BOOK III.

RATIO AND PROPORTION.

DEFINITIONS.

1. The measurement of magnitudes is effected by comparing them with some particular magnitude of the same kind, which is fixed upon as a standard or measuring unit.

This comparison is usually conducted by means of equations, ratios and proportions.

Note. When we ascertain the dimensions of a given magnitude by discovering its equality with some other magnitude of known dimensions, we make use of equations. But we frequently have occasion to compare a given magnitude with another, greater or less than itself, and the question arises how much greater one is than the other, or what part one is of the other. To answer this inquiry we resort to ratio and proportion.

2. RATIO is that relation between two magnitudes of the same kind, which is expressed by the quotient of the one divided by the other.

Thus the ratio of A to B, is

A

8

; that of 8 to 4, is

=2.

B

4

Note. When the two magnitudes compared are equal, the ratio is a unit, and is called a ratio of equality. If the antecedent is greater than the consequent, the ratio is greater than a unit, and is called a ratio of greater inequality. When the antecedent is less than the consequent, the ratio is less than a unit, and is called a ratio of less inequality.

3. The two magnitudes when spoken of separately, are called the terms of the ratio; when spoken of together, they are called a couplet.

The first term is called the antecedent; and the last, the consequent.

4. Ratio is expressed in two ways:

First, in the form of a fraction, putting the antecedent in the place of the numerator, and the consequent in the place of the denominator:

Second, by placing a colon between the two magnitudes compared. Thus, the ratio of A to B, is written,

A

B

; or, A : B.

One of these expressions is equivalent to the other, and may, at any time, be exchanged for the other at pleasure.

5. Inverse or reciprocal ratio is the ratio of the reciprocals of two magnitudes. Thus the reciprocal ratio of

When the direct ratio is expressed in the form of a fraction, the reciprocal ratio of these magnitudes, is the fraction inverted and when the notation is by points, the reciprocal ratio is expressed by inverting the order of the terms. Thus A is to B inversely, as B to A, and 4 is to 8 inversely, as 8 to 4.

6. Compound ratio is the ratio of the products of the corresponding terms of two or more simple ratios.

Thus the ratio of

And the ratio of

4 to 2 is 2.

9 to 3 is 3.

The ratio compounded of these two ratios, is 36: 6 = 6.

Note. Compound ratio is of the same nature as any other ratio. The term is used to denote the origin of the ratio in particular cases.

7. That class of compound ratios produced by multiplying. a simple ratio into itself, or into another equal ratio, i. e. the square of a simple ratio, is called a duplicate ratio. That produced by multiplying three equal ratios together, i. e. the

cube of a simple ratio, is called a triplicate ratio, &c. The ratio of the square roots of two magnitudes, is called a subduplicate ratio; that of the cube roots, a subtriplicate, &c. Thus,

The simple ratio of A to B is
The duplicate "

66

A : B.

"is

A: B2.

Note. From the definition of ratio, as well as the mode of expressing it in the form of a fraction, it is obvious that the ratio of two quantities is the same as the value of a fraction whose numerator and denominator are respectively equal to the antecedent and consequent of the given couplet. Hence the following principles which have been established in fractions, are also true of ratio. 8. To multiply the antecedent of a couplet by any quantity multiplies the ratio by that quantity; and to divide the antecedent, divides the ratio.

Thus, the ratio of
The ratio of

And the ratio of

16 4 is 4.

16×24 is 8, which equals 4×2.

16-2: 4 is 2, which equals 4÷2.

9. To multiply the consequent of a couplet by any quantity, divides the ratio by that quantity; and to divide the consequent, multiplies the ratio.

Thus, the ratio of 16 : 4 is 4.

The ratio of

16: 4x2 is 2, which equals 4÷2.

And the ratio of 16: 4÷2 is 8, which equals 4×2.

10. To multiply, or divide, both the antecedent and consequent of a couplet by the same quantity, does not alter the

ratio.

Thus, the ratio of

The ratio of

And the ratio of

8:4=2;

8x2 4x2=2;

8 2 42=2.

* Abridgment of Day's Algebra, Arts. 311, 112, 132, 135.

11. PROPORTION is an equality of ratios.

ratio of A to B is equal to the ratio of C to D;

A C
B D'

Thus, if the i. e., if

the four magnitudes A, B, C, D, form a proportion. So the numbers, 12, 4, 6, 2, are proportional, because the ratio of 12 4 is equal to 6: 2.

:

12. Proportion is expressed, either by the sign of equality, (=), or by four points, () placed between the two ratios. Thus the expressions,

A C

BD' and A: B:: CD,

are equivalent to each other. The latter is usually read, "A is to B as C is to D."

13. The number of terms in a proportion must at least be four; for the equality is between the ratios of two couplets, and each couplet has an antecedent and a consequent.

There may, however, be a proportion among three magnitudes; for one of the magnitudes may be repeated so as to form two terms. In this case the term repeated, is called a mean proportional between the other two magnitudes.

Note. In common discourse, the term proportion is often confounded with ratio, and used as synonymous with it. But in scientific usage there is a manifest and important difference in their signification.

One ratio may be greater or less than another; thus the ratio of 12 4, is less than the ratio of 24: 6, and greater than that of 10 5. One proportion, on the other hand, cannot be greater, or less than another; for equality does not admit of degrees. Again, in a ratio, there are but two terms, an antecedent and a consequent ; whereas, to form a proportion, there must be at least four terms, or two couplets.

14. When four magnitudes are proportional, the first and

last terms are called the extremes; and the other two, the

means.

15. Homologous terms are either the two antecedents, or the two consequents.

Analogous terms are the antecedent and consequent of the same couplet.

16. Inverse or reciprocal proportion is an equality between a direct ratio and a reciprocal ratio.

or, 8 : 4 :: 6 : 3.

8 is to 4 inversely, as 3 to 6; When expressed as in the last case, the first term has the same ratio to the second, which the fourth has to the third.

17. When there is a series of magnitudes such, that the ratios of the first to the second, of the second to the third, &c., are all equal, the magnitudes are said to be in continued proportion. Thus, 81, 27, 9, 3, 1, are in continued proportion. In this case the consequent of each preceding ratio, is the antecedent of the following one.

PROPOSITION I. THEOREM.

If to the terms of any couplet, two magnitudes having the same ratio be added, the ratio of the sum of the antecedents to the sum of the consequents, will be equal to the ratio of either antecedent to its consequent.

Let the ratio of A: B, be the same as the ratio of C : D; then will the ratio of A+C: B+D, be equal to the ratio A: B, or C: D.

А С

For, since by hypothesis = multiplying each by BD,

B