52 



3Iajor P. A. MacMahon 



[Feb. 14. 



(1) When the top row is readied, the next number is written at 

 the bottom of the next column. 



(2) AVhen 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 1 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. 



Fig. 



Fig. 4. 



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 principal of De la Loubere will also lead 

 to a magic square ("Pig. 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, 1, 2, 3, 4, 5, and the other 

 five numbers 0, 5, 10, 15, 20. "When these squares are properly 

 formed and a third square constructed by adding together the num- 

 bers in corresponding cells, this third square is magic (Fig. 5). Time 

 does not permit me to enter into the exact m.ethod of forming the 

 subsidiary squares, and I will merely mention that each of them j^os- 

 sesses a particular projjerty, viz. only five difi"erent numbers are in- 

 volved, and all five appear in each column and in each row ; in 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 



