Archimedes the Numerical Analyst

Abstract

Let p _N and P _N denote half the lengths of the perimeters of the inscribed and circumscribed regular N-gons of the unit circle. Thus EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa % aaleaacaaIZaaabeaakiabg2da9iaaiodadaGcaaqaaiaaiodaaSqa % baGccaGGVaGaaGOmaiaacYcacaaMi8UaamiCamaaBaaaleaacaaIZa % aabeaakiabg2da9iaaiodadaGcaaqaaiaaiodaaSqabaGccaGGSaGa % aGjcVlaadchadaWgaaWcbaGaaGinaaqabaGccqGH9aqpcaaIYaWaaO % aaaeaacaaIYaaaleqaaaaa!4981! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${p_3} = 3\sqrt 3 /2,{\kern 1pt} {p_3} = 3\sqrt 3 ,{\kern 1pt} {p_4} = 2\sqrt 2 $$, and P ₄ = 4. It is geometrically obvious that the sequences {p _N } and {P _N } are respectively monotonic increasing and monotonic decreasing, with common limit π. This is the basis of Archimedes’ method for approximating to π. (See, for example, Heath [2].) Using elementary geometrical reasoning, Archimedes obtained the following recurrence relation, in which the two sequences remain entwined:

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaac+ % cacaWGqbWaaSbaaSqaaiaaikdacaWGobaabeaakiabg2da9maalaaa % baGaaGymaaqaaiaaikdaaaWaaeWaaeaacaaIXaGaai4laiaadcfada % WgaaWcbaGaamOtaaqabaGccqGHRaWkcaaIXaGaai4laiaadchadaWg % aaWcbaGaamOtaaqabaaakiaawIcacaGLPaaaaaa!45AC! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$1/{P_{2N}} = \frac{1}{2}\left( {1/{P_N} + 1/{p_N}} \right)$$

(1a)

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa % aaleaacaaIYaaabeaakmaaBaaaleaacaWGobaabeaakiabg2da9maa % kaaabaWaaeWaaeaacaWGqbWaaSbaaSqaaiaaikdacaWGobaabeaaki % aadchadaWgaaWcbaGaamOtaaqabaaakiaawIcacaGLPaaaaSqabaaa % aa!4026! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${p_2}_N = \sqrt {\left( {{P_{2N}}{p_N}} \right)} $$

(1b)