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 surface 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 required the greatest insight and is most often associated with his name.
Fig. 4. 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 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 mathematics 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 pasigraphical 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 geometry is his study of the axioms needed for the proof of particular theorems which he has collected in the latest edition of his book.
