two of the circles do not give the constant total of 26; but with this sacrifice, however, we are able to get twelve additional summations of 26, which are shown by the dotted circles in Figs. 34, 35 and 36. Fig. 34 shows the vertical receding plane of eight numbers; Fig. 35, the horizontal plane; and Fig. 36, the plane parallel to the picture, the latter containing the two concentric circles that do not give totals of 26.

In this example all pairs are placed on radial lines with one number in each sphere which satisfies the summations of the twelve dotted circles. The selections for the four concentric circles are shown in Fig. 37. The full lines show the selections for Fig. 34 and the dotted lines for Fig. 35. It is impossible to get constant totals for all six concentric circles.

Fig. 38 is a sphere containing the series 1, 2, 3 .... 98, ar

ranged in fifteen circles of sixteen numbers each, with totals of 792. It contains six 3X3 magic squares, two of which, each form the nucleus of a 5x5 concentric square. Also, the sum of any two diametrically opposite numbers is 99.

To construct this figure, we must select two complementary sets of 25 numbers each, that will form the two concentric squares;

and four sets of 9 numbers each, to form the remaining squares, the four sets to be selected in two complementary pairs.

This selection is shown in Fig. 39, in which the numbers enclosed in full and dotted circles represent the selection for the front and back concentric squares respectively. The numbers marked with T, B, L and R represent the selections for the top, bottom, left and right horizon squares respectively.

After arranging the numbers in the top horizon square, we locate the complementary of each number, diametrically opposite and accordingly form the bottom square. The same method is used in placing the left and right square.

The numbers for the front concentric square are duplicated in Fig. 40. The numbers marked by dot and circle represent the selection for the nucleus square, and the diagram shows the selections

for the sides of the surrounding panel, the numbers 4, 70, 34 and 40 forming the corners.

By placing the complementaries of each of the above 25 numbers, diametrically opposite, we form the rear concentric square.

After forming the six squares, we find there are twelve numbers left, which are shown in Fig. 41. These are used to form the four horizon triads. Two pairs are placed on the central circle, and by selection, as shown in the diagram, we fill in the other two

circles with complementary numbers diametrically opposite. The above selection is such that it forms two groups of numbers, each with a summation of 198; this being the amount necessary to complete the required summations of the horizon circles.

There are many selections, other than those shown in Fig. 39, which could have been taken. A much simpler one would be to select the top 25 pairs for the front and back concentric squares,