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 = 2p2, whence also ʼn is an even number. But we have seen that m and n cannot both be even, therefore, there can b. 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 1000 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 Con1mensurable 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
COMMENSURABLE MAGNITUDES ONLY.

[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+n 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 imes. 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 :—

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.

Take C the mnth multiple of D.

Then C=mnD=n.mD=nA.

In like manner C=mnD=m.nD=mB.

Hence A and B have a common multiple C.

Q.E.D.

DEF. 3. If two or more magnitudes have a common multiple or measure they are said to be commensurable.

DEF. 4.

The ratio of one magnitude to another of the same kind is a certain relation of the former to the latter in respect of quantity, the comparison being made by considering what multiples of the two magnitudes are equal to one another. The ratio of A to B is denoted thus, A: B, and A is called the antecedent, В the consequent.

Thus the ratio of a half-crown to a florin is the relation expressed by stating that 4 half-crowns=5 florins.

DEF. 5. The ratio A: B is said to be equal to the ratio P: Q when like multiples of A and P (mA and mP) are equal respectively to like multiples of B and Q (nB and nQ), and the four magnitudes are said to be proportionals, or to form a proportion.

The equality of the ratios is denoted by the symbol::; and the proportion thus, A: B:: P: Q, which is read, "A is to B as Pis to Q." A and Q are called the extremes, B and P the means, and Q is said to be the fourth proportional to A, B, and P.

The antecedents A, P are said to be homologous to one another, and so also are the consequents B, Q.

Thus because 4 half-crowns=5 florins =5 francs

and 4 shillings