BOOK V. An abridgment of Euclid's Fifth Book-mainly based on De Morgan's Connexion of Number and Magnitude. By the word number, in what follows, we merely convey the notion of times or repetitions, considered independently of the things counted or repeated. By the word magnitude is meant a thing presented to us simply as that which is made up of parts, not differing from the whole in anything but in being less so that, if we consider separately a part and the whole, we have only two inferences The part is less than the whole, The whole is greater than the part. We shall use capital letters, as A, B, C, &c., to represent magnitudes-not as in algebra, the number of units which the magnitudes contain, but the magnitudes themselves—so that if it be, for example, weight of which we are speaking, A is not a number of pounds, but the weight itself. Concerning magnitudes we shall only assume that magnitudes of the same kind may be added together, or that the same magnitude may be repeated any number of times; and that a lesser magnitude may be taken from another of the same kind. We shall use small letters as m, n, p, &c., to denote integer numbers as just defined; and any one of them placed before a capital will denote repetition of the magnitude represented by that capital: thus as 3 A denotes A repeated thrice, so m A denotes A repeated m times. Def. When a greater magnitude contains a lesser magnitude a definite number of times exactly, the greater magnitude is called a multiple of the lesser; and the lesser is called a sub-multiple of the greater. If the greater magnitude is denoted by A, and the lesser by B, then the relation expressed in this definition will be denoted thus— A = mB, which is to be considered merely as an equivalent for these words— 'A is a multiple of B.' So we might have B = n C, C = p D, &c., with similar significations. Def. A sub-multiple is sometimes said to measure, or be a measure of its multiple. Hence the word commensurable, which means having a common measure; so also incommensurable means not having a common measure. We shall assume that the following properties of multiples are evident 1. A >,=, or < B according as m A >,, or < m B. 6. m. n A = mn. A = nm. A = n.m A. From the mutual relationship of two magnitudes of the same kind, there arises a tertium quid, which represents their relative, as distinguished from their absolute greatness, and which is called the ratio of the magnitudes. We cannot define exactly what the word ratio, in its most general sense, means; but we can compare two ratios, and determine whether one ratio is greater than, equal to, or less than another. This relationship of equality, or inequality, between ratios is the subject of Euclid's Fifth Book. That two magnitudes may have a ratio they must be of the same kind: as far as plane geometry is concerned this means that they must be both lines, or both angles, or both areas. This necessary and sufficient condition is expressed by Euclid in the words—‘magnitudes are said to have a ratio to each other which can, being multiplied, exceed the one the other.' In other words they must be such that, if either of them is repeated often enough, the sum of its repetitions will exceed the other. We cannot therefore have the ratio of a line to an area, for we cannot make a line exceed an area, however often we repeat the line. The ratio of one magnitude (A) to another of the same kind (B), may be estimated by examining how the multiples of A are distributed among the multiples of B, when both are arranged in order of magnitude, and the series of multiples continued onwards without limit. Now as the multiplication of a magnitude, being simply its repetition an assigned number of times, is always possible, the preceding mode of estimation must be always possible. This is the geometrical mode set forth in Euclid's Fifth Book. If the ratio of A to B is given, we are not given A and B themselves, but only the answer to this question, for all values of m— Between what multiples of B lies m A ? To put it more fully-If we form the two scales of multiples B, 2B, 3B, &c., also continued indefinitely; we know the ratio of A to B if we know the multiples n B and (n + 1) B between which m A lies, for any value of m. If it should happen that any one of the multiples of B (say n B) is exactly equal to m A, then the quantities A and B are said to be commensurable. In that case their treatment falls within the province of arithmetic. we say-That relation in virtue of which A is a fraction of B-a fraction being defined as the ratio of two numbers-is called the ratio of A to B, when they are commensurable. But if it should not happen that there are any two terms in the scales, which are equal, so that we only know that mA lies between n B and (n + 1) B, then the quantities A and B are said to be incommensurable; and the arithmetical mode of treatment fails entirely. Since when m A lies between n B and (n + 1) B, I and that, as we make m larger, we diminish we say 'That relation in m virtue of which A can be expressed as lying between two fractions of B, which fractions can be brought as near together as we please, is called the ratio of A to B, when they are incommensurable. In the case of commensurables when two ratios are equal to the same fraction, the four magnitudes constituting them are said to be proportionals. then A has to B a ratio which is equal to the ratio of C to D; and A, B, C, D are cailed arithmetic proportionals. But in the case of incommensurables the above test fails, because there is no common fraction n m to which the ratios can be equated; and we have to examine how the two pairs of scales of multiples of the magnitudes are connected. |