portion thus, A: B:: P: Q, which is read "A is to B as P is to Q." A and Q are called the extremes, B and P the means, and Qis 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.

DEF. 6. If A, B, C are three magnitudes of the same kind such that A: B : : B : C, B is said to be the mean proportional between A and C, and C the third proportional to A and B.

DEF. 7. If there are three magnitudes A, B, C, of the same kind, then A is said to have to C the ratio compounded of the ratios A: B, B: C.

DEF. 8. If there are two ratios A: B, P: Q, and C be taken such that B: C:: P: Q, then A is said to have to C the ratio compounded of the ratios A: B, P: Q.

DEF. 9. A ratio compounded of two equal ratios is called the duplicate of either of these ratios.

DEF. 10. When a line AB is divided internally at P and externally at Q in the same ratio, it is said to be divided harmonically by the points P and Q; and the points A, P, B, Q are said to form an harmonic range.

PART II.

FUNDAMENTAL PROPOSITIONS OF PROPORTION FOR MAGNITUDES, WITHOUT RESPECT TO COMMENSURABILITY.

SECTION I.

OF RATIO AND PROPORTION.

[Notation.

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. The symbol > denotes greater than, and ◄ less than.]

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

the multiple said to be the 1st, 2nd, 3rd . mth.

m, so is

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 :1. As A> or < B, so is m A >= or <mB.

The converse necessarily follows (Introd., p. 14), so that 2. As mA> or <mB, so is A> or <B.

3. mA+mB+... =m(A+B+...).

4. mA-mBm(A-B), (A being greater than B).

6. mA-nA=(m-n) A, (m being greater than n). 7. m.nA=mn. A=nm.A=n.mA.

DEF. 3.-The ratio of one magnitude to another of the same kind is the relation of the former to the latter in respect of quantuplicity.

The quantuplicity of A with respect to B is 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.

Thus, if A denotes the diagonal, and B the side of a square, the scale of interdistribution of their multiples begins thus :

each multiple of A being placed over the interval between the multiples of B, between which it lies.

This interdistribution of multiples is definite for two given magnitudes A and B, and is different from that for A and C, if C differ from B by any magnitude however small.

For if B and C differ by a magnitude D, m may be found so great that mD > A, however small D be, and then mB and mC would differ by more than A, and, therefore, could not lie between the same two multiples of A, so that the interdistribution of the multiples of A and B would for high multiples (if not for lower ones) differ from that for A and C.

The ratio of A to B is denoted thus A ; B, and A is called the antecedent, B the consequent, of the ratio.

DEF. 4. The ratio of one magnitude to another is equal to that of a third magnitude to a fourth (whether of the same or of a different kind from the first and second), when any equimultiples whatever of the antecedents of the ratios being taken and likewise any equimultiples whatever of the consequents, the multiple of one antecedent is greater than, equal to, or less than that of its consequent, according as that of the other antecedent is greater than, equal to, or less than that of its consequent.

Or in other words :

The ratio of A to B is equal to that of P to Q, when mA is greater than, equal to, or less than B, according as mP is greater than, equal to, or less than nQ, whatever whole numbers m and n may be.

It is an immediate consequence that :

The ratio of A to B is equal to that of P to Q, when, m being any number whatever, and n another number determined so that either mA is between B and (n + 1) B or equal to nB, according as mA is between nB and (n + 1) B or is equal to nB, so is mP between nQ and (n + 1) Q, or equal to nQ. The definition may also be expressed thus:

The ratio of A to B is equal to that of P to Q when the multiples of A are distributed among those of B in the same manner as the multiples of P are among those of Q.

DEF. 5. The ratio of one magnitude to another is greater than that of a third magnitude to a fourth, when equimultiples of the

antecedents and equimultiples of the consequents can be found such that, while the multiple of the antecedent of the first is greater than, or equal to, that of its consequent, the multiple of the antecedent of the other is not greater, or is less, than that of its consequent.

Or in other words :

The ratio of A to B is greater than that of P to Q, when whole numbers mand n can be found, such that, while mA is greater than nB, mP is not greater than nQ, or while mA = nB, mP is less than nQ.

DEF. 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: B: P: Q,

which is read, "A is to B as P is 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, and so are the consequents, B, Q.

DEF. 7.

Three magnitudes (A, B, C) of the same kind are said to be proportionals, when the ratio of the first to the second is equal to that of the second to the third; that is, when

A: BB: C.

In this case C is said to be the third proportional to A and B, and B the mean proportional between A and C.

DEF. 8. The ratio of any magnitude to an equal magnitude is said to be a ratio of equality. If A be greater than B, the ratio A: B is said to be a ratio of greater inequality, and the ratio B : A a ratio of less inequality. Also the ratios A: B and B: A are said to be reciprocal to one another.