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"The whole moon was eaten": Southeast Asian eclipse calculation.

One might classify earthquakes and volcanic eruptions as "natural disasters", whereas comets and eclipses might be described merely as "natural curiosities". In earlier times, however, all were equally frightening, signs that the gods were angry.

A form of protection against eclipses was available, however, in the sense that prediction provided a means of countering the event.(1) Consequently one main function of the astronomer/astrologer (henceforth "hora") in his society was to warn his patron that the sky would darken and the birds stop singing at such and such a time on such and such a day.

The genre of historical record in Thailand known as chotmaihet hon (Astrologers' Diaries) contains many entries that are ad hoc: the death of a white elephant in one year might be followed by the great cost of rice in the next year and by the collapse of a temple in the third year. If there were no corroboration from elsewhere, it would be impossible to determine what accuracy should be assigned to these records. Solar and lunar eclipses, however, are also a constant point of attention in them and the vast majority of the predictions are correctly timed. How was this done? By modifications to the everyday system of calculating solar and lunar position. The horas, we may say, had one workaday system in place for normal use, and a more demanding and more refined system in place for eclipses.

In what follows we describe in general terms what this more sophisticated procedure involved.

The Southeast Asian system of astronomical reckoning employs a number - the ahargana (Thai: horakhun) - that represents the elapsed days in a given era; and it is this number that is first adopted in the normal calculation process. The Thai Era in common use in the historical record is the "Little Era" (Chulasakarat), which has the same numerical value as the Thekkarit Era of Burma.(2) It takes its origin in 22 March 638 A.D. The Mean Longitude of each planet on this day being known canonically, and the Mean daily motion of each planet also being known canonically, their Mean positions for any subsequent day can then be found by simple arithmetic.(3)

It was the function of the top experts to determine these canonical values, but thereafter the everyday calculations could be made by anyone with a mechanical grasp of the procedures involved. For instance, a workaday hora would know that the first value he had to adopt was the number 292207. He would also know that his astronomical era runs in cycles of 800 years and that his astronomical day is divided into 800 parts. He does not also need to know that if you divide 292207 by 800, the answer is 365.25875 - the length of the Southeast Asian astronomical year. The point we stress here is that the hora memorizes the numerical values and the successive calculation procedures; he does not need to have any theoretical or conceptual understanding of what he is doing in order to arrive at the correct results.

From this single base number, all the rest of the calculation follows in train, though of course the knowledge of how to perform the calculation process (tedious even for day-to-day reckoning) was the special preserve of the horas and would be passed on from father to son or from master to select pupil.

Calculation for the everyday, of course, involved finding the positions of the five planets Mercury to Saturn in addition to the positions of the sun, the moon, and the moon's nodes (the points where the moon crosses the plane of the sun's path). When it came to eclipse prediction, however, only the latter three elements (sun, moon, nodes) were involved, but nonetheless rather more expertise was required - both because the procedure was more complex and because the values employed were more refined. The hora could not simply use the ahargana of date as his calculation base, he had to begin by subtracting 184298 days from it. This number, in turn, would be canonical for him, and he would not need to know its function. We can easily deduce, however, that its effect was to place the epoch of eclipse prediction 184298 days later than the epoch of everyday calculation, and that a new point of departure was used for making eclipse predictions: one that falls on 19 October 1142 A.D.(4)

Having redefined the ahargana value, the hora then had to deal with larger, more complex values. For the everyday it was sufficient in finding the position of the Mean Longitude of the sun to multiply the ahargana by 59[minutes] (minutes of arc). But for eclipse calculations the ahargana was multiplied by 591361716.(5) Not surprisingly, only the first eight digits of this massive result were then taken to be significant.

To the resulting value the hora then adds 12268[minutes] (minutes of arc). Authority is vested in the numbers, which in a long training were memorized and applied routinely. What then is the function here of 12268? This number loses its mystifying quality when it is divided by 60 (converting minutes of arc to degrees of arc) and becomes 204 degrees 28 minutes, or Libra 24 degrees 28 minutes. It can then be conjectured to be (and verified to be) the Mean Longitude of the sun at the time chosen for the commencement of the eclipse epoch - in fact the sun's Mean Longitude at 8:56 hrs on 20 October 1142 A.D. In short a new epochal date requires a different canonical value, and this value (if the hora is to perform his calculations accurately) replaces the one he uses in day-to-day reckoning.(6)

Next in line come the corrections from Mean to True Longitude for the sun and the moon. Here, too, modified values are adopted. In day-to-day procedure the value for the quantity known as the moon's apogee (the point at which the moon is furthest from the earth in its orbit) is treated as a constant with the value 4800[minutes] (80[degrees], as in the Suryasiddhanta): eclipse calculation modifies this value to 4680[minutes] (78[degrees], as in Aryabhata). Similarly the maximum value by which the sun's Mean position could vary from its True position (its "maximum Equation") was normally 134[minutes]; but with eclipse calculation this value was modified to 129[minutes]. Here again the workaday values deriving from an older Indian treatise, the Suryasiddhanta, are supplanted by more meticulous values deriving from an Indian revision of it, ascribed to Aryabhata.

The majority of the remaining calculations are routine (even if dauntingly laborious), but a few elements in them are of interest.

The calculations for the everyday are determined for midnight (ardharatrika, in Indian parlance), but solar and lunar positions on an eclipse day are determined for sunrise (audayika in Indian parlance). The interval from midnight to midnight is fixed; but the interval from sunrise to sunset (and the next sunrise) varies in the course of the year, and the length of the given eclipse day has to be known. The values used for day-length in normal reckoning, however, were (and still are) crude; but in eclipse reckoning day-length was allowed to be affected by difference in terrestrial latitude. Here again is a clear indication that the hora had to be on his best behaviour, so to speak, when dealing with eclipses.

We should take a moment to explain the difference involved here. Volvelles can still be bought in astrology bookshops on which the zodiacal signs Aries (Mesa) and Libra (Tula) are accorded 120 minutes to rise above the horizon, and values for the intervening signs are determined by the addition or subtraction of 24 minutes.(7) The fact that the values inscribed on these volvelles are valid for no place on earth is counter-balanced by the fact that they can be used anywhere in Southeast Asia without their having a sufficient distortion to become unworkable in everyday terms.(8)

For eclipse calculation, on the other hand, greater accuracy was needed, and hence a more elaborate system was developed, in which the values adopted were sensitive to geographical latitude. It is clear that both Wisandarunkorn (our Thai authority) and Faraut (our Cambodian authority) adopted values determined by this more elaborate and sensitive mode of reckoning, but it is demonstrable that Wisandarunkorn used values in fact operative at the latitude of Lavo (Lopburi), a locality in which the Khmer also anciently had a base. And of two different sets of value adopted by Faraut (as reported to him by the Cambodian Astronomer Royal) one set seems to point to Nakhon Si Thammarat on the Malay Peninsula, again anciently a Khmer base; and the other set seems to suggest Rangoon.

In other words, neither author recognized that he was inheriting numbers wholly inappropriate to his particular location.(9) In particular, Faraut (as instructed by his informant) used two different latitude values, but for the same purpose, values that were operative for locations seven degrees distant from each other, neither being valid for Phnom Penh.

We would have been fortunate indeed had we been able to determine how these erroneous values migrated (presumably eastwards) to Bangkok and to Phnom Penh, but we may conclude that the eclipse calculation procedure had a long history and over time could become fossilized, as it were. Indeed it is a characteristic of both eclipse calculation and normal calculation procedures that their theoretical basis disappears totally from sight. The merit of the system is that it can be applied mechanically. Its disadvantage is that once an error creeps in, it has very little chance of being eradicated.

A factor that must have made a major contribution to the "blind", memorial application of the calculation procedures is the terminology it employs. One of the few recognizably Thai words in Wisandarunkorn's terminology is phim, which as a noun means essentially an "image or pattern".(10) In the context of an eclipse calculation, however, the word comes to mean "that part of the moon's disc not obscured at centrality". In one sense, though, the sheer barbarity of the terms (which no native speaker of Thai or of Cambodian would know) works in their favour. It proves, for instance, to be the case that the word that Faraut heard as oicheapol is the word that Wisandarunkorn represents as uccabala, and that both can be referred back to Sanskrit ucca.(11)

With any complex or unfamiliar procedure, however, there is no substitute for a worked example. By using some ingenuity and more perseverance, one can come at the sense of the terms by examining the function of their associated numerical values. The situation is not unlike that of the familiar IQ question: What is the number next in series - 49 1625 ...?

The obscurity in the terminology employed and the absence of any clear sense of theory make it remarkable that the eclipse calculation system, and also the more ancient "normal" system for that matter, have been in successful operation for many centuries across Southeast Asia up to the present. And the "normal" system is indeed in operation up to the present day. In Bangkok one can purchase annually one's "Diari Hon" (Astrologer's Diary), and the values given in it for the sun and the moon are precisely those that an accurate Thai or Burmese inscription of 600 years ago would exhibit.

Rationally speaking, if the changes employed in eclipse prediction gave better results, they would do the same for everyday calculation; but the one system never encroached upon the other. We see two possible explanations for this:

(1) the necessary learning was acquired by a select few and passed on by rote, so that the two systems might be learnt independently of each other, one for one purpose, the other for the other purpose; (2) the hora who learnt both systems recognized their interconnection, but shrewdly saw that the less-demanding system was perfectly adequate for the every-day, and so retained it as his "lazy" option. And we could well imagine that in such a labour-intensive exercise as even the day-to-day calculation required, short-cuts, rules of thumb, and approximations would be adopted wherever possible. And here again a situation that we might project on to the past can be shown to have in fact been operative.

In the copious astronomical detail to be found in the temples of Pagan, for instance, it is common to find the positions of the sun and the moon accurately recorded to the nearest minute of arc; but that the positions allocated to the planets appear not to have the same degree of accuracy. One discovers, however, that the longitudes for, say, Mars make sense if one takes its position at the start and middle of the month, with the intervening positions then being determined by taking an average rate of motion. Since the mean motion of that planet is roughly 31[prime] day, little distortion will arise from calculating its exact position only twice a month instead of thirty times.

Other data manifests this same "laziness". A Burmese calendar running for 75 years (BE 1210-1285) is content to indicate the positions of the slow-moving Rahu, and of Saturn and Jupiter, only once a month; though the positions of the sun, Mars, Mercury and Venus (the moon not being calculated) are given daily. By this expedient a useable ephemeris was produced that required its author to record only 3800 or so positions in 75 years, rather than 6300 positions.

The horas must have been glad that eclipses were not of frequent occurrence, and happy to use the shorter, "lazy" mode of reckoning for most of the time.

There is one particularly notable occasion on which the Western and the Southeast Asian systems of reckoning eclipses come into contact. When King Louis of France sent his Jesuit experts to Siam in the 1680s, he equipped them with the latest in astronomical instrumentation for reasons that had as much to do with religion and commerce as with science.

But with what one would take to be a mixture of pique and surprise, the Jesuits who observed the lunar eclipse of 11 December 1685 (NS) found themselves obliged to report that "a Bramen Astrologer ... foretold [it] to a quarter of an hour almost" (Tachard, Voyage to Siam, p. 242). Their purposes would have been well served if the local astrologer had been a day or even some hours out in his prediction. But provided with the latest equipment and intending to blind with science, Tachard and his colleagues found that the local expert could do, if not better than they could, far too well for comfort.

If we look to the beginning of the partial eclipse as an indicator, Tachard records it as occurring at 3:19 a.m., whereas a computer programme returns a time of 03.11 a.m.(12) The difference is half of the quarter of an hour attributed to the local hora. By this token it is no more than justice to the Southeast Asian system to note that though Tachard also says, contemptuously, that the local astrologer "was mightily mistaken as to the duration" of the eclipse, his own team found itself obliged to record a leeway of eight minutes: "the beginning of the Emersion was at 6h 1m 11sec or rather at 6h 9m [0 sec.] and to that Observation it was concluded we must hold ...."

A working of the eclipse of 18 August 1868 that King Mongkut fatefully went to observe has survived to us in some anonymous calculations of the event.(13) The document contains no expository text, merely an uncompromising set of the Thai-Sanskrit labels one encounters in this context, accompanied by the relevant numerical values. But our computer programme indicates, for instance, that such key values as the size of the sun's disc and of the moon's disc at the time of eclipse, are given correctly here to within one-sixtieth of a unit.(14)

We note that each of the three very disparate sources available to us employs the same terminology and arrives at such critical values as those for the start and duration of the eclipse as are very comfortably inside the Brahman's "quarter of an hour".

A useful consequence of establishing the accuracy observable here is the influence it may be allowed to have upon one's attitude towards the chotmaihet hon. There is, as we noted at the start, a great deal in these sources that cannot be verified from elsewhere and which to this extent has to be taken on trust. In these records, however, there is also mention of over eighty eclipses.(15) Some thirty of these eclipses have starting times assigned to them (in multiples of six minutes), where the bulk of these times can be demonstrated to be correct within the "quarter of an hour almost".

To take one of the earliest cases in which a time is mentioned, the lunar eclipse of 15 February 1794. This is said to have taken place (i.e. commenced rather than culminated) at "9 thum plus 1 bat"; that is at 9 hours and six minutes after the previous 6 p.m. (i.e. at 3:06 a.m., local time). By contrast Gislen's computer programme places the start time for Bangkok at 3:10 a.m. (20:27 hrs UT).(16)

Hitherto, it is fair to say, historians have not had the competence or means to inspect data of this kind. It is also fair to say that when in former times they turned to historians of astronomy for assistance they were ill-served, since it was supposed that the Indian mode of reckoning could simply be superimposed on the South East Asian data.(17) Computer programmes and electronic access to them, however, have entirely removed this barrier.

APPENDIX: A TECHNICAL PRECIS OF THE CALCULATION SYSTEM

The procedures for calculating an eclipse, either solar or lunar, are as follows:

For solar eclipses, the true positions in longitude of the moon and sun at sunrise on the day of eclipse are computed, as are also these positions for sunrise on the day following the eclipse. In the case of a lunar eclipse, the point opposite to the sun on the ecliptic, the shadow centre (Rahu), is used instead of the sun.

One now calculates the true daily respective motions of the two bodies - the amount by which their true longitudes will increase from one day to the next. With this quantity known, it is then also possible to calculate the extent to which the two bodies have shifted position relative to each other.

It is now a simple matter to calculate the exact moment when the two bodies will coincide in longitude, which will therefore define the time of conjunction - the moment when maximum eclipse will take place.

The moon, however, moves in an orbit that is inclined by about five degrees to the ecliptic. Consequently its orbit crosses the ecliptic in two points, the nodes (known to Southeast Asia as Rahu and Ketu).

It follows that the moon's distance from a node as measured along its orbit will define its angular distance perpendicular to the ecliptic, i.e. give the latitude of the moon - an element not needed and not heeded in everyday calculation, but necessary in the precise business of defining an eclipse. Taking this latitude into account is important for eclipse prediction: if the latitude of the moon is too great, it will not eclipse the sun, or the shadow of the earth will not eclipse the moon. On the other hand, if the latitude is zero, the duration of the eclipse will be maximum. Using the latitude at the moment of conjunction thus makes it possible to predict the duration of a particular eclipse.

It is assumed by the procedure that if the separation between the moon's centre and the centre of the shadow cone is greater than 54[prime], there will be no eclipse. The value is determined as follows: the moon's mean diameter is taken to be fixed at 31[prime] and the shadow cone's mean diameter is taken to be fixed at 2.5 times this amount, where (31 + 77)/2 is 54. When the distance between the centres of the moon and of the shadow cone has been found for a particular eclipse event, the value attains an importance in helping to define how long the eclipse will last. Suppose that the distance between the centres has been found to be 33:18[prime]. The complement of this value is 20:42[prime]. Now set the complement's fraction aside and adopt a procedure of successive subtraction, as follows:

20 - 1 = 19 19 - 2 = 17 17 - 3 = 14 14 - 4 = 10 10 - 5 = 5 5 - 6 ...

As soon as the remainder (5 in this case) is smaller than the subtrahend, multiply that remainder by 60 and add it to the fractional part of the complement (42 in this case):

5 * 60 = 300 300 + 42 = 342.

Divide this result by the last subtrahend (6 in this case):

342 / 6 = 57:00.

Regard the remainder of the successive subtraction (here, 5) as representing nadi and regard the integer part of the division by the subtrahend (here, 57) as representing vinadi.(18) The two yield 5:57 nadi. Whereas we have 24 hours in our day, Southeast Asia has 60 hours in its day: 5:57 nadi therefore converts to 2 hrs 22 mins.

From a modern point of view the procedure above is neither very elegant nor very accurate. It would be quite easy to calculate the eclipse duration by simple geometry (which is done in the Hindu scheme). The reason for using these more mechanical procedures might be that they are easy to remember and it is easy to apply the subtraction series 1, 2, 3 ... by rote.

For a solar eclipse, the procedures are a little more complicated.

In contrast to a lunar eclipse, which appears the same to any observer on the earth, the visibility of a solar eclipse depends crucially on the position of the observer on the earth's surface.

The difference in visibility is due to the finite distance between the moon and the earth, giving rise to the phenomenon of parallax.

Point A is directly on the straight line joining the centres of the earth, the moon and the sun. An observer at A will see the moon and sun discs aligned together. But an observer at B, who is not on this line, will see the moon displaced relative to the sun.

There are three ways in which parallax will influence the appearance of a solar eclipse, depending on where the viewer is located.

(1) Horizontal parallax is caused by the observer being displaced in geographic longitude relative to point A. This will cause the time of the eclipse to vary from the time of true conjunction.

(2) Parallax in latitude is caused by the two factors

(2a) by the observer not being on the earth's equator

(2b) because, as the earth rotates, the observer will move in a plane that is inclined with respect to the ecliptic plane. The observer will then sometimes be above, sometimes in, and sometimes below the ecliptic plane.

All these causes of parallax depend on each other in a quite complicated way. But, if the observer is not far from the equator, it is possible to treat them as approximately independent, which is exactly what the Southeast Asian horas did.

It is evident that the original designers of the system knew very well what they were doing and had a sophisticated model of the universe. And there are similarities with the Hindu scheme for eclipse calculation in the Southeast Asian mode of reckoning, but also significant differences. From the eclipse calculation procedures documented for us by Faraut, Wisandarunkorn, and the Chiang Mai document, we can conclude that the original scheme was formulated according to a rigorous set of computational rules which had the advantage that anyone, however ignorant of the true cause of an eclipse, could calculate an eclipse event with quite good accuracy.

This property of the system made it possible for it to propagate itself, intact, for many centuries. The weakness of the system, on the other hand, was that once one of the rules was corrupted, that error would propagate itself unless checked by someone with an unusually clear understanding of the calculation procedure.

There is some clear evidence in the three eclipse calculations that we have studied that such corruption occurred. and to this extent the procedure lost its connection with reality. Even so, these errors in procedure were not of a magnitude to make the system useless or grossly inaccurate. It may be for this reason that they were allowed to survive or perhaps even remained undetected.

1 In popular thinking, an eclipse was a time when the monster Rahu was expected to try and devour either the sun or the moon. This disaster had to be avoided, so at the appropriate time (which it was the duty of the hora to predict in advance) as much noise as possible had to be generated, in order to scare Rahu away.

2 We make a distinction here between Thai Chulasakarat and Burmese Thekkarit. Although the year-values are numerically the same for the two groups, their systems of intercalation are so different that it is misleading to lump the two systems together. For instance the year CS 803 begins on Caitra 7 waxing, but BE 803 begins on Kason 8 waxing (one month and a day, not just one day, later).

3 The standard procedure in finding the position of a planet is first to calculate where it would be if it appeared to circle the Earth always at an even speed (its "Mean" position). Correction formulae are then applied to this position to discover where it actually will be when viewed from the Earth (its "True" position).

4 This date falls within the reign of the Khmer ruler, Suryavarman II, and at a period when the Khmer empire had secure bases in Siam.

5 There is a lesson here for those who suppose that the application of Southeast Asian astronomy had anything to do with real-time observation or with the possibility of making it. The sun's mean motion of slightly less than one degree per day is here multiplied by ten million! In the 86400 seconds of time that make up 24 hours, at this rate the sun would pass through about 120 of these millions of units in one second of time. The hora's power lies in his ability mechanically to calculate numbers on paper.

6 We know of no method of assessing the transmission lines from expert to hora historically. But whatever they were, they were certainly efficient. Our data comes mainly from a Thai expert in the mid-twentieth century and from a French civil servant taught in Cambodia in the early twentieth century. By some accident they both use the same eclipse as a worked example and they both come to essentially the same results. Our third source, a calculation of the Mongkut eclipse (albeit of uncertain date and provenance) follows exactly the same procedures with exactly the same values as one would predict of it. See further below.

7 For those who wish to pursue how the reckoning is made here: the values from Aries (Mesa) to Virgo (Kanya) are given respectively as 120, 96, 72, 120, 144, 168 = 720 minutes = 12 hours. The values in the other half of the circle, from Libra (Tula) to Pisces (Mina) are 168, 144, 120, 72, 96, 120 = 720 minutes = 12 hours.

8 The low geographical latitudes of the region, where the extremes between the shortest day and the longest day experienced, for example, in Europe, do not apply - this proximity to the Equator makes the volvelle values workable. They give tolerable results in Bangkok, Rangoon, or Phnom Penh: they would generate rubbish in London, Pads, or New York.

9 Wisandarunkorn's values for day-length are operative at 15[degrees]45[minutes]N; Faraut's two sets of values are operative at 16[degrees]45[minutes]N and at 9[degrees]30[minutes]N. They are, in short, markedly different from those that should have been adopted for Bangkok (13[degrees]45[minutes]N) or for Phnom Penh (11[degrees]30[minutes]N).

See Luang Wisandarunkorn, Kamphi Horasasat Thai (Bangkok, 1965), p. 180 and F.G. Faraut, Astronomie Cambodgienne. (Saigon, 1910), pp. 166 and 174.

10 See, e.g. the standard work, G.B. McFarland, Thai-English Dictionary (Stanford University Press, 1944), p. 588, col 2, ad init.

11 Monier Monier-Williams, A Sanskrit-English Dictionary, col. 172c: "the apex of the orbit of a planet".

12 The programme is Gislen's "Planets" (available at Sum-Aim Archive either for Macintosh with floating point unit or for Mac Power PC).

13 The manuscript is held, we understand, by the Library of Chiang Mai University.

14 This (Macintosh) computer programme, written by Eade and Gislen, is available, free of charge, from lars.gislen@teorfys.lu.se.

15 I have reproduced the data, together with verifications where possible, in Appendix VII of The Calendrical Systems of Mainland Southeast Asia (E.J. Brill, Leiden, 1995). The sources drawn on are three different examples of the genre known as "chotmaihet hon".

16 This time is replicated exactly by the commercial programme "Starry Night" (Sienna Software).

17 A conspicuous case is where, in the 1920s, the Superintendent of the Archaeological Survey in Burma had to rely in his Reports upon the findings of Pillai, a noted (and justly respectable) expert on Indian chronology, but who had no inkling that Burmese chronology might run on a different basis from the Indian one.

18 The nadi and vinadi are the equivalent of hours and minutes, where there are 60 nadi instead of 24 hours to a day, and 60 vinadi to a nadi.
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Author:Eade, J.C.; Gislen, Lars
Publication:Journal of Southeast Asian Studies
Date:Sep 1, 1998
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