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4. Halve the root and halve the sum. 61-2=30.5

35+2=17,5 5. Add the half root to the half sum to

30.5 produce the greater number.

17.5

The greater number, 48.0 6. Subtract the half root from the half

30.5 sum to produce the lesser number.

17.5

The lesser number, 13.0

CUBE ROOT.
A plain and improved method of extracting the Cube Root.

A TABLE OF CUBES.
Roots, 1 2 3 4 5 6 7 8 9

Cubes, 1 8 2764 125 216 343 5121729 As in the Square Root we are obliged to distinguish the square from the root, so in the Cube, we must know the cube from the cube root ; they being two distinct and very different objects.

EXAMPLE. If the square of 4, that is, 16, be multiplied by 4, the last product 64 will be the cube of 4; and 4 will be the cube root of 64: How to extract this root, is now our task.

RULES. 1. Point off the work as in the Square Root, with this difference; point off three figures instead of two. Call these portions of figures triplets.

2. Find the greatest cube up to the first point on the left, and place its root in the quotient.

3. Place the cube thus found under those figures that are on the left of the first point: Subtract, and to the right of the remainder bring down the next triplet: Call this the dividend.

4. Square the quotient and multiply the square by 300; call this product the triple square.

5. Multiply the quotient by 30, and call the product the triple quotient.

6. Add the triple square and triple quotient, and call their amount a divisor.

7. Seek how oft the divisor is contained in the dividend, and set the result in the quotient.

8. Multiply the last quotient figure by the triple square, and place the product in a memorandum.

9. Multiply the square of the last quotient figure by the triple quotient, and place the product under the product of the triple square; and under the whole place the cube of the last quotient figure.

10. Call their sum total the subtrahend, which place under and subtract from the dividend : Then proceed with the residue of the work as before.

EXAMPLES.

LESSON 1. What is the Cube Root of 12167?

Ans. 23. OPERATION. 12,167(23 root

2 quotient. 8

2

[blocks in formation]

3600 product for memorandum, 3600 9 square of the last quotient figure. 60 triple quotient. 540 product for memorandum,

540 Cube of the last quotient figure,

27 Subtrahend,

4167

LESSON 2. What is the Cube Root of 76.765625 ?

Ans. 4.25

OPERATION,
76.765,625,(4.25 root.
64

4920)12765

10088

4 quotient.
4

530460). 2677625 16 square of the quotient.

2677625 300

4809

4800 triple square,
4 quotient.
30

120

4920

9600

120 triple quotient,

Divisor,
2 last quotient figure.
4800 triple square.
9600 product for memorandum,

2 last quotient figure.
2
4
square

of the same
120 triple quotient.
480 product for memorandum,
8 cube of the last quotient figure,

Subtrahend,
Quotient, 42

42

480

8

10088

[blocks in formation]
[blocks in formation]

125 cube of the ļast figure in the quotient, 125 Subtrahend,

2677625 Note.-The same rule holds good in the Cube Root, as in the Square Root, respecting pointing off decimals: point off so many decimals in the Root as there are triplets in the decimals of the given number. In the las; Lesson there are two triplets, viz. 765,625,; therefore we point off two decituals in the root.

Again : As in the Square root we annex two ciphers to remainders, so in the Cube Root we must annex lbrée ciphers to remainders when running a decimal to a small value.

POSITION POSITION is called the Rule of False, because we can suppose and take false numbers to reason from, and thereby find the true number sought. This rule is divided into two parts, single and double.

SINGLE POSITION.

LESSON 1.

What sum being loaned at 6 per cent. per annum, simple interest, will amount to 1250 dollars in 10 years time?

Ans. $781 25 cts.

RULE.
As the result of the supposed number,

is to the supposed number;
So is the given number,
to the number sought.

OPERATION.
Suppose we lend 600 dollars at 6

per

cent. 600 principal.

36 interest 1 year. 6 rate per cent.

10

36100 interést 1 year.

360 interest for 10 years. 600 principal

960 amount.

960: Now say-As the amount or result is to its supposed number

.600 :: So is the given number 1250 :

to its original true number; that is, the principal put to interest.

Am't. Prin. Am't.
960 : 600 :: 1250 to a fourth number,

600

960)750000(781.25 answer. (Continued.

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