MENSURATION OF LINES AND SUPERFICIES. 460. In taking the measure of any line, surface, or solid, we are always governed by some denomination, a unit of which is called the Unit of Measure. Thus, if any lineal measure be estimated in feet, the unit of measure is 1 foot; if in inches, the unit is 1 inch. If any superficial measure be estimated in feet, the unit of measure is 1 square foot; if in yards, the unit is 1 square yard.

461. If any solid or cubic measure be estimated in feet, the unit of measure is 1 cubic foot; if in yards, the unit is 1 cubic yard.

462. The area of a figure is its superficial contents, or the surface included within any given lines,

without regard to thickness.

463. An Oblique Angle is an angle greater or less than a right angle; thus, A B C and C B D are oblique angles.

B

CASE I.

464. To find the area of a square or a rectangle.

465. A Square is a figure having four equal sides and four right angles.

466. A Rectangle is a figure having four right angles, and its opposite sides equal.

RULE. Multiply the length by the breadth, and the product will be the square contents.

EXAMPLES FOR PRACTICE.

1. How many square inches in a board 3 feet long and 20 inches wide? Ans. 720. 2. A man bought a farm 198 rods long and 150 rods wide, and agreed to give $32 an acre; how much did the farm cost him?

Ans. $5940. 3. A certain rectangular piece of land measures 1000 links by 100; how many acres does it contain? Ans. 1 A.

CASE II.

467. To find the area of a rhombus or a rhomboid. 468. A Rhombus is a figure having four equal sides and four oblique angles.

469. A Rhomboid is a figure having its opposite sides equal and parallel, and its angles oblique.

The square, rectangle, rhombus, and rhomboid, having their opposite sides par. allel, are called by the general name, parallelogram.

It is proved in geometry that any parallelogram is equal to a rect angle of the same length and width.

RULE. Multiply the length by the shortest or perpendicular distance between two opposite sides.

EXAMPLES FOR PRACTICE.

1. A meadow in the form of a rhomboid is 20 chains long, and the shortest distance between its longest sides is 12 chains; how many days of 10 hours each will it take a man to mow the grass on this meadow, at the rate of 1 square rod a minute Ans. 6 da. 4 h.

2. The side of a plat in the form of a rhombus is 15 feet, and a per pendicular drawn from one oblique angle to the side opposite, will meet this side 9 feet from the adjacent angle; what is the area of the plat? Ans. 180 sq. ft.

CASE III.

470. To find the area of a trapezoid.

471. A Trapezoid is a figure having

four sides, of which two are parallel.

The mean length of a trapezoid is one

half the sum of the parallel sides.

RULE. Multiply one half the sum of the parallel sides by the perpendicular distance between them.

EXAMPLES FOR PRACTICE.

Ans. 12 sq. ft.

1. What are the square contents of a board 12 feet long, 16 inches wide at one end, and 9 at the other? 2. What is the area of a board 8 feet long, 16 inches wide at each end, and 8 in the middle? Ans. 8 sq. ft.

3. One side of a field is 40 chains long, the side parallel to it is 22 chains, and the perpendicular distance between these two sides is 25 chains; how many acres in the field? Ans. 77 A. 5 sq. ch.

CASE IV.

472. To find the area of a triangle.

473. The Base of a triangle is the side on which it is supposed to stand.

474. The Altitude of a triangle is the perpendicular distance from the angle opposite the base to the base, or to the base produced or extended.

475. A Triangle is one half of a parallelogram of the same base and altitude.

RULE. Multiply one-half the base by the altitude, or one-half the a'titude by the base. Or, Multiply the base by the altitude, and divide the product by 2.

EXAMPLES FOR PRACTICE.

1. How many square yards in a triangle whose and perpendicular 45 feet?

base is 148 feet, Ans. 370 sq. yds.

2. The gable ends of a barn are each 28 feet wide, and the perpen. dicular height of the ridge above the eaves is 7 feet; how many feet of boards will be required to board up both gables? Ans. 196 feet.

CASE V.

476. To find the circumference or the diameter of a circle.

477. A Circle is a figure bounded by one uniform curved line.

478. The Circumference of a circle is the curved line bounding it.

479. The Diameter of a circle is a straight line passing through the center, and terminating in the circumference.

It is proved in geometry that in every circle the ratio between the diameter and the circumference is 3.1416+.

RULE. I. To find the circumference.-Multiply the diameter by 8.1416. II. To find the diameter.—Multiply the circumference by .3183.

EXAMPLES FOR PRACTICE.

1. What length of tire will it take to band a carriage wheel 5 feet in diameter? Ans. 15 ft. 8.4+in. 2. What is the circumference of a circular lake 721 rods in diameter? Ans. 7 mi. 25 rds. 1.54+ ft.

3. What is the diameter of a circle 33 yards in circumference? Ans. 10.5+yards.

CASE VI.

480. To find the area of a circle.

From the principles of geometry is derived the following

RULE. I. When both diameter and circumference are given;Multiply the diameter by the circumference, and divide the product by 4. II. When the diameter is given;- Multiply the square of the diameter by .7854.

III. When the circumference is given;- Multiply the square of the circumference by .07958.

EXAMPLES FOR PRACTICE.

1. The diameter of a circle is 113, and the circumference 355; what is the area? Ans. 10028.75. 2. What is the diameter of a circular island containing 1 square mile of land? Ans. 1 mi. 41 rd. 1.4+ ft. 3. A man has a circular garden requiring 84 rods of fencing to inclose it; how much land in the garden? Ans. 3 A. 81.5+ P.

481. The Metric System of Weights and Measures has now received the sanction of law among more than half the inhabitants of the civilized world. Up to this date it has been adopted in France, Germany, Austria, the Netherlands, Southern Europe, and South America, and has been legalized in Great Britain, Germany, and the United States.

*The Metric System, as it was presented in all the editions of this book, printed previous to 1877, was useless, because the symbols and applications did not correspond with present usage.

This will be a sufficient reason for substituting in place of the former matter, a condensed and practical treatise of the system, together with some useful miscellaneous tables.

482. The Metric System of weights and measures is based upon the decimal scale.

483. The Meter is the base of the system, and is the one ten millionth part of the distance on the earth's surface from the equator to either pole, or 39.37079 inches.

From the Meter are made the Ar (are), the Liter (leeter), and the Gram; these constitute the primary or principal units of the system, from which all the others are derived.

484. The Multiple Units, or higher denominations, are named by prefixing to the name of the primary units the Greek numerals, Deka (10), Hecto (100), Kilo (1000), and Myra (10000).

485. The Sub-Multiple Units, or lower denominations, are named by prefixing to the names of the primary units the Latin numerals, Deci (†), Centi (¡¦ð), Milli (1000).

Hence, it is apparent from the name of a unit whether it is greater or less than the standard unit, and also how many times.

Meter means measure; and the three principal units are length, capacity or volume, and weight.

The Meter, like our yard, is the unit used in measuring cloths and short distances.

The Kilometer is commonly used for measuring long distances, and is about § of a common mile.