Hinton - The Fourth Dimension.pdf

SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 47
No mathematical processes beyond this simple one of
counting will be necessary.
Let us suppose we have before us in
fig. 19 a plane covered with points at regular
intervals, so placed that every four deter-
Fig. 19.
mine a square.
Now it is evident that as four points
determine a square, so four squares meet in a point.
Thus, considering a point inside a square as
belonging to it, we may say that a point on
the corner of a square belongs to it and to
four others equally: belongs a quarter of it to
Fig. 20.
each square.
Thus the square ACDE (fig. 21) contains one point, and
has four points at the four corners. Since one-fourth of
each of these four belongs to the square, the four together
count as one point, and the point value of the square is
two points—the one inside and the four at the corner
make two points belonging to it exclusively.
E
D
A
C
B
D
C
A B
Fig. 21. Fig. 22.
Now the area of this square is two unit squares, as one
can see by drawing two diagonals in fig. 22.
We also notice that the square in question is equal to
the sum of the squares on the sides AB, BC, of the rightangled
triangle ABC. Thus we recognise the proposition
that the square on the hypothenuse is equal to the sum
of the squares on the two sides of a right-angled triangle.
Now suppose we set ourselves the question of determining
the whereabouts in the ordered system of points,