121

CHAP. VI.

ON RATIOS, PROPORTION, AND VARIABLE

QUANTITIES.

XXVII.

Definitions.

92. BY RATIO is meant the relation which one quantity bears to another, with respect to magnitude. It is evident that this relation can exist only between quantities of a similar kind; thus, a number must be compared with a number; a linė with a line; &c. &c.; and it would be absurd to compare a certain number of feet with a certain number of pounds; &c. &c.

93. There are two ways in which the magnitude of quantities may be compared. In the first place, they may be compared with regard to their difference; and then the question asked, is, "How much one quantity is greater or less than another." The relation which quantities bear to each other in this respect, is called their Arithmetical Ratio. The other way in which they may be compared, is, by inquiring "How often one quantity is contained in the other." This relation between quantities is called their Geometrical Ratio. The term ratio, when simply applied, is generally understood in the latter sense; and it is in this sense that the word will be made use of in the present Chapter.

94. In considering how often one quantity is contained in another, the natural process is to divide the one by the other. Thus, in comparing the number 12 with the numbers 4 and 3,

we know that 4 is contained in 12 three times, and that 3 is contained in the same number four times; from which we infer the ratio of 12: 3 is greater than the ratio of 12 to 4, the magnitude of a ratio being measured by the number of times one quantity is contained in another. For the same reason, the ratio of 11: 7 is said to be less than the ratio of 11:5. When a ratio is thus expressed, the first term of it is called the antecedent, the last term the consequent, of that ratio.

95. From this mode of estimating the magnitude of a ratio, it appears that when the consequent of a ratio is not an aliquot part of the antecedent, the value of the ratio must be expressed by a fraction, whose numerator is the antecedent, and denominator the consequent of that ratio. Thus the magnitude of the ratio of 15:7 is expressed by the fraction and of

4

15

7

the ratio 4:13 by the fraction When the antecedent of a

13

ratio is greater than the consequent, it is called a ratio of greater inequality; when the antecedent is less than the consequent, a ratio of lesser inequality; and if the two terms of a ratio be the same, then it is said to be a ratio of equality.

96. The foregoing definitions evidently apply to those instances only, in which the consequent of a ratio is contained a certain number of times in the antecedent, or in which the magnitude of the ratio may be expressed by some definite fraction. It does not therefore comprehend such ratios as &c. &c.; where the values &c. can only be expressed in

√2:5; √3:27; 4: √1; of the quantities/2, √3, 7 decimal fractions which do not terminate. The ratio which exists between quantities of this latter kind, when the radica

quantity

(a) In expressing the ratio of two quantities, the word "to" is generally supplied by two dots; thus, the ratio of "a to b" is expressed by "a: b."

quantity is expressed by a decimal fraction, is called their approximate ratio.

97. Proportion consists in the equality of ratios; thus, since 4 is contained in 12, the same number of times that 6 is in 18, the ratio of 12: 4 is said to be equal to the ratio of 18: 6, or, in other words, that 12:4:: 18:6. Of the four terms of which every proportion consists, the first and last terms are called the extremes, and the second and third the means of that proportion.

98. If there be a set of quantities related together in the following manner, viz. a:b::b:c::c:d::d: e, &c. where the consequent of every preceding ratio is the antecedent of the following one, then the quantities a, b, c, d, e, &c. are said to be in continued proportion; and if only three quantities be concerned, as in the proportion ab:: bc, then b is said to be a mean proportional between the two extremes a and c.

99. Since the proportion a: b:: c:d expresses the equality of the ratios ab and c:d; and since the magnitude of the ratio a: b is measured by the fraction and that of the ratio

a

Б

"four quantities are proportional, the quotient of the first "divided by the second, is equal to the quotient of the third "divided by the fourth ;" and vice versa, if "there be four r quantities a, b, c, d, such, that, then those four quan"tities are proportional, or a: b::c: d."

XXVIII.

(b) In stating a proportion, the words "is to" and "to" are generally supplied by two dots, and the words "so is" by four dots; thus, the proportion "a is to b so is c to d," is expressed by "a b c : d."

XXVIII.

On the Comparison and Composition of Ratios.

100. On the comparison of Ratios.

1. Since the ratio of a: b may be expressed by the fraction a let the numerator and denominator of this fraction be ī

ma a
=

mb ł

multiplied by any quantity m (m being either integral or fractional), then and .. the ratio of ma: mb is the same with the ratio of a: b; from which we infer, that "if the terms "of a ratio be multiplied or divided by the same quantity, it "does not alter the value of the ratio." From hence also it appears, that a ratio is reduced to its lowest terms by dividing its antecedent and consequent by their greatest common measure.

II. "Ratios are compared together by reducing the frac❝tions by which their values are respectively represented, to a common denominator." Thus, the ratio of 8: 5 is represented by the fraction, and the ratio of 9: 6 by the fraction; reduce these fractions to others of the same value having a

8

III. "A ratio of greater inequality is diminished, and a ratio "of lesser inequality is increased, by adding the same quantity "to both its terms." Let a+b: a represent a ratio of greater inequality, and let x be added to each of its terms, and it becomes the ratio of a+b+x:a+x. Now the ratio of a+b: a = a + b

a

and that of: a+b+x: a+x=a+b+x; let

a + x

these fractions be reduced to others of the same value having

a common

and

a (a+x)

a2+ab+ax respectively; and since a+ab+ax+bx is a(a+x) evidently greater than a'+ab+ax, the ratio of a+ba is greater than the ratio of a+b+x:a+x; i. e. the ratio of a+b: a has been diminished by adding x to each of its terms. Next, let a-b: a represent a ratio of lesser inequality; then a-b a-b+x, as in the proceeding with the fractions

a

and

former instance, the resulting fractions are

and

a + x

a2-ab+ax-bx a(a+x)

a'-ab+ax ; and since a'-ab+ax-bx is less than a(a+) a-ab+ax, the ratio of a-ba is less than the ratio of a−b+x:a+x, and consequently the ratio of a-b: a has been increased by adding x to each of its terms. In the same manner it might be shewn that "a ratio of greater inequality is in"creased, and a ratio of lesser inequality is diminished, by "subtracting the same quantity from each of its terms."

101. On the composition of Ratios.

be

1. Ratios are compounded together by multiplying their antecedents together for a new antecedent, and their consequents together for a new consequent. Thus, if the ratio of a compounded with the ratio of c: d, the resulting ratio is that of ac: bd; or if the ratios 4:3; 5: 2; and 7: 1, be compounded together, there results the ratio of 4 × 5 × 7:3 × 2 × 1, or of 140: 6, or (dividing each term by 2) of 70:3.

II. If the same ratio be compounded with itself once, twice, thrice, &c. the resulting ratios are those of a3 : b3 ; a3 : b3 ; a: b', &c. &c. The ratio of a: 6 is called the duplicate ratio of a : b; a3: b3 the triplicate; a*: b1 the quadruplicate; &c. &c.; and as these ratios receive their denominations from

the