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the odd numbers, 1, 3, 5, 7, 9, &c. as long as division can be made ; that is, divide the first reserved quotient by 1, the second by 3, the third by 5, the fourth by 7, and
3. Add all these last quotients together, and the sum will be the logarithm of b-a; therefore to this logarithm add also the given logarithm of the said next less number ag so will the last sum be the logarithm of the number b proposed. That is, log. of b is log. a't xiita
. ; + &c. where n denotes the constant given decimal •8685889638, &c.
EXAMPLE 1. Let it be required to find the logarithm of the number 2.
Here the given number b is 2, and the next less number a is 1, whose logarithm is o ; also the sum 2+1=3=5, and its square s* =9. Then the operation will be as fol
EXAMPLE 2. To compute the logarithm of the number 3.
Here b=3, the next less number a=2, and the sum atb=5=s, whose square să is 25, to divide by which, always multiply by '04. Then the operation is as follows:
5):868588964 1):173717793('173717793 25):173717793
3) 69487120 2316237 25) 6948712
2779486 55590 25) 277948
11118 1588 25) II118
Log. of } .176091260
Log. of 3 sought '477121255
Then, because the sum of the logarithms of numbers gives the logarithm of their product, and the difference of the logarithms gives the logarithm of the quotient of the numbers, from the above two logarithms, and the logarithm of 10, which is 1, we may raise a great many logarithms, as in the following examples :
To logarithm 2 *301029995
Sum is logarithm 6*778151250.
And thus, computing by this general ruke, the logar ithms to the other prime numbers 7, 11, 13, 17, 19, 23, &c. and then using composition and division, we may ea
sily find as many logarithms as we please, or may speedily examine any logarithm in the table. *
DESCRIPTION AND USE OF THE TABLE OF
Integral numbers are supposed to form a geometrical series, increasing from unity toward the left ; but decimals are supposed to form a like series, decreasing from unity toward the right, and the indices of their logarithms are negative. Thus, +r is the logarithm of 1o, but I is
-I the logarithm of to, or 'I ; and +2 is the logarithm of 100, but -2 is the logarithm of 10 or 'ol; and so on.
Hence it appears in general, that all numbers, which consist of the same figures, whether they be integral, or fractional, or mixed, will have the decimal parts of their logarithms the same, differing only in the index, which will be more or less, and positive or negative, according to the place of the first figure of the number. Thus, the logarithm of 2651 being 3*4234097, the logarithm of io or too or Tool &c. part of it, will be as follows :
04234097 •2651 -14234097 *02651
-2'4234097 *002651 -34234097
* Many other ingenious methods of finding the logarithms of numbers, and peculiar artifices for constructing an entire table of them,
may be seen in Dr. Hutton's Introduction to his Tables, and Baron Maseres' Scriptores Logarithmici.
Hence it appears, that the index, or characteristic, of any logarithm is always less by i than the number of integer figures, which the natural number consists of ; or it is equal to the distance of the first or left hand figure from the place of units, or first place of integers, whether on the left, or.on the right of it: and this index is constantly to be placed on the left of the decimal part of the logarithm. i
When there are integers in the given number, the index is always affirmative ; but when there are no integers, the index is negative, and is to be marked by a short line drawn before, or above, it. Thus, a number having 1, 2, 3, 4, 5, &c. integer places, the index of its logarithm is 0, 1, 2, 3, 4, &c. or 1 less than the number of those places.
And a decimal fraction, having its first figure in the ist, 2d, 3d, 4th, &c. place of decimals, has always --1, -2, -3, -4, &c. for the index of its logarithm.
It may also be observed, that though the indices of fractional quantities be negative, yet the decimal parts of their logarithms are always affirmative.
I, TO FIND, IN THE TABLE, THE LOGARITHM TO ANY
1. If the number do not exceed 100000, the decimal part of the logarithm is found, by inspection in the table, standing against the given number, in this manner, viz. in most tables, the first four figures of the given number
* The Tables, considered as the best, are those of GARDINER in 4to. first published in the year 1742 ; of Dr. Hutton, in 8vo. first printed in 1785 ; of Taylor, in large 4to. published in 1792 ; and in France, those of CALLET, the second edition published in 1795.