712 



SCIENCE. 



[N. S. Vol. XIV. No. 358. 



From the hypothesis that through a given 

 point we can draw an infinity of parallels 

 to a given straight it follows, if we exclude 

 the Archimedes axiom, not that the sum of 

 the angles of a triangle is less than two 

 right angles, but on the contrary that this 

 sum may be (a) greater than two right 

 angles, (6) equal to two right angles. 



The first case (a) is proven by the exist- 

 ence of the non-Legandrean geometry. 



To demonstrate the second case (6), 

 D&hn constructs a geometry wherein the 

 parallel-axiom does not hold good, and 

 wherein nevertheless are verified all the 

 theorems of Euclidean geometry ; the sum 

 of the angles of a triangle is equal to two 

 right angles, similar triangles exist, the 

 extremities of equal perpendiculars to a 

 straight are all situated on the same 

 straight, etc. 



As Dehn states this result : There are 

 non-Archi median geometries, in which the 

 parallel-axiom is not valid and yet the 

 angle-sura in every triangle is equal to two 

 right angles. 



Such a geometry he calls '■ semi Euclidean. ' 

 Therefore, it follows that none of the the- 

 orems, the angle-sum in the triangle is two 

 right angles, the equidistantial is a straight, 

 etc., can be considered as equivalent to the 

 parallel-postulate, and that Euclid in set- 

 ting up the parallel-postulate hit just the 

 right assumption. 



This is a marvelous triumph for Euclid. 

 Finally Dehn arrives at this surprising 

 theorem : 



From the hypothesis that there are no 

 parallels, it follows that the sum of the 

 angles of a triangle is greater than two 

 right angles. 



Thus the two non-Euclidean hypotheses 

 about parallels act in a manner absolutely 

 difi'erent with regard to the Archimedes 

 Axiom. 



The different results obtained may now 

 be tabulated thus : 



Riemann, Helmholtz and Sophus Lie 

 founded geometry on an analytical basis in 

 contradistinction to Euclid's pure synthetic 

 method. 



They elected to conceive of space as a 

 manifold of numbers. In the Columbus 

 report is an account of the Helmholtz-Lie 

 investigation of the essential characteristics 

 of space by a consideration of the move- 

 ments possible therein. 



This is notably simplified if we sup- 

 pose given a priori the graphic concepts of 

 straight and plane, and admit that move- 

 ment transforms a straight or a plane into 

 a straight or respectively a plane. Killing 

 determines analytically the three types of 

 projective groups, but the same results are 

 reached in a way geometric and purely ele- 

 mentary by Roberto Bonola in a beautiful 

 little article entitled, ' Determinazione, per 

 via geometrica, dei tre tipi di spazio : Iper- 

 bolico, Ellittico, Parabolico (Rendiconti del 

 Circolo Matematico di Palermo, Tomo XV., 

 pp. 56-65, April, 1901). 



In 1833 was published in London the 

 fourth edition of an extraordinary book 

 (3d Ed., 1830) by T. Perronet Thompson 

 of Queen's College, Cambridge, with the 

 following title : 



' Geometry without Axioms.' 

 " Being an attempt to get rid of Axioms 

 and Postulates ; and particularly to estab- 

 lish the theory of parallel lines without re- 

 course to any principle not grounded on 

 previous demonstration. 



