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those, which are to be found at present, in most of the common tables of logarithms.

The distinguishing mark of this system of logarithms is, that the index or logarithm of 10 is 1; that of 100 is 2; that of 1000 is 3, &c. And, in decimals, the logarithm of I is —1; that of or is -2; that of 001 is -3, &c. the logarithm of 1 being o in every system.

Whence it follows, that the logarithm of any number between 1 and 10 must be o and some fractional parts; and that of a number between 10 and 100, I and some 'fractional parts; and so on, for any other number what

ever.

And since the integral part of a logarithm, thus readily found, shews the highest place of the corresponding number, it is called the index, or characteristic, and is commonly omitted in the tables; being left to be supplied by the person, who uses them, 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 Nr", 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 o, then N is 1, whatever the value of r is; which shews, that the logarithm of 1 is always o, in every system of logarithms.

When n is, then N is r; so that the radix r is always that number, whose logarithm is 1 in every system.

When the radix r is 2718281828459, &c. the indices n are the hyperbolic or Napier's logarithm of the numbers N; so that n is always the hyperbolic logarithm

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of the number N or 2.718, &c..

But when the radix r is 10, then the index n becomes the common or Briggs' logarithm of the number N; so

that

that the common logarithm of any number 10" 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 106997, then is 169897 the common logarithm of And, in general, the following decuple series of

50. terms,

3

viz. 10, 103, 102, 10', 10°, 1072, 10-2, 1073, 104, OI, 001, 0001,

or 10000, 1000, 100, 10, I, 'I,

3, 2, I, 0,

I,

have 4, 2, -3, 4, for their logarithms, respectively. And from this scale of numbers and logarithms, the same properties easily follow, as before mentioned.

PROBLEM.

To compute the logarithm to any of the natural numbers, 1, 2, 3, 4, 5, &c.

RULE.

Let b be the number, whose logarithm is required to be found; and a the number next less than b, so that b-a1, 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 respectively by

the

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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

so on.

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 a, so will the last sum be the logarithm of the number b proposed.

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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+13=5, and its square 9. Then the operation will be as fol

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9)*289529654

1)*289529654(*289529654

3) 32169962( 10723321

9) 32169962 5) 3574440(

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EXAMPLE 2. To compute the logarithm of the number 3.

Here 3, the next less number a=2, and the sum a+b=5s, whose squares is 25, to divide by which, always multiply by 04. Then the operation is as follows: 5).868588964 1)*173717793(*173717793

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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 :

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And thus, computing by this general rule, the logar→→ ithms to the other prime numbers 7, 11, 13, 17, 19, 23, &c. and then using composition and division, we may ea

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