logarithms of the other terms, and the remainder will be the logarithm of the term sought.

Or, the same may be performed more conveniently thus,

Find the complement of the logarithm of the first term of the proportion, or what it wants of 10, by beginning at the left hand, and taking each of its figures from 9, except the last significant figure, on the right, which must be taken from 10; then add this result and the logarithms of the other two terms together, and the sum, abating 10 in the index, will be the logarithm of the fourth term, as before.

Or, if two or more logarithms are to be subtracted, as in the latter part of the above rule, add their complements and the logarithms of the terms to be multiplied together, and the result, abating as many 10's in the index as there are logarithms to be subtracted, will be the logarithm of the term required: observing, when the index of the logarithm, whose complement is to be taken, is negative, to add it, as if it were affirmative, to 9; and then take the rest of the figures from 9, as before.

EXAMPLES.

1. Find a fourth proportional to 37.125, 14.768, and 135.279, by logarithms.

2. Find a fourth proportional to .05764, 7186,

and .34721, by logarithms,

Log. of .05764

Complement

Log. of .7186

Log. of .34721

An's. 4.328681

2.7607240

11.2392760

1.8564872

1.5405922

0.6363554

4. Find the interest of 2791. 5s. for 274 days, at

4 per cent. per annum, by logarithms.

5. Find a fourth proportional to 12.678, 14.065,

and 100.979, by logarithms.

Ans. 112.0263

6. Find a fourth proportional to 1.9864, .4678,

and 50.4567, by logarithms.

Ans. 11.88262

7. Find a fourth proportional to .09658, .24958, and .008967, by logarithms. Ans. .02317234

8. Find a mean proportional between .498621 and 2.9587, by logarithms. Ans. 17.55623

INVOLUTION, OR THE RAISING OF POWERS,
BY LOGARITHMS.

Take out the logarithm of the given number from the tables, and multiply it by the index of the proposed power; then the natural number, answering to the result, will be the power required.

Observing, if the index of the logarithm be negative, that this part of the product will be negative; but as what is to be carried from the decimal part will be affirmative, the index of the result must be taken accordingly (n).

EXAMPLES.

1. Find the square of 2.7568, by logarithms.

Log. of 2.7568.

Square 7.599946

0.4402477

2

0.8804954

2. Find the cube of 7.0851, by logarithms.

Log. 7.0851

Cube 355.6475 .

0.8503399

3

2.5510197

(n) In some examples of this rule, the resulting logarithm, when the question is wrought in the usual way, may come out wholly negative, in which case, as it cannot be found in that form in the tables, the decimal part of it must be subtracted from 1, and then 1, or I, be put before the remainder, as

3. Find the fifth power of .087451, by logs.

Where 5 times the negative index 1, being -5, and +4 to carry, the index of the power is 1.

4. Find the 365th power of 1.0045, by logs.

a negative index; which will evidently adapt it to the purpose required, without altering its value.

Thus, if it were required to find the value of (.07).7 by logarithms, the operation, agreeably to the above rule, would stand thus:

where the natural number answering to the result is .8301539;

which is the value required.

Or the same answer may be obtained more concisely, in a dif ferent way, by taking .14 from .05915686.

5. Required the square of 6.05987, by logarithms. Ans. 36.72203 6. Required the cube of .176546, by logarithms. Ans. .005502674 7. Required the 4th power of .076543, by logaAns. .0000343259 8. Required the 7th power of 1.09684, by logarithms. Ans. 1.909864

rithms.

EVOLUTION, OR THE EXTRACTION OF ROOTS, BY LOGARITHMS.

Take out the logarithm of the given number from the table, and divide it by 2 for the square root, 3 for the cube root, &c. and the natural number answering to the result will be the root required.

But if it be a compound root, or one that consists both of a root and a power, multiply the logarithm of the given number by the numerator of the index, and divide the product by the denominator, for the logarithm of the root sought.

Observing, in either case, when the index of the logarithm is negative, and cannot be divided without a remainder, to increase it by such a number as will render it exactly divisible; and then carry the units borrowed, as so many tens, to the first figure of the decimal part, and divide the whole accordingly.

EXAMPLES.

1. Find the square root of 27.465, by logs.

Log. of 27.465

Root 5.2407