2. Cylinder, alt. 25 ft.; radius of base 10 ft.

Ans. 1570.8 sq. ft.

3. Cone, slant height 20 ft.; radius of base 5 ft.

Ans. 314.16 sq. ft.

4. Frustum of a cone whose slant height is 36 ft.; radius of upper base 10 ft., of lower base 15 ft. Ans. 2827.44 sq. ft. 5. Find entire surface of frustum of a cone, slant height 20 ft.; radius of upper base 5 ft., and lower base 24 ft.

Ans. 569.415 sq. ft.

6. Find the surface of a sphere whose diameter is 25. Ans. 25 x 3.1416 × 25 = 1963.5.

7. Find the surface of a sphere whose radius is 7 rd. Ans. 615.7536 sq. rd.

VOLUMES.

To find the volume of a prism or cylinder, 563. Multiply the area of the base by the altitude. To find the volume of a pyramid or cone,

564. Multiply the area of the base by the altitude. To find the volume of the frustum of a pyramid or cone, 565. To the sum of the areas of the two bases, add the square root of their product, and multiply this result by 1⁄2 of the altitude.

To find the volume of a sphere,

566. Multiply the cube of the diameter by .5236. Or, Multiply the cube of the radius by 4.1888.

PROBLEMS.

1. Find the capacity of a cylindrical measure 18 inches in diameter, and 8 in. deep.

Ans. 1 bu.

2. Of a cylindrical bucket 5 inches in diameter and 7 in. deep (wine measure).

Ans. 2.38 qt.

3. Find the volume of a triangular bar 15 ft. long, and whose sides are 1 in., 1 in., and 1.414 in. Ans. 90 cu. in.

4. How many conical glasses, each 3 inches in diameter and 4.902 in. deep, can be filled from a qt. bottle? Ans. 5. 5. What is the volume of a quadrangular pyramid, each side of the base being 100 ft. and the altitude 75 ft.?

Ans. 250000 cu. ft. 6. What is the volume of the frustum of a pyramid 24 ft. high, 6 ft. square at one end and 4 ft. square at the other? Ans. 608 cu. ft.

7. How many bbl. will a cistern 12 ft. deep, with upper diameter 9 ft. and lower diameter 3 ft., hold? Ans. 87.3 bbl. 8. How many cubic feet of iron in a cannon ball 36 in. in diameter ? Ans. 14.1372. 9. How many cubic ft. of granite in a cylindrical monument 36 ft. high, with a base whose radius is 2 ft.?

10. How many square feet of sheet-ironin. thick can be made of a cylindrical shaft 20 ft. long and 4 inches in diameter ? Ans. 83.776 sq. ft.

11. What are the relative values of a ball of gold 4 inches in diameter and a cylinder of same diameter and altitude? Ans. 2: 3.

12. What is the capacity of an oval lime-kiln, whose top, bottom, and greatest diameters are 6, 2, and 7 ft.; the distances of the top and bottom diameters from the greatest diameter being 5 and 15 ft.? Ans. 370.26 bu.

Multiply the sum of the square of the top diameter and twice the square of the greatest diameter by the distance between these diameters. Do the same with the bottom and greatest diameter and their distance. Then multiply the sum of these two results by .2618.

[(36+2 × 49) × 5+ (4+2 × 49) × 15] .2618 575.96 cu. ft.

THE METRIC SYSTEM.

567. The Metric System is a decimal system of Weights and Measures, founded on a certain unit of length called the metre 39.37079 inches.

The system is an exceedingly valuable one, but its limited use in this country does not justify our treating the subject as fully as its importance deserves.

The Metric System adopts a standard unit of measure, and then forms lower denominations by the decimal sub-measures of this unit; and higher denominations by the decimal multiples. Thus, taking the metre as the unit, and using the Latin prefixes, deci (tenth), centi (hundredth), milli (thousandth), we have

Deci-metre (dess'-e-meet-ur) =.1 metre (meet'ur).

Centi-metre (cent'-e-meet-ur) = .01 metre.

Milli-metre (mil'-e-meet-ur) = .001 metre.

And by using the Greek prefixes deca (ten), hecta (hundred), kilo (thousand), myria (ten-thousand), we have Deca-metre (dek'-a-meet-ur) = 10 metres.

Hecta-metre (hec'-ta-meet-ur) = 100 metres.

Kilo-metre (kil'-o-meet-ur)

1000 metres.

Myria-metre (mir'-ee-a-meet-ur) = 10000 metres.

In like manner taking the litre as the unit of measure, we have milli-litre, centi-litre, deci-litre, deca-litre, &c.

The Units of Measure are

568. The Metre, the Linear Unit, nearly 1 ten-millionth of the distance from the equator to either pole.

569. The Are (air), which is the Unit of Surface Measure, and is a square whose side is equal to 10 metres, or 1 decametre.

570. The Stere (stair), which is the Unit of Solid Measure, and is a cube whose edge is 1 metre.

571. The Litre (leet-ur), which is the Unit of Measures of Capacity, and is a cube whose edge is .1 metre, or 1 decimetre.

572. Gramme (gram), which is the Unit of Weight. This is a cube of pure water whose edge is .01 metre, or 1 centimetre, weighed in a vacuum, at a temperature, when its density is greatest, 39.2° Fahrenheit's Thermometer.

NUMERATION AND NOTATION.

573. Each standard unit is represented by the initial letter of its name. Thus, M, for metre; A, for are; S, for stere, &c. A multiple is represented by the first letter of its prefix followed by the first letter of the standard unit; and a sub-multiple by the first letter of its prefix followed by the initial letter of the standard unit written in small letters. Thus, MM. stands for myriametres, mm. for millimetres; DM. stands for decametres, dm. for decimetres, &c., &c., though in general, since all denominations are decimal, the name of the standard unit is the only name that is necessary to express a number. Thus, 21.632 M., or M. 21.632, or 21. 632 is read twenty-one and six hundred thirty-two thousandths metres; 32.3 L., or L. 32.3 is read thirty-two and three-tenths litres. TABLES.

Long and Linear Measure.

10 KM.

1 Myriametre.

between M. 31476.341, and $31476.341.

Instead of placing the separatrix between the metres and decimetres, which we do in measuring cloth and ordinary distances, we may place it between the kilometres and hectometres, which we do when measuring long distances, as the length of rivers, &c. The number should then be read 31 and 476341 hundred-thousandths kilometres.

By omitting the separatrix, the number may be read 31476341 millimetres.

Surface or Square Measure.

This measure is used in measuring land and other surfaces. The Unit of Measure is an are = 100 sq. M. = 119.60 sq. yd., or a hectare = 2.471 A.

The denomination centare is sometimes written centiare.

It is customary also to express the area of ordinary surfaces by the square of the linear metre, called the square meter 1550 sq. in.

TABLE.

100 sq. cm. = 1 sq. dm.

100 sq. dm. 1 sq. m.

347.2017 is read three hundred forty-seven and two thousand seventeen ten-thousandths square metres.

Solid or Cubic Measures.

The denominations decastere and decistere are sometimes used when applied to the measure of the volume of firewood and building-timber, but ordinarily the stere = 35.316 cu. ft.

TABLE.

10 decistere = 1 stere.

10 stere = 1 decastere.

37489.004 is read 37489 and 4 thousandths steres.

Volumes, such as excavations, embankments, &c., are expressed by the cube of some denomination of linear measure; thus,