Hello,

I'm sensing that this interesting discussion hasn't yet reached a natural end. Aythan Avior has criticised here an article on this site by Scott Brodie (hence "the article") where the latter criticises Euclid's proof of the Exterior Angle Theorem (http://www.cut-the-knot.org/fta/Eat/EAT.shtml). Avior's post contains a fascinating discussion of geometrical axioms. He takes Brodie's main point to be to recommend Hilbert's proof as an alternative to Euclid's. Against this, Avior demonstrates that Hilbert's proof shares exactly the same status of Euclid's, in terms of its reliance on geometrical assumptions. Brodie's reply (hence "the reply") contains two parts. The first part is that his discussion was meant to stay at a "pre-college level" in the sense that it remains partly intuitive, and does not indulge in a formal discussion of axioms, as in Avior's post. This sounds very plausible, but it doesn't yet face the geometrical issue itself. The second part relates (if briefly) to the issue itself: to "the heart of the problem". But, that part seems to me unconvicing (I'll return to it below). My post also tries to concern itself with the heart of the problem. I'm thinking about two main issues: first, is the article's condemnation of Euclid justified? Second, are the materials that make up the article's counter-examples to Euclid's proof, i.e spherical triangles, adequate to criticizing Euclid's proof?

Here is what I call the article's condemnation of Euclid:

"It is therefore distressing to discover that Euclid's proof of the Exterior Angle Theorem is deeply flawed! It can charitably be described as a glib example of "reasoning from the diagram.""

So this is a central question: is Euclid's proof really deeply flawed? Has he really reasoned from the diagram? I'll try to explain why I find the article's criticism doubtful, without myself invoking formal axiomatics. The article's criticism starts with, and mainly relies on, a consideration of spherical triangles:

"To see what can go wrong, let us recall Euclid's argument, and see what happens if we try to apply it to triangles drawn on the surface of a sphere."

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?

This question is not answered directly in the article, but I think that it is implied, as follows. At the beginning of the article the author applauds Euclid for proving the Exterior Angle Theorem without relying on the Parallel Postulate:

"In Euclid's sequence of propositions, the Exterior Angle Theorem appears before any invocation of the Parallel Postulate. [It is a great credit to Euclid's sense of parsimony that by delaying any use of the Parallel Postulate as long as possible, he helps draw the distinction between those parts of geometry which are valid independent of the Parallel Postulate (so-called "absolute" or "neutral" geometry, and those which depend on it ..."

It seems to be implied here that prior to invoking the Parallel Postulate, Euclid's axioms and theorems apply not only to the flat plane but also to the spherical plane. This would indeed explain why spherical triangles may be legitimately invoked in order to criticize Euclid's proof.

Now this is a delicate point (it perhaps explains why Avior was tempted to deal with it by a careful discussion of axioms). It seems correct that many of the propositions in Book I of the Elements apply to the spherical surface (at least on limited parts of it), and even more so (as Avior mentions) with regard to the related Elliptic geometry, where every two antipodal points on the surface of the sphere are identified as a single Point. Indeed, at D.E.Joyce's online version of the Elements, the latter remarks (http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI16.html) that all (!) the first fifteen propositions of the Elements apply to Elliptic geometry. The 16th proposition - the EAT - however does not apply to it (and a fortiori does not apply to standard spherical geometry).

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. 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. 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:

"Of course, these issues are handled much more clearly in modern treatments of Euclidean geometry, but I don't think they really get to the heart of the "problem" with Euclid's proof of EAT. A "skinny" triangle on the sphere consisting of two meridians running from the North Pole to the equator and a very small segment of the equator can be drawn without encountering difficulties identifying antipodal points or worrying about extending lines indefinitely, but EAT fails for such a triangle."

The narrow triangle that the reply describes contains two right angles, so that its existence does contradict one of the conclusions of the EAT, and hence the EAT itself. It is said that since that triangle is narrow, the fact that is fails the EAT, and Euclid's proof of it, is unrelated to the phenomenon of antipodal points. However, this seems to me to be incorrect. If you try to find out just where Euclid's proof fails for this triangle, you will find out that it again fails for antipodal points: the north and south poles. 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. Is there such a counterexample? I don't see it.

The reply afterwards indicates that the real problem is with the lack of a separation axiom:

"The real problem, as Avior indicates, is the need for an additional "plane" axiom describing how lines "separate" the plane."

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). 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, because of the article's intuitive nature:

"The subject of Plane Separation Axioms and their equivalents is for most students of Geometry tedious and opaque in the extreme -- a quintessential example of how mathematicians take great pains to prove the obvious only be making even more obscure assumptions!"

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". 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". 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.


Ram