Pythagoras' Theorem

Stream: Is there code for X?

[Kevin Buzzard]

Probably some differential geometers think that Pythagoras' theorem is $ds=\sqrt{dx^2+dy^2}$ in some abstract origin-less affine plane. Can we state something which means something like this? One could prove that $dx^2+dy^2$ was rotation-invariant (and reflection invariant, hence an invariant of the affine plane. @Joseph Myers you probably understand what I'm trying to understand here.

[Patrick Massot]

We already have affine spaces and it doesn't require being a differential geometer to think affine geometry doesn't need an origin.

[Kevin Buzzard]

But there's some content to the statement that the integral of ds from 0 to (a,b) is sqrt(a^2+b^2) right?