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In this ach, of vari and Denia
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 afsure you required more Charge, Care, and Trouble, to find out nicely, than I was at first aware of.
Now from hence it will be easy to determine the Weight of any proposed Quantity, of the same Matter and Kind with those in the Table; it's Solid 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 same kind of Matter) the Product will be the Weight of that Quantity in Ounces, &c.
EX A M P L E.
Suppose it were required to find the Weight of a piece of Marble, containing three Solid Feet, and 40 Cubick Inches.
First 1728 x3 = 5184 the Cubick Inches in 3 Solid Feet.
And 5184 +40=5224 the Number of Cubick Inches in the Piece of Marble.
Then 5224 * 1,429411 =7410,066624 Ounces Troy.
The Weight of that Piece of Marble, in Ounces, &c. which is easily brought into Pounds, &c. The like for any of the reft.
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, 8c.
Thus, divide the given Weight of the proposed Quantity (it being first reduced into Ounces, &c.) by the tabular Weight of one Inch (of the same 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 it is immersed or put into Water, you must subtract the Weight of an equal Quantity of Water (with that of the Body) from the Weight of the proposed 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.
their Difference is, = 5,441268 the Weight of a Cubick Inch of Lead in the Water, &c.
c H A P.
c H A P. XI. Evolution, or Extrasting the Roots out of all Single Powers ; by one Geometrical Method.
SECT. I. Velution is the Unravelling, or as it were the Unfolding and
Resolving any proposed Power or Number, into the same Parts of which it was compofed, or supposed to be made up. Now in order to perform that, it will be convenient to consider how those Powers are composed, &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 composed of the two equal Numbers 2 and 2, viz. 2 x2=4. Or the Square Number 9 is composed of the two equal Numbers 3 and 3, viz. 3x3=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. Łu. 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 2 = 8, c. That is, if any Number be multiplied into itself, and that Product be multiplied with the fame Number; the second Product is called a Cube Number.
These two, viz. the Square and Cube Numbers, borrow their Names from Geometrical Extensions or Figures; as from the three Signal Quantities mentioned in page 2. That is, a Root is represented by a Line or Side, having but one Dimension, viz. that of Length only. The Square is a Plane or Figure of two Dimensions, having equal Length and Breadth. The Cube is a Solid Body of three Dimensions ; having equal Length, Breadth, and Thickness: But beyond these three, Nature proceeds not, as to Local Extension. That is, the Nature of Place or Space, admits no Room for other Ways of Extension. than Length, Breadth, and Thickness. Neither is it possible to form, or compose any Figure or Body beyond that of a Solid.
And therefore all the superior Powers above the Cube or third Power ; as the Biquadrat or fourth Power, the Surfolid or fifth Power, &c. are best explained and understood by a Rank or Series of Numbers in Geometrical Proportion. For Instance : Suppose any Rank of Geometrical Proportionals, whose firft Term and Ratio are the same; and to them let there be assigned a Series
This Table plainly shews (by Inspection) 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 Number whole Root or Side is a single Figurę.
But if the Root consists of two, three, or more Places of Figures, then it must be found by piece-meal, or Figure after Figure, at leveral Operations.
The Extraction of all Roots, above the Square (viz. of the Cabe, Biquadrat, Sursolid, &c.) hath heretofore been a very tedious and troublesome Piece of Work: All which is now very much fortened, and rendered easy, 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 set over (or under) their proper Figures; according as the given Power, whose Root is sought doth require ; and that is done by considering 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 so many Places of Figures in the given Power, for each single Figure of the Root, as it's Index denotes ; always beginning thosa Points over the Place of Unity, and ascend towards the LeftHand if the given Number be Integers, and descend towards the Right-Hand in Decimal Parts. As in these following.
Suppose any given Number; as 75640387246 which I shall all along hereafter call the Resolvend.
Then if it be required to extract any of the following Roots, it must be pointed (according to the forementioned Confideration) in this Manner :
ç Square Root Thus 75640387246
75640387246 Viz. For the
Biquadrat Root 75640387246
į Surfolid Root 75640387246 Or suppose the Number to be 0,674035982
rSquare Root Thus 0,6740359820 Then for the Cube Root
0,674035982 Biquadrat Root 0,674035982000 Now the Reason of pointing the given Resolvend in this manner; viz, the allowing two Figures in the Square; three Figuras in the Cube, and four Figures in the Biquadrat, &c. For one Figure in the Root, may be made evident several Ways; but I think it is easily conceived from the Table of single Powers, Wherein you may observe that all the powers of the Figure'
(which is but a single Figure) have the same Number of Places of Figures, as the Index of those Powers denotes: Therefore so many Places of Figures must be taken or assigned for every single Figure in the Root. Consequently by these Points is known how many Places of Figures there will be in the Root, viz. So many Points as there are so many Figures there muft be in the Root, and whether they must be Integers or Decimal Parts, is easily determined by the respective Places of the Points.
Sect. 2. To Ertrag the Square Root.
on the Quotient liplied with bee chat can bedenkt, and to the
AND first how to extract the Square Root, according to the
common Method. Having pointed the given Resolvend into Periods of two Figures as before directed ; then by the Table of Powers (or otherwise) find 'the greatest Square that is contained in the first Period towards the Left-Hand (setting down it's Root, like a Quotient Figure in Di. vifion) and subtract that Square out of the said Period of the Resolvend: To the Remainder bring down the next Period of Figures, for a Dividend, and double the Root of the first Square for a Divisor; enquiring how oft it may be had in that Dividend, so as when the Quotient Figure is annexed to the Divisor, and that increased Divisor multiplied with the same Quotient Figure, the Product may be the greatest Number that can be taken oat of that Dividend; which subtract from the said Dividend, and to the Remainder bring down the next Period of Figures, for another new Dividend: Then see how often the last increased Divisor, can be had in the new Dividend (with the fame: Caution as before, viz.) so as that the Quotient Figure, being annexed to the Divisor, and that increased Divisor multiplied with the fame Quotieft Fi. gure, their Product may be the greatest Number that can be subtracted from the new Dividend. (As before) And fo proceed on from Period to Period (viz. from Point to Point) in the very same Manner, until all be finished.
An Example or two being well observed will render the Work of forming the new Divisors, &c. more plain and cafy than can * be exprested in a Multitude of Words.
Example 1. Let it be required to extract the Square Root out 5572199960721. This Resolvend being prepared or pointed as ore directed, will stand