guished from absolute ones, by setting a negative sine over them, as above.

To find the Number of a given Logarithm.

Look for the given log. amongst the logs. from 1000 to 10000 (not regarding the index or first figure) and if you find the exact log. you want, you have in the margin the required number. But if the index of the given log. be less than 3, cut off from the number found, as many figures as it is less; and the figures so cut off will be decimals, and the others integers. Or if the first figure or index, be greater than 3, add as many cyphers to the number found as it is more, and you have the number required.

EXAMPLES.

Find the numbers correspondent to the following logarithms. Given logarithms. Numbers. 5.53230 Answer 356,00. 4.55230

35670. 3.53230

3567. 2.55230

356.7 1.53230

35.67 0.55234

3.567

But if the exact log. cannot be found in the table, and the number of figures required exceed four, then 1. Find as before (not regarding the index) the log. answering to the first four figures, but less than the given log.

2. Take that from the given one, and if the remainder do not consist of two figures, prefix a cypher to it; and after these two figures annex three cyphers, so will you have five figures for a dividend.

3. Divide that by the difference between the log. found, and the next following, and if

your quotient do not consist of three figures prefix a cypher or cyphers to make it; which three figures place after the first four found.

Then observe the index of the given log. which shews how many figures must be integers, and how many decimals; for the number of integers is one more than the given index as before.

EXAMPLE 1.

1. Required, the number of the log.

4.55241

3.55230

The nearest log. which is less is its number is 3567.

The difference of these with three cyphers is for å dividend

11000

The log. found
The next log.

3.55230 3.55242

12)11000(916 Quotient.

108

20
12

80

Which quotient place after the first four figures found, and you have 3567916; and because the index is 4, the number will be 35679.16 required.

2. Required, the number answering to the log.

5.09901 The nearest log. to which is 3.09899, its No. 1256

Dividend 02000

Log. found 3.09899
Next log. 3.09934

35 Divisor 35)02000(57

175

250
245

5

Because the quotient consists of but two figures, prefix a cypher to it to make it three, and it is 057; which annexed to the first four found, is 1256057; and because the index of the given log. is 5, its number will be 125605.7.

From what has been said on this head, the following problems may be easily solved by logarithms, viz.

PROB. I.

Multiply 134 by 25,6

To log. of 134
Add the log. of 25.6

2.12710 1.40824

Sum 3.53534

The number answering to which sum, viz. 3430, is nearly the product of 134 by 25.6 and is the answer.

Again, what is the quotient of 30550 by 47?

From the log. of 30550 4.48501
Take the log. of 47 1.67210

2.81291 its number

is 650'the quotient required.

PROB, III.

Three numbers in direct proportion given, to find a fourth.

From the sum of the logarithms of the second and third numbers; deduct the logarithm of the first, the remainder will be the logarithm of the fourth required.

EXAMPLE I.

Let the three proposed numbers be 36, 48, 66, to find a fourth proportional.