358. In the series а1x + α2x2 + аz 203 +

in inf. such

a value may be given to , that the value of the whole series shall be less than any proposed quantity p.

Let k be the greatest of the coefficients a1, a2, ɑ3, then the whole series is less than

kx+kx2 + kx3 + in inf.

......

...... 9

<k.

hence that which is required is done, if x be such a value that

COR. Hence also in the series a + ɑ1x + ɑ2x2 +

inf. such a value may always be given to a, that the first term is greater than the sum of all the other terms. This value of

359. DEF. If there be a series of magnitudes

are called the measures of the ratios of those magnitudes to 1, or the Logarithms of the magnitudes, for the reason assigned in Art. 219. Thus, the logarithm of any number n, is such a quantity, that a* = n.

Here a may be assumed at pleasure, and is called the base; and for every different value so assumed a different system of logarithms will be formed. In the common Tabular logarithms a is 10, and consequently 0, 1, 2, 3, ..., are the logarithms of 1, 10, 100, 1000, 10 in that system.

...

360. COR. 1. Since the tabular logarithm of 10 is 1, the logarithm of a number between 1 and 10 is less than 1; and, in the same manner, the logarithm of a number between 10 and 100 is between 1 and 2; of a number between 100 and 1000 is between 2 and 3; &c.

These logarithms are also real quantities, to which approximation, sufficiently accurate for all practical purposes, may be made.

Thus, if a be the logarithm of 5, then 10 = 5; let be substituted for a, and 103 is found to be less than 5, therefore is less than the logarithm of 5; but 10 is greater than 5, or is greater than the logarithm of 5; thus it appears that there is a value of x between and 2, such that 10* = 5; the value set down in the Tables is 0.69897, and 10069897 = 5, nearly.

[361. COR. 2. logarithm of 1 is 0. base is always 1.]

Since ao 1, 6o 1, &c., in any system the

=

=

Also since a1a, the logarithm of the

The method of finding the logarithms of the natural numbers, or forming a Table*, is explained in Treatises on Trigonometry. [See Snowball's Trigonometry, 5th Edition, Appendix 1.]

[DEF. If n be any number, log, n signifies the logarithm of n to base a; and log n the logarithm of n to any base.]

362. In the same system the sum of the logarithms of two numbers is the logarithm of their product; and the difference of the logarithms is the logarithm of their quotient.

Let a logan, and y = log, n'; then a* = n, and a = n' ;

=

n

n

x − y is loga; that is, logann' = logan + logan'; and loga

n

Ex. 2.

Log pqr = log pq + logr = log p + log q + logr.

Ex. 5. Log100'06 = log106 log10 103= 0·77815 — 3.

The last two results are usually written I-77815, 3.77815.]

363. If the logarithm of a number be multiplied by n, the product is the logarithm of that number raised to the nth power.

Let N be the number whose logarithm is a, or a* = N; then an*=N"; that is, na is the log. of N", or log,N"=n.logaN. Exs. Log (13)5 = 5 x log 13. Log b* = × × log b.

* [Tables of Logarithms have been lately published in a very cheap and convenient form by Taylor and Walton, London, under the superintendence of the Society for the Diffusion of Useful Knowledge.]

364. If the logarithm of a number be divided by n, the quotient is the logarithm of the nth root of that number.

Let a = N, then an = N or is the log. of N", that

n

365. The utility of a Table of logarithms in arithmetical calculations will from hence be manifest; the multiplication and division of numbers being performed by the addition and subtraction of these artificial representatives; and the involution or evolution of numbers by multiplying or dividing their logarithms by the indices of the powers or roots required.

Ex. 1. Let the value of 2x3 be required.

The log. of the proposed quantity to base 10 is

{log107+ log10 2 + log10 3}.

And by the Tables log10 7 0.845098

=

INTEREST AND ANNUITIES.

366. DEF. Interest is the consideration paid for the use of money which belongs to another. The rate of interest is the consideration paid for the use of a certain sum for a certain time, as of 1£ for one year.

When the interest of the Principal alone, or sum lent, is taken, it is called Simple Interest; but if the interest, as soon as it becomes due, be added to the principal, and interest be charged upon the whole, it is called Compound Interest.

The Amount is the whole sum due at the end of Interest and Principal together.

any time,

Discount is the abatement made for the payment of money before it becomes due.

SIMPLE INTEREST.

367. To find the Amount of a given sum, in any time, at simple interest.

Let P be the principal, in pounds,

n the No. of years for which the interest is to be calculated*.

r the interest of 1£ for one year†,

M the amount.

Then, since the interest of a given sum, at a given rate, must be proportional to the time, 1 (year) : n (years) :: r : nr, the interest of 1£ for n years; and the interest of P£ must be P times as great, or Pnr; therefore the amount M=P+Pnr.

368. From this simple equation, any three of the quantities P, n, r, M being given, the fourth may be found; thus

* [When days, weeks, or months, not making an exact number of years, enter the calculation, n is fractional.

It must always be borne in mind that r is not the rate per cent. but only the hundredth part of it. Thus for 4 per cent. r = 0·04 £, for 5 per cent. r = 0·05£; and so on.]