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Re: [HM] Snell, Fermat, light rays, and all that
Posted:
Oct 18, 1998 1:31 PM
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Ken:
You might enjoy the demonstration by Christian Huygens at the end of his
chapter three on Refraction:
Treatise on Light.
I have an undated edition rendered into English by Silvanus P. Thompson,
University of Chicago Press, in which the demonstration begins on page 42,
last paragraph:
I will finish this theory of refraction by demonstrating a remarkable
proposition which depends on it; namely, that a ray of light in order to
go from one point to another, when these points are in different media,
is refracted in such wise at the plane surface which joins these two
media that it employs the least possible time: and exactly the same
happens in the case of reflexion against a plane surface. Mr Fermat was
the first to propound this property of refraction holding with us, and
directly counter to the opinion of Mr. Des Cartes, that light passes
more slowly through glass and water than through air. But he assumed
besides this a constant ratio of Sines, which we have just proved by
these different degrees of velocity alone . . .His demonstration,
which may be seen in his printed works, and in the volume of letters
of Mr. Des Cartes, is very long; wherefore I give here another which is
simpler and easier.
What follows is Huygens' solution without equations--just geometry.
Best wishes from Annapolis,
Sam Kutler
> Hi,
>
> I'm piggybacking on my wife's subscription (hope that's OK) and have a
> question that may interest others as well.
>
> I plan to give a short discovery-type lesson demonstrating: If a line
> divides a plane so that a tourist travels at speed v in one half-plane
> and speed w in the other half-plane, and if A and B are in different
> half-planes then the fasted route from A to B is along a bent line
> obeying Snell's law.
>
> I would like to be historically correct.
>
> My encyclopedia tells me that Snell discovered/published his law of
> refraction of light in 1621. This is usually stated v/w=sin theta/sin phi
> where theta and phi are the path angles to the normal at the interface.
> But of course the speeds v and w were totally unknown at the time, or so
> I understand, so it seems unlikely the law was formulated in exactly that
> way: sin theta/sin phi = const seems like a law that could be empirically
> discovered/verified. The constant depending on the two media.
> So Q1: What did Snell discover/assert?
>
> Next, as I understand it, Fermat showed that Snell's result could be
> derived by assuming light follows a minimum time path (local minimum
> anyway) with light speed dependent upon medium. That is, he derived the
> usual formulation.
> Q2: Was there any consensus at the time that the speed of light was even
> finite? Was this another brilliant insight of Fermat or had there been
> some previous speculation along these lines? His proof, without calculus,
> is brilliant mathematics but what was the status of physical thought at
> the time?
>
> Finally, Q3. This is probably an item where the record is easier to
> discover: Who/when/how empirically worked out the speed of light in
> space/air/water/etc.? References will suffice, I'll look it up.
>
> I appreciate any helpful comments and I apologize if I have inadvertently
> misused the list. In particular this is a "reply" since I don't see how to
> just "start". Sorry.
>
> Ken Berg
> Dept. of Math.
> Univ. of MD.
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