THE MAGIC SQUARE. 



45 



the inscription shall be continued from the left-hand margin in the 

 row just above, and on reaching the upper margin shall be continued 

 from the lower margin in the column next adjacent to the right, 

 noting that whenever we are arrested in our progress by a square 

 already occupied we are to fill out the square next beneath the one 

 we have last filled. In this manner, for example, the last preced- 

 ing square of 7 times 7 cells is formed, in which the reader is re- 

 quested to follow the numbers in their natural sequence (Fig. 5). 



For the next further advancements of the theory of magic 

 squares, and of the methods for their construction we are indebted 

 to the Byzantian Greek, Moschopulus, who lived in the fourteenth 

 century ; also, after Albrecht Durer who lived about the year 1500, 

 to the celebrated arithmetician Adam Riese, and to the mathemati- 

 cian Michael Stifel, which two last lived about 1550. In the seven- 

 teenth century Bachet de Meziriac, and Athanasius Kircher em- 

 ployed themselves on magic squares. About 1700, finally, the 

 French mathematicians De la Hire and Sauveur made considerable 

 contributions to the theory. In recent times mathematicians have 

 concerned themselves much less about magic squares, as they have 

 indeed about mathematical recreations generally. But quite recently 

 the Brunswick mathematician Scheffler has put forth his own and 

 other's studies on this subject in an elegant form. 



