ARITHMETIC.

(PART 4.)

COMPOUND NUMBERS.

1. Numbers may be expressed according to a uniform or a varying scale. By scale is meant the relation of a unit of one order to the unit of the next higher or lower order. When the relation is the same for any two consecutive orders, the scale is said to be uniform; otherwise, it is varying. For example, the scale by which numbers are expressed in the Arabic notation is a uniform scale, since a unit of any order is 10 times as great as the unit of the next lower order; for 100 is 10 times 10, and 10 is 10 times 1, etc. The Arabic notation, the metric system, and United States money are the leading examples of the application of a uniform scale. All other numbers in commercial use in English-speaking countries require the use of a varying scale. Thus, to express 4 yards 2 feet 7 inches, it is necessary to write the words yards, feet, and inches, or their abbreviations, since 12 inches equal 1 foot, and 3 feet equal 1 yard.

2. A simple number is one which expresses one or more units of the same name or denomination; as, 5, 6 yards, etc.

3. A compound number is one which expresses units of two or more denominations of the same kind, the denominations increasing or decreasing according to a varying scale; as, 4 yards 2 feet 7 inches. But 4 yards and 5 ounces is not a compound number, since there is no relation between yards and ounces; that is, no number of ounces can equal a yard. Compound numbers are also frequently called denominate numbers.

For notice of copyright, see page immediately following the title page.

4. Compound numbers form a very important section of arithmetic. It is necessary for the student to obtain a clear idea of their use, and to be able to add, subtract, multiply, and divide them. The only real difficulty that arises is the memorizing of the tables. The best way to do this is to read them over very carefully several times, and then, after carefully studying the sections on Reduction, Addition, etc., work the examples (all of them), constantly referring to the tables for help. But, before leaving the subject to take up the next paper, he should thoroughly memorize all of the tables, so that when any one asks him how many square rods there are in an acre, or a similar question, he can answer instantly, without being obliged to stop and think.

5. A measure is a standard unit established by law or custom, by which the length, surface, capacity, and weight of things are estimated.

6. Measures are of six kinds, as follows:

MEASURES OF EXTENSION.

7. Measures of extension are used in measuring lengths (distances), surfaces (areas), and solids (volumes), and are divided, accordingly, into linear measure, square measure, and cubic measure.

LINEAR MEASURE.

8. The standard to which all measures of extension are referred is the yard, which is the distance between two points on a brass bar kept at Washington. The yard is subdivided into feet and inches; and multiples of the yard are termed rods and miles. The relations between the different units are shown in the following table; the letters in Italics are the abbreviations of the names of the units.

9. The inch is usually divided into halves, quarters, eighths, and sixteenths; by civil engineers and scientists, into tenths, hundredths, thousandths, etc., and in other ways. In measuring cloth, ribbons, and other goods that are sold by the yard, the yard is divided into halves, quarters, eighths, and sixteenths.

A furlong is one-eighth of a mile, or 40 rods. The mile of 5,280 feet is a statute mile, so called to distinguish it from the geographical or sea mile, which equals 6,081 feet.

10. Another abbreviation frequently used for inches and feet is (") and ('). Thus, instead of writing 4 feet 6 inches as 4 ft. 6 in., it may be written 4′ 6′′; but when so written, it is customary to place a dash between the feet and inches; thus, 4'-6". Still another way of writing the above is 4 ft. 6′′.

11. Surveyor's linear measure is used in measuring land, roads, etc. The unit is a steel chain 66 feet long, and made of 100 links, all of equal length; therefore, the length of a link is 66 × 12÷100 = 7.92 inches. For railroad survey. ing and other purposes, civil engineers use a steel tape 100 feet long, the feet being divided into tenths and hundredths. In computations, the links are written as so many hundredths of a chain.

SQUARE MEASURE.
TABLE III.

144 square inches (sq. in.)....... 1 square foot...... sq. ft.

9 square feet.

30 square yards.

160 square rods..

= 1 acre ......

= 1 square mile

sq. yd.

sq. rd.

4,840

3,097,600

=

= 102,400 = 640 = 1

12. Square measure is used in estimating the area of

-12-1 ft

FIG. 1.

surfaces. In commercial use, the square yard is the largest unit employed, the square rod, acre, and square mile being used for measuring land. The unit of square measure is a square whose sides are equal in length to the linear unit. The units of square measure are derived units; that is, they depend for their value upon the values of some other units, which,

in this case, are the units of linear measure. If the large in Fig. 1 measures 1 foot on each side, as indicated

square

by the dimension lines, then the space which the square covers, on a flat surface, is 1 square foot. If horizontal and vertical lines be drawn 1 inch apart, as shown, the small squares so formed will measure 1 inch on each side; and if these small squares be counted, they will be found to number 144. Hence, there are 144 square inches in 1 square foot. In the same way, it can be shown that 1 square yard contains 9 square feet. The acre is the only unit that forms an exception-it cannot be expressed as an exact square of any unit. A piece of land 208.71 feet square contains almost exactly an acre.

Roofers, plasterers, and carpenters frequently call 100 square feet a square.

36 square miles (6 miles square).... = 1 township..... Tp.

13. Surveyor's square measure is used only by civil engineers and surveyors. For this reason, no further remarks will be made concerning this measure.

14. Cubic measure is used in measuring the volumes of solids or bodies which have length, breadth, and thickThe units of cubic measure are also derived units, since they depend upon linear measurements for their values.

ness.