MAGIC SQUARES. 



733 



arranged (fig. 863), forms such a square that the product obtained by 

 multiplying the three terms of one row, or one diagonal, is 4,096, which is 



= 65 

 = 65 

 =65 



ii r. n ii 



en u> en en 



Fig. 862. Examples of magic squares formed by terms of arithmetical progression. 



the cube of the mean term 16. The squares have been termed magic y 

 because, according to Ozanam, they were held in great veneration by the 

 Pythagoreans. In the time of alchemy and astrology, certain magic squares 

 were dedicated to the seven planets, and engraved on a metal blade which 

 sympathized with the planet. To give an idea of the combinations to which 



the study of magic squares lends itself, it is suffi- 

 cient to add that mathematicians have written 

 whole treatises on the subject. Frenicle de Bessy, 

 one of the most eminent calculators of the 

 seventeenth century, consecrated a part of his life 

 to the study of magic squares. He discovered new 

 rules, and found out the means of varying them in 

 a multitude of ways. Thus for the magic square, 

 the root of which is 4, only sixteen different 

 arrangements were known. 



Frenicle de Bessy found 880 new solutions. 

 An important work from the pen of this learned mathematician has been 

 published under the title of " Carre's ou Tables Magiques," in the " Memoirs 

 de 1' Academic Royale des Sciences," from 1666 1699, v l* v - Amateurs, 

 therefore, who are accused of occupying themselves with a useless game, 

 unworthy the attention of serious minds, will do well to bear in mind the 

 works of Frenicle, and better still, to consult them. 



We have so far considered only the first part of the puzzle. We may 

 now examine the problem to which specially it has given rise. We arc 

 quite in accord with M. Piarron de Mondesir, who has been so good as to en- 

 lighten us upon the subject, which is really much more difficult than it appears. 



Fig. 863. Magic square formed by 

 terms of geometrical progression. 



