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RULE. Find the nearest less logarithm in the tables to the given one, and take out the four figures answering to it as before. Take the difference between this logarithm and the next greater in the tables, and also between this logarithm and the given one; divide the latter difference, with ciphers annexed to it, by the former, and place the quotient to the right hand of the natural number already found.

Required the natural number answering to the log. 4.59859. Given log. 59859 nearest less log. 59857 next greater 59868

Next less 59857 nat. numb. 3968

diff. 2 annex ciphers.

diff. 11

then 200, &c. 1118, &c. the natural number is therefore 39681.8, the index being 4, there are five whole numbers.

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(L) To find the product of two whole or mixed numbers." RULE. Add the logarithms of the numbers together, the natural number answering to the sum will be the product required.

When several numbers are to be multiplied together, some of which are less than an unit, add the logarithms of the numbers together; when you come to the indices, add the affirmative indices and what you carry into one sum, and the negative indices into another. Take the difference between these sums for the index of the product, prefixing the sign of the greater sum. Required the product of 84 and 56.

1.92428

Logarithm of 84
Logarithm of 56

=

= 1.74819

Product 4704 Log. — 3'67247

Required the product of 76.5 by 5'5.

Logarithm of 76.5

= 1.88366

Logarithm of 5.5

=

0.74036

Product 42075 Log. = 2·62402

-X

a; 10 =b; 10 ~y = c; 10 = d, &e. Then 10-x-y+% Where v is the logarithm of a;

r the logarithm of b;

-y

• Let 10" =abcd, &c. the logarithm of e; x the logarithm of d, &c. And v-x-y+x is the logarithm of a b c d. (A. 1. and C. 2.)

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Logarithm of the product = 0·24067 The number answering to which is 1.7405, the product required.

Required the product of 37 × 426 × 5 × '004 × 275 × 336. Answer 29.128.

PROPOSITION VIII.

(M) To divide one number by another.*

Subtract the logarithm of the divisor from the logarithm of the dividend, and the remainder will be the logarithm of the quotient.

If any of the indices be negative, or if the divisor be greater than the dividend, change the index of the divisor: then if the indices have unlike signs take their difference, and prefix the sign of the greater; if they have like signs take their sum, and prefix the common sign. When there is an unit to carry from the decimal part of the logarithm of the divisor, subtract it from the index of that logarithm if it be negative, otherwise add it, before the signs are changed.

Divide 3450 by 23.

Logarithm of 3450

3.53782

Logarithm of

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(N) To involve a number to any power; that is to square, cube, and number, &c.*

RULE. Multiply the logarithm of the number by the number expressing the power, viz. by 2 for the square, 3 for the cube, 4 for the biquadrate, &c.; and the product will be the logarithm of the power required.

If the index to the given logarithm be negative; multiply this index and the decimal part of the logarithm separately by the index of the given power, and subtract the former product from the latter; when you come to the indices, or whole numbers of these products, their difference must be taken, but if there be an unit to carry from the decimal part of the lower product, it must be added to the index of that product before you take the difference. The result will in every case be negative.

Required the cube of 1.2.

Logarithm of 1.2 = 0.07918

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* Let 10% = a, and n = the index of the power, then 10x is the logarithm of a, and nx the logarithm of a".

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Then 86034 × 3 = 2·58102 product of the decimal.

and 2 × 3 = 6:

Power 000381 log. 4.58102

product of the indices.

Required the 6.25 power of 0032.

The logarithm of 0032 is 3.50515.

Then 50515 × 6·25 3.15719

and 3 x 6.25 = 18.75

Power 00000000000000025538log.-16.40719

PROPOSITION X.

product of the dec. product of the ind.

(0) To extract the square or cube root, &c. of any number.* Divide the logarithm of the number by 2 for the squareroot, 3 for the cube-root, &c. and the quotient will be the logarithm of the root.

If the index to the logarithm be negative, and does not exactly contain the divisor, increase it by such a number as will make it exactly divisible, and increase the logarithm also by the same number before you begin to divide.

What is the square-root of 3.24?

Logarithm of 3.24 is 0.51054, which divided by 2 gives 0.25527, the number answering to which is 1.8. What is the cube-root of 10648?

Logarithm of 10648 is 4.02726, which divided by 3 gives 1.34242, the number answering to which is 22.

What is the cube-root of 0003811?

The logarithm of 0003811 is-4.58104-4+0.58104; and by adding 2 to each part, it is=-6+2.58104, divide by 3, then 2.86034 is the logarithm of the root, the number answering to which is '0725, the root sought.

What is the 72 root of '096?

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The logarithm of '096 is 2.98227-2+0.98227, and by adding 16 to each part (in order that the negative index may divide even by '72) it becomes 2.161.14227, divide

* Let 102 = a, and n = the index of the root. Then 10 =

n

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1

is the logarithm of a, and is the logarithm of an

n

by 72 then 3 + 1·586482.58648 is the logarithm of the root; hence the root sought is '03859.

PROPOSITION XI.

(P) To find the value of a quantity having a vulgar fraction for its exponent.

RULE. Multiply the logarithm of the given number by the numerator of the exponent, and divide the product by the denominator; the quotient will be the logarithm of the quantity required.

The multiplication must be performed as directed in the 9th Proposition, and the division according to the directions given in the 10th Proposition: for, the numerator denotes the power to which the given number is to be raised, and the denominator shows what root of that power is to be extracted.

What is the value of 0961?

The logarithm of '096 — — 2.98227 which multiplied by 25 (Prop. IX.) produces — 26.55675; this divided by 18 (Prop. X.) gives 2.58648; the number answering to which is 03859. Answer.

PROPOSITION XII.

(Q) To find a fourth proportional to three given numbers; or to work a question in the rule of three by logarithms.

RULE. Add the logarithms of the second and third terms together, and from the sum subtract the logarithm of the first term, the remainder will be the logarithm of the fourth term. What is the fourth proportional to 75; 36; and •008? Logarithm of 36 = 1.55630

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3.90309

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answering to which is 384 the fourth proportional required. For, 75: 36 :: 008: 384.

PROMISCUOUS EXAMPLES, EXERCISING ALL THE PROPOSITIONS.

(1.) Find the logarithm of 36 Ans. — 1·88303.

48

(2.) Required the logarithm of 563' or '56. Ans.-1.75076. (3.) Required the logarithm of '06'28803'. Ans.-2.79751.

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