760 



SCIENCE 



[N. S. Vol. XXXIH. No. 855 



made it the subject of a memoir to which 

 we shall return later. 



The fourth group contains onlj^ Euclid's 

 postulate. 



The fifth group comprises two assump- 

 tions; the first and most important is that 

 of Archimedes. 



Let there be any two points A and B on 

 a straight d; let a be anj' sect ; construction 

 d, starting from the point A, and in the 

 sense AB, a series of sects all equal to one 

 another and equal to a : 



AA^, A^A^, ■■■ An.-i^An; we can alwaj^s 

 take n so great that the point B is on one 

 of these sects. 



This is to say that, if we take any two 

 sects I and L, we can always find a whole 

 number n so great that by adding the sect 

 I to itself n times, we obtain a total sect 

 greater than L. 



The second is the assumption of com- 

 pleteness of which I shall explain the mean- 

 ing further on. 



INDEPENDENCE OP THE ASSUMPTIONS 



The list of assumptions once drawn up, 

 it is necessary to see if it is free from con- 

 tradictions. We well know that it is, since 

 geometry exists; and Hilbert first replied 

 j'^es by constructing a geometry. But 

 strange to say, this geometry is not exactly 

 ours, his space is not ours, or at least is 

 only a part of it. In Hilbert 's space are 

 not all the points which are in ours, but 

 only tliose that, starting from two given 

 points, we can construct with ruler and 

 compasses. In this space, for example, 

 there is no angle of 10°. 



In his second edition Hilbert tried to fill 

 out his list so as to obtain our geometry 

 and no other, and so he introduced the as- 

 sumption of completeness which he states 

 as follows : 



To the system of points, straights and 

 planes it is impossible to adjoin another 



system of objects such that the complete 

 system satisfies all the other assumptions. 



It is evident then that this space of 

 which I spoke, which does not contain all 

 the points of our space, does not satisfy 

 this new axiom, because we can adjoin to it 

 those points of our space which it does not 

 contain, without its ceasing to satisfy all 

 the assumptions. 



There is, therefore, an infinity of geom- 

 etries which satisfy all the assumptions 

 except the assumption of completeness, but 

 only one, ours, which satisfies also this lat- 

 ter assumption. 



"We then must ask if the assumptions are 

 independent, that is to say, if we could 

 sacrifice one of the five groups, retaining 

 the other four, and nevertheless obtain a 

 coherent geometry. Thus it is, suppress- 

 ing group IV. (Euclid's postulate), we 

 obtain Bolyai's non-euclidean geometry. 



We can equally suppress group III. 

 Hilbert has succeeded in retaining groups 

 I., II., IV. and v., as also the two sub- 

 groups of the metric assumptions of sects 

 and angles, while rejecting the metric as- 

 sumption of triangles, that is to say, the 

 proposition III., 6. 



Non-arcliimedean Geometry. — But Hil- 

 bert 's most original conception is that of 

 non-archimedean geometry, where all the 

 assumptions remain true save that of 

 Archimedes. For this it is needful first 

 to make a system of non-arcMmedean num- 

 bers, that is to say, a system of elements 

 between which we can conceive relations of 

 equality and inequality, and to which we 

 can apply operations corresponding to 

 arithmetical addition and multiplication, 

 and this in a way to satisfy the following 

 conditions : 



1. The arithmetical rules of addition 

 and of multiplication (eommutativity, as- 

 sociativity, distributivity, etc., arithmetical 



