Where, in geometry, is the notion of "path length of a curve" introduced? This question is inspired by a question (now closed) about proving that a path entering and exiting a circle that intersects all diameters of that circle, must have length at least as great as that of a diameter. The question was asked in the context of neutral geometry, and I started thinking about it, and realized that I don't know, and a definition level, what "curve" and "path length" mean in neutral geometry.
Then I realized that I don't know what those concepts mean even in Euclidean geometry.  Of course, a curve can be described as a locus of points having some property (for example, a circle or an ellipse) and in cases where the points on a curve have some natural ordering, the path length can be defined in a calculus-like way, (a calculous way?) in terms of the sum of lengths of tiny line segments.  (Think Archimedes estimating the circumference of a circle.)
But when I have to prove something about a generic curve (even a generic continuous curve) I have no general definition to start from.  But this must be a well-studied matter.
In the context of the question that had been asked, I suspect that going to a definition that involves open sets and so forth would be way beyond the pale; but if that is where one has to go, so be it.
 A: This is the definition of the length of a curve in a general metric space $(X,d)$.  Define a partition of $[0,1]$ to be a finite set $\{t_0,...,t_n\}$ so that $t_0=0$ and $t_n=1$ and $t_i>t_{i-1}$. A curve $\gamma:[0,1]\to X$ has length
$$\ell(\gamma)=\sup\left\{\sum_{i=1}^n d(\gamma(t_i),\gamma(t_{i-1})):\{t_0,...,t_n\}\text{ a partition of $[0,1]$}\right\}.$$
A curve is rectifiable if $\ell(\gamma)<\infty$. Not all curves are rectifiable.
This definition is used frequently when studying general geodesic spaces, where a curve is a geodesic if $d(\gamma(0),\gamma(1))=\ell(\gamma)$.
A: There is not much place for continuity in the Hilbert axiomatic approach to plane geometry. I published something and got to know Marvin Greenberg as a result. My little article was very much from the viewpoint of the Euclidean plane and the hyperbolic plane as unusually pleasant Riemannian manifolds. Therefore i had distances and such. This was the tradition in the Russian articles I was quoting from about 1930-1955, so it all seemed sensible to me.
You might borrow Hartshorne Geometry:Euclid and Beyond  and Greenberg Euclidean and Non-Euclidean Geometries. I'm in the fourth edition of the latter. In fact, see ARTICLE HERE   you can download the article itself. 
For material closer to the real plane, i used George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane.  Very strong on actual compass and straightedge constructions. 
Hmmm.... Just flipped through Hartshorne, I see lengths of segments, possibly lengths of circular arcs defined according to central angle... A fair amount of work discussing area for polygons, and i do remember some comment about the minimum requirements to be able to define area of a circle at all, something about a limit of polygon areas, so the field in question is not that far short of the real numbers anyway. 
