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The Area of the Bafe is the fame here as in the Tetraedron, which multiplied by 3.023, the Third of the Height, gives the folid Content of one Pyramid; and this multiplied by 20, the Number of Pyramids, gives the Solidity of the Eicofiedron.

62.352

3.023

187056 124704 1870560

188.49loog6

20

3769.86 Solidity.

Then the Area of the Bafe, multiplied by 20, gives the fu• perficial Content.

62.352

20

1247.040 Superficies.

I have omitted the Figures of thefe Bodies, as fome of them would give but a confused Idea to fuch as are entirely unacquainted with the Bodies: But the following Figures, cut out in Pafteboard, and the Lines cut half through, will fold up into the feveral Bodies.

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Dodecaedron.

Eicofiedron.

A TABLE of the folid and fuperficial Content of each of the Five Bodies, the Sides being 1, or Unity.

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All like folid Bodies being in Proportion to one another as the Cubes of their like Sides, and their Superficies as the Squares of their like Sides; the folid Content of any of these Bodies may be found, by multiplying the Cubes of their Sides by the Numbers in the fecond Column under Solidity, and their Superficies, by multiplying the Squares of their Sides into the Numbers in the third Column under Superficies.

The Defcription and Ufe of the Sliding-Rule.

Soon after the Invention of Logarithms, Mr. Edmund Gunter contrived the laying them on a Ruler or Scale, which from him was called Gunter's Scale; on which, with the Help of a Pair of Compalles, any Queftion relating to Proportion may be fpeedily wrought: Since that Mr. Oughtred contrived to make two Scales flide one by the other, which performs the Work without Compaffes, and is called the Sliding-Rule: Of these there are feveral Sorts; but, as that ufually called Coggeshall's is the most common and best known, I fhall give the Defcription and Ufe of that.

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This is ufually put on one of the Sides of a Carpenter's JointRule, by a Piece made to flide in a Groove: This fliding Piece, as alfo the Line above it, is divided and numbered 1, 2, 3, &c. to 9, and again from 1 in the Middle, to 10 at the End: These two are called the Lines of Numbers; each of those Divifions betwixt the Figures is divided into ten Parts; and those Divifions, from 1 at the Beginning and Middle to the 2, are divided into five Parts, and, from each 2, to each 3, into Halves; the Divifions being afterwards too fmall to be divided. The Sliding-Piece has two of those Lines on it; one on the upper Edge, to answer that on the Rule; another on the under Edge, to answer the Girt-Line on the Rule, below the Slider: Thefe four Lines are marked A, B, C, D.

The Girt-Line is a broken Line of Numbers, contrived for measuring Timber at one Operation: It is numbered from 4 at the Beginning, to 5, 6, 7, 8, 9, 10, 20, 30, 40, at the End; each Divifion, from the Beginning to 8, divided into ten, and those again halved; but, from 8 to 9, and from 9 to 10, are only divided into ten Parts; then, from 10 to the End, each of the numbered Divifions is divided, first into ten Parts, and each of those Tenths into four smaller Divifions,

To find a given Number on the Line of Numbers.

In the first Part of the Line, to the left Hand, if the Numbers 1, 2, 3, &c. be accounted as Units, then the larger Di. visions are, each, tenth Parts of an Unit, and the smaller Divifions are Parts of thofe Tenths: The I in the Middle of the Line ftands for 10, and the other Figures 2, 3, &c. to the End, ftand for 20, 30, 40, &c. to the 10 at the End, which ftands for 100. In this Cafe, therefore, all Numbers under 10 are to be look'd for in the firft Part of the Line, and Numbers from 10 to 100 in the laft Half: Thus, if it be required to find 12 on the Line, the 1 in the Middle ftanding for 10, count from thence two of the larger Divifions, and you'll find this mark'd 12: If 37 were requir'd, from 3 on the laft Part of the Line count feven of the Divifions; and fo of any other Number under 100. If it were requir'd to find 2.48, from 2 in the first Part of the Line, count four of the larger Divi fions, this is 2.4; then, as each of these Divifions are here divided into Half, fomewhat more than half Way betwixt this

fmaller

fmaller Divifion and the Fifth of the larger, is the Place of of 8 of the larger Divifions, which is .08 of the number'd Divifions, and is the Place of 2.48, the given Number. The fame Way may other decimal Parts be found, either in the firft or laft Part of the Line.

Of the Ufe of the Line of Numbers.

To multiply one Number by another, fet 1 on the Line B to the Number given for the Multiplier in the Line A; then will the Number given for the Multiplicand in the Line B ftand against the Product in the Line A. Example: To multiply 8 by 7, fet 1 in the Line B against 7 in the Line A; then will 8 in the Line B ftand against 56, the Product, in the Line A; and while, the Rule is in this Pofition, you have the Products of all the Numbers on the Line B, from 1, to fo far as the Line A reaches, multiplied by 7, and the Products in the Line A: Thus 2, 3, 4, 5, &c. on the Line B, ftand against 14, 21, 28, 35, &c. in the Line A. The fame is the Cafe of any mixed Number found on the Line B; thus 1.6 ftands against 11.2 in the Line A.

To divide one Number by another, fet the Divifor on the Line B against 1 in the Line A; then against the Dividend in the Line B, you have the Quotient in the Line A. Example: To divide 96 by 6, fet I in the Line A against 6 on the Line B; then will 96 on the Line B ftand against 16, the Quotient, in the Line A; and, while the Rule is in this Pofition, you have the Quotient on the Line A of all the Numbers on the Line B, divided by 6.

For the Measuring of Timber, the Girt-Line mark'd D is particularly defign'd; for the Ufe of which take the following Examples.

Example 1. A Piece of Timber 10 Inches fquare, 44 Feet long, to find the Content.

Set the 12 on the Line D to 44 on the Line C; then will 10 on the Line D ftand against 33.68 Feet, the Content; which is 33 Feet, 8 Inches.

Example 2. A Piece of Timber 22 Inches fquare, 34 Feet long, to find the Content in Feet.

Set 12 on the Girt-Line D to 34, the Length of the Piece of Timber in Feet, on the Line C; then will 22 on the Line D, ftand againft the Content in Feet on the Line C: But here it is to be obferved, that 22 on the Line D, will reach beyond

Ꮓ 2

the

the Line C, the Content being more than 100 Feet: Therefore, in this Cafe, the 12 in the Girt-Line D must be fet to 34 in the former Part of the Line C; accounting the Figures as Tens, the fame as in the latter Part; then the I in the Middle is 100, the 2, 3, 4, &c. following, 200, 300, 400, &r. to the 10 at the End, which is now 1000. Hence the larger Divilions, which before were Units, are in this Cafe Tens, and the Imaller Divifions increase their Value in the fame Proportion.

Having fet the 12 in the Line D against 34 in the first Part of the Line C, then will 22, the Inches fquare in the Line D, ftand againft 114, and fomewhat more; that is, it will ftand a little beyond the fecond fmall Division, betwixt the first and fecond of the large Divifions, anfwering to 114.277 Feet, or 114 Feet, 3 Inches.

By the Method, used in this Example, of making the Numbers in the firft Part of the Line ftand for Tens (or Hundreds, if neceffary) the Ufe of the Sliding-Rule may be extended to larger Numbers; always remembering, that all the Numbers on the laft Part of the Scale will be of the fame Denomination with the 1 in the Middle; and, though-in large Numbers we cannot come to the Degree of Exactness neceffary in fome Things, yet it may be of great Ufe to correct a Miftake.

I fhall now add fome Queftions, to exercife the Learner, in fome of the foregoing Rules.

Question 1. Seven Men bought a Grinding-Stone of 5 Feet (or 60 Inches) Diameter: How much of the Stone's Diameter muft each grind, to have an equal Share of the Stone, if one first grind his Share, and then another his, till the Stone is ground away?

In order to folve this Queftion, fquare the Diameter, and divide the Square by 7, the Number of Perfons; fubtract the Quotient from the Square of the Diameter, and extract the Square Root of the Remainder, which is the Length of the Diameter, after the first Man has ground his Share; this Work repeated, by fubtracting the fame Quotient, or feventh Part, from the Remainder, for every one, to the laft, extract the Square Root of the Remainders, and fubtract thofe Roots from the Diameter, one after another, and the feveral Remainders are Answers.

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