Here-1, in the lower index, is changed into +1, which is then taken for the index of the result.

4. Divide .27684 by 5.1576, by logarithms.

Here the 1 that is to be carried from the decimals, is taken as 1, and then added to -1, in the upper index, which gives 2 for the index of the result.

5. Divide 6.9875 by .075789, by logarithms.

Here 3, in the lower index, is changed into +3, and this added to 1, the other index, gives +3-1 or 2.

For any single proportion, add the logarithms of the second and third terms together, and subtract the logarithm of the first from their sum, according to the foregoing rules; then the natural number answering to the result will be the fourth term required.

But if the proportion be compound, add together the logarithms of all the terms that are to be multiplied, and from the result take the sum of the 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.

And, if two or more logarithms are to be subtracted, as in the latter part of the above rule, add their com plements 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.

3. Find a third proportional to 12.796 and 3.21718, by

4. Find the interest of 2791. 5s. for 274 days, at 43 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, and a third proportional to 12.796 and 3.24718 by logarithms. Ans. 17.55623 and .8240216

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 affir-mative, the index of the result must be taken accordingly.

EXAMPLES

1. Find the square of 2.7568, by logarithms.

2. Find the cube of 7.0851, by logarithms..