358 



SCIENCE. 



[N. S. Vol. XX. No. 507. 



properly so called does not exist. Here 

 would be an interesting question to examine 

 closely. Is it possible to construct a non- 

 Archimedean geometry where this parallel 

 properly so called exists and to which are 

 applicable the resitlts of Hilbert? 



An analogous question is treated in an- 

 other article by Hilbert, 'Ueber die Gleich- 

 heit der Basiswinkel im gleichschenkligen 

 Dreieck. ' 



In the ordinary plane geometry, the plane 

 is symmetric, which expresses itself by the 

 equality of the angles at the base of the 

 isosceles triangle. 



One should make this symmetry of the 

 ■plane figure in the list of metric axioms. 

 In all the geometries more or less strange of 

 which we have hitherto spoken, in those 

 at least where one admits the metric axioms, 

 in the metric non-Archimedean geometry, 

 in the new geometries of Dehn, in those 

 which have made the subject of the memoir 

 'Ueber eine neue Begriindung * * * ' 

 this symmetry of the plane is always as- 

 sumed. Is it a consequence of the other 

 metric axioms? Yes, as Hilbert shows, if 

 one admits the axiom of Archimedes. No, 

 in the contrary case. 



There are non-Archimedean geometries 

 where all the metric axioms are true, with 

 the exception of this of the symmetry of 

 the plane. Here is an example: 



The non-Archimedean numbers pre- 

 viously defined may be infinite or finite or 

 infinitesimal; but an angle will be always 

 finite or infinitesimal because of the rela- 

 tion 



cos^ <P + sin'' <f — 1. 



An angle may, therefore, always be put 

 under the form 6 -\- r, ft being an ordinary 

 real number and t a non-Archimedean 

 infinitesimal. 



We define then the rectangular coordi- 

 nates of a point, the straights and the 

 translations in the ordinary manner and 



define rotation in the following manner. 

 Let a, 13 be the coordinates of the center of 

 rotation ; 6 -\- t the angle of rotation ; 

 X, y the coordinates of any point before 

 the rotation ; x', y' its coordinates after the 

 rotation ; one will have 



(x' — a) + i(ij' — /3) = e(i9+T+iT) [(a- _ „) 



Consider the group formed by the rotations 

 about the origin. This group will not be 

 permutable for the transformation which 

 changes y into — y, nor for any transfor- 

 mation which retains the origin, which 

 changes straights into straights and of 

 which the square reduces to the identical 

 transformation. The plane is, therefore, 

 not symmetric. 



All the other metric axioms subsist, how- 

 ever, as does also the postulatum of Euclid 

 and even a new axiom which Hilbert calls 

 Axiom der Nachharschaft and which he 

 states thus : 



"Given any sect AB, one can always 

 find a triangle in the interior of which can 

 be found no sect congruent to AB." 



This results easily from the equation 

 of the circle'. The equation of a circle of 

 radius p and center a, /3 is in fact : 



(x-ay+(y — ISy = pV^; 



^-^= tan (^'-l- r). 



X — a 



In return, it is not true that the angles 

 at the base of an isosceles triangle are 

 equal ; it is not true that in a triangle one 

 side is less than the sum of the other two; 

 finally the theorem of Pythagoras on the 

 square of the hypothenuse is not true. 



"It is for this reason that this geometry 

 is called non-Pythagorean. [I may in- 

 terpolate here that Poincare is in error in 

 saying Hilbert shows that the equality of 

 the basal angles can be proved from the 

 other metric axioms if one admits the axiom 



