858 



SCIENCE. 



[N. S. Vol. XX. No. 521. 



I. 



It would appear natural to commence 

 by speaking of the concept itself of whole 

 number; but this subject is not alone of 

 logical order, 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 car- 

 dinal 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 neg- 

 lects too much in these questions psychol- 

 ogy and the lessons uncivilized races give 

 us; it would seem to result from these 

 studies that the priority is with the car- 

 dinal number. 



It may also be there is no general re- 

 sponse to the question, the response vary- 

 ing according to races and according to 

 mentalities. 



I have sometimes thought, on this sub- 

 ject, of the distinction between 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 agree- 

 ment Avith the ethnologists and biologists; 

 these latter in questions of origin are al- 

 ways 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 pos- 

 tulating 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 constructing 

 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 ap- 

 pear to have been its first code, as is ex- 

 pressed 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 with- 

 out 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 geo- 

 metric arithmetic, and one of the most in- 

 teresting 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 ab- 

 stract study of numbers, called by them 

 arithmetic, their speculative spirit showed 

 little taste for practical calculation, which 

 they called logistic. 



In remote antiquity, the Egyptians and 

 the Chaldeans, and later the Hindus and 

 the Arabs, carried far the science of calcu- 

 lation. 



They were led on by practical needs; 

 logistic preceded arithmetic, as land-sur- 

 veying and geodesy opened the way to 

 geometry ; in the same way, trigonometry 

 developed under the influence of the in- 

 creasing needs of astronomy. 



The history of science at its beginnings 

 shows a close relation between pure and 

 applied mathematics; this we will meet 

 again constantly in the course of this study. 



