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are in the first column of the page, and the fifth figure in the uppermost line of it ; then in the angle of meeting are the last four figures of the logarithm, and the first three figures of the same at the beginning of the same line ; to which is to be prefixed the proper index. , 1

So the logarithm of 34'092 is 1'5326525, that is, the decimal 5326525, found in the table, with the index 1 prefixed, because the given number contains two integers.

2. But if the given number contain more than five figures, take out the logarithm of the first five figures by inspec. tion in the table as before, as also the next greater logo arithm, subtracting one logarithm from the other, and al

one of their corresponding numbers from the other,

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Then say,

As the difference between the two numbers
Is to the difference of their logarithms,
So is the remaining part of the given number
To the proportional part of the logarithm.

Which part being added to the less logarithm, before taken out, the whole logarithm sought is obtained very nearly:

EXAMPLE.

To find the logarithm of the number 34:09264: The log. of 3409200, as before, is 5326525, and' log. of 3409300

is

5326652,

the diff.

100

and

127

Then, as 100 : 127 :: 64 : 81, the proportional part, This added to

5326525, the first logarithm,

gives, with the index, 1'5326606 for the logarithm of 34'09264.

Or, in the best tables, the proportional part may often be taken out by inspection, by means of the small tables of proportional parts, placed in the margin.

If

3. If the number consist both of integers and fractions, or be entirely fractional, find the decimal part of the logarithm, as if all its figures were integral; then this, the proper characteristic being prefixed, will give the logarithm required. in

And if the given number be a proper fraction, subtract the logarithm of the denominator from the logarithm of the numerator, and the remainder will be the logarithm sought ; which, being that of a decimal fraction, must always have a negative index.

But if it be a mixed number, reduce it to an improper fraction, and find the difference of the logarithms of the numerator and denominator, in the same manner as before.

EXAMPLES

1. To find the logarithm of 2 Logarithm of 37

1'5682017 Logarithm of 94

1'9731279

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Where the index 1 is negative.
2. Țo find the logarithm of 171 to

First, 177* = 43. Then,
Logarithm of 405 2.6074550
Logarithm of 23

1'3617278

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JI. TO FIND, IN THE TABLE, THE NATURAL NUMBER TO

ANY LOGARITHM.

This is to be found by the reverse method to the former, namely, by searching for the proposed logarithm among those in the table, and taking out the corresponding num

ber

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ber by inspection, in which the proper number of integers is to be pointed off, viz. I more than the units of the affirmative index. For, in finding the number answering to any given logarithm, the index always shews how far the first figure must be removed from the place of units, to the left or in integers, when the index is affirmative ; but to the right or in decimals, when it is negative.

EXAMPLES.

So, the number to the logarithm 1'5326525 is 34'092. And the number of the logarithm-15326525 is *34092.

But if the logarithm cannot be exactly found in the table, take out the next greater and the next less, subtracting one of these logarithms from the other, and also one of their natural numbers from the other, and the less logarithm from the logarithm proposed.' Then say,

As the first difference, or that of the tabular logarithms,
Is to the difference of their natural numbers,
So is the difference of the given logarithm and the last

tabular logarithm
To their corresponding numeral difference.

Which being annexed to the least natural number above taken, the natural number corresponding to the proposed logarithm is obtained.

EXAMPLE.

Find the natural number answering to the given logarithm 1'5326606.

Here the next greater and next less tabular logarithms, with their corresponding numbers, &c. are as below : Next greater 5326652 its num. 3409300 ; giv. log. 5326606 Next less 5326525 its num. 3409200; next less 5326525

Differences

127

100

81

Then,

Then, as 127 : 100 :: 81 ; 64 nearly, the numeral difference.

Therefore 34'09264 is the number sought, two integers being marked off, because the index of the given logarithm is 1. Had the index been negative, thus, -95326606, its corresponding number would have been '3409264, wholly decimal.

Or, the proportional numeral difference may be found, in the best tables, by inspection of the small tables of pro. portional parts, placed in the margin.

MULTIPLICATION Br LOGARITHMS.

RULE.

Take out the logarithms of the factors from the table, then add them together, and their sum will be the logarithm of the product required. Then, by means of the table, take out the natural number answering to the sum,

for the product sought.

Note 1. In every operation, what is carried from the decimal part of a logarithm to its index is affirmative ; and is therefore to be added to the index, when it is affirmative ; but subtracted, when it is negative.

When the indices have like signs, that is, both + or both, they are to be added, and the sum has the common sign ; but when they have unlike signs, that is, onest and the other, their difference, with the sign of the greater, is to be taken for the index of the

NOTE 2.

sum.

EXAMPLES,

EXAMPLES

1. To multiply 23'14 by 5'062.

Numbers. Logarithms.

23:14 1'3643634
5'062 0*7043221

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2. To multiply 2°581926 by 3*457291.
Numbers:

Logarithms.
2'581926 04119438
3457291

6'5387359

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3. To multiply 3.902, 597°16 and .0314728 all to: gether.

Numbers. Logarithms.
3'902

0'5912873
597'16

2.7760907 *0314728

-2.4979353 Product 73'33533

1.8653133

Here the 2 cancels the 2, and the i, to be carried from the decimals, is set down.

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4. To multiply 3.586, 2.1046, 0·8372 and oʻ0294 all together.

Numbers. Logarithms.
3'586

0-5546103
2*1046

0-3231696:
0 8372

I9228292
O'0294

-24683473

Product o'1857618

1°2689564

Here

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