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9. The wall of a town is 25 feet high, which is surrounded by a moat of 30 feet in breadth, I defire to know the length of a ladder that will reach from the outfide of the moat to the top of the wall, Anf. 39,05 feet.

The hypothenufe and perpendicular given to find the bafe. RULE. The fquare root of the difference of the fquares of the hypothenuse and perpendicular is the length of the base.

The bafe and hypothenufe given to find the perpendicular.

RULE. The fquare root of the difference of the hypothenuse and bafe is the heighth of the perpendicular.

N. B. The two laft queftions may be varied for examples to the two laft propofitions.

Any number of men being given to form them into a fquare battle, or to find the number of ranks and files.

RULE. The fquare root of the number of men given, is the number of men either in rank or file.

10.

An army confifting of 331776 men, I defire to know how many in rank and file ?

11.

Ans. 576.

A certain fquare pavement contains 48841 fquare stones, all of the fame fize, I demand how many are contained in one of the fides. Anf. 221.

EXTRACTION OF THE CUBE ROOT.

To extract the Cube Root is to find out a number which being multiplied into itself, and then into that product, produceth the given number.

RULE. Point every third figure of the cube given, beginning at the unit's place, feek the greateft cube to the first point and fubtract it therefrom, put the root in the quotient, and bring down the figures in the next point to the remainder for a refolvend.

2. Find a divifor by multiplying the fquare of the quotient by 3. See how often it is contained in the refolvend, rejecting the units and tens, and put the anfwer in the quotient.

3. To find the fubtrahend. 1. Cube the last figure in the quotient. 2. Multiply all the figures in the quotient by 3, except the laft, and that product by the fquare of the laft. 3. Multiply the divifor by the laft figure. Add thefe products together, gives the fubtrahend, which fubtract from the refolvend; to the remainder bring down the next point and proceed as before.

Roots.
CUBES.

2.

I.

2. 3. 4. 5. 6. 7. 8. 9.
8. 27. 64. 125. 216. 343. 512 729.

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Another new and more concife method of extracting the Cube Root.

RULE. 1. Point every third figure of the cube given, beginning at the unit's place, then find the nearest cube to the first point, and fubtract it therefrom, put the root in the quotient, bring down the figures in the next point to the remainder for a refolvend.

2.

Square the quotient and triple the fquare for a divifor-as 4X4X3=48. Find how often it is contained in the refolvend, rejecting units and tens, and put the answer in the quotient.

3. Square the laft figure in the quotient, and put it on the right hand of the divifor :

As 6×6=36 put to the divifor 48=4836.

4. Triple the laft figure in the quotient, and multiply by the former, put it under the other, units under the tens, add them together, and multiply the fum by the laft figure in the quotient, fubtract that product from the refolvend, bring down the next point and proceed as before.

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EXAMPLES.

What is the cube root of 99252847 ?

Square of 4X3=48 Divifor. Square of 6, put to 48-4836 6X3X4=72

Square of 46 2116X3=6348 Divifor Square of 39 put to 6348*634809 8X3X46 414

3.

4.

5.

99252847(463

5556 X 6

=

35252
33336

1916847

638949 X 3=1916847

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Anf. 4968.

Anf. 6031.

2. What is the cube root of 389017 ? What is the cube root of 5735339 ? What is the cube root of 32461759? What is the cube root of 84604519 ? 6. What is the cube root of 259694072 ? What is the cube root of 48228544 ? What is the cube root of 27054036008 ? What is the cube root of 22069810125? What is the cube root of 122615327232 What is the cube root of 219365327791 ? 12. What is the cube root of 673373097125 ?

7.

8.

9.

10.

11.

Anf. 8765.

When the given number confifts of a whole number and decimal together, make the number of decimals to confist of 3, 6, 9, &c. places, by adding cyphers thereto, fo that there may be a point fall on the unit's place of the whole number.

13.

14.

15.

16.

17.

18.

What is the cube root of 12.977875 ?
What is the cube root of 36155,027576?
What is the cube root of ,001906624 ?
What is the cube root of 33.230979637 ?
What is the cube root of 15926,972504?
What is the cube root of ,053258279 ?

Anf. 2,35 Anf. 33,06+ Anf.,124

Anf. 3,215+

Anf. 25,19+

Anf. 376 +

To extract the Cube Root of a Vulgar Fraction.

RULE. Reduce the fraction to its lowest terms, then extract the cube root of the numerator and denominator for a new numerator and denominator, but if the fraction be a surd, reduce it to a decimal, and then extract the root from it.

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• When the quotient is 2 or 3 there must be a cypher put to fupply the place of tens,

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To extract the Cube Root of a mixt number.

RULE. Reduce the fractional part to its lowest terms, and then the mixt number to an improper fraction, extract the cube roots of the numerator and denominator for a new numerator and denominator; but if the mixt number given be a furd, reduce the fractional part to a decimal, annex it to the whole number, and extract the root therefrom.

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THE APPLICATION.

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1. If a cubical piece of timber be 47 inches long, 47 inches broad, and 47 inches deep, how many cubical inches doth it contain? Anf. 103823.

2. There is a cellar dug that is 12 feet every way, in length, breadth and depth, how many solid feet of earth were taken out of it? Anf. 1728.

3. There is a ftone of a cubic form, which contains 389017 folid feet, what is the superficial content of one of its fides ? Anf. 5329.

Between two numbers given, to find two mean proportionals.

RULE. Divide the greater extreme by the leffer, and the cube root of the quotient multiplied by the leffer extreme, gives the leffer mean; multiply the faid cube root by the leffer mean, and the product will be the greater mean proportional.

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EXAMPLES.

4. What are the two mean Proportionals between 6 and 162? Anf. 18 and 54.

5. What are the two mean Proportionals between 4 and 108? Anf. 12 and 36.

To find the fide of a cube that fhall be equal in folidity to any given folid, as a globe, cylinder, prifm, cone, &c.

RULE. The cube root of the folid content of any solid body given is the fide of the cube of equal folidity.

EXAMPLE.

6. If the folid content of a globe is 10648, what is the fide of a cube of equal folidity? Anf. 22.

The fide of the cube being given, to find the fide of that cube, that fhall be double, treble, &c. in quantity to the given cube.

RULE. Cube the fide given, and multiply it by 2, 3, &c. the cube root of the product is the fide fought.

EXAMPLE.

7. There is a cubical veffel, whose fide is 12 Inches, and it is required to find the fide of another veffel, that is to contain 3 times as much ? Anf. 17,306.

EXTRACTION OF THE .BIQUADRATE ROOT.

To extract the Biquadrate Root is to find out a number, which being involved four times into itself, will produce the given number.

RULE. First extract the Square Root of the given number, and then extract the Square Root of that Square Root, and it will give the Biquadrate Root required.

2.

3.

EXAMPLES.

1. What is the biquadrate of 27 ?
What is the biquadrate of 76 ?
What is the biquadrate of 275 ?
4. What is the biquadrate root of 531441?
5. What is the biquadrate root of 33362176 ?
6. What is the biquadrate root of 5719140625 ?

Anf. 531441 33362176 5719140625

27

76

275

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