A Avior raises some interesting questions regarding the choice of axiom systems for synthetic geometry (the traditional constructions-and-congruences kind) and the proof and/or validity of the Exterior Angle Theorem ("EAT"). Many of these are beyond the scope of my orignal web page, which is addressed to a pre-college audience.

There is a large literature on the "meaning" of Euclid's postulates regarding lines through pairs of distinct points, and how they should be interpreted in an attempt to do synthetic geometry on a sphere. With the understanding that "lines" on a sphere are to be realized as "great circles," it is not clear how we are to interpret Euclid's terms "produce" and "continuously." Euclid does not explicitly postulate that lines can be "produced" or "extended" indefinitely, though this is certainly implicit in the wording of his Fifth Postulate. Similarly, the issue of "uniqueness" is rarely of concern in Euclid, and does not come up in Book I.

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.

To be sure, the appeal to geometry "on the sphere" is probably best done informally, as a Euclidean-style set of axioms for synthetic geometry on the sphere is difficult to design, due precisely to the difficulties with order and extension raised in Avior's letter. (My original web page on EAT was carefully written with this in mind.)

The real problem, as Avior indicates, is the need for an additional "plane" axiom describing how lines "separate" the plane. This can be framed in many different ways, including the "Postulate of Pasch" mentioned in Avior's letter. He is absolutely correct that such an axiom, or its equivalent, is required for both a "salvaged" version of Euclid's proof of EAT and is used implicitly in Hilbert's proof (this aspect of Hilbert's proof was deliberately overlooked on the EAT web page as perhaps too technical a distraction for the intended audience).

The subject of Plane Separation Axioms and their equivalents is well covered in Moise's book cited on the EAT page. The interested reader will find there a careful treatment of Euclid's proof of EAT along these lines. For most students of Geometry, however, the Plane Separation Axioms and their consequences are 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! This subject, "Incidence Geometry" is indeed difficult, mainly because one must carefully suppress one's intuitions so as to avoid assuming "obvious" facts which have not yet been proven formally.

Hope this helps!

Scott.
Scott E. Brodie, MD, PhD
New York
scott.brodie@mssm.edu