4 6 



THE MAGIC SQUARE. 



The best known of the various methods of constructing magic 

 squares of an odd number of cells is the following. First write the 

 numbers in diagonal succession as in the preceding diagram (Fig. 6). 

 After 25 cells of the square of 49 cells which we have to fill out, 

 have thus been occupied, transfer the six figures found outside each 

 side of the square, without changing their configuration, into the 

 empty cells of the side directly opposite. By this method, which 

 we owe to Bachet de Meziriac, we obtain the following magic square 

 of the numbers from i to 49 : 



Fig- 7- 



III. 



MODERN MODES OF CONSTRUCTION OF ODD-NUMBERED 

 SQUARES. 



The reader will justly ask whether there do not exist other cor- 

 rect magic squares which are constructed after a different method 

 from that just given, and whether there do not exist modes of con- 

 struction which will lead to all the imaginable and possible magic 

 squares of a definite number of cells. A general mode of construc- 

 tion of this character was first given for odd-numbered squares by 

 De la Hire, and recently perfected by Professor SchefHer. 



To acquaint ourselves with this general method, let us select 

 as our example a square of 5. First we form two auxiliary squares. 

 In the first we write the numbers from i to 5 five times ; and in the 

 second, five times, the following multiples of five, viz.: o, 5, 10, 15, 

 20. It is clear now that by adding each of the numbers of the series 

 from i to 5 to each of the numbers o, 5, 10, 15, 20, we shall get 

 all the 25 numerals from i to 25. All that additionally remains to 

 be done therefore, is, so to inscribe the numbers that by the addition 



