Hence, the logarithm of any number between 1 and 10 is some number between 0 and 1; i.e., 0+ a fraction; between 10 and 100 is some number between 1 and 2; i.e., 1 + a fraction, etc., etc.

Thus, it appears that the logarithm of any number greater than 1 is positive, and the logarithm of any positive number less than 1 is negative; and in general the logarithm of a number consists of two parts, an integral part and a decimal part.

The integral part is called the characteristic of the logarithm, and may be either positive or negative.

The decimal part is called the mantissa of the logarithm, and is always kept positive, in order that the mantissa of the logarithms of all numbers expressed by the same digits in the same order may always be the same.

3. It is evident from the above table that the characteristic can always be obtained by the following rule:

The characteristic of the logarithm of a number greater than unity is positive, and one less than the number of digits preceding the decimal point.

The characteristic of the logarithm of a number less than unity is negative, and one more than the number of ciphers immediately after the decimal point.

Thus, the characteristics of the logarithms of 3406, 340.6, 34.06, 3.406, .3406, .0003406, are respectively, 3, 2, 1, 0, -1, -4; the mantissæ are the same, being .53224.

Hence, log .0003406 = 4.53224, the minus sign being written over the characteristic to indicate that it only is negative, the mantissa being always positive.

4. In practice it is more common to avoid the use of negative characteristics by increasing them by 10, and then by allowing for it in the interpretation of the results.

NOTE. It is only in rare cases that more than seven places of the mantissa are required; in general, four or five are sufficient; and it is only for the most accurate computations that six or seven are used.

5. A table of logarithms is a table by which the logarithm of any given number, or the number corresponding to any given loga.

TABLE I. LOGARITHMS OF NUMBERS (Pages 1-19).

6. This table gives the mantissæ of the logarithms of the natural numbers from 1 to 10009, calculated to five decimal places.* The characteristics are determined by the rule in Art. 3. On p. 1, both the characteristic and the mantissa are given.

7. To find the logarithm of a given number.

(1) For a number of one, two, or three figures only.

If the number has one or two figures, find it on page 1 in the column headed N. Then in the same horizontal line as the mumber, and in the next column headed Log, will be found its logarithm.

Thus,

log 7 = 0.84510; log 68 = 1.83251.

If the number has three figures, find on one of the pages 2-19, in the column headed N, the given number. Then in the same horizontal line as the number, and in the next vertical column, which is headed 0, will be found the mantissa of its logarithm: prefix the characteristic by the rule in Art. 3.

Thus, log 415 = 2.61805; log 94.8 1.97681.

=

NOTE 1. A dash under a terminal 5 indicates that the true value is less than 5. Thus the logarithm of 415 to seven decimal places is 2.6180481. If only five decimal places are required, we neglect the 81 and increase 4 to 5. If six decimal places are required, the 1 is neglected; thus the above logarithm is written 2.618048.

(2) For a number of four figures.

the

Find on one of the pages 2-19, in the column headed N, first three figures of the given number. Then in the same horizontal line as the first three figures, and in the vertical column which has the fourth figure of the given number at the top, will be found the last three figures of the mantissa of the required logarithm, to which the first two figures in the nearest mantissa above, in the column headed 0, are to be prefixed; supply the characteristic by the rule in Art. 3.

NOTE 2.

To save space, only the last three figures of the mantissæ are given in the columns headed 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and the first two at intervals, in the column under L.

When the first two figures are not

* With five decimal places the numbers will be correct to the one hundred-thousandth part

of a unit, which is near enough for most practical applications.

given in any line, they are to be taken from the first line above containing them, unless the last three are preceded by a star *, in which case they are to be taken from the line immediately below.

Thus, (p. 13) log 6615 = 3.82053, log 67.36 = 1.82840, log 6.764 = 0.83020.

and

(3) For a number of more than four figures.

log 28463.45423

diff. for 1 = 0.00015

Thus, for an increase of 1 in the number there is an increase of .00015 in the logarithm.

Hence, assuming that the increase of the logarithm is pro portional to the increase of the number, then an increase in the number of .672 will correspond to an increase in the logarithm of .672 × .00015= .00010, to the nearest fifth decimal place.

NOTE 3. We assumed in this method that the increase in a logarithmi is proportional to the increase in the number. Although this is not strictly true, yet in most cases it is sufficiently exact for practical purposes.

8. From the above work we have the following rule for a number of more than four figures:

Find the tabular mantissa of the first four significant figures of the number; subtract this mantissa from the next greater tabular mantissa; multiply the difference thus found by the remaining figures of the number, as a decimal; add the product to the mantissa of the first four figures, and prefix the proper characteristic.

NOTE 4. The difference between any mantissa in the table and the mantissa of the next higher number of four figures, is called the tabular dijference; and the corresponding proportional parts are placed in the colum headed P.P. By means of this column of proportional parts the above mutiplication is facilitated.

It will be seen that this difference between the logarithms of two consecutive numbers is not always the same; for instance, those in the upper part of p. 5 differ by .00018, while those in the

middle and the lower parts differ by .00016 and .00014. In the column with the heading 15 we see the difference 9 corresponding to the figure 6, which implies that when the difference between the logarithms of two consecutive numbers is .00015, the increase in the logarithm corresponding to an increase of .6 in the number is .00009.

Thus, the mantissæ of the logarithms of 2845 and 2846 differ by .00015; therefore 15 is the tabular difference. Then in the proportional table for 15, we find

or 10 to the nearest integer, which agrees with the value above.

9. To find the number corresponding to a given logarithm.

By reversing the above operations, the number corresponding to a given logarithm may be found, as will be seen by the following example:

Find the number whose logarithm is 3.47384.

We find that this mantissa does not occur exactly in the table. We therefore take out the next smaller mantissa, .47378 (on p. 5), whose corresponding number is 2977, and the next greater mantissa .47392, whose corresponding number is 2978.

The difference between these two mantissæ

=.00014.

The difference between the smaller and given mantissæ=.00006. Thus, for an increase of 1 in the number, there is an increase of .00014 in the mantissa; hence for an increase of .00006 in the mantissa there will be an increase of of 1 in the number = .43.

6

Hence, the number corresponding = 2977.43.

From the above work we have the following rule:

Find the tabular mantissa next less than the given mantissa, and the corresponding number of four figures; divide the difference of these mantisse by the tabular difference, annex the quotient to the first four figures of the number, and point off the result according to

NOTE. The labor of division may be saved by using the table of proportional parts for 14, as follows:

10. Arithmetic Complement. By the arithmetic complement or the logarithm of a number, or briefly, the cologarithm of the number, is meant the remainder found by subtracting the logarithm from 10. To subtract one logarithm, b, from another, a, is the same as to add the cologarithm, 10 b, and then subtract 10 from the result.

Thus,

When one logarithm is to be subtracted from the sum of several others, it is more convenient to add its cologarithm to the sum, and reject 10. The advantage of using the cologarithm is that it enables us to exhibit the work in a more compact form. The cologarithm is easily taken from the table mentally by subtracting the last significant figure on the right from 10, and all the others from 9.

TABLE II. LOGARITHMS OF SINES, TANGENTS, ETC.

11. This table (pages 21-82) contains the logarithms of the sines, tangents, cotangents, and cosines of all angles from 0° to 90°. If the angle is less than 45°, we look for the name of the function and the number of degrees in the angle at the top of the page, and the minutes in the left-hand column.

If the angle is between 45° and 90°, we look for the name of the function and the number of degrees at the bottom of the page, and the minutes in the right-hand column. In each case the horizontal rows at the top of the pages go with the degrees at the top, and the horizontal rows at the bottom go with the degrees at the bottom.

On pp. 21-33 the minutes and each ten seconds are given in columns at the left and right, and the odd seconds are given in a horizontal row at the top and bottom of each page. On