the mantissas alone are given; the characteristic may be
found by one of the rules of Art. 4.

Before explaining the use of the table, it is to be shown that the mantissa of the logarithm of any number is not changed by multiplying or dividing the number by any exact power of 10.

Let n represent any number whatever, and 10" any power of 10, p being any whole number, either positive or negative. Then, in accordance with the principles of Arts. 5 and 3, we shall have

log (n x 10") log n + log 10" = p + log n;

but p is, by hypothesis, a whole number: hence, the decimal part of the log (n x 10") is the same as that of log n; which was to be proved.

Hence, in finding the mantissa of the logarithm of a number, the position of the decimal point may be changed at pleasure. Thus, the mantissa of the logarithm of 456357, is the same as that of the number 4563.57; and the mantissa of the logarithm of 759 is the same as that of 7590.

MANNER OF USING THE TABLE.

1o. To find the logarithm of a number less than 100.

10. Look on the first page, in the column headed "N," for the given number; the number opposite is the logarithm required. Thus,

log 67 1.826075.

2°.

To find the logarithm of a number between 100 and 10,000.

11. Find the characteristic by the first rule of Art. 4. To determine the mantissa, find in the column headed "N" the left-hand three figures of the given number; then pass along the horizontal line in which these figures. are found, to the column headed by the fourth figure of the given number, and take out the four figures found there; pass back again to the column headed "0," and there will be found in this column, either upon the horizontal line of the first three figures or a few lines above it, a number consisting of six figures, the left-hand two figures of which must be prefixed to the four already taken out. Thus,

log 8979 = 3.953228.

If, however, any dots are found at the place of the four figures first taken out, or if in returning to the "0" column any dots are passed, the two figures to be prefixed are the left-hand two of the six figures of the "0" column immediately below. Dots in the number taken out must be replaced by zeros. Thus,

NOTE. The above method of finding the mantissa assumes that the given number has four places of figures. If, therefore, the number lies between 100 and 1000, and has but three places of figures, find the characteristic by the first rule of Art. 4, and then, to find the mantissa, fill out the given number to four places of figures (or conceive it to be so filled out) by annexing 0 (see Art. 9), and find the mantissa corresponding to the resulting number, as above.

3. To find the logarithm of a number greater than 10,000. 12. Find the characteristic by the first rule of Art. 4. To find the mantissa: set aside all of the given number except the left-hand four figures, and find the mantissa corresponding to these four, as in Art. 11; multiply the corresponding tabular difference, found in column "D," by the part of the number set aside, and discard as many of the right-hand figures of the product as there are figures in the multiplier, and add the result thus obtained to the mantissa already found. If the left-hand figure of those discarded is 5 or more, increase the number added by 1.

NOTE. It is to be observed that the tabular difference, found in column "D," is millionths, and not a whole number; and that, therefore, the result to be added "to the mantissa already found" is millionths.

EXAMPLE.-To find the logarithm of 672887: the characteristic is 5; set aside 87, and the mantissa corresponding to 6728 is .827886; the corresponding tabular difference is 65, which multiplied by 87, the part of the number set aside, gives 5655; as there are two figures in the multiplier, discard the right-hand two figures of this product, leaving 56; but as the left-hand figure of those discarded is 5, call the result 57 (which is millionths); adding this 57 to the mantissa already found, will give .827943 for the required mantissa; hence,

log 672887 = 5.827943.

The explanation of the method just given is briefly this: for the purpose of finding the mantissa, the given number is conceived to be a mixed one, thus, 6728.87, the mantissa not being affected by the position of the decimal point (see Art. 9). The numbers in the column

"D" are the differences between the logarithms of two consecutive whole numbers. In the example just given, the mantissa of the logarithm of 6728 is .827886, and that of 6729 is .827951, and their difference is 65 millionths; 87 hundredths of this difference is 57 millionths; hence, the mantissa of the logarithm of 6728.87 is found by adding 57 millionths to .827886. The principle employed is, that the differences of numbers are proportional to the differences of their logarithms, when these differences are small.

4°. To find the logarithm of a decimal.

13. Find the characteristic by the second rule of Art. 4. To find the mantissa, drop the decimal point, and consider the decimal a whole number. Find the mantissa of the logarithm of this number as in preceding articles, and it will be the mantissa required. Thus,

NOTE. To find the logarithm of a mixed number, find the characteristic by the Note, Art. 4; then drop the decimal point and proceed as above.

5°. To find the number corresponding to a given logarithm.

14. The rule is the reverse of those just given. Look in the table for the mantissa of the given logarithm. If it can not be found, take out the next less mantissa, and also the corresponding number, which set aside. Find the difference between the mantissa taken out and that of the given logarithm; annex any number of 0's, and divide this result by the corresponding number in the column "D." Annex the quotient to the number set aside, and

then, if the characteristic is positive, point off, from the left hand, a number of places of figures equal to the characteristic plus 1; the result will be the number required.

If the characteristic is negative, prefix to the figures obtained a number of 0's one less than the number of units in the negative characteristic and to the whole prefix a decimal point; the result, a pure decimal, will be the number required.

Examples.

1. Let it be required to find the number corresponding to the logarithm 5.233568.

The next less mantissa in the table is 233504; the corresponding number is 1712, and the tabular difference

The required number is 171225.296.

The number corresponding to the logarithm 2.233568 is .0171225.

2. What is the number corresponding to the logarithm 2.785407? Ans. .06101084.

3. What is the number corresponding to the logarithm 1.846741? Ans. .702653.

MULTIPLICATION BY MEANS OF LOGARITHMS.

15. From the principle proved in Art. 5, we deduce the following

RULE. Find the logarithms of the factors, and take their