This rule must be taken on authority. For proof, see Geometry.

DEFINITIONS. A right cylinder is a round body of uniform diameter; its two ends parallel and at right angles with its length. Conceive the surface of such a cylinder cut lengthwise and rolled out, it will form a parallelogram whose width is the circumference of the cylinder; therefore, to find the surface of a cylinder; we have the following

RULE 8. Multiply the length by the circumference.

A right cone may be covered by a multitude of equal isosceles triangles, whose base is the base of the cone, ́and whose altitude is the slant height of the cone; therefore,

To find the surface of a right cone.

RULE 9. Multiply the circumference of the base by half the slant height.

To find the area of an ellipse.

RULE. Multiply the two diameters together, and that product by the decimal ,7854.

EXAMPLES UNDER THE FOREGOING RULES.

1. There is a square of 80 rods on a side, and a parallelogram also of 80 rods each side; but from the base to the opposite side is only 70 rods of perpendicular distance; what is the difference in area of these two figures? Ans. 800 rods.

2. How many yards in a triangle whose base is 148 feet, and perpendicular 45 feet? Ans. 370 yards. 3. The three sides of a triangle are severally 60, 50, and 40 rods; how many acres does it contain?

Ans. 61.

4. On a base of 120 rods, a surveyor wished to lay off a rectangular lot of land containing 40 acres; what distance in rods must he run from his base line?

Ans. 53 rods.

5. How many boards will it require to cover the body of a barn 60 feet long, 40 feet wide and 20 feet high, to

the eaves, the boards being on an average 15 inches wide and 10 feet long? Ans. 320 boards. 6. How many feet in a board 22 inches wide at one end, 8 inches wide at the other, and 14 feet long?

7. A board is 9 inches wide; how will it require to make 6 square feet?

Ans. 17 feet.

much in length Ans. 8 feet. 8. The bottom of a circular cistern is 5 feet in diameter; how many square feet of surface?

Ans. 23,758+-feet. 9. What quantity of surface in a cylinder 30 feet long and 3 feet in diameter, the two ends included?

Ans. 296,88+feet. 10. A man bought a farm 198 rods long, and 150 rods wide, and agreed to give $32 per acre; what did the farm come to ? Ans. $5940.

N. B. Make no attempt to compute the number of acres definitely.

11. If the forward wheels of a coach are 4 feet, and the hind ones 5 feet in diameter, how many more times will the former revolve than the latter, in going a mile? Ans. 84.

N. B. In this problem use 7 to 22. The solutions of the foregoing problems are all very brief by canceling, except the 3d and 9th.

(ART. 121.) The foregoing rules apply to Plasterers', Pavers' and Carpenters' work. These artificers generally compute their work at so much a square yard, or so much for 100 square feet, which last is sometimes called a carpenter's square.

To compute the number of square feet, yards, &c., the length, breadth and heighth of a room is taken in feet and inches, never measuring closer than inches. In books, we often meet with feet, inches, thirds, fourths, &c., descending on a scale of 12, called duodecimals, the foot being the unit; hence inches are fractions, and two fractions multiplied together, that is, taking part of a part, makes a less part. Therefore, inches multiplied by inches, must give less than inches, that is, give thirds, &c.

T

As a study of numbers, a few examples in duodecimals, might be of some advantage; but with them we must retain cumberous and clumsy modes of operations, which are of no practical value, and thus cherish bad habits that ought to be forgotten; we shall, therefore, dispense with duodecimals. All practical men take no account of measures lower than inches.

EXAMPLES.

[NOTE. In the solution of problems, we always call inches part of a foot; 6 inches is, 7 inches 7, 9 inches , &c.]

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1. What will it cost to pave a foot-path 40 feet long, and 71⁄2 feet wide, at 22 cents per square yard?

Ans. 7 dollars.

2. What will it cost to paint a room which measures 82 feet 6 inches in compass and 12 feet 9 inches high, at 16 cents a square yard? Ans. $18,70.

3. A floor is 36 feet 4 inches, by 14 feet 9 inches; what will it cost to lay it at 3 dollars 25 cents per square? Ans. $17,42 nearly.

4. What will it cost to roof a building 42 feet long, and the rafters on each side being 17 feet 6 inches long, at 3 dollars and 25 cents per square of 100 feet? Ans. $47,77!.

5. What will the plastering of a ceiling come to, at 15 cents per yard, it being 21 feet 7 inches long, and 12 feet 6 inches broad? Ans. $4,47 +

6. What will the whitewashing of a room cost at 2 cents per yard, allowing it to be 30 feet 6 inches long, 24 feet 9 inches wide, and 10 feet 6 inches high, no deductions being made for vacuities? Ans. $4,23+

7. A room is 20 feet 6 inches long, and 13 feet 6 inches in width, and 10 feet high; what will it cost to paper the walls, at 27 cents per square yard, deducting a fireplace of 4 feet 4 inches, and 3 windows, each 6 feet by 3 feet 2 inches? Ans. $18,17.

8. A room is 27 feet 6 inches by 22 feet 6 inches, and 10 feet 3 inches from the wash-board up. In said room is a fire-place of 4 feet by 4 feet 6 inches; 2 doors, each 8 feet by 4 feet 4 inches, and 2 windows, each 6 feet by 3 feet 4 inches. What will it cost to plaster said room, at 18 cents per yard, sides and ceiling, deducting for vacuities? Ans. $30,32.

9. What is the cost of smoothing and polishing a marble slab, whose length is 5 feet 7 inches, breadth 2 feet 3 inches, at 35 cents per square foot? Ans. $4,524.

MENSURATION OF SOLIDS AND CAPACITIES.

(ART. 122.) Ir will be recollected that surfaces are measured by squares (Art. 118): so solids are measured by unit-cubes, whatever size cube we take for the unit. A cube is a solid figure of six equal square surfaces, and all its angles right angles. To measure a right angular solid, such as a square stick of timber, for instance, the surface of the end is measured by little squares; and if we saw off a length equal to a side of one of these little squares, it is plain that we shall have as many cubes as the surface of the end contains squares; twice this length will give twice as many cubes; three times the length, three times as many cubes, &c. &c. But to find the number of squares in the end, we multiply the width and depth of the end together; then that product by the length, and we have the number of unit-cubes in the whole solid. All solid figures, of whatever shape, are referred to regular right angular solids, directly or indirectly.

To reduce one solid figure to its equivalent, in another shape, is the office and object of solid geometry, not arithmetic; hence our rules are drawn from geometry; and, as arithmeticians, we must take them on authority. We must advance to geometry to perfectly comprehend them.

Definition 1st. To find the cubical contents of a right angular solid.

RULE 1. Multiply length, breadth and depth together.

[NOTE. No school should be without a set of geometrical solids. A bare inspection of them will give a better idea of the regular solids, than the most exact definition, or the most minute description.]

Def. 2. A prism is a solid, whose ends are any similar equal and parallel surfaces, and its sides rectangles; hence, to find the solidity of a prism,

RULE 2. Multiply the area of one end by the length.

Def. 3. A cylinder is a circular solid, equal at both ends. A right cylinder has circular ends, at right angles to the axis. To find the contents of such a cylinder.

RULE 3. Multiply the area of one of the circular ends by the length.

To estimate the contents of a wedge formed solid. RULE 4. Multiply the area of its end by half of its perpendicular length.

Def. 4. A pyramid is a solid which decreases uniformly from its base, until it comes to a point. The base of a pyramid may be either a square, a triangle, or a circle. Hence it is named a square pyramid, a triangular pyramid, or a cone. The point in which the pyramid ends, is called its vertex; a straight line from the vertex at right angles to the base, is called the altitude. To find the solidity of a pyramid or cone.

RULE 5. Multiply the area of the base by one-third of the altitude.

Def. 5. When the smaller end of a pyramid or cone is cut off parallel to its base, the residue is called the frustrum of a pyramid or cone. When the base is a square, the following rule will give the solidity of a frustrum:

RULE 6. Square both upper and lower base; and, to the sum of these squares, add the product of the sides