556 



SCIENCE 



[N. S. Vol. XLV. No. 1170 



same way. On the contrary, an operation 

 group may have elements enjoying special 

 privileges and hence it has more extensive 

 contact in the actual world of thought. 



A little study of the stated problem re- 

 vealed the interesting fact that when the 

 number of tables is any power of 2 the sub- 

 stitutions of a well-known type of substitu- 

 tion groups and its group of isomorphisms 

 exhibit directly how the players can be ar- 

 ranged so that each one will play once and 

 only once with, and twice and only twice 

 against, each of the others in a certain 

 series of games. To make myself perfectly 

 clear, I may say that if 8 tables, or 32 play- 

 ers, are involved, one can write directly by 

 means of a certain regular substitution 

 group of order 32 a set of possible arrange- 

 ments so that in 31 successive games each 

 one of these 32 players would play once and 

 only once with each of the others and twice 

 and only twice against each of them. This 

 was, however, not the first solution of the 

 general problem in question. In fact, about 

 twenty years ago Professor E. H. Moore 

 published a different solution of it in Vol- 

 ume 18 of the American Journal of Mathe- 

 matics under the title "Tactical Memor- 

 anda. ' ' 



I have referred to this matter here 

 mainly for the purpose of emphasizing the 

 fact that intellectual penetration is often 

 attended bj^-the most unexpected by-prod- 

 ucts, but I should also be pleased to have 

 people know that certain kinds of recrea- 

 tion can easily be enriched bj^ making use 

 of results which the mathematician de- 

 veloped for a totally different purpose. 

 Science should and does enrich both work 

 and play. More than a thousand years ago 

 the Hindu astronomer Brahmagupta said: 



As the sun obscures the stars, so does the pro- 

 ficient eclipse the glory of other astronomers in an 

 assembly of people by the recital of algebraic 

 problems, and stUI more by their solution.6 



« H. T. Colebrooke, ' ' Algebra with Arithmetic 



The playful question. Where do the 

 finger nails find so much dark dirt to put 

 under them 1 may serve to arouse a thought- 

 ful attitude on the part of the boy who has 

 been taught to keep his hands clean. In 

 fact, our play and recreation are perhaps 

 as fundamentally affected by questions of 

 science as our serious work and the vic- 

 trolas and moving pictures should have a 

 marked influence on the popular attitude 

 towards science in view of the fact that they 

 reach so many people. If it is true that the 

 greatest service which science is rendering 

 the human race is the reduction of super- 

 stition, it is clear that the efficiency of sci- 

 ence depends largely upon its popularity. 



The hypothesis that space and the opera- 

 tions of nature are discontinuous clearly 

 excludes the hypothesis that they are con- 

 tinuous, but it is interesting to note that 

 the mathematics relating to the discontinu- 

 ous does not exclude that relating to the 

 continuous. On the contrary, there are the 

 most helpful interrelations between these 

 two types of mathematics. Such a subject 

 as number theory, relating decidedly to dis- 

 crete qiiantities, has been greatly extended 

 by analytic methods relating to continuous 

 quantities, and, on the other hand, processes 

 relating to the study of continuous func- 

 tions are largely based upon those relating 

 to the discontinuous. 



This may perhaps tend to show that even 

 if our hypotheses in regard to the continu- 

 itj'' of space and the operations of nature 

 have to be largely modified, as seems now 

 probable, the mathematical methods of at- 

 tack may require less modification than 

 might at first appear to be necessary. The 

 language which mathematics has provided 

 for science includes not only concepts rela- 

 ting to the continuous and the discontinu- 

 ous, but fortunately it also shows relations 

 between these concepts and these relations 



and Mensuration from the Sanscrit," by Brahma- 

 gupta and Bhaseara, 1817, p. 379. 



