Problem 5th. To Measure a Cylinder. Fig. 31.

Definition. This folid has a round bafe, and is in form of a rolling stone.

Dimenfions. Measure its diameter at the bafe and length.

RULE. Multiply the area of the base by the length, and the product is the folidity.

If the diameter of the bafe of a cylinder be 60 inches and its length 22 feet, required its folidity?

Definition. This folid has a circular bafe and tapers gradually to a point, over its centre.

Dimenfions. Measure the diameter of its bafe and height.

RULE. Multiply the area of the bafe by a third of the height, if the cone be an upright one; if the cone be an oblique one its bafe will be eliptical, and the area of that base must be multiplied by a third of a perpendicular from the apex of the cone, to the plain of the base produced for its folidity.

If the diameter of the bafe of an upright cone be 40 inches, and its height 8 feet, required its folidity ?

40X40X.7854 = 1256.74 the area of the bafe. 1256.64 X 23

144

23.27 feet the folid content.

Problem

Problem 7th. To Measure the lower Fruftum of a Cone or Pyramid. Fig. 33.

Definition. This folid is a part of a cone or pyramid, having its top taken away by a plain parallel to its bafe.

Dimenfions. Measure the fide of each base, and the perpendicnlar height or length.

RULE. For the fruftum of a pyramid; to the areas of both bases add a mean area, multiply that sum by a third of the height, and the product is the folidity.

For the fruftum of a cone; to the fum of the squares of the diameters of both bafes, add the product of the diameters, multiply that fum by the area of unity and by one third of the altitude or height, and that product is the folidity.

If the two diameters of the bases of a fruftrum of a cone be 50 and 30 inches, and the height 9 feet, required the folidity ?

50×50+30X 30+50× 30× .7854 X 3 = 80.1

Problem 8th. To Measure a Globe. Fig 34,

Definition. This folid may be conceived to be generated by a femicircle turning round on its diameter. Dimenfions. Measure its axis or diameter.

RULE. Multiply the cube of the axis by .5236 called the folidity of unity, and that product is the folidity. For all fimilar folids are in the direct ratio of their like fides. Or multiply the area of the diameter, by two thirds of the faid diameter, and that product is the folidity.

If the axis of a globe be 50 inches, query its folidity?

50X 50X50X.5236 = 65450 inches the folidity.

Problem 9th. To Measure the Segment of a Globe. Fig. 34.

Definition. This is a part of a globe made by a plain paffing through it parallel to its axis.

Dimenfions. Measure the diameter of the fegment's bafe, and greatest altitude of the fegment.

RULE. To three times the fquare of the radius of the fruftum's bafe, add the fquare of the altitude of the fruftum, multiply that fum by the altitude or height of the fruftum, and that product multiplied by the folidity of unity viz. .5236, gives the folidity. Or you may compute a diameter at the middle of the fruftum's altitude. Thus

By the diameter of the fruitum's bafe and altitude find the axis of the fphere; from which fubtract half the height of the fruftum; then multiply the remainder by half the height of the fruftum ; and the square root of that product doubled is the diameter at the middle of the altitude. Then to 4 times the fquare of that diameter add the fquare of the diameter at the bafe, and multiply that fum by the area of unity(.7854) and that product by a fixth of the height of the fruftum, and you have the folidity.

N. B. This rule is very exact for the fegment lefs than a femiglobe.

If the diameter of a fegment of a globe's base be 16 inches, and its height 4, query its folidity?

8X8X 3

= 192

4X4 = 16

208

N. B.

the whole globe.

8x8416, and 16+4=20 the axis of

8 X 8X3 192

16X 16256 448 × 16 X .52363753.1648 the folidity of the other fruitum.

448

Then 20 X 20 X 20 X.5236

Also 435.6352 + 3753-16484188.8 Proof.

4188.8

18 X 2 = 6.

Or thus by a middle diameter, which X 212 the middle diameter.

Then 12×12×4+16 × 16 × .7854×4 435.6352 as before

Problem 10th. To Measure the middle Zone of a Globe. Fig. 36.

Definition. This is a part of a globe, fomewhat like a cafk, two equal fegments or flices being wanting, one on each fide of the axis.

Dimensions. Meafure the end diameter, middle diameter, and length of the zone.

RULE. To twice the fquare of the middle diameter, add the fquare of the end diameter, then multiply that fum by the area of unity (.7854) and that product multiplied by one third of the length is the folidity.

Or to four times the fquare of the middle diameter, add twice the fquare of the end diameter, that fum multiplied by the area of unity, and that product by a fxth of the length is the folidity.

N. B. This rule is applicable to the fruftum of a cone or pyramid.

If the middle diameter of a zone be zo inches, he end diameters each 16 inches, and length 12, required its folidity?

20X20X2+16x16 X.7854×4 = 3317.5296 anf,

Problem

1

Problem 11th. To Measure a Spheroid. Fig. 37.

Definition. This folid may be generated by the revolution of a femielipfis about one of its diameters. Hence, if the femielipfis turns about its shorter diameter, the folid will be fomewhat like a turnip. Bnt if the femiarch of the elipfis be turned about the longer diameter, the folid will partly resemble an egg, the former is called a prolate, and the latter an oblate fpheroid.

Dimenfions. Measure both the principal axis of the folid.

RULE. Multiply the fquare of the greateft circle's diameter by .7854 and that product multiplied by two thirds of the axis, is the folidity. This folid being two thirds of its leaft circumfcribing cylinder.

Or you may multiply the fquare of the circular diameter by the other axis, and then as 2626: 1375 :: that product the folidity nearly.

If the principal axis of a fpheroid be 90 and 50 inches, quere the folidity thereof?

50 X 50 X .7854 × 60 = 117810 answer. N. B. 90 X 2 ÷ 3 = 60

Or as 2626: 1375 50 × 50 × 90: 117812 ans. nearly agreeing with the former.

Problem 12th. To Measure the middle Fruftum of the Spheroid. Fig. 38.

Definition. This is a cafk like folid, wanting two equal fegments to compleat the spheroid.

Dimenfions and RULE, the fame as in problem tenth.

If the middle and end diameters of the middle fruftum of a spheroid, or cafk like folid be 40 and 30 inches, and its length 50, required its folidity?

Q3

N. B.