Euclid's Elements start with five Postulates, including the fifth one, the famous Parallel Postulate. Less well, known however is the Postulate that forms the basis for motivation behind the fifth: the fourth one, which states that "all right angles are equal." Students who see this for the first time might find this puzzling, because obviously two angles which are equal to a 90 degree angle are equal to each other, since Common Notion 1 says that "things which are equal to the same thing are are also equal to one another". But then they realize that the matter is so straightforward: the definition of a right angle is an angle produced when two lines intersect each other and produce equal adjacent angles, and it's not clear why an angle produced by one such pair of lines should bear any relation to an angle produced by another such pair of lines.
So Euclid's fourth Postulate is not redundant for the reason that beginning students might think. But my question is, is it nevertheless a redundant postulate, although for far less trivial reasons? David Hilbert, in his Foundations of Geometry (Grundlagen der Geometrie in German), claims to prove Euclid's fourth Postulate in theorem 15 (on page 19 of the PDF or page 13 according to the book's internal page numbering), prefacing the proof by saying "it is possible to deduce the following simple theorem, which Euclid held – although it seems to me wrongly – to be an axiom."
Now it's fair to say that Hilbert was working in a different (and more rigorous) system of axioms than Euclid was, but I think Hilbert's proof should be seriously considered for two reasons. First of all, why would he dub Euclid's decision to call "all right angles are equal" a Postulate as "wrong" if it merely reflected a stylistic difference concerning what you choose as starting assumptions and what you consider theorems? But more importantly, by tracing back all the assumptions used in the proof of theorem 15, it seems to me that only four of Hilbert's axioms are ultimately used: IV 3, IV 4, IV 5, and IV 6. And I don't think Euclid would have objected to any of these statements:
IV 3 follows directly from Euclid's Common Notion 2.
IV 4 is partly stated in Euclid's Book I Proposition 23, which doesn't depend on the fourth postulate, and the part of IV 4 which (I think) is not stated is easily provable in Euclid's system.
IV 5 follows from Euclid's Common Notion 1.
IV 6 is just part of Euclid's Book I Proposition 4, which doesn't depend on the fourth postulate at all.
So could Euclid have proven his fourth Postulate as a theorem instead of just assuming it?
Any help would be greatly appreciated.
Thank You in Advance.
EDIT: It seems to me that the key idea driving Hilbert's proof is that Euclid's Book I Proposition 4, i.e. the Side-Angle-Side (SAS) congruence theorem, implies that the supplements of equal angles are equal. Can anyone confirm or deny that this implication is in fact valid, and if it is valid, that the conclusion can be used to show that all right angles are equal?
Best Answer
Euclid's right-angle postulate excludes the existence of cone points: right angles at the vertex of a cone are smaller than right angles elsewhere on the cone. So this postulate cannot be proved insofar as the other axioms apply on a cone, which one could argue that they do.