1902.] on Magic Squares and other Problems on a Chess-Board. 51 



squares was seen to be fraught with difficulty, and it was considered 

 possible that some new properties of numbers might be disco verecil 

 which mathematicians coulcl turn to account. This has, in fact, 

 proved to be the case ; for from a certain point of view the subject 

 has been found to be algebraical rather than arithmetical, and to be 

 intimately connected with great departments of science, such as the 

 " infinitesimal calculus," " the calculus of operations," and the " theory 

 of groups." 



In the next diagram (Fig. 2) I show you a magic square of order 5 , 

 the sum of the numbers in each row, column, and diagonal being 65. 

 This number 65 is obtained by multi- 

 plying 25, the number of cells, by the 

 next higher number, 26, and then divid- 

 ing by twice the order of the square, 

 viz. 10. A similar rule applies in the 

 case of a magic square .of any order. 

 The formation of these squares has a 

 fascination for many persons, and, as a 

 consequence, a large amount of inge- 

 nuity has been expended in forming 

 particular examples, and in discovering 

 general principles of formation. As an 

 example of the amount of labour that 

 some have expended on this matter, it 

 may be mentioned that in 1693 Frenicle, 



a Frenchman, published a work of more than 600 pages upon magic 

 squares. In this work he showed that 880 magic squares of the 

 fourth order could be constructed, and in an appendix he gave the 

 actual diagrams of the whole of them. The number of magic squares 

 of the order 5 has not been exactly determined, but it has been shown 

 that the number certainly exceeds 60,000. 



As a consequence it is not very difficult to compose particular 

 specimens, and, for the most part, the fascinated individuals to 

 whom I have alluded have devoted their energies to the discovery of 

 principles of formation. Of such principles I will give a few, re- 

 marking that the cases of squares of uneven order, 1, 3, 5, . . . are 

 more simple than those of even order, 4, 6, . . . and that no magic 

 square of order 2 exists at all. The simplest of all methods for an 

 uneven order is shown in the diagram (Fig. 3), where certain addi- 

 tional cells are added to the square, the numbers written as shown in 

 natural order diagonally, and then the numbers which are outside 

 the square projected into the empty compartments according to an 

 easily understood law\ The second method is associated with the 

 name of De la Loubere, though it is stated that he learnt it during 

 a visit to Siam in 1687. The number 1 (see Fig. 2) is placed in the 

 middle cell of the top row, and the successive numbers placed in 

 their natural order in a diagonal line sloping upwards to the right, 

 subject to the laws : — 



Fig. 2. 



