THE MAGIC SQUARE. 



55 



fourthly twenty additional times from each and every pair of any two 

 rows that lie parallel to a diagonal, have together 1 1 cells, and lie 

 on different sides of the diagonal, as for example, 196, 122, 158, 205, 

 131, 167, 214, 140, 187, 223, 149. 



VI. 



CONCENTRIC MAGIC SQUARES. 



The acuteness of mathematicians has also discovered magic 

 squares which possess the peculiar property that if one row after an- 

 other be taken away from each side, the smaller inner squares re- 

 maining will still be magical squares, that is to say, all their rows 

 when added will give the same sum. It will be sufficient to give 

 two examples here of such squares, (the laws for their construction 

 being somewhat more complicated,) of which the first has 7 times 7 

 and the second 8 times 8 places. The numbers within each of the 

 dark-bordered frames form with respect to the centre smaller squares 

 which in their own turn are magical. 



Fig. 25. 



Fig. 26. 



In the first of these two squares the internal square of 3 times 3 

 places contains the numbers from 21 to 29 in such a manner that 

 each row gives when added the sum of 75. This square lies within 

 a larger one of 5 times 5 spaces, which contains the numbers from 

 13 to 37 in such a manner that each row gives the sum of 125. 

 Finally, this last square forms part of a square of 7 times 7 places 

 which contains the numbers from i to 49 so that each row gives the 

 sum of 175. 



In the second square the inner central square of 4 times 4 places 

 contains the cumbers from 25 to 40 in such a manner that each row 



