SPECIAL OR ELEMENTARY GEOMETRY. INTRODUCTION. SECTION I LOGICO-MATHEMATICAL TERMS.* 1. A Proposition is a statement of something to be considered or done. ILL. Thus, the common statement, "Life is short," is a proposition; so, also, we make, or state a proposition, when we say, "Let us seek earnestly after truth."- -"The product of the divisor and quotient, plus the remainder, equals the dividend," and the requirement, " To reduce a fraction to its lowest terms," are examples of Arithmetical propositions. 2. Propositions are distinguished as Axioms, Theorems, Lemmas, Corollaries, Postulates, and Problems. 3. An Axiom is a proposition which states a principle that is so simple, elementary, and evident as to require no proof. ILL. Thus, "A part of a thing is less than the whole of it," "Equimultiples of equals are equal," are examples of axioms. If any one does not admit the truth of axioms, when he understands the terms used, we say that his mind is not sound, and that we cannot reason with him. 4. A Theorem is a proposition which states a real or supposed fact, whose truth or falsity we are to determine by reasoning. ILL." If the same quantity be added to both numerator and denominator of a proper fraction, the value of the fraction will be increased," is a theorem. It is a statement the truth or falsity of which we are to determine by a course of reasoning. * That is, terms used in the science in consequence of its logical character. The science of the Pure Mathematics may be considered as a department of practical logic. |