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part of the logarithm. The index must be placed before it agreeably to the above observation. Thus the log. of 421 is 2.62428, the log. of 4.21 is 0.62428, and the log. of .0421 is —2.62428.

If the given number consists of four figures, find the three left hand figures in the column marked No. as before, and the remaining, or right hand figure at the top of the table ; in the column under this figure, and against the other three, is the decimal part of the logarithm. Thus the log. of 5163 is 3.71290, and the log. of .6387 is -1.80530.

If the given number consist of five or six figures, find the logarithm of the four left hand figures as before ; then take the difference between this logarithm and the next greater in the table. Multiply this difference by the remaining figure or figures of the given number, and cut off one, or two figures to the right hand of the product, according as the multiplier consists of one, or two figures; then add the remaining figure or figures of the product to the logarithm first taken out of the table, and the sum will be the logarithm required. Thus, let it be required to find the logarithm of 59686 ; then,

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The natural number consisting of five integers, the index must be 4; therefore the log. of 59686 is 4.77587.

Again, let it be required to find the log. of .0131755; then, Logarithm of 1317 is

11959 The next greater log.is

11991

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As the given number is a decimal, and has one cipher between the decimal point and first significant figure, the index must be -2; therefore the log. of .0131755 is -2.11977.

EXAMPLES.

1. Required the log. of 4.3 2. Required the log. of 7986 3. Required the log. of .3754 4. Required the log. of 596.87 5. Required the log. of 785925 6. Required the log. of 6543900 7. Required the log. of .0027863

Ans. 0.63347
Ans. 3.90233
Ans. -1.57449
Ans. 2.77588
Ans. 5.89538
Ans. 6.81583
Ans. -3.44503

* Because 17.6 is nearer 18 than 17.

PROBLEM II.

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To find the natural number corresponding to a given

logarithm. If four figures only be required in the answer, look in the table for the decimal part of the given logarithm, and if it cannot be found exactly, take the one nearest to it, whether greater or less; then the three figures in the first column, marked No. which are in a line with the logarithm found, together with the figure at the top of the table directly above it, will form the number required. Observing, that when the index of the given logarithm is affirnative, the integers in the number found, must be one more than the number expressed by the index; but when the index of the given logarithm is negative, the number found will be wholly a decimal, and must have one cypher less, placed between the decimal point and first significant figure on the left hand, than the number expressed by the index. Thus the natural number corresponding to the logarithm 2.90233 is 798.6, the natural number corresponding to the logarithm 3.77055 is 5896, and the natural number corresponding to the logarithm —3.36361 is .00231.

If the exact logarithm be found in the table, and the figures in the number corresponding do not exceed the index by one, annex ciphers to the right hand till they do. Thus the natural number corresponding to the logarithm 6.64068 is 4372000.

If five or six figures be required in the answer, find, in the table, the logarithm next less than the given one, and take out the four figures answering to it as before. Subtract this logarithm from the next greater in the table, and also from the given logarithm; to the latter

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or six figures are required, and divide the number thus produced, by the former difference; annex the quotient to the right hand of the four figures already found, and it will give the natural number required.

Thus let it be required to find the natural number corresponding to the logarithm 2.53899 true to five figures; then,

Given logarithm .53899
Next less

.53895 the natural number

corresponding is 3459 Diff. with one cipher annexed 40 Next less log

.53895 Next greater

.53008

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Divide 40 by 13 and the quotient will be 3, which, annexed to the right hand of 3459, the four figures already found, makes 34593; therefore as the index is 2, the required natural number is 345.93.

Again let it be required to find the natural number corresponding to the logarithm 4.59859, true to six figures; then,

Given logarithm .59859
Next less

.59857, the natural number an

swering to it is 3968. Diff. with two ciphers annexed 200 Next less log.

59857 Next greater

59868

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Divide 200 by 11, and the quotient will be 18, which annexed to the right hand of 3968, the four figures al

ready found, makes 396818; therefore as the index is 4, the required natural number is 39681.8.

EXAMPLES.

1. Required the natural number answering to the logarithm 1.88030. Ans. 75.91.

2. Required the natural number answering to the logarithm 5.37081. Ans. 234861.

3. Required the natural number answering to the logarithm 3.11977. Ans. 1317.56.

4. Required the natural number answering to the logarithm-2.97435. Ans. .094265.

PROBLEM III. To multiply numbers by means of logarithms. Case 1.-When all the factors are whole or mixed numbers.

RULE.

Add together the logarithms of the factors, and the sum will be the logarithm of the product.

EXAMPLES.

1. Required the product of 84 by 56. Logarithm of 84

is 1.92428 Do. of 56

is 1.74819

Product 4704

Sum 3.67247 2. Required the continued product of 17.3, 1.907 and 34.

Logarithm of 17.3 is 1.23805
Do.

1.907 is 0.28035
Do.

34. is 1.53148

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