the three figures, will be found the mantissa of the logarithm sought. Prefix the index according to Rules XVIII and XIX. The result is the logarithm sought.

EXAMPLE.

Ex. 1. Required the logarithms 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) in the line of digits 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 characteristic 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.

Again, the logarithm of 3470 is 3'541330; that of 3°492 is 0'543074; and that of o 3468 is I'540079; that of 74'39 is 1.871515; that of 325600 is 5'512648, in which case the mantissa of 3256 is taken out, since it is the same as the mantissa of 3256000.

NOTE.-The foregoing rule may be used not only in the case of numbers consisting of four places of figures, but may be made to include all numbers consisting of less than four significant digits, and so enable us to dispense with the Rules XXII and XXIII. Thus, if the number consists of less than four figures, make up four by placing cyphers, if not already placed (or by supposing them placed), on the right of the number; cyphers so added being regarded as decimals. Then proceed to find the mantissa of the log. by the

foregoing rule.

Thus, the log. 75 is the same as the log. of 75'00; the log. of 8 is the same as the log. of 8.000; and that of 035 tho same as that of 03500 (3500 in the tables).

90. Although the tables in RAPER, NORIE, and the "Nautical Tables" accompanying this work are constructed so that the mantissa corresponding to more than four figures cannot be taken out directly, yet the mantissa of numbers containing five or six figures can be found from them without much trouble by means of the tabular difference taken out of the extreme right hand column of the page (see 86).

If the number consists of more than four figures other than final cyphers, or if the number be a decimal fraction, cyphers immediately following the decimal point, we use

RULE XXV.

1°. Cut off the first four figures and consider the rest as a decimal.

2°. Find the mantissa corresponding to the first four figures (Rule XXIV). 3°. Multiply the tabular difference by the decimal cut off, i.e., by the remaining figures of the given number, and cut off from the right-hand as many figures as there are in the multiplier, but at the same time adding unity if the highest figure thus cut off is not less than 5.

4°. Add the integer part of this product to the figures of the mantissa just found. These proportional parts are thus compiled on the supposition that the difference between the numbers (nearly equal to each other) is proportional to

the difference between their logarithms. This proportion can be shown to be approximately true.

The result is the mantissa of the required logarithm.

The characteristic or index is found by Rules XVIII and XIX, pages 61 and 62.

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 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.

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 4 stands 941213; then in the right-hand column in a line with this stands the diff. 50, which being multiplied by 57, the remaining digits of the given number, makes 2580; from this we cut off two digits to the right (since we have multiplied by two digits), when it becomes 18; 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 XVIII and XIX.)

91. To find the natural number corresponding to a given logarithm.If the logarithm be given, the number which corresponds to it may be found by the following rules, which are the converse of those last given for finding the logarithm when the number is given.

....

Since the characteristic denotes how many places the first significant figure stands to the right or left of the unit's place; conversely, therefore, if logs. be given having for characteristics 1, 2, 3, 1, 2, 3, .... there are in the integral parts of the number to which these logs. belong, 2, 3, 4, 0, 1, 2, ... digits respectively. In illustration of these remarks take the following:

Log. 4'589950 (in which characteristic 4) gives 38900

....

In which it will be observed that the first answer must consist of five integers, because the index of the given logarithm is 4; that the second answer must contain four integers, because the index of the given logarithm is 3; that the third answer must contain three integers, because the index of the logarithm is 2, &c., &c.; and that the sixth answer must be a decimal fraction having the first significant figure in the place of tenths, because the logarithmic index is ī; and lastly, that the seventh answer must be a decimal fraction having the first significant figure in the place of hundredths, because the logarithmic index is 2.

92. From the foregoing it is evident that when the figures of the natural number have been found, we must place the decimal point so that the number of integral figures may be one more than the characteristic denotes. Cyphers must be supplied to the right, if necessary, to make up the number, hence

I

RULE XXVI.

Add 1 to the characteristic of the given logarithm, and mark off to the left the number of figures for whole numbers; the rest (if any) will be decimals.

If the characteristic is negative place the decimal point to the left of the natural number found, along with as many cyphers as may remove the first significant figure to that place of decimals which the index expresses; that is, one cypher fewer than the number denoted by the characteristic, whence, to find the place of the decimal, we have the following

RULE XXVII.

The number corresponding to a logarithm with a negative index is wholly decimal, and the number of cyphers following the decimal point is one less than the characteristic of the logarithm.

But instead of the negative characteristic its arithmetical complement is sometimes used, in which case we proceed by

RULE XXVIII.

Add to the index, and subtract the number thus found from 10; the remainder is the number of cyphers to be prefixed to the figures taken out of the Tables. Place the dot before the first cypher.

93. To find the natural number corresponding to any given logarithm. When the mantissa or decimal part of the logarithm can be found exactly in the Table, we proceed by

RULE XXIX.

1°. Seck out the mantissa, and take from the column No. the three figures in the same horizontal row.

2°. From the head of the column take the fourth figure.

3°. From the characteristic find by the rules already given the position of the decimal point, and so adjust the local value of the figures. (Rules XXVI, XXVII, and XXVIII, No. 92, page 68).

(a) When the characteristic of the given logarithm requires a greater number of digits to the left of the decimal point than there are in the number found by the above rule, the deficiency is made up by adding a sufficient number of cyphers to the right.

(b) If the natural number is a decimal fraction, and the final figure or figures are cyphers, they need not be written down.

EXAMPLES.

Ex. 1. Given the logarithm 2.698970 to find the natural number.

Entering the Table with the decimal part 698970, we find the natural number corresponding to it to be 5, or 50, or 500, or 5000, &c., but as the index of the logarithm is 2, the natural number must contain three integral figures. Hence the natural number of 2·698970 is 500.

Ex. 2. Given the logarithm 3.539954 or 7*539954: find the number.

Entering the Table with the decimal part, we find the corresponding number is 3467; to this we prefix two cyphers, since the index is 3; or adding 1 to 7 (8), and subtract 8 from 10, we have 2, the number of cyphers to be prefixed, and then the decimal point; hence the number corresponding to 7.539954 is '0034567.

Ex. 3. What number corresponds to the logarithm 4'214314.

The decimal part of the log. being found opposite 163 and under the figure 8 at the top of the page: therefore the digits of the required number are 1638. But as the characteristic is 4, there must be in it 5 places of integers. A cypher is annexed (see Rule XXIX, (a). Hence the required number is 16380.

Ex. 4. Required the natural numbers corresponding to logs. o'176091 and 4'176091. (1) The mantissa 176091 stands in the Table opposite 150, and the column with 0 at the top; and the characteristic 0 shows that one of these is integral, whence the number sought is 1.500 or 1.5 (see Rule XXVI, page 68).

(2) The mantissa of second log. being the same as that of the first, the corresponding number will consist of the same significant figures, but the characteristic 4 shows that the first significant figure (1) must occupy the fourth place to the right of the decimal point, whence the number sought is coo15. (See Rule XXVII or XXVIII, pages 68 and 69.)

Ex. 5. Required the natural number whose logarithms are respectively 1.813514, *303412, 4'996993, 2·299943 or 8·299943, 4′000000, 4000000, 7.816109, we shall find them to be as follows:

Where it will be observed that the first answer must contain only two integers, as the index of the given logarithm is 1; that the second must contain only one integer as the characteristic is o; that the third must consist of five integers, because the index of the given logarithm is 4, and therefore to 9931, the number found in the Table, a cypher is annexed, (see Rule XXIX, (a); and that the fourth answer must be a decimal, having the first significant figure two places to the right of the decimal point because the characteristic is ž; the fifth answer must consist of five integral figures (a cypher being annexed to make up the number) since the characteristic is 4; the mantissa of the sixth log., or 'oo0000, gives the corresponding natural number 1000, but adjusting the decimal punctuation, or the local value of the figures, the characteristic 4 denotes that the first significant figure (1) must stand in the fourth decimal place, and, therefore, three cyphers must be prefixed, and the natural number will be 'ooor-the three final cyphers not being written down. Finally, the mantissa of last log. being found in the table gives the natural number corresponding as 6548, to which annex four cyphers; the characteristic 7 determines the number to consist of 8 integral figures.

94. When, as usually happens, the mantissa cannot be found exactly in the Tables, but lies between two successive records in the Tables, and it is proposed to find the corresponding number correct to six places of figures, other than final cyphers immediately following a decimal point, the number is to be found by the method of proportional parts, on the supposition that, between two successive records in the table, the number advances in proportion to the increase of the logarithm.

95. To find the natural number corresponding to a given logarithm, when more than four figures are required. We proceed by

RULE XXX.

1o. Having found the next lower mantissa in the Tables, note the four figures which correspond to it.

2°. From the given logarithm subtract that taken out of the Tables, divide the remainder (annexing as many cyphers as there are digits required above four) by the tabular difference, and reduce the quotient to the form of a decimal.