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If the Number given be either a Mixt Number, or a Decimal; make the Number of Decimal Places either three, fix, nine, &c. by annexing Cyphers to the Right Hand, that the Point may fall upon the Units Place of the whole Number.

What is the Cube Root of 65.31?

65.310.000.000 (4.027 Root

Cube of 4 = 64

Square of 4 multiplied by 3=48) 13
Subtract the Cube of 40 = 64000

Sq. of 40 multiplied by 3 = 4800) 1310
Subtract the Cube of 402 = 64964808

Sq. of 402 multiplied by 3=161604)3451920
Subtract Cube of 4027

=

65304767683

(5232317)

There were, in this Example, but two Decimal Places; we therefore annexed 7 Cyphers to the given Number, by which Means a Point not only falls on the Unit's Place in the Whole Number, but we gain 3 Decimal Places in the Root.

To extract the Cube Root of a Mulgar Fration.

The best Way will be to reduce it to a Decimal Fraction equal to it; and then proceed in all Refpects to find its Root as before.

Let be a Vulgar Fraction given, whose Cube Root is required.

First, reduce it to a Decimal thus.

As 9: 4 :: 100000000

4

9) 4000000000 (4 over.

444444444 the Decimal Fraction.

Then extract the Cube Root of .444444444 (.763 Root.

Cube of 7

343

Square of 7 multiplied by 3=147) 1014

Subtract the Cube of 76

=

438976

Square of 76 multiplied by 3=17328)54684

Subtract the Cube of 763

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18

In like Manner the Cube Root of will be found to be.912, with a Remainder of 8116138, which the Learner may try at his Leisure.

The Use of the Cube Root.

Problem 1.

To find two Mean Proportionals between any two Numbers given.

Rule.

Divide the greater Extreme by the less; and the Cube Root of that Quotient multiplied by the less Extreme, will give the leffer Mean; then multiply the said Cube Root by the leffer Mean, and the Product will be the greater Mean Proportional required.

Crample.

What are the two Mean Proportionals between 8 and 216?

Operation.

First, 216 divided by 8=27; whose Cube Root is=3; which multiplied by 8 is = 24, the leffer Mean.

And, the Cube Root 3 multiplied by 24 = 72, the greater Mean.

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In like Manner, the two Mean Proportionals between 9 and 243 are found to be 27 and 81.

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Problem 2.

The Solid Content of any Vessel, or other Solid Body being given, to find the Side of a Cube, which shall be equal in Solidity thereto.

Rule.

Extract the Cube Root of the Solid Content of the given Body, and that Root will be the Side of the Cube required.

Example 1.

Suppose the Solid Content of a Globe, or Cylinder, Pyramid, or Cone, &c. be 157464 cubic Inches, what is the Side of a Cube of equal Solidity?

Operation.

157464 (54 Root; the Side of the Cube required.

Cube of 5=12;

5×5×3=75)324 Subtract 54 × 54×54=157464

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Example 2.

_ In digging a Cellar of a Cubic Form, about_3375 Solid Feet of Earth were carried out; how many Feet was it every Way, viz. in Length, Breadth, and Depth.

Answer 15 Feet, as the Learner may try at his Leisure.

Problem 3.

The Dimensions, Capacity, or Weight of a Solid being given; to find the Dimensions, Weight, &c. of a like Solid of a different Capacity and Dimension.

Rule.

Like Solids are in a (Triplicate or) Cubic Proportion to their like Sides; it will therefore always be; as the Cube of the Dimension given, is to its given Weight; so is the Cube of any like Dimension, to the Weight required; and the contrary.

Example 1.

Surpose a Ball or Bomb of 4 Inches Diameter weighs 9 Pounds; what is the Diameter of another of the fame Shape and Metal that weighs 84 Pounds?

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Answer, 8 Inches, 4 Tenths, and something over.

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