62 



Major P. A. MacMahon 



[Feb. 14, 



square are intimately counected with the infinitesimal calculus and 

 the calculus of finite differences is to sum the matter up. Much, 

 however, remains to be done. The present method is not able to deal 

 with diagonal properties, or with arrangements which depend upon 

 the knight's move. The subject is only in its infancy at present. 

 More workers are required who will, without doubt, introduce new 

 ideas and obtain results far transcending those we are in possession 



15 + 10 = 15 + 16 + 6 + I 

 1 + 12= 1 + 6+14+17 



12 + 13= 12 + 1 + 17 + 8 

 8+ 5= 8 + 17+ 4+ 9 

 6 + 17 + 9 + 13 + 10 + 2 = 16 + 



11 + 10 + 15+ 1+ 6+14 = 14 + 

 4 + 18 + 13+ 5 + 8 + 9 



18+13+12+ 1+ 6+ 7= 5 + 



3+11 

 1 + 17 



S + 17 + 14 + 11 + 



18+ 4 + 17+ 1 + 15+ 2= 5+13+ 3+14+ 6 + 16 



Fig. 15. 



of now. The latest work has shown that the method is applicable to 

 boards of triangular and trapezoidal shapes, and also to solid boards 

 in three dimensions, so that the remote ground occupied by magic 

 and Nasik cubes will soon be invaded. 



In conclusion, I bring before you an interesting example of magic 

 arrangement that I found whilst engaged in rummaging amongst the 



