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tegers is one or more than the given index as before.

EXAMFLE I.

1. Required the number of the log. 4.55241

The nearest log. which is less is 3.55230 its number is 3567.

The difference of these with three cyphers is for a dividend

11000

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Which quotient place after the first four figures found, and you have 3567916; and because the index is 4, the number will be 35679.16 required. 2. Required, the number answering to the log.

5.09901 The nearest log. to which is 3.09899, its No. 1256

Dividend 02000

к

Log. found 3.09899
Next log. 3.09934

35 Divisor 35)020006(57

175

250
245

5

Because the quotient consists of but two figures, prefix a cypher to it to make it three, and it is 057; which annexed to the first four found, is 1256057; and because the index of the given log. is 5, its number will be 125605.7

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From what has been said on this head, the following problems may easily be solved by logariihms, viz.

PROBLEM I.

Multiply 134 by 25.6

To the log. of 134
Add the log. of 25.6

2.12810 1.40824

Sum 3.53534

The number answering to which sum, viz. 3430, is nearly the product of 133 by 25.6 and is the an

swer.

Again, multiply 234 by 36.

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Again, what is the quotient of 30550 by 47 ?

From the log. of 30550 4.48501
Take the log. of 47 1.67210

2.81291 its number

is 650 the quotient required.

PROBLEM III.

Three numbers in a direct proportion given, to find a fourth.

From the sum of the logarithms of the second and third numbers; deduct the logarithm of the first, the remainder will be the logarithm of the fourth required.

EXAMPLE.

Let the three proposed numbers be 36, 48, 66, to find a fourth proportional.

To log. of 48
Add log. of 66

1.68124
1.81954

Sum
Take log. of 36

3.50078
1.55630

1.94448 the number is 88 the

fourth required.

Again, let three numbers be 240, 1440, 1230, to find a fourth proportional.

To the log. of 1440 3.15836
Add the log. of 1230 3.08991

From the product 6.24827
Take the log. of 240 2.38021

3.86806 its number 7380

the 4th required.

PROBLEM IV.

To find the square of any given number.

Multiply the given number's logarithm by 2, and the product is the logarithm of its square.

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To extract the square root of any given number.

Take half of the logarithm of the number, and that is the logarithm of its square root.

EXAMPLE.

Required the square root of 1296.
Log. of 1296 3.11260

Its half is

1.55630 its number is 36 the

square root of the number required.

By the manner of projecting the lines of chords, sines, tangents, and secants (being prob. 20 of geometry) it is evident, that if the radius be supposed any number of equal parts (as 1000 or 10000, &c.) the sine, tangent, &c. of every arc, must consist of some number of those equal parts; and by computing them in parts of the ra

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