PROBLEM.

To compute the Logarithm to any of the Natural Numbers 1, 2, 3, 4, 5, &c.

RULE I*.

TAKE the geometric series, 1, 10, 100, 1000, 10000, &c. and apply to it the arithmetic series, 0, 1, 2, 3, 4, &c. as logarithms. Find a geometric mean between 1 and 10, or between 10 and 100, or any other two adjacent terms of the series, between which the number proposed lies.-In like manner, between the mean, thus found, and the nearest extreme, find another geometrical mean; and so on, till you arrive within the proposed limit of the number whose logarithm is sought. Find also as many arithmetical means, in the same order as you found the geometrical ones, and these will be the logarithms answering to the said geometrical

means.

EXAMPLE.

Let it be required to find the logarithm of 9.

Here the proposed number lies between 1 and 10. First, then, the log. of 10 is 1, and the log. of 1 is 0;

theref. 1+02·5 is the arithmetical mean, and 10 x 1√10 3·1622777 the geom. mean ;

hence the log. of 3.1622777 is 5.

Secondly, the log. of 10 is 1, and the log. of 3.1622777 is 5; theref. 15275 is the arithmetical mean,

and 10 x 31622777 5.6234132 is the geom. mean; hence the log. of 5.6234132 is 75.

Thirdly, the log. of 10 is 1, and the log. of 5.6234132 is 75; theref. 1752 875 is the arithmetical mean,

and 10 × 5.6235132 = 7·4989422 the geom. mean; hence the log. of 7·4989422 is ·875.

Fourthly, the log. of 10 is 1, and the log. of 7·4989422 is '875;

theref. 1875 ÷ 2·9375 is the arithmetical mean, and 10 x 74989422 = 8·6596431 the geom. mean, hence the log. of 8·6596431 is ·9375.

*The reader who wishes to inform himself more particularly concerning the history, nature, and construction of Logarithms, may consult the introduction to my Mathematical Tables, lately published, where he will find his curiosity amply gratified.

Fifthly,

Fifthly, the log. of 10 is 1, and the log. of 8.6596431 is 9375; theref. 1+9375÷296875 is the arithmetical mean,

and 10 x 8.6596431 = 9.3057204 the geom. mean ; hence the log. of 9.3057204 is ·96875.

Sixthly, the log. of 86596431 is 9375, and the log. of 9.3057204 is 96875;

theref. 9375968752953125 is the arith.mean, and 8.6596431 x 9.3057204 8.9768713 the geometric mean;

hence the log. of 8.9768713 is 953125.

And proceeding in this manner, after 25 extractions, it will be found that the logarithm of 8.9999998 is 9542425; which may be taken for the logarithm of 9, as it differs so little from it, that it is sufficiently exact for all practical purposes. And in this manner were the logarithms of almost all the prime numbers at first computed.

RULE II*.

LET be the number whose logarithm is required to be found; and a the number next less than b, so that b-a=1, the logarithm of a being known; and let s denote the sum of the two numbers a + b. Then

1. Divide the constant decimal 8685889638 &c, by s, and reserve the quotient: divide the reserved quotient by the square of S, and reserve this quotient: divide this last quotient also by the square of s, and again reserve the quotient and thus proceed, continually dividing the last quotient by the square of s, as long as division can be made.

2. Then write these quotients orderly under one another, the first uppermost, and divide them re pectively by 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 so on.

3. Add all these last quotients together, and the sum will be the logarithm of ba; therefore to this logarithm add also the given logarithm of the said next less number a, so will the last sum be the logarithm of the number 6 proposed.

*For the demonstration of this rule, see my Mathematical Tables, p. 109, &c.

That

where n denotes the constant given decimal 8685889638 &c.

EXAMPLES.

Ex. 1. Let it be required to find the log. of the number 2. Here the given number b is 2, and the next less number a is 1, whose log. is 0; also the sum 2+1=3 = s, and its square s2 9. Then the operation will be as follows:

Ex. 2. To compute the logarithm of the number 3. Here b 3, the next less number a= 2, and the sum a + b = 5 = s, whose square s2 is 25, to divide by which, always multiply by 04. Then the operation is as follows:

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;

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:

And thus, computing, by this general rule, the logarithms to the other prime numbers, 7, 11, 13, 17, 19, 23, &c, and then using composition and division, we may easily find as many logarithms as we please, or may speedily examine any logarithm in the table *.

There are, besides these, many other ingenious methods, which later writers have discovered for finding the logarithms of numbers, in a much easier way than by the original inventor; but, as they cannot be understood without a knowledge of some of the higher branches of the mathematics, it is thought proper to omit them, and to refer the reader to those works which are written expressly on the subject. It would likewise much exceed the limits of this compendium, to point out all the peculiar artifices that are made use of for constructing an entire table of these numbers; but any information of this kind, which the learner may wish to obtain, may be found in my Tables, before mentioned.

Description

Description and Use of the TABLE of Logarithms.

HAVING explained the manner of forming a table of the logarithms of numbers, greater than unity; the next thing to be done is, to show how the logarithms of fractional quantities may be found. In order to this, it may be observed, that as in the former case a geometric series is supposed to increase towards the left, from unity, so in the latter case it is supposed to decrease towards the right hand, still beginning with unit; as exhibited in the general description, page 148, where the indices being made negative, still show the logarithms to which they belong. Whence it appears, that as+1 is the log. of 10, so I is the log. of or 1; and as+2 is the log. of 100, so 2 is the log. of

01 and so on.

or

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, but 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 3.423410, the log. of ro, or to, or roos, &c, part of it; will be as follows:

I

Hence it also appears, that the index of any logarithm, is always less by 1 than the number of integer figures which the natural number consists of; or it is equal to the distance of the first 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-hand side of the decimal part of the logarithm.

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 it, or else above it. Thus,

A number having 1, 2, 3, 4, 5, &c, integer places, the index of its log. is 0, 1, 2, 3, 4, &c. or 1 less than those places.

And