waerden heliocentrism

The Heliocentric System in Greek, Persian and Hindu Astronomy

B . L . VAN DER WAERDEN Wieslacher 5

8053 Zurich, Switzerland

INTRODUCTION

N THE YEARS 295-283 B.c., Timocharis observed positions of the moon and I of Venus with respect to the fixed stars.’ About 280 B.c., Aristarchos of Samos proposed the heliocentric hypothesis.2 At about the same time, a man named Dionysios established an astronomical calendar based on the assumption of a tropical year of 365 y4 days3 An anonymous observer used this calendar to date his observations of Mercury, Mars, and Jupiter made in the years 270-239 B.c .~ The observations of Timocharis and those of the unknown observer complement each other nicely: Ptolemy was able to use the observations of Timocharis for the moon and Venus, and the anonymous observations for Mercury, Mars, and Jupiter.

In another case too we have two series of observations that complement each other. Timocharis observed the declination of twelve fixed stars, and Aristyllos observed the declination of six more stars.5 From independent in- vestigations by Rawlins and Maeyama6 we may conclude that the observations of Aristyllos were made with great accuracy between 280 and 240 B.C. We may suppose that Timocharis observed his declinations between (say) 290 and 270 B.c., and that Aristyllos continued the program of Timocharis.

It seems that here a team was at work, consisting of three observers and a calendar maker (or perhaps of three observers only). What was the purpose of their teamwork?

In all cases in which we know the whole story, the purpose of astronomical observations is always: to determine the constants of an astronomical system, with the ultimate aim of computing tables based on that system. For instance, Ptolemy used the observations of Timocharis and Aristyllos and of our anonymous observer in order to determine the constants in his theory, and next he composed ”handy tables” for the use of astrologers. The same order of events can be observed in Babylonia, in India, in the Islamic realm, and in Western Europe. So it is not unreasonable to suppose that the purpose of the observations made in Alexandria between 295 and 239 B.C. was: to de-

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526 ANNALS NEW YORK ACADEMY OF SCIENCES

termine the constants of an astronomical theory in order to compute tables based on that theory.

Now if this is assumed, we may ask: What kind of theory was it that the observers had in mind? In a recent paper7 I have shown that the most probable candidate for this theory is the heliocentric theory of Aristarchos. Thus the question arises: Can we find traces of tables based on the heliocentric system?

To answer this question I shall first discuss the ancient testimonies concerning Aristarchos of Samos and Seleukos of Seleukia, next the work of the great Hindu astronomer Aryabhata, and next the Persian table set Tables of the Shah.

ARISTARCHOS OF SAMOS

The testimonies about Aristarchos have been collected and commented upon by Thomas Heath.8 I shall use his translations. The earliest, most important testimony is from the Sand-Reckoner of Archimedes:

You (King Gelon) are aware that "universe" is the name given by most astronomers to the sphere, the centre of which is the centre of the earth, while its radius is equal to the straight line between the centre of the sun and the centre of the earth. This is the common account (TU ypacpopwa), as you have heard from astronomers. But Aristarchus brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the "universe" just mentioned. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface.

As Archimedes himself notes, the last statement cannot be taken literally. A point cannot have a ratio to a surface area. What is meant is, that our distance to the fixed stars is so large as compared with the diameter of the orbit of the earth that one can assume that orbit to be concentrated in one single point. For Aristarchos this assumption was necessary, because otherwise the apparent distances between fixed stars would vary in the course of the year.

The next testimony is from Plutarchos, De facie in orbe Lunae, c.6,

Only do not, my good fellow, enter an action against me for impiety in the style of Cleanthes, who thought it was the duty of Greeks to indict Aristarchus of

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VAN DER WAERDEN: THE HELIOCENTRIC SYSTEM 527

Samos on the charge of impiety for putting in motion the Hearth of the Uni- verse, this being the effect of his attempt to save the phenomena by supposing the heaven to remain at rest and the earth to revolve in an oblique circle, while it rotates, at the same time, about his own axis.

Plutarchos is certainly right in saying that Aristarchos assumed a rotation of the earth about its own axis, for in a heliocentric system one is forced to assume as axial rotation of the earth.

SELEUKOS OF SELEUKIA

Most ancient astronomers rejected the idea of a motion of the earth, for reasons explained by Aristotle in De cuelo and by Ptolemy in Almagest 1.7. The only astronomer who is known to have accepted the theory of Aristarchos is Seleukos of Seleukia. What do we know about his life and work?

In Chapter XVI of his Geographia, Strabo mentions several "Chaldaen" astronomers. At the end he adds: "Seleukios of Seleukia was a Chaldaean too."

As Strabo himself notes, the word Chaldaean has two meanings. Firstly, a Chaldaean people existed. A Chaldaean dynasty, to which the great king Nebuchadnezar I1 belonged, reigned in Babylonia from 626 to 539 B.C. Sec- ondly, Babylonian astrologers and astronomers were often called "Chaldaeans." Strabo calls them "the so-called Chaldaeans". Their writings were translated into Greek and used by later authors like G e m i n ~ s . ~

The "Chaldaean" astronomers mentioned by Strabo are Kidenas, Naburianos, Sudines, and Seleukos. The first two are also known from astronomical cuneiform texts under their Akkadian names Nabu-Rimannu and Kidinnu.

Among several cities named Seleukia, the best known is Seleukia on the Tigris, the capital of the Seleucid kingdom. It is possible that the astronomer Seleukos lived or was born in this city, but it is also possible that his native town was Seleukia on the Erythrean Sea, for the doxographer "Aetios," fol- lowed by Stobaios, calls him XFXEUKOS b 'Epu9paioq.10 He is said to have observed the tides. According to Strabo (1.1.9), he stated that the tides are due to the attraction of the moon, and he also knew that the height of the tides depends on the moon's position relative to the sun.l1

As an astronomer, Seleukos seems to have been held in considerable es- teem. According to Stobaios (Eclogae physicue, 1.21), he assumed the universe to be infinite.12

Concerning the relation between Aristarchos and Seleukos we have an important testimony of Plutarchos. In his discussion of a statement of Plato on the motion of the earth (Quuestiones Plutonicu, Qu. 8) , he writes:


XLVI . . . i 6 E I TfiV YqV, ihhOp6WlV K E p i TOV 6lU 7tUVTOV 7lOhOV TETapkVOV, pfi pEpqXavqaOa1 ouv~xopkvqv, ~ a i pkvouoav, bnha orp~cpopkvqv Kai bmhoup6vqv voEiv' bq O D T E ~ O V Apiotapxoq K a i E ~ ~ E U K O ~ hnoSsiKvuoav' 6 ~ Z V , GnoeEp~voq p ~ v o v * b oz C ~ ~ E V K O ~ , K a i hocpaivbpEvog.

In my translation of the first half of this sentence I shall follow Heath.I3 For the last part, in which Aristarchos and Seleukos are quoted, I shall present my own translation, and justify it afterwards. I propose the following translation:

Was it necessary to conceive that the earth "rolling about the axis stretched through the universe" [this is a quotation from Plato] was not represented as being held together and at rest, but as turning and revolving, as Aristarchos and Seleukos afterwards proved, the former stating it only as a hypothesis, the latter showing it by reasoning?

In the last part of this passage, Plutarchos uses three verbs to characterize what Aristarchos and Seleukos did, namely ~ Z O S E ~ K V U W L , h o ~ i e q p t , irnocpaivo.

Heath translated a form of the first verb as "maintained," but the usual meaning is "proved." For the second verb I have adopted the translation of Heath "stating as a hypothesis." The third verb can have the meaning "stating as a definite opinion," but my translation "showing by reasoning" is equally possible. I shall now explain why I prefer the latter translation.

Plutarchos says first that Aristarchos and Seleukos both "proved" ( ~ T O S E ~ K W O ~ V ) the motion of the earth. As Heath rightly remarks, one cannot "prove" an astronomical hypothesis in the strict sense of the word. What one can do, and what Copernicus actually did, is: to deduce consequences from the hypothesis and to verify them empirically. I now suppose that this is what Aristarchos and Seleukos did according to Plutarchos, and that he denoted this activity by the verbs ~ Z O S E ~ K V U ~ ~ and irnocpaivo.

In this connection, I may note that Ptolemy, speaking of the activity of earlier astronomers who composed certain "eternal tables," uses a related verb b ~ t s ~ i ~ v u p t in just this sense. In Almagest IX.2, Ptolemy critizes these astronomers, saying that they "tried to explain the phenomena by assuming eccentric circles or concentric circles carrying epicycles or even - by Zeus! -a combination of the two." They "declared the anomaly connected with the ecliptic to be so and so large, and the anomaly relative to the sun so and so large," and they wanted "to prove the uniform motion on circles" through so- called "eternal tables".

Now the verb ~ S E ~ K W ~ I , which I have translated by "to prove," is a synonym of b ~ o t i ~ i ~ v u p t . I suppose that both verbs, used by Plutarchos and Ptolemy, have the same meaning, namely: to compute planetary positions from a geometrical theory, and to compare them with reality.


Now let's turn to Plutarchos. He first says that Aristarchos and Seleukos both "proved the heliocentric hypothesis. Next he corrects himself and makes a distinction between the two. He says that Aristarchos only stated it as a hypothesis, which implies that he did not prove it in the sense just explained. The first assertion that both proved it is now restricted to Seleukos alone, but instead of ~ O ~ E ~ K V U ~ I he uses the synonym hnocpaivo, or rather the medium bnocpaivovat, which according to Liddell and Scott can be used like the active.

I suppose that Aristarchos did not dispose of a sufficient number of observations to determine the constants of his theory. Also, the accurate determi- nation of these constants and the calculation of planetary positions requires trigonometrical methods, which were not available at the time of Aristarchos. On the other hand, if we suppose that Seleukos was a near-contemporary of Hipparchos (second century B.c.), trigonometrical methods and accurate observations were available at his time, so that he was in a position to "prove" the heliocentric hypothesis in the sense just explained.

The translation adopted by Heath: knocpatvop~voq = "stating as a definite opinion" is philologically possible, but in my opinion improbable. Seleukos was a competent astronomer. He knew that the motion of the earth is a hypothesis and not a certain fact. He knew that other hypotheses explain the phenomena just as well. Aristarchos, the initiator of the heliocentric theory, had called it a hypothesis only. Why should his follower Seleukos enunciate it as a definite assertion? This would be a strange and unusual behavior. The astronomer Geminos, quoted by Simplikios in his commentary on the Physics of Aristotle (ed. Diels, 291-92), says that it is the task of the astronomer to introduce hypotheses to explain the phenomena, leaving it the physicist to investigate "what is by nature suited to a position of rest and what sort of bodies are apt to move."

For these reasons I shall assume that SeIeukos determined the constants in the heliocentric theory and developed methods to compute planetary positions from this theory. Maybe he computed tables himself, or he enabled his successors to compute tables for the need of astrologers.

TRACES OF THE HELIOCENTRIC THEORY I N THE WORK OF ARYABHATA

GENERAL CONSIDERATIONS

The astronomical system of Aryabhatal4 was composed about A.D. 510 or a little later.15 It is based on the assumption of epicycles and eccenters, so it is not heliocentric, but my hypothesis is that it was based on an originally


heliocentric theory. Arguments in favor of this view will be presented in the sequel.

In the history of astronomy, we may observe a tendency to get away from the idea of a motion of the earth. It is always possible to transform a heliocentric theory into a equivalent geocentric theory by putting the earth at rest, while retaining the relative motions of the sun and the planets as seen from the earth. For instance, Tycho Brahe transformed the system of Copernicus into a geocentric system in just this way.

Another example. h b h a t a assumed a rotation of the earth. His followers accepted his methods of calculation, but they put the earth at rest.

Consequently, if we consider a set of tables or a set of rules for computing planetary positions, it is often not possible to see whether the underlying theory is heliocentric or geocentric. However, there are details which may give us more information about the origin of the theory.

THE FIRST TRACE

One such detail is the axial rotation of the earth. In a geocentric system there is no reason to assume such a rotation. The assumption of an axial rotation does not simplify the system, and there are strong arguments against this assumption (see e.g., Almagest 1.7). On the other hand, in a heliocentric system one is forced to assume a rotation of the earth. Now, if a heliocentric theory is transformed into a geocentric theory, one may or may not retain the axial rotation.

Most Hindu astronomers did not assume a rotation of the earth, but Ary- abhata did assume it. We can explain this by supposing that his theory was derived from a heliocentric theory.

Of course, this is not a conclusive proof: it is only an indication of a possible explanation of a curious fact.

THE SECOND TRACE

To explain what I mean by the second trace I must first examine the trans- formation of a heliocentric into a geocentric theory a little more closely.

Let me start with Venus. In the earliest epicycle theory16 the sun rotates on a small epicycle about the "mean sun," and Venus rotates on a larger epicycle in such a way that the centers of the two epicycles are always in one line from the earth. An equivalent theory is explained by Theon of Smyrna in his Ex- positio rerum mathematicarum ad legendum Platonem utilium (ed. Hiller,


FIGURE 1.

186-87): in this the epicycles of Venus, Mercury and the sun are concentric. In both theories the center of the Venus epicycle is in the direction of the mean sun.

Now let us consider the heliocentric theory. I feel we may safely assume that Aristarchos made the earth move in an eccentric circle, for the anomaly of the apparent motion of the sun was known already at the time of Kal- 1ipp0s.l~ On the other hand, the eccentricity of the orbit of Venus is very small, so it is not unreasonable to suppose that Aristarchos made Venus move on a concentric circle about the sun.

Now if this model of the motion of the earth is transformed into a geocentric theory, one had to make the sun move on an eccentric circle and to make Venus move on an epicycle having its center at the center of the sun. Thus, the eccentric cycle carrying the epicycle of Venus will have the same apogee and the same eccentricity as the sun’s orbit (see FIGURE 1).

In a genuine epicycle theory for Venus, the eccenter carrying the epicycle is independent of the sun’s orbit. Its apogee and eccentricity are determined from observations of Venus, whereas the apogee and eccentricity of the sun are determined from eclipse observations. The probability that the apogee and eccentricity of Venus coincide with those of the sun in very small. In fact, in Ptolemy’s theory they do not coincide, nor do they coincide in the Aryabha- tiya of Aryabhata. According to the latter treatise the apogee of the sun is at 78 degrees, and that of Venus at 90 degrees. The eccentricities are also


FIGURE 2.

different: see the table of magnitudes of the "manda epicycles" on page 24 of Shukla's translation of the Aryabhatiya.ls

However, the same h a b h a t a also conceived another astronomical system, the "Midnight System," so called because in this system the Kaliyuga was supposed to begin at midnight. The constants of this system are known from Brahmagupta's treatise Kha~dakh~dyakalg and from the "old Siiryasid- dhiinta" of Varaha Mihira.20

The relation between this Midnight System and the more sophisticated Sun- rise System underlying the Aryabhatiya have been carefully investigated by P.C. Sengupta.21 In the Midnight System the apogees of the sun and Venus are both at SO', and their eccentricities are also equal. This fact can be explained by assuming that the Midnight System was originally derived from a heliocentric system.

THE THIRD TRACE

In an epicycle theory, in which the planets move on epicycles carried by eccentric circles, the position of any planet is determined by two angles x and y (see FIGURE 2). The first angle x is the angle ACS, where A is the apogee of the eccentric circle, C its center (or in Ptolemy's system the equant point), and S the center of the epicycle. The second angle y is the angle BSP, where P is the planet, while SB is the prolongation of the line CS.


FIGURE 3.

The angles x and y are linear functions of time. Their values at any instant may be calculated from tables of mean motions. As soon as they are known, the direction EP can be computed by plane trigonometry.

In the theory of Aryabhata, the eccentric circle is replaced by a second epicycle, the "manda epicycle", with centre M (see FIGURE 3) . If one draws the parallelogram ECSM, one sees that it makes no difference whether the point S moves on an eccentric circle centered at C or on a small epicycle carried by a circle centered at E.

For the trigonometric calculation of the true latitude of any planet, it does not matter whether one starts with a heliocentric or with a geocentric theory. However, there is one characteristic difference. In a genuine geocentric theory the primary quantities determining the configuration at any time are the angles x and y. The period of x is the sidereal period of the planet, and the period of y is the synodic period. On the other hand, in a heliocentric theory of Venus (or Mercury) the primary quantities, which determine the positions of the earth and Venus as seen from the sun, are not x and y, but

L = u + x = mean longitude of sun

and

z = u + x + y = heliocentric longitude of Venus,

where u is the longitude of the apogee. The period of z is the heliocentric period


of Venus. In the heliocentric theory, this is a fundamental quantity. On the other hand, in the geocentric theory, this period has no importance whatever.

Now let us look at the text of the Aryabhatiya. Stanzas 3-4 of the first book read in the translation of Shukla:

3-4. In a yuga, the eastward revolutions of the Sun are 43,20,000; of the Moon, 5,77,53,336; of the Earth, 1,58,22,37,500; of Saturn 1,46,564; of Jupiter, 3,64,224; of Mars, 22,96,824; of Mercury and Venus, the same as those of the Sun; of the Moon's apogee, 4,88,219; of [the Sighrocca of] Mercury, 1,79,37,020; of [the Sighrocca of] Venus, 70,22,388; of [the Sighroccas of] the other planets, the same as those of the Sun; of the moon's ascending node in the opposite direction [ i e . , westward], 2,32,226.

The words "the Sighrocca of" are not in the text: they are added by the translator. Aryabhata himself just says "of Mercury 17 937 020, of Venus 7 022 388." Clark's translation agrees with that of Shukla.

Obviously, the revolutions of Mercury and Venus considered by h a b h a t a are heliocentric revolutions.

Because of these three traces, I think it is highly probable that the system of Aryabhata was derived from a heliocentric theory by setting the center of the earth at rest.

HELIOCENTRIC PERIODS IN PERSIAN SOURCES

Some twenty years ago, E. S. Kennedy and I analyzed an astronomical system described by Al-Biriini and Al-Sijzi and ascribed, in both sources, to the Book of the Thousands of AbU Ma'shar al-Balkhi.22 This system is based on the assumption of a "world-year" of 360 000 years. In this period, all planets are supposed to perform an integer number of revolutions, starting from positions near 0" Aries at the beginning of the period.

The numbers of revolutions of the planets in 360 000 years according to Abii Ma'shar are recorded by Al-Hashimi and Al-Biriini. In the following table I reproduce the parameters from Pingree's studyz3 and compare them with those derived from the "midnight system" of Aryabhata:

Midnight System - Abii Ma'shar ___ Planet Saturn 12 214 12 214 - % Jupiter 30 352 30 352 - % Mars 191 402 191 402 Venus 585 199 585 199 Mercury 1 494 751 1 494 750

4 812 778 Moon 4 812 778 __


It is seen that the numbers ascribed to Abii Ma'shar agree exactly with those of Aryabhata in three cases, and nearly in three other cases. Also, in both systems, a conjuction of all planets in 3102 B.C. is assumed. According to Abii Ma'shar the conjunction took place in the early morning (at midnight for Babylon) on Thursday, February 17, but according to Aryabhata it took place at the end of the same day, at midnight for Lanka.

The system of Abu Ma'shar is more primitive and less accurate than that of Aryabhata, mainly because of the shorter period of 360 000 years. The longer period of Aryabhata allows a better adaptation to observed longitudes.

It is very remarkable that Abti Ma'shar has only one number of revolutions for every planet, and that his revolutions are heliocentric revolutions. In Babylonian astronomy and in the system of Ptolemy every planet has two periods: a sidereal and a synodic period. The only known theory in which every planet has only one period of revolution is the heliocentric theory. There- fore I suppose that the astronomical system adopted by Abti Ma'shar was derived from a heliocentric theory.

Concerning the origin of this system we have several testimonies. Al-Sij- zi, in his summary of Abti Ma'shar's Book of the thousands, ascribes the system to "the Persians and some of the Babyl~nians."~~ In another version of al-Sijzi's summary, only the "Persians" are mentioned. Still another, most important testimony is found in al-Biriini's Chronology. Al-Birtini discusses Abu Ma'shar's "star-cycles" of 360,000 years and says that Abii Ma'shar "had computed these star-cycles only from the motions of the stars, as they had been fixed by the observations of the Persians" (my italics).25

Already elsewhere I have investigated the identity of the "Persians" and the "Babylonians" mentioned in our testimoniesl26 I shall now discuss the two cases separately.

THE PERSIANS

In the extant treatises of Al-Biriini the expression "the Persians" occurs very often. According to E. S. Kennedy, "the Persians" is a generic term com- prising Ya'qtib ibn Tariq and Al-Khwarizmi and the Zij-i Shiih.Z7 The latter is a table set composed in Sassanid Persia under Khusro Aniishirvan (531-579) and revised under Yazdigerd I11 (632-642). In most cases, when Bi- riini refers to "the Persians," he just means the authors of the Tables of the S h d ~ . ~ ~ These tables are lost, but their contents can largely be reconstructed from other sources.29 Burckhardt and I have shown that the Tables of the Shah are based on the assumption of a conjunction of all planets at Babylon, midnight between February 16 and 17, 3102 B.C. The same date was also as-


sumed in the system of Abii Ma'shar. So, we may conclude that Abu Ma'shar's system was based on the Tables of the Shah.

THE BABYLONIANS

According to Al-Sijzi, the world-year of 360 000 years and the cycles of Abii Ma'shar are due to "the Persians and the Babylonians." Is this statement reliable? It occurs within a passage in which several world-years are compared, namely

(1) a period of 4 320 millions of years, ascribed to "those in a region of

(2) a period of 4 320 000 years, ascribed to "the partisans of Arjabhaz," (3) a period of 360 000 years, ascribed to "the Persians and some of the

The first two ascriptions are correct. The period (1) is the "Kalpa" of Brah- magupta, and the period (2) is the Mahsyuga" of Aryabhata (= Arjabhaz). As far as the Persians are concerned, the ascription (3) is confirmed by the testimony of Al-Biriini. So let us see whether we can make sense of the ascription to "the Babylonians."

Elsewhere I have gathered several references to the "Babylonians" in Greek and Arabic s0urces.3~ From these references it results that we have to distin- guish between the "Chaldaeans" and the "Babylonians". For instance, the astrologer Vettius Valens ascribes to the "Babylonians" a certain duration of the year, namely

365 + 1/4 + sari days,

and he ascribes to the "Chaldaeans" another duration of the year. I have also shown that the "Chaldaeans" are early Hellenistic authors, whose methods are closely related to those used in the cuneiform texts, whereas the "Babylo- nians" were late Hellenistic auth0rs.~1

Al-Biriini ascribes to the "Babylonians" a method to compute the rising times of the zodiacal signs. I fully agree with Neugebauer32 that this method is a late Hellenistic invention, based on the insight that the earth is spherical and may be divided into parallel zones having different "climata."

One of the Babylonians was "Teukros the Babylonian," of whom we have extensive fragments in Greek as well as in Arabic. According to Franz Boll,33 who edited the fragments in his book "Sphaera," Teukros must have lived in the first century B.C. or slightly earlier. In Arabic sources he was called "Tinkelos the Babylonian" or "Tinqeros the Babylonian."34

The work of Teukros was well known in Sassanid Persia and in the Islamic

India,"

Babylonians".


world. A treatise of Teukros concerning the shapes of the 36 zodiacal decans was translated from Greek into Persian, and from Persian into Arabic. The Arabic translation is contained in the "Great Introduction" of Abii Ma'shar. The Greek text of the treatise as well as the Arabic text and a Greek translation, was published by B O ~ I . ~ ~

Abii Ma'shar's direct source was, without any doubt, a Persian text, for he mentions several Persian names of decans. In his introduction he ascribes the text to "the old learned men of Persia, Babylonia, and Egypt."

In Abii Ma'shar's text, the author Teukros is mentioned several times, and no other Babylonian author is mentioned. From this, we may tentatively conclude that for Abii Ma'shar the expression "the learned men of Babylon" is just a synonym of "Teukros the Babylonian." If we assume that the same thing holds for the expression "some of the Babylonians" used in al-Sijzi's excerpt from Abii Ma'shar, we may conjecture that the latter derived his "World- Year" of 360 000 years from a lost treatise of Teukros.

Not let us return to Aryabhata.

c o N N E c T I ON s B E T w E E N XRY A B H A T A s "M I D N I G H T s Y s T E M " A N D T H E zri-I S H ~ H

Aryabhata's fundamental period, the Mahiiyuga, is just 12 times the Per- sian "world-year" of 360 000 years, and his numbers of revolutions are exactly or nearly 12 times those of Abii Ma'shar, as we have seen. The multiplica- tion of the period by 12 enabled Aryabhata to attain a better adaptation to the observed phenomena.

The Tables of the Shah, the system of Abii Ma'shar, the Midnight System, and the Sunrise System of Aryabhata all have in common the assumption of a conjunction of all planets in mid-February, 3102 B.C. The assumed dates of this conjunction are slightly different in our sources, namely: midnight February 16/17 Babylon in the system of Abu Ma'shar and in the Tables of the Shuh, midnight February 17/18 Lanka in the "midnight system" of Aryabhata, sunrise February 18 Lanka in the "sunrise system" of the h y a - b hatiya .

The difference between the dates are due to different assumptions concerning the duration of the year. The differences between the durations according to Abii Ma'shar and Aryabhata amount to approximately 1 day in the 3 600 years between the conjunction and the lifetime of Aryabhata. This explains the difference of one day in the dates of the conjunctions.

As Kennedy has shown in his paper on the Tables of the Shah, the maxima of the anomalies of the planets in that work were nearly the same as in the


"midnight system" of h a b h a t a . Al-Biriini informs us that in the case of the sun and the moon these maxima "came from the Hindus to the Persians."36 Thus we may assume the following dependence:

Aryabhata

I Tables of the Shah I This direction of the dependence is chronologically correct. h a b h a t a

reached the age of 23 years in A.D. 499. On the other hand, we know from al-Bifini,j7 that Khosro Anoshirvan convoked an assembly of astronomers in A.D. 556 and ordered them to correct the Tabfes of the Shah. Al-Hashimi informs us38 that Khosro compared the A h a g e s t with the Arkand, that is, with the midnight system of Aryabhata, and found that this system accords better with the observations that the Almagest. So he ordered the astronomers to compute tables according to the Arkand. Thus, all our sources agree that the Tables of the Shah are based on the midnight system of Aryabhata.

However, if we take at its face value Bifini's statement that the astronomers were ordered to "revise" the tables, we must conclude that an earlier version of the tables existed that was not dependent of the Arkand. This conclusion is confirmed by a statement of Ibn Yiinus to the effect that the Persians "observed the solar apogee at 77" 55' about A.D. 450, that is, half a century before the time of A r ~ a b h a t a . ~ ~

The epoch of the Tables of the Shah was midnight Babylon.40 This seems to indicate that the tables were based on a Babylonian treatise (perhaps by Teukros) .

THE CONJUCTION OF 3102 B.C.

As we have seen, Aryabhata and the authors of the Tables of the Shah assumed a conjunction of all planets in mid-February, 3102 B.C. In a recent paper41 I have shown that this alleged conjunction was not observed, but calculated from a planetary theory. It cannot have been observed, because about this time no such conjunction took place.

In the same paper I have argued that the conjunction of 3102 was calculated by a Hellenistic astronomer or astrologer who wanted to date the Deluge. According to al-Bifini,*Z "the sages among the inhabitants of Babylon and the Chaldaeans" have made several attempts to date the Deluge by calculating conjunctions of Jupiter and Saturn. Some of these astrologers assumed 3580


B.C. to be the date of the Deluge. Others, including Abii Ma'shar, supposed that a Deluge would occur once in every 180 000 years, and that the last deluge before their own time had occurred in February 3102 B.C.

All calculations of Jupiter-Saturn conjunctions are based on the calculation of mean longitudes. In ancient Babylonian tables, as we know them from cuneiform texts, the notion "mean longitude" does not occur. On the other hand, in Greek tables such as the Handy tables of Ptolemy the prescription is: First compute mean longitudes and next add or subtract correction terms. Tables of this kind require trigonometry, so they cannot have existed before Apollonios (about 200 B.c.). Only after 200 B.C. has the calculation of not observed conjunctions become possible.

k t me summarize my conclusions. After the discovery of trigonometry, tables were calculated by trigonometrical methods. By the aid of these tables, several Hellenistic authors tried to date the Deluge by calculating conjunctions of Saturn and Jupiter in the fourth millennium B.C. One of these attempts led to the discovery of an approximate mean conjunction of all planets in 3102 B.C. Next, a new theory and new tables were fabricated, based on the assumption of a mean conjunction of all planets in February of this year, and of an exact repetition of all planetary positions at the end of a certain World-Year. These tables were used, with corrections, in Sassanid Persia. Aryabhata cor- rected the theory, replacing the Persian world-year of 360 000 years by a period twelve times as large.

ETERNAL TABLES

The astronomical systems of Brahmagupta, h b h a t a , and Abii Ma'shar are periodic, that is, after the completion of a certain number of years the motions of the planets are repeated. Since the cycles of Aba Ma'shar are derived, according to al-Birtini, from those of the "Persians," we may safely assume that in the earliest version of the Tables of the Shah the motions were also periodic with a period of 360 000 years. A table set based on such a period may well be called "perpetual" or "eternal": If the motions in a world-year are known, the motions, and hence the positions, are known for all times.

Such "eternal tables" are mentioned by Vettius V a l e n ~ ~ ~ and Pt01emy.~~ I feel that the genesis of the system of Abii Ma'shar with its world-year of 360 000 years can best be understood if we assume that the "eternal tables" used by Vettius Valens were already based on the assumption of a similar world-year.

Now let us compare the way in which the anomalies of the planets are computed according to h a b h a t a with the indications of Ptolemy concerning the


A

FIGURE 4.

calculation of the same anomalies in the ”eternal tables.” Although the indications of Ptolemy are rather vague, we shall see that they agree well with what we know from the Aryabhatiya.

To pass from the mean longitude of any planet to its true longitude, one has to add correction terms, the so-called “equations.” In the Aryabhatiya two kinds of equations occur, namely

(1) the mandapala or ”equation of axis” f, a function of the angle x in FIGURE 4,

(2 ) the 4ighrapala or Qighra equation g , a function of the angle y in FIGURE 5 . In Aryabhata‘s “double epicycle theory” we have two kinds of epicycles:

the manda epicycle from which the mandapala is calculated, and the Sighra epicycle from which the Sighrapah is calculated. Both epicycles are carried by concentric circles. Some years I explained the rules by which the corrections from mean to true longitudes are found according to Aryabhata, and I have shown that these rules can be understood as yielding a reasonable approximation if one supposes the center S of the epicycle to move on a eccentric circle with equant point, as in the theory of Ptolemy. The inventor of this theory and of the approximation used by Aryabhata must have been an excellent mathematician; my guess is that it was Apollonios of Perge.

A remarkable feature of this method is the fact that the two equations are


FIGURE 5.

computed separately, not from a combined figure such as FIGURE 2. More pre- cisely, they are found separately in a first approximation. Next they are halved and used to correct x and y, and then the final values of the manda and S i g h equations are found. In this way, the use of two-argument tables is avoided.46

Now let us consider, once more, what Ptolemy says about the authors of the "eternal tables". According to P t ~ l e m y , ~ ~ these authors "tried to explain the phenomena by assuming eccentric circles or concentric circles carrying epicycles or even - by Zeus! -a combination of the two." These words are well in accordance with our FIGURES 4 and 5, for the motion on the manda epicycle of FIGURE 4 is equivalent to the motion on an eccentric circle, and FIGURE 5 shows a concentric circle carrying an epicycle.

Ptolemy continues: "They declared the anomaly connected with the ecliptic to be so and so large, and the anomaly relative to the sun so and so large." This means: they calculated the two anomalies separately, whereas Ptolemy taught how to combine the two anomalies in a geometrically correct way.

But, says Ptolemy, their method was completely wrong, so that "some of them missed their aim completely, whereas others succeeded to a certain extent." In my opinion, h a b h a t a and his predecessors belonged to those who "succeeded to a certain extent," for their methods of calculation yielded a reasonable good approximation, as I have shown previously.48


CONCLUSION

Combining the results of the preceding sections, we may draw a scheme of the probable development:

Seleukos

Tables Based on the Heliocentric Theory

I

Discovery of the Conjunction of 3102 B.C.

Eternal Tables (Periodic)

"Some of the Babylonians" (Teukros the Babylonian?)

1 Early Persian Tables

Tables of the Shcih

Abii Ma'shar

NOTES 1. Almagest, VII.3 and X.4. 2. See Heath 1913 for details. 3. See van der Waerden 1983. 4. Ahagest I X , 264-67, 288-89, 352 and 386 (Heiberg). 5. Ahagest, VII.3. 6. Rawlins, D. (unpublished). See also Maeyama.


7. See van der Waerden 1983. 8. Heath 1913. 9. See van der Waerden 1972. 10. See Cumont. 11. See also Kroll in Pauly-Wissowa’s Real-Encyclopaedie, Supp. 5, cols. 962-63. 12. See also Pines. 13. Heath 1932, 108. 14. On this, see Sengupta, Clark, and Shukla. 15. In Billard it is shown that Aryabhata‘s theory is based on observations made about 510. 16. See van der Waerden 1974. 17. ibid. 18. See Shukla. (I have not seen the translation of Bina Chatterjee.) 19. Listed as Sengupta 1934. 20. Listed as Neugebauer and Pingree. 21. See Sengupta 1929. 22. See Kennedy and van der Waerden. 23. Pingree, 30. 24. See Kennedy and van der Waerden. 25. Sachau, 29. 26. See van der Waerden 1977. 27. See Kennedy. 28. See van der Waerden 1977. 29, See Kennedy and Burchhardt and van der Waerden. 30. See van der Waerden 1977. 31. See van der Waerden 1972. 32. See Neugebauer 1942. 33. See Boll. 34. Boll, 10 and Pingree, 10-11. 35. Boll, 25-30 and 490-539. 36. Burckhardt and van der Waerden, 4. 37. Kennedy and Saffouri and Ifram, Section 24:7-11. 38. Kennedy and Pingree and Haddad, 95, 11. 7-23. 39. Kennedy and van der Waerden, 323. 40. See Kennedy 1958. 41. See van der Waerden 1980. 42. Sachau, 28. 43. See Neugebauer 1975, 789. 44. Alrnagest, IX.2, 211 (Heiberg). 45. See van der Waerden 1961. 46. See Burckhardt and van der Waerden, or van der Waerden 1961. 47. See note 44 above. 48. See van der Waerden 1961.

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