THE FOUNDATIONS OF GEOMETRY 25 



usual propositions about the order in which points lie on a line; the 

 complete line and order itself being defined in terms of the elements 

 and relations mentioned above. He then introduces the plane sur- 

 face by means of some further axioms, among which are : 



1. Every three points are in a plane surface. 



2. If a line segment lies between two points of a plane surface there is a 

 plane surface in which lie all points of the given plane surface and also all 

 points of the line segment. 



4. If A, B, C are three non-collinear points of a plane surface and any 

 segment DE of the surface has a point in common with one of the segments 

 AB, BC, CA, the line DE has a point in common with one of the other segments 

 or one of the points A, B, C. 



This fourth axiom of Pasch is the one 

 that is generally regarded as having re- 

 quired the greatest insight and is most 

 often associated with his name. 



A very great improvement over the 

 work of Pasch was made by the Italian 

 mathematician, Peano, who published in 

 1889 his ' I Principii di Geometria.' The FlG 4 . 



undefined terms of Peano are the elements 



'point and segment and the relations lie on and congruent to. The 

 plane segment of Pasch is defined as a certain set of points. 



In Italy, at this time, there was beginning a great revival of 

 interest, largely due to the influence of Peano, in the purely logical 

 aspects of mathematics. This has resulted in a large number of 

 investigations not only of the foundations of geometry, but of mathe- 

 matics in general. The results are mainly expressed in terms of 

 symbolic logic and proceed a long way toward solving the problem 

 to obtain the smallest number of undefined symbols and unproved 

 propositions that will suffice to build up geometry. Besides Peano 

 one needs to mention chiefly Pieri, who has investigated projective 

 geometry and also the possibility of basing elementary geometry on 

 the concepts, point and motion. Standing aside from the pasigraph- 

 ical school of Peano, there is Veronese, who has done pioneer work in 

 connection with the axioms of continuity. 



In Germany the chief figure at present is D. Hilbert, whose book 

 on ' Foundations of Geometry' (1899) has been translated into several 

 languages, including English. Hilbert's work is the first systematic 

 study that has received widespread attention, and he has therefore been 

 credited with originating a great many ideas that are really due to 

 the Italians. Hilbert's chief contribution to the foundations of geom- 

 etry is his study of the axioms needed for the proof of particular 

 theorems which he has collected in the latest edition of his book. 



