4. In the Philofophical Tranfactions, (Number 169 and 199) there is an Account of a great many Experiments of this Kind; from whence I collected thefe following, viz. Gold 18888 Mercury 14019. Lead 11343. Silver 11087. Copper 8843. Hammered Brass 8349. Caft Brafs 8100. Steel 7852. Iron 7643. Tin 7321 . Pump-water 1000.

These last Proportions being approved of and published by Order of the Royal Society feem to be unquestionably true: Nevertheless, because they differ fo much from the before-mentioned, (and thofe from one another) I have for my own Satisfaction made feveral Experiments of that Kind: And have (I presume) obtained the Proportions of Weight that one Body bears to another of the fame Bulk or Magnitude, as nicely as the Nature of fuch Matter, which may be contracted or brought into a leffer Body (viz. either by Drying, or Hammering, or otherwise) will admit of; which are followeth :

In this Table you have the Specifick Gravity or Weight of a Cubick Inch, of various Sorts of Bodies, both in Troy Ounces and Averdupois Ounces, and Decimal Parts of an Ounce, which I can affure you required more Charge, Care, and Trouble, to find out nicely, than I was at first aware of.

Now from hence it will be eafy to determine the Weight of any propofed Quantity, of the fame Matter and Kind with thofe in the Table; it's folid Content being given in Cubick Inches. For it is plain, that if the Number of Cubick Inches contained in any given Quantity, be multiplied with the tabular Weight of one Inch, of the fame kind of Matter) the Product will be the Weight of that Quantity in Ounces, &c.

EXAMPLE.

Suppose it were required to find the Weight of a Piece of Marble, containing three Solid Feet, and 40 Cubick Inches.

First 1728 x 35184 the Cubick Inches in 3 Solid Feet. And 5184+40=5224 the Number of Cubick Inches in the Piece of Marble.

Then 5224 x 1,4294117410,066624 Ounces Troy.

Or

5224 x 1,568859 8195,719416 Ounces Averdupois.

The Weight of that Piece of Marble, in Ounces, &c. which is eafily brought into Pounds, &c. The like for any of the rest.

The Converse of this Work is as eafy; viz. if the Weight of any propofed Quantity be given, thence to find the Solid Content of that Quantity in Cubick Inches, &c.

Thus, divide the given Weight of the propofed Quantity (it being firft reduced into Ounces, &c.) by the tabular Weight of one Inch (of the fame kind of Matter) and the Quotient will be the Number of Cubick Inches contained in that Quantity.

Note, If you would find what Weight any Quantity of those Bodies mentioned in the Table will have, when immerfed or put into Water, you must fubtract the Weight of an equal Quantity of Water (with that of the Body) from the Weight of the propofed Body (if it be heavier than Water) and there will remain the Weight required. As for Instance,

A Cubick Inch of Lead = 5,984010 Ounces Troy, &c.
A Cubick Inch of Water = 0,542742

Their Difference is, = 5,441268 the Weight of a Cubick

Inch of Lead in the Water, &c..

CHAP.

CHA P. XI.

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

E

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 2 and 2, viz. 2 × 24. Or the Square Number 9 is composed of the two equal Numbers 3 and 3, viz. 3 x3=9: 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 compofed of the three equal Numbers 2 and 2 and 2, viz. 2 x 2 x 2 = 8, &c. That is, if any Number be multiplied into itself, and that Product be multipliedwith the fame Number; the fecond Product is called a Cube Number.

These two, viz. the Square and Cube Numbers, borrow their Names from Geometrical Extenfions or Figures; as from the three Signal Quantities mentioned in page 2. That is, a Root is reprefented by a Line or Side, having but one Dimenfion, 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 thefe 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 Thicknefs. Neither is it poffible to form, or compofe 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: Suppofe any Rank of Geometrical Proportionals, whofe firft Term and Ratio are the fame; and to them let there be affigned a Scries

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

Then are those Numbers in produced by a continued Mul tiplication of the firft Term or Root into itself; and thofe 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 28, or 4 × 2=8, the Cube or third Term; 2x2x2 x 2 = 16, or 8 x 2 = 16, the fourth Term or Biquadrat. And so on for the reft.

L

2x2

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 Powers; wherein the Root is only one fingle Figure.

Table of

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

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 thefe following.

Suppofe any given Number; as 75640387246, which I shall 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.

?

Now the Reafon 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

2

(which