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size of the universe subsection</span> </button> <ul id="toc-Estimation_of_the_size_of_the_universe-sublist" class="vector-toc-list"> <li id="toc-Calculation_of_the_number_of_grains_of_sand_in_the_Aristarchian_Universe" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Calculation_of_the_number_of_grains_of_sand_in_the_Aristarchian_Universe"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Calculation of the number of grains of sand in the Aristarchian Universe</span> </div> </a> <ul id="toc-Calculation_of_the_number_of_grains_of_sand_in_the_Aristarchian_Universe-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Additional_calculations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Additional_calculations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Additional calculations</span> </div> </a> <ul id="toc-Additional_calculations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Quote" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Quote"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Quote</span> </div> </a> <ul id="toc-Quote-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" 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Available in 17 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-17" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">17 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D0%B1%D1%80%D0%BE%D0%B8%D1%82%D0%B5%D0%BB%D1%8F%D1%82_%D0%BD%D0%B0_%D0%BF%D1%8F%D1%81%D1%8A%D0%BA" title="Преброителят на пясък – Bulgarian" lang="bg" hreflang="bg" data-title="Преброителят на пясък" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/L%27Arenari" title="L'Arenari – Catalan" lang="ca" hreflang="ca" data-title="L'Arenari" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Der_Sandrechner" title="Der Sandrechner – German" lang="de" hreflang="de" data-title="Der Sandrechner" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A8%CE%B1%CE%BC%CE%BC%CE%AF%CF%84%CE%B7%CF%82_(%CE%91%CF%81%CF%87%CE%B9%CE%BC%CE%AE%CE%B4%CE%B7%CF%82)" title="Ψαμμίτης (Αρχιμήδης) – Greek" lang="el" hreflang="el" data-title="Ψαμμίτης (Αρχιμήδης)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/El_contador_de_arena" title="El contador de arena – Spanish" lang="es" hreflang="es" data-title="El contador de arena" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/L%27Ar%C3%A9naire" title="L'Arénaire – French" lang="fr" hreflang="fr" data-title="L'Arénaire" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AA%A8%EB%9E%98%EC%95%8C%EC%9D%84_%EC%84%B8%EB%8A%94_%EC%82%AC%EB%9E%8C" title="모래알을 세는 사람 – Korean" lang="ko" hreflang="ko" data-title="모래알을 세는 사람" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/The_Sand_Reckoner" title="The Sand Reckoner – Indonesian" lang="id" hreflang="id" data-title="The Sand Reckoner" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Arenario" title="Arenario – Italian" lang="it" hreflang="it" data-title="Arenario" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9F%D1%81%D0%B0%D0%BC%D0%BC%D0%B8%D1%82" 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Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22The+Sand+Reckoner%22">"The Sand Reckoner"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22The+Sand+Reckoner%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22The+Sand+Reckoner%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22The+Sand+Reckoner%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22The+Sand+Reckoner%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22The+Sand+Reckoner%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">August 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox vcard"><caption class="infobox-title" style="font-size:125%; font-style:italic; padding-bottom:0.2em;">The Sand Reckoner (Arenarius) <span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Sand+Reckoner+%28Arenarius%29&rft.author=%5B%5BArchimedes%5D%5D"></span></caption><tbody><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Arenarius.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Arenarius.jpg/220px-Arenarius.jpg" decoding="async" width="220" height="357" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Arenarius.jpg/330px-Arenarius.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Arenarius.jpg/440px-Arenarius.jpg 2x" data-file-width="552" data-file-height="896" /></a></span></td></tr><tr><th scope="row" class="infobox-label">Author</th><td class="infobox-data"><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></td></tr><tr><th scope="row" class="infobox-label">Language</th><td class="infobox-data">Latin</td></tr><tr><th scope="row" class="infobox-label">Genre</th><td class="infobox-data"><a href="/wiki/Googology" class="mw-redirect" title="Googology">Googology</a>, <a href="/wiki/Astronomy" title="Astronomy">Astronomy</a></td></tr></tbody></table> <p><i><b>The Sand Reckoner</b></i> (<a href="/wiki/Greek_language" title="Greek language">Greek</a>: <span lang="el">Ψαμμίτης</span>, <i>Psammites</i>) is a work by <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a>, an <a href="/wiki/Ancient_Greek" title="Ancient Greek">Ancient Greek</a> mathematician of the <a href="/wiki/3rd_century_BC" title="3rd century BC">3rd century BC</a>, in which he set out to determine an upper bound for the number of grains of sand that fit into the <a href="/wiki/Universe" title="Universe">universe</a>. In order to do this, Archimedes had to estimate the size of the universe according to the contemporary model, and invent a way to talk about extremely large numbers. </p><p>The work, also known in Latin as <i>Arenarius</i>, is about eight pages long in translation and is addressed to the <a href="/wiki/Syracuse,_Sicily" title="Syracuse, Sicily">Syracusan</a> king <a href="/wiki/Gelo,_son_of_Hiero_II" title="Gelo, son of Hiero II">Gelo II</a> (son of <a href="/wiki/Hiero_II_of_Syracuse" title="Hiero II of Syracuse">Hiero II</a>). It is considered the most accessible work of Archimedes.<sup id="cite_ref-v_1-0" class="reference"><a href="#cite_note-v-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Naming_large_numbers">Naming large numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=The_Sand_Reckoner&action=edit&section=1" title="Edit section: Naming large numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Exponentiation#History" title="Exponentiation">Exponentiation § History</a></div> <table class="wikitable collapsible collapsed" style="text-align:center;float:right;margin:0 0 0 1ex;"> <tbody><tr> <th colspan="4" style="font-weight:normal;"><i>Periods</i> and <i>orders</i> with their <a href="/wiki/Interval_(mathematics)#Including_or_excluding_endpoints" title="Interval (mathematics)">intervals</a> in modern notation<sup id="cite_ref-eureka_man_2-0" class="reference"><a href="#cite_note-eureka_man-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </th></tr> <tr> <th>Period</th> <th>Order</th> <th>Interval</th> <th>log<sub>10</sub> of interval </th></tr> <tr> <td rowspan="6"><b>1</b></td> <td>1</td> <td>(1, <i>Ơ</i>], where the<br /><i>unit of the second order</i>,<br /><i>Ơ</i> = 10<sup>8</sup></td> <td>(0, 8] </td></tr> <tr> <td>2</td> <td>(<i>Ơ</i>, <i>Ơ</i><sup>2</sup>]</td> <td>(8, 16] </td></tr> <tr> <td colspan="3" style="padding:0;line-height:1ex;">⋅⋅⋅ </td></tr> <tr> <td><i>k</i></td> <td>(<i>Ơ</i><sup><i>k</i> − 1</sup>, <i>Ơ</i><sup><i>k</i></sup>]</td> <td>(8<i>k</i> − 8, 8<i>k</i>] </td></tr> <tr> <td colspan="3" style="padding:0;line-height:1ex;">⋅⋅⋅ </td></tr> <tr> <td><i>Ơ</i></td> <td>(<i>Ơ</i><sup><i>Ơ</i> − 1</sup>, <i>Ƥ</i>], where the<br /><i>unit of the second period</i>,<br /><i>Ƥ</i> = <i>Ơ</i><sup><i>Ơ</i></sup> = 10<sup>8<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000800000000000000♠"></span>8</span></sup></sup></td> <td>(8<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000800000000000000♠"></span>8</span></sup> − 8, 8<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000800000000000000♠"></span>8</span></sup>]<br />= (799,999,992, 800,000,000] </td></tr> <tr style="border-top:2px solid #999999;"> <td rowspan="6"><b>2</b></td> <td>1</td> <td>(<i>Ƥ</i>, <i>ƤƠ</i>]</td> <td>(8<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000800000000000000♠"></span>8</span></sup>, 8 × (10<sup>8</sup> + 1)]<br />= (800,000,000, 800,000,008] </td></tr> <tr> <td>2</td> <td>(<i>ƤƠ</i>, <i>ƤƠ</i><sup>2</sup>]</td> <td>(8 × (10<sup>8</sup> + 1), 8 × (10<sup>8</sup> + 2)] </td></tr> <tr> <td colspan="3" style="padding:0;line-height:1ex;">⋅⋅⋅ </td></tr> <tr> <td><i>k</i></td> <td>(<i>ƤƠ</i><sup><i>k</i> − 1</sup>, <i>ƤƠ</i><sup><i>k</i></sup>]</td> <td>(8 × (10<sup>8</sup> + <i>k</i> − 1), 8 × (10<sup>8</sup> + <i>k</i>)] </td></tr> <tr> <td colspan="3" style="padding:0;line-height:1ex;">⋅⋅⋅ </td></tr> <tr> <td><i>Ơ</i></td> <td>(<i>ƤƠ</i><sup><i>Ơ</i> − 1</sup>, <i>ƤƠ</i><sup><i>Ơ</i></sup>]<br />= (<i>Ƥ</i><sup>2</sup><i>Ơ</i><sup>−1</sup>, <i>Ƥ</i><sup>2</sup>] </td> <td style="text-align:left;">(8 × (2<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000800000000000000♠"></span>8</span></sup> − 1), 8 × (2<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000800000000000000♠"></span>8</span></sup>)]<br />= (1.6<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000900000000000000♠"></span>9</span></sup> − 8, 1.6<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000900000000000000♠"></span>9</span></sup>]<br />= (1,599,999,992, 1,600,000,000] </td></tr> <tr> <td colspan="4" style="padding:0;">⋅⋅⋅ </td></tr> <tr> <td rowspan="6"><i><b>Ơ</b></i></td> <td>1</td> <td>(<i>Ƥ</i><sup><i>Ơ</i> − 1</sup>, <i>Ƥ</i><sup><i>Ơ</i> − 1</sup><i>Ơ</i>] </td> <td style="text-align:left;">(8<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000800000000000000♠"></span>8</span></sup> × (10<sup>8</sup> − 1),<br /> 8 × (10<sup>8</sup> × (10<sup>8</sup> − 1) + 1)]<br />= (79,999,999,200,000,000,<br />    79,999,999,200,000,008] </td></tr> <tr> <td>2</td> <td>(<i>Ƥ</i><sup><i>Ơ</i> − 1</sup><i>Ơ</i>, <i>Ƥ</i><sup><i>Ơ</i> − 1</sup><i>Ơ</i><sup>2</sup>]</td> <td>(8 × (10<sup>8</sup> × (10<sup>8</sup> − 1) + 1),<br />  8 × (10<sup>8</sup> × (10<sup>8</sup> − 1) + 2)] </td></tr> <tr> <td colspan="3" style="padding:0;line-height:1ex;">⋅⋅⋅ </td></tr> <tr> <td><i>k</i></td> <td>(<i>Ƥ</i><sup><i>Ơ</i> − 1</sup><i>Ơ</i><sup><i>k</i> − 1</sup>, <i>Ƥ</i><sup><i>Ơ</i> − 1</sup><i>Ơ</i><sup><i>k</i></sup>]</td> <td>(8 × (10<sup>8</sup> × (10<sup>8</sup> − 1) + <i>k</i> − 1),<br />8 × (10<sup>8</sup> × (10<sup>8</sup> − 1) + <i>k</i>)] </td></tr> <tr> <td colspan="3" style="padding:0;line-height:1ex;">⋅⋅⋅ </td></tr> <tr> <td><i>Ơ</i></td> <td>(<i>Ƥ</i><sup><i>Ơ</i> − 1</sup><i>Ơ</i><sup><i>Ơ</i> − 1</sup>, <i>Ƥ</i><sup><i>Ơ</i> − 1</sup><i>Ơ</i><sup><i>Ơ</i></sup>]<br />= (<i>Ƥ</i><sup><i>Ơ</i></sup><i>Ơ</i><sup>−1</sup>, <i>Ƥ</i><sup><i>Ơ</i></sup>] </td> <td style="text-align:left;">(8 × (2<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000800000000000000♠"></span>8</span></sup> − 1), 8 × (2<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000800000000000000♠"></span>8</span></sup>)]<br />= (8<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001160000000000000♠"></span>16</span></sup> − 8, 8<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001160000000000000♠"></span>16</span></sup>]<br />= (79,999,999,999,999,992,<br />    80,000,000,000,000,000] </td></tr></tbody></table> <p>First, Archimedes had to invent a system of naming <a href="/wiki/Large_numbers" title="Large numbers">large numbers</a>. The number system in use at that time could express numbers up to a <a href="/wiki/Myriad" title="Myriad">myriad</a> (μυριάς — 10,000), and by utilizing the word <i>myriad</i> itself, one can immediately extend this to naming all numbers up to a myriad myriads (10<sup>8</sup>).<sup id="cite_ref-Analysis_3-0" class="reference"><a href="#cite_note-Analysis-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Archimedes called the numbers up to 10<sup>8</sup> "first order" and called 10<sup>8</sup> itself the "unit of the second order". Multiples of this unit then became the second order, up to this unit taken a myriad-myriad times, 10<sup>8</sup>·10<sup>8</sup>=10<sup>16</sup>. This became the "unit of the third order", whose multiples were the third order, and so on. Archimedes continued naming numbers in this way up to a myriad-myriad times the unit of the 10<sup>8</sup>-th order, i.e., (10<sup>8</sup>)^(10<sup>8</sup>) </p><p>After having done this, Archimedes called the orders he had defined the "orders of the first period", and called the last one, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (10^{8})^{(10^{8})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (10^{8})^{(10^{8})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fccb281bfa71183e0e923fe9679292c6ca76c8d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.176ex; height:3.509ex;" alt="{\displaystyle (10^{8})^{(10^{8})}}"></span>, the "unit of the second period". He then constructed the orders of the second period by taking multiples of this unit in a way analogous to the way in which the orders of the first period were constructed. Continuing in this manner, he eventually arrived at the orders of the myriad-myriadth period. The largest number named by Archimedes was the last number in this period, which is </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\left(10^{8}\right)^{(10^{8})}\right)^{(10^{8})}=10^{8\cdot 10^{16}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\left(10^{8}\right)^{(10^{8})}\right)^{(10^{8})}=10^{8\cdot 10^{16}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/436c3937f34864c082b7f3b05dabddc6fb2447e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.983ex; height:5.843ex;" alt="{\displaystyle \left(\left(10^{8}\right)^{(10^{8})}\right)^{(10^{8})}=10^{8\cdot 10^{16}}.}"></span></dd></dl></dd></dl> <p>Another way of describing this number is a one followed by (<a href="/wiki/Long_and_short_scales" title="Long and short scales">short scale</a>) eighty quadrillion (80·10<sup>15</sup>) zeroes. </p><p>Archimedes' system is reminiscent of a <a href="/wiki/Positional_numeral_system" class="mw-redirect" title="Positional numeral system">positional numeral system</a> with base 10<sup>8</sup>, which is remarkable because the ancient Greeks used a very <a href="/wiki/Greek_numerals" title="Greek numerals">simple system for writing numbers</a>, which employs 27 different letters of the alphabet for the units 1 through 9, the tens 10 through 90 and the hundreds 100 through 900. </p> <div class="mw-heading mw-heading3"><h3 id="Law_of_exponents">Law of exponents</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=The_Sand_Reckoner&action=edit&section=2" title="Edit section: Law of exponents"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Archimedes also discovered and proved the <a href="/wiki/Exponentiation" title="Exponentiation">law of exponents</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{a}10^{b}=10^{a+b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{a}10^{b}=10^{a+b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf5c3052d56540abdd66ce215ffd7f3cdd1189a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.198ex; height:2.676ex;" alt="{\displaystyle 10^{a}10^{b}=10^{a+b}}"></span>, necessary to manipulate powers of 10. </p> <div class="mw-heading mw-heading2"><h2 id="Estimation_of_the_size_of_the_universe">Estimation of the size of the universe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=The_Sand_Reckoner&action=edit&section=3" title="Edit section: Estimation of the size of the universe"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Archimedes then estimated an upper bound for the number of grains of sand required to fill the Universe. To do this, he used the <a href="/wiki/Heliocentric_model" class="mw-redirect" title="Heliocentric model">heliocentric model</a> of <a href="/wiki/Aristarchus_of_Samos" title="Aristarchus of Samos">Aristarchus of Samos</a>. The original work by Aristarchus has been lost. This work by Archimedes however is one of the few surviving references to his theory,<sup id="cite_ref-a_4-0" class="reference"><a href="#cite_note-a-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> whereby the <a href="/wiki/Sun" title="Sun">Sun</a> remains unmoved while the <a href="/wiki/Earth" title="Earth">Earth</a> <a href="/wiki/Orbital_period" title="Orbital period">orbits</a> the Sun. In Archimedes's own words: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>His [Aristarchus'] hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun on the circumference of a circle, the Sun lying in the middle of the orbit, and that the sphere of fixed stars, situated about the same center as the Sun, is so great that the circle in which he supposes the Earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>The reason for the large size of this model is that the Greeks were unable to observe <a href="/wiki/Stellar_parallax" title="Stellar parallax">stellar parallax</a> with available techniques, which implies that any parallax is extremely small and so the stars must be placed at great distances from the Earth (assuming <a href="/wiki/Heliocentrism" title="Heliocentrism">heliocentrism</a> to be true). </p><p>According to Archimedes, Aristarchus did not state how far the stars were from the Earth. Archimedes therefore had to make the following assumptions: </p> <ul><li>The Universe was spherical</li> <li>The ratio of the diameter of the Universe to the diameter of the orbit of the Earth around the Sun equalled the ratio of the diameter of the orbit of the Earth around the Sun to the diameter of the Earth.</li></ul> <p>This assumption can also be expressed by saying that the stellar parallax caused by the motion of the Earth around its orbit equals the solar parallax caused by motion around the Earth. Put in a ratio: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\text{Diameter of Universe}}{\text{Diameter of Earth orbit around the Sun}}}={\frac {\text{Diameter of Earth orbit around the Sun}}{\text{ Diameter of Earth}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>Diameter of Universe</mtext> <mtext>Diameter of Earth orbit around the Sun</mtext> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>Diameter of Earth orbit around the Sun</mtext> <mtext> Diameter of Earth</mtext> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\text{Diameter of Universe}}{\text{Diameter of Earth orbit around the Sun}}}={\frac {\text{Diameter of Earth orbit around the Sun}}{\text{ Diameter of Earth}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29f3360ef670102a5b86f17358a9ccc8bd0fe7a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:84.278ex; height:5.509ex;" alt="{\displaystyle {\frac {\text{Diameter of Universe}}{\text{Diameter of Earth orbit around the Sun}}}={\frac {\text{Diameter of Earth orbit around the Sun}}{\text{ Diameter of Earth}}}}"></span> </p><p>In order to obtain an upper bound, Archimedes made the following assumptions of their dimensions: </p> <ul><li>that the perimeter of the Earth was no bigger than 300 myriad <a href="/wiki/Ancient_Greek_weights_and_measures#Length" class="mw-redirect" title="Ancient Greek weights and measures">stadia</a> (5.55·10<sup>5</sup> km).</li> <li>that the Moon was no larger than the Earth, and that the Sun was no more than thirty times larger than the Moon.</li> <li>that the angular diameter of the Sun, as seen from the Earth, was greater than 1/200 of a right angle (π/400 <a href="/wiki/Radian" title="Radian">radians</a> = 0.45<a href="/wiki/Degree_(angle)" title="Degree (angle)">° degrees</a>).</li></ul> <p>Archimedes then concluded that the diameter of the Universe was no more than 10<sup>14</sup> stadia (in modern units, about 2 <a href="/wiki/Light_year" class="mw-redirect" title="Light year">light years</a>), and that it would require no more than 10<sup>63</sup> grains of sand to fill it. With these measurements, each grain of sand in Archimedes's thought-experiment would have been approximately 19 μm (0.019 mm) in diameter. </p> <div class="mw-heading mw-heading3"><h3 id="Calculation_of_the_number_of_grains_of_sand_in_the_Aristarchian_Universe">Calculation of the number of grains of sand in the Aristarchian Universe</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=The_Sand_Reckoner&action=edit&section=4" title="Edit section: Calculation of the number of grains of sand in the Aristarchian Universe"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Archimedes claims that forty poppy-seeds laid side by side would equal one Greek dactyl (finger-width) which was approximately 19 mm (3/4 inch) in length. Since volume proceeds as the cube of a linear dimension ("For it has been proved that spheres have the triplicate ratio to one another of their diameters") then a sphere one dactyl in diameter would contain (using our current number system) 40<sup>3</sup>, or 64,000 poppy seeds. </p><p>He then claimed (without evidence) that each poppy seed could contain a myriad (10,000) grains of sand. Multiplying the two figures together he proposed 640,000,000 as the number of hypothetical grains of sand in a sphere one dactyl in diameter. </p><p>To make further calculations easier, he rounded up 640 million to one billion, noting only that the first number is smaller than the second, and that therefore the number of grains of sand calculated subsequently will exceed the actual number of grains. Recall that Archimedes's meta-goal with this essay was to show how to calculate with what were previously considered impossibly large numbers, not simply to accurately calculate the number of grains of sand that would fit in the universe. </p><p>A Greek stadium had a length of 600 Greek feet, and each foot was 16 dactyls long, so there were 9,600 dactyls in a stadium. Archimedes rounded this number up to 10,000 (a myriad) to make calculations easier, again, noting that the resulting number will exceed the actual number of grains of sand. </p><p>The cube of 10,000 is a trillion (10<sup>12</sup>); and multiplying a billion (the number of grains of sand in a dactyl-sphere) by a trillion (number of dactyl-spheres in a stadium-sphere) yields 10<sup>21</sup>, the number of grains of sand in a stadium-sphere. </p><p>Archimedes had estimated that the Aristarchian Universe was 10<sup>14</sup> stadia in diameter, so there would accordingly be (10<sup>14</sup>)<sup>3</sup> stadium-spheres in the universe, or 10<sup>42</sup>. Multiplying 10<sup>21</sup> by 10<sup>42</sup> yields 10<sup>63</sup>, the number of grains of sand in the Aristarchian Universe.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Following Archimedes's estimate of a myriad (10,000) grains of sand in a poppy seed; 64,000 poppy seeds in a dactyl-sphere; the length of a stadium as 10,000 dactyls; and accepting 19mm as the width of a dactyl, the diameter of Archimedes's typical sand grain would be 18.3 μm, which today we would call a grain of <a href="/wiki/Silt" title="Silt">silt</a>. Currently, the smallest grain of sand would be defined as 50 μm in diameter. </p> <div class="mw-heading mw-heading3"><h3 id="Additional_calculations">Additional calculations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=The_Sand_Reckoner&action=edit&section=5" title="Edit section: Additional calculations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Archimedes made some interesting experiments and computations along the way. One experiment was to estimate the angular size of the Sun, as seen from the Earth. Archimedes's method is especially interesting as it takes into account the finite size of the eye's pupil,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and therefore may be the first known example of experimentation in <a href="/wiki/Psychophysics" title="Psychophysics">psychophysics</a>, the branch of <a href="/wiki/Psychology" title="Psychology">psychology</a> dealing with the <a href="/wiki/Psychophysics" title="Psychophysics">mechanics of human perception</a>, whose development is generally attributed to <a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Hermann von Helmholtz</a>. Another interesting computation accounts for solar parallax and the different distances between the viewer and the Sun, whether viewed from the center of the Earth or from the surface of the Earth at sunrise. This may be the first known computation dealing with solar parallax.<sup id="cite_ref-v_1-1" class="reference"><a href="#cite_note-v-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Quote">Quote</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=The_Sand_Reckoner&action=edit&section=6" title="Edit section: Quote"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"> <p>There are some, king Gelon, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited. Again there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its magnitude. And it is clear that they who hold this view, if they imagined a mass made up of sand in other respects as large as the mass of the Earth, including in it all the seas and the hollows of the Earth filled up to a height equal to that of the highest of the mountains, would be many times further still from recognizing that any number could be expressed which exceeded the multitude of the sand so taken. </p><p> But I will try to show you by means of geometrical proofs, which you will be able to follow, that, of the numbers named by me and given in the work which I sent to Zeuxippus, some exceed not only the number of the mass of sand equal in magnitude to the Earth filled up in the way described, but also that of the mass equal in magnitude to the universe.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></p><div class="templatequotecite">— <cite><i>Archimedis Syracusani Arenarius & Dimensio Circuli</i></cite></div></blockquote> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=The_Sand_Reckoner&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-v-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-v_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-v_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.lix.polytechnique.fr/Labo/Ilan.Vardi/sand_reckoner.ps">Archimedes, The Sand Reckoner 511 R U, by Ilan Vardi</a>, accessed 28-II-2007.</span> </li> <li id="cite_note-eureka_man-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-eureka_man_2-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAlan_Hirshfeld2009" class="citation book cs1">Alan Hirshfeld (8 September 2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zbcfLoZKDl8C&pg=PA60"><i>Eureka Man: The Life and Legacy of Archimedes</i></a>. Bloomsbury Publishing USA. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780802719799" title="Special:BookSources/9780802719799"><bdi>9780802719799</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">17 February</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Eureka+Man%3A+The+Life+and+Legacy+of+Archimedes&rft.pub=Bloomsbury+Publishing+USA&rft.date=2009-09-08&rft.isbn=9780802719799&rft.au=Alan+Hirshfeld&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzbcfLoZKDl8C%26pg%3DPA60&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThe+Sand+Reckoner" class="Z3988"></span></span> </li> <li id="cite_note-Analysis-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Analysis_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/51607350"><i>A history of analysis</i></a>. H. N. Jahnke. Providence, RI: American Mathematical Society. 2003. p. 22. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-2623-9" title="Special:BookSources/0-8218-2623-9"><bdi>0-8218-2623-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/51607350">51607350</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+analysis&rft.place=Providence%2C+RI&rft.pages=22&rft.pub=American+Mathematical+Society&rft.date=2003&rft_id=info%3Aoclcnum%2F51607350&rft.isbn=0-8218-2623-9&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F51607350&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThe+Sand+Reckoner" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: others (<a href="/wiki/Category:CS1_maint:_others" title="Category:CS1 maint: others">link</a>)</span></span> </li> <li id="cite_note-a-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-a_4-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Aristarchus.html">Aristarchus biography at MacTutor</a>, accessed 26-II-2007.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Arenarius, I., 4–7</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"> Annotated translation of The Sand Reckoner <a rel="nofollow" class="external autonumber" href="https://web.archive.org/web/20040808005307/http://www.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Archimedes/SandReckoner/SandReckoner.html">[1]</a> Cal State University, Los Angeles </span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Smith, William — A Dictionary of Greek and Roman Biography and Mythology (1880), p. 272</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Newman, James R. — The World of Mathematics (2000), p. 420</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=The_Sand_Reckoner&action=edit&section=8" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i>The Sand-Reckoner</i>, by <a href="/wiki/Gillian_Bradshaw" title="Gillian Bradshaw">Gillian Bradshaw</a>. Forge (2000), 348pp, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-312-87581-9" title="Special:BookSources/0-312-87581-9">0-312-87581-9</a>. This is a historical novel about the life and work of Archimedes.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=The_Sand_Reckoner&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.lix.polytechnique.fr/Labo/Ilan.Vardi/psammites.ps">Original Greek text</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20040808005307/http://www.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Archimedes/SandReckoner/SandReckoner.html"><i>The Sand Reckoner</i> (annotated)</a></li> <li><a rel="nofollow" class="external text" href="http://www.heinrichfleck.net/Quaderni/quaderni.html"><i>The Sand Reckoner</i> (Arenario) Italian annotated translation, with notes about Archimedes and Greek mathematical notation and unit of measure</a>. Source file of the Arenarius Greek text (for LaTeX).</li> <li><a rel="nofollow" class="external text" href="http://www.lix.polytechnique.fr/Labo/Ilan.Vardi/sand_reckoner.ps">Archimedes, <i>The Sand Reckoner</i>, by Ilan Vardi; includes a literal English version of the original Greek text</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output 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Elder">Aristaeus the Elder</a></li> <li><a href="/wiki/Aristarchus_of_Samos" title="Aristarchus of Samos">Aristarchus</a></li> <li><a href="/wiki/Aristotle" title="Aristotle">Aristotle</a></li> <li><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li> <li><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></li> <li><a href="/wiki/Autolycus_of_Pitane" title="Autolycus of Pitane">Autolycus</a></li> <li><a href="/wiki/Bion_of_Abdera" title="Bion of Abdera">Bion</a></li> <li><a href="/wiki/Bryson_of_Heraclea" title="Bryson of Heraclea">Bryson</a></li> <li><a href="/wiki/Callippus" title="Callippus">Callippus</a></li> <li><a href="/wiki/Carpus_of_Antioch" title="Carpus of Antioch">Carpus</a></li> <li><a href="/wiki/Chrysippus" title="Chrysippus">Chrysippus</a></li> <li><a href="/wiki/Cleomedes" title="Cleomedes">Cleomedes</a></li> <li><a href="/wiki/Conon_of_Samos" title="Conon of Samos">Conon</a></li> <li><a href="/wiki/Ctesibius" title="Ctesibius">Ctesibius</a></li> <li><a href="/wiki/Democritus" title="Democritus">Democritus</a></li> <li><a href="/wiki/Dicaearchus" title="Dicaearchus">Dicaearchus</a></li> <li><a href="/wiki/Diocles_(mathematician)" title="Diocles (mathematician)">Diocles</a></li> <li><a href="/wiki/Diophantus" title="Diophantus">Diophantus</a></li> <li><a href="/wiki/Dinostratus" title="Dinostratus">Dinostratus</a></li> <li><a href="/wiki/Dionysodorus" title="Dionysodorus">Dionysodorus</a></li> <li><a href="/wiki/Domninus_of_Larissa" title="Domninus of Larissa">Domninus</a></li> <li><a href="/wiki/Eratosthenes" title="Eratosthenes">Eratosthenes</a></li> <li><a href="/wiki/Eudemus_of_Rhodes" title="Eudemus of Rhodes">Eudemus</a></li> <li><a href="/wiki/Euclid" title="Euclid">Euclid</a></li> <li><a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus</a></li> <li><a href="/wiki/Eutocius_of_Ascalon" title="Eutocius of Ascalon">Eutocius</a></li> <li><a href="/wiki/Geminus" title="Geminus">Geminus</a></li> <li><a href="/wiki/Heliodorus_of_Larissa" title="Heliodorus of Larissa">Heliodorus</a></li> <li><a href="/wiki/Hero_of_Alexandria" title="Hero of Alexandria">Heron</a></li> <li><a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a></li> <li><a href="/wiki/Hippasus" title="Hippasus">Hippasus</a></li> <li><a href="/wiki/Hippias" title="Hippias">Hippias</a></li> <li><a href="/wiki/Hippocrates_of_Chios" title="Hippocrates of Chios">Hippocrates</a></li> <li><a href="/wiki/Hypatia" title="Hypatia">Hypatia</a></li> <li><a href="/wiki/Hypsicles" title="Hypsicles">Hypsicles</a></li> <li><a href="/wiki/Isidore_of_Miletus" title="Isidore of Miletus">Isidore of Miletus</a></li> <li><a href="/wiki/Leon_(mathematician)" title="Leon (mathematician)">Leon</a></li> <li><a href="/wiki/Marinus_of_Neapolis" title="Marinus of Neapolis">Marinus</a></li> <li><a href="/wiki/Menaechmus" title="Menaechmus">Menaechmus</a></li> <li><a href="/wiki/Menelaus_of_Alexandria" title="Menelaus of Alexandria">Menelaus</a></li> <li><a href="/wiki/Metrodorus_(grammarian)" title="Metrodorus (grammarian)">Metrodorus</a></li> <li><a href="/wiki/Nicomachus" title="Nicomachus">Nicomachus</a></li> <li><a href="/wiki/Nicomedes_(mathematician)" title="Nicomedes (mathematician)">Nicomedes</a></li> <li><a href="/wiki/Nicoteles_of_Cyrene" title="Nicoteles of Cyrene">Nicoteles</a></li> <li><a href="/wiki/Oenopides" title="Oenopides">Oenopides</a></li> <li><a href="/wiki/Pappus_of_Alexandria" title="Pappus of Alexandria">Pappus</a></li> <li><a href="/wiki/Perseus_(geometer)" title="Perseus (geometer)">Perseus</a></li> <li><a href="/wiki/Philolaus" title="Philolaus">Philolaus</a></li> <li><a href="/wiki/Philon" title="Philon">Philon</a></li> <li><a href="/wiki/Philonides_of_Laodicea" title="Philonides of Laodicea">Philonides</a></li> <li><a href="/wiki/Plato" title="Plato">Plato</a></li> <li><a href="/wiki/Porphyry_(philosopher)" title="Porphyry (philosopher)">Porphyry</a></li> <li><a href="/wiki/Posidonius" title="Posidonius">Posidonius</a></li> <li><a href="/wiki/Proclus" title="Proclus">Proclus</a></li> <li><a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a></li> <li><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a></li> <li><a href="/wiki/Serenus_of_Antino%C3%B6polis" title="Serenus of Antinoöpolis">Serenus </a></li> <li><a href="/wiki/Simplicius_of_Cilicia" title="Simplicius of Cilicia">Simplicius</a></li> <li><a href="/wiki/Sosigenes_of_Alexandria" class="mw-redirect" title="Sosigenes of Alexandria">Sosigenes</a></li> <li><a href="/wiki/Sporus_of_Nicaea" title="Sporus of Nicaea">Sporus</a></li> <li><a href="/wiki/Thales_of_Miletus" title="Thales of Miletus">Thales</a></li> <li><a href="/wiki/Theaetetus_(mathematician)" title="Theaetetus (mathematician)">Theaetetus</a></li> <li><a href="/wiki/Theano_(philosopher)" title="Theano (philosopher)">Theano</a></li> <li><a href="/wiki/Theodorus_of_Cyrene" title="Theodorus of Cyrene">Theodorus</a></li> <li><a href="/wiki/Theodosius_of_Bithynia" title="Theodosius of Bithynia">Theodosius</a></li> <li><a href="/wiki/Theon_of_Alexandria" title="Theon of Alexandria">Theon of Alexandria</a></li> <li><a href="/wiki/Theon_of_Smyrna" title="Theon of Smyrna">Theon of Smyrna</a></li> <li><a href="/wiki/Thymaridas" title="Thymaridas">Thymaridas</a></li> <li><a href="/wiki/Xenocrates" title="Xenocrates">Xenocrates</a></li> <li><a href="/wiki/Zeno_of_Elea" title="Zeno of Elea">Zeno of Elea</a></li> <li><a href="/wiki/Zeno_of_Sidon" title="Zeno of Sidon">Zeno of Sidon</a></li> <li><a href="/wiki/Zenodorus_(mathematician)" title="Zenodorus (mathematician)">Zenodorus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Treatises</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Almagest" title="Almagest">Almagest</a></i></li> <li><a href="/wiki/Archimedes_Palimpsest" title="Archimedes Palimpsest">Archimedes Palimpsest</a></li> <li><i><a href="/wiki/Arithmetica" title="Arithmetica">Arithmetica</a></i></li> <li><a href="/wiki/Apollonius_of_Perga#Conics" title="Apollonius of Perga"><i>Conics</i> <span style="font-size:85%;">(Apollonius)</span></a></li> <li><i><a href="/wiki/Catoptrics" title="Catoptrics">Catoptrics</a></i></li> <li><a href="/wiki/Data_(Euclid)" class="mw-redirect" title="Data (Euclid)"><i>Data</i> <span style="font-size:85%;">(Euclid)</span></a></li> <li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i> <span style="font-size:85%;">(Euclid)</span></a></li> <li><i><a href="/wiki/Measurement_of_a_Circle" title="Measurement of a Circle">Measurement of a Circle</a></i></li> <li><i><a href="/wiki/On_Conoids_and_Spheroids" title="On Conoids and Spheroids">On Conoids and Spheroids</a></i></li> <li><a href="/wiki/On_the_Sizes_and_Distances_(Aristarchus)" title="On the Sizes and Distances (Aristarchus)"><i>On the Sizes and Distances</i> <span style="font-size:85%;">(Aristarchus)</span></a></li> <li><a href="/wiki/On_Sizes_and_Distances_(Hipparchus)" title="On Sizes and Distances (Hipparchus)"><i>On Sizes and Distances</i> <span style="font-size:85%;">(Hipparchus)</span></a></li> <li><a href="/wiki/Autolycus_of_Pitane" title="Autolycus of Pitane"><i>On the Moving Sphere</i> <span style="font-size:85%;">(Autolycus)</span></a></li> <li><a href="/wiki/Euclid%27s_Optics" title="Euclid's Optics"><i>Optics</i> <span style="font-size:85%;">(Euclid)</span></a></li> <li><i><a href="/wiki/On_Spirals" title="On Spirals">On Spirals</a></i></li> <li><i><a href="/wiki/On_the_Sphere_and_Cylinder" title="On the Sphere and Cylinder">On the Sphere and Cylinder</a></i></li> <li><i><a href="/wiki/Ostomachion" title="Ostomachion">Ostomachion</a></i></li> <li><i><a href="/wiki/Planisphaerium" title="Planisphaerium">Planisphaerium</a></i></li> <li><a href="/wiki/Theodosius%27_Spherics" title="Theodosius' Spherics"><i>Spherics</i> <span style="font-size:85%;">(Theodosius)</span></a></li> <li><a href="/wiki/Menelaus_of_Alexandria" title="Menelaus of Alexandria"><i>Spherics</i> <span style="font-size:85%;">(Menelaus)</span></a></li> <li><i><a href="/wiki/The_Quadrature_of_the_Parabola" class="mw-redirect" title="The Quadrature of the Parabola">The Quadrature of the Parabola</a></i></li> <li><i><a class="mw-selflink selflink">The Sand Reckoner</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Problems</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Constructible_number" title="Constructible number">Constructible numbers</a> <ul><li><a href="/wiki/Angle_trisection" title="Angle trisection">Angle trisection</a></li> <li><a href="/wiki/Doubling_the_cube" title="Doubling the cube">Doubling the cube</a></li> <li><a href="/wiki/Squaring_the_circle" title="Squaring the circle">Squaring the circle</a></li></ul></li> <li><a href="/wiki/Problem_of_Apollonius" title="Problem of Apollonius">Problem of Apollonius</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts<br />and definitions</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Angle" title="Angle">Angle</a> <ul><li><a href="/wiki/Central_angle" title="Central angle">Central</a></li> <li><a href="/wiki/Inscribed_angle" title="Inscribed angle">Inscribed</a></li></ul></li> <li><a href="/wiki/Axiomatic_system" title="Axiomatic system">Axiomatic system</a> <ul><li><a href="/wiki/Axiom" title="Axiom">Axiom</a></li></ul></li> <li><a href="/wiki/Chord_(geometry)" title="Chord (geometry)">Chord</a></li> <li><a href="/wiki/Circles_of_Apollonius" title="Circles of Apollonius">Circles of Apollonius</a> <ul><li><a href="/wiki/Apollonian_circles" title="Apollonian circles">Apollonian circles</a></li> <li><a href="/wiki/Apollonian_gasket" title="Apollonian gasket">Apollonian gasket</a></li></ul></li> <li><a href="/wiki/Circumscribed_circle" title="Circumscribed circle">Circumscribed circle</a></li> <li><a href="/wiki/Commensurability_(mathematics)" title="Commensurability (mathematics)">Commensurability</a></li> <li><a href="/wiki/Diophantine_equation" title="Diophantine equation">Diophantine equation</a></li> <li><a href="https://en.wikiquote.org/wiki/Doctrine_of_proportion_(mathematics)" class="extiw" title="wikiquote:Doctrine of proportion (mathematics)">Doctrine of proportionality</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li> <li><a href="/wiki/Golden_ratio" title="Golden ratio">Golden ratio</a></li> <li><a href="/wiki/Greek_numerals" title="Greek numerals">Greek numerals</a></li> <li><a href="/wiki/Incircle_and_excircles_of_a_triangle" class="mw-redirect" title="Incircle and excircles of a triangle">Incircle and excircles of a triangle</a></li> <li><a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">Method of exhaustion</a></li> <li><a href="/wiki/Parallel_postulate" title="Parallel postulate">Parallel postulate</a></li> <li><a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solid</a></li> <li><a href="/wiki/Lune_of_Hippocrates" title="Lune of Hippocrates">Lune of Hippocrates</a></li> <li><a href="/wiki/Quadratrix_of_Hippias" title="Quadratrix of Hippias">Quadratrix of Hippias</a></li> <li><a href="/wiki/Regular_polygon" title="Regular polygon">Regular polygon</a></li> <li><a href="/wiki/Straightedge_and_compass_construction" title="Straightedge and compass construction">Straightedge and compass construction</a></li> <li><a href="/wiki/Triangle_center" title="Triangle center">Triangle center</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">In <a href="/wiki/Euclid%27s_elements" class="mw-redirect" title="Euclid's elements"><i>Elements</i></a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Angle_bisector_theorem" title="Angle bisector theorem">Angle bisector theorem</a></li> <li><a href="/wiki/Exterior_angle_theorem" title="Exterior angle theorem">Exterior angle theorem</a></li> <li><a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a></li> <li><a href="/wiki/Euclid%27s_theorem" title="Euclid's theorem">Euclid's theorem</a></li> <li><a href="/wiki/Geometric_mean_theorem" title="Geometric mean theorem">Geometric mean theorem</a></li> <li><a href="/wiki/Greek_geometric_algebra" class="mw-redirect" title="Greek geometric algebra">Greek geometric algebra</a></li> <li><a href="/wiki/Hinge_theorem" title="Hinge theorem">Hinge theorem</a></li> <li><a href="/wiki/Inscribed_angle_theorem" class="mw-redirect" title="Inscribed angle theorem">Inscribed angle theorem</a></li> <li><a href="/wiki/Intercept_theorem" title="Intercept theorem">Intercept theorem</a></li> <li><a href="/wiki/Intersecting_chords_theorem" title="Intersecting chords theorem">Intersecting chords theorem</a></li> <li><a href="/wiki/Intersecting_secants_theorem" title="Intersecting secants theorem">Intersecting secants theorem</a></li> <li><a href="/wiki/Law_of_cosines" title="Law of cosines">Law of cosines</a></li> <li><a href="/wiki/Pons_asinorum" title="Pons asinorum">Pons asinorum</a></li> <li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li> <li><a href="/wiki/Tangent-secant_theorem" class="mw-redirect" title="Tangent-secant theorem">Tangent-secant theorem</a></li> <li><a href="/wiki/Thales%27s_theorem" title="Thales's theorem">Thales's theorem</a></li> <li><a href="/wiki/Theorem_of_the_gnomon" title="Theorem of the gnomon">Theorem of the gnomon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Apollonius_of_Tyana" title="Apollonius of Tyana">Apollonius</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Apollonius%27s_theorem" title="Apollonius's theorem">Apollonius's theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aristarchus%27s_inequality" title="Aristarchus's inequality">Aristarchus's inequality</a></li> <li><a href="/wiki/Crossbar_theorem" title="Crossbar theorem">Crossbar theorem</a></li> <li><a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a></li> <li><a href="/wiki/Irrational_number" title="Irrational number">Irrational numbers</a></li> <li><a href="/wiki/Law_of_sines" title="Law of sines">Law of sines</a></li> <li><a href="/wiki/Menelaus%27s_theorem" title="Menelaus's theorem">Menelaus's theorem</a></li> <li><a href="/wiki/Pappus%27s_area_theorem" title="Pappus's area theorem">Pappus's area theorem</a></li> <li><a href="/wiki/Diophantus_II.VIII" title="Diophantus II.VIII">Problem II.8 of <i>Arithmetica</i></a></li> <li><a href="/wiki/Ptolemy%27s_inequality" title="Ptolemy's inequality">Ptolemy's inequality</a></li> <li><a href="/wiki/Ptolemy%27s_table_of_chords" title="Ptolemy's table of chords">Ptolemy's table of chords</a></li> <li><a href="/wiki/Ptolemy%27s_theorem" title="Ptolemy's theorem">Ptolemy's theorem</a></li> <li><a href="/wiki/Spiral_of_Theodorus" title="Spiral of Theodorus">Spiral of Theodorus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Centers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cyrene,_Libya" title="Cyrene, Libya">Cyrene</a></li> <li><a href="/wiki/Musaeum" class="mw-redirect" title="Musaeum">Mouseion of Alexandria</a></li> <li><a href="/wiki/Platonic_Academy" title="Platonic Academy">Platonic Academy</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ancient_Greek_astronomy" title="Ancient Greek astronomy">Ancient Greek astronomy</a></li> <li><a href="/wiki/Attic_numerals" title="Attic numerals">Attic numerals</a></li> <li><a href="/wiki/Greek_numerals" title="Greek numerals">Greek numerals</a></li> <li><a href="/wiki/Latin_translations_of_the_12th_century" title="Latin translations of the 12th century">Latin translations of the 12th century</a></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean geometry</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Neusis_construction" title="Neusis construction">Neusis construction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">History of</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/A_History_of_Greek_Mathematics" title="A History of Greek Mathematics">A History of Greek Mathematics</a></i> <ul><li>by <a href="/wiki/Thomas_Heath_(classicist)" title="Thomas Heath (classicist)">Thomas Heath</a></li></ul></li> <li><a href="/wiki/History_of_algebra" title="History of algebra">algebra</a> <ul><li><a href="/wiki/Timeline_of_algebra" title="Timeline of algebra">timeline</a></li></ul></li> <li><a href="/wiki/History_of_arithmetic" class="mw-redirect" title="History of arithmetic">arithmetic</a> <ul><li><a href="/wiki/Timeline_of_numerals_and_arithmetic" title="Timeline of numerals and arithmetic">timeline</a></li></ul></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">calculus</a> <ul><li><a href="/wiki/Timeline_of_calculus_and_mathematical_analysis" title="Timeline of calculus and mathematical analysis">timeline</a></li></ul></li> <li><a href="/wiki/History_of_geometry" title="History of geometry">geometry</a> <ul><li><a href="/wiki/Timeline_of_geometry" title="Timeline of geometry">timeline</a></li></ul></li> <li><a href="/wiki/History_of_logic" title="History of logic">logic</a> <ul><li><a href="/wiki/Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li> <li><a href="/wiki/History_of_mathematics" title="History of mathematics">mathematics</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">timeline</a></li></ul></li> <li><a href="/wiki/History_of_numbers" class="mw-redirect" title="History of numbers">numbers</a> <ul><li><a href="/wiki/Prehistoric_counting" title="Prehistoric counting">prehistoric counting</a></li></ul></li> <li><a href="/wiki/History_of_ancient_numeral_systems" title="History of ancient numeral systems">numeral systems</a> <ul><li><a href="/wiki/List_of_numeral_systems" title="List of numeral systems">list</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other cultures</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematics_in_the_medieval_Islamic_world" title="Mathematics in the medieval Islamic world">Arabian/Islamic</a></li> <li><a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonian</a></li> <li><a href="/wiki/Chinese_mathematics" title="Chinese mathematics">Chinese</a></li> <li><a href="/wiki/Ancient_Egyptian_mathematics" title="Ancient Egyptian mathematics">Egyptian</a></li> <li><a href="/wiki/Mathematics_of_the_Incas" title="Mathematics of the Incas">Incan</a></li> <li><a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian</a></li> <li><a href="/wiki/Japanese_mathematics" title="Japanese mathematics">Japanese</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Parthenon_from_west.jpg/16px-Parthenon_from_west.jpg" decoding="async" width="16" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Parthenon_from_west.jpg/24px-Parthenon_from_west.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Parthenon_from_west.jpg/32px-Parthenon_from_west.jpg 2x" data-file-width="2048" data-file-height="1536" /></span></span> </span><a href="/wiki/Portal:Ancient_Greece" title="Portal:Ancient Greece">Ancient Greece portal</a></b> •  <b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></b></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Archimedes" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Archimedes" title="Template:Archimedes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Archimedes" title="Template talk:Archimedes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Archimedes" title="Special:EditPage/Template:Archimedes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Archimedes" style="font-size:114%;margin:0 4em"><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Written works</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Measurement_of_a_Circle" title="Measurement of a Circle">Measurement of a Circle</a></i></li> <li><i><a class="mw-selflink selflink">The Sand Reckoner</a></i></li> <li><i><a href="/wiki/On_the_Equilibrium_of_Planes" title="On the Equilibrium of Planes">On the Equilibrium of Planes</a></i></li> <li><i><a href="/wiki/Quadrature_of_the_Parabola" title="Quadrature of the Parabola">Quadrature of the Parabola</a></i></li> <li><i><a href="/wiki/On_the_Sphere_and_Cylinder" title="On the Sphere and Cylinder">On the Sphere and Cylinder</a></i></li> <li><i><a href="/wiki/On_Spirals" title="On Spirals">On Spirals</a></i></li> <li><i><a href="/wiki/On_Conoids_and_Spheroids" title="On Conoids and Spheroids">On Conoids and Spheroids</a></i></li> <li><i><a href="/wiki/On_Floating_Bodies" title="On Floating Bodies">On Floating Bodies</a></i></li> <li><i><a href="/wiki/Ostomachion" title="Ostomachion">Ostomachion</a></i></li> <li><i><a href="/wiki/The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></i></li> <li><i><a href="/wiki/Book_of_Lemmas" title="Book of Lemmas">Book of Lemmas</a></i> (apocryphal)</li></ul> </div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Domenico-Fetti_Archimedes_1620.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Domenico-Fetti_Archimedes_1620.jpg/90px-Domenico-Fetti_Archimedes_1620.jpg" decoding="async" width="90" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Domenico-Fetti_Archimedes_1620.jpg/135px-Domenico-Fetti_Archimedes_1620.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Domenico-Fetti_Archimedes_1620.jpg/180px-Domenico-Fetti_Archimedes_1620.jpg 2x" data-file-width="1364" data-file-height="1818" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Discoveries and inventions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solid</a></li> <li><a href="/wiki/Archimedes%27s_cattle_problem" title="Archimedes's cattle problem">Archimedes's cattle problem</a></li> <li><a href="/wiki/Archimedes%27_principle" title="Archimedes' principle">Archimedes' principle</a></li> <li><a href="/wiki/Archimedes%27s_screw" class="mw-redirect" title="Archimedes's screw">Archimedes's screw</a></li> <li><a href="/wiki/Claw_of_Archimedes" title="Claw of Archimedes">Claw of Archimedes</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Archimedes%27_heat_ray" title="Archimedes' heat ray">Archimedes' heat ray</a></li> <li><a href="/wiki/Archimedes_Palimpsest" title="Archimedes Palimpsest">Archimedes Palimpsest</a></li> <li><a href="/wiki/List_of_things_named_after_Archimedes" title="List of things named after Archimedes">List of things named after Archimedes</a></li> <li><a href="/wiki/Pseudo-Archimedes" title="Pseudo-Archimedes">Pseudo-Archimedes</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related people</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Euclid" title="Euclid">Euclid</a></li> <li><a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus of Cnidus</a></li> <li><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius of Perga</a></li> <li><a href="/wiki/Hero_of_Alexandria" title="Hero of Alexandria">Hero of Alexandria</a></li> <li><a href="/wiki/Eutocius_of_Ascalon" title="Eutocius of Ascalon">Eutocius of Ascalon</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Archimedes" 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