878 



SCIENCE. 



[N. S. Vol. IV. No. 102. 



Yet he asserts that in no American text-book 

 ' has a thoroughly satisfactory treatment been 

 given,' and says: "In my opinion, it is not 

 possible to discuss, in an elementary manner, 

 propositions relating to the magnitude of curved 

 lines until after the introduction of Duhamel's 

 well-known postulate. It may therefore be of 

 psychologic as well as geometric interest to point 

 out that I had lived through the mental state in 

 which my honored friend. Prof. Fiske, now finds 

 himself, and had already attained simpler and 

 clearer light before 1893, when there appears 

 in my paper ' The Old and the New Geometry ' 

 in the Educational Revieiv the following : 



' ' That stale stupidity, ' A straight line is the 

 shortest distance between two points,' is equally 

 unavailable for foundation building. 



' 'As Helmholtz says : ' The foundation of all 

 proof, by Euclid's method, consists in establish- 

 ing the congruence of lines, angles, plane 

 figures, solids, etc. 



" ' To make the congruence evident the geo- 

 metrical figures are supposed to be applied to 

 one another, of course without changing their 

 form and dimensions.' 



' 'But since no part of a curve can be congruent 

 to any piece of a straight line, so, for example, 

 no part of a circle can be equivalent to any sect 

 in accordance with the definition of equivalent 

 magnitudes as those which can be cut into 

 pieces congruent in pairs. Thus the whole of 

 Euclid's Elements fails utterly to prove any re- 

 lation as regards size between a sect and an arc 

 joining the same two points. We cannot even 

 affirm that any ratio exists between a circle and 

 its diameter until after we have made extra- 

 Euclidean and post-Euclidean assumptions at 

 least equivalent to the following : 1. No arc is 

 less than its chord. 2. No minor arc is greater 

 than the sum of the tangents at its extremi- 

 ties." 



May I be allowed to state that in the years 

 that have followed my printing of this double 

 postulate I have only been more confirmed in 

 my opinion that it is more elementary and more 

 elegant than the one for which I deliberately 

 substituted it, and which Prof. Fiske has again 

 given on p. 724 of Science. When Prof. Fiske 

 applies these ideas to the geometry of Beman 

 and Smith, I am very forcibly reminded that 



without the slightest word of acknowledgment 

 these professors ' took ' a whole block of prob- 

 lems and a long note from Halsted's Elements 

 of Geometry. 



The section Partition of a Perigon, Elements 

 p. 151, is so peculiarly my own that it was as 

 startling as a ghost to meet it unexpectedly in 

 Beman and Smith p. 179. Then follows my 

 Problem I: To bisect a perigon, with my 

 corollary; then follows as their Problem 2 my 

 Problem II: To trisect a perigon, with my cor- 

 ollary. Then my Problem III: To cut a perigon 

 into five equal parts, and my corollary. Then 

 my Problem IV: To cut a perigon into fifteen 

 equal parts, with my corollary. Then before 

 they go on to my Problem V. and Problem VI. 

 and Problem VII. and Problem VIII. , they in- 

 sert my long note. Elements p. 155; but here 

 they out-Herod Herod, or rather out-Perigon 

 Halsted, for where I say that Gauss, in 1796, 

 found that a regular polygon of 17 sides was in- 

 scriptible, they make it say ' In 1796 Gauss 

 found that a perigon could be divided into 17, 

 etc' But, of course, the whole of the matter here 

 involved is so well known that I accept the im- 

 plied compliment, broad as it is, and dream that 

 even my rather cranky problem to bisect a peri- 

 gon was not really as peculiar as I had thought 

 it. George Bruce Halsted. 



Austin", Texas. 



the date of publication again. 

 Dr. J. A. Allen has not offered any serious 

 objections, to my view of this matter in his re- 

 marks in Science of December 4th, as it seems 

 to me, but he has in one instance misunderstood 

 me, as I now explain. He quotes as follows 

 my remark, that ' ' although some reports issued 

 by our government may bear dates much prior 

 to the dates of issue, it does not follow that the 

 date of printing bears any such relation to the 

 date of issue. ' ' What I meant by this may be 

 illustrated by a concrete case. The ' Report of 

 the Commissioner of Education,' which I last 

 received, bears on its back and title page the 

 dates 1893-4. As it was not printed until 

 1896, I find the date 1896 at the foot of the 

 title page. This will explain my meaning, 

 which would seem to have been misunderstood 

 by Dr. Allen. It also explains my remark 



