Maeoh 11, 1904.] 



SCIENCE. 



413 



uity assumption. But nearly the whole of 

 Euclid can be obtained without any con- 

 tinuity assumption whatever, and this great 

 part it is which forms the real domain of 

 elementary geometry. 



Continuity belongs, with limits and in- 

 finitesimals, in the Calculus. 



Professor W. G. Alexejeff, of Dorpat, in 

 'Die Mathematik als Grundlage der Kritik 

 wissenschaf tlich - philosophischer Weltan- 

 schauung' (1903), shows how men of sci- 

 ence have stultified themselves by igno- 

 rantly presupposing continuity. He calls 

 that a higher, standpoint which takes ac- 

 count of the individuality of the elements, 

 and gives as examples of this discrete or 

 discontinuous mathematics the beautiful 

 enumerative geometry, the invariants of 

 Sylvester and Cayley, and in chemistry the 

 atomic-structure theory of Kekule and the 

 periodic system of the chemical elements 

 by Mendelejev, to which two theories, both 

 exclusively discrete in character, we may 

 safely attribute almost entirely the present 

 standpoint of the science. 



Still more must discontinuity play the 

 chief role in biology and sociology, dealing 

 as they do with differing individuals, cells 

 and persons. How desirable, then, that 

 the new freedom should appear even as 

 early as in elementary geometry. 



After mathematicians all knew that 

 number is in origin and basis entirely in- 

 dependent of measurement or measurable 

 magnitude; after in fact the dominant 

 trend of all pure mathematics was its 

 arithmetization, weeding out as irrelevant 

 any fundamental use of measurement or 

 measurable quantity, there originated in 

 Chicago from the urbane Professor Dewey 

 (whom, in parenthesis, I must thank for his 

 amiable courtesy throughout the article in 

 the Educational Review which he devoted 

 to my paper on the 'Teaching of Geom- 

 etry'), the shocking tumble or reversal that 



the origin, basis and essence of number is 

 measurement. 



Many unfortunate teachers and pro- 

 fessors of pedagogy ran after the new 

 darkness, and even books were issued try- 

 ing to teach how to use these dark lines in 

 the spectrum for illuminating purposes. 



There is a ludicrous element in the 

 parody of all this just now in the domain 

 of geometry. 



After mathematicians all know of the 

 wondrous fruit and outcome of the non- 

 Euclidean geometry in removing all the 

 difficulties of pure elementary geometry, 

 there comes another philosopher, a Mr. 

 Perry, who never having by any chance 

 heard of all this, advises the cure of these 

 troubles by the abolition of rational geom- 

 etry. 



Just as there was a Dewey movement so 

 is there a Perry movement, and books on 

 geometry written by persons who never 

 read 'Alice in Wonderland' or its com- 

 panion volume, 'Euclid and his Modern 

 Rivals. ' 



But the spirits of Bolyai and Loba- 

 chevski smile on this well-meaning strenu- 

 osity, and whisper, 'It is something to 

 know what proof is and what it is not ; and 

 where can this be better learned than in a 

 science which has never had to take one 

 footstep backward?' 



George Bruce Halsted. 



Ken YON College. 



THE SOCIETY FOR PLANT M0RPB0L0G7 

 AND PHYSIOLOGY. 



The seventh regular annual meeting of 

 this society was held, in conjunction with 

 the meetings of several other scientific so- 

 cieties, at the University of Pennsylvania, 

 Philadelphia, Pa., December 28-30, 1903. 

 In the absence of the president and vice- 

 president, the most recent past president. 

 Dr. Erwin F. Smith, presided. Though 

 not large in point of numbers the meeting 



