224. Make by the protractor an angle of 90°, and give a figure to show which you consider the most convenient way of holding the protractor, when, to a line, you wish to raise or let fall a perpendicular.

225. Can you make an isosceles triangle, having its base 1, and the sum of the other two sides 3?

226. Can you determine, by means of the scale, the length of the hypothenuse of a rightangled triangle, whose base is 4, and perpendicular 3?

227. Place a hexagon inside a circle, and another outside, in such positions with regard to each other as to show the ratio the inner one has to the outer.

By the area of a figure is meant its superficial contents, as expressed in the terms of any specified system of measures.

In England, the system of linear measures squared is generally used to express areas; as

1

The terms acres and roods are the exceptions.

square inches, square feet, square yards, square poles, square chains, square miles.

The area of a square whose side is one inch is called a square inch; and a square inch is the unit by a certain number of which the areas of all squares are either expressed or implied.

The area of a square in square inches may be found by multiplying its length in inches by its breadth, or, which is the same thing, its base by its perpendicular height; and as, in the square, the base and perpendicular height are always of equal extent, the area of a square is said to be found by multiplying the base by a number equal to itself, that is, by squaring the

base.

228. Make squares whose sides shall represent respectively, 1, 2, 3, 4, 5, etc., inches, and show that their areas shall represent respectively, 1, 4, 9, 16, 25, etc., square inches; that is, shall represent respectively a number of inches that shall be equal to 12, 22, 3, 4, 52, etc.

229. Make equilateral triangles, whose sides. shall represent 1, 2, 3, 4, 5, etc., inches, respec

tively, and show that their areas (though not actually so much as 1, 4, 9, 16, 25, etc.) are in the ratio of 1, 4, 9, 16, 25, etc.; that is, that their areas are in the ratio of the squares of their sides.

230. How would you express in general terms the relation existing between the sides and areas of similar figures?

231. Show by a figure that a square yard contains 9 square feet; that is, that the area of a square yard is equal to 9 square feet.

232. Give a figure of half a square yard, and another of half a yard square, and say what relation one bears to the other.

233. Show that the area of a square foot is equal to 144 square inches.

234. Can you show that the squares upon the two sides of a right-angled isosceles triangle are together equal to the square upon the hypothenuse?

Geometricians have demonstrated that a tri

angle, whose sides are 3, 4, and 5, is a rightangled triangle.

235. Make a triangle, whose sides are 3, 4, and 5; erect a square on each of such sides, and see how any two of the squares are related to the third square.

236. Can you raise a perpendicular to a line, and from the end of it?

237. Can you find other three numbers, besides 3, 4, and 5, such that the squares of the less two numbers shall together be equal to the square of the greater, and show that the triangles they make, so far as the eye can judge, by the assistance of a protractor, are right-angled triangles?

The area of a rectangle, whose base is 4, and perpendicular 3, is 12.

238. Show by a figure that the area of a right-angled triangle, whose base is 4, and perpendicular 3, is half 4×3; i. e., is 4x3-12=6.

2

A solid bounded by six rectangles, having

only the opposite ones similar, parallel and equal, is called a parallelopiped.

The most common dimensions of the parallelopiped called a building-brick are 9, 42, and 3 inches.

239. Make of one piece of cardboard a parallelopiped of the same form as a common building-brick;' show how you arrange all the sides to fit, and give a sketch of it.

It is now above 2,000 years since geometricians discovered that the square upon the base of any right-angled triangle, together with the square upon the perpendicular, is equal to the square upon the hypothenuse.

You have proved that the squares upon the two sides of a right-angled isosceles triangle are together equal to the square upon the hypothe

nuse.

240. Can you invent any method of proving to the eye that the squares upon the base and perpendicular of any right-angled triangle what

1 When a parallelopiped is long, it takes the name of bar, as a bar of iron,