I was rashly extrapolating from the original statement by Archimedes, which I learned from Andy Gleason when I was an undergraduate. I’m working from memory here, but it went something like this: if you have a line segment from point P to point Q, a convex curve from P to Q, and another convex curve from P to Q, such that the first curve lies between the line segment and the second curve, then the length of the first curve is intermediate between the length of the line segment and the length of the second curve. Jim Propp On Mon, Jun 24, 2019 at 2:44 AM Brad Klee <bradklee@gmail.com> wrote:
Hi Jim,
Is your reference "Archimede's Axioms for Arc-Length and Area" by Scott Brodie [1]? If yes, you need another condition on L.
This example with the two ellipses shows none of the pathology of Brodie's Figure 2, so (unless Adam wants to tell us more about his thesis) we really don't have context to need to worry about Archimede's Axiom 2.
However, Brodie's Figure 2 is a nice challenge to the law of sines technique. Yet if we approximate FBG as a triangle, and ignore small deviations, then FB is shorter than FG + GB by the triangle inequality. As the angles FBG and BFG are significantly greater than the small angle at A, we can be fairly confident that the contained curve is the shorter.
--Brad
[1] https://www.jstor.org/stable/2690029?seq=1#page_scan_tab_contents
On Sun, Jun 23, 2019 at 9:49 PM James Propp <jamespropp@gmail.com> wrote:
In case that wasn’t clear, I was replying to Adam Goucher’s question, saying that I believe that Archimedes said something that in modern language would be that if L is a compact subset of R^2 and K is a convex compact subset of L, then the perimeter of K is bounded above by the perimeter of L, with equality if and only if K = L.
That is: Rather than giving a definition of arc length (as we moderns
would
feel obliged to do), Archimedes gave a characterization of arc length that is just as useful.
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