Magic Squares and Cubes

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1908 Excerpt: ...the result, but it is not probable that he derived his square according to the scheme employed here. Our 16X16 square is not exactly the same as the square of Franklin, but it belongs to the same class. Our method gives the key to the construction, and it is understood that the system here represented will allow us to construct many more squares by simply pushing the square beyond its limits into the opposite row which by this move has to be transferred. There is the same relation between Franklin's 16X16 square and our square constructed by alternation with quaternate transposition, that exists between the corresponding 8X8 squares. REFLECTIONS ON MAGIC SQUARES. MATHEMATICS, especially in the field where it touches philosophy, has always been my foible, and so Mr. W. S. Andrews's article on "Magic Squares" tempted me to seek a graphic key to the interrelation among their figures which should reveal at a glance the mystery of their construction. THE ORDER OF FIGURES. In odd magic squares, 3X3, 5X5, 7X7, etc., there is no difficulty whatever, as Mr. Andrews's diagrams show at a glance (Fig. 213). The consecutive figures run up slantingly in the form of a staircase, so as to let the next higher figure pass over into the next higher or lower cell of the next row, and those figures that according to this method would fall outside of the square, revert into it as if the magic square were for the time (at the moment of crossing its boundary) connected with its opposite side into the shape of a cylinder. This cannot be clone at once with both its two opposite vertical and its two opposite horizontal sides, but the process is easily represented in the plane by having the magic square extended on all its sides, and on passing its limits on one side we must...