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11. Divide 8408 by 9627.

Ans. 8734-.

12. Divide 005,842 by 2382.

Ans. 02453 - .

PROBLEM XII.

To find the fourth Term of a Proportion by Logarithms.

RULE. Add together the logarithms of the second and third terms, and from the sum subtract the logarithm of the first.

EXAMPLE 1. Find the fourth term of the following proportion. As 657 1533 :: 279 :

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NOTE 1. After a little practice, the student will be able to subtract the first logarithm from the sum of the other two at once, without previously setting down the said sum. But, if he find that he cannot do this easily and correctly, his best way is to put down the second and third terms before the first, leaving a space between for the sum of the former, thus:

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NOTE 2. If the first term is a decimal fraction without integers, remove the decimal point as many places to the right as will render the first figure an integer, and make compensation for so doing, by removing the decimal point an equal number of places to the right in the second or third term, or in both together.

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NOTE 3. After the operation directed in the preceding note, or when that is not required, if the second or third term is a decimal fraction without integers, remove the decimal point as far to the right as will render the first figure an integer, and make compensation for it in the fourth term, when found, by returning the point equally far to the left.

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EXAMPLE 6. As 0·336: 0·484 :: 095 :

Here we first proceed by Note 2, changing the first and second terms to 3.36 and 4.84. After that, since the third remains in a fractional form, we make it 9.5, and make compensation by restoring the decimal point two places to the left in the answer.

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NOTE 4. If the removal of the decimal point, as directed in the last note, was necessary both in the second term and in the third, restore the decimal point in the fourth term, by moving it as many places to the left as will be equal to the number of places it was previously removed to the right in the second and third terms together.

EXAMPLE 7. As 5·33 : 0.28 :: 0·055 :

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NOTE 5. When the three given terms commence with integral figures, or when they have been transformed by the directions in the preceding Notes, so as to commence with integers-if the logarithm of the first term is greater than the sum of the logarithms of the other two terms, remove the decimal point to the right, in either the second or the third term, as far as may be necessary to render the sum of the two last-mentioned logarithms greater than the logarithm of the first term. Then, after finding the fourth term, count how many places the decimal point has been removed in the second and third terms together, more than in the first, and restore it that number of places in the fourth term.

EXAMPLE 8. As 0·977 0.146 :: 0·045 :

9.77...............0.989895

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Here we find the decimal point has been removed one place in the first term, and four places in the other two terms together; we must therefore restore it three places in the answer.

Ans. 005342-.

EXERCISES. Compute the fourth term of each of the following proportions, by Logarithms, and prove the first three by common Arithmetic.

1. As 1314: 3066 :: 558 : 1302.

2. As 321-33: 14·282 :: 39-419: 1.752. (See Problem vi, Rule 1.)

3. As 35.04: 7·884 :: 2856: 642.6.

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5. As 0-3504: 78.84: 2·856 642-6. (See Note 2.) 6. As 00098: 0·304 :: 0235 : 7·290—.

7. As 4.38: 0·985 :: 379 : 85.23 +. (See Note 3.) 8. As 0-5072 : 0·4814 :: 0688: 06530+.

9. As 2·69 : 0·548 :: 0-886: 0·1805. (See Note 4.) 10. As 0.9826: 04385 :: 07848: 003502+.

11. As 1314 : 30·66 :: 5·58 : 0·1302. (See Note 5.) 12. As 32.133 : 1·4282 :: 0·39419 : 01752.

PROBLEM XIII.

To Square or Cube a given Number by Logarithms.

RULE. Multiply the logarithm of the number by 2 for the square, or by 3 for the cube; and find the natural number answering to the product.

EXAMPLE 1. What is the cube of 12?

Num. 12.........Log. 1.079181

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NOTE. If the given number is a decimal without integers, remove the decimal point as many places to the right as shall render the first significant figure an integer; and, when the square or cube is found, restore the decimal point in the result to its proper place, by removing it to

the left,-in the case of the square, twice as many places as it was previously removed to the right, and, in the case of the cube, three times as many.

EXAMPLE 2. What is the square of '0836?

Num. 8.36.........Log. 0.922206

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EXAMPLE 3. What is the cube of 270695?

Num. 2.707............Log. 0·432488

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EXERCISE 1. Square the numbers 12, 65.4, 49.59, and

4.732.

Answers 144, 4277+, 2459+, 22.39 +.

2. Cube the numbers 19, 5.832, and 16.28.

Answers: 6859, 1984, and 4315-. 3. Square the number 793, and cube the same number, giving the answers correct to four figures each, and supplying the remaining places with ciphers.

Answers: 628,800, and 498;700,000. 4. Square the numbers 01738, 00256, and 26954. (See the Note.)

Answers: 000,302,1,000,006,554, and '072,65+. 5. Cube the numbers 3462, and 0987.

Answers: 041,49+, and 000,961,5+.

PROBLEM XV.

To extract the Square Root or Cube Root of any given Number.

RULE. Divide the logarithm of the given number by 2 or by 3. The natural number answering to the quotient will be the square root or the cube root required.

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