Greek Astronomical Calendars: III. The Calendar of DionysiosAuthor(s): B. L. van der WaerdenReviewed work(s):Source: Archive for History of Exact Sciences, Vol. 29, No. 2 (1984), pp. 125-130Published by: SpringerStable URL: http://www.jstor.org/stable/41133707 .Accessed: 08/06/2012 06:48

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Greek Astronomical Calendars III. The Calendar of Dionysios

B. L. VAN DER WaERDEN

1. Introduction

In Ptolemy's Almagest we find eight observations of the planets Mercury, Mars and Jupiter dated in the calendar of Dionysios. As an example I shall quote the first observation of Mercury (Almagest IX 9, p. 264 Heiberg):

In the 23rd year of the era of Dionysios, in the morning of the 21st day of Hydron (i.e. in a part of the year corresponding to the zodiacal sign Aquarius) Mercury stood 3 lunar diameters north of the brilliant star (

126 B. L. van der Waerden

Otto Neugebauer2 criticizes Boeckh's reconstruction as follows:

Much speculation has been spent in attempts to reconstruct the complete Dio- nysian calendar, a rather valueless enterprise as long as we have no hope of checking these highly hypothetical reconstructions on additional material.

In my opinion, this harsh judgment is not justified. In what follows it will be shown that Boeckh's reconstruction is not based on "highly hypothetical recon- structions", but on facts. If we accept Ptolemy's double dates, we have no other choice than to agree with Boeckh's conclusions.

2. Ptolemy's Double Dates

Following Boeckh, I shall divide the double dates recorded by Ptolemy into 3 groups:

A. Year numbers An + 1: years 13, 21 and 45. C. Year numbers An + 3: year 23. D. Year numbers An: years 24 and 28.

In Boeckh's group B (years An + 2) there happen to be no dates. The double dates in group A are:

Year 13: Capricorn 25 = Athyr 20/21 morning (Heiberg p. 352) = -271 January 18

Year 21: Scorpio 22 = Thoth 18/19 morning (Heiberg p. 288) = -264 November 15

Year 21: Scorpio 26 = Thoth 22/23 morning (Heiberg p. 289) = -264 November 19

Year 45: Virgo 10 = Epiphi 17/18 morning (Heiberg p. 386) = -240 September 4

In group C we have two entries

Year 23: Aquarius 21 = Choiak 17/18 morning (Heiberg p. 264) = -261 February 12

Year 23 : Taurus 4 = Mechir 30/Phamenoth 1 evening (Heiberg p. 265) = -261 April 25

In group D we have

Year 24: Leo 28 = Payni 30 evening (Heiberg p. 267) = -261 August 23

2 O. Neugebauer: A History of Ancient Mathematical Astronomy (Springer- Verlag 1975), Part Three, p. 1067.

Greek Astronomical Calendars. Ill 127

Year 28: Gemini 7 = Pharmuthi 5/6 evening (Heiberg p. 266) = -256 May 28

From these double dates we see that the years of Dionysios were divided into 12 parts named after the zodiacal signs. In this respect Dionysios followed the tradition of Euctemon and Callippos, who used tropical years beginning with the summer solstice on the day Cancer 1.

The first year of the "Era of Dionysios" began in the Julian year -284. This can be verified by calculating the differences between the year numbers of Diony- sios and the Julian year numbers. In the first half of the Julian years, before the summer solstice, the differences are

13 - (-271) = 284 in January 23 - (-261) = 284 in February and April 28 - (-256) = 284 in May

but after the solstice they are

24 - (-261) = 285 in August 45 - (-240) = 285 in September 21 - (-264) = 285 in November.

Since the years of Dionysios began and ended with the summer solstice, they cannot have been Egyptian years of 365 days. It is easy to check this conclusion by comparing dates in years far apart. For instance we may start with the last double date of group A :

(1) Year 45: Virgo 10 morning = September 4.

Adding 76 days and supposing that the signs Virgo and Libra have 30 days each, as in the parapegmata of Euctemon and Callippos, one obtains:

(2) Year 45: Scorpio 26 morning = November 19

with a possible error of one of two days, because the signs Virgo and Libra might have 29 or 31 days.

Twenty-four years earlier we have the double date

(3) Year 21 : Scorpio 26 morning = November 19.

The time between the dates (2) and (3) is 24 Julian years, or 24 Egyptian years and 6 days, but the Dionysian day numbers in (2) and (3) are exactly the same. It follows that Dionysios did use years of 365-j days, and it also follows that the signe Virgo and Libra had exactly 30 days each.

In group C the difference between the two nights in the Egyptian calendar is 73 days, and the distance between the two Dionysian dates is 3 zodiacal signs minus 17 days. If we assume, tentatively, that the signs Aquarius, Pisces and Aries have 30 days each, and if we assume (with Boeckh) that the Dionysian day did not begin at midnight, but at sunrise or noon or sunset, the agreement between 73 days and 3 signs minus 17 days is perfect.

1 28 B. L. van der Waerden

Still more signs can be shown to have 30 days only. In group D we have the equation

(4) Year 24: Leo 28 evening = August 23.

Adding 279 days and supposing that the signs Leo through Taurus have 30 days each, one obtains

(5) Year 24: Gemini 7 evening = May 28.

Four years later one has from the text of Ptolemy

(6) Year 28 : Gemini 7 evening = May 28.

These two equations are in perfect accordance, since 4 Julian years are equal to 4 Dionysian years, as we have seen. It follows that the ten signs from Leo to Taurus have together exactly 300 days.

This is a very remarkable result. In the parapegma of Euctemon these 10 signs have together 304 days, and in the parapegma of Callippos, which accords well with modern theory, they have together 302 days. If 10 signs have together 300 days, the remaining two signs Gemini and Cancer have together 65 or 66 days : an impossibility in any reasonable theory of the annual motion of the sun. As Neugebauer (note 2, p. 629) rightly notes, the Dionysian zodiacal dates, consid- ered as solar longitudes, "deviate in an irregular fashion both from true and from mean solar longitudes".

Thus we are bound to conclude (with Boeckh) that Dionysios used a division of the year into 1 1 parts (Cancer through Taurus) of 30 days each, and one part (Gemini) of 35 of 36 days. As in the Egyptian and Alexandrian calendars the 5 or 6 "epagomenal days" were placed at the end of the year, which is very con- venient for calendaric calculations.

If this is assumed, we can go back from each of the eight double dates and calculate the beginning of the Dionysian year in each case. This has been done by Boeckh, with the following result ("Sonnenkreise", p. 317):

The years A, B, C, whose Dionysian year numbers are not divisible by 4, begin on June 26, and the years D begin on June 27.

It follows that the years A, B, D have 365 days each, and the years C 366 days. If this is assumed, all double dates are correctly reduced to the Egyptian calen-

dar in Ptolemy's text, with the exception of the first, which should read

Year 13: Capricorn 26 = Athyr 20/21 morning = -271 January 18.

The calendar of Dionysios is not an astronomical calendar in the strict sense, because it is not based on a theory about the sun's course in the zodiac. It is a convenient compromise between the Egyptian calendar with its 12 months of 30 days and 5 epagomenal days, and the more sophisticated zodiacal calendars of Euctemon and Callippos.

Greek Astronomical Calendars. Ill 129

3. Greek Observations in the Third Century B. C.

Many observations of the sun, the moon, the planets and the fixed stars were made in the first half of the third century B.C. in Greece and Hellenistic Egypt:

A. Aristarchos of Samos observed the summer solstice in -279 (Almagest III, 1, p. 207 Heiberg).

B. Timocharis observed Venus, the moon and the fixed stars during the years -294 to -271 (Almagest VII 3 and X 4).

C. The anonymous astronomer who used the calendar of Dionysios observed Mercury, Mars and Jupiter during the years -271 to -240.

D. Aristyllos observed the declinations of fixed stars (Almagest VII 3).

Dennis Rawlins3 has analyzed the observations of Aristyllos by the Method of Least Squares. He found that the observations were extremely accurate, having a standard error of about 4', and that they were made, most probably, later than -275. Whereas Timocharis had observed the declinations of 12 stars, his successor Aristyllos observed 6 other stars. Thus his program was a con- tinuation of that of Timocharis.

The latter also observed occultations of fixed stars by the moon (Almagest VII 3). I suppose that the purpose of those observations was, to study the motion of the moon with respect to the fixed stars. For this purpose, one would need not only the declinations, but also the longitudes of the fixed stars in question. So we may suppose that Timocharis and Aristyllos observed longitudes and declina- tions of fixed stars.

These fixed star data might serve as a reference system for studying the motion of the moon and the planets. In fact, Timocharis himself observed distances between Venus and fixed stars, and our anonymous observer (see under C) observed distances of Mercury, Mars and Jupiter from fixed stars.

Thus one gets the strong impression that the establishment of the calendar of Dionysios and the observations of our four observers all formed part of a pro- gram, directed at the study of the motions of the moon and the planets.

In the history of astronomy, we may observe that the purpose of observations of the moon and the planets always is to determine the values of constants occur- ring in astronomical theories. This is true for Babylonian, Greek, Arabic and western astronomy. Astrologers need panetary tables, and a necessary prerequi- site for computing such tables is the determination of the constants in a theory. Thus it is not too bold to conjecture that the observations made in Samos and Egypt in the third century B.C. had just this purpose.

At that time, two types of theories existed. Linear, or arithmetical methods were used in the Babylonian theories A and B and in Egyptian planetary tables written in the Roman age4. On the other hand, geometrical methods, based on the assumption of uniform circular motions, were developed in Greece and Alex-

3 D. Rawlins: Aristyllos' Date with Vindication, to be published in this Archive. 4 See van der Waerden: Aegyptische Planetenrechnung, Centaurus 16, pp. 65-91 (1972). See also G.Abraham: The Motion of Mars in Egyptian Planetary Tables, to be published in this Archive.

1 30 B. L. van der Waerden

andria from the classical period to the time of Ptolemy. We now may ask: What type of theory did Aristarchos, Timocharis, Aristyllos and our Anony- mous aim at?

For Aristarchos the case is clear. He was the author of the heliocentric theory, in which the planets (including the earth) were supposed to rotate around the sun. If he wanted to determine the constants of this theory, he first had to study the apparent motion of the sun. He could take the durations of the astro- nomical seasons from the parapegma of Callippos, but he had to observe at least one solstitium or equinoctium for his own time, and this he did.

As for Dionysios, his calendar followed the tradition of the Greek parapegma- tists Euctemon and Callippos. His division of the zodiac into 12 signs was a tro- pical division : the sun was supposed to enter Cancer at the time of the summer solstice, whereas in the Babylonian system A the same solstice was located at Cancer 10, and in system B at 8. So the calendar of Dionysios belonged to the Greek and not to the Babylonian tradition.

The unknown author of the observations of Mercury, Mars and Jupiter lived at the time of Aristarchos and used the calendar of Dionysios. He used, like Dionysios, the Greek division of the zodiac, in which the sign Cancer begins with the summer solstice. Later Egyptian planetary texts all used the Babylonian siderial division of the zodiac, and a sidereal year longer than 365-J- days. For these reasons, it is not likely that the planetary observations were made in order to determine the constants of a Babylonian type theory.

What geometrical planetary theories were available between -271 and -240, when these observations were made?

The "primitive epicycle theory" discussed in my paper5, which was probably invented by the Pythagoreans, cannot be used to explain the motion of Mars, as Aaboe6 has shown. The same holds for the homocentric spheres of Eudoxos and Callippos, as Schiaparelli7 proved. Thus, if our observers in Alexandria had a definite geometrical theory in mind, the most probable candidate is the heliocentric theory of Aristarchos of Samos.

Hence my hypothesis: The purpose of the observation of Aristarchos, Timocharis, Aristyllos and our Anonymous was to determine the constants in the heliocentric theory of Aristarchos.

Wiesliacher 5 CH-8053 Zrich

(Received February 2, 1983)

5 Van der Waerden: The Motion of Venus, Mercury and the Sun in Early Greek Astronomy, Archive for History of Exact Sciences 26, pp. 99-113 (1982). 6 A. Aaboe: On a Greek Qualitative Model of the Epicycle Variety, Centaurus 9, pp. 1-10 (1963). 7 G. V. Schiaparelli: Scritti sulla stona della astronomia antica II, pp. 3-112. German translation: Abhandlungen zur Geschichte der Math. I (1877), Supplement to Zeitschrift fr Math, und Phys. 22, pp. 101-198.

Article Contentsp. [125]p. 126p. 127p. 128p. 129p. 130

Issue Table of ContentsArchive for History of Exact Sciences, Vol. 29, No. 2 (1984), pp. 101-199Front MatterGreek Astronomical Calendars: I. The Parapegma of Euctemon [pp. 101-114]Greek Astronomical Calendars: II. Callippos and his Calendar [pp. 115-124]Greek Astronomical Calendars: III. The Calendar of Dionysios [pp. 125-130]The Lemniscate and Fagnano's Contributions to Elliptic Integrals [pp. 131-149]On the History of the Statistical Method in Astronomy [pp. 151-199]