RULE III.

Add to the number of cyphers between the decimal point and the first significant figure, and subtract from 10; the remainder is the index required. Thus the characteristic of the log. of 04 is 7 or 8, since I added to the number of cyphers following the decimal point is 2, then 2 from 10 is 8.

If the index of a vulgar fraction is required, it must first be reduced to an equivalent decimal fraction, and then the index is found by the rule.

An alteration in the position of the decimal point alters only the characteristic, and not the decimal part of the logarithm, if the significant figures remain the same; thus all the following numbers and fractions have the same decimal part in their logarithms, with different characteristics :

This is, for the purposes of numerical calculation, the decisive advantage the base Io possesses over any other. It is plain then, that one calculation gives us the logarithm of the above numbers, and in fact of as many numbers as can be made by shifting the decimal point to different positions in the combination 2, 5, 6,; but if we adopt any other base, we should require a separate calculation for each of them, The student will perceive that the base 10 has this advantage, in consequence of our system of notation being decimal, If our system were duodecimal, our logarithms would then have to be calculated to the base 12 to be possessed of the like advantage, and so on for any other system.

The characteristic may also be found as follows :—

RULE IV.

Place your pen between the first and second figure (not cypher), and count one for each figure or cypher, until you come to the decimal point, the number

thus given will be the characteristic: but observe that if you count to the left you must subtract the number found from 10, and consider the remainder as the characteristic.

4601.7

123

Thus, in finding the log. of 46017, if you place your pen between the first figure (4) and second (6), it falls on the decimal point, in this case the characteristic is o, Next in the case of log. of 46017 place your pen between 4 and 6 and count the characteristic is 3. Next in the case 4601700, here the decimal point falls behind the last cypher (No. 5). Hence, counting as before, we have characteristic is 6.

4.321

4601700
123456

and the

Again in the case of log, '00046017 the first significant figure is 4. Hence, counting, *00046017 we have but here we count to the left, so that the characteristic is negative, or 4, which taken from 10 is 6. Again in the case of log. of '46017, we have 46 ¡6017, and the characteristic is ī, or 9.

LOGARITHMIC TABLES.

The tables containing the logarithms of numbers give them, most frequently, to six places of decimals, as shown in Raper's Navigation, Table 64, and Norie's Epitome, Table 24.

If the number be given, its logarithm may be found as follows :— To find the logarithm of a number consisting of not more than two digits, i.e., does not exceed 100.

RULE V.

Seek for the number in the column at the top of which is No., and the decimal part of the logarithm will be found opposite to it in the next column to the right hand. Prefix the proper characteristic to the mantissa, see Rules I and II, page 17. The result is the logarithm sought.

Ex 1. Required the logarithm of 21, 2'1, 21, and 021.

In the first page of the Table, and in one of the vertical columns marked No., we find the 21, against which stands 1322219, the logarithm sought. Since the mantissa of the logarithm of any number consisting of the same figures is the same whether the number be integral, fractional, or mixed, the logarithms of the numbers 2.1 and 021 will have the same decimal part as 21, consequently the logarithm of 2'1 is 0*322219, and the logarithm of 021 is 8.322219.

Ex. 2.

To find the logarithm of 52, 5*2, '52, and '00052:—

In the Tables we find the log. of 52 is 1716003, and, therefore, simply changing the index, the log. of 5.2 is 0'716003, °52 is 9°*716003, and the log、 of 00052 is 4716003 or 6.716003.

To find the logarithm of a number consisting of not more than three places of figures (from 100 to 1000).

RULE VI.

Find the given number in the vertical column marked No. at the top, and under O will stand the mantissa or decimal part of logarithm. Prefix the characteristic according to Rules I and II. The result is the logarithm sought.

Ex. 1. Required the log. of 476, 4°76, and 00476.

We seek in the left hand column of the Table for 476, against which in the column marked O at the top, stands the mantissa corresponding thereto; and this part by the rule is the same for each of the abové numbers. Now prefixing the index according to the number of integral figures in the natural number, we find the log. of 476 is 2.677607; of 4.76 is 0'677607; and of '00476 is 7.677607.

If the number contains four places of figures, the logarithm is found by

RULE VII,

The first three figures will be found in the vertical column to the left marked No., and the fourth in the horizontal column at the top of the page. Under this last, and opposite the three figures, will be found the mantissa of the logarithm sought. Prefix the index according to Rules I and II. The result is the logarithm sought. •

Ex. 1. Required the logarithm of 4587 and of 0.0004587.

The first three figures (viz. 458) being found in the column to the left marked No., and the fourth (7) at the top of the page, the decimal part of logarithm ('661529) is found in the same horizontal line as the three first figures of the given number, and in the same column as the fourth. The index is 3, being one less than the number of integers in the whole number: whence the completed logarithm is 3.661529. The logarithm of 0004587 is 4·661529, the characteristic being negative, and one more than the number of prefixed cyphers,

If the number consists of more than four figures, we use

1°.

RULE VIII.

2.835247 or 8.835247 3'444669

or 7'444669

Cut off the first four figures and consider the rest as a decimal.

2°. Find the mantissa corresponding to the first four figures.

3. Multiply the tabular difference by the decimal cut off; but at the same

time adding unity if the highest cut off is not less than 5.

4°. Add the integer part of this product to the figures of the mantissa

just found.

The result is the mantissa of the required logarithm.

The characteristic or index is found by Rules I and II, page 17.

We seek in the left hand column of the Table for 284 (the first three digits) and also at the top of the page in one of the horizontal columns we find 3 (the fourth figure), then in a line with the former and in the column with the latter at the top we have 453777, which is the mantissa of 2843. In a line with the quantity and in the right hand column marked Diff., stands tab. diff. 153; which multiplied by 4, the remaining digit of the given number, produces 612; then cutting off one digit from this (since we have multiplied by only one digit) it becomes 61, which being added to 453777 (the mantissa of 2843) makes 453838, and, with the characteristic, 4'453838 the required logarithm.

Tab. diff.
338
X 6

The logarithm of 284 34 is 2'453838, and the log, of .028434 is 2·453838 or 8.453838.

Ex. 2. Required the logarithm of 12806.

The mantissa of the first four figures is found thus :-opposite the 873 and under stands 941213, then in the right column in a line with this stands the Diff. 50, which being multiplied by 57, the remaining digits of the given number, makes 2850; from this we cut off two digits to the right (since we have multiplied by two digits), when it becomes 28; but as the highest digit cut off is 5, we add unity, which makes 29. Then 5941212 (the logarithm of 8734) + 29 = 5'941242 is the required logarithm. Ex, 4. Required the logarithm of 628007.

The log. of 628 067 is 2'798006, and the log. of 00628067 is 3'798006 or 7.798006. The mantissa of the log, of each of these numbers being the same, the index only being varied (See Rules I and II),

The logarithm of a vulgar fraction is the result of the subtraction of the logarithm of its denominator from that of its numerator; because