the fractional form always denotes the division of the quantity above the line by that below it.

Here the learner may require some explanation respecting the index of the answer. When, in subtracting, he comes to the place of tenths, the process is carried to 5 makes 6, which subtracted from 14 leaves 8; then I carried to 1 makes 2, which subtracted from 1 leaves ī, that is, I still to be subtracted; the characteristic of the quotient: but as observed (foot of page 17) to avoid negative characteristics, borrow 10, whence the index is 9.

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 units' 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, o, ī, ž, .... digits respectively. In illustration of these remarks take the following:

....

To find the place of the decimal point proceed as follows:

I

RULE IX.

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

Thus if the index of a logarithm be 2, the natural number contains three integral places, and so on: the rest of the figures (if any) in the natural number taken out are decimals.

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.

Thus if the index is 4, prefix three cyphers to the figures taken out of the table; if z, prefix one cypher; if ī, the decimal point is placed next the figures taken out.

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

RULE X.

Add 1 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.

Thus, if the index is 3 or 7, I added to 7 is 8, then 8 from 10 leaves 2; whence we prefix two cyphers to the number taken out of the table before placing the decimal point.

When the mantissa or decimal part of the logarithm can be found in the table, we proceed by

RULE XI.

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

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

3°. From the characteristic or index find by the rules already given the position of the decimal point.

(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 zeros (cyphers) to the right.

EXAMPLES.

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.

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, 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 003467.

What number corresponds to the logarithm 4214314.

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 XI, (a)). Hence the required number is 16380.

Required the natural numbers corresponding to logs. o 176091, and 4176091.

(1). The mantissa 176091 stands in the table opposite 150, and in the column with O at the top; and the characteristic o shows that one of these is integral, whence the number sought is 1500 or 15 (see Rule IX, page 23).

(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 00015.

Required the natural numbers whose logarithms are respectively 1813514, 30341, 4996993, and 2'299943 or 8·299943, 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; 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 z.

When, as usually happens, the mantissa cannot be found in the tables, we proceed by

RULE XII.

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

20. 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. 3. To the four figures already found, add this decimal, and shift the decimal point to suit the characteristic of proposed logarithm.

The result will be the required number.

Given the logarithm 3'54027 to find the natural number,

Attaching this (62) to the four figures, we have 349162, &c. But as the index is 3, we obtain, by pointing off four figures to the left, 3491'62 the natural number sought. Given the logarithms 5'654329 and 2·654273 to find the natural numbers. 654273, which corresponds with the natural number 4511, is the logarithm next less than the given one; therefore the first four digits of the required number are 4511. Adding two cyphers to 56, the difference between 654273 and the given logarithm, it becomes 5600, which being divided by 96, the tabular difference corresponding with 4511, gives 58 as quotient and 32 as remainder. The integers of the required number (one more than 5, the characteristic) are, therefore, 451158. The mantissa of the second log. being the same as the first one, the natural number will contain the same significant figures, viz., 451158, but the characteristic z shows that the first significant figure of the nat, no. (4) must stand in the second place to the right of the decimal point; therefore, the nat. no, corresponding to Z 654273 is '0451158. Let it be required to find the number of which the logarithm is 3*104831.

Therefore, 3*104831 = log, of 1273009 nearly. In dividing by tab. diff. we take remainder 3 and a cypher, then 341 in 30 goes no times, which we place down in the

quotient, then taking another cypher we have 300, which contains 341 no times, lastly, 341 goes into 3000 eight times with 272 for remainder, The remainder 272 being more than half the quotient the last figure of the quotient (8) is increased by

The arithmetical complement of a number is the number by which it falls short of the unit of the next higher denominator. It is abbreviated into Ar. co.

:

The lowest denominator considered is the unit, thus:

The most expeditious way of finding the arithmetical complement is as follows:Begin from the left, subtract every figure from 9 up to the lowest significant figure, which subtract from 10. Repeat the cyphers at the end, if any.

A subtractive quantity is, by this means, made additive. The process is equivalent to subtracting the number from 10, and the reason of it is evident on considering that to add 3 and subtract 10 is the same as to subtract 7. In like manner, instead of subtracting 42m 10s for example, we may add 17m 50s (the complement to 60m), provided we subtract 1b (or 60m); and thus any number of quantities of which some are additive and some subtractive, may be rendered all additive, provided that the larger numbers which are employed in taking the complements be themselves subtracted.

Degree of Dependence.-The number of places of figures which may be obtained in a result derived from any table of logarithms, is the same usually, rejecting prefixed cypher, as the number of decimals to which the logarithms are carried. But towards