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On the Theorem of Pappus

Published online by Cambridge University Press:  24 October 2008

M. H. A. Newman
Affiliation:
St John's College

Extract

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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1925

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References

* 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