Easy mathematics : chiefly arithmetic, being a collection of hints to teachers, parents, self-taught students and adults, and containing a summary or indication of most things in e by Lodge Oliver Sir 1851-1940

Author:Lodge, Oliver, Sir, 1851-1940
Language: eng
Format: epub, pdf
Tags: Arithmetic
Publisher: London ; New York : Macmillan
Published: 1905-03-25T05:00:00+00:00

Fia. 12.

So also that the square of 4 is 16, and the square of 5 is 25. That the cube of 2 is 8 is illustrated thus, but the best plan of dealing with solids is to use cubical wooden blocks and build them up. 8 blocks will build a cube whose side is 2 27 „ „ „ 3

64 „ „ „ 4

and so on.

Fia. 13.

The same blocks laid flat on the table will serve conveniently for squares and rectangles and commensurable areas generally. They will also serve to outline commensurable triangles: with conspicuous advantage in some cases.

By this kind of practice a reality about square and cube numbers is attained which can be got in no other way.

Naturally also the area of any rectangle can be thus illustrated as the product of length and breadth; and the volume of rectangular solids as the product of length, breadth, and height.

If we try to illustrate fourth or higher powers in this way we shall find ourselves helpless. Space is only of 3 dimensions. There are length and breadth and thickness, and no more. Some have tried to imagine what a fourth dimension would be like, but for the present we will be content with an actually experienced and familiar three dimensions.

So much for powers; now what about roots 1

The few commensurable roots that exist must all be whole numbers, and they will be represented, so far as square and cube roots are concerned, by the length of the sides or edges of the squares or cubes which have so far been drawn or built up. Thus, for instance, the square root of 16 is 4, and the cube root of 27 is 3. But this fact, which is experimentally obvious in the commensurable case, where the square or the cube can be built of blocks, is true also in the general case. The length of a side of a square is the square root of its area always, and the length of the edge of a cube is the cube root of its volume always. This represents the geometrical notion of a root so far as geometry can illustrate it. "We will now proceed a little further. Suppose we take a square and draw a diagonal ^'°- ^*- across it, what is the length of that diagonal 1 It is evidently greater than a side, and not so great as two sides.

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