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.

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 frustrum 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?

20×20×2+16×16ו7854x4 = 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.

A

B

C

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? 50÷3-166, then 40x40x2+30x30x•7854×16·6=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 7854, and the product by one sixth of the altitude, and it will give the solidity.

427

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

30X30X4+36x36x•7851x3•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 7854x21=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.

40x40+30x30x⚫7854x10=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.

20x20x+7854×32 (=60×8÷15)=10053-12 Ans. ART. 43. To measure the middle Zone, or middle Frustum, of a Parabolick 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?

'40x40 1600
30x30 = 900

Sum = 2500

Half sum = 1250

1600-900 700 the difference of the squares. 700x+3=210= three tenths of do. then,

2500+1250+210×20 (=} of 60)=79200 Ans.

ART.

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

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

A prismoid is a solid somewhat like the frustum of a pyramid, but its bases are disproportional.

RULE. The same as for the frustum of a cone or pyramid: or, to the areas of both bases, add a mean area, that is, the square root of the product of the two bases, then multiply that sum by a third of the height or length, and it will give the solidity.

If the diameters of the greater base of a cylinderoid be 30 and 20 inches, the diameter of the less base 12, and length 60 inches; what is the solidity.

30×20=600

12x12 = 144

144x600=293.9

1037.9

1037-9x 7854x20 (=60+3)=

16303-33 Ans.

If the diameters of the greater base of a prismoid be 30 and 20 inches, the less base 20 by 10 inches, and length 30 inches: What is its solidity?

30×20=600
20x10 = 200

600×200=346.4

1146.4

1146.4×10 (= 30÷3) = 11464 solidity. in inches.

ART. 45. To measure a Solid Ring.

RULE.-Measure the internal diameter of the ring, and its girth, or circumference: then multiply the girth by 31831, and the product will be the diameter of the wire, which add to the internal diameter; multiply this sum by 3-1416, and the product will be the length of a cylinder equal to the ring of the same base. Then the area of a section of the ring multiplied by the length of the said cylinder will give the solidity of the ring.

if an iron ring be 12 inches in girth, and its internal diameter be 20 inches; what is its solidity ?

•3_931×12=3·8=ring's diameter. 20+3.8X3-1416-74-77 the length of a cylinder equal to the ring: And

3.8X3.8X-7854x74·77-847-97-solidity. ART. 46. To measure the Solidity of any irregular Body, whose dimensions

cannot be taken.

Take any regular vessel, either square or round, and put the ir regular body into it: pour so much water into the vessel as will exactly cover the body, and measure the dry part from the top of the vessel to the water, then take out the body, and measure again from the top of the vessel to the water, and subtract the first measure from the second, and the difference is the fall of the water: then, if the vessel be square, multiply the side by itself, and that product by the fall of the water, and you will have the content of the body; but if it be a long square, multiply the length by the breadth, and that product by the fall

of

of the water; or, lastly, if it be a round vessel, multiply the square of the diameter by 7854, and that product by the fall of the water, and you will have the content. EXAM. 1. A body) being put into a vessel 18 inches square, on taking out the body, the water sunk 9 inches; required the content of the body

EXAM. 2. A body ( EXAM. 3. A body

being put into a round tub, whose diameter was 1.5 foot, on takre-ing out the body, the water fell 1.5 foot; what was the content of the body?

put into a cistern 4
feet by 3, on taking
it out, the water
fell 6 inches;
quired the content
content
of the body?
4x3x-56 feet, con-

content.

Of the five Regular Bodies.

1.5 x 1.5 x ⚫7854x1.5 =2.65 feet, con

tent.

There are five solids contained under equal regular sides, which by way of distinction, are called the five regular bodies.

These are the Tetraedron, the Hexaedron or Cube, the Odaedron, the Dodecaedron, and the Eicosiedron. The measuring of the cube was shewn at Art. 28. I shall now show how to measure the other four by the following Table, which is the shortest method.

A Table of the solid and superficial content of each of the five bodies, the sides being unity, or 1.

All like solid bodies being in proportion to one another as the cubes of their like sides, the solid content of any of these bodies may be found by multiplying the cubes of their sides by the numbers in the second column under Solidity; and their superficies, by multiplying the squares of their sides into the numbers in the third column, under Superficies.

OF THE TETRAEDRON.

This solid is contained under four equal and equilateral triangles, that is, it is a triangular pyramid of four equal faces, the side of whose base is equal to the slant height of the pyramid, from the angles to the vertex.

ART. 47. The side of the Tetraedron being 3, to find the solid and superficial

content.

=

Cube = 3×3×3 = 27, and 27x-11785 = 3.18195 solidity.
Square = 3x3 = 9, and 9×1.73205 = 15·58845 = superficies.

OF THE OCTAEDRON.

This solid is contained under eight equal and equilateral triangles, which may be conceived to consist of two quadrangular pyramids of equal bases joined together, the sides of whose bases are equal to the given sides of the triangles, under which it is contained.

ART.

ART. 48. The side of an Otaedron being 3, to find the solid and superficial

content.

Cube = 3x3x3 = 27, and 27ו4714 12·7278 solidity.
Square 3x3 = 9, and 9x3-464 31-176

OF THE DODECAEDRON.

superficies.

This solid is contained under 12 equilateral pentagons, and may be conceived to consist of twelve pentagonal pyramids, of equal bases and altitude, whose vertices meet in the centre of the dodecaedron.

ART. 49. The side of a Dodecaedron being 3, to find the solid and superficial

content.

Cube 3x3x3 = 27, and 27x7·663119 = 206·904.
Square 3x3 = 9, and 9×20-6457 185·8113.

=

OF THE EICOSIEDRON.

This solid is contained under twenty equal and equilateral triangles, and may be conceived to consist of twenty equal triangular pyramids, whose vertices all meet in the centre.

ART. 50. The side of an Eicosiedron being 3, to find the solid and superficial

content.

Cube = 3x3x3 = 27, and 27×2∙18169 = 58.90563 = solidity.
Square = 3x3 = 9, and 9×8-66025 = 77.94225= superficies.

As the figures of some of these bodies would give but a confused idea of them, I have omitted them; but the following figures, cut out in pasteboard, and the lines cut half through, will fold up inte the several bodies.

Among the many different canons drawn from Stereometry, for Gauging casks, the following is as exact as any,

Take the dimensions of the cask in inches, viz. the diameter at the bung and head, and length of the cask; subtract the head diameter from the bung diameter, and note the difference.