the logarithm of this number, and it will be the mantissa required. Thus,

log
.0327 =

2.514548
log 378.024 = 2.577520

6o. 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 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 ; an-' nex as many O's as may be necessary, and divide this result by the corresponding number in the 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.

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 is 253.

OPERATION.

Given mantissa,

233568 Next less mantissa,

233504

1712

253) 6400000 ( 25296 :: The required number is 171225.296.

2. What is the number corresponding to the logarithm 2.233568?

Ans...0171225. 3. What is the number corresponding to the logarithm 2.785407?

Ans. .06101084. 4. What is the number corresponding to the logarithm 1.846741 ?

Ans. .702653.

MULTIPLICATION BY LOGARITHMS. 15. From the principle proved in (Art. 5), we deduce the following

RULE.—Find the logarithms of the factors, and take their sum; then find the number corresponding to the resulting logarithm, and it will be the product required.

2. Find the continued product of 3.902, 597.16, and 14728.

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

3. Find the continued product of 3.586, 2.1046, 0.8372, and 0.0294.

Ans. 0.1857615.

DIVISION BY LOGARITHMS.

16. From the principle proved in (Art. 6), we have the following

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

Here, 1 taken from 1, gives 2 for a result. The subtraction, as in this case, is always to be performed in the algebraic sense. 3. To divide 0.06314 by .007241. log 0.06314

2.800305 log 0.007241

3.859799

0.940506 ... 8.7198, quotient.

Here, 1 carried from the decimal part to the 3, changes it to 2, which being taken from 2, leaves 0 for the characteristic.

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

ARITHMETICAL COMPLEMENT.

17. The ARITHMETICAL COMPLEMENT of a logarithm is the remainder obtained by subtracting it from 10. Thus, 8.130456 is the arithmetical complement of 1.869544. The arithmetical complement of a logarithm 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. The arithmetical complement is denoted by the symbol (a. c.).

Let a and b represent any two logarithms whaterer, and

- b their difference. Since we may add 10 to, and snbtract it from, a - b, without altering its value, we have,

,

a

a b = a + (10 – 6) – 10.

(10.) But, 10 - b is, by definition, the arithmetical complement of b: hence, Equation (10) shows that the difference between two logarithms is equal to the first, plus the arithmetical complement of the second, minus 10.

Hence, to divide one number by another by means of the arithmetical complement, we have the following

RULE.—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.

In this example, the sum of the characteristics is 8, from which, taking 10, the remainder is 2.

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