unit, a surface two units long and one unit wide must contain two square units; a surface three units long and one unit wide must contain three square units; and generally, a surface one unit wide must contain as many square units as there are units in length.

A surface two units wide must contain twice as many square units as a surface one unit wide, i. e., twice as many as there are units in length; a surface three units wide must contain three times as many square units as a surface one unit wide, i. e., three times as many as there are units in length; and generally, any surface must contain as many square units as there are in the product obtained by multiplying its length by its breadth. The surfaces here spoken of are, in all cases, supposed to be rectangular ones.

41. Cubic Measure.

Cubic Measure is used in measuring solids.

(a.) A solid is a magnitude which has length, breadth, and thickness.

NOTE. The term "solid," as used in mathematics, refers to space rather than to material substances,

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

(c.) 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. A cubic unit, then, is any solid equivalent to a cube 1 unit long, 1 unit wide, and 1 unit high.

(d.) The solid contents of bodies are measured by cubic measure, i. e., by the number of cubes of a given size which the bodies contain, or to which they are equivalent.

(e.) In this measure, C. stands for cord, Cd. ft. for cord foot, cu. ft. for cubic foot, cu. yd. for cubic yard, and cu. in. for cubic inch.

NOTE. Since a solid one unit long, one unit wide, and one unit high contains a cubic unit, a solid one unit wide, one unit high, and two units long must contain two cubic units; one three units long must contain three cubic units; one four units long must contain four cubic units; and generally, a solid one unit wide and one unit high must contain as many cubic units as there are linear units in its length.

Again. Since a solid two units wide must contain twice as many cubic units as a solid one unit wide, and a solid three units wide must contain three times as many as a solid one unit wide, &c., it follows that a solid one unit high must contain as many cubic units as there are in the product of its length by its breadth, i. e., as many cubic units as there are superficial units in its base.

Again. Since a solid two units high contains twice as many cubic units as a solid one unit high, and a solid three units high contains three times as many cubic units as a solid one unit high, it follows that any solid must contain as many cubic units as there are in the product obtained by multiplying the number of superficial units in its base by the number of linear units in its height, i. e., as many cubic units as there are in the product of its length multiplied by its breadth, multiplied by its height.

42. 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 every where equally distant from a point within called the centre. The boundary line is called the circumference of the circle.

Fig. 6 represents a circle of which C is the centre.

FIG. 6.

C

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

(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 of the circle.

(e.) Any portion of the circumference is called an arc. (f) Every circumference of a circle, whether the circle be large or small, is supposed to be divided into 360 equal parts,

called degrees. Each degree is divided into 60 equal parts, called minutes, and each minute into 60 equal parts, called seconds.

(g.) A degree is to be regarded simply as the 360th part of the circumference of the circle considered. Hence, it is obvious that its length, and that of its subdivisions, must vary with the size of the circle.

(h.) If from the vertex of an angle, as a centre, we should draw a circumference, some portion of that circumference would be included between the sides of the angle. The larger the angle is, the larger will be the arc included between its sides, and the smaller the angle is, the smaller will be the included arc.

(i.) Since the angle and the arc thus vary with each other, the arc is taken as the measure of the angle. If the arc contains 50 degrees, the angle is one of 50 degrees, &c.

Fig. 7 illustrates this.

FIG. 7.

F

(k.) In this figure let C be the centre of the circle and the vertex of the several angles, and let F H and G D be lines perpendicular to each other, and C E and C I be lines drawn at random. Then the arc E D is the measure of the angle ECD; the arc D F, of the angle D C F; the arc D I, of the angle D CI; the arc E F, of the angle E C F, &c.

(1.) Since F H and G D are perpendicular to each other, the angles F C D, D C H, H C G, and G C F are all right angles, and each of them must include of the angular space about the point C. Therefore the arc included between the sides of a right angle equals of the circumference, and the measure of a right angle is of 360 degrees, which equals 90 degrees. The measure of an acute angle is less than 90 degrees, and that of an obtuse angle more than 90 degrees.

TABLE OF CIRCULAR OR ANGULAR MEASURE.

60 seconds 1 minute.

60 minutes = 1 degree.

(m.) Degrees are marked by the character, minutes by ',

seconds by" thus, 13° 27′ 49′′ 13 degrees, 27 minutes, and 49 seconds.

An arc of 90° is called a quadrant.

A sign is an astronomical measure of 30°.

43. Dry Measure.

(a.) Dry Measure is used for measuring all kinds of grain, beans, nuts, salt, &c.

(b.) The chaldron of 36 bushels is sometimes used in measuring coals. ch. stands for chaldron, bu. for bushel, pk. for peck, qt. for quart, and pt. for pint.

(c.) The bushel contains 2150 cubic inches. The quart must, therefore, contain 67 cubic inches.

44. Liquid Measure.

(a.) All kinds of liquids are measured by Liquid Measure.

(b.) In this measure gal. stands for gallon, qt. for quart, pt. for pint, and gi. for gill.

(c.) The hogshead of 63 gallons is used in estimating the . contents of reservoirs or other large bodies of water; but in all other cases, the term hogshead is not a definite measure. Casks containing from 50 or 60 to 100 or 200 gallons are called hogsheads.

(d.) A barrel of cider is usually reckoned at 31 gallons. (e.) The gallon contains 231 cubic inches.

(f) The beer gallon is sometimes used for measuring milk, beer, and ale. It contains 282 cubic inches. must therefore contain 70 cubic inches.

The beer quart

45. Comparison of Dry, Liquid, and Beer Measures.

The following table will be convenient for use in comparing Dry, Liquid, and Beer Measures: —

(a.) The year is divided into 12 months, which differ somewhat in length. In this measure yr. stands for year, mo. for month, wk. for week, da. for day, h. for hour, m. for minute, and sec. for second.

(b.) To avoid the inconvenience of reckoning of a day with each year, every fourth year (called leap year) is reckoned at 366 days, and the others are reckoned at 365 days. A leap year may always be known by this, viz.: Its number can be divided by 4. Thus we know that 1852 is a leap year, because 1852 can be divided by 4 without a remainder.

(c.) The year in reality contains but 365 days, 5 h. 48 m. 48 sec.; so that by reckoning 365 days we make a slight error each year, which in 100 years amounts to about 1 day. The centennial years are not, therefore, reckoned as leap years, unless the number of the year be divisible by 400. Thus the year 1900 will not be a leap year; but the year 2000 will be.