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Science and Calendars in China and the West

From Clavius to Xu Guangqi and Schall

Peter H. Richter

Bremen, April 25, 2008

Abstract

The design of calendars is fundamentally different between China and the West.Chinese calendars stress the correctness of prediction and therefore rely on obser-vational precision. Western calendars, on the other hand, emphasize the ease ofcomputation and are not worried about discrepancies with astronomical reality.The article discusses how the Jesuits who worked out the Chinese calendar reformof 1634/45 may have perceived this cultural difference. It points to the role of XuGuangqi as sensitive supporter of mostly young Jesuits who left Europe becauseat the time they did not see a future there as scientists.

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1 Introduction

This article is an attempt to come to an understanding of a fascinating pe-riod of early scientific interaction between East and West, 400 years ago inChina. It must be pointed out at the very start that I am neither historiannor sinologist; as theoretical physicist with interest in philosophy I have someknowledge of astronomy and the history of science, but I cannot point to anyresearch of mine in these fields. Therefore, to experts in the field it may seemrather jaunty on my part to contribute an article on a subject which has beendiscussed zillions of times in the literature: the Jesuits’ role as mediators ofcultural exchange in the 17th century.

I came to this topic by way of planetarium presentations that I used to givein Bremen, when I decided to discuss the various calendars of the world. Iwas thrilled to learn that the two most widespread calendars, the Gregorianand the traditional Chinese, had been shaped by German Jesuits, Christo-pher Clavius of Bamberg, and Adam Schall von Bell of Cologne, respectively.Dwelling deeper into the matter I found that the two must even have met eachother when Clavius, age 74, organized that memorable reception for Galileiat the Collegio Romano, 1611, and Schall, age 19, was a student there. Ilearnt that Matteo Ricci, the first Jesuit in China, had been a student ofClavius, and that the publication of his diaries in Europe by Nicolas Trigaultin 1615 attracted an impressive crew of jesuitic scientists to China as of 1618.My sources were mostly jesuitic literature, some of them rather lyric. Littledid they talk of Xu Guangqi, except for the fact that he could be claimed asthe first Chinese Christian, Dr. Paul.

I learnt of Xu Guangqi from Andreas Dress during my first visit to Shanghaiin March 2007. And suddenly the story appeared in different colors. Therewas this high ranking Chinese scholar, expert in many fields (philosophy,mathematics, artillery, agriculture) who recognized the potential benefit forChina that might be gained from Ricci’s knowledge of European mathemat-ics, mechanics, and astronomy. There was their joint project of translatingEuclid’s Elements into Chinese which Xu published in 1607, and their com-mon interest in matters of calendars. Perhaps most importantly, there wasXu’s role as political protector of the Jesuits when hostility broke out againstthem in Nanjing around 1617.

It became more and more obvious that the calendar reform cannot have beenthe major reason for the Jesuits to go to China even though the Emperorseems to have had a vested interest in it; after all, the matter was not that

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complicated . But then what? Looking back at the Europe of that time,it becomes obvious that young men would not perceive it as fertile groundfor science: Giordano Bruno’s auto-da-fe of 1600, Galilei’s admonition in1616, the fierce antagonism between catholics and protestants leading to theThirty Year’s War – all this taken together was rather abhorrent. China, incontrast, promised to provide asylum: a country where knowledge was thekey to recognition and success.

If this was the case, then their assignment to do missionary work would nothave been their major interest. Of course one cannot expect to support such aclaim by studying their correspondence with Rome which had to be obedient.But perhaps it is possible to find hints in the work that they produced andleft in China, the calendar work as an example. This article is a superficialattempt to argue in just that direction. It goes to some depth in comparingthe design principles of Western and Chinese calendars, and it stresses thatthe Jesuits did not transfer the Gregorian reform to China. Rather they fullyrespected the Chinese tradition and most probably appreciated it as beingmore rational than the various Western traditions (Babylonian, Egyptian,Hebrew, Roman, Julian, Gregorian, Orthodox) which are so heavily loadedwith religious demands. A stronger argument would require a thorough in-vestigation into the Chinese writings of the Jesuits. So far I did not succeedto find out how much of this has already been done by Sinologists. A prac-tical problem seems to be that the combination of ancient astronomical andsinological knowledge is rare.

The article has four parts. The first, Sec. 2, is a general review of the art ofcalendar making. Sec. 3 explains the Gregorian calendar with its roots andits design as a theoretical construct. Then, in Sec. 4, the Chinese counterpartis presented as an attempt to predict events as correctly as possible. Thehistory of the reforms from Clavius to Xu and Schall, and an evaluation fromtoday’s perspective, are given in Sec. 5.

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2 Science and the art of calendar design

2.1 The challenge of calendar making

The origin of science lies buried in the distant past, but undoubtedly it wasstimulated by the experience that life on earth is closely related to celestialphenomena. These phenomena range from the daily cycles of the Sun andthe starry sky, the yearly cycle of the seasons, the motion of the Moon andthe other “wandering stars”, i. e. the planets Mercuryyh (Shuixing), Venus�h (Jinxing), Mars Ûh (Huoxing), Jupiter ÷h (Muxing), Saturn Hh(Tuxing), but also less regular events like thunderstorms, rainbows, floods,or lunar and solar eclipses. Except for the daily and yearly cycles, it wasnot at all clear how these celestial phenomena influence our lives, and howthey can be predicted. Priest-scientists were assigned the task to find out.They did so by a combination of observation, generalization, calculation, andspeculation. One result of their studies, with immediate practical relevance,were calendars but they were embedded in more comprehensive schemes likeastronomy and astrology, science and religion.

Calendars are therefore an expression of knowledge on matters concerningthe interaction of heaven and earth. Calendar reforms, if worked out by ex-perts in the field, reflect progress of that knowledge. However, to the extentthat political interests interfere with their design they may occasionally losereliability or even become useless.

Of the celestial phenomena mentioned above, some are more predictable thanothers. The length of a day (depending on the season), the solar year, andlunar months present no major difficulties. The motion of planets is alreadymore complicated, but with sufficient effort still predictable. Whether or nottheir constellations have any observable effect on human life as claimed byastrology, has always been a matter of dispute. In ancient times, astrologywas treasured as a serious discipline, but in the end, astronomy and astrol-ogy parted: the former developing into a modern scientific discipline, thelatter into a speculative art of character and fortune telling. On the otherhand, there is no question that the phenomena of weather and climate arehighly important for human activities. Yet, their unpredictability beyondseason-related tendencies was recognized early on; so they are not includedin respectable calendars.

In the end, calendar makers concentrated on the motion of Sun and Moonamong the stars. The art of calendar design is to define rules which solve the

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following problems:

1. How to assign integer numbers of days to years if the period of the Sunis not a rational, let alone integer multiple of a day?

2. How to assign integer numbers of days to months if the period of theMoon is also an irrational multiple of a day, and even worse, not alwaysthe same?

3. How to assign integer numbers of months to years if the solar periodis not a rational multiple of the lunar period? In other words: how to“couple” the Moon to the Sun in spite of their unrelated paces?

Many different solutions were presented in the course of history, some of themtrivial in that they ignored either the Moon or the Sun: The old Egyptiancalendar only referred to the Sun and defined the year to be 365 days; it didnot care about the discrepancy of some 0.25 days which made the calendricalyear drift through the “true” astronomical year with the Sothis period of1460 years. In contrast, the Islamic calendar only refers to the Moon anddefines the year to be 12 lunations; it does not care about the discrepancyof some 11 days to the solar year which results in 33 true solar years beingcounted as 34 Islamic years. Notice that the difference of those two calendarsis not only the shift from purely solar to purely lunar character but also from“mathematical” to “astronomical”, or “theoretical” to “observational”: theEgyptian calendar defined the length of a year, the Islamic calendar insiststhat the beginning of a month (new Moon) be determined by observation (asa result of which the dateline is not fixed on Earth but changes location andshape from month to month).

In the following we concentrate on lunisolar calendars which take on the taskof accommodating lunar months and solar years by offering solutions to allthree problems mentioned above. The different roles of Sun ó (Yang) andMoonR (Yin) in such calendars derive from the facts that the powerful dailylight of the Sun determines the seasons whereas the Moon offers nightly lightfor festivities. Hence the calendars define a solar year to regulate economicactivities, and a lunar year to set the dates for ritual and festive events. Inaddition they formulate rules for the interlocking of the two kinds of years.

Careful observation of Sun and Moon also allows for the prediction of solarand lunar eclipses which therefore might be included in the to-do list of lu-nisolar calendar makers. However, because of the rather singular occurrenceof these events, and the difficulty to understand and predict them, they wereusually treated as specialist’s matter. Eclipses are not considered to be part

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of Western calendars, but it appears that in China the Emperor, as Heaven’sson �� (Tianzi), attached great importance to possessing and using theknowledge of foreseeing them. More about this in Sec. 4.3.

2.2 Basic decisions reflecting cultural traditions

As calendars are the result of early scientific activity, they reflect ancientcultural premises. These come to light when traditional eastern and westerncalendars are compared.

2.2.1 The nature of time

To begin with, the very notion of time seems to be different between Eastand West. Western calendars give years consecutive numbers, starting from acertain event: The AM (anno mundi) of the Jewish calendar counts the yearssince the creation of the world, assumed to have taken place in 3760 BC; theRoman AUC (ab urbe condita) starts with the founding of Rome 753 BC;the Christian AD (anno domini) is the year after Christ’s birth as assumedby Dionysius Exiguus in AD 525; the Islamic AH (anno hegirae) numbersIslamic years starting with Mohammed’s emigration to Medina (the Hejira)in AD 622. In all cases time is perceived as having a beginning but no end.The option to count backward with BC (before Christ) seems to be a rela-tively recent addition to the Christian calendar; it offers the possibility toconceive of a time scale which is open on both sides.

Such kind of sequential numbering of years is alien to old Chinese calendars.Their numbering starts anew with each new Emperor, and as a subsidiaryscheme they use cycles of names which are combinations of ten celestial stems�� (Tiangan) and twelve terrestial branches �| (Dizhi) , see fig. 1.1 Bothstem-cycle and branch-cycle proceed one step from year to year, starting with � (Jiazi) and ending with �2 (Guihai). In this way, the total numberof combinations amounts to 60, the least common multiplier of 10 and 12.After 60 years, the cycle starts all over again. In old days the hexagesimalscheme was also applied to months and days. In recent times, however, onlythe numbering of months within a year (from 1 to 12), and of days within amonth (from 1 to 29 or 30), has remained in use, and the 60-year cycles aregiven sequential numbers, starting with some year during the reign of the

1The interpretation of stem and branch names is a matter of astrology, not of calendardesign. The stem names refer to Yang and Yin varieties of the five Chinese elementswhereas the branches denote 2-hour periods of a day, which were later associated withcertain animals as indicated in the figure.

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legendary Emperor ±� (Huangdi), either at 2637 or 2697 BC, so that theNew Year of 1984 started the cycle number 78 or 79.

Figure 1: The cyles of 10 celestial stems (left) and 12 terrestial branches (right).From year to year, each cycle proceeds on step, the left one counterclockwise, theright one clockwise, as indicated by the arrows. The name �` (Jiazi shu, rat)denotes the years ... 1924, 1984, 2044, ...; in 2008 two branch-cycles have beencompleted, hence this year carries the name �` (Wuzi shu, rat).

2.2.2 Cosmology, science and religion

If calendars regulate matters between Heaven and Earth they must dependon a society’s view of the cosmos: who reigns the Heaven, and who representsit on Earth? In China, the Emperor was considered to be �� (Tianzi), theHeaven’s son; there was no splitting of secular and spiritual authority. Hecontrolled life on Earth and was held responsible for everything that came“from above”. To be successful in this double role, the Emperor needed (i) ascience of the heaven�©¦ (tianwen xue) as rational as possible to producereliable predictions, and (ii) a detailed system of rites to delegate responsi-bility, if something went wrong, to those who did not comply with the rules.

In the West, Heaven was considered to be the home of Gods, or of only oneeternal God – in many ways a mirror image of the ruling actors on Earth –,

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while Earth is governed by conflicting powers: secular and spiritual. Eversince Constantine I elevated the Christian doctrine to the rank of the Ro-man Empire’s religion (around AD 320) did Emperor and Church competefor dominance over people’s lives: the Emperor claiming their bodies (assoldiers, farmers, workers), the Church aiming at their souls (with promisesfor life after death). Heaven became the realm of Church, heavily loadedwith irrational patterns of thinking. If sciences, and astronomy in particu-lar, had reached an impressive level during antiquity, this fell into oblivionuntil, more than a thousand years later and mediated by Arabic scientists,European scholars rediscovered rational thinking and, starting from Aris-totle, Euclid, and Archimedes, paved the way for modern science. Muchof this was achieved in confrontation with the Church, but it is fair to saythat at least the order of the Jesuits, founded in 1540 to promote religionthrough education and discipline, sided with the new scientific developments.

Comparing East and West, it appears that in China the science of the Heavenbenefitted from the Emperor’s interest in accurate information, whereas inEurope nobody really cared because the conflicts between Emperors andPopes absorbed all attention. Several calendar reforms took place in Chinaafter the basic principles had been laid out in terms of the�Ôð»�(TaiChu Calendar) under Emperor GÉ� (Han Wudi of the Western Han dy-nasty) in 104 BC. Their aim was always to improve the agreement betweenprediction and observed motion of Sun and Moon. In Europe, on the otherhand, where science was considered business of the Church, nothing happenedbetween 325 and the 16th century. Not before the Renaissance revived inter-est in scientific matters was the discrepancy between calendar standards andastronomical facts perceived as an annoyance: it had increased to 10 dayswith regard to the Sun, and the Moon was wrong by 3 days.

2.2.3 Design principles

Lunisolar calendars Ró» (Yin Yang Li) define solar years µ (Sui), lunarmonths Û (Yue) and lunar years # (Nian): a solar year extends betweenequivalent points along the Sun’s path – from winter solstice Á� to wintersolsticeÁ� (Dongzhi to Dongzhi) as in China, or from vernal equinox�Ito vernal equinox �I (Chunfen to Chunfen) as in the West; a lunar month(also called a lunation) extends from one new moon to the next; a lunaryear comprises 12 or 13 lunar months, depending on the rules of calendricalinterlocking of solar and lunar motion.

But where do the dates of solstices, equinoxes, and new moons come from?

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Here we see the major divide between East and West. The Chinese Em-peror requires precision, hence he insists that the true astronomical dates betaken, even though their prediction may be difficult. The Western tradition,on the other hand, ranks manageability and computability higher, hence ra-tional approximations to long time averages are taken to define the lengthsor months and years, notwithstanding discrepancies between true and meanmotion. Chinese calendars may therefore be called astronomical calendarswhereas the Western are mathematical or arithmetical calendars [3].

A difference of lesser degree concerns the relative importance of Sun andMoon. If both are treated in an exact way like in China, they are equallyimportant. In the West, however, the Hebrew and Islamic calendars givethe Moon priority over the Sun whereas the Roman and Christian calendarshave it the other way round. As was already mentioned, the Islamic calendargoes to the extreme by virtually ignoring the Sun; in the Hebrew calendarthe Moon’s priority appears more subtly, as will be seen in the next section.The Roman and Christian calendars, in Egyptian tradition, seem to be puresolar calendars on first sight; however, the division of the year into 12 “civilmonths” is a remnant of trying to approximate the length of a year by aninteger number of lunations, and closer inspection of the Julian and Gregoriancalendars shows that indeed they define lunar years based on the Moon’smean motion. They need this to determine the date of Easter, the mostimportant Christian holiday, defined to be the first Sunday after the first fullmoon in spring.

3 The Gregorian calendar °°°°°°»»»of 1582

The Gregorian calendar°°» (Geligao Li) was made official by Pope Gre-gor XIII in 1582. With some delay in the protestant countries of Europe, itwas eventually generally accepted in the entire Western world (Great Britainadopted it in 1752) whereas Eastern Europe assumed a split attitude: thecivil authorities introduced it during or shortly after the first World Warwhereas the Orthodox Church has adhered to the old Julian calendar untiltoday. In China º¥Ì (Sun Yatsen) decreed a change from the traditionalto the Gregorian calendar in 1912, but not until the revolution of 1949 wasit accepted throughout the country, when the lunar part of the traditionalcalendar was reintroduced to determine the New Year c# (Xin Nian) andother festivities.

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3.1 Roots in Julian, Roman, Egyptian, Babylonian,Hebrew calendars

To understand the Gregorian calendar with all its fancy details, it is nec-essary to consider its various roots. The Julian calendar had been in effectsince AD 325 when the Council of Nicaea introduced a lunisolar calendarwhich combined the Roman calendar with a much older Babylonean ruleto couple Sun and Moon. To be precise, the solar part was adopted fromJulius Cesar’s definition of the year to have 365.25 days and to start withJanuary 1; this required an intercalary day every fourth year. The lunarpart appeared in two different forms, a superficial and a somewhat hiddenauthentic version. The superficial definition of a month also continued theRoman scheme and has remained unchanged ever since: it divides the yearinto the 12 months January, February, etc. as we know them. These monthsclearly derive from lunations but have been adjusted to agree with the simpleratio 12:1 of months per year; on the average they are about one day longerthan true lunar months, hence they are of no use for predicting new moons.Such prediction is needed, however, to determine the date of Easter. There-fore the Julian calendar also adopted the so-called Metonic rule, named afterthe Greek astronomer Meton but previously known in Babylon (and China):19 years and 235 lunations happen to be almost the same; they differ byonly two hours. Therefore the length of a lunar month was defined to bethe 19/235-th part of a year. This meant that 235 months had to be dis-tributed over 19 years, and this was done according to an old Babyloneansystem where 12 years have 12 months and 7 years 13 months. The orderin which this happens was also fixed: in a cycle of 19 years, the years withnumbers 2, 5, 7, 10, 13, 16, 18 are long years with 13 months, the othersshort with 12 months. Considering the dates of new moons (or full moons)from one short year to the next year, they come 11 days earlier becausethe solar year is about 11 days longer than 12 lunations. From a long yearto the next year, the dates advance by 19 days because 13 lunations areabout 19 days longer than one year. Now 7 advances by 19 days makes atotal of 133, while 12 recesses by 11 days makes a total of 132. Thus onemore day must be taken back to comply with the 19 year periodicity. Thisis done in the transition from the last year of the cycle (which is short) tothe first year of the next: here the dates come 12 days earlier, rather than 11.

With these rules it was possible by rather simple arithmetic to predict thedates of new or full moons for all times (with a period of 532 years). How-ever, these predictions could not be accurate if the numbers for the lengthsof year and month were not, and this was the problem of the Julian calendar.

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But before we address the Gregorian correction let us comment on the otherWestern calendars. The Roman calendar as established in 45 BC by JuliusCesar was a mild correction of the Egyptian calendar where the year by def-inition had 365 days, divided into 12 months of 30 days each, plus 5 days forend-of-year festivities. This was bad as an approximation to the solar year,and even worse with respect to the lunations. The Romans (actually withadvice from Egyptian astronomers) adjusted the length of a year from 365to 365.25, introducing the leap years, but adopted as practical the divisionof the year into precisely 12 months. This means they had to distribute 11extra days among those months, and this was done in various ways beforeCesar and Augustus fixed the system that was preserved in both Julian andGregorian calendars. It may be asked why the leap day was placed at theend of February; the answer is that an older Roman calendar had the yearbegin with the first of March, and the five days prior to that were reservedfor end-of-year celebrations as in Egypt. Hence the sixth day before the firstof March (calendae Martis) was doubled in a leap year. This is the reasonwhy the leap year is still called annee bissextile in French – it is the yearwith two sixth days before the first of March.

This shows how the Christian calendar is burdened with historical remnantswhich from a distance appear accidental or even ridiculous. The religiouscontext of Easter adds further queerness. According to the biblical account,Jesus Christ’s resurrection took place on the third day after the beginningof the Jewish Pessah which by definition of the Hebrew calendar falls on thefirst full moon in spring – whatever the weekday. Christians wanted to cele-brate Easter on a Sunday, so they converted “the third day after” into “theSunday after”. This would not have caused a problem had the Christian andthe Jewish calendars agreed on the first day of spring and its first full moon.Unfortunately this was not the case: both calendars are arithmetical ratherthan astronomical calendars, but their arithmetics are different. In contrastto the Julian calendar, the Hebrew calendar in Babylonian tradition givespriority to the Moon rather than the Sun. This means the basic period is theaverage length of lunations for which the reform of 360 AD [17] assumes 29.5days plus 793/18 minutes which amounts to the remarkably accurate valueof 29.530 594 days (only half a second longer than the astronomical value).The length of the year is derived from this number by way of the Metonicrule: it is defined to be 235/19 times the length of a month which amountsto 365.246 82 days, about midway between the Julian 365.25 and the correctastronomical value 365.242 19. This implies that the Julian Easter is “on thesafe side” of the Jewish Pessah: the Julian beginning of spring drifts towardsthe real summer about twice as fast as the Hebrew, and in addition the Ju-

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lian full moon drifts, somewhat slower, in the same direction. As a result theJulian first full moon in spring is always later than Pessah, either by just afew days (four days in the present era) or by more than one lunation. TheJulian Easter may occur more than five weeks later than Pessah (as in 2002with Pessah on March 28 and Julian full moon on May 1) but cannot beearlier. For the orthodox Churches this is the reason not to adopt the Grego-rian calendar where Easter may happen before Pessah (like in 2008 where theGregorian Easter was March 23, Pessah April 20, and Julian Easter April 27)because the Gregorian beginning of spring lags behind the Hebrew. So farthe orthodox authorities have not been worried by the fact that after some60 000 years the Julian Easter will have advanced with respect to Pessah bya full year so that it comes just before the next Pessah (which will then besome time in January).

Inasmuch as the Gregorian calendar uses better values for the lengths ofmonths and years, it might serve to eliminate such problems. But neitherthe orthodox Christians nor the orthodox Jews would consider to accepta proposal that comes from the catholic Church. Religion does not favoragreement but dissent. From the perspective of rational thinking this appearschildish. Why is the West not using the real Sun to define a year (via solsticesor equinoxes) and the real Moon to define a month? The answer is twofold.Mathematicians, and theoretical scientists in general, try to outwit Naturewith their intellectual power – a fair game as long as they are willing tocorrect their errors. Religious dogmatism, on the other hand, is not interestedin reality, and immune to rational argument: may the calendrical March 21coincide with summer solstice, so what? The decree of the council of Nicaeais held sacred.

3.2 Clavius and the new calendar of 1582

The European Renaissance in the 15th and 16th century was an era where– through Arabic mediation – pre-Christian antique thinking was rediscov-ered, and cherished for its rational reasoning. It goes without saying that itcould only develop under religious patronage, yet it became the fertile groundwhich bred the modern sciences. Within the catholic Church, the order ofthe Jesuits (Societas Jesu) was particularly devoted to promoting that spirit.One of their most impressive representatives was the mathematician and as-tronomer Christopher Clavius (.°�Wc.n¿, 1537-1612), praised asthe “Euclid of the 16th century”. He wrote the leading mathematical text-books of the time and taught at the Collegio Romano. Based on suggestionsof Aloysius Lilius (1510-1576) he worked out the calendar reform which was

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decreed by Pope Gregor XIII in 1582 [4].

The Gregorian reform was as faithful to the old Julian calendar as possi-ble, while at the same time correcting its discrepancies to the astronomicalknowledge of the time. In line with tradition, the new calendar remainedarithmetical, based on the average lengths of the tropical year and a slightlymodified Metonic cycle; it defined Easter as before, to be the Sunday afterthe first full moon in spring; it allowed modifications of the Julian rules onlyat turns between centuries (the “principle of secularity”). The correctionsconsisted of

1. adjusting the length of a year to 365.2425 days (as compared to theJulian value 365.25, and the astronomical value 365.24219),

2. adjusting the number of months per year to 235/19 − 43/300 000 =12.368 278 which amounts to 29.530 587 days per lunation (as comparedto the Julian value 29.530 851, and the astronomical value 29.530 589).

The new length of the solar year required to have only 97 leap days ratherthan 100 in 400 years, and the new length of the lunation required to resetthe lunar age (i. e. the number of days, from 0 to 29, between January 1stand the preceding new moon, also called the epact of the year) 43 times in10 000 years, in comparison to the Julian rules. The first modification wasimplemented in terms of the rule that century years are not to be takenas leap years, unless their number can be divided by 400. The lunar ruleis somewhat more complicated and consists of two parts: first, in centuryyears which are no leap years, the age of the moon is decreased by one day,amounting to 75 backward resets in 10 000 years; second, starting in 1800, acycle of 2 500 years was set in effect where the age of the Moon is increasedby one day seven times every 300 years, then once after 400 years. Thisamounts to 8 forward resets in 2 500 years, or 32 in 10 000 years, so that thenet total is 75− 32 = 43 backward resets.

In addition to these new leap rules, a one-time correction was decreed for theyear 1582: Thursday, October 4, was to be followed by Friday, October 15,the ten days in between were omitted; at the same time the age of the moonwas increased by three days.

Let us compare the Julian calendar (Fig. 2) and the Gregorian calendar(Fig. 3), for the years 1987-2007, to astronomical reality. First, the Julianbeginning of spring comes 13 or 14 days after the real vernal equinox; in theGregorian calendar the shift is only 0 or 1. The Julian full moon is 3 to5 days later than the real full moon; in the Gregorian calendar the shift is

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Figure 2: Julian Easter and Jewish Pessah for the years 1987-2007. Each lineshows the days March 20 through May 5 according to the Gregorian calendar, thebeginnings of April and May being marked by a thick bar. Sundays are shaded.The numbers at right are the years’ positions in the Metonic cycle. Little crosseson March 20 or 21 mark the astronomical beginning of spring, the letter Y is theJulian beginning of spring and, by definition, March 21 of the Julian calendar.The little black dots are astronomical full moons, the open circles are calendricalfull moons. The black Sundays are Julian Easter dates, the letters X mark thefirst days of Jewish Pessah. Note that Easter comes always later than Pessah.

between plus and minus 1. The respective Easter dates agree in the years1987, 1990, 2001, 2004, and 2007, but in most cases the Julian Easter is one,four, or five weeks later. The Gregorian date is then usually closer to an as-tronomically determined date – except for 1998 where the full moon occurredSunday, April 12, at 01:23 o’clock Jerusalem time, so that the Sunday afterwas April 19 (and the Julian Easter was correct).

As already mentioned, the Julian rules make sure that Easter cannot oc-cur before Pessah whereas in the Gregorian calendar that happened in 1989,1997, and 2005, see Fig. 3. The reason is that the Hebrew calendar’s so-lar year is a little too long, so that their beginning of spring has effectively

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Figure 3: Gregorian Easter and Jewish Pessah for the years 1987-2007. Thesymbols have the same meaning as in Fig.2. Note that in 1989, 1997, and 2005Easter comes earlier than Pessah.

shifted to March 27. Clavius was apparently not sensitive to this issue. Orperhaps he hoped that the Hebrew calendar would also be reformed, in itssolar part.

The Gregorian calendar has served well for the calculation of Easter dates.After the young Gauss in 1800 condensed its rules into a simple algorithmof ten lines [6, 14], every child can now compute the date of Easter for anyyear in the calendar’s cycle of 5 700 000 years. This easy computability is thecalendar’s main advantage. Its arithmetical character, however, and smallremaining discrepancies to the true lengths of years and months, perpetuatethe difficulties of the Julian and Hebrew calendars in principle: in the longrun, on the scale of thousands of years, it will drift away from astronomicalcorrectness in the mean, and on short scales, from year to year, it does notcare about the exact positions of Sun and Moon. Thus from its basic designprinciples, it would be unable to incorporate predictions of eclipses.

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4 The Chinese calendar âââÞÞÞ»»»of 1634

The prediction of eclipses seems to have been an important part of Chinesecalendars. To what extent this is true I am unable to report because myknowledge on traditional Chinese calendars is quite rudimentary. It derivesmainly from the excellent article by Helmer Aslaksen [3] and various otherinternet sources such as Wikipedia [1, 2].

The last reform of the Chinese calendar took place around the transitionfrom the Ò (Ming) to the 8 (Qing) dynasty, under the direction of ��é Xu Guangqi and its successors. It was completed by Adam Schall vonBell (é¥� Tang Ruowang) in 1634 who presented it to the Emperor âÞ(Chongzhen) with the book �âÞ»V�(Chongzhen Lishu) [16]. This cal-endar is based on older Chinese sources, and even though its Jesuit authorshad known and were directly influenced by Clavius, its design is fundamen-tally different from that of the Gregorian calendar: it is an astronomicalcalendar where the emphasis is on the true position of Sun and Moon. Onlyfrom such a perspective is it conceivable to have eclipses included.

The basic principles were developed in the distant past, and in the courseof many reforms the calendar gradually assumed its present design. A sig-nificant progress was made when ��ñ (Kublai Khan), the first Emperorof the à (Yuan) dynasty, assigned (E¹ (Guo Shoujing) and |; (WangXun) to overhaul the old calendar. The result, known as�G�»�(ShoushiCalendar), took effect in 1280. Its precision was about the same as that ofthe Gregorian calendar of 1582, with a year of 365.2425 days, and a month of29.530 593 days, but quite a different internal organization. As it happened,the standards of Chinese astronomy declined during the 350 years betweenGuo Shoujing and Xu Guangqi: observational diligence and precision deteri-orated, and the calendar lost its predictive power, in particular with respectto solar and lunar eclipses. This became obvious in the competitions thatXu Guangqi arranged in connection with the solar eclipses of December 15,1610, and June 21, 1629. The Jesuit astronomers working under his directioncame out on top and thereby won Emperor Chongzhen’s assignment to putthe calendar back in order.

But what did that mean? Did they introduce the Gregorian calendar inChina? Certainly not, the less so as that calendar did not make any ref-erence to eclipses. No, the assignment was to restore precision but not tochange the calendar’s design. And the Jesuits were well equipped to doing

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just that: they possessed the best observational data of the time, those ofthe Danish astronomer Tycho Brahe (�÷Ynb Digu Bulahe), and thegeometry of Euclid (N�°y Oujilide) as applied to the celestial sphere.The new calendar was the old one in more precise terms: the most rationallunisolar calendar (Ró» Yin Yang Li) ever invented. Let us discuss itsvarious parts in turn.

4.1 The solar year µµµ (Sui)

The solar year is taken from one winter solstice Á� (Dongzhi) to the next.Of course, the Gregorian value of 365.2425 days was known at the time, butrather than turning this approximate mean value into a rule of leap yearsequences, it was determined to accept the date at Beijing of the real Sun’swinter solstice as the beginning of a Sui. Thus the succession of years with365 and 366 days is less regular than in the Gregorian calendar, but alwaysin agreement with astronomical reality.

In order to divide the year into seasons, the old scheme of 24 solar terms �í (Jieqi) was adopted: 24 positions of the Sun along its yearly course amongthe stars, the ecliptic – not at equal time intervals, but at equal angular in-tervals of 15◦ each. The 12 positions at multiples of 30◦ are the main solarterms ¥í (Zhongqi). They include the two solstices of winter and sum-mer, and the equinoxes of spring and autumn. The other 12 Jieqi includethe 4 positions midway between solstices and equinoxes which the Chineseperceive as the beginnings of seasons: the Chinese seasons are at their peakwhen the Western seasons begin.

Fig. 4 shows the slightly elliptic orbit of the Sun in Tycho Brahe’s repre-sentation, with the Earth at the center. The direction of the Earth’s axis isindicated by the red arrow pointing towards the point of summer solstice.The symmetry axis of the Sun’s orbit is drawn as a red line from perigee(left) to apogee (right). The red dots on the ellipse are the first days of the12 Western months January through December, starting just below wintersolstice, in anti-clockwise direction. The tickmarks indicate five day inter-vals. The radial lines at angular intervals of 15◦ point to the 24 solar terms.Note that because of the orbit’s eccentricity the number of days betweensuccessive Zhongqi is 31 around the apogee (summer), even 32 from ��(Xiazhi) to L[ (Dashu), and 30 near the perigee (winter), only 29 from Á� (Dongzhi) to L; (Dahan).

The names of the solar terms are noted in fig. 5, and in tab. 1 they are listed

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together with their meaning and the approximate Gregorian dates (whichmay vary, from year to year, by plus or minus one day). They refer to seasonrelated natural phenomena and make perfect sense for a calendar based onthe orbit of the Sun. The numbering J1-J12 for the minor, and Z1-Z12 forthe major terms reflects the coupling of solar and lunar year because thelatter starts with the spring festival �I (Chunfen).

Figure 4: The 24 solar terms of the Chinese calendar are defined by 15◦ intervalsalong the ecliptic.

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Figure 5: Names of the solar terms. The ¥í (Zhongqi) are listed in boldfacealong the inner circle, the other �í (Jieqi) along the outer circle.

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Chinese name Translation approx. date

J1 Á� Lichun beginning of spring February 4

Z1 ¥y Yushui rain water February 19

J2 ¯T Jingzhe awakening of insects March 5

Z2 �I Chunfen spring equinox March 21

J3 8Ò Qingming clear brightness April 5

Z3 ÷¥ Guyu grain rain April 20

J4 Á� Lixia beginning of summer May 6

Z4 Bw Xiaoman grain full May 21

J5 }« Mangzhong grain in ear June 6

Z5 �� Xiazhi summer solstice June 21

J6 B[ Xiaoshu little heat July 7

Z6 L[ Dashu great heat July 23

J7 ÁB Liqiu beginning of fall August 7

Z7 ÿ[ Chushu end of heat August 23

J8 ¸3 Bailu white dew September 8

Z8 BI Qiufen autumnal equinox September 23

J9 ;3 Hanlu cold dew October 8

Z9 u\ Shuangjiang descent of frost October 23

J10 ÁÁ Lidong beginning of winter November 7

Z10 B¨ Xiaoxue little snow November 22

J11 L¨ Daxue great snow December 7

Z11 Á� Dongzhi winter solstice December 22

J12 B; Xiaohan little cold January 6

Z12 L; Dahan great cold January 20

Table 1: List of the 24 solar terms, together with their meaning and approx-imate Gregorian dates. J1-J12 are the minor Jieqi, Z1-Z12 the Zhongqi.

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4.2 The lunar year ### (Nian),and its coupling to the µµµ (Sui)

The lunar year consists of 12 or 13 lunations, depending on the date of thelunar New Yearc# (Xin Nian), see below. A new lunar month begins withthe date of a new moon and has roughly equal chances to span 29 days (BÛxiao yue) or 30 (LÛ da yue), with a slight preference for 30 in the winterhalf between the equinoxes, and 29 in the summer half. The reason is that theSun is faster near perigee (in winter) than near apogee (in summer), so theMoon has to cover a longer path from new moon to new moon in winter thanin summer. Fig. 6 shows the new moon positions for the lunar years 2008(Ñ�` Wuzi shu, starting Feb. 7) and 2009 (�î: Jichou niu, startingJan. 26 and ending Feb. 14, 2010). The year 2008 has 12 months while 2009has 13. The months do not carry names as in the Roman-Julian-Gregoriancalendar but are numbered from 1 through 12. – Yes, 12, even if the yearmay have 13 months. This sounds strange at first, but it is one of the tworules that define the coupling of the solar and the lunar year.

Figure 6: The lunar years 2008 (left) and 2009 (right). The first New Moonis marked as a large dot with white interior, the others follow one by one incounterclockwise direction. 2008 (Wuzi shu) is a short year with 12 months, 2009(Jichou niu) a long year with 13 months.

The rules are the following:2

1. the lunation containing the winter solstice is called the 11th month;

2The exact definition is slightly more intricate; see [3] for details.

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2. a lunation which does not contain a Zhongqi is considered a leap month£Û (run yue) and carries no number of its own – rather it gets thesame number as the previous month.

To understand this, consider again Fig. 6. As a consequence of the first rule,the beginning of a New Year is the second New Moon after winter solstice;it comes more than 29 days (one month) but less than 60 days (two months)later. Therefore it falls between the two zhongqi Z12 L; (Dahan) and Z1¥y (Yushui). A short year like 2008 starts close enough to Z1 so thatits 12 months end after Z12. Therefore each of its 12 months contain oneZhongqi. A long year, on the other hand, starts close enough to Z12 so thatits 13 months end before the Z1 of the next year. This means its 13 monthscontain only 12 zhongqi. Hence there must be one month without a Zhongqi,the leap month according to the second rule. In the right part of Fig. 6 wesee that there is a new moon very close to Z6L[ (Dashu); inspection of theephemeris for 2009 shows that new moon falls on Juli 22, Dashu on Juli 23.Thus the month from June 23 to July 22 is lacking a Zhongqi and hence isthe leap month; it is the “second fifth” month of that year.

The succession of short years and long years is of course in line with theMetonic rule that 12 out of 19 years are short and 7 long. But the order doesnot in general follow the Babylonic rhythm of the Hebrew and Christian cal-endars; it is determined by reality, not theory.

As the example of July 22 and 23, 2009, shows, the Chinese calendar requiresprecise ephemeris, either from observation or from calculation, in practice:from both. In this respect it is much more challenging for astronomers thanthe Gregorian calendar, as the mean periods of Sun and Moon are insufficientto determine the correct dates of New Moons. This was of concern to theChinese Emperor but not to the Churches in Europe.

4.3 Eclipses

The challenge was yet an order of magnitude greater with respect to theeclipses. Their prediction requires knowledge at least of the lunar nodes (theDragon’s points), and for details also of the perigees and apogees of Sun andMoon. Lunar eclipses are easier than solar eclipses because they can be seenfrom any point on the dark side of the Earth, and happen about twice peryear. Solar eclipses happen at about the same rate, but can be seen onlyfrom those regions on Earth which are hit by the Moon’s sweeping shadow,see fig. 7. Whether or not there exists a strip of total occultation depends

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on the relative distances from Earth of Sun and Moon, which in turn dependon their positions along the respective elliptic orbits. The width of the to-tality strip is at best a few hundred kilometers (as in the 1629 eclipse). Thismeans that for a given location on Earth, total solar eclipses occur only oncein some 300-400 years, on the average. Nowadays it is possible to obtaindetailed information about every eclipse from 2000 BC to AD 3000, due tothe dedicated work of Fred Espanak [5]. But in the 17th century a satisfac-tory theory of the lunar orbit did not exist – even the great Isaac Newton(1643-1727) struggled for it all his life and did not really succeed. Observa-tion was therefore all the more important. Another source for prediction arecatalogues of previous eclipses because they occur with certain regularity, asdiscussed below.

Figure 7: The solar eclipses of December 15, 1610 (left), June 21, 1629 (middle)and September 1, 1644 (right). Only the 1629 eclipse was total, the others annular.The strip of totality is bounded by two blue lines, that of annularity by red lines.Between the dotted lines, more than half of the Sun is covered by the Moon. –The pictures are taken from Espanak and Meeus [5].

What is the difficulty? If Sun and Moon moved in the same plane, everyNew Moon would give rise to a solar eclipse. But the planes are inclinedwith respect to each other, by about 5◦. The angular sizes of Sun and Moon,however, are only about 0.5◦. Hence most of the time the New Moon passesabove or below the Sun, and cannot be seen. An eclipse occurs only whenthe New Moon is close to one of the two nodal points where its path crossesthe ecliptic, either ascending (going to north, at the “Dragon’s head”, Rahuin vedic astrology) or descending (going to south, at the “Dragon’s tail”,Ketu in vedic astrology). This usually happens twice a year but occasionallyalso three or even four times. An illustration for the year 2008 is given infig. 8. The new moons’ angular positions are the same as in fig. 6, but now

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their radial distances from the Earth vary in proportion to the real values(with about twofold exaggeration). Each time the Moon completes a syn-odal cycle, from one new moon to the next, it meets the two nodes andthe two apsides (perigee and apogee). For example, during the first monthfrom February 7 to March 8, it crosses the ascending node (rightmost blueline) still on February 7 (so that indeed an eclipse occurred; more aboutthat later), the perigee on February 14 (lowest of the magenta points), thedescending node (leftmost red line) on February 20 (close to the full moonposition on February 21 so that there was a lunar eclipse!), and finally theapogee on February 28 (uppermost blue point). During the next month themoon meets the nodes a little earlier, and the apsides a little later, becausethe nodes move in clockwise direction while the apsides move anticlockwiselike the Moon. The relevant periods underlying these facts are the siderealmonth of 27.32 days – the time for the moon to return to a fixed positionamong the stars; the draconic month of 27.21 days – the time to return to agiven node; and the anomalistic month of 27.56 days – the time from perigeeto perigee. From these periods it is easy to derive that a full cycle of thenodes, relative to the fixed stars, takes 18.6 years, while a cycle of the apsidestakes 8.9 years in the opposite direction.

These numbers were known to ancient astronomers, from long term obser-vation, but notice from the figure that the steps do not all have the samelength. For precise prediction it remains indispensable to continuously followthe positions of Sun and Moon, as well as their sizes. Only then can it bedetermined when and where exactly the Moon’s shadow will fall on the Earth.

Some help comes from the fact that eclipses occur in certain patterns. Thereason are resonances between the various periods of the Moon. Just as theMetonic cycle of 19 tropical years = 235 synodical months reflects a resonance(i. e. a rational ratio of periods) between Sun and Moon, the Saros cycle is aresonance between the different months. As it happens, 223 synodic monthscoincide with 242 draconic months and 239 anomalistic months (up to aboutan hour); this time span is 18 years, 11 days, and 8 hours. It follows that agiven eclipse will recur after this time with almost identical characteristics,except that it is shifted westward by 120◦. For lunar eclipses this shift doesnot matter, hence the cycle was first known for them. For solar eclipses itmeans that only after three Saros periods does the same eclipse return toabout the same region. This requires record keeping over 54 years and 34days.

Because of the slight deviations from resonance, a given Saros cycles lasts

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Figure 8: Positions of lunar nodes and apsides in the year 2008, compare fig. 6.Ascending nodes are shown in blue, descending nodes in red. From one passageof the Moon to the next the nodes proceed in clockwise direction. Perigees areshown in magenta, apogees in blue color; they proceed anticlockwise like the newmoons.

only for about 70 to 80 returns, or some 1200-1300 years. A new cycle ap-pears about every 30 years, so at any given epoch there are some 40 Saroscycles in existence; currently the cycles 117 (first appearance 1262, last ap-pearance 2054) to 155 (born 1928) are “alive”. The fate of an entire Saroscycle is the following. It first appears at one of the poles: the north (south)pole for eclipses at the ascending (descending) node; the last occurrence is atthe opposite pole. During its lifespan the cycle gradually sweeps the Earthfrom pole to pole.

In addition to the Saros cycle there exist a few less important cycles. Theshorter Tritos has 135 synodic = 146.501 draconic months which implies thateclipses at ascending and descending nodes alternate; they are not in reso-nance with the anomalistic month. This cycle of a little less than 11 years

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was known in ancient China. The Inex is longer, 358 synodic = 388.500draconic months, just 20 days less than 29 years.

Much of this must be contained in the Chinese calendar books »�V(Lifashu). Unfortunately, I do not know what exactly Adam Schall wrote down intheâÞ»V (Chongzhen Lishu, 1634) and theÜòc�»V (Xiyang XinfaLishu, 1645), and where his knowledge came from. All I can tell from lookingat the drawings in the books [16] is that a good part of them is devoted toeclipses. I hope to learn more about this in the near future.

5 From Clavius to Xu and Schall

5.1 Jesuits in China

The connection between the calendar reforms in Europe (1582) and China(1634/1645) is fascinating and yet to this very day, in my opinion, not fullyelucidated. It is the story of an early intellectual encounter between Chinaand Europe with considerable benefit for both sides. The encounter tookplace exclusively on Chinese soil and was mediated by Jesuits, i. e. membersof the catholic order Societas Jesu which had been founded in 1540 to pro-mote religion through education and discipline. Even nowadays Jesuits areintensely trained academics: they study not only theology but in additiona science of their choice. This qualifies them for high level intellectual dis-course, and it may be one reason why they were dreaded and sometimes evenpersecuted (for example in protestant Europe). The idea of their founder Ig-natius of Loyola was to send them all over the world to proselytize amongeducated and high-ranking non-catholics. China became one of their pre-ferred targets because it was known in Europe that education played a keyrole there for climbing the social ladder in the Empire’s hierarchy.

Of a large number of Jesuits who went to China, Matteo Ricci (¼g{ LiMadu, 1552-1610) was the first. He had been a student of Clavius in Romeand knew all about Euclid and the Gregorian calendar reform of 1582 whenhe left Europe for ¥  (Aomen/Macao) in that same year. He arrived inð® (Beijing) in 1601 and stayed there until his death. Among a growing groupof Jesuits from all over Europe he was the undisputed leading figure. His en-counter in 1600 with��é (Xu Guangqi, 1562-1633) left a deep impressionin either of them and had far reaching consequences both for Europe andChina. Xu, an outstanding Chinese scholar of the time, made Ricci famil-iar with Confucianism, while Ricci taught Xu mathematics and astronomy.

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The first Chinese translation of Euclid’s Elements (�[Æ® Jihe Yuanli,1607) was their joint work (see various articles in [9]). On the other hand,when Ricci’s diaries were brought to Europe by Nicolas Trigault (��Ä JinNige, 1577-1629) and published in 1615, they had an enormous impact onEuropean intellectuals, as they conveyed the picture of a highly advancedsociety, notably in matters of ethics. This had repercussions until deep intothe 18th century: The renowned philosopher Christian Wolff was expelledfrom his professorship in Halle for praising Chinese ethics on account of theirnon-religious foundation [18].

Still in the year of Ricci’s death, the Jesuit Sabbathin de Ursis (}®¯ XiongSanba, 1575-1620) impressed Xu with his prediction of the solar eclipse inthe evening of December 15, 1610, which was better than any Chinese as-tronomer’s. This reinforced Xu’s conviction of the need for a calendar reformin China, and he was determined to ask the Jesuits for help. Trigault usedthis among other arguments to recruit for the China mission brilliant sci-entists from all over Europe. He described China as a country longing fortheir knowledge and offering attractive working conditions. This must beseen on the background of grim circumstances in Europe. Galilei (�¼Q�¼t) and his discoveries by the new telescope had been celebrated 1611 inthe Collegio Romano, with a triumphant reception that the great old Claviushad organized, but soon the tides had turned against him: an official admo-nition in 1616 had warned Galilei not to teach Copernican ideas any more.Further north, in the heart of Europe, the Thirty Year’s War was about tobreak out (1618): not a good environment for science (even though the greatJohannes Kepler (Õ?�� Ê�) developed his theory of the planets inthe middle of all this).

Trigault headed back to China in 1618, with 22 mostly young jesuitic scien-tists he had been able to hire. The biggest catch among them was proba-bly Johannes Schreck (�¬< Deng Yuhan, 1576-1630) who called himselfTerrentius. A former friend of Galilei, with him an early member of theAccademia dei Lincei in Rome, he was a multi-talented scholar. Born insouthern Germany, he was fluent in many languages and familiar with al-most all sciences of the day: medicine, botany, mathematics, mechanics, andastronomy. He met with Galilei’s disapproval when he joined the SocietasJesu, not long after the above-mentioned event in the Collegio Romano. Itmay be speculated that already at that time he thought of leaving Europefor China, perhaps even on Clavius’ advice who must have been proud ofhis former student Matteo Ricci there. Anyway, Schreck helped Trigault toassemble the group of 22, and Trigault successfully raised funds to buy an

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impressive library which contained the major scientific books of the time.Schreck turned to Galilei for help with the prediction of eclipses but did notget any – probably not because of anger on Galilei’s part as it is often said,but simply because Galilei did not know much about the matter; he neverhad a theory of the celestial bodies of the standard of Kepler’s. As to sci-entific equipment, Schreck turned to the cardinal Borromeo of Milano whosupplied him with a telescope and other things.

The journey to China was not blessed with luck; they lost two of the 22 men,and had Schreck not been on board as an experienced doctor, the loss mighthave been bigger. It took them 15 months to reach Macao where they stayedfor almost two years before they moved north, first to I² (Hangzhou),then to Beijing where they arrived in 1623 and were welcomed by Xu. Im-mediately they set out to translate mathematical, scientific, and engineeringbooks into Chinese. This shows that the calendar reform was certainly nottheir only business. But for their protection by the Emperor it probably wasthe most important.

In 1623 Schreck called on Kepler for help with the prediction of eclipses.He did this by way of his jesuitic colleagues in Ingolstadt. Kepler receivedthe letter in 1627 and immediately set out to formulate an answer (“Com-mentatiuncula”, Regensburg 1627). With an appendix written in 1630, hepublished this in Sagan and sent it to China [10] together with his RudolphineTables – the culmination of his life’s work, and the world’s best astronom-ical data for some time to come. Implicit in these tables is what Keplerknew about the motion of the Moon, hence they contain invaluable infor-mation concerning eclipses. I have not been able to find out whether or notthis knowledge has influenced the Chinese calendar reform, and if so, when.Hammer [7] reports that its existence in Macao is documented for 1646. Butthis would have been too late for the relevant calendar books of 1634 and1645.

At this point I’d like to comment on the nature of the sources that we havefrom that time. They are of two kinds: There are letters that the Jesuitssent to Rome, and there are the books they wrote in China, in Chinese lan-guage. Western accounts of their work seem to be based mostly on theirreports to Rome, but these, I believe, need to be viewed with scepticism. Ifthey wrote they would bless the Chinese with the Gregorian calendar reform,they were clearly lying. If they claimed great efforts in matters of missionfor the Church, they could point to Xu Guangqi who had adopted the Chris-tian religion under the impression of Ricci’s personality, but as the later rite

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controversy showed, they did not operate quite to the Pope’s and other mis-sionaries’ liking. If in order to improve the quality of their predictions theyused Kepler’s, the protestant’s, tables which were based on the Copernicanview of the world, they would not have reported that to Rome. But it wouldbecome apparent from a study of their Chinese books. I tried to find outwhether translations into Western languages exist, but was not successful,and unfortunately I cannot read the books myself, nor can I read what theexpert in the field, Professor TAÆ (Jiang Xiaoyuan) from Jiaotong Uni-versity, writes about it.

I do not know how Schreck and the other Jesuits developed their skills inpredicting eclipses. His telescope could not be of much help (contrary towhat is sometimes claimed). Tycho Brahe’s data from which Kepler’s theoryand tables were derived, were collected before telescopes existed. To followthe orbit of the Moon it is much more important to have (i) a good atlas ofthe stars showing the course of the ecliptic, (ii) day to day measurements ofthe Moon’s height above horizon at culmination (or something equivalent)and (iii) clocks that allow to relate sidereal to solar time. Whichever waythey did it, they succeeded to win the competition of predicting the eclipseof 1629, see fig. 7. This eclipse was spectacular for its date: noontime ofmidsummer day June 21, even though it was only partial in China. Inci-dentally, it was number 39 in the Saros cycle 121, with the longest totalityduration of the entire series, 6:20 minutes somewhat south of China. Thecycle had started at the north pole on April 25, 944, and will end on June 7,2206, near the south pole. It may be interesting to note that the eclipse ofNew Year’s day February 7, 2008, was number 60 of the same series, cf. fig. 8.

The success with that prediction won Xu Guangqi Emperor Chongzhen’sofficial assignment to create a reformed calendar. He called on Schreck todirect the work, but then in May 1630 Schreck died unexpectedly, and AdamSchall von Bell (1592-1666) assumed his position. Schreck’s sudden deathhas remained a mystery ever since, but nothing in the records supports thespeculations of two recent books that an agent of the Inquisition might havepoisoned Schreck as a response for his performing autopsies and other revo-lutionary scientific experiments [8], or that Schall himself might have killedhim to prevent his sympathy for Copernicus to become evident [11] . AdamSchall, a younger Jesuit of German origin, continued the work under Xu’sdirection, and after the latter’s death in 1633, under ¯�² (Li Tianjing).The result was co-authored by Li Tianjiang, Adam Schall, and ¯�þ (LiZhizao) and presented to the Emperor in 1634, under the name âÞ»V(Chongzhen Almanac) [15], but was never officially put in effect. The reason

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was that these were the last years of the Ming dynasty and the Emperorhad more urgent problems to tend to. Not before the turnover to the newQing dynasty in 1644 was the interest in a calender reform renewed. Thesolar eclipse of September 1, 1644, was the third occasion for the Jesuits todemonstrate their superior astronomical knowledge, this time to represen-tatives of the new Qing Emperor. Adam Schall was ordered to finish thecalendar reform, and after its completion in 1645 (with help from GiacomoRho, an Italian Jesuit) he was raised to the position of Imperial astronomer(�){)t (qintian jian de jianzheng). The new calendar appeared underthe name Üòc�»V (Xiyang Xinfa Lishu, New Calendar according tothe Western Method).

But what exactly was the “Western method”? There was the superior pre-cision of Tycho Brahe’s data, possibly in Kepler’s interpretation. Also itseems obvious that Euclid’s geometry as applied to the celestial sphere wasa considerable progress in theory. Another relevant improvement probablycame with the rather precise star maps that the Jesuits brought along. Itis often said that the Jesuits introduced the ecliptic reference system on thecelestial sphere as something new to the Chinese who had used the equatorialsystem before [13]. However, I find it hard to believe that the Shoushi Li of1280 should have been ignorant of the ecliptic system if it was able to predicteclipses. This knowledge might have been forgotten 300 years later, but notnew. The design of the new calendar was certainly the old one, so I wonderwhat the exact nature of the reform really was.

5.2 View from today

The great Joseph Needham [13] was perhaps the most distinguished Westernauthor to express serious criticism of the Jesuits as mediators between Eastand West. He sees them as primarily the Pope’s missionaries whose maininterest was to proselytize, using science only as a vehicle to that end. Heblames them for bringing to China Tycho’s geocentric view of the universe,thereby obstructing the spread of the more modern Copernican heliocentricdoctrine. However, from the point of view of calendar making it makes per-fect sense to put the Earth into focus. Furthermore, Tycho had a strongargument not to believe the heliocentric “hypothesis” (as Galilei was forcedto call the Copernican world view after 1616): he could not detect any stellarparallax in the course of a year, as would be expected from a moving Earth.(The correct order of magnitude of the stars’ distances was inconceivable atthe time. Only in 1838 was Bessel able to measure the parallax to a nearbystar, amounting to no more than 0.3 arcseconds.) Needham sniffs at the

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Jesuits for having been “inspired by religious fervour”, and he scolds themfor, as he sees it, belittling or even not understanding scientific achievementsof the Chinese past.

I think this criticism is not quite fair. The Chongzhen Li is closer in de-sign to the tradition of the Shoushi Li than to the Gregorian calendar, eventhough the Jesuits reported to Rome they would bestow China with thatcalendar. I’d like to advocate that their work should be rated by what theywere doing rather than by what they wrote home in self-defense. With onlya little empathy for the Jesuits of the time, and the conditions of their lives,it is easily conceivable that scientific much more than religious fervor hadattracted them to China. They were well trained scholars with open andscientifically oriented minds, delving into the rich cultural wealth of the Eastand trying to create something new out of the encounter. The enthusias-tic accounts that they sent to Europe made China appear to be a countryof enviable rationality, and culminated in Leibniz [12] siding with the Je-suits in their praise of Chinese ethics and political philosophy. In the otherdirection, their contribution to scientific progress in China by translatingimportant European books from diverse disciplines, adding original writingsof their own, were highly welcome by their educated counterparts, rangingfrom Xu Guangqi to Emperor �Ú (Kangxi, 1654-1722). Had it not beenfor the intrigues of Franciscans and Dominicans (the same kind of peoplewho denigrated Galilei), the cultural East-West interaction could have re-mained harmonic, but the Pope decreed in 1704, and again in 1715, thatthe jesuitic attitude of mutual respect could not be tolerated. In retaliation,Christian missionaries were expelled from China in 1724. Nevertheless theJesuits could remain as astronomers, well beyond the times where their orderhad even been suspended by the Church (1773-1815).

Towards the end of his evaluation Needham becomes somewhat more concil-iatory and praises the “intercourse between civilizations” that seems to haveno parallel in history with words that remain true 50 years later: “It is vitaltoday that the world should recognise that 17th-century Europe did not giverise to essentially ‘European’ or ‘Western’ science, but to universally validworld science, that is to say, ‘modern’ science as opposed to the ancient andmedieval sciences. ... when once the basic technique of discovery had itselfbeen discovered, once the full method of scientific investigation of Nature hadbeen understood, the sciences assumed the absolute universality of mathe-matics, and in their modern form are at home under any meridian ...”.

With regard to the two calendars of Europe 1582 and China 1634/45, it must

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have been clear to the Jesuits involved that from the point of view of universalscience the latter is purer than the former. The Chinese Emperor’s interestin accurate predictions was a better breeding ground for clear scientific so-lutions than the Pope’s interest in manageability. The Chongzhen Li, if Iunderstand it correctly, is a scientific cross-cultural synthesis [9] in the bestsense: the achievement of rational goals (accurate prediction of celestial phe-nomena) with rational means (precise observation combined with advancedmathematics). The same cannot be said of the Gregorian calendar whereaccuracy of prediction is, to some extent, replaced by religious leniency, andprecision of observation by the pragmatism of easy computability.

From this point of view I find it deplorable that China in 1912 so readilyabandoned the Chongzhen Li in favor of the Gregorian calendar, now calledÚ» (Gong Li, Common Calendar), or ó» (Yang Li, Solar Calendar),where the Moon is simply ignored. Luckily this was in part repaired whenthe People’s Republic of China incorporated the traditional Chinese calendarinto the Gong Li. A Chinese calendar now shows the Gregorian date togetherwith its number in the respectiveÛ (Yue, lunar month), and it identifies thejieqi of the solar year, also referred to as @» (Nong Li, farmers’ calendar).

6 Acknowledgements

It is a pleasure to thank Andreas Dress for inviting me to the highly stimulat-ing International Xu Guangqi Conference in the PICB at Shanghai, October2007. I am indebted to |{ (Wang Yong) and K� (Dr. Zhao Jun) forhelping me with Chinese texts.

References

[1] Chinese calendar. Internet,http://en.wikipedia.org/wiki/Chinese Calendar, 2008.

[2] Zhongguo lifa de fazhan shi. Internet,http://baike/baidu.com/view/3965.htm, 2008.

[3] H. Aslaksen. The Mathematics of the Chinese Calendar. Internet,http://www.math.nus.edu.sg/aslaksen/calendar/chinese.html, 2008.

[4] Ch. Clavius. Romani Calendarii a Gregorio XIII. P. M. restituti Ex-plicatio. Opera Mathematica, Tom. V, Rome 1595 and 1603. Mainz,1612.

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[5] F. Espanak. NASA Eclipse Web Site. Internet,http://eclipse.gsfc.nasa.gov/eclipse.html, 2008.

[6] C. F. Gauß. Berechnung des Osterfestes. Monatl. Corresp. zurBeforderung der Erd- und Himmelskunde, Aug. 1800, Werke VI:73–79,Gottingen 1874.

[7] F. Hammer. Keplers Rudolphinische Tafeln in China. Naturwiss. Rund-schau, pages 194–198, 1950.

[8] I. Iannaccone. L’Ami de Galilee. Stock, Paris, 2006.

[9] C. Jami, P. Engelfriet, and G. Blue, editors. Statecraft and IntellectualRenewal in Late Ming China. Brill, Leiden, 2001.

[10] J. Kepler. Calendaria et Prognostica, volume XI,2 of Gesammelte Werke.C. H. Beck, Munchen, 1993.

[11] R.-K. Langner. Kopernikus in der Verbotenen Stadt. S. Fischer, Frank-furt am Main, 2007.

[12] G. W. Leibniz. Novissima Sinica. S. Ammon, Hannover, 1697.

[13] J. Needham and Wang L. Mathematics and the Sciences of the Heavensand Earth, volume III of Science and Civilisation in China. CambridgeUniversity Press, Cambridge, 1959.

[14] P. H. Richter. Kalender und die Gaußsche Osterformel – Was stecktdahinter? Mitt. Gauß-Ges. Gottingen, 44:59–78, 2007.

[15] J. A. Schall von Bell. Chongzhen Lishu, volume 382 of Gugong ZhenbenCongkan. Hainan Publ., Haikou, 2000.

[16] J. A. Schall von Bell. Xiyang Xinfa Lishu, volume 383-387 of GugongZhenben Congkan. Hainan Publ., Haikou, 2000.

[17] C. Tondering. The Calendar FAQ. Internet,http://www.tondering.dk/claus/calendar.html, 2005.

[18] C. Wolff. Oratio de Sinarum philosophia practica. Felix Meiner, Ham-burg, 1985. Latin and German.

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