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.
Sources
- 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.5$: Archimedes (ca. $\text {287}$ – $\text {212}$ B.C.)