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take the Half of that for a mean Side; which is another Error, tho' not of fo much Confequence as thofe relating to round Timber: But, by Means of thefe fucceffive Errors, a Piece of Timber, when fquar'd, contains more Feet than when it was round. It will therefore be proper to work an Example each Way, that the Difference may be seen.

Exam. 1. A Piece of round Timber being 15 Feet long, and Girt in the Middle 42 Inches, how much Timber is contain'd in it?

4) 42.0 (10.5 one Fourth of the Girt.

.875 Decimal of 10 Girt 42 Inches, or 3.5

Feet.

.875

(Inches.

3.5

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According to the common Way of Measuring, this Piece contains 11 Feet and about a Quarter and Half-Quarter; but by fquaring the Circumference, and multiplying the Square by 07958, as taught at Art. 65, and multiplying that Product by the Length, which is the true Method, we have 15 Feet, which is a confiderable Difference; and this Difference would be increased, if a mean Base were to be taken, which the Circumference in the Middle is not.

Exam.

Exam. 2. A Piece of fquared Timber being 9 Inches wide, 10.5 Inches deep, and 16 Feet long, how much Timber is contain'd in it?

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In this Example the Difference is but little, as I obferv'd before, being not one Tenth of a Foot.

To find how much in Length is requir'd to make a Foot of Timber to any given Square, there is on the two-foot JointRules a Table of Timber-Measure, on the contrary End to that of Board-Meafure; the Ufe of which is known to every Carpenter: And to find it to any given Square, not in the Table, divide 1728, the cubic Inches in a Foot, by the given Inches fquare of the Piece, and the Quotient is the Length requir'd to make a Foot.

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Of the Five Regular Bodies.

There are five Solids contain'd under equal, regular Sides, which, by way of Diftinction, are called The five regular Bodies. Thefe are the Tetraedron, the Hexaedron or Cube, the Octaedron, the Dodecaedron and the Eicofiedron. The Measuring of the Cube was fhewn at Art. 78. I fhall now fhew how to Measure the other four.

Of the Tetraedron.

The Tetraedron is a Solid contain'd under four equal and equilateral Triangles; that is, it is a triangular Pyramid of four equal Faces, the Side of whofe Bafe is equal to the flant Height of the Pyramid, from the Angles to the Vertex.

The firft Thing to be done is, to find the perpendicular Height of the Triangles; which is done thus; from the Square of the given Side fubtract the Square of Half the Side, and extract the Square Root of the Remainder, and this will be the perpendicular Height of the Triangle.

The

The Side of a Tetraedron being 12, to find the folid and fuperficial Content.

Side 12

Half the Side 6 108.0000 (10.39

12

144
36

108

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6

I

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479

If four more Cyphers were annex'd, the Root would be 10.3923 for the perpendicular Height of the Triangle; therefore, by Art. 56, Half the Side multiplied into the perpendicular Height gives the Area of the Base.

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To find the folid Content: At Art. 83, we have fhewn the Way to find the Solidity of a Pyramid, by multiplying the Area of the Bafe by one Third of the Height, or, which is the fame, one Third of the Area by the Height of the Pyramid: Now, to find the perpendicular Height of the Tetraedron, fubtract the Square of Half the given Side from the Square of the Triangle's Height before found, and divide the Remainder by the Double of the Triangle's Height, and the Quotient will be the Distance of the Center of the Bafe from the Sides; and this Quotient, fubtracted from the Height of the Triangle, gives the Diftance of the Center from the Angles. 108 The Square of the Triangle's Height. 36 The Square of Half the Side.

20.7846) 72.0000 (3.464 Distance of the Center from

623538

(the Sides,

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If the Square of the Distance of the Center of the Bafe from the Sides be fubtracted from the Square of the Triangle's Height, the Square Root of the Remainder is the perpendicular Height of the Pyramid.

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From 108 Square of the Triangle's Height.

Subt. 12 Square of the Diftance of the Center from the Sides.

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As we took 12 for the Square of the Distance of the Center, inftead of 11.999296, we may call the Height of the Tetradron 9.8, it being near the Truth: This multiply'd into 20.7846, the Third of the Area of the Base, gives the Solidity of the Tetraedron.

20.7846
9.8

1662768

1870614

203.68908 Solidity.

For the Superficies, the Area of the Bafe multiply'd by 4, as the Faces are all equal, gives the Surface of the Tetraedron. If the Square of the Distance of the Center from the Angles had been fubtracted from the Square of 12, the given

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