724 



SCIENCE. 



[N. S. Vol. IV. No. 98. 



DISCUSSION AND COBBESPONDENCE. 

 THE LENGTH OF A CURVED LINE. 



I SHOULD be very sorry to have anyone in- 

 terpret my remarks in a recent number of 

 Science (see page 533) as imputing ignorance 

 of fundamental principles to so distinguished a 

 geometer as Prof. Halsted. In saying that 

 Prof. Halsted ' appears to believe ' that he has 

 given a logically complete discussion, my mean- 

 ing was that he so appears to the unassisted 

 reader of his 'Elements of Geometry.' My 

 criticism was directed at the book rather than 

 at the man. Further, as he says in his reply 

 on page 656 of Science, the criticism is not ap- 

 plicable to his more recent work, ' Elementary 

 Synthetic Geometry.' 



In my opinion, it is not possible to discuss, in 

 an elementary manner, propositions relating to 

 the magnitude of curved lines until after the in- 

 troduction of the following postulate : The 

 magnitude of a curved line is the limit toward 

 which a broken line made up of consecutive chords 

 of that curved line approaches, when the number of 

 chords is increased in such a manner that the 

 chords are all diminished without limit. After the 

 introduction of this postulate it is possible to 

 compare the magnitude of a curved line with 

 that of a straight line. 



To turn again to Prof. Halsted' s 'Elements 

 of Geometry,' not only was it an error of logic 

 to attempt to demonstrate without this postu- 

 late, or its equivalent, that a straight line is the 

 shortest line joining two fixed points ; but it 

 was an error of the same sort to introduce, on 

 pages 162-165 of that work, propositions re- 

 lating to isoperimetric figures, which from their 

 very nature depend on a comparison of non- 

 congruent lines. 



It seems worth while to insist upon the points 

 made in this note and in my preceding note, 

 because they relate to subjects treated in almost 

 every American text-book of geometry; but in 

 none, so far at least as the writer is aware, has 

 a thoroughly satisfactory treatment been given. 



In the very recent text-book of Beman and 

 Smith, of which the writer has expressed a 

 high opinion (See Science, this volume, page 

 203), the following appears on page 187: 



"Postulate op Limits. The circle and its 



circumference are the respective limits which 

 the inscribed and circumscribed regular poly- 

 gons and their perimeters approach if the num- 

 ber of their sides increases indefinitely. 



' ' This statement is so evident that a proof is 

 not considered necessary. Like valid proofs of 

 many fundamental principles, it is too difficult 

 for an elementary text-book." 



The statement consists of two parts, one re- 

 lating to superficial magnitude, the other to 

 linear magnitude. The former is capable of 

 simple proof. The circle is greater than any in- 

 scribed polygon, and any circumscribed polygon 

 is greater than the circle; by the axiom, the whole 

 is greater than any of its parts. Proofs based upon 

 these considerations are older than the text of 

 Euclid. The second part of the statement is 

 a ' postulate ' in a strict sense. It cannot be 

 proved at all except from equivalent assump- 

 tions. Thomas S. Fiske. 



October 31, 1896. 



ON criticisms of organic selection. 



A long absence in Europe has prevented my 

 seeing several criticisms of my papers in this 

 Journal, until very recently ; and although 

 the issues may now be forgotten by the critics 

 as well as by the readers of Science, I venture 

 to write a few lines, if only to express my 

 thanks for the kindly words which have aided 

 me to see where the articles were not clear. 



First, I may say that I have published, in the 

 American Naturalist (June and July, 1896), a 

 paper of some length under the title 'A New 

 Factor in Evolution,' gathering the positions of 

 the Science articles into a single sketch, thus 

 carrying out, to a degree, the suggestion made 

 by Prof. Wesley Mills in Science, May 22 (a 

 suggestion which, however, I did not see until 

 my return in September). Condensed summa- 

 ries of the two main positions involved in the 

 doctrine of Organic Selection (which I ventured 

 to call a 'new factor') were quoted in this 

 Journal for July 31, p. 139, and I need not 

 stop to requote them. 



I am glad to know, both from Prof. Mills' ar- 

 ticle in Science, May 22d, and also from a 

 personal letter from him, that he accepts the 

 class of facts which I have emphasized, and ad- 

 mits their importance (having himself before 



