Similarly, the proposition may be established whatever be the number of ratios.

381. A ratio of greater inequality compounded with another increases it, and a ratio of less inequality compounded with another diminishes it.

according as x is greater or less than y

382. If the difference between the antecedent and the consequent of a ratio be small compared with either of them, the ratio of their squares is nearly obtained by doubling this difference.

Let the proposed ratio be a +x: a, where x is small compared with a; then a2 + 2ax + x2: a2 is the ratio of the squares of the antecedent and consequent. But x is small compared with a, and therefore x2 or xxx is small compared with 2a xx, and much smaller than a × a. Hence a2+2ax: a2, that is, a + 2x : a, will nearly express the ratio (a + x)2: a3.

Thus the ratio of the square of 1001 to the square of 1000 is nearly 1002: 1000. The real ratio is 1002·001 1000, in which the antecedent differs from its approximate value 1002 only by one-thousandth part of unity.

383. Hence we may infer that the ratio of the square root of a + 2x to the square root of a is the ratio a +x: a nearly, when x is small compared with a. That is; if the difference of two quantities be small compared with either of them, the ratio of their square roots is nearly obtained by halving this difference.

In the same manner as in Art. 382 it may be shewn when x is small compared with a, that a + 3x a is nearly equal to the ratio (a + x)3 : a3, and a + 4x : a is nearly equal to the ratio (a + x)* : aa.

These results may be generalised by the student when he is acquainted with the Binomial Theorem.

T. A.

14

384. We will place here a theorem respecting ratios which

where p, q, r, n are any quantities whatever.

therefore, p(kb)" + q (kd)" + r (kf)" = pa" + qc” + re" ;

The same mode of demonstration may be applied, and a similar

result obtained, when there are more than three ratios

a с e

b' d' ƒ

given equal. It may be observed that p, q, r, n are not necessarily positive quantities.

As a particular example we may suppose n = 1, then we see

each of these ratios is equal to pa + qc+ re

pb + qd + rf'

EXAMPLES OF RATIO.

1. Write down the duplicate ratio of 2: 3, and the subduplicate ratio of 100 144.

:

2. Write down the ratio which is compounded of the ratios 35 and 7: 9.

3. Two numbers are in the ratio of 2 to 3, and if 9 be added to each they are in the ratio of 3: 4. Find the numbers.

4. Shew that the ratio a b is the double of the ratio a + c : b + c if c be a mean proportional between a and b.

5. There are two roads from A to B, one of them 14 miles longer than the other, and two roads from B to C, one of them 8 miles longer than the other. The distances from A to B and from B to C along the shorter roads are in the ratio of 1 to 2, and the distances along the longer roads are in the ratio of 2 to 3. Determine the distances.

and equal to each of the former; and that each fraction

XXVII. PROPORTION.

385. Four quantities are said to be proportionals when the first is the same multiple, part, or parts, of the second, as the

third is of the fourth; that is, when

a с

=

b d'

the four quantities

a, b, c, d, are called proportionals. This is usually expressed by saying, a is to b as c to d, and is represented thus, a : b:: c : :d, or thus, a b = c : d.

The terms a and d are called the extremes, and b and c the

means.

386. When four quantities are proportionals, the product of the extremes is equal to the product of the means.

Let a, b, c, d be the four quantities; then since they are pro

(Art. 385); and by multiplying both sides of

the equation by bd, we have ad = bc.

387. Hence if the first be to the second as the second to the third, the product of the extremes is equal to the square of the mean.

388. If any three terms in a proportion are given, the fourth may be determined from the equation ad = bc.

389. If the product of two quantities be equal to the product of two others, the four are proportionals; the terms of either product being taken for the means, and the terms of the other product for the extremes.

390. If a b c d and cd::e:ƒ, then

abe: f.

391. If four quantities be proportionals they are proportionals when taken inversely.

392. If four quantities be proportionals they are proportionals when taken alternately.

Unless the four quantities are of the same kind the alternation cannot take place; because this operation supposes the first to be some multiple, part, or parts, of the third. One line may have to another line the same ratio as one weight has to another weight, but there is no relation, with respect to magnitude, between a line and a weight. In such cases, however, if the four quantities be represented by numbers, or by other quantities which are all of the same kind, the alternation may take place.

393. When four quantities are proportionals, the first together with the second is to the second as the third together with the fourth is to the fourth.