EXAMPLE 2. To compute the logarithm of the number 3. Here 6=3, the next less number a=2, and the sum a+5= 5=*, whose squares is 25, to divide by which, always mul tiply by 04. Then the operation is as follows.

Then, because the sum of the logarithms of numbers gives the logarithm of their product, and the difference of the logarithms gives the logarithm of the quotient of the numbers, from the above two logarithms, and the logarithm of 10, which is 1, we may raise a great many logarithms, as in the following examples*

And thus, computing by this general rule, the logarithms to the other prime numbers 7, 11, 13, 17, 19, 23, &c. and then using composition and division, we may easfty find as many logarithms as we please, or may speedily examine any logarithm in the table.*

* Many other ingenious methods of finding the logarithms of numbers, and peculiar artifices for constructing an entire table of them, may be seen in Dr. Hutton's Introduction to his Tables, and Baron Maseres' Scriptores Logarithmici.

DESCRIPTION AND USE OF THE TABLE OF LOGARITHMS.

Integral numbers are supposed to form a geometrical series, increasing from unity toward the left; but decimals are supposed to form a like series, decreasing from unity toward the right, and the indices of their logarithms are negative. Thus, +1 is the logarithm of 10, but—1 is the logarithm of, or '1; and +2 is the logarithm of 100, but —2 is the logarithm of, or '01; and so on.

Hence it appears in general, that all numbers, which consist of the same figures, whether they be integral, or fractional, or mixed, will have the decimal parts of their logarithms the same, differing only in the index, which will be more or less, and positive or negative, according to the place of the first figure of the number. Thus, the logarithm of 2651 be ing 3'4234097, the logarithm of, TV or or too, &c. part of it will be as follows.

1

Hence it appears, that the index, or characteristic, of any logarithm is always less by 1 than the number of integral figures, which the natural number consists of; or it is equal to the distance of the first or left hand figure from the place of units, or first place of integers, whether on the left, or on the right of it; and this index is constantly to be placed on the left of the decimal part of the logarithm.

When there are integers in the given number, the index is always affirmative; but when there are no integers, the index is negative, and it is to be marked by a short line drawn before, or above it. Thus, a number having 1, 2, 3, 4, 5, &c. integral places, the index of its logarithm is 0, 1, 2, 3, 4, &c, or 1 less than the number of those places.

And a decimal fraction, having its first figure in the 1st) 2d, 3d, 4th, &c. place of decimals, has always—1, -2, -3, 4, &c. for the index of its logarithm.

It may also be observed, that though the indices of fractional quantities be negative, yet the decimal parts of their logarithms are always affirmative.

1. To find, in the Table, the Logarithm to any Number.*

1. If the number do not exceed 100O0O, the decimal part of the logarithm is found, by inspection in the table, standing against the given number, in this manner, viz. in most tables, the first four figures of the given number are in the first column of the page, and the fifth figure in the uppermost line of it; then in the angle of meeting are the last four figures of the logarithm, and the first three figures of the same at the beginning of the same line; to which is to be prefixed the proper index.

So the logarithm of 3l'092 is 1*5326525, that is, the decimal 5326525, found in the table, with the index 1 prefixed, because the given number contains two integers.

2. But if the given number contain more than five figures, take out the logarithm of the first five figures by inspection in the table as before, as also the next greater logarithm, subtracting one logarithm from the other, and also one of their corresponding numbers from the other. Then say, As the difference between the two numbers.

Is to the difference of their logarithms,

So is the remaining part of the given number
To the proportional part of the logarithm.

Which part being added to the less logarithm, before taken out, the whole logarithm sought is obtained very nearly.

* The Tables, considered as the best, are those of Gardiner in 4to. first published in the year 1742; of Dr. Hutton, in 8vo. first printed in 1785; of Taylor, in large 4to. published in 1792; and in France, those of Callet, the second edition published in 1795.

EXAMPLE.

To find the logarithm of the number 34'09264.
The log. of 3409200, as before, is 5326525,

gives, with the index, 1'5326606 for the logarithm of 34 '09264.

Or, in the best tables, the proportional part may often be taken out by inspection, by means of the small tables of proportional parts, placed in the margin.

If the number consist both of integers and fractions, or be entirely fractional, find the decimal part of the logarithm, as if all its figures were integral; then this, the proper characteristic being prefixed, will give the logarithm required.

And if the given number be a proper fraction, subtract the logarithm of the denominator from the logarithm of the numerator, and the remainder will be the logarithm sought; which, being that of a decimal fraction, must always have a negative index.

But if it be a mixed number, reduce it to an improper fraction, and find the difference of the logarithms of the numerator and denominator, in the same manner as before.

1