CH. VII] 



MAGIC SQUARES 



147 



De la Hire showed that by methods known in his day and apart 

 from mere reflexions and rotations, there were 57600 magic 

 squares of the fifth order which could be formed by the methods 

 he enumerated, and taking account of other methods, it is now 

 known that the total number of magic squares of the fifth order 

 considerably exceeds three-quarters of a million. 



Product Squares. Before leaving this part of the subject, 

 I may mention that Montucla suggested the construction of 

 squares whose cells are occupied by numbers such that the 

 product of the numbers in each row, column, and diagonal is 

 constant. The formation of such figures is immediately de- 

 ducible from that of magic squares, for if the consecutive 

 numbers, namely 1, 2, 3, &c, in a magic square are replaced 

 by consecutive powers of any number m, namely m, m 2 , m 3 , &c, 

 the products of the numbers in every line will be magic. This 

 is obvious, for if the numbers in any line of a square are a, a', 

 a", &c, such that %a is constant for every line in the square, 

 then Tlm a is also constant. 



Magic Stars. Some elegant magic constructions on star- 

 shaped figures (pentagons, hexagons, &c.) may be noticed in 



is 

 Figure xiv. A Magic Star. 



passing, though I will not go into details. One instance will 



suffice. Suppose a re-entrant octagon is constructed by the 



10—2 



