December 25, 1896.] 



SCIENCE. 



955 



For a considerable period of its early life 

 every child uses a number system consisting of 

 only three terms, one, two, many, and no count- 

 ing. As datum may be taken a psychical con- 

 tinuum, and distinctness may be found the out- 

 come of a process of differentiation ; but what 

 may be spoken of as the physically originated 

 primitive individuals, however complete in 

 their distinctness, have no numeric suggestion 

 or quality. 



The intuitive but creative apperception and 

 synthesis of a manifold must precede its con- 

 scious analysis, which alone gives number. 



It is only to conceptual unities that the nu- 

 meric quality pertains. Such conceptual unities 

 are of human make, and in a sense are not in 

 nature, while, on the other hand, though the 

 world we consciously perceive is out and out a 

 mental phenomenon, yet the primitive individ- 

 uals, distinct things, while forming part of the 

 artificial unities, exist in another way, in that 

 they are subsisting somehow in nature as well 

 as in conscious perception. 



With the preceding hints as to the reviewer's 

 position in reference to fundamental matters in 

 regard to which some strange blunders have 

 been made of late by eminent philosophers and 

 teachers, not mathematicians, it will be easy to 

 understand why Lefevre's ' Number and its 

 Algebra,' seems to us of exceptional importance 

 just now to American teachers in general and 

 teachers of pedagogy in particular. It is exceed- 

 ingly timely, philosophic, bold, yet withal sound. 



That the book was written down under ex- 

 ceptional difficulties makes only more note- 

 worthy its general and sustained excellence. 



That American teacher who does not read it 

 is certainly doing an injustice to himself. In 

 genuine compliment to the book, some points 

 may be mentioned which the reviewer would 

 have wished otherwise. 



The fundamental idea of measurement comes 

 perilously near to being misconceived. Section 

 12 says: ' ' To the man whose concept of number 

 is only what has been defined as primary num- 

 ber, measurement is hardly to be distinguished 

 from counting. For measurement of discrete 

 magnitude is counting ; and to the intelligence 

 supposed there is no real measurement of con- 

 tinuous magnitude." Now, on the contrary, 



the number or numeric picture of a group is a 

 selective photograph of the group, which takes 

 or represents only one quality of the group, but 

 takes that all at once. 



This picture process only applies primarily to 

 those particular artificial wholes which may be 

 called discrete aggregates. But the overwhelm- 

 ing importance of the number-picture, prima- 

 rily as a means of identification, led, after cen- 

 turies of its use, to a human invention as clearly 

 a device of man for himself as is the telephone. 



This was a device for making a primitive in- 

 dividual thinkable as a recognizable and re- 

 coverable artificial individual of the kind having 

 numeric quality. 



This recondite device is measurement. Meas- 

 urement is an artifice for making a primitive 

 individual conceivable as an artificial individual 

 of the group kind, and so having a number pic- 

 ture. The height of a horse, by use of the 

 unit ' a hand, ' is thinkable as a discrete aggre- 

 gate, and so has a number-picture identifiable 

 by comparison with the standard set of pictures, 

 that is by counting, as say 16. 



And directly contrary to the position of Prof. 

 Lefevre, the measurement of continuous magni- 

 tude surely came at just the very stage of in- 

 telligence to which he denies it, for the fraction 

 had this very origin ; it originated from the in- 

 vention, the device, measurement. 



Number long preceded any measurement, but 

 measurement long preceded any idea of number 

 as continuous. 



In fact, measurement suggested not only frac- 

 tions, but later the finer, more geometric idea, 

 ratio, as is clearly presented in §80. 



In Euclid's wonderful Fifth Book a ratio is 

 never a number, 



Newton, with the purpose of taking in the 

 so-called surds or irrationals of arithmetic and 

 algebra, assumed a ratio to be a number. Any 

 continuity in his number system comes then 

 from the continuity in the magnitude whose 

 ratio to a chosen unit for that magnitude is 

 taken. He never gave any arithmetical or 

 algebraic proof of the continuity of any number 

 system. 



Certain passages in Mr. Lefevre's book might 

 easily suggest that somewhere in it, never cited, 

 he has himself treated this basal problem of 



