and a decimal, such as 36.78, it may be put under the form 3873. But since a fraction is equal to the quotient obtained by dividing the numerator by the denominator, its logarithm will be equal to the logarithm of the numerator minus the logarithm of the denominator. Therefore,

3678

log 3573=log 3678-log 100=3.565612-2=1.565612 from which we see, that a mixed number may be treated as though it were entire, except in fixing the value of the characteristic, which is always one less than the number of the integer figures.

12. The logarithm of a decimal fraction is also readily found. For,

log 0.8=log=log 8-1=-1+log 8. But,

in which, however, the minus sign belongs only to the characte ristic. Hence it appears, that the logarithm of tenths is the same as the logarithm of the corresponding whole number, excepting, that the characteristic instead of being 0, is-1.

If the fraction were of the form 0.06 it might be written taking the logarithms, we have,

loglog 06-2-2+log 06=-2.778151 in which the minus sign, as before, belongs only to the characteristic. If the decimal were 0.006 its logarithm would be the same as before, excepting the characteristic, which would be-3. It is, indeed, evident, that the negative characteristic will always be one greater than the number of ciphers be tween the decimal point and the first significant figure: Therefore, the logarithm of a decimal fraction is found, by considering it as a whole number, and then prefixing to the decimal part of its logarithm a negative characteristic greater by unity than the number of ciphers between the decimal point ana the first significant figure.

That we may not, for a moment, suppose the negative sign to belong to the whole logarithm, when in fact it belongs only to the characteristic, we place the sign above the characte ristic, thus,

To find in the table, the number answering to a given logarithm. 13. Search in the columns of logarithms for the decimal Irt of the given logarithm, and if it can be exactly found, set down the corresponding number. Then, if the characteristic of the given logarithm is positive, point off from the left of the number found, one more place for whole numbers than there are units in the characteristic of the given logarithm, and treat the figures to the right as decimals.

If the characteristic of the given logarithm is 0, there will be one place of whole numbers; if it is-1, the number will be entirely decimal; if it is -2, there will be one cipher between the decimal point and the first significant figure; if it is 3, there will be two, &c

The number whose logarithm is 1.492481, is found at page 5, and is 31.08.

But when the decimal part of the logarithm cannot be exactly found in the table, take the number answering to the nearest less logarithm; take also from the table the corresponding difference in the column D. Then, subtract this less logarithm from the given logarithm, and having annexed any number of ciphers to the remainder, divide it by the dif ference taken from the column D, and annex the quotient to the number answering to the less logarithm: this gives the required number, nearly. This rule, like that for finding the logarithm of a number when the places of figures exceed four, supposes the numbers to be proportional to their corresponding logarithms.

1. Find the number answering to the, logarithm 1.532708.

Given logarithm is

1.532708

Next less tabular logarithm is

1.532627

Their difference is

81

The number corresponding to the tabular logarithm is 34.09 And the tabular difference is

128:

The 63 being annexed to the tabular number 34.09 gives 34.0963 for the number answering to the logarithm 1.532708.

2. Required the number answering to the logarithm 3.233568.

Hence the number sought, is 1712.25, marking four places for integers since the characteristic is 3.

MULTIPLICATION BY LOGARITHMS.

14. When it is required to multiply numbers by means of their logarithms, we first find from the table the logarithms of the numbers to be multiplied; we next add these logarithms together, and their sum is the logarithm of the product of the numbers (Art. 2).

The term sum is to be understood in its algebraic sense; therefore, if any of the logarithms have negative characteristics, the difference between their sum and that of the positive characteristics, is to be taken, and the sign of the greater prefixed.

Here the

1.865314

cancels the +2, and the 1 carried from the

3. Multiply 3.586, 2.1046, 0.8372, and 0.0294, together.

In this example the 2, carried from the decimal part, cancels 2, and there remains I to be set down.

DIVISION OF NUMBERS BY LOGARITHMS.

15. When it is required to divide numbers by means of their logarithms, we have only to recollect, that the subtraction of logarithms corresponds to the division of their numbers (Art. 3). Hence, if we find the logarithm of the dividend, and from it subtract the logarithm of the divisor, the remainder will be the logarithm of the quotient.

This additional caution may be added. The difference of the logarithms, as here used, means the algebraic difference; so that, if the logarithm of the divisor have a negative characteristic its sign must be changed to positive, after diminishing it by the unit, if any, carried in the subtraction from the decimal part of the logarithm. Or, if the characteristic of the logarithm of the dividend is negative, it must. be treated as a negative number.

Here, 1 carried from the decimal part to the 3 changes it to 2, which being taken from 2, leaves o for the characteristic. 3. To divide 37.149 by 523.76

Here, the 1 taken from ī, gives 2 for a result, as set down.

ARITHMETICAL COMPLEMENT.

16. The Arithmetical complement of a logarithm is the number which remains after subtracting this logarithm from 10.

Thus
Hence,

of 9.274687.

10-9.274687=0.725313.

0.725313 is the arithmetical complement

17 We will now show that, the difference between two logarithms is truly found, by adding to the first logarithm the arithmetical complement of the logarithm to be subtracted, and then diminishing the sum by 10.

and

Let a the first logarithm

b the logarithm to be subtracted

c=10-b the arithmetical complement of b. Now the difference between the two logarithms will be expressed by a-b.

But, from the equation c=10-b, we have

c-10=-b

hence, if we place for-b its value, we shall have

a-b=a+c-10

which agrees with the enunciation.

When we wish the arithmetical complement of a logarithm, we may write it directly from the table, by subtracting the left hand figure from 9, then proceeding to the right, subtract each figure from 9 till we reach the last significant figure, which must be taken from 10: this will be the same as taking the logarithm from 10.