tissa being always positive. This is indicated by writing the negative sign over the characteristic: thus, 2.380211 is equivalent to - 2.380211. (See last example, p. 8.)

4. The characteristic of the logarithm of a mixed number is the same as that of its entire part. Thus the mixed number 74.103 lies between 10 and 100; hence its logarithm lies between 1 and 2, as does the logarithm of 74.

5. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.

The logarithm of a quotient is equal to the logarithm of the dividend diminished by that of the divisor.

The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power. The logarithm of any root of a number is equal to the logarithm of the number divided by the index of the root.

6. The preceding principles enable us to abridge labor in arithmetical calculations, by using simple addition and subtraction instead of multiplication and division.

II.

MANNER OF USING THE TABLES.

TO FIND THE LOGARITHM OF ANY NUMBER.

1. First find the characteristic by rule 2, 3, or 4, given above.

2. Then, if the number be less than 100, look in column N of the table for 10 times or 100 times the amount of it; opposite this multiple, in column O, will be found the mantissa.

Thus the logarithm of 6 is 0.778151; that of 84 is 1.924279. 3. If the number lie between 100 and 10000, find the first three figures of it in column N; then pass along a horizontal line until you come to the column headed with the fourth figure of the number. At this place will be found the mantissa.

Thus the logarithm of 7200 is 3.857332; that of 8536 is

4. If the number be greater than 10000, place a decimal point after the fourth figure, thus converting the number into a mixed number. Find the mantissa of the entire part by the method last given. Then take from column D the corresponding tabular difference, multiply this by the decimal part, and add the product to the mantissa just found. The principle employed is that the differences of numbers are proportional to the differences of their logarithms, when these differences are small.

Thus the logarithm of 672887 is 5.827943; that of 43467 is 4.638160.

5. If the number be a decimal, drop the decimal point, thus reducing it to a whole number. Find the mantissa of the logarithm of this number, and it will be the mantissa required. Thus the logarithm of .0327 is 2.514548; that of 378.024 is 2.577520.

TO FIND THE NUMBER CORRESPONDING TO A GIVEN

LOGARITHM.

6. The rule is the reverse of those just given. Look in the able for the mantissa of the given logarithm. If it cannot 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 as many O's as may be necessary, and divide this result by the corresponding number in column D. Annex the quotient to the number set aside, and then 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, the result will be a pure decimal, and the number of O's which immediately follow the decimal point will be one less than the number of units in the characteristic.

Thus the number corresponding to the logarithm 5.233568 is 171225.296; that corresponding to the logarithm 2.233568 is .0171225.

MULTIPLICATION BY MEANS OF LOGARITHMS.

7. Find the logarithms of the factors, and take their sum; then find the number corresponding to the resulting logarithm,

Example.

Find the continued product of 3.902, 597.16, and 0.0314728.

Here the 2 cancels the +2, and the 1 carried from the deci mal part is set down.

DIVISION BY MEANS OF LOGARITHMS.

8. Find the logarithms of the dividend and the divisor, and subtract the latter from the former; then find the number corresponding to the resulting logarithm, and it will be the quotient required.

2.759267=log. 0.057447, the quotient.

Here 1 taken from gives 2 for a result. The subtraction, as in this case, is always to be performed in the algebraic way.

9. The operation of division, particularly when combined with that of multiplication, can often be simplified by using the principle of the arithmetical complement.

is the result obtained by subtracting it from 10: it may be written out by commencing at the left hand, and subtracting each figure from 9 until the last significant figure is reached, which must be taken from 10. Thus 8.130456 is the arithmetical complement of 1.869544.

To divide one number by another by means of the arithmetical complement, find the logarithm of the dividend and the arithmetical complement of the logarithm of the divisor; add them together, and diminish the sum by 10; the number corresponding to the resulting logarithm will be the quotient required.

Example.

Multiply 358884 by 5672, and divide the product by 89721.

TO RAISE A NUMBER TO ANY POWER BY MEANS OF LOGA

RITHMS.

10. Find the logarithm of the number, and multiply it by the exponent of the power; then find the number corresponding to the resulting logarithm, and it will be the power required.

TO EXTRACT ROOTS BY MEANS OF LOGARITHMS.

11. Find the logarithm of the number, and divide it by the index of the root; then find the number corresponding to the

Example.

Find the cube root of 4,096.

Operation.

Log. 4,096, 3.612360; one-third of this is 1.204120, to which the corresponding number is 16, which is the root sought.

12. When the characteristic is negative, and not divisible by the index, add to it the smallest negative number that will make it divisible, and then prefix the same number, with a plus sign, to the mantissa.

Example.

Find the 4th root of .00000081. _The logarithm of this number is 7.908485, which is equal to 8+1 908485, and one-fourth of this is 2.477121; the number corresponding to this logarithm is .03: hence .03 is the root required.

13. Five-figure logarithms are sufficiently accurate for ordinary railroad field-work. The tables in this book may therefore, as a rule, be used without interpolation.