Hinton - The Fourth Dimension.pdf

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 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
point more; hence its point value is 2. The
value is the measure of its area—the size
of this square is two of the unit squares.
Looking now at the sides of this figure
Fig. 17.
we see that there is a unit square on each
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 hypotenuse 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
Fig. 18.
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