>But, while this discussion of spherical triangles is (again)
>independently very interesting, it is not at all clear why
>it is adequate to criticizing Euclid's proof. After all,
>Euclid's axioms attempt to capture the flat plane, not the
>the surface of a sphere. So why would the eccentric behavior
>of spherical triangles be thought of as a problem for
>Euclid?

Because Euclid's construction goes through on the sphere, while his conclusion there does not hold. Brodie thus argues that Euclid's conclusion could not be a logical consequence of his premises.

>Now, the article's relevant point seems to be that the
>failure of Euclid's proof for the spherical plane is not
>only a failure on that eccentric plane but is also - the
>very same failure - on the flat plane.

There is a question of what is a failure and, if present, whose it is. The failure of Euclid's proof on the sphere is an indication of a logical flaw in the argument.

>This seems to be the
>backing of the claim that Euclid's proof is "deeply flawed".
>But is this correct? The article's counterexample seems to
>rely clearly on a feature of the spherical plane - of
>antipodal points - that they coincide with more than one
>line. This is not a feature is the flat plane.

Right. The problem is whether or not Euclid's foundations for I.16 go through on the sphere - and they do. Avior argues that Euclid's axioms admit various interpretations. Some apply to the sphere, some do not. Which shows a way of salvaging Euclid's proof. Brodie argues that Euclid had only one interpretation in mind: the one that follows the common planar intuition supported by the diagramatic view. All interpretations could be looked at as the models for Eulcid's logical argument. And a valid argument must be correct in all interpretations. Euclid's fails this criterion.

>So why is the
>counterexample relevant? Brodie is sensitive to this
>question in his reply to Avior, where he suggests to replace
>the original counterexample, which involved a very "wide"
>triangle (occupying 1/8 of the entire spherical surface)
>with an example consisting of a "skinny" (narrow) triangle:

This is indeed unfortunate.

>It turns out, it seems to me, that it isn't sufficient to
>make the triangle narrow. It also has to be
>short. It has to short on all dimensions, not just
>narrow. A better counterexample would have to consist of a
>triangle whose every side is shorter than 1/4 of the
>sphere's circumference.

This is quite correct.

> Is there such a counterexample? I don't see it.

Why? Open up - if only a little - Scott's 90-90-90 triangle to make it 90-91-90, and then lift the equator slightly so that the base angles do not change by more than .25°. Surely you can do that by continuity.

>However, this indication seems to me inconsistent with the
>original criticism of Euclid's proof, as also to miss
>Avior's main point (which has been, as I mentioned above,
>that Hilbert's proof is not better than Euclid's, in terms
>of reliance on geometrical assumptions).

Except that Hilbert did have a separation axiom (viz., Pasch's).

>It is as if the
>subject has changed. Because, the original (as also the
>modifed) spherical triangle counterexample simply does not
>rely on a lack of plane-separation. Criticizing Euclid for a
>lack of a separation axiom is a different issue, one which
>the original article obviously wished to avoid,

I think Brodie got somewhat carried away trying to fall in line with Avior's argumentation. Euclid bases his proof on the Common Notion 5: a part is smaller than the whole, which appears applicable since CF is between CA and CD. Scott's example shows that the betweenness in this case is not exactly what one might expect.

>To sum: if my above case is sound, then it doesn't seem
>legitimate to invoke spherical triangles in order to
>criticize Euclid's proof of the EAT. Consequently, it
>doesn't seem justified to say that Euclid's proof of the EAT
>is "deeply flawed".

It is flawed, though.

>From the standpoint of precise
>axiomatics, Euclid's system does need improvments (such as
>an additional plane-separation axiom), but this is clearly
>an aspect that the article (with good reasons) did not
>regard when condemning Euclid for his "deep flaw".

Again, Brodie's (and, in fact, Avior's) argument shows that, depending on interpretation, Euclid's conclusion may or may not be correct, hence pointing to a logical flaw in his proof.

>Finally,
>since the article is part of this (dynamic and
>authoritative) site, I think that it ought (again, if the
>above considerations are sound) to be corrected.

Thank you for the kind words. I think that the "dynamic" is the best characterization of the site. I only regret that it is not in the wiki form. I think that a possible way to proceed is to augment Brodie's article with this discussion, or perhaps just modify his example as above.