Euclid: What is the difference between a 'surface' and a plane 'surface'?

Question

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.

Answer 1

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.

Answer 2

Let's take the first definition first. When he says "has length and breadth only", he basically means the same as when a differential geometer defines a 2-manifold. There are at each point two degrees of freedom for movement within the surface itself. For instance, the surface of the earth can be thought of as only having length and breadth (longitude and latitude is mostly enough to specify any point on the planet).

As for when he says "lies evenly with all the lines on itself", he probably means that plane surfaces are convex in the modern sense: Take any two points on a surface. Those define a line. If that line does not lie evenly with (i.e. is not contained in) the surface, then the surface is not a plane surface.

Answer 3

What is the difference between a surface and a plane surface ?

What is the difference between a line and a straight line ?

Answer 4

In differential geometry a 2-D surface is like a sphere which has length and breadth, Gauss curvature K is non-zero. It is doubly curved.

A flat surface has zero K and is a part of a plane or developable to it, at least one of two radii of curvature is infinity.

If you try to flatten a Pringles saddle shaped chip, circumferential lines tend to overlap before they break up. On the other hand when you step on an egg shell crushing it, it splits the boundary circumferentially.

Such differential deformation behavior is at the back of intuitive physical perception of spatial embedding of positive, negative and zero double curvature of surfaces.. while engrossed in the formalism of their mathematical treatment.