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Metonic cycle
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<h2 id="application-in-traditional-calendars">Application in traditional calendars</h2> <p>In the <a href="/facts/Babylonian_calendar/dckoqnCX">Babylonian</a> and <a href="/facts/Hebrew_calendar/s5XBFz4h">Hebrew</a> <a href="/facts/Lunisolar_calendar/L2jRkjyJ">lunisolar calendars</a>, the years 3, 6, 8, 11, 14, 17, and 19 are the long (13-month) years of the Metonic cycle. This cycle forms the basis of the Greek and Hebrew calendars. A 19-year cycle is used for the <a href="/facts/Computus/HUosdPeP">computation of the date of Easter</a> each year. </p><p>The Babylonians applied the 19-year cycle from the late sixth century BC.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p>According to <a href="/facts/Livy/Plc1YwYE">Livy</a>, the second king of Rome, <a href="/facts/Numa_Pompilius/tIac78B9">Numa Pompilius</a> (reigned 715–673 BC), inserted <a href="/facts/Intercalation_(timekeeping)/j60S1XNa">intercalary</a> months in such a way that "in the twentieth year the days should fall in with the same position of the sun from which they had started".<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> As "the twentieth year" takes place nineteen years after "the first year", this seems to indicate that the Metonic cycle was applied to Numa's calendar. </p><p><a href="/facts/Diodorus_Siculus/8u9UIDTz">Diodorus Siculus</a> reports that <a href="/facts/Apollo/eBJV1wsJ">Apollo</a> is said to have visited the <a href="/facts/Hyperborea/yPSWcBla">Hyperboreans</a> once every 19 years.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p>The Metonic cycle was implemented in the 2nd century BC <a href="/facts/Antikythera_mechanism/GHqvxCwR">Antikythera mechanism</a>, which offers unexpected evidence for the popularity of the calendar based on it.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>The (19-year) Metonic cycle is a lunisolar cycle, as is the (76-year) <a href="/facts/Callippic_cycle/Xd8J7mJU">Callippic cycle</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> An important example of an application of the Metonic cycle in the <a href="/facts/Julian_calendar/EQBtA1ML">Julian calendar</a> is the 19-year lunar cycle insofar as provided with a Metonic structure.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> Meton introduced the 19 year cycle to the <a href="/facts/Attic_calendar/oeNA1VNB">Attic calendar</a> in 432 BC. In the following century, <a href="/facts/Callippus/yADECQud">Callippus</a> developed the Callippic cycle of four 19-year periods for a 76-year cycle with a mean year of exactly 365.25 days. </p><p>Around AD 260 the Alexandrian <a href="/facts/Computus/HUosdPeP">computist</a> <a href="/facts/Anatolius_of_Laodicea/JEUpGzxl">Anatolius</a>, who became bishop of Laodicea in AD 268, was the first to devise a method for determining the date of Easter Sunday.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> However, it was some later, somewhat different, version of the Metonic 19-year lunar cycle which, as the basic structure of <a href="/facts/Dionysius_Exiguus/RUJ6gTwj">Dionysius Exiguus</a>' and also of <a href="/facts/Bede/y9yWnIua">Bede</a>'s Easter table, would ultimately prevail throughout <a href="/facts/Christendom/vWEEqosk">Christendom</a>,<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> at least until in the year 1582, when the <a href="/facts/Gregorian_calendar/axJfTXV1">Gregorian calendar</a> was introduced. </p><p>The <a href="/facts/Coligny_calendar/wBfonPeV">Coligny calendar</a> is a Celtic lunisolar calendar using the Metonic cycle. The bronze plaque on which it was found dates from c. AD 200, but the internal evidence points to the calendar itself being several centuries older, created in the Iron Age or late Bronze Age.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p><p>The Metonic cycle is thought to be numerically encoded on the <a href="/facts/Berlin_Gold_Hat/4lqVI2Go">Berlin Gold Hat</a> from central Europe, dating from c. 1000-800 BC.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a><a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p><p>The <a href="/facts/Runic_calendar/dn59Gp9a">Runic calendar</a> is a <a href="/facts/Perpetual_calendar/2GnnaUlM">perpetual calendar</a> based on the 19-year-long Metonic cycle. It is also known as a Rune staff or Runic Almanac. This calendar does not rely on knowledge of the duration of the tropical year or of the occurrence of leap years. It is set at the beginning of each year by observing the first full moon after the winter solstice. The oldest one known, and the only one from the Middle Ages, is the Nyköping staff, which is believed to date from the 13th century. </p><p>The <a href="/facts/Bah%25C3%25A1%2527%25C3%25AD_calendar/u5rBeXbC">Bahá'í calendar</a>, established during the middle of the 19th century, is also based on cycles of 19 solar years. </p> <h3>Hebrew calendar</h3> <p>A Small Maḥzor (<a href="/facts/Hebrew_language/hqKCSy9d">Hebrew</a> מחזור, pronounced [maχˈzor], meaning "cycle") is a 19-year cycle in the <a href="/facts/Lunisolar_calendar/L2jRkjyJ">lunisolar calendar</a> system used by the <a href="/facts/Jewish_people/T9Y4wE6A">Jewish people</a>. It is similar to, but slightly different in usage from, the Greek Metonic cycle (being based on a month of 29+13753⁄25920 days, giving a cycle of 6939+3575⁄5184 ≈ 6939.69 days<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a>), and likely derived from or alongside the much earlier <a href="/facts/Babylonian_calendar/dckoqnCX">Babylonian</a> calendar.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p> <h3>Polynesia</h3> <p>It is possible that the Polynesian kilo-hoku (astronomers) discovered the Metonic cycle in the same way Meton had, by trying to make the month fit the year.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p> <h2 id="mathematical-basis">Mathematical basis</h2> <p>The Metonic cycle is the most accurate cycle of time (in a timespan of less than 100 years) for synchronizing the <a href="/facts/Tropical_year/ASBLWNMY">tropical year</a> and the lunar month (<a href="/facts/Synodic_month/2l4tadsA">synodic month</a>), when the method of synchronizing is the <a href="/facts/Intercalation_(timekeeping)/j60S1XNa">intercalation</a> of a thirteenth lunar month in a calendar year from time to time.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> The traditional lunar year of 12 synodic months is about 354 days, approximately eleven days short of the solar year. Thus, every 2 to 3 years there is a discrepancy of 22 to 33 days, or a full synodic month. For example, if the winter solstice and the new moon coincide, it takes 19 tropical years for the coincidence to recur. The mathematical logic is this: </p> <ul><li>A tropical year lasts 365.2422 days.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> a span of 19 tropical years (365.2422 × 19) lasts 6,939.602 days</li></ul> <p>That duration is almost the same as 235 synodic months: </p> <ul><li>A synodic month lasts 29.53059 days.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> a span of 235 synodic months (29.53059 × 235) lasts 6,939.689 days</li></ul> <p>Thus the algorithm is correct to 0.087 days (2 hours, 5 minutes and 16 seconds). </p><p>For a <a href="/facts/Lunisolar_calendar/L2jRkjyJ">lunisolar calendar</a> to 'catch up' to this discrepancy and thus maintain seasonal consistency, seven <a href="/facts/Intercalation_(timekeeping)/j60S1XNa">intercalary months</a> are added (one at a time), at intervals of every 2–3 years during the course of 19 solar years. Thus twelve of those years have 12 lunar months and seven have 13 months. Of these months, 125 were full (30 days) and 110 were deficient (29 days), to give a total of 6940 days.<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Octaeteris/fmKD3wx3">Octaeteris</a> (8-year cycle of antiquity)</li> <li><a href="/facts/Callippic_cycle/Xd8J7mJU">Callippic cycle</a> (76-year cycle from 330 BC)</li> <li><a href="/facts/Hipparchic_cycle/8XxyNd2d">Hipparchic cycle</a> (304-year cycle from 2nd century BC)</li> <li><a href="/facts/Saros_cycle/M1E5nPoN">Saros cycle</a> of eclipses</li> <li><a href="/facts/Attic_calendar/oeNA1VNB">Attic</a> and <a href="/facts/Byzantine_calendar/2GVsEaMD">Byzantine calendar</a></li> <li><a href="/facts/Julian_day/mLwtKoYQ">Julian day</a></li> <li><a href="/facts/Date_of_Easter/HUosdPeP">Date of Easter</a> ("the Computus")</li></ul> <h3>Notes</h3> <h3>Citations</h3> <h3>Sources</h3> <ul><li>Anon (2003). <a href="https://archive.org/details/goldundkultderbr0000unse"><i>Gold und Kult der Bronzezeit</i></a>. Germanisches Nationalmuseum, Nuremberg. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-926982-95-0.</li> <li>Declercq, Georges (2000). <i>Anno Domini: The Origins of the Christian Era</i>. Turnhout. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9782503510507.</li> <li><a href="https://web.archive.org/web/20201021173505/http://asa.hmnao.com/SecM/Glossary.html">"Glossary"</a>. <i>The Astronomical Almanac Online!</i>. Washington, DC: United States Naval Observatory. 2022. Archived from <a href="http://asa.hmnao.com/SecM/Glossary.html">the original</a> on 21 October 2020. Retrieved 30 March 2022.</li> <li>Freeth, Tony; Jones, Alexander; Steele, John M.; Bitsakis, Yanis (31 July 2008). <a href="http://www.antikythera-mechanism.gr/system/files/Antikythera_Nature2008_submitted.pdf">"Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism"</a> (PDF). <i>Nature</i>. 454 (7204): 614–7. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2008Natur.454..614F">2008Natur.454..614F</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1038%2Fnature07130">10.1038/nature07130</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/18668103">18668103</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:4400693">4400693</a>.</li> <li>Hannah, Robert (2005). <i>Greek & Roman Calendars: Construction of Time in the Classical World</i>. London: Duckworth.</li> <li>Johnson, Rubellite Kawena (2001). <i>Essays in Hawaiian Literature Part 1 Origin Myths and Migration traditions</i>. author.</li> <li>McCarthy, Daniel P.; Breen, Aidan (2003). <i>The ante-Nicene Christian Pasch | </i>De ratione paschali:<i> The Paschal tract of Anatolius, bishop of Laodicea</i>. Dublin: Four Courts Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781851826971. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/367715096">367715096</a>.</li> <li>Menghin, Wilfried (2008). <a href="https://journals.ub.uni-heidelberg.de/index.php/apa/issue/view/5045">"Zahlensymbolik und digitales Rechnersystem in der Ornamentik des Berliner Goldhutes"</a>. <i>Acta Praehistorica et Archaeologica</i> (in German). 40: 157–169. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.11588%2Fapa.2008.0.71505">10.11588/apa.2008.0.71505</a>.</li> <li>Nothaft, C Philipp E. (2012). <i>Dating the Passion: The Life of Jesus and the Emergence of Scientific Chronology (200–1600</i>. Leiden: BRILL. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9789004212190.</li> <li>Olmsted, Garrett (2001). <a href="https://www.academia.edu/62011364">"A Definitive Reconstructed Text of the Coligny Calendar"</a>. <i>Journal of Indo-European Studies</i>.</li> <li>Richards, E. G. (1998). <i>Mapping Time: The Calendar and its History</i>. Oxford University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0192862051.</li> <li>Richards, E. G. (2013). "Calendars". In Urban, Sean E.; Seidelmann, P. Kenneth (eds.). <i>Explanatory Supplement to the Astronomical Almanac</i> (3rd ed.). Mill Valley, CA: University Science Books. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-891389-85-6.</li> <li>Watkins, Harold (1954). <i>Time Counts: The Story of the Calendars</i>. New York: Philosophical Library.</li></ul> <h2 id="external-links">External links</h2> <ul><li> Media related to Metonic cycle at Wikimedia Commons</li> <li><a href="http://jqjacobs.net/astro/eclipse.html#metonic">Eclipses, Cosmic Clockwork of the Ancients</a></li> <li><a href="https://www.thomasweibel.ch/meton/">Meton</a>, Virtual Mechanical Lunar Phase and Eclipse Calculator</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Meton". Oxford Reference. <a href="https://www.oxfordreference.com/display/10.1093/oi/authority.20110803100154128" target="_blank">https://www.oxfordreference.com/display/10.1093/oi/authority.20110803100154128</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"The Babylonian Calendar". Mathematical Institute. Utrecht University. July 2021. Archived from the original on 2 September 2023. <a href="https://webspace.science.uu.nl/~gent0113/babylon/babycal.htm" target="_blank">https://webspace.science.uu.nl/~gent0113/babylon/babycal.htm</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Livy, Ab Urbe Condita, I, XIX, 6. <a href="/wiki/Ab_Urbe_Condita" target="_blank">/wiki/Ab_Urbe_Condita</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Diodorus Siculus, Bibl. Hist. II.47. <a href="https://archive.org/details/diodorusofsicily02dioduoft/page/40/mode/2up" target="_blank">https://archive.org/details/diodorusofsicily02dioduoft/page/40/mode/2up</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Freeth et al. 2008, pp. 614–617. - Freeth, Tony; Jones, Alexander; Steele, John M.; Bitsakis, Yanis (31 July 2008). "Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism" (PDF). Nature. 454 (7204): 614–7. Bibcode:2008Natur.454..614F. doi:10.1038/nature07130. PMID 18668103. S2CID 4400693. <a href="http://www.antikythera-mechanism.gr/system/files/Antikythera_Nature2008_submitted.pdf" target="_blank">http://www.antikythera-mechanism.gr/system/files/Antikythera_Nature2008_submitted.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Nothaft 2012, p. 168. - Nothaft, C Philipp E. (2012). Dating the Passion: The Life of Jesus and the Emergence of Scientific Chronology (200–1600. Leiden: BRILL. ISBN 9789004212190. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>McCarthy & Breen 2003, p. 17. - McCarthy, Daniel P.; Breen, Aidan (2003). The ante-Nicene Christian Pasch | De ratione paschali: The Paschal tract of Anatolius, bishop of Laodicea. Dublin: Four Courts Press. ISBN 9781851826971. OCLC 367715096. <a href="https://search.worldcat.org/oclc/367715096" target="_blank">https://search.worldcat.org/oclc/367715096</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Declercq 2000, pp. 65–66. - Declercq, Georges (2000). Anno Domini: The Origins of the Christian Era. Turnhout. ISBN 9782503510507. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Declercq 2000, p. 66. - Declercq, Georges (2000). Anno Domini: The Origins of the Christian Era. Turnhout. ISBN 9782503510507. <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Olmsted 2001. - Olmsted, Garrett (2001). "A Definitive Reconstructed Text of the Coligny Calendar". Journal of Indo-European Studies. <a href="https://www.academia.edu/62011364" target="_blank">https://www.academia.edu/62011364</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>"Life and Belief During the Bronze Age" Neues Museum, Berlin". Retrieved 13 March 2022. What is especially fascinating is the ornamentation on the [Berlin gold hat] in which a complex counting system is encoded, enabling calendar calculations, especially the 19-year cycle of the sun and the moon. ... The star at the tip symbolises the sun, with the sickles and eye patterns representing the moon and Venus, while the circular ornaments can equally be interpreted as depictions of the sun or the moon. … The cycle of the sun determines day and nigh and the seasons, while the moon determines the division of the year into months and days. But the lunar year is eleven days shorter than the solar year. Even as early as the 2nd millennium BC intercalary days were inserted to bring the solar and lunar cycles into alignment. This knowledge is reflected in the ornamentation of the Gold Hat. The stamped patterns should be read as a calendar. For instance, the number of circles in certain decorative areas equals the twelve lunar periods of 354 days. If the patterns in other decorative areas are added, this gives the 365 days of the solar year. It takes 19 years for the solar year and the lunar year to align again. In the ornamentation of the hat the fact is encoded that seven lunar months need to be inserted into the 19-year cycle. Other calculations can be made as well, such as the dates of eclipses of the moon. (…) The golden hats show that astronomical knowledge was combined with cult activities… They were apparently worn over several generation and at some point buried in the ground in a sacred act to protect them from desecration and to place them in the realm of the gods. It seems that Bronze Age rulers combined worldly and spiritual power. <a href="https://artsandculture.google.com/story/DAVRgpAwHmLsLw?hl=en" target="_blank">https://artsandculture.google.com/story/DAVRgpAwHmLsLw?hl=en</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>"Golden Ceremonial Hat ("Berlin Gold Hat")". Neues Museum Berlin. One particularly impressive piece of evidence for early man's astronomical knowledge is the Bronze Age Berlin gold hat, unique in its size and preservation. The sun, evoked by the gold coloration and the pattern of rays at the top of the hat, creates day, night and the seasons by apparently circling the earth. The moon, represented several times on the hat, marks out months and weeks. The number and arrangement of the ornaments is not random; it allows a nineteen-year lunisolar cycle of 228 solar months and 235 lunar months to be calculated. Someone who knew how to read these ornaments would be able to calculate the shifts between the solar year and the lunar year, predict lunar eclipses, and set fixed dates for significant events. … Over half a millennium before the astronomer and mathematician Meton in 432 BC calculated the shifts in the lunisolar cycle, they were already known to the educated elite of the Bronze Age. The golden hat may have been worn by a ruler with a religious role on ceremonial occasions. Other Bronze Age items prove that astronomical knowledge was often preserved in coded form on valuable and sacred objects. <a href="https://artsandculture.google.com/asset/golden-ceremonial-hat-berlin-gold-hat-artist-unknown/hAGC3knXgLdPZg?hl=en" target="_blank">https://artsandculture.google.com/asset/golden-ceremonial-hat-berlin-gold-hat-artist-unknown/hAGC3knXgLdPZg?hl=en</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Menghin 2008, pp. 157–169. - Menghin, Wilfried (2008). "Zahlensymbolik und digitales Rechnersystem in der Ornamentik des Berliner Goldhutes". Acta Praehistorica et Archaeologica (in German). 40: 157–169. doi:10.11588/apa.2008.0.71505. <a href="https://journals.ub.uni-heidelberg.de/index.php/apa/issue/view/5045" target="_blank">https://journals.ub.uni-heidelberg.de/index.php/apa/issue/view/5045</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Anon 2003, pp. 220–237. - Anon (2003). Gold und Kult der Bronzezeit. Germanisches Nationalmuseum, Nuremberg. ISBN 3-926982-95-0. <a href="https://archive.org/details/goldundkultderbr0000unse" target="_blank">https://archive.org/details/goldundkultderbr0000unse</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>The month is 29.5 days and 793 "parts", where a part is 1/18 of a minute. Tøndering, Trine; Tøndering, Claus. "Calendar FAQ: the Hebrew calendar: New moon". This comes to 29+13753⁄25920 days. Multiplying this by 235 gives the length of the cycle. <a href="https://www.tondering.dk/claus/cal/hebrew.php#newmoon" target="_blank">https://www.tondering.dk/claus/cal/hebrew.php#newmoon</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>"Jewish religious year | Cycle, Holidays, & Facts | Britannica". www.britannica.com. Retrieved 14 November 2021. <a href="https://www.britannica.com/topic/Jewish-religious-year" target="_blank">https://www.britannica.com/topic/Jewish-religious-year</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Johnson 2001, p. 238. - Johnson, Rubellite Kawena (2001). Essays in Hawaiian Literature Part 1 Origin Myths and Migration traditions. author. <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Richards (1998), pp. 94–96. - Richards, E. G. (1998). Mapping Time: The Calendar and its History. Oxford University Press. ISBN 978-0192862051. <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>glossary (2022), s.v. year, tropical. - "Glossary". The Astronomical Almanac Online!. Washington, DC: United States Naval Observatory. 2022. Archived from the original on 21 October 2020. Retrieved 30 March 2022. <a href="https://web.archive.org/web/20201021173505/http://asa.hmnao.com/SecM/Glossary.html" target="_blank">https://web.archive.org/web/20201021173505/http://asa.hmnao.com/SecM/Glossary.html</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Richards (2013), p. 587. - Richards, E. G. (2013). "Calendars". In Urban, Sean E.; Seidelmann, P. Kenneth (eds.). Explanatory Supplement to the Astronomical Almanac (3rd ed.). Mill Valley, CA: University Science Books. ISBN 978-1-891389-85-6. <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Richards 1998, p. 95. - Richards, E. G. (1998). Mapping Time: The Calendar and its History. Oxford University Press. ISBN 978-0192862051. <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> </ol>