Suggestions for Original Problems.

1. Pupils will find suggestions for original problems in the . Miscellaneous Exercises; or, it may be required that they construct problems of their own after models dictated by the teacher.

2. Having obtained reliable information from parents and others in regard to prices, trade customs, etc., they can make out bills, and furnish items for bills to be made by the class.

3. They may draw diagrams showing the forms and dimensions of lots to be fenced, dictate the kinds of fences to be built, prices of boards, posts, labor, nails, etc., and require the whole cost. They may give, in like manner, the information necessary to reckon the cost of digging cellars, building walls, laying board, stone, or brick walks, etc., etc. Pupils may often obtain from each other such information as may be needed.

4. Let illustrations, like the one on page 174, be required, showing .33, 1.27, etc., etc., of given squares.

5. Let pupils obtain where they can, the data necessary to enable them to calculate the cost of papering, carpeting, plastering, the schoolroom.

6. Pupils who have a little constructive skill may make paper boxes, and require their classmates to calculate their contentshow many quarts of blackberries or vinegar they will contain, etc.

7. Try the experiment of ascertaining the height of some tall tree or steeple, by measuring the length of its shadow, and the length of the shadow cast at the same moment by a stick or post, the length of which above ground can be easily measured.

8. Give the dimensions of a pile or load of wood, and ask, How many cords? or of a wood-shed, and ask, How many cords can be piled in it? or the length of a pile of wood, and ask how high it must be to contain some required number of cords.

9. Give the dimensions of a box containing a gross of such crayons as are used at the blackboard, and ask the length and width of a case which will exactly contain a gross of such boxes.

199. The length, breadth, and height of objects are their dimensions.

A line has only one dimension-length.

A surface has two dimensions-length and breadth.

A solid or space has three dimensions-length, breadth, and height or thickness.

Measures of Extension.

200. Measures used to ascertain how long a line is, or in calculating the size (extent) of a surface or solid, are called Measures of Extension. These are the Linear, Square, and Cubic Measures.

Linear or Line Measure.

201. In measuring length or distance, linear or line measure is used. The standard unit is the yard.

1 mile = 320 rods = 1760 yards = 5280 feet = 63360 inches.

Notes.-1. For measuring cloth the yard is divided into halves, fourths, eighths,

and sixteenths. In the United States custom-houses it is divided decimally. 2. A Furlong/ mile.-The rod is also called a Pole or Perch. 3. A Pace is variously estimated from 3 to 3.3 feet.

4. A Line =

1/12 inch.

202. The mile given in the table is the mile used in land measurements. Its length is fixed by law, and is called the statute mile. It is thus distinguished from the geographical mile of the following table, used on shipboard and at sea.

A Knot corresponds to one geographical or nautical mile, and is used to estimate the speed of vessels at sea.

Note. In the absence of a more exact instrument the hand was formerly used as a measure, From this we have the Palm (breadth of four fingers) = about 3 inches; the Hand (the breadth of palm and thumb, used in measuring the height of horses at the shoulder) = 4 inches; the Span (the distance between the tips of the thumb and the little finger, when the hand is extended against a flat surface) = about 9 inches, or 1/4 of a yard.

How many

ORAL EXERCISES.

1. Feet in 3%, 42, 73, 4.4, 11%, 33% yd.?

2. Feet in 25, 16, 30, 39, 14 in.?

3. Yards in 111, 22, 5, 811 rods.?

4. Rods in 2, 1/4, 1/8 mi.; in 121, 49% yd.?

5. Inches in 134, 65%, 32, 55, 77/12 ft.? 6. Feet in 212, 34, 104, 61⁄2 fathoms?

Surveyors' Measure.

203. Gunter's Chain, used in measuring roads and the boundary lines of land, is 4 rods (= 66 ft.) in length. It has 100 links, each 7.92 inches long.

Square or Surface Measure.

204. There is no measure which is directly applied to a surface to find its extent. Even if there were such a measure, it would be difficult to apply it. Suppose, for instance, that we wished to ascertain how many square yards there are in a plot of ground 51/2 yards long and 51/2 yards wide. If we had a squareyard measure we might perhaps mark off 25 square yards and the fractions of a yard, as in the diagram. But it would be much easier to measure the length and breadth with a yardstick, and then compute the number of square yards in the surface.

51, yards.

5', yards.

205. The square inch, foot, yard, rod, and mile are derived from corresponding linear measure.

□ inches.

1=640 = 102400 = 3097600 = 27878400 = 4014489600

The sign is used for the abbreviation "sq." In written exercises, either can be used.

Note. The acre has no corresponding denomination in linear measure. A square, measuring 208.71 + feet on each side, contains 1 acre.

ORAL EXERCISES.

How many

1. Square yards in 12, 1881, 26, 100, 66 ft.?

3. Square feet in a board 6 ft. 6 in. long, 13 ft. wide ?

4. A board 18 in. long contains half a □ ft.; how wide is it? 5. How many rods in 1/2, 1/4, 5/8 3/16 of an acre?

89

6. How many acres in a half section of land? In a quarter?

Cubic Measure.

206. To measure a block of marble, or to find how much a box, a bin, or a room will contain, we have to ascertain its length, breadth, and height or thickness, by a linear measure, as a foot-rule, a yard-stick, or a tape-line; and, with the aid of the dimensions thus found, to calculate the contents of the block, or bin, or room, in Cubic Measure, that is, we calculate how many times the room, or the space occupied by the block, would contain some known cubic unit, such as a cubic inch, cubic foot, etc.

207. A Rectangular Solid is a solid having six rectangular faces.

208. A Cube is a rectangular solid having six equal square faces. (See also page 103.)

The figure at the left represents the outlines of a cubic foot, with a layer or course of cubic inches at the bottom. With this figure before the pupil let him answer the following questions: 1. How many cubic inches in the course represented? 2. How many such courses are needed to complete the foot? 3. How many cubic inches in a cubic foot? In 1/12? 3/4? 1/24? etc.

209. Thus the cubic inch, foot, and yard are derived from the corresponding linear measures.

Table.

1728 Cubic Inches = 1 Cubic Foot.

27 Cubic Feet = 1 Cubic Yard.

Equivalents.

1 Cubic Yard = 27 Cubic Feet = 46656 Cubic Inches.

Note.-Higher denominations than these are seldom referred to.