2. At 20 cts. a sq. yd., what will it cost to plaster a ceil ing 22 ft. 7 in. long, 13 ft. 11 in. wide? Ans. $6.984+

3. A room is 20 ft. 6 in. long, 16 ft. 3 in. broad, 10 ft. 4 in. high: how many yd. of plastering in it, deducting a fireplace 6 ft. 3 in. by 4 ft. 2 in.; a door 7 ft. by 4 ft. 2 in., and two windows, each 6 ft. by 3 ft. 3 in.? Ans. 110 sq. yd. 8,5 sq. ft.

:

4. A room is 20 ft. long, 14 ft. 6 in. broad, and 10 ft. 4 in. high what will the coloring of the walls cost, at 27 cts. per sq. yd., deducting a fireplace 4 ft. by 4 ft. 4 in., and two windows, each 6 ft. by 3 ft. 2 in. ? Ans. $19.73

5. At 18 cts. per sq. yd., find the cost of paving a walk 35 ft. 4 in. long, 8 ft. 3 in. broad. Ans. $5.83

6. What will it cost to pave a rectangular yard, 21 yd. long, and 15 yd. broad, in which a footpath, 5 ft. 3 in. wide, runs the whole length of the yard; the path paved with flags, at 36 cts. per sq. yd., and the rest with bricks, at 24 cts. per sq. yd.? Ans. $80.01

7. At 10 cts. a sq. yd., what the cost to paint the walls of a room 75 ft. 6 in. in compass, 12 ft. 6 in. high? Ans. $10.486+ 8. A house has 3 tiers of windows, 7 in a tier: the height of the first tier is 6 ft. 11 in.; of the 2d, 5 ft. 4 in.; the 3d, 4 ft. 3 in.; each window is 3 ft. 6 in. wide: what cost the glazing, at 16 cts. per sq. ft.? Ans. $64.68 9. A floor is 36 ft. 3 in. long, 16 ft. 6 in. wide: what will it cost to lay it, at $3 a square ? Ans. $17.943+

10., A room is 35 ft. long, and 30 ft. wide: what will the flooring cost, at $5 per square, deducting a fireplace 6 ft. by 4 ft. 6 in., and a stairway, 8 ft. by 10 ft. Gin.? Ans. $46.95

11. At $3.50 per square, what cost a roof 40 ft. long, the rafters on each side 18 ft. 6 in. long? Ans. $51.80

ART. 313. TO FIND THE AREA OF A TRIANGLE. Rule.-Multiply the base by the perpendicular hight, and take half the product for the area.

Or, when the sides are given, the following RULE:

1st. Add the three sides together, and take half the sum. 2d. From the half sum take the 3 sides severally.

3d. Multiply the half sum and the 3 remainders together, and extract the square root of the product, which gives the area.

1. Find the area of the triangle, E F G H, the base, FH, is 15 feet; the perpendicular height, G E, 12 feet. Ans. 90 sq. ft.

2. The contents of a triangular space, the base 44 rd., perpendicular height 18 rd.

Ans. 2 A. 1 R. 36 P.

3. How many acres in a triangular field; perpendicular height, 67 rd. ?

the base 80 rd.; Ans. 16A. 3 R.

NOTE. The area of any field or piece of land may be found by dividing it into triangles, and measuring the base and perpendicular height of each triangle thus formed.

4. What cost the glazing of a triangular skylight, at 12 cts. per sq. ft., the base, 12 ft. 6 in., the perpendicular height, 16 ft. 9 in. ? Ans. $12.56

5. Find the area of a triangle, the sides being 13, 14, and 15 ft. Ans. 84 sq. ft. 6. The area of a triangle, the sides 2, 3, and 4 feet respectively. Ans. 2.9047375+sq. ft.

ART. 314. TO FIND THE AREA OF A TRAPEZOID.

Rule.-Multiply the sum of the parallel sides by the perpendicular breadth, take half the product.

1. The parallel sides of a trapezoid, FCG D, are 35 and 26 inches; its breadth 11 in.; required the area

Ans. 335 sq. in.

C

2. A field is the form of a trapezoid; one of the parallel sides is 25 rd., the other 19 rd.; the width 32 rd. how many acres in it? Ans. 4 A. 1 R. 24 P.

ART. 315. TO FIND THE CIRCUMFERENCE OF A CIRCLE,

WHEN THE DIAMETER IS GIVEN.

Rule.-Multiply the diameter by 3.1416, the product will

the circumference.

1. The diameter A B of the circle A DBE

is 48 feet: what is the circumference?

2. The diameter of a wheel is 4 feet: find

the circumference.

Ans. 12 ft. 6.7968 in.

3d Bk.

20

3. What is the circumference of the earth, the mean di ameter being 7912.4 mi. ? Ans. 24857.59584 mi.

ART. 316. TO FIND THE DIAMETER OF A CIRCLE, WHEN

THE CIRCUMFERENCE IS GIVEN.

Rule. Divide the circumference by 3.1416, the quotient will be the diameter.

1. The circumference of a circle is 15 feet: what is the diameter ? Ans. 4 ft. 9.295+in.

2. If the girt of a tree is 12 feet 5 inches, what its thickness or diameter? Ans. 3 ft. 11.428+in.

ART. 317. TO FIND THE AREA OF A CIRCLE.

Rule.-Multiply the diameter by the circumference, and take one-fourth of the product. Or, Multiply the square of the diameter by .7854; or, for greater accuracy, by .785398 Or, Multiply the square of the radius by 3.1416

1. Find the area of a circle, the diameter being 42 feet. Ans. 1385.4456 sq. ft. 2. Find the area of a space on which a horse may graze, when confined by a cord 7 rods long, one of its ends being fixed at a certain point. Ans. 1A. 16.715P.

ART. 318. TO FIND THE DIAMETER OF A CIRCLE, WHEN
THE AREA IS GIVEN.

Rule.-Divide the area by .7854; the square root of the quotient will be the diameter.

1. The area of a circle is 962.115: what its diameter and circumference? Ans. diam. 35: circum. 109.956

2. What length of halter will fasten a horse to a post in the center of an acre of grass, so that he can graze upon the 1 A. and Lo more? Ans. 7.1364+rd., or 117 ft. 9+in.

ART. 319. MEASUREMENT OF BODIES OR SOLIDS.

DEFINITIONS.-1. A BODY or SOLID, has length, breadth, and thickness or depth.

2. A PRISM is a solid whose ends, or bases, are parallel; its sides, parallelograms. Such a body is termed a RIGHT

P

PRISM when each of its bases is perpendicular to its other sides; and it is TRIANGULAR, QUADRANGULAR, &c., according as its base is a triangle, quadrangle, &c. Thus, P is a triangular prism.

3. A PAR-AL-LEL-O-PI'-PED is a prism whose bases and also its other sides are parallelograms. Thus, B is a parallelopiped.

4. A parallelopiped is RIGHT when each of its faces is a rectangle. A common chest, a bar of iron, brick, &c., are instances of right parallelopipeds. When each face of a right parallelopiped, as A, is a square, it is termed a cube. A cube has 6 equal square faces.

5. A CYLINDER is a round prism, having circles for its ends. Thus, C is a cylinder, of which the line E F passing through the centers of both ends, is called the axis.

6. A PYRAMID is a solid having any plane figure for a base, and its sides triangles, whose vertices meet in a point at the top, called the VERTEX of the pyramid. A pyramid is TRIANGULAR, QUADRANGULAR, &c., according as its base is a triangle, quadrangle, &c. Thus, A is a triangular pyramid.

7. A body which has a circular base, and tapers uniformly to a point named the VERTEX, is called a CONE. The axis of a cone is a line passing through the vertex and the center of the base. Thus, C is a cone of which B V is the axis.

C

B

8. A FRUSTUM of any body, as a pyramid or cone, is what remains when the top is cut off by a plane parallel to the base,

9. A GLOBE, or SPHERE, is a body of such a figure, that all points of the surface are equally distant from a point within, called

the center.

The diameter of a sphere, is a line passing through the center, and terminated both ways by the surface. The radius of a sphere is a line drawn from the center to the surface. Thus, A B is a diameter: CA a radius; C being the center of the sphere.

10. The HEIGHT or altitude of a solid, is a line drawn from its vertex or top, perpendicular to its base.

11. The CONTENTS or SOLIDITY of a body, is the space within it. The magnitude of this space is expressed by the number of times it contains a given space called the measuring unit.

12. The MEASURING UNIT for solids, is a cube whose base is the measuring unit for surfaces; as a cu. in., cu. ft., &c.

ART. 320. To find the SOLID CONTENTS of a PARALLELOPIPED. Rule.-Multiply the length, breadth, and depth together: the product will be the solid contents.

1. Find the solid contents of a parallelopiped: the length, 12 ft.; breadth, 3 ft. 3 in.; depth, 4 ft. 4 in. Ans. 169 cu. ft. 2. The solid contents of a rectangular stone: the length, 6 ft.; breadth, 2 ft. 6 in.; depth, 1 ft. 9 in. Ans. 261 cu. ft. 3. A block of marble, in the form of a parallelopiped, is in length, 3 ft. 2 in., breadth, 2 ft. 8 in.; depth, 2 ft. 6 in. what its cost, at 81 cts. per cu. ft.? Ans. $17.10

4. How many solid feet in a box 4 ft. 10 in. long, 2 ft. 11 in. broad, and 2 ft. 2 in. deep? Ans. 30 cu. ft. 6' 6" 4""

ART. 321. The principles of the preceding rule are applied to the measurement of

MASONS' AND BRICKLAYERS' WORK.

Masons' work is measured by the solid foot, or by the perch, which is 16 ft. long, 18 in. broad, 1 ft. deep;

And multiplying these numbers together, shows that a perch contains 24, or 24.75 cu. ft.