Hence, the preceding series becomes l'. (1+

or

2

2z+1 3 (2z+1)3 5 (2z+1)5

+...).

This series gives the difference between the logarithms of two consecutive numbers, and converges more rapidly than series (6) Making successively, z= 1, 2, 3, 4, 5..., we find

whence we see, that knowing the logarithm of 100, the first term of the series is sufficient for obtaining that of 101 to seven places of decimals.

There are formulas more converging than the above, from which we may obtain a series of logarithms in functions of others already known, but the preceding are sufficient to give an idea of the facility with which tables may be constructed. We may now suppose the Naperian logarithms of all numbers to be known.

The Naperian logarithm of 10 may be deduced from the first. and fourth of the above equations, by simply adding the logarithm of 2 to that of 5 (Art. 258). This number has been calculated with great exactness, and is 2.302585093.

271. We have already observed that the base of the common system of logarithms is 10 (Art. 257). We will now find its modulus. We have,

1. (1+ y): 1. (1+ y) : : 1 : A (Art. 267).
l'.

If now, we multiply the Naperian logarithms before found, by this modulus, we shall obtain a table of common logarithms (Art. 268).

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. 267),

l'.el.e: : 1 : 0.434294482.

But l'e1 (Art. 263): hence,

hence,

1. : l.e : : 1 : 0.434294482,

1.e 0.434294482.

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 logarithm of the Naperian base

Of Interpolation.

272. A table of logarithms is a tabulated series of numbers, showing the value of x in the equation

corresponding to all the integral values of N, between 1 and some higher number which marks the limit of the table. It has already been remarked that in the system in common use, the value of the base а, is 10.

And generally, any mathematical table consists of a series of values of some letter in an algebraic expression, corresponding to equi-dis tant values of the function on which it depends.

The principle of interpolation, which is of great value in prac tical science, has for its object to find from the tabulated numbers

which are given, other similar numbers which shall correspond to intermediate values of the function. For example, suppose p, q, r, s, &c., to be a series of tabulated numbers corresponding to, and written opposite the functions a, a + b, a + 2b, a + 3b, &c., and it were required to find the tabulated number corresponding to the function a+ 26. This is a question of interpolation, and is resolved by taking the successive differences of the tabulated numbers, thus:

From the above equations, we have

q=p dp, r = q + dq = p + dp + dq=p +2dp + d2p;

and by a similar process, we have

s=p+3dp + 3d2p + d3p,

t = p + 4dp + 6d2p + 4d3p + d1p,

in which notation it should be observed, that d2, d3, &c., denote the second, third, &c. differences of the successive tabulated numbers

It is plain, that the above law from which the numerical coefficients for any term may be derived, is similar to that for the co-efficients of a binomial: hence, if T denote the n+1 term of the tabulated numbers, reckoning from p inclusive, we shall have n. (n − 1) (n − 2)

T=p+ndp +

-

n (n
1. 2

1)

d2p +

1.

2. 3

d3p + &c.

to

Let it be required to find the tabulated number corresponding a + 3b. We then have, n = 3: hence,

Tp+3dp + 3d2p + d3p,

the same value as that found above for r.

to the functions a + 3b.

49

Next, let it be required to find the tabulated value answering 3 Then, n = and if we know the tabulated number p, and the successive differences d, d2, &c., the approximate value of T can easily be found.

It is plain from the series that the interpolated values are but approximations, since no order of difference can reduce to zero, and hence, the series will contain an infinite number of terms. Generally, however, the tabulated values are themselves but approximations, and the successive differences decrease so rapidly in value, that the series becomes very converging.

Let us suppose for example, that we have the logarithms of 12, 13, 14, 15, &c., and that it is required to find the logarithm of 12 and a half. Then,

INTEREST.

273. The solution of all questions relating to interest, may be greatly simplified by employing the algebraic formulas.

In treating of this subject, we shall employ the following no

tation :

Let

p=

r

the amount bearing interest, called the principal;

the part of $1, which expresses its interest for one year, called the rate per cent.;

t the time that p draws interest;

= the interest of p dollars for t years;

Sp

the interest which accrues in the time t, which 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 t times as inuch: hence,

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,