That’s not correct. Elliptic geometry fails to satisfy some of the other postulates, depending on how they are phrased. I’m not too familiar with the standard ways of making Euclid’s postulates rigorous, but if you’re looking at Hilbert’s axioms instead, then elliptic geometry fails to satisfy O3 (the third order axiom): if three points A, B, C are on a line, then any of the points is between the other two. Possibly some other axioms are violated as well.
Notably, elliptic geometry does not contain any parallel lines, while it is a theorem of neutral geometry that parallel lines do in fact exist.
Hyperbolic geometry was actually necessary to prove the independence of Euclid’s fifth postulate, and few would call it a “fairly simple counterexample”.
I agree that introducing elliptic geometry (and other simple examples like the Fano plane) earlier on in history would have made the discussion of Euclid’s fifth postulate much more coherent much sooner.
For thousands of years, mathematicians tried proving the parallel postulate from Euclid’s four other postulates, even though there are fairly simple counterexamples which show such a proof to be impossible. I suspect that at least part of the reason for this delay is a failure to appreciate this post’s point : that a “straight line”, like a “number” has to be defined/specified by a set of axioms, and that a great circle is in fact a “straight line” as far as the first four of Euclid’s postulates are concerned.
That’s not correct. Elliptic geometry fails to satisfy some of the other postulates, depending on how they are phrased. I’m not too familiar with the standard ways of making Euclid’s postulates rigorous, but if you’re looking at Hilbert’s axioms instead, then elliptic geometry fails to satisfy O3 (the third order axiom): if three points A, B, C are on a line, then any of the points is between the other two. Possibly some other axioms are violated as well.
Notably, elliptic geometry does not contain any parallel lines, while it is a theorem of neutral geometry that parallel lines do in fact exist.
Hyperbolic geometry was actually necessary to prove the independence of Euclid’s fifth postulate, and few would call it a “fairly simple counterexample”.
I agree that introducing elliptic geometry (and other simple examples like the Fano plane) earlier on in history would have made the discussion of Euclid’s fifth postulate much more coherent much sooner.