I've begun to study Euclid's Elements and i've a few questions regarding the difference between a surface and plane surface.
A surface is said to be "that which has length and breadth only", it then goes on to say "a plane surface is a surface which lies evenly with the straight lines on itself".
What is the difference between the two? What exactly is a plane surface? Can a plane surface be curved, or is it only flat?
I apologise for being a pedant, these definitions should be straight forward but I wish they could be a bit more specific, I want a good understanding of them before I move on.
Best Answer
The first point to make is that Euclid's definitions are not definitions in the modern sense, so that this kind of confusion is common. A surface, for Euclid, is roughly a two-manifold embedded in 3-dimensional space, e.g. the standard 2-sphere or one of its hemispheres, or the graph of certain functions from the plane to the line. A plane surface is just a surface that happens to lie in a plane, e.g. the unit disk. One more modern way to think about "lies evenly with the straight lines on itself" is that the tangent lines to a plane surface lie within the surface, as does not occur in general.