which came first euclidean or non euclidean geometry

Euclid’s Elements is not a classic for nothing. Arguably, the Greeks were more sophisticated foundational geometers than even the Paris elite in 1800. He didn’t make any of those hundred mistakes that later mathematicians did. Top mathematicians who were never wrong about anything else stumbled on this one issue. History shows that there are a hundred ways to make tiny invisible mistakes that fool even the best mathematicians. It’s so easy to make subtle mistakes in the theory of parallels. The Greeks, you know, they were really something else. It cannot be proved from the other axioms, as so many mathematicians during those millennia had mistakenly believed. The parallel postulate really does need to be a separate assumptions, just as Euclid had made it. But the fact is that Euclid was right all along. People tried to improve on Euclid for millennia. It’s astonishing that this was more than 2000 years after Euclid. Elite establishment stuff.īut even these bigwigs were wrong. Lagrange was even buried in the Panthéon in Paris.

Their names are engraved in gold on the Eiffel Tower. Around 1800, people like Lagrange and Legendre proposed so-called proofs of the parallel postulate. From antiquity all the way to the 19th century. Many people tried to “improve” on Euclid in this way. The parallel postulate should be a consequence of the core notions of geometry, not a separate assumption. Many felt that it should not be necessary to assume the parallel postulate as Euclid had done. So, people tried to prove the parallel postulate as a theorem. Very different from the other postulates, which are things like: any line segment can be spun around to make a circle. This particular type of configuration has such-and-such a particular property. It states that two lines will cross if a rather elaborate condition is met. It’s not a primordial intuition like the other postulates. The parallel postulate, by contrast, is not very simple at all. It makes sense that the bedrock axioms of geometry should be the simplest possible things, such as the existence of a line between any two points. The earlier postulates were very straightforward: there’s a line between any two points, stuff like that. Euclid’s postulate had rubbed a lot of people the wrong way even since antiquity. I’m referring to Euclid’s fifth postulate, the parallel postulate.

It’s going to be hell to pay for this, as you can imagine. What had been thought to have been proofs were exposed as fallacies. If this armor is compromised, if proofs are fallible, then what’s left of mathematics? It would ruin everything.īut the nightmare came true in the 19th century. That’s what mathematical proofs are supposed to do: eliminate any risk of being wrong. There are thousands of theorems in Greek geometry, and every last one of them is correct.

The mathematician has only one nightmare: to claim to have proved something that later turns out to be false. Or subscribe with your favorite app by using the address below

Which came first euclidean or non euclidean geometry