We look at how the Babylonians represented information in their astronomical tablets by examining their number system and the unit system. Then we dive into the Babylonian discovery of the Saros cycle, which was a deep regularity in the pattern of lunar eclipses.

Transcript

Good evening, and welcome to the Song of Urania, a podcast about the history of astronomy with new episodes every full moon. My name is Joe Antognini. Before we begin I have to take care of a bit of housekeeping. The show now has a website with a real domain name: www.songofurania.com. If you are an early listener to the show and are listening to this episode around the time it originally came out on April 26, 2021, you likely are subscribed to an RSS feed from a libsyn.com domain name. I recommend that you switch your feed to the new domain name, songofurania.com/rss. The old Libsyn feed will continue to work for a little while, but the new domain name will be the permanent location of the podcast feed, so if you want to keep getting new episodes every full moon, I recommend that you make the switch. If you are a later listener to this show and are going through the back catalog, you are almost certainly already subscribed to the new songofurania.com domain name and have nothing to worry about.

In the last episode we started talking more systematically about the Babylonian astronomical tablets we have available to us. We looked at the older tablets like the Enuma Anu Enlil, the Mul.Apin and the Astronomical Diaries. Then we started getting into the later, more mathematical, astronomical tablets that started appearing during the Neo-Babylonian Empire and especially after the Persian Conquest, but we only had time to talk about the Goal Year texts. You may recall that the idea of the goal year text was to make an ephemeris — a prediction of where a planet would be at a certain point in time. Now Babylonian astronomers weren’t interested in predicting the location of a planet on an some arbitrary date. Instead they limited their ephemerides to determining when and where certain important phenomena happened in the planet’s motion, for example, the first stationary point, where the planet stopped drifting eastward in the sky relative to the background stars, and entered into retrograde motion and started drifting westward. Opposition was another important point in the planet’s motion, when it was on the exact opposite side of the sky from the Sun.

The principle of the goal year text was pretty simple. The Babylonians had kept meticulous records of the observed dates and locations of these phenomena for centuries. And since the motions of the planets are periodic, once the Babylonians had figured out what the periods of different planetary phenomena were, if they wanted to make new predictions about when and where these phenomena would occur next, all they had to do was consult their records and then copy the observed locations that these phenomena had been seen to happen at however many years in the past that the period was.

Now this technique, simple as it is, actually works pretty well. But it was far from being the limit of what the Babylonians figured out about the heavens. Soon enough we will dive into the deep end of the more sophisticated techniques that the Babylonians developed, creatively called System A and System B, but before we can get to that, we should take a look at some of the nuts and bolts of how the Babylonian astronomical worked and what they looked like.

The first thing we need to know is how Babylonian numbers worked. The Babylonian number system was really quite logical and stands today as one of the most elegant number systems devised by any civilization. Even more remarkably, much like Athena sprung from the head of Zeus fully formed, the Babylonian number system appears in the earliest tablets in complete form and underwent little evolution. It is not all that surprising that numbers appear very frequently in Mesopotamian tablets since many of them were astronomical in nature or were contracts, both of which tend to contain a great deal of numbers. And even in the earliest of these tablets, which date back to around 2000 BC, we see the Babylonian number system is essentially the same as it appeared two thousand years later in the sophisticated mathematical astronomy tablets. It just made that much sense.

So what did Babylonian numbers look like? Well this is one of those instances where describing this in a podcast makes things a little tricky because I can’t show you an image. Perhaps in due time I will add some images to the website that you can look at to get a better idea. But for now we’ll just have to use our imagination.

As I mentioned in previous episodes, ancient Mesopotamian records were written in cuneiform, which literally means “wedge-shaped” from the Latin word “cuneus”. Over the millennia a number of different wedge shapes were used in cuneiform, but for their numbers, the Babylonians used just two shapes: the vertical wedge and the corner wedge. The vertical wedge looks like a vertical line with a little upside-down triangle on top, sort of like a martini glass without the base. The corner wedge looks like a less-than sign with a little vertical line drawn near the corner to make a little triangle.

Now you may have heard that the Babylonian number system was sexagesimal, that is, base 60. And indeed it was. But it also had elements of a decimal system built into it. The way it worked was that the number “one” was represented by a single vertical wedge. Makes sense. If you wanted to represent the number “two”, you’d just put two vertical wedges next to each other. Easy enough. So it would continue with the number three. Then when you got to four, you’d put the fourth wedge underneath the first three. For five, you now have three vertical wedges on top, two on the bottom. By the time you get to seven, you put the seventh wedge underneath two rows of three. And so you sort of fill up this little 3x3 grid with vertical wedges. So far so good.

Now when you get to 10 things change. Now you represent the 10 with a corner wedge. Eleven is a corner wedge followed by a vertical wedge. Twelve is a corner wedge followed by two vertical wedges, and so on all the way up to twenty. For twenty you just put two corner wedges next to each other. Thirty is three corner wedges next to each other. Once you get to forty, the corner wedges are arranged in a little triangle shape, and once you get to fifty, the corner wedges get arranged in a sort of diamond shape. By the time you get to fifty-nine you now have five corner wedges in a little diamond next to nine vertical wedges in a square.

When want to write down larger numbers, you just start all over again one place over to the left. The number seventy-one is a vertical wedge followed by a corner wedge followed by a vertical wedge. The leftmost vertical wedge represents the number sixty, the corner wedge represents ten, and the rightmost vertical wedge represents the number one. Add them all together and you get seventy-one. Similarly, 131 is two vertical wedges followed by a corner wedge followed by a vertical wedge. The first two wedges represent two sixties, so 120, and the second two represent the number 11, so 131 in all. So with just two digits this lets you write down numbers all the way up to 3600, which is pretty good for most use cases in the ancient world. And with three digits you’re already at 216,000, which is really larger than anything I personally can conceive of. So with just two shapes they managed to logically and compactly represent arbitrarily large numbers. Beautiful!

Now, you may have noticed that I skipped over the number sixty, which is a bit strange, because sixty really is the number in this whole system. But we actually have a bit of a problem with the number sixty. Because sixty is just a single vertical wedge, which is exactly the same as the representation of the number one. Well if we see a single vertical wedge somewhere in a tablet, how do we know if this number is one or sixty? We cannot. Without a numeral zero, it is impossible for us to distinguish these two numbers except by context. This was the major shortcoming with the Babylonian number system, and indeed, all numeral systems until the development of the Hindu-Arabic numeral system around the seventh century AD.

Now it wasn’t all bad for the Babylonians. They did have a sort of medial zero. If they had three digits in a row and the middle one would have been a zero, they left some space between the two. This spacing allowed them to distinguish between two, which was two vertical wedges connected to each other, 61, which was two vertical wedges next to each other but not connected, and 3601, which was two vertical wedges with some space in between. But they did not or maybe more precisely could not add that extra space at the end of the number to distinguish between sixty and one. By the year 300 BC the medial zero began to be written explicitly with its own separate character — this was the only major development in the Babylonian numeral system in its two thousand year history. But it wasn’t a true zero because they didn’t put it at the end of numbers to distinguish between one and sixty — it only went between other numbers.

Now this wasn’t all there was to the Babylonian number system, because, like our number system, they also had decimals as well. And just like the rest of their number system, the decimals weren’t really decimals but rather sexagesimals — they were all base sixty. They would write their decimals as fractions of the number sixty. So for example the number 1.5 would be written as a vertical wedge representing a one followed by three corner wedges, representing thirty. Thirty is one half of sixty, so this gives you 1.5. But again we have a problem because if you see a vertical wedge followed by three corner wedges, you don’t know if that’s the number 1.5 or the number 90. Both are represented the same way. You have to figure it out from context. Most of the time it’s pretty obvious. For whatever reason the Babylonians only used fractions as intermediate steps in calculations, never as the final result. Nevertheless, when it comes to numbers you’d really rather there be absolutely no ambiguous cases rather than very few ambiguous cases. But maybe I’m being a little harsh — the Babylonian number system was after all among the most sophisticated number systems in the ancient world and it wasn’t until the development of the Hindu-Arabic number system in India around 700 AD, the predecessor of our modern number system, that a number system appears anywhere that is unequivocally superior to the Babylonian number system.

Understanding the Babylonian number system is important if you want to understand the mathematical astronomical tablets because those tablets were chock full of numbers. But numbers alone aren’t enough to do scientific calculations. Suppose you ask me to calculate the radius of the Earth and I go off for a little bit and later come back and confidently tell you that I’ve calculated it to be 3209. Well, you will say 3209 what? 3209 meters? Miles? Kilometers? Furlongs? You have to provide a unit. Without a unit, the number is meaningless. So the quantities in the mathematical tablets generally have units associated with them as well.

The Babylonian tablets have three types of units. The first of these are units of time. The principal unit here is, unsurprisingly, the day. The day is then divided into 12 intervals called a beru, which is equivalent to two hours. Then each beru is divided into 30 ush, each of which is four minutes long. Then the ush is divided into 60 ninda, each of which is four seconds long. This unit system is actually a little surprising. You may have heard that our system of hours, minutes, and seconds is a relic of the base-60 Babylonian number system. And it is true that it is thanks to the influence of the Babylonian obsession with the number sixty that we divide our hours into sixty minutes, but the Babylonians themselves actually did not do this for their own timekeeping system! The division of an hour into sixty minutes came much later in the 10th century AD during the Islamic Golden Age by the Iranian astronomer Al-Biruni.

The second type of unit that shows up in the astronomical tablets is length. Only one unit shows up, the cubit. The cubit also appears throughout the Hebrew Bible, most famously as the dimensions of Noah’s Ark. Now, systems of weights and measures were not particularly standardized in the ancient world, and really weren’t standardized across large regions in the Western world in general until perhaps the 18th century, so how long a cubit really was is hard to say and almost certainly varied from place to place and time to time. But it would have been somewhere around half a meter. Cubits generally show up in the astronomical tablets in reference to the lengths of shadows. At noon on the summer solstice the length of the shadow cast by a stick would be shortest and on the winter solstice the length of the shadow would be longest. The texts would provide ways to calculate how long the shadow would be at different times throughout the day at different times of the year. This, in effect, provided the Babylonians a way to measure and calculate the elevation of the Sun at different times of the day throughout the year.

The final type of unit that we see in the astronomical tablets is weight. It’s pretty easy to see how units of time show up in astronomical tablets — so-and-so star rises this many hours after that star, etc. And as we just saw, units of length were used to measure the elevation of the Sun. But it’s a bit strange that weight would come up in an astronomical text. It turns out that units of weight end up effectively being a unit of time as well because the Babylonians used water clocks to measure time.

Unfortunately no Babylonian water clocks survive, so we have to go off of descriptions of the devices in tablets. But water clocks are in principle pretty simple. You basically take a jug with a small hole in the bottom and then fill it up with water. Time is measured by how long it takes for the water to drain from the jug, basically like an hourglass but with water instead of sand. Since a certain amount of elapsed time corresponded to a certain weight of water, we see why the Babylonians sometimes measured time in terms of weight.

Water clocks were used by many civilizations throughout the ancient world, but while the Babylonians were ahead of the curve on their astronomy, it seems their water clocks were quite rudimentary. The Egyptians figured out that by adding gradations in the jug they could read out intermediate times without needing to wait for the entire jug to empty. The Chinese and Greeks developed much more sophisticated water clocks which we will talk about in future episodes.

Now I may be being a bit unfair to the Babylonians because, after all, no water clocks actually survive. So maybe the actual clocks were more sophisticated than we think. The tablets we are reconstructing this from may have neglected to describe the operation of these devices in complete detail. Regardless, water clocks see to have played an important role in Babylonian society. We learn from the tablets that water clocks were used not just for astronomical measurements, but for payment as well. Babylon, like any respectable city-state in the ancient world, needed watchmen to be on the lookout for invaders, both day and night. Someone had to sound the alarm if the Kassites, Hittites, or Elamites started storming in.

Well, being a watchman probably wasn’t the most thrilling of jobs. You’d have to stay up all night in the freezing cold, peering into darkness, bored out of your mind. The plight of the watchman even makes it into one of the psalms of the Hebrew Bible where the psalmist writes, “my soul waits for the Lord more than watchmen for the morning”. Well needless to say the watchmen didn’t do this out of the goodness of their hearts. They expected to be paid. But how much to pay them? Well the Babylonians had an interesting strategy. The pay would vary over the course of the year. After all, the length of the night varies. You should get paid more as a night watchman during the winter when the nights were long, than in the summer when the nights were short. Now, of course, the threat of invasion didn’t end at dawn. The city had to be guarded during the day as well, and the pay of the day watchmen varied inversely. They were paid more during the summer and less during the winter. In effect, watchmen were paid by the hour, and if you’re going to pay them by the hour, you need a way to keep track of the hours. So measuring the length of day and night was actually of prime social importance, and this was one of the purposes of the water clock, and the measurement of the heavens.

The Mul.Apin, which I talked about in the last episode, actually goes into some detail about this and describes a scheme whereby on the summer solstice, the length of a day is twice the length of the night, and vice versa on the winter solstice. Then, at each month the proportion changes by two hours so that there is a completely linear increase in the length of the night until the winter solstice, and then a completely linear decrease back. This technique that the Mul.Apin describes has actually generated quite a lot of controversy in the academic literature. Because it’s way off. There is somewhere on Earth where the length of the day is twice as long as the length of the night on the summer solstice, but it’s nowhere near Babylon. It’s closer to the latitude of Paris or Vancouver. At Babylon the length of the day is only about 50% longer than the length of the night. Was this just a bad approximation? What makes things even more confusing is that in a different spot in the Mul.Apin, the Babylonian author seems to use the much more correct ratio of 3:2 for the length of daylight to nighttime on the summer solstice.

One theory proposed by Otto Neugebauer was essentially that the measurements of weight used by the Babylonians were not uniform in time. After all, if you pour two pounds of water into a water clock, the first pound will come out much faster than the second. Initially the stream will be fast, but once most of the water has already been lost, it will be reduced to a trickle. If you were managing a team of watchmen around the summer solstice, at sunset you might need to pour, say, one pound of water into your water clock in order for it to be finished at sunrise. Now, if the weight of water you put in your clock were just proportional to time, you would need to put in 1.5 pounds of water at dawn in order for it to be done at sunset, since the length of the day is 1.5 times as long as the length of the night. But it won’t work that way in practice because the water will flow faster at first since there’s more water in the jug. So you need to put more water in the jug. In the middle of the 20th century, Otto Neugebauer did some calculations and found that you would need to put in about two pounds of water in in order for it to run out at around sunset. So perhaps that explains why the Babylonians said that the length of daylight was twice the length of nighttime on the summer solstice.

Well that is a convenient story, but much like a collection of water clocks, it has some holes. Jens Hoyrup later repeated this calculation and found that Neugebauer’s results were really pretty far off and that even accounting for the extra weight from the additional water, a 2:1 ratio between the length of daylight and nighttime was really far off. Perhaps the Babylonians weren’t especially interested in accuracy in this measurement. Maybe the watchmen were unionized and it was union reps making these measurements.

I jest, but this isn’t too far off from another theory of why the Babylonians had this weird 2:1 ratio between the length of daytime and nighttime at the summer solstice. The idea is that this ratio wasn’t used for scientific purposes, but was only used for the payment of watchmen. They bumped the ratio from the correct 3:2 ratio to a 2:1 ratio as a sort of bonus. During the winter, not only were the nights longer, but they were also colder, so the night watchmen got some extra pay to compensate them for their trouble. And similarly during the summer not only were the days longer, but they were hotter, too, so then the day watchmen were the ones who would get the bonus. But this just a theory. The truth is that we don’t really know why the Babylonians recorded such an inaccurate ratio between the maximum length of day and night in the Mul.Apin.

At any rate I have digressed quite a bit. You may recall we started off talking about units. I’ve tried to explain in maybe a bit too much detail why units of weight show up and are important in Babylonian astronomical records, but just briefly, there are three units of weight in the Babylonian records: the bilta, manu, and shikulu. Just like the cubit, these weights all have correspondences in Hebrew, some of which survive to this day. Units of weight also corresponded to units of currency, just as they have in many countries throughout Europe: the Italian lira, the Spanish peso, and the English pound to name a few. A bilta was heavy, weighing somewhere between 75 to 100 pounds, and represented quite a substantial sum of money in the ancient world, perhaps the total wages a laborer would earn in 10 or 20 years. By a miracle of linguistic evolution, the word “bilta” eventually morphed into the word “talent”, which shows up in the Gospel of Matthew as the parable of the talents. If you haven’t read it, in brief, a master leaves on a journey and entrusts his property to three servants. One receives five talents, another receives two talents, and the last receives one. Upon his return, the master asks his servants what they did with the money he left them. The first two say that they had used his money to engage in prudent trades and doubled. The third, however, says that he had kept his master’s money safe by burying it in the ground. The master then praises the first two servants and turns to the third and declares him to be a “wicked and slothful servant”.

The evangelist then expounds the so-called “Matthew principle”:

For unto every one that hath shall be given, and he shall have abundance: but from him that hath not shall be taken away even that which he hath.

In other words, the rich get richer, the poor get poorer. The master concludes his upbraiding of the third servant by saying “cast ye the unprofitable servant into outer darkness: there shall be weeping and gnashing of teeth.”

It would not be a good Biblical story if it didn’t end with the gnashing of teeth. As is typical with Jesus’s parables, He tells a story that has a rather strange moral if you simply read it as investment advice, but which has rich theological implications. If we read the story as a recommendation from the Son of God that wealth managers should yolo all their clients money into bitcoin, we’ve probably done something wrong. But diving into the theological implications of this parable is even further outside the quite liberal scope of this podcast than I typically venture.

Nevertheless, I bring this up because the modern meaning of the word talent originates from this parable. The word used to refer exclusively to the unit of weight. But it was through this parable that it came to develop a connotation of a “natural gift” that we associate with it today.

Clearly a talent was a somewhat unwieldy unit of measurement, so the talent was divided into 60 mina. The version of the parable told in the Gospel of Luke actually uses minas rather than talents, but otherwise the use of the word mina has not survived to present day. A mina was still a lot of money for most people. A laborer might expect to earn about 4 minas per year in wages. So the mina was in turn divided into the third unit of weight, the shekel, which survives to this day as Israel’s unit of currency. Following the base 60 number system, one mina was divided into 60 shekels. A shekel was roughly the daily wage of a laborer.

So those are the units that we encounter in the Babylonian astronomical tablets. Three units of time, the beru, about 2 hours, the ush, about 4 minutes, and the ninda, about 4 seconds. One unit of distance, the cubit, corresponding to about half a meter. And three units of weight, the bilta, somewhere around 40 kilograms, or 75 to 100 pounds, the mina, a bit more than half a kilogram, or around a pound, and the shekel, somewhere around 10 grams or half an ounce.

Now we’re ready to dive into some of the more sophisticated tablets. Before we get to System A and System B as a promised in the previous episode, we’ll warm up a little bit with the more advanced lunar eclipse predictions that the Babylonians produced.

You may recall that back in episode 3 we talked about how the Babylonians were able to predict when lunar eclipses occurred. This was important because lunar eclipses were among the most significant and malevolent omens in Babylonian astrology. Kings would abdicate the throne and commoners were likely executed as a consequence of lunar eclipses. Throughout most of the history of Babylonia we don’t actually have a very good understanding of how they actually did this, but as I described in episode 3, it was probably pretty simple. They had just recognized that lunar eclipses tend to come in series of five or six, spaced by six month intervals. So if you observed one lunar eclipse, you would know that there would be four or five more every six months. Now it seems that the Babylonians also had recognized some patterns that allowed them to guess when the next series might start, it was usually a month before the last eclipse of the previous series, but it was just a guess and it didn’t always happen. At those times they would be on alert that an eclipse was possible, but they didn’t know for certain.

Sometime during the rule of Babylon by the Persians, Babylonian eclipse predictions had become considerably more sophisticated. To understand what the Babylonians had figured out, we need to remember how lunar eclipses work. A lunar eclipse happens when the Earth comes directly between the Moon and the Sun and casts a shadow on the Moon. Now, this arrangement can only happen during a full moon, but it doesn’t happen during most full moons because the moon’s orbit is slightly inclined relative to the Earth’s. Usually the moon is too far above or below the Earth’s orbit to be in the Earth’s shadow and we just see a normal full moon.

There are, however, two special points in the Moon’s orbit, called nodes, where the plane of the Moon’s orbit intersects the Earth’s. From our perspective here on Earth, this is where the Moon’s path on the sky intersects the ecliptic, which is the Sun’s path on the sky. Now, of course, the Moon goes through one orbit every month, so it passes through the nodes twice a month and usually nothing special happens. But if the Moon happens to be full as it passes through a node, this means that the Sun, Earth, and Moon will all be lined up in just the right way to produce a lunar eclipse.

As I explained in more detail in episode 3, we see the pattern of lunar eclipses separated by six months because if the moon is full as it passes through a node, the next month it will be 30 degrees away from a node, and the next month 60 degrees, etc., until six months later it will be 180 degrees away from the first node, which is exactly where the second node is. So you get an eclipse again.

Now you don’t need a perfect alignment of the Sun, Earth, and Moon to get an eclipse. The Earth’s shadow is fairly large so even if the Moon is a little bit off the eclipse can still happen. In practice, if the full moon is within five or six degrees of the node you can still get a total lunar eclipse, though it won’t last as long as when the full moon happens to be closer to the node. And as long as the moon is within 10 or 12 degrees of the node you’ll see a partial eclipse.

Now the reason I put some range on these numbers like between five or six degrees for a total eclipse and 10 or 12 degrees for a partial eclipse is not because the true value in unknown, but that it actually varies. Sometimes the Moon is closer to the Earth and it can be further away from the node and still be in the Earth’s shadow and produce a lunar eclipse; other times it’s farther away and the alignment needs to be more precise. This is a consequence of the fact that the Moon’s orbit is slightly elliptical. There is a point on the orbit called perigee, where the Moon is closest to the Earth and moves fastest, and another called apogee where it is farthest from the Earth and moves slowest.

But this setup is not static. Due to a torque exerted by the Sun, the Moon’s orbit precesses. The location of perigee drifts eastward by about 3.3 degrees per month, and the nodes also drift in the opposite direction, westward, by about 1.5 degrees per month. It’s because of this drift of the nodes that you only see five or six eclipses in a row, and it’s where the full moon is relative to apogee or perigee that tells you whether you’ll get five or six. So the next time a set of eclipses starts up again, the pattern will look a little bit different.

One way of thinking about this is that there are a number of different ways we can define a “month”. There is the synodic month, which was what the Babylonians meant when they said “month.” This is the time it takes the Sun, Earth, and Moon, to have the same arrangement, or in other words, the time from new moon to new moon. Then there is the sidereal month, which is the time it takes the Moon to return to the same background star on the sky. Because the Sun is moving eastwards on the sky relative to the background stars, just like the Moon only slower, the sidereal month is shorter than the synodic month by a little more than two days.

But then there is also the so-called “draconic month”, which is the time it takes the Moon to return back to a node in its orbit. It takes its name from the Latin word “draco” meaning “dragon” because lunar eclipses only happen when the Moon passes through a node and medieval astronomers conceptualized a dragon as living at the node and consuming the Moon. The draconic month is slightly shorter than the sidereal month because the nodes are moving westwards relative to the background stars, so it takes the Moon less time to catch up to the node.

Finally, there is the anomalistic month, which is the time it takes the moon to go from perigee to perigee. Because perigee moves eastwards on the sky, in other words, in the same direction as the Moon, the anomalistic month is slightly longer than the sidereal month. Where does the name “anomalistic” month come from? Well, if you imagine that the Moon’s orbit was a perfect circle, the Moon’s speed around the Earth would be constant. From here on the ground, we would observe it drift eastwards relative to the background stars at a constant rate of time every night of the month. But because the Moon’s orbit is actually slightly elliptical, it moves a little bit faster when it’s at perigee and a little bit slower at apogee. What we see from Earth is that sometimes the Moon moves a little bit faster than average, and sometimes a little bit slower. The difference between the Moon’s current speed and its average speed is called an “anomaly” in astronomy, and its use is not just limited to the Moon. Any object that has an elliptical orbit will have anomalous motion. So one way to think of the anomalistic month is that it’s the time it takes the Moon to return to the point where it is moving fastest in its orbit and the anomaly is the largest.

Of these four months, only three are actually relevant to lunar eclipses: the synodic, draconic, and anomalistic. The sidereal month just tells you about the Moon’s location with respect to the background stars, but the background stars don’t figure into the presence or character of a lunar eclipse. By a wonderful coincidence, it turns out that there is a period of time which is approximately commensurate with these three kinds of month: this is a period of 18 years and 11 days and is called a “saros”. One saros contains, by definition, 223 synodic months, and that happens to equal 241.999 draconic months, and 238.992 anomalistic months. Very, very nearly 242 draconic months and 239 anomalistic months exactly.

What this means is that after one Saros cycle, the configuration of the Earth, Sun, and Moon will be almost exactly the same. So if you saw an eclipse series where there was a partial eclipse followed by three total eclipses followed by a partial eclipse, 18 years and 11 days later you will see exactly the same series. Even the duration of those eclipses would be almost exactly the same.

Now an important thing to remember about Babylonian astronomy was that they had no physical model for their predictions. They had simply uncovered a long-term pattern in the appearance of eclipses. Now I don’t want to trivialize this. Discovering this pattern was no simple matter because as I mentioned in episode 3, not all eclipses were observed. Some of them, actually about half of them, happened during the day and couldn’t be observed. Or it was cloudy. So uncovering this pattern was not as straightforward as going back over the records and noticing that the same things had happened every 18 years and 11 days. There were a lot of holes in their historical records. Nevertheless they were able to infer the presence of eclipses that hadn’t been observed to fit the pattern they were developing.

One wonderful record of the Saros comes in the form of the so-called Saros Canon, a tablet produced maybe in the early third century BC. The tablet is simple. It’s just a series of triplets of columns. The first column in each triplet contains a list of years and the second column contains a list of months. Periodically there is a horizontal line across the tablet that breaks it into groups of seven or eight lines, and in the first line of each group there is a marking in the third column that says “5 months”.

To an astronomer with a careful eye, there could be no better proof of the Babylonian knowledge of the Saros cycle than a tablet like this. If you count the number of lines in each column, you find that there are 38 and that this corresponds to 223 months — this is exactly the number of synodic months in a Saros cycle. The months listed in each grouping are all six months apart, and there are seven or eight. Now this part is a bit odd, because there are only five or six eclipses in a cycle. The author of this tablet apparently included the month six months before the first partial eclipse in the series and the month six months after the last partial eclipse in the series. But this makes sense for computational purposes. If you do this, then each grouping starts five months after the last month listed in the previous grouping, hence the note saying “5 months” in the third column.

So it seems pretty clear that this tablet is a list of lunar eclipse dates. And indeed we can go back and look at the dates listed and check that there were, in fact, lunar eclipses in the listed months.

Now there is one other oddity about the Saros Canon tablet. It contains five complete sets of these triplets, and a sixth set that has only 14 lines, so we have five complete Saros cycles represented, and about a third of another. But these columns don’t actually begin at the beginning of an eclipse cycle. The columns begin in the middle of an eclipse cycle, list three eclipses, then draw a horizontal line, start listing the complete set of seven or eight months in an eclipse cycle, and then end with another incomplete cycle containing four months, which then gets continued at the top of the next column. Why did they start the columns in the middle of the action? Well, one answer is that that’s just what they did — maybe they started from some eclipse that was particularly noteworthy for historical reasons that have been lost to the sands of time.

However, one of the original researchers to study this tablet, the Jesuit scholar Strassmeier, proposed another idea. Strassmeier noticed that the tablet is broken off at the left hand side. So it’s possible that the piece we have is just the rightmost set of a longer series of columns. Now, the Saros cycle is great and all, but it isn’t perfect. The locations of eclipses end up getting offset a bit less than half a degree from one cycle to the next. Over a long enough time, about eight or nine cycles, or 150 years, this has the effect of causing a new eclipse month to get added to the end of a saros cycle, and the next cycle to lose its first month. This shifts all the eclipse months of the next cycle so that what was the second eclipse of the cycle now becomes the first, the third becomes the second and so forth.

In the tablet, this would have been easily represented by starting the first line of the cycle one line up from the previous cycle. So if you started by listing the beginning of an eclipse series, after eight or nine columns, you’d now start with a lone final month from one eclipse series. So if the original tablet went back 450 years or so, there would have been three of these shifts, and by the time you got to the rightmost edge of the tablet, each column would begin with the last three months of an eclipse series. Well it’s a plausible theory anyway. It is easy to speculate endlessly about ancient records, but without more evidence it’s hard to say anything conclusive.

Well, the discovery of the Saros cycle was a triumph, but it wasn’t actually the height of Babylonian mathematical astronomy when it came to eclipses. I had promised you in the previous episode that we would talk about System A and System B, which were the most sophisticated computational techniques the Babylonians developed, but I’m afraid I have lied to you because we are out of time for this episode. In the next episode I will introduce System A and System B in the context of computing planetary positions, but later I will also describe how they were used to make even more refined eclipse predictions that specified not only the date that an eclipse would occur, but also where it would occur. But that is a tale for the next month. Thank you for listening, and until the next full moon, good night and clear skies.