954 



SGIENGE. 



[N. S. Vol. IV. No. 104. 



Ms earlier papers being included neither in 

 Agassiz and Strickland's Bibliographia zoologise, 

 wMch ends with 1854 ; nor in Taschenberg's, 

 which begins with 1861 ; nor in the catalogue 

 of the Eoyal Society. A. S. Packard. 



Number and its Algebra; Syllabus of Lectures on the 

 Theory of Numb er and its A Igebra. By Arthur 



Lefevre. Boston, Heath. 1896. pp. 230. 



In June, 1891 was published the first piece of 

 Sb treatise entitled ' Number, Discrete and Con- 

 tinuous,' in which were set forth some doc- 

 trines which seemed to the writer, the present 

 reviewer, as new as they were fundamental. 



It was there maintained that counting is es- 

 sentially prior to measuring, but also that the 

 primary number concept is essentially prior to 

 counting and necessary to explain the meaning, 

 cause and aim of counting. 



It was there maintained that integral number 

 had not a metric origin, nor was metric in its 

 original purpose ; that integral number did not 

 involve the idea of ratio, that in fact it was 

 enormously simpler than that very delicate con- 

 cept, ratio. 



Number is primarily a quality of an artificial 

 individual. 



The stress laid upon it, the importance at- 

 tached to this quality, comes first from the ad- 

 vantage of being able to identify one of these 

 artificial individuals. By artificial is meant ' of 

 human make.' 



The characteristic of these artificial individ- 

 uals is that each, though made an individual, is 

 conceived as consisting of other individuals. 



This explanation was set forth again con- 

 cisely in an article entitled ' The Essence of 

 Number,' in Science, Vol. III., pp. 470, 471. 



The primitive function of number is to serve 

 the purposes of identification. But again, 

 counting, which consists in associating with 

 each primitive individual in an artificial indi- 

 vidual a distinct primitive individual in a fami- 

 liar artificial individual, is thus itself essentially 

 the identification, by a one-to-one correspond- 

 ence, of an unfamiliar with a familiar thing. 

 Thus primitive counting decides which of the 

 familiar groups of fingers is to have its numeric 

 quality attached to the unfamiliar group 

 counted. 



This primitive use of number in defining by 

 identification is illustrated by an ordinary pack 

 of playing cards, where the identification of 

 King, Queen and Knave is not more clearly 

 qualitative and opposed to every mode of mea- 

 surement than is the identification of ace, deuce 

 and tray ; and, indeed, that the King outval- 

 ues the Knave has more to do with measure- 

 ment than the fact that the ace outvalues the 

 tray. 



Counting implies, first, a known series of 

 groups, mental wholes each made up of distinct 

 wholes ; secondly, an unfamiliar mental whole ; 

 thirdly, the identification of the unfamiliar 

 group by its one-to-one correspondence with a 

 familiar group of the known series. 



Absolutely no idea of a unit, of measure- 

 ment, of amount, of value, or even of equality, is 

 necessarily involved or, indeed, ordinarily used. 

 One counts when one wishes to find out 

 whether the same group of horses has been 

 driven back at night that were taken out in the 

 morning ; where counting is a process of identi- 

 fication which it would seem intentionally 

 humorous or comical to try to connect funda- 

 mentally with any idea of a unit of reference or 

 of some value to be ascertained, or of the setting 

 oflF of a horse as a sample unit of value and 

 then equating the total value to the number 

 of such units. Such an argumentum in circulo 

 may perhaps be funny, but it is neither fact nor 

 mathematics. Mathematics afterwards defines 

 numerical equality by means of one-to-one cor- 

 respondence, which is absolutely distinct and 

 away apart from the . idea of ratio. We may 

 say with perfect certainty that there is no im- 

 plicit presence of the ratio idea in primitive 

 number. 



From the contemplation of the primitive in- 

 dividual in relation to the artificial individual 

 spring the related ideas ' one ' and ' many. ' An 

 individual thought of in contrast to ' a many ' 

 as not-many gives the idea of ' a one.' A many 

 composed of ' a one ' and another * one ' is char- 

 acterized as 'two.' A many composed of 'a 

 one ' and the special many * a two ' is charac- 

 terized as * three. ' 



And so on ; at first absolutely without count- 

 ing, in fact before the invention of that patent 

 process of identification now called counting. 



