International Journal of Pure and Applied Mathematics Volume 7 No ,

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1 Iteratioal Joural of Pure ad Applied Mathematics Volume 7 No. 003, 07-1 AN IMPROVEMENT OF ARCHIMEDES METHOD OF APPROXIMATING π Gopal Chakrabarti 1, Richard Hudso 1 Uiversity of South Carolia Columbia SC 908, USA Chakrab@egr.sc.edu Departmet of Mathematics Uiversity of South Carolia Columbia SC 908, USA hudso@math.sc.edu Abstract: About 55 B.C., Archimedes used the perimeters of iscribed ad circumscribed polygos of 6, 1, 4, 38, ad 96 sides to fid upper ad lower bouds for π. Usig 96 sides, he showed that < π < I the followig 1900 years, may improvemets of Archimedes bouds were obtaied usig polygos with more sides, but little progress was made i improvig his method. I Cyclometricus (161, Willebrord Sell (Sellius obtaied a dramatic improvemet by suggestig that the perimeter of the iscribed polygo of sides coverges to π twice as fast as the perimeter of the circumscribed polygo, though this was oly first proved by Christia Huyges [] i Usig this, Sell obtaied 7 digits of π usig a 96 sided polygo, ad obtaied 34 digits of π with = 30. I this paper, we improve the Sell-Huyges method by provig that areas of iscribed ad circumscribed polygos of sides, whe averaged i much the same way that Huyges averaged perimeters, coverge exactly 3 8 as fast (i the limit. Usig this, we obtai 10 digits of π whe = 96, ad whe = 6 we obtai 110 digits of π. The table i Sectio 3 ad aalysis of the proof i Sectio suggest for polygos of sides, the Sell-Huyges method gives about twice as may digits of precisio as Archimedes method, our method gives three times as may digits, ad uses oly regular polygos. Received: April 19, 003 c 003, Academic Publicatios Ltd. Correspodece author

2 08 G. Chakrabarti, R. Hudso AMS Subject Classificatio: 11A03, 11A04, 01A08 Key Words: Archimedes method of approximatig π 1. Itroductio ad Summary About 55 B.C., Archimedes (87-1 B.C., i his famous treatise O the Measuremet of the Circle, obtaied a method of approximatig π. Usig a recursive algorithm he foud the perimeters of iscribed ad circumscribed polygos of 6, 1, 4, 48, ad 96 sides. Whe = 96, he obtaied the lower ad upper bouds < π < Sice these upper ad lower bouds have oly two digits of precisio i the approximatio of π ( digits of precisio are defied to be a error of less tha 10, Archimedes is ofte credited with oly two digits of accuracy i his approximatio of π, but simply averagig of his upper ad lower bouds gives a error of about , so we credit him with 3 digits of precisio i our table i Sectio 3. I the succeedig years from 55 B.C. to 1654 A.D., there were may otable improvemets i the bouds obtaied by Archimedes, but almost all were obtaied by usig polygos with more ad more sides. I 63 A.D., Liu Hui [, p. 7] used a 307-go to obtai π About 480 A.D., Tsu Chug-Chih ad his so Tsu-Keg-Chih obtaied the impressive bouds < π < , ad the best approximatio of π usig fractios with 3 or fewer digits i the umerator ad deomiator, π I 144, Al Kashi obtaied 14 decimal places usig a polygo of 6 7 sides, ad i 1593, Romaus [, p. 10] obtaied 15 decimal places with a polygo of 30 sides. I 1609, Ludolph va Ceule, who was Sell teacher ad who devoted his life to this problem, obtaied 35 decimal places usig a polygo of 6 sides, though some authors say that Sell actually completed the computatio of the last 3 decimal places. The 35 decimal places were published posthumously, ad it is said that his widow, at his request, egraved the digits o his tombstoe, though other historias say oly the last 3 digits were egraved [, p. 10]. The stoe has log sice bee lost. I ay case, π became kow i Germay as the Ludolphie umber (the symbol for π was ot actually used util The brilliat mathematicia ad physicist Willebrord Sell was a studet of Ludolph va Ceule ad became iterested i 1617 i fidig a sigificat acceleratio of Archimedes method which would make it possible to fid as may digits of π as possible usig a polygo of at most sides. I Cyclometricus

3 AN IMPROVEMENT OF ARCHIMEDES i 161 he gave such a result. He idicated (without formal proof that the perimeter of a iscribed polygo of sides coverges to π twice as fast (i the limit as the perimiter of the circumscribed polygo. He ad others also used, see, e.g. [, p ], irregular polygos to obtai eve sharper lower ad upper bouds. I this paper we restrict ourselves to the regular polygos used by Archimedes. Christias Huyges [, p. 114] gave the first formal proof of Sell result i 1654 i De circuli magitude iveta. Usig this result, which follows immediately from the first two terms of the MacClauri expasios give i Sectio, he showed empirically that 3 si ta π (1.1 faster tha the simple average of si ( ( π ad ta π, (the perimeters of the iscribed ad circumscribed regular polygos of sides, respectively. Ideed, usig a 96-go, Sell obtaied 7 digits of precisio, whereas Archimedes method yields oly 3 digits. Moreover, Sell showed that usig (1.1 with = 30 yields 34 decimal places of π, greatly reducig the 6 sides required by Ludolph va Ceule to obtai 35 places. I this paper we improve the Sell-Huyges method. Of course, i the last 400 years, may powerful methods have bee developed to compute π so that our improvemet is, as Sell was, oly a improvemet i the sese that oe obtais as may digits of π as possible usig iformatio from polygos of sides. Recetly, Kaada formulas have bee used to compute digits of π usig more powerful methods. The key to our improvemet is ot to igore the iformatio obtaied from lookig at the areas of these polygos. It has log bee kow that areas are less useful tha perimeters i approximatig π as they coverge less rapidly. It seem ot to have bee observed, however, the covergece is exactly 3 8 as rapid whe areas are treated i the same way that Sell treated perimeters. Usig this result, we prove i Sectio that ( ( π π 30 si + 4ta 3si π (1. much faster tha 3 si( π + 3 ta. Ideed, whe = 6 (hexago, we obtai 3 digits of precisio with a error of oly , ad whe = 96, we obtai 10 digits of precisio i the approximatio of π i compariso to 3 for Archimedes ad 7 for Sell-Huyges. Whe = 6, we obtai 110 digits of π. See the table i Sectio 3 for further comparisos of Archimedes method, Sell-Huyges method, ad our method. I the ext sectio we prove

4 10 G. Chakrabarti, R. Hudso that our method has a error of O ( 1 ; see (.3-(.5. I light of this, ad the 6 correspodig O ( ( 1 for the Sell-Huyges method ad O 1 4 for Archimedes method, it is ot surpirisig that the values i the table i Sectio 3 idicate that for a give value of, the Sell-Huyges methods yields about twice as may digits of precisio as Archimedes method, ad our method yields three times as may.. Proof of the Theorem Let I P ad O P deote the perimeters of iscribed ad circumscribed polygos of sides (resp. o a circle of circumferece π. It is easy to show that I P = si ad O P = ta. (.1 Similarly, let I A ad O A deote the areas of these polygos. The, it is ay easy exercise i trigoometry to show that I A = ( π si ad O A = ta. (. Let U = 3 I P O P ad V = 3 O A I A. We prove the folowig theorem. V π Theorem.1. lim U π = 8 3. Proof. Usig well-kow MacClauri series expasios for si x ad ta x we have si = I P = π π3 ta = O P = π + π3 si ( π 6 π O 3 + π O = I A = π π3 3 + π O ( 1 6 ( 1 6 ( 1 6, (.3, (.4. (.5 From the first two terms we see that I P π twice as fast as O P as was observed by Sell ad proved by Huyges, ad O A π twice as fast as I A (with error O ( 1 4. For large we have O P π π I P or 3 O P I P π.

5 AN IMPROVEMENT OF ARCHIMEDES Similarly, π O A O A π or 3 O A I A π. Now usig (.3, (.4, ad (.5, we have after cacellatio ad lettig terms which ted to zero vaish, that ( ( 4π 5 1 V π lim U π = 45 + π π ( 1 4 = 8 3. We ote that V π 8U 3V is equivalet to 8U 3V 5 π. Moreover, 8U 3V 5 simplifies easily to ( ( ( π π 3si + 4ta 3si. ( Numerical Results The followig table gives the umber of digits of precisio obtaied i the evaluatio of π usig Archimedes method, Sell-Huyges method ad our method. Archimedes Sell-Huyges Chakrabarti-Hudso The umbers i the last three colums give the digits of precisio. A umber has precisio if π < 10. The values for Archimedes are calculated by takig the simple average of his lower ad upper bouds. Ludolph va Ceule obtaied 0 digits of π usig a polygo of sides. He used several methods to accelerate covergece to π, but simple averagig, I P +O P gives the 35 digits of π which were reputedly egraved o his tombstoe (i fact the simple average gives 36 digits. Sell-Huyges obtai 37 more digits ad we obtai 70 more digits of precisio usig the same iscribed ad circumscribed polygos of 6

6 1 G. Chakrabarti, R. Hudso sides used by va Ceule. Whe = 6 (hexagos, the error usig Archimedes method is , usig Sell-Huyges method, is , ad usig our method, is , ad the digits of precisio are 1,, ad 3 respectively. Ackowledgemets We would like to thak Erie Croot ad Michael Filaseta who used Maple to compute the values i the table i Sectio 3 whe > Refereces [1] Le Petit Archimede, No. Hors Serie, Le Nombre π (1980. [] P. Beckma, A History of Pi, St. Marti s Press, New York (1971. [3] J. Ardt, C. Haeel, π-uleashed, Spriger (001. [4] J.P. Delahaye, Le fasciat omber π, Bibliotheque pour la Sciece, Berli (1997. [5] H. Egels, Quadrature of the circle i Aciet Egypt, Historia Mathematica, 4 (1977, [6] T.L. Heath, The Works of Archimedes, Cambridge Uiversity Press (1897. [7] L.Y. Lam, T.S. Ag, Circle measuremets i Aciet Chia, Historia Mathematica, 13 (1986, [8] G.M. Phillips, Archimedes the umerical aalyst, The America Mathematical Mothly, 88 (1981, [9] H.C. Schelper, The chroology of Pi, Mathematics Magazie (1950. [10] F. Viete, Opera Mathematica, Reprited, Georg Olms Verlag, Hildesheim, New York (1970. [11] E. Zebrowski, A History of the Circle: Mathematical Reasoig ad the Physical Uiverse, Rutgers Uiversity Press, New Bruswick, New Jersey (1999.