Septembek 16, 1904.] 



SCIENCE. 



355 



space and the equality of the angles at the 

 base in an isosceles triangle. Hilbert does 

 not here treat this question, but he has 

 made it the object of a memoir to which we 

 will recur." This is a mistake. Hilbert 

 explicitly assumes AB = BA, but the equal- 

 ity of the basal angles does not follow there- 

 from. 



It used to be supposed that the Euclidean 

 straight was of essence continuous, and this 

 putative continuity was rested upon to give 

 continuity to the real number system. This 

 mistake is made by Professor H. B. Fine in 

 his book 'The Number-system of Algebra.' 

 For example, on page 43 he says: "The 

 entire system of real numbers, however, 

 inasmuch at it contains an individual num- 

 ber to correspond to every individual point 

 in the continuous series of points forming 

 a right line, is continuous." 



Dedekind had long ago called attention 

 to the fact that Euclid's space had no need 

 of continuity. In an article of my own 

 'How the new mathematics interprets the 

 old,' March 4, 1893, is a quotation of his 

 construction of a discrete space, which goes 

 on, "yet despite the discontinuity, the 

 perforation, of this space, all constructions 

 occurring in Euclid are in it just as 

 achievable as in perfectly continuous space. 

 The discontinuity of this space would, 

 therefore, never be noticed, never be 

 discovered, in Euclid's science. Um so 

 schooner erscheint es mir, das der Mensch 

 ohne jede Vorstellung von messbaren 

 Groessen, und zwar dureh ein endliches 

 System einfacher Denkschritte sich zur 

 Schoepfung des reinen, stetigen Zahlen- 

 reiehes aufschAvingen kann; und erst mit 

 diesem Hiilfsmittel wird es ihm nach 

 meiner Ansieht moeglich, die Vorstellung 

 vom stetigen Raume zu einer deutlich 

 auszubilden. ' ' 



There are naturally no points on the 

 Euclidean straight to correspond to the 

 series of irrational numbers, and Euclid 



felt no more ambition to have them there 

 than he did to have a set of automobiles, 

 and for the same reason, irrational numbers 

 and automobiles had not yet been created. 



Hubert's axioms, analyzing Euclid's 

 space, did not make it continuous. Poin- 

 eare called attention to this, and spoke of 

 the space burdened with these irrational 

 points as our space in contrast to Euclid's 

 space, as if we were debarred by modernity 

 from living in the splendidly free and dis- 

 jointed space of the glorious old Alex- 

 andrian, who spurned the idea of any other 

 way even for kings. "In Hilbert 's space," 

 he says, "there are not all the points which 

 are in ours, but only those that one could 

 construct, starting from two given points, 

 by means of the ruler and compasses. In 

 this space, for example, there would not 

 exist, in general, an angle which would be 

 the third part of a given angle. 



"I have no doubt that this conception 

 would have been regarded by Euclid as 

 more rational than ours." 



He then proceeds, following Dedekind, 

 to give an assumption which will lug 

 in these irrational points. Hilbert in 

 Laugel's translation did the same, but by 

 a quite different assumption, which he calls 

 the 'Axiom der VoUstandigkeit. ' 



"Note. We remark that to the five pre- 

 ceding groups of axioms one may still add 

 the following axiom which is not of a purely 

 geometric nature and which, from the 

 theoretical point of view, merits particular 

 attention. 



"axiom of completeness. 

 "To the system of points, straights mid 

 planes, it is impossible to add other beings 

 (etres) so that the system thus generalized 

 forms a neiu geometry where the axioms of 

 the five groups I.-V. are all verified. In 

 other words: the elements of geometry form, 

 a system of beings which, if one maintains 



