CH.A P. XI.

Evolution, or Extracting the Roots out of all Single
Powers; by one Geometrical Method.
SECT. I.

Volution is the Unravelling, or as it were the Unfolding and Refolving any propofed Power or Number, into the fame Parts of which it was compofed, or fuppofed to be made up. Now in order to perform that, it will be convenient to confider how those Powers are compofed, &c.

A Square Number is that which is equally equal; or which is contained under two equal Numbers. Euclid. 7. Def. 18. Thus the Square Number 4 is compofed of the two equal Numbers z and 2, viz. 2 x 24. Or the Square Number 9 is compofed of the two equal Numbers 3 and 3, viz. 3 x 39: according to Euclid. That is, if any Number be multiplied into itself; that Product is called a Square Number.

A Cube is that Number, which is equally equally equal, or which is contained under three equal Numbers. Eu. 7. Def. 19. Thus the Cube Number 8 is composed of the three equal Numbers 2 and 2 and 2, viz. 2 x 2 x 28, &c. That is, if any Number be multiplied into itself, and that Product be multiplied with the fame Number; the fecond Product is called a Cube Number.

Thefe two, viz. the Square and Cube Numbers, borrow their Names from Geometrical Extenfions or Figures; as from the three Signal Quantities mentioned in page z. That is, a Root is reprefented by a Line or Side, having but one Dimension, viz. that of Length only. The Square is a Plane or Figure of two Dimenfions, having equal Length and Breadth. The Cube is a Solid Body of three Dimenfions; having equal Length, Breadth, and Thickness: But beyond these three, Nature proceeds not, as to Local Extenfion. That is, the Nature of Place or Space, admits no Room for other Ways of Extenfion. than Length, Breadth, and Thickness. Neither is it poffible to form, or compose any Figure or Body beyond that of a Solid.

And therefore all the fuperior Powers above the Cube or third Power; as the Biquadrat or fourth Power, the Surfolid or fifth Power, &c. are beft explained and understood by a Rank or Series of Numbers in Geometrical Proportion. For Inftance: Suppose any Rank of Geometrical Proportionals, whofe firft Term and Ratio are the fame; and to them let there be affigned a Series

Roof,

or fingle Side.

of Numbers in Arithmetical Progreffion, beginning with an Unit or 1, whose common Difference is also 1, as in page 79.

Then are thofe Numbers in produced by a continued Multiplication of the firft Term or Root into itfelf; and those in Arithmetical Progreffion or Indices, do fhew what Degree or Power each Term in the Geometrical Proportion is of. For Example; In this Series of 2 is both the firft Term or Root, and common Ratio of the Series. Then 2 x 2 = 4 the fecond Term or Square; and 2 x 2 x 2 = 8, or 4×28, the Cube or third Term; 2x2x2 X2= 16, or 8 x 2 = 16 the fourth Term or Biquadrat. And fo on for the rest.

Note, This is called Involution, viz. When any Number is drawn into itself, and afterwards into that Product, &c. it is faid to be fo often involved into itself; and the Indices are the Exponents of their respective Powers fo involved.

And according to thefe Involutions, is formed the following Table of Powers; wherein the Root is only one fingle Figure..

This Table plainly fhews (by Infpection) any Power (under the Tenth) of all the nine Figures; and from thence may be taken the nearest Root of any Square, Cube, Biquadrat, &c. of any Num ber whofe Root or Side is a fingle Figure.

1953125

But

But if the Root confifts of two, three, or more Places of Figures, then it must be found by piece-meal, or Figure after Figure, at feveral Operations.

The Extraction of all Roots, above the Square (viz. of the Cube, Biquadrat, Surfolid, &c.) hath heretofore been a very tedious and troublesome Piece of Work: All which is now very much fhortened, and rendered eafy, as will appear further on.

When any Number is proposed to have it's Root extracted, the firft Work is to prepare it by Points fet over (or under) their proper Figures; according as the given Power, whofe Root is fought doth require; and that is done by confidering the Index of the given Power, which for the Square is 2, for the Cube 3, for the Biquadrat is 4, &c. (as in the precedent Table) Then allow fo many Places of Figures in the given Power, for each fingle Figure of the Root, as it's Index denotes; always beginning those Points over the Place of Unity, and afcend towards the LeftHand if the given Number be Integers, and defcend towards the Right-Hand in Decimal Parts. As in these following.

Suppofe any given Number; as 75640387246 which I fhall all along hereafter call the Refolvend.

Then if it be required to extract any of the following Roots, it must be pointed (according to the forementioned Confideration) in this Manner:

Square Root Thus 75640387246

Now the Reason of pointing the given Refolvend in this manner; viz. the allowing two Figures in the Square; three Figures in the Cube, and four Figures in the Biquadrat, &c. For one Figure in the Root, may be made evident feveral Ways; but I think it is eafily conceived from the Table of fingle Powers, wherein you may obferve that all the Powers of the Figure 9

(which is but a fingle Figure) have the fame Number of Places of Figures, as the Index of thofe Powers denotes: Therefore fo many Places of Figures muft be taken or affigned for every fingle Figure in the Root. Confequently by these Points is known how many Places of Figures there will be in the Root, viz. So many Points as there are, fo many Figures there must be in the Root, and whether they must be Integers or Decimal Parts, is eafily determined by the refpective Places of the Points.

Sect. 2. To Extract the Square Root.

AND first how to extract the Square Root, according to the

common Method.

Having pointed the given Refolvend into Periods of two Figures as before directed; then by the Table of Powers (or otherwife) find the greatest Square that is contained in the firft Period towards the Left-Hand (fetting down it's Root, like a Quotient Figure in Divifion) and fubtract that Square out of the faid Period of the Refolvend: To the Remainder bring down the next Period of Figures, for a Dividend, and double the Root of the firft Square for a Divifor; enquiring how oft it may be had in that Dividend, fo as when the Quotient Figure is annexed to the Divifor, and that increased Divifor multiplied with the fame Quotient Figure, the Product may be the greatest Number that can be taken out of that Dividend; which fubtract from the faid Dividend, and to the Remainder bring down the next Period of Figures, for another new Dividend: Then fee how often the laft increased Divifor, can be had in the new Dividend (with the fame Caution as before, viz.) fo as that the Quotient Figure, being annexed to the Divifor, and that increased Divifor multiplied with the fame Quotient Figure, their Product may be the greatest Number that can be fubtracted from the new Dividend. (As before) And fo proceed on from Period to Period (viz. from Point to Point) in the very fame Manner, until all be finifhed.

An Example or two being well obferved will render the Work of forming the new Divifors, &c. more plain and easy than can be expreffed in a Multitude of Words.

Example 1. Let it be required to extract the Square Root out of 572199960721. This Refolvend being prepared or pointed as Before directed, will stand

Thus,

Thus, 572199960721 (756439 the Root.
49 equal the greatest Square in 57.

Proof 756439 x 756439572199960721 the Refolvend.
Example 2. What is the Square Root of 1850701,764025?

Operation 1850701,764025 (1360,405

(0)

Hence 1360,405 is the
Root required.

Ex. 3. What is the Square Root of 0,06076225 Decimal Parts?

Operation 0,06076225 (0,2465 the Root required.