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THe Cube is a Solid, and hath three dimen∣sions, length, breadth, and depth, and is inclosed by six plain square Superficies.
Example.
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Let the Side a, b, or c, d, &c. be 125: To find the Content in Solid Feet or Inches, is the Involution of the Side or Root. Thus: 〈 math 〉〈 math 〉
The Solidity.
And this is called the Third Power.
The Evolution hereof, is also termed the Extraction of the Cube Root, wherein observe first your punctation, omitting two, point every third Figure.
Example.
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The first Figure in the Root is found by ta∣king the greatest Cube Number, contained in the Figure or Figures that stand under the first Point towards the left hand, here 71, whose Root is 4, therefore that 4 must be placed in the Quotient as the first Figure in the Root, and the Example will stand thus: 〈 math 〉〈 math 〉
Then the Cube of 4 is 64, which subduct out of the first Figures, and set down the re∣mainder if any be. The first Figure found in this peculiar manner, the rest are found by Di∣vision thus: The Dividend consists of the re∣mainder, if any be, and the three Figures under the next Point postponed; the Divisor is always three times the Square of the Root, and three times the Root it self: These two Numbers being so to be added together, as that the Unites of the first stand over the Tens of the second.
Three times the q. of the √ = 48
Three times the √ = 12
Divisor 492
Then will the Example stand thus: 〈 math 〉〈 math 〉
Then proceed to Division, always supposing the last Figure in the Divisor to stand under the last save one in the Dividend, and enquire, how many times 4 in 7? place 1 in the Quotient. Then for your Number to be subducted out of the Dividend, it always consists of three Num∣bers, viz.
Three times the q. of the first Figure = 48
Multiplied by the second 1
Product of the first by the second 48
3 times the q. of the second by the first 12
And the Cube of the second 1
The Subducend 4921 〈 math 〉〈 math 〉
The q. of 41 = 1681
3
q of 41 × 3 = 5043
Then for a new Dividend, bring down the three next Figures, postponing them as before.
The Divisor thrice the q. of the √ 41 = 5043
And three time the √ 41 = 123
50553
Which being set on the left hand the Divi∣dend, stands thus: 〈 math 〉〈 math 〉
Then enquire, how many times 5 in 30? you will find •• times, which place in the Quotient. Your Subducend is as before,
3 times the Square of 41 × 5 = 25215
3 times the Square of 5 × 41 = 3075
And the Cube of 5 = 125
The Subducend 2552375
Which being subducted from 3032125
There will remain 479750
Which shews the Number was not a Cube Number; if you add three Cyphers, and work as before, you may have as many Decimals Fra∣ctions as you please.
In this Extraction I have not taken the same Number the Cube first mentioned did produce, but by adding another Figure, made the Number greater, that it might take in all Cases; but in the following Extraction it is explicated.
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Three times the Square of the Root = 3
Three times the Root is also = 3
The first Divisor = 33
Three times the Square of 1 = 3
2
Three times the Square of 1 × 2 = 6
Three times the Square of 2 × 1 = 12
The Cube of 2 = 8
The first Subducend = 728
〈 math 〉〈 math 〉
Three times the Square of the Root = 432
Three times the Root = 36
The second Divisor = 4356
Three times the Square of the √ 12 = 432
5
Three time the Square of 12 × 5 = 2160
Three times the Square of 5 × 12 = 900
The Cube of 5 = 125
The second Subducend = 225125