# Why is sphere non-euclidean space?

There are 5 axioms that define euclidean space and I believe that all hold also for a sphere. The definition of axioms from wikipedia:

• "To draw a straight line from any point to any point."

Definition of "straight" might be tricky, but I think that it's simply a shortest path on the surface.

• "To produce [extend] a finite straight line continuously in a straight line."

Hmm... "Continuosly". Does it mean that I can't draw the extension over the original line? I'm not sure about that, but the axiom doesn't forbid that.

• "To describe a circle with any centre and distance [radius]."

If circle is defined as a set of points with the same distance from the center, no problem there.

• "That all right angles are equal to one another."

• The parallel postulate: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

All straight lines on the sphere intersect, so this is a no-brainer. I see no if and only if.

So where is the problem? Which axioms don't work on a sphere? Is there a problem with basic words like "line" and "straight"?

• That's a rather odd formulation of the parallel postulate... Aug 29, 2017 at 18:27
• Euclid's axioms does not hold up to modern scrutiny. Use some more recent collection of axioms, like Hilbert's. Aug 29, 2017 at 18:28
• @Arthur What is that supposed to mean. Aug 29, 2017 at 18:28
• @NeplatnyUdaj Think again about the last point and your answer to it. I don't see how those two are related. Aug 29, 2017 at 18:29
• @azureai It means that Euclid's axioms, while generally held in high regard for the approach to mathematics that they represent, are hopelessly outdated and full of holes. There have been better axioms for plane geometry available for at least a century. Aug 29, 2017 at 18:30

The basic premise of your question is incorrect: the five axioms you listed do not characterize the Euclidean plane (and in any case need to be formulated a lot more precisely for that question to even make sense). They are the axioms Euclid listed, but actually he implicitly assumed several other axioms.

In particular, one concept that is crucial to Euclidean geometry is order: given three points $A$, $B$, and $C$ on a line, we can say one of them is between the other two and this notion of "betweenness" satisfies a certain list of axioms (see Hilbert's axioms, for instance). There is no appropriate notion of "betweenness" on the sphere. Intuitively, since a "line" on the sphere is a great circle, you can't say one point is between the other two because which points are between $A$ and $B$ depends on which side of the circle you use to travel from $A$ to $B$.

• Thanks. OT: The reason I asked this is that I was reading an article(different language, so I won't link it) about euclidean and non-euclidean space and it implied that these five axioms were sufficient and sound to define. And at the end, there was a strange statement about non-euclidean spaces which was in contradiction with the rest of the article. Now I have a better understanding... these 5 axioms are not all there is to the euclidean space. Aug 29, 2017 at 19:39

While Eric Wofsey is correct that Euclid's axioms in and of themselves aren't a complete formulation of the Euclidean plane as Euclid used it, he did include enough "definitions" and "common notions" in his Elements to exclude spherical geometry. Namely: the converse of the parallel postulate is a theorem of the other four plus the common notions.

Where elliptic geometry fails is the second axiom, or rather Euclid's idea of the second axiom: "continuously" does indeed forbid the idea of the line circling back on itself - it would be "indefinitely" in a more modern translation.