138 MAGIC SQUARES [CH. VII 



order ; and figure ix on page 142 represents a magic square of 

 the tenth order. 



The formation of these squares is an old amusement, and 

 in times when mystical ideas were associated with particular 

 numbers it was natural that such arrangements should be 

 studied. Magic squares were constructed in India before the 

 Christian era : their introduction into Europe appears to have 

 been due to Moschopulus, who lived at Constantinople in the 

 early part of the fifteenth century. The famous Cornelius 

 Agrippa (1486 — 1535) constructed magic squares of the orders 

 3, 4, 5, 6, 7, 8, 9, which were associated by him with the seven 

 astrological "planets": namely, Saturn, Jupiter, Mars, the Sun, 

 Venus, Mercury, and the Moon. A magic square engraved on 

 a silver plate was sometimes prescribed as a charm against the 

 plague, and one, namely, that represented in figure i on the last 

 page, is drawn in the picture of Melancholy, painted in 1514 by 

 Albert Diirer : the numbers in the middle cells of the bottom 

 row giving the date of the work. The mathematical theory of 

 the construction of these squares was taken up in France in the 

 seventeenth century, and later has been a favourite subject with 

 writers in many countries*. 



It is convenient to use the following terms. The spaces or 

 small squares occupied by the numbers are called cells. It is 

 customary to call the rows first, second, etc., reckoning from the 

 top, and the columns first, second, etc., reckoning from the left. 

 The hth and (n + 1 — h)th rows (or columns) are said to be com- 

 plementary. The kth cell in the hth row is said to be shewly 

 related to the (n + 1 — k)th cell in the (n + 1 — h)ih row. Skewly 

 related cells are situated symmetrically to the centre of the 

 square. 



Magic Squares of any order higher than two can be con- 

 structed at sight. The rule to be used varies according as the 

 order n is odd, that is, of the form 2m + 1 ; or singly-even, that 

 is, of the form 2 (2m + 1) ; or doubly-even, that is, of the form 



* For a sketch of the history of the subject and its bibliography see 

 9. Giinther's Geschichte der mathematischen Wissenschaften, Leipzig, 1876, 

 chapter iv; and W. Ahrens, Mathematische Unterhaltungen und Spiele, Leipzig, 

 1901, chapter xii. 



