http://dharmawiki.org/api.php?action=feedcontributions&feedformat=atom&user=BharataPuru 2026-05-05T20:33:07Z User contributions MediaWiki 1.35.4 http://dharmawiki.org/index.php?title=Indus_Valley_DNA&diff=133586 2022-05-02T11:33:49Z

<p>BharataPuru: /* Indus Periphery Cline */</p> <hr /> <div>[[File:Shinde et al 2019.png|thumb|400x400px|Indus Valley Civilisation (IVC) Cline is a mixture of the Iranian-related ancestry as major component with a small component of Andemanese Hunter-Gatherer (AHG) related ancestry. The Iranian-related ancestry in IVC cline and the ancient Iranians had a common ancestor who lived more than 12,000 years ago.|alt=]]<br /> In 2019, for the first time several ancient DNA samples related to the people of Indus-Saraswati Civilisation were published. Shinde et al 2019 published the only bronze age DNA samples found in India. This DNA sample was extracted from a skeleton found at archaeological site of Rakhigarhi located in the ancient Saraswati river plains. It was dated to ~2500 BCE and the skeleton belonged to a woman. Narasimhan et al 2019 published hundreds of DNA samples along with thirteen samples belonging to Indus valley migrants found in the Turan region which includes modern day Iran and Turkmenistan regions. These DNA samples were dated variously between 3300 to 2000 BCE and changed our understanding of the modern Indian population. <br /> <br /> The Iranian related component that was once hypothesised to have arrived in India along with farming has turned out to be much more ancient. As the Shinde et al 2019 paper says, &quot;Iranian-related ancestry in South Asia split from Iranian plateau lineages &gt;12,000 years ago&quot;, implying that the people of Indus Valley and ancient Iranians had a common ancestor who lived more than 12,000 years ago. The IVC's Iranian related ancestry itself contains ~10% WSHG ancestry, which is the Western Siberian Hunter-Gatherer ancestry. This discovery lead the scientists to make the conclusion that farming is indigenous to the Indian subcontinent and was not brought in from the western first farmers of Anatolia.<br /> <br /> Narasimhan paper shows the '''''Out of India migration''''' of Harappans into Iran and into Central Asia. ~50% of the total samples DNA samples found at the Iranian site of ''Shahr-i-sokhta'' belongs to the Harappan migrants and the locals of ''Shahr-i-sokhta'' also derive ~20% of ancestry from the Harappans. Narasimhan paper also find out the main population of BMAC harboured up to 5% ancestry related (deeply in time) to Andemanese hunter-gatherers (AHG). This provides evidence of gene flow from South Asia to Iran and Central Asia. Narasimhan paper finds out that BMAC is not the source of ancestry to the modern South Asians instead South Asia is the source of ancestry to BMAC main population. This disproved the earlier claim made by Damgard et al 2018 which proposed BMAC to be a source of modern South Asian based on only two ancient DNA samples from BMAC and none from South Asia where as Narasimhan et al 2019 published 86 ancient DNA samples from BMAC to show '''''Out of India''''' migration into BMAC.<br /> <br /> == IVC cline ==<br /> Indus Valley DNA is characterised by the combination of one DNA sample found at Rakhigarhi published by Shinde et al 2019 and 11 Indus periphery cline DNA samples extracted from the skeletons found at archaeological sites in Iran and Turkmenistan region where they are identified as outliers published in Narasimhan et al 2019. The excessive AHG related ancestry in those 11 samples had indicated that those samples could belong to the migrants from the IVC. These 12 samples together are referred as IVC cline samples. <br /> <br /> The authors of Shinde et al says, &lt;blockquote&gt;&quot;''While our study is sufficient to demonstrate that this ancestry profile was a common feature of the IVC, a single sample—or even the gradient of 12 likely IVC samples we have identified cannot fully characterise a cosmopolitan ancient civilisation. An important direction for future work will be to carry out ancient DNA analysis of additional individuals across the IVC range to obtain a quantitative understanding of how the ancestry of IVC people was distributed and to characterise other features of its population structure.''&quot;&lt;/blockquote&gt;''IVC cline'' is derived distally as AHG-, WSHG- and Iranian- related groups where IVC's Iranian related ancestry splits from the Iranian plateau lineages more than 12,000 years ago. <br /> {| class=&quot;wikitable&quot;<br /> !<br /> !Sample ID<br /> !Location<br /> !Radiocarbon age<br /> !Sex<br /> !mtDNA <br /> haplogroup<br /> !Y chromosome <br /> haplogroup<br /> |-<br /> |1<br /> |I8726<br /> |Shahr-i Sokhta, Iran<br /> |3100-3000 BCE<br /> |M<br /> |U2c1<br /> |J2a1h<br /> |-<br /> |2<br /> |I11480<br /> |Shahr-i Sokhta, Iran<br /> |2900-2800 BCE<br /> |M<br /> |W6<br /> |J<br /> |-<br /> |3<br /> |I11459<br /> |Shahr-i Sokhta, Iran<br /> |2875-2631 calBCE<br /> |M<br /> |W3a1b<br /> |H1a1d2<br /> |-<br /> |4<br /> |I11460 ''&lt;sub&gt;(Low-coverage)&lt;/sub&gt;''<br /> |Shahr-i Sokhta, Iran<br /> |3200-2100 BCE<br /> |M<br /> |H2a<br /> |BT<br /> |-<br /> |5<br /> |I11458<br /> |Shahr-i Sokhta, Iran<br /> |3200-2100 BCE<br /> |M<br /> |..<br /> |F<br /> |-<br /> |6<br /> |I8728<br /> |Shahr-i Sokhta, Iran<br /> |2600-2500 BCE<br /> |M<br /> |R7<br /> |J2a<br /> |-<br /> |7<br /> |I10409<br /> |Gonur, Turkmenistan<br /> |2280-2044 calBCE<br /> |M<br /> |R5a2b<br /> |H1a1d2<br /> |-<br /> |8<br /> |I11041<br /> |Gonur, Turkmenistan<br /> |2140-1972 calBCE<br /> |M<br /> |R2<br /> |R<br /> |-<br /> |9<br /> |I11471<br /> |Shahr-i Sokhta, Iran<br /> |3328-3022 calBCE<br /> |F<br /> |U1a1c1<br /> |n/a (female)<br /> |-<br /> |10<br /> |I11456<br /> |Shahr-i Sokhta, Iran<br /> |2600-2500 BCE<br /> |F<br /> |W3a1b<br /> |n/a (female)<br /> |-<br /> |11<br /> |I11488 ''&lt;sub&gt;(Low-coverage)&lt;/sub&gt;''<br /> |Shahr-i Sokhta, Iran<br /> |3200-2100 BCE<br /> |F<br /> |R<br /> |n/a (female)<br /> |-<br /> |12<br /> |I11466<br /> |Shahr-i Sokhta, Iran<br /> |2500-2000 BCE<br /> |F<br /> |M30b<br /> |n/a (female)<br /> |-<br /> |13<br /> |I2123<br /> |Gonur, Turkmenistan<br /> |2452-2140 calBCE<br /> |F<br /> |..<br /> |n/a (female)<br /> |-<br /> |14<br /> |I6113<br /> |Rakhigarhi, India<br /> |2500 BCE<br /> |F<br /> |U2b2<br /> |n/a (female)<br /> |}<br /> == Rakhigarhi DNA ==<br /> [[File:Rakhigarhi skeleton I6113.png|thumb|Skeleton of a Rakhigarhi woman I6113 dated to ~2500 BC.]]<br /> Rakhigarhi is a village in Hisar district in the state of Harayana in India. Rakhigarhi is now the largest site of the Indus-Saraswati Civilisation popularly known as Indus Valley Civilisation. For the first time the scientists were able to successfully extract DNA sample from bronze archaeological sites of Indus Valley Civilisation. The scientists attempted to generate genome-wide data from skeletal remains of 61 individuals from Rakhigarhi. But only a single sample yielded enough authentic ancient DNA for analysis, that sample is named I6113. The reason for this scarcity of ancient DNA samples available from Indian subcontinent is the tropical weather of India which has caused contamination to the ancient skeletons thus rendering the DNA samples useless for analysis. Despite of hundreds of excavated skeletons from the bronze age sites of IVC, so far the scientists have only been able to extract one low coverage samples from Rakhigarhi.<br /> The skeletal samples from Rakhigarhi were excavated by the archaeological team led by Dr. Vasant Shinde at the Deccan College Post-Graduate and Research Institute in Pune India and sampled by the ancient DNA group led by Dr. Niraj Rai at the Birbal Sahni Institute of Palaeosciences in Lucknow India.<br /> <br /> === Rakhigarhi DNA analysis ===<br /> The analysis of the DNA has changed the understanding of the modern Indian population and also helped understanding the genetic make up of the people that created the metropolitan cities of the Indus-Saraswati Civilisation, the advancement of which are incomparable to rest of the contemporary world. The scientists identified Rakhigarhi skeleton with that of a female and determined that its mitochondrial haplogroup is U2b2, which is absent in whole mitochondrial genomes sequences available from about 400 ancient Central Asians; today, this specific haplogroup is nearly exclusive to South Asia. The major observation made by the scientists using the DNA analysis are<br /> <br /> * The IVC ancestry is the major source of ancestry for the modern South Asian population.<br /> * The IVC and ancient Iranians come from a common ancestry who lived more than 12,000 years ago.<br /> * Farming is indigenous to the Indian subcontinent and developed by local foragers.<br /> * The Iranian-related or the Indian neolithic ancestry is the major source of ancestry (~80%) for the IVC people.<br /> * The IVC population has no ancestry from Central Asian populations.<br /> * Anatolian or Iranians farmer do not contribute to the IVC genome.<br /> <br /> == Indus Periphery Cline ==<br /> Indus periphery cline is the name given to the 11 ancient DNA samples published in the Narasimhan et al 2019. Paper documented 11 outliers - three with radiocarbon dates between 2500 and 2000 BCE from the BMAC site of Gonur in Turkmenistan and eight with radiocarbon dates between 3300 and 2000 BCE from the Iranian site of Shahr-i-Sokhta. These outliers harboured elevated levels of AHG-related ancestry with a major component of ancestry being the Indian neolithic ancestry that shares a common ancestor with the ancient Iranians who lived more than 12,000 years ago. The table contains two low coverage samples - I11460 and I11488 along with 11 ''Indus Periphery cline'' samples and ''Rakhigarhi'' sample. <br /> <br /> ''Indus Periphery'' samples are labeled as as ''Shahr-i-Sokhta_BA2'' and ''Gonur2_BA''. ''Shahr-i-Sokhta_BA2'' samples form ~50% of the samples found at archaeological site of Shahr-i-Sokhta'','' Iran, which shows a significant migration from IVC to Iran. <br /> <br /> == Out of India migration ==<br /> Narasimhan paper shows '''''Out of India''''' migration from South Asia to Iran and Central Asia. <br /> <br /> Narasimhan et al 2019 supplementary report, Table S30 provides a proximal model for the ''Shahr-i-Sokhta_BA1,'' dated 3100 to 2600 BCE, derives 20% of their ancestry from the Harappans ''(Shahr-i-Sokhta_BA2).'' Several proximal models are provided in the paper, each with Indus periphery cline as source peaking up to ~23% in one of the proximal model. About 1,000 graves have been excavated at ''Shahr-i-Sokhta'' and the archaeologists finds &lt;blockquote&gt;''&quot;Extraordinary conservation of organic materials, as well as early evidence of long-distance contacts with southern Central Asia, Afghanistan (especially trade in lapis lazuli) and Baluchistan in South Asia are its main archaeological features.&quot;''&lt;/blockquote&gt;A total of 17 DNA samples were extracted from ''Shahr-i-Sokhta'' out of which 8 samples which is ~50% of the samples belonged to the Harappan migrants to ''Shahr-i-Sokhta''. The Shahr-i-sokhta individuals, labelled ''Shahr-i-sokhta_BA1'' are dated from 3100 BCE to 2600 BCE where the Harappan migrants ''(Shahr-i-sokhta_BA2)'' are dated from 3300 BCE to 2000 BCE. <br /> <br /> Narasimhan et 2019 supplementary reports says&lt;blockquote&gt;&quot;The archaeological and genetic evidence thus suggest that a flow of migrants from the northwestern borderlands of South Asia was active at the beginning of the local settlement, and that the same flow, different from the earlier one from northwest, intensified in the following centuries. We unfortunately do not have ancient DNA from Period III and the later centuries of the Shahr-i-Sokhta sequence when cultural influence from the Indus Valley Civilization appears to become stronger&quot;&lt;/blockquote&gt;The migration from Indus Valley Civilisation to Iran starts in the later half of 4th millennium BCE then continues through third millennium BCE which appears to become stronger genetically and archaeologically towards the end of third millennium BCE but unfortunately there are no ancient DNA available yet to from the following period III from ''Shahr-i-sokhta, Iran.''</div>

BharataPuru http://dharmawiki.org/index.php?title=File:Distal_plot_for_Shahr-i-Sokhta_BA1.png&diff=133585 2022-05-02T11:17:19Z

<p>BharataPuru: Source: Narasimhan et al 2019 Supplementary report</p> <hr /> <div>== Summary ==<br /> Source: Narasimhan et al 2019 Supplementary report</div>

BharataPuru http://dharmawiki.org/index.php?title=Indus_Valley_DNA&diff=133584 2022-05-02T10:09:02Z

<p>BharataPuru: /* Indus Periphery Cline */</p> <hr /> <div>[[File:Shinde et al 2019.png|thumb|390x390px|Indus Valley Civilisation (IVC) Cline is a mixture of the Iranian-related ancestry as major component with a small component of Andemanese Hunter-Gatherer (AHG) related ancestry. The Iranian-related ancestry in IVC cline and the ancient Iranians had a common ancestor who lived more than 12,000 years ago.]]<br /> In 2019, for the first time several ancient DNA samples related to the people of Indus-Saraswati Civilisation were published. Shinde et al 2019 published the only bronze age DNA samples found in India. This DNA sample was extracted from a skeleton found at archaeological site of Rakhigarhi located in the ancient Saraswati river plains. It was dated to ~2500 BCE and the skeleton belonged to a woman. Narasimhan et al 2019 published hundreds of DNA samples along with thirteen samples belonging to Indus valley migrants found in the Turan region which includes modern day Iran and Turkmenistan regions. These DNA samples were dated variously between 3300 to 2000 BCE and changed our understanding of the modern Indian population. <br /> <br /> The Iranian related component that was once hypothesised to have arrived in India along with farming has turned out to be much more ancient. As the Shinde et al 2019 paper says, &quot;Iranian-related ancestry in South Asia split from Iranian plateau lineages &gt;12,000 years ago&quot;, implying that the people of Indus Valley and ancient Iranians had a common ancestor who lived more than 12,000 years ago. The IVC's Iranian related ancestry itself contains ~10% WSHG ancestry, which is the Western Siberian Hunter-Gatherer ancestry. This discovery lead the scientists to make the conclusion that farming is indigenous to the Indian subcontinent and was not brought in from the western first farmers of Anatolia.<br /> <br /> Narasimhan paper disproved the older assertion made by Damgard et al 2018 which claimed that BMAC is a source of ancestry to modern South Asians. Narasimhan paper published large number of bronze age DNA samples from BMAC region and found that BMAC is not the source of ancestry to the modern South Asians but in contrast BMAC main population has South Asian ancestry deeply rooted in time. Rather than showing migration into the Indian subcontinent, the bronze age DNA shows migration of people from South Asia to BMAC regions.<br /> <br /> == IVC cline ==<br /> Indus Valley DNA is characterised by the combination of one DNA sample found at Rakhigarhi published by Shinde et al 2019 and 11 Indus periphery cline DNA samples extracted from the skeletons found at archaeological sites in Iran and Turkmenistan region where they are identified as outliers published in Narasimhan et al 2019. The excessive AHG related ancestry in those 11 samples had indicated that those samples could belong to the migrants from the IVC. These 12 samples together are referred as IVC cline samples. <br /> <br /> The authors of Shinde et al says, &lt;blockquote&gt;&quot;''While our study is sufficient to demonstrate that this ancestry profile was a common feature of the IVC, a single sample—or even the gradient of 12 likely IVC samples we have identified cannot fully characterise a cosmopolitan ancient civilisation. An important direction for future work will be to carry out ancient DNA analysis of additional individuals across the IVC range to obtain a quantitative understanding of how the ancestry of IVC people was distributed and to characterise other features of its population structure.''&quot;&lt;/blockquote&gt;<br /> <br /> == Rakhigarhi DNA ==<br /> [[File:Rakhigarhi skeleton I6113.png|thumb|Skeleton of a Rakhigarhi woman I6113 dated to ~2500 BC.]]<br /> Rakhigarhi is a village in Hisar district in the state of Harayana in India. Rakhigarhi is now the largest site of the Indus-Saraswati Civilisation popularly known as Indus Valley Civilisation. For the first time the scientists were able to successfully extract DNA sample from bronze archaeological sites of Indus Valley Civilisation. The scientists attempted to generate genome-wide data from skeletal remains of 61 individuals from Rakhigarhi. But only a single sample yielded enough authentic ancient DNA for analysis, that sample is named I6113. The reason for this scarcity of ancient DNA samples available from Indian subcontinent is the tropical weather of India which has caused contamination to the ancient skeletons thus rendering the DNA samples useless for analysis. Despite of hundreds of excavated skeletons from the bronze age sites of IVC, so far the scientists have only been able to extract one low coverage samples from Rakhigarhi.<br /> The skeletal samples from Rakhigarhi were excavated by the archaeological team led by Dr. Vasant Shinde at the Deccan College Post-Graduate and Research Institute in Pune India and sampled by the ancient DNA group led by Dr. Niraj Rai at the Birbal Sahni Institute of Palaeosciences in Lucknow India.<br /> <br /> === DNA analysis ===<br /> The analysis of the DNA has changed the understanding of the modern Indian population and also helped understanding the genetic make up of the people that created the metropolitan cities of the Indus-Saraswati Civilisation, the advancement of which are incomparable to rest of the contemporary world. The scientists identified Rakhigarhi skeleton with that of a female and determined that its mitochondrial haplogroup is U2b2, which is absent in whole mitochondrial genomes sequences available from about 400 ancient Central Asians; today, this specific haplogroup is nearly exclusive to South Asia. The major observation made by the scientists using the DNA analysis are<br /> <br /> * The IVC ancestry is the major source of ancestry for the modern South Asian population.<br /> * The IVC and ancient Iranians come from a common ancestry who lived more than 12,000 years ago.<br /> * Farming is indigenous to the Indian subcontinent and developed by local foragers.<br /> * The Iranian-related or the Indian neolithic ancestry is the major source of ancestry (~80%) for the IVC people.<br /> * The IVC population has no ancestry from Central Asian populations.<br /> * Anatolian or Iranians farmer do not contribute to the IVC genome.<br /> <br /> == Indus Periphery Cline ==<br /> Indus periphery cline is the name given to the 11 ancient DNA samples published in the Narasimhan et al 2019. Paper documented 11 outliers - three with radiocarbon dates between 2500 and 2000 BCE from the BMAC site of Gonur in Turkmenistan and eight with radiocarbon dates between 3300 and 2000 BCE from the Iranian site of Shahr-i-Sokhta. These outliers harboured elevated levels of AHG-related ancestry with a major component of ancestry being the Indian neolithic ancestry that shares a common ancestor with the ancient Iranians who lived more than 12,000 years ago. The table contains two low coverage samples along with 11 ''Indus Periphery cline'' samples - I11460 and I11488 and ''Rakhigarhi'' sample. <br /> {| class=&quot;wikitable&quot;<br /> !<br /> !Sample ID<br /> !Location<br /> !Radiocarbon age<br /> !Sex<br /> !mtDNA <br /> haplogroup<br /> !Y chromosome <br /> haplogroup<br /> |-<br /> |1<br /> |I8726<br /> |Shahr-i Sokhta, Iran<br /> |3100-3000 BCE<br /> |M<br /> |U2c1<br /> |J2a1h<br /> |-<br /> |2<br /> |I11480<br /> |Shahr-i Sokhta, Iran<br /> |2900-2800 BCE<br /> |M<br /> |W6<br /> |J<br /> |-<br /> |3<br /> |I11459<br /> |Shahr-i Sokhta, Iran<br /> |2875-2631 calBCE<br /> |M<br /> |W3a1b<br /> |H1a1d2<br /> |-<br /> |4<br /> |I11460<br /> |Shahr-i Sokhta, Iran<br /> |3200-2100 BCE<br /> |M<br /> |H2a<br /> |BT<br /> |-<br /> |5<br /> |I11458<br /> |Shahr-i Sokhta, Iran<br /> |3200-2100 BCE<br /> |M<br /> |..<br /> |F<br /> |-<br /> |6<br /> |I8728<br /> |Shahr-i Sokhta, Iran<br /> |2600-2500 BCE<br /> |M<br /> |R7<br /> |J2a<br /> |-<br /> |7<br /> |I10409<br /> |Gonur, Turkmenistan<br /> |2280-2044 calBCE<br /> |M<br /> |R5a2b<br /> |H1a1d2<br /> |-<br /> |8<br /> |I11041<br /> |Gonur, Turkmenistan<br /> |2140-1972 calBCE<br /> |M<br /> |R2<br /> |R<br /> |-<br /> |9<br /> |I11471<br /> |Shahr-i Sokhta, Iran<br /> |3328-3022 calBCE<br /> |F<br /> |U1a1c1<br /> |n/a (female)<br /> |-<br /> |10<br /> |I11456<br /> |Shahr-i Sokhta, Iran<br /> |2600-2500 BCE<br /> |F<br /> |W3a1b<br /> |n/a (female)<br /> |-<br /> |11<br /> |I11488<br /> |Shahr-i Sokhta, Iran<br /> |3200-2100 BCE<br /> |F<br /> |R<br /> |n/a (female)<br /> |-<br /> |12<br /> |I11466<br /> |Shahr-i Sokhta, Iran<br /> |2500-2000 BCE<br /> |F<br /> |M30b<br /> |n/a (female)<br /> |-<br /> |13<br /> |I2123<br /> |Gonur, Turkmenistan<br /> |2452-2140 calBCE<br /> |F<br /> |..<br /> |n/a (female)<br /> |-<br /> |14<br /> |I6113<br /> |Rakhigarhi, India<br /> |2500 BCE<br /> |F<br /> |U2b2<br /> |n/a (female)<br /> |}</div>

BharataPuru http://dharmawiki.org/index.php?title=Indus_Valley_DNA&diff=133581 2022-05-02T07:59:58Z

<p>BharataPuru: /* Indus Periphery Cline */</p> <hr /> <div>[[File:Shinde et al 2019.png|thumb|390x390px|Indus Valley Civilisation (IVC) Cline is a mixture of the Iranian-related ancestry as major component with a small component of Andemanese Hunter-Gatherer (AHG) related ancestry. The Iranian-related ancestry in IVC cline and the ancient Iranians had a common ancestor who lived more than 12,000 years ago.]]<br /> In 2019, for the first time several ancient DNA samples related to the people of Indus-Saraswati Civilisation were published. Shinde et al 2019 published the only bronze age DNA samples found in India. This DNA sample was extracted from a skeleton found at archaeological site of Rakhigarhi located in the ancient Saraswati river plains. It was dated to ~2500 BCE and the skeleton belonged to a woman. Narasimhan et al 2019 published hundreds of DNA samples along with thirteen samples belonging to Indus valley migrants found in the Turan region which includes modern day Iran and Turkmenistan regions. These DNA samples were dated variously between 3300 to 2000 BCE and changed our understanding of the modern Indian population. <br /> <br /> The Iranian related component that was once hypothesised to have arrived in India along with farming has turned out to be much more ancient. As the Shinde et al 2019 paper says, &quot;Iranian-related ancestry in South Asia split from Iranian plateau lineages &gt;12,000 years ago&quot;, implying that the people of Indus Valley and ancient Iranians had a common ancestor who lived more than 12,000 years ago. The IVC's Iranian related ancestry itself contains ~10% WSHG ancestry, which is the Western Siberian Hunter-Gatherer ancestry. This discovery lead the scientists to make the conclusion that farming is indigenous to the Indian subcontinent and was not brought in from the western first farmers of Anatolia.<br /> <br /> Narasimhan paper disproved the older assertion made by Damgard et al 2018 which claimed that BMAC is a source of ancestry to modern South Asians. Narasimhan paper published large number of bronze age DNA samples from BMAC region and found that BMAC is not the source of ancestry to the modern South Asians but in contrast BMAC main population has South Asian ancestry deeply rooted in time. Rather than showing migration into the Indian subcontinent, the bronze age DNA shows migration of people from South Asia to BMAC regions.<br /> <br /> == IVC cline ==<br /> Indus Valley DNA is characterised by the combination of one DNA sample found at Rakhigarhi published by Shinde et al 2019 and 11 Indus periphery cline DNA samples extracted from the skeletons found at archaeological sites in Iran and Turkmenistan region where they are identified as outliers published in Narasimhan et al 2019. The excessive AHG related ancestry in those 11 samples had indicated that those samples could belong to the migrants from the IVC. These 12 samples together are referred as IVC cline samples. <br /> <br /> The authors of Shinde et al says, &lt;blockquote&gt;&quot;''While our study is sufficient to demonstrate that this ancestry profile was a common feature of the IVC, a single sample—or even the gradient of 12 likely IVC samples we have identified cannot fully characterise a cosmopolitan ancient civilisation. An important direction for future work will be to carry out ancient DNA analysis of additional individuals across the IVC range to obtain a quantitative understanding of how the ancestry of IVC people was distributed and to characterise other features of its population structure.''&quot;&lt;/blockquote&gt;<br /> <br /> == Rakhigarhi DNA ==<br /> [[File:Rakhigarhi skeleton I6113.png|thumb|Skeleton of a Rakhigarhi woman I6113 dated to ~2500 BC.]]<br /> Rakhigarhi is a village in Hisar district in the state of Harayana in India. Rakhigarhi is now the largest site of the Indus-Saraswati Civilisation popularly known as Indus Valley Civilisation. For the first time the scientists were able to successfully extract DNA sample from bronze archaeological sites of Indus Valley Civilisation. The scientists attempted to generate genome-wide data from skeletal remains of 61 individuals from Rakhigarhi. But only a single sample yielded enough authentic ancient DNA for analysis, that sample is named I6113. The reason for this scarcity of ancient DNA samples available from Indian subcontinent is the tropical weather of India which has caused contamination to the ancient skeletons thus rendering the DNA samples useless for analysis. Despite of hundreds of excavated skeletons from the bronze age sites of IVC, so far the scientists have only been able to extract one low coverage samples from Rakhigarhi.<br /> The skeletal samples from Rakhigarhi were excavated by the archaeological team led by Dr. Vasant Shinde at the Deccan College Post-Graduate and Research Institute in Pune India and sampled by the ancient DNA group led by Dr. Niraj Rai at the Birbal Sahni Institute of Palaeosciences in Lucknow India.<br /> <br /> === DNA analysis ===<br /> The analysis of the DNA has changed the understanding of the modern Indian population and also helped understanding the genetic make up of the people that created the metropolitan cities of the Indus-Saraswati Civilisation, the advancement of which are incomparable to rest of the contemporary world. The scientists identified Rakhigarhi skeleton with that of a female and determined that its mitochondrial haplogroup is U2b2, which is absent in whole mitochondrial genomes sequences available from about 400 ancient Central Asians; today, this specific haplogroup is nearly exclusive to South Asia. The major observation made by the scientists using the DNA analysis are<br /> <br /> * The IVC ancestry is the major source of ancestry for the modern South Asian population.<br /> * The IVC and ancient Iranians come from a common ancestry who lived more than 12,000 years ago.<br /> * Farming is indigenous to the Indian subcontinent and developed by local foragers.<br /> * The Iranian-related or the Indian neolithic ancestry is the major source of ancestry (~80%) for the IVC people.<br /> * The IVC population has no ancestry from Central Asian populations.<br /> * Anatolian or Iranians farmer do not contribute to the IVC genome.<br /> <br /> == Indus Periphery Cline ==<br /> Indus periphery cline is the name given to the 11 ancient DNA samples published in the Narasimhan et al 2019. Paper documented 11 outliers - three with radiocarbon dates between 2500 and 2000 BCE from the BMAC site of Gonur in Turkmenistan and eight with radiocarbon dates between 3300 and 2000 BCE from the Iranian site of Shahr-i-Sokhta. These outliers harboured elevated levels of AHG-related ancestry with a major component of ancestry being the Indian neolithic ancestry that shares a common ancestor with the ancient Iranians who lived more than 12,000 years ago. The table contains two low coverage samples along with 11 ''Indus Periphery cline'' samples - I11460 and I11488. <br /> {| class=&quot;wikitable&quot;<br /> |+<br /> !Sample ID<br /> !Radiocarbon age<br /> !Sex<br /> !mtDNA <br /> haplogroup<br /> !Y chromosome <br /> haplogroup<br /> |-<br /> |I11471<br /> |3328-3022 calBCE<br /> |F<br /> |U1a1c1<br /> |n/a (female)<br /> |-<br /> |I8726<br /> |3100-3000 BCE<br /> |M<br /> |U2c1<br /> |J2a1h<br /> |-<br /> |I11480<br /> |2900-2800 BCE<br /> |M<br /> |W6<br /> |J<br /> |-<br /> |I11459<br /> |2875-2631 calBCE<br /> |M<br /> |W3a1b<br /> |H1a1d2<br /> |-<br /> |I11460<br /> |3200-2100 BCE<br /> |M<br /> |H2a<br /> |BT<br /> |-<br /> |I11488<br /> |3200-2100 BCE<br /> |F<br /> |R<br /> |n/a (female)<br /> |-<br /> |I11458<br /> |3200-2100 BCE<br /> |M<br /> |..<br /> |F<br /> |-<br /> |I11456<br /> |2600-2500 BCE<br /> |F<br /> |W3a1b<br /> |n/a (female)<br /> |-<br /> |I8728<br /> |2600-2500 BCE<br /> |M<br /> |R7<br /> |J2a<br /> |-<br /> |I2123<br /> |2452-2140 calBCE<br /> |F<br /> |..<br /> |n/a (female)<br /> |-<br /> |I10409<br /> |2280-2044 calBCE<br /> |M<br /> |R5a2b<br /> |H1a1d2<br /> |-<br /> |I11041<br /> |2140-1972 calBCE <br /> |M<br /> |R2<br /> |R<br /> |-<br /> |I11466<br /> |2500-2000 BCE<br /> |F<br /> |M30b<br /> |n/a (female)<br /> |}</div>

BharataPuru http://dharmawiki.org/index.php?title=Indus_Valley_DNA&diff=133580 2022-04-30T05:26:45Z

<p>BharataPuru: </p> <hr /> <div>[[File:Shinde et al 2019.png|thumb|390x390px|Indus Valley Civilisation (IVC) Cline is a mixture of Indian neolithic (India_N) ancestry as major component with a small component of Andemanese Hunter-Gatherer (AHG) related ancestry.]]<br /> In 2019, for the first time several ancient DNA samples related to the people of Indus-Saraswati Civilisation were published. Shinde et al 2019 published the only bronze age DNA samples found in India. This DNA sample was extracted from a skeleton found at archaeological site of Rakhigarhi located in the ancient Saraswati river plains. It was dated to ~2500 BC and the skeleton belonged to a woman. Narasimhan et al 2019 published hundreds of DNA samples along with thirteen samples belonging to Indus valley migrants found in the Turan region which includes modern day Iran and Turkmenistan regions. These DNA samples were dated variously between 3200 BC to 2000 BC and changed our understanding of the modern Indian population. <br /> <br /> The Iranian related component that was once hypothesised to have arrived in India along with farming has turned out to be much more ancient. As the Shinde et al 2019 paper says, &quot;Iranian-related ancestry in South Asia split from Iranian plateau lineages &gt;12,000 years ago&quot;, implying that the people of Indus Valley and ancient Iranians had a common ancestor who lived more than 12,000 years ago. This ancestry is sometimes referred to as Indian neolithic ancestry (India_N). This discovery lead the scientists to make the conclusion that farming is indigenous to the Indian subcontinent and was not brought in from the western first farmers of Anatolia.<br /> <br /> Narasimhan paper disproved the older assertion made by Damgard et al 2018 which claimed that BMAC is a source of ancestry to modern South Asians. Narasimhan paper published large number of bronze age DNA samples from BMAC region and found that BMAC is not the source of ancestry to the modern South Asians but in contrast BMAC main population has South Asian ancestry deeply rooted in time. Rather than showing migration into the Indian subcontinent, the bronze age DNA shows migration of people from South Asia to BMAC regions.<br /> <br /> == IVC cline ==<br /> Indus Valley DNA is characterised by the combination of one DNA sample found at Rakhigarhi published by Shinde et al 2019 and 11 Indus periphery cline DNA samples extracted from the skeletons found at archaeological sites in Iran and Turkmenistan region where they are identified as outliers published in Narasimhan et al 2019. The excessive AHG related ancestry in those 11 samples had indicated that those samples could belong to the migrants from the IVC. These 12 samples together are referred as IVC cline samples. <br /> <br /> The authors of Shinde et al says, &lt;blockquote&gt;&quot;''While our study is sufficient to demonstrate that this ancestry profile was a common feature of the IVC, a single sample—or even the gradient of 12 likely IVC samples we have identified cannot fully characterise a cosmopolitan ancient civilisation. An important direction for future work will be to carry out ancient DNA analysis of additional individuals across the IVC range to obtain a quantitative understanding of how the ancestry of IVC people was distributed and to characterise other features of its population structure.''&quot;&lt;/blockquote&gt;<br /> <br /> == Rakhigarhi DNA ==<br /> [[File:Rakhigarhi skeleton I6113.png|thumb|Skeleton of a Rakhigarhi woman I6113 dated to ~2500 BC.]]<br /> Rakhigarhi is a village in Hisar district in the state of Harayana in India. Rakhigarhi is now the largest site of the Indus-Saraswati Civilisation popularly known as Indus Valley Civilisation. For the first time the scientists were able to successfully extract DNA sample from bronze archaeological sites of Indus Valley Civilisation. The scientists attempted to generate genome-wide data from skeletal remains of 61 individuals from Rakhigarhi. But only a single sample yielded enough authentic ancient DNA for analysis, that sample is named I6113. The reason for this scarcity of ancient DNA samples available from Indian subcontinent is the tropical weather of India which has caused contamination to the ancient skeletons thus rendering the DNA samples useless for analysis. Despite of hundreds of excavated skeletons from the bronze age sites of IVC, so far the scientists have only been able to extract one low coverage samples from Rakhigarhi.<br /> The skeletal samples from Rakhigarhi were excavated by the archaeological team led by Dr. Vasant Shinde at the Deccan College Post-Graduate and Research Institute in Pune India and sampled by the ancient DNA group led by Dr. Niraj Rai at the Birbal Sahni Institute of Palaeosciences in Lucknow India.<br /> <br /> === DNA analysis ===<br /> The analysis of the DNA has changed the understanding of the modern Indian population and also helped understanding the genetic make up of the people that created the metropolitan cities of the Indus-Saraswati Civilisation, the advancement of which are incomparable to rest of the contemporary world. The scientists identified Rakhigarhi skeleton with that of a female and determined that its mitochondrial haplogroup is U2b2, which is absent in whole mitochondrial genomes sequences available from about 400 ancient Central Asians; today, this specific haplogroup is nearly exclusive to South Asia. The major observation made by the scientists using the DNA analysis are<br /> <br /> * The IVC ancestry is the major source of ancestry for the modern South Asian population.<br /> * The IVC and ancient Iranians come from a common ancestry who lived more than 12,000 years ago.<br /> * Farming is indigenous to the Indian subcontinent and developed by local foragers.<br /> * The Iranian-related or the Indian neolithic ancestry is the major source of ancestry (~80%) for the IVC people.<br /> * The IVC population has no ancestry from Central Asian populations.<br /> * Anatolian or Iranians farmer do not contribute to the IVC genome.<br /> <br /> == Indus Periphery Cline ==</div>

BharataPuru http://dharmawiki.org/index.php?title=Indus_Valley_DNA&diff=133579 2022-04-30T05:25:34Z

<p>BharataPuru: Created page with &quot;File:Shinde et al 2019.png|thumb|390x390px|Indus Valley Civilisation (IVC) Cline is a mixture of Indian neolithic (India_N) ancestry as major component with a small componen...&quot;</p> <hr /> <div>[[File:Shinde et al 2019.png|thumb|390x390px|Indus Valley Civilisation (IVC) Cline is a mixture of Indian neolithic (India_N) ancestry as major component with a small component of Andemanese Hunter-Gatherer (AHG) related ancestry.]]<br /> In 2019, for the first time several ancient DNA samples related to the people of Indus-Saraswati Civilisation were published. Shinde et al 2019 published the only bronze age DNA samples found in India. This DNA sample was extracted from a skeleton found at archaeological site of Rakhigarhi located in the ancient Saraswati river plains. It was dated to ~2500 BC and the skeleton belonged to a woman. Narasimhan et al 2019 published hundreds of DNA samples along with thirteen samples belonging to Indus valley migrants found in the Turan region which includes modern day Iran and Turkmenistan regions. These DNA samples were dated variously between 3200 BC to 2000 BC and changed our understanding of the modern Indian population. <br /> <br /> The Iranian related component that was once hypothesised to have arrived in India along with farming has turned out to be much more ancient. As the Shinde et al 2019 paper says, &quot;Iranian-related ancestry in South Asia split from Iranian plateau lineages &gt;12,000 years ago&quot;, implying that the people of Indus Valley and ancient Iranians had a common ancestor who lived more than 12,000 years ago. This ancestry is sometimes referred to as Indian neolithic ancestry (India_N). This discovery lead the scientists to make the conclusion that farming is indigenous to the Indian subcontinent and was not brought in from the western first farmers of Anatolia.<br /> <br /> Narasimhan paper disproved the older assertion made by Damgard et al 2018 which claimed that BMAC is a source of ancestry to modern South Asians. Narasimhan paper published large number of bronze age DNA samples from BMAC region and found that BMAC is not the source of ancestry to the modern South Asians but in contrast BMAC main population has South Asian ancestry deeply rooted in time. Rather than showing migration into the Indian subcontinent, the bronze age DNA shows migration of people from South Asia to BMAC regions.<br /> <br /> == IVC cline ==<br /> Indus Valley DNA is characterised by the combination of one DNA sample found at Rakhigarhi published by Shinde et al 2019 and 11 Indus periphery cline DNA samples extracted from the skeletons found at archaeological sites in Iran and Turkmenistan region where they are identified as outliers published in Narasimhan et al 2019. The excessive AHG related ancestry in those 11 samples had indicated that those samples could belong to the migrants from the IVC. These 12 samples together are referred as IVC cline samples. <br /> <br /> The authors of Shinde et al says, &lt;blockquote&gt;&quot;''While our study is sufficient to demonstrate that this ancestry profile was a common feature of the IVC, a single sample—or even the gradient of 12 likely IVC samples we have identified cannot fully characterise a cosmopolitan ancient civilisation. An important direction for future work will be to carry out ancient DNA analysis of additional individuals across the IVC range to obtain a quantitative understanding of how the ancestry of IVC people was distributed and to characterise other features of its population structure.''&quot;&lt;/blockquote&gt;<br /> <br /> == Rakhigarhi DNA ==<br /> [[File:Rakhigarhi skeleton I6113.png|thumb|Skeleton of a Rakhigarhi woman I6113 dated to ~2500 BC.]]<br /> Rakhigarhi is a village in Hisar district in the state of Harayana in India. Rakhigarhi is now the largest site of the Indus-Saraswati Civilisation popularly known as Indus Valley Civilisation. For the first time the scientists were able to successfully extract DNA sample from bronze archaeological sites of Indus Valley Civilisation. The scientists attempted to generate genome-wide data from skeletal remains of 61 individuals from Rakhigarhi. But only a single sample yielded enough authentic ancient DNA for analysis, that sample is named I6113. The reason for this scarcity of ancient DNA samples available from Indian subcontinent is the tropical weather of India which has caused contamination to the ancient skeletons thus rendering the DNA samples useless for analysis. Despite of hundreds of excavated skeletons from the bronze age sites of IVC, so far the scientists have only been able to extract one low coverage samples from Rakhigarhi.<br /> The skeletal samples from Rakhigarhi were excavated by the archaeological team led by Dr. Vasant Shinde at the Deccan College Post-Graduate and Research Institute in Pune India and sampled by the ancient DNA group led by Dr. Niraj Rai at the Birbal Sahni Institute of Palaeosciences in Lucknow India.<br /> <br /> === DNA analysis ===<br /> The analysis of the DNA has changed the understanding of the modern Indian population and also helped understanding the genetic make up of the people that created the metropolitan cities of the Indus-Saraswati Civilisation, the advancement of which are incomparable to rest of the contemporary world. The scientists identified Rakhigarhi skeleton with that of a female and determined that its mitochondrial haplogroup was U2b2, which is absent in whole mitochondrial genomes sequences available from about 400 ancient Central Asians; today, this specific haplogroup is nearly exclusive to South Asia. The major observation made by the scientists using the DNA analysis are<br /> <br /> * The IVC ancestry is the major source of ancestry for the modern South Asian population.<br /> * The IVC and ancient Iranians come from a common ancestry who lived more than 12,000 years ago.<br /> * Farming is indigenous to the Indian subcontinent and developed by local foragers.<br /> * The Iranian-related or the Indian neolithic ancestry is the major source of ancestry (~80%) for the IVC people.<br /> * The IVC population has no ancestry from Central Asian populations.<br /> * Anatolian or Iranians farmer do not contribute to the IVC genome.<br /> <br /> == Indus Periphery Cline ==</div>

BharataPuru http://dharmawiki.org/index.php?title=File:Rakhigarhi_skeleton_I6113.png&diff=133578 2022-04-30T04:37:54Z

<p>BharataPuru: Source: Shinde et al 2019 Cell journal</p> <hr /> <div>== Summary ==<br /> Source: Shinde et al 2019 Cell journal</div>

BharataPuru http://dharmawiki.org/index.php?title=File:Shinde_et_al_2019.png&diff=133577 2022-04-30T03:13:01Z

<p>BharataPuru: This picture is taken from Shinde et al 2019 published in cell journal. The Indus Valley people and ancient Iranians come from a common ancestor who lived more than 12,000 years ago. Where that ancestor lived is not known because of the lack of ancient DNA samples.</p> <hr /> <div>== Summary ==<br /> This picture is taken from Shinde et al 2019 published in cell journal.<br /> The Indus Valley people and ancient Iranians come from a common ancestor who lived more than 12,000 years ago. Where that ancestor lived is not known because of the lack of ancient DNA samples.</div>

BharataPuru http://dharmawiki.org/index.php?title=Surya_Siddhanta_(%E0%A4%B8%E0%A5%82%E0%A4%B0%E0%A5%8D%E0%A4%AF_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A4%BE)&diff=131686 2021-09-17T19:35:37Z

<p>BharataPuru: /* Indian origin of seconds, minutes and degrees */</p> <hr /> <div>'''''Sūrya Siddhānta''''' is an ancient Indian treatise in Astronomy. Like many classical Indian works, the ''Sūrya Siddhānta'' is a poem in [[Sanskrit]] language. It has fourteen chapter and 500 verses. It is composed in ''śloka'' metrical style of Sanskrit. It contain works on Indian sine tables, cosmology, eclipses, planetary motions, conjunctions, star positions, geography, instrumentation, concepts of time and mathematics. Unlike conventional books ''Sūrya Siddhānta'' contains advanced calculation and methods which are not easily comprehensible for a rank beginner. &lt;ref&gt;[https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol45_4_1_ANarayan.pdf]&lt;/ref&gt; In second chapter, the text contains the calculation of Earth's obliquity of ''1397 jya (R.sine) 23.975°'' modern units. The text describes the observation of two pole stars one each at the north pole and the south pole seen at the equatorial region, such phenomena was last seen in the 3rd millennium BCE. This indicates the antiquity of the concepts recorded in the text.&lt;seo title=&quot;Surya Siddhanta&quot; titlemode=&quot;append&quot; keywords=&quot;Surya Siddhanta, surya, Surya Siddhanta dharma, dhammawiki&quot; description=&quot;Sūrya Siddhānta is an ancient Indian treatise in Astronomy. Like many classical Indian works, the Sūrya Siddhānta is a poem in Sanskrit language. It has fourteen chapter and 500 verses. It is composed in śloka metrical style of Sanskrit. It contain works on Indian sine tables, cosmology, eclipses, planetary motions, conjunctions, star positions, geography, instrumentation, concepts of time and mathematics.&quot;&gt;&lt;/seo&gt;<br /> == History ==<br /> &lt;nowiki&gt;''Sūrya Siddhānta'' is well known, most referred and most esteemed. The original author of ''Sūrya Siddhānta'' is ''Mayasura'' as described in the story in the first chapter that ''Mayasura'' obtained his knowledge from ''Sūrya'' (the Sun). ''Siddhānta'' in Sanskrit means ''treatise'' and it usually has author'&lt;/nowiki&gt;s name prefixed to it. There were several other works on Astronomy in ancient India, many of which have since been lost.<br /> <br /> {| class=&quot;wikitable&quot;<br /> |-<br /> | ''Surya Siddhānta'' || ''Brahma Siddhānta'' || ''Soma Siddhānta''<br /> |-<br /> | ''Vyasa Siddhānta'' || ''Vashishtha Siddhānta'' || ''Atri Siddhānta''<br /> |-<br /> | ''Parashira Siddhānta'' || ''Kashyap Siddhānta'' || ''Nārad Siddhānta''<br /> |-<br /> | ''Garga Siddhānta'' || ''Marici Siddhānta'' || ''Manu Siddhānta''<br /> |-<br /> | ''Angiras Siddhānta'' || ''Lomasha Siddhānta'' || ''Pulisha Siddhānta''<br /> |-<br /> | ''Cyavana Siddhānta'' || ''Yavana Siddhānta'' || ''Bhrigu Siddhānta''<br /> |}<br /> <br /> == Content ==<br /> ''Sūrya Siddhānta'' contains 14 chapters and 500 verses. The chapters contain observations, methods, instruments and calculations of various astronomical phenomenas. There is a scarcity of scientific analysis done on the text of ''Surya Siddhanta''. Majority western work is based on Indology dates which in itself is controversial and based on their biased opinion of granting the origin of any science or mathematics to the ancient Greek or babylonians despite of immense textual evidence pointing otherwise. Their analysis of Surya Siddhanta primarily avoids the study of actual data and observations recorded within the ''Surya Siddhanta''.<br /> <br /> === Indian origin of seconds, minutes and degrees ===<br /> ''Surya Siddhanta'' in chapter 2 describes the units of seconds, minutes and degrees. These units of measurement are primary basis of the calculations of earth's obliquity and sine tables of ''Surya Siddhanta''. Essentially ''Surya Siddhanta'' describes that Rashi has 30 degrees (Ansh) implying it has 1800 minutes (Kala). The data is as as described in the table<br /> <br /> {| class=&quot;wikitable&quot;<br /> |+ ''Surya Siddhanta'' units: seconds, minutes and degrees&lt;ref&gt;Pundit Bapu Deva Shastri, &quot;English Translation of Surya Siddhanta&quot;,p11, 1861&lt;/ref&gt; <br /> |-<br /> ! Modern SI units !! Surya Siddhanta units !! Value<br /> |-<br /> | Second || Vikala || -<br /> |-<br /> | Minute || Kala || 60 Vikala/seconds<br /> |-<br /> | Degree || Ansh || 60 Kala/minutes or 3600 Vikala/seconds<br /> |-<br /> | Zodiac Sign || Rashi || 30 Ansh/degrees or 1800 Kala/minutes<br /> |-<br /> | Revolution || Bhagan || 12 Rashi/zodiac signs<br /> |}<br /> <br /> These units are used in several calculations done through out the text of ''Surya Siddhanta''. In the sine tables of ''Surya Siddhanta'' the first sine or Jyā is described as the value equal to 1/8th of the number of minutes (Kalas) in a zodiac sign (Rashi).<br /> <br /> === Indian standard circle ===<br /> The ''Surya Siddhanta'' is using the [[Indian standard circle]] in various calculations through out the text. This standard circle is based on radius of 3,438 minutes. The significance lies in the precision of 1/3438 that the ancient Indian astronomers were able to work with. It is evident from the calculation of obliquity of the earth's axis in chapter 2 where 1397 units is the measured R-sine value. <br /> Another interesting outcome of this radius of 3,438 minutes is that the circumference of the standard Indian circle is calculated as 21,600 minutes using the formula of Pi multiply by diameter (twice the radius).<br /> <br /> === Nakshatra (Asterism) System ===<br /> <br /> The ''Surya Siddhanta'' uses the 27 [[Nakshatra system]] throughout the text. The Nakshatra is a smaller constellation typically consisting of 1 to 5 stars. The brightest star is called as Yogtara. Each Nakshatra spans 13° 20' on the ecliptic. Each Nakshatra has its own primary star which is usually the junction star but not always.<br /> <br /> === Longitudinal updates ===<br /> Chapter 8 of ''Surya Siddhanta'' primarily focuses on the stellar data. It provides the longitudinal data for the Asterisms. In comparison to the present day longitudinal values of these stars and the data of Surya Siddhanta, it becomes clear that this update to Surya Siddhanta was made around 580 AD. THe longitude of the stars change by 1° in every 71 years. From the data it is clear that the data does not represent observation but rather is obtained by adding precessional increment to each of the previously calculated data.<br /> <br /> === Obliquity (tilt) of the Earth's axis ===<br /> Obliquity or the axial tilt of earth is the angle which the earth's axis of rotation makes with the perpendicular of orbital plane. This angle varies between 22.1° and 24.5° and it is cyclic phenomena over a period of 41,000 years. Currently the obliquity is 23.4 degrees.&lt;ref&gt;Alan Buis, &quot;Milankovitch (Orbital) Cycles and Their Role in Earth's Climate&quot;, &quot;NASA's Jet Propulsion Laboratory&quot; https://climate.nasa.gov/news/2948/milankovitch-orbital-cycles-and-their-role-in-earths-climate/&lt;/ref&gt; ''Sūrya Siddhānta'' in two different chapters calculate and provide the value of obliquity.<br /> <br /> Chapter 2, verse 28 translates as<br /> {{Quote <br /> |text = ''The sine of the greatest declination is 1397 units; Multiply the sine by the said sine 1397; Divide the product by the radius 3438 units; Find the arc whose sine is equal to the quotient. This arc is the mean declination of the planet''&lt;ref&gt;E. Burgess, &quot;Translation of Surya Siddhanta&quot;, p26, Accessible at https://www.jstor.org/stable/pdf/592174.pdf&lt;/ref&gt;<br /> }}<br /> This way we obtain the obliquity as Sin&lt;sup&gt;-1&lt;/sup&gt;(1397/3438) = 23.975°<br /> <br /> <br /> Chapter 12, verse 68 translates as <br /> {{Quote <br /> |text = ''At the distance of the fifteenth part of the Earth's circumference (from the equator) in the regions of the Gods or the Asuras (i.e. at the north and south terrestrial tropic) the sun passes through the zenith when it arrives at the north or south solstitial point (respectively)''''&lt;ref&gt;Pundit Bapu Deva Shastri, &quot;Translation of Surya Siddhanta&quot;, &quot;Baptist Mission Press&quot;, 1861, Accessible at https://www.wilbourhall.org/pdfs/suryaEnglish.pdf&lt;/ref&gt;<br /> }}<br /> It essentially provides information to calculate the axial tilt of earth which in this case can be calculated as 360°/15 = 24°.<br /> <br /> The significance of these verses is that they pin points the exact time when the obliquity calculations were made by ancient Indian astronomers and added into the ''Sūrya Siddhānta''. <br /> <br /> === North Pole Star and South Pole Star ===<br /> ''Surya Siddhanta'' contains an observation of the presence of pole stars at both north celestial pole and south celestial pole. Because of the precession of the earth's axis it is known that the pole star changes over a period of time which is normally more than thousand years. In present times our North Pole star is Polaris.&lt;ref&gt;Bruce McClure, &quot;Polaris is the North Pole Star&quot;, &quot;Earthsky&quot;, 21 May 2019, Accessible at https://earthsky.org/brightest-stars/polaris-the-present-day-north-star&lt;/ref&gt; This observation is recorded in chapter 12, verse 43-44 and translates as <br /> {{Quote<br /> |text = ''There are two pole stars, one each, near North Celestial Pole (NCP) and near South Celestial Pole (SCP). From equatorial regions, these stars are seen along the horizon. The pole stars are seen along the horizon, thus the place latitude is close to zero, while declination of NCP and SCP is 90 degrees.''<br /> }}<br /> <br /> Such phenomena was last seen around 3rd millennium BCE when Thuban was the North Pole Star and Alpha Hydri was the South Pole star.&lt;ref&gt;Nilesh N Oak and Rupa Bhatty, &quot;Ancient Updates to Surya Siddhanta&quot;, 09 March 2019, &quot;India Facts&quot; Accessible at http://indiafacts.org/ancient-updates-to-surya-siddhanta/&lt;/ref&gt; &lt;ref&gt;Anil Narayanan, &quot;Wonders, Mysteries and Misconceptions in Indian Astronomy – I&quot;, 'India facts&quot;, 09 Sept 2019, Accessible at http://indiafacts.org/wonders-mysteries-and-misconceptions-in-indian-astronomy-i/&lt;/ref&gt; This indicates the antiquity of the concepts written in the text.<br /> <br /> === ''Surya Siddhanta'' sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It is among the earliest form of [[Indian sine tables]]. The sine tables had been improved upon by many ancient Indian mathematicians. ''Surya Siddhanta'' uses an ''Indian standard circle'' of radius 3438 minutes. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and an arc length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 minutes)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The ''Surya Siddhanta'' derived Sin(θ) or Sine values show astonishing precision of 3 to 4 decimal places in comparison to the modern Sine values. The 1st order difference is the value by which each successive sine increases from the previous and similarly 2nd order difference is the increment in the 1st order difference values. ''Burgess'' notes that it is remarkable to see that the 2nd order differences increase as the sines and each, in fact, is about 1/225th part of the corresponding sine.&lt;ref&gt;Burgess, Rev. Ebenezer (1860). Translation of the Surya Siddhanta. p. 115.&lt;/ref&gt;<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle (in degrees, arcminutes)<br /> ![[Surya Siddhanta]] value of ''[[Indian_sine_tables|Jyā]]'' (R.sine)<br /> !Surya Siddhanta versed sines [[Indian_sine_tables#Terminology|Utkramā-jyā]] (R - R.cosine)<br /> !Modern value of ''[[Indian_sine_tables|Jyā]]'' (R.sine)<br /> !SS derived sine values (''[[Indian_sine_tables|Jyā]]'' / 3438)<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |{{center|0.06544503}}<br /> |{{center|0.06540313}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |{{center|0.13059919}}<br /> |{{center|0.13052619}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |{{center|0.19517161}}<br /> |{{center|0.19509032}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |{{center|0.25887144}}<br /> |{{center|0.25881905}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |{{center|0.3212078}}<br /> |{{center|0.32143947}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |{{center|0.38248982}}<br /> |{{center|0.38268343}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |{{center|0.44211751}}<br /> |{{center|0.44228869}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |{{center|0.50000000}}<br /> |{{center|0.50000000}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |{{center|0.55555556}}<br /> |{{center|0.55557023}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |{{center|0.60878418}}<br /> |{{center|0.60876143}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |{{center|0.65939500}}<br /> |{{center|0.65934582}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |{{center|0.70709715}}<br /> |{{center|0.70710678}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |{{center|0.75189063}}<br /> |{{center|0.75183981}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |{{center|0.79348458}}<br /> |{{center|0.79335334}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |{{center|0.83158813}}<br /> |{{center|0.83146961}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |{{center|0.86620128}}<br /> |{{center|0.86602540}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |{{center|0.89703316}}<br /> |{{center|0.89687274}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |{{center|0.92408377}}<br /> |{{center|0.92387953}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |{{center|0.94706225}}<br /> |{{center|0.94693013}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |{{center|0.96596859}}<br /> |{{center|0.96592583}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |{{center|0.98080279}}<br /> |{{center|0.98078528}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |{{center|0.99156486}}<br /> |{{center|0.99144486}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |{{center|0.99796393}}<br /> |{{center|0.99785892}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |{{center|1.00000000}}<br /> |{{center|1.00000000}}<br /> |-<br /> |}<br /> <br /> == See Also ==<br /> *[[Indian sine tables]]<br /> *[[Indian standard circle]]<br /> *[[Madhava's sine table]]<br /> <br /> == References and notes==<br /> &lt;references /&gt;</div>

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<p>BharataPuru: </p> <hr /> <div>'''''Sūrya Siddhānta''''' is an ancient Indian treatise in Astronomy. Like many classical Indian works, the ''Sūrya Siddhānta'' is a poem in [[Sanskrit]] language. It has fourteen chapter and 500 verses. It is composed in ''śloka'' metrical style of Sanskrit. It contain works on Indian sine tables, cosmology, eclipses, planetary motions, conjunctions, star positions, geography, instrumentation, concepts of time and mathematics. Unlike conventional books ''Sūrya Siddhānta'' contains advanced calculation and methods which are not easily comprehensible for a rank beginner. &lt;ref&gt;[https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol45_4_1_ANarayan.pdf]&lt;/ref&gt; In second chapter, the text contains the calculation of Earth's obliquity of ''1397 jya (R.sine) 23.975°'' modern units. The text describes the observation of two pole stars one each at the north pole and the south pole seen at the equatorial region, such phenomena was last seen in the 3rd millennium BCE. This indicates the antiquity of the concepts recorded in the text.&lt;seo title=&quot;Surya Siddhanta&quot; titlemode=&quot;append&quot; keywords=&quot;Surya Siddhanta, surya, Surya Siddhanta dharma, dhammawiki&quot; description=&quot;Sūrya Siddhānta is an ancient Indian treatise in Astronomy. Like many classical Indian works, the Sūrya Siddhānta is a poem in Sanskrit language. It has fourteen chapter and 500 verses. It is composed in śloka metrical style of Sanskrit. It contain works on Indian sine tables, cosmology, eclipses, planetary motions, conjunctions, star positions, geography, instrumentation, concepts of time and mathematics.&quot;&gt;&lt;/seo&gt;<br /> == History ==<br /> &lt;nowiki&gt;''Sūrya Siddhānta'' is well known, most referred and most esteemed. The original author of ''Sūrya Siddhānta'' is ''Mayasura'' as described in the story in the first chapter that ''Mayasura'' obtained his knowledge from ''Sūrya'' (the Sun). ''Siddhānta'' in Sanskrit means ''treatise'' and it usually has author'&lt;/nowiki&gt;s name prefixed to it. There were several other works on Astronomy in ancient India, many of which have since been lost.<br /> <br /> {| class=&quot;wikitable&quot;<br /> |-<br /> | ''Surya Siddhānta'' || ''Brahma Siddhānta'' || ''Soma Siddhānta''<br /> |-<br /> | ''Vyasa Siddhānta'' || ''Vashishtha Siddhānta'' || ''Atri Siddhānta''<br /> |-<br /> | ''Parashira Siddhānta'' || ''Kashyap Siddhānta'' || ''Nārad Siddhānta''<br /> |-<br /> | ''Garga Siddhānta'' || ''Marici Siddhānta'' || ''Manu Siddhānta''<br /> |-<br /> | ''Angiras Siddhānta'' || ''Lomasha Siddhānta'' || ''Pulisha Siddhānta''<br /> |-<br /> | ''Cyavana Siddhānta'' || ''Yavana Siddhānta'' || ''Bhrigu Siddhānta''<br /> |}<br /> <br /> == Content ==<br /> ''Sūrya Siddhānta'' contains 14 chapters and 500 verses. The chapters contain observations, methods, instruments and calculations of various astronomical phenomenas. There is a scarcity of scientific analysis done on the text of ''Surya Siddhanta''. Majority western work is based on Indology dates which in itself is controversial and based on their biased opinion of granting the origin of any science or mathematics to the ancient Greek or babylonians despite of immense textual evidence pointing otherwise. Their analysis of Surya Siddhanta primarily avoids the study of actual data and observations recorded within the ''Surya Siddhanta''.<br /> <br /> === Indian origin of seconds, minutes and degrees ===<br /> ''Surya Siddhanta'' in chapter 2 describes the units of seconds, minutes and degrees. These units of measurement are primary basis of the calculations of earth's obliquity and sine tables of ''Surya Siddhanta''. It is reasonable to think that these units or concepts had been in existence prior to other calculations and observation made in the epoch of 6th millennium BC as discussed in this article. The descriptions are<br /> <br /> {| class=&quot;wikitable&quot;<br /> |+ ''Surya Siddhanta'' units: seconds, minutes and degrees&lt;ref&gt;Pundit Bapu Deva Shastri, &quot;English Translation of Surya Siddhanta&quot;,p11, 1861&lt;/ref&gt; <br /> |-<br /> ! Modern SI units !! Surya Siddhanta units !! Value<br /> |-<br /> | Second || Vikala || -<br /> |-<br /> | Minute || Kala || 60 seconds<br /> |-<br /> | Degree || Ansh || 60 minutes<br /> |-<br /> | Zodiac Sign || Rashi || 30 degrees<br /> |-<br /> | Revolution || Bhagan || 12 zodiac signs<br /> |}<br /> <br /> These units are used in several calculations done through out the text of ''Surya Siddhanta''. In the sine tables of ''Surya Siddhanta'' the first sine or Jyā is described as the value equal to 1/8th of the number of minutes (Kalas) in a zodiac sign (Rashi).<br /> <br /> === Indian standard circle ===<br /> The ''Surya Siddhanta'' is using the [[Indian standard circle]] in various calculations through out the text. This standard circle is based on radius of 3,438 minutes. The significance lies in the precision of 1/3438 that the ancient Indian astronomers were able to work with. It is evident from the calculation of obliquity of the earth's axis in chapter 2 where 1397 units is the measured R-sine value. <br /> Another interesting outcome of this radius of 3,438 minutes is that the circumference of the standard Indian circle is calculated as 21,600 minutes using the formula of Pi multiply by diameter (twice the radius).<br /> <br /> === Nakshatra (Asterism) System ===<br /> <br /> The ''Surya Siddhanta'' uses the 27 [[Nakshatra system]] throughout the text. The Nakshatra is a smaller constellation typically consisting of 1 to 5 stars. The brightest star is called as Yogtara. Each Nakshatra spans 13° 20' on the ecliptic. Each Nakshatra has its own primary star which is usually the junction star but not always.<br /> <br /> === Longitudinal updates ===<br /> Chapter 8 of ''Surya Siddhanta'' primarily focuses on the stellar data. It provides the longitudinal data for the Asterisms. In comparison to the present day longitudinal values of these stars and the data of Surya Siddhanta, it becomes clear that this update to Surya Siddhanta was made around 580 AD. THe longitude of the stars change by 1° in every 71 years. From the data it is clear that the data does not represent observation but rather is obtained by adding precessional increment to each of the previously calculated data.<br /> <br /> === Obliquity (tilt) of the Earth's axis ===<br /> Obliquity or the axial tilt of earth is the angle which the earth's axis of rotation makes with the perpendicular of orbital plane. This angle varies between 22.1° and 24.5° and it is cyclic phenomena over a period of 41,000 years. Currently the obliquity is 23.4 degrees.&lt;ref&gt;Alan Buis, &quot;Milankovitch (Orbital) Cycles and Their Role in Earth's Climate&quot;, &quot;NASA's Jet Propulsion Laboratory&quot; https://climate.nasa.gov/news/2948/milankovitch-orbital-cycles-and-their-role-in-earths-climate/&lt;/ref&gt; ''Sūrya Siddhānta'' in two different chapters calculate and provide the value of obliquity.<br /> <br /> Chapter 2, verse 28 translates as<br /> {{Quote <br /> |text = ''The sine of the greatest declination is 1397 units; Multiply the sine by the said sine 1397; Divide the product by the radius 3438 units; Find the arc whose sine is equal to the quotient. This arc is the mean declination of the planet''&lt;ref&gt;E. Burgess, &quot;Translation of Surya Siddhanta&quot;, p26, Accessible at https://www.jstor.org/stable/pdf/592174.pdf&lt;/ref&gt;<br /> }}<br /> This way we obtain the obliquity as Sin&lt;sup&gt;-1&lt;/sup&gt;(1397/3438) = 23.975°<br /> <br /> <br /> Chapter 12, verse 68 translates as <br /> {{Quote <br /> |text = ''At the distance of the fifteenth part of the Earth's circumference (from the equator) in the regions of the Gods or the Asuras (i.e. at the north and south terrestrial tropic) the sun passes through the zenith when it arrives at the north or south solstitial point (respectively)''''&lt;ref&gt;Pundit Bapu Deva Shastri, &quot;Translation of Surya Siddhanta&quot;, &quot;Baptist Mission Press&quot;, 1861, Accessible at https://www.wilbourhall.org/pdfs/suryaEnglish.pdf&lt;/ref&gt;<br /> }}<br /> It essentially provides information to calculate the axial tilt of earth which in this case can be calculated as 360°/15 = 24°.<br /> <br /> The significance of these verses is that they pin points the exact time when the obliquity calculations were made by ancient Indian astronomers and added into the ''Sūrya Siddhānta''. <br /> <br /> === North Pole Star and South Pole Star ===<br /> ''Surya Siddhanta'' contains an observation of the presence of pole stars at both north celestial pole and south celestial pole. Because of the precession of the earth's axis it is known that the pole star changes over a period of time which is normally more than thousand years. In present times our North Pole star is Polaris.&lt;ref&gt;Bruce McClure, &quot;Polaris is the North Pole Star&quot;, &quot;Earthsky&quot;, 21 May 2019, Accessible at https://earthsky.org/brightest-stars/polaris-the-present-day-north-star&lt;/ref&gt; This observation is recorded in chapter 12, verse 43-44 and translates as <br /> {{Quote<br /> |text = ''There are two pole stars, one each, near North Celestial Pole (NCP) and near South Celestial Pole (SCP). From equatorial regions, these stars are seen along the horizon. The pole stars are seen along the horizon, thus the place latitude is close to zero, while declination of NCP and SCP is 90 degrees.''<br /> }}<br /> <br /> Such phenomena was last seen around 3rd millennium BCE when Thuban was the North Pole Star and Alpha Hydri was the South Pole star.&lt;ref&gt;Nilesh N Oak and Rupa Bhatty, &quot;Ancient Updates to Surya Siddhanta&quot;, 09 March 2019, &quot;India Facts&quot; Accessible at http://indiafacts.org/ancient-updates-to-surya-siddhanta/&lt;/ref&gt; &lt;ref&gt;Anil Narayanan, &quot;Wonders, Mysteries and Misconceptions in Indian Astronomy – I&quot;, 'India facts&quot;, 09 Sept 2019, Accessible at http://indiafacts.org/wonders-mysteries-and-misconceptions-in-indian-astronomy-i/&lt;/ref&gt; This indicates the antiquity of the concepts written in the text.<br /> <br /> === ''Surya Siddhanta'' sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It is among the earliest form of [[Indian sine tables]]. The sine tables had been improved upon by many ancient Indian mathematicians. ''Surya Siddhanta'' uses an ''Indian standard circle'' of radius 3438 minutes. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and an arc length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 minutes)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The ''Surya Siddhanta'' derived Sin(θ) or Sine values show astonishing precision of 3 to 4 decimal places in comparison to the modern Sine values. The 1st order difference is the value by which each successive sine increases from the previous and similarly 2nd order difference is the increment in the 1st order difference values. ''Burgess'' notes that it is remarkable to see that the 2nd order differences increase as the sines and each, in fact, is about 1/225th part of the corresponding sine.&lt;ref&gt;Burgess, Rev. Ebenezer (1860). Translation of the Surya Siddhanta. p. 115.&lt;/ref&gt;<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle (in degrees, arcminutes)<br /> ![[Surya Siddhanta]] value of ''[[Indian_sine_tables|Jyā]]'' (R.sine)<br /> !Surya Siddhanta versed sines [[Indian_sine_tables#Terminology|Utkramā-jyā]] (R - R.cosine)<br /> !Modern value of ''[[Indian_sine_tables|Jyā]]'' (R.sine)<br /> !SS derived sine values (''[[Indian_sine_tables|Jyā]]'' / 3438)<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |{{center|0.06544503}}<br /> |{{center|0.06540313}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |{{center|0.13059919}}<br /> |{{center|0.13052619}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |{{center|0.19517161}}<br /> |{{center|0.19509032}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |{{center|0.25887144}}<br /> |{{center|0.25881905}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |{{center|0.3212078}}<br /> |{{center|0.32143947}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |{{center|0.38248982}}<br /> |{{center|0.38268343}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |{{center|0.44211751}}<br /> |{{center|0.44228869}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |{{center|0.50000000}}<br /> |{{center|0.50000000}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |{{center|0.55555556}}<br /> |{{center|0.55557023}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |{{center|0.60878418}}<br /> |{{center|0.60876143}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |{{center|0.65939500}}<br /> |{{center|0.65934582}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |{{center|0.70709715}}<br /> |{{center|0.70710678}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |{{center|0.75189063}}<br /> |{{center|0.75183981}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |{{center|0.79348458}}<br /> |{{center|0.79335334}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |{{center|0.83158813}}<br /> |{{center|0.83146961}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |{{center|0.86620128}}<br /> |{{center|0.86602540}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |{{center|0.89703316}}<br /> |{{center|0.89687274}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |{{center|0.92408377}}<br /> |{{center|0.92387953}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |{{center|0.94706225}}<br /> |{{center|0.94693013}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |{{center|0.96596859}}<br /> |{{center|0.96592583}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |{{center|0.98080279}}<br /> |{{center|0.98078528}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |{{center|0.99156486}}<br /> |{{center|0.99144486}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |{{center|0.99796393}}<br /> |{{center|0.99785892}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |{{center|1.00000000}}<br /> |{{center|1.00000000}}<br /> |-<br /> |}<br /> <br /> == See Also ==<br /> *[[Indian sine tables]]<br /> *[[Indian standard circle]]<br /> *[[Madhava's sine table]]<br /> <br /> == References and notes==<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_sine_tables&diff=130700 2021-07-03T07:12:30Z

<p>BharataPuru: /* Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) */</p> <hr /> <div>Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the [[Indian mathematician]] and astronomer [[Aryabhata|Āryabhaṭa]] (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.&lt;ref&gt;Āryabhat·a's Rule and Table for Sine-Differences - ScienceDirect&lt;/ref&gt;<br /> <br /> Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics. Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.<br /> <br /> == Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==<br /> '''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three trigonometric functions introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta. These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of &quot;sine&quot; and &quot;cosine&quot; have been traced back to the [[Sanskrit]] words jyā and kotijyā.<br /> <br /> ===Definition===<br /> Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:<br /> <br /> * ''Vyāsardhā'' = Radius (R)<br /> * ''jyā'' of arc AB = MB<br /> * ''koti-jyā'' of arc AB = CM = R cos θ<br /> * ''utkrama-jyā'' of arc AB = MA<br /> <br /> If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:<br /> <br /> * ''jyā'' ( arc AB ) = ''R'' sin θ<br /> * ''koti-jyā'' ( arc AB ) = ''R'' cos θ<br /> * ''utkrama-jyā'' ( arc AB ) = R - R cos θ<br /> <br /> An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means &quot;a bow&quot;. <br /> The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means &quot;a bow-string&quot;.<br /> The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.<br /> At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs. The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant &quot;half of&quot;.<br /> <br /> The Sanskrit word ''koṭi'' has the meaning of &quot;point, cusp&quot;, and specifically &quot;the [[Recurve bow|curved end of a bow]]&quot;.<br /> In trigonometry, it came to denote &quot;the complement of an arc to 90°&quot;. Thus <br /> ''koṭi-jyā'' is &quot;the ''jyā'' of the complementary arc&quot;. In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes &quot;the side of a right angled triangle&quot;. Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .<br /> <br /> ''Utkrama'' means &quot;inverted&quot;, thus ''utkrama-jyā'' means &quot;inverted chord&quot;. The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning &quot;arrow&quot;.<br /> <br /> An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of &quot;the ''jyā'' of three signs&quot;. The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning &quot;semi-diameter&quot;.<br /> <br /> According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by &quot;Rsin&quot; and &quot;Rcos&quot; treated as single words. Others denote ''jyā'' and ''koti-jyā'' respectively by &quot;Sin&quot; and &quot;Cos&quot; (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).<br /> <br /> ===From jyā to sine===<br /> The origins of the modern term sine have been traced to the Sanskrit word ''jyā'', or more specifically to its synonym ''jīva''. This term was adopted in medieval Islamic mathematics, transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means &quot;bosom&quot;. The text's 12th-century Latin translator used the Latin equivalent for &quot;bosom&quot;, ''[[wikt:sinus|sinus]]''. When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.<br /> <br /> ==The Indian sine tables==<br /> === Surya Siddhanta sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> ===Āryabhaṭa's sine table===<br /> The stanza in Āryabhaṭiya describing the sine table is reproduced below:<br /> <br /> मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |<br /> घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||<br /> <br /> The second section of Āryabhaṭiya titled Ganitapādd<br /> a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.<br /> * &quot;When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.&quot;<br /> This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. <br /> <br /> The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''jya'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.<br /> <br /> ===Comparing different sine tables===<br /> <br /> [[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle&lt;br /&gt;(in degrees,&lt;br /&gt;arcminutes)<br /> !Āryabhaṭa's &lt;br /&gt;value of Jyā (R.sine)<br /> ![[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] value of Jyā (R.sine)<br /> !Surya Siddhanta versed sines<br /> !Modern value &lt;br /&gt;of ''Jyā'' R.sine<br /> !Madhava's sine values<br /> !Madhava's derived sine values<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |&lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.06540314<br /> |0.06540313<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |&lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> |0.13052623<br /> |0.13052619<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |&lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> |0.19509032<br /> |0.19509032<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |&lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.25881900<br /> |0.25881905<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |&lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> |0.32143947<br /> |0.32143947<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |&lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> |0.38268340<br /> |0.38268343<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |&lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> |0.44228865<br /> |0.44228869<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.49999998<br /> |0.50000000<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.55557022<br /> |0.55557023<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |&lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> |0.60876139<br /> |0.60876143<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |&lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.65934580<br /> |0.65934582<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |&lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.70710681<br /> |0.70710678<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |&lt;center&gt;2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> |0.75183985<br /> |0.75183981<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |&lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> |0.79335331<br /> |0.79335334<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |&lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> |0.83146960<br /> |0.83146961<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |&lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> |0.86602543<br /> |0.86602540<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |&lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt;<br /> |0.89687275<br /> |0.89687274<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |&lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.92387954<br /> |0.92387953<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |&lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.94693016<br /> |0.94693013<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |&lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> |0.96592581<br /> |0.96592583<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |&lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> |0.98078527<br /> |0.98078528<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |&lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99144487<br /> |0.99144486<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |&lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99785895<br /> |0.99785892<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |&lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt;<br /> |0.99999997<br /> |1.00000000<br /> |-<br /> |}<br /> <br /> ==See also==<br /> * [[Madhava's sine table]]<br /> * [[Bhaskara I's sine approximation formula]]<br /> * [[Indian standard circle|Indian Standard Circle]]<br /> * [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]]<br /> <br /> ==References==<br /> {{reflist}}<br /> <br /> [[Category:Indian Astronomy]]<br /> [[Category:Indian Mathematics]]<br /> __NOINDEX__<br /> __NONEWSECTIONLINK__<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_standard_circle&diff=130699 2021-07-03T07:10:48Z

<p>BharataPuru: </p> <hr /> <div>''Indian standard circle'' is a name given to the standard circle first used in [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and later used by several ancient Indian mathematicians and astronomers to improve the [[Indian sine tables]] and for various other calculations. [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] provides methods for calculating the Jyā (R.sine) values. The circle uses a radius of 3,438 minutes. ''Surya Siddhanta'' calculates the first Jyā (R.sine) as 1/8th of the number of minutes (kalās) in a Rashi (zodiac sign). It says a Rashi (zodiac sign) has 1800 minutes (kalās) and thus calculates the first Jyā to a value of 225 minutes (kalā कला ).&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). &quot;Translation of the Surya Siddhanta&quot;. Ch2 Ve15, pp. 15–16.&lt;/ref&gt;<br /> <br /> &lt;br&gt;The Indian standard circles holds significance as it is based on number of minutes in circle thus leads to 360 degrees in a circle which is the basis of modern trigonometry. Although the [[Indian sine tables]] are not based on the angles but rather on the R.sine (Jyā) values. The [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] data reflect highly sophisticated outcomes of the R.sine values. ''Burgess'' notes that it is remarkable to see that the 2nd order differences increase as the sines and each, in fact, is about 1/225th part of the corresponding sine.&lt;ref&gt;Burgess, Rev. Ebenezer (1860). Translation of the Surya Siddhanta. p. 115.&lt;/ref&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Modern units and Indian units ''(Sanskrit)''<br /> !Modern units<br /> !Indian units ''(Sanskrit)''<br /> !value<br /> |-<br /> |Zodiac sign<br /> |Rashi (राशी)<br /> |30 degrees<br /> |-<br /> |Degree<br /> |Ansh (अंश )<br /> |60 minutes<br /> |-<br /> |Minute<br /> |Kalā (कला )<br /> |60 seconds<br /> |-<br /> |Second<br /> |Vikalā (विकला )<br /> | -<br /> |}<br /> <br /> == See Also ==<br /> * [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]]<br /> * [[Indian sine tables|Indian Sine Tables]]<br /> <br /> == References ==<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Topic:W76dqa6f1xqvlw6j&topic_postId=wbgirg6bv79vsxt7&topic_revId=wbgirg6bv79vsxt7&action=single-view 2021-06-24T08:47:11Z

<span class="plainlinks"><a href="/index.php?title=User:BharataPuru&amp;action=edit&amp;redlink=1" class="new mw-userlink" title="User:BharataPuru (page does not exist)"><bdi>BharataPuru</bdi></a> <span class="mw-usertoollinks">(<a href="/index.php?title=User_talk:BharataPuru&amp;action=edit&amp;redlink=1" class="new mw-usertoollinks-talk" title="User talk:BharataPuru (page does not exist)">talk</a> | <a href="/index.php/Special:Contributions/BharataPuru" class="mw-usertoollinks-contribs" title="Special:Contributions/BharataPuru">contribs</a>)</span> <a rel="nofollow" class="external text" href="http://dharmawiki.org/index.php?title=Topic:W76dqa6f1xqvlw6j&amp;topic_showPostId=wbgirg6bv79vsxt7#flow-post-wbgirg6bv79vsxt7">commented</a> on "Remove the devnagri from the title" (<em>Hi Can you please remove metadata information from the following links or advise me how to do it. Thanks https://dharmawiki.org/index.php...</em>)</span>

BharataPuru http://dharmawiki.org/index.php?title=Topic:W76dqa6f1xqvlw6j&topic_postId=w79pfe7wtuq1djjf&topic_revId=w79pfe7wtuq1djjf&action=single-view 2021-04-17T15:55:57Z

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BharataPuru http://dharmawiki.org/index.php?title=Topic:W76dqa6f1xqvlw6j&topic_postId=w79pcepcznxc79rf&topic_revId=w79pcepcznxc79rf&action=single-view 2021-04-17T15:54:27Z

<span class="plainlinks"><a href="/index.php?title=User:BharataPuru&amp;action=edit&amp;redlink=1" class="new mw-userlink" title="User:BharataPuru (page does not exist)"><bdi>BharataPuru</bdi></a> <span class="mw-usertoollinks">(<a href="/index.php?title=User_talk:BharataPuru&amp;action=edit&amp;redlink=1" class="new mw-usertoollinks-talk" title="User talk:BharataPuru (page does not exist)">talk</a> | <a href="/index.php/Special:Contributions/BharataPuru" class="mw-usertoollinks-contribs" title="Special:Contributions/BharataPuru">contribs</a>)</span> <a rel="nofollow" class="external text" href="http://dharmawiki.org/index.php?title=Topic:W76dqa6f1xqvlw6j&amp;topic_showPostId=w79pcepcznxc79rf#flow-post-w79pcepcznxc79rf">commented</a> on "Remove the devnagri from the title" (<em>No problem, if it doesnot impact the search. Its fine. Thanks</em>)</span>

BharataPuru http://dharmawiki.org/index.php?title=Topic:W76dqa6f1xqvlw6j&topic_postId=w77dfxsu4dfzdicr&topic_revId=w77dfxsu4dfzdicr&action=single-view 2021-04-16T14:47:53Z

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BharataPuru http://dharmawiki.org/index.php?title=Indian_sine_tables&diff=129878 2021-04-16T06:12:35Z

<p>BharataPuru: </p> <hr /> <div>Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the [[Indian mathematician]] and astronomer [[Aryabhata|Āryabhaṭa]] (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.&lt;ref&gt;Āryabhat·a's Rule and Table for Sine-Differences - ScienceDirect&lt;/ref&gt;<br /> <br /> Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics. Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.<br /> <br /> == Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==<br /> '''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three trigonometric functions introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta. These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of &quot;sine&quot; and &quot;cosine&quot; have been traced back to the [[Sanskrit]] words jyā and kotijyā.<br /> [[File:Jya Figure.jpg|thumb|419x419px]]<br /> <br /> ===Definition===<br /> Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:<br /> <br /> * ''Vyāsardhā'' = Radius (R)<br /> * ''jyā'' of arc AB = MB<br /> * ''koti-jyā'' of arc AB = CM = R cos θ<br /> * ''utkrama-jyā'' of arc AB = MA<br /> <br /> If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:<br /> <br /> * ''jyā'' ( arc AB ) = ''R'' sin θ<br /> * ''koti-jyā'' ( arc AB ) = ''R'' cos θ<br /> * ''utkrama-jyā'' ( arc AB ) = R - R cos θ<br /> <br /> An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means &quot;a bow&quot;. <br /> The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means &quot;a bow-string&quot;.<br /> The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.<br /> At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs. The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant &quot;half of&quot;.<br /> <br /> The Sanskrit word ''koṭi'' has the meaning of &quot;point, cusp&quot;, and specifically &quot;the [[Recurve bow|curved end of a bow]]&quot;.<br /> In trigonometry, it came to denote &quot;the complement of an arc to 90°&quot;. Thus <br /> ''koṭi-jyā'' is &quot;the ''jyā'' of the complementary arc&quot;. In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes &quot;the side of a right angled triangle&quot;. Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .<br /> <br /> ''Utkrama'' means &quot;inverted&quot;, thus ''utkrama-jyā'' means &quot;inverted chord&quot;. The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning &quot;arrow&quot;.<br /> <br /> An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of &quot;the ''jyā'' of three signs&quot;. The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning &quot;semi-diameter&quot;.<br /> <br /> According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by &quot;Rsin&quot; and &quot;Rcos&quot; treated as single words. Others denote ''jyā'' and ''koti-jyā'' respectively by &quot;Sin&quot; and &quot;Cos&quot; (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).<br /> <br /> ===From jyā to sine===<br /> The origins of the modern term sine have been traced to the Sanskrit word ''jyā'', or more specifically to its synonym ''jīva''. This term was adopted in medieval Islamic mathematics, transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means &quot;bosom&quot;. The text's 12th-century Latin translator used the Latin equivalent for &quot;bosom&quot;, ''[[wikt:sinus|sinus]]''. When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.<br /> <br /> ==The Indian sine tables==<br /> === Surya Siddhanta sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> ===Āryabhaṭa's sine table===<br /> The stanza in Āryabhaṭiya describing the sine table is reproduced below:<br /> <br /> मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |<br /> घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||<br /> <br /> The second section of Āryabhaṭiya titled Ganitapādd<br /> a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.<br /> * &quot;When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.&quot;<br /> This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. <br /> <br /> The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''jya'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.<br /> <br /> ===Comparing different sine tables===<br /> <br /> [[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle&lt;br /&gt;(in degrees,&lt;br /&gt;arcminutes)<br /> !Āryabhaṭa's &lt;br /&gt;value of Jyā (R.sine)<br /> ![[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] value of Jyā (R.sine)<br /> !Surya Siddhanta versed sines<br /> !Modern value &lt;br /&gt;of ''Jyā'' R.sine<br /> !Madhava's sine values<br /> !Madhava's derived sine values<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |&lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.06540314<br /> |0.06540313<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |&lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> |0.13052623<br /> |0.13052619<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |&lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> |0.19509032<br /> |0.19509032<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |&lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.25881900<br /> |0.25881905<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |&lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> |0.32143947<br /> |0.32143947<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |&lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> |0.38268340<br /> |0.38268343<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |&lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> |0.44228865<br /> |0.44228869<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.49999998<br /> |0.50000000<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.55557022<br /> |0.55557023<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |&lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> |0.60876139<br /> |0.60876143<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |&lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.65934580<br /> |0.65934582<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |&lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.70710681<br /> |0.70710678<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |&lt;center&gt;2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> |0.75183985<br /> |0.75183981<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |&lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> |0.79335331<br /> |0.79335334<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |&lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> |0.83146960<br /> |0.83146961<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |&lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> |0.86602543<br /> |0.86602540<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |&lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt;<br /> |0.89687275<br /> |0.89687274<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |&lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.92387954<br /> |0.92387953<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |&lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.94693016<br /> |0.94693013<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |&lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> |0.96592581<br /> |0.96592583<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |&lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> |0.98078527<br /> |0.98078528<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |&lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99144487<br /> |0.99144486<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |&lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99785895<br /> |0.99785892<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |&lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt;<br /> |0.99999997<br /> |1.00000000<br /> |-<br /> |}<br /> <br /> ==See also==<br /> * [[Madhava's sine table]]<br /> * [[Bhaskara I's sine approximation formula]]<br /> * [[Indian standard circle|Indian Standard Circle]]<br /> * [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]]<br /> <br /> ==References==<br /> {{reflist}}<br /> <br /> [[Category:Indian Astronomy]]<br /> [[Category:Indian Mathematics]]<br /> __NOINDEX__<br /> __NONEWSECTIONLINK__<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_sine_tables&diff=129877 2021-04-16T05:48:09Z

<p>BharataPuru: </p> <hr /> <div>Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the [[Indian mathematician]] and astronomer [[Aryabhata|Āryabhaṭa]] (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.<br /> <br /> Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics. Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.<br /> <br /> == Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==<br /> '''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three trigonometric functions introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta. These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of &quot;sine&quot; and &quot;cosine&quot; have been traced back to the [[Sanskrit]] words jyā and kotijyā.<br /> [[File:Jya Figure.jpg|thumb|419x419px]]<br /> <br /> ===Definition===<br /> Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:<br /> <br /> * ''Vyāsardhā'' = Radius (R)<br /> * ''jyā'' of arc AB = MB<br /> * ''koti-jyā'' of arc AB = CM = R cos θ<br /> * ''utkrama-jyā'' of arc AB = MA<br /> <br /> If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:<br /> <br /> * ''jyā'' ( arc AB ) = ''R'' sin θ<br /> * ''koti-jyā'' ( arc AB ) = ''R'' cos θ<br /> * ''utkrama-jyā'' ( arc AB ) = R - R cos θ<br /> <br /> An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means &quot;a bow&quot;. <br /> The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means &quot;a bow-string&quot;.<br /> The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.<br /> At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs. The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant &quot;half of&quot;.<br /> <br /> The Sanskrit word ''koṭi'' has the meaning of &quot;point, cusp&quot;, and specifically &quot;the [[Recurve bow|curved end of a bow]]&quot;.<br /> In trigonometry, it came to denote &quot;the complement of an arc to 90°&quot;. Thus <br /> ''koṭi-jyā'' is &quot;the ''jyā'' of the complementary arc&quot;. In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes &quot;the side of a right angled triangle&quot;. Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .<br /> <br /> ''Utkrama'' means &quot;inverted&quot;, thus ''utkrama-jyā'' means &quot;inverted chord&quot;. The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning &quot;arrow&quot;.<br /> <br /> An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of &quot;the ''jyā'' of three signs&quot;. The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning &quot;semi-diameter&quot;.<br /> <br /> According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by &quot;Rsin&quot; and &quot;Rcos&quot; treated as single words. Others denote ''jyā'' and ''koti-jyā'' respectively by &quot;Sin&quot; and &quot;Cos&quot; (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).<br /> <br /> ===From jyā to sine===<br /> The origins of the modern term sine have been traced to the Sanskrit word ''jyā'', or more specifically to its synonym ''jīva''. This term was adopted in medieval Islamic mathematics, transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means &quot;bosom&quot;. The text's 12th-century Latin translator used the Latin equivalent for &quot;bosom&quot;, ''[[wikt:sinus|sinus]]''. When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.<br /> <br /> ==The Indian sine tables==<br /> === Surya Siddhanta sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> ===Āryabhaṭa's sine table===<br /> The stanza in Āryabhaṭiya describing the sine table is reproduced below:<br /> <br /> मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |<br /> घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||<br /> <br /> The second section of Āryabhaṭiya titled Ganitapādd<br /> a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.<br /> * &quot;When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.&quot;<br /> This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. <br /> <br /> The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''jya'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.<br /> <br /> ===Comparing different sine tables===<br /> <br /> [[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle&lt;br /&gt;(in degrees,&lt;br /&gt;arcminutes)<br /> !Āryabhaṭa's &lt;br /&gt;value of Jyā (R.sine)<br /> ![[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] value of Jyā (R.sine)<br /> !Surya Siddhanta versed sines<br /> !Modern value &lt;br /&gt;of ''Jyā'' R.sine<br /> !Madhava's sine values<br /> !Madhava's derived sine values<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |&lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.06540314<br /> |0.06540313<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |&lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> |0.13052623<br /> |0.13052619<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |&lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> |0.19509032<br /> |0.19509032<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |&lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.25881900<br /> |0.25881905<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |&lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> |0.32143947<br /> |0.32143947<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |&lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> |0.38268340<br /> |0.38268343<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |&lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> |0.44228865<br /> |0.44228869<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.49999998<br /> |0.50000000<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.55557022<br /> |0.55557023<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |&lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> |0.60876139<br /> |0.60876143<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |&lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.65934580<br /> |0.65934582<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |&lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.70710681<br /> |0.70710678<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |&lt;center&gt;2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> |0.75183985<br /> |0.75183981<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |&lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> |0.79335331<br /> |0.79335334<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |&lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> |0.83146960<br /> |0.83146961<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |&lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> |0.86602543<br /> |0.86602540<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |&lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt;<br /> |0.89687275<br /> |0.89687274<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |&lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.92387954<br /> |0.92387953<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |&lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.94693016<br /> |0.94693013<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |&lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> |0.96592581<br /> |0.96592583<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |&lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> |0.98078527<br /> |0.98078528<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |&lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99144487<br /> |0.99144486<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |&lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99785895<br /> |0.99785892<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |&lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt;<br /> |0.99999997<br /> |1.00000000<br /> |-<br /> |}<br /> <br /> ==See also==<br /> * [[Madhava's sine table]]<br /> * [[Bhaskara I's sine approximation formula]]<br /> * [[Indian standard circle|Indian Standard Circle]]<br /> * [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]]<br /> <br /> ==References==<br /> {{reflist}}<br /> <br /> [[Category:Indian Astronomy]]<br /> [[Category:Indian Mathematics]]<br /> __NOINDEX__<br /> __NONEWSECTIONLINK__<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_sine_tables&diff=129876 2021-04-16T05:47:37Z

<p>BharataPuru: /* References */</p> <hr /> <div>Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the [[Indian mathematician]] and astronomer [[Aryabhata|Āryabhaṭa]] (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.<br /> <br /> Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics. Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.<br /> <br /> == Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==<br /> '''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three trigonometric functions introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta. These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of &quot;sine&quot; and &quot;cosine&quot; have been traced back to the [[Sanskrit]] words jyā and kotijyā.&lt;ref name=&quot;Datta&quot; /&gt;<br /> [[File:Jya Figure.jpg|thumb|419x419px]]<br /> <br /> ===Definition===<br /> Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:<br /> <br /> * ''Vyāsardhā'' = Radius (R)<br /> * ''jyā'' of arc AB = MB<br /> * ''koti-jyā'' of arc AB = CM = R cos θ<br /> * ''utkrama-jyā'' of arc AB = MA<br /> <br /> If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:<br /> <br /> * ''jyā'' ( arc AB ) = ''R'' sin θ<br /> * ''koti-jyā'' ( arc AB ) = ''R'' cos θ<br /> * ''utkrama-jyā'' ( arc AB ) = R - R cos θ<br /> <br /> An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means &quot;a bow&quot;. <br /> The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means &quot;a bow-string&quot;.<br /> The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.<br /> At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs. The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant &quot;half of&quot;.<br /> <br /> The Sanskrit word ''koṭi'' has the meaning of &quot;point, cusp&quot;, and specifically &quot;the [[Recurve bow|curved end of a bow]]&quot;.<br /> In trigonometry, it came to denote &quot;the complement of an arc to 90°&quot;. Thus <br /> ''koṭi-jyā'' is &quot;the ''jyā'' of the complementary arc&quot;. In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes &quot;the side of a right angled triangle&quot;. Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .<br /> <br /> ''Utkrama'' means &quot;inverted&quot;, thus ''utkrama-jyā'' means &quot;inverted chord&quot;. The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning &quot;arrow&quot;.<br /> <br /> An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of &quot;the ''jyā'' of three signs&quot;. The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning &quot;semi-diameter&quot;.<br /> <br /> According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by &quot;Rsin&quot; and &quot;Rcos&quot; treated as single words. Others denote ''jyā'' and ''koti-jyā'' respectively by &quot;Sin&quot; and &quot;Cos&quot; (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).<br /> <br /> ===From jyā to sine===<br /> The origins of the modern term sine have been traced to the Sanskrit word ''jyā'', or more specifically to its synonym ''jīva''. This term was adopted in medieval Islamic mathematics, transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means &quot;bosom&quot;. The text's 12th-century Latin translator used the Latin equivalent for &quot;bosom&quot;, ''[[wikt:sinus|sinus]]''. When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.<br /> <br /> ==The Indian sine tables==<br /> === Surya Siddhanta sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> ===Āryabhaṭa's sine table===<br /> The stanza in Āryabhaṭiya describing the sine table is reproduced below:<br /> <br /> मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |<br /> घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||<br /> <br /> The second section of Āryabhaṭiya titled Ganitapādd<br /> a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.<br /> * &quot;When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.&quot;<br /> This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. <br /> <br /> The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''jya'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.<br /> <br /> ===Comparing different sine tables===<br /> <br /> [[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle&lt;br /&gt;(in degrees,&lt;br /&gt;arcminutes)<br /> !Āryabhaṭa's &lt;br /&gt;value of Jyā (R.sine)<br /> ![[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] value of Jyā (R.sine)<br /> !Surya Siddhanta versed sines<br /> !Modern value &lt;br /&gt;of ''Jyā'' R.sine<br /> !Madhava's sine values<br /> !Madhava's derived sine values<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |&lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.06540314<br /> |0.06540313<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |&lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> |0.13052623<br /> |0.13052619<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |&lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> |0.19509032<br /> |0.19509032<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |&lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.25881900<br /> |0.25881905<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |&lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> |0.32143947<br /> |0.32143947<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |&lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> |0.38268340<br /> |0.38268343<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |&lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> |0.44228865<br /> |0.44228869<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.49999998<br /> |0.50000000<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.55557022<br /> |0.55557023<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |&lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> |0.60876139<br /> |0.60876143<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |&lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.65934580<br /> |0.65934582<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |&lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.70710681<br /> |0.70710678<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |&lt;center&gt;2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> |0.75183985<br /> |0.75183981<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |&lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> |0.79335331<br /> |0.79335334<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |&lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> |0.83146960<br /> |0.83146961<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |&lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> |0.86602543<br /> |0.86602540<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |&lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt;<br /> |0.89687275<br /> |0.89687274<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |&lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.92387954<br /> |0.92387953<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |&lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.94693016<br /> |0.94693013<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |&lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> |0.96592581<br /> |0.96592583<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |&lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> |0.98078527<br /> |0.98078528<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |&lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99144487<br /> |0.99144486<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |&lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99785895<br /> |0.99785892<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |&lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt;<br /> |0.99999997<br /> |1.00000000<br /> |-<br /> |}<br /> <br /> ==See also==<br /> * [[Madhava's sine table]]<br /> * [[Bhaskara I's sine approximation formula]]<br /> * [[Indian standard circle|Indian Standard Circle]]<br /> * [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]]<br /> <br /> ==References==<br /> {{reflist}}<br /> <br /> [[Category:Indian Astronomy]]<br /> [[Category:Indian Mathematics]]<br /> __NOINDEX__<br /> __NONEWSECTIONLINK__<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_sine_tables&diff=129875 2021-04-16T05:46:41Z

<p>BharataPuru: /* References */</p> <hr /> <div>Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the [[Indian mathematician]] and astronomer [[Aryabhata|Āryabhaṭa]] (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.<br /> <br /> Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics. Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.<br /> <br /> == Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==<br /> '''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three trigonometric functions introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta. These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of &quot;sine&quot; and &quot;cosine&quot; have been traced back to the [[Sanskrit]] words jyā and kotijyā.&lt;ref name=&quot;Datta&quot; /&gt;<br /> [[File:Jya Figure.jpg|thumb|419x419px]]<br /> <br /> ===Definition===<br /> Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:<br /> <br /> * ''Vyāsardhā'' = Radius (R)<br /> * ''jyā'' of arc AB = MB<br /> * ''koti-jyā'' of arc AB = CM = R cos θ<br /> * ''utkrama-jyā'' of arc AB = MA<br /> <br /> If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:<br /> <br /> * ''jyā'' ( arc AB ) = ''R'' sin θ<br /> * ''koti-jyā'' ( arc AB ) = ''R'' cos θ<br /> * ''utkrama-jyā'' ( arc AB ) = R - R cos θ<br /> <br /> An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means &quot;a bow&quot;. <br /> The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means &quot;a bow-string&quot;.<br /> The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.<br /> At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs.&lt;ref name=&quot;Datta&quot; /&gt; The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant &quot;half of&quot;.<br /> <br /> The Sanskrit word ''koṭi'' has the meaning of &quot;point, cusp&quot;, and specifically &quot;the [[Recurve bow|curved end of a bow]]&quot;.<br /> In trigonometry, it came to denote &quot;the complement of an arc to 90°&quot;. Thus <br /> ''koṭi-jyā'' is &quot;the ''jyā'' of the complementary arc&quot;. In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes &quot;the side of a right angled triangle&quot;. Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> ''Utkrama'' means &quot;inverted&quot;, thus ''utkrama-jyā'' means &quot;inverted chord&quot;. The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning &quot;arrow&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of &quot;the ''jyā'' of three signs&quot;. The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning &quot;semi-diameter&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by &quot;Rsin&quot; and &quot;Rcos&quot; treated as single words.&lt;ref name=&quot;Datta&quot; /&gt; Others denote ''jyā'' and ''koti-jyā'' respectively by &quot;Sin&quot; and &quot;Cos&quot; (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).&lt;ref name=&quot;Glen&quot; /&gt;<br /> <br /> ===From jyā to sine===<br /> The origins of the modern term sine have been traced to the Sanskrit word ''jyā'', or more specifically to its synonym ''jīva''. This term was adopted in medieval Islamic mathematics, transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means &quot;bosom&quot;. The text's 12th-century Latin translator used the Latin equivalent for &quot;bosom&quot;, ''[[wikt:sinus|sinus]]''. When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.<br /> <br /> ==The Indian sine tables==<br /> === Surya Siddhanta sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> ===Āryabhaṭa's sine table===<br /> The stanza in Āryabhaṭiya describing the sine table is reproduced below:<br /> <br /> मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |<br /> घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||<br /> <br /> The second section of Āryabhaṭiya titled Ganitapādd<br /> a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.<br /> * &quot;When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.&quot;<br /> This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. <br /> <br /> The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''jya'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.<br /> <br /> ===Comparing different sine tables===<br /> <br /> [[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle&lt;br /&gt;(in degrees,&lt;br /&gt;arcminutes)<br /> !Āryabhaṭa's &lt;br /&gt;value of Jyā (R.sine)<br /> ![[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] value of Jyā (R.sine)<br /> !Surya Siddhanta versed sines<br /> !Modern value &lt;br /&gt;of ''Jyā'' R.sine<br /> !Madhava's sine values<br /> !Madhava's derived sine values<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |&lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.06540314<br /> |0.06540313<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |&lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> |0.13052623<br /> |0.13052619<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |&lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> |0.19509032<br /> |0.19509032<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |&lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.25881900<br /> |0.25881905<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |&lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> |0.32143947<br /> |0.32143947<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |&lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> |0.38268340<br /> |0.38268343<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |&lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> |0.44228865<br /> |0.44228869<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.49999998<br /> |0.50000000<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.55557022<br /> |0.55557023<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |&lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> |0.60876139<br /> |0.60876143<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |&lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.65934580<br /> |0.65934582<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |&lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.70710681<br /> |0.70710678<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |&lt;center&gt;2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> |0.75183985<br /> |0.75183981<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |&lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> |0.79335331<br /> |0.79335334<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |&lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> |0.83146960<br /> |0.83146961<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |&lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> |0.86602543<br /> |0.86602540<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |&lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt;<br /> |0.89687275<br /> |0.89687274<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |&lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.92387954<br /> |0.92387953<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |&lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.94693016<br /> |0.94693013<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |&lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> |0.96592581<br /> |0.96592583<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |&lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> |0.98078527<br /> |0.98078528<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |&lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99144487<br /> |0.99144486<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |&lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99785895<br /> |0.99785892<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |&lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt;<br /> |0.99999997<br /> |1.00000000<br /> |-<br /> |}<br /> <br /> ==See also==<br /> * [[Madhava's sine table]]<br /> * [[Bhaskara I's sine approximation formula]]<br /> * [[Indian standard circle|Indian Standard Circle]]<br /> * [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]]<br /> <br /> ==References==<br /> {{reflist}}<br /> <br /> [[Category:Indian Astronomy]]<br /> [[Category:Indian Mathematics]]<br /> __NOINDEX__<br /> __NONEWSECTIONLINK__</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_sine_tables&diff=129874 2021-04-16T05:45:41Z

<p>BharataPuru: /* References */</p> <hr /> <div>Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the [[Indian mathematician]] and astronomer [[Aryabhata|Āryabhaṭa]] (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.<br /> <br /> Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics. Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.<br /> <br /> == Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==<br /> '''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three trigonometric functions introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta. These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of &quot;sine&quot; and &quot;cosine&quot; have been traced back to the [[Sanskrit]] words jyā and kotijyā.&lt;ref name=&quot;Datta&quot; /&gt;<br /> [[File:Jya Figure.jpg|thumb|419x419px]]<br /> <br /> ===Definition===<br /> Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:<br /> <br /> * ''Vyāsardhā'' = Radius (R)<br /> * ''jyā'' of arc AB = MB<br /> * ''koti-jyā'' of arc AB = CM = R cos θ<br /> * ''utkrama-jyā'' of arc AB = MA<br /> <br /> If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:<br /> <br /> * ''jyā'' ( arc AB ) = ''R'' sin θ<br /> * ''koti-jyā'' ( arc AB ) = ''R'' cos θ<br /> * ''utkrama-jyā'' ( arc AB ) = R - R cos θ<br /> <br /> An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means &quot;a bow&quot;. <br /> The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means &quot;a bow-string&quot;.<br /> The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.<br /> At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs.&lt;ref name=&quot;Datta&quot; /&gt; The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant &quot;half of&quot;.<br /> <br /> The Sanskrit word ''koṭi'' has the meaning of &quot;point, cusp&quot;, and specifically &quot;the [[Recurve bow|curved end of a bow]]&quot;.<br /> In trigonometry, it came to denote &quot;the complement of an arc to 90°&quot;. Thus <br /> ''koṭi-jyā'' is &quot;the ''jyā'' of the complementary arc&quot;. In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes &quot;the side of a right angled triangle&quot;. Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> ''Utkrama'' means &quot;inverted&quot;, thus ''utkrama-jyā'' means &quot;inverted chord&quot;. The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning &quot;arrow&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of &quot;the ''jyā'' of three signs&quot;. The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning &quot;semi-diameter&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by &quot;Rsin&quot; and &quot;Rcos&quot; treated as single words.&lt;ref name=&quot;Datta&quot; /&gt; Others denote ''jyā'' and ''koti-jyā'' respectively by &quot;Sin&quot; and &quot;Cos&quot; (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).&lt;ref name=&quot;Glen&quot; /&gt;<br /> <br /> ===From jyā to sine===<br /> The origins of the modern term sine have been traced to the Sanskrit word ''jyā'', or more specifically to its synonym ''jīva''. This term was adopted in medieval Islamic mathematics, transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means &quot;bosom&quot;. The text's 12th-century Latin translator used the Latin equivalent for &quot;bosom&quot;, ''[[wikt:sinus|sinus]]''. When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.<br /> <br /> ==The Indian sine tables==<br /> === Surya Siddhanta sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> ===Āryabhaṭa's sine table===<br /> The stanza in Āryabhaṭiya describing the sine table is reproduced below:<br /> <br /> मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |<br /> घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||<br /> <br /> The second section of Āryabhaṭiya titled Ganitapādd<br /> a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.<br /> * &quot;When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.&quot;<br /> This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. <br /> <br /> The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''jya'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.<br /> <br /> ===Comparing different sine tables===<br /> <br /> [[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle&lt;br /&gt;(in degrees,&lt;br /&gt;arcminutes)<br /> !Āryabhaṭa's &lt;br /&gt;value of Jyā (R.sine)<br /> ![[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] value of Jyā (R.sine)<br /> !Surya Siddhanta versed sines<br /> !Modern value &lt;br /&gt;of ''Jyā'' R.sine<br /> !Madhava's sine values<br /> !Madhava's derived sine values<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |&lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.06540314<br /> |0.06540313<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |&lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> |0.13052623<br /> |0.13052619<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |&lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> |0.19509032<br /> |0.19509032<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |&lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.25881900<br /> |0.25881905<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |&lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> |0.32143947<br /> |0.32143947<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |&lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> |0.38268340<br /> |0.38268343<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |&lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> |0.44228865<br /> |0.44228869<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.49999998<br /> |0.50000000<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.55557022<br /> |0.55557023<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |&lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> |0.60876139<br /> |0.60876143<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |&lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.65934580<br /> |0.65934582<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |&lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.70710681<br /> |0.70710678<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |&lt;center&gt;2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> |0.75183985<br /> |0.75183981<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |&lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> |0.79335331<br /> |0.79335334<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |&lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> |0.83146960<br /> |0.83146961<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |&lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> |0.86602543<br /> |0.86602540<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |&lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt;<br /> |0.89687275<br /> |0.89687274<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |&lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.92387954<br /> |0.92387953<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |&lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.94693016<br /> |0.94693013<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |&lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> |0.96592581<br /> |0.96592583<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |&lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> |0.98078527<br /> |0.98078528<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |&lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99144487<br /> |0.99144486<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |&lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99785895<br /> |0.99785892<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |&lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt;<br /> |0.99999997<br /> |1.00000000<br /> |-<br /> |}<br /> <br /> ==See also==<br /> * [[Madhava's sine table]]<br /> * [[Bhaskara I's sine approximation formula]]<br /> * [[Indian standard circle|Indian Standard Circle]]<br /> * [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]]<br /> <br /> ==References==<br /> <br /> [[Category:Indian Astronomy]]<br /> [[Category:Indian Mathematics]]<br /> __NOINDEX__<br /> __NONEWSECTIONLINK__</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_sine_tables&diff=129873 2021-04-16T05:44:15Z

<p>BharataPuru: </p> <hr /> <div>Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the [[Indian mathematician]] and astronomer [[Aryabhata|Āryabhaṭa]] (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.<br /> <br /> Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics.&lt;ref name=&quot;mcs&quot;&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last=J J O'Connor and E F Robertson|date=June 1996 |accessdate=4 March 2010}}&lt;/ref&gt; Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.<br /> <br /> == Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==<br /> '''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three trigonometric functions introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta.&lt;ref name=&quot;Datta&quot;&gt;{{cite journal|last=B.B. Datta and A.N. Singh|date=1983|title=Hindu Trigonometry|journal=Indian Journal of History of Science|volume=18|issue=1|pages=39&amp;ndash;108|url=http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol18_1_5_BDatta.pdf|accessdate=1 March 2010}}&lt;/ref&gt; These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of &quot;sine&quot; and &quot;cosine&quot; have been traced back to the [[Sanskrit]] words jyā and kotijyā.&lt;ref name=&quot;Datta&quot; /&gt;<br /> [[File:Jya Figure.jpg|thumb|419x419px]]<br /> <br /> ===Definition===<br /> Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:<br /> <br /> * ''Vyāsardhā'' = Radius (R)<br /> * ''jyā'' of arc AB = MB<br /> * ''koti-jyā'' of arc AB = CM = R cos θ<br /> * ''utkrama-jyā'' of arc AB = MA<br /> <br /> If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:<br /> <br /> * ''jyā'' ( arc AB ) = ''R'' sin θ<br /> * ''koti-jyā'' ( arc AB ) = ''R'' cos θ<br /> * ''utkrama-jyā'' ( arc AB ) = R - R cos θ<br /> <br /> An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means &quot;a bow&quot;. <br /> The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means &quot;a bow-string&quot;.<br /> The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.&lt;ref&gt;According to lexicographers, it is a synonym also meaning &quot;bow-string&quot;, but only its <br /> geometrical meaning is attested in literature. Monier-Williams, ''A Sanskrit Dictionary'' (1899): &quot;'' jīvá'' n. (in geom. = ''jyā'') the chord of an arc; the sine of an arc ''Suryasiddhanta'' 2.57&quot;; <br /> ''jīvá'' as a generic adjective has the meaning of &quot;living, alive&quot; ([[:wikt:Appendix:Proto-Indo-European/gʷih₃wós|cognate]] with English ''[[:wikt:quick|quick]]'')&lt;/ref&gt;<br /> At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs.&lt;ref name=&quot;Datta&quot; /&gt;&lt;ref name=&quot;Glen&quot;&gt;{{cite book|last=Glen Van Brummelen|title=The mathematics of the heavens and the earth : the early history of trigonometry|publisher=[[Princeton University Press]]|date=2009|pages=95&amp;ndash;97|isbn=978-0-691-12973-0}}&lt;/ref&gt; The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant &quot;half of&quot;.<br /> <br /> The Sanskrit word ''koṭi'' has the meaning of &quot;point, cusp&quot;, and specifically &quot;the [[Recurve bow|curved end of a bow]]&quot;.<br /> In trigonometry, it came to denote &quot;the complement of an arc to 90°&quot;. Thus <br /> ''koṭi-jyā'' is &quot;the ''jyā'' of the complementary arc&quot;. In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes &quot;the side of a right angled triangle&quot;. Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> ''Utkrama'' means &quot;inverted&quot;, thus ''utkrama-jyā'' means &quot;inverted chord&quot;. The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning &quot;arrow&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of &quot;the ''jyā'' of three signs&quot;. The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning &quot;semi-diameter&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by &quot;Rsin&quot; and &quot;Rcos&quot; treated as single words.&lt;ref name=&quot;Datta&quot; /&gt; Others denote ''jyā'' and ''koti-jyā'' respectively by &quot;Sin&quot; and &quot;Cos&quot; (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).&lt;ref name=&quot;Glen&quot; /&gt;<br /> <br /> ===From jyā to sine===<br /> The origins of the modern term sine have been traced to the Sanskrit word ''jyā'',&lt;ref&gt;{{cite web|url=http://mathforum.org/library/drmath/view/54053.html|title=How the Trig Functions Got their Names|work=Ask Dr. Math|publisher=[[Drexel University]]|accessdate=2 March 2010}}&lt;/ref&gt;&lt;ref&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last= J J O'Connor and E F Robertson|date=June 1996 |accessdate=2 March 2010}}&lt;/ref&gt; or more specifically to its synonym ''jīva''. This term was adopted in medieval Islamic mathematics, transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means &quot;bosom&quot;. The text's 12th-century Latin translator used the Latin equivalent for &quot;bosom&quot;, ''[[wikt:sinus|sinus]]''.&lt;ref&gt;Various sources credit the first use of ''sinus'' to either: <br /> * [[Plato Tiburtinus]]'s 1116 translation of the ''Astronomy'' of [[Al-Battani]]<br /> * [[Gerard of Cremona]]'s c. 1150 translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]<br /> * [[Robert of Chester]]'s 1145 translation of the tables of al-Khwārizmī<br /> See Merlet, [https://link.springer.com/chapter/10.1007/1-4020-2204-2_16#page-1 ''A Note on the History of the Trigonometric Functions''] in Ceccarelli (ed.), ''International Symposium on History of Machines and Mechanisms'', Springer, 2004&lt;br&gt;See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.&lt;br&gt;See {{cite book |last=Katx |first=Victor |date=July 2008 |title=A history of mathematics |edition=3rd |location=Boston |publisher=Pearson |page=210 (sidebar) |isbn= 978-0321387004 |language=English }}&lt;/ref&gt; When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.<br /> <br /> ==The Indian sine tables==<br /> === Surya Siddhanta sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> ===Āryabhaṭa's sine table===<br /> The stanza in Āryabhaṭiya describing the sine table is reproduced below:<br /> <br /> मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |<br /> घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||<br /> <br /> The second section of Āryabhaṭiya titled Ganitapādd<br /> a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.<br /> * &quot;When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.&quot;<br /> This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. <br /> <br /> The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''jya'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.<br /> <br /> ===Comparing different sine tables===<br /> <br /> [[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle&lt;br /&gt;(in degrees,&lt;br /&gt;arcminutes)<br /> !Āryabhaṭa's &lt;br /&gt;value of Jyā (R.sine)<br /> ![[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] value of Jyā (R.sine)<br /> !Surya Siddhanta versed sines<br /> !Modern value &lt;br /&gt;of ''Jyā'' R.sine<br /> !Madhava's sine values<br /> !Madhava's derived sine values<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |&lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.06540314<br /> |0.06540313<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |&lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> |0.13052623<br /> |0.13052619<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |&lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> |0.19509032<br /> |0.19509032<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |&lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.25881900<br /> |0.25881905<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |&lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> |0.32143947<br /> |0.32143947<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |&lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> |0.38268340<br /> |0.38268343<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |&lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> |0.44228865<br /> |0.44228869<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.49999998<br /> |0.50000000<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.55557022<br /> |0.55557023<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |&lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> |0.60876139<br /> |0.60876143<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |&lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.65934580<br /> |0.65934582<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |&lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.70710681<br /> |0.70710678<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |&lt;center&gt;2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> |0.75183985<br /> |0.75183981<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |&lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> |0.79335331<br /> |0.79335334<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |&lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> |0.83146960<br /> |0.83146961<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |&lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> |0.86602543<br /> |0.86602540<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |&lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt;<br /> |0.89687275<br /> |0.89687274<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |&lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.92387954<br /> |0.92387953<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |&lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.94693016<br /> |0.94693013<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |&lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> |0.96592581<br /> |0.96592583<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |&lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> |0.98078527<br /> |0.98078528<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |&lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99144487<br /> |0.99144486<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |&lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99785895<br /> |0.99785892<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |&lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt;<br /> |0.99999997<br /> |1.00000000<br /> |-<br /> |}<br /> <br /> ==See also==<br /> * [[Madhava's sine table]]<br /> * [[Bhaskara I's sine approximation formula]]<br /> * [[Indian standard circle|Indian Standard Circle]]<br /> * [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]]<br /> <br /> ==References==<br /> <br /> {{Reference needed}}<br /> &lt;references /&gt;<br /> [[Category:Indian Astronomy]]<br /> [[Category:Indian Mathematics]]<br /> __NOINDEX__<br /> __NONEWSECTIONLINK__</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_sine_tables&diff=129872 2021-04-16T05:39:16Z

<p>BharataPuru: </p> <hr /> <div>Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the [[Indian mathematician]] and astronomer [[Aryabhata|Āryabhaṭa]] (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.<br /> <br /> Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics.&lt;ref name=&quot;mcs&quot;&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last=J J O'Connor and E F Robertson|date=June 1996 |accessdate=4 March 2010}}&lt;/ref&gt; Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.<br /> <br /> == Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==<br /> '''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three trigonometric functions introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta.&lt;ref name=&quot;Datta&quot;&gt;{{cite journal|last=B.B. Datta and A.N. Singh|date=1983|title=Hindu Trigonometry|journal=Indian Journal of History of Science|volume=18|issue=1|pages=39&amp;ndash;108|url=http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol18_1_5_BDatta.pdf|accessdate=1 March 2010}}&lt;/ref&gt; These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of &quot;sine&quot; and &quot;cosine&quot; have been traced back to the [[Sanskrit]] words jyā and kotijyā.&lt;ref name=&quot;Datta&quot; /&gt;<br /> [[File:Jya Figure.jpg|thumb|419x419px]]<br /> <br /> ===Definition===<br /> Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:<br /> <br /> * ''Vyāsardhā'' = Radius (R)<br /> * ''jyā'' of arc AB = MB<br /> * ''koti-jyā'' of arc AB = CM = R cos θ<br /> * ''utkrama-jyā'' of arc AB = MA<br /> <br /> If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:<br /> <br /> * ''jyā'' ( arc AB ) = ''R'' sin θ<br /> * ''koti-jyā'' ( arc AB ) = ''R'' cos θ<br /> * ''utkrama-jyā'' ( arc AB ) = R - R cos θ<br /> <br /> An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means &quot;a bow&quot;. <br /> The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means &quot;a bow-string&quot;.<br /> The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.&lt;ref&gt;According to lexicographers, it is a synonym also meaning &quot;bow-string&quot;, but only its <br /> geometrical meaning is attested in literature. Monier-Williams, ''A Sanskrit Dictionary'' (1899): &quot;'' jīvá'' n. (in geom. = ''jyā'') the chord of an arc; the sine of an arc ''Suryasiddhanta'' 2.57&quot;; <br /> ''jīvá'' as a generic adjective has the meaning of &quot;living, alive&quot; ([[:wikt:Appendix:Proto-Indo-European/gʷih₃wós|cognate]] with English ''[[:wikt:quick|quick]]'')&lt;/ref&gt;<br /> At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs.&lt;ref name=&quot;Datta&quot; /&gt;&lt;ref name=&quot;Glen&quot;&gt;{{cite book|last=Glen Van Brummelen|title=The mathematics of the heavens and the earth : the early history of trigonometry|publisher=[[Princeton University Press]]|date=2009|pages=95&amp;ndash;97|isbn=978-0-691-12973-0}}&lt;/ref&gt; The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant &quot;half of&quot;.<br /> <br /> The Sanskrit word ''koṭi'' has the meaning of &quot;point, cusp&quot;, and specifically &quot;the [[Recurve bow|curved end of a bow]]&quot;.<br /> In trigonometry, it came to denote &quot;the complement of an arc to 90°&quot;. Thus <br /> ''koṭi-jyā'' is &quot;the ''jyā'' of the complementary arc&quot;. In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes &quot;the side of a right angled triangle&quot;. Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> ''Utkrama'' means &quot;inverted&quot;, thus ''utkrama-jyā'' means &quot;inverted chord&quot;. The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning &quot;arrow&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of &quot;the ''jyā'' of three signs&quot;. The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning &quot;semi-diameter&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by &quot;Rsin&quot; and &quot;Rcos&quot; treated as single words.&lt;ref name=&quot;Datta&quot; /&gt; Others denote ''jyā'' and ''koti-jyā'' respectively by &quot;Sin&quot; and &quot;Cos&quot; (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).&lt;ref name=&quot;Glen&quot; /&gt;<br /> <br /> ===From jyā to sine===<br /> The origins of the modern term sine have been traced to the Sanskrit word ''jyā'',&lt;ref&gt;{{cite web|url=http://mathforum.org/library/drmath/view/54053.html|title=How the Trig Functions Got their Names|work=Ask Dr. Math|publisher=[[Drexel University]]|accessdate=2 March 2010}}&lt;/ref&gt;&lt;ref&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last= J J O'Connor and E F Robertson|date=June 1996 |accessdate=2 March 2010}}&lt;/ref&gt; or more specifically to its synonym ''jīva''. This term was adopted in medieval Islamic mathematics, transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means &quot;bosom&quot;. The text's 12th-century Latin translator used the Latin equivalent for &quot;bosom&quot;, ''[[wikt:sinus|sinus]]''.&lt;ref&gt;Various sources credit the first use of ''sinus'' to either: <br /> * [[Plato Tiburtinus]]'s 1116 translation of the ''Astronomy'' of [[Al-Battani]]<br /> * [[Gerard of Cremona]]'s c. 1150 translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]<br /> * [[Robert of Chester]]'s 1145 translation of the tables of al-Khwārizmī<br /> See Merlet, [https://link.springer.com/chapter/10.1007/1-4020-2204-2_16#page-1 ''A Note on the History of the Trigonometric Functions''] in Ceccarelli (ed.), ''International Symposium on History of Machines and Mechanisms'', Springer, 2004&lt;br&gt;See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.&lt;br&gt;See {{cite book |last=Katx |first=Victor |date=July 2008 |title=A history of mathematics |edition=3rd |location=Boston |publisher=Pearson |page=210 (sidebar) |isbn= 978-0321387004 |language=English }}&lt;/ref&gt; When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.<br /> <br /> ==The Indian sine tables==<br /> === Surya Siddhanta sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> ===Āryabhaṭa's sine table===<br /> The stanza in Āryabhaṭiya describing the sine table is reproduced below:<br /> <br /> मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |<br /> घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||<br /> <br /> The second section of Āryabhaṭiya titled Ganitapādd<br /> a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.<br /> * &quot;When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.&quot;<br /> This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. <br /> <br /> The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''jya'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.<br /> <br /> ===Comparing different sine tables===<br /> <br /> [[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle&lt;br /&gt;(in degrees,&lt;br /&gt;arcminutes)<br /> !Āryabhaṭa's &lt;br /&gt;value of Jyā (R.sine)<br /> ![[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] value of Jyā (R.sine)<br /> !Surya Siddhanta versed sines<br /> !Modern value &lt;br /&gt;of ''Jyā'' R.sine<br /> !Madhava's sine values<br /> !Madhava's derived sine values<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |&lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.06540314<br /> |0.06540313<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |&lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> |0.13052623<br /> |0.13052619<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |&lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> |0.19509032<br /> |0.19509032<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |&lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.25881900<br /> |0.25881905<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |&lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> |0.32143947<br /> |0.32143947<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |&lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> |0.38268340<br /> |0.38268343<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |&lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> |0.44228865<br /> |0.44228869<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.49999998<br /> |0.50000000<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.55557022<br /> |0.55557023<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |&lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> |0.60876139<br /> |0.60876143<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |&lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.65934580<br /> |0.65934582<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |&lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.70710681<br /> |0.70710678<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |&lt;center&gt;2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> |0.75183985<br /> |0.75183981<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |&lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> |0.79335331<br /> |0.79335334<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |&lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> |0.83146960<br /> |0.83146961<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |&lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> |0.86602543<br /> |0.86602540<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |&lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt;<br /> |0.89687275<br /> |0.89687274<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |&lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.92387954<br /> |0.92387953<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |&lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.94693016<br /> |0.94693013<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |&lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> |0.96592581<br /> |0.96592583<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |&lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> |0.98078527<br /> |0.98078528<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |&lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99144487<br /> |0.99144486<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |&lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99785895<br /> |0.99785892<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |&lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt;<br /> |0.99999997<br /> |1.00000000<br /> |-<br /> |}<br /> <br /> ==See also==<br /> * [[Madhava's sine table]]<br /> * [[Bhaskara I's sine approximation formula]]<br /> * [[Indian standard circle|Indian Standard Circle]]<br /> * [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]]<br /> <br /> ==References==<br /> <br /> {{reflist}}<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_sine_tables&diff=129871 2021-04-16T05:38:23Z

<p>BharataPuru: </p> <hr /> <div>Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the [[Indian mathematician]] and astronomer [[Aryabhata|Āryabhaṭa]] (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.&lt;ref&gt;{{cite journal|doi=10.1006/hmat.1997.2160|last=Takao Hayashi|first1=T|date=November 1997|title=Āryabhaṭa's rule and table for sine-differences|journal=Historia Mathematica |volume=24|issue=4|pages=396–406 }}&lt;/ref&gt;&lt;ref&gt;{{cite journal|doi=10.1007/BF00329978|last=B. L. van der Waerden|date=March 1988|first1=B. L.|title=Reconstruction of a Greek table of chords|journal=Archive for History of Exact Sciences|volume=38|issue=1|pages=23–38|title-link=table of chords}}&lt;/ref&gt;<br /> <br /> Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics.&lt;ref name=&quot;mcs&quot;&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last=J J O'Connor and E F Robertson|date=June 1996 |accessdate=4 March 2010}}&lt;/ref&gt; Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.<br /> <br /> == Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==<br /> '''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three trigonometric functions introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta.&lt;ref name=&quot;Datta&quot;&gt;{{cite journal|last=B.B. Datta and A.N. Singh|date=1983|title=Hindu Trigonometry|journal=Indian Journal of History of Science|volume=18|issue=1|pages=39&amp;ndash;108|url=http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol18_1_5_BDatta.pdf|accessdate=1 March 2010}}&lt;/ref&gt; These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of &quot;sine&quot; and &quot;cosine&quot; have been traced back to the [[Sanskrit]] words jyā and kotijyā.&lt;ref name=&quot;Datta&quot; /&gt;<br /> [[File:Jya Figure.jpg|thumb|419x419px]]<br /> <br /> ===Definition===<br /> Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:<br /> <br /> * ''Vyāsardhā'' = Radius (R)<br /> * ''jyā'' of arc AB = MB<br /> * ''koti-jyā'' of arc AB = CM = R cos θ<br /> * ''utkrama-jyā'' of arc AB = MA<br /> <br /> If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:<br /> <br /> * ''jyā'' ( arc AB ) = ''R'' sin θ<br /> * ''koti-jyā'' ( arc AB ) = ''R'' cos θ<br /> * ''utkrama-jyā'' ( arc AB ) = R - R cos θ<br /> <br /> An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means &quot;a bow&quot;. <br /> The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means &quot;a bow-string&quot;.<br /> The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.&lt;ref&gt;According to lexicographers, it is a synonym also meaning &quot;bow-string&quot;, but only its <br /> geometrical meaning is attested in literature. Monier-Williams, ''A Sanskrit Dictionary'' (1899): &quot;'' jīvá'' n. (in geom. = ''jyā'') the chord of an arc; the sine of an arc ''Suryasiddhanta'' 2.57&quot;; <br /> ''jīvá'' as a generic adjective has the meaning of &quot;living, alive&quot; ([[:wikt:Appendix:Proto-Indo-European/gʷih₃wós|cognate]] with English ''[[:wikt:quick|quick]]'')&lt;/ref&gt;<br /> At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs.&lt;ref name=&quot;Datta&quot; /&gt;&lt;ref name=&quot;Glen&quot;&gt;{{cite book|last=Glen Van Brummelen|title=The mathematics of the heavens and the earth : the early history of trigonometry|publisher=[[Princeton University Press]]|date=2009|pages=95&amp;ndash;97|isbn=978-0-691-12973-0}}&lt;/ref&gt; The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant &quot;half of&quot;.<br /> <br /> The Sanskrit word ''koṭi'' has the meaning of &quot;point, cusp&quot;, and specifically &quot;the [[Recurve bow|curved end of a bow]]&quot;.<br /> In trigonometry, it came to denote &quot;the complement of an arc to 90°&quot;. Thus <br /> ''koṭi-jyā'' is &quot;the ''jyā'' of the complementary arc&quot;. In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes &quot;the side of a right angled triangle&quot;. Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> ''Utkrama'' means &quot;inverted&quot;, thus ''utkrama-jyā'' means &quot;inverted chord&quot;. The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning &quot;arrow&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of &quot;the ''jyā'' of three signs&quot;. The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning &quot;semi-diameter&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by &quot;Rsin&quot; and &quot;Rcos&quot; treated as single words.&lt;ref name=&quot;Datta&quot; /&gt; Others denote ''jyā'' and ''koti-jyā'' respectively by &quot;Sin&quot; and &quot;Cos&quot; (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).&lt;ref name=&quot;Glen&quot; /&gt;<br /> <br /> ===From jyā to sine===<br /> The origins of the modern term sine have been traced to the Sanskrit word ''jyā'',&lt;ref&gt;{{cite web|url=http://mathforum.org/library/drmath/view/54053.html|title=How the Trig Functions Got their Names|work=Ask Dr. Math|publisher=[[Drexel University]]|accessdate=2 March 2010}}&lt;/ref&gt;&lt;ref&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last= J J O'Connor and E F Robertson|date=June 1996 |accessdate=2 March 2010}}&lt;/ref&gt; or more specifically to its synonym ''jīva''. This term was adopted in medieval Islamic mathematics, transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means &quot;bosom&quot;. The text's 12th-century Latin translator used the Latin equivalent for &quot;bosom&quot;, ''[[wikt:sinus|sinus]]''.&lt;ref&gt;Various sources credit the first use of ''sinus'' to either: <br /> * [[Plato Tiburtinus]]'s 1116 translation of the ''Astronomy'' of [[Al-Battani]]<br /> * [[Gerard of Cremona]]'s c. 1150 translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]<br /> * [[Robert of Chester]]'s 1145 translation of the tables of al-Khwārizmī<br /> See Merlet, [https://link.springer.com/chapter/10.1007/1-4020-2204-2_16#page-1 ''A Note on the History of the Trigonometric Functions''] in Ceccarelli (ed.), ''International Symposium on History of Machines and Mechanisms'', Springer, 2004&lt;br&gt;See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.&lt;br&gt;See {{cite book |last=Katx |first=Victor |date=July 2008 |title=A history of mathematics |edition=3rd |location=Boston |publisher=Pearson |page=210 (sidebar) |isbn= 978-0321387004 |language=English }}&lt;/ref&gt; When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.<br /> <br /> ==The Indian sine tables==<br /> === Surya Siddhanta sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> ===Āryabhaṭa's sine table===<br /> The stanza in Āryabhaṭiya describing the sine table is reproduced below:<br /> <br /> मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |<br /> घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||<br /> <br /> The second section of Āryabhaṭiya titled Ganitapādd<br /> a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.&lt;ref name=&quot;Katz&quot; /&gt;<br /> * &quot;When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.&quot;<br /> This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. <br /> <br /> The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''jya'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.<br /> <br /> ===Comparing different sine tables===<br /> <br /> [[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle&lt;br /&gt;(in degrees,&lt;br /&gt;arcminutes)<br /> !Āryabhaṭa's &lt;br /&gt;value of Jyā (R.sine)<br /> ![[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] value of Jyā (R.sine)<br /> !Surya Siddhanta versed sines<br /> !Modern value &lt;br /&gt;of ''Jyā'' R.sine<br /> !Madhava's sine values<br /> !Madhava's derived sine values<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |&lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.06540314<br /> |0.06540313<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |&lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> |0.13052623<br /> |0.13052619<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |&lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> |0.19509032<br /> |0.19509032<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |&lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.25881900<br /> |0.25881905<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |&lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> |0.32143947<br /> |0.32143947<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |&lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> |0.38268340<br /> |0.38268343<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |&lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> |0.44228865<br /> |0.44228869<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.49999998<br /> |0.50000000<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.55557022<br /> |0.55557023<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |&lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> |0.60876139<br /> |0.60876143<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |&lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.65934580<br /> |0.65934582<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |&lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.70710681<br /> |0.70710678<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |&lt;center&gt;2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> |0.75183985<br /> |0.75183981<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |&lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> |0.79335331<br /> |0.79335334<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |&lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> |0.83146960<br /> |0.83146961<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |&lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> |0.86602543<br /> |0.86602540<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |&lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt;<br /> |0.89687275<br /> |0.89687274<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |&lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.92387954<br /> |0.92387953<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |&lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.94693016<br /> |0.94693013<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |&lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> |0.96592581<br /> |0.96592583<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |&lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> |0.98078527<br /> |0.98078528<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |&lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99144487<br /> |0.99144486<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |&lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99785895<br /> |0.99785892<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |&lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt;<br /> |0.99999997<br /> |1.00000000<br /> |-<br /> |}<br /> <br /> ==See also==<br /> * [[Madhava's sine table]]<br /> * [[Bhaskara I's sine approximation formula]]<br /> * [[Indian standard circle|Indian Standard Circle]]<br /> * [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]]<br /> <br /> ==References==<br /> <br /> {{reflist}}</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_sine_tables&diff=129870 2021-04-16T05:36:41Z

<p>BharataPuru: </p> <hr /> <div>Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the [[Indian mathematician]] and astronomer [[Aryabhata|Āryabhaṭa]] (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.&lt;ref&gt;{{cite journal|doi=10.1006/hmat.1997.2160|last=Takao Hayashi|first1=T|date=November 1997|title=Āryabhaṭa's rule and table for sine-differences|journal=Historia Mathematica |volume=24|issue=4|pages=396–406 }}&lt;/ref&gt;&lt;ref&gt;{{cite journal|doi=10.1007/BF00329978|last=B. L. van der Waerden|date=March 1988|first1=B. L.|title=Reconstruction of a Greek table of chords|journal=Archive for History of Exact Sciences|volume=38|issue=1|pages=23–38|title-link=table of chords}}&lt;/ref&gt;<br /> <br /> Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics.&lt;ref name=&quot;mcs&quot;&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last=J J O'Connor and E F Robertson|date=June 1996 |accessdate=4 March 2010}}&lt;/ref&gt; Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.<br /> <br /> == Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==<br /> '''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three trigonometric functions introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta.&lt;ref name=&quot;Datta&quot;&gt;{{cite journal|last=B.B. Datta and A.N. Singh|date=1983|title=Hindu Trigonometry|journal=Indian Journal of History of Science|volume=18|issue=1|pages=39&amp;ndash;108|url=http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol18_1_5_BDatta.pdf|accessdate=1 March 2010}}&lt;/ref&gt; These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of &quot;sine&quot; and &quot;cosine&quot; have been traced back to the [[Sanskrit]] words jyā and kotijyā.&lt;ref name=&quot;Datta&quot; /&gt;<br /> [[File:Jya Figure.jpg|thumb|419x419px]]<br /> <br /> ===Definition===<br /> Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:<br /> <br /> * ''Vyāsardhā'' = Radius (R)<br /> * ''jyā'' of arc AB = MB<br /> * ''koti-jyā'' of arc AB = CM = R cos θ<br /> * ''utkrama-jyā'' of arc AB = MA<br /> <br /> If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:<br /> <br /> * ''jyā'' ( arc AB ) = ''R'' sin θ<br /> * ''koti-jyā'' ( arc AB ) = ''R'' cos θ<br /> * ''utkrama-jyā'' ( arc AB ) = R - R cos θ<br /> <br /> An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means &quot;a bow&quot;. <br /> The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means &quot;a bow-string&quot;.<br /> The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.&lt;ref&gt;According to lexicographers, it is a synonym also meaning &quot;bow-string&quot;, but only its <br /> geometrical meaning is attested in literature. Monier-Williams, ''A Sanskrit Dictionary'' (1899): &quot;'' jīvá'' n. (in geom. = ''jyā'') the chord of an arc; the sine of an arc ''Suryasiddhanta'' 2.57&quot;; <br /> ''jīvá'' as a generic adjective has the meaning of &quot;living, alive&quot; ([[:wikt:Appendix:Proto-Indo-European/gʷih₃wós|cognate]] with English ''[[:wikt:quick|quick]]'')&lt;/ref&gt;<br /> At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs.&lt;ref name=&quot;Datta&quot; /&gt;&lt;ref name=&quot;Glen&quot;&gt;{{cite book|last=Glen Van Brummelen|title=The mathematics of the heavens and the earth : the early history of trigonometry|publisher=[[Princeton University Press]]|date=2009|pages=95&amp;ndash;97|isbn=978-0-691-12973-0}}&lt;/ref&gt; The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant &quot;half of&quot;.<br /> <br /> The Sanskrit word ''koṭi'' has the meaning of &quot;point, cusp&quot;, and specifically &quot;the [[Recurve bow|curved end of a bow]]&quot;.<br /> In trigonometry, it came to denote &quot;the complement of an arc to 90°&quot;. Thus <br /> ''koṭi-jyā'' is &quot;the ''jyā'' of the complementary arc&quot;. In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes &quot;the side of a right angled triangle&quot;. Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> ''Utkrama'' means &quot;inverted&quot;, thus ''utkrama-jyā'' means &quot;inverted chord&quot;. The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning &quot;arrow&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of &quot;the ''jyā'' of three signs&quot;. The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning &quot;semi-diameter&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by &quot;Rsin&quot; and &quot;Rcos&quot; treated as single words.&lt;ref name=&quot;Datta&quot; /&gt; Others denote ''jyā'' and ''koti-jyā'' respectively by &quot;Sin&quot; and &quot;Cos&quot; (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).&lt;ref name=&quot;Glen&quot; /&gt;<br /> <br /> ===From jyā to sine===<br /> The origins of the modern term sine have been traced to the Sanskrit word ''jyā'',&lt;ref&gt;{{cite web|url=http://mathforum.org/library/drmath/view/54053.html|title=How the Trig Functions Got their Names|work=Ask Dr. Math|publisher=[[Drexel University]]|accessdate=2 March 2010}}&lt;/ref&gt;&lt;ref&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last= J J O'Connor and E F Robertson|date=June 1996 |accessdate=2 March 2010}}&lt;/ref&gt; or more specifically to its synonym ''jīva''. This term was adopted in medieval Islamic mathematics, transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means &quot;bosom&quot;. The text's 12th-century Latin translator used the Latin equivalent for &quot;bosom&quot;, ''[[wikt:sinus|sinus]]''.&lt;ref&gt;Various sources credit the first use of ''sinus'' to either: <br /> * [[Plato Tiburtinus]]'s 1116 translation of the ''Astronomy'' of [[Al-Battani]]<br /> * [[Gerard of Cremona]]'s c. 1150 translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]<br /> * [[Robert of Chester]]'s 1145 translation of the tables of al-Khwārizmī<br /> See Merlet, [https://link.springer.com/chapter/10.1007/1-4020-2204-2_16#page-1 ''A Note on the History of the Trigonometric Functions''] in Ceccarelli (ed.), ''International Symposium on History of Machines and Mechanisms'', Springer, 2004&lt;br&gt;See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.&lt;br&gt;See {{cite book |last=Katx |first=Victor |date=July 2008 |title=A history of mathematics |edition=3rd |location=Boston |publisher=Pearson |page=210 (sidebar) |isbn= 978-0321387004 |language=English }}&lt;/ref&gt; When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.<br /> <br /> ==The Indian sine tables==<br /> === Surya Siddhanta sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> ===Āryabhaṭa's sine table===<br /> The stanza in Āryabhaṭiya describing the sine table is reproduced below:<br /> <br /> मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |<br /> घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||<br /> <br /> The second section of Āryabhaṭiya titled Ganitapādd<br /> a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.&lt;ref name=&quot;Katz&quot; /&gt;<br /> * &quot;When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.&quot;<br /> This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. <br /> <br /> The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''jya'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.<br /> <br /> ===Comparing different sine tables===<br /> <br /> [[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle&lt;br /&gt;(in degrees,&lt;br /&gt;arcminutes)<br /> !Āryabhaṭa's &lt;br /&gt;value of Jyā (R.sine)<br /> ![[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] value of Jyā (R.sine)<br /> !Surya Siddhanta versed sines<br /> !Modern value &lt;br /&gt;of ''Jyā'' R.sine<br /> !Madhava's sine values<br /> !Madhava's derived sine values<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |&lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.06540314<br /> |0.06540313<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |&lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> |0.13052623<br /> |0.13052619<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |&lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> |0.19509032<br /> |0.19509032<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |&lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.25881900<br /> |0.25881905<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |&lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> |0.32143947<br /> |0.32143947<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |&lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> |0.38268340<br /> |0.38268343<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |&lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> |0.44228865<br /> |0.44228869<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.49999998<br /> |0.50000000<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.55557022<br /> |0.55557023<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |&lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> |0.60876139<br /> |0.60876143<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |&lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.65934580<br /> |0.65934582<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |&lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.70710681<br /> |0.70710678<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |&lt;center&gt;2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> |0.75183985<br /> |0.75183981<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |&lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> |0.79335331<br /> |0.79335334<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |&lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> |0.83146960<br /> |0.83146961<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |&lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> |0.86602543<br /> |0.86602540<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |&lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt;<br /> |0.89687275<br /> |0.89687274<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |&lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.92387954<br /> |0.92387953<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |&lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.94693016<br /> |0.94693013<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |&lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> |0.96592581<br /> |0.96592583<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |&lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> |0.98078527<br /> |0.98078528<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |&lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99144487<br /> |0.99144486<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |&lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99785895<br /> |0.99785892<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |&lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt;<br /> |0.99999997<br /> |1.00000000<br /> |-<br /> |}<br /> <br /> ==See also==<br /> * [[Madhava's sine table]]<br /> * [[Bhaskara I's sine approximation formula]]<br /> * [[Indian standard circle|Indian Standard Circle]]<br /> * [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]]</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_sine_tables&diff=129869 2021-04-16T05:35:03Z

<p>BharataPuru: /* References */</p> <hr /> <div>Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the [[Indian mathematician]] and astronomer [[Aryabhata|Āryabhaṭa]] (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.&lt;ref&gt;{{cite journal|doi=10.1006/hmat.1997.2160|last=Takao Hayashi|first1=T|date=November 1997|title=Āryabhaṭa's rule and table for sine-differences|journal=Historia Mathematica |volume=24|issue=4|pages=396–406 }}&lt;/ref&gt;&lt;ref&gt;{{cite journal|doi=10.1007/BF00329978|last=B. L. van der Waerden|date=March 1988|first1=B. L.|title=Reconstruction of a Greek table of chords|journal=Archive for History of Exact Sciences|volume=38|issue=1|pages=23–38|title-link=table of chords}}&lt;/ref&gt;<br /> <br /> Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics.&lt;ref name=&quot;mcs&quot;&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last=J J O'Connor and E F Robertson|date=June 1996 |accessdate=4 March 2010}}&lt;/ref&gt; Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.<br /> <br /> == Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==<br /> '''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three trigonometric functions introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta.&lt;ref name=&quot;Datta&quot;&gt;{{cite journal|last=B.B. Datta and A.N. Singh|date=1983|title=Hindu Trigonometry|journal=Indian Journal of History of Science|volume=18|issue=1|pages=39&amp;ndash;108|url=http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol18_1_5_BDatta.pdf|accessdate=1 March 2010}}&lt;/ref&gt; These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of &quot;sine&quot; and &quot;cosine&quot; have been traced back to the [[Sanskrit]] words jyā and kotijyā.&lt;ref name=&quot;Datta&quot; /&gt;<br /> [[File:Jya Figure.jpg|thumb|419x419px]]<br /> <br /> ===Definition===<br /> Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:<br /> <br /> * ''Vyāsardhā'' = Radius (R)<br /> * ''jyā'' of arc AB = MB<br /> * ''koti-jyā'' of arc AB = CM = R cos θ<br /> * ''utkrama-jyā'' of arc AB = MA<br /> <br /> If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:<br /> <br /> * ''jyā'' ( arc AB ) = ''R'' sin θ<br /> * ''koti-jyā'' ( arc AB ) = ''R'' cos θ<br /> * ''utkrama-jyā'' ( arc AB ) = R - R cos θ<br /> <br /> An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means &quot;a bow&quot;. <br /> The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means &quot;a bow-string&quot;.<br /> The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.&lt;ref&gt;According to lexicographers, it is a synonym also meaning &quot;bow-string&quot;, but only its <br /> geometrical meaning is attested in literature. Monier-Williams, ''A Sanskrit Dictionary'' (1899): &quot;'' jīvá'' n. (in geom. = ''jyā'') the chord of an arc; the sine of an arc ''Suryasiddhanta'' 2.57&quot;; <br /> ''jīvá'' as a generic adjective has the meaning of &quot;living, alive&quot; ([[:wikt:Appendix:Proto-Indo-European/gʷih₃wós|cognate]] with English ''[[:wikt:quick|quick]]'')&lt;/ref&gt;<br /> At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs.&lt;ref name=&quot;Datta&quot; /&gt;&lt;ref name=&quot;Glen&quot;&gt;{{cite book|last=Glen Van Brummelen|title=The mathematics of the heavens and the earth : the early history of trigonometry|publisher=[[Princeton University Press]]|date=2009|pages=95&amp;ndash;97|isbn=978-0-691-12973-0}}&lt;/ref&gt; The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant &quot;half of&quot;.<br /> <br /> The Sanskrit word ''koṭi'' has the meaning of &quot;point, cusp&quot;, and specifically &quot;the [[Recurve bow|curved end of a bow]]&quot;.<br /> In trigonometry, it came to denote &quot;the complement of an arc to 90°&quot;. Thus <br /> ''koṭi-jyā'' is &quot;the ''jyā'' of the complementary arc&quot;. In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes &quot;the side of a right angled triangle&quot;. Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> ''Utkrama'' means &quot;inverted&quot;, thus ''utkrama-jyā'' means &quot;inverted chord&quot;. The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning &quot;arrow&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of &quot;the ''jyā'' of three signs&quot;. The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning &quot;semi-diameter&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by &quot;Rsin&quot; and &quot;Rcos&quot; treated as single words.&lt;ref name=&quot;Datta&quot; /&gt; Others denote ''jyā'' and ''koti-jyā'' respectively by &quot;Sin&quot; and &quot;Cos&quot; (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).&lt;ref name=&quot;Glen&quot; /&gt;<br /> <br /> ===From jyā to sine===<br /> The origins of the modern term sine have been traced to the Sanskrit word ''jyā'',&lt;ref&gt;{{cite web|url=http://mathforum.org/library/drmath/view/54053.html|title=How the Trig Functions Got their Names|work=Ask Dr. Math|publisher=[[Drexel University]]|accessdate=2 March 2010}}&lt;/ref&gt;&lt;ref&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last= J J O'Connor and E F Robertson|date=June 1996 |accessdate=2 March 2010}}&lt;/ref&gt; or more specifically to its synonym ''jīva''. This term was adopted in medieval Islamic mathematics, transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means &quot;bosom&quot;. The text's 12th-century Latin translator used the Latin equivalent for &quot;bosom&quot;, ''[[wikt:sinus|sinus]]''.&lt;ref&gt;Various sources credit the first use of ''sinus'' to either: <br /> * [[Plato Tiburtinus]]'s 1116 translation of the ''Astronomy'' of [[Al-Battani]]<br /> * [[Gerard of Cremona]]'s c. 1150 translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]<br /> * [[Robert of Chester]]'s 1145 translation of the tables of al-Khwārizmī<br /> See Merlet, [https://link.springer.com/chapter/10.1007/1-4020-2204-2_16#page-1 ''A Note on the History of the Trigonometric Functions''] in Ceccarelli (ed.), ''International Symposium on History of Machines and Mechanisms'', Springer, 2004&lt;br&gt;See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.&lt;br&gt;See {{cite book |last=Katx |first=Victor |date=July 2008 |title=A history of mathematics |edition=3rd |location=Boston |publisher=Pearson |page=210 (sidebar) |isbn= 978-0321387004 |language=English }}&lt;/ref&gt; When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.<br /> <br /> ==The Indian sine tables==<br /> === Surya Siddhanta sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> ===Āryabhaṭa's sine table===<br /> The stanza in Āryabhaṭiya describing the sine table is reproduced below:<br /> <br /> मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |<br /> घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||<br /> <br /> The second section of Āryabhaṭiya titled Ganitapādd<br /> a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.&lt;ref name=&quot;Katz&quot; /&gt;<br /> * &quot;When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.&quot;<br /> This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. <br /> <br /> The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''jya'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.<br /> <br /> ===Comparing different sine tables===<br /> <br /> [[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle&lt;br /&gt;(in degrees,&lt;br /&gt;arcminutes)<br /> !Āryabhaṭa's &lt;br /&gt;value of Jyā (R.sine)<br /> ![[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] value of Jyā (R.sine)<br /> !Surya Siddhanta versed sines<br /> !Modern value &lt;br /&gt;of ''Jyā'' R.sine<br /> !Madhava's sine values<br /> !Madhava's derived sine values<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |&lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.06540314<br /> |0.06540313<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |&lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> |0.13052623<br /> |0.13052619<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |&lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> |0.19509032<br /> |0.19509032<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |&lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.25881900<br /> |0.25881905<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |&lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> |0.32143947<br /> |0.32143947<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |&lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> |0.38268340<br /> |0.38268343<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |&lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> |0.44228865<br /> |0.44228869<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.49999998<br /> |0.50000000<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.55557022<br /> |0.55557023<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |&lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> |0.60876139<br /> |0.60876143<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |&lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.65934580<br /> |0.65934582<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |&lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.70710681<br /> |0.70710678<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |&lt;center&gt;2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> |0.75183985<br /> |0.75183981<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |&lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> |0.79335331<br /> |0.79335334<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |&lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> |0.83146960<br /> |0.83146961<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |&lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> |0.86602543<br /> |0.86602540<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |&lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt;<br /> |0.89687275<br /> |0.89687274<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |&lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.92387954<br /> |0.92387953<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |&lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.94693016<br /> |0.94693013<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |&lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> |0.96592581<br /> |0.96592583<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |&lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> |0.98078527<br /> |0.98078528<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |&lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99144487<br /> |0.99144486<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |&lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99785895<br /> |0.99785892<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |&lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt;<br /> |0.99999997<br /> |1.00000000<br /> |-<br /> |}<br /> <br /> ==See also==<br /> * [[Madhava's sine table]]<br /> * [[Bhaskara I's sine approximation formula]]<br /> * [[Indian standard circle|Indian Standard Circle]]<br /> * [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]]<br /> <br /> ==References==</div>

BharataPuru http://dharmawiki.org/index.php?title=Nakshatra_system&diff=129868 2021-04-16T04:12:21Z

<p>BharataPuru: </p> <hr /> <div>In [[Hindu astrology|Ancient Indian Astronogy]] or Hindu Astronomy, there are 27 '''''[[nakshatra]]s''''' (&quot;lunar mansions&quot;, Sanskrit: नक्षत्र, IAST: ''Nakṣatra''), or sectors along the ecliptic. A list of them is first found in the ''[[Vedanga Jyotisha]]'', a text dated to the final centuries BCE. The ''Nakṣatra'' system predates the influence of [[Hellenistic astronomy]] on Vedic tradition, which became prevalent from about the 2nd century CE. The following list gives the corresponding regions of sky.&lt;ref&gt;{{cite book|first=Arthur Llewellyn|last=Basham|authorlink=Arthur Llewellyn Basham|title=The Wonder That Was India|chapter=Appendix II: Astronomy|p=490|year=1954|publisher=[[Sidgwick and Jackson]]|title-link=The Wonder That Was India}}&lt;/ref&gt;<br /> ==Indian Nakshatras==<br /> {| class=&quot;wikitable&quot; align=&quot;center&quot; cellspacing=&quot;2&quot; cellpadding=&quot;&quot;<br /> |- bgcolor=&quot;#cccccc&quot;<br /> !No.!! Name !!Associated stars!!Description<br /> |-<br /> | 1<br /> ||'''[[Ashvini]]'''&lt;br /&gt;&quot;physician to the Gods&quot;<br /> ||[[Beta Arietis|β]] and [[Gamma Arietis|γ]] [[Aries (constellation)|Arietis]]<br /> ||<br /> *''Lord'': Ketu (South lunar node)<br /> *''Symbol'' : Horse's head<br /> *''Deity'' : [[Ashvins]], the horse-headed twins who are physicians to the gods<br /> *''Indian zodiac'': 0° - 13°20' Mesha<br /> *''west zodiac'' 23°46 Aries - 7°06' Taurus<br /> |-<br /> | 2<br /> ||'''[[Bharani]]'''&lt;br /&gt;&quot;the bearer&quot;<br /> ||[[35 Arietis|35]], [[39 Arietis|39]], and [[41 Arietis|41]] [[Aries (constellation)|Arietis]]<br /> ||<br /> *''Lord'': Shukra (Venus)<br /> *''Symbol'': Yoni, the female organ of reproduction<br /> *''Deity'': [[Yama]], [[god of death]] or [[Dharma]]<br /> *''Indian zodiac'': 13° 20' - 26°40' Mesha<br /> *''Western zodiac'' 7°06' - 20°26' Taurus<br /> |-<br /> | 3<br /> ||'''[[Krittika]]'''&lt;br /&gt;an old name of the Pleiades; personified as the nurses of ''Kārttikeya'', a son of Shiva.<br /> ||[[Pleiades]]<br /> ||<br /> *''Lord'': [[Surya]] (Sun)<br /> *''Symbol'': [[Knife]] or [[spear]]<br /> *''Deity'' : [[Agni]], god of fire<br /> *''Indian zodiac'': 26°40' [[Mesha]] - 10° Vrishabha<br /> *''Western zodiac'' 20°26' Taurus - 3°46’ Gemini<br /> |-<br /> | 4<br /> ||'''[[Rohini (nakshatra)|Rohini]]'''&lt;br&gt;&quot;the red one&quot;, a name of [[Aldebaran]]. Also known as ''brāhmī''<br /> ||[[Aldebaran]]<br /> ||<br /> *''Lord'': [[Chandra]] ([[Moon]])<br /> *''Symbol'': Cart or [[chariot]], [[temple]], [[banyan tree]]<br /> *''Deity'' : [[Brahma]] or [[Prajapati]], [[Creator deity|the Creator]]<br /> *''Indian zodiac'': 10° - 23°20' Vrishabha<br /> *''Western zodiac'' 3°46 - 17°06' Gemini<br /> |-<br /> | 5<br /> ||'''[[Mrigashīrsha]]'''&lt;br /&gt;&quot;the deer's head&quot;. Also known as ''āgrahāyaṇī''<br /> ||[[Lambda Orionis|λ]], [[Phi Orionis|φ]] Orionis<br /> ||<br /> *''Lord'': Mangala (Mars)<br /> *''Symbol'': Deer's head<br /> *''Deity'': [[Soma (deity)|Soma]], [[Chandra]], the Moon god<br /> *''Indian zodiac'': 23° 20' Vrishabha - 6° 40' Mithuna<br /> *''Western zodiac'': 17°06' Gemini - 0°26' Cancer<br /> |-<br /> | 6<br /> ||'''[[Ardra (nakshatra)|Ardra]]'''&lt;br /&gt;&quot;the storm god&quot;<br /> ||[[Betelgeuse]]<br /> ||<br /> *''Lord'': Rahu (North lunar node)<br /> *''Symbol'': Teardrop, diamond, a human head<br /> *''Deity'' : [[Rudra]], the storm god<br /> *''Indian zodiac'': 6° 40' - 20° Mithuna<br /> *''Western zodiac'': 0°26' - 13°46 Cancer<br /> |-<br /> | 7<br /> ||'''[[Punarvasu]]''' (dual)&lt;br /&gt;&quot;the two restorers of goods&quot;, also known as ''yamakau'' &quot;the two chariots&quot;<br /> ||[[Castor (star)|Castor]] and [[Pollux (star)|Pollux]]<br /> ||<br /> *''Lord'': Guru (Jupiter)<br /> *''Symbol'' : Bow and quiver<br /> *''Deity'' : [[Aditi]], mother of the gods<br /> *''Indian zodiac'': 20° Mithuna - 3°20' Karka<br /> *''Western zodiac'' 13°46 - 27°06' Cancer<br /> |-<br /> | 8<br /> ||'''[[Pushya]]'''&lt;br /&gt;&quot;the nourisher&quot;, also known as ''sidhya'' or ''tiṣya''<br /> ||[[Gamma Cancri|γ]], [[Delta Cancri|δ]] and [[Theta Cancri|θ]] [[Cancer (constellation)|Cancri]]<br /> ||<br /> *''Lord'': [[Shani]] ([[Saturn]])<br /> *''Symbol'' : Cow's udder, [[Lotus Flower|lotus]], arrow and circle<br /> *''Deity'' : [[Bṛhaspati]], priest of the gods<br /> *''Indian zodiac'': 3°20' -16°40' [[Karka]]<br /> *''Western zodiac'' 27°06' Cancer - 10°26' [[Leo (constellation)|Leo]]<br /> |-<br /> | 9<br /> ||'''[[Āshleshā]]'''&lt;br /&gt;&quot;the embrace&quot;<br /> ||δ, ε, η, ρ, and [[Sigma Hydrae|σ]] [[Hydra (constellation)|Hydrae]]<br /> ||<br /> *''Lord'': Budh (Mercury)<br /> *''Symbol'': Serpent<br /> *''Deity'' : Sarpas or [[Nāga|Nagas]], deified snakes<br /> *''Indian zodiac'': 16°40' - 30° Karka<br /> *''Western zodiac'' 10°26' - 23°46 Leo<br /> |-<br /> | 10<br /> ||'''[[Maghā]]'''&lt;br /&gt;&quot;the bountiful&quot;<br /> ||[[Regulus]]<br /> ||<br /> *''Lord'': Ketu (south lunar node)<br /> *''Symbol'' : Royal Throne<br /> *''Deity'' : [[Pitrs]], 'The Fathers', family ancestors<br /> *''Indian zodiac'': 0° - 13°20' Simha<br /> *''Western zodiac'' 23°46 Leo - 7°06' Virgo<br /> |-<br /> | 11<br /> ||'''[[Pūrva Phalgunī]]'''&lt;br&gt;&quot;first reddish one&quot;<br /> ||[[Delta Leonis|δ]] and [[Theta Leonis|θ]] [[Leo (constellation)|Leonis]]<br /> ||<br /> *''Lord'': Shukra (Venus)<br /> *''Symbol'' : Front legs of bed, hammock, fig tree<br /> *''Deity'' : [[Bhaga]], god of marital bliss and prosperity<br /> *''Indian zodiac'': 13°20' - 26°40' Simha<br /> *''Western zodiac'' 7°06' - 20°26' Virgo<br /> |-<br /> | 12<br /> ||'''[[Uttara Phalgunī]]'''&lt;br /&gt;&quot;second reddish one&quot;<br /> ||[[Denebola]]<br /> ||<br /> *''Lord'': Surya (Sun)<br /> *''Symbol'': Four legs of bed, hammock<br /> *''Deity'' : [[Aryaman]], god of patronage and favours<br /> *''Indian zodiac'': 26°40' Simha- 10° Kanya<br /> *''Western zodiac'' 20°26' Virgo - 3°46 Libra<br /> |-<br /> | 13<br /> ||'''[[Nakshatra|Hasta]]'''&lt;br /&gt;&quot;the hand&quot;<br /> ||[[Alpha Corvi|α]], [[Beta Corvi|β]], [[Gamma Corvi|γ]], [[Delta Corvi|δ]] and [[Epsilon Corvi|ε]] [[Corvus (constellation)|Corvi]]<br /> ||<br /> *''Lord'': Chandra (Moon)<br /> *''Symbol'': Hand or fist<br /> *''Deity'' : [[Surya|Saviti]] or [[Surya]], the Sun god<br /> *''Indian zodiac'': 10° - 23°20' Kanya<br /> *''Western zodiac'' 3°46 - 17°06' Libra<br /> |-<br /> | 14<br /> ||'''Chitra'''&lt;br /&gt;&quot;the bright one&quot;, a name of Spica<br /> ||[[Spica]]<br /> ||<br /> *''Lord'': Mangala (Mars)<br /> *''Symbol'': Bright jewel or pearl<br /> *''Deity'' : [[Tvastar]] or [[Vishvakarman]], the celestial architect<br /> *''Indian zodiac'': 23°20' Kanya - 6°40' Tula<br /> *''Western zodiac'': 17°06' Libra - 0°26' Scorpio<br /> |-<br /> | 15<br /> ||'''[[Swati the Star|Swāti]]'''&lt;br /&gt;&quot;Su-Ati (sanskrit) Very good&quot; name of Arcturus<br /> ||[[Arcturus]]<br /> ||<br /> *''Lord'': Rahu (north lunar node)<br /> *''Symbol'': Shoot of plant, coral<br /> *''Deity'' : [[Vayu]], the Wind god<br /> *''Indian zodiac'': 6°40' - 20° Tula<br /> *''Western zodiac'' 0°26' - 13°46 Scorpio<br /> |-<br /> | 16<br /> ||'''Vishakha'''&lt;br /&gt;&quot;forked, having branches&quot;; also known as ''rādhā'' &quot;the gift&quot;<br /> || [[Alpha Librae|α]], [[Beta Librae|β]], [[Gamma Librae|γ]] and ι [[Libra (constellation)|Librae]]<br /> ||<br /> *''Lord'': Guru (Jupiter)<br /> *''Symbol'' : Triumphal arch, potter's wheel<br /> *''Deity'' : [[Indra]], chief of the gods; [[Agni]], god of Fire<br /> *''Indian zodiac'': 20° Tula - 3°20' Vrishchika<br /> *''Western zodiac'' 13°46 - 27°06' Scorpio<br /> |-<br /> | 17<br /> ||'''[[Anuradha (nakshatra)|Anuradha]]'''&lt;br /&gt; &quot;following ''rādhā''&quot;<br /> || [[Beta Scorpii|β]], [[Delta Scorpii|δ]] and [[Pi Scorpii|π]] [[Scorpius (constellation)|Scorpionis]]<br /> ||<br /> *''Lord'': Shani (Saturn)<br /> *''Symbol'' : Triumphal archway, lotus<br /> *''Deity'' : [[Mitra (Vedic)|Mitra]], one of Adityas of friendship and partnership<br /> *''Indian zodiac'': 3°20' - 16°40' Vrishchika<br /> *''Western zodiac'' 27°06' Scorpio - 10°26' Sagittarius<br /> |-<br /> | 18<br /> ||'''[[Jyeshtha (nakshatra)|Jyeshtha]]'''&lt;br /&gt;&quot;the eldest, most excellent&quot;<br /> || [[Alpha Scorpii|α]], [[Sigma Scorpii|σ]], and [[Tau Scorpii|τ]] [[Scorpius (constellation)|Scorpionis]]<br /> ||<br /> *''Lord'': Budh (Mercury)<br /> *''Symbol'' : circular amulet, umbrella, earring<br /> *''Deity'' : [[Indra]], chief of the gods<br /> *''Indian zodiac'': 16°40' - 30° Vrishchika<br /> *''Western zodiac'' 10°26' - 23°46 Sagittarius<br /> |-<br /> | 19<br /> ||'''[[Mula (nakshatra)|Mula]]'''&lt;br /&gt;&quot;the root&quot;<br /> || [[Epsilon Scorpii|ε]], ζ, [[Eta Scorpii|η]], [[Theta Scorpii|θ]], ι, [[Kappa Scorpii|κ]], [[Lambda Scorpii|λ]], [[Mu Scorpii|μ]] and [[Nu Scorpii|ν]] [[Scorpius (constellation)|Scorpionis]]<br /> ||<br /> *''Lord'': Ketu (south lunar node)<br /> *''Symbol'' : Bunch of roots tied together, elephant goad<br /> *''Deity'' : [[Nirrti]], goddess of dissolution and destruction<br /> *''Indian zodiac'': 0° - 13°20' Dhanus<br /> *''Western zodiac'' 23°46 Sagittarius - 7°06' Capricorn<br /> |-<br /> | 20<br /> ||'''[[Purva Ashadha]]'''&lt;br /&gt;&quot;first of the ''āṣāḍhā''&quot;, ''āṣāḍhā'' &quot;the invincible one&quot; being the name of a constellation<br /> ||[[Delta Sagittarii|δ]] and [[Epsilon Sagittarii|ε]] [[Sagittarius (constellation)|Sagittarii]]<br /> ||<br /> *''Lord'': Shukra (Venus)<br /> *''Symbol'': Elephant tusk, fan, winnowing basket<br /> *''Deity'' : [[Ap (water)|Apah]], god of Water<br /> *''Indian zodiac'': 13°20' - 26°40' Dhanus<br /> *''Western zodiac'' 7°06' - 20°26' Capricorn<br /> |-<br /> | 21<br /> ||'''[[Uttara Ashadha]]'''&lt;br /&gt;&quot;second of the ''āṣāḍhā''&quot;<br /> ||[[Zeta Sagittarii|ζ]] and [[Sigma Sagittarii|σ]] [[Sagittarius (constellation)|Sagittarii]]<br /> ||<br /> *''Lord'': Surya (Sun)<br /> *''Symbol'' : Elephant tusk, small bed<br /> *''Deity'' : [[Visvedevas]], universal gods<br /> *''Indian zodiac'': 26°40' Dhanus - 10° Makara<br /> *''Western zodiac'' 20°26' Capricorn - 3°46 Aquarius<br /> |-<br /> | 22<br /> ||'''[[Shravana|Sravana]]'''&lt;br /&gt;<br /> ||[[Alpha Aquilae|α]], [[Beta Aquilae|β]] and [[Gamma Aquilae|γ]] [[Aquila (constellation)|Aquilae]]<br /> ||<br /> *''Lord'': Chandra (Moon)<br /> *''Symbol'' : Ear or Three Footprints<br /> *''Deity'' : [[Vishnu]], preserver of universe<br /> *''Indian zodiac'': 10° - 23°20' Makara<br /> *''Western zodiac'' 3°46 - 17°06' Aquarius<br /> |-<br /> | 23<br /> ||'''[[Dhanishta]]''' &lt;br /&gt;&quot;most famous&quot;, also ''Shravishthā'' &quot;swiftest&quot;<br /> || [[Alpha Delphini|α]] to [[Delta Delphini|δ Delphini]]<br /> ||<br /> *''Lord'': Mangala (Mars)<br /> *''Symbol'' : Drum or flute<br /> *''Deity'' : Eight [[vasu]]s, deities of earthly abundance<br /> *''Indian zodiac'': 23°20' Makara - 6°40' Kumbha<br /> *''Western zodiac'' 17°06' Aquarius - 0°26' Pisces<br /> |-<br /> | 24<br /> ||'''[[Shatabhisha]]&lt;br /&gt;&quot;'''requiring a hundred physicians'''&quot;'''<br /> || [[Gamma Aquarii|γ Aquarii]]<br /> ||<br /> *''Lord'': Rahu (north lunar node)<br /> *''Symbol'' : Empty circle, 1,000 flowers or stars<br /> *''Deity'' : [[Varuna]], god of cosmic waters, sky and earth<br /> *''Indian zodiac'': 6°40' - 20° Kumbha ; ''Western zodiac'' 0°26' - 13°46 Pisces<br /> |-<br /> | 25<br /> ||'''[[Purva Bhadrapada]]'''&lt;br /&gt;&quot;the first of the blessed feet&quot;<br /> || [[Alpha Pegasi|α]] and [[Beta Pegasi|β]] [[Pegasus (constellation)|Pegasi]]<br /> ||<br /> *''Lord'': Guru (Jupiter)<br /> *''Symbol'' : Swords or two front legs of funeral cot, man with two faces<br /> *''Deity'' : '''Ajaikapada''', an ancient fire dragon<br /> *''Indian zodiac'': 20° Kumbha - 3°20' Meena ; ''Western zodiac'' 13°46 - 27°06' Pisces<br /> |-<br /> | 26<br /> ||'''[[Uttara Bhādrapadā]]'''&lt;br /&gt;&quot;the second of the blessed feet&quot;<br /> || [[Gamma Pegasi|γ]] [[Pegasus (constellation)|Pegasi]] and [[Alpha Andromedae|α]] [[Andromeda (constellation)|Andromedae]]<br /> ||<br /> *''Lord'': Shani (Saturn)<br /> *''Symbol'' : Twins, back legs of funeral cot, snake in the water<br /> *''Deity'' : '''Ahirbudhnya''', serpent or dragon of the deep<br /> *''Indian zodiac'': 3°20' - 16°40' Meena ; ''Western zodiac'' 27°06' Pisces - 10°26' Aries<br /> |-<br /> | 27<br /> ||'''[[Revati (nakshatra)|Revati]]'''&lt;br /&gt;&quot;prosperous&quot;<br /> || ζ [[Pisces (constellation)|Piscium]]<br /> ||<br /> *''Lord'': Budh (Mercury)<br /> *''Symbol'' : Fish or a pair of fish, drum<br /> *''Deity'' : [[Pushan]], nourisher, the protective deity<br /> *''Indian zodiac'': 16°40' - 30° Meena<br /> *''Western zodiac'' 10°26' - 23°46 Aries<br /> |-<br /> | 28<br /> ||'''[[Abhijit (nakshatra)|Abhijit]]'''&lt;br /&gt; &quot;victorious&quot;&lt;ref&gt;Its longitude starts from 06° 40' to 10° 53' 40'' in sidereal Capricorn i.e. from the last quarter of Uttra Ashadha to first 1/15 th part of Shravana.{{Citation needed|date=October 2011}}''<br /> Its span is 4° 13' 40''.{{Citation needed|date=October 2011}} The span of 27 mean daily lunar motions totals 355.76167 degrees, and together these total 359.98945 degrees. Considered an &quot;intercalary&quot; lunar mansion.{{Citation needed|date=October 2011}}''&lt;/ref&gt;<br /> || [[Alpha Lyrae|α]], [[Epsilon Lyrae|ε]] and ζ [[Lyra (constellation)|Lyrae]] - [[Vega]]<br /> || ''Lord'': [[Brahma]] (creator)<br /> *''Indian zodiac'': 06° 40' - 10° 53' 40 Makara<br /> |}<br /> <br /> ==Padas (quarters)==<br /> The 27 Nakshatras cover 13°20’ of the [[ecliptic]] each. Each Nakshatra is also divided into quarters or ''padas'' of 3°20’, and the below table lists the appropriate starting sound to name the child. The 27 nakshatras, each with 4 padas, give 108, which is the number of beads in a japa mala, indicating all the elements (ansh) of Vishnu:<br /> <br /> {| class=&quot;wikitable&quot; align=&quot;center&quot; cellspacing=&quot;2&quot; cellpadding=&quot;&quot;<br /> |- bgcolor=&quot;#cccccc&quot;<br /> !#!! Name !! Pada 1 !! Pada 2 !! Pada 3 !! Pada 4<br /> |-<br /> | 1|| Ashwini (अश्विनि)|| चु Chu || चे Che || चो Cho || ला Laa<br /> |-<br /> | 2|| [[Bharani]] (भरणी)|| ली Lii || लू Luu || ले Le || लो Lo<br /> |-<br /> | 3 || [[Krittika]] (कृत्तिका)|| अ A || ई I || उ U || ए E<br /> |-<br /> | 4 || Rohini(रोहिणी)|| ओ O || वा Vaa/Baa || वी Vii/Bii || वु Vuu/Buu<br /> |-<br /> | 5 || [[Mrigashīrsha]](मृगशीर्ष)|| वे Ve/Be || वो Vo/Bo || का Kaa || की Kii<br /> |-<br /> | 6 || [[Ardra (nakshatra)|Ārdrā]] (आर्द्रा)|| कु Ku || घ Gha || ङ Ng/Na || छ Chha<br /> |-<br /> | 7 || [[Punarvasu]] (पुनर्वसु)|| के Ke || को Ko || हा Haa || ही Hii<br /> |-<br /> | 8 || [[Pushya]] (पुष्य) || हु Hu || हे He || हो Ho || ड ḍa<br /> |-<br /> | 9 || [[Āshleshā]] (अश्लेषा)|| डी ḍii || डू ḍuu || डे ḍe || डो ḍo<br /> |-<br /> | 10 || [[Maghā]] (मघा)|| मा Maa || मी Mii || मू Muu || मे Me<br /> |-<br /> | 11 || Pūrva or [[Pūrva Phalgunī]] (पूर्व फल्गुनी) || मो Mo || टा ṭaa || टी ṭii || टू ṭuu<br /> |-<br /> | 12 || Uttara or [[Uttara Phalgunī]] (उत्तर फल्गुनी)|| टे ṭe || टो ṭo || पा Paa || पी Pii<br /> |-<br /> | 13 || [[Hasta (nakshatra)|Hasta]] (हस्त)|| पू Puu || ष Sha || ण Na || ठ ṭha<br /> |-<br /> | 14 || Chitra (चित्रा)|| पे Pe || पो Po || रा Raa || री Rii<br /> |-<br /> | 15 || [[Svātī]] (स्वाति) || रू Ruu || रे Re || रो Ro || ता Taa<br /> |-<br /> | 16 || [[Viśākhā]] (विशाखा)|| ती Tii || तू Tuu || ते Te || तो To<br /> |-<br /> | 17 || Anurādhā (अनुराधा)|| ना Naa || नी Nii || नू Nuu || ने Ne<br /> |-<br /> | 18 || [[Jyeshtha]] (ज्येष्ठा)|| नो No || या Yaa || यी Yii || यू Yuu<br /> |-<br /> | 19 || [[Mula (astrology)|Mula]] (मूल)|| ये Ye || यो Yo || भा Bhaa || भी Bhii<br /> |-<br /> | 20 || [[Pūrva Āshādhā]] (पूर्व आषाढ़)|| भू Bhuu || धा Dhaa || फा Bhaa/Phaa || ढा Daa<br /> |-<br /> | 21 || [[Uttara Āshadha|Uttara Āṣāḍhā]] (उत्तर आषाढ़)|| भे Bhe || भो Bho || जा Jaa || जी Jii<br /> |-<br /> | 22 || [[Śrāvaṇa]] (श्र‌ावण)|| खी Ju/Khii || खू Je/Khuu || खे Jo/Khe || खो Gha/Kho<br /> |-<br /> | 23 || Śrāviṣṭha (श्रविष्ठा) or [[Dhanishta]] || गा Gaa || गी Gii || गु Gu || गे Ge<br /> |-<br /> | 24 || [[Shatabhisha]] (शतभिषा)or Śatataraka || गो Go || सा Saa || सी Sii || सू Suu<br /> |-<br /> | 25 || [[Pūrva Bhādrapadā]] (पूर्व भाद्रपद)|| से Se || सो So || दा Daa || दी Dii<br /> |-<br /> | 26 || [[Uttara Bhādrapadā]] (उत्तर भाद्रपद)|| दू Duu || थ Tha || झ Jha || ञ ña <br /> |-<br /> | 27 || [[Revati]] (रेवती)|| दे De || दो Do || च Cha || ची Chii<br /> |}<br /> <br /> ==Names in Indian and Asian languages==<br /> The names of nakshatras in other languages are adapted from the Sanskrit variation (apabhramsa) through Pali or Prakrit. The variations evolved for easier pronunciation in popular usage.<br /> <br /> {| class=&quot;wikitable&quot; align=&quot;center&quot; cellspacing=&quot;2&quot; cellpadding=&quot;&quot;<br /> |- bgcolor=&quot;#cccccc&quot;<br /> !#!![[Sanskrit]]&lt;br&gt;संस्कृतम्!![[Odia language|Odia]]&lt;br&gt;ଓଡିଆ!![[Malayalam]]&lt;br&gt;മലയാളം!![[Tamil language|Tamil]]&lt;br&gt;தமிழ்!![[Sinhala language|Sinhala]]&lt;br&gt;සිoහල!![[Dhivehi language|Dhivehi]]&lt;br&gt;ދިވެހި!![[Telugu language|Telugu]]&lt;br&gt;తెలుగు!![[Kannada]]&lt;br&gt;ಕನ್ನಡ!![[Bengali language|Bengali]]&lt;br&gt;বাংলা!! Mongolian!! [[Chinese language|Chinese]]&lt;br&gt;宿!![[Classical Tibetan|Tibetan]]&lt;br&gt;བོད་སྐད། [http://tibetancalculator.com/tibetan-stars/ རྒྱུ་སྐར་ཉེ་བདུན།] <br /> |-<br /> | 1|| [[Ashvinī]]&lt;br&gt;अश्विनी||Aswini&lt;br&gt;ଅଶ୍ଵିନୀ||Ashvati&lt;br&gt;അശ്വതി ||Ashwini&lt;br&gt;அசுவினி|| Aswida&lt;br&gt;අස්විද|| A'sidha&lt;br&gt;އައްސިދަ|| Aswini&lt;br&gt;అశ్విని || Ashwini&lt;br&gt;ಅಶ್ವಿನಿ ||Ashwini&lt;br&gt;অশ্বিনী || Шийдэм || 婁|| ཐ་སྐར།<br /> |-<br /> | 2|| [[Bharanī]]&lt;br&gt;भरणी ||Dwijaa&lt;br&gt;ଦ୍ଵିଜା|| Bharaṇi&lt;br&gt;ഭരണി ||Baraṇi&lt;br&gt;பரணி || Berana&lt;br&gt;බෙරණ|| Burunu&lt;br&gt;ބުރުނު|| Bharani&lt;br&gt;భరణి|| Bharani&lt;br&gt;ಭರಣಿ ||Bharaṇi&lt;br&gt;ভরণী|| Гоё хүүхэн|| 胃|| བྲ་ཉེ།<br /> |-<br /> | 3 || [[Krittikā]]&lt;br&gt;कृत्तिका ||Krutikaa&lt;br&gt;କୃତିକା|| Kārttika&lt;br&gt;കാർത്തിക || Kārthikai&lt;br&gt;கார்த்திகை || Kethi&lt;br&gt;කැති|| Kethi&lt;br&gt;ކެތި|| Kritika&lt;br&gt;కృత్తిక|| Kritika&lt;br&gt;ಕೃತಿಕ|| Krittika&lt;br&gt;কৃত্তিকা ||Нэг эхт зургаан хөвгүүн || 昴|| སྨིན་དྲུག<br /> |-<br /> | 4 || [[Rohini (nakshatra)|Rohini]]&lt;br&gt;रोहिणी ||Rohini&lt;br&gt;ରୋହିଣୀ|| Rōhiṇi&lt;br&gt;രോഹിണി || Rōhiṇi&lt;br&gt;ரோகிணி || Rehena&lt;br&gt;රෙහෙණ|| Roanu&lt;br&gt;ރޯނު|| Rohini&lt;br&gt;రోహిణి|| Rohini&lt;br&gt;ರೋಹಿಣಿ ||Rohiṇi&lt;br&gt;রোহিণী || Чөлөөт эх || 畢|| སྣར་མ།<br /> |-<br /> | 5 || [[Mrigashīrsha]]&lt;br&gt;म्रृगशीर्षा ||Mrugasiraa&lt;br&gt;ମୃଗଶିରା|| Makayiram&lt;br&gt;മകയിരം || Mirugasīridam&lt;br&gt;மிருகசீரிடம் || Muwasirasa&lt;br&gt;මුවසිරස|| Miyaheli&lt;br&gt;މިޔަހެލި|| Mrigasiraa&lt;br&gt;మృగశిర|| Mrigashira&lt;br&gt;ಮೃಗಶಿರ ||Mrigashira&lt;br&gt;মৃগশিরা ||Гөрөөсөн толгой || 觜|| མགོ<br /> |-<br /> | 6 || [[Ardra (nakshatra)|Ārdrā]]&lt;br&gt;आर्द्रा ||Adra&lt;br&gt;ଆଦ୍ରା|| Ātira or Tiruvātira&lt;br&gt;ആതിര (തിരുവാതിര) || Thiruvādhirai&lt;br&gt;திருவாதிரை || Ada&lt;br&gt;අද|| Adha&lt;br&gt;އަދަ|| Arudra&lt;br&gt;ఆరుద్ర || Ardra&lt;br&gt;ಆರ್ದ್ರ|| Ardra&lt;br&gt;আর্দ্রা || Хэрцгий охин || 參|| ལག<br /> |-<br /> | 7 || [[Punarvasu]]&lt;br&gt;पुनर्वसु ||punarbasu&lt;br&gt;ପୁନର୍ବସୁ|| Puṇartam&lt;br&gt;പുണർതം || Punarpoosam&lt;br&gt;புனர்பூசம் || Punavasa&lt;br&gt;පුනාවස|| Funoas&lt;br&gt;ފުނޯސް|| Punarvasu&lt;br&gt;పునర్వసు|| Punarvasu&lt;br&gt;ಪುನರ್ವಸು ||Punarbasu&lt;br&gt;পুনর্বসু ||Өглөгт охин || 井|| ནབས་སོ།<br /> |-<br /> | 8 || [[Pushya]]&lt;br&gt;पुष्य ||Pushyaa&lt;br&gt;ପୁଷ୍ୟା|| Pūyam&lt;br&gt;പൂയം || Poosam&lt;br&gt;பூசம் || Pusha&lt;br&gt;පුෂ|| Fus&lt;br&gt;ފުސް|| Pushya&lt;br&gt;పుష్య|| Pushyami&lt;br&gt;ಪುಷ್ಯ ||Pushya&lt;br&gt;পুষ্যা || Найралт эх || 鬼|| རྒྱལ།<br /> |-<br /> | 9 || [[Āshleshā]]&lt;br&gt;आश्ळेषा / आश्लेषा||Ashleshaa&lt;br&gt;ଆଶ୍ଳେଷା||Āyilyam&lt;br&gt;ആയില്യം || Ayilyam&lt;br&gt;ஆயில்யம் || Aslisa&lt;br&gt;අස්ලිස|| Ahuliha&lt;br&gt;އަހުލިހަ|| Ashlesha&lt;br&gt;ఆశ్లేష|| Ashlesha&lt;br&gt;ಆಶ್ಲೇಷ ||Ashleshā&lt;br&gt;অশ্লেষা || Үнэг || 柳|| སྐག<br /> |-<br /> | 10 || [[Maghā]]&lt;br&gt;मघा ||Magaa&lt;br&gt;ମଘା|| Makam&lt;br&gt;മകം || Magam&lt;br&gt;மகம் || Maa&lt;br&gt;මා|| Maa&lt;br&gt;މާ|| Maghaa&lt;br&gt;మఘ|| Magha&lt;br&gt;ಮಘ ||Magha&lt;br&gt;মঘা ||Их морь || 星|| མཆུ།<br /> |-<br /> | 11 || Pūrva or [[Pūrva Phalgunī]]&lt;br&gt;पूर्व फल्गुनी ||Purba Falguni&lt;br&gt;ପୂର୍ବ ଫାଲ୍ଗୁନୀ|| Pūram&lt;br&gt;പൂരം || Pooram&lt;br&gt;பூரம் || Puwapal&lt;br&gt;පුවපල්|| Fura&lt;br&gt;ފުރަ|| Pubba&lt;br&gt;పూర్వ ఫల్గుని|| Poorva&lt;br&gt;ಪೂರ್ವ ||Purbaphālguni&lt;br&gt;পূর্বফাল্গুনী || Бага морь || 張|| གྲེ།<br /> |-<br /> | 12 || Uttara or [[Uttara Phalgunī]]&lt;br&gt;उत्तर फल्गुनी ||Uttara falguni&lt;br&gt;ଉତ୍ତର ଫାଲ୍ଗୁନୀ|| Utram&lt;br&gt;ഉത്രം ||Uthiram&lt;br&gt;உத்திரம் || Uttrapal&lt;br&gt;උත්රපල්|| Uthura&lt;br&gt;އުތުރަ|| Uttara&lt;br&gt;ఉత్తర ఫల్గుని|| Uttara&lt;br&gt;ಉತ್ತರ ||Uttar Phālguni&lt;br&gt;উত্তরফাল্গুনী || Харцага || 翼|| དབོ།<br /> |-<br /> | 13 || [[Nakshatra|Hasta]]&lt;br&gt;हस्त ||Hastaa&lt;br&gt;ହସ୍ତା||Attam&lt;br&gt;അത്തം || Ashtham&lt;br&gt;அசுதம் || Hatha&lt;br&gt;හත|| Atha&lt;br&gt;އަތަ|| Hasta&lt;br&gt;హస్త|| Hastaa&lt;br&gt;ಹಸ್ತ ||Hastā &lt;br&gt; হস্তা ||Тугчин || 軫|| མེ་བཞི།<br /> |-<br /> | 14 || [[Nakshatra|Chitrā]]&lt;br&gt;चित्रा ||Chitraa&lt;br&gt;ଚିତ୍ରା||Chittira&lt;br&gt;ചിത്തിര (ചിത്ര) || Chithirai&lt;br&gt;சித்திரை || Sitha&lt;br&gt;සිත|| Hitha&lt;br&gt;ހިތަ|| Chittaa&lt;br&gt;చిత్త|| Chitra&lt;br&gt;ಚಿತ್ರ ||Chitrā &lt;br&gt; চিত্রা ||Тэргүүн дагуул || 角|| ནག་པ།<br /> |-<br /> | 15 || [[Svātī]]&lt;br&gt;स्वाती ||Swati&lt;br&gt;ସ୍ଵାତୀ||Chōti&lt;br&gt;ചോതി || Swathi&lt;br&gt;சுவாதி || Saa&lt;br&gt;සා|| Hey&lt;br&gt;ހޭ|| Swaati&lt;br&gt;స్వాతి|| Swathi&lt;br&gt;ಸ್ವಾತಿ ||Swāti &lt;br&gt; স্বাতী || Салхины эх || 亢|| ས་རི།<br /> |-<br /> | 16 || [[Vishākhā]]&lt;br&gt;विशाखा ||Bishakha&lt;br&gt;ବିଶାଖା||Vishākham&lt;br&gt;വിശാഖം || Visakam&lt;br&gt;விசாகம்|| Wisa&lt;br&gt;විසා|| Vihaa&lt;br&gt;ވިހާ|| Vishaakha&lt;br&gt;విశాఖ|| Vishakha&lt;br&gt;ವಿಶಾಖ ||Bishakha &lt;br&gt; বিশাখা || Эрхтний тэнгэрт || 氐|| ས་ག<br /> |-<br /> | 17 ||[https://aaps.space/docs/nakshatra/Anuradha-Nakshatra/ Anurādhā]&lt;br&gt;अनुराधा ||Anuradha&lt;br&gt;ଅନୁରାଧା|| Anizham&lt;br&gt;അനിഴം || Anusham&lt;br&gt;அனுசம் || Anura&lt;br&gt;අනුර|| Nora&lt;br&gt;ނޮރަ|| Anuraadha&lt;br&gt;అనురాధ|| Anuradha&lt;br&gt;ಅನುರಾಧಾ ||Anuradha &lt;br&gt; অনুরাধা || Гар од || 房||ལྷ་མཚམས།<br /> |-<br /> | 18 || [[Jyeshtha]]&lt;br&gt;ज्येष्ठा ||Jyosthaa&lt;br&gt;ଜ୍ୟୋଷ୍ଠା||Kēṭṭa (Trikkēṭṭa)&lt;br&gt;തൃക്കേട്ട|| Kettai&lt;br&gt;கேட்டை || Deta&lt;br&gt;දෙට|| Dhosha&lt;br&gt;ދޮށަ|| Jyesthaa&lt;br&gt;జ్యేష్ఠ|| Jyestha&lt;br&gt;ಜ್ಯೇಷ್ಠ ||Jyestha &lt;br&gt; জ্যেষ্ঠা || Хонгорцог || 心|| སྣྲོན།<br /> |-<br /> | 19 || [[Mula (astrology)|Mūla]]&lt;br&gt;मूल ||Mulaa&lt;br&gt;ମୂଳା|| Mūlam&lt;br&gt;മൂലം|| Mūlam&lt;br&gt;மூலம் || Moola&lt;br&gt;මුල|| Mula&lt;br&gt;މުލަ|| Moolaa&lt;br&gt;మూల|| Moola&lt;br&gt;ಮೂಲ ||Mula &lt;br&gt; মূলা || Онгоц || 尾|| སྣུབས།<br /> |-<br /> | 20 || [[Pūrva Ashādhā]]&lt;br&gt;पूर्वाषाढा ||Purbasaadhaa&lt;br&gt;ପୂର୍ବାଷାଢା|| Pūrāṭam&lt;br&gt;പൂരാടം || Pūradam&lt;br&gt;பூராடம் || Puwasala&lt;br&gt;පුවසල|| Furahalha&lt;br&gt;ފުރަހަޅަ|| Poorvashaadha&lt;br&gt;పూర్వాషాడ|| Poorvashadha&lt;br&gt;ಪುರ್ವಾಷಾಡ ||Poorbashada&lt;br&gt;পূর্বাষাঢ়া || Суулга || 箕||ཆུ་སྟོད།<br /> |-<br /> | 21 || [[Uttara Ashādhā]]&lt;br&gt;उत्तराषाढा ||Uttarasaadha&lt;br&gt;ଉତ୍ତରଷାଢା|| Utrāṭam&lt;br&gt;ഉത്രാടം || Uthirādam&lt;br&gt;உத்திராடம் || Uttrasala&lt;br&gt;උත්රසල|| Uthurahalha&lt;br&gt;އުތުރަހަޅަ|| Uttarashaadha&lt;br&gt;ఉత్తరాషాడ|| Uttarashadha&lt;br&gt;ಉತ್ತರಾಷಾಡ ||Uttarashada]]&lt;br&gt;উত্তরাষাঢ়া || Элдэв тэнгэртэн || 斗|| ཆུ་སྨད།<br /> |-<br /> | 22 || [[Shravana]]&lt;br&gt;श्रवण ||Sravana&lt;br&gt;ଶ୍ରବଣା|| Tiruvōnam &lt;br&gt;ഓണം (തിരുവോണം)|| Tiruvōnam&lt;br&gt;திருவோணம் || Suvana&lt;br&gt;සුවණ|| Huvan&lt;br&gt;ހުވަން|| Shravana&lt;br&gt;శ్రవణ|| Shravana&lt;br&gt;ಶ್ರವಣ||Shraban&lt;br&gt;শ্রবণা || Булаагч/Яруу эгшигт || 女|| གྲོ་བཞིན།<br /> |-<br /> | 23 || Shravishthā or [[Dhanishta]] &lt;br&gt;श्रविष्ठा or धनिष्ठा ||Dhanishathaa&lt;br&gt;ଧନିଷ୍ଠା|| Aviṭṭam&lt;br&gt;അവിട്ടം || Aviṭṭam&lt;br&gt;அவிட்டம் || Denata&lt;br&gt;දෙණට|| Dhinasha&lt;br&gt;ދިނަށަ|| Dhanishta&lt;br&gt;ధనిష్ఠ|| Dhanishta&lt;br&gt;ಧನಿಷ್ಠ||Dhanishta&lt;br&gt;ধনিষ্ঠা ||Тооно || 虛||མོན་གྲེ།<br /> |-<br /> | 24 || [[Shatabhishā]] or Shatataraka&lt;br&gt;शतभिषक् / शततारका||Satavisaa&lt;br&gt;ସତଭିଷା|| Chatayam&lt;br&gt;ചതയം || Sadayam&lt;br&gt;சதயம் || Siyawasa&lt;br&gt;සියාවස|| Hiyavihaa&lt;br&gt;ހިޔަވިހާ|| Shatabhisha&lt;br&gt;శతభిష|| Shatabhisha&lt;br&gt;ಶತಭಿಷ ||Shatabhisha &lt;br&gt; শতভিষা || Чөдөр || 危|| མོན་གྲུ<br /> |-<br /> | 25 || [[Pūrva Bhādrapadā]]&lt;br&gt;पूर्वभाद्रपदा / पूर्वप्रोष्ठपदा ||Purba vadrapada&lt;br&gt;ପୂର୍ବଭାଦ୍ରପଦ|| Pūruruṭṭāti&lt;br&gt;പൂരുരുട്ടാതി || Pūraṭṭādhi&lt;br&gt;பூரட்டாதி || Puvaputupa&lt;br&gt;පුවපුටුප|| Furabadhuruva&lt;br&gt;ފުރަބަދުރުވަ|| Poorvabhadraa&lt;br&gt;పూర్వ భాద్రపద|| Poorva Bhadrapada&lt;br&gt;ಪೂರ್ವ ಭಾದ್ರಪದ ||Poorbabhadra &lt;br&gt; পূর্বভাদ্রপদ || Шувуун хошуут || 室||ཁྲུམས་སྟོད།<br /> |-<br /> | 26 || [[Uttara Bhādrapadā]]&lt;br&gt;उत्तरभाद्रपदा / उत्तरप्रोष्ठपदा ||Uttara vadrapada&lt;br&gt;ଉତ୍ତରାଭାଦ୍ରପଦ|| Uttṛṭṭāti&lt;br&gt;ഉത്രട്ടാതി || Uttṛṭṭādhi&lt;br&gt;உத்திரட்டாதி || Uttraputupa&lt;br&gt;උත්රපුටුප|| Fasbadhuruva&lt;br&gt;ފަސްބަދުރުވަ|| Uttaraa bhadra&lt;br&gt;ఉత్తర భాద్రపద|| Uttara Bhadrapada&lt;br&gt;ಉತ್ತರ ಭಾದ್ರಪದ||Uttarbabhadra &lt;br&gt; উত্তরভাদ্রপদ || Могой хүлэгч || 壁|| ཁྲུམས་སྨད།<br /> |<br /> |-<br /> | 27 || [[Revati (nakshatra)|Revati]]&lt;br&gt;रेवती ||Revati&lt;br&gt;ରେବତୀ|| Rēvati&lt;br&gt;രേവതി || Rēvathi&lt;br&gt;ரேவதி || Revathi&lt;br&gt;රේවතී|| Reyva&lt;br&gt;ރޭވަ|| Revati&lt;br&gt;రేవతి|| Revati&lt;br&gt;ರೇವತಿ ||Rebati &lt;br&gt; রেবতী || Дэлгэрүүлэгч || 奎|| ནམ་གྲུ།<br /> |}<br /> <br /> ==References==<br /> <br /> {{reflist}}<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Nakshatra_system&diff=129867 2021-04-16T04:11:28Z

<p>BharataPuru: </p> <hr /> <div>In [[Hindu astrology|Ancient Indian Astronogy]] or Hindu Astronomy, there are 27 '''''[[nakshatra]]s''''' (&quot;lunar mansions&quot;, Sanskrit: नक्षत्र, IAST: ''Nakṣatra''), or sectors along the ecliptic. A list of them is first found in the ''[[Vedanga Jyotisha]]'', a text dated to the final centuries BCE. The ''Nakṣatra'' system predates the influence of [[Hellenistic astronomy]] on Vedic tradition, which became prevalent from about the 2nd century CE. The following list gives the corresponding regions of sky.&lt;ref&gt;{{cite book|first=Arthur Llewellyn|last=Basham|authorlink=Arthur Llewellyn Basham|title=The Wonder That Was India|chapter=Appendix II: Astronomy|p=490|year=1954|publisher=[[Sidgwick and Jackson]]|title-link=The Wonder That Was India}}&lt;/ref&gt;<br /> ==Indian Nakshatras==<br /> {| class=&quot;wikitable&quot; align=&quot;center&quot; cellspacing=&quot;2&quot; cellpadding=&quot;&quot;<br /> |- bgcolor=&quot;#cccccc&quot;<br /> !No.!! Name !!Associated stars!!Description!! Image<br /> |-<br /> | 1<br /> ||'''[[Ashvini]]'''&lt;br /&gt;&quot;physician to the Gods&quot;<br /> ||[[Beta Arietis|β]] and [[Gamma Arietis|γ]] [[Aries (constellation)|Arietis]]<br /> ||<br /> *''Lord'': Ketu (South lunar node)<br /> *''Symbol'' : Horse's head<br /> *''Deity'' : [[Ashvins]], the horse-headed twins who are physicians to the gods<br /> *''Indian zodiac'': 0° - 13°20' Mesha<br /> *''west zodiac'' 23°46 Aries - 7°06' Taurus<br /> ||[[File:Aries constellation map.svg|100px]]<br /> |-<br /> | 2<br /> ||'''[[Bharani]]'''&lt;br /&gt;&quot;the bearer&quot;<br /> ||[[35 Arietis|35]], [[39 Arietis|39]], and [[41 Arietis|41]] [[Aries (constellation)|Arietis]]<br /> ||<br /> *''Lord'': Shukra (Venus)<br /> *''Symbol'': Yoni, the female organ of reproduction<br /> *''Deity'': [[Yama]], [[god of death]] or [[Dharma]]<br /> *''Indian zodiac'': 13° 20' - 26°40' Mesha<br /> *''Western zodiac'' 7°06' - 20°26' Taurus<br /> ||[[File:Aries constellation map.svg|100px]]<br /> |-<br /> | 3<br /> ||'''[[Krittika]]'''&lt;br /&gt;an old name of the Pleiades; personified as the nurses of ''Kārttikeya'', a son of Shiva.<br /> ||[[Pleiades]]<br /> ||<br /> *''Lord'': [[Surya]] (Sun)<br /> *''Symbol'': [[Knife]] or [[spear]]<br /> *''Deity'' : [[Agni]], god of fire<br /> *''Indian zodiac'': 26°40' [[Mesha]] - 10° Vrishabha<br /> *''Western zodiac'' 20°26' Taurus - 3°46’ Gemini<br /> ||[[File:Taurus constellation map.png|100px]]<br /> |-<br /> | 4<br /> ||'''[[Rohini (nakshatra)|Rohini]]'''&lt;br&gt;&quot;the red one&quot;, a name of [[Aldebaran]]. Also known as ''brāhmī''<br /> ||[[Aldebaran]]<br /> ||<br /> *''Lord'': [[Chandra]] ([[Moon]])<br /> *''Symbol'': Cart or [[chariot]], [[temple]], [[banyan tree]]<br /> *''Deity'' : [[Brahma]] or [[Prajapati]], [[Creator deity|the Creator]]<br /> *''Indian zodiac'': 10° - 23°20' Vrishabha<br /> *''Western zodiac'' 3°46 - 17°06' Gemini<br /> ||[[File:Taurus constellation map.png|100px]]<br /> |-<br /> | 5<br /> ||'''[[Mrigashīrsha]]'''&lt;br /&gt;&quot;the deer's head&quot;. Also known as ''āgrahāyaṇī''<br /> ||[[Lambda Orionis|λ]], [[Phi Orionis|φ]] Orionis<br /> ||<br /> *''Lord'': Mangala (Mars)<br /> *''Symbol'': Deer's head<br /> *''Deity'': [[Soma (deity)|Soma]], [[Chandra]], the Moon god<br /> *''Indian zodiac'': 23° 20' Vrishabha - 6° 40' Mithuna<br /> *''Western zodiac'': 17°06' Gemini - 0°26' Cancer<br /> ||[[File:Orion constellation map.png|100px]]<br /> |-<br /> | 6<br /> ||'''[[Ardra (nakshatra)|Ardra]]'''&lt;br /&gt;&quot;the storm god&quot;<br /> ||[[Betelgeuse]]<br /> ||<br /> *''Lord'': Rahu (North lunar node)<br /> *''Symbol'': Teardrop, diamond, a human head<br /> *''Deity'' : [[Rudra]], the storm god<br /> *''Indian zodiac'': 6° 40' - 20° Mithuna<br /> *''Western zodiac'': 0°26' - 13°46 Cancer<br /> ||[[File:Orion constellation map.png|100px]]<br /> |-<br /> | 7<br /> ||'''[[Punarvasu]]''' (dual)&lt;br /&gt;&quot;the two restorers of goods&quot;, also known as ''yamakau'' &quot;the two chariots&quot;<br /> ||[[Castor (star)|Castor]] and [[Pollux (star)|Pollux]]<br /> ||<br /> *''Lord'': Guru (Jupiter)<br /> *''Symbol'' : Bow and quiver<br /> *''Deity'' : [[Aditi]], mother of the gods<br /> *''Indian zodiac'': 20° Mithuna - 3°20' Karka<br /> *''Western zodiac'' 13°46 - 27°06' Cancer<br /> ||[[File:Gemini constellation map.svg|100px]]<br /> |-<br /> | 8<br /> ||'''[[Pushya]]'''&lt;br /&gt;&quot;the nourisher&quot;, also known as ''sidhya'' or ''tiṣya''<br /> ||[[Gamma Cancri|γ]], [[Delta Cancri|δ]] and [[Theta Cancri|θ]] [[Cancer (constellation)|Cancri]]<br /> ||<br /> *''Lord'': [[Shani]] ([[Saturn]])<br /> *''Symbol'' : Cow's udder, [[Lotus Flower|lotus]], arrow and circle<br /> *''Deity'' : [[Bṛhaspati]], priest of the gods<br /> *''Indian zodiac'': 3°20' -16°40' [[Karka]]<br /> *''Western zodiac'' 27°06' Cancer - 10°26' [[Leo (constellation)|Leo]]<br /> ||[[File:Cancer constellation map.png|100px]]<br /> |-<br /> | 9<br /> ||'''[[Āshleshā]]'''&lt;br /&gt;&quot;the embrace&quot;<br /> ||δ, ε, η, ρ, and [[Sigma Hydrae|σ]] [[Hydra (constellation)|Hydrae]]<br /> ||<br /> *''Lord'': Budh (Mercury)<br /> *''Symbol'': Serpent<br /> *''Deity'' : Sarpas or [[Nāga|Nagas]], deified snakes<br /> *''Indian zodiac'': 16°40' - 30° Karka<br /> *''Western zodiac'' 10°26' - 23°46 Leo<br /> ||[[File:Hydra constellation map.png|100px]]<br /> |-<br /> | 10<br /> ||'''[[Maghā]]'''&lt;br /&gt;&quot;the bountiful&quot;<br /> ||[[Regulus]]<br /> ||<br /> *''Lord'': Ketu (south lunar node)<br /> *''Symbol'' : Royal Throne<br /> *''Deity'' : [[Pitrs]], 'The Fathers', family ancestors<br /> *''Indian zodiac'': 0° - 13°20' Simha<br /> *''Western zodiac'' 23°46 Leo - 7°06' Virgo<br /> ||[[File:Leo constellation map.png|100px]]<br /> |-<br /> | 11<br /> ||'''[[Pūrva Phalgunī]]'''&lt;br&gt;&quot;first reddish one&quot;<br /> ||[[Delta Leonis|δ]] and [[Theta Leonis|θ]] [[Leo (constellation)|Leonis]]<br /> ||<br /> *''Lord'': Shukra (Venus)<br /> *''Symbol'' : Front legs of bed, hammock, fig tree<br /> *''Deity'' : [[Bhaga]], god of marital bliss and prosperity<br /> *''Indian zodiac'': 13°20' - 26°40' Simha<br /> *''Western zodiac'' 7°06' - 20°26' Virgo<br /> || [[File:Leo constellation map.png|100px]]<br /> |-<br /> | 12<br /> ||'''[[Uttara Phalgunī]]'''&lt;br /&gt;&quot;second reddish one&quot;<br /> ||[[Denebola]]<br /> ||<br /> *''Lord'': Surya (Sun)<br /> *''Symbol'': Four legs of bed, hammock<br /> *''Deity'' : [[Aryaman]], god of patronage and favours<br /> *''Indian zodiac'': 26°40' Simha- 10° Kanya<br /> *''Western zodiac'' 20°26' Virgo - 3°46 Libra<br /> || [[File:Leo constellation map.png|100px]]<br /> |-<br /> | 13<br /> ||'''[[Nakshatra|Hasta]]'''&lt;br /&gt;&quot;the hand&quot;<br /> ||[[Alpha Corvi|α]], [[Beta Corvi|β]], [[Gamma Corvi|γ]], [[Delta Corvi|δ]] and [[Epsilon Corvi|ε]] [[Corvus (constellation)|Corvi]]<br /> ||<br /> *''Lord'': Chandra (Moon)<br /> *''Symbol'': Hand or fist<br /> *''Deity'' : [[Surya|Saviti]] or [[Surya]], the Sun god<br /> *''Indian zodiac'': 10° - 23°20' Kanya<br /> *''Western zodiac'' 3°46 - 17°06' Libra<br /> ||[[File:Corvus constellation map.png|100px]]<br /> |-<br /> | 14<br /> ||'''Chitra'''&lt;br /&gt;&quot;the bright one&quot;, a name of Spica<br /> ||[[Spica]]<br /> ||<br /> *''Lord'': Mangala (Mars)<br /> *''Symbol'': Bright jewel or pearl<br /> *''Deity'' : [[Tvastar]] or [[Vishvakarman]], the celestial architect<br /> *''Indian zodiac'': 23°20' Kanya - 6°40' Tula<br /> *''Western zodiac'': 17°06' Libra - 0°26' Scorpio<br /> || [[File:Virgo constellation map.png|100px]]<br /> |-<br /> | 15<br /> ||'''[[Swati the Star|Swāti]]'''&lt;br /&gt;&quot;Su-Ati (sanskrit) Very good&quot; name of Arcturus<br /> ||[[Arcturus]]<br /> ||<br /> *''Lord'': Rahu (north lunar node)<br /> *''Symbol'': Shoot of plant, coral<br /> *''Deity'' : [[Vayu]], the Wind god<br /> *''Indian zodiac'': 6°40' - 20° Tula<br /> *''Western zodiac'' 0°26' - 13°46 Scorpio<br /> || [[File:Bootes constellation map.png|100px]]<br /> |-<br /> | 16<br /> ||'''Vishakha'''&lt;br /&gt;&quot;forked, having branches&quot;; also known as ''rādhā'' &quot;the gift&quot;<br /> || [[Alpha Librae|α]], [[Beta Librae|β]], [[Gamma Librae|γ]] and ι [[Libra (constellation)|Librae]]<br /> ||<br /> *''Lord'': Guru (Jupiter)<br /> *''Symbol'' : Triumphal arch, potter's wheel<br /> *''Deity'' : [[Indra]], chief of the gods; [[Agni]], god of Fire<br /> *''Indian zodiac'': 20° Tula - 3°20' Vrishchika<br /> *''Western zodiac'' 13°46 - 27°06' Scorpio<br /> ||[[File:Libra constellation map.png|100px]]<br /> |-<br /> | 17<br /> ||'''[[Anuradha (nakshatra)|Anuradha]]'''&lt;br /&gt; &quot;following ''rādhā''&quot;<br /> || [[Beta Scorpii|β]], [[Delta Scorpii|δ]] and [[Pi Scorpii|π]] [[Scorpius (constellation)|Scorpionis]]<br /> ||<br /> *''Lord'': Shani (Saturn)<br /> *''Symbol'' : Triumphal archway, lotus<br /> *''Deity'' : [[Mitra (Vedic)|Mitra]], one of Adityas of friendship and partnership<br /> *''Indian zodiac'': 3°20' - 16°40' Vrishchika<br /> *''Western zodiac'' 27°06' Scorpio - 10°26' Sagittarius<br /> || [[File:Scorpius constellation map.svg|100px]]<br /> |-<br /> | 18<br /> ||'''[[Jyeshtha (nakshatra)|Jyeshtha]]'''&lt;br /&gt;&quot;the eldest, most excellent&quot;<br /> || [[Alpha Scorpii|α]], [[Sigma Scorpii|σ]], and [[Tau Scorpii|τ]] [[Scorpius (constellation)|Scorpionis]]<br /> ||<br /> *''Lord'': Budh (Mercury)<br /> *''Symbol'' : circular amulet, umbrella, earring<br /> *''Deity'' : [[Indra]], chief of the gods<br /> *''Indian zodiac'': 16°40' - 30° Vrishchika<br /> *''Western zodiac'' 10°26' - 23°46 Sagittarius<br /> || [[File:Scorpius constellation map.png|100px]]<br /> |-<br /> | 19<br /> ||'''[[Mula (nakshatra)|Mula]]'''&lt;br /&gt;&quot;the root&quot;<br /> || [[Epsilon Scorpii|ε]], ζ, [[Eta Scorpii|η]], [[Theta Scorpii|θ]], ι, [[Kappa Scorpii|κ]], [[Lambda Scorpii|λ]], [[Mu Scorpii|μ]] and [[Nu Scorpii|ν]] [[Scorpius (constellation)|Scorpionis]]<br /> ||<br /> *''Lord'': Ketu (south lunar node)<br /> *''Symbol'' : Bunch of roots tied together, elephant goad<br /> *''Deity'' : [[Nirrti]], goddess of dissolution and destruction<br /> *''Indian zodiac'': 0° - 13°20' Dhanus<br /> *''Western zodiac'' 23°46 Sagittarius - 7°06' Capricorn<br /> || [[File:Scorpius constellation map.png|100px]]<br /> |-<br /> | 20<br /> ||'''[[Purva Ashadha]]'''&lt;br /&gt;&quot;first of the ''āṣāḍhā''&quot;, ''āṣāḍhā'' &quot;the invincible one&quot; being the name of a constellation<br /> ||[[Delta Sagittarii|δ]] and [[Epsilon Sagittarii|ε]] [[Sagittarius (constellation)|Sagittarii]]<br /> ||<br /> *''Lord'': Shukra (Venus)<br /> *''Symbol'': Elephant tusk, fan, winnowing basket<br /> *''Deity'' : [[Ap (water)|Apah]], god of Water<br /> *''Indian zodiac'': 13°20' - 26°40' Dhanus<br /> *''Western zodiac'' 7°06' - 20°26' Capricorn<br /> || [[File:Sagittarius constellation map.png|100px]]<br /> |-<br /> | 21<br /> ||'''[[Uttara Ashadha]]'''&lt;br /&gt;&quot;second of the ''āṣāḍhā''&quot;<br /> ||[[Zeta Sagittarii|ζ]] and [[Sigma Sagittarii|σ]] [[Sagittarius (constellation)|Sagittarii]]<br /> ||<br /> *''Lord'': Surya (Sun)<br /> *''Symbol'' : Elephant tusk, small bed<br /> *''Deity'' : [[Visvedevas]], universal gods<br /> *''Indian zodiac'': 26°40' Dhanus - 10° Makara<br /> *''Western zodiac'' 20°26' Capricorn - 3°46 Aquarius<br /> || [[File:Sagittarius constellation map.png|100px]]<br /> |-<br /> | 22<br /> ||'''[[Shravana|Sravana]]'''&lt;br /&gt;<br /> ||[[Alpha Aquilae|α]], [[Beta Aquilae|β]] and [[Gamma Aquilae|γ]] [[Aquila (constellation)|Aquilae]]<br /> ||<br /> *''Lord'': Chandra (Moon)<br /> *''Symbol'' : Ear or Three Footprints<br /> *''Deity'' : [[Vishnu]], preserver of universe<br /> *''Indian zodiac'': 10° - 23°20' Makara<br /> *''Western zodiac'' 3°46 - 17°06' Aquarius<br /> || [[File:Aquila constellation map.png|100px]]<br /> |-<br /> | 23<br /> ||'''[[Dhanishta]]''' &lt;br /&gt;&quot;most famous&quot;, also ''Shravishthā'' &quot;swiftest&quot;<br /> || [[Alpha Delphini|α]] to [[Delta Delphini|δ Delphini]]<br /> ||<br /> *''Lord'': Mangala (Mars)<br /> *''Symbol'' : Drum or flute<br /> *''Deity'' : Eight [[vasu]]s, deities of earthly abundance<br /> *''Indian zodiac'': 23°20' Makara - 6°40' Kumbha<br /> *''Western zodiac'' 17°06' Aquarius - 0°26' Pisces<br /> || [[File:Delphinus constellation map.png|100px]]<br /> |-<br /> | 24<br /> ||'''[[Shatabhisha]]&lt;br /&gt;&quot;'''requiring a hundred physicians'''&quot;'''<br /> || [[Gamma Aquarii|γ Aquarii]]<br /> ||<br /> *''Lord'': Rahu (north lunar node)<br /> *''Symbol'' : Empty circle, 1,000 flowers or stars<br /> *''Deity'' : [[Varuna]], god of cosmic waters, sky and earth<br /> *''Indian zodiac'': 6°40' - 20° Kumbha ; ''Western zodiac'' 0°26' - 13°46 Pisces<br /> || [[File:Aquarius constellation map.svg|100px]]<br /> |-<br /> | 25<br /> ||'''[[Purva Bhadrapada]]'''&lt;br /&gt;&quot;the first of the blessed feet&quot;<br /> || [[Alpha Pegasi|α]] and [[Beta Pegasi|β]] [[Pegasus (constellation)|Pegasi]]<br /> ||<br /> *''Lord'': Guru (Jupiter)<br /> *''Symbol'' : Swords or two front legs of funeral cot, man with two faces<br /> *''Deity'' : '''Ajaikapada''', an ancient fire dragon<br /> *''Indian zodiac'': 20° Kumbha - 3°20' Meena ; ''Western zodiac'' 13°46 - 27°06' Pisces<br /> || [[File:Pegasus constellation map.png|100px]]<br /> |-<br /> | 26<br /> ||'''[[Uttara Bhādrapadā]]'''&lt;br /&gt;&quot;the second of the blessed feet&quot;<br /> || [[Gamma Pegasi|γ]] [[Pegasus (constellation)|Pegasi]] and [[Alpha Andromedae|α]] [[Andromeda (constellation)|Andromedae]]<br /> ||<br /> *''Lord'': Shani (Saturn)<br /> *''Symbol'' : Twins, back legs of funeral cot, snake in the water<br /> *''Deity'' : '''Ahirbudhnya''', serpent or dragon of the deep<br /> *''Indian zodiac'': 3°20' - 16°40' Meena ; ''Western zodiac'' 27°06' Pisces - 10°26' Aries<br /> || [[File:Andromeda constellation map (1).png|100px]]<br /> |-<br /> | 27<br /> ||'''[[Revati (nakshatra)|Revati]]'''&lt;br /&gt;&quot;prosperous&quot;<br /> || ζ [[Pisces (constellation)|Piscium]]<br /> ||<br /> *''Lord'': Budh (Mercury)<br /> *''Symbol'' : Fish or a pair of fish, drum<br /> *''Deity'' : [[Pushan]], nourisher, the protective deity<br /> *''Indian zodiac'': 16°40' - 30° Meena<br /> *''Western zodiac'' 10°26' - 23°46 Aries<br /> || [[File:Pisces constellation map.png|100px]]<br /> |-<br /> | 28<br /> ||'''[[Abhijit (nakshatra)|Abhijit]]'''&lt;br /&gt; &quot;victorious&quot;&lt;ref&gt;Its longitude starts from 06° 40' to 10° 53' 40'' in sidereal Capricorn i.e. from the last quarter of Uttra Ashadha to first 1/15 th part of Shravana.{{Citation needed|date=October 2011}}''<br /> Its span is 4° 13' 40''.{{Citation needed|date=October 2011}} The span of 27 mean daily lunar motions totals 355.76167 degrees, and together these total 359.98945 degrees. Considered an &quot;intercalary&quot; lunar mansion.{{Citation needed|date=October 2011}}''&lt;/ref&gt;<br /> || [[Alpha Lyrae|α]], [[Epsilon Lyrae|ε]] and ζ [[Lyra (constellation)|Lyrae]] - [[Vega]]<br /> || ''Lord'': [[Brahma]] (creator)<br /> *''Indian zodiac'': 06° 40' - 10° 53' 40 Makara<br /> || [[File:Lyra constellation map.png|100px]]<br /> |}<br /> <br /> ==Padas (quarters)==<br /> The 27 Nakshatras cover 13°20’ of the [[ecliptic]] each. Each Nakshatra is also divided into quarters or ''padas'' of 3°20’, and the below table lists the appropriate starting sound to name the child. The 27 nakshatras, each with 4 padas, give 108, which is the number of beads in a japa mala, indicating all the elements (ansh) of Vishnu:<br /> <br /> {| class=&quot;wikitable&quot; align=&quot;center&quot; cellspacing=&quot;2&quot; cellpadding=&quot;&quot;<br /> |- bgcolor=&quot;#cccccc&quot;<br /> !#!! Name !! Pada 1 !! Pada 2 !! Pada 3 !! Pada 4<br /> |-<br /> | 1|| Ashwini (अश्विनि)|| चु Chu || चे Che || चो Cho || ला Laa<br /> |-<br /> | 2|| [[Bharani]] (भरणी)|| ली Lii || लू Luu || ले Le || लो Lo<br /> |-<br /> | 3 || [[Krittika]] (कृत्तिका)|| अ A || ई I || उ U || ए E<br /> |-<br /> | 4 || Rohini(रोहिणी)|| ओ O || वा Vaa/Baa || वी Vii/Bii || वु Vuu/Buu<br /> |-<br /> | 5 || [[Mrigashīrsha]](मृगशीर्ष)|| वे Ve/Be || वो Vo/Bo || का Kaa || की Kii<br /> |-<br /> | 6 || [[Ardra (nakshatra)|Ārdrā]] (आर्द्रा)|| कु Ku || घ Gha || ङ Ng/Na || छ Chha<br /> |-<br /> | 7 || [[Punarvasu]] (पुनर्वसु)|| के Ke || को Ko || हा Haa || ही Hii<br /> |-<br /> | 8 || [[Pushya]] (पुष्य) || हु Hu || हे He || हो Ho || ड ḍa<br /> |-<br /> | 9 || [[Āshleshā]] (अश्लेषा)|| डी ḍii || डू ḍuu || डे ḍe || डो ḍo<br /> |-<br /> | 10 || [[Maghā]] (मघा)|| मा Maa || मी Mii || मू Muu || मे Me<br /> |-<br /> | 11 || Pūrva or [[Pūrva Phalgunī]] (पूर्व फल्गुनी) || मो Mo || टा ṭaa || टी ṭii || टू ṭuu<br /> |-<br /> | 12 || Uttara or [[Uttara Phalgunī]] (उत्तर फल्गुनी)|| टे ṭe || टो ṭo || पा Paa || पी Pii<br /> |-<br /> | 13 || [[Hasta (nakshatra)|Hasta]] (हस्त)|| पू Puu || ष Sha || ण Na || ठ ṭha<br /> |-<br /> | 14 || Chitra (चित्रा)|| पे Pe || पो Po || रा Raa || री Rii<br /> |-<br /> | 15 || [[Svātī]] (स्वाति) || रू Ruu || रे Re || रो Ro || ता Taa<br /> |-<br /> | 16 || [[Viśākhā]] (विशाखा)|| ती Tii || तू Tuu || ते Te || तो To<br /> |-<br /> | 17 || Anurādhā (अनुराधा)|| ना Naa || नी Nii || नू Nuu || ने Ne<br /> |-<br /> | 18 || [[Jyeshtha]] (ज्येष्ठा)|| नो No || या Yaa || यी Yii || यू Yuu<br /> |-<br /> | 19 || [[Mula (astrology)|Mula]] (मूल)|| ये Ye || यो Yo || भा Bhaa || भी Bhii<br /> |-<br /> | 20 || [[Pūrva Āshādhā]] (पूर्व आषाढ़)|| भू Bhuu || धा Dhaa || फा Bhaa/Phaa || ढा Daa<br /> |-<br /> | 21 || [[Uttara Āshadha|Uttara Āṣāḍhā]] (उत्तर आषाढ़)|| भे Bhe || भो Bho || जा Jaa || जी Jii<br /> |-<br /> | 22 || [[Śrāvaṇa]] (श्र‌ावण)|| खी Ju/Khii || खू Je/Khuu || खे Jo/Khe || खो Gha/Kho<br /> |-<br /> | 23 || Śrāviṣṭha (श्रविष्ठा) or [[Dhanishta]] || गा Gaa || गी Gii || गु Gu || गे Ge<br /> |-<br /> | 24 || [[Shatabhisha]] (शतभिषा)or Śatataraka || गो Go || सा Saa || सी Sii || सू Suu<br /> |-<br /> | 25 || [[Pūrva Bhādrapadā]] (पूर्व भाद्रपद)|| से Se || सो So || दा Daa || दी Dii<br /> |-<br /> | 26 || [[Uttara Bhādrapadā]] (उत्तर भाद्रपद)|| दू Duu || थ Tha || झ Jha || ञ ña <br /> |-<br /> | 27 || [[Revati]] (रेवती)|| दे De || दो Do || च Cha || ची Chii<br /> |}<br /> <br /> ==Names in Indian and Asian languages==<br /> The names of nakshatras in other languages are adapted from the Sanskrit variation (apabhramsa) through Pali or Prakrit. The variations evolved for easier pronunciation in popular usage.<br /> <br /> {| class=&quot;wikitable&quot; align=&quot;center&quot; cellspacing=&quot;2&quot; cellpadding=&quot;&quot;<br /> |- bgcolor=&quot;#cccccc&quot;<br /> !#!![[Sanskrit]]&lt;br&gt;संस्कृतम्!![[Odia language|Odia]]&lt;br&gt;ଓଡିଆ!![[Malayalam]]&lt;br&gt;മലയാളം!![[Tamil language|Tamil]]&lt;br&gt;தமிழ்!![[Sinhala language|Sinhala]]&lt;br&gt;සිoහල!![[Dhivehi language|Dhivehi]]&lt;br&gt;ދިވެހި!![[Telugu language|Telugu]]&lt;br&gt;తెలుగు!![[Kannada]]&lt;br&gt;ಕನ್ನಡ!![[Bengali language|Bengali]]&lt;br&gt;বাংলা!! Mongolian!! [[Chinese language|Chinese]]&lt;br&gt;宿!![[Classical Tibetan|Tibetan]]&lt;br&gt;བོད་སྐད། [http://tibetancalculator.com/tibetan-stars/ རྒྱུ་སྐར་ཉེ་བདུན།] <br /> |-<br /> | 1|| [[Ashvinī]]&lt;br&gt;अश्विनी||Aswini&lt;br&gt;ଅଶ୍ଵିନୀ||Ashvati&lt;br&gt;അശ്വതി ||Ashwini&lt;br&gt;அசுவினி|| Aswida&lt;br&gt;අස්විද|| A'sidha&lt;br&gt;އައްސިދަ|| Aswini&lt;br&gt;అశ్విని || Ashwini&lt;br&gt;ಅಶ್ವಿನಿ ||Ashwini&lt;br&gt;অশ্বিনী || Шийдэм || 婁|| ཐ་སྐར།<br /> |-<br /> | 2|| [[Bharanī]]&lt;br&gt;भरणी ||Dwijaa&lt;br&gt;ଦ୍ଵିଜା|| Bharaṇi&lt;br&gt;ഭരണി ||Baraṇi&lt;br&gt;பரணி || Berana&lt;br&gt;බෙරණ|| Burunu&lt;br&gt;ބުރުނު|| Bharani&lt;br&gt;భరణి|| Bharani&lt;br&gt;ಭರಣಿ ||Bharaṇi&lt;br&gt;ভরণী|| Гоё хүүхэн|| 胃|| བྲ་ཉེ།<br /> |-<br /> | 3 || [[Krittikā]]&lt;br&gt;कृत्तिका ||Krutikaa&lt;br&gt;କୃତିକା|| Kārttika&lt;br&gt;കാർത്തിക || Kārthikai&lt;br&gt;கார்த்திகை || Kethi&lt;br&gt;කැති|| Kethi&lt;br&gt;ކެތި|| Kritika&lt;br&gt;కృత్తిక|| Kritika&lt;br&gt;ಕೃತಿಕ|| Krittika&lt;br&gt;কৃত্তিকা ||Нэг эхт зургаан хөвгүүн || 昴|| སྨིན་དྲུག<br /> |-<br /> | 4 || [[Rohini (nakshatra)|Rohini]]&lt;br&gt;रोहिणी ||Rohini&lt;br&gt;ରୋହିଣୀ|| Rōhiṇi&lt;br&gt;രോഹിണി || Rōhiṇi&lt;br&gt;ரோகிணி || Rehena&lt;br&gt;රෙහෙණ|| Roanu&lt;br&gt;ރޯނު|| Rohini&lt;br&gt;రోహిణి|| Rohini&lt;br&gt;ರೋಹಿಣಿ ||Rohiṇi&lt;br&gt;রোহিণী || Чөлөөт эх || 畢|| སྣར་མ།<br /> |-<br /> | 5 || [[Mrigashīrsha]]&lt;br&gt;म्रृगशीर्षा ||Mrugasiraa&lt;br&gt;ମୃଗଶିରା|| Makayiram&lt;br&gt;മകയിരം || Mirugasīridam&lt;br&gt;மிருகசீரிடம் || Muwasirasa&lt;br&gt;මුවසිරස|| Miyaheli&lt;br&gt;މިޔަހެލި|| Mrigasiraa&lt;br&gt;మృగశిర|| Mrigashira&lt;br&gt;ಮೃಗಶಿರ ||Mrigashira&lt;br&gt;মৃগশিরা ||Гөрөөсөн толгой || 觜|| མགོ<br /> |-<br /> | 6 || [[Ardra (nakshatra)|Ārdrā]]&lt;br&gt;आर्द्रा ||Adra&lt;br&gt;ଆଦ୍ରା|| Ātira or Tiruvātira&lt;br&gt;ആതിര (തിരുവാതിര) || Thiruvādhirai&lt;br&gt;திருவாதிரை || Ada&lt;br&gt;අද|| Adha&lt;br&gt;އަދަ|| Arudra&lt;br&gt;ఆరుద్ర || Ardra&lt;br&gt;ಆರ್ದ್ರ|| Ardra&lt;br&gt;আর্দ্রা || Хэрцгий охин || 參|| ལག<br /> |-<br /> | 7 || [[Punarvasu]]&lt;br&gt;पुनर्वसु ||punarbasu&lt;br&gt;ପୁନର୍ବସୁ|| Puṇartam&lt;br&gt;പുണർതം || Punarpoosam&lt;br&gt;புனர்பூசம் || Punavasa&lt;br&gt;පුනාවස|| Funoas&lt;br&gt;ފުނޯސް|| Punarvasu&lt;br&gt;పునర్వసు|| Punarvasu&lt;br&gt;ಪುನರ್ವಸು ||Punarbasu&lt;br&gt;পুনর্বসু ||Өглөгт охин || 井|| ནབས་སོ།<br /> |-<br /> | 8 || [[Pushya]]&lt;br&gt;पुष्य ||Pushyaa&lt;br&gt;ପୁଷ୍ୟା|| Pūyam&lt;br&gt;പൂയം || Poosam&lt;br&gt;பூசம் || Pusha&lt;br&gt;පුෂ|| Fus&lt;br&gt;ފުސް|| Pushya&lt;br&gt;పుష్య|| Pushyami&lt;br&gt;ಪುಷ್ಯ ||Pushya&lt;br&gt;পুষ্যা || Найралт эх || 鬼|| རྒྱལ།<br /> |-<br /> | 9 || [[Āshleshā]]&lt;br&gt;आश्ळेषा / आश्लेषा||Ashleshaa&lt;br&gt;ଆଶ୍ଳେଷା||Āyilyam&lt;br&gt;ആയില്യം || Ayilyam&lt;br&gt;ஆயில்யம் || Aslisa&lt;br&gt;අස්ලිස|| Ahuliha&lt;br&gt;އަހުލިހަ|| Ashlesha&lt;br&gt;ఆశ్లేష|| Ashlesha&lt;br&gt;ಆಶ್ಲೇಷ ||Ashleshā&lt;br&gt;অশ্লেষা || Үнэг || 柳|| སྐག<br /> |-<br /> | 10 || [[Maghā]]&lt;br&gt;मघा ||Magaa&lt;br&gt;ମଘା|| Makam&lt;br&gt;മകം || Magam&lt;br&gt;மகம் || Maa&lt;br&gt;මා|| Maa&lt;br&gt;މާ|| Maghaa&lt;br&gt;మఘ|| Magha&lt;br&gt;ಮಘ ||Magha&lt;br&gt;মঘা ||Их морь || 星|| མཆུ།<br /> |-<br /> | 11 || Pūrva or [[Pūrva Phalgunī]]&lt;br&gt;पूर्व फल्गुनी ||Purba Falguni&lt;br&gt;ପୂର୍ବ ଫାଲ୍ଗୁନୀ|| Pūram&lt;br&gt;പൂരം || Pooram&lt;br&gt;பூரம் || Puwapal&lt;br&gt;පුවපල්|| Fura&lt;br&gt;ފުރަ|| Pubba&lt;br&gt;పూర్వ ఫల్గుని|| Poorva&lt;br&gt;ಪೂರ್ವ ||Purbaphālguni&lt;br&gt;পূর্বফাল্গুনী || Бага морь || 張|| གྲེ།<br /> |-<br /> | 12 || Uttara or [[Uttara Phalgunī]]&lt;br&gt;उत्तर फल्गुनी ||Uttara falguni&lt;br&gt;ଉତ୍ତର ଫାଲ୍ଗୁନୀ|| Utram&lt;br&gt;ഉത്രം ||Uthiram&lt;br&gt;உத்திரம் || Uttrapal&lt;br&gt;උත්රපල්|| Uthura&lt;br&gt;އުތުރަ|| Uttara&lt;br&gt;ఉత్తర ఫల్గుని|| Uttara&lt;br&gt;ಉತ್ತರ ||Uttar Phālguni&lt;br&gt;উত্তরফাল্গুনী || Харцага || 翼|| དབོ།<br /> |-<br /> | 13 || [[Nakshatra|Hasta]]&lt;br&gt;हस्त ||Hastaa&lt;br&gt;ହସ୍ତା||Attam&lt;br&gt;അത്തം || Ashtham&lt;br&gt;அசுதம் || Hatha&lt;br&gt;හත|| Atha&lt;br&gt;އަތަ|| Hasta&lt;br&gt;హస్త|| Hastaa&lt;br&gt;ಹಸ್ತ ||Hastā &lt;br&gt; হস্তা ||Тугчин || 軫|| མེ་བཞི།<br /> |-<br /> | 14 || [[Nakshatra|Chitrā]]&lt;br&gt;चित्रा ||Chitraa&lt;br&gt;ଚିତ୍ରା||Chittira&lt;br&gt;ചിത്തിര (ചിത്ര) || Chithirai&lt;br&gt;சித்திரை || Sitha&lt;br&gt;සිත|| Hitha&lt;br&gt;ހިތަ|| Chittaa&lt;br&gt;చిత్త|| Chitra&lt;br&gt;ಚಿತ್ರ ||Chitrā &lt;br&gt; চিত্রা ||Тэргүүн дагуул || 角|| ནག་པ།<br /> |-<br /> | 15 || [[Svātī]]&lt;br&gt;स्वाती ||Swati&lt;br&gt;ସ୍ଵାତୀ||Chōti&lt;br&gt;ചോതി || Swathi&lt;br&gt;சுவாதி || Saa&lt;br&gt;සා|| Hey&lt;br&gt;ހޭ|| Swaati&lt;br&gt;స్వాతి|| Swathi&lt;br&gt;ಸ್ವಾತಿ ||Swāti &lt;br&gt; স্বাতী || Салхины эх || 亢|| ས་རི།<br /> |-<br /> | 16 || [[Vishākhā]]&lt;br&gt;विशाखा ||Bishakha&lt;br&gt;ବିଶାଖା||Vishākham&lt;br&gt;വിശാഖം || Visakam&lt;br&gt;விசாகம்|| Wisa&lt;br&gt;විසා|| Vihaa&lt;br&gt;ވިހާ|| Vishaakha&lt;br&gt;విశాఖ|| Vishakha&lt;br&gt;ವಿಶಾಖ ||Bishakha &lt;br&gt; বিশাখা || Эрхтний тэнгэрт || 氐|| ས་ག<br /> |-<br /> | 17 ||[https://aaps.space/docs/nakshatra/Anuradha-Nakshatra/ Anurādhā]&lt;br&gt;अनुराधा ||Anuradha&lt;br&gt;ଅନୁରାଧା|| Anizham&lt;br&gt;അനിഴം || Anusham&lt;br&gt;அனுசம் || Anura&lt;br&gt;අනුර|| Nora&lt;br&gt;ނޮރަ|| Anuraadha&lt;br&gt;అనురాధ|| Anuradha&lt;br&gt;ಅನುರಾಧಾ ||Anuradha &lt;br&gt; অনুরাধা || Гар од || 房||ལྷ་མཚམས།<br /> |-<br /> | 18 || [[Jyeshtha]]&lt;br&gt;ज्येष्ठा ||Jyosthaa&lt;br&gt;ଜ୍ୟୋଷ୍ଠା||Kēṭṭa (Trikkēṭṭa)&lt;br&gt;തൃക്കേട്ട|| Kettai&lt;br&gt;கேட்டை || Deta&lt;br&gt;දෙට|| Dhosha&lt;br&gt;ދޮށަ|| Jyesthaa&lt;br&gt;జ్యేష్ఠ|| Jyestha&lt;br&gt;ಜ್ಯೇಷ್ಠ ||Jyestha &lt;br&gt; জ্যেষ্ঠা || Хонгорцог || 心|| སྣྲོན།<br /> |-<br /> | 19 || [[Mula (astrology)|Mūla]]&lt;br&gt;मूल ||Mulaa&lt;br&gt;ମୂଳା|| Mūlam&lt;br&gt;മൂലം|| Mūlam&lt;br&gt;மூலம் || Moola&lt;br&gt;මුල|| Mula&lt;br&gt;މުލަ|| Moolaa&lt;br&gt;మూల|| Moola&lt;br&gt;ಮೂಲ ||Mula &lt;br&gt; মূলা || Онгоц || 尾|| སྣུབས།<br /> |-<br /> | 20 || [[Pūrva Ashādhā]]&lt;br&gt;पूर्वाषाढा ||Purbasaadhaa&lt;br&gt;ପୂର୍ବାଷାଢା|| Pūrāṭam&lt;br&gt;പൂരാടം || Pūradam&lt;br&gt;பூராடம் || Puwasala&lt;br&gt;පුවසල|| Furahalha&lt;br&gt;ފުރަހަޅަ|| Poorvashaadha&lt;br&gt;పూర్వాషాడ|| Poorvashadha&lt;br&gt;ಪುರ್ವಾಷಾಡ ||Poorbashada&lt;br&gt;পূর্বাষাঢ়া || Суулга || 箕||ཆུ་སྟོད།<br /> |-<br /> | 21 || [[Uttara Ashādhā]]&lt;br&gt;उत्तराषाढा ||Uttarasaadha&lt;br&gt;ଉତ୍ତରଷାଢା|| Utrāṭam&lt;br&gt;ഉത്രാടം || Uthirādam&lt;br&gt;உத்திராடம் || Uttrasala&lt;br&gt;උත්රසල|| Uthurahalha&lt;br&gt;އުތުރަހަޅަ|| Uttarashaadha&lt;br&gt;ఉత్తరాషాడ|| Uttarashadha&lt;br&gt;ಉತ್ತರಾಷಾಡ ||Uttarashada]]&lt;br&gt;উত্তরাষাঢ়া || Элдэв тэнгэртэн || 斗|| ཆུ་སྨད།<br /> |-<br /> | 22 || [[Shravana]]&lt;br&gt;श्रवण ||Sravana&lt;br&gt;ଶ୍ରବଣା|| Tiruvōnam &lt;br&gt;ഓണം (തിരുവോണം)|| Tiruvōnam&lt;br&gt;திருவோணம் || Suvana&lt;br&gt;සුවණ|| Huvan&lt;br&gt;ހުވަން|| Shravana&lt;br&gt;శ్రవణ|| Shravana&lt;br&gt;ಶ್ರವಣ||Shraban&lt;br&gt;শ্রবণা || Булаагч/Яруу эгшигт || 女|| གྲོ་བཞིན།<br /> |-<br /> | 23 || Shravishthā or [[Dhanishta]] &lt;br&gt;श्रविष्ठा or धनिष्ठा ||Dhanishathaa&lt;br&gt;ଧନିଷ୍ଠା|| Aviṭṭam&lt;br&gt;അവിട്ടം || Aviṭṭam&lt;br&gt;அவிட்டம் || Denata&lt;br&gt;දෙණට|| Dhinasha&lt;br&gt;ދިނަށަ|| Dhanishta&lt;br&gt;ధనిష్ఠ|| Dhanishta&lt;br&gt;ಧನಿಷ್ಠ||Dhanishta&lt;br&gt;ধনিষ্ঠা ||Тооно || 虛||མོན་གྲེ།<br /> |-<br /> | 24 || [[Shatabhishā]] or Shatataraka&lt;br&gt;शतभिषक् / शततारका||Satavisaa&lt;br&gt;ସତଭିଷା|| Chatayam&lt;br&gt;ചതയം || Sadayam&lt;br&gt;சதயம் || Siyawasa&lt;br&gt;සියාවස|| Hiyavihaa&lt;br&gt;ހިޔަވިހާ|| Shatabhisha&lt;br&gt;శతభిష|| Shatabhisha&lt;br&gt;ಶತಭಿಷ ||Shatabhisha &lt;br&gt; শতভিষা || Чөдөр || 危|| མོན་གྲུ<br /> |-<br /> | 25 || [[Pūrva Bhādrapadā]]&lt;br&gt;पूर्वभाद्रपदा / पूर्वप्रोष्ठपदा ||Purba vadrapada&lt;br&gt;ପୂର୍ବଭାଦ୍ରପଦ|| Pūruruṭṭāti&lt;br&gt;പൂരുരുട്ടാതി || Pūraṭṭādhi&lt;br&gt;பூரட்டாதி || Puvaputupa&lt;br&gt;පුවපුටුප|| Furabadhuruva&lt;br&gt;ފުރަބަދުރުވަ|| Poorvabhadraa&lt;br&gt;పూర్వ భాద్రపద|| Poorva Bhadrapada&lt;br&gt;ಪೂರ್ವ ಭಾದ್ರಪದ ||Poorbabhadra &lt;br&gt; পূর্বভাদ্রপদ || Шувуун хошуут || 室||ཁྲུམས་སྟོད།<br /> |-<br /> | 26 || [[Uttara Bhādrapadā]]&lt;br&gt;उत्तरभाद्रपदा / उत्तरप्रोष्ठपदा ||Uttara vadrapada&lt;br&gt;ଉତ୍ତରାଭାଦ୍ରପଦ|| Uttṛṭṭāti&lt;br&gt;ഉത്രട്ടാതി || Uttṛṭṭādhi&lt;br&gt;உத்திரட்டாதி || Uttraputupa&lt;br&gt;උත්රපුටුප|| Fasbadhuruva&lt;br&gt;ފަސްބަދުރުވަ|| Uttaraa bhadra&lt;br&gt;ఉత్తర భాద్రపద|| Uttara Bhadrapada&lt;br&gt;ಉತ್ತರ ಭಾದ್ರಪದ||Uttarbabhadra &lt;br&gt; উত্তরভাদ্রপদ || Могой хүлэгч || 壁|| ཁྲུམས་སྨད།<br /> |<br /> |-<br /> | 27 || [[Revati (nakshatra)|Revati]]&lt;br&gt;रेवती ||Revati&lt;br&gt;ରେବତୀ|| Rēvati&lt;br&gt;രേവതി || Rēvathi&lt;br&gt;ரேவதி || Revathi&lt;br&gt;රේවතී|| Reyva&lt;br&gt;ރޭވަ|| Revati&lt;br&gt;రేవతి|| Revati&lt;br&gt;ರೇವತಿ ||Rebati &lt;br&gt; রেবতী || Дэлгэрүүлэгч || 奎|| ནམ་གྲུ།<br /> |}<br /> <br /> ==References==<br /> <br /> {{reflist}}<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Nakshatra_system&diff=129866 2021-04-16T04:08:37Z

<p>BharataPuru: Initial creation of the page</p> <hr /> <div>{{short description|Wikipedia list article}}In [[Hindu astrology]], there are 27 '''''[[nakshatra]]s''''' (&quot;lunar mansions&quot;, Sanskrit: नक्षत्र, IAST: ''Nakṣatra''), or sectors along the [[ecliptic]]. A list of them is first found in the ''[[Vedanga Jyotisha]]'', a text dated to the final centuries BCE{{fact|date=January 2020}}. The ''Nakṣatra'' system predates the influence of [[Hellenistic astronomy]] on Vedic tradition, which became prevalent from about the 2nd century CE. There are various systems of enumerating the ''Nakṣatra''-s;{{Clarify|date=October 2011}} although there are 27-28 days to a sidereal month, by custom only 27 days are used. The following list gives the corresponding regions of sky.&lt;ref&gt;{{cite book|first=Arthur Llewellyn|last=Basham|authorlink=Arthur Llewellyn Basham|title=The Wonder That Was India|chapter=Appendix II: Astronomy|p=490|year=1954|publisher=[[Sidgwick and Jackson]]|title-link=The Wonder That Was India}}&lt;/ref&gt;<br /> <br /> ==Nakshatras==<br /> {| class=&quot;wikitable&quot; align=&quot;center&quot; cellspacing=&quot;2&quot; cellpadding=&quot;&quot;<br /> |- bgcolor=#cccccc<br /> !No.!! Name !!Associated stars!!Description!! Image<br /> |-<br /> | 1<br /> ||'''[[Ashvini]]'''&lt;br/&gt;&quot;physician to the Gods&quot;<br /> ||[[Beta Arietis|β]] and [[Gamma Arietis|γ]] [[Aries (constellation)|Arietis]]<br /> ||<br /> *''Lord'': Ketu (South lunar node)<br /> *''Symbol'' : Horse's head<br /> *''Deity'' : [[Ashvins]], the horse-headed twins who are physicians to the gods<br /> *''Indian zodiac'': 0° - 13°20' Mesha<br /> *''west zodiac'' 23°46 Aries - 7°06' Taurus<br /> ||[[File:Aries constellation map.svg|100px]]<br /> |-<br /> | 2<br /> ||'''[[Bharani]]'''&lt;br/&gt;&quot;the bearer&quot;<br /> ||[[35 Arietis|35]], [[39 Arietis|39]], and [[41 Arietis|41]] [[Aries (constellation)|Arietis]]<br /> ||<br /> *''Lord'': Shukra (Venus)<br /> *''Symbol'': Yoni, the female organ of reproduction<br /> *''Deity'': [[Yama]], [[god of death]] or [[Dharma]]<br /> *''Indian zodiac'': 13° 20' - 26°40' Mesha<br /> *''Western zodiac'' 7°06' - 20°26' Taurus<br /> ||[[File:Aries constellation map.svg|100px]]<br /> |-<br /> | 3<br /> ||'''[[Krittika]]'''&lt;br/&gt;an old name of the Pleiades; personified as the nurses of ''Kārttikeya'', a son of Shiva.<br /> ||[[Pleiades]]<br /> ||<br /> *''Lord'': [[Surya]] (Sun)<br /> *''Symbol'': [[Knife]] or [[spear]]<br /> *''Deity'' : [[Agni]], god of fire<br /> *''Indian zodiac'': 26°40' [[Mesha]] - 10° Vrishabha<br /> *''Western zodiac'' 20°26' Taurus - 3°46’ Gemini<br /> ||[[File:Taurus constellation map.png|100px]]<br /> |-<br /> | 4<br /> ||'''[[Rohini (nakshatra)|Rohini]]'''&lt;br&gt;&quot;the red one&quot;, a name of [[Aldebaran]]. Also known as ''brāhmī''<br /> ||[[Aldebaran]]<br /> ||<br /> *''Lord'': [[Chandra]] ([[Moon]])<br /> *''Symbol'': Cart or [[chariot]], [[temple]], [[banyan tree]]<br /> *''Deity'' : [[Brahma]] or [[Prajapati]], [[Creator deity|the Creator]]<br /> *''Indian zodiac'': 10° - 23°20' Vrishabha<br /> *''Western zodiac'' 3°46 - 17°06' Gemini<br /> ||[[File:Taurus constellation map.png|100px]]<br /> |-<br /> | 5<br /> ||'''[[Mrigashīrsha]]'''&lt;br/&gt;&quot;the deer's head&quot;. Also known as ''āgrahāyaṇī''<br /> ||[[Lambda Orionis|λ]], [[Phi Orionis|φ]] Orionis<br /> ||<br /> *''Lord'': Mangala (Mars)<br /> *''Symbol'': Deer's head<br /> *''Deity'': [[Soma (deity)|Soma]], [[Chandra]], the Moon god<br /> *''Indian zodiac'': 23° 20' Vrishabha - 6° 40' Mithuna<br /> *''Western zodiac'': 17°06' Gemini - 0°26' Cancer<br /> ||[[File:Orion constellation map.png|100px]]<br /> |-<br /> | 6<br /> ||'''[[Ardra (nakshatra)|Ardra]]'''&lt;br/&gt;&quot;the storm god&quot;<br /> ||[[Betelgeuse]]<br /> ||<br /> *''Lord'': Rahu (North lunar node)<br /> *''Symbol'': Teardrop, diamond, a human head<br /> *''Deity'' : [[Rudra]], the storm god<br /> *''Indian zodiac'': 6° 40' - 20° Mithuna<br /> *''Western zodiac'': 0°26' - 13°46 Cancer<br /> ||[[File:Orion constellation map.png|100px]]<br /> |-<br /> | 7<br /> ||'''[[Punarvasu]]''' (dual)&lt;br/&gt;&quot;the two restorers of goods&quot;, also known as ''yamakau'' &quot;the two chariots&quot;<br /> ||[[Castor (star)|Castor]] and [[Pollux (star)|Pollux]]<br /> ||<br /> *''Lord'': Guru (Jupiter)<br /> *''Symbol'' : Bow and quiver<br /> *''Deity'' : [[Aditi]], mother of the gods<br /> *''Indian zodiac'': 20° Mithuna - 3°20' Karka<br /> *''Western zodiac'' 13°46 - 27°06' Cancer<br /> ||[[File:Gemini constellation map.svg|100px]]<br /> |-<br /> | 8<br /> ||'''[[Pushya]]'''&lt;br/&gt;&quot;the nourisher&quot;, also known as ''sidhya'' or ''tiṣya''<br /> ||[[Gamma Cancri|γ]], [[Delta Cancri|δ]] and [[Theta Cancri|θ]] [[Cancer (constellation)|Cancri]]<br /> ||<br /> *''Lord'': [[Shani]] ([[Saturn]])<br /> *''Symbol'' : Cow's udder, [[Lotus Flower|lotus]], arrow and circle<br /> *''Deity'' : [[Bṛhaspati]], priest of the gods<br /> *''Indian zodiac'': 3°20' -16°40' [[Karka]]<br /> *''Western zodiac'' 27°06' Cancer - 10°26' [[Leo (constellation)|Leo]]<br /> ||[[File:Cancer constellation map.png|100px]]<br /> |-<br /> | 9<br /> ||'''[[Āshleshā]]'''&lt;br/&gt;&quot;the embrace&quot;<br /> ||δ, ε, η, ρ, and [[Sigma Hydrae|σ]] [[Hydra (constellation)|Hydrae]]<br /> ||<br /> *''Lord'': Budh (Mercury)<br /> *''Symbol'': Serpent<br /> *''Deity'' : Sarpas or [[Nāga|Nagas]], deified snakes<br /> *''Indian zodiac'': 16°40' - 30° Karka<br /> *''Western zodiac'' 10°26' - 23°46 Leo<br /> ||[[File:Hydra constellation map.png|100px]]<br /> |-<br /> | 10<br /> ||'''[[Maghā]]'''&lt;br/&gt;&quot;the bountiful&quot;<br /> ||[[Regulus]]<br /> ||<br /> *''Lord'': Ketu (south lunar node)<br /> *''Symbol'' : Royal Throne<br /> *''Deity'' : [[Pitrs]], 'The Fathers', family ancestors<br /> *''Indian zodiac'': 0° - 13°20' Simha<br /> *''Western zodiac'' 23°46 Leo - 7°06' Virgo<br /> ||[[File:Leo constellation map.png|100px]]<br /> |-<br /> | 11<br /> ||'''[[Pūrva Phalgunī]]'''&lt;br&gt;&quot;first reddish one&quot;<br /> ||[[Delta Leonis|δ]] and [[Theta Leonis|θ]] [[Leo (constellation)|Leonis]]<br /> ||<br /> *''Lord'': Shukra (Venus)<br /> *''Symbol'' : Front legs of bed, hammock, fig tree<br /> *''Deity'' : [[Bhaga]], god of marital bliss and prosperity<br /> *''Indian zodiac'': 13°20' - 26°40' Simha<br /> *''Western zodiac'' 7°06' - 20°26' Virgo<br /> || [[File:Leo constellation map.png|100px]]<br /> |-<br /> | 12<br /> ||'''[[Uttara Phalgunī]]'''&lt;br/&gt;&quot;second reddish one&quot;<br /> ||[[Denebola]]<br /> ||<br /> *''Lord'': Surya (Sun)<br /> *''Symbol'': Four legs of bed, hammock<br /> *''Deity'' : [[Aryaman]], god of patronage and favours<br /> *''Indian zodiac'': 26°40' Simha- 10° Kanya<br /> *''Western zodiac'' 20°26' Virgo - 3°46 Libra<br /> || [[File:Leo constellation map.png|100px]]<br /> |-<br /> | 13<br /> ||'''[[Nakshatra|Hasta]]'''&lt;br/&gt;&quot;the hand&quot;<br /> ||[[Alpha Corvi|α]], [[Beta Corvi|β]], [[Gamma Corvi|γ]], [[Delta Corvi|δ]] and [[Epsilon Corvi|ε]] [[Corvus (constellation)|Corvi]]<br /> ||<br /> *''Lord'': Chandra (Moon)<br /> *''Symbol'': Hand or fist<br /> *''Deity'' : [[Surya|Saviti]] or [[Surya]], the Sun god<br /> *''Indian zodiac'': 10° - 23°20' Kanya<br /> *''Western zodiac'' 3°46 - 17°06' Libra<br /> ||[[File:Corvus constellation map.png|100px]]<br /> |-<br /> | 14<br /> ||'''Chitra'''&lt;br/&gt;&quot;the bright one&quot;, a name of Spica<br /> ||[[Spica]]<br /> ||<br /> *''Lord'': Mangala (Mars)<br /> *''Symbol'': Bright jewel or pearl<br /> *''Deity'' : [[Tvastar]] or [[Vishvakarman]], the celestial architect<br /> *''Indian zodiac'': 23°20' Kanya - 6°40' Tula<br /> *''Western zodiac'': 17°06' Libra - 0°26' Scorpio<br /> || [[File:Virgo constellation map.png|100px]]<br /> |-<br /> | 15<br /> ||'''[[Swati the Star|Swāti]]'''&lt;br/&gt;&quot;Su-Ati (sanskrit) Very good&quot; name of Arcturus<br /> ||[[Arcturus]]<br /> ||<br /> *''Lord'': Rahu (north lunar node)<br /> *''Symbol'': Shoot of plant, coral<br /> *''Deity'' : [[Vayu]], the Wind god<br /> *''Indian zodiac'': 6°40' - 20° Tula<br /> *''Western zodiac'' 0°26' - 13°46 Scorpio<br /> || [[File:Bootes constellation map.png|100px]]<br /> |-<br /> | 16<br /> ||'''Vishakha'''&lt;br/&gt;&quot;forked, having branches&quot;; also known as ''rādhā'' &quot;the gift&quot;<br /> || [[Alpha Librae|α]], [[Beta Librae|β]], [[Gamma Librae|γ]] and ι [[Libra (constellation)|Librae]]<br /> ||<br /> *''Lord'': Guru (Jupiter)<br /> *''Symbol'' : Triumphal arch, potter's wheel<br /> *''Deity'' : [[Indra]], chief of the gods; [[Agni]], god of Fire<br /> *''Indian zodiac'': 20° Tula - 3°20' Vrishchika<br /> *''Western zodiac'' 13°46 - 27°06' Scorpio<br /> ||[[File:Libra constellation map.png|100px]]<br /> |-<br /> | 17<br /> ||'''[[Anuradha (nakshatra)|Anuradha]]'''&lt;br/&gt; &quot;following ''rādhā''&quot;<br /> || [[Beta Scorpii|β]], [[Delta Scorpii|δ]] and [[Pi Scorpii|π]] [[Scorpius (constellation)|Scorpionis]]<br /> ||<br /> *''Lord'': Shani (Saturn)<br /> *''Symbol'' : Triumphal archway, lotus<br /> *''Deity'' : [[Mitra (Vedic)|Mitra]], one of Adityas of friendship and partnership<br /> *''Indian zodiac'': 3°20' - 16°40' Vrishchika<br /> *''Western zodiac'' 27°06' Scorpio - 10°26' Sagittarius<br /> || [[File:Scorpius constellation map.svg|100px]]<br /> |-<br /> | 18<br /> ||'''[[Jyeshtha (nakshatra)|Jyeshtha]]'''&lt;br/&gt;&quot;the eldest, most excellent&quot;<br /> || [[Alpha Scorpii|α]], [[Sigma Scorpii|σ]], and [[Tau Scorpii|τ]] [[Scorpius (constellation)|Scorpionis]]<br /> ||<br /> *''Lord'': Budh (Mercury)<br /> *''Symbol'' : circular amulet, umbrella, earring<br /> *''Deity'' : [[Indra]], chief of the gods<br /> *''Indian zodiac'': 16°40' - 30° Vrishchika<br /> *''Western zodiac'' 10°26' - 23°46 Sagittarius<br /> || [[File:Scorpius constellation map.png|100px]]<br /> |-<br /> | 19<br /> ||'''[[Mula (nakshatra)|Mula]]'''&lt;br/&gt;&quot;the root&quot;<br /> || [[Epsilon Scorpii|ε]], ζ, [[Eta Scorpii|η]], [[Theta Scorpii|θ]], ι, [[Kappa Scorpii|κ]], [[Lambda Scorpii|λ]], [[Mu Scorpii|μ]] and [[Nu Scorpii|ν]] [[Scorpius (constellation)|Scorpionis]]<br /> ||<br /> *''Lord'': Ketu (south lunar node)<br /> *''Symbol'' : Bunch of roots tied together, elephant goad<br /> *''Deity'' : [[Nirrti]], goddess of dissolution and destruction<br /> *''Indian zodiac'': 0° - 13°20' Dhanus<br /> *''Western zodiac'' 23°46 Sagittarius - 7°06' Capricorn<br /> || [[File:Scorpius constellation map.png|100px]]<br /> |-<br /> | 20<br /> ||'''[[Purva Ashadha]]'''&lt;br/&gt;&quot;first of the ''āṣāḍhā''&quot;, ''āṣāḍhā'' &quot;the invincible one&quot; being the name of a constellation<br /> ||[[Delta Sagittarii|δ]] and [[Epsilon Sagittarii|ε]] [[Sagittarius (constellation)|Sagittarii]]<br /> ||<br /> *''Lord'': Shukra (Venus)<br /> *''Symbol'': Elephant tusk, fan, winnowing basket<br /> *''Deity'' : [[Ap (water)|Apah]], god of Water<br /> *''Indian zodiac'': 13°20' - 26°40' Dhanus<br /> *''Western zodiac'' 7°06' - 20°26' Capricorn<br /> || [[File:Sagittarius constellation map.png|100px]]<br /> |-<br /> | 21<br /> ||'''[[Uttara Ashadha]]'''&lt;br/&gt;&quot;second of the ''āṣāḍhā''&quot;<br /> ||[[Zeta Sagittarii|ζ]] and [[Sigma Sagittarii|σ]] [[Sagittarius (constellation)|Sagittarii]]<br /> ||<br /> *''Lord'': Surya (Sun)<br /> *''Symbol'' : Elephant tusk, small bed<br /> *''Deity'' : [[Visvedevas]], universal gods<br /> *''Indian zodiac'': 26°40' Dhanus - 10° Makara<br /> *''Western zodiac'' 20°26' Capricorn - 3°46 Aquarius<br /> || [[File:Sagittarius constellation map.png|100px]]<br /> |-<br /> | 22<br /> ||'''[[Shravana|Sravana]]'''&lt;br/&gt;<br /> ||[[Alpha Aquilae|α]], [[Beta Aquilae|β]] and [[Gamma Aquilae|γ]] [[Aquila (constellation)|Aquilae]]<br /> ||<br /> *''Lord'': Chandra (Moon)<br /> *''Symbol'' : Ear or Three Footprints<br /> *''Deity'' : [[Vishnu]], preserver of universe<br /> *''Indian zodiac'': 10° - 23°20' Makara<br /> *''Western zodiac'' 3°46 - 17°06' Aquarius<br /> || [[File:Aquila constellation map.png|100px]]<br /> |-<br /> | 23<br /> ||'''[[Dhanishta]]''' &lt;br/&gt;&quot;most famous&quot;, also ''Shravishthā'' &quot;swiftest&quot;<br /> || [[Alpha Delphini|α]] to [[Delta Delphini|δ Delphini]]<br /> ||<br /> *''Lord'': Mangala (Mars)<br /> *''Symbol'' : Drum or flute<br /> *''Deity'' : Eight [[vasu]]s, deities of earthly abundance<br /> *''Indian zodiac'': 23°20' Makara - 6°40' Kumbha<br /> *''Western zodiac'' 17°06' Aquarius - 0°26' Pisces<br /> || [[File:Delphinus constellation map.png|100px]]<br /> |-<br /> | 24<br /> ||'''[[Shatabhisha]]&lt;br/&gt;&quot;'''requiring a hundred physicians'''&quot;<br /> || [[Gamma Aquarii|γ Aquarii]]<br /> ||<br /> *''Lord'': Rahu (north lunar node)<br /> *''Symbol'' : Empty circle, 1,000 flowers or stars<br /> *''Deity'' : [[Varuna]], god of cosmic waters, sky and earth<br /> *''Indian zodiac'': 6°40' - 20° Kumbha ; ''Western zodiac'' 0°26' - 13°46 Pisces<br /> || [[File:Aquarius constellation map.svg|100px]]<br /> |-<br /> | 25<br /> ||'''[[Purva Bhadrapada]]'''&lt;br/&gt;&quot;the first of the blessed feet&quot;<br /> || [[Alpha Pegasi|α]] and [[Beta Pegasi|β]] [[Pegasus (constellation)|Pegasi]]<br /> ||<br /> *''Lord'': Guru (Jupiter)<br /> *''Symbol'' : Swords or two front legs of funeral cot, man with two faces<br /> *''Deity'' : '''Ajaikapada''', an ancient fire dragon<br /> *''Indian zodiac'': 20° Kumbha - 3°20' Meena ; ''Western zodiac'' 13°46 - 27°06' Pisces<br /> || [[File:Pegasus constellation map.png|100px]]<br /> |-<br /> | 26<br /> ||'''[[Uttara Bhādrapadā]]'''&lt;br/&gt;&quot;the second of the blessed feet&quot;<br /> || [[Gamma Pegasi|γ]] [[Pegasus (constellation)|Pegasi]] and [[Alpha Andromedae|α]] [[Andromeda (constellation)|Andromedae]]<br /> ||<br /> *''Lord'': Shani (Saturn)<br /> *''Symbol'' : Twins, back legs of funeral cot, snake in the water<br /> *''Deity'' : '''Ahirbudhnya''', serpent or dragon of the deep<br /> *''Indian zodiac'': 3°20' - 16°40' Meena ; ''Western zodiac'' 27°06' Pisces - 10°26' Aries<br /> || [[File:Andromeda constellation map (1).png|100px]]<br /> |-<br /> | 27<br /> ||'''[[Revati (nakshatra)|Revati]]'''&lt;br/&gt;&quot;prosperous&quot;<br /> || ζ [[Pisces (constellation)|Piscium]]<br /> ||<br /> *''Lord'': Budh (Mercury)<br /> *''Symbol'' : Fish or a pair of fish, drum<br /> *''Deity'' : [[Pushan]], nourisher, the protective deity<br /> *''Indian zodiac'': 16°40' - 30° Meena<br /> *''Western zodiac'' 10°26' - 23°46 Aries<br /> || [[File:Pisces constellation map.png|100px]]<br /> |-<br /> | 28<br /> ||'''[[Abhijit (nakshatra)|Abhijit]]'''&lt;br/&gt; &quot;victorious&quot;&lt;ref&gt;<br /> Its longitude starts from 06° 40' to 10° 53' 40'' in sidereal Capricorn i.e. from the last quarter of Uttra Ashadha to first 1/15 th part of Shravana.{{Citation needed|date=October 2011}}<br /> Its span is 4° 13' 40''.{{Citation needed|date=October 2011}} The span of 27 mean daily lunar motions totals 355.76167 degrees, and together these total 359.98945 degrees. Considered an &quot;intercalary&quot; lunar mansion.{{Citation needed|date=October 2011}}&lt;/ref&gt;<br /> || [[Alpha Lyrae|α]], [[Epsilon Lyrae|ε]] and ζ [[Lyra (constellation)|Lyrae]] - [[Vega]]<br /> || ''Lord'': [[Brahma]] (creator)<br /> *''Indian zodiac'': 06° 40' - 10° 53' 40 Makara<br /> || [[File:Lyra constellation map.png|100px]]<br /> |}<br /> <br /> ==Padas (quarters)==<br /> The 27 Nakshatras cover 13°20’ of the [[ecliptic]] each. Each Nakshatra is also divided into quarters or ''padas'' of 3°20’, and the below table lists the appropriate starting sound to name the child. The 27 nakshatras, each with 4 padas, give 108, which is the number of beads in a japa mala, indicating all the elements (ansh) of Vishnu:<br /> <br /> {| class=&quot;wikitable&quot; align=&quot;center&quot; cellspacing=&quot;2&quot; cellpadding=&quot;&quot;<br /> |- bgcolor=#cccccc<br /> !#!! Name !! Pada 1 !! Pada 2 !! Pada 3 !! Pada 4<br /> |-<br /> | 1|| Ashwini (अश्विनि)|| चु Chu || चे Che || चो Cho || ला Laa<br /> |-<br /> | 2|| [[Bharani]] (भरणी)|| ली Lii || लू Luu || ले Le || लो Lo<br /> |-<br /> | 3 || [[Krittika]] (कृत्तिका)|| अ A || ई I || उ U || ए E<br /> |-<br /> | 4 || Rohini(रोहिणी)|| ओ O || वा Vaa/Baa || वी Vii/Bii || वु Vuu/Buu<br /> |-<br /> | 5 || [[Mrigashīrsha]](मृगशीर्ष)|| वे Ve/Be || वो Vo/Bo || का Kaa || की Kii<br /> |-<br /> | 6 || [[Ardra (nakshatra)|Ārdrā]] (आर्द्रा)|| कु Ku || घ Gha || ङ Ng/Na || छ Chha<br /> |-<br /> | 7 || [[Punarvasu]] (पुनर्वसु)|| के Ke || को Ko || हा Haa || ही Hii<br /> |-<br /> | 8 || [[Pushya]] (पुष्य) || हु Hu || हे He || हो Ho || ड ḍa<br /> |-<br /> | 9 || [[Āshleshā]] (अश्लेषा)|| डी ḍii || डू ḍuu || डे ḍe || डो ḍo<br /> |-<br /> | 10 || [[Maghā]] (मघा)|| मा Maa || मी Mii || मू Muu || मे Me<br /> |-<br /> | 11 || Pūrva or [[Pūrva Phalgunī]] (पूर्व फल्गुनी) || मो Mo || टा ṭaa || टी ṭii || टू ṭuu<br /> |-<br /> | 12 || Uttara or [[Uttara Phalgunī]] (उत्तर फल्गुनी)|| टे ṭe || टो ṭo || पा Paa || पी Pii<br /> |-<br /> | 13 || [[Hasta (nakshatra)|Hasta]] (हस्त)|| पू Puu || ष Sha || ण Na || ठ ṭha<br /> |-<br /> | 14 || Chitra (चित्रा)|| पे Pe || पो Po || रा Raa || री Rii<br /> |-<br /> | 15 || [[Svātī]] (स्वाति) || रू Ruu || रे Re || रो Ro || ता Taa<br /> |-<br /> | 16 || [[Viśākhā]] (विशाखा)|| ती Tii || तू Tuu || ते Te || तो To<br /> |-<br /> | 17 || Anurādhā (अनुराधा)|| ना Naa || नी Nii || नू Nuu || ने Ne<br /> |-<br /> | 18 || [[Jyeshtha]] (ज्येष्ठा)|| नो No || या Yaa || यी Yii || यू Yuu<br /> |-<br /> | 19 || [[Mula (astrology)|Mula]] (मूल)|| ये Ye || यो Yo || भा Bhaa || भी Bhii<br /> |-<br /> | 20 || [[Pūrva Āshādhā]] (पूर्व आषाढ़)|| भू Bhuu || धा Dhaa || फा Bhaa/Phaa || ढा Daa<br /> |-<br /> | 21 || [[Uttara Āshadha|Uttara Āṣāḍhā]] (उत्तर आषाढ़)|| भे Bhe || भो Bho || जा Jaa || जी Jii<br /> |-<br /> | 22 || [[Śrāvaṇa]] (श्र‌ावण)|| खी Ju/Khii || खू Je/Khuu || खे Jo/Khe || खो Gha/Kho<br /> |-<br /> | 23 || Śrāviṣṭha (श्रविष्ठा) or [[Dhanishta]] || गा Gaa || गी Gii || गु Gu || गे Ge<br /> |-<br /> | 24 || [[Shatabhisha]] (शतभिषा)or Śatataraka || गो Go || सा Saa || सी Sii || सू Suu<br /> |-<br /> | 25 || [[Pūrva Bhādrapadā]] (पूर्व भाद्रपद)|| से Se || सो So || दा Daa || दी Dii<br /> |-<br /> | 26 || [[Uttara Bhādrapadā]] (उत्तर भाद्रपद)|| दू Duu || थ Tha || झ Jha || ञ ña <br /> |-<br /> | 27 || [[Revati]] (रेवती)|| दे De || दो Do || च Cha || ची Chii<br /> |}<br /> <br /> ==Names in Indian and Asian languages==<br /> The names of nakshatras in other languages are adapted from the Sanskrit variation (apabhramsa) through Pali or Prakrit. The variations evolved for easier pronunciation in popular usage.<br /> <br /> {| class=&quot;wikitable&quot; align=&quot;center&quot; cellspacing=&quot;2&quot; cellpadding=&quot;&quot;<br /> |- bgcolor=#cccccc<br /> !#!![[Sanskrit]]&lt;br&gt;संस्कृतम्!![[Odia language|Odia]]&lt;br&gt;ଓଡିଆ!![[Malayalam]]&lt;br&gt;മലയാളം!![[Tamil language|Tamil]]&lt;br&gt;தமிழ்!![[Sinhala language|Sinhala]]&lt;br&gt;සිoහල!![[Dhivehi language|Dhivehi]]&lt;br&gt;ދިވެހި!![[Telugu language|Telugu]]&lt;br&gt;తెలుగు!![[Kannada]]&lt;br&gt;ಕನ್ನಡ!![[Bengali language|Bengali]]&lt;br&gt;বাংলা!! Mongolian!! [[Chinese language|Chinese]]&lt;br&gt;宿!![[Classical Tibetan|Tibetan]]&lt;br&gt;བོད་སྐད། [http://tibetancalculator.com/tibetan-stars/ རྒྱུ་སྐར་ཉེ་བདུན།] <br /> |-<br /> | 1|| [[Ashvinī]]&lt;br&gt;अश्विनी||Aswini&lt;br&gt;ଅଶ୍ଵିନୀ||Ashvati&lt;br&gt;അശ്വതി ||Ashwini&lt;br&gt;அசுவினி|| Aswida&lt;br&gt;අස්විද|| A'sidha&lt;br&gt;އައްސިދަ|| Aswini&lt;br&gt;అశ్విని || Ashwini&lt;br&gt;ಅಶ್ವಿನಿ ||Ashwini&lt;br&gt;অশ্বিনী || Шийдэм || 婁|| ཐ་སྐར།<br /> |-<br /> | 2|| [[Bharanī]]&lt;br&gt;भरणी ||Dwijaa&lt;br&gt;ଦ୍ଵିଜା|| Bharaṇi&lt;br&gt;ഭരണി ||Baraṇi&lt;br&gt;பரணி || Berana&lt;br&gt;බෙරණ|| Burunu&lt;br&gt;ބުރުނު|| Bharani&lt;br&gt;భరణి|| Bharani&lt;br&gt;ಭರಣಿ ||Bharaṇi&lt;br&gt;ভরণী|| Гоё хүүхэн|| 胃|| བྲ་ཉེ།<br /> |-<br /> | 3 || [[Krittikā]]&lt;br&gt;कृत्तिका ||Krutikaa&lt;br&gt;କୃତିକା|| Kārttika&lt;br&gt;കാർത്തിക || Kārthikai&lt;br&gt;கார்த்திகை || Kethi&lt;br&gt;කැති|| Kethi&lt;br&gt;ކެތި|| Kritika&lt;br&gt;కృత్తిక|| Kritika&lt;br&gt;ಕೃತಿಕ|| Krittika&lt;br&gt;কৃত্তিকা ||Нэг эхт зургаан хөвгүүн || 昴|| སྨིན་དྲུག<br /> |-<br /> | 4 || [[Rohini (nakshatra)|Rohini]]&lt;br&gt;रोहिणी ||Rohini&lt;br&gt;ରୋହିଣୀ|| Rōhiṇi&lt;br&gt;രോഹിണി || Rōhiṇi&lt;br&gt;ரோகிணி || Rehena&lt;br&gt;රෙහෙණ|| Roanu&lt;br&gt;ރޯނު|| Rohini&lt;br&gt;రోహిణి|| Rohini&lt;br&gt;ರೋಹಿಣಿ ||Rohiṇi&lt;br&gt;রোহিণী || Чөлөөт эх || 畢|| སྣར་མ།<br /> |-<br /> | 5 || [[Mrigashīrsha]]&lt;br&gt;म्रृगशीर्षा ||Mrugasiraa&lt;br&gt;ମୃଗଶିରା|| Makayiram&lt;br&gt;മകയിരം || Mirugasīridam&lt;br&gt;மிருகசீரிடம் || Muwasirasa&lt;br&gt;මුවසිරස|| Miyaheli&lt;br&gt;މިޔަހެލި|| Mrigasiraa&lt;br&gt;మృగశిర|| Mrigashira&lt;br&gt;ಮೃಗಶಿರ ||Mrigashira&lt;br&gt;মৃগশিরা ||Гөрөөсөн толгой || 觜|| མགོ<br /> |-<br /> | 6 || [[Ardra (nakshatra)|Ārdrā]]&lt;br&gt;आर्द्रा ||Adra&lt;br&gt;ଆଦ୍ରା|| Ātira or Tiruvātira&lt;br&gt;ആതിര (തിരുവാതിര) || Thiruvādhirai&lt;br&gt;திருவாதிரை || Ada&lt;br&gt;අද|| Adha&lt;br&gt;އަދަ|| Arudra&lt;br&gt;ఆరుద్ర || Ardra&lt;br&gt;ಆರ್ದ್ರ|| Ardra&lt;br&gt;আর্দ্রা || Хэрцгий охин || 參|| ལག<br /> |-<br /> | 7 || [[Punarvasu]]&lt;br&gt;पुनर्वसु ||punarbasu&lt;br&gt;ପୁନର୍ବସୁ|| Puṇartam&lt;br&gt;പുണർതം || Punarpoosam&lt;br&gt;புனர்பூசம் || Punavasa&lt;br&gt;පුනාවස|| Funoas&lt;br&gt;ފުނޯސް|| Punarvasu&lt;br&gt;పునర్వసు|| Punarvasu&lt;br&gt;ಪುನರ್ವಸು ||Punarbasu&lt;br&gt;পুনর্বসু ||Өглөгт охин || 井|| ནབས་སོ།<br /> |-<br /> | 8 || [[Pushya]]&lt;br&gt;पुष्य ||Pushyaa&lt;br&gt;ପୁଷ୍ୟା|| Pūyam&lt;br&gt;പൂയം || Poosam&lt;br&gt;பூசம் || Pusha&lt;br&gt;පුෂ|| Fus&lt;br&gt;ފުސް|| Pushya&lt;br&gt;పుష్య|| Pushyami&lt;br&gt;ಪುಷ್ಯ ||Pushya&lt;br&gt;পুষ্যা || Найралт эх || 鬼|| རྒྱལ།<br /> |-<br /> | 9 || [[Āshleshā]]&lt;br&gt;आश्ळेषा / आश्लेषा||Ashleshaa&lt;br&gt;ଆଶ୍ଳେଷା||Āyilyam&lt;br&gt;ആയില്യം || Ayilyam&lt;br&gt;ஆயில்யம் || Aslisa&lt;br&gt;අස්ලිස|| Ahuliha&lt;br&gt;އަހުލިހަ|| Ashlesha&lt;br&gt;ఆశ్లేష|| Ashlesha&lt;br&gt;ಆಶ್ಲೇಷ ||Ashleshā&lt;br&gt;অশ্লেষা || Үнэг || 柳|| སྐག<br /> |-<br /> | 10 || [[Maghā]]&lt;br&gt;मघा ||Magaa&lt;br&gt;ମଘା|| Makam&lt;br&gt;മകം || Magam&lt;br&gt;மகம் || Maa&lt;br&gt;මා|| Maa&lt;br&gt;މާ|| Maghaa&lt;br&gt;మఘ|| Magha&lt;br&gt;ಮಘ ||Magha&lt;br&gt;মঘা ||Их морь || 星|| མཆུ།<br /> |-<br /> | 11 || Pūrva or [[Pūrva Phalgunī]]&lt;br&gt;पूर्व फल्गुनी ||Purba Falguni&lt;br&gt;ପୂର୍ବ ଫାଲ୍ଗୁନୀ|| Pūram&lt;br&gt;പൂരം || Pooram&lt;br&gt;பூரம் || Puwapal&lt;br&gt;පුවපල්|| Fura&lt;br&gt;ފުރަ|| Pubba&lt;br&gt;పూర్వ ఫల్గుని|| Poorva&lt;br&gt;ಪೂರ್ವ ||Purbaphālguni&lt;br&gt;পূর্বফাল্গুনী || Бага морь || 張|| གྲེ།<br /> |-<br /> | 12 || Uttara or [[Uttara Phalgunī]]&lt;br&gt;उत्तर फल्गुनी ||Uttara falguni&lt;br&gt;ଉତ୍ତର ଫାଲ୍ଗୁନୀ|| Utram&lt;br&gt;ഉത്രം ||Uthiram&lt;br&gt;உத்திரம் || Uttrapal&lt;br&gt;උත්රපල්|| Uthura&lt;br&gt;އުތުރަ|| Uttara&lt;br&gt;ఉత్తర ఫల్గుని|| Uttara&lt;br&gt;ಉತ್ತರ ||Uttar Phālguni&lt;br&gt;উত্তরফাল্গুনী || Харцага || 翼|| དབོ།<br /> |-<br /> | 13 || [[Nakshatra|Hasta]]&lt;br&gt;हस्त ||Hastaa&lt;br&gt;ହସ୍ତା||Attam&lt;br&gt;അത്തം || Ashtham&lt;br&gt;அசுதம் || Hatha&lt;br&gt;හත|| Atha&lt;br&gt;އަތަ|| Hasta&lt;br&gt;హస్త|| Hastaa&lt;br&gt;ಹಸ್ತ ||Hastā &lt;br&gt; হস্তা ||Тугчин || 軫|| མེ་བཞི།<br /> |-<br /> | 14 || [[Nakshatra|Chitrā]]&lt;br&gt;चित्रा ||Chitraa&lt;br&gt;ଚିତ୍ରା||Chittira&lt;br&gt;ചിത്തിര (ചിത്ര) || Chithirai&lt;br&gt;சித்திரை || Sitha&lt;br&gt;සිත|| Hitha&lt;br&gt;ހިތަ|| Chittaa&lt;br&gt;చిత్త|| Chitra&lt;br&gt;ಚಿತ್ರ ||Chitrā &lt;br&gt; চিত্রা ||Тэргүүн дагуул || 角|| ནག་པ།<br /> |-<br /> | 15 || [[Svātī]]&lt;br&gt;स्वाती ||Swati&lt;br&gt;ସ୍ଵାତୀ||Chōti&lt;br&gt;ചോതി || Swathi&lt;br&gt;சுவாதி || Saa&lt;br&gt;සා|| Hey&lt;br&gt;ހޭ|| Swaati&lt;br&gt;స్వాతి|| Swathi&lt;br&gt;ಸ್ವಾತಿ ||Swāti &lt;br&gt; স্বাতী || Салхины эх || 亢|| ས་རི།<br /> |-<br /> | 16 || [[Vishākhā]]&lt;br&gt;विशाखा ||Bishakha&lt;br&gt;ବିଶାଖା||Vishākham&lt;br&gt;വിശാഖം || Visakam&lt;br&gt;விசாகம்|| Wisa&lt;br&gt;විසා|| Vihaa&lt;br&gt;ވިހާ|| Vishaakha&lt;br&gt;విశాఖ|| Vishakha&lt;br&gt;ವಿಶಾಖ ||Bishakha &lt;br&gt; বিশাখা || Эрхтний тэнгэрт || 氐|| ས་ག<br /> |-<br /> | 17 ||[https://aaps.space/docs/nakshatra/Anuradha-Nakshatra/ Anurādhā]&lt;br&gt;अनुराधा ||Anuradha&lt;br&gt;ଅନୁରାଧା|| Anizham&lt;br&gt;അനിഴം || Anusham&lt;br&gt;அனுசம் || Anura&lt;br&gt;අනුර|| Nora&lt;br&gt;ނޮރަ|| Anuraadha&lt;br&gt;అనురాధ|| Anuradha&lt;br&gt;ಅನುರಾಧಾ ||Anuradha &lt;br&gt; অনুরাধা || Гар од || 房||ལྷ་མཚམས།<br /> |-<br /> | 18 || [[Jyeshtha]]&lt;br&gt;ज्येष्ठा ||Jyosthaa&lt;br&gt;ଜ୍ୟୋଷ୍ଠା||Kēṭṭa (Trikkēṭṭa)&lt;br&gt;തൃക്കേട്ട|| Kettai&lt;br&gt;கேட்டை || Deta&lt;br&gt;දෙට|| Dhosha&lt;br&gt;ދޮށަ|| Jyesthaa&lt;br&gt;జ్యేష్ఠ|| Jyestha&lt;br&gt;ಜ್ಯೇಷ್ಠ ||Jyestha &lt;br&gt; জ্যেষ্ঠা || Хонгорцог || 心|| སྣྲོན།<br /> |-<br /> | 19 || [[Mula (astrology)|Mūla]]&lt;br&gt;मूल ||Mulaa&lt;br&gt;ମୂଳା|| Mūlam&lt;br&gt;മൂലം|| Mūlam&lt;br&gt;மூலம் || Moola&lt;br&gt;මුල|| Mula&lt;br&gt;މުލަ|| Moolaa&lt;br&gt;మూల|| Moola&lt;br&gt;ಮೂಲ ||Mula &lt;br&gt; মূলা || Онгоц || 尾|| སྣུབས།<br /> |-<br /> | 20 || [[Pūrva Ashādhā]]&lt;br&gt;पूर्वाषाढा ||Purbasaadhaa&lt;br&gt;ପୂର୍ବାଷାଢା|| Pūrāṭam&lt;br&gt;പൂരാടം || Pūradam&lt;br&gt;பூராடம் || Puwasala&lt;br&gt;පුවසල|| Furahalha&lt;br&gt;ފުރަހަޅަ|| Poorvashaadha&lt;br&gt;పూర్వాషాడ|| Poorvashadha&lt;br&gt;ಪುರ್ವಾಷಾಡ ||Poorbashada&lt;br&gt;পূর্বাষাঢ়া || Суулга || 箕||ཆུ་སྟོད།<br /> |-<br /> | 21 || [[Uttara Ashādhā]]&lt;br&gt;उत्तराषाढा ||Uttarasaadha&lt;br&gt;ଉତ୍ତରଷାଢା|| Utrāṭam&lt;br&gt;ഉത്രാടം || Uthirādam&lt;br&gt;உத்திராடம் || Uttrasala&lt;br&gt;උත්රසල|| Uthurahalha&lt;br&gt;އުތުރަހަޅަ|| Uttarashaadha&lt;br&gt;ఉత్తరాషాడ|| Uttarashadha&lt;br&gt;ಉತ್ತರಾಷಾಡ ||Uttarashada]]&lt;br&gt;উত্তরাষাঢ়া || Элдэв тэнгэртэн || 斗|| ཆུ་སྨད།<br /> |-<br /> | 22 || [[Shravana]]&lt;br&gt;श्रवण ||Sravana&lt;br&gt;ଶ୍ରବଣା|| Tiruvōnam &lt;br&gt;ഓണം (തിരുവോണം)|| Tiruvōnam&lt;br&gt;திருவோணம் || Suvana&lt;br&gt;සුවණ|| Huvan&lt;br&gt;ހުވަން|| Shravana&lt;br&gt;శ్రవణ|| Shravana&lt;br&gt;ಶ್ರವಣ||Shraban&lt;br&gt;শ্রবণা || Булаагч/Яруу эгшигт || 女|| གྲོ་བཞིན།<br /> |-<br /> | 23 || Shravishthā or [[Dhanishta]] &lt;br&gt;श्रविष्ठा or धनिष्ठा ||Dhanishathaa&lt;br&gt;ଧନିଷ୍ଠା|| Aviṭṭam&lt;br&gt;അവിട്ടം || Aviṭṭam&lt;br&gt;அவிட்டம் || Denata&lt;br&gt;දෙණට|| Dhinasha&lt;br&gt;ދިނަށަ|| Dhanishta&lt;br&gt;ధనిష్ఠ|| Dhanishta&lt;br&gt;ಧನಿಷ್ಠ||Dhanishta&lt;br&gt;ধনিষ্ঠা ||Тооно || 虛||མོན་གྲེ།<br /> |-<br /> | 24 || [[Shatabhishā]] or Shatataraka&lt;br&gt;शतभिषक् / शततारका||Satavisaa&lt;br&gt;ସତଭିଷା|| Chatayam&lt;br&gt;ചതയം || Sadayam&lt;br&gt;சதயம் || Siyawasa&lt;br&gt;සියාවස|| Hiyavihaa&lt;br&gt;ހިޔަވިހާ|| Shatabhisha&lt;br&gt;శతభిష|| Shatabhisha&lt;br&gt;ಶತಭಿಷ ||Shatabhisha &lt;br&gt; শতভিষা || Чөдөр || 危|| མོན་གྲུ<br /> |-<br /> | 25 || [[Pūrva Bhādrapadā]]&lt;br&gt;पूर्वभाद्रपदा / पूर्वप्रोष्ठपदा ||Purba vadrapada&lt;br&gt;ପୂର୍ବଭାଦ୍ରପଦ|| Pūruruṭṭāti&lt;br&gt;പൂരുരുട്ടാതി || Pūraṭṭādhi&lt;br&gt;பூரட்டாதி || Puvaputupa&lt;br&gt;පුවපුටුප|| Furabadhuruva&lt;br&gt;ފުރަބަދުރުވަ|| Poorvabhadraa&lt;br&gt;పూర్వ భాద్రపద|| Poorva Bhadrapada&lt;br&gt;ಪೂರ್ವ ಭಾದ್ರಪದ ||Poorbabhadra &lt;br&gt; পূর্বভাদ্রপদ || Шувуун хошуут || 室||ཁྲུམས་སྟོད།<br /> |-<br /> | 26 || [[Uttara Bhādrapadā]]&lt;br&gt;उत्तरभाद्रपदा / उत्तरप्रोष्ठपदा ||Uttara vadrapada&lt;br&gt;ଉତ୍ତରାଭାଦ୍ରପଦ|| Uttṛṭṭāti&lt;br&gt;ഉത്രട്ടാതി || Uttṛṭṭādhi&lt;br&gt;உத்திரட்டாதி || Uttraputupa&lt;br&gt;උත්රපුටුප|| Fasbadhuruva&lt;br&gt;ފަސްބަދުރުވަ|| Uttaraa bhadra&lt;br&gt;ఉత్తర భాద్రపద|| Uttara Bhadrapada&lt;br&gt;ಉತ್ತರ ಭಾದ್ರಪದ||Uttarbabhadra &lt;br&gt; উত্তরভাদ্রপদ || Могой хүлэгч || 壁|| ཁྲུམས་སྨད།<br /> |<br /> |-<br /> | 27 || [[Revati (nakshatra)|Revati]]&lt;br&gt;रेवती ||Revati&lt;br&gt;ରେବତୀ|| Rēvati&lt;br&gt;രേവതി || Rēvathi&lt;br&gt;ரேவதி || Revathi&lt;br&gt;රේවතී|| Reyva&lt;br&gt;ރޭވަ|| Revati&lt;br&gt;రేవతి|| Revati&lt;br&gt;ರೇವತಿ ||Rebati &lt;br&gt; রেবতী || Дэлгэрүүлэгч || 奎|| ནམ་གྲུ།<br /> |}<br /> <br /> ==References==<br /> <br /> {{reflist}}</div>

BharataPuru http://dharmawiki.org/index.php?title=Topic:W76dqa6f1xqvlw6j&topic_postId=w76dqa6i66m6a9m3&topic_revId=w76dqa6i66m6a9m3&action=single-view 2021-04-16T04:06:37Z

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BharataPuru http://dharmawiki.org/index.php?title=Indian_standard_circle&diff=129863 2021-04-16T04:03:09Z

<p>BharataPuru: </p> <hr /> <div>[[File:Indian Standard circle.jpg|thumb|320x320px]]''Indian standard circle'' is a name given to the standard circle first used in [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and later used by several ancient Indian mathematicians and astronomers to improve the [[Indian sine tables]] and for various other calculations. [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] provides methods for calculating the Jyā (R.sine) values. The circle uses a radius of 3,438 minutes. ''Surya Siddhanta'' calculates the first Jyā (R.sine) as 1/8th of the number of minutes (kalās) in a Rashi (zodiac sign). It says a Rashi (zodiac sign) has 1800 minutes (kalās) and thus calculates the first Jyā to a value of 225 minutes (kalā कला ).&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). &quot;Translation of the Surya Siddhanta&quot;. Ch2 Ve15, pp. 15–16.&lt;/ref&gt;<br /> <br /> &lt;br&gt;The Indian standard circles holds significance as it is based on number of minutes in circle thus leads to 360 degrees in a circle which is the basis of modern trigonometry. Although the [[Indian sine tables]] are not based on the angles but rather on the R.sine (Jyā) values. The [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] data reflect highly sophisticated outcomes of the R.sine values. ''Burgess'' notes that it is remarkable to see that the 2nd order differences increase as the sines and each, in fact, is about 1/225th part of the corresponding sine.&lt;ref&gt;Burgess, Rev. Ebenezer (1860). Translation of the Surya Siddhanta. p. 115.&lt;/ref&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Modern units and Indian units ''(Sanskrit)''<br /> !Modern units<br /> !Indian units ''(Sanskrit)''<br /> !value<br /> |-<br /> |Zodiac sign<br /> |Rashi (राशी)<br /> |30 degrees<br /> |-<br /> |Degree<br /> |Ansh (अंश )<br /> |60 minutes<br /> |-<br /> |Minute<br /> |Kalā (कला )<br /> |60 seconds<br /> |-<br /> |Second<br /> |Vikalā (विकला )<br /> | -<br /> |}<br /> <br /> == See Also ==<br /> * [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]]<br /> * [[Indian sine tables|Indian Sine Tables]]<br /> <br /> == References ==<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_standard_circle&diff=129862 2021-04-16T04:02:32Z

<p>BharataPuru: </p> <hr /> <div>[[File:Indian Standard circle.jpg|thumb|320x320px]]''Indian standard circle'' is a name given to the standard circle first used in [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and later used by several ancient Indian mathematicians and astronomers to improve the [[Indian sine tables]] and for various other calculations. [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] provides methods for calculating the Jyā (R.sine) values. The circle uses a radius of 3,438 minutes. ''Surya Siddhanta'' calculates the first Jyā (R.sine) as 1/8th of the number of minutes (kalās) in a Rashi (zodiac sign). It says a Rashi (zodiac sign) has 1800 minutes (kalās) and thus calculates the first Jyā to a value of 225 minutes (kalā कला ).&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). &quot;Translation of the Surya Siddhanta&quot;. Ch2 Ve15, pp. 15–16.&lt;/ref&gt;<br /> <br /> &lt;br&gt;The Indian standard circles holds significance as it is based on number of minutes in circle thus leads to 360 degrees in a circle which is the basis of modern trigonometry. Although the [[Indian sine tables]] are not based on the angles but rather on the R.sine (Jyā) values. The [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] data reflect highly sophisticated outcomes of the R.sine values. ''Burgess'' notes that it is remarkable to see that the 2nd order differences increase as the sines and each, in fact, is about 1/225th part of the corresponding sine.&lt;ref&gt;Burgess, Rev. Ebenezer (1860). Translation of the Surya Siddhanta. p. 115.&lt;/ref&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Modern units and Indian units ''(Sanskrit)''<br /> !Modern units<br /> !Indian units<br /> !value<br /> |-<br /> |Zodiac sign<br /> |Rashi (राशी)<br /> |30 degrees<br /> |-<br /> |Degree<br /> |Ansh (अंश )<br /> |60 minutes<br /> |-<br /> |Minute<br /> |Kalā (कला )<br /> |60 seconds<br /> |-<br /> |Second<br /> |Vikalā (विकला )<br /> | -<br /> |}<br /> <br /> == See Also ==<br /> * [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]]<br /> * [[Indian sine tables|Indian Sine Tables]]<br /> <br /> == References ==<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_sine_tables&diff=129861 2021-04-16T04:00:10Z

<p>BharataPuru: /* See also */</p> <hr /> <div>Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the [[Indian mathematician]] and astronomer [[Aryabhata|Āryabhaṭa]] (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.&lt;ref&gt;{{cite journal|doi=10.1006/hmat.1997.2160|last=Takao Hayashi|first1=T|date=November 1997|title=Āryabhaṭa's rule and table for sine-differences|journal=Historia Mathematica |volume=24|issue=4|pages=396–406 }}&lt;/ref&gt;&lt;ref&gt;{{cite journal|doi=10.1007/BF00329978|last=B. L. van der Waerden|date=March 1988|first1=B. L.|title=Reconstruction of a Greek table of chords|journal=Archive for History of Exact Sciences|volume=38|issue=1|pages=23–38|title-link=table of chords}}&lt;/ref&gt;<br /> <br /> Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics.&lt;ref name=&quot;mcs&quot;&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last=J J O'Connor and E F Robertson|date=June 1996 |accessdate=4 March 2010}}&lt;/ref&gt; Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.<br /> <br /> == Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==<br /> '''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three trigonometric functions introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta.&lt;ref name=&quot;Datta&quot;&gt;{{cite journal|last=B.B. Datta and A.N. Singh|date=1983|title=Hindu Trigonometry|journal=Indian Journal of History of Science|volume=18|issue=1|pages=39&amp;ndash;108|url=http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol18_1_5_BDatta.pdf|accessdate=1 March 2010}}&lt;/ref&gt; These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of &quot;sine&quot; and &quot;cosine&quot; have been traced back to the [[Sanskrit]] words jyā and kotijyā.&lt;ref name=&quot;Datta&quot; /&gt;<br /> [[File:Jya Figure.jpg|thumb|419x419px]]<br /> <br /> ===Definition===<br /> Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:<br /> <br /> * ''Vyāsardhā'' = Radius (R)<br /> * ''jyā'' of arc AB = MB<br /> * ''koti-jyā'' of arc AB = CM = R cos θ<br /> * ''utkrama-jyā'' of arc AB = MA<br /> <br /> If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:<br /> <br /> * ''jyā'' ( arc AB ) = ''R'' sin θ<br /> * ''koti-jyā'' ( arc AB ) = ''R'' cos θ<br /> * ''utkrama-jyā'' ( arc AB ) = R - R cos θ<br /> <br /> An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means &quot;a bow&quot;. <br /> The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means &quot;a bow-string&quot;.<br /> The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.&lt;ref&gt;According to lexicographers, it is a synonym also meaning &quot;bow-string&quot;, but only its <br /> geometrical meaning is attested in literature. Monier-Williams, ''A Sanskrit Dictionary'' (1899): &quot;'' jīvá'' n. (in geom. = ''jyā'') the chord of an arc; the sine of an arc ''Suryasiddhanta'' 2.57&quot;; <br /> ''jīvá'' as a generic adjective has the meaning of &quot;living, alive&quot; ([[:wikt:Appendix:Proto-Indo-European/gʷih₃wós|cognate]] with English ''[[:wikt:quick|quick]]'')&lt;/ref&gt;<br /> At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs.&lt;ref name=&quot;Datta&quot; /&gt;&lt;ref name=&quot;Glen&quot;&gt;{{cite book|last=Glen Van Brummelen|title=The mathematics of the heavens and the earth : the early history of trigonometry|publisher=[[Princeton University Press]]|date=2009|pages=95&amp;ndash;97|isbn=978-0-691-12973-0}}&lt;/ref&gt; The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant &quot;half of&quot;.<br /> <br /> The Sanskrit word ''koṭi'' has the meaning of &quot;point, cusp&quot;, and specifically &quot;the [[Recurve bow|curved end of a bow]]&quot;.<br /> In trigonometry, it came to denote &quot;the complement of an arc to 90°&quot;. Thus <br /> ''koṭi-jyā'' is &quot;the ''jyā'' of the complementary arc&quot;. In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes &quot;the side of a right angled triangle&quot;. Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> ''Utkrama'' means &quot;inverted&quot;, thus ''utkrama-jyā'' means &quot;inverted chord&quot;. The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning &quot;arrow&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of &quot;the ''jyā'' of three signs&quot;. The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning &quot;semi-diameter&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by &quot;Rsin&quot; and &quot;Rcos&quot; treated as single words.&lt;ref name=&quot;Datta&quot; /&gt; Others denote ''jyā'' and ''koti-jyā'' respectively by &quot;Sin&quot; and &quot;Cos&quot; (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).&lt;ref name=&quot;Glen&quot; /&gt;<br /> <br /> ===From jyā to sine===<br /> The origins of the modern term sine have been traced to the Sanskrit word ''jyā'',&lt;ref&gt;{{cite web|url=http://mathforum.org/library/drmath/view/54053.html|title=How the Trig Functions Got their Names|work=Ask Dr. Math|publisher=[[Drexel University]]|accessdate=2 March 2010}}&lt;/ref&gt;&lt;ref&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last= J J O'Connor and E F Robertson|date=June 1996 |accessdate=2 March 2010}}&lt;/ref&gt; or more specifically to its synonym ''jīva''. This term was adopted in medieval Islamic mathematics, transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means &quot;bosom&quot;. The text's 12th-century Latin translator used the Latin equivalent for &quot;bosom&quot;, ''[[wikt:sinus|sinus]]''.&lt;ref&gt;Various sources credit the first use of ''sinus'' to either: <br /> * [[Plato Tiburtinus]]'s 1116 translation of the ''Astronomy'' of [[Al-Battani]]<br /> * [[Gerard of Cremona]]'s c. 1150 translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]<br /> * [[Robert of Chester]]'s 1145 translation of the tables of al-Khwārizmī<br /> See Merlet, [https://link.springer.com/chapter/10.1007/1-4020-2204-2_16#page-1 ''A Note on the History of the Trigonometric Functions''] in Ceccarelli (ed.), ''International Symposium on History of Machines and Mechanisms'', Springer, 2004&lt;br&gt;See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.&lt;br&gt;See {{cite book |last=Katx |first=Victor |date=July 2008 |title=A history of mathematics |edition=3rd |location=Boston |publisher=Pearson |page=210 (sidebar) |isbn= 978-0321387004 |language=English }}&lt;/ref&gt; When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.<br /> <br /> ==The Indian sine tables==<br /> === Surya Siddhanta sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> ===Āryabhaṭa's sine table===<br /> The stanza in Āryabhaṭiya describing the sine table is reproduced below:<br /> <br /> मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |<br /> घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||<br /> <br /> The second section of Āryabhaṭiya titled Ganitapādd<br /> a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.&lt;ref name=&quot;Katz&quot; /&gt;<br /> * &quot;When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.&quot;<br /> This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. <br /> <br /> The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''jya'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.<br /> <br /> ===Comparing different sine tables===<br /> <br /> [[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle&lt;br /&gt;(in degrees,&lt;br /&gt;arcminutes)<br /> !Āryabhaṭa's &lt;br /&gt;value of Jyā (R.sine)<br /> ![[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] value of Jyā (R.sine)<br /> !Surya Siddhanta versed sines<br /> !Modern value &lt;br /&gt;of ''Jyā'' R.sine<br /> !Madhava's sine values<br /> !Madhava's derived sine values<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |&lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.06540314<br /> |0.06540313<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |&lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> |0.13052623<br /> |0.13052619<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |&lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> |0.19509032<br /> |0.19509032<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |&lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.25881900<br /> |0.25881905<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |&lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> |0.32143947<br /> |0.32143947<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |&lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> |0.38268340<br /> |0.38268343<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |&lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> |0.44228865<br /> |0.44228869<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.49999998<br /> |0.50000000<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.55557022<br /> |0.55557023<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |&lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> |0.60876139<br /> |0.60876143<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |&lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.65934580<br /> |0.65934582<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |&lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.70710681<br /> |0.70710678<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |&lt;center&gt;2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> |0.75183985<br /> |0.75183981<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |&lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> |0.79335331<br /> |0.79335334<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |&lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> |0.83146960<br /> |0.83146961<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |&lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> |0.86602543<br /> |0.86602540<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |&lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt;<br /> |0.89687275<br /> |0.89687274<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |&lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.92387954<br /> |0.92387953<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |&lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.94693016<br /> |0.94693013<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |&lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> |0.96592581<br /> |0.96592583<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |&lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> |0.98078527<br /> |0.98078528<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |&lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99144487<br /> |0.99144486<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |&lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99785895<br /> |0.99785892<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |&lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt;<br /> |0.99999997<br /> |1.00000000<br /> |-<br /> |}<br /> <br /> ==See also==<br /> * [[Madhava's sine table]]<br /> * [[Bhaskara I's sine approximation formula]]<br /> * [[Indian standard circle|Indian Standard Circle]]<br /> * [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]]<br /> <br /> ==References==<br /> <br /> {{reflist}}<br /> <br /> {{DEFAULTSORT:Aryabhata's Sine Table}}<br /> [[Category:Trigonometry]]<br /> [[Category:Indian mathematics]]<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_standard_circle&diff=129860 2021-04-16T03:57:27Z

<p>BharataPuru: addition of diagram and a table</p> <hr /> <div>''Indian standard circle'' is a name given to the standard circle first used in [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and later used by several ancient Indian mathematicians and astronomers to improve the [[Indian sine tables]] and for various other calculations. [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] provides methods for calculating the Jyā (R.sine) values. The circle uses a radius of 3,438 minutes. ''Surya Siddhanta'' calculates the first Jyā (R.sine) as 1/8th of the number of minutes (kalās) in a Rashi (zodiac sign). It says a Rashi (zodiac sign) has 1800 minutes (kalās) and thus calculates the first Jyā to a value of 225 minutes (kalā कला ).&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). &quot;Translation of the Surya Siddhanta&quot;. Ch2 Ve15, pp. 15–16.&lt;/ref&gt; <br /> [[File:Indian Standard circle.jpg|thumb|400x400px]]<br /> {| class=&quot;wikitable&quot;<br /> |+Modern units and Indian units ''(Sanskrit)''<br /> !Modern units<br /> !Indian units<br /> !value<br /> |-<br /> |Zodiac sign<br /> |Rashi (राशी)<br /> |30 degrees<br /> |-<br /> |Degree<br /> |Ansh (अंश )<br /> |60 minutes<br /> |-<br /> |Minute<br /> |Kalā (कला )<br /> |60 seconds<br /> |-<br /> |Second<br /> |Vikalā (विकला )<br /> | -<br /> |}<br /> &lt;br&gt;<br /> The Indian standard circles holds significance as it is based on number of minutes in circle thus leads to 360 degrees in a circle which is the basis of modern trigonometry. Although the [[Indian sine tables]] are not based on the angles but rather on the R.sine (Jyā) values. The [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] data reflect highly sophisticated outcomes of the R.sine values. ''Burgess'' notes that it is remarkable to see that the 2nd order differences increase as the sines and each, in fact, is about 1/225th part of the corresponding sine.&lt;ref&gt;Burgess, Rev. Ebenezer (1860). Translation of the Surya Siddhanta. p. 115.&lt;/ref&gt;<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=File:Indian_Standard_circle.jpg&diff=129859 2021-04-16T03:56:47Z

<p>BharataPuru: </p> <hr /> <div>Indian standard circle of radius 3438 minutes and circumference of 21,600 minutes</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_sine_tables&diff=129858 2021-04-16T03:15:42Z

<p>BharataPuru: /* See also */</p> <hr /> <div>Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the [[Indian mathematician]] and astronomer [[Aryabhata|Āryabhaṭa]] (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.&lt;ref&gt;{{cite journal|doi=10.1006/hmat.1997.2160|last=Takao Hayashi|first1=T|date=November 1997|title=Āryabhaṭa's rule and table for sine-differences|journal=Historia Mathematica |volume=24|issue=4|pages=396–406 }}&lt;/ref&gt;&lt;ref&gt;{{cite journal|doi=10.1007/BF00329978|last=B. L. van der Waerden|date=March 1988|first1=B. L.|title=Reconstruction of a Greek table of chords|journal=Archive for History of Exact Sciences|volume=38|issue=1|pages=23–38|title-link=table of chords}}&lt;/ref&gt;<br /> <br /> Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics.&lt;ref name=&quot;mcs&quot;&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last=J J O'Connor and E F Robertson|date=June 1996 |accessdate=4 March 2010}}&lt;/ref&gt; Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.<br /> <br /> == Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==<br /> '''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three trigonometric functions introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta.&lt;ref name=&quot;Datta&quot;&gt;{{cite journal|last=B.B. Datta and A.N. Singh|date=1983|title=Hindu Trigonometry|journal=Indian Journal of History of Science|volume=18|issue=1|pages=39&amp;ndash;108|url=http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol18_1_5_BDatta.pdf|accessdate=1 March 2010}}&lt;/ref&gt; These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of &quot;sine&quot; and &quot;cosine&quot; have been traced back to the [[Sanskrit]] words jyā and kotijyā.&lt;ref name=&quot;Datta&quot; /&gt;<br /> [[File:Jya Figure.jpg|thumb|419x419px]]<br /> <br /> ===Definition===<br /> Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:<br /> <br /> * ''Vyāsardhā'' = Radius (R)<br /> * ''jyā'' of arc AB = MB<br /> * ''koti-jyā'' of arc AB = CM = R cos θ<br /> * ''utkrama-jyā'' of arc AB = MA<br /> <br /> If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:<br /> <br /> * ''jyā'' ( arc AB ) = ''R'' sin θ<br /> * ''koti-jyā'' ( arc AB ) = ''R'' cos θ<br /> * ''utkrama-jyā'' ( arc AB ) = R - R cos θ<br /> <br /> An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means &quot;a bow&quot;. <br /> The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means &quot;a bow-string&quot;.<br /> The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.&lt;ref&gt;According to lexicographers, it is a synonym also meaning &quot;bow-string&quot;, but only its <br /> geometrical meaning is attested in literature. Monier-Williams, ''A Sanskrit Dictionary'' (1899): &quot;'' jīvá'' n. (in geom. = ''jyā'') the chord of an arc; the sine of an arc ''Suryasiddhanta'' 2.57&quot;; <br /> ''jīvá'' as a generic adjective has the meaning of &quot;living, alive&quot; ([[:wikt:Appendix:Proto-Indo-European/gʷih₃wós|cognate]] with English ''[[:wikt:quick|quick]]'')&lt;/ref&gt;<br /> At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs.&lt;ref name=&quot;Datta&quot; /&gt;&lt;ref name=&quot;Glen&quot;&gt;{{cite book|last=Glen Van Brummelen|title=The mathematics of the heavens and the earth : the early history of trigonometry|publisher=[[Princeton University Press]]|date=2009|pages=95&amp;ndash;97|isbn=978-0-691-12973-0}}&lt;/ref&gt; The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant &quot;half of&quot;.<br /> <br /> The Sanskrit word ''koṭi'' has the meaning of &quot;point, cusp&quot;, and specifically &quot;the [[Recurve bow|curved end of a bow]]&quot;.<br /> In trigonometry, it came to denote &quot;the complement of an arc to 90°&quot;. Thus <br /> ''koṭi-jyā'' is &quot;the ''jyā'' of the complementary arc&quot;. In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes &quot;the side of a right angled triangle&quot;. Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> ''Utkrama'' means &quot;inverted&quot;, thus ''utkrama-jyā'' means &quot;inverted chord&quot;. The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning &quot;arrow&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of &quot;the ''jyā'' of three signs&quot;. The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning &quot;semi-diameter&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by &quot;Rsin&quot; and &quot;Rcos&quot; treated as single words.&lt;ref name=&quot;Datta&quot; /&gt; Others denote ''jyā'' and ''koti-jyā'' respectively by &quot;Sin&quot; and &quot;Cos&quot; (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).&lt;ref name=&quot;Glen&quot; /&gt;<br /> <br /> ===From jyā to sine===<br /> The origins of the modern term sine have been traced to the Sanskrit word ''jyā'',&lt;ref&gt;{{cite web|url=http://mathforum.org/library/drmath/view/54053.html|title=How the Trig Functions Got their Names|work=Ask Dr. Math|publisher=[[Drexel University]]|accessdate=2 March 2010}}&lt;/ref&gt;&lt;ref&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last= J J O'Connor and E F Robertson|date=June 1996 |accessdate=2 March 2010}}&lt;/ref&gt; or more specifically to its synonym ''jīva''. This term was adopted in medieval Islamic mathematics, transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means &quot;bosom&quot;. The text's 12th-century Latin translator used the Latin equivalent for &quot;bosom&quot;, ''[[wikt:sinus|sinus]]''.&lt;ref&gt;Various sources credit the first use of ''sinus'' to either: <br /> * [[Plato Tiburtinus]]'s 1116 translation of the ''Astronomy'' of [[Al-Battani]]<br /> * [[Gerard of Cremona]]'s c. 1150 translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]<br /> * [[Robert of Chester]]'s 1145 translation of the tables of al-Khwārizmī<br /> See Merlet, [https://link.springer.com/chapter/10.1007/1-4020-2204-2_16#page-1 ''A Note on the History of the Trigonometric Functions''] in Ceccarelli (ed.), ''International Symposium on History of Machines and Mechanisms'', Springer, 2004&lt;br&gt;See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.&lt;br&gt;See {{cite book |last=Katx |first=Victor |date=July 2008 |title=A history of mathematics |edition=3rd |location=Boston |publisher=Pearson |page=210 (sidebar) |isbn= 978-0321387004 |language=English }}&lt;/ref&gt; When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.<br /> <br /> ==The Indian sine tables==<br /> === Surya Siddhanta sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> ===Āryabhaṭa's sine table===<br /> The stanza in Āryabhaṭiya describing the sine table is reproduced below:<br /> <br /> मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |<br /> घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||<br /> <br /> The second section of Āryabhaṭiya titled Ganitapādd<br /> a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.&lt;ref name=&quot;Katz&quot; /&gt;<br /> * &quot;When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.&quot;<br /> This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. <br /> <br /> The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''jya'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.<br /> <br /> ===Comparing different sine tables===<br /> <br /> [[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle&lt;br /&gt;(in degrees,&lt;br /&gt;arcminutes)<br /> !Āryabhaṭa's &lt;br /&gt;value of Jyā (R.sine)<br /> ![[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] value of Jyā (R.sine)<br /> !Surya Siddhanta versed sines<br /> !Modern value &lt;br /&gt;of ''Jyā'' R.sine<br /> !Madhava's sine values<br /> !Madhava's derived sine values<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |&lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.06540314<br /> |0.06540313<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |&lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> |0.13052623<br /> |0.13052619<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |&lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> |0.19509032<br /> |0.19509032<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |&lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.25881900<br /> |0.25881905<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |&lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> |0.32143947<br /> |0.32143947<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |&lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> |0.38268340<br /> |0.38268343<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |&lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> |0.44228865<br /> |0.44228869<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.49999998<br /> |0.50000000<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.55557022<br /> |0.55557023<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |&lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> |0.60876139<br /> |0.60876143<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |&lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.65934580<br /> |0.65934582<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |&lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.70710681<br /> |0.70710678<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |&lt;center&gt;2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> |0.75183985<br /> |0.75183981<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |&lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> |0.79335331<br /> |0.79335334<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |&lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> |0.83146960<br /> |0.83146961<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |&lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> |0.86602543<br /> |0.86602540<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |&lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt;<br /> |0.89687275<br /> |0.89687274<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |&lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.92387954<br /> |0.92387953<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |&lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.94693016<br /> |0.94693013<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |&lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> |0.96592581<br /> |0.96592583<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |&lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> |0.98078527<br /> |0.98078528<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |&lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99144487<br /> |0.99144486<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |&lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99785895<br /> |0.99785892<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |&lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt;<br /> |0.99999997<br /> |1.00000000<br /> |-<br /> |}<br /> <br /> ==See also==<br /> * [[Madhava's sine table]]<br /> * [[Bhaskara I's sine approximation formula]]<br /> * [[Indian Standard Circle]]<br /> * [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]]<br /> <br /> ==References==<br /> <br /> {{reflist}}<br /> <br /> {{DEFAULTSORT:Aryabhata's Sine Table}}<br /> [[Category:Trigonometry]]<br /> [[Category:Indian mathematics]]<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_sine_tables&diff=129857 2021-04-16T03:14:45Z

<p>BharataPuru: made minor changes</p> <hr /> <div>Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the [[Indian mathematician]] and astronomer [[Aryabhata|Āryabhaṭa]] (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.&lt;ref&gt;{{cite journal|doi=10.1006/hmat.1997.2160|last=Takao Hayashi|first1=T|date=November 1997|title=Āryabhaṭa's rule and table for sine-differences|journal=Historia Mathematica |volume=24|issue=4|pages=396–406 }}&lt;/ref&gt;&lt;ref&gt;{{cite journal|doi=10.1007/BF00329978|last=B. L. van der Waerden|date=March 1988|first1=B. L.|title=Reconstruction of a Greek table of chords|journal=Archive for History of Exact Sciences|volume=38|issue=1|pages=23–38|title-link=table of chords}}&lt;/ref&gt;<br /> <br /> Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics.&lt;ref name=&quot;mcs&quot;&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last=J J O'Connor and E F Robertson|date=June 1996 |accessdate=4 March 2010}}&lt;/ref&gt; Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.<br /> <br /> == Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==<br /> '''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three trigonometric functions introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta.&lt;ref name=&quot;Datta&quot;&gt;{{cite journal|last=B.B. Datta and A.N. Singh|date=1983|title=Hindu Trigonometry|journal=Indian Journal of History of Science|volume=18|issue=1|pages=39&amp;ndash;108|url=http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol18_1_5_BDatta.pdf|accessdate=1 March 2010}}&lt;/ref&gt; These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of &quot;sine&quot; and &quot;cosine&quot; have been traced back to the [[Sanskrit]] words jyā and kotijyā.&lt;ref name=&quot;Datta&quot; /&gt;<br /> [[File:Jya Figure.jpg|thumb|419x419px]]<br /> <br /> ===Definition===<br /> Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:<br /> <br /> * ''Vyāsardhā'' = Radius (R)<br /> * ''jyā'' of arc AB = MB<br /> * ''koti-jyā'' of arc AB = CM = R cos θ<br /> * ''utkrama-jyā'' of arc AB = MA<br /> <br /> If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:<br /> <br /> * ''jyā'' ( arc AB ) = ''R'' sin θ<br /> * ''koti-jyā'' ( arc AB ) = ''R'' cos θ<br /> * ''utkrama-jyā'' ( arc AB ) = R - R cos θ<br /> <br /> An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means &quot;a bow&quot;. <br /> The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means &quot;a bow-string&quot;.<br /> The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.&lt;ref&gt;According to lexicographers, it is a synonym also meaning &quot;bow-string&quot;, but only its <br /> geometrical meaning is attested in literature. Monier-Williams, ''A Sanskrit Dictionary'' (1899): &quot;'' jīvá'' n. (in geom. = ''jyā'') the chord of an arc; the sine of an arc ''Suryasiddhanta'' 2.57&quot;; <br /> ''jīvá'' as a generic adjective has the meaning of &quot;living, alive&quot; ([[:wikt:Appendix:Proto-Indo-European/gʷih₃wós|cognate]] with English ''[[:wikt:quick|quick]]'')&lt;/ref&gt;<br /> At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs.&lt;ref name=&quot;Datta&quot; /&gt;&lt;ref name=&quot;Glen&quot;&gt;{{cite book|last=Glen Van Brummelen|title=The mathematics of the heavens and the earth : the early history of trigonometry|publisher=[[Princeton University Press]]|date=2009|pages=95&amp;ndash;97|isbn=978-0-691-12973-0}}&lt;/ref&gt; The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant &quot;half of&quot;.<br /> <br /> The Sanskrit word ''koṭi'' has the meaning of &quot;point, cusp&quot;, and specifically &quot;the [[Recurve bow|curved end of a bow]]&quot;.<br /> In trigonometry, it came to denote &quot;the complement of an arc to 90°&quot;. Thus <br /> ''koṭi-jyā'' is &quot;the ''jyā'' of the complementary arc&quot;. In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes &quot;the side of a right angled triangle&quot;. Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> ''Utkrama'' means &quot;inverted&quot;, thus ''utkrama-jyā'' means &quot;inverted chord&quot;. The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning &quot;arrow&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of &quot;the ''jyā'' of three signs&quot;. The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning &quot;semi-diameter&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by &quot;Rsin&quot; and &quot;Rcos&quot; treated as single words.&lt;ref name=&quot;Datta&quot; /&gt; Others denote ''jyā'' and ''koti-jyā'' respectively by &quot;Sin&quot; and &quot;Cos&quot; (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).&lt;ref name=&quot;Glen&quot; /&gt;<br /> <br /> ===From jyā to sine===<br /> The origins of the modern term sine have been traced to the Sanskrit word ''jyā'',&lt;ref&gt;{{cite web|url=http://mathforum.org/library/drmath/view/54053.html|title=How the Trig Functions Got their Names|work=Ask Dr. Math|publisher=[[Drexel University]]|accessdate=2 March 2010}}&lt;/ref&gt;&lt;ref&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last= J J O'Connor and E F Robertson|date=June 1996 |accessdate=2 March 2010}}&lt;/ref&gt; or more specifically to its synonym ''jīva''. This term was adopted in medieval Islamic mathematics, transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means &quot;bosom&quot;. The text's 12th-century Latin translator used the Latin equivalent for &quot;bosom&quot;, ''[[wikt:sinus|sinus]]''.&lt;ref&gt;Various sources credit the first use of ''sinus'' to either: <br /> * [[Plato Tiburtinus]]'s 1116 translation of the ''Astronomy'' of [[Al-Battani]]<br /> * [[Gerard of Cremona]]'s c. 1150 translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]<br /> * [[Robert of Chester]]'s 1145 translation of the tables of al-Khwārizmī<br /> See Merlet, [https://link.springer.com/chapter/10.1007/1-4020-2204-2_16#page-1 ''A Note on the History of the Trigonometric Functions''] in Ceccarelli (ed.), ''International Symposium on History of Machines and Mechanisms'', Springer, 2004&lt;br&gt;See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.&lt;br&gt;See {{cite book |last=Katx |first=Victor |date=July 2008 |title=A history of mathematics |edition=3rd |location=Boston |publisher=Pearson |page=210 (sidebar) |isbn= 978-0321387004 |language=English }}&lt;/ref&gt; When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.<br /> <br /> ==The Indian sine tables==<br /> === Surya Siddhanta sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> ===Āryabhaṭa's sine table===<br /> The stanza in Āryabhaṭiya describing the sine table is reproduced below:<br /> <br /> मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |<br /> घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||<br /> <br /> The second section of Āryabhaṭiya titled Ganitapādd<br /> a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.&lt;ref name=&quot;Katz&quot; /&gt;<br /> * &quot;When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.&quot;<br /> This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. <br /> <br /> The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''jya'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.<br /> <br /> ===Comparing different sine tables===<br /> <br /> [[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle&lt;br /&gt;(in degrees,&lt;br /&gt;arcminutes)<br /> !Āryabhaṭa's &lt;br /&gt;value of Jyā (R.sine)<br /> ![[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] value of Jyā (R.sine)<br /> !Surya Siddhanta versed sines<br /> !Modern value &lt;br /&gt;of ''Jyā'' R.sine<br /> !Madhava's sine values<br /> !Madhava's derived sine values<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |&lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.06540314<br /> |0.06540313<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |&lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> |0.13052623<br /> |0.13052619<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |&lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> |0.19509032<br /> |0.19509032<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |&lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.25881900<br /> |0.25881905<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |&lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> |0.32143947<br /> |0.32143947<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |&lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> |0.38268340<br /> |0.38268343<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |&lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> |0.44228865<br /> |0.44228869<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.49999998<br /> |0.50000000<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.55557022<br /> |0.55557023<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |&lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> |0.60876139<br /> |0.60876143<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |&lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.65934580<br /> |0.65934582<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |&lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.70710681<br /> |0.70710678<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |&lt;center&gt;2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> |0.75183985<br /> |0.75183981<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |&lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> |0.79335331<br /> |0.79335334<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |&lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> |0.83146960<br /> |0.83146961<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |&lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> |0.86602543<br /> |0.86602540<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |&lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt;<br /> |0.89687275<br /> |0.89687274<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |&lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.92387954<br /> |0.92387953<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |&lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.94693016<br /> |0.94693013<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |&lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> |0.96592581<br /> |0.96592583<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |&lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> |0.98078527<br /> |0.98078528<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |&lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99144487<br /> |0.99144486<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |&lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99785895<br /> |0.99785892<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |&lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt;<br /> |0.99999997<br /> |1.00000000<br /> |-<br /> |}<br /> <br /> ==See also==<br /> * [[Madhava's sine table]]<br /> * [[Bhaskara I's sine approximation formula]]<br /> <br /> ==References==<br /> <br /> {{reflist}}<br /> <br /> {{DEFAULTSORT:Aryabhata's Sine Table}}<br /> [[Category:Trigonometry]]<br /> [[Category:Indian mathematics]]<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_sine_tables&diff=129856 2021-04-16T02:19:03Z

<p>BharataPuru: added the picture and made few improvements</p> <hr /> <div>Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in [[Surya Siddhanta]] and another text is the astronomical treatise [[Āryabhaṭīya]] which was composed during the fifth century by the [[Indian mathematician]] and astronomer [[Aryabhata|Āryabhaṭa]] (476–550 CE), for the computation of the [[Jya|half-chords]] of certain set of arcs of a circle. The table found in [[Surya Siddhanta]] is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the [[Finite difference|first differences]] of the values of [[Trigonometric functions|trigonometric sines]] expressed in [[arcminutes]], and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.&lt;ref&gt;{{cite journal|doi=10.1006/hmat.1997.2160|last=Takao Hayashi|first1=T|date=November 1997|title=Āryabhaṭa's rule and table for sine-differences|journal=Historia Mathematica |volume=24|issue=4|pages=396–406 }}&lt;/ref&gt;&lt;ref&gt;{{cite journal|doi=10.1007/BF00329978|last=B. L. van der Waerden|date=March 1988|first1=B. L.|title=Reconstruction of a Greek table of chords|journal=Archive for History of Exact Sciences|volume=38|issue=1|pages=23–38|title-link=table of chords}}&lt;/ref&gt;<br /> <br /> Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the [[history of mathematics]].&lt;ref name=&quot;mcs&quot;&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last=J J O'Connor and E F Robertson|date=June 1996 |accessdate=4 March 2010}}&lt;/ref&gt; Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the [[Madhava series|power series expansions]] of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the [[Kerala school of astronomy and mathematics]], and the tabulation of a [[Madhava's sine table|sine table by Madhava]] with values accurate to seven or eight decimal places.<br /> <br /> == Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==<br /> '''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three [[trigonometric functions]] introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is [[Surya Siddhanta]].&lt;ref name=&quot;Datta&quot;&gt;{{cite journal|last=B.B. Datta and A.N. Singh|date=1983|title=Hindu Trigonometry|journal=Indian Journal of History of Science|volume=18|issue=1|pages=39&amp;ndash;108|url=http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol18_1_5_BDatta.pdf|accessdate=1 March 2010}}&lt;/ref&gt; These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern [[trigonometric functions]] of [[sine]] and [[cosine]]. In fact, the origins of the modern terms of &quot;sine&quot; and &quot;cosine&quot; have been traced back to the [[Sanskrit]] words jyā and kotijyā.&lt;ref name=&quot;Datta&quot; /&gt;<br /> [[File:Jya Figure.jpg|thumb|419x419px]]<br /> <br /> ===Definition===<br /> Let 'arc AB' denote an [[Arc (geometry)|arc]] whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:<br /> <br /> * ''Vyāsardhā'' = Radius (R)<br /> * ''jyā'' of arc AB = MB<br /> * ''koti-jyā'' of arc AB = CM = R cos θ<br /> * ''utkrama-jyā'' of arc AB = MA<br /> <br /> If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:<br /> <br /> * ''jyā'' ( arc AB ) = ''R'' sin θ<br /> * ''koti-jyā'' ( arc AB ) = ''R'' cos θ<br /> * ''utkrama-jyā'' ( arc AB ) = R - R cos θ<br /> <br /> An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means &quot;a bow&quot;. <br /> The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means &quot;a bow-string&quot;.<br /> The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.&lt;ref&gt;According to lexicographers, it is a synonym also meaning &quot;bow-string&quot;, but only its <br /> geometrical meaning is attested in literature. Monier-Williams, ''A Sanskrit Dictionary'' (1899): &quot;'' jīvá'' n. (in geom. = ''jyā'') the chord of an arc; the sine of an arc ''Suryasiddhanta'' 2.57&quot;; <br /> ''jīvá'' as a generic adjective has the meaning of &quot;living, alive&quot; ([[:wikt:Appendix:Proto-Indo-European/gʷih₃wós|cognate]] with English ''[[:wikt:quick|quick]]'')&lt;/ref&gt;<br /> At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs.&lt;ref name=&quot;Datta&quot; /&gt;&lt;ref name=&quot;Glen&quot;&gt;{{cite book|last=Glen Van Brummelen|title=The mathematics of the heavens and the earth : the early history of trigonometry|publisher=[[Princeton University Press]]|date=2009|pages=95&amp;ndash;97|isbn=978-0-691-12973-0}}&lt;/ref&gt; The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant &quot;half of&quot;.<br /> <br /> The Sanskrit word ''koṭi'' has the meaning of &quot;point, cusp&quot;, and specifically &quot;the [[Recurve bow|curved end of a bow]]&quot;.<br /> In trigonometry, it came to denote &quot;the complement of an arc to 90°&quot;. Thus <br /> ''koṭi-jyā'' is &quot;the ''jyā'' of the complementary arc&quot;. In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes &quot;the side of a right angled triangle&quot;. Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> ''Utkrama'' means &quot;inverted&quot;, thus ''utkrama-jyā'' means &quot;inverted chord&quot;. The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning &quot;arrow&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of &quot;the ''jyā'' of three signs&quot;. The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning &quot;semi-diameter&quot;.&lt;ref name=&quot;Datta&quot; /&gt;<br /> <br /> According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by &quot;Rsin&quot; and &quot;Rcos&quot; treated as single words.&lt;ref name=&quot;Datta&quot; /&gt; Others denote ''jyā'' and ''koti-jyā'' respectively by &quot;Sin&quot; and &quot;Cos&quot; (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).&lt;ref name=&quot;Glen&quot; /&gt;<br /> <br /> ===From jyā to sine===<br /> The origins of the modern term sine have been traced to the Sanskrit word ''jyā'',&lt;ref&gt;{{cite web|url=http://mathforum.org/library/drmath/view/54053.html|title=How the Trig Functions Got their Names|work=Ask Dr. Math|publisher=[[Drexel University]]|accessdate=2 March 2010}}&lt;/ref&gt;&lt;ref&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last= J J O'Connor and E F Robertson|date=June 1996 |accessdate=2 March 2010}}&lt;/ref&gt; <br /> or more specifically to its synonym ''jīva''.<br /> This term was [[Indian influence on Islamic science|adopted in medieval Islamic mathematics]], transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means &quot;bosom&quot;. The text's 12th-century [[Medieval Latin|Latin]] translator used the Latin equivalent for &quot;bosom&quot;, ''[[wikt:sinus|sinus]]''.&lt;ref&gt;Various sources credit the first use of ''sinus'' to either: <br /> * [[Plato Tiburtinus]]'s 1116 translation of the ''Astronomy'' of [[Al-Battani]]<br /> * [[Gerard of Cremona]]'s c. 1150 translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]<br /> * [[Robert of Chester]]'s 1145 translation of the tables of al-Khwārizmī<br /> See Merlet, [https://link.springer.com/chapter/10.1007/1-4020-2204-2_16#page-1 ''A Note on the History of the Trigonometric Functions''] in Ceccarelli (ed.), ''International Symposium on History of Machines and Mechanisms'', Springer, 2004&lt;br&gt;See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.&lt;br&gt;See {{cite book |last=Katx |first=Victor |date=July 2008 |title=A history of mathematics |edition=3rd |location=Boston |publisher=Pearson |page=210 (sidebar) |isbn= 978-0321387004 |language=English }}&lt;/ref&gt; When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.<br /> <br /> ==The Indian sine tables==<br /> === Surya Siddhanta sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> ===Āryabhaṭa's sine table===<br /> The stanza in Āryabhaṭiya describing the sine table is reproduced below:<br /> <br /> मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |<br /> घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||<br /> <br /> The second section of Āryabhaṭiya titled Ganitapādd<br /> a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.&lt;ref name=&quot;Katz&quot; /&gt;<br /> * &quot;When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.&quot;<br /> This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function.<br /> Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in [[Degree (angle)|degrees]], [[arcminutes|minutes]], [[arcseconds|seconds]], etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. <br /> <br /> The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the [[Aryabhata numeration|numerical scheme]] explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in [[ISO 15919]] transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''[[jya]]'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.<br /> <br /> ===Comparing different sine tables===<br /> <br /> [[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle&lt;br /&gt;(in [[Degree (angle)|degrees]],&lt;br /&gt;[[arcminutes]])<br /> !Āryabhaṭa's &lt;br /&gt;value of Jyā (R.sine)<br /> ![[Surya Siddhanta]] value of Jyā (R.sine)<br /> !Surya Siddhanta versed sines<br /> !Modern value &lt;br /&gt;of ''[[Jyā]]'' R.sine<br /> !Madhava's sine values<br /> !Madhava's derived sine values<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |&lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.06540314<br /> |0.06540313<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |&lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> |0.13052623<br /> |0.13052619<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |&lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> |0.19509032<br /> |0.19509032<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |&lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.25881900<br /> |0.25881905<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |&lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> |0.32143947<br /> |0.32143947<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |&lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> |0.38268340<br /> |0.38268343<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |&lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> |0.44228865<br /> |0.44228869<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.49999998<br /> |0.50000000<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.55557022<br /> |0.55557023<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |&lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> |0.60876139<br /> |0.60876143<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |&lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.65934580<br /> |0.65934582<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |&lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.70710681<br /> |0.70710678<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |&lt;center&gt;2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> |0.75183985<br /> |0.75183981<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |&lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> |0.79335331<br /> |0.79335334<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |&lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> |0.83146960<br /> |0.83146961<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |&lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> |0.86602543<br /> |0.86602540<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |&lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt;<br /> |0.89687275<br /> |0.89687274<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |&lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.92387954<br /> |0.92387953<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |&lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.94693016<br /> |0.94693013<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |&lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> |0.96592581<br /> |0.96592583<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |&lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> |0.98078527<br /> |0.98078528<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |&lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99144487<br /> |0.99144486<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |&lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99785895<br /> |0.99785892<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |&lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt;<br /> |0.99999997<br /> |1.00000000<br /> |-<br /> |}<br /> <br /> ==See also==<br /> * [[Madhava's sine table]]<br /> * [[Bhaskara I's sine approximation formula]]<br /> <br /> ==References==<br /> <br /> {{reflist}}<br /> <br /> {{DEFAULTSORT:Aryabhata's Sine Table}}<br /> [[Category:Trigonometry]]<br /> [[Category:Indian mathematics]]<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=File:Jya_Figure.jpg&diff=129855 2021-04-16T02:12:59Z

<p>BharataPuru: </p> <hr /> <div>Jya (R sin) and Utkrama-Jya (R-Rcos) as shown in the figure is an Indian system using which the world's first sine tables were developed</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_standard_circle&diff=129665 2021-03-21T09:23:04Z

<p>BharataPuru: Created page with &quot;''Indian standard circle'' is a name given to the standard circle first used in Surya Siddhanta and later used by several ancient Indian mathematicians and astronomers to...&quot;</p> <hr /> <div>''Indian standard circle'' is a name given to the standard circle first used in [[Surya Siddhanta]] and later used by several ancient Indian mathematicians and astronomers to improve the [[Indian sine tables]] and for various other calculations. [[Surya Siddhanta]] provides methods for calculating the Jyā (R.sine) values. The circle uses a radius of 3,438 minutes. ''Surya Siddhanta'' calculates the first Jyā (R.sine) as 1/8th of the number of minutes(kalās) in a Rashi (zodiac sign). It says a Rashi (zodiac sign) has 1800 minutes (kalās) and thus calculates the first Jyā to a value of 225 minutes (kalās).&lt;ref&gt; Deva Shastri, Pundit Bapu (1861). &quot;Translation of the Surya Siddhanta&quot;. Ch2 Ve15, pp. 15–16.&lt;/ref&gt; &lt;br&gt;<br /> The Indian standard circles holds significance as it is based on number of minutes in circle thus leads to 360 degrees in a circle which is the basis of modern trigonometry. Although the [[Indian sine tables]] are not based on the degrees but rather on the R.sine (Jyā) values. The [[Surya Siddhanta]] data reflect highly sophisticated outcomes of the R.sine values. ''Burgess'' notes that it is remarkable to see that the 2nd order differences increase as the sines and each, in fact, is about 1/225th part of the corresponding sine.&lt;ref&gt;Burgess, Rev. Ebenezer (1860). Translation of the Surya Siddhanta. p. 115.&lt;/ref&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Madhava_of_Sangamagrama&diff=129664 2021-03-21T09:22:33Z

<p>BharataPuru: </p> <hr /> <div>'''Madhava's sine table''' is the table of sines of various angles constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama. The table lists the trigonometric sines of the twenty-four angles 3.75°, 7.50°, 11.25°, ..., and 90.00° (angles that are integral multiples of 3.75°, i.e. 1/24 of a right angle, beginning with 3.75 and ending with 90.00). The table is encoded in the letters of [[Devanagari]] using the [[Katapayadi system]]. This gives the entries in the table an appearance of the verses of a poem in [[Sanskrit]].<br /> <br /> Madhava's original work containing the sine table has not yet been traced. The table is seen reproduced in the ''Aryabhatiyabhashya'' of [[Nilakantha Somayaji]]&lt;ref name=&quot;:0&quot;&gt;''The Aryabhatiam of Aryabhattacharya with the Bhashya of Nilakantha Somasutvan, Part1-Gaṇitapāda,'' Edited by K. Sambasiva Sastri, Trivandrum Sanskrit Series No.101. p. 55.<br /> <br /> https://ia601902.us.archive.org/28/items/Trivandrum_Sanskrit_Series_TSS/TSS-101_Aryabhatiya_With_the_Commentary_of_Nilakanta_Somasutvan_Part_1_-_KS_Sastri_1930.pdf<br /> <br /> http://www.sanskritebooks.org/2013/02/trivandrum-sanskrit-series-anantasayana-samskrita-granthavali/&lt;/ref&gt;(1444–1544) and also in the ''Yuktidipika/Laghuvivrti'' commentary of [[Tantrasamgraha]] by [[Sankara Variar]] (circa. 1500-1560).&lt;ref name=&quot;Raju&quot;&gt;{{cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|pages=114–123}}&lt;/ref&gt;<br /> <br /> == The table ==<br /> The image below gives Madhava's sine table in [[Devanagari]] as reproduced in ''Cultural foundations of mathematics'' by C.K. Raju.&lt;ref&gt;{{cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|pages=120}}&lt;/ref&gt; The first twelve lines constitute the entries in the table. The last word in the thirteenth line indicates that these are &quot;as told by Madhava&quot;.<br /> <br /> [[File:Madhava sine tabe in Devanagari.jpg|thumb|center|350px]]<br /> <br /> ==Values in Madhava's table==<br /> [[File:Madhavasine.jpeg.jpg|thumb]]<br /> <br /> To understand the meaning of the values tabulated by [[Madhava of Sangamagrama|Madhava]], consider some angle whose measure is A. Consider a circle of unit radius and center O. Let the arc PQ of the circle subtend an angle A at the center O. Drop the perpendicular QR from Q to OP; then the length of the line segment RQ is the value of the trigonometric sine of the angle A. Let PS be an arc of the circle whose length is equal to the length of the segment RQ. For various angles A, Madhava's table gives the measures of the corresponding angles &lt;math&gt;\angle&lt;/math&gt;POS in [[Minute of arc|arcminutes]], [[arcsecond]]s and sixtieths of an [[arcsecond]].<br /> <br /> As an example, let A 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 arcseconds of the angle whose radian measure is the modern value of sin 22.50°. The modern numerical value of sin 22.50° is 0.382683432363 and,<br /> <br /> :0.382683432363 radians = 180 / &amp;pi; &amp;times; 0.382683432363 degrees = 21.926145564094 degrees.<br /> <br /> and<br /> <br /> :21.926145564094 degrees = 1315 arcminutes 34 arcseconds 07 sixtieths of arcsecond.<br /> <br /> In the [[Katapayadi system]] the digits are written in the reverse order. Thus in Madhava's table, the entry corresponding to 22.50° is 70435131.<br /> <br /> ==Derivation of trigonometric sines from Madhava's table==<br /> <br /> For an angle whose measure is ''A'', let<br /> <br /> :&lt;math&gt;\angle POS = m \text{ arcminutes, } s \text{ arcseconds, } t \text{ sixtieths of an arcsecond}&lt;/math&gt;<br /> <br /> Then<br /> <br /> :&lt;math&gt;<br /> \begin{align}<br /> \sin (A) &amp; = RQ \\<br /> &amp; = \text{length of arc } PS \\<br /> &amp; = \angle POS \text{ in radians}\\<br /> &amp; = \frac{\pi}{180\times 60}\left( m + \frac{s}{60}+ \frac{t}{60\times 60}\right).<br /> \end{align}<br /> &lt;/math&gt;<br /> <br /> Each of the lines in the table specifies eight digits. Let the digits corresponding to angle A (read from left to right) be<br /> <br /> :&lt;math&gt; d_1\quad d_2\quad d_3\quad d_4\quad d_5\quad d_6\quad d_7\quad d_8 &lt;/math&gt;<br /> <br /> Then according to the rules of the [[Katapayadi system]] of Kerala mathematicians we have<br /> <br /> :&lt;math&gt;<br /> \begin{align}<br /> m &amp; = d_8\times 1000 + d_7\times 100 + d_6 \times 10 +d_5\\<br /> s &amp; = d_4\times 10 + d_3\\<br /> t &amp; = d_2\times 10 + d_1<br /> \end{align}<br /> &lt;/math&gt;<br /> <br /> == Madhava's value of pi ==<br /> <br /> To complete the numerical computations one must have a knowledge of the value of [[pi]] (π). It is appropriate that we use the value of [[pi|&amp;pi;]] computed by Madhava himself. [[Nilakantha Somayaji]] has given this value of π in his [[Āryabhaṭīya]]-Bhashya as follows:&lt;ref&gt;{{cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|pages=119}}&lt;/ref&gt;<br /> <br /> &lt;br&gt;<br /> A transliteration of the last two lines:<br /> <br /> vibudha-netra-gaja-ahi-hutāśana&lt;br&gt;<br /> tri-guṇa-veda-bha-vāraṇa-bāhavaḥ&lt;br&gt;<br /> nava-nikharva-mite vr̥tivistare&lt;br&gt;<br /> paridhi-mānam idaṁ jagadur budhāḥ<br /> <br /> &lt;br&gt;<br /> The various words indicate certain numbers encoded in a scheme known as the [[bhūtasaṃkhyā system]]. The meaning of the words and the numbers encoded by them (beginning with the units place) are detailed in the following translation of the verse: <br /> &quot;Gods (vibudha : 33), eyes (netra : 2), elephants (gaja : 8), snakes (ahi : 8), fires (hutāśana : 3), three (tri : 3), qualities (guṇa : 3), vedas (veda : 4), nakṣatras (bha : 27), elephants (vāraṇa : 8), and arms (bāhavaḥ : 2) - the wise say that this is the measure of the circumference when the diameter of a circle is nava-nikharva (900,000,000,000).&quot;<br /> <br /> So, the translation of the poem using the [[bhūtasaṃkhyā system]] will simply read &quot;2827433388233 is, as the wise say, the circumference of a circle whose diameter is nava-nikharva (900,000,000,000)&quot;. That is, divide 2827433388233 (the number from the first two lines of the poem in reverse order) by nava-nikharva (900,000,000,000) to get the value of pi (π). This calculation yields the value π = 3.1415926535922. This is the value of π used by Madhava in his further calculations and is accurate to 11 decimal places.<br /> <br /> ==Example==<br /> <br /> Madhava's table lists the following digits corresponding to the angle 45.00°:<br /> <br /> :&lt;math&gt;5\quad 1\quad 1\quad 5\quad 0\quad 3\quad 4\quad 2&lt;/math&gt;<br /> <br /> This yields the angle with measure<br /> <br /> :&lt;math&gt;<br /> \begin{align}<br /> m &amp; = 2\times 1000 + 4\times 100 + 3\times 10 + 0 \text{ arcminutes}\\<br /> &amp; = 2430 \text{ arcminutes} \\<br /> s &amp; = 5\times 10 + 1 \text{ arcseconds}\\<br /> &amp; = 51 \text{ arcseconds}\\<br /> t &amp; = 1\times 10 + 5 \text{ sixtieths of an arcsecond}\\<br /> &amp; = 15 \text{ sixtieths of an arcsecond}<br /> \end{align}<br /> &lt;/math&gt;<br /> The value of the trigonometric sine of 45.00° as given in Madhava's table is<br /> <br /> :&lt;math&gt;<br /> \sin 45^\circ = \frac{\pi}{180\times 60}\left( 2430 + \frac{51}{60} + \frac{15}{60\times 60}\right)<br /> &lt;/math&gt;<br /> <br /> Substituting the value of π computed by Madhava in the above expression, one gets sin 45° as 0.70710681.<br /> <br /> This value may be compared with the modern exact value of sin 45.00°, namely, 0.70710678.<br /> <br /> == Comparison of Madhava's and modern sine values ==<br /> In table below the first column contains the list of the twenty-four angles beginning with 3.75 and ending with 90.00. The second column contains the values tabulated by Madhava in [[Devanagari]] in the form in which it was given by Madhava. (These are taken from ''Malayalam Commentary of [[Karanapaddhati]]'' by P.K. Koru&lt;ref&gt;{{cite book|last=[[Puthumana Somayaji]]|title=[[Karanapaddhati]] (with a commentary in [[Malayalam]] by P.K. Koru)|publisher=Astro Printing and Publishing Company|location= [[Cherpu]], [[Kerala]], [[India]]}} (Published in 1953)&lt;/ref&gt; and are slightly different from the table given in ''Cultural foundations of mathematics'' by C.K. Raju.&lt;ref name=&quot;Raju&quot; /&gt;) The third column contains ISO 15919 transliterations of the lines given in the second column. The digits encoded by the lines in second column are given in [[Arabic numeral]]s in the fourth column. The values of the trigonometric sines derived from the numbers specified in Madhava's table are listed in the fifth column. These values are computed using the approximate value 3.1415926535922 for π obtained by Madhava. For comparison, the exact values of the trigonometric sines of the angles are given in the sixth column.<br /> <br /> &lt;center&gt;<br /> <br /> {| align=&quot;center&quot; border=&quot;1&quot; cellpadding=&quot;5&quot; cellspacing=&quot;1&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Angle A &lt;br /&gt;in degrees <br /> ! colspan=&quot;3&quot; | Madhava's numbers for specifying sin A <br /> ! rowspan=&quot;2&quot; | Value of sin A &lt;br&gt; derived from &lt;br&gt; Madhava's table<br /> ! rowspan=&quot;2&quot; | Modern value &lt;br /&gt; of sin A <br /> |-<br /> | '''in [[Devanagari|Devanagari script]] &lt;br /&gt; using [[Katapayadi system]] &lt;br /&gt;(as in Madhava's &lt;br /&gt; original table) '''<br /> | '''in ISO 15919 transliteration''' &lt;br /&gt; scheme<br /> | '''Decoded Values in'''&lt;br&gt;'''minutes seconds thirds'''<br /> |-<br /> | &lt;center&gt;'''(1)'''&lt;/center&gt;<br /> | &lt;center&gt;'''(2)'''&lt;/center&gt;<br /> | &lt;center&gt;'''(3)'''&lt;/center&gt;<br /> | &lt;center&gt;'''(4)'''&lt;/center&gt;<br /> | &lt;center&gt;'''(5)'''&lt;/center&gt;<br /> | &lt;center&gt;'''(6)'''&lt;/center&gt;<br /> |-<br /> | &lt;center&gt;03.75&lt;/center&gt;<br /> | श्रेष्ठं नाम वरिष्ठानां<br /> | śreṣṭhaṁ nāma variṣṭhānāṁ<br /> | &lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> | 0.06540314 <br /> | 0.06540313 <br /> |-<br /> | &lt;center&gt;07.50&lt;/center&gt;<br /> | हिमाद्रिर्वेदभावनः<br /> | himādrirvēdabhāvanaḥ <br /> | &lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> | 0.13052623 <br /> | 0.13052619 <br /> |-<br /> | &lt;center&gt;11.25&lt;/center&gt;<br /> | तपनो भानु सूक्तज्ञो<br /> | tapanō bhānu sūktajñō <br /> | &lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> | 0.19509032 <br /> | 0.19509032 <br /> |-<br /> | &lt;center&gt;15.00&lt;/center&gt;<br /> | मध्यमं विद्धि दोहनं<br /> | maddhyamaṁ viddhi dōhanaṁ <br /> | &lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> | 0.25881900 <br /> | 0.25881905 <br /> |-<br /> | &lt;center&gt;18.75&lt;/center&gt;<br /> | धिगाज्यो नाशनं कष्टं<br /> | dhigājyō nāśanaṁ kaṣṭaṁ <br /> | &lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> | 0.32143947 <br /> | 0.32143947 <br /> |-<br /> | &lt;center&gt;22.50&lt;/center&gt;<br /> | छन्नभोगाशयाम्बिका<br /> | channabhōgāśayāmbikā<br /> | &lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> | 0.38268340 <br /> | 0.38268343 <br /> |-<br /> | &lt;center&gt;26.25&lt;/center&gt;<br /> | मृगाहारो नरेशोयं<br /> | mr̥gāhārō narēśōyaṁ<br /> | &lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> | 0.44228865 <br /> | 0.44228869 <br /> |-<br /> | &lt;center&gt;30.00&lt;/center&gt;<br /> | वीरो रणजयोत्सुकः<br /> | vīrō raṇajayōtsukaḥ<br /> | &lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> | 0.49999998 <br /> | 0.50000000<br /> |- <br /> | &lt;center&gt;33.75&lt;/center&gt;<br /> | मूलं विशुद्धं नाळस्य<br /> | mūlaṁ viṣuddhaṁ nāḷasya <br /> | &lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> | 0.55557022 <br /> | 0.55557023 <br /> |-<br /> | &lt;center&gt;37.50&lt;/center&gt;<br /> | गानेषु विरळा नराः<br /> | gāneṣu viraḷā narāḥ<br /> | &lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> | 0.60876139 <br /> | 0.60876143 <br /> |-<br /> | &lt;center&gt;41.25&lt;/center&gt;<br /> | अशुद्धिगुप्ता चोरश्रीः <br /> | aśuddhiguptā cōraśrīḥ <br /> | &lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> | 0.65934580<br /> | 0.65934582 <br /> |-<br /> | &lt;center&gt;45.00&lt;/center&gt;<br /> | शङ्कुकर्णो नगेश्वरः<br /> | śaṅkukarṇō nageśvaraḥ <br /> | &lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt; <br /> | 0.70710681 <br /> | 0.70710678 <br /> |-<br /> | &lt;center&gt;48.75&lt;/center&gt;<br /> | तनुजो गर्भजो मित्रं<br /> | tanujō garbhajō mitraṃ <br /> |&lt;center&gt; 2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> | 0.75183985 <br /> | 0.75183981 <br /> |-<br /> | &lt;center&gt;52.50&lt;/center&gt;<br /> | श्रीमानत्र सुखी सखे<br /> | śrīmānatra sukhī sakhē <br /> | &lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> | 0.79335331 <br /> | 0.79335334 <br /> |-<br /> | &lt;center&gt;56.25&lt;/center&gt;<br /> | शशी रात्रौ हिमाहारौ<br /> | śaśī rātrou himāhārou<br /> | &lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> | 0.83146960 <br /> | 0.83146961 <br /> |-<br /> | &lt;center&gt;60.00&lt;/center&gt;<br /> | वेगज्ञः पथि सिन्धुरः<br /> | vēgajñaḥ pathi sindhuraḥ <br /> | &lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> | 0.86602543 <br /> | 0.86602540 <br /> |-<br /> | &lt;center&gt;63.25&lt;/center&gt;<br /> | छाया लयो गजो नीलो<br /> | chāya layō gajō nīlō<br /> | &lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt; <br /> | 0.89687275 <br /> | 0.89687274 <br /> |-<br /> | &lt;center&gt;67.50&lt;/center&gt;<br /> | निर्मलो नास्ति सत्कुले<br /> | nirmalō nāsti satkulē <br /> | &lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> | 0.92387954 <br /> | 0.92387953 <br /> |- <br /> | &lt;center&gt;71.25&lt;/center&gt;<br /> | रात्रौ दर्पणमभ्राङ्गं<br /> | rātrou darpaṇamabhrāṅgaṁ <br /> | &lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> | 0.94693016 <br /> | 0.94693013 <br /> |-<br /> | &lt;center&gt;75.00&lt;/center&gt;<br /> | नागस्तुङ्ग नखो बली<br /> | nāgastuṅga nakhō balī<br /> | &lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> | 0.96592581 <br /> | 0.96592583 <br /> |-<br /> | &lt;center&gt;78.75&lt;/center&gt;<br /> | धीरो युवा कथालोलः<br /> | dhīrō yuvā kathālōlaḥ <br /> | &lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> | 0.98078527 <br /> | 0.98078528 <br /> |-<br /> | &lt;center&gt;82.50&lt;/center&gt;<br /> | पूज्यो नारीजनैर्भगः<br /> | pūjyō nārījanairbhagaḥ<br /> | &lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> | 0.99144487 <br /> | 0.99144486 <br /> |-<br /> | &lt;center&gt;86.25&lt;/center&gt;<br /> | कन्यागारे नागवल्ली<br /> | kanyāgārē nāgavallī <br /> | &lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> | 0.99785895 <br /> | 0.99785892 <br /> |-<br /> | &lt;center&gt;90.00&lt;/center&gt;<br /> | देवो विश्वस्थली भृगुः<br /> | devō viśvasthalī bhr̥ guḥ<br /> | &lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt; <br /> | 0.99999997 <br /> | 1.00000000 <br /> |}<br /> &lt;/center&gt;<br /> <br /> ==Madhava's method of computation==<br /> <br /> 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 like [[Nilakantha Somayaji]] ([[Tantrasangraha]]) and [[Jyeshtadeva]] ([[Yuktibhāṣā]]) that give ample references to Madhava's accomplishments, it is conjectured that Madhava computed his sine table using the power series expansion of sin ''x''.<br /> <br /> :&lt;math&gt;<br /> \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots<br /> &lt;/math&gt;<br /> <br /> ==See also==<br /> <br /> *[[Indian sine tables]]<br /> *[[Surya Siddhanta]]<br /> *[[Indian standard circle]]<br /> <br /> ==References==<br /> {{reflist}}<br /> <br /> ==Further references==<br /> *{{cite journal|last=Bag|first=A.K.|year=1976|title=Madhava's sine and cosine series|journal=Indian Journal of History of Science|publisher=Indian National Academy of Science|volume=11|issue=1|pages=54&amp;ndash;57|url=http://www.dli.gov.in/rawdataupload/upload/insa/INSA_1/20005af4_54.pdf|accessdate=21 August 2016|archive-url=https://web.archive.org/web/20150705200732/http://www.dli.gov.in/rawdataupload/upload/insa/INSA_1/20005af4_54.pdf|archive-date=5 July 2015|url-status=dead}}<br /> *For an account of Madhava's computation of the sine table see : {{cite book|last=Van Brummelen|first=Glen |title=The mathematics of the heavens and the earth : the early history of trigonometry|publisher=[[Princeton University Press]]|location=Princeton|year=2009|pages=113–120|isbn=978-0-691-12973-0|url=http://press.princeton.edu/titles/8956.html}}<br /> *For a thorough discussion of the computation of Madhava's sine table with historical references : {{cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|pages=114–123}}<br /> <br /> {{Indian mathematics}}<br /> <br /> {{DEFAULTSORT:Madhava's Sine Table}}<br /> [[Category:Trigonometry]]<br /> [[Category:Indian mathematics]]<br /> [[Category:Kerala school of astronomy and mathematics]]<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Madhava_of_Sangamagrama&diff=129663 2021-03-21T09:08:52Z

<p>BharataPuru: Creating a new page</p> <hr /> <div>'''Madhava's sine table''' is the table of sines of various angles constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama. The table lists the trigonometric sines of the twenty-four angles 3.75°, 7.50°, 11.25°, ..., and 90.00° (angles that are [[integer|integral]] [[Multiple (mathematics)|multiples]] of 3.75°, i.e. 1/24 of a right angle, beginning with 3.75 and ending with 90.00). The table is [[character encoding|encoded]] in the [[Letter (alphabet)|letters]] of [[Devanagari]] using the [[Katapayadi system]]. This gives the entries in the table an appearance of the [[Verse (poetry)|verses]] of a [[poem]] in [[Sanskrit]].<br /> <br /> Madhava's original work containing the sine table has not yet been traced. The table is seen reproduced in the ''Aryabhatiyabhashya'' of [[Nilakantha Somayaji]]&lt;ref name=&quot;:0&quot;&gt;''The Aryabhatiam of Aryabhattacharya with the Bhashya of Nilakantha Somasutvan, Part1-Gaṇitapāda,'' Edited by K. Sambasiva Sastri, Trivandrum Sanskrit Series No.101. p. 55.<br /> <br /> https://ia601902.us.archive.org/28/items/Trivandrum_Sanskrit_Series_TSS/TSS-101_Aryabhatiya_With_the_Commentary_of_Nilakanta_Somasutvan_Part_1_-_KS_Sastri_1930.pdf<br /> <br /> http://www.sanskritebooks.org/2013/02/trivandrum-sanskrit-series-anantasayana-samskrita-granthavali/&lt;/ref&gt;(1444–1544) and also in the ''Yuktidipika/Laghuvivrti'' commentary of [[Tantrasamgraha]] by [[Sankara Variar]] (circa. 1500-1560).&lt;ref name=&quot;Raju&quot;&gt;{{cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|pages=114–123}}&lt;/ref&gt;<br /> <br /> == The table ==<br /> The image below gives Madhava's sine table in [[Devanagari]] as reproduced in ''Cultural foundations of mathematics'' by C.K. Raju.&lt;ref&gt;{{cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|pages=120}}&lt;/ref&gt; The first twelve lines constitute the entries in the table. The last word in the thirteenth line indicates that these are &quot;as told by Madhava&quot;.<br /> <br /> [[File:Madhava sine tabe in Devanagari.jpg|thumb|center|350px]]<br /> <br /> ==Values in Madhava's table==<br /> [[File:Madhavasine.jpeg.jpg|thumb]]<br /> <br /> To understand the meaning of the values tabulated by [[Madhava of Sangamagrama|Madhava]], consider some angle whose measure is A. Consider a [[circle]] of unit radius and center O. Let the arc PQ of the circle subtend an angle A at the center O. Drop the [[perpendicular]] QR from Q to OP; then the length of the line segment RQ is the value of the trigonometric sine of the angle A. Let PS be an arc of the circle whose length is equal to the length of the segment RQ. For various angles A, Madhava's table gives the measures of the corresponding angles &lt;math&gt;\angle&lt;/math&gt;POS in [[Minute of arc|arcminutes]], [[arcsecond]]s and sixtieths of an [[arcsecond]].<br /> <br /> As an example, let A 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 arcseconds of the angle whose radian measure is the modern value of sin 22.50°. The modern numerical value of sin 22.50° is 0.382683432363 and,<br /> <br /> :0.382683432363 radians = 180 / &amp;pi; &amp;times; 0.382683432363 degrees = 21.926145564094 degrees.<br /> <br /> and<br /> <br /> :21.926145564094 degrees = 1315 arcminutes 34 arcseconds 07 sixtieths of arcsecond.<br /> <br /> In the [[Katapayadi system]] the digits are written in the reverse order. Thus in Madhava's table, the entry corresponding to 22.50° is 70435131.<br /> <br /> ==Derivation of trigonometric sines from Madhava's table==<br /> <br /> For an angle whose measure is ''A'', let<br /> <br /> :&lt;math&gt;\angle POS = m \text{ arcminutes, } s \text{ arcseconds, } t \text{ sixtieths of an arcsecond}&lt;/math&gt;<br /> <br /> Then<br /> <br /> :&lt;math&gt;<br /> \begin{align}<br /> \sin (A) &amp; = RQ \\<br /> &amp; = \text{length of arc } PS \\<br /> &amp; = \angle POS \text{ in radians}\\<br /> &amp; = \frac{\pi}{180\times 60}\left( m + \frac{s}{60}+ \frac{t}{60\times 60}\right).<br /> \end{align}<br /> &lt;/math&gt;<br /> <br /> Each of the lines in the table specifies eight digits. Let the digits corresponding to angle A (read from left to right) be<br /> <br /> :&lt;math&gt; d_1\quad d_2\quad d_3\quad d_4\quad d_5\quad d_6\quad d_7\quad d_8 &lt;/math&gt;<br /> <br /> Then according to the rules of the [[Katapayadi system]] of Kerala mathematicians we have<br /> <br /> :&lt;math&gt;<br /> \begin{align}<br /> m &amp; = d_8\times 1000 + d_7\times 100 + d_6 \times 10 +d_5\\<br /> s &amp; = d_4\times 10 + d_3\\<br /> t &amp; = d_2\times 10 + d_1<br /> \end{align}<br /> &lt;/math&gt;<br /> <br /> == Madhava's value of pi ==<br /> <br /> To complete the numerical computations one must have a knowledge of the value of [[pi]] (π). It is appropriate that we use the value of [[pi|&amp;pi;]] computed by Madhava himself. [[Nilakantha Somayaji]] has given this value of π in his [[Āryabhaṭīya]]-Bhashya as follows:&lt;ref&gt;{{cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|pages=119}}&lt;/ref&gt;<br /> &lt;br&gt;<br /> <br /> [[File:Madhava value of pi.jpg|thumb|left|350px]]&lt;br&gt;<br /> &lt;br&gt;<br /> &lt;br&gt;<br /> &lt;br&gt;<br /> &lt;br&gt;<br /> &lt;br&gt;<br /> A transliteration of the last two lines:<br /> <br /> vibudha-netra-gaja-ahi-hutāśana&lt;br&gt;<br /> tri-guṇa-veda-bha-vāraṇa-bāhavaḥ&lt;br&gt;<br /> nava-nikharva-mite vr̥tivistare&lt;br&gt;<br /> paridhi-mānam idaṁ jagadur budhāḥ<br /> <br /> &lt;br&gt;<br /> The various words indicate certain numbers encoded in a scheme known as the [[bhūtasaṃkhyā system]]. The meaning of the words and the numbers encoded by them (beginning with the units place) are detailed in the following translation of the verse: <br /> &quot;Gods (vibudha : 33), eyes (netra : 2), elephants (gaja : 8), snakes (ahi : 8), fires (hutāśana : 3), three (tri : 3), qualities (guṇa : 3), vedas (veda : 4), nakṣatras (bha : 27), elephants (vāraṇa : 8), and arms (bāhavaḥ : 2) - the wise say that this is the measure of the circumference when the diameter of a circle is nava-nikharva (900,000,000,000).&quot;<br /> <br /> So, the translation of the poem using the [[bhūtasaṃkhyā system]] will simply read &quot;2827433388233 is, as the wise say, the circumference of a circle whose diameter is nava-nikharva (900,000,000,000)&quot;. That is, divide 2827433388233 (the number from the first two lines of the poem in reverse order) by nava-nikharva (900,000,000,000) to get the value of pi (π). This calculation yields the value π = 3.1415926535922. This is the value of π used by Madhava in his further calculations and is accurate to 11 decimal places.<br /> <br /> ==Example==<br /> <br /> Madhava's table lists the following digits corresponding to the angle 45.00°:<br /> <br /> :&lt;math&gt;5\quad 1\quad 1\quad 5\quad 0\quad 3\quad 4\quad 2&lt;/math&gt;<br /> <br /> This yields the angle with measure<br /> <br /> :&lt;math&gt;<br /> \begin{align}<br /> m &amp; = 2\times 1000 + 4\times 100 + 3\times 10 + 0 \text{ arcminutes}\\<br /> &amp; = 2430 \text{ arcminutes} \\<br /> s &amp; = 5\times 10 + 1 \text{ arcseconds}\\<br /> &amp; = 51 \text{ arcseconds}\\<br /> t &amp; = 1\times 10 + 5 \text{ sixtieths of an arcsecond}\\<br /> &amp; = 15 \text{ sixtieths of an arcsecond}<br /> \end{align}<br /> &lt;/math&gt;<br /> The value of the trigonometric sine of 45.00° as given in Madhava's table is<br /> <br /> :&lt;math&gt;<br /> \sin 45^\circ = \frac{\pi}{180\times 60}\left( 2430 + \frac{51}{60} + \frac{15}{60\times 60}\right)<br /> &lt;/math&gt;<br /> <br /> Substituting the value of π computed by Madhava in the above expression, one gets sin 45° as 0.70710681.<br /> <br /> This value may be compared with the modern exact value of sin 45.00°, namely, 0.70710678.<br /> <br /> == Comparison of Madhava's and modern sine values ==<br /> In table below the first column contains the list of the twenty-four angles beginning with 3.75 and ending with 90.00. The second column contains the values tabulated by Madhava in [[Devanagari]] in the form in which it was given by Madhava. (These are taken from ''Malayalam Commentary of [[Karanapaddhati]]'' by P.K. Koru&lt;ref&gt;{{cite book|last=[[Puthumana Somayaji]]|title=[[Karanapaddhati]] (with a commentary in [[Malayalam]] by P.K. Koru)|publisher=Astro Printing and Publishing Company|location= [[Cherpu]], [[Kerala]], [[India]]}} (Published in 1953)&lt;/ref&gt; and are slightly different from the table given in ''Cultural foundations of mathematics'' by C.K. Raju.&lt;ref name=&quot;Raju&quot;/&gt;) The third column contains [[ISO 15919|ISO 15919 transliterations]] of the lines given in the second column. The digits encoded by the lines in second column are given in [[Arabic numeral]]s in the fourth column. The values of the trigonometric sines derived from the numbers specified in Madhava's table are listed in the fifth column. These values are computed using the approximate value 3.1415926535922 for π obtained by Madhava. For comparison, the exact values of the trigonometric sines of the angles are given in the sixth column.<br /> <br /> &lt;center&gt;<br /> <br /> {| align=&quot;center&quot; border=&quot;1&quot; cellpadding=&quot;5&quot; cellspacing=&quot;1&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Angle A &lt;br/&gt;in degrees <br /> ! colspan=&quot;3&quot; | Madhava's numbers for specifying sin A <br /> ! rowspan=&quot;2&quot; | Value of sin A &lt;br&gt; derived from &lt;br&gt; Madhava's table<br /> ! rowspan=&quot;2&quot; | Modern value &lt;br/&gt; of sin A <br /> |-<br /> | '''in [[Devanagari|Devanagari script]] &lt;br/&gt; using [[Katapayadi system]] &lt;br/&gt;(as in Madhava's &lt;br/&gt; original table) '''<br /> | '''in [[ISO 15919|ISO 15919 transliteration]]''' &lt;br/&gt; scheme<br /> | '''Decoded Values in'''&lt;br&gt;'''minutes seconds thirds'''<br /> |-<br /> | &lt;center&gt;'''(1)'''&lt;/center&gt;<br /> | &lt;center&gt;'''(2)'''&lt;/center&gt;<br /> | &lt;center&gt;'''(3)'''&lt;/center&gt;<br /> | &lt;center&gt;'''(4)'''&lt;/center&gt;<br /> | &lt;center&gt;'''(5)'''&lt;/center&gt;<br /> | &lt;center&gt;'''(6)'''&lt;/center&gt;<br /> |-<br /> | &lt;center&gt;03.75&lt;/center&gt;<br /> | श्रेष्ठं नाम वरिष्ठानां<br /> | śreṣṭhaṁ nāma variṣṭhānāṁ<br /> | &lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> | 0.06540314 <br /> | 0.06540313 <br /> |-<br /> | &lt;center&gt;07.50&lt;/center&gt;<br /> | हिमाद्रिर्वेदभावनः<br /> | himādrirvēdabhāvanaḥ <br /> | &lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> | 0.13052623 <br /> | 0.13052619 <br /> |-<br /> | &lt;center&gt;11.25&lt;/center&gt;<br /> | तपनो भानु सूक्तज्ञो<br /> | tapanō bhānu sūktajñō <br /> | &lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> | 0.19509032 <br /> | 0.19509032 <br /> |-<br /> | &lt;center&gt;15.00&lt;/center&gt;<br /> | मध्यमं विद्धि दोहनं<br /> | maddhyamaṁ viddhi dōhanaṁ <br /> | &lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> | 0.25881900 <br /> | 0.25881905 <br /> |-<br /> | &lt;center&gt;18.75&lt;/center&gt;<br /> | धिगाज्यो नाशनं कष्टं<br /> | dhigājyō nāśanaṁ kaṣṭaṁ <br /> | &lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> | 0.32143947 <br /> | 0.32143947 <br /> |-<br /> | &lt;center&gt;22.50&lt;/center&gt;<br /> | छन्नभोगाशयाम्बिका<br /> | channabhōgāśayāmbikā<br /> | &lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> | 0.38268340 <br /> | 0.38268343 <br /> |-<br /> | &lt;center&gt;26.25&lt;/center&gt;<br /> | मृगाहारो नरेशोयं<br /> | mr̥gāhārō narēśōyaṁ<br /> | &lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> | 0.44228865 <br /> | 0.44228869 <br /> |-<br /> | &lt;center&gt;30.00&lt;/center&gt;<br /> | वीरो रणजयोत्सुकः<br /> | vīrō raṇajayōtsukaḥ<br /> | &lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> | 0.49999998 <br /> | 0.50000000<br /> |- <br /> | &lt;center&gt;33.75&lt;/center&gt;<br /> | मूलं विशुद्धं नाळस्य<br /> | mūlaṁ viṣuddhaṁ nāḷasya <br /> | &lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> | 0.55557022 <br /> | 0.55557023 <br /> |-<br /> | &lt;center&gt;37.50&lt;/center&gt;<br /> | गानेषु विरळा नराः<br /> | gāneṣu viraḷā narāḥ<br /> | &lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> | 0.60876139 <br /> | 0.60876143 <br /> |-<br /> | &lt;center&gt;41.25&lt;/center&gt;<br /> | अशुद्धिगुप्ता चोरश्रीः <br /> | aśuddhiguptā cōraśrīḥ <br /> | &lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> | 0.65934580<br /> | 0.65934582 <br /> |-<br /> | &lt;center&gt;45.00&lt;/center&gt;<br /> | शङ्कुकर्णो नगेश्वरः<br /> | śaṅkukarṇō nageśvaraḥ <br /> | &lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt; <br /> | 0.70710681 <br /> | 0.70710678 <br /> |-<br /> | &lt;center&gt;48.75&lt;/center&gt;<br /> | तनुजो गर्भजो मित्रं<br /> | tanujō garbhajō mitraṃ <br /> |&lt;center&gt; 2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> | 0.75183985 <br /> | 0.75183981 <br /> |-<br /> | &lt;center&gt;52.50&lt;/center&gt;<br /> | श्रीमानत्र सुखी सखे<br /> | śrīmānatra sukhī sakhē <br /> | &lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> | 0.79335331 <br /> | 0.79335334 <br /> |-<br /> | &lt;center&gt;56.25&lt;/center&gt;<br /> | शशी रात्रौ हिमाहारौ<br /> | śaśī rātrou himāhārou<br /> | &lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> | 0.83146960 <br /> | 0.83146961 <br /> |-<br /> | &lt;center&gt;60.00&lt;/center&gt;<br /> | वेगज्ञः पथि सिन्धुरः<br /> | vēgajñaḥ pathi sindhuraḥ <br /> | &lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> | 0.86602543 <br /> | 0.86602540 <br /> |-<br /> | &lt;center&gt;63.25&lt;/center&gt;<br /> | छाया लयो गजो नीलो<br /> | chāya layō gajō nīlō<br /> | &lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt; <br /> | 0.89687275 <br /> | 0.89687274 <br /> |-<br /> | &lt;center&gt;67.50&lt;/center&gt;<br /> | निर्मलो नास्ति सत्कुले<br /> | nirmalō nāsti satkulē <br /> | &lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> | 0.92387954 <br /> | 0.92387953 <br /> |- <br /> | &lt;center&gt;71.25&lt;/center&gt;<br /> | रात्रौ दर्पणमभ्राङ्गं<br /> | rātrou darpaṇamabhrāṅgaṁ <br /> | &lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> | 0.94693016 <br /> | 0.94693013 <br /> |-<br /> | &lt;center&gt;75.00&lt;/center&gt;<br /> | नागस्तुङ्ग नखो बली<br /> | nāgastuṅga nakhō balī<br /> | &lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> | 0.96592581 <br /> | 0.96592583 <br /> |-<br /> | &lt;center&gt;78.75&lt;/center&gt;<br /> | धीरो युवा कथालोलः<br /> | dhīrō yuvā kathālōlaḥ <br /> | &lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> | 0.98078527 <br /> | 0.98078528 <br /> |-<br /> | &lt;center&gt;82.50&lt;/center&gt;<br /> | पूज्यो नारीजनैर्भगः<br /> | pūjyō nārījanairbhagaḥ<br /> | &lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> | 0.99144487 <br /> | 0.99144486 <br /> |-<br /> | &lt;center&gt;86.25&lt;/center&gt;<br /> | कन्यागारे नागवल्ली<br /> | kanyāgārē nāgavallī <br /> | &lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> | 0.99785895 <br /> | 0.99785892 <br /> |-<br /> | &lt;center&gt;90.00&lt;/center&gt;<br /> | देवो विश्वस्थली भृगुः<br /> | devō viśvasthalī bhr̥ guḥ<br /> | &lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt; <br /> | 0.99999997 <br /> | 1.00000000 <br /> |}<br /> &lt;/center&gt;<br /> <br /> ==Madhava's method of computation==<br /> <br /> 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 like [[Nilakantha Somayaji]] ([[Tantrasangraha]]) and [[Jyeshtadeva]] ([[Yuktibhāṣā]]) that give ample references to Madhava's accomplishments, it is conjectured that Madhava computed his sine table using the power series expansion of sin ''x''.<br /> <br /> :&lt;math&gt;<br /> \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots<br /> &lt;/math&gt;<br /> <br /> ==See also==<br /> <br /> *[[Indian sine tables]]<br /> *[[Surya Siddhanta]]<br /> *[[Indian standard circle]]<br /> <br /> ==References==<br /> {{reflist}}<br /> <br /> ==Further references==<br /> *{{cite journal|last=Bag|first=A.K.|year=1976|title=Madhava's sine and cosine series|journal=Indian Journal of History of Science|publisher=Indian National Academy of Science|volume=11|issue=1|pages=54&amp;ndash;57|url=http://www.dli.gov.in/rawdataupload/upload/insa/INSA_1/20005af4_54.pdf|accessdate=21 August 2016|archive-url=https://web.archive.org/web/20150705200732/http://www.dli.gov.in/rawdataupload/upload/insa/INSA_1/20005af4_54.pdf|archive-date=5 July 2015|url-status=dead}}<br /> *For an account of Madhava's computation of the sine table see : {{cite book|last=Van Brummelen|first=Glen |title=The mathematics of the heavens and the earth : the early history of trigonometry|publisher=[[Princeton University Press]]|location=Princeton|year=2009|pages=113–120|isbn=978-0-691-12973-0|url=http://press.princeton.edu/titles/8956.html}}<br /> *For a thorough discussion of the computation of Madhava's sine table with historical references : {{cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|pages=114–123}}<br /> <br /> {{Indian mathematics}}<br /> <br /> {{DEFAULTSORT:Madhava's Sine Table}}<br /> [[Category:Trigonometry]]<br /> [[Category:Indian mathematics]]<br /> [[Category:Kerala school of astronomy and mathematics]]</div>

BharataPuru http://dharmawiki.org/index.php?title=Indian_sine_tables&diff=129662 2021-03-21T09:06:02Z

<p>BharataPuru: Creating a new page</p> <hr /> <div>Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in [[Surya Siddhanta]] and another text is the astronomical treatise [[Āryabhaṭīya]] which was composed during the fifth century by the [[Indian mathematician]] and astronomer [[Aryabhata|Āryabhaṭa]] (476–550 CE), for the computation of the [[Jya|half-chords]] of certain set of arcs of a circle. The table found in [[Surya Siddhanta]] is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the [[Finite difference|first differences]] of the values of [[Trigonometric functions|trigonometric sines]] expressed in [[arcminutes]], and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.&lt;ref&gt;{{cite journal|doi=10.1006/hmat.1997.2160|last=Takao Hayashi|first1=T|date=November 1997|title=Āryabhaṭa's rule and table for sine-differences|journal=Historia Mathematica |volume=24|issue=4|pages=396–406 }}&lt;/ref&gt;&lt;ref&gt;{{cite journal|doi=10.1007/BF00329978|last=B. L. van der Waerden|date=March 1988|first1=B. L.|title=Reconstruction of a Greek table of chords|journal=Archive for History of Exact Sciences|volume=38|issue=1|pages=23–38|title-link=table of chords}}&lt;/ref&gt;<br /> [[File:Jya and ardhajya.jpg|thumb]]<br /> <br /> Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the [[history of mathematics]].&lt;ref name=&quot;mcs&quot;&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last=J J O'Connor and E F Robertson|date=June 1996 |accessdate=4 March 2010}}&lt;/ref&gt; Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the [[Madhava series|power series expansions]] of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the [[Kerala school of astronomy and mathematics]], and the tabulation of a [[Madhava's sine table|sine table by Madhava]] with values accurate to seven or eight decimal places.<br /> <br /> == Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==<br /> '''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three [[trigonometric functions]] introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is [[Surya Siddhanta]].&lt;ref name=&quot;Datta&quot;&gt;{{cite journal|last=B.B. Datta and A.N. Singh|date=1983|title=Hindu Trigonometry|journal=Indian Journal of History of Science|volume=18|issue=1|pages=39&amp;ndash;108|url=http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol18_1_5_BDatta.pdf|accessdate=1 March 2010}}&lt;/ref&gt; These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern [[trigonometric functions]] of [[sine]] and [[cosine]]. In fact, the origins of the modern terms of &quot;sine&quot; and &quot;cosine&quot; have been traced back to the [[Sanskrit]] words jyā and kotijyā.&lt;ref name=&quot;Datta&quot;/&gt;<br /> <br /> ===Definition===<br /> [[File:1920px-Modern diagram for jya and kojya.svg.png|thumb]]<br /> <br /> Let 'arc AB' denote an [[Arc (geometry)|arc]] whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:<br /> <br /> * ''jyā'' of arc AB = BM<br /> * ''koti-jyā'' of arc AB = OM<br /> * ''utkrama-jyā'' of arc AB = MA<br /> <br /> If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:<br /> <br /> * ''jyā'' ( arc AB ) = ''R'' sin ( ''s'' / ''R'' )<br /> * ''koti-jyā'' ( arc AB ) = ''R'' cos ( ''s'' / ''R'' )<br /> * ''utkrama-jyā'' ( arc AB ) = ''R'' ( 1 - cos ( ''s'' / ''R'' ) ) = ''R'' [[Versine|versin]] ( ''s'' / ''R'' )<br /> <br /> ==={{anchor|Rsin|Rcos}}Terminology===<br /> [[File:Jya and ardhajya.jpg|thumb]]<br /> [[File:Jya and kotijya.jpg|thumb]]<br /> <br /> An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means &quot;a bow&quot;. <br /> The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means &quot;a bow-string&quot;.<br /> The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.&lt;ref&gt;According to lexicographers, it is a synonym also meaning &quot;bow-string&quot;, but only its <br /> geometrical meaning is attested in literature. Monier-Williams, ''A Sanskrit Dictionary'' (1899): &quot;'' jīvá'' n. (in geom. = ''jyā'') the chord of an arc; the sine of an arc ''Suryasiddhanta'' 2.57&quot;; <br /> ''jīvá'' as a generic adjective has the meaning of &quot;living, alive&quot; ([[:wikt:Appendix:Proto-Indo-European/gʷih₃wós|cognate]] with English ''[[:wikt:quick|quick]]'')&lt;/ref&gt;<br /> At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs.&lt;ref name=&quot;Datta&quot;/&gt;&lt;ref name=&quot;Glen&quot;&gt;{{cite book|last=Glen Van Brummelen|title=The mathematics of the heavens and the earth : the early history of trigonometry|publisher=[[Princeton University Press]]|date=2009|pages=95&amp;ndash;97|isbn=978-0-691-12973-0}}&lt;/ref&gt; The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant &quot;half of&quot;.<br /> <br /> The Sanskrit word ''koṭi'' has the meaning of &quot;point, cusp&quot;, and specifically &quot;the [[Recurve bow|curved end of a bow]]&quot;.<br /> In trigonometry, it came to denote &quot;the complement of an arc to 90°&quot;. Thus <br /> ''koṭi-jyā'' is &quot;the ''jyā'' of the complementary arc&quot;. In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes &quot;the side of a right angled triangle&quot;. Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .&lt;ref name=&quot;Datta&quot;/&gt;<br /> <br /> ''Utkrama'' means &quot;inverted&quot;, thus ''utkrama-jyā'' means &quot;inverted chord&quot;. <br /> The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning &quot;arrow&quot;.&lt;ref name=&quot;Datta&quot;/&gt;<br /> <br /> An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of &quot;the ''jyā'' of three signs&quot;. The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning &quot;semi-diameter&quot;.&lt;ref name=&quot;Datta&quot;/&gt;<br /> <br /> According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by &quot;Rsin&quot; and &quot;Rcos&quot; treated as single words.&lt;ref name=&quot;Datta&quot;/&gt; Others denote ''jyā'' and ''koti-jyā'' respectively by &quot;Sin&quot; and &quot;Cos&quot; (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).&lt;ref name=&quot;Glen&quot;/&gt;<br /> <br /> ===From jyā to sine===<br /> The origins of the modern term sine have been traced to the Sanskrit word ''jyā'',&lt;ref&gt;{{cite web|url=http://mathforum.org/library/drmath/view/54053.html|title=How the Trig Functions Got their Names|work=Ask Dr. Math|publisher=[[Drexel University]]|accessdate=2 March 2010}}&lt;/ref&gt;&lt;ref&gt;{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last= J J O'Connor and E F Robertson|date=June 1996 |accessdate=2 March 2010}}&lt;/ref&gt; <br /> or more specifically to its synonym ''jīva''.<br /> This term was [[Indian influence on Islamic science|adopted in medieval Islamic mathematics]], transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means &quot;bosom&quot;. The text's 12th-century [[Medieval Latin|Latin]] translator used the Latin equivalent for &quot;bosom&quot;, ''[[wikt:sinus|sinus]]''.&lt;ref&gt; Various sources credit the first use of ''sinus'' to either: <br /> * [[Plato Tiburtinus]]'s 1116 translation of the ''Astronomy'' of [[Al-Battani]]<br /> * [[Gerard of Cremona]]'s c. 1150 translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]<br /> * [[Robert of Chester]]'s 1145 translation of the tables of al-Khwārizmī<br /> See Merlet, [https://link.springer.com/chapter/10.1007/1-4020-2204-2_16#page-1 ''A Note on the History of the Trigonometric Functions''] in Ceccarelli (ed.), ''International Symposium on History of Machines and Mechanisms'', Springer, 2004&lt;br&gt;See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.&lt;br&gt;See {{cite book |last=Katx |first=Victor |date=July 2008 |title=A history of mathematics |edition=3rd |location=Boston |publisher=Pearson |page=210 (sidebar) |isbn= 978-0321387004 |language=English }}&lt;/ref&gt; When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.<br /> <br /> ==The Indian sine tables==<br /> === Surya Siddhanta sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> ===Āryabhaṭa's sine table===<br /> The stanza in Āryabhaṭiya describing the sine table is reproduced below:<br /> <br /> मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |<br /> घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||<br /> <br /> The second section of Āryabhaṭiya titled Ganitapādd<br /> a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.&lt;ref name=&quot;Katz&quot;/&gt;<br /> * &quot;When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.&quot;<br /> This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function.<br /> Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in [[Degree (angle)|degrees]], [[arcminutes|minutes]], [[arcseconds|seconds]], etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. <br /> <br /> The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the [[Aryabhata numeration|numerical scheme]] explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in [[ISO 15919]] transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''[[jya]]'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.<br /> <br /> ===Comparing different sine tables===<br /> <br /> [[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle&lt;br /&gt;(in [[Degree (angle)|degrees]],&lt;br /&gt;[[arcminutes]])<br /> !Āryabhaṭa's &lt;br /&gt;value of Jyā (R.sine)<br /> ![[Surya Siddhanta]] value of Jyā (R.sine)<br /> !Surya Siddhanta versed sines<br /> !Modern value &lt;br /&gt;of ''[[Jyā]]'' R.sine<br /> !Madhava's sine values<br /> !Madhava's derived sine values<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |&lt;center&gt;224ʹ 50ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.06540314<br /> |0.06540313<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |&lt;center&gt;448ʹ 42ʹʹ 58ʹʹʹ&lt;/center&gt;<br /> |0.13052623<br /> |0.13052619<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |&lt;center&gt;670ʹ 40ʹʹ 16ʹʹʹ&lt;/center&gt;<br /> |0.19509032<br /> |0.19509032<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |&lt;center&gt;889ʹ 45ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.25881900<br /> |0.25881905<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |&lt;center&gt;1105ʹ 01ʹʹ 39ʹʹʹ&lt;/center&gt;<br /> |0.32143947<br /> |0.32143947<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |&lt;center&gt;1315ʹ 34ʹʹ 07ʹʹʹ&lt;/center&gt;<br /> |0.38268340<br /> |0.38268343<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |&lt;center&gt;1520ʹ 28ʹʹ 35ʹʹʹ&lt;/center&gt;<br /> |0.44228865<br /> |0.44228869<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.49999998<br /> |0.50000000<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |&lt;center&gt;1718ʹ 52ʹʹ 24ʹʹʹ&lt;/center&gt;<br /> |0.55557022<br /> |0.55557023<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |&lt;center&gt;2092ʹ 46ʹʹ 03ʹʹʹ&lt;/center&gt;<br /> |0.60876139<br /> |0.60876143<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |&lt;center&gt;2266ʹ 39ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.65934580<br /> |0.65934582<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |&lt;center&gt;2430ʹ 51ʹʹ 15ʹʹʹ&lt;/center&gt;<br /> |0.70710681<br /> |0.70710678<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |&lt;center&gt;2584ʹ 38ʹʹ 06ʹʹʹ&lt;/center&gt;<br /> |0.75183985<br /> |0.75183981<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |&lt;center&gt;2727ʹ 20ʹʹ 52ʹʹʹ&lt;/center&gt;<br /> |0.79335331<br /> |0.79335334<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |&lt;center&gt;2858ʹ 22ʹʹ 55ʹʹʹ&lt;/center&gt;<br /> |0.83146960<br /> |0.83146961<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |&lt;center&gt;2977ʹ 10ʹʹ 34ʹʹʹ&lt;/center&gt;<br /> |0.86602543<br /> |0.86602540<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |&lt;center&gt;3083ʹ 13ʹʹ 17ʹʹʹ&lt;/center&gt;<br /> |0.89687275<br /> |0.89687274<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |&lt;center&gt;3176ʹ 03ʹʹ 50ʹʹʹ&lt;/center&gt;<br /> |0.92387954<br /> |0.92387953<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |&lt;center&gt;3255ʹ 18ʹʹ 22ʹʹʹ&lt;/center&gt;<br /> |0.94693016<br /> |0.94693013<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |&lt;center&gt;3320ʹ 36ʹʹ 30ʹʹʹ&lt;/center&gt;<br /> |0.96592581<br /> |0.96592583<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |&lt;center&gt;3371ʹ 41ʹʹ 29ʹʹʹ&lt;/center&gt;<br /> |0.98078527<br /> |0.98078528<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |&lt;center&gt;3408ʹ 20ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99144487<br /> |0.99144486<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |&lt;center&gt;3430ʹ 23ʹʹ 11ʹʹʹ&lt;/center&gt;<br /> |0.99785895<br /> |0.99785892<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |&lt;center&gt;3437ʹ 44ʹʹ 48ʹʹʹ&lt;/center&gt;<br /> |0.99999997<br /> |1.00000000<br /> |-<br /> |}<br /> <br /> ==See also==<br /> * [[Madhava's sine table]]<br /> * [[Bhaskara I's sine approximation formula]]<br /> <br /> ==References==<br /> <br /> {{reflist}}<br /> <br /> {{DEFAULTSORT:Aryabhata's Sine Table}}<br /> [[Category:Trigonometry]]<br /> [[Category:Indian mathematics]]</div>

BharataPuru http://dharmawiki.org/index.php?title=Surya_Siddhanta_(%E0%A4%B8%E0%A5%82%E0%A4%B0%E0%A5%8D%E0%A4%AF_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A4%BE)&diff=129661 2021-03-21T08:59:21Z

<p>BharataPuru: /* History */</p> <hr /> <div>'''''Sūrya Siddhānta''''' is an ancient Indian treatise in Astronomy. Like many classical Indian works, the ''Sūrya Siddhānta'' is a poem in [[Sanskrit]] language. It has fourteen chapter and 500 verses. It is composed in ''śloka'' metrical style of Sanskrit. It contain works on Indian sine tables, cosmology, eclipses, planetary motions, conjunctions, star positions, geography, instrumentation, concepts of time and mathematics. Unlike conventional books ''Sūrya Siddhānta'' contains advanced calculation and methods which are not easily comprehensible for a rank beginner. &lt;ref&gt;[https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol45_4_1_ANarayan.pdf]&lt;/ref&gt; The text had been updated several times and the last update appears to have been made around 580 CE to the ''Nakshatra'' longitudes. In second chapter, the text contains the calculation of Earth's obliquity of ''1397 jya (R.sine) 23.975°'' modern units indicating the time of calculation around 3000 BCE. There are several other observations in the tradition of ''Indian Astronomy'' that were also recorded in the vicinity of 3000 BCE. This could possibly be the time of the origin of ''Sūrya Siddhānta'' although scholars do not seem to have a consensus on the origin of this text of ''Indian Astronomy''.<br /> <br /> == History ==<br /> &lt;nowiki&gt;''Sūrya Siddhānta'' is well known, most referred and most esteemed. The original author of ''Sūrya Siddhānta'' is ''Mayasura'' as described in the story in the first chapter that ''Mayasura'' obtained his knowledge from ''Sūrya'' (the Sun). ''Siddhānta'' in Sanskrit means ''treatise'' and it usually has author'&lt;/nowiki&gt;s name prefixed to it. There were several other works on Astronomy in ancient India, many of which have since been lost.<br /> <br /> {| class=&quot;wikitable&quot;<br /> |-<br /> | ''Surya Siddhānta'' || ''Brahma Siddhānta'' || ''Soma Siddhānta''<br /> |-<br /> | ''Vyasa Siddhānta'' || ''Vashishtha Siddhānta'' || ''Atri Siddhānta''<br /> |-<br /> | ''Parashira Siddhānta'' || ''Kashyap Siddhānta'' || ''Nārad Siddhānta''<br /> |-<br /> | ''Garga Siddhānta'' || ''Marici Siddhānta'' || ''Manu Siddhānta''<br /> |-<br /> | ''Angiras Siddhānta'' || ''Lomasha Siddhānta'' || ''Pulisha Siddhānta''<br /> |-<br /> | ''Cyavana Siddhānta'' || ''Yavana Siddhānta'' || ''Bhrigu Siddhānta''<br /> |}<br /> <br /> == Content ==<br /> ''Sūrya Siddhānta'' contains 14 chapters and 500 verses. The chapters contain observations, methods, instruments and calculations of various astronomical phenomenas. There is a scarcity of scientific analysis done on the text of ''Surya Siddhanta''. Majority western work is based on Indology dates which in itself is controversial and based on their biased opinion of granting the origin of any science or mathematics to the ancient Greek or babylonians despite of immense textual evidence pointing otherwise. Their analysis of Surya Siddhanta primarily avoids the study of actual data and observations recorded within the ''Surya Siddhanta''.<br /> <br /> === Indian origin of seconds, minutes and degrees ===<br /> ''Surya Siddhanta'' in chapter 2 describes the units of seconds, minutes and degrees. These units of measurement are primary basis of the calculations of earth's obliquity and sine tables of ''Surya Siddhanta''. It is reasonable to think that these units or concepts had been in existence prior to other calculations and observation made in the epoch of 6th millennium BC as discussed in this article. The descriptions are<br /> <br /> {| class=&quot;wikitable&quot;<br /> |+ ''Surya Siddhanta'' units: seconds, minutes and degrees&lt;ref&gt;Pundit Bapu Deva Shastri, &quot;English Translation of Surya Siddhanta&quot;,p11, 1861&lt;/ref&gt; <br /> |-<br /> ! Modern SI units !! Surya Siddhanta units !! Value<br /> |-<br /> | Second || Vikala || -<br /> |-<br /> | Minute || Kala || 60 seconds<br /> |-<br /> | Degree || Ansh || 60 minutes<br /> |-<br /> | Zodiac Sign || Rashi || 30 degrees<br /> |-<br /> | Revolution || Bhagan || 12 zodiac signs<br /> |}<br /> <br /> These units are used in several calculations done through out the text of ''Surya Siddhanta''. In the sine tables of ''Surya Siddhanta'' the first sine or Jyā is described as the value equal to 1/8th of the number of minutes (Kalas) in a zodiac sign (Rashi).<br /> <br /> === Indian standard circle ===<br /> The ''Surya Siddhanta'' is using the [[Indian standard circle]] in various calculations through out the text. This standard circle is based on radius of 3,438 minutes. The significance lies in the precision of 1/3438 that the ancient Indian astronomers were able to work with. It is evident from the calculation of obliquity of the earth's axis in chapter 2 where 1397 units is the measured R-sine value. <br /> Another interesting outcome of this radius of 3,438 minutes is that the circumference of the standard Indian circle is calculated as 21,600 minutes using the formula of Pi multiply by diameter (twice the radius).<br /> <br /> === Nakshatra (Asterism) System ===<br /> <br /> The ''Surya Siddhanta'' uses the 27 [[Nakshatra system]] throughout the text. The Nakshatra is a smaller constellation typically consisting of 1 to 5 stars. The brightest star is called as Yogtara. Each Nakshatra spans 13° 20' on the ecliptic. Each Nakshatra has its own primary star which is usually the junction star but not always.<br /> <br /> === Longitudinal updates - 580 AD ===<br /> Chapter 8 of ''Surya Siddhanta'' primarily focuses on the stellar data. It provides the longitudinal data for the Asterisms. In comparison to the present day longitudinal values of these stars and the data of Surya Siddhanta, it becomes clear that this update to Surya Siddhanta was made around 580 AD. THe longitude of the stars change by 1° in every 71 years. From the data it is clear that the data does not represent observation but rather is obtained by adding precessional increment to each of the previously calculated data.<br /> <br /> === Obliquity (tilt) of the Earth's axis - 3000 BC ===<br /> Obliquity or the axial tilt of earth is the angle which the earth's axis of rotation makes with the perpendicular of orbital plane. This angle varies between 22.1° and 24.5° and it is cyclic phenomena over a period of 41,000 years. Currently the obliquity is 23.4 degrees.&lt;ref&gt;Alan Buis, &quot;Milankovitch (Orbital) Cycles and Their Role in Earth's Climate&quot;, &quot;NASA's Jet Propulsion Laboratory&quot; https://climate.nasa.gov/news/2948/milankovitch-orbital-cycles-and-their-role-in-earths-climate/&lt;/ref&gt; ''Sūrya Siddhānta'' in two different chapters calculate and provide the value of obliquity.<br /> <br /> Chapter 2, verse 28 translates as<br /> {{Quote <br /> |text = ''The sine of the greatest declination is 1397 units; Multiply the sine by the said sine 1397; Divide the product by the radius 3438 units; Find the arc whose sine is equal to the quotient. This arc is the mean declination of the planet''&lt;ref&gt;E. Burgess, &quot;Translation of Surya Siddhanta&quot;, p26, Accessible at https://www.jstor.org/stable/pdf/592174.pdf&lt;/ref&gt;<br /> }} This way we obtain the obliquity as Sin&lt;sup&gt;-1&lt;/sup&gt;(1397/3438) = 23.975°<br /> <br /> <br /> Chapter 12, verse 68 translates as <br /> {{Quote <br /> |text = ''At the distance of the fifteenth part of the Earth's circumference (from the equator) in the regions of the Gods or the Asuras (i.e. at the north and south terrestrial tropic) the sun passes through the zenith when it arrives at the north or south solstitial point (respectively)''''&lt;ref&gt;Pundit Bapu Deva Shastri, &quot;Translation of Surya Siddhanta&quot;, &quot;Baptist Mission Press&quot;, 1861, Accessible at https://www.wilbourhall.org/pdfs/suryaEnglish.pdf&lt;/ref&gt;<br /> }} It essentially provides information to calculate the axial tilt of earth which in this case can be calculated as 360°/15 = 24°.<br /> <br /> The significance of these verses is that they pin points the exact time when the obliquity calculations were made by ancient Indian astronomers and added into the ''Sūrya Siddhānta''. The epoch this obliquity calculation provides is around 3000BC.&lt;ref&gt;Anil Narayanan, &quot;Dating the Surya Siddhanta using Computer simulation of Proper Motions and Ecliptic variations&quot;, ''Indian Journal of History of Science'', Volume 45, issue 4, 23 March 2010.&lt;/ref&gt;<br /> <br /> === North Pole Star and South Pole Star - 3000 BC ===<br /> ''Surya Siddhanta'' contains an observation of the presence of pole stars at both north celestial pole and south celestial pole. Because of the precession of the earth's axis it is known that the pole star changes over a period of time which is normally more than thousand years. In present times our North Pole star is Polaris.&lt;ref&gt;Bruce McClure, &quot;Polaris is the North Pole Star&quot;, &quot;Earthsky&quot;, 21 May 2019, Accessible at https://earthsky.org/brightest-stars/polaris-the-present-day-north-star&lt;/ref&gt; This observation is recorded in chapter 12, verse 43-44 and translates as <br /> {{Quote<br /> |text = ''There are two pole stars, one each, near North Celestial Pole (NCP) and near South Celestial Pole (SCP). From equatorial regions, these stars are seen along the horizon. The pole stars are seen along the horizon, thus the place latitude is close to zero, while declination of NCP and SCP is 90 degrees.''<br /> }}<br /> <br /> Such phenomena was last seen around 3000 BC when Thuban was the North Pole Star and Alpha Hydri was the South Pole star.&lt;ref&gt;Nilesh N Oak and Rupa Bhatty, &quot;Ancient Updates to Surya Siddhanta&quot;, 09 March 2019, &quot;India Facts&quot; Accessible at http://indiafacts.org/ancient-updates-to-surya-siddhanta/&lt;/ref&gt; &lt;ref&gt; Anil Narayanan, &quot;Wonders, Mysteries and Misconceptions in Indian Astronomy – I&quot;, 'India facts&quot;, 09 Sept 2019, Accessible at http://indiafacts.org/wonders-mysteries-and-misconceptions-in-indian-astronomy-i/&lt;/ref&gt;<br /> <br /> === The Pulsating Indian Epicycle of the Sun - 5000-5500 BC ===<br /> For determining the Sun’s longitude, the pulsating Indian epicycle is far more accurate than the Greek eccentric-epicycle model. The pulsating Indian epicycle for the Sun becomes progressively more accurate as one goes back in time. Peak accuracy, of about 1 minute of arc, is reached around 5200 BC. The current values of the Surya Siddhanta’s pulsating epicycle parameters for the Sun appear to have been set in the 5000-5500 BC timeframe.&lt;ref&gt;Anil Narayanan, &quot;The Pulsating Indian Epicycle of the Sun&quot;, ''Indian Journal of History of Science'', Volume 46, issue 3, p15, 30 June 2011.&lt;/ref&gt;<br /> <br /> === The Latitudinal data - 7300-7500BC ===<br /> Using computer simulation of nakshatra latitudinal data by varying ecliptic obliquity, ecliptic-node-location and ecliptic-sink together with proper motion, a match for the Surya Siddhanta latitudinal data was obtained in the timeframe 7300-7800 BC.&lt;ref&gt;Anil Narayanan, &quot;Dating the Surya Siddhanta using Computer simulation of Proper Motions and Ecliptic variations&quot;, ''Indian Journal of History of Science'', Volume 45, issue 4, p21, 23 March 2010.&lt;/ref&gt; Although the author notes that a major assumption made in this investigation is that star proper motion is fairly constant over several thousands of years. The results may be adversely affected if this were found untrue for the star set under consideration. It should also be noted that this time frame matches with the establishment of the oldest archaeological site of Bhirrana found along the Saraswati river paleochannel. In the 8th millennium BC this site shows that the people were living in the dwelling pits.&lt;ref&gt;Bhirrana, &quot;Archaeological Survey of India&quot;, http://excnagasi.in/excavation_bhirrana.html&lt;/ref&gt; This stands in contrast with the above time frame, the question arises whether people could be that scientifically advanced while they were inhabiting the dwelling pits. Although this view is subject to change given older more advanced archaeological sites are found within the Indian subcontinent.<br /> <br /> === ''Surya Siddhanta'' sine table ===<br /> The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It is among the earliest form of [[Indian sine tables]]. The sine tables had been improved upon by many ancient Indian mathematicians. ''Surya Siddhanta'' uses an ''Indian standard circle'' of radius 3438 minutes. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and an arc length of 225 minutes. The verse 15-16 translates as <br /> {{Quote<br /> |text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 minutes)''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 17-22 translates as <br /> {{Quote<br /> |text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.<br /> Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The verse 23-27 translates as <br /> {{Quote<br /> |text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''&lt;ref&gt;Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.&lt;/ref&gt;<br /> }}<br /> <br /> The ''Surya Siddhanta'' derived Sin(θ) or Sine values show astonishing precision of 3 to 4 decimal places in comparison to the modern Sine values. The 1st order difference is the value by which each successive sine increases from the previous and similarly 2nd order difference is the increment in the 1st order difference values. ''Burgess'' notes that it is remarkable to see that the 2nd order differences increase as the sines and each, in fact, is about 1/225th part of the corresponding sine.&lt;ref&gt;Burgess, Rev. Ebenezer (1860). Translation of the Surya Siddhanta. p. 115.&lt;/ref&gt;<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto;&quot;<br /> !Sl. No<br /> !Angle (in degrees, arcminutes)<br /> ![[Surya Siddhanta]] value of ''[[Indian_sine_tables|Jyā]]'' (R.sine)<br /> !Surya Siddhanta versed sines [[Indian_sine_tables#Terminology|Utkramā-jyā]] (R - R.cosine)<br /> !Modern value of ''[[Indian_sine_tables|Jyā]]'' (R.sine)<br /> !SS derived sine values (''[[Indian_sine_tables|Jyā]]'' / 3438)<br /> !Modern sine values<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 1<br /> |{{center|03° &amp;nbsp; 45′}}<br /> |{{center|225′}}<br /> |{{center|7'}}<br /> |{{center|224.8560}}<br /> |{{center|0.06544503}}<br /> |{{center|0.06540313}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 2<br /> |{{center|07° &amp;nbsp; 30′}}<br /> |{{center|449′}}<br /> |{{center|29'}}<br /> |{{center|448.7490}}<br /> |{{center|0.13059919}}<br /> |{{center|0.13052619}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 3<br /> |{{center|11° &amp;nbsp; 15′}}<br /> |{{center|671′}}<br /> |{{center|66'}}<br /> |{{center|670.7205}}<br /> |{{center|0.19517161}}<br /> |{{center|0.19509032}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 4<br /> |{{center|15° &amp;nbsp; 00′}}<br /> |{{center|890′}}<br /> |{{center|117′}}<br /> |{{center|889.8199}}<br /> |{{center|0.25887144}}<br /> |{{center|0.25881905}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 5<br /> |{{center|18° &amp;nbsp; 45′}}<br /> |{{center|1105′}}<br /> |{{center|182′}}<br /> |{{center|1105.1089}}<br /> |{{center|0.3212078}}<br /> |{{center|0.32143947}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 6<br /> |{{center|22° &amp;nbsp; 30′}}<br /> |{{center|1315′}}<br /> |{{center|261′}}<br /> |{{center|1315.6656}}<br /> |{{center|0.38248982}}<br /> |{{center|0.38268343}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 7<br /> |{{center|26° &amp;nbsp; 15′}}<br /> |{{center|1520′}}<br /> |{{center|354′}}<br /> |{{center|1520.5885}}<br /> |{{center|0.44211751}}<br /> |{{center|0.44228869}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 8<br /> |{{center|30° &amp;nbsp; 00′}}<br /> |{{center|1719′}}<br /> |{{center|460′}}<br /> |{{center|1719.0000}}<br /> |{{center|0.50000000}}<br /> |{{center|0.50000000}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 9<br /> |{{center|33° &amp;nbsp; 45′}}<br /> |{{center|1910′}}<br /> |{{center|579′}}<br /> |{{center|1910.0505}}<br /> |{{center|0.55555556}}<br /> |{{center|0.55557023}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 10<br /> |{{center|37° &amp;nbsp; 30′}}<br /> |{{center|2093′}}<br /> |{{center|710′}}<br /> |{{center|2092.9218}}<br /> |{{center|0.60878418}}<br /> |{{center|0.60876143}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 11<br /> |{{center|41° &amp;nbsp; 15′}}<br /> |{{center|2267′}}<br /> |{{center|853′}}<br /> |{{center|2266.8309}}<br /> |{{center|0.65939500}}<br /> |{{center|0.65934582}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 12<br /> |{{center|45° &amp;nbsp; 00′}}<br /> |{{center|2431′}}<br /> |{{center|1007′}}<br /> |{{center|2431.0331}}<br /> |{{center|0.70709715}}<br /> |{{center|0.70710678}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 13<br /> |{{center|48° &amp;nbsp; 45′}}<br /> |{{center|2585′}}<br /> |{{center|1171′}}<br /> |{{center|2584.8253}}<br /> |{{center|0.75189063}}<br /> |{{center|0.75183981}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 14<br /> |{{center|52° &amp;nbsp; 30′}}<br /> |{{center|2728′}}<br /> |{{center|1345′}}<br /> |{{center|2727.5488}}<br /> |{{center|0.79348458}}<br /> |{{center|0.79335334}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 15<br /> |{{center|56° &amp;nbsp; 15′}}<br /> |{{center|2859′}}<br /> |{{center|1528′}}<br /> |{{center|2858.5925}}<br /> |{{center|0.83158813}}<br /> |{{center|0.83146961}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 16<br /> |{{center|60° &amp;nbsp; 00′}}<br /> |{{center|2978′}}<br /> |{{center|1719′}}<br /> |{{center|2977.3953}}<br /> |{{center|0.86620128}}<br /> |{{center|0.86602540}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 17<br /> |{{center|63° &amp;nbsp; 45′}}<br /> |{{center|3084′}}<br /> |{{center|1918′}}<br /> |{{center|3083.4485}}<br /> |{{center|0.89703316}}<br /> |{{center|0.89687274}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 18<br /> |{{center|67° &amp;nbsp; 30′}}<br /> |{{center|3177′}}<br /> |{{center|2123′}}<br /> |{{center|3176.2978}}<br /> |{{center|0.92408377}}<br /> |{{center|0.92387953}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 19<br /> |{{center|71° &amp;nbsp; 15′}}<br /> |{{center|3256′}}<br /> |{{center|2333′}}<br /> |{{center|3255.5458}}<br /> |{{center|0.94706225}}<br /> |{{center|0.94693013}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 20<br /> |{{center|75° &amp;nbsp; 00′}}<br /> |{{center|3321′}}<br /> |{{center|2548′}}<br /> |{{center|3320.8530}}<br /> |{{center|0.96596859}}<br /> |{{center|0.96592583}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 21<br /> |{{center|78° &amp;nbsp; 45′}}<br /> |{{center|3372′}}<br /> |{{center|2767′}}<br /> |{{center|3371.9398}}<br /> |{{center|0.98080279}}<br /> |{{center|0.98078528}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 22<br /> |{{center|82° &amp;nbsp; 30′}}<br /> |{{center|3409′}}<br /> |{{center|2989′}}<br /> |{{center|3408.5874}}<br /> |{{center|0.99156486}}<br /> |{{center|0.99144486}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 23<br /> |{{center|86° &amp;nbsp; 15′}}<br /> |{{center|3431′}}<br /> |{{center|3213′}}<br /> |{{center|3430.6390}}<br /> |{{center|0.99796393}}<br /> |{{center|0.99785892}}<br /> |-<br /> |&amp;nbsp;&amp;nbsp; 24<br /> |{{center|90° &amp;nbsp; 00′}}<br /> |{{center|3438′}}<br /> |{{center|3438′}}<br /> |{{center|3438.0000}}<br /> |{{center|1.00000000}}<br /> |{{center|1.00000000}}<br /> |-<br /> |}<br /> <br /> == See Also ==<br /> *[[Indian sine tables]]<br /> *[[Indian standard circle]]<br /> *[[Madhava's sine table]]<br /> <br /> == References and notes==<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Surya_Siddhanta_(%E0%A4%B8%E0%A5%82%E0%A4%B0%E0%A5%8D%E0%A4%AF_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A4%BE)&diff=129660 2021-03-20T08:13:44Z

<p>BharataPuru: </p> <hr /> <div>'''''Sūrya Siddhānta''''' is an ancient Indian treatise in Astronomy. Like many classical Indian works, the ''Sūrya Siddhānta'' is a poem in [[Sanskrit]] language. It has fourteen chapter and 500 verses. It is composed in ''śloka'' metrical style of Sanskrit. It contain works on Indian sine tables, cosmology, eclipses, planetary motions, conjunctions, star positions, geography, instrumentation, concepts of time and mathematics. Unlike conventional books ''Sūrya Siddhānta'' contains advanced calculation and methods which are not easily comprehensible for a rank beginner. &lt;ref&gt;[https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol45_4_1_ANarayan.pdf]&lt;/ref&gt; The text had been updated several times and the last update appears to have been made around 580 CE to the ''Nakshatra'' longitudes. In second chapter, the text contains the calculation of Earth's obliquity of ''1397 jya (R.sine) 23.975°'' modern units indicating the time of calculation around 3000 BCE. There are several other observations in the tradition of ''Indian Astronomy'' that were also recorded in the vicinity of 3000 BCE. This could possibly be the time of the origin of ''Sūrya Siddhānta'' although scholars do not seem to have a consensus on the origin of this text of ''Indian Astronomy''.<br /> <br /> == History ==<br /> &lt;nowiki&gt;''Sūrya Siddhānta'' is well known, most referred and most esteemed. The original author of ''Sūrya Siddhānta'' is ''Mayasura'' as described in the story in the first chapter that ''Mayasura'' obtained his knowledge from ''Sūrya'' (the Sun). ''Siddhānta'' in Sanskrit means ''treatise'' and it usually has author'&lt;/nowiki&gt;s name prefixed to it. There were several other works on Astronomy in ancient India, many of which have since been lost.<br /> <br /> {| class=&quot;wikitable&quot;<br /> |-<br /> | ''Surya Siddhānta'' || ''Brahma Siddhānta'' || Soma Siddhānta<br /> |-<br /> | Vyasa Siddhānta || Vashishtha Siddhānta || Atri Siddhānta<br /> |-<br /> | Parashira Siddhānta || Kashyap Siddhānta || Nārad Siddhānta<br /> |-<br /> | Garga Siddhānta || Marici Siddhānta || Manu Siddhānta<br /> |-<br /> | Angiras Siddhānta || Lomasha Siddhānta || Pulisha Siddhānta<br /> |-<br /> | Cyavana Siddhānta || Yavana Siddhānta || Bhrigu Siddhānta<br /> |}<br /> <br /> == References ==<br /> &lt;references /&gt;</div>

BharataPuru http://dharmawiki.org/index.php?title=Surya_Siddhanta_(%E0%A4%B8%E0%A5%82%E0%A4%B0%E0%A5%8D%E0%A4%AF_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A4%BE)&diff=129659 2021-03-20T08:09:17Z

<p>BharataPuru: Creation of the page</p> <hr /> <div>'''''Sūrya Siddhānta''''' is an ancient Indian treatise in Astronomy. Like many classical Indian works, the ''Sūrya Siddhānta'' is a poem in [[Sanskrit]] language. It has fourteen chapter and 500 verses. It is composed in ''śloka'' metrical style of Sanskrit. It contain works on Indian sine tables, cosmology, eclipses, planetary motions, conjunctions, star positions, geography, instrumentation, concepts of time and mathematics. Unlike conventional books ''Sūrya Siddhānta'' contains advanced calculation and methods which are not easily comprehensible for a rank beginner. &lt;ref&gt;[https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol45_4_1_ANarayan.pdf]&lt;/ref&gt; The text had been updated several times and the last update appears to have been made around 580 CE to the ''Nakshatra'' longitudes. In second chapter, the text contains the calculation of Earth's obliquity of ''1397 jya (R.sine) 23.975°'' modern units indicating the time of calculation around 3000 BCE. There are several other observations in the tradition of ''Indian Astronomy'' that were also recorded in the vicinity of 3000 BCE. This could possibly be the time of the origin of ''Sūrya Siddhānta'' although scholars do not seem to have a consensus on the origin of this text of ''Indian Astronomy''.<br /> <br /> == History ==<br /> &lt;nowiki&gt;''Sūrya Siddhānta'' is well known, most referred and most esteemed. The original author of ''Sūrya Siddhānta'' is ''Mayasura'' as described in the story in the first chapter that ''Mayasura'' obtained his knowledge from ''Sūrya'' (the Sun). ''Siddhānta'' in Sanskrit means ''treatise'' and it usually has author'&lt;/nowiki&gt;s name prefixed to it. There were several other works on Astronomy in ancient India, many of which have since been lost.<br /> <br /> <br /> {| class=&quot;wikitable&quot;<br /> |-<br /> | ''Surya Siddhānta'' || ''Brahma Siddhānta'' || Soma Siddhānta<br /> |-<br /> | Vyasa Siddhānta || Vashishtha Siddhānta || Atri Siddhānta<br /> |-<br /> | Parashira Siddhānta || Kashyap Siddhānta || Nārad Siddhānta<br /> |-<br /> | Garga Siddhānta || Marici Siddhānta || Manu Siddhānta<br /> |-<br /> | Angiras Siddhānta || Lomasha Siddhānta || Pulisha Siddhānta<br /> |-<br /> | Cyavana Siddhānta || Yavana Siddhānta || Bhrigu Siddhānta<br /> |}</div>

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