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Er. 5. What is the solidity of a cylinder, when the circumference of the base is 7.85 feet, and the height 12 feet?

(7.85)* x 0758 x 12=58.847022600 the solidity of the cylinder. Prob. 16. To find the solidity of a pyramid.

Multiply the area of a base or end by the perpendicular height, and one-third of the product will give the solidity.

Er. 1. What is the solidity of a square pyramid, of which the height is gf. 6in, and each side of the base 2f. 3in. ?

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Et. 2. Required the solidity of a cone, the diameter of the baso being 2f. 6in, and the height 12f.

f. in.
2 6=2:5

25 x 25 x 7854 x 12=58.905 the solidity of a cylinder of the same base and altitude.

58.905 And = 19*635 the solidity of the cone.

3 Prob. 17. To measure the frustrum of a square pyranid.

To the rectangle of the sides of the two ends, add the sum of their squares; that sum being multiplied by the heigbt, one-third of the product will give the solidity.

Er. In the frustrum of a square pyramid, one side of the greater base being 3f. 6in.; each side of the lesser end or top 2f. 3in.; and the perpendicular height 6f. gin. ; required the solidity.

f. in.
3 6535
2 3=225

6 9=6.75
35
2.25

2.25 side of the lesser base. X35 X 2.25 x35 side of the greater base,

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56.671875 solidity of the frustrum.; Method To the rectangle of the sides of the two bases, add one third of the square of their difference ; that sum being multiplied by the height, will give the solidity.

Ex. In the frustrum of a square pyramid, let one side of the greater base be 3f. 6in.; each side of the top 2f. 3in.; and the perpendicular height 6f. gin.; required the solidity. f. in.

f. in. 3 6

3 6 X 2 3

- 2 3

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6 3 6 9 30 4 6

50 8 0 9 an wer.

Prob. 18. To measure the frustrum of a cone..

To the rectangle of the two diameters, add the sum of the squares of these diameters ; multiply the sum by •7854, and that product by the length; then one-third of the last product will give the solidity:

Note. If the circumferences are given, proceed in the same manner, only multiply by *07958, instead of 7854.

Er. What is the solidity of the frustrum of a' cone, the diameter of the greater end being 3 feet, and that of the lesser end 2 feet, and the atitude 9 feet? 3x3=

•7854 2x2=

x 19
2 X3= 6

70686
7854

19

14 9226

9

3) 134:3084

! 44.7678 the solidity of the frustrum. Dof. A PRIsmoiD is a solid contained under six planes, the ende being parallel, but unlike rectangles; and the other four sides, each opposite, two are equal trapeziums.

Prob. 19. To measure a prismoid.

Multiply the length at the greater end by the breadth at the lesser end, and the length at the lesser end by the breadth at the greater end,

To half the sum of the two products, add the areas of the two ends ; that sum multiplied by one-third of the height, gives the solidity.

Ex. What is the solidity of a prismoid, whose greater end is 12 inches by 8, and the lesser end 8 inches by 6, and the length or height 5 feet?

12x6=72
8x8=64

2)136

68 half the sum of the producte.
12x8=96 the area of the greater end.
8 x 6=48 the area of the lesser end.

212 sum.

20 one-third of the height.

1728)4240(2 solid ft. and 784 solid in tho Ans,

3450

784

Prob. 20. To find the solidity of a sphere or globe.

Multiply the cube of the diameter by 5236, and the product is the solidity. Ex. What is the solidity of a globe, whose diameter is 3 feet?

3x3 x3=27
And •5236

27

36652 10472

14:1372 the solidity. Prob. 21. To find the solidity of a segment of a globe.

To three times the square of half the diameter of the base, add the square of the height; multiply the sum by the height, then the product multiplied by '5236, will give the solidity.

Er. What is the solidity of a spherical segment, the diarneter of the base being 4 feet, and the height of the segment 3 feel?

j=2 balf the diameter.
2
x2

3 x3=9 the square of the height.

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Prob. 22. To find the solidity of a spherical zone, the radii of the two parallel circles being given, and the distance between them.

To the squares of the two radiuses, add one-third of the square of the height; multiply the sum by the height, and the product by 1.5708, will give the solidity.

Er. What is the solid content of a spherical zone, whose greater radius is 12 inches, and the lesser 10 inches ; and the height or distance of the ends 4 inches? 12° +10° +49X 4x1.5708=15666113 the solidity required.

3

Prob. 23. To find the solidity of a wedge.

Multiply the area of the base by the perpendicular height, and half the product will give the solidity.

Er. Required the solidity of a wedge, the dimensions of the base being 1f. 3in, and 2f. 6in. and the height 4f.

2.5 X 1.25 X4 Then

= 625 the solidity.

2 Nole.-The solidity of any prismatic ungnla will be found in the same inanner; that is, half the product of the area of the base multiplied into the height, will give the solidity.

Prob. 23. To find the solidity of a hoof, or ungula, from the frustrum of a square pyramid.

To the square of the side of the base, or that end which is complete, add one half of the product of the sides of the two ends ; this being multiplied by otre-third of the height, gives the solidity.

And if the hoofs are any other than that of a square pyrainid, find the square root of the area of each end, which will give the side of a square equal in area; then proceed as above.

Er. Required the solidity of an ungula, from the frustrum of a square pyramid, the side of the greater end, which is complete, being if. 6in. that of the lesser end 1f. 3in. and the height 5f. ?

1.5 X 1.25

) * =5'3125 which is the solidity required. 1.5

2 Prob. 24. To find the solidity of a spheroid ; the fixt axis and the revolving axis being given.

Multiply the fixt axis by the square of the revolving axis, and the product by '5236, will give the solidity.

Ex. What is the solidity of a prolate spheroid, the major axis of which is 100 feet, and the minor 60 feet?

100 x 60 x 5236=188496 the solidity required. Er. What is the solidity of an oblate spheroid, whose transverse axis is 100 feet, and the shortest axis 60 feet?

60 x 100% X •5236=314160 the solidity required. Prob. 25. To find the solidity of a parabolic conoid, the diameter of the base being given, and the perpendicular height.

Multiply the square of the diameter of the base by 3927, and the product by the height will give the solidity.

Er. What is the solidity of a par bolic conoid, the height being 50 feet, and the diameter of the base 30 feet ?

30' x 3927 x 50=17671.5 the solidity required. Prob. 26. To find the solidity of the frustrum of a parabolic conoid the greater diameter, the lesser, and the perpendicular height being given.

Then (ist

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