Mat 19, 1911] 



SCIENCE 



r59 



lead finally to the complete proof of the 

 theorem. We can not doubt that these con- 

 siderations, which allow also the obtaining 

 of arithmetical relations in making them 

 come from identities where definite inte- 

 grals figure, can some day, when we shall 

 have grasped their meaning, be applied to 

 problems much more extended than that of 

 Waring. 



GEOMETRY 



I come to Hilbert's works so very orig- 

 inal on the foundations of geometry. 



There are in the history of this geometric 

 philosophy three principal epochs ; the first 

 is that where thinkers at whose head we 

 should cite Bolyai founded the non-euclid- 

 ean geometry; the second is that in which 

 Helmholtz and Lie showed the role in 

 geometry of the notion of motion and of 

 group; the third was inaugurated by Hil- 

 bert. 



The German author takes the logical 

 point of view. What are the axioms 

 enunciated and those unconsciously as- 

 sumed; what is their real logical content 

 and what may be deduced from them by 

 the simple application of the rules of logic 

 and without new appeal to intuition? 

 Finally, are they independent, or can we 

 on the contrary, deduce them from one 

 another? These are the questions to face. 



Hilbert commences, therefore, by estab- 

 lishing the complete list of assumptions, 

 striving not to forget a single one. That is 

 not as easy as one might think, and Euclid 

 himself uses some he does not state. 

 Geometric intuition is so familiar to us 

 that we use intuitive verities, so to speak, 

 without our perceiving them; hence to 

 attain the aim Hilbert proposed to him- 

 self, the necessity of not according to intui- 

 tion the least place. 



The savant professor divides the as- 

 sumptions into five groups : 



I. Assumptions of Association (I shall 



translate by projective assumptions in 

 place of seeking a literal translation, as, for 

 example, assumptions of connection, which 

 would not be satisfactory). 



II. Betweenness assumptions (assump- 

 tions of order). 



III. Congruence assumptions or metric 

 assumptions. 



IV. Euclid's postulate. \^ 



V. The Archimedes assumption. 

 Among the projective assumptions we 



distinguished those of the plane and those 

 of space; the first come from the well- 

 known proposition: through two points 

 passes one straight, and only one. 



Going on to the second group, the order 

 assumptions, here is the statement of the 

 first two: 



"If three points are on the same 

 straight, they have a certain relation which 

 we express by saying that one of the points, 

 and only one, is between the other two. 

 If C is between A and B, if D is between 

 A and C, D will also be between A and B, 

 etc. ' ' 



Here still we note that intuition is not 

 brought in; we seek not to fathom the 

 meaning of the word hetween, every rela- 

 tion satisfying the assumptions may be 

 designated by this same word. 



The third group comprises the metric 

 assumptions where we distinguish three 

 subgroups, relative respectively to sects, to 

 angles and to triangles. 



An important point here was not 

 stressed (in the first German edition, 

 though it appears in the French transla- 

 tion). To complete the list of assumptions 

 it needs to be said that the sect AB is con- 

 gruent to the inverse sect BA. This as- 

 sumption implies the symmetry of space 

 and the equality of the angles at the base 

 in an isosceles triangle. Hilbert does 

 not here treat this question, but he has 



