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endless variety of systems of logarithms, to the same common

rumbers, by only changing the second term, 2, 3, dr 10, &c. of the geometrical series of whole numbers ; and by interpolation the whole system of numbers may be made to enter ihe geometric series, and receive their proportional logarithms, whether integers or decimals.

It is also apparent, from the nature of these series, that if any two indices be added together, their sum will be the index of that number which is equal to the product of the two terms, in the geometric progression, to which those indices belong. Thus, the indices 2 and 3, being added to. gether, make 5; and the numbers 4 and 8, or the terms corresponding to those indices, being multiplied together, make 32, which is the number answering to the index 5.

In like manner, if any one index be subtracted from another, the difference will be the index of that number which is equal to the quotient of the two terms to which those indices belong. Thus, the index 6, minus the index 4, is

2 ; and the terms corresponding to those indices are 64 and 16, whose quotient is = 4, which is the number answering to the index 2.

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For the same reason, if the logarithm of any number be multiplied by the index of its power, the product will be equal to the logarithm of that power. Thus, the index or logarithm of 4, in the above series, is 2 ; and if this number be multiplied by 3, the product will be = 6; which is the logarithm of 64, or the third power of 4.

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And, if the logarithm of any number be divided by the index of its root, the quotient will be equal to the logarithm. of that root. Thus, the index or logarithm of 64 is 6; and if this number be divided by 2, the quotient will be = 3; which is the logarithm of 8, or the square root of 64.

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The logarithms most convenient for practice, are such as are adapted to a geometric series increasing in a tenfold proportion, as in the last of the above forms; and are those which are to be found, at present, in most of the common tables on this subject. The distinguishing mark of this system of logarithms is, that the index or logarithm of 10 is !; that of 100 is 2 ; that of 1000 is 3 ; &c.

decimals,

And, in

deciñhals, the logarithm of 1 is -1 ; that_of of is – %; that of .001 is = 3; &c. The log. of 1 being o in every system Whence it follows, that the logarithm of any number between 1 and 10, must be 0 and some fractional parts ; and that of a number between 10 and 100, will be 1 and some fractional parts; and so on, for any other number whatever. And since the integral part of a logarithm, usually called the Index, or Characteristic, is always thus readily found, it is commonly omitted in the tables; being left to be supplied by the operator himself, as.occasion requires.

Another Definition of Logarithms is, that the logarithm of any number is the index of that power of some other number, which is equal to the given number. So, if there' be. Nagh, then n is the log. of N ; where n may be either positive or negative, or nothing, and the root r any number whatever, according to the different systems of logarithms. When n is = 0, then N is = 1, whatever the value of r is; which shows, that the log. of 1 is always 0, in every system of logarithms. When n is = 1, then N is =r; so that the radix , is always that number whose log. is 1, in every system. When the radix is 2.71828 1828459 &c. the indices n are the hyperbolic or Napier's log of the numbers N; so that n is always the hyp. log. of the nuinber N or (2.719 &c.)".

But when the radix p is 10, then the index n becomes the common or Briggs's log, of the number N: so that the common log. of any number 1on or N, is n the index of that power of 10 which is equal to the said number. Thus 100, being the second power of 10, will have 2 for its logarithm ; and 1000, being the third power of 10, will have 3 for its logarithm : hence also, if 50 be 101.69897, then is 1•69897 the common log. of 50. And, in general, the following decuple series of terms, viz. 10", 103, 102, 10", 10°, 10-, 10-, 10-5, 10-" or 10000, 1000, 100, 10, 1, "l, 01, ·001, .0001, have 4, 3, 2, 1, 0,, -2, -3, -4, for their logarithms, respectively. And from this scale of numbers and logarithms, the same propertiei easily follow, as above mentioned.

PROBLEM

PROBLEM.

compiute the Logarithm to any of the Natural Numbers

1, 2, 3, 4, 5, &c.

RULE I

TẢke 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+0=2=} = 5 is the arithmetical mean, and ✓ 10x1 Ņ 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. 1 + 5 = 2

75 is the arithmetical mean, and ✓ 10 X3.1622777 = 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. 1 + 75 – 2

.875 is the arithmetical mean, and ✓ 10 X 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 7.4989423 is 875; theref. 1 + •875 * 2 =

.9375 is the arithmetical mean, and ✓ 10 X 7.4989422 = 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 con. sult 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 :9875;

theref. 1+.9375 - 2 .96875 is the arithmetical nean, and Ņ 10 X 8.6596431 9.3057 204 the geom. mean ;

hence the log. of 9.3057204 is 96875. Sixthly, the log. of 8 6596431 is :9375, and the log. of

9.3057204 is .96875 ;
theref, .9375 +.96875 = 2 = .953125 is the arith. mean,
and ✓ 8.6596431 x 9.3057204 = 8.9768713 the geo.

metric 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 frem 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 o be the number whose logarithm is required to be found; and a the number next less than b, so that 6

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 8, 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 18, as long as division can be made

2. Then write these quotients orderly under one another, the first uppermost, and divide them respectively 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

Then, because the sum of the logarithms of numbers, gives the logarithm of their product ; and the difference of

The logarithms, gives the logarithin of the quotient of the

That is,

n

1

1

+

1 Log. of b is log. a t - *(1+ +

+&c.) where na denotes the constant given decimal •8685889638 &c.

8

382

5.4

786

EXAMPLES

= 8, and its

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Ex. 1. Let it be required 10 find the log of number 2.
Here the given number bis 2, and the next less number a
is 1, whose log. is 0; also the sum 2 + 1 : 3 =
square 9. Then the operation will be as fo:lows :

3 ) •868588964 1) 289529654 ( 289529654
9

) •289529654 3) 32169962 ( 10723321
9

32169962 5) 3 1744 10 ( 714888
3574440 7) 397160

56737
397.60

44129

4903
9
44129
11
4903

446
9
4903

545

42
9
545

61 1
61

log of i 301029995
add log. 1 . '000000000

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13 ) 15 )

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to = 5

he ast be

Ex. 2. To compute the logarithm of the number 3,
Here 6 = 3, the next less number a = 2, and the sum

= 8, whose square so is 25, to divide by which; always multiply by •04. Then the operation is as follows:

5) 868588964 1) 1737 17793 ( 1737 17793 25 ):173717793 3) 6948712 231637 6948712 5) 277948

55590 277948 7) 11118

1588 I1018

445

50 445 11

18 (

2
18

log. of •17609 1960
log, of 2 add •301029995

cr,

25 ) 25 ) 25 ) 25 )

De;

bas

Till add

$0 ed

log. of 3 sought .477121255

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