I understand that Pasch's Axiom is missing from the Euclidean set of axioms, as Moritz Pasch first showed and David Hilbert told the world about. This means that there will be some theorems in the Elements which implicitly assumes that axiom. Which is the first one?

(Here Pasch's theorem is depicted, not Pasch's axiom)


1 Answer 1


Proposition 7, Book I

Given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there cannot be constructed on the same straight line (from its extremities) and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same extremity with it. [Heath's translation.]

refers to "the same side" of a straight line.

That a straight line in a plane has two sides can be be proved using Pasch's axiom in conjunction with some order axioms (postulates) for points on a straight line (see e.g. Hilbert's Grundlagen; English translation here: http://www.gutenberg.org/ebooks/17384). Euclid's postulates don't contain any mention of a side of a straight line, so you could take the view that Pasch's axiom is somehow implicitly assumed in this proposition.

However, Pasch's axiom would not be the only candidate assumption.

Propositions 1-6 also contain assertions which don't follow from Euclid's definitions, postulates and common notions (on any reading - the fact is that the exact meaning of these is in most cases difficult to discern). Shoring up the gaps would require extra postulates. It is conceivable that Pasch's axiom could be involved as an extra postulate in some possible fixes, so I wouldn't be too dogmatic that Proposition 7 is necessarily the first.


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