448 



NA TURE 



[March 13, 1902 



diagonal being 65. This number 65 is obtained by multiplying 

 25, the number of cells, by the next higher number, 26, and then 

 dividing 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 ingenuity has been ex- 

 pended 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 Krenicle, a Frenchman, published a 

 work of more than 500 pages upon magic squares. In this 

 work he showed that SSo magic squares of the fourth order 

 could be constructed, and in an appendix he gave the actual 

 diagrams of the whole of ihcm. 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. 



Asa consequence it is not very difficult to compose particular 

 specimens, and, for the most ])art, the fascinated individuals, to 

 whom I have alluded, have devoted their energies to the dis- 

 covery of principles of formation. Of such principles I will 

 give a few, remarking that the cases of squares of uneven order 

 I, 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 additional cells are added to the 

 square, the numbers written as shown in natural order dia- 

 gonally, and then ihe numbers which are outside the square 



Fig. 3. 



projected into the empty compartments according to an ea.si!y 

 understood law. The second method is associated with the 

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

 during a visit to Siam in 1687. The number i (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 : — 



(i) When the top row is reached, the next number is written 

 at the bottom of the next column. 



(2) When the right-hand column is reached, the next number 

 is written on the left of the row above. 



(3) When it is impossible to proceed according to the above 

 rules, the number is placed in the cell immediately below the 

 last number written. 



If we commence by writing the number i in any cell except 

 that above indicated, a square is reached which is magic in 

 regard to rows and columns, but not in regard to diagonals. 



Subsequent writers have shown that starting with the left- 

 hand bottom cell and using the move of the knight instead of 

 that of the bishop, the general principle of De la Loubere 

 will also lead to a magic square (Fig. 4). The next method is 

 that of De la Hire, and dates from 1705. Two subsidiary 

 squares are constructed as shown, the one involving five 

 numbers I, 2, 3, 4, 5, and the other five numbers o, 5, 10, 15, 

 20. When these squares are properly formed and a third 

 square constructed by adding together the numbers in cor- 

 responding cells, this third square is magic (Fig. 5). Time 

 does not permit me to enter into the exact method of forming 

 the subsidiary squares, and I will merely mention that each of 

 them possesses a particular property, viz., only five different 



NO. 1689, VOL. 65] 



numbers are involved, and all five appear in each column and 

 in each row ; ip other words, no row and no column contains 

 two numbers of the same kind, but no diagonal property is 

 necessarily involved. Such squares are of a great scientific 

 importance, and have been termed by Euler and subsequent 

 writers " Latin squares," for a reason that will presently 

 appear. From a scientific point of view, the chief interest of all 

 arrangements such as I consider this evening lies, not in their 

 actual formation, but in the enumeration of all jiossible ways of 

 forming them, and in this respect very little has been hitherto 



achieved by mathematicians. No person living knows in how 

 many ways it is pos,sible to form a magic square of any order 

 exceeding 4. The fact is, that before we can attempt to 

 enumerate magic squares we must see our way to solve problems 

 of a far more simple character. For example, before we can 

 enumerate the squares that can be formed by De la Hire's 

 method we must take a first step by finding out how many 

 Latin squares can be formed of the ditTerent orders. For the 

 order 5 the question is, "In how many ways can five different 

 objects be placed in the cells so that each column and each 



row contains each object ? " It may occur to some here this 

 evening that such a discussion might be interesting or curious, 

 but could not possibly be of any scientific value. But such is 

 not the case. A department of mathematics that is universally 

 acknowledged to be of fundamental importance is the " theory 

 of groups." Operations of this theory and those connected with 

 logical and other algebras possess what is termeil a " multiplica- 

 tion table," which denotes the laws to which the operations 

 are subject. In Fig. 6 you see such a table of order 6 

 slightly modified from Burnside's "Treatise on the Theory of 

 Groups " ; it is, as you see, a Latin square, and the chief problem 

 that awaits solution is the enumeration of such tables ; the 



