# Euclid's Elements missing axiom of M. Pasch examples

In The Mathematical Experience, Study Edition by Philip J. Davis, Reuben Hersh, and Elena Anne Marchisotto it states pp.175-176:

Just what constitutes the "straightness" of the straight line? There is undoubtedly more in this notion than we know and more than we can state in words or formulas. Here is an instance of this "more." Suppose $a$, $b$, $c$, $d$ are four points on a line. Suppose $b$ is between $a$ and $c$, and $c$ is between $b$ and $d$. Then what can we conclude about $a$, $b$, and $d$? It will not take you long to conclude that $b$ must lie between $a$ and $d$. This fact, surprisingly, cannot be proved from Euclid's axioms; it has to be added as an additional axiom in geometry. This omission of Euclid was first noticed 2000 years after Euclid, by M. Pasch in 1882! Moreover, there are important theorems in Euclid whose complete proof requires Pasch's axiom; without it, the proofs are not valid.

See pp.21-22 for a description of Pasch's axiom and a picture of Pasch from the linked seminar slides: StanfordLogicSeminarApril2014.pdf

From wikipedia: Pasch's axiom

If a line, not passing through any vertex of a triangle, meets one side of the triangle then it meets another side.

I wanted to know which theorems in the Elements are considered in worst shape as a consequence of Pasch's missing axiom? Also have there been any more axioms found to be missing like Paschs?