>interpretation: the principle of charity (see e.g.
>http://en.wikipedia.org/wiki/Principle_of_charity).

This is a very sinsible approach, yes.

>On the other hand I find the "formalistic" approach:
>
>odd. It would mean that Euclid's argument is to be judged as
>a purely syntactic one, which is then subjected to many
>semantic interpretations.

I believe that was Scott's intention and that of many others before him.

>But this is surely not what Euclid
>himself meant (the clean separation of syntax and semantics
>wasn't seriously conceived before the 20th century). And
>surely there are better interpretations of any text, and
>less fortunate ones. We never demand that any text ought to
>satisfy all interpretations.

I do not know about that. This generalization may take too far afield.

>Again, substantially, I think that the above is a marginal
>issue in the present context. Because I don't see that
>Scott's counterexample against the EAT describes a
>situtation where a plane-separation axiom does not hold.

Let's not go in circles. Scott has not originally mentioned any separation axioms. He brought an example which could be constructed according to Eulid's rules and in which the ETA did not hold. Avior found weaknesses in this example. This was based on his analysis and interpretation of Euclid's axioms. Scott gave us to understand that he had no intention to go that deep.

>Instead, it relied on the occurrence of antipodal points,
>which connect many different lines. So to criticize Euclid
>for not providing such an axiom is not strictly relevant.

I am sure he will excuse me if I suggest that he was just trying to fall in line with Avior's argument that dwelt on axioms. He did not actually put a fight. The essence of his reply was that all he wanted was to demonstrate unreliability of the "from diagram" reasoning.

I believe his page does achieve this purpose even if itself may be subject of similar critique.

>sphere). I indicated that I don't see such an example. You
>replied:
>
>"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."
>
>I don't see how this gives what I asked for.

We've been talking not of the side but of angles. This what EAT is about, right?

>Scott's
>90-90-90 triangle has three sides whose length is 1/4 the
>circumference. If we "open it up" - widen it - we make one
>of the sides - the equatorial one - even longer. If we then
>"lift" that side, its length stays the same.

No, of course not. If in an isosceles triangle you lift the base, the base becomes smaller.

>So how is this
>supposed to describe a triangle whose every side is less
>than 1/4 the circumference?

Once, again: I did not related to the 1/4 or other fractions at all, but was looking for a triangle that violated EAT.

>The question can also be raised from the other side (so to
>speak), as I put it in the previous post: where
>exactly does Euclid's proof (construction) fail for
>this triangle? If its failure depends on antipodal points,
>then it will confirm my argument.

This is an intersting question: to find where and why Eulid's argument breaks down in my modification of Scott's example.

>So I stiil don't see what justifies the judgment that
>Euclid's proof the the EAT is flawed. Scott's
>counterexamples seem to rely on a feature (antipodal points)
>which we can surely reasonably say the Euclid assumed
>was impossible,

Yes, and this is exactly the weak point to which Scott wanted to draw our attention. Euclid no doubt made an implicit assumption that was justified by his plane diagram. The trouble was that the construction goes through on the sphere which does not confirm to his assumption, the implicatino being that without that assumption Euclid's argument becomes faulty.

>axiom does not apply. Also, we anyway seem to have to
>attribute such an implicit axiom to Euclid, based of the
>principle of charity.

Well, we may of course. Not every one does, however. There is a gain in having a second view. And even if one decides to extend Euclid a well deserved charity, this only can be done after a realization that the charity is called for has been attained.