and no one comparable with another of a different kind as to quantity or intensity. On the other hand, position, direction, relation, etc., are not magnitudes, inasmuch as it is unmeaning to speak of a greater or less position, direction, etc., unless in some conventional or specially defined sense of the word. In order that any magnitude may be made the subject of exact mathematical treatment, it must be possible to define with exactness the equality of two amounts of such magnitude. In Geometry the test of equality is given in the Axiom (I, Ax. 1), "Magnitudes that can be made to coincide are equal." Corresponding tests of equality form the foundation of the sciences which deal with value, probability, time, speed, weight, etc. It is the impossibility of giving exact tests of the equality of different amounts of hunger, love, courage, talent, etc., or any moral qualities, which prevents their forming the subject of exact measurement, and so of mathematical treatment. Number and magnitudes that are completely expressible by number differ from other magnitudes in that they are essentially discrete or discontinuous. Fundamentally, number is counting, and to count is to add unit after unit, and so to increase by jumps or finite additions, while magnitudes in general are continuous, and increase insensibly from one value to another through every intermediate value. Thus the population of a country is a magnitude which increases or diminishes by the addition or removal of individual human beings, while the height or weight of a man is a magnitude which increases or diminishes by continuous change or growth. It is impossible, therefore, to represent continuous magnitude exactly by number, though number can obviously be represented by continuous magnitude. Magnitudes which are exactly expressible by numbers of the same common unit, are said to be commensurable: while, if there is no common unit by which they can be exactly expressed, they are said to be incommensurable. To prove the existence of incommensurable quantities, let us consider the diagonal and side of a square. If the diagonal and side of a square can be measured or expressed in numbers of a common unit, suppose that the one contains the greatest unit by which they can be so measured m times and the other n times. Then m and n cannot both be even numbers, for in that case the two lines could be measured by double the supposed greatest unit. But since the square on the diagonal contains m2 square units, and that on the side n2 square units, and we know (II. 9) that the square on the diagonal is double the square on the side, therefore m2 = 2n2. Hence m2 is an even number, and, therefore, it is the square of an even number, so that m is even. Let m = 2; then, since 2n2 = m2 = 4p2, therefore n2 = 202, whence also n is an even number. But we have seen that m and n cannot both be even, therefore, there can be no common unit by which the diagonal and side of a square can be expressed, so that they are incommensurable. In fact, it should be observed that with regard to different quantities of the same continuous magnitude, incommensurability is the rule, and commensurability the exception. For practical purposes, by taking a sufficiently small unit, every continuous magnitude may be expressed by means of a number, but it must be clearly understood that such expression is not exact, but approximate only. Thus it is true that of the parts of which the side of a square contains 100, the diagonal contains more than 141, and less than 142, and if of the side be regarded as sufficiently small to be neglected, the diagonal is measured approximately by 141. If greater accuracy is required, but may be neglected, it is measured approximately by 1,414 parts, of which the side contains 1,000 and so on to any degree of accuracy that may be required. A theory of Proportion, incomplete but sufficient for practical applications, is given in Part I. of this book, which treats of Commensurable Magnitudes only; but the student who aims at mastering a complete theory for magnitudes in general, without regard to the distinction of commensurable and incommensurable, must study Part II., which, though different in the form of expression, is essentially the theory contained in the Fifth Book of Euclid's Elements. PART I. FUNDAMENTAL PROPOSITIONS OF PROPORTION FOR [Notation. SECTION I. OF RATIO AND PROPORTION. In what follows, large Roman letters, A, B, etc., are used to denote magnitudes, and where 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 where they are, or may be, of different kinds, from different parts of the alphabet, as A, B compared with P, Q or X, Y. Small italic letters, m, n, etc., denote whole numbers. By m.A or mA is denoted the mth multiple of A, and it may be read as m times A. The product of the numbers m and n is denoted by mn, and it is assumed that mn=nm. The combination m.nA denotes the mth multiple of the nth multiple of A, and may be read as m times nA, and mnA or mn.A as mn times A. By (m+n) A is denoted m+n2 times A.] DEF. 1. One magnitude is said to be a multiple of another magnitude when the former contains the latter an exact number of times. According as the number of times is 1, 2, 3...m, so is the multiple said to be the 1st, 2nd, 3rd,...mth. DEF. 2. One magnitude is said to be a measure or part of another magnitude when the former is contained an exact number of times in the latter. The following properties of multiples will be assumed : 2. If mA=mB, then A=B. 3. mA+mB+ ... =m (A+B+ ... .). 4. mA-mB=m (A-B), (A being greater than B). 5. mA+nA=(m+n)A. 6. mA-nA=(m-n)A, (m being greater than n). 7. m.nA=mn.A=nm.A=n.mA. THEOR. I. If two magnitudes have a common multiple they have also a common measure. Conversely, if two magnitudes have a common measure they have also a common multiple. Let A and B have a common multiple C, i.e., let C=mA or nB. Suppose C to contain mn parts each equal to D. Because C=mA, and C=mnD, therefore mA=mnD =m.nD. Therefore A=nD. In like manner B=mD. Hence A and B have a common measure D. Props. of Mults. 2. Conversely, let A and B have a common measure D, i.e., let A=mD, and B=nD. |