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in the British Museum for nearly twenty years, patiently and tirelessly copying into his notebooks the contents of unpublished clay tablets from Babylon.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).

**References:**

Otto Neugebauer, The Exact Sciences in Antiquity, 2d edition, Dover, 1969.

Otto Neugebauer, A History of Ancient Mathematical Astronomy, 3 volumes, Springer-Verlag, 1975.

Otto Neugebauer, Astronomy and History: Selected Essays, Springer-Verlag, 1983.

Otto Neugebauer, "ACT": Astronomical Cuneiform Texts, 3 volumes, Springer-Verlag, 1983.

Stacy G. Langton (langton@acusd.edu) is Professor of Mathematics and Computer Science at the University of San Diego. At present, he is at work on a translation of some of the writings of Euler.