THE MAGIC SQUARE. 



It will be seen that we are able thus to construct a i ;y large 

 number of magic squares of 5 times 5 spaces by varying in every 

 possible manner, the numbers in the two auxiliary squares. Further- 

 more, the squares thus formed possess the additional peculiarity, 

 that every 5 numbers which fill out two rows that are parallel to a 

 diagonal and lie on different sides of the diagonal also give the con- 

 stant sum of 65. For example: 3 and 7, n, 20, 24; or 10, 14 and 18, 

 22, i. Altogether then the sum 65 is produced out of 20 rows or 

 pairs of rows. On this peculiarity is dependent the fact that if we 

 imagine an unlimited number of such squares placed by the side of, 

 above, or beneath an initial one, we shall be able to obtain as many 

 quadratic cells as we choose, so arranged that the square composed 

 of any 25 of these cells will form a correct magic square, as the fol- 

 lowing figure will show : 



Fig. ii. 



Every square of every 25 of these numbers, as for example the 

 two dark-bordered ones, possesses the property that the-addition of 

 the horizontal, vertical, and diagonal rows gives each the same 

 sum, 65. 



As an example of a higher number of cells we will append here 

 a magic square of ii times ii spaces formed by the general method 

 of De la Hire from the two auxiliary squares of Figs. 12 and 13. 

 From these two auxiliary squares we obtain by the addition of the 



