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class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>A simple way of understanding the table</span> </div> </a> <ul id="toc-A_simple_way_of_understanding_the_table-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Another_(more_difficult)_way_to_understand_the_values" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Another_(more_difficult)_way_to_understand_the_values"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Another (more difficult) way to understand the values</span> </div> </a> <ul id="toc-Another_(more_difficult)_way_to_understand_the_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation_of_trigonometric_sines_from_the_table" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Derivation_of_trigonometric_sines_from_the_table"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Derivation of trigonometric sines from the table</span> </div> </a> <ul id="toc-Derivation_of_trigonometric_sines_from_the_table-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Madhava's_method_of_computation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Madhava's_method_of_computation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Madhava's method of computation</span> </div> </a> <ul id="toc-Madhava's_method_of_computation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_references" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_references"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further references</span> </div> </a> <ul id="toc-Further_references-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div 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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">Mathematical table</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Example.of.complex.text.rendering.svg/60px-Example.of.complex.text.rendering.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/Example.of.complex.text.rendering.svg/80px-Example.of.complex.text.rendering.svg.png 2x" data-file-width="58" data-file-height="28" /></span></span></div> <div class="side-box-text plainlist"><b>This article contains <a href="/wiki/Brahmic_scripts" title="Brahmic scripts">Indic text</a>.</b> Without proper <a href="/wiki/Help:Multilingual_support_(Indic)" title="Help:Multilingual support (Indic)">rendering support</a>, you may see <a href="/wiki/Mojibake" title="Mojibake">question marks or boxes</a>, misplaced vowels or missing conjuncts instead of Indic text.</div></div> </div> <p><b>Madhava's sine table</b> is the <a href="/wiki/Table_(information)" title="Table (information)">table</a> of <a href="/wiki/Sine" class="mw-redirect" title="Sine">trigonometric sines</a> constructed by the 14th century <a href="/wiki/Kerala" title="Kerala">Kerala</a> <a href="/wiki/Mathematician" title="Mathematician">mathematician</a>-<a href="/wiki/Astronomer" title="Astronomer">astronomer</a> <a href="/wiki/Madhava_of_Sangamagrama" title="Madhava of Sangamagrama">Madhava of Sangamagrama</a> (c. 1340 – c. 1425). The table lists the <a href="/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81" title="Jyā, koti-jyā and utkrama-jyā">jya-s</a> or Rsines of the twenty-four <a href="/wiki/Angle" title="Angle">angles</a> from 3.75<a href="/wiki/Degree_(angle)" title="Degree (angle)">°</a> to 90° in steps of 3.75° (1/24 of a <a href="/wiki/Right_angle" title="Right angle">right angle</a>, 90°). Rsine is just the sine multiplied by a selected radius and given as an integer. In this table, as in <a href="/wiki/Aryabhata%27s_sine_table" class="mw-redirect" title="Aryabhata's sine table">Aryabhata's earlier table</a>, <i>R</i> is taken as 21600 ÷ 2<span class="texhtml mvar" style="font-style:italic;">π</span> ≈ 3437.75. </p><p>The table is <a href="/wiki/Character_encoding" title="Character encoding">encoded</a> in the letters of the <a href="/wiki/Sanskrit" title="Sanskrit">Sanskrit</a> alphabet using the <a href="/wiki/Katapayadi_system" title="Katapayadi system">Katapayadi system</a>, giving entries the appearance of the verses of a poem. </p><p>Madhava's original work containing the table has not been found. The table is reproduced in the <i>Aryabhatiyabhashya</i> of <a href="/wiki/Nilakantha_Somayaji" title="Nilakantha Somayaji">Nilakantha Somayaji</a><sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>(1444–1544) and also in the <i>Yuktidipika/Laghuvivrti</i> commentary of <a href="/wiki/Tantrasamgraha" title="Tantrasamgraha">Tantrasamgraha</a> by <a href="/wiki/Sankara_Variar" title="Sankara Variar">Sankara Variar</a> (circa. 1500–1560).<sup id="cite_ref-Raju_2-0" class="reference"><a href="#cite_note-Raju-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 114–123">: 114–123 </span></sup> </p><p>The verses below are given as in <i>Cultural foundations of mathematics</i> by C.K. Raju.<sup id="cite_ref-Raju_2-1" class="reference"><a href="#cite_note-Raju-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 114–123">: 114–123 </span></sup> They are also given in the <i>Malayalam Commentary of <a href="/wiki/Karanapaddhati" title="Karanapaddhati">Karanapaddhati</a></i> by P.K. Koru<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> but slightly differently. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="The_table">The table</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Madhava%27s_sine_table&action=edit&section=1" title="Edit section: The table"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The verses are: </p> <div class="poem"> <p>श्रेष्ठं नाम वरिष्ठानां हिमाद्रिर्वेदभावनः ।<br /> तपनो भानु सूक्तज्ञो मध्यमं विद्धि दोहनम् ॥ १ ॥<br /> धिगाज्यो नाशनं कष्टं छन्नभोगाशयाम्बिका ।<br /> मृगाहारो नरेशोयं वीरो रणजयोत्सुकः ॥ २ ॥<br /> मूलं विशुद्धं नाळस्य गानेषु विरळा नराः ।<br /> अशुद्धिगुप्ता चोरश्रीः शङ्कुकर्णो नगेश्वरः ॥ ३ ॥<br /> तनुजो गर्भजो मित्रं श्रीमानत्र सुखी सखे ।<br /> शशी रात्रौ हिमाहारौ वेगज्ञः पथि सिन्धुरः ॥ ४ ॥<br /> छाया लयो गजो नीलो निर्मलो नास्ति सत्कुले ।<br /> रात्रौ दर्पणमभ्राङ्गं नागस्तुङ्गनखो बली ॥ ५ ॥<br /> धीरो युवा कथालोलः पूज्यो नारीजनैर्भगः ।<br /> कन्यागारे नागवल्ली देवो विश्वस्थली भृगुः ॥ ६ ॥<br /> तत्परादिकलान्तास्तु महाज्या माधवोदिताः ।<br /> स्वस्वपूर्वविशुद्धे तु शिष्टास्तत्खण्डमौर्विकाः ॥ ७ ॥ </p> </div> <p>The quarters of the first six verses represent entries for the twenty-four angles from 3.75° to 90° in steps of 3.75° (first column). The second column contains the Rsine values encoded as Sanskrit words (in Devanagari). The third column contains the same in <a href="/wiki/ISO_15919" title="ISO 15919">ISO 15919 transliterations</a>. The fourth column contains the numbers decoded into arcminutes, arcseconds, and arcthirds in modern numerals. The modern values scaled by the traditional “radius” (21600 ÷ 2<span class="texhtml mvar" style="font-style:italic;">π</span>, with the modern value of <span class="texhtml mvar" style="font-style:italic;">π</span> with two decimals in the arcthirds are given in the fifth column. </p> <table align="center" border="1" cellpadding="5" cellspacing="1"> <tbody><tr> <th rowspan="2">Angle <i>A</i>, degrees </th> <th colspan="3"><i>R</i> sin <i>A</i> given by Madhava </th> <th rowspan="2">Modern <br /> sin <i>A</i> × (21600 ÷ 2<span class="texhtml mvar" style="font-style:italic;">π</span>) <br /> to 2 decimals </th></tr> <tr> <td><b>In Devanagari script</b> </td> <td><b>ISO 15919 transliteration</b> </td> <td><b>Decoded angle in</b><br /><b>minutes′ seconds″ thirds‴</b> </td></tr> <tr> <td style="text-align:center;"><b>(1)</b> </td> <td style="text-align:center;"><b>(2)</b> </td> <td style="text-align:center;"><b>(3)</b> </td> <td style="text-align:center;"><b>(4)</b> </td> <td style="text-align:center;"><b>(5)</b> </td></tr> <tr> <td style="text-align:center;">03.75 </td> <td>श्रेष्ठं नाम वरिष्ठानां </td> <td>śreṣṭhaṁ nāma variṣṭhānāṁ </td> <td style="text-align:center;">0224′50″22‴ </td> <td style="text-align:center;">0224′50″21.83‴ </td></tr> <tr> <td style="text-align:center;">07.50 </td> <td>हिमाद्रिर्वेदभावनः </td> <td>himādrirvēdabhāvanaḥ </td> <td style="text-align:center;">0448′42″58‴ </td> <td style="text-align:center;">0448′42″57.58‴ </td></tr> <tr> <td style="text-align:center;">11.25 </td> <td>तपनो भानुसूक्तज्ञो </td> <td>tapanō bhānusūktajñō </td> <td style="text-align:center;">0670′40″16‴ </td> <td style="text-align:center;">0670′40″16.05‴ </td></tr> <tr> <td style="text-align:center;">15.00 </td> <td>मध्यमं विद्धि दोहनम् </td> <td>madhyamaṁ viddhi dōhanam </td> <td style="text-align:center;">0889′45″15‴ </td> <td style="text-align:center;">0889′45″15.61‴ </td></tr> <tr> <td style="text-align:center;">18.75 </td> <td>धिगाज्यो नाशनं कष्टं </td> <td>dhigājyō nāśanaṁ kaṣṭaṁ </td> <td style="text-align:center;">1105′01″39‴ </td> <td style="text-align:center;">1105′01″38.94‴ </td></tr> <tr> <td style="text-align:center;">22.50 </td> <td>छन्नभोगाशयाम्बिका </td> <td>channabhōgāśayāmbikā </td> <td style="text-align:center;">1315′34″07‴ </td> <td style="text-align:center;">1315′34″07.44‴ </td></tr> <tr> <td style="text-align:center;">26.25 </td> <td>मृगाहारो नरेशोयं </td> <td>mr̥gāhārō narēśōyaṁ </td> <td style="text-align:center;">1520′28″35‴ </td> <td style="text-align:center;">1520′28″35.46‴ </td></tr> <tr> <td style="text-align:center;">30.00 </td> <td>वीरो रणजयोत्सुकः </td> <td>vīrō raṇajayōtsukaḥ </td> <td style="text-align:center;">1718′52″24‴ </td> <td style="text-align:center;">1718′52″24.19‴ </td></tr> <tr> <td style="text-align:center;">33.75 </td> <td>मूलं विशुद्धं नाळस्य </td> <td>mūlaṁ viśuddhaṁ nāḷasya </td> <td style="text-align:center;">1909′54″35‴ </td> <td style="text-align:center;">1909′54″35.19‴ </td></tr> <tr> <td style="text-align:center;">37.50 </td> <td>गानेषु विरळा नराः </td> <td>gāneṣu viraḷā narāḥ </td> <td style="text-align:center;">2092′46″03‴ </td> <td style="text-align:center;">2092′46″03.49‴ </td></tr> <tr> <td style="text-align:center;">41.25 </td> <td>अशुद्धिगुप्ता चोरश्रीः </td> <td>aśuddhiguptā cōraśrīḥ </td> <td style="text-align:center;">2266′39″50‴ </td> <td style="text-align:center;">2266′39″50.21‴ </td></tr> <tr> <td style="text-align:center;">45.00 </td> <td>शङ्कुकर्णो नगेश्वरः </td> <td>śaṅkukarṇō nageśvaraḥ </td> <td style="text-align:center;">2430′51″15‴ </td> <td style="text-align:center;">2430′51″14.59‴ </td></tr> <tr> <td style="text-align:center;">48.75 </td> <td>तनुजो गर्भजो मित्रं </td> <td>tanujō garbhajō mitraṃ </td> <td style="text-align:center;">2584′38″06‴ </td> <td style="text-align:center;">2584′38″05.53‴ </td></tr> <tr> <td style="text-align:center;">52.50 </td> <td>श्रीमानत्र सुखी सखे </td> <td>śrīmānatra sukhī sakhē </td> <td style="text-align:center;">2727′20″52‴ </td> <td style="text-align:center;">2727′20″52.38‴ </td></tr> <tr> <td style="text-align:center;">56.25 </td> <td>शशी रात्रौ हिमाहारौ </td> <td>śaśī rātrou himāhārou </td> <td style="text-align:center;">2858′22″55‴ </td> <td style="text-align:center;">2858′22″55.11‴ </td></tr> <tr> <td style="text-align:center;">60.00 </td> <td>वेगज्ञः पथि सिन्धुरः </td> <td>vēgajñaḥ pathi sindhuraḥ </td> <td style="text-align:center;">2977′10″34‴ </td> <td style="text-align:center;">2977′10″33.73‴ </td></tr> <tr> <td style="text-align:center;">63.25 </td> <td>छाया लयो गजो नीलो </td> <td>chāya layō gajō nīlō </td> <td style="text-align:center;">3083′13″17‴ </td> <td style="text-align:center;">3083′13″16.94‴ </td></tr> <tr> <td style="text-align:center;">67.50 </td> <td>निर्मलो नास्ति सत्कुले </td> <td>nirmalō nāsti satkulē </td> <td style="text-align:center;">3176′03″50‴ </td> <td style="text-align:center;">3176′03″49.97‴ </td></tr> <tr> <td style="text-align:center;">71.25 </td> <td>रात्रौ दर्पणमभ्राङ्गं </td> <td>rātrou darpaṇamabhrāṅgaṁ </td> <td style="text-align:center;">3255′18″22‴ </td> <td style="text-align:center;">3255′18″21.58‴ </td></tr> <tr> <td style="text-align:center;">75.00 </td> <td>नागस्तुङ्गनखो बली </td> <td>nāgastuṅganakhō balī </td> <td style="text-align:center;">3320′36″30‴ </td> <td style="text-align:center;">3320′36″30.20‴ </td></tr> <tr> <td style="text-align:center;">78.75 </td> <td>धीरो युवा कथालोलः </td> <td>dhīrō yuvā kathālōlaḥ </td> <td style="text-align:center;">3371′41″29‴ </td> <td style="text-align:center;">3371′41″29.15‴ </td></tr> <tr> <td style="text-align:center;">82.50 </td> <td>पूज्यो नारीजनैर्भगः </td> <td>pūjyō nārījanairbhagaḥ </td> <td style="text-align:center;">3408′20″11‴ </td> <td style="text-align:center;">3408′20″10.93‴ </td></tr> <tr> <td style="text-align:center;">86.25 </td> <td>कन्यागारे नागवल्ली </td> <td>kanyāgārē nāgavallī </td> <td style="text-align:center;">3430′23″11‴ </td> <td style="text-align:center;">3430′23″10.65‴ </td></tr> <tr> <td style="text-align:center;">90.00 </td> <td>देवो विश्वस्थली भृगुः </td> <td>devō viśvasthalī bhr̥ guḥ </td> <td style="text-align:center;">3437′44″48‴ </td> <td style="text-align:center;">3437′44″48.37‴ </td></tr></tbody></table> <p>The last verse means: “These are the great R-sines as said by Madhava, comprising arcminutes, seconds and thirds. Subtracting from each the previous will give the R-sine-differences.” </p><p>By comparing, one can note that Madhava's values are accurately given rounded to the declared precision of thirds except for Rsin(15°) where one feels he should have rounded up to 889′45″16‴ instead. </p><p>Note that in the <a href="/wiki/Katapayadi_system" title="Katapayadi system">Katapayadi system</a> the digits are written in the reverse order, so for example the literal entry corresponding to 15° is 51549880 which is reversed and then read as 0889′45″15‴. Note that the 0 does not carry a value but is used for the metre of the poem alone. </p> <div class="mw-heading mw-heading2"><h2 id="A_simple_way_of_understanding_the_table">A simple way of understanding the table</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Madhava%27s_sine_table&action=edit&section=2" title="Edit section: A simple way of understanding the table"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Without going into the philosophy of why the value of <i>R</i> = 21600 ÷ 2<span class="texhtml mvar" style="font-style:italic;">π</span> was chosen etc, the simplest way to relate the jya tables to our modern concept of sine tables is as follows: </p><p>Even today sine tables are given as decimals to a certain precision. If sin(15°) is given as 0.1736, it means the rational 1736 ÷ 10000 is a good approximation of the actual infinite precision number. The only difference is that in the earlier days they had not standardized on decimal values (or powers of ten as denominator) for fractions. Hence they used other denominators based on other considerations (which are not discussed here). </p><p>Hence the sine values represented in the tables may simply be taken as approximated by the given integer values divided by the <i>R</i> chosen for the table. </p><p>Another possible confusion point is the usage of angle measures like arcminute etc in expressing the R-sines. Modern sines are unitless ratios. Jya-s or R-sines are the same multiplied by a measure of length or distance. However, since these tables were mostly used for astronomy, and distance on the celestial sphere is expressed in angle measures, these values are also given likewise. However, the unit is not really important and need not be taken too seriously, as the value will anyhow be used as part of a rational and the unit will cancel out. </p><p>However, this also leads to the usage of sexagesimal subdivisions in Madhava's refining the earlier table of Aryabhata. Instead of choosing a larger <i>R</i>, he gave the extra precision determined by him on top of the earlier given minutes by using seconds and thirds. As before, these may simply be taken as a different way of expressing fractions and not necessarily as angle measures. </p> <div class="mw-heading mw-heading2"><h2 id="Another_(more_difficult)_way_to_understand_the_values"><span id="Another_.28more_difficult.29_way_to_understand_the_values"></span>Another (more difficult) way to understand the values</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Madhava%27s_sine_table&action=edit&section=3" title="Edit section: Another (more difficult) way to understand the values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Madhavasine.jpeg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Madhavasine.jpeg/300px-Madhavasine.jpeg" decoding="async" width="300" height="238" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Madhavasine.jpeg/450px-Madhavasine.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Madhavasine.jpeg/600px-Madhavasine.jpeg 2x" data-file-width="621" data-file-height="493" /></a><figcaption>Diagram explaining the meaning of the values in Madhava's table</figcaption></figure> <p>Consider some angle whose measure is <i>A</i>. Consider a <a href="/wiki/Circle" title="Circle">circle</a> of unit radius and center O. Let the arc PQ of the circle subtend an angle <i>A</i> at the center O. Drop the <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> QR from Q to OP; then the length of the line segment RQ is the value of the trigonometric sine of the angle <i>A</i>. Let PS be an arc of the circle whose length is equal to the length of the segment RQ. For various angles <i>A</i>, Madhava's table gives the measures of the corresponding angles <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 \angle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb772c0cb6f32e9359da3c1a0a26f8ac8f66301f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \angle }"></span>POS in <a href="/wiki/Minute_of_arc" class="mw-redirect" title="Minute of arc">arcminutes</a>, <a href="/wiki/Arcsecond" class="mw-redirect" title="Arcsecond">arcseconds</a> and sixtieths of an <a href="/wiki/Arcsecond" class="mw-redirect" title="Arcsecond">arcsecond</a>. </p><p>As an example, let <i>A</i> be an angle whose measure is 22.50°. In Madhava's table, the entry corresponding to 22.50° is the measure in arcminutes, arcseconds and sixtieths of an arcsecond of the angle whose radian measure is the value of <code>sin 22.50°</code>, which is 0.3826834; </p> <dl><dd>multiply 0.3826834 radians by 180/<span class="texhtml mvar" style="font-style:italic;">π</span> to convert to 21.92614 degrees, which is</dd> <dd>1315 arcminutes 34 arcseconds 07 sixtieths of an arcsecond, abbreviated 13153407.</dd></dl> <p>For an angle whose measure is <i>A</i>, let </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 \angle POS=m{\text{ arcminutes, }}s{\text{ arcseconds, }}t{\text{ sixtieths of an arcsecond}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>P</mi> <mi>O</mi> <mi>S</mi> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> arcminutes, </mtext> </mrow> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> arcseconds, </mtext> </mrow> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> sixtieths of an arcsecond</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle POS=m{\text{ arcminutes, }}s{\text{ arcseconds, }}t{\text{ sixtieths of an arcsecond}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40ec5888b7624a3d65be140dfc225d6034884c9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:64.235ex; height:2.509ex;" alt="{\displaystyle \angle POS=m{\text{ arcminutes, }}s{\text{ arcseconds, }}t{\text{ sixtieths of an arcsecond}}}"></span></dd></dl> <p>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 {\begin{aligned}\sin(A)&=RQ\\&={\text{length of arc }}PS\\&=\angle POS{\text{ in radians}}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>R</mi> <mi>Q</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>length of arc </mtext> </mrow> <mi>P</mi> <mi>S</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>P</mi> <mi>O</mi> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> in radians</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(A)&=RQ\\&={\text{length of arc }}PS\\&=\angle POS{\text{ in radians}}\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25725635ed6509d3fc3e97a496f35856c4f4d8e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:27.439ex; height:9.009ex;" alt="{\displaystyle {\begin{aligned}\sin(A)&=RQ\\&={\text{length of arc }}PS\\&=\angle POS{\text{ in radians}}\\\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Derivation_of_trigonometric_sines_from_the_table">Derivation of trigonometric sines from the table</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Madhava%27s_sine_table&action=edit&section=4" title="Edit section: Derivation of trigonometric sines from the table"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Each of the lines in the table specifies eight digits. Let the digits corresponding to angle <i>A</i> (read from left to right) be: </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 d_{1}\quad d_{2}\quad d_{3}\quad d_{4}\quad d_{5}\quad d_{6}\quad d_{7}\quad d_{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="1em" /> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="1em" /> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="1em" /> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mspace width="1em" /> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mspace width="1em" /> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mspace width="1em" /> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mspace width="1em" /> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}\quad d_{2}\quad d_{3}\quad d_{4}\quad d_{5}\quad d_{6}\quad d_{7}\quad d_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c632c71b0ab165ab57c3ea3e3a3e0cde7d9c2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.363ex; height:2.509ex;" alt="{\displaystyle d_{1}\quad d_{2}\quad d_{3}\quad d_{4}\quad d_{5}\quad d_{6}\quad d_{7}\quad d_{8}}"></span></dd></dl> <p>Then according to the rules of the <a href="/wiki/Katapayadi_system" title="Katapayadi system">Katapayadi system</a> they should be taken from right to left and we have: </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 {\begin{aligned}m&=d_{8}\times 1000+d_{7}\times 100+d_{6}\times 10+d_{5}\\s&=d_{4}\times 10+d_{3}\\t&=d_{2}\times 10+d_{1}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>m</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>×</mo> <mn>1000</mn> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>×</mo> <mn>100</mn> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>×</mo> <mn>10</mn> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>×</mo> <mn>10</mn> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>×</mo> <mn>10</mn> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}m&=d_{8}\times 1000+d_{7}\times 100+d_{6}\times 10+d_{5}\\s&=d_{4}\times 10+d_{3}\\t&=d_{2}\times 10+d_{1}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b815a4b5e4c428e383b05d1800301bdcc202c98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:42.447ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}m&=d_{8}\times 1000+d_{7}\times 100+d_{6}\times 10+d_{5}\\s&=d_{4}\times 10+d_{3}\\t&=d_{2}\times 10+d_{1}\end{aligned}}}"></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 B=m^{\prime }s^{\prime \prime }t^{\prime \prime \prime }={\frac {1^{\circ }}{60}}\left(m+{\frac {s}{60}}+{\frac {t}{60\times 60}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mn>60</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>m</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>60</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mrow> <mn>60</mn> <mo>×</mo> <mn>60</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=m^{\prime }s^{\prime \prime }t^{\prime \prime \prime }={\frac {1^{\circ }}{60}}\left(m+{\frac {s}{60}}+{\frac {t}{60\times 60}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1772c6a44617d04cb049413b2cf5f092bdf2db40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.521ex; height:6.176ex;" alt="{\displaystyle B=m^{\prime }s^{\prime \prime }t^{\prime \prime \prime }={\frac {1^{\circ }}{60}}\left(m+{\frac {s}{60}}+{\frac {t}{60\times 60}}\right)}"></span></dd></dl> <p>The value of the above angle <i>B</i> expressed in radians will correspond to the sine value of <i>A</i>. </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 \sin A={\frac {\pi }{180}}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>180</mn> </mfrac> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin A={\frac {\pi }{180}}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fd702df270224605d21e378ac0d4c867199025b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.172ex; height:4.676ex;" alt="{\displaystyle \sin A={\frac {\pi }{180}}B}"></span></dd></dl> <p>As said earlier, this is the same as dividing the encoded value by the taken <i>R</i> value: </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 \sin A={\frac {B}{\frac {21600^{\prime }}{2\pi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>B</mi> <mfrac> <msup> <mn>21600</mn> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin A={\frac {B}{\frac {21600^{\prime }}{2\pi }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a00f51038c745a63a8921a788ff8278cf2f58e32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:14.398ex; height:7.009ex;" alt="{\displaystyle \sin A={\frac {B}{\frac {21600^{\prime }}{2\pi }}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Madhava%27s_sine_table&action=edit&section=5" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The table lists the following digits corresponding to the angle <i>A</i> = 45.00°: </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 5\quad 1\quad 1\quad 5\quad 0\quad 3\quad 4\quad 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mspace width="1em" /> <mn>1</mn> <mspace width="1em" /> <mn>1</mn> <mspace width="1em" /> <mn>5</mn> <mspace width="1em" /> <mn>0</mn> <mspace width="1em" /> <mn>3</mn> <mspace width="1em" /> <mn>4</mn> <mspace width="1em" /> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\quad 1\quad 1\quad 5\quad 0\quad 3\quad 4\quad 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3940bd2b684ea241102eee95c0406b118ab0e4c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:25.558ex; height:2.176ex;" alt="{\displaystyle 5\quad 1\quad 1\quad 5\quad 0\quad 3\quad 4\quad 2}"></span></dd></dl> <p>This yields the angle with measure: </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 {\begin{aligned}m&=2\times 1000+4\times 100+3\times 10+0{\text{ arcminutes}}\\&=2430{\text{ arcminutes}}\\s&=5\times 10+1{\text{ arcseconds}}\\&=51{\text{ arcseconds}}\\t&=1\times 10+5{\text{ sixtieths of an arcsecond}}\\&=15{\text{ sixtieths of an arcsecond}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>m</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo>×</mo> <mn>1000</mn> <mo>+</mo> <mn>4</mn> <mo>×</mo> <mn>100</mn> <mo>+</mo> <mn>3</mn> <mo>×</mo> <mn>10</mn> <mo>+</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> arcminutes</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2430</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> arcminutes</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>5</mn> <mo>×</mo> <mn>10</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> arcseconds</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>51</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> arcseconds</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>×</mo> <mn>10</mn> <mo>+</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> sixtieths of an arcsecond</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> sixtieths of an arcsecond</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}m&=2\times 1000+4\times 100+3\times 10+0{\text{ arcminutes}}\\&=2430{\text{ arcminutes}}\\s&=5\times 10+1{\text{ arcseconds}}\\&=51{\text{ arcseconds}}\\t&=1\times 10+5{\text{ sixtieths of an arcsecond}}\\&=15{\text{ sixtieths of an arcsecond}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ca0befb922e44208b3172ed7e8914b67afe57a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:49.753ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}m&=2\times 1000+4\times 100+3\times 10+0{\text{ arcminutes}}\\&=2430{\text{ arcminutes}}\\s&=5\times 10+1{\text{ arcseconds}}\\&=51{\text{ arcseconds}}\\t&=1\times 10+5{\text{ sixtieths of an arcsecond}}\\&=15{\text{ sixtieths of an arcsecond}}\end{aligned}}}"></span></dd></dl> <p>From which we get: </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 B={\frac {1^{\circ }}{60}}\left(2430+{\frac {51}{60}}+{\frac {15}{60\times 60}}\right)={\frac {116681}{2880}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mn>60</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2430</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>51</mn> <mn>60</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>15</mn> <mrow> <mn>60</mn> <mo>×</mo> <mn>60</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>116681</mn> <mn>2880</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B={\frac {1^{\circ }}{60}}\left(2430+{\frac {51}{60}}+{\frac {15}{60\times 60}}\right)={\frac {116681}{2880}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33fd96783c61cd3250ff58c0bb871652443d5dd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.559ex; height:6.176ex;" alt="{\displaystyle B={\frac {1^{\circ }}{60}}\left(2430+{\frac {51}{60}}+{\frac {15}{60\times 60}}\right)={\frac {116681}{2880}}}"></span></dd></dl> <p>The value of the sine of <i>A</i> = 45.00° as given in Madhava's table is then just <i>B</i> converted to radians: </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 \sin 45^{\circ }={\frac {\pi }{180}}B={\frac {\pi }{180}}\times {\frac {116681}{2880}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>180</mn> </mfrac> </mrow> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>180</mn> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>116681</mn> <mn>2880</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin 45^{\circ }={\frac {\pi }{180}}B={\frac {\pi }{180}}\times {\frac {116681}{2880}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8812a1fb5e8cdd1506dcd118cf692c4c469956e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.881ex; height:5.176ex;" alt="{\displaystyle \sin 45^{\circ }={\frac {\pi }{180}}B={\frac {\pi }{180}}\times {\frac {116681}{2880}}}"></span></dd></dl> <p>Evaluating the above, one can find that sin 45° is 0.70710681… This is accurate to 6 decimal places. </p> <div class="mw-heading mw-heading2"><h2 id="Madhava's_method_of_computation"><span id="Madhava.27s_method_of_computation"></span>Madhava's method of computation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Madhava%27s_sine_table&action=edit&section=6" title="Edit section: Madhava's method of computation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>No work of Madhava detailing the methods used by him for the computation of the sine table has survived. However from the writings of later Kerala mathematicians including <a href="/wiki/Nilakantha_Somayaji" title="Nilakantha Somayaji">Nilakantha Somayaji</a> (<a href="/wiki/Tantrasangraha" class="mw-redirect" title="Tantrasangraha">Tantrasangraha</a>) and <a href="/wiki/Jyeshtadeva" class="mw-redirect" title="Jyeshtadeva">Jyeshtadeva</a> (<a href="/wiki/Yuktibh%C4%81%E1%B9%A3%C4%81" title="Yuktibhāṣā">Yuktibhāṣā</a>) that give ample references to Madhava's accomplishments, it is conjectured that Madhava computed his sine table using the <a href="/wiki/Sine_and_cosine#Series_definitions" title="Sine and cosine">power series expansion of sin <i>x</i></a>: </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 \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15a9027c77b90e40215c01281b39d37730e7e537" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.746ex; height:5.843ex;" alt="{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots }"></span></dd></dl> <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=Madhava%27s_sine_table&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/Madhava_series" title="Madhava series">Madhava series</a></li> <li><a href="/wiki/Madhava%27s_correction_term" title="Madhava's correction term">Madhava's correction term</a></li> <li><a href="/wiki/Madhava%27s_value_of_%CF%80" class="mw-redirect" title="Madhava's value of π">Madhava's value of <span class="texhtml mvar" style="font-style:italic;">π</span></a></li> <li><a href="/wiki/%C4%80ryabha%E1%B9%ADa%27s_sine_table" title="Āryabhaṭa's sine table">Āryabhaṭa's sine table</a></li> <li><a href="/wiki/Ptolemy%27s_table_of_chords" title="Ptolemy's table of chords">Ptolemy's table of chords</a></li></ul> <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=Madhava%27s_sine_table&action=edit&section=8" 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-:0-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-:0_1-0">^</a></b></span> <span class="reference-text"><i>The Aryabhatiam of Aryabhattacharya with the Bhashya of Nilakantha Somasutvan, Part 1 - Gaṇitapāda,</i> Edited by K. Sambasiva Sastri, Trivandrum Sanskrit Series No.101. p. 55. <a rel="nofollow" class="external free" href="https://archive.org/details/Trivandrum_Sanskrit_Series_TSS">https://archive.org/details/Trivandrum_Sanskrit_Series_TSS</a> <a rel="nofollow" class="external free" href="http://www.sanskritebooks.org/2013/02/trivandrum-sanskrit-series-anantasayana-samskrita-granthavali/">http://www.sanskritebooks.org/2013/02/trivandrum-sanskrit-series-anantasayana-samskrita-granthavali/</a></span> </li> <li id="cite_note-Raju-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Raju_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Raju_2-1"><sup><i><b>b</b></i></sup></a></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="CITEREFC.K._Raju2007" class="citation book cs1 cs1-prop-long-vol">C.K. Raju (2007). <i>Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16th c. CE</i>. History of Philosophy, Science and Culture in Indian Civilization. Vol. X Part 4. New Delhi: Centre for Studies in Civilizations and Pearson Education in South Asia. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-317-0871-2" title="Special:BookSources/978-81-317-0871-2"><bdi>978-81-317-0871-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cultural+foundations+of+mathematics%3A+The+nature+of+mathematical+proof+and+the+transmission+of+calculus+from+India+to+Europe+in+the+16th+c.+CE&rft.place=New+Delhi&rft.series=History+of+Philosophy%2C+Science+and+Culture+in+Indian+Civilization&rft.pub=Centre+for+Studies+in+Civilizations+and+Pearson+Education+in+South+Asia&rft.date=2007&rft.isbn=978-81-317-0871-2&rft.au=C.K.+Raju&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMadhava%27s+sine+table" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPuthumana_Somayaji" class="citation book cs1"><a href="/wiki/Puthumana_Somayaji" title="Puthumana Somayaji">Puthumana Somayaji</a>. <i>Karanapaddhati (with a commentary in Malayalam by P.K. Koru)</i>. <a href="/wiki/Cherpu" title="Cherpu">Cherpu</a>, <a href="/wiki/Kerala" title="Kerala">Kerala</a>, <a href="/wiki/India" title="India">India</a>: Astro Printing and Publishing Company.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Karanapaddhati+%28with+a+commentary+in+Malayalam+by+P.K.+Koru%29&rft.place=Cherpu%2C+Kerala%2C+India&rft.pub=Astro+Printing+and+Publishing+Company&rft.au=Puthumana+Somayaji&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMadhava%27s+sine+table" class="Z3988"></span> (Published in 1953)</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_references">Further references</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Madhava%27s_sine_table&action=edit&section=9" title="Edit section: Further references"><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="CITEREFBag1976" class="citation journal cs1">Bag, A.K. (1976). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150705200732/http://www.dli.gov.in/rawdataupload/upload/insa/INSA_1/20005af4_54.pdf">"Madhava's sine and cosine series"</a> <span class="cs1-format">(PDF)</span>. <i>Indian Journal of History of Science</i>. <b>11</b> (1). Indian National Academy of Science: 54–57. Archived from <a rel="nofollow" class="external text" href="http://www.dli.gov.in/rawdataupload/upload/insa/INSA_1/20005af4_54.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 5 July 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">21 August</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Indian+Journal+of+History+of+Science&rft.atitle=Madhava%27s+sine+and+cosine+series&rft.volume=11&rft.issue=1&rft.pages=54-57&rft.date=1976&rft.aulast=Bag&rft.aufirst=A.K.&rft_id=http%3A%2F%2Fwww.dli.gov.in%2Frawdataupload%2Fupload%2Finsa%2FINSA_1%2F20005af4_54.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMadhava%27s+sine+table" class="Z3988"></span></li> <li>For an account of Madhava's computation of the sine table see : <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_Brummelen2009" class="citation book cs1">Van Brummelen, Glen (2009). <a rel="nofollow" class="external text" href="http://press.princeton.edu/titles/8956.html"><i>The mathematics of the heavens and the earth : the early history of trigonometry</i></a>. Princeton: <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. pp. 113–120. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-12973-0" title="Special:BookSources/978-0-691-12973-0"><bdi>978-0-691-12973-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+mathematics+of+the+heavens+and+the+earth+%3A+the+early+history+of+trigonometry&rft.place=Princeton&rft.pages=113-120&rft.pub=Princeton+University+Press&rft.date=2009&rft.isbn=978-0-691-12973-0&rft.aulast=Van+Brummelen&rft.aufirst=Glen&rft_id=http%3A%2F%2Fpress.princeton.edu%2Ftitles%2F8956.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMadhava%27s+sine+table" class="Z3988"></span></li> <li>For a thorough discussion of the computation of Madhava's sine table with historical references : <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFC.K._Raju2007" class="citation book cs1 cs1-prop-long-vol">C.K. Raju (2007). <i>Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE</i>. History of Philosophy, Science and Culture in Indian Civilization. Vol. X Part 4. Delhi: Centre for Studies in Civilizations. pp. 114–123.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cultural+foundations+of+mathematics%3A+The+nature+of+mathematical+proof+and+the+transmission+of+calculus+from+India+to+Europe+in+the+16+thc.+CE&rft.place=Delhi&rft.series=History+of+Philosophy%2C+Science+and+Culture+in+Indian+Civilization&rft.pages=114-123&rft.pub=Centre+for+Studies+in+Civilizations&rft.date=2007&rft.au=C.K.+Raju&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMadhava%27s+sine+table" class="Z3988"></span></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 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href="/wiki/List_of_astronomers_and_mathematicians_of_the_Kerala_school" title="List of astronomers and mathematicians of the Kerala school">(Complete list)</a></li> <li><a href="/wiki/Achyutha_Pisharadi" class="mw-redirect" title="Achyutha Pisharadi">Acyuta Piṣāraṭi</a></li> <li><a href="/wiki/Azhvanchery_Thamprakkal" title="Azhvanchery Thamprakkal">Azhvāñceri Taṃprākkaḷ</a></li> <li><a href="/wiki/Chitrabhanu_(mathematician)" title="Chitrabhanu (mathematician)">Citrabhānu</a></li> <li><a href="/wiki/Damodara" title="Damodara">Dāmodara of Vațaśreņi</a></li> <li><a href="/wiki/Dev%C4%81c%C4%81rya" title="Devācārya">Devācārya</a></li> <li><a href="/wiki/Govinda_Bhattathiri" title="Govinda Bhattathiri">Govinda Bhattathiri</a></li> <li><a href="/wiki/Govindasvamin" class="mw-redirect" title="Govindasvamin">Govindasvāmin</a></li> <li><a href="/wiki/Haridatta" title="Haridatta">Haridatta</a></li> <li><a href="/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li> <li><a href="/wiki/Kochukrishnan_Asan" title="Kochukrishnan Asan">Kochukrishnan Asan</a></li> <li><a href="/wiki/Madhava_of_Sangamagrama" title="Madhava of Sangamagrama">Madhava of Sangamagrama</a></li> <li><a href="/wiki/Mazhama%E1%B9%85gala%E1%B9%83_N%C4%81r%C4%81ya%E1%B9%87an_Na%E1%B9%83p%C5%ABtiri" title="Mazhamaṅgalaṃ Nārāyaṇan Naṃpūtiri">Mazhamaṅgalaṃ Nārāyaṇan Naṃpūtiri</a></li> <li><a href="/wiki/Nilakantha_Somayaji" title="Nilakantha Somayaji">Nīlakaņțha Sōmayāji</a></li> <li><a href="/wiki/Kanakkusaram" title="Kanakkusaram">Nīlakaṇṭha II</a></li> <li><a href="/wiki/Parameshvara_Nambudiri" title="Parameshvara Nambudiri">Parameśvara of Vațaśśeri</a></li> <li><a href="/wiki/%C5%9Aa%E1%B9%85karan%C4%81r%C4%81ya%E1%B9%87a" title="Śaṅkaranārāyaṇa">Śaṅkaranārāyaṇa</a></li> <li><a href="/wiki/Mazhama%E1%B9%85gala%E1%B9%83_%C5%9Aa%E1%B9%85karan_Na%E1%B9%83p%C5%ABtiri" title="Mazhamaṅgalaṃ Śaṅkaran Naṃpūtiri">Śaṅkara of Mahiṣamaṅgalam</a></li> <li><a href="/wiki/Puthumana_Somayaji" title="Puthumana Somayaji">Putumana Somayāji</a></li> <li><a href="/wiki/Sankara_Variar" title="Sankara Variar">Śankara Vāriyar</a></li> <li><a href="/wiki/Sankara_Varman" title="Sankara Varman">Śaṅkara Varma of Kaṭattanāṭ</a></li> <li><a href="/wiki/Udayadiv%C4%81kara" title="Udayadivākara">Udayadivākara</a></li> <li><a href="/wiki/Vararuci" class="mw-redirect" title="Vararuci">Vararuci I</a></li> <li><a href="/wiki/Vidyamadhava" title="Vidyamadhava">Vidyamadhava</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>Aryabhatiya-bhashya (<a href="/wiki/Nilakantha_Somayaji" title="Nilakantha Somayaji">Nīlakaņțha Sōmayāji</a>)</li> <li>Candrachayaganita</li> <li>Dasādhyāyi</li> <li><a href="/wiki/Drigganita" title="Drigganita">Drigganita</a></li> <li>Dṛkkaraṇa</li> <li>Gaṇitayuktayah</li> <li>Golavada</li> <li>Golasara</li> <li>Grahacāraṇinibandhana</li> <li>Jātaka-sāara-saṃgraha</li> <li><a href="/wiki/Jyotirmimamsa" title="Jyotirmimamsa">Jyotirmimamsa</a></li> <li>Jyotiśśāstra-saṃgraha</li> <li><a href="/wiki/Kanakkusaram" title="Kanakkusaram">Kanakkusāraṃ</a></li> <li><a href="/wiki/Karanapaddhati" title="Karanapaddhati">Karaṇapaddhati</a></li> <li><a href="/wiki/Kriyakramakari" title="Kriyakramakari">Kriyakramakari</a></li> <li>Laghuvivṛti (commentary on <a href="/wiki/Tantrasamgraha" title="Tantrasamgraha">Tantrasamgraha</a>)</li> <li>Madhyamanayanaprakara</li> <li>Mahābhāskarīya Bhāshya</li> <li><a href="/wiki/Pa%C3%B1cabodha" title="Pañcabodha">Pañcabodha</a></li> <li><a href="/wiki/Sadratnamala" title="Sadratnamala">Sadratnamālā</a></li> <li>Sidhhantadarpana</li> <li>Spuṭanirṇaya</li> <li><a href="/wiki/Tantrasamgraha" title="Tantrasamgraha">Tantrasamgraha</a></li> <li><a href="/wiki/Venvaroha" title="Venvaroha">Venvaroha</a></li> <li><a href="/wiki/Yuktibh%C4%81%E1%B9%A3%C4%81" title="Yuktibhāṣā">Yuktibhāṣā</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts/Topics</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/Chandravakyas" title="Chandravakyas">Chandravakyas</a></li> <li><a href="/wiki/Drigganita" title="Drigganita">Drigganita system</a></li> <li><a href="/wiki/Katapayadi_system" title="Katapayadi system">Kaṭapayādi system</a></li> <li><a href="/wiki/Madhava%27s_correction_terms" class="mw-redirect" title="Madhava's correction terms">Madhava's correction terms</a></li> <li><a href="/wiki/Madhava_series" title="Madhava series">Madhava series</a></li> <li><a class="mw-selflink selflink">Madhava's sine table</a></li> <li><a href="/wiki/Parahita" title="Parahita">Parahita system</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Places associated with<br />members of the school</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/Athavanad" title="Athavanad">Athavanad</a> (Azhvanchery Thamprakkal)</li> <li><a href="/wiki/Alattiyur,_Malappuram" title="Alattiyur, Malappuram">Alathiyur</a> (Govinda Bhattathiri, etc)</li> <li><a href="/wiki/Kadathanadu" title="Kadathanadu">Kadathanadu</a> (Sankara Varman)</li> <li><a href="/wiki/Kodungallur" title="Kodungallur">Mahodayapuram (Kodungallur)</a> (Sankaranarayana)</li> <li><a href="/wiki/Kudallur" title="Kudallur">Kudallur</a> (Madhava of Sangamagrama)</li> <li><a href="/wiki/Peruvanam" title="Peruvanam">Peruvanam</a> (Mahishamangalam Sankaran Namputhiri)</li> <li><a href="/wiki/Thrikkandiyur" class="mw-redirect" title="Thrikkandiyur">Thrikkandiyur</a> (Nilakantha Somayaji, etc)</li> <li><a href="/wiki/Thrissur" title="Thrissur">Sivapuram (Thrissur)</a> (Putumana Somayaji, etc)</li> <li><a href="/wiki/Tirunavaya" title="Tirunavaya">Tirunavaya</a> (Haridatta)</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="Indian_mathematics" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed 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:Indian_mathematics" title="Template:Indian mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Indian_mathematics" title="Template talk:Indian mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Indian_mathematics" title="Special:EditPage/Template:Indian mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Indian_mathematics" style="font-size:114%;margin:0 4em"><a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_Indian_mathematicians" title="List of Indian mathematicians">Mathematicians</a></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%">Ancient</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/Apastamba_Dharmasutra" title="Apastamba Dharmasutra">Apastamba</a></li> <li><a href="/wiki/Baudhayana_sutras" title="Baudhayana sutras">Baudhayana</a></li> <li><a href="/wiki/K%C4%81ty%C4%81yana" title="Kātyāyana">Katyayana</a></li> <li><a href="/wiki/Manava" title="Manava">Manava</a></li> <li><a href="/wiki/P%C4%81%E1%B9%87ini" title="Pāṇini">Pāṇini</a></li> <li><a href="/wiki/Pingala" title="Pingala">Pingala</a></li> <li><a href="/wiki/Yajnavalkya" title="Yajnavalkya">Yajnavalkya</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classical</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/Aryabhata" title="Aryabhata">Āryabhaṭa I</a></li> <li><a href="/wiki/Aryabhata_II" title="Aryabhata II">Āryabhaṭa II</a></li> <li><a href="/wiki/Bh%C4%81skara_I" title="Bhāskara I">Bhāskara I</a></li> <li><a href="/wiki/Bh%C4%81skara_II" title="Bhāskara II">Bhāskara II</a></li> <li><a href="/wiki/Melpathur_Narayana_Bhattathiri" title="Melpathur Narayana Bhattathiri">Melpathur Narayana Bhattathiri</a></li> <li><a href="/wiki/Brahmadeva" title="Brahmadeva">Brahmadeva</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/Govindasv%C4%81mi" title="Govindasvāmi">Govindasvāmi</a></li> <li><a href="/wiki/Halayudha" title="Halayudha">Halayudha</a></li> <li><a href="/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li> <li><a href="/wiki/Kamalakara" title="Kamalakara">Kamalakara</a></li> <li><a href="/wiki/Madhava_of_Sangamagrama" title="Madhava of Sangamagrama">Mādhava of Saṅgamagrāma</a></li> <li><a href="/wiki/Mah%C4%81v%C4%ABra_(mathematician)" title="Mahāvīra (mathematician)">Mahāvīra</a></li> <li><a href="/wiki/Mahendra_S%C5%ABri" title="Mahendra Sūri">Mahendra Sūri</a></li> <li><a href="/wiki/Munishvara" title="Munishvara">Munishvara</a></li> <li><a href="/wiki/Narayana_Pandit" class="mw-redirect" title="Narayana Pandit">Narayana</a></li> <li><a href="/wiki/Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li> <li><a href="/wiki/Achyuta_Pisharati" class="mw-redirect" title="Achyuta Pisharati">Achyuta Pisharati</a></li> <li><a href="/wiki/Jagannatha_Samrat" title="Jagannatha Samrat">Jagannatha Samrat</a></li> <li><a href="/wiki/Nilakantha_Somayaji" title="Nilakantha Somayaji">Nilakantha Somayaji</a></li> <li><a href="/wiki/%C5%9Ar%C4%ABpati" title="Śrīpati">Śrīpati</a></li> <li><a href="/wiki/Sridhara" title="Sridhara">Sridhara</a></li> <li><a href="/wiki/Gangesha_Upadhyaya" class="mw-redirect" title="Gangesha Upadhyaya">Gangesha Upadhyaya</a></li> <li><a href="/wiki/Var%C4%81hamihira" title="Varāhamihira">Varāhamihira</a></li> <li><a href="/wiki/Sankara_Variar" title="Sankara Variar">Sankara Variar</a></li> <li><a href="/wiki/Virasena" title="Virasena">Virasena</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Modern</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/Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Srinivasa Ramanujan</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Satyendra Nath Bose</a></li> <li><a href="/wiki/P.C._Mahalanobis" class="mw-redirect" title="P.C. Mahalanobis">P.C. Mahalanobis</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Subrahmanyan Chandrasekhar</a></li> <li><a href="/wiki/C.R._Rao" class="mw-redirect" title="C.R. Rao">C.R. Rao</a></li> <li><a href="/wiki/Veeravalli_S._Varadarajan" title="Veeravalli S. Varadarajan">Veeravalli S. Varadarajan</a></li> <li><a href="/wiki/S._R._Srinivasa_Varadhan" title="S. R. Srinivasa Varadhan">S. R. Srinivasa Varadhan</a></li> <li><a href="/wiki/K._R._Parthasarathy_(probabilist)" title="K. R. Parthasarathy (probabilist)">K. R. Parthasarathy (probabilist)</a></li> <li><a href="/wiki/M._S._Narasimhan" title="M. S. Narasimhan">M. S. Narasimhan</a></li> <li><a href="/wiki/C._S._Seshadri" title="C. S. Seshadri">C. S. Seshadri</a></li> <li><a href="/wiki/Harish-Chandra" title="Harish-Chandra">Harish-Chandra</a></li> <li><a href="/wiki/Subbayya_Sivasankaranarayana_Pillai" title="Subbayya Sivasankaranarayana Pillai">Subbayya Sivasankaranarayana Pillai</a></li> <li><a href="/wiki/Tilak_Raj_Prabhakar" title="Tilak Raj Prabhakar">Tilak Raj Prabhakar</a></li> <li><a href="/wiki/Manjul_Bhargava" title="Manjul Bhargava">Manjul Bhargava</a></li> <li><a href="/wiki/Akshay_Venkatesh" title="Akshay Venkatesh">Akshay Venkatesh</a></li> <li><a href="/wiki/Ravi_Vakil" title="Ravi Vakil">Ravi Vakil</a></li> <li><a href="/wiki/Kannan_Soundararajan" title="Kannan Soundararajan">Kannan Soundararajan</a></li> <li><a href="/wiki/Template:SSBPST_recipients_in_Mathematical_Science" title="Template:SSBPST recipients in Mathematical Science">Shanti Swarup Bhatnagar Prize recipients in Mathematical Science</a></li></ul> </div></td></tr></tbody></table><div></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/Aryabhatiya" title="Aryabhatiya">Aryabhatiya</a></i></li> <li><a href="/wiki/Bakhshali_manuscript" title="Bakhshali manuscript">Bakhshali manuscript</a></li> <li><i><a href="/wiki/Bijaganita" title="Bijaganita">Bijaganita</a></i></li> <li><i><a href="/wiki/Br%C4%81hmasphu%E1%B9%ADasiddh%C4%81nta" title="Brāhmasphuṭasiddhānta">Brāhmasphuṭasiddhānta</a></i></li> <li><i><a href="/wiki/Ganita_Kaumudi" title="Ganita Kaumudi">Ganita Kaumudi</a></i></li> <li><i><a href="/wiki/Kanakkusaram" title="Kanakkusaram">Kanakkusaram</a></i></li> <li><i><a href="/wiki/Karanapaddhati" title="Karanapaddhati">Karanapaddhati</a></i></li> <li><i><a href="/wiki/L%C4%ABl%C4%81vat%C4%AB" title="Līlāvatī">Līlāvatī</a></i></li> <li><i><a href="/wiki/Lokavibhaga" title="Lokavibhaga">Lokavibhaga</a></i></li> <li><i><a href="/wiki/P%C4%81t%C4%ABga%E1%B9%87ita" title="Pātīgaṇita">Pātīgaṇita</a></i></li> <li><i><a href="/wiki/Paulisa_Siddhanta" title="Paulisa Siddhanta">Paulisa Siddhanta</a></i></li> <li><i><a href="/wiki/Var%C4%81hamihira#Pancha-Siddhantika" title="Varāhamihira">Paitamaha Siddhanta</a></i></li> <li><i><a href="/wiki/Romaka_Siddhanta" title="Romaka Siddhanta">Romaka Siddhanta</a></i></li> <li><i><a href="/wiki/Sadratnamala" title="Sadratnamala">Sadratnamala</a></i></li> <li><i><a href="/wiki/Siddh%C4%81nta_Shiromani" title="Siddhānta Shiromani">Siddhānta Shiromani</a></i></li> <li><i><a href="/wiki/Shulba_Sutras" title="Shulba Sutras">Śulba Sūtras</a></i></li> <li><i><a href="/wiki/Surya_Siddhanta" title="Surya Siddhanta">Surya Siddhanta</a></i></li> <li><i><a href="/wiki/Tantrasamgraha" title="Tantrasamgraha">Tantrasamgraha</a></i></li> <li><i><a href="/wiki/Vasishtha_Siddhanta" title="Vasishtha Siddhanta">Vasishtha Siddhanta</a></i></li> <li><i><a href="/wiki/Venvaroha" title="Venvaroha">Veṇvāroha</a></i></li> <li><i><a href="/wiki/Yuktibh%C4%81%E1%B9%A3%C4%81" title="Yuktibhāṣā">Yuktibhāṣā</a></i></li> <li><i><a href="/wiki/Yavanajataka" title="Yavanajataka">Yavanajataka</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Pioneering<br /> innovations</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/Brahmi_numerals" title="Brahmi numerals">Brahmi numerals</a></li> <li><a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a></li> <li>Symbol for <a href="/wiki/0" title="0">zero (0)</a></li> <li><a href="/wiki/Madhava_series" title="Madhava series">Infinite series expansions for the trigonometric functions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Centres</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/Kerala_school_of_astronomy_and_mathematics" title="Kerala school of astronomy and mathematics">Kerala school of astronomy and mathematics</a></li> <li><a href="/wiki/Jantar_Mantar" title="Jantar Mantar">Jantar Mantar</a> (<a href="/wiki/Jantar_Mantar,_Jaipur" title="Jantar Mantar, Jaipur">Jaipur</a>, <a href="/wiki/Jantar_Mantar,_New_Delhi" title="Jantar Mantar, New Delhi">New Delhi</a>, <a href="/wiki/Jantar_Mantar,_Ujjain" title="Jantar Mantar, Ujjain">Ujjain</a>, <a href="/wiki/Jantar_Mantar,_Varanasi" title="Jantar Mantar, Varanasi">Varanasi</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Historians of<br /> mathematics</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/Bapudeva_Sastri" title="Bapudeva Sastri">Bapudeva Sastri</a> (1821–1900)</li> <li><a href="/w/index.php?title=Shankar_Balakrishna_Dikshit&action=edit&redlink=1" class="new" title="Shankar Balakrishna Dikshit (page does not exist)">Shankar Balakrishna Dikshit</a> (1853–1898)</li> <li><a href="/wiki/Sudhakara_Dvivedi" title="Sudhakara Dvivedi">Sudhakara Dvivedi</a> (1855–1910)</li> <li><a href="/w/index.php?title=M._Rangacarya&action=edit&redlink=1" class="new" title="M. Rangacarya (page does not exist)">M. Rangacarya</a> (1861–1916)</li> <li><a href="/wiki/P._C._Sengupta" class="mw-redirect" title="P. C. Sengupta">P. C. Sengupta</a> (1876–1962)</li> <li><a href="/wiki/B._B._Datta" class="mw-redirect" title="B. B. Datta">B. B. Datta</a> (1888–1958)</li> <li><a href="/wiki/Takao_Hayashi" title="Takao Hayashi">T. Hayashi</a></li> <li><a href="/w/index.php?title=A._A._Krishnaswamy_Ayyangar&action=edit&redlink=1" class="new" title="A. A. Krishnaswamy Ayyangar (page does not exist)">A. A. Krishnaswamy Ayyangar</a> (1892– 1953)</li> <li><a href="/wiki/Avadhesh_Narayan_Singh" title="Avadhesh Narayan Singh">A. N. Singh</a> (1901–1954)</li> <li><a href="/wiki/C._T._Rajagopal" class="mw-redirect" title="C. T. Rajagopal">C. T. Rajagopal</a> (1903–1978)</li> <li><a href="/wiki/T._A._Saraswati_Amma" class="mw-redirect" title="T. A. Saraswati Amma">T. A. Saraswati Amma</a> (1918–2000)</li> <li><a href="/w/index.php?title=S._N._Sen&action=edit&redlink=1" class="new" title="S. N. Sen (page does not exist)">S. N. Sen</a> (1918–1992)</li> <li><a href="/wiki/K._S._Shukla" class="mw-redirect" title="K. S. Shukla">K. S. Shukla</a> (1918–2007)</li> <li><a href="/wiki/K._V._Sarma" title="K. V. Sarma">K. V. Sarma</a> (1919–2005)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Translators</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/Walter_Eugene_Clark" title="Walter Eugene Clark">Walter Eugene Clark</a></li> <li><a href="/wiki/Henry_Thomas_Colebrooke" title="Henry Thomas Colebrooke">Henry Thomas Colebrooke </a></li> <li><a href="/wiki/David_Pingree" title="David Pingree">David Pingree</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other regions</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_mathematics" title="Babylonian mathematics">Babylon</a></li> <li><a href="/wiki/Chinese_mathematics" title="Chinese mathematics">China</a></li> <li><a href="/wiki/Greek_mathematics" title="Greek mathematics">Greece</a></li> <li><a href="/wiki/Mathematics_in_the_medieval_Islamic_world" title="Mathematics in the medieval Islamic world">Islamic mathematics</a></li> <li><a href="/wiki/History_of_mathematics#Medieval_European_mathematics" title="History of mathematics">Europe</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Modern<br /> institutions</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/Indian_Statistical_Institute" title="Indian Statistical Institute">Indian Statistical Institute</a></li> <li><a href="/wiki/Bhaskaracharya_Pratishthana" title="Bhaskaracharya Pratishthana">Bhaskaracharya Pratishthana</a></li> <li><a href="/wiki/Chennai_Mathematical_Institute" title="Chennai Mathematical Institute">Chennai Mathematical Institute</a></li> <li><a href="/wiki/Institute_of_Mathematical_Sciences,_Chennai" title="Institute of Mathematical Sciences, Chennai">Institute of Mathematical Sciences</a></li> <li><a href="/wiki/Indian_Institute_of_Science" title="Indian Institute of Science">Indian Institute of Science</a></li> <li><a href="/wiki/Harish-Chandra_Research_Institute" title="Harish-Chandra Research Institute">Harish-Chandra Research Institute</a></li> <li><a href="/wiki/Homi_Bhabha_Centre_for_Science_Education" title="Homi Bhabha Centre for Science Education">Homi Bhabha Centre for Science Education</a></li> <li><a href="/wiki/Ramanujan_Institute_for_Advanced_Study_in_Mathematics" title="Ramanujan Institute for Advanced Study in Mathematics">Ramanujan Institute for Advanced Study in Mathematics</a></li> <li><a href="/wiki/Tata_Institute_of_Fundamental_Research" title="Tata Institute of Fundamental Research">TIFR</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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