380 



THE POPULAR EDUCATOR. 



Fig. 1. 



Fig. 2. 



16 square inches ; a square, of which the side is 5 feet, contains 

 25 square feet. The truth of this will appear from the following 

 diagram : 



Draw a square, each of the sides of 

 which suppose to be 4 inches long ; divide 

 the sides into lengths of 1 inch, and com- 

 plete the figure by drawing parallel lines, 

 as in the margin. This divides the square 

 into small squares, each of whose sides is 

 an inch in length. Now, in any one row, 

 such as we have indicated by the figures, 

 there are 4 such squares, and there are 4 

 rows. Hence, there are 16 square inches 

 in the given square. 



Suppose that two opposite sides be lengthened to 6 inches, so 

 that the figure is no longer a square, but a rectangle. Dividing 

 the figure as before into square inches, we see that there are 

 necessarily six rows, each containing 4 

 square inches. Hence the number of square 

 inches in a rectangle, two of whose sides 

 are 4 inches long, and the other two 6 

 inches, is 6 X 4, or 24 square inches. The 

 same method is evidently true for any 

 other rectangle, so that, to obtain the 

 number of square units in any rectangle, 

 we must multiply the number expressing 

 the number of linear units in the length by 

 the number expressing the number of linear 

 units in the breadth. 



The same is true if the lengths of the 

 sides be fractional parts of the unit of 

 length. For instance, to find the area of a rectangle J of a foot 

 long and J a foot wide. Referring back to Fig. 1, suppose now 

 that it is a square, each side of which is 1 foot. Then, dividing, 

 as in the figure, each foot into 4 parts, the square contains 16 

 square parts, each of which, therefore, is <fo of a square foot. 

 Now the dotted line encloses a rectangle, one side of which is 

 | and the other f or | of a foot, and this rectangle contains 6 

 of the 16 parts into which the square is divided; or the area 

 of ^ of a square foot, i.e., f X by i of a square foot. 



Obs. It must be observed that, in multiplying together the 

 numbers, fractional or otherwise, which express the number of 

 units in the sides of a rectangle, only one denomination must be 

 used. The fact is, that we cannot talk of multiplying two 

 geometrical magnitudes together. We cannot, for example, talk 

 of multiplying 3 feet by an inch, or by 2 feet ; but we can 

 multiply two numbers together which indicate the lengths of the 

 two lines, with reference to some one standard unit, and then 

 deduce the geometrical result which corresponds to the nume- 

 rical result thus obtained. 



8. The following table of Square Measure is by the above 

 principle deduced from that of the .Measures of Length. The 

 learner is recommended to do this for himself. 



SQUARE MEASURE. 



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



9 square feet = 1 square yard 1 sq. yd. 



SO^ square yards, or ) _ 1 square rod, perch, 

 272^ square feet f or pole 



40 square perches = 1 rood 1 ro. 



4 roods s= 1 acre 1 ac. 



640 acres = 1 square mile 1 sq. m. 



The acre contains, as will be found by calculation, 10 square 

 chains, or 100,000 square links, or 4,840 square yards. 



Flooring, roofing, plastering, etc., are often calculated by a 

 " square " of 100 square feet. 



A hide of land is 100 acres. 



MEASURES OF SOLIDITY OR VOLUME. CUBIC MEASURE. 



9. Definitions. A solid figure is that which has length, 

 breadth, and thickness. A cube is a solid contained by six 

 squares, of which every opposite two are parallel. The sides of 

 bhe squares are called the edges of the cube. 



All solids, or spaces which could be filled by solids, are 

 measured by means of the number of cid)ic inches, cubic feet, etc., 

 ffhich they contain, .i.e., by cubes, the edges of which are 

 respectively 1 inch, 1 foot, etc., in length. 



The magnitude of any solid figure is sometimes called its 

 volume. 



10. To find the magnitude of a Cube, the length of an edge 

 being given. 



Eaise the number expressing the number of linear units in the 

 edge to the third power. This will give the number of cubic 

 units of the same kind in the given cube. 



For instance, a cube of which the edge is 4 inches long con- 

 tains 64 cubic inches ; a cube of which the edge is 5 feet long 

 contains 125 cubic feet. 



The truth of this will appear from the following diagram : 



Take a cube, as in the diagram, 

 of which the edge is supposed to 

 be 4 inches long, and divide each 

 edge into lengths of one inch. 

 Then, by drawing parallel planes, 

 as indicated in the figure, we can 

 divide the cube into a number of 

 cubes, each of which is a cubic 

 inch. Now, any one slice such as 

 that which is shaded clearly con- 

 tains 4 X 4, or 16 cubic inches, and there are 4 such slices. 

 Hence the cube contains 4 x 4 X 4, or 64 cubic inches. 



11. Definitions. A rectangular parallelepiped is a solid figure 

 contained by six rectangular figures, of which every opposite 

 two are parallel. 



This differs from a cube in the fact that the length, breadth, 

 and thickness are not equal. 



The volume of (i.e., the number of cubic units in) a parallele- 

 piped is obtained by multiplying the numbers together which 

 express the number of linear units in the length, breadth, and 

 thickness respectively. 



This will perhaps be sufficiently apparent from the accom- 

 panying diagram of a rectangular parallelepiped, of which the 

 length, breadth, and height are supposed to be 6, 5, 4 inches 

 respectively. 



There will evidently be six such slices as that we have shaded, 

 each containing 5 x 4, or 20 cubic inches. 



The volume of the solid will therefore be 6 X 5 X 4, or 120 

 cubic inches. 



CUBIC MEASURE. 



1728 cubic inches f 1 cubic foot, written 1 c. ft. 

 27 cubic feet = 1 cubic yard 1 c. yd. 



This measure is used in estimating the magnitude of timber, 

 stone, boxes of goods, the capacity of rooms, ships, the solid 

 mass of earth in railway cuttings, etc. 



For example, 42 cubic feet are defined to be one ton of 

 shipping. 



For liquids and dry commodities other systems are adopted, 

 which wo will give after we have explained the measures of 

 weight. 



LESSONS IN PENMANSHIP. XXIV. 



ALTHOUGH it is not possible to give a detailed scheme of 

 elementary forms of which the capital letters of the writing 

 alphabet are composed, as was done with regard to the small 

 letters, it may be as well, for the benefit and instruction of the 

 self -teacher, to make a few remarks on the method of forming 

 each of the capital letters. 



In the capital letters of the writing alphabet the letter I is 

 the key, and forms the principal part of most of the letters ; it 

 consists of a nicely tapering black stroke, commencing with a 

 hair-stroke, and ending in a hair-stroke with a full point or a 

 scroll. The head or top of this letter is variously made; a 

 common form is seen in the capital letters in page 357 ; some- 

 times the head is formed like that of the capital J, which is the 

 same letter in writing, with the black-stroke and the bottom 

 hair-stroke carried below the level and terminated in a loop to 

 the left. 



