(17) A cellar, which measures 12 ft. long and 9 ft. wide, is flooded to a depth of 4 in.; find the weight of the water, supposing that 1 cubic foot of water weighs 1000 oz. avoirdupois.

(18**) What weight of water will a rectangular cistern contain, the length being 4 ft., the breadth 2 ft. 6 in., and the depth 3 ft. 3 in., when a cubic foot of water weighs 1000 oz. avoirdupois ?

(19*) How much lead will be required to line a cistern, open at the top, which is 4 ft. 6 in. long, 2 ft. 8 in. wide, and contains 42 cubic feet?

(20*) How many bricks will be required to build a wall 90 ft. long, 18 in. thick, and 8 ft. high; a brick being 9 in. long, 4 in. wide, and 3 in. deep?

(21) A quantity of earth, in the shape of a rectangular parallelopiped, which measures 6 ft. long, 5 ft. wide, and 4 ft. deep, is spread uniformly over a rectangular court which measures 48 ft. long, and 10 ft. wide; find the depth of the earth.

(22) A box is 4 ft. 6 in. long, 3 ft. 6 in. broad, and 2 ft. 6 in. deep; how many books will it contain, when each book measures 7 in. long, 6 in. wide, and 11⁄2 in. thick?

(23) A book has the following dimensions: 8 in. long, 6 in. wide, and 12 in. thick; find the depth of the box, whose length and breadth are 3 ft. 4 in. and 2 ft. 6 in., that it may contain 400 such books.

(24) The water in a large rectangular cistern, which is 15 ft. 6 in. long and 12 ft. wide, has sunk 3 in.; find how many cubic feet of water have been drawn off.

(25) How many bricks will be required for a wall 25 yds. long, 15 ft. high, and 1 ft. 10 in. thick; each brick being 9 in. long, 4 in. wide, and 3 in. deep?

(26) The internal dimensions of an open cistern, made of iron 2 in. thick, in the shape of a rectangular parallelopiped, are length 4 ft. 6 in., breadth 3 ft., and depth 2 ft. 6in.;

find how many cubic feet of iron were used in its construction.

(27) Find the expense of covering a flat roof, 17 ft. 4 in. 、long and 13 ft. 4 in. wide, with sheet-lead of an inch thick, supposing that a cubic inch of lead weighs 6 oz. avoirdupois, and that 1 lb. costs 3d.

(28) If a cubic foot of marble weighs 2.716 times as much as a cubic foot of water, find the weight of a block of marble 9 ft. 6 in. long, 2 ft. 3 in. broad, and 2 ft. thick, supposing a cubic foot of water weighs 1000 oz. avoirdupois.

(29*) The beams of wood used in building a house are 3in. thick and 10 in. wide: 200 of them are used, which together amount to 1000 cub. ft. What is the length of each beam ?

(30) The length and breadth of a rectangular cistern are 10 ft. and 5 ft.; what must be its depth that it may contain 1000 gallons of water, supposing that a gallon contains 277.274 cub. in. ?

(31) An open rectangular cistern is made of cast iron 1 in. thick, and has for its external length, breadth, and depth 5 ft., 4 ft., and 3 ft. respectively; find its weight when empty, when iron weighs 7.2 times as much as water, and 1 cubic foot of water weighs 1000 oz. avoirdupois.

(32**) How many gallons of water will a cistern hold whose length, breadth, and depth are 5 ft. 6 in., 3 ft. 9 in., and 1 ft. 3 in. ? What would be the weight of water contained (a gallon contains 277 cub. in., and a cub. ft. of water weighs 1000 oz. avoirdupois)?

(33) An open cistern, which is 6 ft. long and 4 ft. wide, contains 108 cub. ft. of water; find how many cubic feet of lead will be required for lining its sides and bottom, when the lead is in. thick.

(34) If a cubic foot of gold may be made to cover uniformly and perfectly 432000000 sq. in., find the thickness of the coating of gold.

(35) Rain has fallen to the depth of half an inch; find how many cubic feet of water have fallen on an acre of land.

(36) The length, breadth, and depth of a rectangular cistern are 8 ft., 6 ft., and 4 ft.; find the length, breadth, and depth (proportional to the former) of another rectangular cistern that shall contain eight times the quantity of water.

(37) The length of a rectangular cistern is 10 ft., its breadth is 8 ft., and its depth is 6 ft.; find the length, breadth, and depth (proportional to the former) of another cistern that shall contain only half the quantity of water.

(38) The water in a large rectangular cistern, whose length is 12 ft. and breadth 10 ft., has sunk 8 in. on account of a leakage; find how many gallons will have been wasted, when there are 6 gallons in a cubic foot.

XX. THE RIGHT-PRISM AND THE RIGHT

CYLINDER.

Definitions.-I. A prism is a solid whose two ends ABC, DEF are similar, equal, and parallel plane rectilineal figures, and whose side faces ABED, BCFE, &c. are parallelograms.

The ends of a prism A may be triangles, as fig. 1; trapezoids, or pentagons, as in fig. 2; hence a prism is called a triangular prism, pentagonal c prism, &c., according to the figure that it has for its ends.

The line drawn from the centre of one end to the centre of the

B

D

Fig. 1.

F

A

B

A

E

E

other end is called the axis of the prism,

C

Fig. 2.

F

A right-prism has the axis at right angles to the ends of the prism; or it may be defined to be a prism that has all its side faces rectangles.

In an oblique prism the axis is not at right angles to the ends, and its side faces are not rectangles.

The whole surface of a prism will consist of the areas of the several side faces ABED, BCFE, &c., and the areas of the two ends ABC and DEF.

II. A cylinder is a solid having its ends two equal and parallel circles.

The straight line CE between the two ends, from centre to centre, is called the axis of the cylinder.

In the right-cylinder (fig. 3) this axis CE is perpendicular to

the ends.

In the oblique cylinder (fig. 4) the axis CE is not perpendicular to the ends.

A

C

B

F

Ꭰ

E

B

The whole surface of a cylinder will consist of the areas of the two circular ends (ABC and DEF) and the curved or round surface.

The height, or altitude, or length of a cylinder or prism is the perpendicular drawn from one end to the other end; and in the case of a right-prism or right-cylinder, the height or length of the figure is the same as the axis.

N.B. The cylinders and the prisms mentioned in the following Examples are right-cylinders and right-prisms.

RULES.-(1) To find the volume of a right-cylinder or a right-prism.

Multiply the area of the base by the height or length,

(2) To find the curved surface of a right-cylinder, or the side faces of a right-prism.

Multiply the circumference or perimeter of the end by the height or length.

(3) To find the whole surface of a right-cylinder or right-prism.

To the curved surface of the cylinder, or to the area of the side faces of the prism (found by Rule 2), add the areas of the two ends.

Note 1.-To find the length or height of a right-cylinder or right-prism.-Divide the volume by the area of the base or end.

Note 2.-To find the area of the base or end of a rightcylinder or right-prism.-Divide the volume by the height. or length.

Note 3.-The end of a prism being any plane figure, its area, of course, will have to be found by the rules given for finding the area of that particular figure in the Mensuration of Superficies.

Note 4.-The volume of an oblique cylinder or oblique prism will be the same as that of a right-cylinder or rightprism having the

CDEH, if they have the same area DE for the base, and lie between the same parallel planes.