Archimedes' Limits to Value of Pi/Historical Note

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Historical Note on Archimedes' Limits to Value of Pi

Archimedes demonstrated the limits to the value of $\pi$ in his Measurement of a Circle.

He does not say where the two estimates $\dfrac {265} {153}$ and $\dfrac {1351} {780}$ for $\sqrt 3$ come from.

They can be easily derived from the continued fraction representation, namely, $\sqbrk {1, \sequence {1, 2} }$, but whether he knew this, or had another method, is unclear.