THE MAGIC SQUARE. 



emblems, which the reader will notice in our reproduction of the cut, 

 the subjoined square. This arrangement of the sixteen natural num- 



Fig. i. 



bers from i to 16 possesses the remarkable property that the same 

 sum 34 will always be obtained whether we add together the four 

 figures of any of the horizontal rows or the four of any vertical row 

 or the four which lie in either of the two diagonals. Such an ar- 

 rangement of numbers is termed a magic square, and the square 

 which we have reproduced above is the first magic square which is 

 met with in the Christian Occident. 



Like chess and many of the problems founded on the figure of 

 the chess-board, the problem of constructing a magic square also 

 probably traces its origin to Indian soil. From there the problem 

 found its way among the Arabs, and by them it was brought to the 

 Roman Orient. Finally, since Albrecht Dtirer's time, the scholars of 

 Western Europe also have occupied themselves with methods for 

 the construction of squares of this character. 



The oldest and the simplest magic square consists of the quad- 

 ratic arrangement of the nine numbers from i to 9 in such a man 

 ner that the sum of each horizontal, vertical, or diagonal row, al- 

 ways remains the same, namely 15. This square is the adjoined. 



Fig. 2. 



Here, we will find, 15 always comes out whether we add 2 and 7 and 

 6, or 9 and 5 and i, or 4 and 3 and 8, or 2 and 9 and 4, or 7 and 5 

 Bind 3, or 6 and i and 8, Of 2 and 5 and 8, or 6 and 5 and 4. 



The question naturally presents itself, whether this condition 

 of the constant equality of the added sum also remains fulfilled when 

 the numbers are assigned different places. It may be easily shown 



