EXPLANATION OF THE TABLES.

TABLE I.

COMMON LOGARITHMS.

N.B. The meaning and properties of logarithms are explained in works on algebra.

1. The first page of the table gives the characteristics and mantissas of numbers from 1 up to 100. The remainder of the table gives only mantissas. The characteristics are obtained by the following rule, which is deduced in algebra:

*

When the number is greater than 1, the characteristic is positive, and is one less than the number of figures to the left of the decimal point; when the number is less than 1, the characteristic is negative, and is one more than the number of zeros between the decimal point and the first significant figure.

The first three figures of a number of four figures are found in the left-hand column marked N; the fourth figure of the number is found in the lines at the top and the foot of the page. The last three figures of the mantissa are found in the same line as the first three figures of the number, and in the same column as the fourth figure of the number. The first two figures of the mantissa are in the column headed 0, and are printed only once. They are found either in the same line as the last three figures, or in the first line above which contains a whole mantissa. If, however, a precedes the last three figures of the mantissa, the first two figures are found in the following line.

*

*This rule may be easily deduced in arithmetic.

2. To find the logarithm of a number.

RULE: Write the characteristic, and then annex the mantissa found by means of the table.

(a) A number of four figures.

=

log 3552 = 3.55047; log 355.7 2.55108; log 35.74 = 1.55315; log 36.34 1.56038; log 536.2=2.72933; log 5.371 = 0.73006.

=

(b) A number of less than four figures. In this case, annex ciphers, or suppose them to be annexed, and proceed as in case (a). log.213 = 1.32838; log 47.6 = 1.67761; log .0375 = 2.57403.

(c) A number of more than four figures.

To find log 47653. The characteristic is 4. The mantissa, as shown in algebra, is the same as the mantissa of log 4765.3. Log 4765.3 lies between log 4765 and log 4766. Hence the mantissa of log 4765.3 is between the mantissas of log 4765 and log 4766. It is assumed that the change in the mantissa is proportional to the change in the number, as the latter increases from 4765 to 4766; that is,

mantissa of log 4765.3 = mantissa of log 4765 + .3

× (mantissa of log 4766 — mantissa of log 4765).*

NOTE 1. By general agreement, a number with six or more decimal places is reduced to a number with five in the following way:

If a number less than 5 is in the sixth decimal place, then the number in the fifth place is left unchanged; if a number greater than 5 is in the sixth place, or if there is a 5 in the sixth place and it is followed by figures other

*It is assumed that when a number varies from one value to another, the change in the mantissa is proportional to the change in the number if the latter change is small in comparison with the number. This is not strictly correct, but is accurate enough for practical purposes.

than zeros only, then the number in the fifth place is increased by unity; if the number in the sixth place is 5 and it is followed by zeros only, then an even number in the fifth place is left unchanged, and an odd number in the fifth place is increased by unity.

NOTE 2. The difference between the mantissas for two consecutive numbers of four figures is called their tabular difference, and is printed in the column marked D. At the lower parts of the first three pages of the table the tabular differences for the mantissas on these pages are multiplied by the nine digits expressed as tenths. The results, which are called proportional parts, are the amounts to be added in obtaining the logarithms of five-figure numbers. It is better for the beginner in logarithmic computation to find the tabular differences by subtraction, and make the calculations for himself. The process described above for finding the logarithms of numbers of five or more figures, is called interpolation.

To find log 476.532.

log 476.5

difference for .32 = .32 × 9=

2.67806

288

..log 476.5322.67809 [See Ex. above.]

NOTE 3. A five-place table of logarithms is not used, in general, with numbers of more than five figures. In numbers having more than five figures the digits beyond the fifth have little effect on logarithms that are calculated no farther than to five places of decimals.

RULE: Find the mantissa corresponding to the first four figures of the number; multiply the tabular difference at that place in the table by the fifth and following figures treated as a decimal; and add the product to the mantissa just found.

NOTE 4. The logarithm of the reciprocal of a number is called the cologarithm of the number, or the arithmetical complement of the logarithm of the number. For instance, log=colog 325. Now

log = log 1-log 3250 -2.51188 (10-2.51188) 10 = 7.48812 - 10.

=

Thus the cologarithm of a number is equal to the negative logarithm of the number. The cologarithm can be written directly from the logarithm in the table. The use of cologarithms sometimes helps in computation. For 23.41 x 375 log log 23.41 + log 375 +colog 92.83. 92.83

example,

=

3. To find the number corresponding to a given logarithm. This operation is the reverse of the preceding. The position of the decimal point in the required number is shown by the characteristic. The number of figures before the decimal point is one more than the characteristic when the latter is positive; when the characteristic is negative the number is a decimal, and the number of ciphers between the decimal point and the first significant digit is one less than the figure in the characteristic. (See the rule for finding the characteristic.)

The sequence of figures in the number is found from the mantissa. (a) When the given mantissa is in the tables. The first two figures of the mantissa will be found in the column headed 0; the last three figures will be found in the same line as the first two, or in the line above (where it will be preceded by *), or in one of the lines following. The first three figures of the number are in the column headed N, and are in the same line as the last three figures of the mantissa; the fourth figure of the number is at the top of the page in the same column as the last three figures of the mantissa.

To find the number whose logarithm is 2.55047. On turning in the table to the mantissa 55047 it is found that the corresponding sequence of figures is 3552. The characteristic 2 shows that the required number is 355.2. The number having 2.55047 for its logarithm is .03552.

Given log N=5.67815, find N. The sequence of figures in the required number, as found on turning in the table to the mantissa 67815, is 4766. The characteristic 5 shows that the required number is 476600. The number having 1.67815 for its logarithm is .4766.

(b) When the given mantissa is not in the tables. In this case the process of interpolation is employed.

To find the number whose logarithm is 2.57072. Inspection of the table shows that the given mantissa lies between the tabu

lated mantissas, 57066 and 57078. Hence the required number

If 12 is the difference for 1, for what is 6 the difference? Obviously for of 1, i.e. .5. Hence the required number is 372.15.

12

RULE: Find the number corresponding to the mantissa in the table next less than the given mantissa; find the difference between these mantissas; divide this difference by the tabular difference; and annex the quotient to the four figures already found.

TABLE II.

LOGARITHMS OF CERTAIN TRIGONOMETRIC RATIOS.

4. The numbers given in this table are sometimes called logarithmic sines, logarithmic cosines, etc., or the tabular logarithms of the sines, cosines, etc. These terms are considered necessary because these numbers, with the exception of those in one column on each page, are not the logarithms of the sines, cosines, etc., but are these logarithms increased by 10. Hence, in working examples these numbers should be diminished by 10. In the column headed L. Cot., however, the logarithms are given correctly.

The degrees from 0° to 44° are given at the top of the page, and the minutes to be taken with any of these degrees are given from O down to 60 in the column on the left. The degrees from 45° to 89° are given at the foot of the page, and the minutes to be taken with any of these degrees are given from 0 up to 60 in the column on the right. For the degrees printed at the top of the page the contents of the columns are indicated at the top of the page; for the degrees printed at the foot of the page the contents of the columns are indicated at the foot of the page. A ratio is printed at the top of each column (excepting the columns for minutes), and the corresponding co-ratio is at the foot. This convenient arrangement of the table is possible, because, as shown