After briefly examining the astronomy of Timocharis and Aristyllus, who developed the first known stellar catalog, we turn our attention to Hipparchus, who I claim was the greatest astronomer of ancient times. Hipparchus made major developments in virtually every area of astronomy known to him. His measurement of the lengths of the year and month were of unprecedented accuracy, he measured the distance to the Moon, and he developed a star catalog that was an order of magnitude larger than the earlier catalog of Timocharis and Aristyllus, which he was able to use to discover the precession of the equinoxes.

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 at the end of the last episode I said that there was just one more major astronomer of the Hellenistic Era, Hipparchus, and that I would devote an entire episode to him. But it seems that once again I have bamboozled the poor listeners of this podcast. Because as I was researching this episode I felt that I just could not avoid mentioning two other earlier astronomers of the era, Timocharis and Aristyllus.

Never fear, the main focus of this episode will still be Hipparchus, and I still stand by the claim that he was the greatest astronomer of the ancient world. But he will have to share the spotlight, at least briefly, this episode.

So who are these two interlopers to this episode, Timocharis and Aristyllus? Well as with all the other obscure astronomers of the ancient world, there is not very much that we do know about the two. The years of their birth and death are unknown, all we know is that they were active and doing their astronomy early in the 3rd century BC in Alexandria, so this puts them as being roughly contemporaneous with Aristarchus. It is believed that Aristyllus was the student of Timocharis, so he was probably about one generation younger, but apart from that, that is all we can say about their biographical details. In fact, since the only records of Aristyllus and Timocharis survive to us through Ptolemy’s great treatise, the Almagest, the only reason we can place the approximate years of their work is by from the astronomical events that they observed.

Ptolemy writes in the Almagest that in the thirteenth year of Philadelphos and on the 17th day of the month of Mesore in the Egyptian calendar at around midnight Timocharis observed Venus to pass in front of a star in the constellation Virgo. In our modern calendar, this works out to be the evening of the 11th of October, 272 BC. Ptolemy describes the star that Venus occults as “the star opposite Vindemiatrix.” This is a somewhat vague designation, but Vindemiamtrix is a star in the constellation Virgo, more formally called epsilon Virginis since it is the fifth brightest star in that constellation. Based on modern calculations we can see that the star in question is today called eta Virginis, the seventh brightest star in Virgo.

Now, I’ve been saying that Timocharis observed an occultation of this star by Venus, and that’s how the original Greek word “kateilephos” is usually translated — Venus passing in front of the star. But planets and stars are small enough on the sky that it’s fairly rare for a planet to actually pass directly in front of a bright star and block out its light. On average this happens around once or twice a decade and even still won’t be visible from everywhere on the Earth. At the time that I am recording this podcast in 2022, the next occultation of a relatively bright star will be in 13 years, in February of 2035 when Venus will pass in front of Pi Sagittarii, the 16th brightest star in the constellation Sagittarius. In fact, we can see from modern calculations that in 272 BC, Venus only passed very close to the star eta Virginis, getting within about 12 arcminutes south of it, but did not pass right in front of it.

At any rate, Timocharis’s observation of this event at least places him to being active in the year 272 BC. Ptolemy also records that Timocharis observed four occultations of background stars by the Moon in the years between 295 and 283 BC, two of which were of Spica, one was of Acrab, and the last of which was of part of the Pleiades.

These are the first observations of occultations that we have records of in Greek astronomy and it raises a natural question — why did anyone care? Why did Ptolemy preserve these observations in the Almagest and why did Timocharis make these observations in the first place? The former question has a straightforward answer which we will get to in much more detail later on in the episode — Ptolemy includes Timocharis’s observations in the Almagest because they were critical data points to demonstrate the existence of the precession of the equinoxes. But Timocharis didn’t know about precession, so why did he bother to record these observations? Well, here we end up more in the realm of speculation since we’re attempting to delve into the mental state of an astronomer whose only traces are a few lines in a work written three centuries after his death. But that is not enough to deter the intrepid historian of science, and two historians of science in particular Bernard Goldstein and Alan Bowen couldn’t help but be struck by the similarity of these observations with the kind of observations that are recorded in the astronomical diaries of the Babylonians that I talked about way back in Episode 4. Given that the Greeks had previously demonstrated no interest in recording things like occultations or planetary positions, at least as far as we can tell from the surviving records, the sudden interest in making these observations during the Hellenistic Era has been taken to be evidence that around this time the Greeks must have come into contact with the astronomical records of the Babylonians and started to reproduce the kinds of observations that the Babylonians had been making for centuries.

At least in the view of Goldstein and Bowen, Timocharis probably wasn’t merely aping the observations of the Babylonians, but may have been inspired by this new type of observation to make a more precise measurement of something that had long interested Greek astronomers, the length of the sidereal month, that is, the time it takes the Moon to rotate a full 360 degrees around the Earth and return to the same position with respect to the background stars. To this end, he was probably trying to track the position of the Moon as precisely as possible relative to the background stars.

Well, all these occultations aside, what history has principally remembered Timocharis and Aristyllus for, even if only marginally, is having created the first known star catalog. Now, as I’m sure you’ve come to expect from these obscure Greek astronomers, the catalog itself does not survive directly, but Ptolemy included their measurements of the positions of 18 stars in the Almagest. As we’ll get to in a little bit, the reason Ptolemy did this was that he was comparing the positions of stars recorded by Timocharis and Aristyllus with the positions of those same stars as they were measured by Hipparchus to show how they had shifted over the centuries due to the precession of the equinoxes. But because he only needed the positions of a handful of stars to demonstrate this phenomenon, he had no reason to include their complete catalog in his work, so the entire list of stars in their catalog has been lost to time.

Well with that we can now turn to the astronomer who really is the only reason why Timocharis and Aristyllus have been remembered at all, Hipparchus, who was, I will once again claim, perhaps the greatest astronomer in all of antiquity. He was born in the city of Nicea which was in northwestern Anatolia. His birthplace is probably most famous today for hosting a series of councils of the bishops of the early Christian Church several centuries later, or if you happen to be an Arian, infamous for hosting those councils since the main item on the agenda was to denounce Arianism as a heresy.

We don’t really know very much at all about Hipparchus’s life since most of our sources about him come through Ptolemy who was a fairly no-nonsense kind of guy and just presented Hipparchus’s measurements without any of the frippery about his biographical details. A few other details about his work come to us through Strabo’s Geography and Pliny’s Natural History, but again not much in terms of his life.

In fact, as with Timocharis and Aristyllus, our estimates of the years of his birth and death are just based on the dates of the observations he made. So his death is assumed to be not too long after his last recorded observation in 127 BC, and his birth is assumed to be some 20 or 30 years before his first observation. Now unfortunately, the earliest observation we can definitively attribute to him is one recorded in Ptolemy’s Almagest in 147 BC, but it’s believed that there are even earlier observations due to him going back to 162 BC, so his birth is usually given as being around 190 BC and his death around 120 BC, but we cannot really say anything definitive except that he was certainly active between 147 and 127 BC.

Likewise with where he lived, all we can say for sure is that he was born in Nicea and later in life moved to Rhodes because at least some of his observations recorded in Ptolemy’s Almagest are stated to have been taken at Rhodes. Since by the end of the Hellenistic Era Rhodes was an extremely prosperous city, it’s usually assumed that Hipparchus moved there in his youth and spent the rest of his life there, though really this is still just an educated guess based on the scant data we do have.

Nevertheless, it is worth saying a few words about Hipparchus’s purported home of Rhodes because it was one of the more dynamic cities of the Mediterranean during the Hellenistic Era and was a focal point for the geopolitics of the region. You may recall from Episode 19 that after Alexander the Great died, his sprawling empire immediately splintered with his various generals seizing what regions they could for themselves. In the eastern Mediterranean there ended up being three main players in this war of succession. One was the general Ptolemy Soter, who later was also called Ptolemy I, who seized Egypt, which was at the time one of the richest regions in the known world, both in trade and agriculture, and had the cultural prestige of being heir to the most ancient civilization known. After all, at that time, some of the Pyramids were older to the people of the Hellenistic Era as the people of the Hellenistic Era are to us today.

The second major player in the region was Antigonus Monophthalmus, whose name literally means the “one-eyed.” The reason for the moniker is pretty self explanatory, and according to Plutarch he lost an eye in his late 30s during the Siege of Perinthos, though sadly for him it was all for nought as the siege failed. After Alexander’s death, Antigonus Monophthalmus managed to seize Alexander’s original territory, the areas around Greece with Cyprus to boot.

And the last mover and shaker in the region was Seleucus, who took the eastern portions of Alexander’s empire: Syria, Mesopotamia, and parts of Persia. Seleucus’s territory ultimately became the Seleucid Empire and lasted about two and a half centuries.

Well, Hipparchus’s adopted home of Rhodes is an island in the Mediterranean just off the coast of southwestern Anatolia, so if you look at in on a map it’s basically right where these three powers meet. And, rather remarkably, like a radio broadcast antenna can stay upright when three support wires are pulling it in three different directions, Rhodes managed to maintain its independence as a republic by sitting right in the middle of these three great powers. Now, of course, this was not some agreed-upon zone of neutrality, all the powers were constantly vying to take Rhodes into their own empires, but by deftly playing one empire against the other, the Rhodesians managed to maintain their independence, and even beyond that managed to retain some elements of democratic governance.

Now, I don’t want to overstate the neutrality of Rhodes too much. Although Rhodes was an independent republic, through the Hellenistic Era they were generally most closely aligned with Ptolemaic Egypt. Thanks to their unique position, Rhodes in effect controlled access to the Aegean Sea and as a consequence became a hub for trade and commerce in the eastern Mediterranean over the 3rd century BC. Thanks to this flourishing economy, Rhodes was able to support a vibrant intellectual culture. Not quite as vibrant as Alexandria or Athens a century or two earlier, but nevertheless nothing to sneeze at. One of the great sculptors of the ancient world, Agesander, worked at Rhodes around this time, though as with Homer there is some debate among the scholars as to whether or not he was a single individual or several. But certainly the most visible symbol of the city’s worldly success and cultural relevance was its enormous statue of the sun god Helios, commonly known as the Colossus of Rhodes. It was about as large as the Statue of Liberty and was so impressive that the Greek poet Antipater of Sidon listed it among his seven wonders of the ancient world. The statue came about after Antigonus Monophthalmus staged a siege of Rhodes which failed, just as his siege of Perinthos failed where he had lost his eye all those years before. To celebrate their victory, the Rhodesians took all the siege equipment that had been left behind by the Macedonians and sold it and used the money to fund a gigantic statue to their patron god.

The statue lasted about 50 years, but in 226 BC, so about 30 years before Hipparchus’s birth, Rhodes was hit by an earthquake and according to the author Strabo, the statue broke at the knees and fell to its side. As a gesture of goodwill, Ptolemy III in Egypt offered to pay for its reconstruction, but Strabo writes that, “in accordance with a certain oracle, the people did not raise it again.” So by the time of Hipparchus’s birth and throughout his life, the great Colossus of Rhodes would have been broken and lying on its side, where it would remain until the 7th century AD, at which point the ruins were sold off for scrap metal by Islamic invaders.

But enough with the sights of the city where Hipparchus worked, let’s turn to something about Hipparchus that we actually do have some real knowledge of — his science. Now I think we are quite fortunate to know as much about Hipparchus’s work as we do because although he wrote a great deal, at least 14 works, only one of them survives directly. You may recall from back in Episode 16 that Eudoxus had written two works, the Mirror and the Phaenomena and later on a poet by the name of Aratus had written a poetic rendition of them. Although Eudoxus’s original texts did not survive, Aratus’s poem became extremely popular throughout the ancient Mediterranean and was even briefly quoted in the Acts of the Apostles. Hipparchus’s only surviving work was a commentary on Aratus’s poem, basically explaining everything that it got wrong.

But Hipparchus’s real astronomical legacy comes to us indirectly, and mostly through Ptolemy. Ptolemy’s Almagest mentions three of Hipparchus’s works in particular: On the Length of the Year, On the Intercalation of Months and Days, and On the Change of the Solstices and Equinoxes.

But, although Hipparchus is best known for his astronomy, and deservedly so, he also had some thoughts on physics which have survived to us through a passage in Simplicius. So before we dive straight into his astronomy, it’s worth taking a look at has ideas about how objects move. Now, before I do this I have to once again clear my throat and mention the usual caveats about how we are getting his ideas indirectly, through a later author, and in a fairly brief passage, so there is some debate about what exactly the nature of his views was. But the general opinion is that Hipparchus broke from the Aristotelian orthodoxy on the physics of motion. You may remember from back in Episode 18 that Aristotle’s theory of motion was essentially that it is the natural state of all matter to remain at rest, at least all matter in the sublunary realm. Consequently, according to Aristotle, any motion had to be imparted by a collision with some other object, and, what’s more, this external force had to be continuous to maintain the motion — once it stopped, the object would stop as well. So, if I throw a rock, according to Aristotle, my hand initially imparts motion to the rock, but once it leaves my hand, the reason that it continues to move through the air is that the surrounding air forms a sort of vortex which continually pushes the rock forward.

Well, Hipparchus rejected this theory of motion and was, if we are interpreting this passage correctly, the first person to propose a theory of internal impetus. According to Hipparchus, if I throw a rock straight up in the air, the rock becomes imbued with the property of motion as my hand pushes it up. And once it leaves my hand, it continues to retain this property of motion, though it slowly dissipates. Unlike in Aristotle’s theory, this dissipation is not instantaneous — it only happens gradually. Eventually, the object loses its property of motion and returns to the ground. Hipparchus’s theory of motion did not have an immediate impact, no pun intended, and the theory of impetus was not really considered again until the Middle Ages when the Byzantine scholar John Philiponus proposed it again in the 6th century AD and it was later taken up by Islamic scholars in the 10th century.

A half a millennium after that, Galileo Galilei also came to reject Aristotle’s theory of motion and subscribe to an impetus theory like Hipparchus’s. Galileo argued that motion was analogous to heat. If you stick a poker into a fire, the poker does not immediately become hot, but gradually becomes hotter and hotter as it remains in the fire. And what’s more the poker does not become infinitely hot if you leave it in the fire indefinitely, but arrives at the temperature of the fire. Then if you take it out, it does not instantaneously cool down, but cools down only gradually. So it was with motion in Hipparchus’s theory. As you throw the rock up, you imbue the rock with the quality of motion, but if you could somehow continue to push the rock upwards, you won’t push it to move infinitely fast just as a poker left in a fire won’t become infinitely hot. And once you release the rock, it retains this quality of upward motion and continues to move upward of it own accord, but slowly this quality dissipates and the rock returns to rest.

Now Hipparchus also analyzed another situation, where you’re holding a rock in the air and then you release it. In Aristotelian physics this situation is just explained teleologically. The natural place of the element Earth is the center of the universe so the rock moves in that direction until it is prevented from doing so by either the ground or your hand. In Hipparchus’s explanation, when you are holding the rock aloft, you are continuously imbuing it with motion contrary to its inherent downward motion. Once you release it, the rock maintains some of that internal upward motion, but again it starts to dissipate and so the rock accelerates downward.

Now, in modern physics we use terms like velocity, acceleration, force, and momentum, with very precise, quantitative meanings, but conceptually, Hipparchus’s theory of impetus is a qualitative precursor to the modern idea of momentum and in any case was a far more accurate model than Aristotle’s picture.

Well okay, I have been touting Hipparchus as being the greatest astronomer of antiquity, so let’s get to his astronomy. I’ll start with a result of his that might seem a bit banal at first blush, his measurement of the length of the year. But even something as apparently straightforward as this starts to showcase his talents as an astronomer and reveal some of the subtleties that start to crop up when making measurements at the precision he was capable of.

Now, up to this point, the most accurate measurement of the length of the year in Greek astronomy had been done by Callippus, and I talked about it a bit back in Episode 17. Callippus had noticed that the old Metonic cycle of about 19 years had an integer number of days, 6940 to be exact. But by the time of Callippus it was known that the year was not an integer number of days. So Callippus combined four Metonic cycles and subtracted one day from that, which produced a year that was 365 days and a quarter. This was really quite good, and in point of fact, after the calendar reforms of Julius Caesar in 46 BC, the Western world adopted 365 and a quarter days as being the length of a year by definition and this more or less worked out okay for about 16 centuries. So that is not too bad for Callippus.

But in the second century BC, decades before Julius Caesar was even a gleam in the eye of Gaius Caesar, Hipparchus had already discovered that taking the year to be 365 and a quarter days was not quite right. In the year 135 BC, Hipparchus had made a measurement of the date and time of the summer solstice. He compared the date of the solstice he measured to the date of a summer solstice measured 145 years earlier by Aristarchus in 280 BC. If the year had been exactly 365 and a quarter days in length, the summer solstice would have come half a day later. This meant that the year was about 1/290th of a day shorter than 365.25 days, which Hipparchus rounded to 1/300th of a day. This corresponded to a length of the year of 365.2467 days, or 365 days, 5 hours 55 minutes and 12 seconds.

Now this measurement is so accurate that we now actually have to be pretty careful when we are trying to assess how accurate it is. Firstly, it is accurate enough that we have to ask just what kind of a year Hipparchus was trying to measure. For the most part, we can just say a year without thinking too hard about what that means, but there are in fact several different definitions of a year, all of which have slightly different lengths. The standard definition, normally called the sidereal year, is the time it takes the Earth to revolve 360 degrees around the Sun, or, equivalently, the time it takes the Sun to move to the same position with respect to the background stars. But an alternative definition of the year is the time to go from one summer solstice to the next, or one vernal equinox to the next. This is what is called the tropical year, since “tropics” comes from the Greek word for “turning,” and the solstices were called “turning points” in ancient Greek since that was when the Sun stopped moving northward or southward, turned around, and started going in the other direction.

These two kinds of years are very similar, but they’re not exactly the same, the sidereal year is longer than the tropical year by about 20 and a half minutes. The reason for this difference is due to the precession of the equinoxes which I’ll say more about in a bit. At any rate, since Hipparchus was measuring the time between summer solstices, he was measuring the length of the tropical year and his result was only six minutes off from the correct value. If we had mistakenly compared his measurement of the year to length of the sidereal year, we would have accused him of being more than twice as far off as he actually was.

Another subtlety that we have to consider here when assessing the accuracy of his measurement of the length of the year, as with his measurement of other quantities like the length of the month, is that the length of a day has not remained constant. So, if we compare the number of days he claimed were in a tropical year to the modern measurement of the number of days in a tropical year, his result will look worse than it really was because the length of a day has changed between his time and ours. Over the eons, the Earth’s rotation has been gradually slowing down and so the length of the day has gradually increased. Now, the primary reason for this deceleration in the Earth’s rotation is tidal friction from the Moon. The differential gravity from the Moon generates tides in Earth’s oceans. The ocean nearest to the Moon feels a slightly stronger gravitational pull from the Moon than the center of the Earth does, which in turn feels a slightly stronger gravitational pull from the point on the Earth most distant from the Moon. Likewise the points on the Earth perpendicular from the axis to the Moon feel a differential force that pushes them towards the axis between the Earth and the Moon. The result of this is a stretching in the direction along the axis to the Moon and a squishing in the directions perpendicular to this which is a phenomenon called spaghettification. In the extreme case of an unfortunate astronaut falling into a black hole, the tidal forces become so extreme that the astronaut would be stretched thin head-to-toe and squeezed tight around the waist like a strand of spaghetti.

Now, to take a slight digression here it’s a bit of a surprise that the tides in the Earth’s oceans are so noticeable that you can easily see the difference between high and low tide with your eyes just by spending a day at the beach. If you were to calculate the amount by which the tidal force would raise the part of the ocean closest to the Moon, the increase in the height of the ocean would be completely negligible, of order a few hundred microns. In essence, the tidal force would be trying to stretch the water up and make it less dense. But the tidal force just isn’t strong enough to do that. The reason we see the sea level change by about half a meter or so rather than half a millimeter or so is due to points on the Earth 45 degrees away from the axis with the Moon. There, the tidal force is parallel to the Earth’s surface and the tidal force can then generate a current along the Earth’s surface and collect water at the point below the Moon. So the tides are caused not so much by the Moon stretching the Earth’s ocean out, but by pooling water together from the component of the tidal force which is parallel to the Earth’s surface.

At any rate, tidal effects from the Moon generate large scale currents in the Earth’s oceans, and these currents generate friction against the Earth’s rotation and slow it down. The effect is small, but it accumulates over the millennia. As it happens, at one point the Moon freely spun around so that both sides were visible from the Earth, but the tidal force of the Earth on the Moon is about 20 times larger than the tidal force of the Moon on the Earth, so the Moon became tidally locked to the Earth much more quickly than the Earth will become tidally locked to the Moon. Now, you may be thinking that if the Earth’s rotation is slowing down, where does all that angular momentum go? After all, angular momentum is conserved, so if the Earth rotates more slowly, that angular momentum has to go somewhere. And it turns out that it goes into increasing the size of the Moon’s orbit. It’s not an enormous effect, only 3 centimeters per year, but it accumulates over the years. So, assuming that the Artemis program goes according to plan and NASA lands a group of astronauts on the Moon in 2025, they will have to travel about one and a half meters farther than Neil Armstrong and Buzz Aldrin did in 1969.

Well, to get back to the main point of all of this, the length of the day is increasing, and the biggest contributor to this increase long term is the tidal friction from the Moon. But, sadly for astronomers, programmers, and anyone else interested in precision timekeeping, the rate at which the Earth’s rotation slows down is not constant. In the short term other factors can cause changes in the Earth’s rotation that are much larger than the gradual slowing due to tidal locking. Climate change in particular can have a big effect. As the Earth goes through cycles of warming and cooling, the formation of ice sheets in northern latitudes and their subsequent melting changes the Earth’s moment of inertia of the Earth, which in turn changes the Earth’s rotation speed. And on even shorter timescales, big earthquakes shift the Earth’s tectonic plates which also changes its moment of inertia and in turn changes the length of a day by a detectable amount.

Now over short time periods, these changes in the Earth’s rotation period are pretty small. On average, the length of a day increases by around 2 ms per century. But because the days just keep passing us by, this effect is cumulative and it doesn’t take all that many years before these differences accumulate into a difference of a second between where the Earth is supposed to be if the day were exactly 86,400 seconds long and where it actually is. So every couple of years a leap second has to get added to the calendar to bring the clocks back into sync with the Earth’s rotation. Depending on how your computer’s clock decides to display this leap second this can occasionally lead to a disturbing New Year’s Eve phenomenon, where you watch the seconds tick up to midnight and then after 11:59:59 rather than seeing the clock hit midnight and the new year begin you see it rather ominously display 11:59:60, leading you to briefly wonder if you have somehow become trapped in a time warp that has left you stuck in the same year for all eternity..

Anyway, the fact that the changes to the Earth’s rotation are fairly random on short timescales makes life a little difficult for the historian of astronomy. Suppose you want to know where a particular solar eclipse was visible in the ancient world. Figuring out when the Moon passes in front of the Sun is a very easy thing to do. But figuring out where on Earth the eclipse was visible from depends on you knowing the exact rotation angle of the Earth thousands of years in the past. But these variations in the length of the day add up over the thousands of years to the point where the angle of the Earth can be uncertain by up to 75 degrees or so. So in practice it usually goes the other way round. Astronomers or geologists will take ancient reports of solar eclipses and then use those to constrain the rate at which the Earth’s rotation has changed.

But to get back to the main subject of the show, to appreciate just how accurate Hipparchus’s measurement of the length of the year was, we also have to factor in the changes in the Earth’s rate of rotation, since these changes over the centuries added up to a difference in the length of the year of around 20 seconds, at least when the year was expressed as a number of days. So once we account for this correction we find that Hipparchus was only six minutes off from the correct value.

In my description of Hipparchus’s measurement, I did rather gloss over the way that he got such an accurate measurement. I just said that he measured the time of the summer solstice and compared it to a measurement by Aristarchus 145 years earlier and found that it was half a day off. But how did he figure out when the summer solstice was? Now, unfortunately we don’t have a direct description of the technique he used, so we have to speculate a bit as to how he could have gotten the measurement he did with the tools he had available to him.

Now, figuring out what day a solstice occurs on is in principle fairly straightforward. Not easy, necessarily, but straightforward. There are two ways you can do it. The first is to observe where on the horizon the Sun rises or sets. In the northern hemisphere, the Sun will rise at the northernmost spot on the horizon on the summer solstice and on the southernmost spot on the horizon on the winter solstice. So if you can track where on the horizon the Sun is rising using an instrument like a diopter, you can find out what day it stops rising at points further to the north and starts rising at points further to the south, and that day is when the solstice is. Another technique, the one that was more likely to actually have been used, is to measure the altitude of the Sun at noon. The Sun will have the highest noontime altitude on the summer solstice and the lowest noontime altitude on the winter solstice. How do you measure the Sun’s altitude? It’s pretty easy, you use an instrument called a gnomon, which is a fancy word for a stick in the ground. The stick casts a shadow and you can trace this shadow over the course of a day. The shadow will be shortest at noon, and the length of the shadow at this time will tell you the Sun’s altitude. The longer the shadow is, the lower the altitude. The day where the gnomon’s noontime shadow is shortest is the summer solstice and the day where its noontime shadow is longest is the winter solstice.

But Hipparchus clearly had to be doing something a little more sophisticated than this since his whole result was predicated on the solstice being just half a day off from where it was supposed to be. This meant that he had to be able to measure the time of the solstice to better than half a day. Just finding the day of the year where the noontime shadow is shortest won’t be good enough because you won’t be able to know exactly when during the day the Sun reaches its highest declination. What Hipparchus probably did was to measure the length of the shadow of the gnomon at noon on several days around the solstice. By finding the days on either side of the solstice where the length of the shadow at noon was equal, he could average the two and determine when the solstice occurred to a higher precision than just a day. Now, in practice this is a tricky thing to do, because around the solstice the maximum altitude of the Sun does not change very much, so you really have to get pretty far away from the solstice before you can be confident that you’ve found two days where the lengths of the shadow at noon are equal. It would probably take measurements about a month and a half away from the solstice in either direction, and even then the precision you could get is only to within about a quarter of a day. Nevertheless, this was good enough for Hipparchus to realize that the standard length of the year of 365 and a quarter days was off and to make his correction of 1/300th of a day.

Now, Hipparchus measured the equinoxes as well, and for these measurements we have more detail as to how he did it thanks to Ptolemy. The measurement of when the equinox was was more precise than the measurement of when the solstice was. Hipparchus constructed a device called an equatorial ring, which is really a pretty simple thing. It’s just a big hoop oriented at a precise angle — 90 degrees minus the latitude. The idea was that the ring would cast a shadow from the Sun. However, when the altitude of the Sun was exactly the same as the angle of the ring, which only happened at the equinox, the shadow from the uppermost part of the ring would be cast precisely on the lower half of the ring, and the lower half would be completely shaded. The neat thing about this device is that it didn’t matter what time of day it was when the equinox occurred. As long as the Sun was up, the entire bottom half of the ring would be shaded only at the exact moment of the equinox, so the time of the equinox could be determined fairly precisely.

Unfortunately, the ancient astronomers did not trust the reliability of the device as much as they perhaps should have. Ptolemy writes that the time of the equinox usually could not be determined to a greater precision than about half a day because what would sometimes happen is that early in the morning the shadow would creep forward until it covered the bottom half of the ring, which seemed to imply that it was the equinox, proceed further, then recede back to the original side of the ring, and then later in the day cover the bottom half of the ring once again, which made it look like the equinox happened twice. Ptolemy attributed these fluctuations to an instability in the device — that maybe small changes in the orientation of the ring were causing the shadow to fluctuate onto either side of the bottom half of the ring. But we now know today that what Hipparchus was seeing was refraction. Early in the morning when the Sun is low on the sky, the atmosphere refracts the light, which causes the Sun to appear slightly higher in the sky than it really is. This small change in elevation pushes the apparent location of the Sun from one hemisphere into the other, so the shadow crosses the lower half of the ring. But as the Sun rises in the sky, the refraction rapidly decreases and the Sun’s apparent position goes back to matching its true position back on the original hemisphere. Then the Sun proceeds to cross from one hemisphere to the other as the equinox actually occurs and the shadow from the upper half of the ring falls completely on the lower half of the ring a second time.

Hipparchus’s measurements of the solstices and equinoxes allowed him to measure the length of the seasons. Now it had by this point already been well established that the seasons were of unequal length. Callippus had figured this out centuries ago and I talked about his result back in Episode 17. But Hipparchus’s measurements were the most accurate to date and were good enough that they agree with the modern values to within half a day, which was about the precision he had available to him with his techniques.

But beyond just measuring the inequality in the length of the seasons, Hipparchus was the first to provide a theoretical explanation of their inequality. He proposed an eccentric circle model for the motion of the Sun. So instead of the Sun revolving directly around the Earth, it revolved around a point somewhat offset from the Earth. Specifically, this point was 1/24th the distance of the Sun away from the Earth, and its direction was at 65 and a half degrees to the east of the vernal equinox.

Now I had mentioned in last month’s episode that Apollonius of Perga had already demonstrated that the eccentric circle model was mathematically equivalent to the epicycle model. So why did Hipparchus stick with the old eccentric circle model rather than develop an epicycle model for the motion of the Sun? In truth we don’t really know, but to speculate a bit, Hipparchus may have considered the eccentric circle model of the Sun’s orbit to be a simpler and more intuitive model to him.

Hipparchus also adopted an eccentric circle model of the Moon’s orbit. Now this is a fairly interesting development because it required him to make relatively extensive observations of the Moon. The Greeks had been interested in the motions of the planets for centuries and by this point had developed several successive models to try to explain their motion, and, in particular, that puzzling retrograde motion that they saw every so often. But the Moon displays no retrograde motion, so it had been something of an afterthought in Greek astronomy, at least compared to the Babylonians who had over the past several centuries developed extremely sophisticated models of the Moon’s motion.

To arrive at his model of lunar motion, Hipparchus observed several lunar eclipses in the years between 146 and 135 BC. By comparing his observations against the records of earlier lunar eclipses, he was able to come up with an improvement on the Saros cycle. The Saros cycle has come up a couple of times now, but to refresh your memory, it’s a period of 223 synodic months, that is, 223 cycles of new moon to new moon. This period happens to be extremely close to 242 draconic months, or the number of times the moon crosses from the southern side of the ecliptic to the northern side over the point where the Moon’s orbit intersects the ecliptic, called a node. In order for a lunar eclipse to occur, the Moon has to both be full and it has to be very close to the ecliptic, crossing through a node so that it is exactly opposite the Sun and falls into the Earth’s shadow. Since a lunar eclipse then depends both on the phase of the Moon, and so the synodic month, along with the location of the nodes, and hence the draconic month, lunar eclipses repeat almost exactly on this 18 year Saros cycle.

By comparing the date, time, and location of lunar eclipses he observed with records of lunar eclipses from an earlier century, probably from the Babylonians, he was able to extend the Saros cycle to include the sidereal month as well, that is, the time it takes the moon to orbit a full 360 degrees around the earth. Hipparchus found that a period of 126,007 days and 1 hour is almost exactly 4267 synodic months and also 4612 sidereal months plus 7 and a half degrees. This works out to be a cycle of about 345 years and it implies that the length of a synodic month is 29 days 12 hours 44 minutes and 3.3 seconds, only 0.4 seconds larger than its true value. And likewise this meant that the sidereal month was 27 days 7 hours 43 minutes and 13.1 seconds long, only 1.3 seconds too large. So we see that by combining his own observations with those of earlier astronomers, probably from Babylonia, he was able to measure the Moon’s orbital period, both synodic and sidereal, to an accuracy of about a second, more than one and a half millennia before the second hand on a clock hand even been invented. This is, in fact, something of a characteristic feature of astronomy. Many of the phenomena that astronomers study, certainly the phenomena they were predominantly interested in prior to the 20th century, are periodic in nature. The orbits of the planets and moon may be complicated, but they are an interplay of a series of regularly repeating events like a full moon or the moon passing through a node. Even without accurate timepieces, it ends up being possible to measure their period to extremely high accuracy just by observing their periodicities for a very long time. And, of course, until the development of high quality mechanical clocks in the 18th century, the most accurate means of timekeeping was astronomy. We today don’t think of astronomy as having too much to do with timekeeping, but throughout most of it’s history that was it’s primary function. As an aside, this facet of astronomy made a reappearance in the late 20th century when the accuracy of pulsar rotations was such that there were a few decades where pulsar timing measurements competed with atomic clocks for being the highest accuracy way of keeping time.

In addition to measuring the length of the synodic and sidereal month, Hipparchus also measured the length of yet another kind of month, the anomalistic month. If you observe the moon’s motion carefully, you’ll notice that it does not move uniformly across the sky. At some times during the month it moves a little faster and at other times it moves a little slower. This variation in the moon’s speed, or any celestial body’s speed, is called the anomaly, and today we explain it as being due to the fact that the moon has a slightly eccentric orbit, so sometimes it’s closer to the earth and moves faster, and other times it’s farther away and moves slower. But this point in the moon’s orbit, called the perigee, is not static, but rotates around over time as the moon’s orbit wobbles due to the torque from the sun, so the length of the anomalistic month is not the same as the length of the sidereal or synodic months. Hipparchus was able to show that this 126,007 day and 1 hour period was 4573 anomalistic months. This works out to be 27 days 13 hours 18 minutes and 34.7 seconds, off by around 2 seconds, though, again, we have to be careful in assessing its accuracy given the variation in the Earth’s rotational speed and perturbations to the moon’s orbit from the other planets over this time period. Well because the point in the moon’s orbit that moved fastest changes over time, this meant that Hipparchus couldn’t use quite as simple a model for the moon’s motion as he could for the Sun. For the Sun, Hipparchus could model its motion with an eccentric circle where the center of that circle sat at a static location. For the Moon, Hipparchus still chose to model its motion with a circle offset from the Earth, but now he had to allow the offset to move. It didn’t move very fast with a period of about 18 and a half years, but the fact that it had to move at all was a complication relative to his model of the Sun’s motion.

Well Hipparchus was not simply interested in the position of the Moon on the sky, like Aristarchus a century earlier he was also interested in how far away it was from the Earth. But once again Hipparchus outshone his predecessor because when he set out to try to estimate the distance to the Moon, he developed not one but two separate methods to do so. The first was to take advantage of a total solar eclipse. As with all of these things, there is some debate as to which year exactly this solar eclipse occurred in, but the consensus opinion is that it was in 129 BC, though one historian of science, Gerald Toomer, argued that it occurred in 190 BC, around the time Hipparchus was born, and that Hipparchus had just heard stories of the eclipse. Now, in the Hellespont, the traditional border between Europe and Asia Minor, this eclipse was total. However, according to Hipparchus, either from his direct measurements or from reports from other astronomers, this eclipse was observed to be only partial in Alexandria, and specifically the Sun was measured to be only 4/5 eclipsed. Hipparchus knew what the difference in latitude was between Alexandria and the Hellespont and seems to have assumed that they were directly north and south of each other so that the difference in longitude was negligible, which is a reasonable enough assumption for a basic estimate of the Moon’s distance. With some trigonometry, he could then figure out that the Moon was somewhere between 71 and 83 Earth radii away, with the uncertainty probably coming from the sophistication of the trigonometry that Hipparchus had available to him. And, in fact, it wasn’t so much the trigonometry that was available to him in a passive sense, but the trigonometry that he made available to him by inventing it himself. I’m mostly focused in this episode on Hipparchus’s astronomy, but Hipparchus made substantial developments in trigonometry in order to make progress on the astronomical problems he was trying to solve.

Now, the true distance to the Moon is about 60 Earth radii, so Hipparchus’s estimate of somewhere between 71 and 83 Earth radii really wasn’t so far off, the major source of error probably being the accuracy with which the astronomers in Alexandria could measure the fraction that the Sun was eclipsed.

But Hipparchus wasn’t content with just measuring the distance to the Moon one single way, he proposed a second, independent method. With this technique we’ll once again run into the limitations of describing geometrical constructions on a podcast, but the basic idea was that during a lunar eclipse you can measure the angular width of the Earth’s shadow by seeing where the Moon is relative to the background stars when the lunar eclipse begins versus when it ends. Likewise you can also measure the angular diameter of the Sun. Now, let’s suppose the moon is halfway into the Earth’s shadow, so it is half eclipsed and imagine that we’re observing it from a point on the earth where the moon is just on the horizon. We can now draw two triangles, one from this point to the center of the Earth and then to the edge of the Sun at the highest elevation; and another triangle from this point on the Earth to the center of the Earth to the center of the Moon. Both of these triangles are right triangles with the right angle at the point on the Earth’s surface, and both have a very small angle at the more distant points on the Sun or Moon. These two angles are the parallax of the Sun and the parallax of the Moon, respectively. So it is the amount that the Sun or Moon’s position appears to change if you were to observe them from the center of the Earth versus if you observe them from a point on the Earth’s surface where they are on the horizon. The whole question is to measure these parallax angles. If you can measure these parallaxes, you’ll be in business since this tells you directly how far away these bodies are in terms of the Earth’s radius. Now if you draw out a diagram of this situation you will see that there is a basic relationship between all these angles — the parallax of the Moon plus the angular radius of the Sun is equal to the parallax of the Sun plus the angular radius of the Earth’s shadow. Hipparchus could measure the angular width of the Earth’s shadow and he could measure the angular diameter of the Sun, but it seems that he would then be stuck. He had a single equation with four variables, two of which were unknown. But then Hipparchus made a crucial assumption. He had earlier tried to measure the parallax of the Sun directly and had found that he could not detect any parallax. He estimated that the smallest parallax he could hope to measure would be about 7 arcminutes, so whatever the parallax of the Sun was, it had to be smaller than that, which then implied that the Sun was at least 490 Earth radii away.

But then, in a stroke of genius, he realized that because the parallax of the Sun was so small anyway, it could just be omitted from the equation entirely. He could just drop that term from the equation and say that the parallax of the Moon was equal to the angular radius of the Earth’s shadow minus the angular radius of the Sun. Technically, of course, this was only an upper bound on the parallax, and so a lower bound on the distance to the Moon, and effectively made the assumption that the Sun was infinitely far away, but when the math all came out, the error due to this assumption was pretty small. Hipparchus ended up estimating that the Moon was somewhere between 62 and 72 and 2/3 Earth radii away. And since the true value is about 63, which is in the range that he had estimated using this second technique.

This was not only a clever and accurate way of measuring the distance to the Moon, but also an important development in the history of mathematical physics. It’s the first instance I’m aware of in the sciences of a scientist deliberately neglecting a small term in an equation that would otherwise be intractable. This technique of neglecting extremely small terms is really at the heart of calculus and these days physicists are notorious for reaching into their equations and tossing out any terms that they consider to be irrelevant left and right. So anytime that you neglect a term in one of your equations remember that you’re in good company, Hipparchus did it first.

One other thing to note about this technique was that Hipparchus had to measure the angular diameter of the Sun. Now, we heard in Episode 19 that Aristarchus had already done this in order to arrive at his estimate of the distances to the Sun and Moon. But according to the author Proclus, Hipparchus built an instrument for this purpose. Unfortunately we don’t know in detail how this instrument worked, but the idea seems to be that there was a rod with a set of movable slits at the end of it, and by moving the slits closer together or further apart until it just spanned the width of the object in question, he could measure its angular diameter.

Well, perhaps Hipparchus’s longest lived legacy is his stellar catalog. Now, Hipparchus wasn’t the first astronomer to compile a catalog of the positions of the stars on the sky. We’ve already seen how that distinction belongs to Timocharis and Aristyllus. But Hipparchus’s catalog was around an order of magnitude larger and contained about 850 stars. Unlike the catalog of Timocharis and Aristyllus, we have a fairly good idea of what his catalog looked like because it formed the basis of Ptolemy’s stellar catalog a few centuries later.

Now, to the modern mind, compiling a catalog of the positions of the stars may seem like the kind of a fundamental first step for any program of astronomical research, but compiling a catalog of the stars is, let’s face it, not the most interesting work. I mean there’s just so many of them and measuring they’re positions is not exactly trivial. So why did Hipparchus bother doing this? At least according to Pliny the Elder, the motivation for his stellar catalog was that he had noticed a new star in the sky one night. Now, as I discussed in Episode 18, this was generally incompatible with Aristotle’s model of the world, in which the superlunary realm was largely unchangeable, with the possible exception of superlunary comets. But it seems that Hipparchus couldn’t quite convince himself that the star was new rather than just having been there all along without his noticing it. So he determined to produce a catalog of all the stars in the sky, or at least all the bright ones, so the next time a new star did seem to show up, he could compare it to his catalog and see if it was really new or not.

Now this story from Pliny the Elder is very intriguing to modern astronomers because today we do know that new stars can appear on the sky, or at least objects that look like stars can. Historically they were all called novae, though today things that were once called novae are understood to be a variety of different astrophysical phenomena. But in many cases these events produce some sort of a remnant that can be observed up to tens of thousands of years later. So over the decades, astronomers have periodically tried to connect a supernova remnant to this mysterious new object that Hipparchus thought he had seen. However, a more prosaic possibility is that Hipparchus had simply observed a variable star like Mira, whose brightness fluctuates fairly dramatically. Now, unfortunately because the original catalog does not survive, it only comes to us filtered through Ptolemy, we don’t know precisely how he compiled it. In particular we don’t know what coordinate system he used to record the positions of the stars. Ptolemy presents the catalog with an ecliptic coordinate system, where the position of the stars is given by the ecliptic longitude, the angle along the ecliptic, and the ecliptic latitude, the angle above the ecliptic. But Hipparchus may well have used another coordinate system, or have used one coordinate system for the stars in the zodiac and another coordinate system for the stars closer to the celestial pole. Intriguingly, Pliny the Elder says that Hipparchus used several instruments to measure the positions of the stars in his catalog, so it seems that as with measuring the distance to the Moon he was not content with limiting himself to a single technique. And the fact that he used multiple instruments, though we don’t know what exactly they were, has led some modern historians of science to the conclusion that he probably represented his catalog in multiple coordinate systems since different instruments would naturally lend themselves to measurements in different coordinate systems. In addition to listing the positions of the stars, Hipparchus also recorded how bright they appeared to him. For this, he developed a fairly rudimentary scale. He simply rated each star’s brightness on a scale from one to six, with one being the brightest and six being the faintest. As any professional astronomer will know, this seemingly innocuous decision of Hipparchus’s was to have profound consequences for thousands upon thousands of befuddled students of astronomy centuries later, because Hipparchus’s choice of scale forms the basis of the modern day magnitude system which is one of the more baroque and perplexing unit systems in a field riddled with baroque and perplexing unit systems. In order to make modern photometric measurements of the brightnesses of stars roughly map onto Hipparchus’s original scale, astronomers ended up with a system where an increase by one in the magnitude of a star corresponds to a decrease in its brightness by a factor of the fifth root of 100. But, believe it or not, this is really the easiest part of the modern magnitude system to wrap your mind around because the scale then needs to be defined relative to a reference point and here you get into whether or not you are using the Vega magnitude system, in which case your reference point is the star Vega, which was chosen long before our photodetectors were any good, and turns out to be an extremely poor star to choose as your reference point because you want your reference to be constant and Vega really isn’t very constant in brightness at all, or the AB magnitude system, which is more rational but has the disadvantage of not having any true reference stars that you can use. But I digress.

At any rate, Hipparchus’s catalog enabled what is perhaps his greatest discovery — the precession of the equinoxes. What Hipparchus had done was measure the distance of several stars from the Moon during a lunar eclipse, in particular the star Spica in the constellation Virgo. Now what does the distance of a star from the Moon during a lunar eclipse tell you? Well, if the Moon is being eclipsed, that must mean that it is on the ecliptic. So by measuring this distance that immediately tells you the distance of the star to one point on the ecliptic. But, the Sun at this time is 180 degrees away from the Moon during a lunar eclipse. Now, any astronomer worth their salt should be able to tell you what the ecliptic longitude of the Sun is on a given day of the year. It’s at an ecliptic longitude of zero degrees on the vernal equinox and increases by about 1 degree per day throughout the year. So Hipparchus knew how far away the Sun was from the vernal equinox on the day of this lunar eclipse. But this then meant that he could compute how far away the Moon was from the autumnal equinox, since that was 180 degrees away from the vernal equinox. And since he could measure the distance of Spica from the Moon, from that he could compute how far away Spica was from the autumnal equinox.

But here Hipparchus found something strange. He had found that Spica was 6 degrees away from the autumnal equinox. But when he compared this to the records from Timocharis, Timocharis had observed Spica to be 8 degrees away from the vernal equinox. Two degrees was easily within the observational capabilities of both Hipparchus and Timocharis, so what was going on? And this didn’t seem to be limited to Spica alone. Other stars that Hipparchus observed also seemed to move with respect to the location of the equinoxes. Or, put another way, the location of the equinoxes seemed to be moving with respect to the stars.

Once again, Hipparchus was not content with looking at a single line of evidence. He also looked at the declination of stars close to the north celestial pole, that is, the distance of these stars from the celestial equator. And here when he compared the declinations he measured with the declinations from the catalog of Timocharis and Aristyllus, he found that on one side of the pole, the declinations had all decreased, and on the other side they had all increased. So this seemed to imply that the celestial pole was moving very slowly as well.

This was the first really conclusive evidence of the precession of the equinoxes. I mentioned way back in Episode 6 that Babylonian records indicate the existence of this phenomenon, but there’s no evidence that the Babylonians recognized it for what it was. For this, Hipparchus appears to be the first. Now, Hipparchus had noticed a drift of 2 degrees between his measurements and those of Timocharis and Aristyllus, which were about 160 years earlier. This then implies a yearly precession of 45 arcseconds per year which is really in very remarkable agreement with the modern value of 50 arcseconds per year. We now know that the location of the equinoxes on the ecliptic moves by about 1 degree every 72 years and similarly the north celestial pole traces out a slow circle on the sky with a period of about 26,000 years, leading to the north star changing over the millennia.

Well, the only area of astronomy that Hipparchus did not quite make a breakthrough in was in modeling planetary motions. Here we know that he had used the epicycle model of Apollonius, but he did not develop any refinements to this model. However, this isn’t to say that he was exactly satisfied with it. Apollonius’s model was fairly simple, much simpler than the model of Callippus and Aristotle. Each planet orbited in a circular epicycle around a deferent, and the deferent revolved in a circle around the Earth. This reproduced the retrograde motions of the planets and so was a pretty decent alternative to the baroque model of Callippus. But Hipparchus seems to have noticed that this model had a shortcoming. It predicts that a planet’s retrograde motions should all be identical. The times between the first and second stationary points should all be the same and the distances between the first and second stationary points should also all be the same. Now, they are fairly close to all being the same, especially for the outer planets. But by comparing his observations of the planetary positions to those of earlier astronomers, Hipparchus seems to have realized that there was some variation here and recognized that this could not be explained through the basic model of epicycles. Particularly for Mars, sometimes the distance between the stationary points was smaller and sometimes it was larger, and the time it took Mars to go between these two points varied as well. But he also recognized that he lacked enough data to put forward a competing model that could explain this phenomenon. So he was forced to leave it as a problem for future astronomers to solve, which, another 200 years or so later, another one of the greats of the field, Claudius Ptolemy, was to try to do.

So, I’ve talked about the astronomy of dozens of astronomers of the ancient world at this point, virtually all of them Greek because very few of the names of the Babylonian astronomers were preserved. But I think it is fair to rate Hipparchus as being the greatest of the ancient astronomers due to the precision, creativity, and sheer variety in his astronomy. Other Greek astronomers made major developments in one or maybe two areas of astronomy, Eudoxus for example proposed the first serious model of planetary motion. Or Aristarchus measured the distance to the Moon. But Hipparchus was the only astronomer who made major advances in virtually every area of astronomy available to him. He measured the distance to the Moon, not just one, but two ways; he made by far the most accurate measurement of the length of the year and the motion of the Moon. His stellar catalog was far more complete than the one that came before him. And, of course, the quality of his observations was enough that he could detect one of the subtlest astronomical phenomena discovered in the ancient world, the precession of the equinoxes. And Hipparchus is unique among the ancient astronomers in that he is the only one whose work still influences the way modern astronomy is done in the form of the magnitude system. Maybe thanks to that some students of astronomy might wish that he had been a slightly less capable astronomer, but whether it is Stockholm syndrome or not, many astronomers, observational astronomers in particular, have learned to stop worrying and love the magnitude system. A good life lesson is that you don’t always get the unit system that you want to, but if you try sometime, you just might find you get the unit system you need.

If we compare the astronomy of Hipparchus to the astronomy of the earliest Greek astronomer, Thales, we have clearly come a long way. Hipparchus’s astronomy looks a lot like modern science. Gone are the idle speculations about the fundamental nature of matter. Instead careful observation and rigorous mathematical analysis are the rule of the day. Or at least, rigorous enough mathematics — just like in the modern sciences, Hipparchus knew when it was appropriate to fudge things slightly and throw away a term or two in his equations.

And, what also strikes Hipparchus’s astronomy as being very modern is its reliance on the work of other astronomers. Hipparchus was a very good observer, of course, but he was not a lone genius working in isolation. Many of his greatest discoveries, the precession of the equinoxes being foremost among them, only came about because he combined his own careful observations with the careful observations of other astronomers from the past and had the ability to interpret the anomalies in these comparisons.

Well, as I mentioned, the only area of astronomy that he did not make a major breakthrough in was in modeling planetary motions. And here he could see that something was amiss, but didn’t have enough data to develop a better model. That would have to wait for Claudius Ptolemy, who is the last of the great Greek astronomers of antiquity. But this is not foreshadowing what is to come in the next episode because it’s unlikely that I will talk about Ptolemy next month. Next month’s episode will likely be a little shorter, but I want to talk a bit about the astronomical instruments that the ancient Greeks used. One, in particular, stands out for its sheer sophistication, and this is the famous Antikythera mechanism. Hipparchus himself in fact may have had a hand in designing this remarkable instrument, though this episode has already gone on long enough that I can’t give it the attention it deserves here.

So it will have to wait until next month. I hope you’ll join me then. Until the next full moon, good night and clear skies.

Additional references

• Goldstein & Bower, 1991, Dated Observations in Greek Astronomy
• Yavetz, Ido, Hipparchus on the Theory of Prolonged Motion, 2015