OF CALIFORNIA 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. 5. A Demonstration is the course of reasoning by means of which the truth or falsity of a theorem is made to appear. The term is also applied to a logical statement of the reasons for the processes of a rule. A solution tells how a thing is done: a demonstration tells why it is so done. A demonstration is often called proof. 6. A Lemma is a theorem demonstrated for the purpose of using it in the demonstration of another theorem. ILL. Thus, in order to demonstrate the rule for finding the greatest common divisor of two or more numbers, it may be best first to prove that A divisor of two numbers is a divisor of their sum, and also of their difference." This theorem, when proved for such a purpose, is called a Lemma. The term Lemma is not much used, and is not very important, since most theorems, once proved, become in turn auxiliary to the proof of others, and hence might be called lemmas. 7. A Corollary is a subordinate theorem which is suggested, or the truth of which is made evident, in the course of the demonstration of a more general theorem, or which is a direct inference from a proposition. ILL. Thus, by the discussion of the ordinary process of performing subtraction in Arithmetic, the following Corollary might be suggested: "Subtraction may also be performed by addition, as we can readily observe what number must be added to the subtrahend to produce the minuend." 8. A Postulate is a proposition which states that something can be done, and which is so evidently true as to require no process of reasoning to show that it is possible to be done. We may or may not know how to perform the operation. ILL. Quantities of the same kind can be added together. 9. A Problem is a proposition to do some specified thing, and is stated with reference to developing the method of doing it. ILL-A problem is often stated as an incomplete sentence, as, "To reduce fractions to a common denominator."-This incomplete statement means that "We propose to show how to reduce fractions to a common denominator." Again, the problem "To construct a square," means that "We propose to draw a figure which is called a square, and to tell how it is done." 10. A Rule is a formal statement of the method of solving a general problem, and is designed for practical application in solving special examples of the same class. Of course a rule requires a demonstration. |