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<div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Work by Aristarchus of Samos, Greek astronomer</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the work by Hipparchus, see <a href="/wiki/On_Sizes_and_Distances_(Hipparchus)" title="On Sizes and Distances (Hipparchus)">On Sizes and Distances (Hipparchus)</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Aristarchus_working.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Aristarchus_working.jpg/220px-Aristarchus_working.jpg" decoding="async" width="220" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Aristarchus_working.jpg/330px-Aristarchus_working.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Aristarchus_working.jpg/440px-Aristarchus_working.jpg 2x" data-file-width="508" data-file-height="338" /></a><figcaption><a href="/wiki/Aristarchus_of_Samos" title="Aristarchus of Samos">Aristarchus's</a> 3rd century BCE calculations on the relative sizes of, from left, the Sun, Earth and Moon, from a 10th-century CE Greek copy</figcaption></figure> <p><i><b>On the Sizes and Distances (of the Sun and Moon)</b></i> (<a href="/wiki/Ancient_Greek_language" class="mw-redirect" title="Ancient Greek language">Ancient Greek</a>: <span lang="grc">Περὶ μεγεθῶν καὶ ἀποστημάτων [ἡλίου καὶ σελήνης]</span>, <small><a href="/wiki/Romanization_of_Ancient_Greek" class="mw-redirect" title="Romanization of Ancient Greek">romanized</a>: </small><span title="Ancient Greek-language romanization"><i lang="grc-Latn"><span title="Ancient Greek transliteration" lang="grc-Latn"><i>Perì megethôn kaì apostēmátōn [hēlíou kaì selḗnēs]</i></span></i></span>) is widely accepted as the only extant work written by <a href="/wiki/Aristarchus_of_Samos" title="Aristarchus of Samos">Aristarchus of Samos</a>, an <a href="/wiki/Ancient_Greek_astronomy" title="Ancient Greek astronomy">ancient Greek astronomer</a> who lived circa 310–230 BCE. This work calculates the sizes of the <a href="/wiki/Sun" title="Sun">Sun</a> and <a href="/wiki/Moon" title="Moon">Moon</a>, as well as their distances from the <a href="/wiki/Earth" title="Earth">Earth</a> in terms of Earth's radius. </p><p>The book was presumably preserved by students of <a href="/wiki/Pappus_of_Alexandria" title="Pappus of Alexandria">Pappus of Alexandria</a>'s course in mathematics, although there is no evidence of this. The <i><a href="/wiki/Editio_princeps" title="Editio princeps">editio princeps</a></i> was published by <a href="/wiki/John_Wallis" title="John Wallis">John Wallis</a> in 1688, using several medieval manuscripts compiled by Sir <a href="/wiki/Henry_Savile_(Bible_translator)" title="Henry Savile (Bible translator)">Henry Savile</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The earliest Latin translation was made by <a href="/wiki/Giorgio_Valla" title="Giorgio Valla">Giorgio Valla</a> in 1488. There is also a <a rel="nofollow" class="external text" href="https://www.loc.gov/resource/rbctos.2017gen18687">1572 Latin translation and commentary</a> by <a href="/wiki/Frederico_Commandino" class="mw-redirect" title="Frederico Commandino">Frederico Commandino</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Symbols">Symbols</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=On_the_Sizes_and_Distances_(Aristarchus)&action=edit&section=1" title="Edit section: Symbols"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The work's method relied on several observations: </p> <ul><li>The apparent size of the Sun and the Moon in the sky.</li> <li>The size of the Earth's shadow in relation to the Moon during a <a href="/wiki/Lunar_eclipse" title="Lunar eclipse">lunar eclipse</a></li> <li>The angle between the Sun and Moon during a <a href="/wiki/Lunar_phase" title="Lunar phase">half moon</a> is 90°.</li></ul> <p>The rest of the article details a reconstruction of Aristarchus' method and results.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The reconstruction uses the following variables: </p> <table class="wikitable"> <tbody><tr> <th>Symbol</th> <th>Meaning </th></tr> <tr> <td align="center"><i>φ</i></td> <td>Angle between the Moon and the Sun during a half moon (directly measurable) </td></tr> <tr> <td align="center"><i>L</i></td> <td>Distance from the Earth to the <a href="/wiki/Moon" title="Moon">Moon</a> </td></tr> <tr> <td align="center"><i>S</i></td> <td>Distance from the Earth to the <a href="/wiki/Sun" title="Sun">Sun</a> </td></tr> <tr> <td align="center"><i>ℓ</i></td> <td>Radius of the <a href="/wiki/Moon" title="Moon">Moon</a> </td></tr> <tr> <td align="center"><i>s</i></td> <td>Radius of the <a href="/wiki/Sun" title="Sun">Sun</a> </td></tr> <tr> <td align="center"><i>t</i></td> <td>Radius of the <a href="/wiki/Earth" title="Earth">Earth</a> </td></tr> <tr> <td align="center"><i>D</i></td> <td>Distance from the center of Earth to the vertex of Earth's shadow cone </td></tr> <tr> <td align="center"><i>d</i></td> <td>Radius of the Earth's shadow at the location of the Moon </td></tr> <tr> <td align="center"><i>n</i></td> <td>Ratio, <i>d/ℓ</i> (a directly observable quantity during a <a href="/wiki/Lunar_eclipse" title="Lunar eclipse">lunar eclipse</a>) </td></tr> <tr> <td align="center"><i>x</i></td> <td>Ratio, <i>S/L</i> = <i>s/ℓ</i> (which is calculated from <i>φ</i>) </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Half_Moon">Half Moon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=On_the_Sizes_and_Distances_(Aristarchus)&action=edit&section=2" title="Edit section: Half Moon"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Aristarchus began with the premise that, during a <a href="/wiki/Lunar_phase" title="Lunar phase">half moon</a>, the moon forms a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> with the Sun and Earth. By observing the angle between the Sun and Moon, <i>φ</i>, the ratio of the distances to the Sun and Moon could be deduced using a form of <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a>. </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:AristarchusHalfLitMoon2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/f3/AristarchusHalfLitMoon2.png" decoding="async" width="484" height="255" class="mw-file-element" data-file-width="484" data-file-height="255" /></a></span> </p><p>From the diagram and trigonometry, we can calculate that </p> <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 {\frac {S}{L}}={\frac {1}{\cos \varphi }}=\sec \varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <mi>L</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>sec</mi> <mo>⁡</mo> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {S}{L}}={\frac {1}{\cos \varphi }}=\sec \varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c150c64a4bc546581590a8e53962f7c7e65f94aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.005ex; height:5.843ex;" alt="{\displaystyle {\frac {S}{L}}={\frac {1}{\cos \varphi }}=\sec \varphi .}"></span></dd></dl> <p>The diagram is greatly exaggerated, because in reality, <i>S = 390 L</i>, and <i>φ</i> is extremely close to 90°. Aristarchus determined <i>φ</i> to be a thirtieth of a quadrant (in modern terms, 3°) less than a right angle: in current terminology, 87°. Trigonometric functions had not yet been invented, but using geometrical analysis in the style of <a href="/wiki/Euclid" title="Euclid">Euclid</a>, Aristarchus determined that </p> <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 18<{\frac {S}{L}}<20.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>18</mn> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <mi>L</mi> </mfrac> </mrow> <mo><</mo> <mn>20.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 18<{\frac {S}{L}}<20.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d623129c9da7990f06e9fcc7452caa0bd546efd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.913ex; height:5.343ex;" alt="{\displaystyle 18<{\frac {S}{L}}<20.}"></span></dd></dl> <p>In other words, the distance to the Sun was somewhere between 18 and 20 times greater than the distance to the Moon. This value (or values close to it) was accepted by astronomers for the next two thousand years, until the invention of the telescope permitted a more precise estimate of <a href="/wiki/Solar_parallax" class="mw-redirect" title="Solar parallax">solar parallax</a>. </p><p>Aristarchus also reasoned that as the <a href="/wiki/Angular_size" class="mw-redirect" title="Angular size">angular size</a> of the Sun and the Moon were the same, but the distance to the Sun was between 18 and 20 times further than the Moon, the Sun must therefore be 18–20 times larger. </p> <div class="mw-heading mw-heading2"><h2 id="Lunar_eclipse">Lunar eclipse</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=On_the_Sizes_and_Distances_(Aristarchus)&action=edit&section=3" title="Edit section: Lunar eclipse"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Aristarchus then used another construction based on a lunar eclipse: </p><p><span typeof="mw:File"><a href="/wiki/File:AristarchusLunar_Eclipse2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/AristarchusLunar_Eclipse2.png/600px-AristarchusLunar_Eclipse2.png" decoding="async" width="600" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/72/AristarchusLunar_Eclipse2.png 1.5x" data-file-width="874" data-file-height="263" /></a></span> </p><p>By similarity of the triangles, <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 {D}{L}}={\frac {t}{t-d}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mi>L</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mrow> <mi>t</mi> <mo>−</mo> <mi>d</mi> </mrow> </mfrac> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {D}{L}}={\frac {t}{t-d}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb8887920380524e0e5b943073b15881162ebd30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.913ex; height:5.509ex;" alt="{\displaystyle {\frac {D}{L}}={\frac {t}{t-d}}\quad }"></span> and <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 \quad {\frac {D}{S}}={\frac {t}{s-t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mi>S</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mrow> <mi>s</mi> <mo>−</mo> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\frac {D}{S}}={\frac {t}{s-t}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4d2fb7f3c279e2c0c549e3a7388d63a7c99000" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.435ex; height:5.343ex;" alt="{\displaystyle \quad {\frac {D}{S}}={\frac {t}{s-t}}.}"></span> </p><p>Dividing these two equations and using the observation that the Sun and Moon appear the same size to people on Earth, <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 s/S=\ell /L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>S</mi> <mo>=</mo> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s/S=\ell /L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2de44feac39791df2197492bddc66c3defe5114" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.566ex; height:2.843ex;" alt="{\displaystyle s/S=\ell /L}"></span>, yields </p> <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 {\frac {\ell }{s}}={\frac {t-d}{s-t}}\ \ \implies \ \ {\frac {s-t}{s}}={\frac {t-d}{\ell }}\ \ \implies \ \ {\frac {t}{\ell }}+{\frac {t}{s}}=1+{\frac {d}{\ell }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ℓ</mi> <mi>s</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mo>−</mo> <mi>d</mi> </mrow> <mrow> <mi>s</mi> <mo>−</mo> <mi>t</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mtext> </mtext> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo>−</mo> <mi>t</mi> </mrow> <mi>s</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mo>−</mo> <mi>d</mi> </mrow> <mi>ℓ</mi> </mfrac> </mrow> <mtext> </mtext> <mtext> </mtext> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mi>ℓ</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mi>s</mi> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mi>ℓ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\ell }{s}}={\frac {t-d}{s-t}}\ \ \implies \ \ {\frac {s-t}{s}}={\frac {t-d}{\ell }}\ \ \implies \ \ {\frac {t}{\ell }}+{\frac {t}{s}}=1+{\frac {d}{\ell }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42d79772fb78c0ac26c12376aa5f9224eef08107" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:58.985ex; height:5.509ex;" alt="{\displaystyle {\frac {\ell }{s}}={\frac {t-d}{s-t}}\ \ \implies \ \ {\frac {s-t}{s}}={\frac {t-d}{\ell }}\ \ \implies \ \ {\frac {t}{\ell }}+{\frac {t}{s}}=1+{\frac {d}{\ell }}.}"></span></dd></dl> <p>The rightmost equation can either be solved for <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 \ell /t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell /t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5689d13b6962ac014fbe9531d8b8d91cb0d407cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle \ell /t}"></span> or <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 s/t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s/t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c14f2a59b9c0fba49ba3bc52b04f00d8f3dea2b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.093ex; height:2.843ex;" alt="{\displaystyle s/t}"></span> </p> <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 {\frac {\ell }{t}}={\frac {1+{\dfrac {\ell }{s}}}{1+{\dfrac {d}{\ell }}}},\qquad {\frac {s}{t}}={\frac {1+{\dfrac {s}{\ell }}}{1+{\dfrac {d}{\ell }}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ℓ</mi> <mi>t</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>ℓ</mi> <mi>s</mi> </mfrac> </mstyle> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>d</mi> <mi>ℓ</mi> </mfrac> </mstyle> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>t</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>s</mi> <mi>ℓ</mi> </mfrac> </mstyle> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>d</mi> <mi>ℓ</mi> </mfrac> </mstyle> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\ell }{t}}={\frac {1+{\dfrac {\ell }{s}}}{1+{\dfrac {d}{\ell }}}},\qquad {\frac {s}{t}}={\frac {1+{\dfrac {s}{\ell }}}{1+{\dfrac {d}{\ell }}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/065a37663712a5edf6bc53f6cc8b40cb1dabe81f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:30.037ex; height:11.509ex;" alt="{\displaystyle {\frac {\ell }{t}}={\frac {1+{\dfrac {\ell }{s}}}{1+{\dfrac {d}{\ell }}}},\qquad {\frac {s}{t}}={\frac {1+{\dfrac {s}{\ell }}}{1+{\dfrac {d}{\ell }}}}.}"></span></dd></dl> <p>These equations can be made to appear simpler by expressing the lengths <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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> and <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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> in terms of the moon's radius <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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span> as a unit, defining <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 {\hat {d}}=d/\ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {d}}=d/\ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/811c0282f9044aa001cb27d9dc4fa45120162596" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.028ex; height:3.343ex;" alt="{\displaystyle {\hat {d}}=d/\ell }"></span> and <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 {\hat {s}}=s/\ell .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ℓ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {s}}=s/\ell .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1833a11d0c969babac05f5592877385f320a8af2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.259ex; height:2.843ex;" alt="{\displaystyle {\hat {s}}=s/\ell .}"></span> Then </p> <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 {\frac {\ell }{t}}={\frac {1+{\hat {s}}}{{\hat {s}}(1+{\hat {d}})}},\qquad {\frac {s}{t}}={\frac {1+{\hat {s}}}{1+{\hat {d}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ℓ</mi> <mi>t</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>t</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\ell }{t}}={\frac {1+{\hat {s}}}{{\hat {s}}(1+{\hat {d}})}},\qquad {\frac {s}{t}}={\frac {1+{\hat {s}}}{1+{\hat {d}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfccbf537d54b449e4b808b3e315c71bfa7415b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.551ex; height:6.676ex;" alt="{\displaystyle {\frac {\ell }{t}}={\frac {1+{\hat {s}}}{{\hat {s}}(1+{\hat {d}})}},\qquad {\frac {s}{t}}={\frac {1+{\hat {s}}}{1+{\hat {d}}}}}"></span></dd></dl> <p>The above equations give the radii of the Moon and Sun entirely in terms of observable quantities. </p><p>The following formulae give the distances to the Sun and Moon in terrestrial units: </p> <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 {\frac {L}{t}}=\left({\frac {\ell }{t}}\right)\left({\frac {180}{\pi \theta }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>t</mi> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ℓ</mi> <mi>t</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>180</mn> <mrow> <mi>π</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L}{t}}=\left({\frac {\ell }{t}}\right)\left({\frac {180}{\pi \theta }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/457ef3cf999e4a1fdcf2ea1f9d0bad44f8be3c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.876ex; height:6.176ex;" alt="{\displaystyle {\frac {L}{t}}=\left({\frac {\ell }{t}}\right)\left({\frac {180}{\pi \theta }}\right)}"></span></dd></dl> <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 {\frac {S}{t}}={\biggl (}{\frac {s}{t}}{\biggr )}\left({\frac {180}{\pi \theta }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <mi>t</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>t</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>180</mn> <mrow> <mi>π</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {S}{t}}={\biggl (}{\frac {s}{t}}{\biggr )}\left({\frac {180}{\pi \theta }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39f2fa230fe0bdd04cc9e43870b9d28ccae277e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.913ex; height:6.176ex;" alt="{\displaystyle {\frac {S}{t}}={\biggl (}{\frac {s}{t}}{\biggr )}\left({\frac {180}{\pi \theta }}\right)}"></span></dd></dl> <p>where <i>θ</i> is the apparent radius of the Moon and Sun measured in degrees. </p><p>Aristarchus did not use these exact formulae, yet these formulae are likely a good approximation for those of Aristarchus. </p> <div class="mw-heading mw-heading2"><h2 id="Results">Results</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=On_the_Sizes_and_Distances_(Aristarchus)&action=edit&section=4" title="Edit section: Results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above formulae can be used to reconstruct the results of Aristarchus. The following table shows the results of a long-standing (but dubious) reconstruction using <i>n</i> = 2, <i>x</i> = 19.1 (<i>φ</i> = 87°) and <i>θ</i> = 1°, alongside the modern day accepted values. </p> <table class="wikitable"> <tbody><tr> <th>Quantity</th> <th>Relation</th> <th>Reconstruction</th> <th>Modern </th></tr> <tr align="center"> <td><i>s/t</i></td> <td>Sun's radius in Earth radii</td> <td>6.7</td> <td>109 </td></tr> <tr align="center"> <td><i>t/ℓ</i></td> <td>Earth's radius in Moon radii</td> <td>2.85</td> <td>3.67 </td></tr> <tr align="center"> <td><i>L/t</i></td> <td>Earth-Moon distance in Earth radii</td> <td>20</td> <td>60.34 </td></tr> <tr align="center"> <td><i>S/t</i></td> <td>Earth-Sun distance in Earth radii</td> <td>380</td> <td>23481 </td></tr></tbody></table><p><sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="needs credible references for modern values (February 2016)">citation needed</span></a></i>]</sup> </p><p>The error in this calculation comes primarily from the poor values for <i>x</i> and <i>θ</i>. The poor value for <i>θ</i> is especially surprising, since <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> writes that Aristarchus was the first to determine that the Sun and Moon had an apparent diameter of half a degree. This would give a value of <i>θ</i> = 0.25, and a corresponding distance to the Moon of 80 Earth radii, a much better estimate. The disagreement of the work with Archimedes seems to be due to its taking an Aristarchus statement that the lunisolar diameter is 1/15 of a "meros" of the zodiac to mean 1/15 of a zodiacal sign (30°), unaware that the Greek word "meros" meant either "portion" or 7°1/2; and 1/15 of the latter amount is 1°/2, in agreement with Archimedes' testimony. </p><p>A <a href="/wiki/Hipparchus_On_Sizes_and_Distances" class="mw-redirect" title="Hipparchus On Sizes and Distances">similar procedure</a> was later used by <a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a>, who estimated the mean distance to the Moon as 67 Earth radii, and <a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a>, who took 59 Earth radii for this value. </p> <div class="mw-heading mw-heading2"><h2 id="Illustrations">Illustrations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=On_the_Sizes_and_Distances_(Aristarchus)&action=edit&section=5" title="Edit section: Illustrations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some interactive illustrations of the propositions in <i>On Sizes</i> can be found here: </p> <ul><li><a rel="nofollow" class="external text" href="https://www.geogebra.org/m/ZYFQqZa5"><b>Hypothesis 4</b></a> states that when the Moon appears to us halved, its distance from the Sun is then less than a quadrant by one-thirtieth of a quadrant [that is, it is less than 90° by 1/30th of 90° or 3°, and is therefore equal to 87°] (Heath 1913:353).</li> <li><a rel="nofollow" class="external text" href="https://www.geogebra.org/m/BcVB8g23"><b>Proposition 1</b></a> states that two equal spheres are comprehended by one and the same cylinder, and two unequal spheres by one and the same cone which has its vertex in the direction of the lesser sphere; and the straight line drawn through the centres of the spheres is at right angles to each of the circles in which the surface of the cylinder, or of the cone, touches the spheres (Heath 1913:354).</li> <li><a rel="nofollow" class="external text" href="https://www.geogebra.org/m/EsGKE5yA"><b>Proposition 2</b></a> states that if a sphere be illuminated by a sphere greater than itself, the illuminated portion of the former sphere will be greater than a hemisphere (Heath 1913:358).</li> <li><a rel="nofollow" class="external text" href="https://www.geogebra.org/m/YEJ8GyaR"><b>Proposition 3</b></a> states that the circle in the Moon which divides the dark and the bright portions is least when the cone comprehending both the Sun and the Moon has its vertex at our eye (Heath 1913:362).</li> <li><a rel="nofollow" class="external text" href="https://www.geogebra.org/m/dxJPXypp"><b>Proposition 4</b></a> states that the circle which divides the dark and the bright portions in the Moon is not perceptibly different from a great circle in the Moon (Heath 1913:365).</li> <li><a rel="nofollow" class="external text" href="https://www.geogebra.org/m/MYdPsEku"><b>Proposition 6</b></a> states that the Moon moves [in an orbit] lower than [that of] the Sun, and, when it is halved, is distant less than a quadrant from the Sun (Heath 1913:372).</li> <li><a rel="nofollow" class="external text" href="https://www.geogebra.org/m/u4EKJ5uQ"><b>Proposition 7</b></a> states that the distance of the Sun from the Earth is greater than 18 times, but less than 20 times, the distance of the Moon from the Earth (Heath 1913:377). In other words, the Sun is 18 to 20 times farther away and wider than the Moon.</li> <li><a rel="nofollow" class="external text" href="https://www.geogebra.org/m/rDNQhNx9"><b>Proposition 13</b></a> states that the straight line subtending the portion intercepted within the earth's shadow of the circumference of the circle in which the extremities of the diameter of the circle dividing the dark and the bright portions in the Moon move is less than double of the diameter of the Moon, but has to it a ratio greater than that which 88 has to 45; and it is less than 1/9th part of the diameter of the Sun, but has to it a ratio greater than that which 21 has to 225. But it has to the straight line drawn from the centre of the Sun at right angles to the axis and meeting the sides of the cone a ratio greater than that which 979 has to 10 125 (Heath 1913:394).</li> <li><a rel="nofollow" class="external text" href="https://www.geogebra.org/m/nCqsrFSQ"><b>Proposition 14</b></a> states that the straight line joined from the centre of the Earth to the centre of the Moon has to the straight line cut off from the axis towards the centre of the Moon by the straight line subtending the [circumference] within the Earth's shadow a ratio greater than that which 675 has to 1 (Heath 1913:400).</li> <li><a rel="nofollow" class="external text" href="https://www.geogebra.org/m/eChNpgEj"><b>Proposition 15</b></a> states that the diameter of the Sun has to the diameter of the Earth a ratio greater than 19/3, but less than 43/6 (Heath 1913:403). This means that the Sun is (a mean of) <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac">6<span class="sr-only">+</span><span class="num">3</span>⁄<span class="den">4</span></span> times wider than the Earth, or that the Sun is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac">13<span class="sr-only">+</span><span class="num">1</span>⁄<span class="den">2</span></span> Earth-radii wide. The Moon and Sun must then be <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac">20<span class="sr-only">+</span><span class="num">1</span>⁄<span class="den">4</span></span> and 387 Earth-radii away from us in order to subtend an angular size of 2º.</li> <li><a rel="nofollow" class="external text" href="https://www.geogebra.org/m/GrQBA2pf"><b>Proposition 17a</b></a> in al-Tusi's medieval Arabic version of the book <i>On Sizes</i> states that the ratio of the distance of the vertex of the shadow cone from the center of the Moon (when the Moon is on the axis [that is, at the middle of an eclipse] of the cone containing the Earth and the Sun) to the distance of the center of the Moon from the center of the Earth is greater than the ratio 71 to 37 and less than the ratio 3 to one (Berggren & Sidoli 2007:218).<sup id="cite_ref-Berggren_5-0" class="reference"><a href="#cite_note-Berggren-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In other words, that the tip of the Earth's shadow cone is between 108/37 and four times farther away than the Moon.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Known_copies">Known copies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=On_the_Sizes_and_Distances_(Aristarchus)&action=edit&section=6" title="Edit section: Known copies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Library of Congress Vatican Exhibit.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=On_the_Sizes_and_Distances_(Aristarchus)&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Aristarchus_of_Samos" title="Aristarchus of Samos">Aristarchus of Samos</a></li> <li><a href="/wiki/Eratosthenes" title="Eratosthenes">Eratosthenes</a> (<a href="https://en.wiktionary.org/wiki/circa#English" class="extiw" title="wikt:circa">c.</a><span style="white-space:nowrap;"> 276</span> – c.<span style="white-space:nowrap;"> 194/195 BC</span>), a Greek mathematician who <a href="/wiki/On_the_measure_of_the_Earth" class="mw-redirect" title="On the measure of the Earth">calculated</a> the circumference of the Earth and also the distance from the Earth to the Sun.</li> <li><a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a> (<a href="https://en.wiktionary.org/wiki/circa#English" class="extiw" title="wikt:circa">c.</a><span style="white-space:nowrap;"> 190</span> – c.<span style="white-space:nowrap;"> 120 BC</span>), a Greek mathematician who <a href="/wiki/On_Sizes_and_Distances_(Hipparchus)" title="On Sizes and Distances (Hipparchus)">measured</a> the radii of the Sun and the Moon as well as their distances from the Earth.</li> <li><a href="/wiki/On_Sizes_and_Distances_(Hipparchus)" title="On Sizes and Distances (Hipparchus)"><i>On the Sizes and Distances</i> (Hipparchus)</a></li> <li><a href="/wiki/Posidonius" title="Posidonius">Posidonius</a> (<a href="https://en.wiktionary.org/wiki/circa#English" class="extiw" title="wikt:circa">c.</a><span style="white-space:nowrap;"> 135</span> – c.<span style="white-space:nowrap;"> 51 BC</span>), a Greek astronomer and mathematician who <a href="/wiki/Posidonius#Calculation_of_Earth's_circumference" title="Posidonius">calculated</a> the circumference of the Earth.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=On_the_Sizes_and_Distances_(Aristarchus)&action=edit&section=8" title="Edit section: Notes"><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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</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="CITEREFHeath1913" class="citation book cs1"><a href="/wiki/Thomas_Little_Heath" class="mw-redirect" title="Thomas Little Heath">Heath, Thomas</a> (1913). <a rel="nofollow" class="external text" href="https://archive.org/details/aristarchusofsam00heat"><i>Aristarchus of Samos, the Ancient Copernicus</i></a>. Oxford: Clarendon. p. <a rel="nofollow" class="external text" href="https://archive.org/stream/aristarchusofsam00heat#page/323/mode/1up">323</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Aristarchus+of+Samos%2C+the+Ancient+Copernicus&rft.place=Oxford&rft.pages=323&rft.pub=Clarendon&rft.date=1913&rft.aulast=Heath&rft.aufirst=Thomas&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Faristarchusofsam00heat&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn+the+Sizes+and+Distances+%28Aristarchus%29" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Berggren and Sidoli. 2007. 'Aristarchus's On the Sizes and Distances of the Sun and the Moon: Greek and Arabic Texts'. Arch. Hist. Exact Sci. 61(3), pp. 213–54. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00407-006-0118-4">10.1007/s00407-006-0118-4</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Noack B. (1992) Aristarch von Samos: Untersuchungen zur Überlieferungsgeschichte der Schrif <i>Περὶ μεγεθῶν καὶ ἀποστημάτων ἡλίου καὶ σελήνης</i>, Wiesbaden.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=LXHxx8L9YFo">A video on reconstruction of Aristarchus' method</a> (in Turkish, no subtitles)</span> </li> <li id="cite_note-Berggren-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Berggren_5-0">^</a></b></span> <span class="reference-text">Berggren, J. L. & N. Sidoli (2007) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110428064826/http://individual.utoronto.ca/acephalous/Berggren_Sidoli_2007a.pdf">"<span class="cs1-kern-left"></span>'Aristarchus's On the Sizes and Distances of the Sun and the Moon: Greek and Arabic Texts', <i>Archive for History of Exact Sciences,</i> Vol. 61, no. 3, 213–254"</a> <span class="cs1-format">(PDF)</span>. Archived from the original on April 28, 2011<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-11-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=%27Aristarchus%27s+On+the+Sizes+and+Distances+of+the+Sun+and+the+Moon%3A+Greek+and+Arabic+Texts%27%2C+Archive+for+History+of+Exact+Sciences%2C+Vol.+61%2C+no.+3%2C+213%E2%80%93254&rft_id=http%3A%2F%2Findividual.utoronto.ca%2Facephalous%2FBerggren_Sidoli_2007a.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn+the+Sizes+and+Distances+%28Aristarchus%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_web" title="Template:Cite web">cite web</a>}}</code>: CS1 maint: bot: original URL status unknown (<a href="/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown" title="Category:CS1 maint: bot: original URL status unknown">link</a>)</span>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=On_the_Sizes_and_Distances_(Aristarchus)&action=edit&section=9" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGomez2023" class="citation book cs1">Gomez, Alberto (2023). <a rel="nofollow" class="external text" href="https://www.peterlang.com/document/1288830"><i>Decoding Aristarchus</i></a>. <a href="/wiki/Berlin" title="Berlin">Berlin</a>: <a href="/wiki/Peter_Lang_Verlag" class="mw-redirect" title="Peter Lang Verlag">Peter Lang Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783631892619" title="Special:BookSources/9783631892619"><bdi>9783631892619</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Decoding+Aristarchus&rft.place=Berlin&rft.pub=Peter+Lang+Verlag&rft.date=2023&rft.isbn=9783631892619&rft.aulast=Gomez&rft.aufirst=Alberto&rft_id=https%3A%2F%2Fwww.peterlang.com%2Fdocument%2F1288830&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn+the+Sizes+and+Distances+%28Aristarchus%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeath1913" class="citation book cs1"><a href="/wiki/Thomas_Little_Heath" class="mw-redirect" title="Thomas Little Heath">Heath, Thomas</a> (1913). <a rel="nofollow" class="external text" href="https://archive.org/details/aristarchusofsam00heat"><i>Aristarchus of Samos, the Ancient Copernicus</i></a>. Oxford: Clarendon.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Aristarchus+of+Samos%2C+the+Ancient+Copernicus&rft.place=Oxford&rft.pub=Clarendon&rft.date=1913&rft.aulast=Heath&rft.aufirst=Thomas&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Faristarchusofsam00heat&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn+the+Sizes+and+Distances+%28Aristarchus%29" class="Z3988"></span> This was later reprinted, see (<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/0-486-43886-4" title="Special:BookSources/0-486-43886-4">0-486-43886-4</a>).</li> <li>van Helden, A. <i>Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley</i>. Chicago: Univ. of Chicago Pr., 1985. <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/0-226-84882-5" title="Special:BookSources/0-226-84882-5">0-226-84882-5</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 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Perga">Apollonius</a></li> <li><a href="/wiki/Aratus" title="Aratus">Aratus</a></li> <li><a href="/wiki/Aristarchus_of_Samos" title="Aristarchus of Samos">Aristarchus</a></li> <li><a href="/wiki/Aristyllus" title="Aristyllus">Aristyllus</a></li> <li><a href="/wiki/Attalus_of_Rhodes" title="Attalus of Rhodes">Attalus</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/Callippus" title="Callippus">Callippus</a></li> <li><a href="/wiki/Cleomedes" title="Cleomedes">Cleomedes</a></li> <li><a href="/wiki/Cleostratus" title="Cleostratus">Cleostratus</a></li> <li><a href="/wiki/Conon_of_Samos" title="Conon of Samos">Conon</a></li> <li><a href="/wiki/Eratosthenes" title="Eratosthenes">Eratosthenes</a></li> <li><a href="/wiki/Euctemon" title="Euctemon">Euctemon</a></li> <li><a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus</a></li> <li><a href="/wiki/Geminus" title="Geminus">Geminus</a></li> <li><a href="/wiki/Heraclides_Ponticus" title="Heraclides Ponticus">Heraclides</a></li> <li><a href="/wiki/Hicetas" title="Hicetas">Hicetas</a></li> <li><a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a></li> <li><a href="/wiki/Hippocrates_of_Chios" title="Hippocrates of Chios">Hippocrates of Chios</a></li> <li><a href="/wiki/Hypsicles" title="Hypsicles">Hypsicles</a></li> <li><a href="/wiki/Menelaus_of_Alexandria" title="Menelaus of Alexandria">Menelaus</a></li> <li><a href="/wiki/Meton_of_Athens" title="Meton of Athens">Meton</a></li> <li><a href="/wiki/Oenopides" title="Oenopides">Oenopides</a></li> <li><a href="/wiki/Philip_of_Opus" title="Philip of Opus">Philip of Opus</a></li> <li><a href="/wiki/Philolaus" title="Philolaus">Philolaus</a></li> <li><a href="/wiki/Posidonius" title="Posidonius">Posidonius</a></li> <li><a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a></li> <li><a href="/wiki/Pytheas" title="Pytheas">Pytheas</a></li> <li><a href="/wiki/Seleucus_of_Seleucia" title="Seleucus of Seleucia">Seleucus</a></li> <li><a href="/wiki/Sosigenes_of_Alexandria" class="mw-redirect" title="Sosigenes of Alexandria">Sosigenes of Alexandria</a></li> <li><a href="/wiki/Sosigenes_the_Peripatetic" title="Sosigenes the Peripatetic">Sosigenes the Peripatetic</a></li> <li><a href="/wiki/Strabo" title="Strabo">Strabo</a></li> <li><a href="/wiki/Thales_of_Miletus" title="Thales of Miletus">Thales</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/Timocharis" title="Timocharis">Timocharis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Works</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/Almagest" title="Almagest"><i>Almagest</i> <span style="font-size:85%;">(Ptolemy)</span></a></li> <li><a href="/wiki/Catasterismi" title="Catasterismi"><i>Catasterismi</i> <span style="font-size:85%;">(Eratosthenes)</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 class="mw-selflink selflink"><i>On the Sizes and Distances</i> <span style="font-size:85%;">(Aristarchus)</span></a></li> <li><a href="/wiki/On_the_Heavens" title="On the Heavens"><i>On the Heavens</i> <span style="font-size:85%;">(Aristotle)</span></a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Instruments</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/Antikythera_mechanism" title="Antikythera mechanism">Antikythera mechanism</a></li> <li><a href="/wiki/Armillary_sphere" title="Armillary sphere">Armillary sphere</a></li> <li><a href="/wiki/Astrolabe" title="Astrolabe">Astrolabe</a></li> <li><a href="/wiki/Dioptra" title="Dioptra">Dioptra</a></li> <li><a href="/wiki/Equatorial_ring" title="Equatorial ring">Equatorial ring</a></li> <li><a href="/wiki/Gnomon" title="Gnomon">Gnomon</a></li> <li><a href="/wiki/Mural_instrument" title="Mural instrument">Mural instrument</a></li> <li><a href="/wiki/Triquetrum_(astronomy)" title="Triquetrum (astronomy)">Triquetrum</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</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/Callippic_cycle" title="Callippic cycle">Callippic cycle</a></li> <li><a href="/wiki/Celestial_spheres" title="Celestial spheres">Celestial spheres</a></li> <li><a href="/wiki/Circle_of_latitude" title="Circle of latitude">Circle of latitude</a></li> <li><a href="/wiki/Counter-Earth" title="Counter-Earth">Counter-Earth</a></li> <li><a href="/wiki/Deferent_and_epicycle" title="Deferent and epicycle">Deferent and epicycle</a></li> <li><a href="/wiki/Equant" title="Equant">Equant</a></li> <li><a href="/wiki/Geocentric_model" title="Geocentric model">Geocentrism</a></li> <li><a href="/wiki/Heliocentrism" title="Heliocentrism">Heliocentrism</a></li> <li><a href="/wiki/Hipparchic_cycle" title="Hipparchic cycle">Hipparchic cycle</a></li> <li><a href="/wiki/Inferior_and_superior_planets" title="Inferior and superior planets">Inferior and superior planets</a></li> <li><a href="/wiki/Metonic_cycle" title="Metonic cycle">Metonic cycle</a></li> <li><a href="/wiki/Octaeteris" title="Octaeteris">Octaeteris</a></li> <li><a href="/wiki/Solstice" title="Solstice">Solstice</a></li> <li><a href="/wiki/Spherical_Earth" title="Spherical Earth">Spherical Earth</a></li> <li><a href="/wiki/Sublunary_sphere" title="Sublunary sphere">Sublunary sphere</a></li> <li><a href="/wiki/Zodiac" title="Zodiac">Zodiac</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Influences</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/Babylonian_astronomy" title="Babylonian astronomy">Babylonian astronomy</a></li> <li><a href="/wiki/Egyptian_astronomy" title="Egyptian astronomy">Egyptian astronomy</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Influenced</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/European_science_in_the_Middle_Ages" title="European science in the Middle Ages">Medieval European science</a></li> <li><a href="/wiki/Indian_astronomy" title="Indian astronomy">Indian astronomy</a></li> <li><a href="/wiki/Astronomy_in_the_medieval_Islamic_world" title="Astronomy in the medieval Islamic world">Medieval Islamic astronomy</a></li></ul> </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="Ancient_Greek_mathematics" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><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:Ancient_Greek_mathematics" title="Template:Ancient Greek mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Ancient_Greek_mathematics" title="Template talk:Ancient Greek mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Ancient_Greek_mathematics" title="Special:EditPage/Template:Ancient Greek mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Ancient_Greek_mathematics" style="font-size:114%;margin:0 4em"><a href="/wiki/Greek_mathematics" title="Greek mathematics">Ancient Greek mathematics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_Greek_mathematicians" title="List of Greek mathematicians">Mathematicians</a><br /><a href="/wiki/Timeline_of_ancient_Greek_mathematicians" title="Timeline of ancient Greek mathematicians">(timeline)</a></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/Anaxagoras" title="Anaxagoras">Anaxagoras</a></li> <li><a href="/wiki/Anthemius_of_Tralles" title="Anthemius of Tralles">Anthemius</a></li> <li><a href="/wiki/Archytas" title="Archytas">Archytas</a></li> <li><a href="/wiki/Aristaeus_the_Elder" title="Aristaeus the 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 class="mw-selflink selflink"><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 href="/wiki/The_Sand_Reckoner" title="The Sand Reckoner">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" 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