DEVELOPMENT OF MATHEMATICAL THOUGHT. 679 



where the factors are the same, as in Newton's bi- 

 nomial theorem, to combinations with permutation ; and 

 consequently the doctrine of chances and of arrange- 

 ments in triangular, pyramidal, or other figures is closely 

 connected with the doctrine of series and algebraical 

 expressions. In this country the interest in the subject 

 has been stimulated and kept alive by isolated problems 

 and puzzles in older popular periodicals, such as the 

 ' Gentleman's Magazine ' and the ' Ladies' Diary ' ; in Ger- 

 many as we noticed before a school of mathematicians 

 arose who attempted a systematic treatment of the whole 

 subject, which, owing to its barrenness in practical re- 

 sults, brought this line of research somewhat into dis- 

 repute. What was wanted was a problem of real 

 scientific interest and a method of abbreviation and 

 condensation. Both were supplied from unexpected 1 



1 The theory of arrangement or 

 of order, also called the " Ars Coin- 

 binatoria," has exerted a great 

 fascination on some master minds, 

 as it has also given endless oppor- 

 tunities for the practical ingenuity 

 of smaller talents ; among the 

 former we must count in the first 

 place Leibniz, and in recent times 

 J. J. Sylvester, who conceived the 

 "sole proper business of mathe- 

 matics to be the development of 

 the three germinal ideas of which 

 continuity is one, and order and 

 number the other two" ('Philo- 

 sophical Transactions,' vol. clix. p. 

 613). This idea has been dwelt on 

 by Major MacMahon in his address 

 (Brit. Assoc., 1901, p. 526), who says: 

 " The combinatorial analysis may 

 be described as occupying an ex- 

 tensive region between the algebras 

 of discontinuous and continuous 

 quantity. It is to a certain extent 

 a science of enumeration, of mea- 



surement by means of integers as 

 opposed to measurement of quan- 

 tities which vary by infinitesimal 

 increments. It is also concerned 

 with arrangements in which differ- 

 ences of quality and relative position 

 in one, two, or three dimensions are 

 factors. Its chief problem is the 

 formation of connecting roads be- 

 tween the sciences of discontinu- 

 ous and continuous quantity. To 

 enable, on the one hand, the 

 treatment of quantities which vary 

 per saltvm, either in magnitude 

 or position, by the methods of the 

 science of continuously varying 

 quantity and position, and, on the 

 other hand, to reduce problems of 

 continuity to the resources avail- 

 able for the management of dis- 

 continuity. These two roads of 

 research should be regarded as pene- 

 trating deeply into the domains 

 which they connect." 



