We shall employ the following notation.

Capital letters, A, B, C,... will be used to denote the magnitudes themselves, not any numerical or algebraical measures of them, and small letters, m, n, p,... will be used to denote whole numbers. Also it will be assumed that multiplication, in the sense of repeated addition, can be applied to any magnitude, so that m. A or mА will denote the magnitude A taken m times.

The symbol > will be used for the words greater than, and <for less than.

DEFINITIONS.

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 following properties of multiples will be assumed as self-evident. (1) MA > = or < mВ according as A > = or < B; and conversely.

3.

(2) mA+mB+.
· ... = m (A + B + ...).

(3) If A> B, then mA - mB=m (A – B).

(4) mA+NA+. = (m + n + ...) A.

(5) If m>n, then mĀ – nĀ= (m − n) A.

(6) m.nAmn. Anm. A= n. mĀ.

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 term quantuplicity denotes the capacity of the first magnitude to contain the second with or without remainder. If the magnitudes are commensurable, their quantuplicity may be expressed numerically by observing what multiples of the two magnitudes are equal to one another.

A

Thus if Ama, and B=na, it follows that nAmB. In this case B, and the quantuplicity of A with respect to B is the arith

m

n

But if the magnitudes are incommensurable, no multiple of the first can be equal to any multiple of the second, and therefore the quantuplicity of one with respect to the other cannot exactly be expressed numerically: in this case it is determined by examining how the multiples of one magnitude are distributed among the multiples of the other.

Thus, let all the multiples of A be formed, the scale extending ad infinitum; also let all the multiples of B be formed and placed in their proper order of magnitude among the multiples of A. This forms the relative scale of the two magnitudes, and the quantuplicity of A with respect to B is estimated by examining how the multiples of A are distributed among those of B in their relative scale.

In other words, the ratio of A to B is known, if for all integral values of m we know the multiples nB and (n+1) B between which mA lies.

In the case of two given magnitudes A and B, the relative scale of multiples is definite, and is different from that of A to C, if C differs from B by any magnitude however small.

For let D be the difference between B and C; then however small D may be, it will be possible to find a number m such that mD>A. In this case, mB and mC would differ by a magnitude greater than A, and therefore could not lie between the same two multiples of A; so that after a certain point the relative scale of A and B would differ from that of A and C.

[It is worthy of notice that we can always estimate the arithmetical ratio of two incommensurable magnitudes within any required degree of accuracy.

For suppose that A and B are incommensurable; divide B into m equal parts each equal to ß, so that B=mß, where m is an integer. Also suppose ẞ is contained in A more than n times and less than (n+1) times; then

A

so that

B

n

1

m

m

differs from by a quantity less than And since we

can choose ẞ (our unit of measurement) as small as we please, m can

1

be made as great as we please. Hence can be made as small as we

m

please, and two integers n and m can be found whose ratio will express that of a and b to any required degree of accuracy.]

4.

The ratio of one magnitude to another is equal to that of a third magnitude to a fourth, when if any equimultiples whatever of the antecedents of the ratios are taken, and also 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 the multiple of the other antecedent is greater than, equal to, or less than that of its consequent.

Thus the ratio A to B is equal to that of C to D when mC > = or < nD according as mĀ >= or < nB, whatever whole numbers m and n may be.

Again, let m be any whole number whatever, and n another whole number determined in such a way that either mA is equal to nB, or mA lies between nB and (n+1) B; then the definition asserts that the ratio of A to B is equal to that of C to D if mC=nD when mА=nB; or if mC lies between nD and (n+1) D when mA lies between nB and (n+1) B.

In other words, the ratio of A to B is equal to that of C to D when the multiples of A are distributed among those of B in the same manner as the multiples of C are distributed among those of D.

5. When the ratio of A to B is equal to that of C to D the four magnitudes are called proportionals. This is expressed by saying "A is to B as C is to D", and the proportion is written

A and D are called the extremes, B and C the means; also D is said to be a fourth proportional to A, B, and C.

Two terms in a proportion are said to be homologous when they are both antecedents, or both consequents of the ratios.

[It will be useful here to compare the algebraical and geometrical definitions of proportion, and to shew that each may be deduced from the other.

According to the geometrical definition A, B, C, D are in propor tion, when mC>=<nD according as mA>=<nB, m and n being any positive integers whatever.

According to the algebraical definition A, B, C, D are in proportion A C

when

B D'

(i) To deduce the geometrical definition of proportion from the algebraical definition.

hence from the nature of fractions,

mC>=<nD according as mĀ>
>=

which is the geometrical test of proportion.

<nB,

(ii) To deduce the algebraical definition of proportion from

the geometrical definition.

<nB, to prove

Given that mC>= <nD according as mĀ>
>=<N

; then it will be possible to find some fraction

m

which lies between them, n and m being positive integers.

6. 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.

H. .E

19

This definition asserts that if whole numbers m and n can be found such that while mA is greater than nB, mC is not greater than D, or while mA=nB, mC is less than nD, then the ratio of A to B is greater than that of C to D.

7. If A is equal to B, the ratio of A to B is called a ratio of equality.

If A is greater than B, the ratio of A to B is called a ratio of greater inequality.

If A is less than B, the ratio of A to B is called a ratio of less inequality.

8. Two ratios are said to be reciprocal when the antecedent and consequent of one are the consequent and antecedent of the other respectively; thus B: A is the reciprocal of A: B.

9. Three magnitudes 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.

Thus A, B, C are proportionals if

AB: B: C.

B is called a mean proportional to A and C, and C is called a third proportional to A and B.

10. Three or more magnitudes are said to be in continued proportion when the ratio of the first to the second is equal to that of the second to the third, and the ratio of the second to the third is equal to that of the third to the fourth, and so on.

11. When there are any number of magnitudes of the same kind, the first is said to have to the last the ratio compounded of the ratios of the first to the second, of the second to the third, and so on up to the ratio of the last but one to the last magnitude.

For example, if A, B, C, D, E be magnitudes of the same kind, AE is the ratio compounded of the ratios A : B, BC, CD, and D : E.