does not, therefore, comprehend such ratios as 2:5; √3:7; 4: 10, &c. &c. in which the values of the quan tities 2, 3, 7, &c. can only be expressed in decimal frac tions that do not terminate. The ratio that exists between quantities of this latter kind, when the radical quantity is expressed by a decimal fraction, is called their approximate ratio.

(101.) Proportion may be defined the equality of ratios; for, 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 186, or, in other words, that 12: 4 :: 18: 6. The first and last terms of every proportion are called the extremes, and the second and third the means of that proportion.

(102.) A continued proportion, is that in which a set of quantities are related to each other in the following manner, viz. a:b::bc::cd::d: e, &c. where the consequent of every preceding ratio is the antecedent of the following one; and if only three quantities be concerned, as in the proportion a:b:: bc, then b is called a mean proportional between the two extremes a and c.

Note. 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."

(103.) Since the proportion a : b:: c d denotes the equality of the ratios a b and c : d; and since the magnitude of the

ratio a : b is measured by the fraction, and that of the ratio

a C

cd by the fraction, it follows that = that is, when four b d'

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 quantities a, b, c, d, such, that

a

C then those four quantities are proportional, or a : b :: c : d.

On the Comparison and Composition of Ratios.

(104.) On the comparison of Ratios.

1. Since the ratio of a b may be expressed by the fraction if both the numerator and denominator of this fraction be

multiplied by any quantity m (m being either integral or frac

ma

onal), then = mbb'

and, consequently, 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 whence, also, it appears, that a ratio is reduced to its lowest terms by dividing both its antecedent and consequent by their greatest common

measure.

2. Ratios are compared together by reducing to a common denominator the fractions by which their values are respectively represented.

Thus, we represent the ratio of 8: 5, by the fraction, and

the ratio of 9 : 6 by the fraction; reduce these fractions to others of the same value, having a common denominator, and they become

ratio 8 5 is greater than the ratio of 9 : 6.

(105.) 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 represent a ratio of greater inequality, and let z be added to each of its terms, and it then becomes the ratio of a+b+x: a+r. But the ratio of

a+b: a=a+, and that of a+b+x: a+x=

a

a+b+x
a + x

Now reduce these fractions to others of the same value, having

a common denominator, and they become

a2 + ab + ax + br

a(a+x)

and

a2+ab+ax respectively; and since a2+ab+ax+bx is evidently

a(a+x)

3

greater than a2+ab+ax, the ratio of a+b: a is greater than the ratio of a+b+x: a+x. In other words, the ratio of a+b: a has been diminished by adding ≈ to each of its terms.

Again, let a- -b: a represent a ratio of lesser inequality; then pro

d—ab+ax—br is less than a2-ab+ax, the ratio of a-b: a is less than the ratio of a―b+x: a+x, and consequently the ratio of a-b :a has been increased by the addition of a to each of its terms.

In the same manner it might be shewn that a ratio of greater inequality is INCREASED, and a ratio of lesser inequality is DIMINISHED, by SUBTRACTING the same quantity from each of its

terms.

(106.) On the composition of Ratios.

I. Ratios are compounded together by multiplying their antecedents for a new antecedent, and their consequents for a new consequent.

Thus, if the ratio of a: b be 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, there results the ratio of 4 × 5 × 7:3 x 2 x 1, or of 140: 6, or of 70: 3, dividing each term by 2.

II. When the same ratio is compounded with itself once, twice, thrice, &c. the resulting ratios are those of a bo; a3 : b3 ; a': b', &c. &c.

:

:

The ratio of a b2 is called the duplicate ratio of a b; the ratio of ab3 the triplicate; the ratio of a: b the quadruplicate; &c. &c.; and as these ratios receive their denominations from the indices of the several powers of a and b, the ratio of a b is called the subduplicate ratio of a b; the ratio of Va: Vb, the subtriplicate; &c. &c.

:

III. When a set of ratios, of which the consequent of the preceding ratio is the same with the antecedent of the succeeding one, is compounded together, the resulting ratio is that of the first antecedent to the last consequent.

Thus, when the ratios of ab; of be; of cd; of d: e; &c. are compounded together, the resulting ratio is that of abcd, &c.; of bede, &c. or (dividing by bcd) that of a: e, or of the first antecedent: the last consequent.

IV. A ratio, compounded with a ratio of greater inequality, will be increased; but, if it be compounded with a ratio of lesser inequality, it will be diminished.

:

Thus, let 1+n: 1 represent a ratio of greater inequality, and let it be compounded with the ratio a: b, the resulting ratio is that of a+na: b, which is evidently greater than the ratio of a b. On the other hand, let 1-n: 1 represent a ratio of lesser inequality, and let it be compounded with the ratio of a: b, then the resulting ratio is that of a—na": b, which is evidently less than the ratio of a: b.

EXAMPLES.

1. Reduce to their lowest terms the ratio of 360: 315, and 1595

667.

2. Reduce the ratio of a'+2ar: a2 to its lowest terms.

3. Whether is the ratio of 16: 15, or that of 17: 14, the greater ? 4. Which is the least of the three ratios, 20: 17, 22: 18, or 25 : 23 ? and of the three ratios 8:7; 6: 5; and 10: 9, which is the greatest ? 5. Which is the greater, the ratio of a+2: a+4, or that of a+4: fa+5? Ans. the ratio of a +4: a+5. 6. Compound together the ratios of 11: 3, of 7: 2, and of 5: 9. Ans. 385:54.

7. Compound together the ratios of 15: 12, of 6: 7, and of 9:4; and afterwards reduce the resulting ratio to its lowest terms.

Ans. 135: 56.

8. Express in its simplest terms the ratio compounded of a2x2, : a2, a+x: b and ba-x.

9. If the ratios of x+y: a, x—y: b, and b :

Ans. (a+x): a2.

pounded, shew that the resulting ratio is a ratio of equality. 10. If the ratios of 3a+2: 6a+1, and of 2a+3: a+2, be compounded, is the resulting ratio a ratio of greater or lesser inequality? Ans. A ratio of greater inequality.

Ans. 14 and 15.

11. What are the least numbers which will represent the ratio compounded of the three following ratios, viz. the ratio 7: 5, the duplicate ratio of 4: 9, and the triplicate ratio of 3:2 ? 12. Compound the subduplicate ratio of xy, plicate ratio of /x: √y.

Of Proportion.

:

with the quadruAns. a3: y3.

(107.) The following are the most useful Theorems relating to proportional quantities.

(108.) THEOREM 1. If four quantities be proportional, the product of the extremes will be equal to the product of the means.

a C

For let a b c : d, then, (by Art. 103) = therefore ad == bc.

bd

COROL. From hence it follows, that if any three terms of a proportion be known, the fourth may be found; for from the equation

(109.) THEOREM 2. The converse of the foregoing Theorem is also true; viz. If the product of any two quantities be equal to the product of two others, those four quantities will constitute a proportion, provided that the terms of one product be made the MEANS, and

terms of the other product be made the EXTREMES, of such proportion.

1

Thus, if the four quantities a, b, c, d be such that a'd` = b' c′,

a

then =

୯

ď — dividing by b'd', therefore, d' : b :· d' : ď, by Art. 103. (110.) THEOREM 3. If three quantities be proportional, the product of the two extremes is equal to the square of the mean;

Or, if a b :: bc, then (by THEOR. 2,) ac = b2. From hence also it follows, that a mean proportional between any two quantities is equal to the square root of their product; for let x be a mean proportional between a and c, then a x:x:c, therefore ac, and x

= √ac.

(111.) THEOREM 4. If four quantities be proportional, they will also be proportional when taken inversely or alternately;

a

C

Thus, if a : b :: c: d, then =; but invert the fractions, and

b d

then; therefore bad: c. Again, since (by THEOR. 1,)

a C

(112.) THEOREM 5. If there be six proportional quantities, and the first be to the second as the third is to the fourth; and the third to the fourth as the fifth to the sixth; then will the first be to the second as the fifth to the sixth.

For let a : b::c:d, and c:d::e:f; then; and

e

(113.) THEOREM 6. If four quantities be proportional, then the SUM OF DIFFERENCE of the first and second will be to the second as the SUM or DIFFERENCE of the third and fourth is to the fourth.

For let a b::c:d, then

:

C

=

add or subtract 1 from each

side of the equation; then +1=1, therefore, a+c+d

consequently, a+bb::c+d: d, by Art. 103.

(114.) THEOREM 7. If four quantities be proportional, the