Journal der indischen Mathematical Society, v. 1913, S. 132

Es sei PQR ein Kreis mit dem Mittelpunkt O, von dem PR der Durchmesser ist. Halbiere PO in H, und T sei der Punkt aus der Dreiteilung von OR nahe R. Zeichne TQ senkrecht zu PR und setze die Sehne RS = TQ.

Verbinde P mit S und zeichne OM und TN parallel zu RS. Setze eine Sehne PK = PM und zeichne die Tangente PL = MN. Verbinde R mit L, R mit K und K mit L. Abschnitt RC = RH. Zeichne CD parallel zu KL, [CD] trifft auf RL in D.

Dann ist das Quadrat über RD annähernd gleich dem Kreis PQR.

Denn ${\displaystyle RS^{2}={\frac {5}{36}}d^{2},}$

worin ${\displaystyle d}$ der Durchmesser des Kreises ist.

Somit ${\displaystyle PS^{2}={\frac {31}{36}}d^{2}.}$

Aber ${\displaystyle PL}$ und ${\displaystyle PK}$ sind gleich ${\displaystyle MN}$ bzw. ${\displaystyle PM.}$

Somit ${\displaystyle PK^{2}={\frac {31}{144}}d^{2},}$ und ${\displaystyle PL^{2}={\frac {31}{324}}d^{2}.}$

Folglich ${\displaystyle RK^{2}=PR^{2}-PK^{2}={\frac {113}{144}}d^{2},}$

und ${\displaystyle RL^{2}=PR^{2}+PL^{2}={\frac {355}{324}}d^{2}.}$

Aber ${\displaystyle {\frac {RK}{RL}}={\frac {RC}{RD}}={\frac {3}{2}}{\sqrt {\frac {113}{355}}},}$

und ${\displaystyle RC={\frac {3}{4}}d.}$

Darum ${\displaystyle RD={\frac {d}{2}}{\sqrt {\frac {355}{113}}}=r{\sqrt {\pi }},}$ nahezu gleich.

Anmerkung: Wenn die Fläche des Kreises 140.000 Quadratmeilen ist, dann ist RD um etwa einen Zoll größer als die wahre Länge.

Deshalb ${\displaystyle \textstyle RD={\frac {d}{2}}\cdot {\sqrt {\frac {355}{113}}}}$ dies ist ${\displaystyle r\cdot {\sqrt {\pi }}}$ sehr nahe. Ergänzung: Abweichung = 7,52...E-8 [LE]

Bemerkung

Wenn die Fläche des Kreises 140.000 Quadratmeilen ist, dann ist RD etwa einen Zoll größer als die wahre Länge.

Ergänzung zur Bemerkung

1 Meile [mile] = 1.609,344 m

${\displaystyle \textstyle {\sqrt {140.000}}}$ mile² = 374,165738... mile

RD = 1,772453926158302 m

${\displaystyle \textstyle 374,165738...\cdot {\frac {1.609,344}{1,772453926158302...}}=339.733{,}1668...\;[o.D.]}$

Abweichung${\displaystyle \textstyle =7{,}52...E-8\;m\cdot 339.733{,}1668...\cdot 1000\approx 25{,}55\;mm>1\;Zoll=25{,}4\;mm}$

Ergänzung

RD² = 3,14159292035398 (sechs Nachkommastellen sind gleich denen von ${\displaystyle \pi }$)

(-) ${\displaystyle \pi }$ = 3,14159265358979...

Abweichnung zu ${\displaystyle \pi }$ ≈ 2,667E-7 mm

Bu çalışma ABD'de veya yazarın yaşamının sona ermiş olmasından 70 veya daha fazla süre geçtiğinde bu duruma uygun telif yasaları olan tüm ülkelerde kamu malıdır. Ayrıca bu çalışma ABD'de neden kamu malı olduğuna dair ABD için kamu malı etiketlerini bulundurmalıdır. Note that a few countries have copyright terms longer than 70 years: Mexico has 100 years, Jamaica has 95 years, Colombia has 80 years, and Guatemala and Samoa have 75 years. This image may not be in the public domain in these countries, which moreover do not implement the rule of the shorter term. Honduras has a general copyright term of 75 years, but it does implement the rule of the shorter term. Copyright may extend on works created by French who died for France in World War II (more information), Russians who served in the Eastern Front of World War II (known as the Great Patriotic War in Russia) and posthumously rehabilitated victims of Soviet repressions (more information).

Bu dosyanın, tüm ilgili ve komşu haklar da dâhil olmak üzere, telif hakkı yasası kapsamında bilinen kısıtlamalardan arınmış olduğu belirlendi.

https://creativecommons.org/publicdomain/mark/1.0/PDMCreative Commons Public Domain Mark 1.0falsefalse

This file is in DjVu, a computer file format designed primarily to store scanned documents. You may view this DjVu file here online. If the document is multi-page you may use the controls on the right of the image to change pages. You may also view this DjVu file in your web browser with a browser plugin/add-on, or use a desktop DjVu viewer for your operating system. You can choose suitable software from this list. See Help:DjVu for more information. অসমীয়া ∙ català ∙ čeština ∙ Deutsch ∙ Deutsch (Sie-Form) ∙ English ∙ Esperanto ∙ español ∙ français ∙ galego ∙ magyar ∙ italiano ∙ 日本語 ∙ македонски ∙ Nederlands ∙ polski ∙ português ∙ русский ∙ sicilianu ∙ українська ∙ 简体中文 ∙ 繁體中文 ∙ +/−