Now if x be small when compared with a,

x

is a small fraction;

: 1, therefore since is small compared with 1,

is small compared with

a2

is very small compared with 1

a

x

3x2

a

X3

3x

1 ± : 1, or a± 3x: a, is a near approximation to (a ±x)3 : a3, if x be

a

small when compared with a.

Similarly it may be shewn that a± 4x : a; a± 5x : a; &c. are approximations respectively to (ax)* : a',

Again, since √ax: √a = √1 +2:

The utility of the rules here proved will be sufficiently manifest from the following Examples, when it is observed by what a troublesome process the several proposed ratios would be found without the rules.

Ex. 1. (1·5241)1: (1·524)* = 1·5240 + 4 × 0·0001: 1·524 nearly 1.5244 : 1·524 nearly.

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

third is of the fourth; that is, when

a, b, c, d, are called proportionals. This is usually expressed by

saying a is to b as c is to d, and is thus represented, a b c d; or, sometimes, a : b = c : d.

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

means.

237. 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

portionals,

a с

=

b d

(Art. 236); and by multiplying both sides of the equation by bd, ad = bc.

238. COR. 1. 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.

239. COR. 2. Any three terms in a proportion being given, the fourth may be determined from the equation ad = bc; for d=

bc

=

a

Hence we have the Single Rule of Three in

240. If the product of two quantities be equal to the product of two others, the four are proportionals, making the terms of one product the means, and the terms of the other the extremes.

241. If a b:: c: d, and c d :: e: f, then also a : b :: e : f. (EUCLID, B. v. Prop. x1.)

242. If four quantities be proportionals, they are also proportionals when taken inversely. (EUCLID, B. v. Prop. B.)

If a b c d, then bad c. For

- =

and di

viding unity by each of these equal quantities, or taking their

243. If four quantities be proportionals, they are proportionals when taken alternately. (EUCLID, B. v. Prop. xvI.)

If a b c d, then a cb d.

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 that one weight has to another weight, but a line has no relation in respect of magnitude to a weight. In cases of this kind, if the four quantities be represented by numbers, or other quantities which are similar, the alternation may take place, and the conclusions drawn from it will be just.

244. 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. This operation is called componendo.

245. Also, dividendo, the excess of the first above the second is to the second, as the excess of the third above the fourth is to the fourth. (EUCLID, B. v. Prop. xv11.)

246. Again, convertendo, the first is to its excess above the second, as the third to its excess above the fourth. (EUCLID,

B. v. Prop. E.)

247. When four quantities are proportionals, the sum of the first and second is to their difference, as the sum of the third and fourth is to their difference.

If a b :: cd; then a + b: a

bc+d: c-d.

b c d c-d.

248.

When any number of quantities are proportionals, as

one antecedent is to its consequent, so is the sum of all the antecedents to the sum of all the consequents. (EUCLID, B. v. Prop. XII.)

then ab: a + c + e + &c. : b+d+f + &c.

abba; hence ab + ad + af = ba + bc + be,

.. by Art. 240, a b :: a + c + e : b + d +f;

and similarly when more quantities are taken.

249. When four quantities are proportionals, if the first and second be multiplied, or divided, by any quantity, as also the third and fourth, the resulting quantities will be proportionals.

250. If the first and third be multiplied, or divided, by any quantity, and also the second and fourth, the resulting quantities

251.

COR. Hence, in any proportion, if instead of the second and fourth terms quantities proportional to them be substituted, we