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A Cube is that Number which is equally equal, or, which is contained under three equal Numbers.
THus (8) is a Cube which is contained under (2) and (2) and (2).
| Three equal Numbers | Cube, | ||||
| 1 | into | 1 | into | 1 | 1 |
| 2 | into | 2 | into | 2 | 8 |
| 3 | into | 3 | into | 1 | 27 |
| 4 | into | 5 | into | 4 | 64 |
| 5 | into | 5 | into | 5 | 125 |
| 6 | into | 6 | into | 6 | 216 |
| 7 | into | 7 | into | 7 | 347 |
| 8 | into | 8 | into | 8 | 512 |
| 9 | into | 9 | into | 9 | 729 |
Sectio CUBI in Octo Solida, a duobus Lateribus {Act B} i. e. {3 et 2} effecta
Quorum Quatuor ordinatim sumpta, sunt continue Proportionalia; nimirum
Thus (3) and (3) and (3) make the Cube (27) for (3) multiplied by (3) makes the Square (9) and the Square (9) multiplied again by (3) makes the Cube (27) as is more clear by the foregoing Table.
And so doe (5) and (5) and (5) make the Cube (125) That is, (5) into (5) makes the Square (25) and the Square (25) into (5) makes the Cube (125) &c.
Wherefore to Extract a Cube-Root, is nothing else then to find out a number, which being first multipli∣ed into it self, and then into the Product, produces the given Cube-Number. Thus to extract the Cube-Root of (15̇625̇) is to find out the number (25) which be∣ing first multiplied into its self (makes 625) and then multiplied into that (625) makes the given Number (15̇625̇)
Now because this construction of the Cube from a sin∣gle Root, contributes nothing towards the finding out that Root from a given Cube-Number, therefore was found out by the Antients, that admirable Art of cutting or dividing the Root into two parts, which they therefore called a Binomial Root; and from those two parts they erected 8 solid numbers, whereof the great∣est and the least are always two pure Cube numbers, of those two distinct parts, and of the other six Paral∣lelepipedons, by which I mean solid Numbers made by multiplying the Square of one Number into another Number, in imitation of the Geometrical Parallelepi∣pedons defined by Euclid. lib. 11. Defin. 30. to be a solid Figure contained under six Equilateral Figures, whereof those which are opposite are Parallel.
The three greatest Parallelepipedons are equal one to another, and each of them made by multiplying the Square of the greatest part of the Binomial-Root into the lesser part.
And the three lesser Parallelepipedons are equal to one another, and each of them made, by multiplying the Square of the lesser of the Binomial Root into the great∣er part.
Thus in the opposite Figure the whole Root is (5) and divided or cut into a greater part A= 3 and a lesser part B=2
| The Cube of the greater A=3 is equal to | 27 |
| The Cube of the lesser B=2 is equal to | 8 |
| One of the greater Parallelepipedons, or (Aq into B) is equal to | 18 |
| One of the lesser Parallelepipedons or (A into Bq) is equal to | 12 |
| Then To those two Cubes, namely | The greater or Ac | 27 |
| The lesser or Bc | 8 | |
| Adding Three of the greater Paralle∣lepipedons, viz. | (Aq) into (B) | 18 |
| (Aq) into (B) | 18 | |
| (Aq) into (B) | 18 | |
| And Three of the lesser Parallele∣pipedons, viz. | (A) into (Bq) | 12 |
| (A) into (Bq) | 12 | |
| (A) into (Bq) | 12 | |
| The total summe is the entire Cube Num∣ber of the Binomial-Root A=3 more by B=2 that is to say (5) and amounteth to | 125 |
Now the Practitioner is to conceive the Unites of the Cube-Number (125) to be as so many Dice, or Cu∣bical Unites, and 27 of these being piled one upon another orderly and equally to make up (Ac) and 8 of them to make (Bc) and 18 of them to make one of the greater Parallelepipedons (Aq into B) and 12 of them to make one of the lesser Parallelepipedons (A into Bq) and then all these eight solid Numbers being orderly put together, to make up the entire Cube or (125) And this is the Genesis of a Cube. vid. Fig.
After the very same manner, let the Root (25) be made Binomial and cut into two parts, viz. (20) and (5) and the greater called A, and the lesser B
Then
| The Cube of the greater A=20 is equal to | 8000 |
| The Cube of the lesser B=5 is equal to | 125 |
| The greater Parallelepipedon, or (Aq into B) is e∣qual to | 2000 |
| The lesser Parallelepipedon, or (A into Bq) is equal to | 500 |
| Then to these two Cubes, namely | The greater or (Ac) 8000 |
| The lesser, or (Bc) 125 | |
| Adding Three of the greater Parallelepi∣pedons, viz. | (Aq) into (B) 2000 |
| (Aq) into (B) | 2000 |
| (Aq) into (B) | 2000 |
| And Three of the lesser Parallelepipe∣dons, viz. | (A) into (Bq) 500 |
| (A) into (Bq) | 500 |
| (A) into (Bq) | 500 |
| Summe | 15625 |
| The total Summe of all the eight Solids, is the Summe of the entire Cube which amounteth to | 15625 |
All which is consonant to that Theorem of Ramus, (which is in imitation of that of Euclid concerning a Square Number.