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Firft, find the Superficial Content of the Base: Then multiply that by of the Heighth ; and it produceth the Solid Content of the Pyramid.

Prob. 27

To Measure the Fruftrum of a Pj

ramid or Cone.

Fig. 83. The Fruftrum here given to be measured is ABCD, the fide of the greater Base A, being 24 Inches, and the side of the lefser Base at B, 8 Inches, and the lengih of it I M 20 Feet = B = CK 20 Foot.

It is evident that if I find the Solidity of the whole Pyramid AED, and also the Sce lidity of the leffer Pyramid BEC, and then fubftract the Content of BEC, from the Content of A ED, there will remain the Solidity of the Frustrum ABCD

;

and certainly this way of measuring the Fru. ftrum of a Pyramid or Cone, is the most exact of any, and it may be easily mea. fured thus: First of all find out the heighrh of the whole Pyramid EM, which you may do by the following proportion, viz.

As the Sumi-difference of the Bares,
Is to the heighth of the Fruftrum,
So is the greater Base,
To the heighth of the whole Pyramid.

Which proportion will hold good in Cones as well as Pyramids.

Let AD be the Diameter of the greater Bar., and BC the Diameter of the lefler Base'; from B, and C, let fall the Perpendiculars BO and CK, then fhall OK be equal to BC, and the sum of AO and KD are the difference of the Diameters of the Bares AD and BC; and confequently All the Semi difference, and BO the heighth of the Frftrum, and AM is the side of the greater Base, and EM is the heighth of the whole Pyramid. Then, AS AO the Semi-difference of the Diameters, Is to BO the heighth of the Fruftrum,

So is AM (the great Diameter, To EM the heighth of the whole Pyramid:

So the heighth of the whole Pyramid AED, will be found to be 30 Foot; for the greater Diameter AD is 24 Inches, the leffer 8, the difference 16, the Semi-dil ference 8, therefore fhall Me be 30 foot; for, 8:201 : 123 30.

1 Now having found the heighth of the whole Pyximid to be 30 Feet, I thereby find the conteat of the wholeo Pyramid to be 40 Foot, then in theleffer Pyramid BCE there is given the file of its Base BC = 8, and its length IE 10 Inches for EM 30 - IM 20 ** IE-10, 'fo I find the solid Content of it to be 1.48. Foot, which being subftracted

front from 40 the content of the greater Pyramid there will remain 38. 52 Feet for the true folid Content of the Fruftrum ABCD.

After the same manner is found the folidity of the Fruftrum of a Cope. And this is also useful in measuring of Tapering Timber, Round or Square, aod for finding the Liquid Capacity of Brewers Conical, os Pyramidal Tuns.

Of Measuring Artificer's Work. Because inost, if not all Workmen in cafts ing up their Dimenfions, make use of cross Multiplication, I think it will not be amifs. to give you an Example, or two of it, before I enter upon their several methods of measuring their work.

Note, Feet multiply'd by Feet produce Feet ; Feet by Inches produce Feet and Incbes į and Inches by Inches, produce Incbes. and twelfths of Inches.

Feet Inches.
Therefore te. multiply 3

by 44

:3 First multiply the Feet by the 12 : 0 Feet and the product is 12 2:12! Feet, then multiply cross-wise 0.: 9 Feet by Inches, viz.. 4 by 6_14: 10. which makes 29 Inches or 2 Feet and 3 by 3, which makes 9 Inches. Laftly, Multiply

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the Inches 6 by 3 and the product is 18 twelfths of an Inch, or i. _Inch; all which products set down and add together, as in the Operation:

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55:11:8 1 80: 8:11 235: 6:8

Fird, Glafiers Work, and rub'd and gauged Brick-work are measur'd by the Foot Square.

Example If a Window be 6 Feet 6 Inches high and 3 feet 4 Inches broad; how many Square Feet of Glazing is there. Ans. 21 : 8.

6 3

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21 : 8

Secondly, Secondly, Painting, Paving, Plastering, and Wainscoting,are measur’d by the Square Yard.

Ex.imple. It a Ceiling, be 13 Feet broad, and 17 Feet. 4. In long, how many Yards doth it ccntain? Anf. 25 Yards.

13 17

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221 . 0

4.4

9) 225:41

25:

Thirdly, Tyling, Raftering, and Flooring measur'd by the Square, containing 100 Square Feet.

Example If a piece of Tyling, be 40 Feet long and 13 Feet 5 Inches broad: How many Squares are there in it? Ans. 5 Sg. 36 Feet

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