Two definitions of $\pi$

Asked Mar 14 '19 at 11:04

Active Mar 14 '19 at 11:55

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I have the feeling that ancient mathematicians (like Greek or Chinese), trying to find good approximations of $\pi$ used two definitions:

Had they a proof that these definitions are of the same number?

edited Mar 14 '19 at 11:05

asked Mar 14 '19 at 11:04

ajotatxe

2

This question belongs on hsm.se, but I presume they used this to explain why the circumference being $2\pi r$ implies the area is $\pi r^2$. Update: no, Archimedes used this instead. See also this question. – J.G. Mar 14 '19 at 11:05
@J.G.Aren't they basically the same? – tomasz Mar 14 '19 at 11:13
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IMO, Ancient Greeks do not think at a "constant" (they do not named it). See Archimedes' Measurement : they proved that $A=\dfrac 1 2 (\ell r)$ and calculated approximations for the ratio $\mathcal \ell : d$. But this ratio was not a number for ancient Greeks. – Mauro ALLEGRANZA Mar 14 '19 at 11:30
Maybe useful : AJEM Smeur, On the Value Equivalent to pi in Ancient Mathematical Texts, (AHES, 1970). See also Pi: A Source Book. – Mauro ALLEGRANZA Mar 14 '19 at 11:40

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