UNIV. OF VINYOJITVO 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 m n = 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 : = (i). mA> 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 antecedent, 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: O A 2A 3A 4A 5A 6A 7A 8A Multiples of A: Multiples of B: B 2B 3B 4B 5B 6B Fig. 1. 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 ot 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.) incommensurable if no such coincidence takes place, however far the process of comparison is carried. 5. The ratio of two magnitudes is said to be equal to a second ratio of two other magnitudes (whether of the same or of a different kind from the former), when the multiples of the antecedent of the first ratio are distributed among those of its consequent in the same order as the multiples. of the antecedent of the second ratio among those of its consequent. As tests of the equality of two ratios either of the following criteria may be employed, m and n being integers : (i). The ratio of A to B is equal to that of P to Q, when m A = or < n B according as mP> or << nQ. = (ii). If m be any integer whatever and n another integer so determined that either mA is between n B and (n+1) B or is equal to n B, then the ratio of A to B is equal to that of P to Q, when mP is between nQ and (n+1)Q or equal to nQ according as mA is between nB and (n+1)B or equal to nB. (iii). It should be remarked that the rule of identity* is applicable to this definition, and that therefore, if the ratio. of A to B be equal to that of P to Q, then also according as mA = or < n B m P = or < n Q. 6. "When the ratio of A to B is equal to that of P to Q, the four magnitudes are said to be proportionals, or to form a proportion. The proportion is denoted thus : A: BP: Q, * RULE OF IDENTITY. If there be only one fact or state of things X, and only one fact or state of things Y, then from X is Y the converse Y is X of necessity follows. which is read: A is to B as P is to Q. A and Q are called the extremes, B and P the means.' The proportion A: B:: P: Q may be represented graphically thus: In a diagram of this kind it is obvious that in general, of the two figures thus compared, representing two equal ratios, one will be an enlarged copy of the other, but in particular, if the antecedents and consequents be respectively equal to one another, the two figures will be congruent. 7. "The ratio of one magnitude to another is greater than that of a third magnitude to a fourth, when it is possible to find equimultiples of the antecedents and equimultiples of the consequents such that while the multiple of the antecedent of the first ratio is greater than, or equal to, that of its consequent, the multiple of the antecedent of the second is not greater, or is less, than that of its consequent." That is, A: B> P:Q, if integers m, n'can be found such that if or if mAn B, then m PnQ, mA 8. "If A is equal to B, the ratio of A to B is called a ratio of equality. |