The fourth dimension

30 THE FOURTH DIMENSION

Thus we may say that the point value of the square shown is one point, for if we take the square in fig. 16 (1) it has four points, but each of these belong equally to four other squares. Hence one fourth of each of them belongs to the square (1) in fig. 16. Thus the point value of the square is one point.

The result of counting the points is the same as that arrived at by reckoning the square units enclosed.

Hence, if we wish to measure the area of any square we can take the number of points it encloses, count these as one each, and take one-fourth of the number of points at its corners.

Now draw a diagonal square as shown in fig. 17. It contains one point and the four corners count for one

. e + e point more; hence its point value is 2. os « The value is the measure of its area—the « \4-1..4 © size of this square is two of the unit squares.

» i.4 .%. Looking now at the sides of this figure « e « « we see that there is a unit square on each Fig. 17. of them—the two squares contain no points, but have four corner points each, which gives the point value of each as one point.

Hence we see that the square on the diagonal is equal to the squares on the two sides; or as it is generally expressed, the square on the hypothenuse is equal to the sum of the squares on the sides.

Noticing this fact we can proceed to ask if it is always true. Drawing the square shown in fig. 18, we can count * ° © * © the number of its points. There are five “altogether. There are four points inside the square on the diagonal, and hence, with * the four points at its corners the point * value is 5—that is, the area is 5. Now

the squares on the sides are respectively of the area 4 and 1. Hence in this case also the square

Fig. 18.