How did Pasch show his axiom was independent of Euclid's? Pasch's axiom is independent of Euclids'.  How did Pasch show this? Surprisingly, I can find no reference about this, nor can I can find an (English) copy of Pasch's work.  And, none of the approaches I can think of are possible:
According to this post, models of geometry violating Pasch's axiom (but otherwise meeting the axioms) require AoC and are not constructible.  So, Pasch didn't construct one.
Likewise, since a formal axiomitization of Euclid wasn't done till after Pasch by Hilbert, who used Pasch as a start, Pasch couldn't have proven this using formal logic over Euclid's axioms.
How did Pasch show his axiom was independent of Euclid's?
 A: A partial answer, from what I've been able to determine so far:
Pasch's Axiom is equivalent to the Plane Separation Axiom (PSA).  There are several ways a model may violate the Plane Separation Axiom:
\1. Gaps or holes where there are no points.  The simplest model of this is a grid, where points exist only at the grid intersections.  A line is a dotted line, a circle a dotted circle.  Segments and circles have finite numbers of points.
In this model, a line at a non-perpendicular angle can pass through the gaps between points, violating PSA.
A grid model can be precluded by a simple axiom stating between any two distinct points there exists a third point.  But gaps of a similar nature could still exist, e.g. if we admit points only at the rationals.  Completeness precludes gaps of this nature entirely (as do weaker things, e.g. admitting points at all constructible irrational numbers).
\2. Adding completeness, we can still violate PSP if lines wrap around the plane back to the other side.  Imagine a plane wrapped into an infinite cylinder.
Lines might wrap around back to the same point, or to a different point.
Eliminating these models can be done via simpler axioms than PSP: e.g. for any two distinct points, there is exactly one segment.
\3. Finally, we can preclude both the above, and still violate PSP, by having parts of the plane mixed up in very bewildering ways.  These ways are bewildering enough to be non-constructible, requiring AoC, and relate to non-linear solutions of $f(x+y) = f(x) + f(y)$.
Are Euclid's axioms independent of the above? Euclid wrote in language different than today's, and so we need to translate them appropriately.  I'd argue that the first two postulates, along with the definition of line and plane, preclude #1 above.  Likewise, #2 is precluded if we take Postulate 1 to mean "a unique segment", as is commonly understood.  They don't preclude #3 - but, even Tarski didn't accept AoC (to my understanding).  So it's not surprising that Euclid would see no need to preclude it.
These are my thoughts.  I'd be grateful if an expert would opine on this.
