《天原發微》Vol. 12

Chapter 3 of Volume 12 of the Tianyuan Fawei discusses the calendrical concepts of "surplus and deficit." The solar calendar accumulates an annual surplus of six days, called "qi surplus" (qiyi), equivalent to five and a quarter days. The lunar calendar runs six days short annually, termed "new moon deficit" (shuoxu), meaning the synodic month cycle is actually six days shorter (during minor months). This creates a combined annual discrepancy of twelve days, requiring intercalary months for adjustment.

Ancient calendrical systems featured two principal theories: "prior heaven" (xiantian) and "posterior heaven" (houtian). The "prior heaven" theory posited that the sun, moon, and five planets all rotated leftward with the celestial sphere—a view endorsed by Confucian scholars. The "posterior heaven" theory maintained they rotated rightward against the celestial sphere, with the sun moving slower and the moon faster. While more practical for calendrical calculations, this latter approach found little Confucian acceptance. Both theories were preserved for reference.

The Classic of Documents' "Canon of Yao" records a 366-day year adjusted through intercalary months to harmonize the seasons. Specifically, one complete cycle runs from winter solstice to the next winter solstice, typically spanning 366 days or slightly over 365 days. Each celestial conjunction of the sun marks one year, with the twenty-four solar terms maintaining consistent biennial intervals regardless of intercalary months.

Ancient calendars divided each day into 940 units (called "day measure" (rifa)). When multiplied by the celestial circumference degrees, this still leaves a quarter-day remainder—because the 940 units were subdivided into four sets of 235, with one unit allocated per degree. Zhu Xi explained that while the solar term count yields 360 days and the lunar phase count 354 days, the median 360-day figure keeps discrepancies within six days. The "Canon of Yao" rounded this to 366 days.

Hu Shen further noted: seasonal calculations allot 90 days per season (360 total); six qi divisions assign 60 days per qi (360 total); five phase systems allocate 72 days per phase (again 360 days). All these computational methods converge on the fundamental 360-day cycle.

The calculation principle of the Qian (Heaven) and Kun (Earth) hexagrams follows the same pattern, representing the standard law of celestial numbers. Exceeding this standard results in surplus, while falling short leads to deficiency. When in surplus, the movement of Yang Qi always runs ahead—hence the sun's complete revolution along the ecliptic (one celestial cycle) as a year extends beyond 360 days by approximately five extra days annually. When deficient, the movement of Yin Qi always lags behind—thus the moon's celestial cycle as a month often falls short of 30 days, explaining the existence of shorter months (29 days) and the annual surplus of about five days that accumulates into leap years. The phrase "eleven days minus" refers precisely to this discrepancy.

Accumulated over years, this discrepancy is adjusted in ratios of 3:1 and 5:1. After 19 years, a complete intercalary cycle of seven leap months is formed, achieving perfect harmony in the celestial Dao's movement. The sun and moon cannot operate without any surplus or deficiency, nor can calendar-makers avoid all errors. Therefore, leap months are instituted to regulate time and stabilize the annual cycle.

The Huangji Jingshi (Supreme Principles Governing the World) also takes 360 as its base unit: one "Yuan" (grand cycle) contains 360 "Yun," one "Hui" contains 360 "Shi," one "Yun" spans 360 years, one "Shi" spans 360 months, one year spans 360 days, and one month spans 360 two-hour periods (Shichen). The movement of Yang Qi completes its full cycle in six sets of 360.

Master Zhang Zai (Hengqu) stated that both the sun and moon rotate leftward in their motion. His explanation was: Heaven moves with extreme swiftness, completing a full celestial circuit of 365¼ degrees in one day and night, with an additional degree left over. The sun moves slightly slower than Heaven, traveling exactly 365¼ degrees in the same time. Because Heaven moves one degree farther, the sun appears to retreat by one degree. After two days, Heaven moves two degrees ahead, and the sun appears to retreat by two degrees. Accumulated over 365¼ days, the extra degrees traversed by Heaven complete a full cycle, while the sun's apparent retreat also completes a full cycle, reuniting with Heaven to form a complete year.

The moon moves even more slowly, traveling less than 365¼ degrees in a day and night, retreating about 13 degrees compared to Heaven's motion. Zhu Xi questioned this: If Heaven completes one rotation per day while the sun lags by one degree daily, why do the celestial phenomena observed in the four seasons differ? If this were truly the case, every day would appear identical—how then could solar terms be distinguished? Timekeeping would descend into chaos, where midday might mistakenly be called the third watch. Here, references are drawn from two annotations in the Yueling (Monthly Ordinaries) commentary to explain this phenomenon.

Regarding the movement patterns of celestial bodies, no other calendrical texts explain them as clearly as this account. From the perspective of Earth's interior, celestial bodies traverse 365 and 1/4 degrees daily. However, when observed from the cosmic void, one discovers these bodies shift positions each day without remaining in their original locations.

Both the sun and moon commence their cycles from the Jiao constellation (one of the twenty-eight lunar mansions). While the sun completes its daily revolution and returns to Jiao, celestial movement slightly overshoots this point. Accumulated over time, the solar and celestial trajectories realign after one year. According to two scholars' theories: celestial movement proves slightly faster, lunar motion somewhat slower, with only the sun maintaining perfect accuracy - hence the 360-day standard for annual calculation.

Master Shao Yong observed: After 360 days of yang energy's movement, an excess of 5 days and 235 fen remains - termed "qi surplus." Specifically, this amounts to 13 fen, 7 li, 8 si, and 3 hu daily. From winter solstice to spring commencement, this accumulates to 4,935 fen, equivalent to 5 days and 235 fen. Calculated in whole numbers, celestial movement exceeds solar motion by 6 days, while solar motion falls short by 6 days.

Master Zhu Xi explained: The celestial sphere forms a perfect circle with a circumference of 365 and 1/4 degrees, rotating leftward around Earth daily while advancing 1 extra degree. Though attached to this sphere, the sun moves slightly slower - completing its daily Earth revolution while lagging 1 degree relative to the celestial sphere. After 365 days and 235/940 days, their trajectories realign, constituting one annual cycle.

Kong Yingda's commentary reveals: The cosmic void itself lacks fixed form - what we call "heaven" actually refers to stellar trajectories. Heaven envelops Earth like egg white surrounding yolk. Stars move east to west, returning to original positions after 365 and 1/4 days. While stars rotate leftward, the sun progresses rightward, likewise realigning with stellar positions after 365 and 1/4 days. The twenty-eight mansions distributed across four directions form heaven's structural framework. Each mansion spans 13 degrees and 402 fen, collectively matching heaven's 365 and 1/4-degree circumference.

This passage discusses the patterns of solar and lunar motion in traditional calendrical astronomy. Ancient astronomers used "degrees" to calculate celestial positions, dividing the year into 360 degrees corresponding to 360 days. However, the actual speeds of the sun and moon are not perfectly uniform, sometimes faster and sometimes slower, creating what was termed "surplus" (盈) and "deficit" (亏) in calendrical calculations.

If we were to assume perfectly regular motion where both luminaries traveled equal daily distances, completing exactly 360 days without remainder, there would be no seasonal variations, no distinction between long and short months, nor occurrences of droughts or floods. Yet such artificial regularity would violate the natural order (造化) and contradict the heavenly principle that "spirits bless the virtuous and punish the licentious" - the cosmic law of moral retribution.

Specifically, the moon moves slower than the sun. In a lunar month (synodic cycle), the moon lags behind by 441 parts (ancient astronomers divided each day into 940 parts). From the eleventh to twelfth lunar months, this accumulates to 5,292 parts, equivalent to 5 days and 592 parts - approximately six days of lag. This represents the moon's cumulative deficit relative to the sun.

Zhu Xi explained that the moon falls behind the sun by 13 and 7/19 degrees daily (the celestial sphere being divided into 19 sections, each about 49.4736 parts). After 29 and 499/940 days, the moon completes one conjunction cycle (a synodic month). Twelve such cycles total 354 and 348/940 days - constituting a lunar year.

The author's calculations show: the moon lags 12 degrees and 346 parts behind the sun each day. After 29 days and 499 parts, this accumulates to 365 degrees and 235 parts - completing a full cycle where moon and sun realign, forming a complete month. Twelve such cycles compose one year.

This passage explains the ancient calendrical methods for calculating intercalary months (added to reconcile lunar and solar calendars) and the patterns of solar and lunar movements.

Ancient astronomers observed that the sun moves faster than the moon, accumulating about 10 days and 827 fen (an ancient time unit) annually. They adopted the "one intercalary month every three years, two every five years" method for balance: three years accumulate approximately 32 days and 601 fen (rounded to 36 days), while five years accumulate about 54 days and 275 fen (rounded to 60 days). After 29 years of this cycle, the solar terms and new moon days (the first day of each lunar month) realign, ensuring accurate correspondence between celestial phenomena and human affairs.

Regarding lunar phases: if the crescent moon becomes visible on the second day, it indicates the previous month was a long month (30 days); visibility on the third day means it was a short month (29 days). At dusk on the first day, the moon sets immediately after sunset. From the second day onward, the distance between sun and moon gradually increases. At the full moon on the fifteenth day, they stand opposite each other across the sky. Between the sixteenth day and month's end, their movement discrepancy reaches maximum before converging again.

Shao Yong (a Northern Song Dynasty Neo-Confucian scholar) summarized: annual differences between six yin months and six yang months accumulate to 36 days over three years (requiring one intercalary month) and 60 days over five years (requiring two). Though not perfectly precise, these calculations sufficiently guided agricultural activities and daily life.

(Note: "Fen" was an ancient time unit where 1 day = 940 fen; "mystical bird" refers to swallows, using migratory patterns to illustrate phenological cycles; "last day" denotes month-end, while "new moon day" means the first day.)

This passage explains the ancient method of intercalary month calculation in traditional calendars. The ancients observed that the solar and lunar cycles were not perfectly synchronized—the sun moves approximately 1 degree per day across the sky, while the moon moves 13 and 7/19 degrees daily. This results in the moon gaining 7/19 degrees annually over the solar cycle.

To reconcile this discrepancy, they developed the "7 intercalary months in 19 years" principle. The calculation method works as follows: taking 19 years as one cycle (called a "zhang"), seven intercalary months are inserted within this period. This totals 235 months (19 years × 12 months + 7 intercalary months).

A more refined calculation method involves multiplying 12 (months per year) by 7 (intercalary months) to get 84, then by 19 to obtain 228. This creates proportional relationships where 84 corresponds to 7 parts and 228 to 19 parts, allowing precise determination of intercalary month placement.

For example, dividing a day into 12 double-hours (shichen) using this ratio shows that 19 years yield 132 shichen (equivalent to 11 days) with a remainder of 12 fen. Calculated over the full 19-year cycle, this accumulates to 47,880 fen, resulting in an annual surplus of 11 days and 12 fen. Over 19 years, this sums to 209 days and 228 fen surplus.

Two calculation approaches emerged: the coarser method uses the 7:19 ratio, while the refined version employs 84:228 proportions. Through these mathematical models, ancient astronomers could accurately predict intercalary month insertions, maintaining synchronization between the lunar calendar and solar motion.

This passage explains the astronomical principles and calculation methods behind the intercalary month system in ancient calendars. The core concept is that while the sun's movement appears slow (迟) it actually progresses faster (进), whereas the moon's movement seems swift (疾) but in reality lags behind (退). This differential speed necessitates intercalary months to harmonize the solar and lunar calendars.

Specifically, the sun outpaces the moon by approximately six days annually (the solar year being six days longer than twelve lunar months). Through the calculation method of "adding half a day to solar reckoning while subtracting half a day from lunar reckoning," this discrepancy accumulates as "intercalary surplus." Every three years, the difference amounts to roughly one month, hence the need for an intercalary month ("three years, one intercalation"). Over a nineteen-year cycle, seven intercalary months accumulate ("nineteen years, seven intercalations"), forming a complete period called a "zhang" (章).

Key numerical relationships emerge: 7 multiplied by 19 yields 133 (the total months in nineteen years with seven intercalations), while 228 represents the fundamental cycle number for solar-lunar synchronization. These figures scale up to form larger calendrical cycles, such as the 45.6 million-day "grand cycle" (一元之数), all derived from actual astronomical observations of solar and lunar movements.

Zhu Xi (Master Zhu) further clarifies that intercalary months should be determined by the "middle qi" (中气) – those solar terms occupying even positions in the twenty-four-term system. This demonstrates the ancient scientific method of using solar positional observations to establish intercalations.

The text reveals how ancient astronomers employed precise celestial observations and mathematical computations to ingeniously place intercalary months within the lunisolar calendar, ensuring lunar phases aligned with seasonal cycles. The statement "the yang follows the yin as its rhythm, yin and yang serving as each other's substance and function" embodies the Daoist cosmological view of "mutual nourishment between yin and yang."

This passage explains the principles of solar terms, intercalary months, and celestial motion in ancient Chinese calendrical astronomy.

Regarding intercalary months: When a lunar month lacks a "Zhongqi" (the even-numbered solar terms in the 24-term system, such as Rain Water or Spring Equinox), that month is designated as an intercalary month. The Big Dipper's handle points to specific directions corresponding to particular Zhongqi. When the sun and moon pass between two celestial positions indicated by the Dipper's handle without encountering a Zhongqi, an intercalary month must be inserted. The Dipper's handle rotates leftward while the sun and moon move rightward, with timekeeping and positional markers always coordinated.

On celestial division: Ancient astronomers divided the celestial sphere into twelve palaces, each with precise measurements in degrees, minutes and seconds (100 seconds make a minute, 100 minutes make a degree). The solar year of 365.25 days corresponds exactly to the celestial circumference; a lunar month of approximately 30.42 days matches one palace's span. Here "degrees" function as boundaries marking celestial motion. The sun moves 1 degree daily, requiring over 365 days to complete its circuit; the moon moves about 12+ degrees daily (astrologers rounded to 13 degrees), completing its cycle monthly. Since lunar cycles don't align with whole days, months alternate between 29 and 30 days - two cycles complete in 59 days, averaging about 29.5 days per cycle.

Scholar Wang Yan (Song dynasty) observed: Calendrical experts throughout history essentially calculated the boundaries of Yin-Yang transformation. Thus Confucius avoided specific calendrical discussions, simply advocating "follow the Xia calendar."

Celestial mechanics can be understood through a four-wheel model: The outermost wheel represents heaven's slow leftward rotation (encompassing earth), while inner wheels bearing the sun and moon also rotate leftward at differing speeds. Because outer wheels move faster than inner ones, the sun and moon appear to move rightward. Specifically: The great wheel symbolizes heaven's expansive, measured motion; smaller inner wheels carrying the sun and moon have more limited rotation. The moon lags behind the sun by about 0.25 degrees daily, so the sun trails heaven by 1 degree daily while the moon trails the sun by about 12+ degrees. The outer wheel's greater circumference naturally produces faster apparent motion than the narrower inner wheels.

This text reflects ancient Chinese wisdom in developing calendars through Dipper and celestial observations, using mechanical wheel models to explain complex astronomical phenomena. The concept of "heaven encompassing earth" embodies the traditional "Canopy Heaven" (Gaitian) cosmology.

This passage primarily discusses the calculation methods of ancient astronomical calendars, particularly focusing on the principles behind determining the cycles of solar and lunar movements and the intercalation of leap months.

It mentions that a friend named Wu Jiuxia created an astronomical diagram using four concentric circles to represent celestial motion: The first circle marks the degrees of the celestial sphere (the full division of the heavens); the second shows that the heavens complete a full rotation starting from the Jiao constellation (one of the Twenty-Eight Mansions) each day, yet still exceed their original position by one degree; the third indicates that the sun also begins its daily journey from Jiao but does not surpass its starting point, meaning it lags behind the heavens by one degree each day; the fourth circle demonstrates that the moon moves even more slowly, falling behind the sun by roughly twelve degrees daily.

Regarding leap month calculations, the text explains that the moon lags behind the sun by about 365 degrees each month, completing a synodic month in 29.5 days. Over twelve months, this accumulates to roughly five days of discrepancy, and when combined with seasonal variations, an annual surplus of about ten days arises. These differences are reconciled through the insertion of leap months. The passage references Shao Yong (Kangjie), who proposed two methods of intercalation: "hidden leap" (藏闰) and "manifest leap" (显闰). The hidden leap disperses the extra six days annually within the sexagenary cycle, while the manifest leap explicitly marks twelve leap days from the "opening of things" (开物) to the "closing of things" (闭物), referring to a specific temporal framework.

Finally, it cites Zhu Xi's evaluation of historical calendars: Sima Qian’s Records of the Calendar documented the Taichu Calendar and the Zhuanxu Quarter-Day Calendar, which Liu Xin later revised into the Triple Concordance Calendar. The Great Expansion Calendar (大衍历) by the Tang dynasty monk Yixing was the most precise. The calendars of the Five Dynasties period, though concise, were prone to errors. Only Shao Yong’s system, using a large base unit of 129,600 parts, achieved the utmost accuracy and rigor. The text concludes by noting that the Quarter-Day Calendar originated from Emperor Zhuanxu’s deductions based on the Lianshan Yi (an ancient version of the Yijing) and the numerical principles of the River Chart (河图).

This passage discusses the evolution and characteristics of several important calendars in ancient China. It first mentions that the Qin dynasty adopted the Sifen calendar (a system dividing the year into 365¼ days), with the tenth lunar month as the year's beginning. During Emperor Wu of Han's reign, astronomers like Tang Du and Luoxia Hong established the Taichu calendar, shifting the year's start to the first month (Yin month) and changing the era name from Yuanfeng 7 to Taichu 1. This year corresponded to the sexagenary cycle's Jiayin year (referred to as "Yanfeng Shetige" in ancient celestial stem-branch terminology).

The text clarifies that Emperor Wu's calendar reform aimed to align with the Jiayin starting point of the ancient Zhuanxu calendar, seeking harmony with the auspicious celestial phenomenon called "hebi lianzhu" (when the sun, moon, and five planets align). However, the actual year wasn't a true Jiayin year. Later, Liu Xin revised the Taichu calendar to create the Santong calendar, dividing time into three systems (heaven, earth, and humanity) corresponding to the musical pitches Huangzhong, Linzhong, and Taicu, using the sixty-year sexagenary cycle.

Yang Xiong's Taixuan calendar from the Han dynasty resonated with both the Taichu and Zhuanxu systems. While the Taichu calendar used 81 (9×9) as its day-counting base, the Taixuan adopted 72 (8×9). For smaller time units, Taichu employed 32 (8×4) seconds, whereas Taixuan used 36 (9×4). Despite different calculation methods, both ultimately arrived at 2,592 seconds, demonstrating diverse paths reaching the same destination.

Finally, the passage describes the Dayan calendar compiled by the Tang dynasty monk Yixing. Unlike previous music-based systems, this calendar drew from the Yijing's (Book of Changes) "hexagram qi" theory and Hetu numerology. It used combinations of the Hetu's directional numbers to symbolize seasons, took the sum of its central numbers (5+10) as the base, and correlated 9, 6, 7, and 8 with hexagram line transformations. Through the sexagenary cycle and 420 strategic calculations, it established an immutable cosmic operating principle for later generations.

This passage primarily discusses the evolution and characteristics of ancient Chinese calendrical systems, particularly comparing the differences between the Pre-Heaven (先天) and Post-Heaven (后天) calendars.

The Xiande Tongtian Calendar (显德通天历), created by Wang Pu during the Five Dynasties period and recorded by Ouyang Xiu in the "Sitian Kao" (司天考), forms its own systematic approach. It employs 36 yang counting rods and 24 yin counting rods, with their combination producing the foundational number 72. Through progressive layering calculations: 72 layers yield 720, another layer produces 720,000, and further layering reaches 72 million, thereby generating the Yuanji (元纪, calendrical cycles). When all temporal units (years, months, days, and hours) return to the starting point of the Jiazi (甲子) cycle, and when the sun, moon, and five planets all align at the Zi (子) position, perfect coordination of the "Seven Administrators" (七政, referring to the sun, moon, and five visible planets) is achieved.

The author considers the Han Dynasty's Taichu Calendar (太初历) and the Tang Dynasty's Dayan Calendar (大衍历) as the most outstanding calendrical systems in Chinese history, with other calendars essentially being adjustments and improvements based on these two. The six pitch-pipes (六律) use Luo Shu (洛书) numbers beginning with odd numerals, while the eight trigrams (八卦) employ He Tu (河图) numbers starting with even numerals. Though their numerical derivation methods differ, both belong to the Post-Heaven Yi (后天易) system referring to the Later Heaven arrangement of trigrams in the Zhouyi. At that time, the Pre-Heaven Diagram (先天图, Fu Xi's primordial trigram arrangement) had not yet been discovered.

Shao Yong (邵雍, styled Kangjie) created the Pre-Heaven Calendar in his Huangji Jingshi (皇极经世), based on the multiplicative combination of heavenly stems and earthly branches: with day-stem as Jia (甲), month-branch as Zi (子), star-stem as Jia (甲), and hour-branch as Zi (子), collectively termed the "Four Images" (四象). The grand cycle was called "Yuan-Hui-Yun-Shi" (元会运世), while smaller cycles were termed "Sui-Yue-Ri-Chen" (岁月日辰). The specific calculations were: day-Jia as one unit, month-Zi as twelve, star-Jia as 360 units, and hour-Zi as 4,320. Thirty-six Jiazi cycles were evenly distributed among the twelve double-hours, with each double-hour corresponding to 360 numerical values.

The most significant difference between Pre-Heaven and Post-Heaven calendars lies in the exclusion of intercalary months. The Pre-Heaven calendar divides one year into twelve Hui (会), 360 Yun (运), and 4,320 Shi (世), with the complete cycle spanning 129,600 years. Zhu Xi particularly esteemed Shao Yong's calendrical system. Cheng Hao also commented that Shao's "difference method" (差法, for calculating celestial motion discrepancies) surpassed all ancient and contemporary methods. By observing the waxing and waning of yang and yin (阳盈阴亏) during solar-lunar conjunctions (moon phases) to correct errors, his method achieved remarkable precision, though being prone to cumulative deviations.

(Note: The "He Tu and Luo Shu" (河图洛书) refer to legendary mystical diagrams considered foundational to the numerical principles of the Yijing; "Yuan-Hui-Yun-Shi" represents Shao Yong's cosmic cycle theory using four temporal units to describe grand universal cycles.)