498 ALGEBRA AND ANALYSIS 



it is also of order historic and psychologic, and would draw us away 

 into too many discussions. 



Since the concept of number has been sifted, in it have been found 

 unfathomable depths; thus, it is a question still pending to know, 

 between the two forms, the cardinal number and the ordinal number, 

 under which the idea of number presents itself, which of the two is 

 anterior to the other, that is to say, whether the idea of number 

 properly so called is anterior to that of order, or if it is the inverse. 



It seems that the geometer-logician neglects too much in these 

 questions psychology and the lessons uncivilized races give us; it 

 would seem to result from these studies that the priority is with the 

 cardinal number. 



It may also be there is no general response to the question, the 

 response varying according to races and according to mentalities. 



I have sometimes thought, on this subject, of the distinction be- 

 tween auditives and visuals, auditives favoring the ordinal theory, 

 visuals the cardinal. 



But I will not linger on this ground full of snares; I fear that our 

 modern school of logicians with difficulty comes to agreement with 

 the ethnologists and biologists; these latter in questions of origin 

 are always dominated by the evolution idea, and, for more than one 

 of them, logic is only the resume of ancestral experience. Mathe- 

 maticians are even reproached with postulating in principle that 

 there is a human mind in some way exterior to things, and that it 

 has its logic. We must, however, submit to this, on pain of con- 

 structing nothing. We need this point of departure, and certainly, 

 supposing it to have evolved during the course of prehistoric time, 

 this logic of the human mind was perfectly fixed at the time of the 

 oldest geometric schools, those of Greece; their works appear to 

 have been its first code, as is expressed by the story of Plato writing 

 over the door of his school, " Let no one not a geometer enter 

 here." 



Long before the bizarre word algebra was derived from the Arabic, 

 expressing, it would seem, the operation by which equalities are 

 reduced to a certain canonic form, the Greeks had made algebra 

 without knowing it; relations more intimate could not be imagined 

 than those binding together their algebra and their geometry, or 

 rather, one would be embarrassed to classify, if there were occasion, 

 their geometric algebra, in which they reason not on numbers but on 

 magnitudes. 



Among the Greeks also we find a geometric arithmetic, and one of 

 the most interesting phases of its development is the conflict which, 

 among the Pythagoreans, arose in this subject between number and 

 magnitude, apropos of irrationals. 



Though the Greeks cultivated the abstract study of numbers, called 



