convenient system is that in which the ratio of the geometrical series is 10; and this is the one in common use. Thus,

0. 1. 2. 3.

4.

5. &c. indices or logar. 1. 10. 100. 1000. 10000. 100000. &c. natural numbers

In this system the log. of 1. is 0, the log. of 10 is 1. the log. of 100, is 2, &c. Hence it is plain that the log. of any number between 1 and 10, will be expressed by a decimal, the log. of any number between 10, and 100, by 1 and a decimal, the log. of any number between 100 and 1000, by 2 and a decimal, &c.

The numbers, 0, 1, 2, 3, &c. that stand before the decimal part of logarithms, are called indices and are always less by unity, than the number of figures in the integral part of the corresponding natural 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 decimal, the index is negative, and is mark ed by a negative sign (-) placed either before or above it. If the first significant figure of the decimal be adjacent to the decimal point, the index is,-1, or 1; if there be one cipher between them, the index is-2, or 2; if there be two ciphers between them, the index is ―3 or 3, &c.

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

The method of finding logarithms in the tables, and of multiplying, dividing, &c. by them, is contained in the following problems.

PROBLEM I.

To find the Logarithm of a given number.

If the given number consists of one or two figures only, find it in the column marked No. in the first page of the table, and against it, in the next column, marked log. is the logarithm. Thus the log. of 7 will be found 0.84510, and the log. of 85 will be found 1.92942.

But if the given number be either wholly or in part decimal, the index must be changed accordingly. Observing that the index must always be one less than the number of figures in the integral part of the given number; also, when the given number is wholly a decimal, the index is negative, and must be one more than the number of the ciphers between the decimal point and first significant figure on the left hand. Thus the log. of .7 is -1.84510, and the log. of .0085 is -3.92942.

If the given number consists of three figures, find it in one of the other pages of the table, in the column marked

part of the logarithm. The index must be placed before it agreeably to the above observation. Thus the log. of 421 is 2.62428, the log. of 4.21 is 0.62428, and the log. of .0421 is -2.62428.

If the given number consists of four figures, find the three left hand figures in the column marked No. as before, and the remaining, or right hand figure at the top of the table; in the column under this figure, and against the other three, is the decimal part of the logarithm. Thus the log. of 5163 is 3.71290, and the log. of .6387 is-1.80530.

If the given number consist of five or six figures, find the logarithm of the four left hand figures as before; then take the difference between this logarithm and the next greater in the table. Multiply this difference by the remaining figure or figures of the given number, and cut off one, or two figures to the right hand of the product, according as the multiplier consists of one, or two figures; then add the remaining figure or figures of the product to the logarithm first taken out of the table, and the sum will be the logarithm required. Thus, let it be required to find the logarithm of 59686; then,

The natural number consisting of five integers, the index must be 4; therefore the log. of 59686 is 4.77587.

Again, let it be required to find the log. of .0131755;

As the given number is a decimal, and has one cipher between the decimal point and first significant figure, the index must be -2; therefore the log. of .0131755 is -2.11977.

PROBLEM II.

To find the natural number corresponding to a given logarithm.

If four figures only be required in the answer, look in the table for the decimal part of the given logarithm, and if it cannot be found exactly, take the one nearest to it, whether greater or less; then the three figures in the first column, marked No. which are in a line with the logarithm found, together with the figure at the top of the table directly above it, will form the number required. Observing, that when the index of the given logarithm is affirmative, the integers in the number found, must be one more than the number expressed by the index; but when the index of the given logarithm is negative, the number found will be wholly a decimal, and must have one cypher less, placed between the decimal point and first significant figure on the left hand, than the number expressed by the index. Thus the natural number corresponding to the logarithm 2.90233 is 798.6, the natural number corresponding to the logarithm 3.77055 is 5896, and the natural number corresponding to the logarithm -3.36361 is .00231.

If the exact logarithm be found in the table, and the figures in the number corresponding do not exceed the index by one, annex ciphers to the right hand till they do. Thus the natural number corresponding to the logarithm 6.64068 is 4372000.

If five or six figures be required in the answer, find, in the table, the logarithm next less than the given one, and take out the four figures answering to it as before. Subtract this logarithm from the next greater in the table, and also from the given logarithm; to the latter