Euclid built the whole of classical geometry on top of five postulates — starting assumptions taken without proof, because a system of logic has to start somewhere. Four of them are almost boringly self-evident: a straight line can be drawn between any two points, a straight line can be extended indefinitely, and so on. The fifth is not like the others. Stated in its original form, it says that if a line crosses two other lines and the interior angles on one side add up to less than two right angles, those two lines will eventually meet on that side. It's true, but it reads like a claim that needs an argument, not a starting assumption you're simply handed. For roughly two thousand years, mathematicians agreed on that discomfort and tried to fix it by proving the fifth postulate as a theorem, derivable from the first four alone.
Every proof had a hidden postulate stitched inside it
Generations of mathematicians produced "proofs" of the fifth postulate, and generations of later mathematicians found the same flaw in nearly all of them: somewhere in the argument, often disguised as an innocuous-looking step, the proof quietly assumed something logically equivalent to the fifth postulate itself — commonly the version taught in school today, that through a point not on a given line, exactly one line can be drawn parallel to it. Assuming that to prove the fifth postulate is circular, not a proof. Two thousand years of attempts kept rediscovering the same trap in slightly different disguises, which is itself a strange kind of evidence: if the postulate really followed from the other four, it's odd that nobody, across two millennia of serious mathematical effort, ever managed to show it without smuggling in an equivalent assumption.
The proof that never came, because it couldn't
In the early 19th century, János Bolyai and Nikolai Lobachevsky, working independently, took a different approach: instead of trying once more to prove the fifth postulate, they assumed it was false and built out the consequences anyway, to see whether the result was nonsense or something coherent. It was coherent — a fully consistent geometry in which more than one line through a point can be parallel to a given line, angles of a triangle sum to less than 180 degrees, and the whole system holds together without contradiction. Bernhard Riemann later constructed a second consistent alternative, in which no parallel lines exist at all and triangle angles sum to more than 180 degrees. Two internally consistent geometries, each built on a version of the fifth postulate different from Euclid's, settled the two-thousand-year question decisively: since a coherent alternative geometry exists without Euclid's version of the postulate, that postulate genuinely cannot be derived from the other four. The thing everyone had tried to prove wasn't secretly provable after all — it was, as it had always looked, a separate assumption.
What we're still unsure about
It's a satisfying twist to say this "failed" two-thousand-year project quietly built the mathematics general relativity would later need to describe curved spacetime, and there's something real underneath that — non-Euclidean geometry is indeed part of the mathematical toolkit differential geometry provides. But stating it that simply skips nearly a century of separate mathematical development, largely through Riemann's broader work on curved manifolds and the later tensor calculus Einstein needed, between the 19th-century parallel-postulate debates and general relativity in 1915. The two threads are genuinely connected, but treating 19th-century non-Euclidean geometry as if it were built with relativity already in mind overstates a link that was, historically, discovered rather than intended.
This sits inside Euclid's Postulates, one of eight topics in Geometry, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.