54 



Major P. A, MacMahon 



[Feb. 14, 



to tables in the theory of groups ; but still, we must walk before we 

 can run, and a step in the right direction is the enumeration of all 

 Latin squares. When I call to mind that the theory of groups 

 has an important bearing upon many branches of physical science, 

 notably upon dynamics, I consider that I have made good my 

 point. 



I now concentrate attention on these Latin squares, and observe 

 that the theory of the enumeration has nothing to do with the par- 

 ticular numbers that occupy the compartments ; the only essential 

 is that the numbers shall be different one from another. My atten- 

 tion was first called to the subject of the Latin square by a work of 

 the renowned mathematician Euler, written in 1782, 'Eecherches 

 sur une nouvelle espece de Quarres Magiques.' I may say that 

 Euler seems to have been the first to grasp the necessity of consider- 

 ing squares possessing what may be termed a magical property of a 

 far less recondite character than that possessed by the magic squares 



Fig. 6. 



Fig. 7. 



of the ancients, and, as we shall see presently, he might have gone a 

 step further in the same direction with advantage and have com- 

 menced with arrangements of a more simple character than that of 

 the Latin square, with arrangements, in fact, which present no difii- 

 culties of enumeration, but which supply the key to the unlocking 

 of the secrets of which we are in search. He commences by re- 

 marking that a curious problem had been exercising the wits of many 

 persons. He describes it as follows : There are 36 officers of six 

 dififerent ranks drawn from six different regiments, and the problem 

 is to arrange them in a square of order 6, one officer in each com- 

 partment, in such wise that in each row, as well as in each column, 

 there appears an officer of each rank and also an officer of each 

 regiment. Of a single regiment we have, suppose, a colonel, lieu- 

 tenant-colonel, major, captain, first lieutenant and second lieutenant, 

 and similarly for five other regiments, so that there are in all 36 

 officers who must be so placed that in each row and in each column 

 each rank is represented, and also each regiment. Euler denotes 



