# The Annual Difference: How a Four-Century Debate between Confucian, Daoist, and Buddhist Thinkers on a Problem of Classical Exegesis Shaped a Predictive Mathematical Construct

## Abstract

**:**

## 1. Introduction

## 2. Scripture, Science, and the Measure of the Invisible

Thus it is that in the Annals there are solar eclipses on consecutive months and that there are long years without any at all: there is no single principle, and holding to constant values in computation is why there is no astronomy (li) that does not slip ahead or behind. 故春秋日有頻月而蝕者，有曠年不蝕者，理不得一，而算守恒數，故曆無不有先後也.3

When the day is [of] medium [length] the star is Bird, and this is used to determine mid-spring. … When the day is long[est] the star is Fire, and this is used to fix mid-summer. … When the night is [of] medium [length] the star is Barrens, and this is used to determine mid-autumn. … When the day is short[est] the star is Mane, and this is used to fix mid-winter. 日中星鳥，以殷仲春⋯⋯日永星火，以正仲夏⋯⋯宵中星虛，以殷仲秋⋯⋯日短星昴，以正仲冬.7.

In the late autumn month, the sun is in Chamber; Barrens is centered at dusk, and Willow is centered at dawn. 季秋之月，日在房，昏虛中，旦柳中.8

## 3. Men of the Southern Sky

Barrens is the cent[ral] star of the Dark Warrior, which is also to say that [its] seven stars (lodges) are all visible on the date of the autumn equinox. 虛，玄武之中星，亦言七星皆以秋分日見.

Yu Xi said, “On winter solstice in the time of Yao, ‘The day is at its shortest, and the star is Mane’. Today, more than 2700 years [later], Eastern Wall is centered, so we know that this was brought about by a gradual difference every year”. 虞喜云：「堯時冬至『日短星昴』，今二千七百餘年，乃東壁中，則知每歲漸差之所至」.13

The “Yaodian” states that “The day is at its longest, and the star is Fire;—you may thus exactly determine the mid-summer”. Today, Fire (i.e., Heart) is centered in the late summer month. [It] further [states] that “The night is of medium length, and the star is Barrens;—you may thus exactly determine the mid-autumn”. Today, Barrens is centered in the late autumn month. Investigating this through centered stars more than 2700 years hence, [one finds that] the difference (cha) is 27 to 28 du, so on the winter solstice decreed by Yao, the sun was around 10 du in[to] Maid. 堯典云「日永星火，以正仲夏」。今季夏則火中。又「宵中星虛，以殷仲秋」。今季秋則虛中。爾來二千七百餘年，以中星檢之，所差二十七八度。則堯令冬至，日在須女十度左右也.18

_{B01}子 in the middle of the northern quadrant (Figure 2), is the natural start point of the winter solstice at high origin; (2) that jiazi.

_{01}甲子 is the natural start point in the sexagenary cycle for both the year and day at high origin; and (3) that, as per tradition, all astronomical cycles must start at zero at high origin.24 In addition to the symbolic perfection of these choices, Zu had also developed a method for interpolating the exact moment of the solstice from multiple shadow measures, which led to a massive improvement in contemporary data. It is worth emphasizing, however, that the alignment of the burgeoning number of astronomical cycles to a single high origin required compromise, and as the most invisible and experimental of them all, his value for the “annual difference” was likely stretched to fit to his choice of origin (Qu 2008, pp. 175–77, 182).

## 4. Isolation, Conquest, and the New Underdogs

## 5. New Stakes, New Positions

It was mostly greatly looser than [Li] Chunfeng (below). Still, the two were at odds (as concerns predictions), and Chunfeng was unable to exceed the hits that it got. 其大最疎於淳風。然更相出入，其有所中，淳風亦不能逾之.36

“When the day is short[est] the star is Mane, and this is used to fix mid-winter”. The seven lodges are all visible, but only the central lodge is evoked. If the central lodge is evoked, then the remaining stars can be known. 「日短星昴，以正仲冬。」七宿畢見，舉中宿言耳。舉中宿，則餘星可知.38

_{01}on 19 December 640. Li had developed “new procedures” 新術 that happened to nudge the new moon past midnight, onto the more auspicious day, and they won the unconditional support of both Directorate staff and Kong Yingda, then chancellor of the National University and an influential voice on the Rites and imperial ceremonial. The very next year, Li was appointed as an erudite at the Ministry of Rites and, shortly after, saw himself promoted to assistant director of the Astronomical Directorate. The third petition, in 644, involved the symbolic “unfixing” of the civil calendar to the true, astronomical moon to avoid four “big” 30-day months in a row. As it happened, by the time that the throne grew tired of amending the Wuyin yuan li in 664, Li was the director of the Astronomical Directorate, he was ready in waiting with his own Linde li 麟德曆, and this too was immediately adopted without trial or debate.40

If Mane was centered at the winter solstice, then neither the Stellar Fire nor the Stellar Barrens would yet be due west at the summer solstice and autumn equinox. And if Fire was centered at the summer solstice, and Barrens centered at the autumn equinox, then Mane would already be due east at the winter solstice. There is [some] expansion and contraction between these, and [thus] this is insufficient proof of the annual difference. 若冬至昴中，則夏至秋分星火、星虛，皆在未正之西。若以夏至火中，秋分虛中，則冬至昴在巳正之東。互有盈縮，不足以為歲差證.44

Mr. Wang’s explanation (that the “centered stars” are those of the following month) violates the flow of the text. The way that Mr. Kong [Anguo] directly gets “they (the lodges of the appropriate quadrant) are all visible” is slightly tortuous, [but] it is superior to Wang [Su] and Ma [Rong and Zheng Xuan] in terms of its reasoning (i.e., that these are modern-style “centered stars”). 王氏之說非文勢也。孔氏直取「畢見」，稍為迂闊，比諸王、馬，於理最優.51

It is calculated (ji 計) that in mid-spring the sun is in Straddler–Harvester, and it sets at you._{B10}(18:00, due west) on the horizon. At the time that dusk begins, Well–Ghost is at wu._{B07}(12:00, the meridian); Willow–Stars–Spread is at si._{B06}(10:00, two hours from culmination); and Axletree–Wings is at chen._{B05}(08:00, four hours from culmination). This is to say that the seven lodges of the Vermillion Bird can all be seen… 計仲春日在奎、婁而入於酉地，則初昏之時井、鬼在午，柳、星、張在巳，軫、翼在辰，是朱鳥七宿皆得見也⋯⋯

It is calculated that in mid-summer the sun is in Eastern Well, and it sets at you._{B10}on the horizon. At the time that dusk begins, Horn–Gullet is at wu._{B07}; Base–Chamber–Heart is at si._{B06}; and Tail–Basket is at chen._{B05}. This is to say that the seven lodges of the east are all visible… 計仲夏日在東井而入於酉地，即初昏之時角、亢在午，氐、房、心在巳，尾、箕在辰，是東方七宿皆得見也⋯⋯

It is calculated that in mid-autumn the sun is in Horn–Gullet, and it sets at you._{B10}on the horizon. At the time that dusk begins, Dipper–Ox is at wu._{B07}; Maid–Barrens–Rooftop is at si._{B06}; Hall–Wall is at chen._{B05}. Speaking of this by raising the cent[ral] star Barrens is also to say that all seven stars (lodges) are visible together at the hour of dusk on the day of the autumn equinox. 計仲秋日在角、亢而入於酉地，初昏之時斗、牛在午，女、虛、危在巳，室、壁在辰，舉虛中星言之，亦言七星皆以秋分之日昏時並見.52

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SKQS | Siku quanshu 四庫全書. Rpt. of Wenyuange Siku quanshu 文淵閣四庫全書. 1500 vols. Taibei: Taiwan Shangwu yinshuguan, 1983. |

SSJ | Chongkan Song ben Shisanjing zhushu fu jiaokan ji 重刊宋本十三經注疏附校勘記. Ruan Yuan 阮元, ed. Taibei: Yiwen yinshuguan, 1965. |

T | Taishō shinshū daizō kyō 大正新脩大藏經. 100 vols. Takakusu Junjirō 高楠順次郎 and Watanabe Kaikyoku 渡邊海旭 et al., eds. Tōkyō: Taishō issaikyō kankō-kai, 1924–1934. |

## Notes

1 | |

2 | I thank Reader 1 and Guo Jinsong 郭津嵩 for both bringing Wang Yutong’s 王雨桐 work to my attention during the review process. |

3 | |

4 | Shiji 史記 (Zhonghua shuju edn.), j. 27, p. 1349. |

5 | |

6 | See for example their use in the evaluation of the Shoushi li 授時曆 of 1280 (Sivin 2009, pp. 311–66) and in the study of the earth’s rotation (Stephenson 1997, pp. 221–27). |

7 | Shangshu zhushu 尚書注疏 (SSJ), j. 2, pp. 21a–b |

8 | Liji zhushu 禮記注疏 (SSJ), j. 17, p. 337a. |

9 | |

10 | Xin Tangshu, j. 27A, p. 600. |

11 | These four commentaries are quoted and summarized in Shangshu zhushu, j. 2, pp. 21b, 24b. |

12 | |

13 | Songshi 宋史 (Zhonghua shuju edn.), j. 74, p. 1689. |

14 | Xushu zhengji 續述征記, cited in Taiping yulan 太平御覽 (Sibu congkan edn), j. 80, pp. 8a–b. |

15 | Liji zhushu, j. 17, p. 344b. |

16 | For its classification, see Suishu 隋書 (Zhonghua shuju edn.), j. 34, p. 999. Fragments of the lost text are collected in Quan shanggu sandai Qin Han Sanguo Liuchao wen 全上古三代秦漢三國六朝文 (Hebei jiaoyu chubanshe, 1997), vol. 5, j. 82, pp. 853–54. I thank Johan Rols for directing me to his dissertation (Rols 2021) and helping me better understand the nature and contents of this work in a personal communication 15–17 June 2023. |

17 | See the Yus’ biographies in Jinshu, j. 82, pp. 2143–2147; j. 91, pp. 2348–49, Sanguo zhi 三國志 (Zhonghua shuju edn.), j. 57, pp. 1317–27, and Martin and Chaussende (2020, pp. 601–2). On their scientific work, see M. Chen (2003, pp. 237–38). |

18 | Songshu 宋書 (Zhonghua shuju edn.), j. 12, p. 261. |

19 | |

20 | He Chengtian’s solar table of 443 has the centered star at Greater Heat (: late summer) as “Tail $8\frac{8}{12}$” 尾八太弱 and that at Frost Settles (: late autumn) at “Maid $11\frac{5}{12}$” 女十一半弱 (Songshu, j. 13, p. 280). Compare this to “In the late summer month, the sun is in Willow, and Fire is centered at dusk”季夏之月，日在柳，昏火中 (Liji zhushu, j. 16, p. 318b) and “In the late autumn month, the sun is in Chamber, and Barrens is centered at dusk” (above). |

21 | |

22 | Songshu, j. 13, pp. 304–17, Nanshi 南史 (Zhonghua shuju edn.), j. 72, p. 1774, and Suishu, j. 17, pp. 416–17. |

23 | |

24 | Songshu, j. 13, pp. 289–90. |

25 | Nanshi, j. 72, p. 1774. |

26 | Suishu, j. 17, pp. 420–21. From Ye, Zhang Bin’s team included the academician and failed astronomical reformer Zheng Yuanwei 鄭元偉 (fl. 576–581). The famous Northern Qi scholar Li Delin 李德林 (532–592) is also attributed with a Kaihuang li (Jiu Tangshu 舊唐書 [Zhonghua shuju edn.], j. 47, p. 2038), suggesting that he too was part of the picture. The southerner Yu Jicai remained the acting director of the Astronomical Directorate at this time, and while he and his clique did not participate in the authorship of the Kaihuang li, they were later convicted for failing to “investigate its accuracy” 審疎密 and “vainly operating this astronomy” 虛行此曆 [upon imperial order] despite their expertise and position (Suishu, j. 17, pp. 434–35). |

27 | Suishu, j. 17, pp. 421–22, and Kaiyuan zhanjing 開元占經 (SKQS), j. 105, pp. 12a–14a. |

28 | Qian (1998), pp. 334–37. For a recent study of the Wujing suanshu and math problems in Confucian commentary, see Chemla and Zhu (2022). |

29 | Kaiyuan zhanjing, j. 105, pp. 11a–b. |

30 | |

31 | Fozu tongji 佛祖統紀 (T no. 2035), j. 42, p. 386b, and Fozu lidai tongzai 佛祖歷代通載 (T no. 2036), j. 9, p. 554c. |

32 | See for example Xu Gaoseng zhuan 續高僧傳 (T no. 2060), j. 16, p. 559, j. 19, p. 584, j. 20, p. 590, j. 23, p. 626, j. 28, p. 686, and Sandong qunxian lu 三洞群仙錄 (Zhengtong Daozang Hanfenlou 正統道藏 涵芬樓 edn.), j. 13, p.8a. |

33 | Suishu, j. 17, pp. 435. |

34 | Mengxi bitan jiaozheng 夢溪筆談校證 (Shanghai chuban gongsi, 1956), j. 7, p. 294. |

35 | Xin Tangshu, j. 25, p. 536. |

36 | Xin Tangshu, j. 25, p. 536. |

37 | These are lunar eclipse 06316 and solar eclipses 06240 and 06241 in Espenak and Meeus (2007) and Espenak and Meeus (2009). |

38 | Xin Tangshu, j. 25, p. 534. Compare Zu and Wang’s language to the earlier quotation from Kong Anguo above and “Fire is the central asterism (zhongxing) of the Cerulean Dragon, and we know by evoking its center that its seven stars (lodges) are visible” 火，蒼龍之中星，舉中則七星見可知 (Shangshu zhushu, j. 2, p. 21b; cf. Wang (2022, p. 249). |

39 | Xin Tangshu, j. 25, pp. 534–36. |

40 | |

41 | |

42 | This point ties into the position on human progress developed in Li Chunfeng’s historical writings, studied in Morgan (2017, pp. 177–12). |

43 | I have yet to reconstruct the calculations necessary to give an exact number of matches between the “Yueling” and Li Chunfeng’s Linde li. However, a comparison of its 36 monthly data points with those for the medial qi in He Chengtian’s solar table of 443 shows 16 matches, 17 mismatches, and 3 additional points that one can consider matches despite the fact that they use the defunct lodges Establishment Star (Jianxing 建星, in Dipper) and Bow (Hu 弧, in Well). On solar tables and the procedures by which they were replaced, see Zhang et al. (2008, pp. 27–49). |

44 | |

45 | Jiu Tangshu, j. 25, p. 534. |

46 | Jiu Tangshu, j. 79, pp. 2714–17; cf. Fozu lidai tongzai (T no. 2036), j. 11, pp. 564a–68b. |

47 | On the mathematical canon and its role in later education, see (Keller and Volkov 2014, esp. pp. 59–64). |

48 | Morgan (2019, p. 151; 2017, pp. 188–96). Note that Wang (2022, p. 257n3) disagrees with this reading, affirming that Li Chunfeng was consistent on the armillary sphere’s sage-time origins. |

49 | Shangshu zhushu, preface, p. 4a. |

50 | See the works cited in Note 56. |

51 | Shangshu zhushu, j. 2, p. 24b. |

52 | Shangshu zhushu, j. 2, p. 24a–25a. |

53 | Gautama Siddhārtha was named after the Buddha, he translated an astronomical work from Sanskrit, we know that his ancestors Prajñāruci and Dharmajñāna learned, taught, and translated Buddhist texts, and Wang Pu 王朴 (d. 959) erroneously identifies Gautama Siddhārtha as “a foreign monk from India” 天竺番僧 (Sen 1995). It thus seems safe to assume that their contemporaries would have associated the Gautamas with Buddhism in some sense. |

54 | |

55 | Shangshu tongkao 尚書通考 (SKQS), j. 1, pp. 44a–54b. |

56 | Shu jizhuan huowen 書集傳或問 (SKQS), j. 1, p. 12b. A similar statement is to be found in Lin Zhiqi’s 林之奇 (1112–1176) commentary in Shangshu quanjie 尚書全解 (Zhonghua shuju edn.), j. 1, pp. 23a–b. |

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**Figure 2.**(

**a**) The 28 lodges and 24 solar terms. (

**b**) “Lodge entry-du” (ruxiu du 入宿度). (

**c**) A “centered star” (zhongxing 中星). The du 度is a linear measure of the circumference of an astronomical great circle defined as the daily displacement of the mean sun, meaning that there are as many du in a circle as there are days in the sidereal year and, roughly speaking, $365\frac{1}{4}\phantom{\rule{0.222222em}{0ex}}du={360}^{\circ}$, and $1\phantom{\rule{0.222222em}{0ex}}du=0.{99}^{\circ}$. The specific lodge widths and solar term association are as per He Chengtian 何承天 (370–447).

**Figure 3.**Map. Note that peoples’ places of origin are given throughout at the level of the commandery (jun 郡) or prefecture (zhou 州).

**Figure 4.**Distribution of solar positions and centered stars according to He Chengtian’s solar table for 443 (

**A**) compared to the “Yaodian”, as per Liu Chunfeng’s analysis (

**B**). As such, either the centered stars for the autumn equinox (AE) and summer solstice (SS) work, and Mane is too far east, or the winter solstice (WS) works and Barrens and Fire are too far west. Diagram based on Qu (2008, p. 188, Figures 3–14).

**Figure 5.**Illustration of Kong Yingda’s description of the position of the sun and “centered stars” in the “Yaodian”.

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Morgan, D.P.
The Annual Difference: How a Four-Century Debate between Confucian, Daoist, and Buddhist Thinkers on a Problem of Classical Exegesis Shaped a Predictive Mathematical Construct. *Religions* **2024**, *15*, 70.
https://doi.org/10.3390/rel15010070

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Morgan DP.
The Annual Difference: How a Four-Century Debate between Confucian, Daoist, and Buddhist Thinkers on a Problem of Classical Exegesis Shaped a Predictive Mathematical Construct. *Religions*. 2024; 15(1):70.
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2024. "The Annual Difference: How a Four-Century Debate between Confucian, Daoist, and Buddhist Thinkers on a Problem of Classical Exegesis Shaped a Predictive Mathematical Construct" *Religions* 15, no. 1: 70.
https://doi.org/10.3390/rel15010070