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Published online by Cambridge University Press: 24 October 2008
1. The axioms necessary for the construction of a proof of Pappus' Theorem, regarded as a theorem in the geometry of projective space of three dimensions, fall into three groups:
I. Axioms of Incidence;
II. Axioms of Order, giving the properties of the relation “between”, and establishing the order type of the projective line as cyclical and dense in itself;
III. An Axiom of Continuity.
It is customary in treatises on projective geometry to adopt in Group III the Axiom of Dedekind, which states that if the points of a segment are divided into two classes, L and R, which have each at least one member, and are such that no member of L lies between two points of R, nor vice versa, then there is a point of the segment, not an end-point, which is neither between two points of L nor between two points of R. This axiom, however, assumes considerably more than is necessary for the proof of the theorem.
* Cf. e.g. Whitehead, The Axioms of Projective Geometry; Enriques, Geometria Proiettiva;Google ScholarBaker, , Principles of Geometry, vol. I.Google Scholar
† Cf. Hilbert, Grundlagen der Geometrie, 5te Auflage, § 14.Google Scholar
* Cf. Baker, op. cit. pp. 47 and 128.Google Scholar
‡ In the strict sense.Google Scholar