PROPORTION.

(136.) PROPORTION consists in an equality of the ratios of two or more antecedents to their respective consequents-but is usually confined to four terms.

(137.) Four quantities are in Proportion when the ratio of the first to the second is equal to the ratio of the third to the fourth; that is, when the first divided by the second is equal to the third divided by the fourth.

Thus the numbers 6, 3, 8, 4 are in proportion,

since the ratio equals the ratio .

And the quantities a, b, x, y are in proportion,

a

when the ratio equals the ratio, (129).

b

The first and third terms are the antecedents of the ratios; the second and fourth are the consequents. The first and fourth are the two extremes; the second and third are the two means.

The fourth term is called a fourth proportional to the other three taken in order; thus 4 is a fourth proportional to 6, 3, and 8.

(138.) Three quantities are in Proportion when the ratio of the first to the second is equal to the ratio of the second to the third,—the second term being called a mean proportional between the other two.

Thus the numbers 8, 4, 2 are in proportion,

since the ratio equals the ratio;

and 4 is a mean proportional between 8 and 2.

Direct and Inverse Proportion.

(139.) A direct Proportion consists in an equality between two direct ratios, and an inverse or reciprocal Proportion in an equality between a direct and an inverse ratio.

Thus the numbers 6, 3, 8, 4 are in direct proportion; (137).

The same numbers in the order 6, 3, 4, 8 are in inverse proportion, since the direct ratio is equal to the inverse ratio &, (130).

The term proportion used alone always means direct proportion.

Sign of Proportion.

(140.) A Proportion is denoted by a double colon (: :), or the sign between the equal ratios of the proportion.

Thus 6 38: 4, or 6: 38: 4, or =

:

denotes that these numbers are in proportion, and is read 6 is to 3 as 8 is to 4.

To denote an inverse Proportion we employ the sign between the two ratios of such proportion.

Thus 6 34 8, denotes that 6 is to 3 inversely as 4 is to 8.

Inverse Converted into Direct Proportion.

(141.) An inverse is converted into a direct Proportion by interchanging either antecedent and its consequent; or by substituting the reciprocals of either antecedent and its consequent.

Thus from the inverse proportion 6: 34: 8,

:

we get the direct proportion 3: 64: 8, by interchanging 6 and 3, or 48, by substituting and. The reason of this is evident from the nature of inverse ratio, (130).

VARIATION.

(142.) VARIATION is such a dependence of one term or quantity on another, that any new value of one of them will produce a new value of the other, in a constant ratio of increase or diminution.

1. One quantity varies directly as another when their dependence is such that if one of them be multiplied, the other must be multiplied, by the same quantity.

For example, the Interest on money, for a given time and rate per cent., varies directly as the Principal, since the Interest will be doubled, or tripled, &c., if the Principal be doubled, or tripled, &c.

2. One quantity varies inversely as another when their dependence is such that if one of them be multiplied, the other must be divided, by the same quantity.

For example, the Time in which a given amount of interest will accrue on a given principal, varies inversely as the Rate per cent., since the Time will be doubled, &c. if the Rate be halved, &c.

(143.) When one quantity varies inversely as another, the product of the two is always the same constant quantity.

For as one of the two quantities is multiplied, the other is divided by the same number; the product of the two will therefore be multiplied and divided by the same number; hence its value will remain unchanged.

Variation-an Abbreviated Proportion.

(144.) The two terms of a variation are the two antecedents in a Proportion in which the two consequents are not expressed, but may be understood, to complete the proportion.

Thus when we say that the Interest varies as the Principal, for a given time and rate per cent., it is understood, that

The Interest on any principal is to the Interest on any other Principal, for the same time and rate, as the first Principal is to the second.

Instead of saying "the Interest varies as the Principal," we may say, the Interest is proportional to the Principal; which is a brief method of expressing a Proportion by means of its antecedents,—the consequents being understood.

The character

2

Sign of Variation.

placed between two terms, denotes that one of them varies as the other. Thus xy, x varies as y, or x is proportional to y.

1

x~ denotes that x varies as the reciprocal of y; or x varies rey

ciprocally or inversely as y.

y

2~ denotes that x varies directly as y, and inversely as z; that is, a varies as the quotient of y divided by z.

THEOREMS IN PROPORTION.

(145.) A Theorem is a proposition to be demonstrated or proved.A Corollary is an inference drawn from a preceding proposition of demonstration.

THEOREM I.

(146.) Two Fractions having a common denominator, are to each other as their numerators; and two fractions having a common numerator are to each other inversely as their denominators.

First. Let d be the common denominator;

a

d

and is the ratio of the numerator a to the numerator c.

с

Secondly. Let n be the common numerator;

C

and is the inverse ratio of the denominator a to the denominato

a

c, (130).

Therefore, two fractions having a common denominator, &c.

(147.) Corollary. The value of a Fraction varies directly as its numerator, and inversely as its denominator.

THEOREM II.

(148.) In any Proportion, if one antecedent be greater than its consequent, the other antecedent will be greater than its consequent; if equal, equal; and if less, less.

Now if a be greater than 6 the first ratio will be greater than a unit, and consequently the second ratio will be greater than a unit, and therefore x will be greater than y. In like manner if a be equal to b, x will be equal to y, &c.

Hence, in any Proportion, if one antecedent, &c.

THEOREM III.

(149.) When four quantities are in Proportion, the product of the two extremes is equal to the product of the two means.

Clearing the Equation of fractions.
ay = bx, (115).

Therefore, when four quantities are in Proportion, &c.

(150.) Cor. 1. A fourth proportional to three given quantities, is find by dividing the product of the second and third by the first. Thus from the equation ay=bx, we find y=

bx

a

(151.) Cor. 2. When three quantities are in Proportion, the produt of the two extremes is equal to the square of the mean. For let abb: x; then ax=bb=b2.

(152.) Cor. 3. A mean proportional between two given quantities, is equal to the square root of their product. Thus from the equation ax=b2, we find b=(ax)*.

THEOREM IV.

(153.) When the product of two quantities is equal to the product of two other quantities, either pair of factors may be made the extremes, and the other the means, of a Proportion.

Let ab=xy;

then will ax:: y: b.

Dividing both sides of the given equation by b,

xy

α= b

In like manner a and b may be taken for the means, and x and y Therefore, when the product of two quantities, &c

for the extremės.