CHAP. XXXVI.

OF THE EXPANSION OF NEGATIVE POWERS.

1

291. Ir has been already shown that =a~1, and it is

a

in like manner be repre

sented by a+b-1; so that the fraction

1

a+b may be considered as a power of a+b, of which the exponent is -1;

and it follows that the formula for expanding at a+b" will apply equally to this case.

292. Considering therefore

as a+b-1, we shall have

a+b

n=-1, and consequently the coefficients will be as follow, viz.

n-3

1;

=-1, &c.

4

3

a+b

+

a2 α

as

a4

a5

b5

a6

+ &c.

a series which exactly corresponds with that which we have

before found by the actual division of 1 by a+b.

being equivalent to a+b-2, may in

like manner be expanded into an infinite series, for n=-2,

1

and

294. In reviewing these different cases, we are enabled to arrive at the following general formula for the expansion of any negative power of a binomial, viz.

nb
an+1+

Adopting this formula for the expansion of a+b we shall be enabled at once to write,

295. By means of this formula we are likewise enabled to expand the negative powers of a binomial, whose exponents are fractional, by substituting such fractions in the place of n, in order to express irrational quantities.

Let

Let it be required, for example, to find the value of

OF ARITHMETICAL RATIO, OR THE DIFFERENCE BETWEEN TWO NUMBERS.

296. Two quantities are either equal to one another, or they are not. In the latter case, where one is greater than another, we may consider their inequality in two different points of view; we may either enquire, how much one of the quantities is greater than the other? Or how many times the one is contained in the other. The results which furnish the answers to these two questions are both termed ratios or relations. The first however is usually termed Arithmetical Ratio, and the second Geometrical Ratio.

It is evident that quantities of which we are now speaking must be those of the same kind, as we could not otherwise determine any thing with respect to their equality or inequality. In the following observations, however, it is of numbers only that we propose to treat.

297. When

297. When of two given numbers, therefore, it is required to know how much the one is greater than the other, the answer to this question determines the arithmetical ratio of those numbers; but since this answer consists in giving the difference of the two numbers, it follows that an arithmetical ratio is nothing more than the difference of two numbers; and as this appears to be the better expression, we shall reserve the terms ratio and relation to express geometrical ratios, of which we shall hereafter treat.

298. Now the difference between two numbers we know to be determined by subtracting the one from the other; and consequently, when the difference between 5 and 3 is sought, we are immediately enabled to answer that the difference is 2.

299. We have, therefore, here three things to consider; first, the greater of two given numbers; 2dly, the less; and 3dly, the difference and these three quantities are so connected, that any two being given, the third may always be determined from them.

Let a be the greater number, b the less, and d the difference, we shall then find the difference d by subtracting b from a, so that da-b, from which we perceive that a and b being given, d may be found.

Again, if the difference and the lesser number be given to find the greater, then will the greater be found by adding together the difference and the lesser number, that is to say, a=b+d.

Lastly, if the difference d and the greater number a be given, b will be found by subtracting the difference d from the greater number a, so that ba-d. Hence we have the three following equations, da−b; a=b+d; b=a-d.

300. Now it must be observed, with respect to arithmetical ratios, that if to the two numbers a and b, any other number

C be

▸ be added or subtracted, the difference will remain the same; that is to say, that if d be the difference between a and b, it will also be the difference between a + c and b+c, or between a-c and b-c. For instance, the difference between 20 and 12 being 8, this difference will be the same whatever number be added to or subtracted from 12 and 20. If 5 be added to them they will become 17 and 25, and if 5 be subtracted from them they will become 7 and 15, but 8 is still the difference between them. The proof of this is evident, for if a-b-d, then will (a+c) — (b+c)=d,

or, (a-c)-(b-c)=d.

301. If the numbers be doubled, the difference will also be doubled; for if a-b=d, then will 2a-2b-2d, and in general na-nbnd, whatever number n may be.

CHAP. XXXVIII.

OF ARITHMETICAL PROPORTION.

302. WHEN two arithmetical ratios are equal, this equality is termed an arithmetical proportion. Thus when a-b-d, and p-q=d, so that the difference between p and q is equal to the difference between a and b, these numbers are said to form an arithmetical proportion, which we express thus:

a-b=p-q.

303. An arithmetical proportion therefore consists of four terms, which must be such that if the second be subtracted from the first, the remainder may be equal to that which is. obtained by subtracting the fourth from the third: thus the four numbers, 12, 7, 9, 4 form an arithmetical proportion because 12-7 = 9−4.

304. In