CH. VII] 



MAGIC SQUARES 



157 



order in such a way that if the number in each cell is replaced 

 by its square the resulting square shall also be magic*. Here 

 are two examples. Figure xxii represents a magic square of 

 the eighth order (the sum of the numbers in each line being 



Figure xxii. A Doubly-Magic Square, n=8. 



Figure xxiii. A Doubly-Magic Square, n = 9. 



equal to 260) so constructed that if the number in each cell is 



replaced by its square the resulting square is also magic (the 



sum of the numbers in each line being equal to 11180). Figure 



xxiii, where n stands for 9, represents a doubly-magic square of 



* See M. Ooccoz in L' Illustration, May 29, 1897. The subject has been 

 studied by Messieurs G. Tarry, B. Portier, M. Coocoz, A. Billy, E. Barbette 

 and W. S. Andrews. More than 200 such squares have been given by Billy in 

 his Eludes sur le$ Triangles et Us Carris Magiques aux deux premiers degree, 

 Troyes, 1901. 



