tiplier; then add the remainder to the logarithm first taken out: the sum will be the required logarithm.

Let the logarithm of 6475.48 be required.

32 added to 811240 gives .811272;

and the index being 3, the complete logarithm is 3.811272.

Next let the logarithm of .0026579 be required.

The logarithm of 2657 is

The next greater

Difference

.424392

4555

163

9

146,7

424392 + 147.424539, and the index being -3, the complete logarithm is -3.424539.

NOTE. In this last example, the product is 1467: the figure stricken off being 7, which is more than 5, 147 is taken instead of 146.

12. To find the natural number corresponding to a given Logarithm. If four figures only be needed in the answer, seek in the columns of logarithms for the one nearest to the decimal part of the given logarithm: the first three figures of the natural number will be found in the column marked N; and the fourth, at the top of the column in which the logarithm is found.

When the index is positive, the number of integral

figures will be one greater than the number expressed by the index; but, if the index is negative, the number will be wholly decimal, and have one less cipher between the decimal point and the first significant figure than the number expressed by the index. Thus, the natural number corresponding to the logarithm 2.860996 is 726.1; and that corresponding to -2.860996 is .07261.

If the logarithm be found exactly in the tables, and there be not enough figures in the corresponding number, the deficiency must be supplied by ciphers. Thus, the natural number corresponding to 6.891649 is 7792000.

But, if five or six figures be required, find in the table the logarithm next less than the given one, and take out the corresponding number as before; subtract this logarithm from the next greater in the table, and also from the given logarithm; annex one or two ciphers to the latter remainder, according as five or six figures are required, and divide the result by the former. The quotient annexed to the figures first taken out will give the figures required, the decimal point being placed as before.

Required the number corresponding to 2.649378, to six figures

ON THE USE OF LOGARITHMS.

13. Multiplication. To multiply numbers by means of logarithms. Add together the logarithms of the factors, and take out the natural number corresponding to the sum. If any of the indices be negative, the figure to be carried from the sum of the decimal portions must be considered positive, and added to the sum of the positive, or subtracted from the sum of the negative indices. Then collect the affirmative indices into one sum, and the negative into another, take the difference between these sums, and prefix thereto the sign of the greater sum.

Ex. 2. Required the product of 764.3, .8175, .04729, and .00125.

Ex. 3. Required the product of 87.5 and 6.7.

Ans. 586.25.

Ex. 4. Required the continued product of .0625, 41.67, .81427, and 2.1463.

Ans. 4.5516.

Ex. 5. Multiply 67.594, .8739, and 463.92 together.

Ans. 27404.

Ex. 6. Multiply 46.75, .841, .037654, and .5273 together. Ans. .780633.

Ex. 7. Multiply .00314, 16.2587, .32734, .05642, and 1.7638 together. Ans. .001663.

14. Division. To divide numbers by logarithms. Subtract the logarithm of the divisor from that of the dividend: the remainder will be the logarithm of the quotient.

If one or both of the indices are negative, subtract the decimal portions of the logarithm as before; and, if there be one to carry from the last figure, add it to the index of the divisor, if this be positive, but subtract if it be negative; then conceive the sign of the result to be changed, and if, when so changed, the two indices have the same sign, add them together; but, if they have different signs, take their difference and prefix the sign of the greater.

15. To involve a number to a power. Multiply the logarithm of the number by the index of the power to which it is to be raised.

If the index of the logarithm is negative, and there is any thing to be carried from the product of the decimal part by the multiplier, instead of adding this to the product of the index, subtract it: the difference will be the index of the product, and will always be negative. Ex. 1. Required the fourth power of 5.5.

Ex. 2. Required the fifth power of .63.

Ex. 3. Required the fourth power of 7.639.

Ans. 3405.24.

Ex. 4. Required the third power of .03275.

Ans. .00003513.

Ex. 5. What is the fifteenth power of 1.06?

Ex. 6. What is the sixth power of .1362?

Ans. 2.3966.

Ans. .0000063836.

Ex. 7. What is the tenth power of .9637?

Ans. .69091.

16. To extract a given root of a number. Divide the logarithm of the number by the degree of the root to be extracted: the quotient will be the logarithm of the root.

If the index of the logarithm is negative, and does not