We turn back the clock and see how a variety of Greek astronomers over the centuries contributed to the Greek calendar, and how Greek politicians ignored their developments. Then we see how the discovery that the seasons are not of equal lengths posed a problem for Eudoxus's theory of planetary motion.

Transcript

Good evening, and welcome to the Song of Urania, a podcast about the history of astronomy from antiquity to the present, with new episodes every full moon. My name is Joe Antognini.

Well I’m afraid that I have to open this episode with an erratum. Last month the star of the show was the astronomer Eudoxus. Or at least, that is what I led you to believe. But I was mispronouncing his name the entire episode like some sort of rube. In fact his name is properly pronounced Eudoxus. The correct order of operations when it comes to Greek names is to first look up the pronunciation and then say it, but in this case I did it backwards. Mea culpa.

Well, in the grand narrative of the Greek astronomy we’ve been following so far, the early astronomy began with what we might call cosmological speculations. The questions that the earliest Greek astronomers were trying to answer were: what is the structure of the universe, what’s the shape of the Earth, how are the planets arranged, and, above all, what is the arche, the fundamental nature of matter? Over the centuries there had been a lot of thought that went into answering these questions, but from a modern scientific perspective, there was not too much to show for it. But in Eudoxus, our star of the last episode, we see something really different — the first serious attempt at a mathematical model of planetary motions.

Now, this episode is going to be a little unusual in that we will spend some time seeing how students of Eudoxus’s school developed his model of homocentric spheres, but I also want to go back and fill in a few gaps before we do. In studying Greek astronomy, we are like the drunk searching for his keys under the street lamp. As I’ve tried to emphasize, not too much of early Greek intellectual life survives, and almost all of what does is filtered through summaries of quotations of later authors. So our picture of what ancient Greek astronomy was like is inevitably colored by the interests of these later authors.

Now, one of the things that these later authors were interested in was questions about the fundamental nature of matter and the structure of the universe. Because they hadn’t really made much progress in answering these questions, the viewpoints of the ancient philosophers going all the way back to the Ionian School and Thales were still relevant to them — those philosophers might have something to teach them, so they listened to what they had to say. But when it came to modeling planetary motions, things were a little different. Those later astronomers were also interested in modeling the motions of the planets, but their later models were far more accurate than those of Eudoxus, so they really didn’t have much use for Eudoxus’s model. It was just obsolete. Fortunately for those of us interested in the history of astronomy the basic outline of the model survived, but only just barely. As I discussed in last month’s episode, there are only a few references to it, and it wasn’t until the late 1800s AD that anyone reconstructed it in detail.

Well, there is another important use for astronomy that I have almost completely neglected until now. When I was talking about Babylonian astronomy I went into great detail about how important astronomy was for their calendar. But I haven’t really talked about the Greek calendar at all. You might be forgiven for assuming that Greek astronomers just weren’t interested in timekeeping. And, really, if you read through the surviving texts, you might still come to that conclusion. But, like any good astronomer, we have to beware of selection biases. Later astronomers didn’t really have any interest in the calendars from centuries before, so the work of earlier astronomers around timekeeping makes hardly a trace in the surviving literature. But that doesn’t mean that it didn’t exist. There are enough oblique references, maybe only a single sentence for some astronomers, that we can at least be sure that the work on using astronomy for timekeeping was a part of Greek astronomy even if we can’t know nearly as much about it as we would like.

So, at this point we’re going to turn the clock back considerably. When we left off last episode, we had finished up with Eudoxus, which had brought us to the middle of the 4th century BC — the 350s, or so, when Eudoxus died. We’ll start by going back a bit more than 150 years, to the late 6th century BC. Now, Greek society, like many others across the globe, found the Moon to be extremely convenient for timekeeping. Way back in the second episode I talked about the three natural clocks that we have available to us: the daily rising and setting of the Sun, the phases of the Moon, and the annual change of the seasons. In the course of ordinary life in an agricultural society it’s necessary to keep track of the days — you have to be up when the Sun is up and without cheap lighting, there wasn’t much that could be done after it was down. So then, to a much greater degree than today, the Sun’s daily rising and setting regulated human activity. Then of course it was necessary to keep track of the years. Different crops needed to be sown and harvested at different times of the year. So the annual change of the seasons also regulated human activity.

But what about the Moon? Why keep track of the phases of the Moon? Well, one good reason is that the Moon tracked a regular rhythm for half of all humans. After all, by a remarkable coincidence the 29 and a half day period of the Moon’s phases corresponds fairly closely to the average length of the menstrual cycle. In fact the word menstruation has the same etymological root as the word Moon, as does the word month. It’s perhaps no surprise then, that lunar deities were feminine in most cultures across the world, the Greeks being no exception, though the Babylonians were one of the minority cultures who had a male lunar deity. And as an aside, the close correspondence between the cycles seen in the heavens and the rhythms on Earth was one of the reasons that astrology was so attractive in the ancient world. Clearly the position of the Sun on the sky had an effect on our daily life, as did the Moon. Is it really so preposterous to suppose that the other features of the sky might affect us, too? The planets were fainter, so presumably the effects were weaker, but perhaps with careful study the learned astrologer could understand what those effects were and what that would mean for the affairs of Earth. I haven’t said much about Greek astrology yet, but that’s not to say that it didn’t exist. It certainly did, and I’ll have a lot more to say about the Greeks and astrology in a few episodes once we get to Ptolemy, because although today he’s known for his sophisticated planetary model, in his own time and throughout the Middle Ages, he was better known for his astrology

But to get back to the original topic, why a society would bother with keeping track of the cycles of the Moon, one reason to do this was that it seemed to more or less regulate one of the rhythms of human activity, the menstrual cycle. But beyond that, the monthly changes in the Moon’s phases is just blindingly obvious. It is easy for those of us in the modern world to live as though the night sky doesn’t exist. We can go weeks without really looking at it and just forget that it’s even there. We spend most of our time indoors, we live in cities — even those of us who live in small towns would be living in the equivalent of a metropolis in the ancient world. And our cities are filled with artificial lighting. If we ever do bother to look up, there’s just not a lot to see. But it wasn’t so in the ancient world. The only artificial lighting came from fires and was expensive and limited. Unless you’ve been in a really dark site, it’s hard to describe just how prominent the Moon is in a truly dark sky. A full moon can actually feel blinding. So one reason to keep track of the phases of the Moon is for the simple reason that that you really couldn’t help it.

So, if you lived in an ancient society and you want to keep track of time, turning to the phases of the Moon was an obvious choice. Of course you’ll keep track of days as well, but there are really no other obvious units of time. Sure there’s the year as well, but the year is not a very precise unit of time. You can generally tell if you’re in the middle of summer or winter, but figuring out exactly how many days are in the year is not trivial. It requires very careful observation of the locations of sunrises or sunsets or solar elevations, and even then it’s easy to get off by a day or two. But the phases of the Moon are very easy to observe. So if you want any coarser unit of time than a day you’re stuck with months. Now, another option would be to just come up with some arbitrary grouping of days — say, every twelve days or every ten days, or any other random number like seven. A seven day grouping is actually a bit more natural than it would initially appear, since it’s about the length of a single phase of the Moon and one fourth of a month. But for the next several centuries the concept of a seven day grouping would be confined to Near East. And different cultures chose different groupings. The Etruscans used an eight day week, the Aztecs and Mayans used 13 day weeks, the Javanese used a five day week, and the Egyptians used a 10 day week.

At any rate, all this is to say that really the only principled choice of timekeeping in the ancient world was the month, governed by the lunar cycle. And the Greeks were no different than other societies in this regard. They kept track of the months. Now, as I said perhaps ad nauseum when I was talking about Babylonian astronomy, really the fundamental problem in astronomy that societies had to deal with was the fact that the length of the month does not neatly divide the length of the year. So you keep track of the months, but over time the months that used to be in summer start to drift earlier and earlier in the year and start to come during spring.

The usual solution to this problem is called intercalation, which is a fancy way of saying that every now and again you stick an extra 13th month into the year, and this pushes the months back to where they’re supposed to be with respect to the seasons.

Well, with all that prelude we can start to look at some of the astronomers that we have hitherto neglected during our journey through ancient Greece. The first of these is Cleostratus of Tenedos. Now, going back to the drunk searching under the streetlight, part of the reason that these astronomers are neglected by modern historians of astronomy is that they were neglected by early historians of astronomy, so we really just don’t have that much to say about them. In the case of Cleostratus, we know that he lived during the early 5th century BC, so at the beginning of the Classical period of Ancient Greece, roughly contemporaneous with Anaxagoras. Pliny the Elder stated that he was the first to specify the signs of the zodiac. And just to remind you, the signs of the zodiac are distinct from the constellations of the zodiac. The constellations of the zodiac are nice to look at, but they’re kind of useless for astronomy because they’re of all different sizes. But as a way to observe the progression of the planets along the ecliptic, it’s useful to divide up the ecliptic into 12 equal regions of 30 degrees. These regions roughly match up with the corresponding constellation in the zodiac, but all else being equal each planet spends an equal amount of time progressing through each sign since they’re all 30 degrees wide. This idea was not original to the Greeks, the Babylonians did this as well, as did other cultures like the Chinese which we’ll discuss in future episodes. But whether or not Cleostratus developed this idea independently or got the idea from the Babylonians we don’t know.

Cleostratus apparently also made observations of the solstices from Mount Ida, which is in northwestern Anatolia. Now this is an interesting thing to know because the reason you would do this is to figure out when the solstice was, and this would allow you to figure out how long the year was. So Cleostratus was probably the first Greek astronomer to make the crucial measurement of the length of a year. Cleostratus’s major innovation, though, was a cycle called the octaeteris, which was an intercalation cycle of eight years. This was really the first attempt to solve the fundamental problem of practical astronomy, how to figure out how many months there would be in a year. Cleostratus had found that over eight years there are about 2922 days, and this is very close to 99 lunar months. There are 2923 and a half days in those 99 months, so there is only a one and a half day difference between the two. This worked out very nicely with the Greek system of Olympiads, as well. So you would add one extra month in the first Olympiad, giving you 49 months, and then you would add two extra months in the following Olympiad, giving you 50 months. Then the cycle would repeat.

So by the early Classical Period, we have some evidence that Greek astronomers were attempting to quantify the durations of months and years and determine how to make these two periods commensurate with each other.

Sometime around the end of Cleostratus’s life there came of age another, even shadowier astronomer, by the name of Phaeinos. We have no evidence that Phaeinos interacted with Cleostratus at all, but like Cleostratus he took measurements of the solstices from a mountain, although in his case it was Mt. Lycabettus instead of Mt. Ida. For those of you not up on your Greek geography, Mt. Lycabettus is a big hill in the center of Athens, calling it Mt. Lycabettus is maybe a bit of an overstatement. But at any rate, Phaeinos lived in Athens though he was not actually an Athenian citizen. He was what was called a metic, or a foreign resident of Athens. Now, this does not necessarily mean he wasn’t born in Athens. In ancient Greece citizenship was inherited rather than determined by your place of birth, so a metic could have moved to Athens a few months ago, or could be the descendent of a family of metics who had lived in Athens for generations. Well, in the case of Phaeinos we really can’t say because exactly one line survives about his life. But that line from Theophrastus tells us that he was a metic who measured the solstices on Mt. Lycabettus, and importantly, taught Meton of a particular cycle.

So Phaeinos managed to escape consignment to historical oblivion by the skin of his teeth by being a teacher, at least in some capacity to the astronomer Meton. Now, while we haven’t talked about Meton the astronomer yet, his name has already come up in another form in this podcast in the Metonic Cycle that I talked about in Episodes 2 and 4. Well, Meton was an Athenian and, evidently, a pupil of Phaeinos, and while we don’t really know the dates of his birth and death, we do know that he was active around 430 BC or so, so a bit after Anaxagoras, and a bit before Plato, so maybe around the same time as Socrates. Incidentally, there was another astronomer in Athens at the time who worked with Meton by the name of Euctemon, but sadly for him we don’t really know anything about Euctemon except that he worked with Meton. Given the modern scientific habit of attaching everyone’s name to a phenomenon no matter how tenuous the association, we should perhaps more properly call the Metonic Cycle the Meton-Euctemon-Phaeinic Cycle instead. Or, if we want to reject the Great Man theory of scientific discovery and erase names from phenomena entirely, we can also use the alternate name for the cycle, the enneadecaeteris, which is more faithful to the original Greek. But frankly that is a mouthful and it took me enough takes just to say that correctly the one time, so I will stick with the more popular and traditional term, the Metonic cycle. And, just to lay all my cards on the table, I do rather prefer naming phenomena after people because it does force us to recognize that science is a human endeavor that has taken place in the broader context of history, and that these laws of nature were not handed to us atop of Mt. Sinai. They were the product of the efforts of real men and women who lived in specific times and places. Of course, we should at the same time be conscious of Stigler’s Law of Eponymy, which states that no scientific discovery is named after the person who originally discovered it. This law was, of course, discovered by the sociologist Robert K. Merton.

Well, unlike Cleostratus, Phaeinos, or Euctemon, Meton apparently made a bigger splash in the broader culture because a few authors writing centuries later told some tales about his life which have survived to today. Now, of course, whether or not these tales have any basis in fact is hard to say, but the fact that later authors like Plutarch writing in the first century AD bothered to say anything at all about Meton speaks to his cultural stature. Well, after that disclaimer I feel like I can’t pique your interest and then leave you hanging by not telling you the story, even if it may not have happened. What can I say, I report, you decide.

The story that Plutarch tells occurs around the year 416 BC. Now, I won’t get into all the Greek geopolitics of the time, but the big event leading up to all this was, of course, the First Peloponnesian War, when Athens and Sparta went to war with each other. Or really, more properly it was the Delian League, an alliance of Greek city states led by Athens, fighting against the Peloponnesian League which was another alliance of Greek city states led by Sparta. A few years back two of the main agitators for war had been killed in action and, under the diplomacy of the Athenian general Nicias, Athens and Sparta agreed to a truce. Now, the fact that I called this the First Peloponnesian War is a hint that this was probably something of an uneasy peace. After a few years of this, the people of Athens had become bellicose and were demanding war again. The cause for this was reports that their allies in Sicily had been attacked by Syracuse, the biggest city on Sicily, which was allied with the Spartans.

Well, in Plutarch’s telling, the Athenian leaders were skeptical of the idea of attacking Syracuse because they believed that such a campaign, against such a powerful city state so far away, would be doomed to disaster. Nevertheless, the city leaders were spineless and didn’t speak out against war to avoid enraging the masses who were calling for Syracusean blood and being goaded on by a nobleman named Alcibiades. Just one nobleman spoke out against the war, and this was Nicias, the same general who had ended the Peloponnesian war. But, although the people rejected his arguments against going to war, they nevertheless appreciated his battlefield experience and decided to appoint him the commander of the invasion that he opposed.

Well Nicias was wily and had one more trick up his sleeve to try to convince the people of Athens not to go to war. He said that he would accept command of the operation but only if Athens provided him with a much, much larger force than they had originally requested. His gambit was that this proposal would be so extraordinarily expensive that the people would reconsider the war. But not all gambits work and the people enthusiastically appropriated the funds needed to raise the extra ships and soldiers.

Well, if the way I’m telling this story is giving you a sense of foreboding, it is for good reason, because, and apologies for the spoilers here, this expedition ended in utter disaster for Athens, and Nicias did not make things better with his attempt at 4D chess. The Athenian army was annihilated in Syracuse and the fact that they lost almost all of their army rather than the smaller force that they had originally planned to send, essentially left the city-state defenseless. As it happens, Alcibiades, the man who helped goad the people into war, then defected to Sparta and helped to convince them to restart hostilities with Athens. So, all in all, this incident was not the finest hour for Athens.

Well, Plutarch, too, adds to the sense of foreboding before the campaign begins by saying that the night before the army was to set sail, a series of evil omens occurred throughout the land. Over in Delphi, an icon of Minerva was destroyed by crows. And in Athens itself, the herms placed throughout the city had been defaced. I mentioned these in Episode 9, you might recall that these are rather strange statues, basically a square pillar with a head representing the god Hermes, and then, just sort of stuck on at the anatomically correct level, is a stone carving of male genitalia. These herms were a sort of talisman for the Athenians, so their mutilation was a very bad omen indeed.

At least in Plutarch’s telling, anyone of any intelligence could see that this military campaign was doomed from the start, and this included Socrates, as well as the astronomer that this long-winded tale is all about, Meton. Plutarch actually tells two versions of this story. In one of them, Meton had been appointed to be a commander in the army. But, knowing that it was going to be a fiasco, he feigned insanity and, to make the act look genuine, grabbed a torch and threatened to burn down his own house. In the other version of the story Meton’s son was pressed into service into the military and to get his son out of it, he really did burn his own house down and then beseeched the Assembly for an exception for his son due to the disaster that had befallen him, neglecting to tell them of course that the disaster was self-inflicted.

Meton is also parodied in Aristophanes’s play, The Birds. The plot of this play centers around two men who attempt to convince a group of birds to build a great city in the sky. As they discuss this plan, a series of interlopers come through who offer their services in the construction of the city, and many of these interlopers are caricatures of well known figures in Athenian public life. Well, Meton is among those who show up to offer advice for the city the birds are trying to build and he says that he can survey this city in the sky and divide up the air into plots using his fancy instruments.

Okay, well enough with the stories about Meton, what did he do for astronomy, why bother with him? Now, one of the reasons that he may have attained such a prominent place in Athenian society was his role in administering the calendar for Athens, which is called the Attic calendar, Attica being the region around Athens.

As I mentioned when we first started getting into Greek astronomy, Greek civilization was highly decentralized. Greek city-states were largely independent political entities who occasionally formed alliances with each other for some particular political goal, but these alliances were generally not very long lived. Naturally, different city-states maintained their own calendars, and for a long time, reckoned years with respect to their own kings until the method of referencing Olympiads started to become popular in the fourth century BC. Another source of variance was when the year started. You’ll recall that the Babylonians started their new year at the beginning of spring on the vernal equinox. Several Greek city-states also followed this practice, but Athens began their new year on the summer solstice. Furthermore, different city-states had different names for the months, some of which overlapped with the months of other city-states, and others which didn’t. You may remember that the names of the Babylonian months generally corresponded to the agricultural seasons. The fourth month was the time for sowing, so it had the symbols for seed and hand. The 10th month was the time for reaping, so it had the symbols for corn and cutting, and in the 11th month the harvest was stored so it had the symbols for corn and house. But agriculture didn’t play quite as prominent a role in Greece as it did in Babylon because the land is not as well suited to large scale agricultural activity. Instead, the Greek months generally referenced festivals or the Gods that were celebrated in that month. Evidently, though, the names of months are stickier than the festivals that get celebrated, because over time the month names got a bit out of date. By the time of Meton, many of the month names referred to festivals that had either become minor holidays, or had been forgotten entirely. But, the first month of the year, Hekatombaion, was an exception, and referenced the Hekatomb, which was the sacrifice of 100 oxen which took place at the end of the Panathenaic games. Now the Panathenaic games only took place every four years, so the Hekatomb wasn’t always performed, but there was a celebration of some kind every year to ring in the new year and the name of the month reflected that.

Like the Babylonians, the Greeks began their month on the first day that a crescent moon was visible after the new moon. From an astronomical standpoint, this is a pretty annoying choice, because it’s not extremely well defined when precisely the crescent moon is visible — is it when the Moon is 5% illuminated, 2% illuminated, 10% illuminated? Furthermore it’s not even constant. At some times during the year the Moon rises more or less vertically, and at other times during the year the ecliptic is more inclined, so the Moon rises more horizontally and the fractional illumination that’s necessary to see a visible crescent is different. If we were basing the lunar month purely on astronomical calculations, it would make much more sense to begin it at the new moon or the full moon since that’s way easier to calculate. But that’s not the way it happened. The calendar came first and the astronomy came later, to support the calendar. By the time astronomers started to make rigorous measurements of the Moon and planets, Greek civilization had been starting their months at the first visible crescent for centuries. A few astronomers weren’t going to change that just so that their calculations would be easier.

And this really is as it should be. Calendars are not just astronomical constructs, they are principally social constructs, and social constructs change very slowly. There’s a great deal of inertia particularly when it comes to timekeeping because any changes necessarily throw things out of sync when they take effect. We’ll talk in some detail later about the Gregorian calendar reform in the 16th century, during which a whole 10 days disappeared from the calendar. This causes all sorts of headaches. Are you supposed to pay your full month’s rent for a month that is missing 10 days? You might think the answer is obvious, but then again, your landlord might think the answer is obvious, too, but their obvious answer might be different from yours. This social inertia around timekeeping is in part why we, enlightened moderners we believe we are, have to remember a little rhyme to figure out whether or not March has 30 or 31 days, and why September seems to be telling us it’s the seventh month when in fact it’s the ninth.

So, long before Greek astronomers were measuring equinoxes and solstices, people had noticed that the Moon would periodically disappear for a few days, and then it would reappear again in the night sky as a thin crescent visible briefly after sunset. Given what they knew, this was a perfectly reasonable time to start a new month. Now, you might ask, why not start on the full moon, and this is also a fairly reasonable choice to make. The Purnimanta tradition in the Hindu calendar, for example, marks new months at the full moon. But, the re-appearance of the Moon after its absence for a day or two is a fairly conspicuous transition which is why most societies that kept lunar calendars marked new months around the new moon. And since they didn’t know that this was a much harder thing to calculate, many societies ended up going with the first visible crescent as the beginning of the new month. Even the Chinese, who from a remarkably early age were starting their months on the new moon rather than the first visible crescent, seem to have initially used the observation of the first visible crescent and switched to the calculated date later on, probably sometime in the 7th century BC.

At any rate, in ancient Greece as in Babylonia the new month began on the night when the first crescent was visible after the new moon. Now, one big difference between Greece and Babylonia was that in Babylonia, the astronomers were priests and they were the ones who regulated the calendar. But in Greece, practical astronomy developed much later and the calendar was controlled by political officials. Now, giving politicians control of anything is an invitation for mischief, and they also managed to make a hash out of something as banal as the calendar. We have a few records of Greek writers complaining that their political leaders were tweaking the calendar for political gain. In particular, the Attic calendar was filled with festivals and holidays. There wasn’t really a concept of a week, so there was no concept of the weekend. Since people don’t like to work all day every day, the calendar was stuffed with all sorts of holidays during which people wouldn’t work. Naturally, there was a festival at the beginning of every month. But there was another festival on the seventh day of every month, dedicated to Apollo, and similarly there was a festival dedicated to Athena on the fourth day of the month. In fact, the beginning of every month was fairly front-loaded with a series of recurring festivals so people could take some time off to relax. Then there was usually something else going on in the middle of the month around the time of the full moon. The ends of the months were generally a little sparser, relatively speaking, so that the Greeks could get a few hours of work in, but the specific holidays varied month to month. In all, there was a festival or holiday of some kind or another on about a third of the days of the year. Now, the members of the Athenian Assembly being well-bred and respectable members of the aristocracy, naturally didn’t meet on any holidays. So, a conniving political leader who needed the Assembly to meet earlier than they otherwise would, or prevent them from meeting to buy himself some more time to negotiate a deal, could futz around with the calendar by adding in an extra month when convenient, or declaring the start of the new month a day or two later than someone with keener eyes might have. We have a report that the preparations for a particular festival were running behind and a certain play wasn’t ready yet, so a whole four days got added to a month to give the actors enough time to learn their lines. Aristophanes makes fun of the politicians’ meddling in his play Clouds, when the Moon complains that the Athenians have been screwing around with the months. Over time the temptation to fiddle with the calendar for political purposes grew, and apparently by the 2nd century BC the calendar had become so off kilter that sometimes two dates would have to be published: the official government date, and then the date that it would be according to the Moon if the politicians hadn’t been screwing around with the calendar for centuries.

Well, what with the number of months in a year and the length of those months changing arbitrarily, it was necessary to publicize information about the calendar so that people would know when the festivals would be so they could take some well deserved time off. One of the ways that this was done in Athens was with a system of what are called “parapegm”. The parapegm were a set of pillars erected around Athens with a set of 30 holes drilled into them. Then the authorities could insert various pegs into the different holes to indicate the dates about different festivals, when stars would rise, when the solstice or equinox would be, and so on. Meton is the first astronomer we are aware of who published this information on the parapegms, starting with the summer solstice of 432 BC, and this may be one of the reasons that he was such a prominent figure in Athenian society — just walking around through your local plaza, you would see the parapegm with the results of Meton’s work all over it. He might have played a cultural role somewhat like the weatherman on your local news station.

Well, it’s no surprise that the thing Meton was best known for is the Metonic Cycle. As I mentioned earlier, at least one author claimed that Meton did not discover this himself, but instead learned of it from his teacher Phaeinos, but many other authors attribute the discovery of the Metonic cycle to him directly. Now, the big problem that the Metonic cycle is trying to solve is what to do about the seemingly random insertion of months into the calendar. The lunar month is about 29 and a half days, and the year is about 365 and a quarter days. So if you just have 12 months every year, that works out to 354 days, 11 and a quarter days short of a full year, so the months will start drifting earlier and earlier with respect to the seasons. This means that you periodically have to add an extra month to reset the months so that they always happen at around the same time of the year. But this intercalation, this insertion of an extra month, was for centuries done on an ad hoc basis so it was hard to figure out what the right pattern of extra months should be. The octaeteris developed by Cleostratus was an early attempt at a solution, but Meton — or Phaeinos, or someone else — found something better.

Now, I already described the Metonic cycle back in Episode 4, but, believe it or not, that was more than a year ago now, so you’ll be forgiven for maybe having forgotten a detail or two about it. As a quick refresher, the basic idea is to find a number of years that contains an approximately integer number of synodic months, a synodic month being the time from new moon to new moon. This means that at after a full cycle, you have full moons occurring at the same time of the year as they did in the previous cycle, and quarter moons happening at the same time of the year, new moons happening at the same time of the year, etc. The octaeteris that Cleostratus developed with eight years was pretty good, especially when it came to where the Moon was with respect to the zodiac — there it was off only by 2 hours and 39 minutes over an 8 year period — but when it came to the phase of the Moon relative to the year, it was off by about 2 days. Not bad, but it turns out that another period works much better. For the Metonic cycle, that period is 19 years, and it turns out that this is 6939 days, 14 hours and 27 minutes. And 235 synodic months, is 6939 days, 16 hours and 32 minutes. So the synchronization between the lunar phases and the year drifts by only 2 hours and 5 minutes over a 19 year period. So, it’s not perfect and eventually the cycle would need to be reset since a full moon would drift slightly later in the year than it should according the cycle but it would take 12 of these 19 year periods, or 228 years before it would even be off by a day, so it was certainly good enough for the entirety of anyone’s lifetime. The main advantage of figuring out this cycle, is that you could establish a pattern of these extra leap months that sporadically got added to the calendar, and then as long as you had seven leap months in a 19 year period, you could just repeat that pattern essentially indefinitely. At long last the calendar could be liberated from the tyrannical whims of the Greek politicians. Or at least, this discovery made such a liberation possible.

The Metonic cycle came to be of critical importance in Western civilization, though not so much for its original purpose of determining leap months because the Greco-Roman world eventually transitioned to a purely solar calendar and no longer bothered to synchronize their months with the phases of the moon, just as we continue to do today. But as Christianity began to spread through the Roman world, one vestige of the lunar calendar of the Near East persisted in the date of Easter. The date of Easter is the Sunday after the first full moon after the vernal equinox. Since the entire liturgical calendar of the Catholic Church revolves around the date of Easter, figuring out this date well in advance was a big honking deal and ultimately was the impetus for a great deal of astronomical research in the Middle Ages. The Metonic cycle turned out to be a great tool for this purpose, but a consequence of having an institution like the Catholic Church being so long lived is that even though a drift of 2 hours and 5 minutes every 19 years doesn’t sound like a lot, over 15 centuries, it does start to add up and the Church had to figure out some improvements to the Metonic cycle.

But I am getting ahead of myself. We will have a lot more to say about Easter, and, of course, the holiday that originated it, Passover, in future episodes. Although the Metonic cycle would come to play a huge role in the Western calendar centuries in the future, in Meton’s day this cycle was not actually used in practice. Although the main application of the Metonic cycle is to add leap months to the calendar in a regular predictable way, there is no evidence that this ever happened in practice. Unlike in Babylonia, where the priests stopped adding leap months based on whether or not they felt like it was just too early in the year for it already to be the month of Tishritum already and switched to the more rational Metonic cycle soon after they discovered it, it seems like the Greek politicians simply couldn’t stomach letting go of their power over the calendar. To paraphrase George Orwell, if you want a picture of the Greek chronometrical future, imagine a boot stamping on a clock face forever.

One open question in all this is how exactly Meton or Phaeinos had discovered the cycle. Unlike the Babylonians, it seems as though he did not have centuries of detailed records to draw from. It’s possible that he travelled to Egypt or Babylonia at some point and got the chance to peak at their records. We don’t have any stories about Meton doing that, but we really don’t know much for sure about Meton’s life anyway and the trope of the Greek astronomer traveling to the east and learning the secrets of geometry and astronomy from the orient is common enough that we have to believe that at least some astronomers did, in fact, make this journey and some knowledge did transfer from Babylonia to Greece. The alternative is that he, or, again Phaeinos, whoever it was who discovered the cycle did so by measuring the length of a year and the length of a month somewhat accurately. The accuracy necessary to hit upon the Metonic cycle is not incredibly high, so this is not out of the question. Meton would basically have just needed to figure out that the length of the year is 365 and a quarter days, and that the month was slightly over 29 and a half days.

Given that we know that he made measurements of the solstices and equinoxes, it seems that he had the means to make the discovery of the Metonic cycle independently, even if he may have actually learned of it from elsewhere. In fact, he built an observatory in Athens for the purpose of measuring the length of the year called the heliotropion, the remains of which can still be seen today and is, at least for some definition of observatory, the oldest one in the world which still survives. It was in an important part of town, next to a building called the Pnyx, which was where the legislature met, and just west of the Altar of Zeus. Back in those days, and really right up until the early 20th century, there was no need to put observatories in remote locations because no one had any light to ruin astronomer’s observations, so the sky in a metropolis was about as dark as a sky out in the countryside. So in those days astronomers could enjoy the comforts of city life as they probed the secrets of the universe. The story is different today and astronomers now have to journey to remote mountaintops in Hawaii or Chile and load up with a month’s supply of Doritos, Cheeze-Its, and other non-perishable foods if they want to observe at a world-class observatory. Sure, they get to enjoy the pristine skies of Hawaii, but at what cost?

But I digress. The name of the observatory, the heliotropion, tells us what its primary use was. Helio for Sun, and tropion, for turning. The turning points of the Sun were the solstices, when the Sun stopped rising on more northern points on the horizon and started to rise at more southern points or vice versa. This root is still with us today in the word “tropics” or “tropical.” Although our association with that word is that it’s a warm sunny place near the equator, strictly speaking there are two lines of latitude called the tropics, the tropic of Cancer and the tropic of Capricorn, where the Sun is overhead at noon on the solstices. So so the tropics are where the Sun turns around so to speak and starts moving in the other direction north or south depending on the line, and it so happens that the region between these two latitudes is very hot on account of the Sun being high overhead all year long, hence the image in your head of a sunny beach with a palm tree when I say the word “tropical.” At any rate, the heliotropion of Meton was most importantly used for determining the dates of the solstices. Meton, and I should add, in collaboration with Euctemon, who really does get short shrift, not only measured the dates of the solstices, but the equinoxes as well. The equinoxes could be measured either by determining the day when the Sun rose directly east and set directly west, or by measuring the elevation of the Sun at noon. And here they found something remarkable. The lengths of the seasons were not all the same. They found that spring was longer than winter, which was in turn longer than summer and autumn. Specifically, they found that spring, or the time from the vernal equinox to the summer solstice, was 93 days long, winter, from the winter solstice to the vernal equinox was 92 days long, and both summer and autumn were 90 days long. This measurement was not quite right, really spring is 94.1 days long, not 93, and summer is 92.2, not 90, autumn is 88.6 rather than 90, and winter is 90.4 rather than 92. So evidently their measurements were fairly primitive. But even if their measured values were off by a few days, the important thing is that they correctly discovered that the seasons were not all equally long.

This turned out to be an important problem for Eudoxus’s model of the planetary motions, and really was one of the first pieces of data that the Greeks had that ultimately turned them to the theory of epicycles. But I will talk a little bit more about the consequence of the unequal length of the seasons for Eudoxus’s model later in just a bit.

First we need to turn back to Eudoxus and the school he founded in Cyzicus. As I mentioned last episode, over the course of his travels, after he studied from the priests in Egypt, he ended up in northwestern Anatolia in the city of Cyzicus and there established a school and collected a following. Three of his students in Cyzicus were of some note. Two of them were Menaechmus and Dinostratus, who were brothers. They were better known for their mathematics than their astronomy, and as far as I’m aware they were the first sibling duo of note in mathematics, although they would certainly not be the last. Both of them tackled and solved those famous impossible problems of geometry that I mentioned in last month’s episode. Menaechmus found another solution to the Delian problem of doubling the cube and Dinostratus managed to square the circle, although of course neither of their solutions were straightedge and compass solutions. In Dinostratus’s method of squaring the circle, he developed a curve called the quadratix, which cannot be produced from a straightedge and compass and relies on the intersection of the curve generated by translating an Archimedean spiral through space with a right circular cone. Menaechmus’s work was related to this as he is credited with being the first mathematician to study conic sections, which, we will one day see, became essential for the study of orbits by the 17th century. Incidentally, Menaechmus is also attributed as being the geometry tutor to Alexander the Great. Evidently young Alex became frustrated with the difficulty of the proofs and asked if surely there was not some shortcut. Menaechmus is said to have responded that when it comes to traveling over a countryside there are the royal roads and the roads for the commoners. But in geometry there is but one road for all. As with any of these ancient quotes it’s hard to know how literally to believe the story. The later authors who wrote them down certainly believed the quote was true in the sense that the sentiment behind it is true, but whether or not Menaechmus literally said this to Alexander the Great is more of a modern concern. At any rate the saying that there is “no royal road to geometry” is also attributed to Euclid, though this may have been in part because over the centuries Euclid had a much higher stature thanks to his book The Elements, and later authors might not have wanted to waste such a banger of a quote on the more obscure Menaechmus and attributed it to the much better known Euclid instead. But as always this is all speculation.

Well Menaechmus and Dinostratus were both supporters of Eudoxus’s astronomical theory of homocentric spheres, but they didn’t contribute much to it. Their contribution to astronomy, the discovery of conic sections, would lay dormant for about 2000 years before bursting onto the scene again with the work of Johannes Kepler. So stay tuned for the year 2040 or so when we get to the episode about that.

Well I mentioned that there were three students of Eudoxus, and the third was Polemarchus. We have to be a bit careful here, because there were probably two Polemarchuses of note around this time, or maybe the proper plural is Polemarchi. One of these is best known as the host of the dinner party setting in which the dialog for Plato’s Republic takes place. This Polemarchus came to a tragic end as he was caught up in the reign of terror during the period of the Thirty Tyrants and was among those executed. He was well known enough that the tyrants even went so far as to prohibit his family from having a funeral for him. This prohibition was a big deal. If you think back to Horace’s ode about Archytas in last month’s episode, the whole thing was about a sailor feeling compelled to pour some sand on top of the shipwrecked body of Archytas so as to give him a proper burial so that the dead philosopher would not wander the Earth as a ghost for 100 years. The necessity of giving the dead proper funeral rites ran deep in Greek culture. Indeed, the entire plot of Sophocles’s play Antigone revolves around a conflict between a king who orders that the body of Polynices not be buried because he had led a civil war, and Antigone, the sister of Polynices who defies the king and buries him anyway, and, in good Greek dramaturgical fashion, results in death and tragedy for all involved. But the death of this particular Polemarchus would have come too early and he would not have overlapped with Eudoxus, so he was probably a separate figure.

The Polemarchus we are interested in had a happier ending. Rather than wander the Earth for a hundred years after his death, during his life he wandered from Cyzicus, where he had met and learned from Eudoxus, to Athens and brought Eudoxus’s theory along with him. While he probably contributed something to mathematics or astronomy in his day, it doesn’t survive, and so in our story, he acts as a sort of pollinator, being the bridge that carried the ideas of Eudoxus back to the intellectual center of the Greek world, Athens.

But Polemarchus brought with him more than just Eudoxus’s ideas. A young man by the name of Callippus also tagged along with him. Callippus was born in Cyzicus, but probably was too young to have interacted much with Eudoxus before Eudoxus died. Nevertheless, the spirit of Eudoxus lived on in Cyzicus and Callippus learned astronomy from Eudoxus’s pupils. He ultimately came to write a book of his own, but like so many others it was lost sometime in antiquity.

Although we don’t know much about him personally, Callippus was an accomplished observer. One of his discoveries is an improvement on the Metonic Cycle, which is known, appropriately enough as the Callippic Cycle. It’s simple enough, just a period of four Metonic cycles, or 76 years. The Metonic cycle was pretty good, it only drifted by a few hours every 19 years. But the essential problem with the Metonic cycle is that the year has 365 and a quarter days, but the Metonic cycle is 19 years long, which doesn’t divide by four. It’s about 6939 days and 3/4. So even though there’s an integer number of synodic months, there’s not an integer number of days. But if you stick four Metonic cycles together, all of a sudden this quarter day isn’t a problem anymore and days, years, and synodic months all line up very neatly. It was a clever tweak on top of the Metonic cycle, but for whatever reason it never achieved the same widespread usage as the Metonic cycle did. But timekeeping, like love and war, is not always fair.

Now, at some point in his career, Callippus replicated the measurement of Meton and Euctemon of the length of the seasons. We don’t know if he did this in Cyzicus using Eudoxus’s observatory, or if he did it when he went to Athens and performed the measurement at Meton’s observatory, but the odds are that it was the latter since he probably made the move to Athens when he was fairly young. At any rate, he verified Meton’s core result, that the seasons are not of equal length, but he got slightly different numbers. He measured the length of spring to be 94 days, summer to be 92 days, fall to be 89 days, and winter to be 90 days. Evidently Callippus was a better observer than Meton and Euctemon because his measurements are all correct to within a day.

Now I had hinted earlier that this was a problem for Eudoxus’s model of the planets. You may remember that Eudoxus imagined that the Sun’s motion was due to the superposition of three spheres. One sphere described its daily east to west motion as it rises and sets. Another sphere described its annual motion through the zodiac. And then a final third sphere ended up adding in a variation in the Sun’s ecliptic latitude, though this motion doesn’t really exist, so why he added it is not really known. But what is not possible in this model, is for the seasons to be of unequal length. Because the essential motion along the zodiac is uniform, Eudoxus’s model cannot explain the unequal length of the seasons.

Callippus proposed a modification to Eudoxus’s model to address this issue, as well as a few other problems that Eudoxus’s model had: namely that it completely failed at describing the motion of Venus and Mars. Callippus’s idea was maybe not especially creative. He just added more spheres to Eudoxus’s model. But what it lacked in creativity it made up for in effectiveness. Eudoxus’s original model had 27 spheres, three apiece for the Moon and the Sun, and then 4 apiece for the five planets, along with one sphere for the fixed stars. Callippus gave the Sun and Moon each two extra spheres, so that they had five in all, and added one extra sphere for Mercury, Venus, and Mars. Jupiter and Saturn got no extra spheres. Evidently he thought that Eudoxus’s theory described the motions of Jupiter and Saturn well enough, which in fact it did, so there was no need to fix something that wasn’t broken.

Now in Eudoxus’s original model, you may remember that each of the planets had four spheres, one of these drove its daily rising and setting, and another drove its drift through the zodiac, which was a year for Mercury and Venus, and quite long for Saturn, close to 30 years. But then the other two spheres produced a sort of figure eight motion superimposed on top of that. So sometimes the planet would be a little ahead of where it should be on the ecliptic, and other times a little bit behind, and sometimes it would be above the ecliptic and sometimes below it. And with the right parameters, this could produce the retrograde motion that was observed, at least for some of the planets. But Callippus must have been quite a good geometer, because adding in a fifth sphere makes the motion a lot more complicated. It’s hard, as ever, to describe this motion on a podcast but it’s sort of like you took this figure eight and attached two little bow ties on either end of it. Ultimately this has the effect of allowing the planet to sort of hang out at each end of the figure eight much longer than it otherwise would, and if you flip that around, it essentially means that if the overall period is the same, the motion along the figure eight has to be faster. This allows you to get more retrograde motion bang for your buck.

As with Eudoxus, because so little of the theory survives, we have to do a bit of guesswork. We don’t know exactly what parameters he used for his model, but if we use the best possible values, the model actually works quite well. It fixes the essential problems that Eudoxus’s model had with Mars and Venus and is able to produce retrograde motion that more or less agrees with what he could have observed, and the two extra spheres for the Sun can also more or less reproduce seasons of unequal length with the values that Callippus measured.

So there you have it. Sometime in the fourth century BC the Greeks figured out a perfect model of planetary motion. Astronomy had been solved and there was nothing left to do. It was time for astronomers everywhere to pack it up and get honest jobs.

Unfortunately it wouldn’t be quite so easy. There is one last astronomer I will mention, by the name of Autolycus. Autolycus has the good fortune of being the very first Greek mathematician who wrote a work which survives in its entirety to today, a text called On the Moving Sphere, which is naturally of great relevance to astronomy. Autolycus overlapped with Callippus in Athens and was a supporter of the theory of homocentric spheres. But he did notice one detail which the theory had trouble explaining. Sometimes the planets were brighter and sometimes the planets were dimmer. Mars and Jupiter in particular could vary in brightness by an astonishing amount. Why was that? In both Eudoxus’s theory and Callippus’s modification to it, the planets all move on spheres which are all centered on the Earth — it’s right there in the name, the theory of homocentric spheres. This means that they are always the same distance from the Earth. But then why did their brightnesses vary? Explaining this peculiar observation set the Greeks down the path towards an alternate model, the theory of epicycles, which dominated Western astronomy for the next 2000 years.

But we will not get to epicycles quite yet. I think I have set a record for myself with the number of astronomers I’ve mentioned in one episode, so in next month’s episode I’m going to have to swing the pendulum back hard in the other direction and focus on just one. We’ll return to another one of of Plato’s students, one who was a lesser astronomer than Eudoxus, but of greater influence. He was a man with a vast and, at least in the physical sciences, complicated intellectual legacy. Next month I’ll be talking about none other than the party pooper everyone loves to hate, Aristotle. I hope you’ll join me then. Until the next full moon, good night, and clear skies.

Additional references

• Goldstein & Bowen, 1983, A New View of Early Greek Astronomy
• Mikalson, Jon, 2015, The Sacred and Civil Calendar of the Athenian Year