INTRODUCTION. EUCLID'S DOCTRINE OF PROPORTION. A. NOTATION. In Sections B and C of this Introduction capital letters denote magnitudes, and when the pairs of magnitudes compared are both of the same kind they are denoted by letters taken from the early part of the alphabet, as A, B compared with C, D; but when they are, or may be, of different kinds, by letters taken from different parts of the alphabet, as A, B compared with P, Q or X, Y. Small italic letters m, n, p, q denote integers. By m. A or mA is meant the mth multiple of A, and it may be read m times A; by mn is meant the arithmetical product of the integers m and n, and it is assumed that mn=nm. The combination m. n A denotes the mth multiple of the 7th multiple of A, and it is assumed that m. n An.mA = mn. A. B.-DEFINITIONS AND AXIOMS.* 1. "A greater magnitude is said to be a multiple of a less, when the greater contains the less an exact number of times." 2. "A less magnitude is said to be a submultiple of a greater, when the less is contained an exact number of times in the greater." *The quoted paragraphs of Section B are transcribed in part from the Syllabus of Plane Geometry, published by the Association for the Improvement of Geometrical Teaching, in part from Hall and Stevens' Text Book of Euclid's Elements, Book V. The following property of multiples is assumed as axiomatic : or <m B according as A> or < B. (Euc. Axioms 1 and 3.) The converse necessarily follows: * = (ii). A> == or < B according as m A >= or < m B. (Euc. Axioms 2 and 4.) 3. "The ratio of one magnitude to another of the same kind is the relation which the first bears to the second in respect of quantuplicity:" "The ratio of A to B is denoted thus, A: B; and A is called the anteceden!, B the consequent of the ratio." "The quantuplicity of A with respect to B may be estimated by examining how the multiples of A are distributed among the multiples of B, when both are arranged in ascending order of magnitude and the series of multiples continued without limit." This distribution may be represented graphically thus: 4. If, in this comparison of the multiples of two magnitudes, any multiple, as n A, of the one coincides with (is equal to) any multiple, as m B, of the other, the two magnitudes bear the same ratio to one another as the two numbers m, n, and are said to be commensurable, but * "RULE OF CONVERSION. If the hypotheses of a group of demonstrated theorems be exhaustive (that is, form a set of alternatives of which one must be true), and if the conclusions be mutually exclusive (that is, be such that no two of them can be true at the same time), then the converse of every theorem of the group will necessarily be true." (Syllabus of Plane Geometry, p. 5.) |