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10, and its perpendic. 81; sides of the smallest triangle 3.5 and the perpendicular of its triangle 3; and length

14.?

10x41=42.5 area largest end.

3:5X1:5=5.25 area smallest end. Areas 42-5X 5.25=223-1250=14:93 root added.

2

sum 62-68 sum 62.68x4=292.5 solidity, Ans. 3. What is the solidity of the frustum A B C D fig... ure third, case 5th ; sides of its fargest square 9, and, the smallest 3, and length 14?

Samne question by rule second, Sides of the two squares 9X3327 product. 9-3—6 dif. X6=3612 add.

Solidity.

39x143546 Ans. * What is the solidity of the frus tum ABCD figure fourth, case fifth; sides of the largest end 15 by 9; sides of the smallest end 5 by 3.2; and 14 long?

Sides of the largest end 15x9.5=142-5 area.

Sides of the smallest end 5X3.2 = 16•0 area. Areas. 142.5X165V2280.0=+7.77 root 47.7 add.

2

sum 206•2 206.2x142886-6=1=962-21 Anse 5. What is the solidity of the frustum of a conc that is 10 feet square at one end, and 4 at the other, and is 32 feet long?

Ans. 1664 ft. 6. What is the solidity of the frustum of a cone that is 16 in. square at one end, and 12 at the other, and is 16 ft. long?

Ans. 2134 7. What is the solidity of the frestum of a cone that is 24 in. square at one end and 21 at the oth er, and is 10 feet long?

Ans. 3575

CASE VII.

To find the solidity of the segment of a cone, as the

parts above A B in the figures, case 5th. RULE.-This part forms a cone of itself, and must be measured by case fifth.

CASE VIII.

To find the solidity of a wedge, when the edge

and large end are of equal width. RULE.--Multiply the length and width together and that product by half the thickness, the last product is the solidity,

1: 1

Examples 1. What is the solidity of a wedge that is 12 ft. long, 12 wide and 6 thick, at the large end?

12X12=144x6 =432 solidity, Ans. 2. What is the solidity of a wedge that is 10 ftlong, 6-inches wide, and 6 thick ? Ans. 1. ft.

hi 21

CASE IX.

To find the solidity of a wedge when the edge is nar

rower than the large end,
Rule. To the width of the edge add twice the
width of the large end, and reserve the sum ; mul-
tiply the length of the wedge by the thickness of
the large end ; multiply this last product and the
reserved sum together, and divide by 6; the quo-
tient is the solidity.

Examples
1. What is the solidity of a wedge that is 12 ft

. long, and 18 in. wide at the largest end, and 12

in. wide at the edge ; and 6 in. thick at the large end ? Width of the edge

12 Width of large end 18 x 236

48 reserved sum. Length of wedge

144 inches. Thickness of large end 6 inches.

864 product. Pro. 864 X 4841472-66912 in. or 4 ft. Ans. 2. What is the solidity of a wedge that is 20 long, 15 wide at the large end, and 10 at the edge ; and 12 thick ?

Ans. 1600.

Note: This rule will give the solidity of a cone when it is squa re, or in form of a parallelogram.

Example. What is the solidity of the cone, figure third, case fifth ; width of the large end 9, thickness 9, and length 21 ? Width of the edge

O
Width of large end 9.X2 18

18 reserved sum. 21 X9 189 X18-3402-6567 Ans.

(S:e Ans. to Quust. 3, Case Fifth.)

CASE X. To find the solidity of a globe, or sphere. DEFINITION.-A globe or sphere is a round body bounded by a surface every point of which is equally distant from a point within called the centre ; a line passing from one side to the other through the centre is called the diameter, or axis.

RULE.-Cube the diameter, or axis, and multiply its cube by .5236 the last product is the solidity.

Eramples. 1. What is the solidity of a globe, or sphere whose diameter is 113?

#113x113X113 X.523637355500.8692 solidity.

2. What is the solidity of the globe which we inhabit, in solid miles; allowing its circumference to be 25000 miles ? As 22 is to 7 so is 25000 to 7954

diam. nearly. (Fractions omitted.) 263485304337 sol miles Ans.

3. What is the solidity of a cannon ball that is 9 inches on its diameter ? Ans. 381.7044 sol. in.

4. What is the solidity of a sphere whose diameter is 2 ft, 8 in.

Ans. 9.927 ft. solidity.

CASE XI. To find the solidity of a segment, or part of a globe, or sphere ; or part of a globe cut off parallel to the diameter, as part DGF.

See the figure.

M

E

А

B diam. A B.14

D

RULE.Square the radius of its base, (as D o, or FO) multiply its square by 3, and reserve the product ; square the depth o G, add the square and reserved product together ; and multiply the sum by the depth; and the last product by 5236 ;. the product is the solidity..

Examples. 1. What is the solidity of the segment D GF; radius D o, or F 0,7; and depth o G, 5?

Radius 7x7x3=147 reserved product.
Depth 5X525 add.

172x depth 5. 860 X 5236.450.296 Ans.

che

:- 2. Required the solidity of a segment of a globe whose semidiameter, is 9 in. depth 9 in.

Ans. 1526.8176 in.

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To find the solidity of the middle zone of a sphere, or globe ; or the part of a sphere after two segments have been cut off, parallel to the diameter or

ašis, as CD E F figure in case eleventh.

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Rule.--Square the semidiameter of both ends and add the squares together; and reserve the sum ; square the height, or distance of the two ends as o M, and add of its square to the reserved sum ; multiply the sum by the height, or distance o M, and this product again by 1.5708, the last product is the solidity.

Examples.

1. What is the solidity of the middle zone C D E F, (case eleventh); diameter CE, or DF, 14, and height 0 M, 3 ?

NOTE.-In this example the diameters are alike which is not always the case.

Diam. 14 semi.=7X7349

Diam. 14 semi.=7X7=49
Height o M 3X 3=9=*= 3 add.

ht.

Ans. 101X3=303x1.5708==475.9524

2. What is the solidity of the middle zone of a sphere whose greatest diameter is 12, and least. 8, and height or length 10?

Ans. 1340°41t.

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