Examples.

1. How many solid feet are in a stone that is. 21 feet long, 2 feet wide at one end, and 3 feet at the other, and 1 foot 6 inches thick.

2+3=5÷1=2•5×21×1·5=78·75, Ans.

2. How many solid feet in a stone wall that is 51 feet long, and 7 feet high, and mean thickness 2 feet 6'? Ans. 892 feet.

DEFINITION.-A cylinder is a long round body, all its length of equal bigness.

RULE Find the area of one end, and multiply it by the length, the last product is the solidity. To find the area of one end apply the rule for measuring a circle. (See Circles).

Examples.

1. What is the solidity of a cylinder that has a diameter 22 inches, and is 20 feet long?

diam. circum.

22×69.14=1521.08380 27 area. 380·27×20 ft.7605·414452-81 ft. Ans.

2. What is the solidity of a cylinder that has a diameter 9 feet, and is 21 feet long?

Ans. 1488 69. 3. What is the solidity of a cylinder, whose diameter is 7 inches, and circumference 22, and is 20. Ans. 54 ft.

feet long?

CASE V.

To find the solidity of a cone or pyramid. DEFINITION. A cone is a figure standing upon a base, and of a true slant or taper from the larg est end, to a point or vertex.

The base is either

circular, square, triangular, or in form of a parallelogram; the base of the first figure is a circle; the second is a triangle; the third is a square; the fourth is a parallelogram.

A frustum of a cone is a piece cut off parallel to the base; that part between A B and C D in the following figures is called a frustum; the length of the frustum is represented by the perpendicular line, proceeding from the centre of the base, to the line A B.

RULE. Find the superficial area of the base, and multiply it into one third of the perpendicular height of the cone, and the product will be the solidity.

Examples.

1. What is the solidity of figure first; perpendicular 21 feet, and the diameter of the circle at the base 9 feet?

9.5 diameter, 29.85 circumference..

Half diam. and half circum. 475x14-925-70-89† area of base. Area 70 89x21-1488-69-496.23+ feet, solidity.

2. What is the solidity of figure 2d; perpendicular 21 feet, sides of its triangle 10, and perpendicular of its triangle 8?

10×8-85-42·5X21÷297.5 solidity Ans,

3. What is the solidity of figure third; perpendicular height 21, and the sides of its base 9 feet? -9×981X21÷=567 ft. Ans.

4. What is the solidity of figure fourth; perpendicular height equal 21, and sides 15 by 91? Ans. 997.5 ft.

CASE VI.

To find the solidity of the frustum of a cone, &c.

RULE I. Find the superficial area of both ends, add the two areas together, and reserve the sum ; multiply the two areas together, extract the square root of the product, and add the root to the reserved sum, and multiply the sum by one third the pendicular height, the product is the solidity required.

per

RULE II. If the cone is exactly square at both ends; multiply a side of the greatest, by a side of the least square; also find the difference between the two. sides; square the difference; add one third of the square to the product of the two sides; multiply the last sum by the length, the product is the solidity.

Examples.

1. What is the solidity of the frustum A B C D. figure first, case 5th; largest diameter 9; smallest 3; and length 14?

Largest diam. 9.5X9.5X7854-70-88+ area largest end, Smallest diam. 3·0X30X 7854 706† area least end.

2

Areas 70-88×7·06=500·4128-22·36 root added

len. Ans. 100.30×14= 468·06†

2. What is the solidity of the frustum A B C D figure second, case fifth; the sides of its largest triangle

10, and its perpendic. 8; sides of the smallest triangle 3.5 and the perpendicular of its triangle 3; and length 14?

2

10×4 42.5 area largest end.

3.5×15525 area smallest end.

Areas 425×5 25=223 1250 1493 root added.

sum 62.68

sum 62.68×43-292.5 solidity, Ans.

3. What is the solidity of the frustum A B C D figure third, case 5th; sides of its largest square 9, and the smallest 3, and length 14?

Same question by rule second.

Sides of the two squares 9×327 product. 9-36 dif. x6=36+1 = 12 add.

Solidity. 39×14=546Ans.

4. What is the solidity of the frustum A B C D 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 15×9.5=142·5 area. Sides of the smallest end 5X3.2 160 area. Areas 142.5×162280-047.71 root 47.7 add

2

sum 206.2

206.2×14=2886·6÷÷1962.21 Ans.

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 cone that is 16 in. square at one end, other, and is 16 ft. long?

frustum of a and 12 at the Ans. 214

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

Ans. 35

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.

Examples.

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

12X12=144×6÷432 solidity, Ans.

2. What is the solidity of a wedge that is 10 ft.. Ans. 14 ft. long, 6 inches wide, and 6 thick?

CASE IX.

To find the solidity of a wedge when the edge is narrower than the large end.

RULE. To the width of the edge add twice the width of the large end, and reserve the sum; multiply 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 quotient 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