U 6 SCIENCE PROGRESS 



senting fingers. These original signs display the relation of similarity, the 

 one-one correspondence between members of groups, that underlies all numbers 

 and numbering. The appearance of signs, including written symbols and words, 

 that contained no explicit reference to one-one relations, marked the establishment 

 of numbers as mental concepts when, as an abstraction, five, six, etc., could be 

 thought of as numbers only, without reference to sheep or other objects. All 

 particular numbers were finally conceived under the general concept of number. 

 This rough outline will be recognised as a logicised version of the development of 

 the notion of number ; but, though schematic, it may possibly be accepted as a 

 reasonable account. 



Number still remains a mystery, as, in the last resort, all things remain 

 mysterious. Mr. Russell speaks of any number, such as the number 2, as 

 " a metaphysical entity about which we can never feel sure that we have tracked 

 it down." The preceding enquiry seems to lead to a definition of the number of 

 a class as a standard class to which the numbered class is similar. A cluster 

 of cherries is numbered six if the number six is the standard class to which the 

 cluster is similar. Peano's definition of number as the common property of 

 similar classes seems to adopt a cognate point of view. The common property 

 of collections of six cherries, six apples, etc., is their similarity, their one-one 

 relation to the group of the first six natural numbers. It seems to be in con- 

 formity with our mental habit, and with the order of development of this habit, to 

 regard six as a single entity to which all classes numbered six are similar, seven 

 as an entity to which all classes of seven are similar, and so on. Mr. Russell's 

 definition of the number of a class as "the class of all those classes that are 

 similar to it," appears to obscure the standard of reference contained in number. 

 It seems to reduce any number to the simple status of one of the collections 

 similar to the numbered collection. Thus, six cherries, six elephants, and the 

 number six would be co-equal members of the class of sixes. If this be so it is 

 difficult to understand why we can speak of six elephants and not of elephant 

 sixes or of cherry elephants. The adjectival function of the number six that 

 permits its predication of the cherries, of the elephants, and of all other collections 

 of six seems to entitle it to the more prominent position accorded to it when it is 

 defined as a standard class. 



The general definition of number as "anything which is the number of some 

 class " is independent of the definition preferred for particular numbers. 



AGRICULTURAL ECONOMICS IN ENGLAND AND DEN- 

 MARK, by Walter Stiles, M.A. : on 



(i) Agriculture in Oxfordshire, a Survey made on Behalf of the Institute 

 for Research in Agricultural Economics, by John Orr, with a 

 Chapter on Soils by C. G. T. Morison. [Pp. xii + 239.] (Oxford : 

 at the Clarendon Press, 1916. Price 8j-. 6d. net.) 



(2) Co-operation in Danish Agriculture, by Harold Faber, an English 

 Adaptation of Andelsbevcegelsen i Danmark, by H. Hertel, with a 

 Foreword by E. J. Russell, D.Sc, F.R.S. [Pp. xxii + 176.] 

 (London : Longmans, Green & Co., 19 18. Price Ss. 6d. net.) 



Agriculture in Oxfordshire is the first of a series of monographs on Agriculture 

 in England which the Institute for Research in Agricultural Economics in Oxford 

 proposes to issue. This intention to make a formal economic survey of English 



