THE CHRISTMAS LIE: It's Bigger Than You Think



Astronomy and Astrology Among the Jews in Antiquity
Meir Bar-Ilan

The prophets and the sages were aware of at least seven different stars and constellations (Jer. 44:17-19; Amos 5:8, 5:26; Job 9:9; 38:32).

The way the luminaries are said to have been created (Genesis 1:14) shows awareness of the role of both luminaries in the calendar. The story of the Flood (Gen. 7-8) reveals calendrical thinking, as of an awareness of 11 days gap between a lunar and a solar year (according to the Massoretic text). The five “stops” in the Flood narrative shows awareness of (a solar) calendar.

Theophrastos (372-288/7 BCE), a disciple of Aristotle, wrote that the Jews watch the stars at night and pray to God. Since Theophrastos wrote in the context of sacrifices, it is evident that he was writing about priests in the Temple where the exact date had significance for the ritual, sacrifices and festivals (=calendar). Though there is no other evidence that priests were engaged in Astronomy (Psa. 8:4 is not sufficient), it goes well with the Mesopotamian tradition where there was a connection between astronomers and priests. Moreover, before the destruction of the Temple (in 70 CE) there was a priestly court in Jerusalem where testimonies of the sighting new moon were analyzed (m. Rosh Hashana 1:7), so there is no reason to doubt Theophrastos’ evidence

in 1 Henoch, a Priestly Aramaic book that was edited out of (at least) 5 books, there is an astronomical work (ch. 72-82), from the 3rd century BCE. The Astronomy book is based on Mesopotamian astronomy as seen by its measuring the movements the Sun’s rising point at the along the horizon as well as from other aspects. The book shows a unique combination of Astronomy and religious mystical cognition, based on a 364 day solar year, beginning on Nissan 1.

The people in Qumran had (or: tried to have) the “Jubilees” calendar of a solar year, made of 364 days, as in the Temple Scroll, but it is most likely that they used other calendars (though not simultaneously). The day began probably at sunrise as in Lev. 7:15 and Rabbinic Priestly traditions as in t. Zebahim 6:18. Some 20 texts found in Qumran show a deep consciousness of calendrical issues that, although they do not support a new calendar, they do exemplify the importance of the role of the calendar in Qumran. 2 Enoch (that was not found in Qumran) comes from another priestly milieu, most probably: from Jerusalem, where Enoch was a hero. The author writes of a more complex calendar than that in 1 Enoch, of 365 ¼ days a year, with 7 years of intercalation in a cycle of 19 years, and a great cycle of (19*28=) 532 years (2 Enoch 6:21-26). The author gives the order of the planets in a non-Hellenistic system: Saturn, Venus, Mars, Sun, Jupiter, Mercury, Moon. It is quite obvious that this calendar shows a better knowledge of astronomy (and more ‘modern’) than the one in 1 Enoch, reflecting Hellenistic, combined with Mesopotamian, influence.

...the Rabbis described the world as a ball (p. Aboda Zara 3:1, 42c). Rava (3rd-4th centuries in Babylon) said that the whole world is 6,000 parsah (b. Pesahim 94a), which is about 24,872 miles, close to modern estimates.

Those who saw the new moon appeared before a court, first of priests and then of Rabbis, who checked the witnesses and evaluated their testimonies (m. Rosh Hashana 1:7). The different courts had different rules and it is highly probable that they had different calendars at the same period. R. Yohanan b. Zakkai (1st century), one of the founders of the Rabbinic tradition, was said to know all the Torah, Tequfuth (seasons, the moments of equinoxes and solstices) and Geometry included (b. Sukkah 28a), that is calendrical calculations and astronomy.

 While he stated the opinions of the sages of Israel as against the sages of the nations, we are told that the sages of Israel contend that the sphere is fixed while the zodiac signs are cycling, but the sages of the nations say the opposite. R. Judah the Prince, as the last referee, added an argument to prove the Jewish opinion (b. Pesahim 94b). However, in another dispute, stated at the same place, concerning the path of the sun at night, R. Judah the Prince said that the opinion of the sages of the nations seems to be superior. According to Maimonides (Guide 2:8), the Rabbis held the Pythagorean opinion, while the opposing sages were Aristotelians, and they were victorious.

Shemuel said: “the paths of the sky are known to me like the paths of Nehardea, except for a comet whose nature I do not know” (b. Berakhot 58b). After Shemuel’s statement the Talmud continued: “we have learnt that it (a comet) never passes through Kesil, and if it does – the world will be destroyed” (Kesil is mentioned in Amos 5:8; Job 38:31 and identified by ibn Ezra as “the heart of the Crab”). Shemuel was asked: “(Is it true), we see that it passes?! And he answered: “This is impossible, either the comet goes above it or beneath it. (p. Berakhot 9:2, 13c). The Rabbis said that there is a star that rises once in 70 years and deceives sailors (b. Horayot 10a). It is assumed they meant the comet later named after Halley, a comet that appeared around 140 CE (and 70 is taken as a round number for a cycle of around 78 years).

In a sermon that R. Afes (3rd century) gave in Antioch (Genesis Raba 10:1) there appear data concerning the times of the planets’ circling the earth in a very scientific way (compare to Vitruvius, On Architecture, 9:1). This sermon was given on the occasion of reading the beginning of Genesis, and the whole sermon has a flavor of combining Torah and science (as in modern times).

Bar Kappara (Land of Israel, 2-3 centuries CE) said: “one who knows how to calculate the seasons and zodiacal signs (in calendrical issues), but he doesn’t do so – of him Scripture says: ‘But they regard not the work of the Lord, Neither have they considered the work of His hands’ (Isa. 5:12)”. R. Johanan said that there is a religious commandment to calculate the seasons and the zodiac signs (=months), (b. Shabbat 75a). These opinions are a continuation of a long tradition that sees the study of astronomy as a religious practice.

Shemuel is the first Rabbi to give the duration of the solar year more precisely than the Tannaim. Shemuel said: “in between every tequfa (season) there are 91 days and 7 ½ hours” (B. Erubin 56a), that is an average that yields a solar year of 365 ¼ days, in accordance with the Julian year. He showed his astronomical knowledge by sending to the Land of Israel a calendar with intercalations for 60 years in advance, a calculation that was not enough however to impress R. Yohanan (b. Hulin 95b). Shemuel was also sure he could make calculations of the calendar (without the need for witnesses) for the whole Diaspora (Babylonia), but he was shown that he wasn’t aware of some of the rules of the calendar (b. Rosh Hashana 20b), or else this rule had no Babylonian origin.

In the lost Midrash of 49 Middot, from the Land of Israel, probably from the 4th-5th centuries, there is a sermon that has fortunately been preserved (Yalkut Shimoni, Shemot 418). The Rabbi found a correspondence between the zodiacal signs in the heavens and the way the tribes of Israel dwelled in the desert. He stated: “Tribe of Judah in the East, Isachar and Zebulun with him. Against them in heavens are Aries, Taurus and Gemini. With the Sun, together they serve five parts out of eight”. Probably he meant that out of the 1800 in the Eastern horizon, the Sun on the longest day, rises at azimuth of 67.50. However his words are understood, it is evident that the Rabbi (and his audience) knew something about Astronomy, which played a role in the synagogue.

Baraita deMazalot is a unique astronomical and astrological tractate. It gives astronomical data in unprecedently scientific ways. The author describes two different systems, an Egyptian and a Babylonian one, regarding the position of the stars in the zodiacal signs at the moment of creation. The author gives the exact distances from the Earth to the Moon, from the Moon to Mercury and so forth in a manner resembling the Mesopotamian beru ina same, x heavenly units equal to y earthly units. The author quotes not only Rabbinic sources but states the opinions of “the sages of the Gentiles”, “the sages of Egypt, Chaldeans and the Babylonians”. There is no doubt that the writer drew some of his data from Ptolemy (not necessarily directly), but his other sources remain to be analyzed. The author used Greek terminology, such as trigon (triangle), Stirigamos (standing) and diametron (diameter). The writer was probably a Rabbi who lived in the Land of Israel in 5th-6th centuries, and one of his sermons shows his competence very well.

R. Elazar Ha-Qalir, in the 6-7 centuries, was a master of the Hebrew language, a sage who knew all Rabbinic traditions and a prolific poet who wrote hundreds, if not more, Piyyutim, many of them still in use until this very day. In a famous Piyyut of his (“az raita ve-Safarta”, to parashat Sheqalim), his abilities as a mathematician and geographer are clear. However, in a Piyyut, only recently discovered (“or Hama u-Lebana”), it becomes apparent that he was an astronomer as well. In his Piyyut Ha-Qalir gives data of astronomical value: a solar year is 365 ¼ days (like Shemuel), and a lunar year is 354 1/3 days (without explicit precedent). His intercalation cycle was of 19 years (“Ehad beEhad Gashu”), but he had another solar cycle of 28 years (similar to b. Berakhot 59b), and more data in his Piyyutim has not yet been evaluated (e.g. his Piyyut: “abi kol Hoze”). It is assumed he was the head of the Yeshiva, the Jewish academy, in Tiberias and played a role in the process of intercalations.

Pirqe de Rabbi Eliezer is a midrash from the Land of Israel, from the 7th or the beginning of the 8th century (though it claims to be older with pseudepigraphic beginning and other statements). Chapters 5-7 discuss the calendar in prosaic Hebrew in a way similar to Ha-Qalir’s poetics. The author’s solar year is 365 ¼ days and his lunar month is 29:12:793 (assumed to be a Talmudic tradition). The author continues by stating that the length of a lunar year (as precisely multiplied by 12) is 354:8:876, exactly like in Midrash Agada, Buber ed., Genesis 1:14, Midrash Sod Ha-‘Ibur, and in our “modern” data (though very slightly bigger than the ‘real’ time).

The author gives his calculations in regard to ecliptical limits: A solar eclipse will occur if the moon’s latitude at conjunction is at most 60 “ma’alot” (units), and a lunar eclipse at most 40 “ma‘alot” (units). The moon’s great cycle is of 21 years while the sun cycles every 28 years and this leads to a great cycle of 84 years (21*4=28*3), that is (approximately) one hour of God (based on Psa. 90:4), though this cycle is not easy to explain. However, his system of intercalation is based on a 19 year-cycle, like the “modern” Jewish calendar, while adding one lunar month in the years 3, 6, 8, 11, 14, 17, 19. From the way the author portrays Biblical heroes as having secret knowledge of intercalation, and from his own description of the process of making decisions on this issue, it is evident that the author played an active role in calendar decisions. That is to say the author, Rabbi and astronomer, was one of the heads of the Yeshiva in Tiberias where they met at least once in 2-3 years to consider whether to intercalate the year or not. The author’s data, together with Talmudic halakhah, reveal that the author’s calendar could have been as exact as ours.

Baraita deShmuel is an Astronomical and Astrological tractate, based on Biblical knowledge combined with Greek science, as is clear from its inclusion of a few Greek words, such as ametron (without measure). The author uses some of his own terminology. Some of his Hebrew words are mere translations. A degree is called “hail”, that is an army-corps (following b. Berakhot 32b). The author uses “large hour” as does Pirqe de Rabbi Eliezer. The author shows his ability to use a “reed”, Qane (a precursor of the theodolite, already known in Egyptian Astronomy), which makes him the first Jewish author to do so. The author gives the values of the noon-shadow during the year (Cancer 0; Leo and Gemini 2; Virgo and Taurus 4… Capricorn 12). The author gives the oblique ascension of the zodiac signs such as “Aries 200, Libra (the cane of Libra) 400, from Aries to the cane (of Libra) one should add 40 for each sign” (=Taurus 240; Gemini 280; Cancer 320; Leo 360; Virgo 400; Libra 400; Scorpio 360; etc.). This is precisely a scheme of the Babylonian System A. Some Biblical verses are quoted in a midrashic manner. However, the text is full of astronomical data such as the time it takes each of the planets to circle the Earth, and the apogee and nadir of the zodiac signs. There is a special treatment of the Teli, that is Draco (mentioned no less than 17 times), and other stars are mentioned as well. The author mentions the year 4536 Anno Mundi (= 776 CE), and this is used to conclude that the text was composed that year, though there are doubts concerning the authenticity of that text. The author gives the limit of 120 21’ for both solar and lunar eclipses. The tractate was attributed to Shemuel the Babylonian (sharing a solar year of 365 ¼ days), though its dependence on Baraita deMazalot is very clear. This derivation leads to the probability that the text was composed in the Land of Israel with dependence on Babylonian-type arithmetical methods in Hellenistic culture.

‘Astrology in Ancient Judaism’, ‘Astronomy in Ancient Judaism’, J. Neusner, A. Avery-Peck and W. S. Green (eds.),  The Encyclopaedia of Judaism, V, Supplement Two, Leiden - Boston: Brill, 2004, pp. 2031-2044.


Astronomy in Israel:
From Og's Circle to the Wise Observatory

Yuval Ne'eman#+
Tel-Aviv University, Tel-Aviv, Israel

Several of the Mishnaic scholars ** were versed in Astronomy, such as the "Tannaim" Yehoshua ben Zakkai [7], the Patriarch Gamliel II and in particular Yehoshua (=Joshua) ben Hananiah [8]. In the tractate Horayoth, dealing with errors of Justice, the following anecdote is related [9]:

 "Rabbi Gamliel and Rabbi Yehoshua went together on a voyage at sea. Rabbi Gamliel carried a supply of bread, Rabbi Yehoshua carried a similar amount of bread and in addition a reserve of flour. At sea, they used up the entire supply of bread and had to utilize Rabbi Yehoshua's flour reserve. Rabbi Gamliel then asked Rabbi Yehoshua - "Did you know that this trip would last longer than usual, when you decided to carry this flour reserve?" Rabbi Yehoshua answered - "There is a star that appears every 70 years and induces navigational errors. I thought it might appear and cause us to go astray." Rabbi Gamliel then exclaimed "You are so knowledgeable and you nevertheless have to travel to make a living?"

This observation is generally interpreted as relating to Halley's Comet  [10] , with a period aproximating 76 years. Observing a comet's periodicity, with such a long period, requires records covering many centuries; it is possible that the Mishnaic scholars did inherit such records from the Great Knesset scholars (before 300 BC) who received them during the Babylonian exile (586-537 BC) from the "Chaldeans" (i.e. from Sumer, Akad etc. going back to the IIIrd Millenium BC).

Omar Khayyam, whose Rubayat was just a hobby, his professional creativity having yielded methods for solving factorizable cubic equations etc.) and spread all over the new Mohammedon Empire. The Jews were active participants, and the first Arabic-languate treatise on the Astrolabe was written by the Jew Joel, known as Masha-Allah of Basra (Iraq) around the year 800. This is the treatise that was translated into English by Geoffrey Chaucer ("The Treatise on the Astrolabe") around 1380, from a prior Latin translation.

Sind ben Ali, a heretic Jew, was the main contributor (~830) to the astronomical tables of the Caliph Maimum. The scene now shifts to Spain, where Abraham bar Hiyya Hanasi ("The Prince") of Barcelona (d. 1336) improved on these tables, using calculations performed by the Arab astronomer Al-Battani (d. 929). Abraham bar Hiyya was a prominent mathematician and astronomer, and wrote famous textboks in both fields. He introduced Europe to (Arab) trigonometry in his "Treatise on Mensuration and Calculations". The Hebrew was translated by Plast of Tivoli into Latin in 1145 and his book served as main source material for that later work of Leonardo Fibonacci of Pisa. In Astronomy, his book "The Shape of the Earth" is based upon the Ptolemaic system, contains a roughly correct estimate of the distance to the Moon

His student, Abraham Ibn Ezra (1089-1164), poet, philosopher, Biblical commentator and Astronomer, spent the last part of this life travelling in Italy and France, ending up in Eretz-Israel. He continued the publication of tables, mostly on the movement of the planets. The "Toledo Tables" were compiled by 12 Jewish astronomers led by the Cordovan Arab astronomer Ibn Arzarkali ("Azarchel"). The Latin version (translated by John of Brescia and Jacob Ibn Tibbon) was further improved in 1272 by a group of astronomers led by Isaac Ibn Said, and is known as the "Alphonsine Tables".

Rabbi Moshe ben Maimon's (=Maimonides) main contribution to Astronomy in his complete rejection of Astrology (1194). He is is unique, throughout the Centuries, in making this clear-cut. Remember that Kepler was still drawing horoscopes! Perhaps this should justify a visit to Maimonides' tomb in Tiberias.......

Rabbi Levi ben Gershom ( = Gersonides, also Maestre Leo de Bagnols, Maestre Leo Hebraeus; 1288-1344), was one of the greatest of Medieval astronomers. He lived in Provence, mostly at Orange. As a mathematician, he rediscovered the law of sines and published a sine table, correct to the 5th decimal. As an astronomer (he wrote 136 "chapters"!), he is the first to have relied on his own observations (in his studies of eclipses) rather than on Ptolemy's. He invented "Jacob's staff", a navigational instrument which was widely used for 3 centuries, and was the first person kown to have used a Camera Obscura for his observations.

Rabbi Levi is also the first scientist to derive more realistice estimates of the distance to the fixed stars. Ptolemy's estimate was of the order of 10-5 light years ( a million times smaller than the distance to the nearest star), whereas Rabbi Levi reached a figure of about 105 light years, 10 times our present estimate for the distance to an average star in the Galaxy. Gersonides was also one of the greatest Medieval philosophers and published Commentaries to the Bible.

The Zohar, a compilation of Jewish mystic writings drawn in Spain in the XIIIth Century anticipates Copernicus by stating that "the whole earth spins in a circle like a ball; the one part is up when the other part is down; the one part is light when the other is dark; it is day in the one part and night in the other".

Jewish astronomers played a key role in the theoretical preparation of the great voyages of discovery in the XVth Century. Judah Cresques, forced to adopt Christianity in the massacres of 1391, later became the Director of the Prince Henry of Portugal's ("The Navigator") Nautical Academy of Sagres. Abraham Zacuto ("Zacut", 1452-1515) worked first at Salamanca but moved to Portugal after the expulsion from Spain. As Court Astronomer to Kings John II and Manuel I, he prepared the voyage of Vasco da Gama (1496) and supplied instumentation (include his newly perfected copper astrolabe), improved tables, charts, intruction and briefs. He developed the first copper astrolabe. His very precise predictions of eclipses were used by Columbus to threaten the natives at a dangerous moment. Like all Jews, Zacut had to flee Portugal in 1497 and went to Tunis. He died in Eretz-Israel

The XVI-XVIIth Centuries were centuries of Jewish sufferings, and contributions to Astronomy are less prominent, except perhaps for the Herschel family, of Jewish origin. It was only when Alexander von Humboldt became President of the Prussian Academy of Sciences, that he abolished the requirement of a Christian Oath by a Professor at his ordination. Karl-Gustav Jacobi was the first Jew who did not have to abjure his faith to become a Professor. The oath was re-establised by von Humbold successor, but it was no more in existence when Einstein, Minkowski and Schwarzschild were ordinated.

ASTROLOGY IN THE LAND OF THE BIBLE - Center for ArcheoAstronomy

Number 24 September Equinox 1996

The State of Archaeo/Ethnoastronomy and the Land of the Bible
by Sara L. Gardner, Univ of Arizona

The use of astronomy by the inhabitants of Palestine has interested scholars as early as the turn of this century, when their examination of biblical texts led them to make comparisons with Greek and Mesopotamian astronomy and mythology. About twenty years later mythological tablets (13th century BCE) were found at Ras Shamra (ancient Ugarit) on the coast of Syria in 1929. The language and syntax of the Ugaritic tablets are similar to the Bible, and mythological motifs of the Ugaritic texts reflect motifs of biblical and extra-biblical texts. For example, they recorded how the gods influenced the changing seasons, and how the god of the Hebrews created the seasons in Genesis. Extra- biblical texts are used to study the development of Jewish astronomy and its calendar in the second half of the 1st millennium BCE. The list is lengthy, but the Astronomical Books of Enoch should be mentioned. They were written in the 2nd century BCE, and described the heavens as they were perceived by the ancient Jews. The ancient Hebrew literature as well as archaeological artifacts are rich in astral imaging.

Astral images began in the Chalcolithic period (4500-3300 BCE) at Teleilat Ghassul in Jordan near the Jordan River. The Ghassul Star, the image of a luminary rising above the mountains, and a faint image of a rayed sun or star behind a worshiper testify to an understanding of astronomy in this early period. Astral images continued throughout the history of ancient Palestine at many sites such as Hazor where instruments that marked the path of the sun, moon or star were depicted; for example, an altar showed a star image rising above two columns, and a stele showed the moon rising above two arms and hands that were postured as columns. The data collection of astral images from literature and artifacts is extensive, but nonetheless, this information has not been collected into a study on the practice of astronomy and/or its role in society. Nonetheless, these studies are prolific when compared to studies on the relationship between architecture and the movement of the sun, moon, and stars. Research to date has established one site, Rujim el-Hiri (Golan Heights in northern Israel), was built for the primary purpose of marking the June solstice from approximately 3000-2000 BCE, but its potential as an astronomical observatory has not been fully developed (Mizrachi 1993: 112-18).

The primary interest of biblical scholars is to establish a precise orientation of the entrances of temples to the eastern horizon, or to establish that there is not an orientation. Excavators are cavalier in their interpretations, and casually remark that the entrance of cult building or its cult object caught the rays of the rising sun--without any substantiating studies. Unfortunately, excavation records do not consistently record precise directions, either True or magnetic north (see Taylor 1993). While numerous scholars insist that orientation of temples should be studied, these calls have not mentioned what the premise for orientation should be-- beyond an assumption that the cardinal directions are somehow important.

Problems for scholarship exist beyond the collection of data and its interpretation for the land of the Bible, because it conflicts with current religious beliefs. The primary sources for data come from sacred Hebrew and Christian writings--the Torah and the Bible--and from sites mentioned in these texts. Research has in the past--and continues now--to be an exercise in proving that the modern interpretation of the Torah and the Bible is literal and written in stone, so to speak, while a few scholars approach the subject from an academic perspective and not religious. If research in archaeoastronomy in Palestine is to be initiated, it must be from a secular perspective such as Syro-Palestinian archaeology led by William G. Dever (1990). Through the combination of biblical and extra-biblical texts, astral images found on artifacts, archaeology, and astronomy the potential exists to open a new rich field of study for archaeo/ethnoastronomy as well as add another chapter in the history of astronomy.


Dever, W. G., Recent Archaeological Discoveries and Biblical Research, Seattle: University of Washington Press, 1990.


The Babylonian Theory of the Planets
by N. M. Swerdlow
Reviewed by Stacy G. Langton

 One of the great discoveries of the nineteenth century was a discovery about the past ---the existence of a highly sophisticated mathematical astronomy among the ancient Babylonians. Otto Neugebauer tells the story in The Exact Sciences in Antiquity, pp. 103--105 (full references are given below). The discoverers were three Jesuit priests. The first, Johann Nepomuk Strassmaier, an Assyriologist, worked in the British Museum for nearly twenty years, patiently and tirelessly copying into his notebooks the contents of unpublished clay tablets from Babylon. Among these were many with astronomical contents, which Strassmaier was unable to comprehend. He invoked the help of Joseph Epping, a professor of mathematics and astronomy, then at Quito, Ecuador. Epping's first results were published in 1881, in an obscure Catholic theological periodical. He had been able to decipher the names of the planets and zodiacal signs and had uncovered the main aspects of the Babylonian lunar theory. After Epping's death, his work was continued by Franz Xaver Kugler.

The Babylonian astronomers who created these tablets ---the "Chaldeans" of the Book of Daniel: "The king [Belshazzar] cried aloud to bring in the astrologers, the Chaldeans, and the soothsayers," (Daniel 5:7)--- are referred to by Noel Swerdlow, in the book under review, as the "Scribes of Enuma Anu Enlil". The "Enuma Anu Enlil" is a vast series of seventy tablets containing thousands of omens, originally from the second millenium B.C., referring to the appearances of the sun, moon and planets, as well as meteorological phenomena. ("Enuma Anu Enlil" just means "when Anu and Enlil"; these are the opening words of the first tablet. Anu and Enlil were Sumerian gods.) For example, "If Jupiter [rises] in the path of the [Enlil] stars: the king of Akkad will become strong and [overthrow] his enemies in all lands in battle" (BTP, p. 94).

Because of the importance of celestial phenomena for the understanding of events in Babylonian society, the Scribes, by the neo-Babylonian period (7th century B.C.), had begun to keep records of systematic observations of the sky. These are now known as the "Astronomical Diaries". "Indeed, the Diaries, originally extending from the eighth or seventh to the first century, are by far the longest continuous scientific record, or should we say, the record of the longest continuous scientific research, of any kind in all of history, for modern science itself has existed for only half as long. And of course it is the Diaries, or the records from which the Diaries were compiled, that provided the observations that were later used as the empirical foundation of the mathematical astronomy of the ephemerides, in which the same phenomena of the moon and planets recorded in the omen texts were reduced to calculation" (BTP, p. 17).

The final, mathematical, phase of Babylonian astronomy dates mainly from the third to the first centuries B.C. From this period we have ephemerides, tablets containing tables of the computed positions of the sun, moon, or planets, day by day, or over longer periods, such as month by month. (Strictly speaking, an ephemeris, from the Greek hemera, "day", should mean a daily record; but Neugebauer has applied the term in the more general sense.) There are also tablets called procedure texts, which give schematically the rules for computing ephemerides, much like a modern computer program.

The object of the book under review, Noel Swerdlow's The Babylonian Theory of the Planets (here abbreviated BTP), is to explain how the Scribes could have determined the parameters of their planetary theories. (The lunar theory is not dealt with at all.) What does this mean? The Scribes were mainly interested in planetary phenomena corresponding to those in the omens, particularly heliacal risings and settings. Take, for example, the planet Jupiter. It drifts eastward (most of the time) through the Zodiac, as does the sun (always). But the sun's motion is faster than Jupiter's. Thus the sun will eventually pass Jupiter. After it does, Jupiter will be westward of the sun, and will therefore rise earlier. When the angular distance between Jupiter and the sun is great enough so that the sky is still dark when Jupiter rises, its rising will be visible. Its rising on the first day of visibility is called its "heliacal rising".

Babylonian ephemerides of Jupiter give the dates and positions in the Zodiac of its heliacal risings, which occur, on average, roughly every 399 days; this is called Jupiter's "mean synodic period". Now suppose, for example, that in some year Jupiter's heliacal rising occurs when Jupiter (and also the sun, which must be only a few degrees ---in fact, possibly as much as 17°--- away) is in the constellation Leo. Then the next heliacal rising will occur in the constellation Virgo. On average, each successive heliacal rising will occur about 33° further along the Zodiac, just over one zodiacal sign; this interval is called the "mean synodic arc" of the phenomenon.

Now the Scribes were able to recognize (using the extensive observational records in the Astronomical Diaries) that, after many synodic periods, both the sun and planet would return to essentially the same positions in the Zodiac. For Jupiter this happens, for example, after 427 years (the so-called "ACT period"), during which time there have been 391 heliacal risings, and the position of the heliacal rising has itself gone through the Zodiac 36 times. Since the earth, sun, and Jupiter are now again in the same relative positions as at the beginning of the cycle, the speeds of the sun and Jupiter through the Zodiac will be the same as before, and the whole cycle will essentially repeat.

It follows that the synodic arc from one heliacal rising to the next ---the actual synodic arc, as opposed to the mean synodic arc mentioned above--- will be (as we would say) a function of the location of the phenomenon in the Zodiac. Now the mathematical astronomy of the Scribes was quite different from that of the Greek astronomers, such as Hipparchus and Ptolemy, which was based on a geometric model, representing namely the sky as a sphere, with the earth at the center. The Scribes used no geometric model at all. Rather, they used numerical schemes for computing the function giving the synodic arc of the phenomenon in terms of its location in the Zodiac.

In one of these schemes for Jupiter (known as "System A"), the rule is simple. The Zodiac is divided into two zones, the first running from 25° of Gemini to 0° of Sagittarius, the second from 0° of Sagittarius back to 25° of Gemini. In the first zone, the synodic arc from one heliacal rising to the next is taken to be 30°; in the second, it is taken to be 36°. (There are auxiliary rules for the cases in which the synodic arc overlaps both zones.) For this system, then, the parameters are the lengths and locations in the Zodiac of the zones, and the corresponding synodic arcs.

The question, then, is how the Scribes fell upon these particular values for the parameters. Of course, they must have used their recorded observations, but how? Swerdlow has compiled tables of relevant observational records from the Astronomical Diaries, and has also computed the synodic arcs and times from modern theory. (It is interesting that in some cases ---for example, for Saturn--- the computational procedures of the Scribes give results which fit the modern computed values much better than their own observations do.)

The problem is that the observational records, though extensive, are crude. The date of the heliacal rising can be observed (provided the weather is favorable), but the Scribes had neither the instruments for observing its position precisely nor a coordinate system for recording it. The Diaries give merely the sign: "in Leo", say; sometimes they remark whether it was near the beginning or the end of the sign.

Swerdlow argues, then, that there must have been a way to determine the parameters from the synodic times alone, independent of the synodic arcs. And in fact this is possible. For there is a fundamental relation between mean synodic arc and time: the mean synodic arc can be found from the mean synodic time just by subtracting a certain constant value, characteristic of the given planet.

(It may strike the reader, as it did me, that this relation is rather odd. Though Swerdlow gives a formal derivation, pp. 66--68, he does not give an intuitive explanation of why it should be so. Actually there are two factors involved. First, the synodic time is the time for the sun to go once around the Zodiac and more, until it catches up with Jupiter again; the synodic arc is just the comparatively short arc from one occurrence of the phenomenon to the next. Second, there is the conversion from units of angle to units of time. The angular units are degrees ---a unit which we have inherited from the Babylonians! The units of time are "mean lunar days", nowadays called by the Sanskrit term "tithi". There are precisely 30 tithis in a mean lunation, or synodic month. Now the sun travels roughly one degree per day. More precisely, the conversion factor is 1 plus 0.03 tithis per degree. Since the conversion factor is so close to 1, it is convenient, instead of multiplying by the conversion factor, to add the extra amount corresponding to the increment 0.03.)

The Babylonian Theory of the Planets has been reproduced photographically by Princeton University Press from copy supplied by the author. Consequently, the type-face is rather ugly, and the appearance of the printed page is unpleasant. On the other hand, the copy-text has clearly been prepared with great care. I noticed only a few typographical errors in this very complex book. (There are trivial errors on p. 119, line -3 and p. 154, line 20. On p. 158, line 16, "Table 2.4" should be "Table 3.4". On p. 172, line 9, "principle" should be "principal". Also, on p. 5 there is a reference to a 1990 article of Brown which is not listed in the bibliography.)

The Introduction reviews the three stages of Babylonian astronomy: the omens of the Enuma Anu Enlil and similar texts, the observations of the Astronomical Diaries, and the mathematical astronomy of the ephemerides and procedure texts. Swerdlow obviously considers these stages as arising naturally one out of the other. (It is amusing to contrast this point of view with Neugebauer's remark: "Nor does the interest in celestial omens---as one class of omens among many---lead to astronomy," Astronomy and History, p. 160.) Part 1 of BTP explains the basic period relations for planetary phenomena and works out the relation between synodic arc and time. Part 2, the heart of the book, discusses the various systems which the Scribes used for each of the planets (except Venus), and shows in detail how the Scribes could plausibly have arrived at the parameters they adopted, on the basis of synodic times extracted from the Diaries.

In fact, the Scribes did not explain (at least in any tablets that we have) how they arrived at their parameters; thus, any reconstruction such as Swerdlow's can only be speculative. The important thing (I think) is that he has given a believable account of how it could have been done. Perhaps one could improve on the details, but it seems likely that the Scribes must have done something like this. (Swerdlow himself discusses a couple of alternate approaches in an Appendix.)

(The Babylonian theory of Venus is different from that of the other planets, because Venus has a very short period of 8 synodic periods in 5 years less just 4 tithis. The Scribes, for whatever reason, preferred to use this short near-period, letting successive cycles drift slowly backward through the Zodiac, rather than the long periods which they used for the other planets.)

Part 3 of BTP considers the method of fitting the arithmetical schemes to actual positions in the Zodiac, and the relations between different synodic phenomena: for example, the arc between an heliacal setting and the following heliacal rising.

In principle, Swerdlow's treatment does not assume any previous acquaintance with Babylonian astronomy. He admits, however, that the reader would do well to look first at the relevant chapter in Neugebauer's Exact Sciences in Antiquity. I found BTP to be pretty tough going, nevertheless. Swerdlow's exposition is not always clear. For example, I found his discussion of the basic period relations on pp. 57--59 to be both confusing and incomplete. Here the reader will find a more perspicuous account in Neugebauer's History of Ancient Mathematical Astronomy, pp. 388--390. Of course, the Scribes did all their numerical calculations in sexagesimal (base 60) notation, and Swerdlow naturally follows their lead. (I found it convenient to program my little pocket calculator to operate in sexagesimal ---but I have not checked all the calculations.)

Perhaps few readers of MAA Online will find themselves able to take time from teaching calculus and doing research on Banach spaces (or whatever) to work through this difficult and complex book. I think, however, that they will be rewarded if they do. As Swerdlow insists, Babylonian astronomy is (apart from arithmetic and geometry) the first of the natural sciences. The Scribes were the first to see the possibility and usefulness of applying mathematics to describe and understand the complex phenomena in the natural world. Much of our heritage as mathematicians starts with them.

What was it in fact that led the Scribes to develop a mathematical theory? Swerdlow has a remarkable answer: bad weather! "All those nights of rain and clouds and poor visibility reported in the Diaries turned out to be good for something after all. When it is clear, observe; when it is cloudy, compute (...) [T]he principle first understood by the Scribes of [E]numa Anu Enlil in order to take account of adverse weather has remained, with ever increasing sophistication, the foundation of observational and experimental science, whether in measuring distances of galaxies or masses of subatomic particles. Here above all, science was born in Babylon. From bad weather was born good science. And the reduction of periodic natural phenomena, however great their irregularities, to a precise mathematical description that may be applied to both prospective and retrospective calculation, that is, to mathematical science, was also the achievement of the Babylonians" (BTP, p. 56).

"They have left no record of their theoretical analyses and discussions, but to judge from the works they have left us, the Diaries and ephemerides, the goal-year texts and almanacs, the discussions of two Scribes of Enuma Anu Enlil contained more rigorous science than the speculations of twenty philosophers speaking Greek, not even Aristotle excepted. I say this seriously, not as provocation, and further, I believe it is due precisely to the scientific and technical character of Babylonian astronomy that most historians and philosophers remain without comprehension of it, still preferring to dote upon childish fables and Delphic fragments of Pre-Socratics, requiring no knowledge of mathematics and less taxing to the intellect. (...) The origin of rigorous, technical science was not Greek but Babylonian, not Indo-European but Semitic, something I believe no one who has read Kugler and Neugebauer with understanding can doubt, and, my God, those Scribes were smart" (BTP, pp. 181--182).

Publication Data: The Babylonian Theory of the Planets, by N. M. Swerdlow. Princeton University Press, 1998.  ISBN: 0-691-01196-6.