THE MAGIC SQUARE. 



Fig. 23- 



The magic squares of even numbers thus constructed are not 

 the only possible ones. On the contrary, there are very many others 

 possible, which obey different laws of formation. It has been cal- 

 culated, for example, that with the square of 4 it is possible to con- 

 struct 880, and with the square of 6, several million, different magic 

 squares. The number of odd-numbered magic squares constructible 

 by the method of De la Hire is also very great. With the square of 

 7, the possible constructions amount to 363,916,800. With the 

 squares of higher numbers the multitude of the possibilities increases 

 in the same enormous ratio. 



v. 



MAGIC SQUARES WHOSE SUMMATION GIVES THE NUMBER 

 OF A YEAR. 



The magic squares which we have so far considered contain 

 only the natural numbers from i upwards. It is possible, however, 

 easily to deduce from a correct magic square other squares in which 

 a different law controls the sequence of the numbers to be inscribed. 

 Of the squares obtained in this manner, we shall devote our atten- 

 tion here only to such in which, although formed by the inscription 

 of successive numbers, the sum obtained from the addition of the 

 rows is a determinate number which we have fixed upon beforehand, 

 as the number of a year. In such a case we have simply to add to 

 the numbers of the original square a determinate number so to be 

 calculated, that the required sum shall each time appear. If this 



