Mat 19, 1911] 



SCIENCE 



761 



assumptions of combination) remain with- 

 out change. 



2. The rules of the calculus and trans- 

 formation of inequalities (arithmetical as- 

 sumptions of ordering) likewise remain. 



3. The Archimedes assumption is not 

 true. 



"We may attain this result by choosing 

 for elements series of the following form: 



A^tm -j- A^tm-l -[- AJI,m-2 + . . . , 



where m is an integer positive or negative 

 and where the coefSeients A are real, and 

 convening to apply to these series the ordi- 

 nary rules of addition and of multiplica- 

 tion. It is necessary then to define the 

 conditions of inequality of these series so 

 as to arrange our elements in a determined 

 order. We shall attain this by the follow- 

 ing convention: we will attribute to our 

 series the sign of A^ and we will say that a 

 series is smaller than another when, if 

 taken away from this, it gives a positive 

 difference. 



It is clear that with this convention the 

 rules of the calculus of inequalities hold 

 good, but the Archimedes assumption is no 

 longer true. 



Our common numbers come in as par- 

 ticular cases among these non-arcJiimedean 

 numbers. The new numbers intercalate 

 themselves, so to speak, in the series of our 

 common numbers, in such a way that there 

 is for example an infinity of new numbers 

 less than a given common number A and 

 greater than all the common numbers less 

 than A. 



That settled, imagine a tri-dimensional 

 space wherein the coordinates of a point 

 are measured not by common numbers, but 

 by non-archimedean numbers, but where 

 the usual equations of the straight and of 

 the plane hold good, as also the analytic 

 expressions for angles and sects. It is 

 clear that in this space all the assumptions 

 remain true, save that of Archimedes. 



On any straight between our common 

 points would intercalate themselves new 

 points. Likewise there will be on this 

 straight an infinity of new points to the 

 right of all the common points. In a word, 

 our common space is only a part of non- 

 archimedean space. 



"We see what is the bearing of this in- 

 vention and wherein it constitutes in the 

 progress of our ideas a step almost as bold 

 as that which Bolyai made us take; the 

 geometry non-euclidean respected, so to 

 speak, oar qualitative conception of the 

 geometric continuum while it overturned 

 our ideas on the measure of this con- 

 tinuum. The non-archimedean geometry 

 destroys this conception ; it dissects the con- 

 tinuum to introduce into it new elements. 



In this conception so audacious Hilbert 

 had had a precursor. In his foundations 

 of geometry Veronese had had an analo- 

 gous idea. Chapter "VI. of his introduc- 

 tion is the development of a veritable 

 non-archimedean arithmetic and geometry 

 where the transfinite numbers of Cantor 

 play a preponderant role. Nevertheless, 

 by the elegance and simplicity of his expo- 

 sition, by the depth of his philosophic 

 views, by the advantage he has derived 

 from the fundamental idea, Hilbert has 

 made the new geometry his own. 



Non-arguesian Geometry. — The funda- 

 mental theorem of projective geometry is 

 the theorem of Desargues. 



Two triangles are called homologs when, 

 the straights joining each to each, the cor- 

 responding vertices are copunctal. Des- 

 argues proved that the intersection points 

 of the corresponding sides of two homol- 

 ogous triangles are costraight; the dual 

 is equally true. 



The theorem of Desargues can be estab- 

 lished in two ways : 



1. By using the projective assumptions 

 of the plane and the metric assumptions 

 of the plane. 



