THE MAGIC SQUARE. 63 



In this manner the preceding magic cube of 5 times 5 times 5 

 compartments is formed, in which, it may be additionally noticed, 

 the middle number between i and 125, namely 63, is placed in the 

 central compartment ; by which arrangement the attainment of the 

 sum of 315 is assured in the four principal diagonals and the 30 sub- 

 diagonals. The condition attained in the magic squares, that the 

 diagonal-pairs parallel to the sub-diagonals also shall give the sum 

 315 is not attainable in this case but is so in the case of higher num- 

 bers of compartments. 



CONCLUSION. 



Musing on such problems as are the magic squares is fascinat- 

 ing to thinkers of a mathematical turn of mind. We take de- 

 light in discovering a harmony that abides as an intrinsic quality in 

 the forms of our thought. The problems of the magic squares are 

 playful puzzles, invented as it seems for mere pastime and sport. 

 But there is a deeper problem underlying all these little riddles, and 

 this deeper problem is of a sweeping significance. It is the phil- 

 osophical problem of the world-order. 



The formal sciences are creations of the mind. We build the 

 sciences of mathematics, geometry, and algebra with our conception 

 of pure forms which are abstract ideas. And the same order that 

 prevails in these mental constructions permeates the universe, so that 

 an old philosopher, overwhelmed with the grandeur of law, imagined 

 ne neard its rhythm in a cosmic harmony of the spheres. 



