Hope I am on the right forum (and that my question makes some sense-so here goes. Imagine we are a race of people living on a sphere (not hard because we are) However , rather than buying into the idea that lines are ideally straight we are and have always been well aware of how "parallel " lines meet ,effectively being "great circles". It has never occurred to this race that parallel lines could be imagined as lines that never meet and so all their geometrical measurements are made on the basis of lengths along this curved surface. To make this "more realistic" (well ,maybe less) this race lives on the event horizon of a black hole so that even light seems /does follow the surface around. So how could their geometry have evolved from its first uses? Obviously we might start with the Pythagoras formula. What would have been their alternative formula? If you draw 2 lines from an origin how do you calculate the distance between their end points? Everything must be done in 2D. So it will have to be something along the lines of f(a) G(fb)=h(c) where "G "indicates some process like adding ,multiplying or somehow combining in another way. and a,b and c are the first ,second and (hopefully ) 3rd calculated measurements. (f and h are different , appropriate functions) In other words ,is this race which lives on a curved surface condemned to see the idealised Euclidean view of the world in order to invent mathematics/geometry or can they (in a simple way) plough their own furrow?