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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. 24. To Measure the Fruftrum of a Py
a Tamid or Cone.
Fig. 83. The Fruftrum here given to be mealured is ABCD, the fide of the greater Base A, being 24 Inches, and the side of the lesser Bife at B, 8 Inches, and the lengih of it I M 20 Feet = B = CK 20 Foot.
BO It is evident that if I find the Solidity the whole Pyramid AED, and also the Sclidity 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 meafured 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 Frustrumn, · 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 shall OK be equal to BC, and the sum of AO and KD are the difference of the Dameters of the Bares AD and BC; and consequently AD 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, --ás AO the Sami-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-diterence 8, therefore fhall Me be 30 foot; for,
&:201 : 123 30. Now having found the heighth of the whole Pytámid to bé 30 Feet, I thereby find the content of the wholeo Pysamid 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 * to be 1648 And the folid Content of it
-I to be 148. Eoota which being fubftracted
fron: 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 foo lidity of the Fruftrum of a Cone. And this is also useful in measuring of Tapering Timber, Round or Square, and for finding the Liquid Capacity of Brewers Conical, os Pyramidal Tuns.
Of Measuring Artificer's Work. Because most, if not all Workmen in caft ing up their Diinenfions, 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
and Inches by Inches, produce Incbes. and twelfths of Inches..
Feet Inches. Therefore 10. multiply X;
4 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
the Inches 6 by 3 and the product is 18 twelfths of an Inch, or 1. Inch; all which products set down and add together, as in the Operation:
55:11:8 1.80 :8:ļI 235:6:8
First, Glasiers Work, and rub’d and gauged Brick-work are ineasur'd by the Foot Square.
21 : 8
Secondly, Secondly, Painting, Paving, Plastering, and Wainstcoting,are measur'd by the Square Yard.
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 Sq. 36 Feet