Why did mathematicians feel the need to deduce the parallel postulate from the other axioms of geometry? After all, if you are going to start from some axioms, it doesn't much matter how many there are. Nobody seemed to mind that the other axioms were independent of each other.

Suspicion of the parallel postulate goes back to Euclid, who was the first person to notice (in writing at any rate) that it was needed for some arguments. Whenever he could, he avoided using it, even if that meant producing longer proofs. Did people have some inkling of non-Euclidean geometry, some premonition that the parallel postulate might be false?

No, they certainly did not. However, they felt uneasy about the parallel postulate because it was more complicated to state than the other axioms, and not quite as obviously true. If you have a line L and a point x not on it and claim that there is a line M through x that does not meet L, then you are making a statement about the whole, infinite line M and are therefore on dodgier ground than you are with the other axioms. It seems strange to have to deduce that the angles of a triangle add to 180 by appealing to what goes on unboundedly far away.

Incidentally, there were some serious attempts at proofs of the parallel postulate, but they all turned out to depend on hidden assumptions that were themselves equivalent to the parallel postulate (as is obvious if one bears hyperbolic geometry in mind). For example, one proof used the fact that for every triangle there is a similar triangle of any given size - which is false in the hyperbolic plane.

Click here to return to the main geometry page.