September 12, (901] 



NA TURE 



is another case in point. Again, in physics, a given mass of 

 gas under the operation of varying pressure and temperature has 

 the well-known invariant, pressure multiplied by volume and 

 divided by absolute temperature. Examples might be multiplied. 

 In mathematics the entities under examination may be arithme- 

 tical, algebraical, or geometrical ; the processes to which they are 

 subjected may be any of those which are met with in mathe- 

 matical work. It is the principle which is so valuable. It is 

 the idea of invariance that pervades today all branches of 

 mathematics. It is found that in investigations the invariantive 

 fr.actions are those which persist in presenting themselves, even 

 when the processes involved are not such as to ensure the in- 

 variance of those functions. Guided by analogy may we not 

 anticipate similar phenomena in other fields of work ? 



The combinatorial analysis may be described as occupying an 

 extensive region between the algebras of discontinuous and 

 continuous quantity. It is to a certain extent a science of 

 enumeration, of measurement by means of integers as opposed 

 to measurement of quantities which vary by infinitesimal incre- 

 ments. It is also concerned with arrangements in which 

 difference of quality and relative position in one, two, or three 

 dimensions are factors. Its chief problem is the formation of 

 connecting roads between the sciences of discontinuous and 

 continuous quantity. To enable, on the one hand, the treat- 

 ment of quantities which vary per salttim, either in magnitude 

 or position, by the methods of the science of continuously vary- 

 ing quantity and position, and on the other hand to reduce 

 problems of continuity to the resources available for the manage- 

 ment of discontinuity. These two roads of research should be 

 regarded as penetrating deeply into the domains which they 

 connect. 



In the early days of the revival of mathematical learning in 

 Europe the subject of " combinations " cannot be said to have 

 rested upon a scientific basis. It was brought forward in the 

 shape of a number of isolated questions of arrangement, which 

 were solved by mere counting. Their solutions did not further 

 the general progress, but were merely valuable in connection 

 with the special problems. Life and form, however, were in- 

 fused when it was recognised by De Moivre, Bernoulli and 

 others that it was possible to create a science of probability 

 on the basis of enumeration and arrangement. Jacob Ber- 

 noulli, in his " Ars Conjectandi," 1713, established the 

 fundamental principles of the Calculus of Probabilities. A 

 systematic advance in certain questions which depend upon the 

 partitions of numbers was only possible when Euler showed 

 that the identity .v".v'' = ,v''+' reduced arithmetical addition to alge- 

 braical multiplication and vice versa. Starting with this notion, 

 Euler developed a theory of generating functions on the ex- 

 pansion of which depended the formal solutions of many 

 problems. The subsequent work of Cayley and Sylvester 

 rested on the same idea, and gave rise to many improvements. 

 The combinations under enumeration had all to do with what 

 may be termed arrangements on a line subject to certain laws. 

 The results were important algebraically, as throwing light on 

 the theory of Algebraic series, but another large class of problems 

 remained untouched, and was considered as being both outside 

 of the scope and beyond the power of the method. I propose 

 to give some account of these problems, and to give a short 

 history of the way in which a method of solution has been 

 reached. It will be gathered from remarks made above that I 

 regard any department of scientific work which seems to be 

 narrow or isolated as a proper subject for research. I do not 

 believe in any branch of science or subject of scientific work 

 being destitute of connection with other branches. If it appears 

 to be so, it is especially marked out for investigation by the 

 very unity of science. There is no necessarily pathless desert 

 separating different regions. Now a department of pure mathe- 

 matics which appeared to be somewhat in this forlorn condition 

 a few years ago was that which included problems of the nature 

 of the magic square of the ancients. Conceive a rectangular 

 lattice or generalised chess board Icf. " Gitter," Klein), whose 

 compartments are situations for given numbers or quantities, so 

 that there is a rectangular array of certain entities. The general 

 problem is the enumeration of the arrays when both the rows 

 and the columns of the lattice satisfy certain conditions. With 

 the simplest of such problems certain progress had undoubtedly 

 been made. The article on Magic Squares in the " Encyclo- 

 p-Tidia Britannica" and others on the same subject in various 

 scientific publications are examples of such progress, but the 

 position of isolation was not sensibly ameliorated. Again, the 



NO. 1663, VOL. 64 J 



well-known " problem e des rencontres" is an instance in 

 point. Here the problem is to place a number of different 

 entities in an assigned order in a line and beneath them the 

 same entities in a different order subject to the condition that 

 the entities in the same vertical line are to be different. This 

 easy question has been solved by generating functions, finite 

 differences, and in many other ways. In fact, when the number 

 of rows is restricted to two the difficulties inherent in the problem 

 when more than two rows are in question do not present them- 

 selves. The problem of the Latin Square is concerned with a 

 square of order n and n different quantities which have to be 

 placed one in each of the n- compartments in such wise that each 

 row and each column contains each of the quantities. The 

 enumeration of such arrangements was studied by mathemati- 

 cians from Euler to Cayley without any real progress being 

 made. In reply to the remark " Cui bono ? " I should say that 

 such arrangements have presented themselves for investigation 

 in other branches of mathematics. Symbolical algebras and in 

 particular the theory of discontinuous groups of operations have 

 their laws defined by what Cayley has termed a multiplication 

 table. Such multiplication tables are necessarily Latin Squares, 

 though it is not conversely true that every Latin Square corre- 

 sponds to a multiplication table. One of the most important 

 questions awaiting solution in connection with the theory of 

 finite discontinuous groups is the enumeration of the types of 

 groups of given order or of Latin Squares which satisfy additional 

 conditions. It thus comes about that the subject of Latin 

 Squares is important in mathematics, and some new method of 

 dealing with them seemed imperative. 



A fundamental idea was that it might be possible to find some 

 mathematical operation of which a particular Latin Square might 

 be the diagrammatic representative. If, then, a one-to-one 

 correspondence could be established between such mathematical 

 operations and the Latin Squares, the enumeration might con- 

 cei%ably follow. Bearing this notion in mind, consider the 

 differentiation of -i" with regard to x. Noticing that the result 

 is «.«■""' (« an integer), let us inquire whether we can break 

 up the operation of difl'erentiation into n elementary portions, 

 each of which will contribute a unit to the resulting co- 

 efficient n. If we write down -v" as the product of >i letters, viz. 

 xxxx . . ., it is obvious that if we substitute unity in place of 

 a single -v in all possible ways, and add together the results, we 

 shall obtain h.v"-'. We have, therefore, n different elementary 

 operations, each of which consists in substituting unity for -v. 

 We may denote these diagrammatically by 



and from this point of view ^- is a combinatorial symbol, and 



dx 

 denotes by the coefficient n the number of ways of selecting one 

 out of It different things. 



Similarly, the higher differentiations give rise to diagrams of 

 two or more rows, the numbers of which are given by the co- 

 efficients which result from such differentiations. Following 

 up this clue much progress has been made. For a particular 

 problem success depends upon the design, on the one hand, of 

 a function, on the other hand, of an operation such that 

 diagrams make their appearance which have a one-to-one corre- 

 spondence with the entities whose enumeration is sought. For 

 a general investigation, however, it is more scientific to start by 

 designing functions and operations, and to then ascertain the 

 problems of which the solution is furnished. The difficulties 

 connected with the Latin Square and with other more general 

 questions have in this way been completely overcome. 



The second new method in analysis that I desire to bring 

 before the Section had its origin in the theory of partition 

 Diophantus was accustomed to consider algebraical questions in 

 which the symbols of quantity were subject to certain con 

 ditions, such, for instance, that they must denote positive num 

 bers or integer numbers. A usual condition with him was thai 

 the quantities must denote positive integers. All such problems 

 and particularly those last specified, are qualified by the adjec 

 tive Diophantine. The partition of numbers is ahen on al 

 fours with the Diophantine equation 



1 + ,8-f 7-f- 



+ v = ) 



