EXAMP. 1. The former cone in Art. 32, Examp. 3, being cut off in the middle, the greater diameter AC is 13, the less BD 61, and height EF 24, to find the content of the frustum.

AC 13 inches.

BD 6.5 inches.

-

13
6.5

A

BE

the differ.

.7854

720

EXAMP. 2. What number of barrels, each 32 gallons of Ale measure, is contained in a cistern whose largest diameter is 6 feet, and smallest diameter 5 feet, and whose depth is 8 feet?

6x5=30

329340 2112 cubic inches, which divided by 9024, the cubic inches in a barrel or 32 gallons, gives 36·5 barrels nearly, Aus.

If the answer had been required in Beer Measure, where the barrel contains 36 gallons, the answer would have been 32-4 barrels.

Note. If when the end diameters of a conical cistern are given, it is required to find the length of the cistern to contain a certain number of barrels; divide the cubic feet contained in the number of barrels by the mean area, and the quotient will be the height.

Let the mean area be as in the last Ex. to find the length of the cistern to contain 50 barrels of 32 gallons of Ale measure.

261.11111 &c. cubic feet in 50 barrels, which divided by 23 8938, the mean area, gives 10.95 feet, for the length of the cistern, Ans.

To find the diameters of the cistern, when the content, and length, and difference of the diameters, are given, see Art. 53.

ART. 34. To measure a Sphere or Globe.

Definition. A sphere or globe is a round solid body, in the middle of which is a point, from which all lines drawn to the surface are equal.

RULE. Multiply the cube of the diameter by 5236, and the product will be the solid content.

Or, multiply the circumferenee by the diameter, which will give the superficial content; then multiply the surface by one sixth of the diameter, and it will give the solidity.

Or, multiply the cube of the diameter by 11, and the product divided by 21, will give the solidity.

EXAMP. The diameter, AB, of a globe, is 4.5 feet; to find the solid content.

Note. If the circumference, or greatest circle of the sphere, be given, multiply the cube of it by 016887 for the content.

The surface of the globe may be found by multiplying the square of the diameter by 3-1416; or by multiplying the area of its greatest circle by 4, or the square of the circumference by ⚫3183.

When the solidity of a globe is given, the diameter may be found by dividing the solidity by 5236, and extracting the cube root of the quotient.

Or, if the circumference be required, divide the solidity by 016887, and the cube root of the quotient will give it.

ART. 35. To measure the Solidity of a Frustum or Segment of a Globe.
Definition. The frustum of a globe is any part cut off by a

plane.

RULE. To three times the square of the semidiameter of the base, add the square of the height; then multiplying that sum by the height, and the product by 5236, you will have the solid

content.

EXAMP. The height BD being 9 inches, and the diameter of the base AC 24 inches: to find the content.

To measure the Surface of a Frustum or Segment of a Globe.

RULE. Find the diameter of the globe by Art. 24, and the surface of the whole globe, by Art. 34; then, as the diameter of the globe is to the height of the frustum; so is the surface of the globe to the surface of the frustum; then, by Art. 15, find the area of the base; add these two together, and the sum will be the whole surface of the frustum.

ART. 36. To measure the middle Zone of a Globe.

Definition. This part of a globe is somewhat like a cask, two equal segments being wanting, one on each side of the axis.

RULE. To twice the square of the middle diameter, add the square of the end diameter; multiply that sum by 7854, and that product, multiplied by one third of the length, will give the solidity.

M S

Or, to four times the square of the middle diameter add twice the square of the end diameter; that sum multiplied by 7854, and that product by one sixth of the length, will give the solidity.

Note. This rule is applicable to the frustum of a cone or pyramid.

If the middle diameter of a zone be 20 inches, the end diameters each 16 inches, and length 12 inches: Required its solidity? 20x20x2+16x16x 7854x4=3317-5296, Ans.

ART. 37. To measure a Spheroid.

Definition. A spheroid is a solid body like an egg, only both its ends are the same.

RULE. Multiply the square of the diameter of the greatest circle, viz. the diameter of the middle (DB in the figure) by the length AC, and that product by 5236, and you will have the solidity.

EXAMP. The diameter BD being 20, and the length AC 30, to find the

content.

20X20X30X 5236-6283-2, Ans.

ART. 38. To measure the middle Frustum of the Spheroid. Definition. This is a cask like solid, wanting two equal segments to complete the spheroid.

RULE. The same as in Article 36.

If the middle and end diameters of the middle frustum of a spheroid be 40 and 30 inches, and its length 50; what is its solidity? 503-166, then 40x40x2+30x30x7854x16 6=53454-324,Ans.

ART. 39. To measure a Segment, or Frustum of a Spheroid. Definition. This is a part of a spheroid made by a plane, parallel to its greatest circular diameter.

RULE. To four times the square of the middle diameter add the square of the base diameter, then multiply that sum by 7354, and the product by one sixth of the altitude, and it will give the solidity.

If the base diameter of the end frustum of a spheroid be 36, diameter at the middle of the height 30, and the height 20 inches required its solidity?

30X30X4+36X36X 7854×3 3=12689-55+, Ans.

ART. 40. To measure a Parabolick Conoid.

Definition. This solid may be generated by turning a semiparabola about its abscissa or altitude.

RULE. As a parabolick conoid is half of its circumscribing cylinder, of the same base and altitude; multiply the area of the base by half the height for the solidity.

If the diameter of the base of a parabolick conoid be 40 inches, and its height 42; what is the solidity?

40X40X 7854×21=26389-44, Ans.

ART. 41. To measure the lower Frustum of a Parabolick Conoid. Definition. This solid is made by a plane passing through the conoid parallel to its base.

RULE. Multiply the sum of the squares of the diameters of the bases by 7854, and that product by half the height, for the solidity. If the diameters of a frustum of a parabolick conoid be 40 and 30 inches, and its height 20 inches; required its solidity.

40×40+30×30× 7854×10=19635, Ans.

ART. 42. To measure a Parabolick Spindle.

Definition. This solid is formed by an obtuse parabola, turned about its greatest ordinate.

RULE. This solid being eight fifteenths of its least circumscribing cylinder, multiply the area of its middle or greatest diameter by eight fifteenths of its perpendicular length, and it will give its solidity.

If the diameter at the middle of a parabolick spindle be 20 inches, and its length 60; required its solidity.

20×20X 7854×32 (=60×8-15)=10053.12, Ans.

ART. 43. To measure the middle Zone, or middle Frustum, of a Par. abolick Spindle.

Definition. This is a cask like solid, wanting two equal ends of said spindle.

RULE. To the sum and half sum of the squares of the two diameters add three tenths of the difference of their squares, which multiply by a third of the length, and the product will be the solidity.

If the middle and end diameters of the middle frustum of a parabolick spindle be 40 and 30 inches, and its length 60; what is its solidity?

40×40 1600 1600-900-700 the difference of the squares. 30X30 900 700×·3=210=three tenths of do. then,

Sum=2500 2500+1250+210 × 20 (=1 of 60)=79200, Ans. Half sum=1250

ART. 44. To measure a Cylindroid, or Prismoid.

Definition. A cylindroid is a solid somewhat like the frustum of a cone, one base may be an ellipsis, and the other a disproportion. al ellipsis or circle.