more rapidly converging than the one above given, but this serves to show the facility with which logarithms may be computed.

241*. We have already observed, that the base of the common system of logarithms is 10. We will now find its modulus.

We have,

(1+ y): log (1+ y) : : 1 : M (Art. 238).

If we make y9, we shall have,

710: log 10 : : 1: M.

But the 710 2.302585093, and log 101 (Art. 228);

1

hence, M =

= 0.434294482 the modulus of the

2.302585093

common system.

If now, we multiply the Naperian logarithms before found, by this modulus, we shall obtain a table of common logarithms (Art. 238).

All that now remains to be done, is to find the base of the Naperian system. If we designate that base by e, we shall have (Art. 237),

But as we have already explained the method of calculating the common tables, we may use them to find the number whose logarithm is 0.434294482, which we shall find to be 2.718281828; hence,

e 2.718281828......

We see from the last equation but one, that

The modulus of the common system is equal to the common loga rithm of the Naperian base.

Of Interpolation.

242. When the law of a series is given, and several terms taken at equal distances are known, we may, by means of

already deduced, (Art. 209), introduce other terms between them, which terms shall conform to the law of the series This operation is called interpolation.

In most cases, the law of the series is not given, but only numerical values of certain terms of the series, taken at fixed intervals; in this case we can only approximate to the law of the series, or to the value of any intermediate term, by the aid of formula (1).

To illustrate the use of formula (1) in interpolating a term in a tabulated series of numbers, let us suppose that we have the logarithms of 12, 13, 14, 15, and that it is required to find the logarithm of 124. Forming the orders of differences from the logarithms of 12, 13, 14 and 15 respectively, and taking the first terms of each,

we find

d1

=

+ 0.000355,

0.034762, d=0.002577, d=0.000355.

If we consider log 12 as the first term, we have also

Making these several substitutions in the formula, and neglecting the terms after the fourth, since they are inappreciable,

or, by substituting for d, da, &c., their values, and for a its

Had it been required to find the logarithm of 12.39, we should have made n = .39, and the process would have been the same as.above. In like manner we may interpolate terms between the tabulated terms of any mathematical table.

INTEREST.

243. The solution of all problems relating to interest, may be greatly simplified by employing algebraic formulas.

In treating of this subject, we shall employ the following notation:

66

Let p denote the amount bearing interest, called the principal; the part of $1, which expresses its interest for one year, called the rate per cent.;

the time, in years, that p draws interest;

the interest of p dollars for t years;

p

the interest which accrues in the time t. This sum is called the amount.

Simple Interest.

To find the interest of a sum p for t years, at the rate r, and the amount then due.

Sincer denotes the part of a dollar which expresses its interest for a single year, the interest of p dollars for the same

time will be expressed by pr; and for t years it will be times

1. What is the interest, and what the amount of $365 for three years and a half, at the rate of 4 per cent. per annum. Here,

The present value of any sum S, due t years hence, is the prin cipal p, which put at interest for the time t, will produce the amount S.

The discount on any sum due t years hence, is the difference between that sum and the present value.

To find the present value of a sum of dollars denoted by S, due t years hence, at simple interest, at the rate r; also, the discount.

and for the discount, which we will denote by D, we have

1. Required the discount on $100, due 3 months hence, at the rate of 5 per cent. per annum.

Compound interest is when the interest on a sum of money becoming due, and not paid, is added to the principal, and the interest then calculated on this amount as on a new principal.

To find the amount of a sum p placed at interest for t years, compound interest being allowed annually at the rate r.

At the end of one year the amount will be,

Sp+pr = p(1 + r).

Since compound interest is allowed, this sum now becomes the principal, and hence, at the end of the second year, the amount will be,

S′ = p(1 + r) + pr(1 + r) = p(1 + r)2.

Regard p(1+r)2 as a new principal; we have, at the end of the third year,

S'' = p(1+r)2+ pr(1 + r)2= p(1 + r)3 ;