Logarithms may be defined to be a series of quantities expressing the measures of the ratios of another series of quantities; or, they are the exponents of a series of numbers in geometrical progression. Or, logarithms may be conceived to be a series of numbers so contrived, that the sum of the logarithms of any two numbers, is the logarithm of the product of these numbers. Hence, it is inferred, that if a rank, or series of numbers in arithmetical progression, be adapted to a series of numbers in geometrical progression, any term in the arithmetical progression will be the logarithm of the corre sponding term in the geometrical progression.

Let, therefore, the following be a series of numbers in geometrical

d

2d 3d 4d 5d

6d 7d

progression, namely, 1, r, T, T, r, r, ", r, &c. and the Od 2d 3d 4d 5d 6d 7d, &c. will be

exponents, or

the corresponding terms each of each.

By a comparison of the several terms of these serieses, it is evident that 2d+5d or 7d, answers to the logarithm of r24X75d or î‍ď. And 7d+2d, or 5d, answers to, or rd Also

Ar2d Xr4d = √ röd = r3d, &c.

2d+4d
2

or 3d, answers to

And hence all the various rules for performing multiplication, division, proportion, involution, evolution, &c. by logarithms are obvious.

Sincer and d may be expounded by any numbers whatever, it is hence, evident, there may be an infinite variety of different kinds of logarithms. Let 10 be substituted for r, and 9 for d, and the serieses will become;

1, 10, 100, 1000, 10000, 100000, 1000000,
1, 2,

3,

4,

5,

6,

10000000

7, &c.

These are the logarithms in common use, and are called common logarithms, in order to distinguish them from other kinds of logarithms.

Hence, in this form of logarithms, the logarithm of 1 is 0, the log. of 10 is 1, the log. of 100 is 2, the log. of 1000 is 1, &c. Whence, the logarithm of any term between 1 and 10, being greater than O, but less than 1, is a proper fraction, and is expressed decimally.The logarithm of each term between 10 and 100 is 1, with a decimal annexed; the logarithm of each term between 100 and 1000 is 2, with a decimal annexed, and so on. The integral part of the logarithm is called the index, or characteristic, and the other the decimal part. The index is, evidently, always one less than the number of figures in the natural number, exclusive of fractions, if there are any in that number.

The index of the logarithm of a number, consisting in whole, or in part of integers, is affirmative; but if the number be a proper

fraction,

fraction, the index is negative, and is usually marked by the sign (-) placed either before or above the index. If the first effective figure of the decimal fraction be adjacent to the decimal point, the index is 1, or 1; or if there is one cypher between them, the index is 2 or 2; if two cyphers, the index is 3 or 3. Instead of negative indices, the arithmetical complements are used by many; as by this means the computations are rendered easier, especially to those unacquainted with the first principles of algebra.

The decimal parts of the logarithms of numbers, consisting of the same figures, are the same, whether the number be integral, fractional, or mixed. This is illustrated as follows:

This table contains the logarithms of all numbers under 10000, to six decimal places; being sufficiently accurate for the purpose to which it is principally intended; namely, for reducing the apparent distance between the Moon and the Sun or a fixed star, to the true distance, by the method given in Prob. 1, page 150.

In the left hand column, and in the upper horizontal row, are the natural numbers proceeding in a regular order. In the body of the table are the corresponding logarithms; and in the right hand column, under Diff. is the difference between two adjacent logarithms, opposite to which it is placed. As the index is not prefixed, it must, therefore, be supplied; which is easily done, being always one less than the number of figures in the corresponding natural number.

The logarithm answering to any given number consisting of four figures, or under, and conversely, is found directly by the table; but, because, in the first method of reducing the apparent to the true distance, the logarithm of a natural number consisting of more than four figures is frequently wanted, and conversely, the two following problems, therefore, become necessary.

PROBLEM I.

Given a Number, consisting of more than four Figures, to find the corresponding Logarithm.

RULE.

Multiply the number, from the right hand column, marked Diff. opposite to the three first figures of the given number, by the unit figure of the given natural number, if it consists of five figures; by the two right hand figures, if it consists of six; by the three right hand figures, if it consists of seven, &c.; then point off as many figures to the right hand of the product, as there are in the multiplier; and the remainder being added to the logarithm answering to the first four figures of the natural number, will be the required logarithm nearly. If the figures pointed off to the right hand exceed a half, the unit figure of the remainder is to be increased by 1.

To find the natural Number answering to a given Logarithm.

RULE.

Find the next less logarithm answering to that given in the column marked O at the top, and continue the sight along the horizontal line, and a logarithm, either the same as that given, or very near it, will be found; then the three first figures of the corresponding natural number will be found opposite thereto in the side column, and the fourth figure immediately above it, at the top of the page. If the index of the given logarithm is 3, the four figures thus found are integers; if the index is 2, the three first figures are integers, and the fourth is a decimal, and so on.

If the given logarithm cannot be exactly found in the table, and if more than four figures be wanted in the corresponding natural number, then find the difference between the given, and next less logarithm, to which, annex as many cyphers as there are figures above four, required in the natural number; which being divided by the tabular difference, and the quotient annexed to the four figures formerly found, will give the required natural number.

EXAMPLE.

Required the natural number answering to the logarithm 5.890197?

The next less logarithm in the table is 890086, answering to the number 7794; the difference between which and the given logarithm is 21, to which two cyphers being annexed, because six places of figures are wanted in the natural number, gives 2100; now, this being divided by the tabular difference 56, the quotient is 37; which, an nexed to 7764, gives 776437, the natural number required.

PROBLEM III.

To perform Multiplication by Logarithms.

RULE.

Add the logarithms of the factors, and the sum will be the logarithm of the product.

If there are negative and affirmative indices, the difference is to be taken; or, rather use the arithmetical complements of the negative indices, and reject tens from the sum of these indices.

From the logarithm of the dividend subtract the logarithm of the divisor, the remainder will be the logarithm of the quotient. If any of the quantities is a decimal, or a mixed number, either the negative index of that quantity, or its arithmetical compliment, is to be used.

Otherwise, if one or both of the given terms are decimals, remove the decimal points till the factors contain whole numbers, and the dividend the greatest; then, if the dividend be more places removed than the divisor, remove the decimal point of the quotient as many places to the left hand; but if the divisor be more places removed, then remove the decimal point of the quotient as many places to the right hand. If the dividend and divisor be equally removed, the quotient is not to be altered.