In that famous paper http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WKCGeom.html, Riemann writes the below. I get lost at the part in bold. Can someone explain what he means with an example. I'm thinking of level-surfaces...
§ 3. I shall show how conversely one may resolve a variability whose region is given into a variability of one dimension and a variability of fewer dimensions. To this end let us suppose a variable piece of a manifoldness of one dimension - reckoned from a fixed origin, that the values of it may be comparable with one another - which has for every point of the given manifoldness a definite value, varying continuously with the point; or, in other words, let us take a continuous function of position within the given manifoldness, which, moreover, is not constant throughout any part of that manifoldness. Every system of points where the function has a constant value, forms then a continuous manifoldness of fewer dimensions than the given one. These manifoldnesses pass over continuously into one another as the function changes; we may therefore assume that out of one of them the others proceed, and speaking generally this may occur in such a way that each point passes over into a definite point of the other;
After rereading this a bunch of times, I'm fairly certain that he's describing how to construct an arbitrary n-dimensional coordinate system over an n-dimensional manifold.
The most confusing part is where he introduces the continuous, non-constant function of position (1) and in the next sentence starts off about points where that function is constant (2). It sounds like a contradiction.
But when I think of how this must apply to an ordinary Euclidean 2D plane with Cartesian coordinates, I imagine (1) to be like walking from the origin along some radial line and recording the x coordinate at each point, and then walking along another radial line and doing the same. After walking all radial lines I can then draw lines thru sets of points (2) where I discovered the same x coordinate, and those are the grid lines of the x axis.