INTRODUCTION.

OF THE USE OF THE TABLES.

In the table of the logarithms of numbers all the logarithms are supposed to have the decimal point prefixed to them, and the corresponding natural numbers may be either integers, or decimals, or mixed numbers; the decimal part of the logarithm being the same for all numbers expressed by the same significant digits placed in the same order.

The characteristic or index of a logarithm is a whole number placed before the decimal point, for the purpose of characterising the natural number, or indicating the denominations of its several digits, which it does by expressing the distance of its first or left hand figure from the units' place.

The characteristic may be positive or negative, and must be supplied according to the following rules:

1. The characteristic of the logarithm of a number greater than unity is positive, and equal to one less than the number of integer places.

2. The characteristic of the logarithm of a number less than unity is negative, and equal to one added to the number of zeros following the decimal point.

The decimal part of the logarithm being always positive, it is usual to distinguish a negative characteristic by putting the sign — over it instead of before it. Thus the logarithm of '0692 is written 2·840106, instead of — 2·840106.

Instead of employing negative characteristics it is generally more convenient to use their arithmetical complements to 10; thus, instead of the three last logarithms it is more usual to write 9.570426, 8.570426, 7.570426 respectively. This is equivalent to adding 10 to the characteristic, and consequently when a characteristic is thus written, it is necessary, at some subsequent stage of the computation, to diminish the characteristic by the ten (or tens) supposed to have been previously added.

PROBLEM I. To find the logarithm of a given number.

Case 1. If the given number consists of four figures, look for the three first figures of the number in the column of numbers headed N., and for the last figure in the line at the top or foot of the page. Then the two first figures of the logarithm will be found in the column headed O, either in the same line with the three first figures of the number, or in a line immediately above, and the four last figures in the column at the head of which the last figure of the number stands, and in the same line with the three first figures of the number. To this must be prefixed the characteristic according to the preceding rules. For example, let the proposed number be

5327; the decimal part of the logarithm found in the table is 726483, to which prefixing the characteristic 2, the complete logarithm is 2.726483.

If the proposed number consists of only three figures, a zero preceded by the decimal point may be supposed to be placed after it; and if it consist of only two figures, its logarithm will be found in the table on the first

page.

PROBLEM II. To find the logarithm of a number consisting of five or six figures.

Find the logarithm of the four first figures as directed in the last problem; multiply the difference which stands in the same line in the last column (headed D.) of the page by the fifth figure, and add the tenth part of the product to the logarithm of the four first figures; multiply the same difference by the sixth figure, and add also the 100th part of the product, and to the sum thus obtained prefix the proper characteristic.

In taking the 10th or 100th part of the product of the difference by the given figure, if the first rejected figure is 5 or a higher number, care must be taken to add 1 to the preceding figure.

Example 1. Required the logarithm of 13647.

Towards the latter end of the table, the last figure of the

logarithm will sometimes differ by unity (in excess or defect) from its true value, when the natural number consists of six figures. In the present case, the true logarithm, to seven places, is 2.9303182.

PROBLEM III. To find the number corresponding to a given logarithm.

Find the two first figures of the proposed logarithm in the second column, headed O, and look for the remaining four figures in the same line, or one of the lines immediately following. If the logarithm is found exactly, then the three first figures of the number sought will be found in the same line in the first column, headed N., and the fourth figure at the head or foot of the column in which the logarithm was found. But if the proposed logarithm is not found in the table, then take the number corresponding to the tabular logarithm next less, and in order to find the fifth and sixth figures proceed as follows:

From the given logarithm subtract the tabular logarithm, which is next less; add two zeros to the remainder, divide the remainder thus increased by the number which stands in the same line in the column of differences, and annex the quotient to the number corresponding to the next less logarithm in the table.

Example. Required the number corresponding to the logarithm 2.148179.

PROBLEM IV. To multiply together two or more numbers. Add together the logarithms of the several numbers, and the sum will be the logarithm of their product.

Example 1. Required the product of 74, 119, and 7.236.

Which is true to 5 places. The exact product is 63720.216. Example 2. Required the product of 1.4735, 0.82561, and 0.002673.

Here 10 was added to the characteristic both of the second and third factor; therefore, subtracting 20 from the sum 17 (the 7 only is written), the characteristic becomes 3, or its complement +7, indicating that the first significant figure must occupy the third place to the right of the decimal point.

PROBLEM V. To divide one number by another.

Subtract the logarithm of the divisor from that of the dividend, and the remainder is the logarithm of the