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21. Comparison of Weights. (a.) A pound Troy is equal to a pound Apothecaries', but is smaller than a pound Avoirdupois; while an ounce Troy is equal to an ounce Apothecaries', and is greater than an ounce Avoirdupois.

(b.) The following table shows the value, in Troy grains, of a unit of each denomination named in the last three tables, and thus furnishes ready means of comparing the different weights with each other.

TABLE OF COMPARISON.
1 lb. Avoirdupois = 7000 grains Troy.
1 lb. Troy = 1 lb. = 5760 grains Troy.
1 oz. Avoirdupois = 4377 grains Troy.

1 oz. Troy = 13 = 480 grains Troy.
1 dr. Avoirdupois = 27 grains Troy.

13 = 60 grains Troy.

17 = 20 grains Troy.

1 dwt. = 24 grains Troy. 1 gr. Apothecaries' =1 grain Troy. (c.) It follows from the above, that —

144 lb. Avoirdupois = 175 lb. Troy.
192 oz. " = 175 oz. Troy.

1 lb. " = 115 of 1 lb. Troy.
1 oz. " = 175 of 1 oz. Troy.

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(b.) Surveyors usually measure distances by means of a chain 4 rods in length, called GUNTER'S CHAIN, or the SURVEYOR'S CHAIN. This chain contains 100 equal links; 25 links will, therefore, equal 1 rod, and I link will equal 7.92 inches.

23. Cloth Measure.

(a.) Cloth MEASURE is used for measuring cloths, silks, etc. FULL TABLE.

ABBREVIATED. 24 inches = 1 nail.

21 in. = 1 na. 4 nails = 1 quarter.

4 na. = 1qr. 4 quarters = 1 yard.

4 qr. = 1 yd. (b.) The yard and inch are the same in length as the yard and inch in Long Measure.

24. Square Measure. (a.) SQUARE MEASURE is used in measuring land, and all kinds of surfaces.

(b.) Preliminary Definitions. — An Angle is the difference in direction of two lines. The point where the lines meet is called the VERTEX OF THE ANGLE.

(c.) When the two angles formed by one straight line meeting another are equal to each other, they are RIGHT ANGLES.

(d.) One line is PERPENDICULAR to another when it makes right angles with it.

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The angle A C B is equal to the angle B C D, and hence they are right angles. Therefore, B C is perpendicular to A D.

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(e.) An angle greater than a right angle is called an OBTUSE ANGLE, and an angle less than a right angle is called an ACUTE ANGLE.

A RIGHT ANGLE.

AN ACUTE ANGLE.

AN OBTUSS ANGLE.

(f.) A four-sided figure, having all its angles right angles, is called a RECTANGLE.

(g.) A rectangle, having all its sides equal, is called a SQUARE. A SQUARE, then, has four equal sides, and four equal angles.

(h.) A SQUARE FOor is a square, 1 ft. or 12 in. long, and 1 ft. or 12 in. wide, and hence contains 12 times 12, or 144 sq. in. In like manner, a SQUARE YARD is a square measuring 1 yd., or 3 ft., on each side, and hence contains 3 times 3, or 9 sq. ft.

This figure represents a square yard.

(i.) To measure a surface, we find how many squares of a given size it equals. Thus, a surface contains 5 sq. ft. when it is equivalent to 5 squares, each measuring 1 foot on a side.

(j.) Any rectangular surface contains as many square units as there are in the product obtained by multiplying its length by its breadth.

Table of Square Measure.
FULL TABLE.

ABBREVIATED. 144 square inches = 1 square foot. 144 sq. in.= lsq. ft.

9 square feet = 1 square yard. 9 sq. ft. = 1 sq. yd. 301 squareyards, )

304 sq. yd. or _ or 2724 square } = 1 square rod.

2724 sq. ft. 5 34. feet 40 square rods = 1 rood.

40 sq. rd. =1R. 4 roods = 1 acre.

4R. =1A. 640 acres = 1 square mile.

640 A. = 1 sq. m. 25. Cubic Measure. (a.) CUBIC MEASURE is used in measuring solide. (b.) A SOLID is whatever has length, breadth, and thickness. NOTE.--The term “solid,as used in mathematics, refers to space, rather than to material substances.

(c.) A CUBE is a rectangular solid, whose length, breadth, and height are equal. It may also be defined as a solid bounded by six equal squares.

(d.) A cube 1 foot long, 1 foot wide, and 1 foot high, would be a cubic foot; a cube 1 yard long, 1 yard wide, and 1 yard high, would be a cubic yard, etc.

This figure represents a cubic yard.

(e.) Any rectangular solid contains as many cubic units as there are in the product of its length multiplied by its breadth, multiplied by its height. FULL TABLE.

ABBREVIATED. 1728 cubic inches = 1 cubic foot. 1728 cu. in. = 1 cu. ft. 27 cubic feet =1 cubic yard.

27 cu. ft. = 1 cu. yd. 16 cubic feet = 1 cord foot.

16 cu. ft. = 1 cd. ft. 8 cord feet, or_1 cord of 8 cd. ft. or)

Ş=1 cd. 128 cubic feet wood.

128 cu. ft.

26. Circular or Angular Measure. (a.) CIRCULAR or AngulaR MEASURE is used to measure angles, and the circumferences of circles.

(b.) A CIRCLE is a surface bounded by a curved line, which is everywhere equally distant from a point within, called the CENTRE. The boundary line is called the CIRCUMFERENCE of the circle.

The figure represents a circle, of which C is the centre.

(c.) The distance from the centre of a circle to the circumference is called the RADIUS.

(d.) The distance from a point on one side of a circle through the centre to a point on the opposite side, is called the DIAMETER. Any portion of the circumference is called an ARC.

(e.) Every circumference of a circle, whether large or small, is supposed to contain 360 equal parts, called DEGREES ; each degree is divided into 60 equal parts, called MINUTES; and each minute into 60 equal parts, called SECONDS.

NOTE. - The length of a degree, as well as that of its subdivisions, varies with the size of the circle.

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27. Dry Measure. (a.) DRY MEASURE is used for measuring grain, nuts, salt, etc. FULL TABLE.

ABBREVIATED. 2 pints =l quart.

2 pt. = 1 qt. 8 quarts = 1 peck.

8 qt. = 1 pk. 4 pecks = 1 bushel.

4 pk.= 1 bu. (b.) The chaldron of 36 bushels is sometimes used for measuring coal. Ch. is the sign for chaldron.

(c.) The bushel contains 2150 cubic inches, and the quart contains 67 cubic inches.

28. Liquid Measure. (a.) All kinds of liquids are measured by LIQUID MEASURE. FULL TABLE.

ABBREVIATED. 4 gills = 1 pint.

4 gi.= 1 pt. 2 pints=1 quart.

2 pt. = 1 qt. 4 quarts = 1 gallon.

4 qt. = 1 gal.

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