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Archimedes famously determined that $223/71 < \pi < 22/7$ using the 96-gons circumscribed by and circumscribing a circle of unit diameter. But I haven't found a reference that explains the final step, making the rational approximation. For example, the exact perimeter of the circumscribed 96-gon is:

$$ 48\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}} $$

How did Archimedes come up with 223/71 from that?

Jerry Guern
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    http://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html –  Jun 13 '16 at 10:25
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    @selfawareuser, thank you for the link. Deleted my earlier comment since it had nothing to do with reality. Makes sence that Archimedes never actually used square roots, just invented an algorithm for approximating perimeters of $2n6$-gons – Yuriy S Jun 14 '16 at 06:54
  • The joy of having a question answered by an anonymous link, now rotted, by an account that's now deleted... :-/ – theHigherGeometer Jul 21 '22 at 23:42
  • Here's an archived version of that page https://web.archive.org/web/20190101123108/https://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html – theHigherGeometer Jul 21 '22 at 23:45

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