986 



SCIENCE. 



[N. S. Vol. XV. No. 390. 



line,' Barbariii puts as an elementary proposi- 

 tion! 



Manning also, p. 2, assumes it, thus in- 

 validating and making ephemeral his pretty 

 little 'Non-Euclidean Geometry' (Ginn & Co., 

 1901). Barbarin then proceeds to classify 

 geometries by Saccheri's three hypotheses, the 

 hypothesis of obtuse angle, the hypothesis of 

 right angle, the hypothesis of acute angle, 

 or that the angle sum of a rectilinear tri- 

 angle is greater than, equal to, less than two 

 right angles. 



But the remarkable discoveries of Dehn 

 have now shown that this classification is 

 invalid. 



Barbarin says, p. 16, 'Sacclieri proves that 

 the hypothesis of the obtuse angle is incom- 

 patible with postulate 6' of Euclid. 



Dehn dissipates this supposed incompati- 

 bility by actually exhibiting a new geometry 

 in which they amicably blend, which he calls 

 the non-Legendrean geometry. 



In the same way, the hypothesis of right 

 angle amalgamates with the contradiction of 

 Euclid's parallel-postulate in a geometry 

 which Dehn calls semi-Euclidean. As Dehn 

 states this result: There are non-Archimiedian 

 geometries in which the parallel-axiom is not 

 valid and yet the angle-sum in every triangle 

 is equal to two right angles. Thus the 

 theorem (Legendre, 12th Ed., I., 23; Bar- 

 barin, p. 25): 'If the sum of the angles of 

 every triangle is equal to two right angles the 

 fifth postulate is true,' is seen to break down. 



Manning's ' Non-Euclidean Geometry,' 

 though it says (p. 93), 'The elliptic geometry 

 was left to be discovered by Riemann,' gives 

 only the single elliptic. 



It never even mentions the double elliptic, 

 or spherical or Riemannian geometry, which 

 Killing maintains was the only form which 

 ever came before Riemann's mind. If so, 

 then Barbarin's book is like Riemann's mind. 

 The Riemannian, as distinguished from the 

 single elliptic, is the only form which ap- 

 pears in it. Killing was the first who (1879, 

 Crelle's Journal^ Bd. 83) made clear the dif- 

 ference between the Riemannian and the 

 single elliptic space (or as he calls it, the 

 polar form of the Riemannian). 



Klein championed the single elliptic. 

 Manning knows no other. 



Professor Simon Newcomb, like Manning, 

 deals only with the single elliptic in his 

 treatise: 'Elementary theorems, relating to 

 the geometry of a space of three dimensions 

 and of uniform positive curvature in the 

 fourth dimension.' 



The last four words F. S. Woods replaces 

 by seven dots in his article 'Space of constant 

 curvature' {Annals of Math., Vol. 3, p. 72), 

 though blaming Professor E. S. Crawley for 

 the error they contain. 



Newcomb's also was the unfortunate con- 

 ceit which dubbed this 'A Eairy-tale of Geom- 

 etry,' a point of view from which he is still 

 suffering in his latest little unburdening in 

 Harper's Magazine. 



Just so Lobachevski had the misfortune 

 to call his creation 'Imaginary Geometry.' 



Contrast John Bolyai's 'The Science Abso- 

 lute of Space.' 



In single elliptic space every complete 

 straight line is of finite constant length n-fc. 



Every pair of straight lines intersect and 

 return again to their point of intersection, 

 but have no other point in common. 



In the so-called spherical space, that is the 

 Riemannian space, two straight lines always 

 meet in two points (opposites, or antipodal 

 points) which are ivk from each other. 



The single elliptic makes the plane a uni- 

 lateral or double surface, so that two antip- 

 odal points would correspond to one point, 

 but to opposite sides of this one-sided plane 

 with reference to surrounding three-dimen- 

 sional elliptic space. 



The geometry for two-dimensional Rie- 

 mannian space coincides completely with pure 

 spherics, that is with spherics established 

 from postulates which make no reference to 

 anything off of the sphere, inside or outside 

 the sphere. ITence the great desirability of 

 a treatise on pure spherics. It would at the 

 same time be true and available for Euclidean 

 and for Riemannian geometry. 



Yet its relations to three-dimensional Eu- 

 clidean and three-dimensional Riemannian 

 space would differ radically. 



Through every Riemannian straight line 



