THE MAGIC SQUARE. ^j 



know whether twice two might not be five in other spheres- of tha 

 universe. 



The author of the short article on " Magic Squares " in the Eng- 

 lish Cyclopaedia (Vol. Ill, p. 415), presumably Prof. De Morgan, 

 says: 



"Though the question of magic squares be in itself of no use, yet it belongs to 

 a class of problems which call into action a beneficial species of investigation. With- 

 out laying down any rules for their construction, we shall content ourselves with 

 destroying their magic quality, and showing that the non-existence of such squares 

 would be much more surprising than their existence." 



This is the point. There obtains a symphonic harmony in 

 mathematics which is the more startling the more obvious and self- 

 evident it appears to him who understands the laws that produce this 



symphonic harmony. 



* 

 * * 



On the wood-cut named "Melancholia"* of the famous Nurem- 

 berg painter, Albrecht Diirer, is found among a number of other 



* The term melancholy meant in Durer's time, as it did also in Shakespeare's 

 and Milton's, " thought or thoughtfulness." Says Milton in // Penseroso ; 



" Hail, thou Goddess, sage and holy, 

 Hail divinest melancholy 



Whose saintly visage is too bright 



To hit the sense of human sight, 

 And therefore to our weaker view 

 O'erlaid with black, staid Wisdom's hue. I, ia. 



"Bought that does not lead to action produces a gloomy state of mind. Thought- 

 ttilness which cannot find a way out of itself is that melancholy which engenders 

 weakness, a truth which is illustrated in Hamlet. Shakespeare still uses the words 

 thought and melancholy as synonyms, saying : 



" The native hue of resolution 

 Is sicklied o'er with the pale cast of thought." 



Durer's melancholy does not represent the gloominess of thought, but the power 

 of invention. Soberness and even a certain sadness are considered only as an ele- 

 ment of this melancholy, but on the whole the genius of thought appears bright, 

 self-possessed, and strong. 



Diirer represents the Science of Mechanical Invention as a winged female figure 

 musing over some problem. Scattered on the floor around her lie some of the sim- 

 ple tools used in the sixteenth century. The ladder leaning against the house as- 

 sists in climbing otherwise inaccessible heights. A scale, an hour-glass, a bell, and 

 the magic square are hanging on the wall behind her. 



At a distance a bat-like creature, being the gloom of melancholy, hovers in the 

 air like a dark cloud, but the sun rises above the horizon, and at the happy middle 

 between these two extremes stands the rainbow of serene hope and cheerful confi- 

 dence. 



