The fourth dimension

THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 49

In the square on ac there are—

24 points inside . . . . . 24 4atthecormers . .» .« e« -« JI

or 25 altogether.

Hence we see again that the square on the hypothenuse is equal to the squares on the sides.

Now take the square AFHG, which is larger than the square on AB. It contains 265 points.

16 inside . : : é - . . 16 16 on the sides, counting as . A nS 4onthecorners . ° ; A oe

making 25 altogether.

If two squares are equal we conclude the sides are equal. Hence, the line aF turning round A would move so that it would after a certain turning coincide with ac.

This is preliminary, but it involves all the mathematical difficulties that will present themselves.

There are two alterations of a body by which its volume is not changed.

One is the one we have just considered, rotation, the other is what is called shear.

Consider a book, or heap of loose pages. They can be

slid so that each one slips | Lo over the preceding one, a and the whole assumes

the shape b in fig. 24.

This deformation is not shear alone, but shear accompanied by rotation.

Shear can be considered as produced in another way.

Take the square axBcp (fig. 25), and suppose that it is pulled out from along one of its diagonals both ways, and proportionately compressed along the other diagonal. It will assume the shape in fig. 26.

Fig. 24.

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