[baker] the foundations OF GEOMETRY US 



that Hubert's geometry is as abstract as the preceding suggests, but I 

 do say that it must be clearly borne in mind that all our knowledge of 

 the elements in Hilbert's geometry must be derived from the assump- 

 tion we make regarding them, and not in any way from our physical 

 erperience of points, lines, etc. 



Again, since Hilbert proposes to create by his assumptions (not 

 axioms, which here I feel to be an unsuitable word) a geometrical 

 universe, it seems reasonable to anticipate that his assumptions will be 

 laore numerous than the axioms of Euclid who, in formulating these 

 axioms but incompletely analyzed a universe already in existence. 



Still further, — Our knowledge of the external universe is a know- 

 ledge of relations. The universe defines itself to us by means of 

 relations. We might anticipate then that Hilbert in presenting to 

 us the universe he brings into existence, and in seeking to makie us 

 conceive it, would not begin by attempting to define such elements as 

 the point, line and plane, but would confine himself to making assump- 

 tions respecting their relations. 



"With this preface I proceed to state Hilbert's assumptions, giving 

 also, by way of illustration, as such illustration seems necessary for a 

 proper understanding and appreciation of the assumptions, certain of 

 Halsted's deductions. 



Hilbert begins by saying, let us consider three distinct systems of 

 things, calling them respectively, points, straigTit lines, and planes. 

 We think of these as having certain relations, and the complete and 

 exact description of these relations are the consequences of the assump- 

 tions of geometry. He then makes these assumptions, dividing them 

 into five groups: I. Assumptions of connection or association; 

 II. Assumptions of order or betweenness ; III. Assumptions of con- 

 gruence: IV. Assumption of parallels; V. Assumption of continuity, 

 or Arcliimedes' axiom. 



I, The assumptions of association are: — 



(1). Two points determine a straight line. 



(2). Any two points on a straight line determine it. 



(3). On every straight line there are at least two points. 



(4). Three points determine a plane. 



(5). Any three points (not co-straight) on a plane determine it. 



(6). On every plane there are least three points (not co-straight). 



(7). If two planes have one point in common they have another. 



(8). If two points on a straight line lie on a plane, then every 

 point on the strai_ght line lies on the plane. 



