To be fair changing the 5th postulate requires some creative redefining of what a straight line is. In my experience when explaining non-euclidean geometries to muggles the hardest part is not the 5th postulate or its consequences but making people accept that on a sphere a straight line is really a great circle (the easier way being through the concept of a geodesic line but this was invented after non-euclidean geometries if I’m not mistaken.
Yeah, there are two equivalent definitions of a straight line, “don’t turn” and “shortest path”, both known to the ancient Greeks, I’m sure, but not formalizable in any easy way until differential calculus was invented, and not well until Riemann. Still, if someone actually asked Euclid “what do you think the closest thing to a straight line might be on a sphere, and which postulates hold there?” he would probably have done the rest.
To be fair changing the 5th postulate requires some creative redefining of what a straight line is. In my experience when explaining non-euclidean geometries to muggles the hardest part is not the 5th postulate or its consequences but making people accept that on a sphere a straight line is really a great circle (the easier way being through the concept of a geodesic line but this was invented after non-euclidean geometries if I’m not mistaken.
Yeah, there are two equivalent definitions of a straight line, “don’t turn” and “shortest path”, both known to the ancient Greeks, I’m sure, but not formalizable in any easy way until differential calculus was invented, and not well until Riemann. Still, if someone actually asked Euclid “what do you think the closest thing to a straight line might be on a sphere, and which postulates hold there?” he would probably have done the rest.