-
Notifications
You must be signed in to change notification settings - Fork 45
/
Copy pathchunqiu.html
executable file
·1 lines (1 loc) · 67.4 KB
/
chunqiu.html
1
<!DOCTYPE html><html lang="en-US"><meta name="keywords" content="Chinese calendar, Chunqiu Lu calendar"><meta charset="UTF-8"><meta name="viewport" content="width=device-width,initial-scale=1"><head><title>Reconstruction of the Chunqiu Calendar</title><link rel="stylesheet" href="calendar_min.css"><style>body{counter-reset:sec-counter}sect::before{counter-increment:sec-counter 1;content:counter(sec-counter) ". "}</style><script src="header_min.js"></script><base href="https://ytliu0.github.io/ChineseCalendar/"><script async src="https://www.googletagmanager.com/gtag/js?id=G-SN0QJRDXXT"></script><script>function gtag(){dataLayer.push(arguments)}window.dataLayer=window.dataLayer||[],gtag("js",new Date),gtag("config","G-SN0QJRDXXT")</script></head><body onload='header(0,"","chunqiu")'><div id="wrapper0"><div id="menu"></div><div id="language"></div><h1>Reconstruction of the Chunqiu Calendar</h1><p>First draft: March, 2019</p><br><p>In ancient time, Chinese calendars were made based on astronomical observations. In the Spring and Autumn period (722 BCE – 481 BCE), people developed a simple method to predict the moon phases. Calendars began to be calculated in advance. China was divided into many states at that time and each state used its own calendar. Unfortunately, none of these calendars is preserved. Today we only have fragmented information about the calendar used by the Lu state from the chronicle <i>Chunqiu</i> (《春秋》) revised by Confucius. This calendar is called Chunqiu here.</p><p>Here I describe the Chunqiu calendar reconstructed using the method and table in Section 3.5 of the book <i>Zhōng Guó Gǔ Dài Lì Fǎ</i> (《中国古代历法》 or <i>Ancient Chinese Calendars and Almanacs</i>) written by Zhāng Péiyú (張培瑜), Chén Měidōng (陳美東), Bó Shùrén (薄樹人), and Hú Tiězhū (胡鐵珠), published by China Science Press (Beijing) in March 2008. Hereafter, I will refer to the book as <i>Ancient Chinese Calendars and Almanacs</i>. According to the preface, chapters in the book were written separately by the four authors. Chapter 3 was written by Zhāng Péiyú, a researcher in the Purple Mountain Observatory in Nanjing, China.</p><br><div style="border:1px solid #000;padding:10px"><p><b>Contents</b></p><ol><li><a href="chunqiu.html#computation">Computation of Lunar Conjunctions</a></li><li><a href="chunqiu.html#leapMons">Leap Months and Accumulated Months</a></li><li><a href="chunqiu.html#typos">Typos in Table 3-7 in <i>Ancient Chinese Calendars and Almanacs</i></a></li><li><a href="chunqiu.html#firstMonth">Leap Month and <i>Jiàn</i> in the Chunqiu Calendar</a></li><li><a href="chunqiu.html#reliability">Reliability of the Reconstructed Chunqiu Calendar</a></li><li><a href="chunqiu.html#julian">Julian Days</a></li></ol> <a href="chunqiu.html#footnotes">Footnotes</a></div><br><sect id="computation">Computation of Lunar Conjunctions</sect><p>According to Zhāng's research, the lunar cycle adopted by the Chunqiu calendar was between 29.5306703 days and 29.5306755 days. For practical calculation, the number 30328/1027 days = (29 + 545/1027) days = 29.53067186 days may be used. This number is quite close to the lunar cycle at that time (29.530583 days) and is more accurate than (29 + 499/940) days adopted by the calendars in the Warring States period (480 BCE – 222 BCE). The accumulated error of the adopted lunar cycle would reach one day in 900 years. Chinese calendars before the 7th century CE used <i>píngshuò</i> for the calculation of lunar conjunction, which only takes into account the mean motions of the moon and Sun. If we know a particular conjunction time, the other conjunction times can be determined by simply adding integral multiples of the adopted lunar cycle.</p><p>For convenience, I use the notation N<sub>y</sub> to denote the year of the Chunqiu calendar whose New Year day was closest to January 1 of the proleptic Julian calendar of year y. For example, N<sub>-721</sub> began on January 16, -721 (722 BCE) and ended on January 4, -720. According to Zhāng's reconstructed Chunqiu calendar, the predicted lunar conjunction time associated with the New Year day of N<sub>-721</sub> occurred on the first <i>xīn sì</i> (sexagenary) day after the winter solstice. The conjunction time was 268/1027 days from the midnight. Expressed in the Western calendar and the modern time system, the conjunction time was on January 16, -721 at 06:16. In ancient time, it was not known that different places had different local times. We can regard the time as being the local time in the state of Lu, whose capital was at the present-day city Qūfù in Shāndōng province in northeast China.</p><p>Decimals were not used in the calendar calculations in ancient China. The fractional part of a number was called <i>xiǎoyú</i> (小餘) and was usually expressed as a fraction. "Fractional day" may be used as an English translation of <i>xiǎoyú</i>. As a result, 268/1027 was the <i>xiǎoyú</i> of the conjunction time. The denominator 1027 is usually omitted and we simply refer the number 268 as the <i>xiǎoyú</i>. <i>Xiǎoyú</i> can be thought of as the time in the day expressed as a fraction of a day. Using today's 24-hour system, a <i>xiǎoyú</i> of 268/1027 is the same 6:15:46. According to the <i>píngshuò</i> rule, the next conjunction would occur (29 + 545/1027) days later, which was (268/1027 + 29 + 545/1027) days, or (29 + 813/1027) days from the midnight of the first conjunction day. Here the integer part was 29 and <i>xiǎoyú</i> was 813. The sexagenary day cycle was 29 days after <i>xīn sì</i>. The cycle of the heavenly stems is 10 and the cycle of earthly branches is 12. Since 29 = 30-1, the heavenly stem was one stem before <i>xīn</i>, which was <i>gēng</i>; mod(29, 12) = 5 and so the earthly branch was 5 branches after <i>sì</i>, which was <i>xū</i>. Here mod(X, Y) denotes the remainder of X divided by Y. The sexagenary day of the first day of month 2 was <i>gēng xū</i>. It is also easy to calculate the date in the proleptic Julian calendar: January (16+29) = February (16+29-31) = February 14.</p><p>In general, the lunar conjunction times predicted by the Chunqiu calendar are given by the following formula.</p><p>M<sub>i</sub> = M<sub>0</sub> + i (29 + 545/1027) days (1)</p><p>Here M<sub>0</sub> is a particular lunar conjunction time, which we will take as the conjunction time associated with the New Year day of N<sub>-721</sub>, i is the number of accumulated months from M<sub>0</sub>. Let A(y) be the accumulated months associated with the New Year day of N<sub>y</sub>. Then the conjunction time associated with the New Year day is given by</p><p>Z(y) = M<sub>0</sub> + A(y) (29 + 545/1027) days (2)</p><p>Once we know A(y), other months in N<sub>y</sub> can be obtained by adding integer multiples of (29 + 545/1027) days. The accumulated months A(y) can be calculated by a recurrence relation. Suppose there was no leap month in N<sub>y</sub>, then A(y+1) = A(y) + 12. If, on the other hand, there was a leap month in N<sub>y</sub>, then A(y+1) = A(y) + 13. By definition, A(-721) = 0. Hence, if we know all the leap months in the Chunqiu calendar, we can easily reconstruct the calendar in any given year.</p><br><sect id="leapMons">Leap Months and Accumulated Months</sect><p>There was no fixed rule for inserting the leap months in the Chunqiu calendar. Instead, the insertion of leap months was determined by astronomical observations. Leap months in the Chunqiu calendar can only be inferred from the chronicle <i>Chunqiu</i>. Based on his research and studies of other scholars, Zhāng constructed a table of the New Year days in the Chunqiu calendar. The table was reproduced in Table 3-7 of <i>Ancient Chinese Calendars and Almanacs</i>. Table 3-7 lists the sexagenary days of the New Year days, the <i>jiàn</i> (i.e. the branch name, see <a href="sexagenary.html#jian">sexagenary cycle page</a>) of the first month, the <i>xiǎoyú</i> of the conjunction associated with the New Year day, the dates of winter solstices in proleptic Julian calendar, and also the Chunqiu calendar data reconstructed by Wáng Tāo (王韜), a scholar in the 19th century, for comparison. It is possible to determine if there was leap month from Table 3-7. The simplest method is to compare the <i>xiǎoyú</i>s in two successive years.</p><p>As mentioned above, <i>xiǎoyú</i> is the conjunction time expressed as a fraction of a day from midnight, with the denominator 1027 omitted. If there was no leap month in N<sub>y</sub>, the <i>xiǎoyú</i> of the new year conjunction K(y) is related to the <i>xiǎoyú</i> of the new year conjunction in N<sub>y+1</sub>, K(y+1), by the following equation.<br>K(y+1) = mod(K(y) + 12×545, 1027) = mod(K(y) + 378, 1027) (3)<br>If there was a leap month in N<sub>y</sub>, then<br>K(y+1) = mod(K(y) + 13×545, 1027) = mod(K(y) - 104, 1027) (4)<br>We can use these two equations to determine from Table 3-7 whether or not a given year had a leap month. The accumulated months A(y) can then be calculated from the recurrence relation mentioned above. The result is listed in the following table, which also lists the <i>jiàn</i> of the first month, the sexagenary day cycle of the New Year day, the date in the proleptic Julian calendar, and the <i>xiǎoyú</i> of the new year day conjunction.</p><br><input type="checkbox" class="accordion" id="table"> <label class="accordion" for="table">Table of Accumulated Months (click to show/hide table)</label><div class="content"><br><table><tr><th>Chunqiu Cal. year</th><th>Acc. months</th><th>Jiàn</th><th>Sexagenary date</th><th>Julian Cal. Date</th><th>Xiǎoyú</th></tr><tr><td>N<sub>-721</sub></td><td>0</td><td>chǒu</td><td>Xīn sì</td><td>Jan. 16, 722 BCE</td><td>268</td></tr><tr><td><sup>*</sup>N<sub>-720</sub></td><td>12</td><td>chǒu</td><td>Yǐ hài</td><td>Jan. 5, 721 BCE</td><td>646</td></tr><tr><td>N<sub>-719</sub></td><td>25</td><td>chǒu</td><td>Jǐ hài</td><td>Jan. 23, 720 BCE</td><td>542</td></tr><tr><td>N<sub>-718</sub></td><td>37</td><td>chǒu</td><td>Guǐ sì</td><td>Jan. 12, 719 BCE</td><td>920</td></tr><tr><td><sup>*</sup>N<sub>-717</sub></td><td>49</td><td>chǒu</td><td>Wù zǐ</td><td>Jan. 2, 718 BCE</td><td>271</td></tr><tr><td>N<sub>-716</sub></td><td>62</td><td>chǒu</td><td>Rén zǐ</td><td>Jan. 21, 717 BCE</td><td>167</td></tr><tr><td><sup>*</sup>N<sub>-715</sub></td><td>74</td><td>chǒu</td><td>Bǐng wǔ</td><td>Jan. 9, 716 BCE</td><td>545</td></tr><tr><td>N<sub>-714</sub></td><td>87</td><td>yín</td><td>Gēng wǔ</td><td>Jan. 28, 715 BCE</td><td>441</td></tr><tr><td><sup>*</sup>N<sub>-713</sub></td><td>99</td><td>chǒu</td><td>Jiǎ zǐ</td><td>Jan. 17, 714 BCE</td><td>819</td></tr><tr><td>N<sub>-712</sub></td><td>112</td><td>yín</td><td>Wù zǐ</td><td>Feb. 5, 713 BCE</td><td>715</td></tr><tr><td>N<sub>-711</sub></td><td>124</td><td>chǒu</td><td>Guǐ wèi</td><td>Jan. 25, 712 BCE</td><td>66</td></tr><tr><td><sup>*</sup>N<sub>-710</sub></td><td>136</td><td>chǒu</td><td>Dīng chǒu</td><td>Jan. 14, 711 BCE</td><td>444</td></tr><tr><td>N<sub>-709</sub></td><td>149</td><td>yín</td><td>Xīn chǒu</td><td>Feb. 2, 710 BCE</td><td>340</td></tr><tr><td>N<sub>-708</sub></td><td>161</td><td>chǒu</td><td>Yǐ wèi</td><td>Jan. 22, 709 BCE</td><td>718</td></tr><tr><td><sup>*</sup>N<sub>-707</sub></td><td>173</td><td>chǒu</td><td>Gēng yín</td><td>Jan. 11, 708 BCE</td><td>69</td></tr></table><p><sup>*</sup> There was a leap month in this year.</p><br><br><table><tr><th>Chunqiu Cal. year</th><th>Acc. months</th><th>Jiàn</th><th>Sexagenary date</th><th>Julian Cal. Date</th><th>Xiǎoyú</th></tr><tr><td>N<sub>-706</sub></td><td>186</td><td>yín</td><td>Guǐ chǒu</td><td>Jan. 29, 707 BCE</td><td>992</td></tr><tr><td>N<sub>-705</sub></td><td>198</td><td>chǒu</td><td>Wù shēn</td><td>Jan. 19, 706 BCE</td><td>343</td></tr><tr><td><sup>*</sup>N<sub>-704</sub></td><td>210</td><td>chǒu</td><td>Rén yín</td><td>Jan. 8, 705 BCE</td><td>721</td></tr><tr><td>N<sub>-703</sub></td><td>223</td><td>chǒu</td><td>Bǐng yín</td><td>Jan. 26, 704 BCE</td><td>617</td></tr><tr><td>N<sub>-702</sub></td><td>235</td><td>chǒu</td><td>Gēng shēn</td><td>Jan. 15, 703 BCE</td><td>995</td></tr><tr><td><sup>*</sup>N<sub>-701</sub></td><td>247</td><td>chǒu</td><td>Yǐ mǎo</td><td>Jan. 5, 702 BCE</td><td>346</td></tr><tr><td>N<sub>-700</sub></td><td>260</td><td>chǒu</td><td>Jǐ mǎo</td><td>Jan. 24, 701 BCE</td><td>242</td></tr><tr><td><sup>*</sup>N<sub>-699</sub></td><td>272</td><td>chǒu</td><td>Guǐ yǒu</td><td>Jan. 12, 700 BCE</td><td>620</td></tr><tr><td>N<sub>-698</sub></td><td>285</td><td>yín</td><td>Dīng yǒu</td><td>Jan. 31, 699 BCE</td><td>516</td></tr><tr><td>N<sub>-697</sub></td><td>297</td><td>chǒu</td><td>Xīn mǎo</td><td>Jan. 20, 698 BCE</td><td>894</td></tr><tr><td>N<sub>-696</sub></td><td>309</td><td>chǒu</td><td>Bǐng xū</td><td>Jan. 10, 697 BCE</td><td>245</td></tr><tr><td><sup>*</sup>N<sub>-695</sub></td><td>321</td><td>chǒu</td><td>Gēng chén</td><td>Dec. 29, 697 BCE</td><td>623</td></tr><tr><td>N<sub>-694</sub></td><td>334</td><td>chǒu</td><td>Jiǎ chén</td><td>Jan. 17, 695 BCE</td><td>519</td></tr><tr><td>N<sub>-693</sub></td><td>346</td><td>chǒu</td><td>Wù xū</td><td>Jan. 6, 694 BCE</td><td>897</td></tr><tr><td><sup>*</sup>N<sub>-692</sub></td><td>358</td><td>zǐ</td><td>Guǐ sì</td><td>Dec. 27, 694 BCE</td><td>248</td></tr></table><p><sup>*</sup> There was a leap month in this year.</p><br><br><table><tr><th>Chunqiu Cal. year</th><th>Acc. months</th><th>Jiàn</th><th>Sexagenary date</th><th>Julian Cal. Date</th><th>Xiǎoyú</th></tr><tr><td>N<sub>-691</sub></td><td>371</td><td>chǒu</td><td>Dīng sì</td><td>Jan. 14, 692 BCE</td><td>144</td></tr><tr><td><sup>*</sup>N<sub>-690</sub></td><td>383</td><td>chǒu</td><td>Xīn hài</td><td>Jan. 3, 691 BCE</td><td>522</td></tr><tr><td>N<sub>-689</sub></td><td>396</td><td>chǒu</td><td>Yǐ hài</td><td>Jan. 22, 690 BCE</td><td>418</td></tr><tr><td>N<sub>-688</sub></td><td>408</td><td>chǒu</td><td>Jǐ sì</td><td>Jan. 11, 689 BCE</td><td>796</td></tr><tr><td>N<sub>-687</sub></td><td>420</td><td>chǒu</td><td>Jiǎ zǐ</td><td>Dec. 31, 689 BCE</td><td>147</td></tr><tr><td><sup>*</sup>N<sub>-686</sub></td><td>432</td><td>zǐ</td><td>Wù wǔ</td><td>Dec. 20, 688 BCE</td><td>525</td></tr><tr><td>N<sub>-685</sub></td><td>445</td><td>chǒu</td><td>Rén wǔ</td><td>Jan. 8, 686 BCE</td><td>421</td></tr><tr><td>N<sub>-684</sub></td><td>457</td><td>zǐ</td><td>Bǐng zǐ</td><td>Dec. 28, 686 BCE</td><td>799</td></tr><tr><td>N<sub>-683</sub></td><td>469</td><td>zǐ</td><td>Xīn wèi</td><td>Dec. 17, 685 BCE</td><td>150</td></tr><tr><td>N<sub>-682</sub></td><td>481</td><td>zǐ</td><td>Yǐ chǒu</td><td>Dec. 6, 684 BCE</td><td>528</td></tr><tr><td><sup>*</sup>N<sub>-681</sub></td><td>493</td><td>hài</td><td>Jǐ wèi</td><td>Nov. 25, 683 BCE</td><td>906</td></tr><tr><td><sup>*</sup>N<sub>-680</sub></td><td>506</td><td>zǐ</td><td>Guǐ wèi</td><td>Dec. 14, 682 BCE</td><td>802</td></tr><tr><td>N<sub>-679</sub></td><td>519</td><td>chǒu</td><td>Dīng wèi</td><td>Jan. 1, 680 BCE</td><td>698</td></tr><tr><td><sup>*</sup>N<sub>-678</sub></td><td>531</td><td>zǐ</td><td>Rén yín</td><td>Dec. 22, 680 BCE</td><td>49</td></tr><tr><td><sup>*</sup>N<sub>-677</sub></td><td>544</td><td>chǒu</td><td>Yǐ chǒu</td><td>Jan. 9, 678 BCE</td><td>972</td></tr></table><p><sup>*</sup> There was a leap month in this year.</p><br><br><table><tr><th>Chunqiu Cal. year</th><th>Acc. months</th><th>Jiàn</th><th>Sexagenary date</th><th>Julian Cal. Date</th><th>Xiǎoyú</th></tr><tr><td>N<sub>-676</sub></td><td>557</td><td>yín</td><td>Jǐ chǒu</td><td>Jan. 28, 677 BCE</td><td>868</td></tr><tr><td>N<sub>-675</sub></td><td>569</td><td>chǒu</td><td>Jiǎ shēn</td><td>Jan. 17, 676 BCE</td><td>219</td></tr><tr><td><sup>*</sup>N<sub>-674</sub></td><td>581</td><td>chǒu</td><td>Wù yín</td><td>Jan. 6, 675 BCE</td><td>597</td></tr><tr><td>N<sub>-673</sub></td><td>594</td><td>chǒu</td><td>Rén yín</td><td>Jan. 25, 674 BCE</td><td>493</td></tr><tr><td>N<sub>-672</sub></td><td>606</td><td>chǒu</td><td>Bǐng shēn</td><td>Jan. 14, 673 BCE</td><td>871</td></tr><tr><td>N<sub>-671</sub></td><td>618</td><td>chǒu</td><td>Xīn mǎo</td><td>Jan. 3, 672 BCE</td><td>222</td></tr><tr><td><sup>*</sup>N<sub>-670</sub></td><td>630</td><td>zǐ</td><td>Yǐ yǒu</td><td>Dec. 23, 672 BCE</td><td>600</td></tr><tr><td>N<sub>-669</sub></td><td>643</td><td>chǒu</td><td>Jǐ yǒu</td><td>Jan. 11, 670 BCE</td><td>496</td></tr><tr><td>N<sub>-668</sub></td><td>655</td><td>chǒu</td><td>Guǐ mǎo</td><td>Dec. 31, 670 BCE</td><td>874</td></tr><tr><td><sup>*</sup>N<sub>-667</sub></td><td>667</td><td>zǐ</td><td>Wù xū</td><td>Dec. 20, 669 BCE</td><td>225</td></tr><tr><td>N<sub>-666</sub></td><td>680</td><td>chǒu</td><td>Rén xū</td><td>Jan. 8, 667 BCE</td><td>121</td></tr><tr><td><sup>*</sup>N<sub>-665</sub></td><td>692</td><td>zǐ</td><td>Bǐng chén</td><td>Dec. 28, 667 BCE</td><td>499</td></tr><tr><td>N<sub>-664</sub></td><td>705</td><td>chǒu</td><td>Gēng chén</td><td>Jan. 16, 665 BCE</td><td>395</td></tr><tr><td>N<sub>-663</sub></td><td>717</td><td>chǒu</td><td>Jiǎ xū</td><td>Jan. 4, 664 BCE</td><td>773</td></tr><tr><td><sup>*</sup>N<sub>-662</sub></td><td>729</td><td>zǐ</td><td>Jǐ sì</td><td>Dec. 25, 664 BCE</td><td>124</td></tr></table><p><sup>*</sup> There was a leap month in this year.</p><br><br><table><tr><th>Chunqiu Cal. year</th><th>Acc. months</th><th>Jiàn</th><th>Sexagenary date</th><th>Julian Cal. Date</th><th>Xiǎoyú</th></tr><tr><td>N<sub>-661</sub></td><td>742</td><td>chǒu</td><td>Guǐ sì</td><td>Jan. 13, 662 BCE</td><td>20</td></tr><tr><td>N<sub>-660</sub></td><td>754</td><td>chǒu</td><td>Dīng hài</td><td>Jan. 2, 661 BCE</td><td>398</td></tr><tr><td><sup>*</sup>N<sub>-659</sub></td><td>766</td><td>zǐ</td><td>Xīn sì</td><td>Dec. 21, 661 BCE</td><td>776</td></tr><tr><td><sup>*</sup>N<sub>-658</sub></td><td>779</td><td>chǒu</td><td>Yǐ sì</td><td>Jan. 9, 659 BCE</td><td>672</td></tr><tr><td>N<sub>-657</sub></td><td>792</td><td>yín</td><td>Jǐ sì</td><td>Jan. 28, 658 BCE</td><td>568</td></tr><tr><td>N<sub>-656</sub></td><td>804</td><td>chǒu</td><td>Guǐ hài</td><td>Jan. 17, 657 BCE</td><td>946</td></tr><tr><td>N<sub>-655</sub></td><td>816</td><td>chǒu</td><td>Wù wǔ</td><td>Jan. 6, 656 BCE</td><td>297</td></tr><tr><td>N<sub>-654</sub></td><td>828</td><td>zǐ</td><td>Rén zǐ</td><td>Dec. 26, 656 BCE</td><td>675</td></tr><tr><td>N<sub>-653</sub></td><td>840</td><td>zǐ</td><td>Dīng wèi</td><td>Dec. 16, 655 BCE</td><td>26</td></tr><tr><td><sup>*</sup>N<sub>-652</sub></td><td>852</td><td>zǐ</td><td>Xīn chǒu</td><td>Dec. 5, 654 BCE</td><td>404</td></tr><tr><td>N<sub>-651</sub></td><td>865</td><td>zǐ</td><td>Yǐ chǒu</td><td>Dec. 23, 653 BCE</td><td>300</td></tr><tr><td><sup>*</sup>N<sub>-650</sub></td><td>877</td><td>zǐ</td><td>Jǐ wèi</td><td>Dec. 12, 652 BCE</td><td>678</td></tr><tr><td>N<sub>-649</sub></td><td>890</td><td>chǒu</td><td>Guǐ wèi</td><td>Dec. 31, 651 BCE</td><td>574</td></tr><tr><td><sup>*</sup>N<sub>-648</sub></td><td>902</td><td>zǐ</td><td>Dīng chǒu</td><td>Dec. 20, 650 BCE</td><td>952</td></tr><tr><td>N<sub>-647</sub></td><td>915</td><td>chǒu</td><td>Xīn chǒu</td><td>Jan. 7, 648 BCE</td><td>848</td></tr></table><p><sup>*</sup> There was a leap month in this year.</p><br><br><table><tr><th>Chunqiu Cal. year</th><th>Acc. months</th><th>Jiàn</th><th>Sexagenary date</th><th>Julian Cal. Date</th><th>Xiǎoyú</th></tr><tr><td><sup>*</sup>N<sub>-646</sub></td><td>927</td><td>chǒu</td><td>Bǐng shēn</td><td>Dec. 28, 648 BCE</td><td>199</td></tr><tr><td>N<sub>-645</sub></td><td>940</td><td>chǒu</td><td>Gēng shēn</td><td>Jan. 16, 646 BCE</td><td>95</td></tr><tr><td>N<sub>-644</sub></td><td>952</td><td>chǒu</td><td>Jiǎ yín</td><td>Jan. 5, 645 BCE</td><td>473</td></tr><tr><td>N<sub>-643</sub></td><td>964</td><td>zǐ</td><td>Wù shēn</td><td>Dec. 24, 645 BCE</td><td>851</td></tr><tr><td><sup>*</sup>N<sub>-642</sub></td><td>976</td><td>zǐ</td><td>Guǐ mǎo</td><td>Dec. 14, 644 BCE</td><td>202</td></tr><tr><td>N<sub>-641</sub></td><td>989</td><td>chǒu</td><td>Dīng mǎo</td><td>Jan. 2, 642 BCE</td><td>98</td></tr><tr><td><sup>*</sup>N<sub>-640</sub></td><td>1001</td><td>zǐ</td><td>Xīn yǒu</td><td>Dec. 22, 642 BCE</td><td>476</td></tr><tr><td>N<sub>-639</sub></td><td>1014</td><td>chǒu</td><td>Yǐ yǒu</td><td>Jan. 9, 640 BCE</td><td>372</td></tr><tr><td>N<sub>-638</sub></td><td>1026</td><td>chǒu</td><td>Jǐ mǎo</td><td>Dec. 29, 640 BCE</td><td>750</td></tr><tr><td>N<sub>-637</sub></td><td>1038</td><td>zǐ</td><td>Jiǎ xū</td><td>Dec. 19, 639 BCE</td><td>101</td></tr><tr><td><sup>*</sup>N<sub>-636</sub></td><td>1050</td><td>zǐ</td><td>Wù chén</td><td>Dec. 8, 638 BCE</td><td>479</td></tr><tr><td>N<sub>-635</sub></td><td>1063</td><td>zǐ</td><td>Rén chén</td><td>Dec. 26, 637 BCE</td><td>375</td></tr><tr><td><sup>*</sup>N<sub>-634</sub></td><td>1075</td><td>zǐ</td><td>Bǐng xū</td><td>Dec. 15, 636 BCE</td><td>753</td></tr><tr><td>N<sub>-633</sub></td><td>1088</td><td>chǒu</td><td>Gēng xū</td><td>Jan. 3, 634 BCE</td><td>649</td></tr><tr><td>N<sub>-632</sub></td><td>1100</td><td>zǐ</td><td>Yǐ sì</td><td>Dec. 24, 634 BCE</td><td>0</td></tr></table><p><sup>*</sup> There was a leap month in this year.</p><br><br><table><tr><th>Chunqiu Cal. year</th><th>Acc. months</th><th>Jiàn</th><th>Sexagenary date</th><th>Julian Cal. Date</th><th>Xiǎoyú</th></tr><tr><td>N<sub>-631</sub></td><td>1112</td><td>zǐ</td><td>Jǐ hài</td><td>Dec. 12, 633 BCE</td><td>378</td></tr><tr><td>N<sub>-630</sub></td><td>1124</td><td>zǐ</td><td>Guǐ sì</td><td>Dec. 1, 632 BCE</td><td>756</td></tr><tr><td><sup>*</sup>N<sub>-629</sub></td><td>1136</td><td>hài</td><td>Wù zǐ</td><td>Nov. 21, 631 BCE</td><td>107</td></tr><tr><td>N<sub>-628</sub></td><td>1149</td><td>zǐ</td><td>Rén zǐ</td><td>Dec. 10, 630 BCE</td><td>3</td></tr><tr><td>N<sub>-627</sub></td><td>1161</td><td>hài</td><td>Bǐng wǔ</td><td>Nov. 28, 629 BCE</td><td>381</td></tr><tr><td><sup>*</sup>N<sub>-626</sub></td><td>1173</td><td>hài</td><td>Gēng zǐ</td><td>Nov. 17, 628 BCE</td><td>759</td></tr><tr><td><sup>*</sup>N<sub>-625</sub></td><td>1186</td><td>zǐ</td><td>Jiǎ zǐ</td><td>Dec. 6, 627 BCE</td><td>655</td></tr><tr><td>N<sub>-624</sub></td><td>1199</td><td>zǐ</td><td>Wù zǐ</td><td>Dec. 25, 626 BCE</td><td>551</td></tr><tr><td><sup>*</sup>N<sub>-623</sub></td><td>1211</td><td>zǐ</td><td>Rén wǔ</td><td>Dec. 13, 625 BCE</td><td>929</td></tr><tr><td>N<sub>-622</sub></td><td>1224</td><td>chǒu</td><td>Bǐng wǔ</td><td>Jan. 1, 623 BCE</td><td>825</td></tr><tr><td>N<sub>-621</sub></td><td>1236</td><td>zǐ</td><td>Xīn chǒu</td><td>Dec. 22, 623 BCE</td><td>176</td></tr><tr><td><sup>*</sup>N<sub>-620</sub></td><td>1248</td><td>zǐ</td><td>Yǐ wèi</td><td>Dec. 11, 622 BCE</td><td>554</td></tr><tr><td>N<sub>-619</sub></td><td>1261</td><td>chǒu</td><td>Jǐ wèi</td><td>Dec. 29, 621 BCE</td><td>450</td></tr><tr><td>N<sub>-618</sub></td><td>1273</td><td>zǐ</td><td>Guǐ chǒu</td><td>Dec. 18, 620 BCE</td><td>828</td></tr><tr><td><sup>*</sup>N<sub>-617</sub></td><td>1285</td><td>zǐ</td><td>Wù shēn</td><td>Dec. 8, 619 BCE</td><td>179</td></tr></table><p><sup>*</sup> There was a leap month in this year.</p><br><br><table><tr><th>Chunqiu Cal. year</th><th>Acc. months</th><th>Jiàn</th><th>Sexagenary date</th><th>Julian Cal. Date</th><th>Xiǎoyú</th></tr><tr><td>N<sub>-616</sub></td><td>1298</td><td>zǐ</td><td>Rén shēn</td><td>Dec. 27, 618 BCE</td><td>75</td></tr><tr><td>N<sub>-615</sub></td><td>1310</td><td>zǐ</td><td>Bǐng yín</td><td>Dec. 15, 617 BCE</td><td>453</td></tr><tr><td><sup>*</sup>N<sub>-614</sub></td><td>1322</td><td>zǐ</td><td>Gēng shēn</td><td>Dec. 4, 616 BCE</td><td>831</td></tr><tr><td>N<sub>-613</sub></td><td>1335</td><td>zǐ</td><td>Jiǎ shēn</td><td>Dec. 23, 615 BCE</td><td>727</td></tr><tr><td>N<sub>-612</sub></td><td>1347</td><td>zǐ</td><td>Jǐ mǎo</td><td>Dec. 13, 614 BCE</td><td>78</td></tr><tr><td><sup>*</sup>N<sub>-611</sub></td><td>1359</td><td>zǐ</td><td>Guǐ yǒu</td><td>Dec. 1, 613 BCE</td><td>456</td></tr><tr><td>N<sub>-610</sub></td><td>1372</td><td>zǐ</td><td>Dīng yǒu</td><td>Dec. 20, 612 BCE</td><td>352</td></tr><tr><td>N<sub>-609</sub></td><td>1384</td><td>zǐ</td><td>Xīn mǎo</td><td>Dec. 9, 611 BCE</td><td>730</td></tr><tr><td><sup>*</sup>N<sub>-608</sub></td><td>1396</td><td>hài</td><td>Bǐng xū</td><td>Nov. 29, 610 BCE</td><td>81</td></tr><tr><td>N<sub>-607</sub></td><td>1409</td><td>zǐ</td><td>Jǐ yǒu</td><td>Dec. 16, 609 BCE</td><td>1004</td></tr><tr><td>N<sub>-606</sub></td><td>1421</td><td>zǐ</td><td>Jiǎ chén</td><td>Dec. 6, 608 BCE</td><td>355</td></tr><tr><td>N<sub>-605</sub></td><td>1433</td><td>hài</td><td>Wù xū</td><td>Nov. 25, 607 BCE</td><td>733</td></tr><tr><td><sup>*</sup>N<sub>-604</sub></td><td>1445</td><td>hài</td><td>Guǐ sì</td><td>Nov. 15, 606 BCE</td><td>84</td></tr><tr><td>N<sub>-603</sub></td><td>1458</td><td>zǐ</td><td>Bǐng chén</td><td>Dec. 2, 605 BCE</td><td>1007</td></tr><tr><td><sup>*</sup>N<sub>-602</sub></td><td>1470</td><td>hài</td><td>Xīn hài</td><td>Nov. 22, 604 BCE</td><td>358</td></tr></table><p><sup>*</sup> There was a leap month in this year.</p><br><br><table><tr><th>Chunqiu Cal. year</th><th>Acc. months</th><th>Jiàn</th><th>Sexagenary date</th><th>Julian Cal. Date</th><th>Xiǎoyú</th></tr><tr><td><sup>*</sup>N<sub>-601</sub></td><td>1483</td><td>zǐ</td><td>Yǐ hài</td><td>Dec. 11, 603 BCE</td><td>254</td></tr><tr><td>N<sub>-600</sub></td><td>1496</td><td>chǒu</td><td>Jǐ hài</td><td>Dec. 30, 602 BCE</td><td>150</td></tr><tr><td>N<sub>-599</sub></td><td>1508</td><td>zǐ</td><td>Guǐ sì</td><td>Dec. 18, 601 BCE</td><td>528</td></tr><tr><td><sup>*</sup>N<sub>-598</sub></td><td>1520</td><td>zǐ</td><td>Dīng hài</td><td>Dec. 7, 600 BCE</td><td>906</td></tr><tr><td>N<sub>-597</sub></td><td>1533</td><td>zǐ</td><td>Xīn hài</td><td>Dec. 26, 599 BCE</td><td>802</td></tr><tr><td>N<sub>-596</sub></td><td>1545</td><td>zǐ</td><td>Bǐng wǔ</td><td>Dec. 16, 598 BCE</td><td>153</td></tr><tr><td><sup>*</sup>N<sub>-595</sub></td><td>1557</td><td>zǐ</td><td>Gēng zǐ</td><td>Dec. 4, 597 BCE</td><td>531</td></tr><tr><td>N<sub>-594</sub></td><td>1570</td><td>zǐ</td><td>Jiǎ zǐ</td><td>Dec. 23, 596 BCE</td><td>427</td></tr><tr><td><sup>*</sup>N<sub>-593</sub></td><td>1582</td><td>zǐ</td><td>Wù wǔ</td><td>Dec. 12, 595 BCE</td><td>805</td></tr><tr><td>N<sub>-592</sub></td><td>1595</td><td>chǒu</td><td>Rén wǔ</td><td>Dec. 31, 594 BCE</td><td>701</td></tr><tr><td>N<sub>-591</sub></td><td>1607</td><td>zǐ</td><td>Dīng chǒu</td><td>Dec. 20, 593 BCE</td><td>52</td></tr><tr><td>N<sub>-590</sub></td><td>1619</td><td>zǐ</td><td>Xīn wèi</td><td>Dec. 9, 592 BCE</td><td>430</td></tr><tr><td><sup>*</sup>N<sub>-589</sub></td><td>1631</td><td>zǐ</td><td>Yǐ chǒu</td><td>Nov. 28, 591 BCE</td><td>808</td></tr><tr><td>N<sub>-588</sub></td><td>1644</td><td>zǐ</td><td>Jǐ chǒu</td><td>Dec. 17, 590 BCE</td><td>704</td></tr><tr><td>N<sub>-587</sub></td><td>1656</td><td>zǐ</td><td>Jiǎ shēn</td><td>Dec. 6, 589 BCE</td><td>55</td></tr></table><p><sup>*</sup> There was a leap month in this year.</p><br><br><table><tr><th>Chunqiu Cal. year</th><th>Acc. months</th><th>Jiàn</th><th>Sexagenary date</th><th>Julian Cal. Date</th><th>Xiǎoyú</th></tr><tr><td><sup>*</sup>N<sub>-586</sub></td><td>1668</td><td>hài</td><td>Wù yín</td><td>Nov. 25, 588 BCE</td><td>433</td></tr><tr><td>N<sub>-585</sub></td><td>1681</td><td>zǐ</td><td>Rén yín</td><td>Dec. 14, 587 BCE</td><td>329</td></tr><tr><td>N<sub>-584</sub></td><td>1693</td><td>zǐ</td><td>Bǐng shēn</td><td>Dec. 3, 586 BCE</td><td>707</td></tr><tr><td><sup>*</sup>N<sub>-583</sub></td><td>1705</td><td>hài</td><td>Xīn mǎo</td><td>Nov. 22, 585 BCE</td><td>58</td></tr><tr><td>N<sub>-582</sub></td><td>1718</td><td>zǐ</td><td>Jiǎ yín</td><td>Dec. 10, 584 BCE</td><td>981</td></tr><tr><td>N<sub>-581</sub></td><td>1730</td><td>zǐ</td><td>Jǐ yǒu</td><td>Nov. 30, 583 BCE</td><td>332</td></tr><tr><td><sup>*</sup>N<sub>-580</sub></td><td>1742</td><td>hài</td><td>Guǐ mǎo</td><td>Nov. 19, 582 BCE</td><td>710</td></tr><tr><td>N<sub>-579</sub></td><td>1755</td><td>zǐ</td><td>Dīng mǎo</td><td>Dec. 7, 581 BCE</td><td>606</td></tr><tr><td><sup>*</sup>N<sub>-578</sub></td><td>1767</td><td>hài</td><td>Xīn yǒu</td><td>Nov. 26, 580 BCE</td><td>984</td></tr><tr><td><sup>*</sup>N<sub>-577</sub></td><td>1780</td><td>zǐ</td><td>Yǐ yǒu</td><td>Dec. 15, 579 BCE</td><td>880</td></tr><tr><td>N<sub>-576</sub></td><td>1793</td><td>chǒu</td><td>Jǐ yǒu</td><td>Jan. 3, 577 BCE</td><td>776</td></tr><tr><td>N<sub>-575</sub></td><td>1805</td><td>zǐ</td><td>Jiǎ chén</td><td>Dec. 23, 577 BCE</td><td>127</td></tr><tr><td>N<sub>-574</sub></td><td>1817</td><td>zǐ</td><td>Wù xū</td><td>Dec. 12, 576 BCE</td><td>505</td></tr><tr><td><sup>*</sup>N<sub>-573</sub></td><td>1829</td><td>zǐ</td><td>Rén chén</td><td>Dec. 1, 575 BCE</td><td>883</td></tr><tr><td>N<sub>-572</sub></td><td>1842</td><td>zǐ</td><td>Bǐng chén</td><td>Dec. 20, 574 BCE</td><td>779</td></tr></table><p><sup>*</sup> There was a leap month in this year.</p><br><br><table><tr><th>Chunqiu Cal. year</th><th>Acc. months</th><th>Jiàn</th><th>Sexagenary date</th><th>Julian Cal. Date</th><th>Xiǎoyú</th></tr><tr><td><sup>*</sup>N<sub>-571</sub></td><td>1854</td><td>zǐ</td><td>Xīn hài</td><td>Dec. 9, 573 BCE</td><td>130</td></tr><tr><td>N<sub>-570</sub></td><td>1867</td><td>chǒu</td><td>Yǐ hài</td><td>Dec. 28, 572 BCE</td><td>26</td></tr><tr><td>N<sub>-569</sub></td><td>1879</td><td>zǐ</td><td>Jǐ sì</td><td>Dec. 17, 571 BCE</td><td>404</td></tr><tr><td><sup>*</sup>N<sub>-568</sub></td><td>1891</td><td>zǐ</td><td>Guǐ hài</td><td>Dec. 6, 570 BCE</td><td>782</td></tr><tr><td>N<sub>-567</sub></td><td>1904</td><td>zǐ</td><td>Dīng hài</td><td>Dec. 24, 569 BCE</td><td>678</td></tr><tr><td>N<sub>-566</sub></td><td>1916</td><td>zǐ</td><td>Rén wǔ</td><td>Dec. 14, 568 BCE</td><td>29</td></tr><tr><td><sup>*</sup>N<sub>-565</sub></td><td>1928</td><td>zǐ</td><td>Bǐng zǐ</td><td>Dec. 3, 567 BCE</td><td>407</td></tr><tr><td>N<sub>-564</sub></td><td>1941</td><td>zǐ</td><td>Gēng zǐ</td><td>Dec. 22, 566 BCE</td><td>303</td></tr><tr><td>N<sub>-563</sub></td><td>1953</td><td>zǐ</td><td>Jiǎ wǔ</td><td>Dec. 10, 565 BCE</td><td>681</td></tr><tr><td><sup>*</sup>N<sub>-562</sub></td><td>1965</td><td>zǐ</td><td>Jǐ chǒu</td><td>Nov. 30, 564 BCE</td><td>32</td></tr><tr><td>N<sub>-561</sub></td><td>1978</td><td>zǐ</td><td>Rén zǐ</td><td>Dec. 18, 563 BCE</td><td>955</td></tr><tr><td><sup>*</sup>N<sub>-560</sub></td><td>1990</td><td>zǐ</td><td>Dīng wèi</td><td>Dec. 8, 562 BCE</td><td>306</td></tr><tr><td>N<sub>-559</sub></td><td>2003</td><td>zǐ</td><td>Xīn wèi</td><td>Dec. 26, 561 BCE</td><td>202</td></tr><tr><td>N<sub>-558</sub></td><td>2015</td><td>zǐ</td><td>Yǐ chǒu</td><td>Dec. 15, 560 BCE</td><td>580</td></tr><tr><td>N<sub>-557</sub></td><td>2027</td><td>zǐ</td><td>Jǐ wèi</td><td>Dec. 4, 559 BCE</td><td>958</td></tr></table><p><sup>*</sup> There was a leap month in this year.</p><br><br><table><tr><th>Chunqiu Cal. year</th><th>Acc. months</th><th>Jiàn</th><th>Sexagenary date</th><th>Julian Cal. Date</th><th>Xiǎoyú</th></tr><tr><td><sup>*</sup>N<sub>-556</sub></td><td>2039</td><td>hài</td><td>Jiǎ yín</td><td>Nov. 24, 558 BCE</td><td>309</td></tr><tr><td>N<sub>-555</sub></td><td>2052</td><td>zǐ</td><td>Wù yín</td><td>Dec. 12, 557 BCE</td><td>205</td></tr><tr><td>N<sub>-554</sub></td><td>2064</td><td>zǐ</td><td>Rén shēn</td><td>Dec. 1, 556 BCE</td><td>583</td></tr><tr><td><sup>*</sup>N<sub>-553</sub></td><td>2076</td><td>hài</td><td>Bǐng yín</td><td>Nov. 20, 555 BCE</td><td>961</td></tr><tr><td><sup>*</sup>N<sub>-552</sub></td><td>2089</td><td>zǐ</td><td>Gēng yín</td><td>Dec. 9, 554 BCE</td><td>857</td></tr><tr><td>N<sub>-551</sub></td><td>2102</td><td>chǒu</td><td>Jiǎ yín</td><td>Dec. 27, 553 BCE</td><td>753</td></tr><tr><td>N<sub>-550</sub></td><td>2114</td><td>zǐ</td><td>Jǐ yǒu</td><td>Dec. 17, 552 BCE</td><td>104</td></tr><tr><td><sup>*</sup>N<sub>-549</sub></td><td>2126</td><td>zǐ</td><td>Guǐ mǎo</td><td>Dec. 6, 551 BCE</td><td>482</td></tr><tr><td>N<sub>-548</sub></td><td>2139</td><td>zǐ</td><td>Dīng mǎo</td><td>Dec. 25, 550 BCE</td><td>378</td></tr><tr><td>N<sub>-547</sub></td><td>2151</td><td>zǐ</td><td>Xīn yǒu</td><td>Dec. 13, 549 BCE</td><td>756</td></tr><tr><td>N<sub>-546</sub></td><td>2163</td><td>zǐ</td><td>Bǐng chén</td><td>Dec. 3, 548 BCE</td><td>107</td></tr><tr><td><sup>*</sup>N<sub>-545</sub></td><td>2175</td><td>hài</td><td>Gēng xū</td><td>Nov. 22, 547 BCE</td><td>485</td></tr><tr><td><sup>*</sup>N<sub>-544</sub></td><td>2188</td><td>zǐ</td><td>Jiǎ xū</td><td>Dec. 11, 546 BCE</td><td>381</td></tr><tr><td>N<sub>-543</sub></td><td>2201</td><td>chǒu</td><td>Wù xū</td><td>Dec. 29, 545 BCE</td><td>277</td></tr><tr><td>N<sub>-542</sub></td><td>2213</td><td>zǐ</td><td>Rén chén</td><td>Dec. 18, 544 BCE</td><td>655</td></tr></table><p><sup>*</sup> There was a leap month in this year.</p><br><br><table><tr><th>Chunqiu Cal. year</th><th>Acc. months</th><th>Jiàn</th><th>Sexagenary date</th><th>Julian Cal. Date</th><th>Xiǎoyú</th></tr><tr><td>N<sub>-541</sub></td><td>2225</td><td>zǐ</td><td>Dīng hài</td><td>Dec. 8, 543 BCE</td><td>6</td></tr><tr><td><sup>*</sup>N<sub>-540</sub></td><td>2237</td><td>hài</td><td>Xīn sì</td><td>Nov. 27, 542 BCE</td><td>384</td></tr><tr><td>N<sub>-539</sub></td><td>2250</td><td>zǐ</td><td>Yǐ sì</td><td>Dec. 15, 541 BCE</td><td>280</td></tr><tr><td><sup>*</sup>N<sub>-538</sub></td><td>2262</td><td>zǐ</td><td>Jǐ hài</td><td>Dec. 4, 540 BCE</td><td>658</td></tr><tr><td>N<sub>-537</sub></td><td>2275</td><td>zǐ</td><td>Guǐ hài</td><td>Dec. 23, 539 BCE</td><td>554</td></tr><tr><td>N<sub>-536</sub></td><td>2287</td><td>zǐ</td><td>Dīng sì</td><td>Dec. 12, 538 BCE</td><td>932</td></tr><tr><td><sup>*</sup>N<sub>-535</sub></td><td>2299</td><td>zǐ</td><td>Rén zǐ</td><td>Dec. 1, 537 BCE</td><td>283</td></tr><tr><td>N<sub>-534</sub></td><td>2312</td><td>zǐ</td><td>Bǐng zǐ</td><td>Dec. 20, 536 BCE</td><td>179</td></tr><tr><td><sup>*</sup>N<sub>-533</sub></td><td>2324</td><td>zǐ</td><td>Gēng wǔ</td><td>Dec. 9, 535 BCE</td><td>557</td></tr><tr><td>N<sub>-532</sub></td><td>2337</td><td>chǒu</td><td>Jiǎ wǔ</td><td>Dec. 28, 534 BCE</td><td>453</td></tr><tr><td>N<sub>-531</sub></td><td>2349</td><td>zǐ</td><td>Wù zǐ</td><td>Dec. 16, 533 BCE</td><td>831</td></tr><tr><td><sup>*</sup>N<sub>-530</sub></td><td>2361</td><td>zǐ</td><td>Guǐ wèi</td><td>Dec. 6, 532 BCE</td><td>182</td></tr><tr><td>N<sub>-529</sub></td><td>2374</td><td>zǐ</td><td>Dīng wèi</td><td>Dec. 25, 531 BCE</td><td>78</td></tr><tr><td>N<sub>-528</sub></td><td>2386</td><td>zǐ</td><td>Xīn chǒu</td><td>Dec. 14, 530 BCE</td><td>456</td></tr><tr><td>N<sub>-527</sub></td><td>2398</td><td>zǐ</td><td>Yǐ wèi</td><td>Dec. 2, 529 BCE</td><td>834</td></tr></table><p><sup>*</sup> There was a leap month in this year.</p><br><br><table><tr><th>Chunqiu Cal. year</th><th>Acc. months</th><th>Jiàn</th><th>Sexagenary date</th><th>Julian Cal. Date</th><th>Xiǎoyú</th></tr><tr><td><sup>*</sup>N<sub>-526</sub></td><td>2410</td><td>hài</td><td>Gēng yín</td><td>Nov. 22, 528 BCE</td><td>185</td></tr><tr><td>N<sub>-525</sub></td><td>2423</td><td>zǐ</td><td>Jiǎ yín</td><td>Dec. 11, 527 BCE</td><td>81</td></tr><tr><td><sup>*</sup>N<sub>-524</sub></td><td>2435</td><td>zǐ</td><td>Wù shēn</td><td>Nov. 30, 526 BCE</td><td>459</td></tr><tr><td>N<sub>-523</sub></td><td>2448</td><td>zǐ</td><td>Rén shēn</td><td>Dec. 18, 525 BCE</td><td>355</td></tr><tr><td><sup>*</sup>N<sub>-522</sub></td><td>2460</td><td>zǐ</td><td>Bǐng yín</td><td>Dec. 7, 524 BCE</td><td>733</td></tr><tr><td>N<sub>-521</sub></td><td>2473</td><td>zǐ</td><td>Gēng yín</td><td>Dec. 26, 523 BCE</td><td>629</td></tr><tr><td><sup>*</sup>N<sub>-520</sub></td><td>2485</td><td>zǐ</td><td>Jiǎ shēn</td><td>Dec. 15, 522 BCE</td><td>1007</td></tr><tr><td>N<sub>-519</sub></td><td>2498</td><td>chǒu</td><td>Wù shēn</td><td>Jan. 2, 520 BCE</td><td>903</td></tr><tr><td>N<sub>-518</sub></td><td>2510</td><td>zǐ</td><td>Guǐ mǎo</td><td>Dec. 23, 520 BCE</td><td>254</td></tr><tr><td>N<sub>-517</sub></td><td>2522</td><td>zǐ</td><td>Dīng yǒu</td><td>Dec. 12, 519 BCE</td><td>632</td></tr><tr><td><sup>*</sup>N<sub>-516</sub></td><td>2534</td><td>zǐ</td><td>Xīn mǎo</td><td>Dec. 1, 518 BCE</td><td>1010</td></tr><tr><td>N<sub>-515</sub></td><td>2547</td><td>zǐ</td><td>Yǐ mǎo</td><td>Dec. 19, 517 BCE</td><td>906</td></tr><tr><td>N<sub>-514</sub></td><td>2559</td><td>zǐ</td><td>Gēng xū</td><td>Dec. 9, 516 BCE</td><td>257</td></tr><tr><td><sup>*</sup>N<sub>-513</sub></td><td>2571</td><td>zǐ</td><td>Jiǎ chén</td><td>Nov. 28, 515 BCE</td><td>635</td></tr><tr><td><sup>*</sup>N<sub>-512</sub></td><td>2584</td><td>zǐ</td><td>Wù chén</td><td>Dec. 17, 514 BCE</td><td>531</td></tr></table><p><sup>*</sup> There was a leap month in this year.</p><br><br><table><tr><th>Chunqiu Cal. year</th><th>Acc. months</th><th>Jiàn</th><th>Sexagenary date</th><th>Julian Cal. Date</th><th>Xiǎoyú</th></tr><tr><td>N<sub>-511</sub></td><td>2597</td><td>chǒu</td><td>Rén chén</td><td>Jan. 4, 512 BCE</td><td>427</td></tr><tr><td>N<sub>-510</sub></td><td>2609</td><td>zǐ</td><td>Bǐng xū</td><td>Dec. 24, 512 BCE</td><td>805</td></tr><tr><td>N<sub>-509</sub></td><td>2621</td><td>zǐ</td><td>Xīn sì</td><td>Dec. 14, 511 BCE</td><td>156</td></tr><tr><td>N<sub>-508</sub></td><td>2633</td><td>zǐ</td><td>Yǐ hài</td><td>Dec. 3, 510 BCE</td><td>534</td></tr><tr><td><sup>*</sup>N<sub>-507</sub></td><td>2645</td><td>hài</td><td>Jǐ sì</td><td>Nov. 21, 509 BCE</td><td>912</td></tr><tr><td>N<sub>-506</sub></td><td>2658</td><td>zǐ</td><td>Guǐ sì</td><td>Dec. 10, 508 BCE</td><td>808</td></tr><tr><td><sup>*</sup>N<sub>-505</sub></td><td>2670</td><td>zǐ</td><td>Wù zǐ</td><td>Nov. 30, 507 BCE</td><td>159</td></tr><tr><td>N<sub>-504</sub></td><td>2683</td><td>zǐ</td><td>Rén zǐ</td><td>Dec. 19, 506 BCE</td><td>55</td></tr><tr><td><sup>*</sup>N<sub>-503</sub></td><td>2695</td><td>zǐ</td><td>Bǐng wǔ</td><td>Dec. 7, 505 BCE</td><td>433</td></tr><tr><td>N<sub>-502</sub></td><td>2708</td><td>zǐ</td><td>Gēng wǔ</td><td>Dec. 26, 504 BCE</td><td>329</td></tr><tr><td>N<sub>-501</sub></td><td>2720</td><td>zǐ</td><td>Jiǎ zǐ</td><td>Dec. 15, 503 BCE</td><td>707</td></tr><tr><td>N<sub>-500</sub></td><td>2732</td><td>zǐ</td><td>Jǐ wèi</td><td>Dec. 5, 502 BCE</td><td>58</td></tr><tr><td><sup>*</sup>N<sub>-499</sub></td><td>2744</td><td>hài</td><td>Guǐ chǒu</td><td>Nov. 23, 501 BCE</td><td>436</td></tr><tr><td>N<sub>-498</sub></td><td>2757</td><td>zǐ</td><td>Dīng chǒu</td><td>Dec. 12, 500 BCE</td><td>332</td></tr><tr><td><sup>*</sup>N<sub>-497</sub></td><td>2769</td><td>zǐ</td><td>Xīn wèi</td><td>Dec. 1, 499 BCE</td><td>710</td></tr></table><p><sup>*</sup> There was a leap month in this year.</p><br><br><table><tr><th>Chunqiu Cal. year</th><th>Acc. months</th><th>Jiàn</th><th>Sexagenary date</th><th>Julian Cal. Date</th><th>Xiǎoyú</th></tr><tr><td>N<sub>-496</sub></td><td>2782</td><td>zǐ</td><td>Yǐ wèi</td><td>Dec. 20, 498 BCE</td><td>606</td></tr><tr><td><sup>*</sup>N<sub>-495</sub></td><td>2794</td><td>zǐ</td><td>Jǐ chǒu</td><td>Dec. 8, 497 BCE</td><td>984</td></tr><tr><td>N<sub>-494</sub></td><td>2807</td><td>chǒu</td><td>Guǐ chǒu</td><td>Dec. 27, 496 BCE</td><td>880</td></tr><tr><td>N<sub>-493</sub></td><td>2819</td><td>zǐ</td><td>Wù shēn</td><td>Dec. 17, 495 BCE</td><td>231</td></tr><tr><td><sup>*</sup>N<sub>-492</sub></td><td>2831</td><td>zǐ</td><td>Rén yín</td><td>Dec. 6, 494 BCE</td><td>609</td></tr><tr><td>N<sub>-491</sub></td><td>2844</td><td>zǐ</td><td>Bǐng yín</td><td>Dec. 24, 493 BCE</td><td>505</td></tr><tr><td>N<sub>-490</sub></td><td>2856</td><td>zǐ</td><td>Gēng shēn</td><td>Dec. 13, 492 BCE</td><td>883</td></tr><tr><td><sup>*</sup>N<sub>-489</sub></td><td>2868</td><td>zǐ</td><td>Yǐ mǎo</td><td>Dec. 3, 491 BCE</td><td>234</td></tr><tr><td>N<sub>-488</sub></td><td>2881</td><td>zǐ</td><td>Jǐ mǎo</td><td>Dec. 22, 490 BCE</td><td>130</td></tr><tr><td><sup>*</sup>N<sub>-487</sub></td><td>2893</td><td>zǐ</td><td>Guǐ yǒu</td><td>Dec. 10, 489 BCE</td><td>508</td></tr><tr><td>N<sub>-486</sub></td><td>2906</td><td>chǒu</td><td>Dīng yǒu</td><td>Dec. 29, 488 BCE</td><td>404</td></tr><tr><td>N<sub>-485</sub></td><td>2918</td><td>zǐ</td><td>Xīn mǎo</td><td>Dec. 18, 487 BCE</td><td>782</td></tr><tr><td><sup>*</sup>N<sub>-484</sub></td><td>2930</td><td>zǐ</td><td>Bǐng xū</td><td>Dec. 8, 486 BCE</td><td>133</td></tr><tr><td>N<sub>-483</sub></td><td>2943</td><td>zǐ</td><td>Gēng xū</td><td>Dec. 26, 485 BCE</td><td>29</td></tr><tr><td>N<sub>-482</sub></td><td>2955</td><td>zǐ</td><td>Jiǎ chén</td><td>Dec. 15, 484 BCE</td><td>407</td></tr></table><p><sup>*</sup> There was a leap month in this year.</p><br></div><p>Once A(y) is known, it is very easy to compute the Chunqiu calendar. The data in other columns are also easy to reproduce, except in some cases the <i>jiàn</i> of the first month. Here I demonstrate the calculation by an example for the year <span class="nowrap">N<sub>-649</sub></span>. From the table above, A(-649) = 890. It follows that the number of days between the new year day conjunction time in N<sub>-649</sub> and midnight of Jan. 16, 722 BCE was<br>[268/1027 + 890×(29 + 545/1027)] days= (26282 + 574/1027) days.<br>The integer part was 26282 and the <i>xiǎoyú</i> was 574, consistent with the <i>xiǎoyú</i> listed in the table. It is easy to calculate the sexagenary day cycle using modular arithmetic: the heavenly stems have a cycle of 10 days and the earthly branches have a cycle of 12 days. Simple calculation shows that 26282 days after <i>xīn sì</i> was <i>guǐ wèi</i>, which is also the same as the one listed in the table. The New Year day in the proleptic Julian calendar can be determined by first computing the number of days between Jan. 16, -721 and Jan. 16, -649. There are 72 years between -721 and -649. Years divisible by 4 have 366 days and others have 365 days. The leap years between -721 and -649 were -720, -716, ..., and -652, a total of 18 leap years. Hence the number of days between Jan. 16, -721 and Jan. 16, -649 was (72×365 + 18) = 26298 days = (26282 + 16) days. This means that the New Year day of N<sub>-649</sub> was 16 days before Jan. 16, -649, which was Dec. 31, -650 and agrees with the date listed in the table. As for the <i>jiàn</i>, the <i>zǐ</i> month is defined to be the month that contains the winter solstice (see <a href="sexagenary.html#jian">sexagenary cycle page</a>). The winter solstice in <span class="nowrap">-650</span> can be looked up from the <a href="sunMoon.html?y=-650">Sun & Moon phenomena page</a> and it was on Dec. 28 in the proleptic Julian calendar<sup><a href="chunqiu.html#fn1">fn1</a></sup>. The New Year day was 3 days after the winter solstice. Thus the previous month was the <i>zǐ</i> month and the first month of N<sub>-649</sub> was one month after the <i>zǐ</i> month, so it was the <i>chǒu</i> month. This also agrees with the <i>jiàn</i> listed in the table.</p><p>Now that we know the New Year day of N<sub>-649</sub>, the first days of the other months in the year can be calculated by adding (29 + 545/1027) days successively to the time of the new year day conjunction. There was no leap month in N<sub>-649</sub> according to the table. The first days of each of the 12 months in N<sub>-649</sub> are listed in the following table.</p><table><tr><td>Month in the Chunqiu Cal.</td><td>First day in Jul. Cal.</td><td>Sexagenary Day</td><td><i>Xiǎoyú</i></td></tr><tr><td>1 (L)</td><td>Dec. 31, 651 BCE</td><td>Guǐ Wèi</td><td>574</td></tr><tr><td>2 (S)</td><td>Jan. 30, 650 BCE</td><td>Guǐ Chǒu</td><td>92</td></tr><tr><td>3 (L)</td><td>Feb. 28</td><td>Rén Wǔ</td><td>637</td></tr><tr><td>4 (S)</td><td>Mar. 30</td><td>Rén Zǐ</td><td>155</td></tr><tr><td>5 (L)</td><td>Apr. 28</td><td>Xīn Sì</td><td>700</td></tr><tr><td>6 (S)</td><td>May 28</td><td>Xīn Hài</td><td>218</td></tr><tr><td>7 (L)</td><td>June 26</td><td>Gēng Chén</td><td>763</td></tr><tr><td>8 (S)</td><td>July 26</td><td>Gēng Xū</td><td>281</td></tr><tr><td>9 (L)</td><td>Aug. 24</td><td>Jǐ Mǎo</td><td>826</td></tr><tr><td>10 (S)</td><td>Sep. 23</td><td>Jǐ Yǒu</td><td>344</td></tr><tr><td>11 (L)</td><td>Oct. 22</td><td>Wù Yín</td><td>889</td></tr><tr><td>12 (S)</td><td>Nov. 21</td><td>Wù Shēn</td><td>407</td></tr></table><p>In the table, L means the month had 30 days (long month); S means the month had 29 days (short months). The number of days in a month is determined by the number of days between two successive lunar conjunctions, which can be calculated easily by the <i>xiǎoyú</i>. There is a simple relationship between the <i>xiǎoyú</i>s of two successive months: the <i>xiǎoyú</i> of a month is equal to the <i>xiǎoyú</i> of the previous month plus 545, and then subtract 1027 if necessary. It is probably easier to understand the logic using today's 24-hour time system. The lunar cycle is 29 days 12 hours 44 minutes and 10 seconds. The time in the day of a conjunction can be obtained by adding 12 hours 44 minutes and 10 seconds from the previous conjunction time. If the number exceeds 24, subtract 24 from it. If the <i>xiǎoyú</i> of a month is smaller than 482, adding 545 to it will result in a number smaller than 1027. The number of days in the month will be 29 in this case and the month is a short month. If the <i>xiǎoyú</i> of the month is equal to or greater than 482, adding 545 to it will result in a number that is at least 1027 and the number has to be subtracted by 1027 to get the <i>xiǎoyú</i> of the following month. There will be 30 days in the month in this case. The conclusion is that a month with <i>xiǎoyú</i> less than 482 has 29 days and a month with <i>xiǎoyú</i> ≥482 has 30 days. It is also easy to show that if <i>xiǎoyú</i> exceeds 963, there will be two long months in a row. It is impossible to have two successive short months in the Chunqiu calendar. Since the synodic month is slightly greater than 29.5 days, it is in general impossible to have two successive short months for calendars based on the <i>píngshuò</i> algorithm.</p><br><sect id="typos">Typos in Table 3-7 in <i>Ancient Chinese Calendars and Almanacs</i></sect><p>I used the calculated A(y) to compute the sexagenary cycle of the New Year day and the <i>jiàn</i> of the first month in each year, and then compared with those listed in Table 3-7 in <i>Ancient Chinese Calendars and Almanacs</i>. I find agreement in most years, but there are discrepancies in 5 years. After further analysis, I find that the data in Table 3-7 for those 5 years are inconsistent with the <i>xiǎoyú</i>s, which means that at least one of them is incorrect. My analysis suggests that the data should be corrected as shown below.</p><table><tr><th>Year</th><th>Correction to Table 3-7 in <i>Ancient Chinese Calendars and Almanacs</i></th></tr><tr><td>N<sub>-720</sub></td><td>Sexagenary day: Jǐ Hài (己亥) <span style="font-size:150%">→</span> Yǐ Hài (乙亥)</td></tr><tr><td>N<sub>-681</sub></td><td><i>Jiàn</i>: Chǒu <span style="font-size:150%">→</span> Hài</td></tr><tr><td>N<sub>-680</sub></td><td><i>Jiàn</i>: Yín <span style="font-size:150%">→</span> Zǐ</td></tr><tr><td>N<sub>-653</sub></td><td>Sexagenary day: Bǐng Wǔ <span style="font-size:150%">→</span> Dīng Wèi</td></tr><tr><td>N<sub>-623</sub></td><td><i>Xiǎoyú</i>: 614 <span style="font-size:150%">→</span> 929</td></tr></table><br><p>Let's look at each of them. The typo in N<sub>-720</sub> is easy to see. If there was no leap month in N<sub>-721</sub>, the <i>xiǎoyú</i> of the new year day conjunction in N<sub>-720</sub> would be 646, which agrees with Table 3-7, but the sexagenary day would be yǐ hài. If there was a leap month in N<sub>-721</sub>, the <i>xiǎoyú</i> would be 164, which disagrees with the value listed in Table 3-7, and the sexagenary day cycle would be yǐ sì. Therefore, the sexagenary day would not be jǐ hài no matter whether there was a leap month in N<sub>-721</sub>. Clearly, jǐ hài is a typo.</p><p>Comparing the <i>xiǎoyú</i>s between N<sub>-682</sub> and N<sub>-681</sub>, I conclude that there was no leap month in N<sub>-682</sub> and the New Year day of N<sub>-681</sub> was Nov. 25, -682 in the proleptic Julian calendar. This was clearly before the winter solstice, which occurred on Dec. 28 in -682. Hence, it is impossible for the first month to be a chǒu month. There was 33 days before Nov. 25 and Dec. 28, which also makes it impossible for the <i>jiàn</i> to be zǐ. The first month was clearly a month before the zǐ month, making the <i>jiàn</i> to be hài. The sexagenary day of Nov. 25, -682 was jǐ wèi, which agrees with Table 3-7. The sexagenary day of the first day of the chǒu month was also jǐ wèi. However, there would be a serious problem if the <i>jiàn</i> ware really chǒu. In addition to the discrepancy of the <i>xiǎoyú</i>, N<sub>-682</sub> would have 14 months, which was impossible. Hence changing the <i>jiàn</i> from chǒu to hài is appropriate.</p><p>Having corrected the typo in N<sub>-681</sub>, it is not difficult to see that the <i>jiàn</i> in N<sub>-680</sub> in Table 3-7 is also incorrect. It follows from the <i>xiǎoyú</i>s in N<sub>-681</sub> and N<sub>-680</sub> that there was a leap month in N<sub>-681</sub> and the New Year day of N<sub>-680</sub> was Dec. 14, -681. The first month clearly contained the winter solstice and so was the zǐ month. The sexagenary day on Dec. 14, -681 was guǐ wèi, which agrees with Table 3-7. The sexagenary day of the conjunction day associated with the yín month was rén wǔ, in contradiction with Table 3-7. The <i>xiǎoyú</i> of the yín month conjunction was also different from the number listed in Table 3-7. Changing the <i>jiàn</i> from yín to zǐ removes the discrepancies.</p><p>Table 3-7 lists the sexagenary day of the New Year day of N<sub>-654</sub> as rén zǐ, and the conjunction <i>xiǎoyú</i> was 675; the <i>xiǎoyú</i> of the new year day conjunction of N<sub>-653</sub> was 26 = mod(675+378, 1027). It follows that there was no leap month in N<sub>-654</sub>, and the number of days between the new year day conjunction of N<sub>-653</sub> and the midnight of the New Year day of N<sub>-654</sub> was<br>[675/1027 + 12×(29+545/1027)] days = (355 + 26/1027) days.<br>The integer part was 355 and the <i>xiǎoyú</i> was 26. It is easy to show, using modular arithmetic, that the sexagenary day 355 days after the day of rén zǐ is dīng wèi, which is one day after the day of bǐng wǔ. The bǐng wǔ listed in Table 3-7 is probably caused by a mistake in arithmetic calculation.</p><p>Table 3-7 lists the New Year day of <span class="nowrap">N<sub>-624</sub></span> as the wù zǐ day in the zǐ month (December 25, 626 BCE) and the <i>xiǎoyú</i> was 551. If there was no leap month in <span class="nowrap">N<sub>-624</sub></span>, the New Year day of <span class="nowrap">N<sub>-623</sub></span> would be the rén wǔ day in the zǐ month (December 13, 625 BCE) and the <i>xiǎoyú</i> would be 929. If there was a leap month in <span class="nowrap">N<sub>-624</sub></span>, the New Year day of <span class="nowrap">N<sub>-623</sub></span> would be the rén zǐ day in the chǒu month (January 12, 624 BCE) and the <i>xiǎoyú</i> would be 447. It is therefore impossible for the <i>xiǎoyú</i> in <span class="nowrap">N<sub>-624</sub></span> to be 614. Table 3-7 lists the New Year day of <span class="nowrap">N<sub>-623</sub></span> as the rén wǔ day in the zǐ month, so the <i>xiǎoyú</i> should be 929.</p><p>Zhāng Péiyú has another book titled <i>Sānqiān Wǔbǎiniǎn Lìrì Tiānxiàng</i> (《三千五百年历日天象》 or <i>3500 Years of Calendars and Astronomical Phenomena</i>), in which there are also calendar dates for the Chunqiu calendar. The New Year days of N<sub>-721</sub>, N<sub>-681</sub> and N<sub>-680</sub> agree with my corrected dates. However, the New Year days of N<sub>-675</sub>, N<sub>-674</sub>, N<sub>-600</sub>, N<sub>-519</sub> and N<sub>-502</sub> differ by one month compared to those in Table 3-7 of <i>Ancient Chinese Calendars and Almanacs</i>. I do not use the data of <i>3500 Years of Calendars and Astronomical Phenomena</i> on this website in the Spring and Autumn period, as the reconstruction method is not stated and so it is difficult to assess the reliability of the data.<sup><a href="chunqiu.html#fn2">fn2</a></sup></p><br><sect id="firstMonth">Leap Month and <i>Jiàn</i> in the Chunqiu Calendar</sect><p>Scholars have not come to an agreement on the position of the leap months in the Chunqiu calendar. The 19th century scholar Wáng Tāo placed some of the leap months in the middle of the years in order to account for some of the dates recorded by the chronicle <i>Chunqiu</i>, which were otherwise irreconcilable with his reconstructed calendar. Zhāng points out that leap months are only mentioned twice in <i>Chunqiu</i> and they were all placed at the end of the years. He also doesn't think that there was any natural algorithm in the Chunqiu calendar to place a leap month in the middle of a year. Thus he thinks all leap months were placed at the end of the years. I adopt Zhāng's view and place all leap months at the end of the years on this website.</p><p>As there were no fixed rules for inserting leap months, the <i>jiàn</i>s of the first months were not fixed either. Looking at the table above, we see that the <i>jiàn</i>s wandered between hài (present-day month 10) and yín (present-day month 1), with chǒu (present-day month 12) occurring more often in the early years and zǐ (present-day month 11) occurring more often in the later years. Zhāng speculates that the Chunqiu calendar makers probably discovered the 19-year cycle (known as the Metonic cycle in ancient Greece) and the cycle of the tropical year in later years, and intentionally shifted the first month's <i>jiàn</i> from chǒu to zǐ. Sometime in the Warring States period (480 BCE – 222 BCE), the Lu state used the Lu calendar, one of the <i>gǔliùlì</i> (<i>ancient six calendars</i>) used in that period, in which a rule for inserting leap months was developed and the first month was fixed in the zǐ month.</p><br><sect id="reliability">Reliability of the Reconstructed Chunqiu Calendar</sect><p>Zhāng's reconstructed Chunqiu calendar is based on a lunar cycle and leap months deduced from the dates recorded in the chronicle <i>Chunqiu</i>. We can only say that it is a model constructed to fit the data. The lunar conjunction is computed by adding integer multiples of the lunar cycle to a constant, which is consistent with the <i>píngshuò</i> rule. However, using one lunar cycle for the entire period of 242 years implies that the rules of calendar did not change over 200+ years, which does not seem to be possible. However, using a simple model to fit the data is reasonable given the lack of information on the calendar evolution at that time. One way to assess the reliability of the reconstructed calendar is to examine how well the model fits the data.</p><p>According to Zhāng's description in Section 3.5 in <i>Ancient Chinese Calendars and Almanacs</i>, there are 393 pieces of calendrical dates mentioned in the chronicle <i>Chunqiu</i>. Most of the dates are consistent with the reconstructed calendar, but 45 of them are inconsistent. Of these 45, 11 of them can be resolved if leap months were inserted in the middle of the years. The current version of <i>Chunqiu</i> was passed down over many generations by hand copying the texts over 2000+ years. Mistakes are inevitable. It is also found that there are incompatible texts in <i>Chunqiu</i>. Of course, some discripancies may be explained by typos. However, it is difficult to decide whether a descrpancy is caused by typos or a true discrpancy between the reconstructed calendar and the texts. If we assume that all of the 45 mismatches are true descrpancies, the reconstructed calendar can explain about 89% of the calendrical dates mentioned in <i>Chunqiu</i>. Comparing the two reconstructed calendars between Zhāng and Wáng Tāo listed in Table 3-7 in <i>Ancient Chinese Calendars and Almanacs</i>, we see that sometimes their New Year days can differ by one month. This is caused by their different opinions on whether or not a leap month could appear in the middle of a year. Based on these pieces of information, I speculate it is very likely that there are true discrepancies between Zhāng's reconstructed calendar and the true calendar used at that time, but the amount of deviations may be small. The reconstructed calendar is of great value for historians.</p><br><sect id="julian">Julian Days</sect><p>In the example demonstrated above, time is measured from the midnight of the New Year day of N<sub>-721</sub>. In modern calendar calculations, time is often measured from noon on January 1, 4713 BCE in the proleptic Julian calendar. This day-counting system is known as the Julian days. Here for convenience, the time origin is assumed to be the local time for the Lu state. The Julian day at midnight on the New Year day of N<sub>-721</sub> is 1457727.5. Therefore, the Julian day of any given time (measured from the midnight of the New Year day of N<sub>-721</sub>) can be obtained by adding 1457727.5.</p><p>The advantage of using Julian days is that there are standard algorithms for converting Julian days to dates in the Julian/Gregorian calendar. Our <a href="sexagenary.html">sexagenary cycle page</a> also lists formulae for converting Julian days to sexagenary days. However, using Julian days for the computation of the Chunqiu calendar alone is not simpler than the calculation demonstrated above. Dates in the Julian calendar are so regular that we can easily invent a conversion method ourselves without relying on the standard algorithms. On the other hand, Julian days are convenient if we want to compute <i>gǔliùlì</i> (<a href="guliuli.html"><i>ancient six calendars</i></a>), and <a href="QinHanCalendars.html">calendars used by the Qin and early Han dynasties</a> in addition to the Chunqiu calendars. Each of these calendars have their own natural time origins. Using Julian days is just to unify the time origin at noon on Jan. 1, -4712. We can then do the calendar date conversion and sexagenary day calculation using a single procedure. This is very useful when writing a computer program for calendar calculations.</p><p>Here I introduce a method to calculate the Chunqiu calendar using Julian days. Equation (1) for the lunar conjunction time can be written in terms of Julian days by the following equation.<br>JD(M<sub>i</sub>) = 1457727.5 + 268/1027 + i(29 + 545/1027) (5)<br>Let's look at the example of computing the new year day conjunction of N<sub>-649</sub> again. From the table above, we have A(y) = 890. It follows from equation (5) that the Julian day of the conjunction was<br>JD = 1457727.5 + 268/1027 + 890× (29 + 545/1027) = 1484009.5 + 574/1027.<br>The Julian day at noon on the New Year day was 1484010 and the <i>xiǎoyú</i> of the conjunction was 574. Using a standard Julian day-Julian calendar conversion algorithm (e.g. <a href="https://en.wikipedia.org/wiki/Julian_day#Julian_or_Gregorian_calendar_from_Julian_day_number" target="_blank">algorithm by Richards</a>), Julian day 1484010 corresponds to Dec. 31, <span class="nowrap">-650</span> in the proleptic Julian calendar. According to the formulae in our <a herf="sexagenary.html">sexagenary cycle page</a>, the heavenly stem number of the New Year day was<br>1 + mod(1484010-1, 10) = 10, which means the heavenly stem of the day was guǐ; the earthly branch number was<br>1+ mod(1484010+1, 12) = 8 and the earthly branch was wèi. Thus the sexagenary cycle of the New Year day was guǐ wèi. All these results agree with the above calculation.</p><p>I use fractions in the above calculation. Calculations are now usually done on a computer and it is more complicated to write a program to handle fractions. Thus, using floating point numbers is easier even though the computational efficiency is slightly reduced. Modern computers are so fast that the slight decrease in efficiency is hardly noticeable. However, there is one thing we should be careful. When the <i>xiǎoyú</i> of a conjunction is exactly 0, the conjunction occurs exactly at midnight. The decimal part of the Julian day is exactly 0.5, but floating point roundoff error may give 0.49999999999 instead, resulting in an off-by-one-day error. This can be prevented by adding 0.0001 to the Julian day. Specifically, equation (1) is modified as follows.<br>JD(M<sub>i</sub>) = 1457727.761054236 + i · 29.53067185978578 (6)<br>Since 0.0001 = 0.1027/1027, adding it to equation (5) results in the <i>xiǎoyú</i> changing from the range between 0 and 1026 to the range between 0.1027 and 1026.1027. Now the conjunction can never occur at midnight and so the floating point roundoff error will not produce a one-day error. Now that we are dealing with decimals, it is no longer necessary to calculate <i>xiǎoyú</i>, but we can still calculate it if we want:<br><i>xiǎoyú</i> (omitting the denominator 1027) = floor(1027(JD + 0.5 - floor(JD + 0.5)) (7)<br>Here floor(x) denotes the largest integer smaller than or equal to x. JD + 0.5 - floor(JD + 0.5) is <i>xiǎoyú</i> expressed in decimals. Since 0.0001 is added to the Julian day, when the number is multiplied by 1027 it becomes an integer plus 0.1027. The floor() function then removes the decimals and the result is <i>xiǎoyú</i> with the denominator 1027 omitted. As mentioned, we no longer need to care about <i>xiǎoyú</i>s when using decimals. The only reason to calculate <i>xiǎoyú</i>s is to compare them with the values listed in Table 3-7 in <i>Ancient Chinese Calendars and Almanacs</i> for code validation.</p><br><hr><h2 id="footnotes">Footnotes</h2><ul style="list-style-type:none"><li id="fn1"><p>[<span class="ref">fn1</span>] The average number of days in the Julian calendar is 365.25 days, which is 0.0078 days longer than the tropical year (365.2422 days). As a result, the dates of the winter solstice gradually drifted earlier and earlier in the Julian calendar. The average number of days in a Gregorian year is 365.2425 days, much closer to the tropical year, so the dates of the winter solstice have not drifted much between now and the Gregorian calendar reform in 1582. Winter solstice occurs around Dec. 21 today. The Gregorian reform in 1582 skipped 10 days by specifying that the date following Oct 4, 1582 was Oct 15, 1582. Hence the date of the winter solstice before the calendar reform was around Dec. 11. There are 2232 years between 1582 and -650 and the winter solstice drifted by 0.0078×2232 days = 17.4 days. It follows that the winter solstice in -650 was around Dec. 28 in the proleptic Julian calendar.</p></li><li id="fn2"><p>[<span class="ref">fn2</span>] The New Year days of N<sub>-675</sub> and N<sub>-674</sub> listed in the book are very dubious. If the data were correct, there would be 3 consecutive leap months occurring in N<sub>-677</sub>, N<sub>-676</sub> and N<sub>-675</sub>, and there would be no leap months in the next 6 years. I have not studied the Chunqiu calendar and cannot say for sure such unusual leap month pattern was impossible, but I find the data in Table 3-7 of <i>Ancient Chinese Calendars and Almanacs</i> to be more reasonable.</p></li></ul></div><br><br></body></html>