The advantages in using the base 10 are that the characteristic can be determined by inspection, and that the mantissa remains unchanged for the same sequence of figures.

From the

8. Rules for determining the characteristic.

foregoing considerations the following rules for determining the characteristic are evident:

(1) When the number is greater than 1, the characteristic is positive, and is one less than the number of digits to the left of the decimal point.

(2) When the number is less than 1 and expressed decimally the characteristic is negative, and is one more than the number of zeros immediately at the right of the decimal point.

When the characteristic is negative the minus sign is placed above the characteristic to show that it alone is negative. Thus, in log 0.009347 3.97067, the 3.97067 means -3+0.97067. It should not be written -3.97067 for then the minus sign would indicate that both characteristic and mantissa were negative, while we have agreed that the mantissa shall always be considered positive.

In computations involving negative characteristics, to avoid the use of the negative, 10 is usually added to the characteristic and subtracted at the right of the mantissa. In writing logarithms in this form, the characteristic, when 10 is added, is 9 minus the number of zeros immediately at the right of the decimal point. Thus, in the above, log 0.009347 7.97067 - 10. The characteristics of the following are as given: of 3426 is 3 by rule (1), of 3.2364 is 0 by rule (1), of 0.00639 is -3, or 7 by rule (2), of 2.04 is 0 by rule (1), of 0.000067 is -5, or 5

=

10,

10,

by rule (2).

In each of the following state the characteristic to the base 10:

9. The mantissa. The determination of the decimal part of a logarithm, the mantissa, is more difficult than the determination of the characteristic. Because of this difficulty the mantissas have been carefully determined and arranged in tables of logarithms.* They are given to three, four, five, or more places of decimals.

The degree of accuracy in computations made by logarithms depends upon the number of places in the table used; the more places in the table the greater the degree of accuracy. The tables generally used are those having from four to seven places.

10. Tables. Upon examining a five place table of logarithms (see Table I), it is noticed that the first column has the letter N at the top and the bottom. This is an abbreviation for number. The other columns have at top and bottom the numbers 0, 1, 2, 3. ... 9. Table I contains the integers from 1000 to 11,009. Pages 32 to 49 have the numbers from 1000 to 10,009. Here the first three figures are printed in the column marked N and the fourth figure at the top and bottom of another column. Thus, to locate 4756, 475 is found in the N-column on page 39 and 6 in the column headed 6. Pages 50 and 51 contain the numbers from 10,000 to 11,009, where the first four figures are printed in the N-column.

The columns of numbers after the first consist of the mantissas of the numbers located in the N-column and at the top, or bottom, of another column. These mantissas are printed correct to five decimals except on pages 50 and 51, where they are given to seven places. To save space, the first two figures of the mantissas are printed in the 0-column only. Any such two figures go with the other figures to the right and below until another two figures is found in the O-column. Except that when an asterisk (*) is found

* Professor Briggs' tables were computed to fourteen places, but were not finished by him. They were completed by Adrian Vlacq (1628), who shortened them to ten places, and finished a table including the numbers from 1 to 100,000.

Briggs' and Vlacq's tables are essentially the same as those in use now. They have been checked and re-computed in part many times. At present, errors found in tables are typographical. The most complete check was undertaken by the French authorities in 1784. It required the labors of nearly one hundred mathematicians and computers for over two years. They computed to fourteen places the logarithms of all integers from 1 to 200,000, besides natural and logarithmic trigonometric functions. These tables were never printed. Two manuscript copies are preserved.

before the three figures given in the other columns, the first two figures of the mantissa are taken from the next line below.

When a mantissa ends in a figure 5 it is printed 5 when it is really less than printed; otherwise a mantissa when ending in 5 is larger than printed.

Thus, if the mantissa is 0.0273496, in contracting it to five places, it is printed 0.02735. This is to guide one wishing to write the mantissas correct to four places.

For the meaning of the Prop. Parts, see Art. 21. For the meaning of the numbers at the foot of the pages and connected with S and T, see Art. 30.

Notice that when advancing in the table, the mantissas increase. The difference between two consecutive mantissas is called the tabular difference.

11. To find the mantissa of the logarithm of a number. - Use Table I, pages 32 to 49.

(1) When the number consists of four significant figures.

Example. Find the mantissa of log 4673.

Find the first three figures, 467, of the number in the N-column and the 3 at the top of the page. The mantissa of log 4673 is found to the right of 467 and in the column headed 3.

(2) When the number consists of one, two, or three significant figures. The number is found in the N-column and the mantissa to the right in the 0-column.

Thus,

Mant. of log 4.78 Mant. of log 4780 = 0.67943.

=

Mant. of log 39 Mant. of log 3900 = 0.59106.
Mant. of log 4 = Mant. of log 4000 0.60206.

(3) When the number consists of five or more significant figures. Example 1. Find the mantissa of log 39,467.

Since 39,467 lies between 39,460 and 39,470 its mantissa must lie between the mantissas of these numbers.

The difference between these mantissas is 0.00011, which is the tabular difference. Since an increase of 10 in the number increases the mantissa 0.00011, an increase of 7 in the number will increase the mantissa 0.7 as much, or the increase is

The process of finding the mantissa as above is called interpolation. As carried out, it is assumed that the increase of the logarithm is proportional to the increase of the number. This assumption is not strictly true as will be seen in Art. 35. Example 2. Find the mantissa of log 792,836.

Since an increase of 100 in the number increases the mantissa 0.00006, an increase of 36 in the number increases the mantissa

correct to the nearest fifth decimal place.

.. Mant. of log 792,836 = 0.89916 +0.00002 = 0.89918.

These processes should seem reasonable; but, since they are to be performed so frequently, it is best to work by rule.

12. Rules for finding the mantissa. (1) For a number consisting of four figures, find the first three figures of the number in the N-column and the fourth figure at the head of a column; then read the mantissa in the column under the last figure and at the right of the first three figures.

(2) For a number consisting of one, two, or three figures, find the number in the N-column and the mantissa to the right in the column headed 0.

(3) For a number consisting of more than four figures, find the mantissa for the first four figures by rule (1) and add to this the product of the tabular difference by the remaining figures of the number considered as a decimal number.

13. Finding the logarithm of a number. In finding the logarithm of a number, it is best to determine the characteristic first and then look up the mantissa. Perform all the interpolations without the aid of a pencil if possible. The use of the proportional parts is explained in Art. 21; but the student is advised to become familiar with interpolating without their help.

Example 1. Find the logarithm of 92.36.

The characteristic is 1, by rule (1) for characteristics. The mantissa is 0.96548, by rule (1) for mantissas.

Example 3. Find the logarithm of 0.00039724.
Rule (2) for characteristic gives 4.

Rule (3) for mantissa gives 0.59905.

.. log 0.00039724

4.59905

=

6.59905 10.

14. To find the number corresponding to a logarithm. If log 31.416 1.49715, then 31.416 is the number corresponding to the logarithm 1.49715. It is sometimes called the antilogarithm and is written 31.416 = log-1 1.49715.