My article in the August 2021 issue of *Sky & Telescope* magazine under the title *Alas Babylon! When Mars Draws Near*. . . is about the Babylonian Goal-Year periods, with Mars as the center of attention. As with my October 2020 post on the subject, the *Sky & Telescope* article reviews some of the clever error-canceling methods used by Babylonian astronomers to find amazingly accurate synodic recurrence periods over centuries. Now, as Mars slips toward the sun, we’ll say farewell to the ruddy planet for a while. Our October post greeted Mars with Babylon, so it seemed appropriate to have Babylon at the send-off, but this time with some notes on some appealing mathematical symmetries in Babylonian Goal-Year astronomy.

Of course it's a vast and specialized field and I have fully relied on the sources mentioned below, especially Otto Neugebauer’s towering *History of Ancient Mathematical Astronomy* (HAMA), the writings of Teije de Jong, and a variety of papers and background materials not mentioned. All contemporary scholarship, we should remember, rests upon a handful of scholars who, beginning at the end of the 19th century, examined fragments of clay tablets found in Mesopotamia (such as shown below) and with astonishing ingenuity, deciphered them. It was an incredible feat of linguistic and mathematical code-breaking. Credit in particular goes to three brilliant Jesuit fathers, Johann Strassmaier (1846-1920), Joseph Epping (1835-1894), and Franz Kugler (1862-1929). Believe it or not, the process of discovery and decipherment – usually from long-slumbering tablets exhumed from the back rooms of museums – is still ongoing.

Here we'll briefly look at some of the elegant mathematical symmetries scholars have deduced from these ancient cuneiform impressions. I’ll give you some tools so you can explore them for yourself. Because the vocabulary is unique, I’ve included a glossary of terms one finds in the literature; it's my attempt at distilling a complex subject into a small space and is of course open to improvement. Then follows a box called ‘Babylonian Math on Your Calculator.’ The subject may sound esoteric, but the math is indeed doable on a calculator (or in your head if you please).

Since the blog and the article are about Mars at opposition, I will continue that focus – it remains the (now-fading) star of the show! But the concepts I’ve talked about for Mars apply equally well to Saturn and Jupiter. Now here's the glossary to get you up to speed. Once you get the hang of the words, the reading is easier!

**A Glossary of Babylonian Planetary Mathematics**

**Exact periods: **The exceptionally long – multi-century – periods discovered by the Babylonians that bring the planet to its exact (or near-exact) spot in the sky where it was (or will be) from any other given opposition, past or future. Exact periods were derived from combinations of the shorter Goal-Year periods that most completely cancelled out their errors, so they are more accurate than the Goal-Year periods. These exact periods were used for the creation of long-term ephemerides. According to recovered cuneiform tablets, the exact period for Mars was 133 synodic cycles in 284 years; for Jupiter, 391 synodic cycles in 427 years; and for Saturn, 256 cycles in 265 years.

**Goal-Year periods, or Goal-Year interval**: For the outer planets, the Goal-Year periods are intervals between those exceedingly rare oppositions where a planet arrives at (very nearly) the same place in the sky as it was at a prior opposition, and where it very nearly will be at the next opposition. For Babylonian astronomers, there were only two oppositions of Mars in a century’s time that qualified, occurring at intervals of 47 years (after 22 synodic cycles) and 79 years (37 synodic cycles).

**Goal-Year errors:** The Babylonians knew their Goal-Year periods were not exact, and through long observation determined their errors. The errors are the number of days by which Mars was either early or late to the exact place in the sky where it was at its last predicted Goal-Year opposition. I sometimes refer to the errors as ‘day variances.’ The clever means by which the Babylonians used these errors to advantage was the main subject of my October Mars post and the *Sky & Telescope* article.

**Period:** In this context, the number of mean synodic arcs that can be fitted into 360° of longitude. It is typically represented by the letter P. For Mars, P = 7.3888, meaning that seven oppositions will occur before Mars has traversed the entire ecliptic circle, plus about 1/3 of a mean synodic arc left over.

**Recurrence period(s)**: The number of years or synodic cycles that must elapse before Mars at opposition will appear at exactly (or near-exactly) the same place in the sky as it was before or will be in the future.

**Sexagesimal notation:** in both Babylonian and Greek mathematics of the Hellenistic era, it’s a convenient way of expressing fractions, and is the first place-value system in history. The number before a colon is the whole number; those following are separated by a comma and are fractions of increasing powers of sixty in the denominator. Hence, 48;43,18,30° means 48° + 43/60 + 18/3600 + 30/21,600 or about 48.722°. Without any colon, it means multiply by 60. Thus, the term 6, 0° corresponds to 360°. It’s easy to use once you are used to it. It is common in all the literature about ancient astronomy. Test your skills by checking with an online calculator, https://planetcalc.com/9216/. The site includes cuneiform versions of the sexagesimal numbers you can learn if you really want to impress your friends!

**Sidereal rotation(s)**: The number of orbits of the planet in question. Each ‘rotation’ of Mars, for example is one loop around the sun, one Martian year.

**Synodic Arc or Mean Synodic Arc: **The mean number of degrees the planet appears to travel across the sky between synodic occurrences of the same type. For example, the Babylonians discovered that on average, Mars traverses about 48.72° of the sky between oppositions. That angle is the mean synodic arc, whether or not the term 'mean' qualifies it. The Babylonians were interested in knowing how many complete sets of synodic arcs are required for Mars (and other planets) to return to exactly the same place in the sky. The reason the synodic arc is a *mean* value is because all their calculations assume circular motion of the planets, when in fact (unknown to the Babylonians) the orbits of the planets are ellipses.

**Synodic occurrence(s), or synodic cycle(s)**: While there are many types of synodic occurrences, such as risings, settings, and oppositions, we have focused on oppositions, where the planet is opposite the sun from our perspective. Each opposition is a ‘synodic occurrence.’ Scholars have represented the number of synodic occurrences in any given interval with the symbol Π. For Mars, the Babylonians discovered that there are about 7.3888 of these occurrences in each 360-degree passage of Mars around the ecliptic. There are 22 synodic occurrences of Mars in 47 years, 37 synodic occurrences in 79 years, and 133 synodic occurrences in 284 years.

**Synodic period**: The mean time in days or years between synodic occurrences, e.g., there are on average 779.94 days between Martian oppositions.

**Wave**: Counting only oppositions, every 360° passage of Mars around ecliptic, hopping like on steppingstones from opposition-to-opposition, has been called a ‘wave’ by some scholars. In 133 synodic cycles, for example, Mars traverses the complete ecliptic 18 times. It is typically represented by the letter Z.

With those definitions, look at the chart below; if you play around with some of the relationships in the third row, you will notice some nice symmetries.

This chart presents a few of the key parameters of Babylonian planetary arithmetic, applicable here to the mean periodic phenomena of the planet Mars, summarized partly from O. Neugebauer (1975), Book II, including Table 9 of his *System A, Parameters*. The algebraic (and sexagesimal) notation is adopted from Neugebauer. The relationships in the chart should become apparent as you go through it. Begin with the first three columns of the 133 Martian synodic recurrence period. They mean that Mars *completes 133 synodic occurrences in 151 orbital revolutions of that planet over the course of 284 years.* Similarly, with the Goal-Year periods just below it. The simple additive relationships among the columns are an important foundation for later astronomy:

“Thus it holds for an outer planet [according to Neugebauer:] number of occurrences + number of sidereal rotations = number of years. This is the well-known relation which underlies the Greek epicyclic models for the outer planets.”

And you probably noticed this:

133 + 18 = 151

Consider also these relationships for Mars, on the third row: 133 synodic arcs will fit in 151 Martian orbits over the course of 284 Earth years. Now, 133 complete synodic arcs comprise 6,480° which happens to be 18 whole revolutions around the 360° ecliptic. We can express this symmetry generally in one of the most elegantly simple equations, again drawn from Neugebauer:

In the case of Mars’s ‘exact’ 133-cycle period, an integer number of mean synodic arcs fit tidily into 18 complete circuits of ecliptic longitude:

133 · 48.722° = 18 · 360° = 6,480°

With the Goal-Year periods, we don’t have the accuracy of the exact, long-term period (offsetting their errors is what allowed the Babylonians to generate the more exact long-term period). The fit isn’t as good. With the 37-cycle/79-year Goal-Year period, we have for the left and right sides of the equation, respectively:

37 · 48.722° = 1,803°

5 · 360° = 1,800°

The difference between the top and bottom is the Goal-Year inaccuracy we explained in the *S&T* article and in our October post. For the 22-cycle/47-year period, the difference is greater and in the other direction:

22 · 48.722° = 1,072°

3 · 48.722° = 1,080°

An even simpler equation involves the period defined above in the glossary. The period, again, is how many mean synodic arcs can be fitted into 360° of longitude. In the customary notation,

P = Π/Z

Using the exact parameters for Mars, the period is,

P = 133/18 = 7.3888

Let’s now can calculate the length of the mean synodic cycle in years, à la Babylonia, by finding the number of synodic occurrences in the corresponding number of earth-years. This would be C/A in the above chart. Using the exact period for Mars, we have,

284/133 = 2.1353 years

which squares with the modern value and underscores how amazingly accurate the Babylonian exact periods were.

Lastly, we can find how many years elapse between close oppositions, on the pathway up to the exact one after 133 oppositions in 284 years:

284/Z = 15.78 years

I still haven’t explored all the variations. Feel free to explore some more and send me your results!

**References**

Below are sources that have been helpful to me, including most especially Neugebauer’s HAMA, the most dominant authority on the subject:

de Jong, Teije. 2018. *A Study of Babylonian Planetary Theory I. The Outer Planets*. Archive for History of Exact Sciences (2019) 73:1–37. https://doi.org/10.1007/s00407-018-0216-0.

Jones, Alexander. 2007.* A Portable Cosmos: Revealing the Antikythera Mechanism, Scientific Wonder of the Ancient World*. Oxford: Oxford University Press.

Lindberg, David C. 2007. *The Beginnings of Western Science: the European Scientific Tradition in Philosophical, Religious, and Institutional Context, Prehistory to A.D*. 1450. Chicago: The University of Chicago Press.

Neugebauer, O., ed. 1955. *Astronomical Cuneiform Texts: Babylonian Ephemerides of the Seleucid Period for the Motion of the Sun, Moon, and the Planets*. 3 vols. London: Lund Humphries.

___. (1957) 1969. *The Exact Sciences in Antiquity*. Reprint, New York: Dover Publications, Inc.

___. 1975. *A History of Ancient Mathematical Astronomy.* 3 vols. Studies in the History of Mathematics and Physical Sciences 1. Berlin: Springer (usually abbreviated as HAMA).

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