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In this lecture about Fermat's principle, Feynman derives the refraction law using the principle of least action. He finds the shortest path between $A$ and $B$ by defining a variable point $X$ and setting its distance to the point corresponding to the shortest path, $C$, to zero. He said,

If we were to plot the time it takes against the position of point X, we would get a curve something like that shown below, where point C corresponds to the shortest of all possible times. This means that if we move the point X to points near C, in the first approximation there is essentially no change in time. enter image description here

In deriving Snell's law, he used the following approximation:
$$ \angle XCF\approx \angle BCN' ~~~(\text{when $X$ is near $C$}),$$

enter image description here

Where did this approximation exactly come from?

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If the lines BC and BF were parallel, this would be exact.

Is X is close to C, I.e. F is close to C, those lines are close to parallel and the approximation is good. What is “close”? Much closer than B is far away, so the lines become parallel. That’s a bit circular, but this proof relies on infinitesimals: “if it’s not a good enough approximation, pick a closer point and repeat until done”

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