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Firft, find the Superficial Content of the Bafe: 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
ramid or Cone.
Fig. 83. The Fruftrum here given to be measured is ABCD, the fide of the greater Bafe A, being 24 Inches, and the fide of the leffer Bafe at B, 8 Inches, and the length of it I M 20 Feet BO CK 20 Foot.
It is evident that if I find the Solidity of the whole Pyramid AED, and alfo 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 Fruftrum ABCD; and certainly this way of meafuring the Fruftrum of a Pyramid or Cone, is the moft. exact of any, and it may be eafily 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 Semi-difference of the Bafes,
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 Baf, and BC the Diameter of the leffer Bafe from B, and C, let fall the Perpendi culars 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 Bafes AD and BC; and confequently AO the Semi difference, and BO the heighth of the Frftrum, and AM is the fide of the greater Bafe, 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-dif ference 8, therefore fhall ME be 30 foɔt; for,
8:20: 128 301
Now having found the heighth of the whole Pyramid to be 30 Feet, thereby find the content of the whole Pyramid to be 40. Foot, then in the leffer Pyrainid BCE there is given the file of its Bafe BC= 8, and its length IE 10 Inches for EM 30-IM 20
IE10, fo I find the folid Content of it to be 148. Foot, which being fubftracted
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 fame manner is found the folidity of the Fruftrum of a Cone. And this is alfo ufeful in meafuring of Tapering Timber, Round or Square, and for finding the Liquid Capacity of Brewers Conical, os Pyramidal Tuns.
Of Meafuring Artificers Work.
Because moft, if not all Workmen in cafting up their Dimenfions, make ufe of cross Multiplication, I think it will not be amifs to give you an Example, or two of it, before I enter upon their feveral methods of measuring their Work.
Note, Feet multiply'd by Feet produce Feet Feet by Inches produce Feet and Inches; and Inches by Inches, produce Inches. and twelfths of Inches...
Therefore to multiply X
Firft multiply the Feet by the 12:0
Feet by Inches, viz. 4 by 6 14:10 — which makes 29 Inches or 2 Feet and 3 by 3, which makes 9 Inches. Lastly, Multiply
the Inches 6 by 3 and the product is 18 twelfths of an Inch, or 1. Inch; all which products fet down and add together, as in the Operation:
55:11:8 1 80:8:11 235:6:8
Firft, Glafiers Work, and rub'd and gauged Brick-work are meafur'd by the Foot Square.
If a Window be 6 Feet 6 Inches high and 2 feet 4 Inches broad; how many Square Feet of Glazing is there. Anf. 21: 8.
2 : 0
Secondly, Painting, Paving, Plastering, and Wainftcoting,are meafur'd by the Square Yard.
If a Ceiling, be 13 Feet broad, and 17 Feet. 4 In long, how many Yards doth it contain? Anf. 25 Yards.
Thirdly, Tyling, Raftering, and Flooring meafur'd by the Square, containing 100
If a piece of Tyling,
be 40 Feet long
and 13 Feet 5 Inches broad: How many
Squares are there in it?
Anf. 5 Sq. 36 Feet.