450 



NA TURE 



[March 13, 1902 



letters in the second row so that each column contains a pair of 

 different letters ? This is a famous question, of which the solu- 

 tion is well known ; it is known to mathematicians as the "pro- 

 blcme des rencontres." It may be staled in a variety of ways ; 

 one of the most interesting is as follows : — A person writes a 

 number of letters and addresses the corresponding envelopes ; 

 if he now put the letters at random into the envelopes, what is 

 the probability that not a single letter is in the right envelope ? 

 Passing on to the problem of determining the number of ways 

 of arranging the first three rows so that each column contains 

 three different letters, it may be staled that up to 1S9S no 

 solution of it had been given ; while it is obvious that as the 

 number of the rows is increased the resulting problems will be 

 of enhanced difficulty. Aparticular case of the three-row problem 

 had, however, been considered under the title " problcme des 

 mcnages" and a solution obtained. It may be stated as 

 follows : — 



A given number of married ladies take their seats at a round 

 table in given positions ; in how many ways can their husbands 

 be seated so that each is between two ladies, but not next to his 

 own wife? For order 5, that is 5 ladies, the question comes to 

 this : — Write down 5 letters and underneath them the same letters 

 shifted one place to the left ; in how many ways can the third 

 row be written so that each column contains three different 

 letters ? This particular case of the three-row problem for any 

 order presents no real difficulty. The results are that in the 

 cases of 3, 4, 5, 6 . . . married couples there are I, 2, 13, 80, 

 &c., ways. 



Since the year 1S90, the problem of the Latin square has 

 been completely solved by an entirely new method, which has 

 also proved successful in solving similar questions of a far more 

 recondite character, and I am here this evening to attempt to 

 give you some notion of the method and some account of the 

 series of problems to which that method has been found to be 

 applicable. 



There is, as viewed mathematically, a fundamental difference 

 between arithmetic and algebra; the former may be regarded as 

 an algebra in which the numerical magnitudes under considera- 

 tion are restricted to be integers ; the two branches contem- 

 plate discontinuous and continuous magnitude respectively. 

 Similarly, in geometry we have the continuous theory, which con- 

 templates figures generated by points moving from one place to 

 another and in doing so passing over an infinite succession of 

 points, tracing a line in a plane or in space, and also a dis- 

 continuous theory, in which the position of a point varies suddenly, 

 per saltitm, and we are not concerned with any continuously 

 varying motion or position. The present problems are concerned 

 sometimes with this discontinuous geometry and sometimes with 

 an additional discontinuity in regard to numerical magnitude, 

 and the object is to count and not to measure. Far removed 

 as these questions are, apparently, from the subject-matter of a 

 calculus of infinitely small quantities and the variation of 

 quantities by infinitesimal increments, my purpose is to show 

 that they are intimately connected with them and that success 

 is a necessary consequence of the relationship. I must first 

 take you to a much simpler problem than that of the Latin 

 square, to one which in a variety of ways is very easy of solu- 

 tion, but which happens to be perhaps the simplest illustration 

 of the method. In the game of chess a castle can move either 

 horizontally or vertically, and it is easy to place S castles on the 

 board so that no piece can be taken by any other piece. One 

 such arrangement is shown in Fig. 10. The condition is 

 simply that one castle must be in each row and also in each 

 column. Every such arrangement is a diagrammatic representa- 

 tion of a certain mathematical process performed upon a certain 

 algebraical function. For consider the process of differentiating 

 r' ; it may be performed as follows : — Write down ,i" as the 

 product of 8 x's, 



X X X X X X X X, 



and now substitute unity for x in all possible ways and add the 

 results ; the substitution can take place in eight different ways, 

 and the addition results in Sa", which will be recognised as the 

 differential coefficient. Observe that the process of ditierentia. 

 tion is thus broken up into eight minor processes, each of which 

 may be diagrammatically represented on the first row of the 

 chess-board by a unit placed in the compartment corresponding 

 10 the particular x for which unity has been substituted. If we 

 now perform differentiation a second time, we may take the 

 results of the above minor processes and in each of them again 



Substitute unity for x in all possible ways ; since in each the 

 'Substitution can take place in seven different ways, it is seen 

 'hat we can regard the process of differentiating twice as com- 

 posed of S X 7 = 56 minor processes, each of which can be 

 diagrammatically represented by two units, one in each of the 

 first two rows of the chess-board, in positions corresponding to 

 the substitutions of unity for x that have been carried out. 

 Proceeding in this manner in regular order up to the eighth 

 differentiation, we find that the whole process of differentiat- 

 ing .t^ eight times in succession can be decomposed into 

 8 X 7 >; 6 X 5 X 4 ;< 3 X 2 X I = 40,320 minor processes, each 

 of which is denoted by a diagram which slight retlection shows 

 is a solution of the castle problem (Fig. 11). There are, in 

 fact, no more solutions, and the whole series of 40,320 diagrams 



constitutes a picture in detail of the differentiations. Simple 

 differentiations of integral powers thus yield the enumerative 

 solutions of the castle problem on chess-boards of any size. 



We have here a clue to a method for the investigation of 

 these chess-board problems ; it is the grain of mustard seed 

 which has grown up into a tree of vigorous growth, throwing 

 out branches and roots in all sorts of unexpected directions. 

 The above illustrations of differentiation gave birth to the idea 

 that it might be possible to design pairs of mathematical pro- 

 cesses and functions which would yield the solution of chess- 

 board problems of a more difficult character. Two plans of 

 operation present themselves. In the first place we may take 

 up a particular question, the Latin square for instance, and 

 attempt to design, on the one hand, a process, and, on the 



•A. •X «Aj t\j t\j Jj^ i\j ^^ 



X'V "^ 'Y* ''i* "^ '¥* '\* 



«A^ »^ %Kj Jl »Ju •A^ 9\. 



t ^ t X. t ^ X X 



2J:x4^x4:4:x4: 



*^ •X. »^ *A/ *X^ JL, aJu JL 



•if,. *Xj %^ «^ ^ *|l *|l/ JL 



4: 4^ :j:. .t :): :|: i :j.- 



other hand, a function the combination of which will lead to 

 the series of diagrams. In the second place, we may have no 

 particular problem in view, but simply start by designing a 

 process and a function, and examine the properties of the series 

 of diagrams to which the combination leads. The first of these 

 plans is the more dilhcult, but was actually accomplished in the 

 case of the Latin square and some other questions ; but the 

 second plan, which is the proper method of investigation, met 

 with great success, and the Latin square was one of its first 

 victims, a solution of a more elegant nature being obtained than 

 that which had resulted from the first plan of operations. There 

 is such an extensive choice of processes and functions that many 

 solutions arc obtainable of any particular problem. I will now 





NO. 1689. VOL. 65] 



