Episode 6: System of a B

June 24, 2021

We dive into the most sophisticated model of planetary and lunar motions that the Babylonians developed: System A and System B and see how the Babylonians used this method to predict eclipses, the length of months, and even measure the precession of the perihelion.


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.

We’ve been diving into the history of Babylonian astronomy for the last few episodes and have more or less traced its development from the earliest records we have from around 2000 BC to the much more numerous records that appear in the last five or so centuries BC. But I have held off on describing the pinnacle of Babylonian astronomy, two techniques today called System A and System B. But no more. It is high time we dive into the best that the Babylonians have to offer us.

To reorient ourselves in the broader history of the Near East, after centuries under the rule of the Assyrians, in the late 600s BC, Babylonia had finally regained its independence and formed what is today called the Neo-Babylonian Empire, to distinguish it from earlier Babylonian empires prior to the conquest of the Assyrians. The fortunes of the Babylonians turned very quickly as they went from centuries of subjugation to conquering most of the land in the fertile crescent within just a few decades. At this time the city of Babylon was a center of trade in the region and very wealthy. In fact, the prophet Ezekiel describes the conquest of Judea by the Babylonians and the exile of the Jews in metaphorical terms. He writes of a great eagle who came to Lebanon and broke off twigs from the top of a cedar and carried them to land of trade and set in a city of many merchants. The eagle was long a symbol of Babylonia, and the land of trade and city of merchants was none other than the city of Babylon.

But as another biblical author writes, those who live by the sword die by the sword, and the Neo-Babylonian empire fell to the Persian empire of Cyril the Great in 539 BC, and the Babylonians never regained independence after that. But while the Persian conquest was a disaster for Babylonian self-rule, it wasn’t so bad for astronomy. Babylonian astronomy really thrived during the period under Persian rule. It is in this period that Systems A and B develop.

As we discussed in episode 4, there were a number of different types of astronomical tablets that appear in Babylonian history: astronomical diaries, astronomical omens, goal year texts, and so on. But the most sophisticated of these are the tables that calculate ephemerides for the locations of the planets, moon, and the Sun using System A and System B. It is a little unclear when Systems A and B were originally developed. It’s actually not known with certainty which of the two came first. System A is a little simpler so it’s usually assumed that it was developed first, but the earliest tablets with System A appear later than the earliest tablets with System B. But the evidence suggests that these two methods of calculating planetary positions show up around 400 BC.

Now the Babylonian astronomers were mostly anonymous. The one main exception was the letters astronomers wrote to the king interpreting astronomical omens, which could be signed. And I mentioned in episode 4 the astronomer Berossus who recorded the story of the Persian conquest of Babylonia. Berossus is known today for the historical texts he wrote around the early 200s BC, but in his day he was known as an astronomer. Unfortunately none of his astronomical works survive, only the historical texts.

But one of the few other Babylonian astronomers whose name we know is an astronomer by the name of Kidinnu who lived during the 4th century BC. According to later texts, Kidinnu was recognized as the inventor of System A and System B. Kidinnu is a bit of a quasi-mythical figure, probably a real person, but also shrouded in a bit of legend. In this way he is a somewhat like the Greek poet Homer — just as there probably was some Greek poet by the name of Homer, there probably also was some Babylonian astronomer by the name of Kidinnu, but just as it’s hard to say whether this poet named Homer wrote everything that was later attributed to him like the Iliad and Odyssey, it is hard to say whether or not Kidinnu himself developed everything that was later attributed to him. Unfortunately nearly all the surviving texts that mention him were written centuries later and many of them were written by Greek astronomers. The only apparently contemporaneous mention of Kidinnu was an astronomical diary which mentions that someone named “Kidinnu” was “killed by the sword” on a date which works out to be the 14th of August 330 BC, less than a year after Alexander the Great took Babylon from the Persians.

So what did Kidinnu do, according to this tradition? Well when we started talking about Babylonian astronomy I described the essential problem of Babylonian civilization as being how to figure out how long a month is and how many of them there are in a year since each month starts when you first see a crescent after the new moon, but the length of each month is variable, and the length of the month does not neatly divide the length of a year. Along the way, the Babylonians created a bunch of new problems for themselves like when lunar eclipses happen and where the planets will be. System A and System B solved all these problems.

In order to start getting into this, we’ll begin with the planetary predictions since they’re somewhat simpler than the lunar predictions. Now you may remember that the Babylonians weren’t really interested in where the planets were in general. Today when we make planetary predictions, we generally want to know where a planet will be at some particular time. But the Babylonians only cared about knowing when and where certain special events occurred like opposition or the first stationary point, when the planet entered into retrograde motion. Now, specifying when this will happen is easy. You just need to say what day it happens on. Calculating that day might be difficult, but what it is you want to calculate is pretty straightforward. In fact, because the period of planetary motion is so long, it’s actually fairly hard to measure the time that something like the first stationary point occurs to within a day. You are basically trying to observe, by eye, when the location of a planet stops drifting eastward and starts drifting westward. But when the planet is close to its stationary point it’s hardly moving at all, so this is a very difficult measurement to make. Centuries later Ptolemy complained about the Babylonian predilection with only recording these special events in a planet’s orbit because it’s hard to measure exactly when they occur. In fact today we can go back through Babylonian records and determine that Babylonian observations of these events tended to be biased slightly early.

So measuring when these events occurred was tricky, but what it was that you wanted to measure, namely the date, was obvious. But you also want to know where the event occurred. And how do you measure that? For this you need a coordinate system. You need some way to connect a location on the sky with a pair of numbers.

Now the simplest coordinate for the sky is what is called the system of altitude and azimuth, or an alt-az coordinate system for short. Here if you want to record the position of a planet on the sky, you measure its altitude as the angle above the horizon that the planet makes. And the azimuth is the angle the star makes relative to north. The advantage of the alt-az system is that it’s really easy to measure the altitude and azimuth of a planet or star and if someone tells you that there’s something spectacular going on at some altitude and azimuth, it’s also really easy to figure out where in the sky to look. But the alt-az system has a big problem. Because the altitude and azimuth of a planet or star is not constant. Every night, stars rise in the east, culminate across the meridian, and then set in the west before doing the same thing all over again the next night. Well when the star rises in the east, its altitude is 0 degrees because it’s on the horizon. Then it will climb in the sky and its altitude will increase and its azimuth increase until it crosses the meridian, at which point its altitude starts decreasing while its azimuth continues to increase. So if you want to record the altitude and azimuth of a star or planet, it really only tells you anything useful right now. After a few hours those numbers will be completely different.

So the system of altitude and azimuth has some conveniences, but also has some major shortcomings as well. But there is an alternative that avoids these problems. For any astronomers listening to this show, you probably already have a guess as to what this alternative is — the famous system of right ascension and declination. But you would be wrong. The system of right ascension and declination is a relatively modern innovation and was unknown in the ancient world. If you don’t know what right ascension and declination are that’s okay — I’ll describe it in detail when it first appears in the history of astronomy in the 18th century. But at the rate we’re going, that will be many moons from now. So check back in maybe the year 2031 when we finally get to Enlightenment era astronomy.

Instead the system that the Babylonians developed is the system of ecliptic latitude and longitude. The idea here is that rather than reference the location of a star or planet with respect to the horizon, since that location is constantly changing, we’ll reference the location of the star or planet with respect to other stars on the sky. Since the stars themselves are fixed and the planets change their locations only very slowly, this gives us a way of saying where a star is that’s much less dependent on the exact time you made the measurement. And, to be complete, it also doesn’t depend on your exact location on Earth, although this wasn’t so much of an issue for the Babylonians since they pretty much did all of their astronomy from one location, namely Babylon.

Okay, so we want to reference the location of a star relative to other stars on the sky. Now how do we do that? The system that the Babylonians used was to pick out the ecliptic as being a special point of reference on the sky. Remember that the ecliptic is the path that the Sun traces out against the background stars over the course of a year. Then they measured how high a star or planet was above the ecliptic, called the ecliptic latitude, and, if you went down to the ecliptic, where the star or planet was along the ecliptic. In other words, what constellation of the zodiac was the object in and where would it be in that constellation if you did that projection.

This is a very convenient coordinate system for measuring the locations of the planets because all the planets move very close to the ecliptic. They only wander a few degrees away from the ecliptic. The Moon gets the farthest, but even the Moon never gets more than about five degrees away from the ecliptic. So most of the motion of the planets is along ecliptic longitude, with some small variation in ecliptic latitude superimposed on the longitudinal motion.

So, to put this all together, the location of a star or planet could be given by the ecliptic latitude, which was the angle above the ecliptic, and the ecliptic longitude, which was how far along the ecliptic it was. The ecliptic latitude was just measured in degrees, but the ecliptic longitude was a little different. The Babylonians divided the ecliptic into twelve signs of the zodiac, each sign being 30 degrees across. Then to specify the ecliptic longitude of an object they would just say which sign the object was in, and how many degrees the object was from the beginning of the sign from 0 to 30.

Based on the accuracy of the measurements of the Babylonian astronomers it seems likely that they weren’t doing it by eye and that they must have had some sort of equipment to help them measure these angles accurately. But unfortunately, as with so much else in the ancient world, none of these instruments survive and they made no records of these instruments in any of the tablets we have today. So we don’t know exactly how these measurements were taken.

Okay, so we now have a coordinate system where we can nicely represent the positions of the planets at different points in time. Let’s now start by taking a look at how the Babylonians would use System A to calculate, say, the time and location when Jupiter would be at opposition. In the completed tablet there would be one column with a set of years and a second column with a set of months and days. Now these days weren’t actual days, but a sort of formal astronomical day called a “tithi”, which was defined to be exactly equal to 1/30 of a month to make the calculations easier. But this formal tithi was always within half a day of some real day that could be computed later, taking into account which months had 30 days and which had 29 days and which years had extra months and all the rest.

In between these two columns there was another column that had the difference in time from one opposition to the next, though by convention they subtracted off one year from this. This is because the period of opposition for a planet like Jupiter is always about a year and a bit, so to save space they just wrote down the extra bit. Finally, in a fourth column, they listed the longitude opposition would occur at.

So how did they figure out these values? The key is to look at how the longitudes change over time. By this point the Babylonians had realized that the planets don’t always move at the same velocity. Sometimes they move faster relative to the background stars and other times slower. If you just assumed that the planet Jupiter moved the same amount from one year to the next, you would find that your predictions for when opposition would be would be systematically too early and other times systematically too late.

The modern day explanation for this is that Jupiter’s orbit is elliptical. Not very elliptical, but elliptical enough that when it is closest to the Sun at perihelion, it moves noticeably faster than when it is farthest from the Sun at aphelion. If Jupiter is passing through perihelion and is moving quickly, it will take longer to reach opposition because the Earth will need to move further to catch up. But if Jupiter is passing through aphelion and is moving slowly, it won’t take as long to reach opposition because it hasn’t moved as far.

So what the Babylonians did is break the ecliptic into two arcs: a fast arc and a slow arc. If Jupiter was in the fast arc, they would add 36 degrees to the previous longitude of opposition to determine the longitude of the next opposition. But if Jupiter was in the slow arc, they would add only 30 degrees.

Well what happened when Jupiter moved from the fast arc to the slow arc or vice versa? As we shall see, during this period the Babylonian astronomer priests became masters of linear interpolation. They would blend the two arcs together by computing an intermediate value that was linearly interpolated between the values on the slow and fast arcs.

The slow arc measured 155 degrees on the sky and the fast arc measured 205 degrees and was centered at about 12 and half degrees into the sign of Pisces, which is close to the actual location of Jupiter’s perihelion of 20 degrees in Pisces. Now the width of the slow and fast arcs was not chosen arbitrarily. The Babylonians derived these widths in such a way that if you ran this algorithm repeatedly, you would find that for every 391 oppositions, Jupiter would travel around the ecliptic 36 times and this would all happen in 391 + 36 = 427 years. This is very close to the true periodicity of this phenomenon for Jupiter’s orbit and acted as a sort of check on the accuracy of the method and prevented the approximations in their method from growing without bound.

So using the slow and fast arcs allowed the Babylonians to figure out where on the sky the next opposition would be, but then there’s the other question of when it will be. For this the Babylonians used a very simple method. They just said that the difference in time from one opposition to the next was equal to the difference in longitude plus a constant, which, in this case, happened to be equal to about 12.08. To modern eyes this is a very odd thing. Because they were adding together numbers with completely different units. They were adding an angle to some constant and somehow getting out a time. In modern notation this just doesn’t work. But it worked for the Babylonians because during one tithis, the artificial day that they used for their calculations, the Sun moves almost exactly one degree in the sky. So to them, time measured in tithis was interchangeable with degrees and the two could be added or subtracted at will. This special constant can be derived by taking the average time between oppositions in units of the tithis and subtracting off the average change in ecliptic longitude. In practice this constant seems to have been determined empirically because it doesn’t quite match the correct value, but it is close.

So that is System A for planetary motions. We dwelled on the mechanics of it in some detail, but the basic idea is that you can model the variable motion of the planets by breaking up the ecliptic into two regions: one region where the planet moves at a slow speed, and another region where the planet moves at a fast speed. In essence, we take this variable motion, which is roughly sinusoidal, and approximate it as a step function or rectangular wave.

Some tablets display a slightly more sophisticated method where the ecliptic gets broken up into several arcs where the planet has different speeds, but the real improvement comes with System B.

System B takes the same idea as System A and makes a tweak. A rectangular wave is better than nothing, to be sure, but it’s still not a great approximation, so what the Babylonians did is now approximate this sinusoid as a triangular wave, or, to use the actual technical term, a zig-zag line. They basically had a line that zig-zagged between a maximum and minimum speed and whose average was the observed average speed of the planet. Other than this refined approximation, though, System B worked on the same principle as System A. Because the positions were calculated by adding together the distances traversed from successive velocities, which in turn increased regularly, in modern terms, the Babylonians were essentially fitting the positions of the planets to a pair of parabolas.

So that, in brief, is System A and System B for planetary motions. The Babylonian tablets are the most complete for Jupiter, but records for all the other planets exist as well. Mars is particularly interesting because it’s so close to the Earth, its apparent speed on the sky is much more variable than the other planets. Consequently, the Babylonians actually broke the ecliptic into six arcs with different speeds to try to model its motion. Especially remarkable were the Babylonian ephemerides for Mercury. Mercury, being closest to the Sun, is by far the hardest planet to observe because it is only visible close to the horizon shortly after sunset or before sunrise. Nevertheless, the Babylonians were generally able to predict its position to within two degrees, although occasionally they were off by as much as 10 degrees. Even more remarkably, the Babylonians somehow measured its synodic period to within 3 and a half minutes of its true value

Okay, so we now see how System A and System B worked to predict the motions of the planets. But the Babylonians also used these systems to predict lunar phenomena as well. There were two reasons the Babylonians had to predict lunar phenomena, which we are by now quite familiar with — the Babylonians wanted to know when lunar eclipses would be, and they also wanted to know when the next month would be.

Predicting the time and location of lunar eclipses is a harder problem than predicting when and where a planet will reach opposition or some other station. In order to accurately predict a lunar eclipse you need to do a few things: you need to be able to figure out when and where a full moon will be, and then you need to figure out the latitude of the moon when it’s full. To make things even trickier, in order to figure out when full moon is, you need to take into account both the motion of the Moon and the Sun, since both are moving.

What the Babylonians did was recognize that the Sun moves a lot slower than the Moon. So you can use the Sun to figure out where the full moon will be and the Moon to figure out when it will be. Since the Sun doesn’t move too fast, as long as you roughly can figure out when the full moon will be, you’ll have a pretty accurate measure of its location, and because the Moon moves fairly rapidly, if you know more or less where it will be when it’s full, you’ll have an accurate measure of its time. So, when you’re figuring out where the full moon will be, you ignore any variation in the motion of the Moon and only account for the variation in the motion of the Sun to set its location. Then when you want to figure out when exactly it will be, you fix the location of the Sun and then take into account the variable motion of the Moon.

In order to take into account this variation, they used the same trick they did with the planetary motions. In the System A tablets, they broke the ecliptic into a fast region and a slow region and had one speed in the slow region and another speed in the faster region. In the System B tablets the speed linearly bounced between two extreme values in the usual zig-zag line. If you take the average between the maximum and minimum velocities of the moon you get the average length of a synodic month, and, incredibly, the Babylonians got 29d 12h 44m 3s 1/3, which is less than half a second off from the correct value. But that’s not all you can learn from this zig-zag line. If you take the period of the zig-zag line, this tells you the ratio between the length of an anomalistic month, remember, the time from perigee to perigee, and the length of the synodic month, the time from new moon to new moon. The period of the zig-zag line tells us that the Babylonians had measured this to be 269 anomalistic months to 251 synodic months, and this means that the length of the anomalistic month is 27d 13h 18m 34s 3/4, which is only 2.7 seconds off from its correct value.

What is perhaps even more incredible than the Babylonians measuring the variable motions of the Moon so accurately, is that when it came to the Sun, they recognized that its motion was variable at all. After all, it is really hard to observe the Sun. I mean, you know it’s there, but it’s so bright that it’s hard to measure exactly where it is on the sky, and after all, it washes out all the background stars so you don’t really have much to compare it against except the horizon. Well the Babylonians didn’t leave us with records as to how they figured this out, but as usual, the Greeks couldn’t keep their mouths shut and blabbed the Babylonian secrets. According to the Greeks, the Babylonians had figured out the variable motion of the Sun by measuring the length of a shadow cast by a stick at noon. On the summer solstice the shadow will be the shortest and on the winter solstice the shadow will be the longest. Surprisingly, the time from the summer solstice to the winter solstice is not the same as the time from the winter solstice to the summer solstice. If you figure out when the equinoxes are, say, by observing when the Sun rises at exactly due east, the discrepancies become even more extreme. The length of summer, from the summer solstice to the autumnal equinox, is 93 2/3 days. But the length of winter, the time from the winter solstice to the vernal equinox, is only 89 days.

Our modern explanation for this phenomenon is that the orbit of the Earth is elliptical and the Earth moves faster when it passes through perihelion, the closest point to the Sun, and slower when it passes through aphelion, the furthest point from the Sun. Today, perihelion occurs in early January and aphelion in the beginning of July. This means that the Earth passes through aphelion during the summer and is moving slowest then and is passing through perihelion during the winter and is moving quickest. It seems the Babylonians noticed this variation as well.

These shadow length measurements were also used to calculate how the length of the day and night varied over the course of the year. We discussed these calculations in the context of the Mul.Apin text in the last episode because the Mul.Apin has this mysterious result that claims that the length of the day on the summer solstice is twice the length of the night. We discussed at some length the various reasons scholars have conjectured why the Babylonians were so far off in this measurement, but in this episode, we’re more interested in the later Babylonian texts and the Mul.Apin was relatively old. By the time we are in the Persian and Hellenistic era, we find that the newer procedure texts use a ratio of 3:2. Now this is much better than 2:1 but is still something of a mystery. Because clearly the Babylonians were very good at measuring astronomical phenomena, but a 3:2 ratio is still noticeably far off. A 3:2 ratio meant that the maximum length of the day was taken to be a full 13 minutes longer than it actually was and would imply that Babylon is 2 degrees north of its true location. As always, scholars have devised a whole host of different explanations for why this might be. Perhaps there was a sacred site some two hundred miles north. Perhaps the combination of atmospheric refraction and tall astronomical towers from which the astronomers took their measurements conspired to bias the measurement of the length of a day slightly longer. Or perhaps there were numerological or even computational reasons for choosing a nice, round, if slightly inaccurate ratio of 3:2.

Well, at any rate, from these measurements of the length of a shadow cast at noon they figured out that the motion of the Sun was variable and developed a zig-zag line to model this variation over the course of a year. Now just as with the variable motion of the Moon, we can use the average between the two extreme values of this zig-zag line to figure out what the Babylonians measured the average speed of the Sun’s motion to be, which tells us the length of a sidereal year, or the time it takes the Sun to return to the same part of the sky. They measured this to be 365d 6h 15m 19s, off by only six minutes. And then we can use the period of the zig-zag line to figure out the length of the anomalistic year, or the time it takes the Earth to go from perihelion to perihelion. Stunningly these two values are not the same. This implied that the location of perihelion changed by 30 arcseconds every year. This phenomenon is called the precession of the perihelion — not to be confused with its more famous cousin the precession of the equinoxes — but precession of the perihelion. Mercury is the planet which is better known for the precession of its perihelion because it was used as evidence for Einstein’s general theory of relativity in the early 20th century, but we’ll get to that story maybe in the 2050s. At any rate, the perihelia of all planets changes over time, and the principal reason for this is gravitational perturbations from other planets, though mostly those perturbations are coming from Jupiter. Now in reality, the precession of Earth’s perihelion is closer to 11 arcseconds per year rather than 30, but the fact that the Babylonians measured this effect at all is really incredible. Of course they had no mechanism to explain this, and whether or not they even noticed this as a particularly interesting phenomenon is not obvious from their tablets. But such is the story of Babylonian astronomy as I’ll say a little more about at the end of the episode.

Well let’s try to keep our eye on the ball here. The Babylonians weren’t really interested in measuring things like the precession of the perihelion, they were interested in sensible things like when the next month was and when lunar eclipses would occur. And if you want to do these things, you can’t only know the location of the Moon in terms of the ecliptic longitude, you also need to know its ecliptic latitude. When it came to the planetary motions, the Babylonians didn’t care too much about the ecliptic latitude. The planets don’t move very far from the ecliptic, and the variation doesn’t help you figure out anything interesting like when opposition will occur anyway. But when it comes to the Moon, the ecliptic latitude is critical for calculating a lunar eclipse. And the Babylonians did this the same way they did everything else in this time period — they used zig-zag lines and linear interpolation. They constructed a zig-zag line to model the Moon’s motion above and below the ecliptic. Unfortunately, for this particular application the zig-zag line failed them. The Moon’s motion, like all the other motions of the planets, is approximately sinusoidal, but when it comes to getting an eclipse prediction exactly right, you really need to know the Moon’s motion as it’s crossing the ecliptic. With a straight line, you have a slope that’s too steep near the maximum and minimum of the Moon’s latitude, and a slope that’s too shallow as it’s crossing the ecliptic.

So they did what any competent modern-day astronomer would do — they took their one line and broke it into two lines. Now instead of a single line zigging up, the line started zigging up, then when it got close to the ecliptic, it zigged even more with a steeper slope, until it got a little ways past the ecliptic, and then it continued zigging at its original, leisurely pace. Then it would hit the maximum latitude start zagging and, once it got close to the ecliptic it would zag more intensely, and then afterwards resume its more leisurely zag.

Because the Babylonians had so many centuries of eclipse records they figured out a way to fit these lines that resulted in exceedingly accurate eclipse predictions and ephemerides of the new and full moons. Though saying something like “fit a line” is admittedly a bit of an anachronism. They did not draw any lines, they just developed numerical algorithms that performed this procedure, though I want to emphasize the air quotes on the “just” because that’s a very impressive thing to do!

Since they had good predictions of the Moon’s ecliptic latitude at full moon, they could also predict what is called an “eclipse index”, which measured how strong of an eclipse there would be. Eclipse indices in a certain range implied a long, total eclipse; in a different range a shorter, but still total eclipse; and other indices implied a partial eclipse or no eclipse at all.

Finally, the Babylonians also used these predictions to solve the problem that had bedevilled them since the dawn of their civilization — when would the next month be? The new month was on the first evening that a crescent moon was observed after new moon and this was a very difficult thing to figure out. Even knowing when and where the new moon is going to be — in itself a very hard problem — that only gets you halfway there. Because you also need to take into account the angle that the ecliptic makes with respect to the horizon, which changes over the course of the year, the latitude of the Moon, and the time between new moon and the next sunset. I won’t walk you through all the details, but the Babylonians managed all that. This was an astronomical problem of tremendous difficulty to solve and the Babylonians were the only civilization of the ancient world that developed the necessary tools to solve it.

Well I would be remiss if I didn’t mention one more thing that the Babylonians measured — the precession of the equinoxes. I mentioned the precession of the perihelion earlier, but the precession of the equinoxes is the better known of the pair. This is a very slow change in the direction that the Earth’s axis of rotation with a period of about 26,000 years, or about one degree every 72 years. This manifests itself as a change in the north star, but also as a change in the location of the vernal equinox — the location of the Sun on the sky at the beginning of spring when the length of the day is equal to the length of the night. In some of their earlier tablets, they noted that the position of the Sun on the equinox was 10 degrees into the Ram. But in their later tablets they recorded this position as 8 1/4 degrees, and in still later tablets they recorded it as 8 degrees.

Now this is a big deal to us modern astronomers because our modern celestial coordinate system, the system of right ascension and declination is based on the location of the vernal equinox. The fact that it changes causes all sorts of headaches because it means that the coordinates of the stars change over time — the same problem I was telling you about with the system of altitude and azimuth. It’s not nearly as big a problem since the drift is much slower, but it’s still there. You need to fix the time that you made the measurements at, these days that’s conventionally January 1, 2000, and then figure out how much time has elapsed since then and then calculate the drift in order to compensate for it. Fortunately we have computers so it’s not a huge hassle unless you’re the poor astronomer who’s stuck with programming those computers, but in a way, the Babylonians were ahead of us here. They fixed their coordinate system to the background stars in a more rigid way. Consequently, the fact that the location of the vernal equinox changed just wasn’t a big deal to them. They noted it as being whatever it happened to be at the time they measured it and then got on with the important work of figuring out when the next month would start.

And this gets at the essential character of Babylonian astronomy. The Babylonians developed an impressive system to predict the positions of celestial bodies, but theirs was an essentially mathematical astronomy. They had no physical model of the phenomena they observed. They simply saw lights move on the sky and came up with methods to predict where those lights would go in the future. And as intricate as the Babylonian mathematical models of the planetary motions became, they were still intimately linked to the Babylonian priestly class. To the end, the Babylonian astronomers began their tablets with the inscription ‘in the name of god Bel and goddess Beltis, my mistress, an omen’, and the motion of the Sun was called “zi sha Shamash”, literally, “life-force of the Sun-god” and the motion of the moon “zi sha Sin”, or “life-force of the Moon-god”.

To get the first truly physical models of the Solar System we will need to leave Babylonia and turn our attention to the Greeks. And that will be the subject of our next episode and, I’m sure, many episodes after that. So until the next full moon, thank you for listening and clear skies.