The fourth dimension

;

: THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 47

No mathematical processes beyond this simple one of + « « « e counting will be necessary. » + «+ + Let us suppose we have before us in + » . « « fig. 19aplane covered with points at regular » + « « « intervals, so placed that every four deter* + « e « mine a square. 11g. 4, 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 4 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 Fig. 20. to each square. Thus the square AcDE (fig. 21) contains one point, and

. . Qo ° ° e<->e. » x ‘B . . 7. £ © 8 «© 7 © 8 @ : Fig, 21. Fig. 22.

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.

Now the area of this square is two unit squares, as can be seen by drawing two diagonals in fig. 22.

We also notice that the square in question 1s equal to the sum of the squares on the sides AB, BC, of the rightangled triangle apc. 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,