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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 produce the greater number.

The greater number, 48.0

30.5

17.5

6. Subtract the half root from the half sum to produce the lesser number.

30.5

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 89
Cubes, 1 8 27 64 125216343512729,

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.

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125 cube of the last figure in the quotient,

Subtrahend,

125

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 last Lesson there are two triplets, viz. 765,625,; therefore we point off two decimals in the root.

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Again: As in the Square root we annex two ciphers to remainders, so in the Cube Root we must annex three 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.

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to its original true number;

that is, the principal put to interest.

Am't. Prin. Am't.

960 :

600 ::

1250 :

960 600 :: 1250 to a fourth number.

600

960)750000(781.25 answer. [Continued.

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