Feynman's statement of Fermat's Principle regarding optics is the following,
"a ray going in a certain particular path has the property that if we make a small change (say a one percent shift) in the ray in any manner whatever, say in the location at which it comes to the mirror, or the shape of the curve, or anything, there will be no first-order change in the time; there will be only a second-order change in the time. In other words, the principle is that light takes a path such that there are many other paths nearby which take almost exactly the same time."
Quote taken from http://www.feynmanlectures.caltech.edu/I_26.html
My question is this, doesn't this principle apply to any path, sensible or crazy, between two points? For every possible path, aren't there "many other paths nearby which take almost the same time?"
I can't see why there are limited paths for which Fermat's principle applies, since I can imagine creating many tiny "nearby" variations in any path, leading to a small change in time taken to traverse them.
Why are there limited paths which only have "second-order" changes in time when small variations are applied?