2nd. To divide the consequent, or to multiply the antecedent of a ratio by any number, divides the ratio by that number. (Arith., Part 3rd., Arts. 143, 144.)

3rd. To multiply, or divide, both the antecedent and consequent of a ratio by any number, does not alter the ratio. (Art. 118.)

ART. 260. When the terms of a ratio are equal to each other, the ratio is said to be a ratio of equality. When the second term is greater than the first, the ratio is said to be a ratio of greater inequality, and when it is less, the ratio is said to be a ratio of less inequality.

Thus, the ratio of 2 to 2 is a ratio of equality.

The ratio of 2 to 3 is a ratio of greater inequality.
The ratio of 3 to 2 is a ratio of less inequality.

Hence, a ratio of equality may be expressed by 1; a ratio of greater inequality, by a number greater than 1; and a ratio of less inequality, by a number less than 1.

ART. 261. When the corresponding terms of two or more ratios are multiplied together, the ratios are said to be compounded, and the result is termed a compound ratio. Thus, the bd bd

ratio of a to b, compounded with the ratio of c to d is

α C ac

A ratio compounded of two equal ratios is called a duplicate ratio. b bb b2

Thus, the duplicate ratio of is

α

A ratio compounded of three

ratio. Thus, the triplicate ratio

α α

The ratio of the square roots of two quantities is called a subduplicate ratio. Thus, the subduplicate ratio of 4 to 9 is 3; and

the subduplicate ratio of a to b is

√a

The ratio of the cube roots of two quantities is called a subtriplicate ratio. Thus, the subtriplicate ratio of 8 to 27 is ; and

the subtriplicate ratio of a to b is

ART. 262. Ratios may be compared with each other by reducing the fractions which represent them to a common denominator. Thus, to ascertain whether the ratio of 2 to 7 is less or greater than the ratio of 3 to 10, we have the two fractions, and 10,

which being reduced to a common denominator are 2 and 20; and since the first is greater than the second, we conclude that the ratio of 2 to 7 is greater than the ratio of 3 to 10.

PROPORTION.

That is,

ART. 263. Proportion is an equality of ratios. when two ratios are equal, their terms are said to be proportional. Thus, if the ratio of a to b, is equal to the ratio of c to d, that is,

then a, b, c, d, form a proportion, and we say that a is to

b as c is to d.

Proportion is written in two ways:

1st. By placing a double colon between the ratios; thus,

a:b::c:d.

2nd. By placing the sign of equality between the ratios; thus, ab-c: d.

The first method is the one commonly used.

From the preceding definition, it follows, that when four quantities are in proportion, the second, divided by the first, must give the same quotient as the fourth divided by the third. This is the primary test of the proportionality of four quantities. Thus, if 3,5,6,10, are the four terms of a proportion, so that 3:5:: 6:10, we must have =0.

10

If these fractions are equal to each other, the proportion is true; if they are not equal to each other, it is false. Thus, let it be required to determine whether 3:8:25. The ratios are 8 and, or 16 and 15; hence, the proportion is false.

REMARK. In common language the words ratio and proportion are sometimes confounded with each other. Thus, two quantities are said to be in the proportion of 2 to 3, instead of in the ratio of 2 to 3. A ratio subsists between two quantities, a proportion only between four. It requires two equal ratios to form a proportion.

ART. 264. Each of the four quantities in a proportion is called a term. The first and last terms are called the extremes; and the second and third terms, the means.

The terms of a proportion may be either monomials or polynomials.

ART. 265. Of four quantities in proportion, the first and third are called the antecedents, and the second and fourth, the conse

quents (Art. 257); and the last is said to be a fourth proportional to the other three taken in their order.

ART. 266. Three quantities are in proportion when the first has the same ratio to the second, that the second has to the third. In this case the middle term is called a mean proportional between the other two. Thus, if we have the proportion

a:b::b:c,

then b is called a mean proportional between a and c; and c is called a third proportional to a and b.

When several quantities have the same ratio between each two that are consecutive, they are said to form a continued proportion. ART. 267. PROPOSITION I.- In every proportion, the product of the means is equal to the product of the extremes.

Let a:b::c:
::d.

Since this is a true proportion, the ratio of the first term to the second, is equal to the ratio of the third term to the fourth (Art. 263). Therefore, we must have

b_d

tions, we have

α

a с

Multiplying both sides of this equality by ac, to clear it of frac

abc__adc
= ;
C

From the equation bc=ad, we find d=1c, c=ad, b=

b=ad

and

C

Illustration by numbers. 2:6:5:15; and 6×5=2×15.

d'

bc a

Hence, if any three terms of a proportion are given, the

fourth may be found.

Ex. 1. The first three terms of a proportion are x+y, x2-y2, and x-y; what is the fourth? Ans. x2-2xy+y2.

2. The first, third, and fourth terms of a proportion are (m—n)2, m2—n2, and m+n; required the second.

Ans. m-n.

REMARK.-This proposition furnishes a more convenient test of the proportionality of four quantities, than the method given in Art. 263. Thus, to ascertain whether 2:35:8, it is merely necessary to compare the product of the means and extremes; and since 3x5 is not equal to 2×8, we infer that 2, 3, 5, and 8, are not in proportion.

be

ART. 268. PROPOSITION II. Conversely, If the product of two quantities is equal to the product of two others, two of them may made the means, and the other two the extremes of a proportion. Let bc-ad.

Dividing each of these equals by ac, we have

By dividing each of the equals by cd, we may prove that

a:c:: b: d.

Illust. 3×12=4X9, and 3:4:9:12; also, 3:9:: 4:12. Art. 269. PrOPOSITION III. If three quantities are in proportion, the product of the extremes is equal to the square of the mean. If

then (Art. 267),

b::b: : C, ac-bb-b2.

It follows from Art. 268, that the converse of this proposition is also true. Thus, if

then,

ac=b2,

a:b::b:c.

That is, if the product of the first and third of three quantities is equal to the square of the second, the first is to the second, as the second to the third.

NOTE. It is recommended to the teacher to require the pupils to illustrate all the propositions by numbers. (See Ray's Algebra, Part 1st., Proportion.)

ART. 270. PROPOSITION IV.- If four quantities are in proportion, they will be in proportion by ALTERNATION; that is, the first will be to the third, as the second to the fourth.

NOTE.

a

a:c::b: d.

This proposition is true, only when the four quantities are of the same kind.

ART. 271. PROPOSITION V.- If four quantities are in proportion, they will be in proportion by INVERSION; that is, the second will be to the first, as the fourth to the third.

ART. 272. PROPOSITION VI.—If two sets of proportions have an antecedent and consequent in the one, equal to an antecedent and consequent in the other, the remaining terms will be proportional.

ART. 273. PROPOSITION VII.-If four quantities are in proportion, they will be in proportion by COMPOSITION; that is, the sum of the first and second will be to the second, as the sum of the third and fourth is to the fourth.

From the 1st proportion, bc=ad, (Art. 267);

bd=bd;

Adding the two equations together, bc+bdad+bd;

NOTE. In a similar manner let the student prove that the sum of the

first and second of two quantities is to the first, as the sum of the third and fourth is to the third.