764 



SCIENCE 



[N. 8. Vol. XXXIII. No. 855 



Pascal's theorem; I mean the theorem 

 about the hexagon inscribed in a conic, 

 supposing that this conic reduces to two 

 straights. So Pascal 's theorem will be true 

 or false according as the system S is,pas- 

 calean or non-pascalean ; and since there 

 are non-pascalean systems, there likewise 

 are non-pascalean geometries. The the- 

 orem of Pascal can be deduced from the 

 metric axioms; it therefore will be true if 

 we suppose figures may be transformed not 

 only by homothety and translation, as we 

 have done, but also by rotation. Pascal's 

 theorem can likewise be deduced from the 

 Archimedes axiom, since we have seen that 

 every system of numbers arguesian and 

 archimedean is at the same time pascalean ; 

 every non-pascalean geometry is therefore 

 at the same time non-archimedean. 



The Sect-carrier. — "We cite still another 

 conception of Hilbert's. He studies the 

 constructions we can make, not with ruler 

 and compasses, but with ruler and a spe- 

 cial instrument which he calls the sect- 

 carrier, and which enables us to set off on 

 a straight a sect equal to another sect taken 

 on another straight. The sect-carrier is 

 not the equivalent of the compasses; this 

 latter instrument enables us to construct 

 the intersection of two circles, or of a 

 circle and any straight; the sect-carrier 

 will only give us the intersection of a circle 

 and a straight passing through the center 

 of this circle. Hilbert seeks therefore what 

 are the constructions which are possible 

 with these two instruments, and he reaches 

 a very remarkable conclusion. 



The constructions which can be achieved 

 with ruler and compasses can likewise be 

 made with the ruler and the sect-carrier, 

 provided these constructions are such that 

 their result is always real. 



It is evident in fact that this condition 

 is necessary, because a circle is always cut 

 in two real points by a straight drawn 



through its center. But it was hard to 

 foresee that this condition would likewise 

 be sufficient. 



But this is not all ; in all these construc- 

 tions, as Kiirschak first noticed, it is pos- 

 sible to replace the sect-carrier by the unit- 

 sect carrier, an instrument which enables 

 us to set off on any straight from any point 

 of it, no longer any sect, but a sect equal 

 to unity. 



An analogous question is treated in an- 

 other article of Hilbert's: On the equality 

 of the angles at the base of an isosceles 

 triangle. 



In the ordinary plane geometry, the 

 plane is symmetric, which expresses itself 

 in the equality of the angles at the base of 

 the isosceles triangle. 



"We should make this symmetry of the 

 plane appear in the list of metric assump- 

 tions. In all the geometries more or less 

 strange of which we have spoken hitherto, 

 in those at least where we admit the metric 

 assumptions, in the non-archimedean met- 

 ric geometry, in the new geometries of 

 Dehn, in those which are the subject of the 

 memoir "On a New Foundation, etc.," 

 this symmetry of the plane is always sup- 

 posed. Is it a consequence of the other 

 metric assimiptions ? Yes, as Hilbert 

 shows, if we admit the Archimedes assump- 

 tion. No, in the contrary case. There are 

 non-archimedean geometries where all the 

 metric assumptions are true with the ex- 

 ception of this of the symmetry of the 

 plane. 



In this geometry 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; the theorem of Pythagoras about the 

 square on the hypothenuse is not true. 

 That is why this geometry is called non- 

 pythagorean. 



I come to an important memoir of Hil- 



