Even when first learning geometry, I felt that the traditional proof of this theorem was, essentially, “cheating”. I was never satisfied that Euclid’s axioms and/or postulates necessarily permitted the ‘method of superposition’, the method upon which the proof hinged.
I eventually came to the conclusion that the notion of superposition must, itself, be taken as a postulate. Later, I was told that others felt much the same way and some people did, in fact, view I-IV in that way.
My question, then, is what is the “official” status of this theorem nowadays? Is it just tacitly assumed that ‘superposition’ is a postulate? Is the entire theorem viewed as such? Or, perhaps, nobody really cares since there are no repercussions either way (BTW, is that true - does it make any difference whether a postulate of the legitimacy of ‘superposition’ is assumed or not?)
Euclid’s axiomatization of geometry is not complete – he makes quite a few implicit assumptions. Certainly, there are assumptions made in his notion of superposition, such as that figures can be moved without changing their size or shape. Because these assumptions aren’t included in Euclid’s axioms, his proof is invalid. However, I believe that similar proofs that are valid have been constructed using a more modern definition of congruence.
At any rate, there is more than one equally correct way to axiomatize geometry, some of which assume SAS as a postulate, and others of which do not. So far as I know, the first complete axiomatization of geometry was that of Hilbert.
Google also turns up this discussion that seems to relate to your question.
There’s also an approach to geometry known as the Klein view where a geometry is viewed as a space and a set of mappings from that space to itself. Whatever properties of a set of points are preserved by those mappings are the properties of the geometry.
The upshot of this is that you don’t need any geometric axioms to do geometry. n-dimensional Euclidean geometry is just R[sup]n[/sup] and set of matrices M such that |x - y| = |Ax - Ay| for any x, y in R[sup]n[/sup] and A in M.
[incredibly petty nitpicking]
Not matrices, functions. Restricting yourself to matrices will exclude translations. (Unless you do that trick with using (n+1)x(n+1) dimensional matrices and tacking a ‘1’ onto the end of every vector, but then they’re not vectors in R[sup]n[/sup] anymore…)
More on topic, if you want a nice modern treatment of axiomatic geometry it’s hard to go wrong with H.S.M. Coxeter. I’ve got my hands on a copy of his book “Non-Euclidean Geometry” right now, which discusses Euler’s axioms in the introductory chapter and includes other axiomatic description in later chapters.
Even if we accept the principle of superposition, there’s at least one more flaw or hidden axiom in Euclid’s proof. He takes as an axiom that given two points, there’s a line joining them, but he does not explicitly postulate that there’s only one line between them. Even once you superimpose your two angles, it’s not a given that the same line is joining the endpoints in both cases.
There are plenty of these hidden axioms in Euclid, some of which have more far-reaching implications. For instance, Euclid also claims to prove that the intersection of two planes is a straight line. This is not true, since it’s also possible for two planes to intersect at a single point (provided that you’re in four or more dimensions, which Euclid doesn’t specify). What Euclid actually did in that proposition was to assume that the intersection was a line, and prove that it was straight.
Oh, if you want something readable and more likely to be right (at the edges I’ve found both of those to be spotty at best and what there is is often enough wrong), you could read through John Baez’ This Week’s Finds in Mathematical Physics.