June 20, 1902.] 



SCIENCE. 



987 



passes an infinity of planes also Eiemannian, 

 and in each of these this straight has a deter- 

 mined and distinct center; but the straight 

 is independent of the planes, and is defined 

 by the postulates. 



Now in the sphere the great circle and the 

 one pseudo-jAane which contains and fixes it, 

 namely the sphere, are inseparable, since any 

 portion, however minute, of either determines 

 all the other as well as its center and radius. 



In the single elliptic geometry the elliptic 

 straight line does not divide the elliptic 

 plane into two separated regions. We can 

 pass from any one point of the plane to any 

 other point without crossing a given straight 

 in it. Starting from the point or intersec- 

 tion of two straights and passing along one 

 of them a certain finite length, we come to 

 the intersection point again without having 

 crossed the other straight. Hence we can 

 pass from what seems one side of the straight 

 line to what seems the other without crossing 

 it, that is, it is uni-lateral or double. 



This single elliptic geometry is never men- 

 tioned in Barbarin's book; just as the Eie- 

 mannian is never mentioned in Manning's 

 book. First take your choice, then buy your 

 non-Euclidean geometry. 



On p. 36, Barbarin gives to Gauss the honor 

 which belongs to Wallis of being the first to 

 remark that the existence of unequal similar 

 figures is equivalent, in continuous space, to 

 the parallel postulate. 



In Chapter VII., 'Les Contradicteurs de la 

 geometrie non-euclidienne,' Professor Bar- 

 barin makes with unanswerable vigor the 

 argument which I gave in my 'Report on 

 Progress in Non-Euclidean Geometry' (Sci- 

 E?fCE, N. S., Vol. X., pp. 545-557). 



There I quoted Whitehead who was the first 

 to publish (March 10, 1898) "the extension 

 of Bolyai's theorem by investigating the prop- 

 erties of the general class of surfaces in any 

 non-Euclidean space, elliptic or hyperbolic, 

 which are such that their geodesic geometry 

 is that of straight lines in a Euclidean plane. 



"Such surfaces are proved to be real in 

 elliptic as well as in hyperbolic space, and 

 their general equations are found for the 

 case when they are surfaces of revolution. 



"In hyperbolic space, Bolyai's limit-sur- 

 faces are shown to be a particular case of 

 such surfaces of revolution. 



"The same principles would enable the 

 problem to be solved of the discovery in any 

 kind of space of surfaces with their 'geodesic' 

 geometry identical with that of [ilanes in 

 any other kind of space." 



Now not only the strikingly important 

 problem solved by Whitehead, but also the 

 analogous problem indicated had both been 

 solved by Barbarin and presented three 

 months before to the Academie Eoyale de 

 Belgique; but these investigations were only 

 published after the appearance of my Ee- 

 port (October 20, 1899). They, as Barbarin 

 says, p. 63, 'bring out in a striking manner 

 the absolute independence of the three sys- 

 tems of geometry, which are able each to get 

 everything from its own resources without 

 need of borrowing anything from the others.' 

 In each of the three spaces, Euclidean, 

 Bolyaian, Eiemannian, there exist surfaces 

 whose geodesies have the metric properties 

 of the straights of the two other spaces. 



But the book in which these beautiful re- 

 searches are published: 'Etudes de geometrie 

 analytique non euclidienne par P. Barbarin, 

 Bruxelles,' 1900, Hayez, pp. 168, has other 

 titles to universal recognition. 



Notwithstanding the ever-present example 

 of Euclid, who never uses a construction or 

 a figure which he has not shown to follow 

 deductively from his two postulated figures, 

 the straight and the circle, an insidious error 

 crept into geometry, taught by Beman and 

 Smith, who should know better, in the follow- 

 ing words: (See their 'Geometry,' 1899, p. 70, 

 § 112) "Note on Assumed Constructions. — 

 It has been assumed that all constructions 

 were made as required for the theorems. 



"Thus an equilateral triangle has been fre- 

 quently mentioned, although the method of 

 constructing one has not yet been indicated, 

 a regular heptagon has been mentioned, and 

 reference might be made to certain results 

 following from the trisection of an angle, 

 although the solutions of the problems, to 

 construct a regular heptagon, and to trisect 

 any angle, are impossible by elementary geom- 



