponding parts or portions of