# Episode 20: The Theory of Epicycles & Deferents

August 13, 2022

In the Hellenistic Era the astronomer Apollonius of Perga (maybe) developed the model of epicycles and deferents that was to dominate Western astronomy for more than 1500 years. Around the same time, Eratosthenes, who was the head librarian at the Library of Alexandria, developed a novel technique to measure the circumference of the Earth and arrived at a suspiciously accurate result.

## 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 in the last episode we finally made that irreversible plunge into the astronomy of the Hellenistic Era of Ancient Greece. At least kind of for Heraclides of Pontica, who technically did make it into the Hellenistic Era, but just barely. But Aristarchus of Samos was well and truly in the Hellenistic Era so now we’re stuck here. As we entered this era, you might have noticed one way that the podcast suddenly changed — that word, the “sources,” suddenly made a reprise to the show. When I first started talking about the astronomy of ancient Greece with Thales and Anaximander and Anaximenes, almost every other word out of my mouth was a caveat about how we don’t really know what they thought and we’re just making educated guesses because the sources are very poor. Nothing from the time survives and so we’re relying on summaries of quotations of excerpts from lost histories. But as we made our way through the centuries, those caveats got fewer and fewer until we got to Aristotle. To our good fortune, many works of Aristotle survive — and to our chagrin not all of them do, but we have enough that we can be pretty confident in the structure of his philosophy, including the structure of his astronomy. So things were looking pretty good.

But then we got to Heraclides and Aristarchus and suddenly all that stopped. Now, to be clear, it’s not quite as bad as it was for the very earliest Greek astronomers like Thales. But we were once again back to closely reading quotations of fragments. So what happened here? It would make for a nice story if there was some catastrophe I could point to around this time, a war or fire or something that destroyed all the works from this time. But there wasn’t one, and if you think about it it wouldn’t be a very compelling explanation even if it were true because if that catastrophe destroyed the works from an astronomer in the early Hellenistic Age, why did it not destroy the earlier works of the astronomers in the late Classical Age? Instead it seems a big reason that it suddenly becomes harder to understand the astronomy of the early Hellenistic Age is that one of the most important sources for the earlier astronomers was a work by Theophrastus that I’ve referenced a few times, his History of Physics. The work itself didn’t survive, but it was influential enough that many later authors pulled from big sections of it so that we have a reasonably complete understanding of what it contained. But Theophrastus was a student of Aristotle and unfortunately, being a mere mortal, he was unable to include in his history any of the astronomers who lived after he did. So now that we have passed Theophrastus chronologically, we’re once again stuck with rather sub par sources for a little bit. Two of the major sources we have to use are a bit dicey. There is some work by an astronomer named Geminus who lived in the first century BC, and then another by the name of Cleomedes, whom we know so little about that he could have lived anywhere from the first century BC to 400 AD, so we can’t even place him to less than 450 years or so.

So, with all that in mind, we’ll just dive right in to the next two most significant astronomers of the Hellenistic Age: Apollonius of Perga and Eratosthenes.

Given that I’ve been harping on the lack of sources we have it won’t be too much of a surprise to you that we know basically nothing about Apollonius’s life. There is a brief biography from an author named Eutocius, who, in turn, was relying on an author named Heraclius, and we know absolutely nothing about this Heraclius except that he was someone called Heraclius. According to Eutocius, Apollonius was born during the reign of Ptolemy III, also called Ptolemy Euergetes. This places his birth sometime between 246 and 221 BC. Another author, Ptolemaeus Chennus, says that a famous astronomer named Apollonius lived during the reign of Ptolemy IV, known as Ptolemy Philopater, who reigned from 221 – 205 BC. Since he had to be at least in his twenties, and probably closer to his forties to be considered a famous astronomer by this point, this means that Apollonius was probably born pretty early on in Ptolemy III’s reign, so most scholars place his birth at around 240 BC, about 80 years after the death of Alexander and Aristotle.

As his name implies, Apollonius of Perga was born in Perga, which is on the southern coast of Anatolia, but like most of the other philosophers of the time it’s assumed that he left his hometown when he came of age to study. Where he went we don’t exactly know, but it’s thought that he lived and worked in Ephesus, Pergamum, and Alexandria. Since Alexandria was the major intellectual center by this point it’s generally assumed that he spent most of his time in the city, but again we don’t really know for sure.

The mathematician Pappus does say, however, that he studied in Alexandria. By this point Euclid had founded his famous mathematical school in Alexandria and Apollonius learned mathematics at this school. Pappus writes, “Apollonius spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought.” Now as an aside, this might sound like a nice thing of Pappus to say about Apollonius, but the surrounding context in Pappus’s text is a lot less generous. In the broader context Pappus is criticizing Apollonius for not giving Euclid sufficient credit in solving a particular mathematical problem. Petty academic disputes over credit are by no means unique to the modern age.

Apollonius was as much a mathematician as he was an astronomer, if not more so, and his surviving works are all on mathematics. We’ve seen this pattern before with Aristarchus and Archimedes, who both had interests in mathematics and astronomy, and we will see this again in the future, too. The considerable overlap between astronomy and mathematics goes right back to Plato, who considered astronomy to simply be the highest form of mathematics. Just as plane geometry is the mathematics of two dimensions and solid geometry was the mathematics of three dimensions, astronomy was the mathematics of four dimensions, how solid objects move through time.

By far and away Apollonius’s most important work was a text called Conica which was about conic sections. The book is a detailed exploration of a special class of curves that are generated by the intersection of a flat plane with a right cone. If the plane intersects the cone parallel to one of the edges, for example, you get a parabola; but if it instead intersects it at an oblique angle you get an ellipse; if it intersects perpendicularly to the axis of the cone you get a circle; and if it intersects it at a more extreme angle you get a hyperbola. Apollonius’s work firstly demonstrated a way to generate these various curves, but more importantly, showed how, despite their various shapes, they were all intimately related through this geometrical construction. Today we tend to think of these curves in algebraic terms where the connection is more obvious, namely that they are a family of curves that are quadratic in their x and y variables, but Apollonius was the first to see that they were connected at all, even if in geometric terms rather than algebraic terms.

This work, Conica, turned out to be tremendously influential in the West. It was highly influential in Greek mathematics, but it was a very intimidating text, very dense and at the cutting edge of what was possible with the mathematics of the day, so by the Roman Era, it had become a text that was more admired than read, maybe with something like the status of Newton’s Principia or Darwin’s On the Origin of Species. Everyone had heard of it, but hardly anyone had actually read it. As we’ll get to when we talk about the astronomy of Rome, or lack thereof, the Romans did not have a very intellectual culture, so there wasn’t demand for a high octane book like this. The Romans more preferred the ancient equivalent of Reader’s Digest versions of Greek philosophy. In modern terms, they were big on pop science, not so big on actual science textbooks. The Islamic scholars of the Islamic Golden Age, by contrast, were a much more sophisticated bunch, and were very interested in pursuing advanced mathematics, so they translated Conica into Arabic and it was through them that the work survived out of antiquity.

Now, conic sections are of more than purely mathematical interest. It will be many episodes before we arrive at this revelation in the chronology of the history of astronomy, but it turns out that the orbits of celestial bodies under gravitational attraction are all conic sections. We’re most familiar with the planetary orbits, which are ellipses. This is what is called the bound case because the two objects are gravitationally bound to each other. They don’t have enough energy to escape out to infinity. But if a celestial body does have enough energy that it can escape to infinity, its trajectory is described by a hyperbola, one of the other conic sections. And in the intermediate case where an object is just barely unbound, where in theory it would escape out to infinity but asymptotically approach zero velocity, its trajectory is described by a parabola, yet another conic section.

It’s therefore maybe not too shocking that the man who popularized Apollonius’s Conica in the West was an astronomer. In the early 1700s, Edmond Halley, who we all know and love from Halley’s comet, had found copies of the Arabic translation of Apollonius’s Conica at the Bodlein Library at Oxford and took it upon himself to learn how to read Arabic so that he could translate the work into Latin and make it accessible to Western scholars.

Incidentally, you may recall from last month that we heard about how both Aristarchus and Archimedes developed methods to write down very large numbers, and along these same lines, Apollonius also wrote a work, unfortunately lost, which presented another method of representing large numbers, and what’s more, he demonstrated a technique to multiply them. Now these days, of course, we learn how to multiply numbers of arbitrary size in elementary school this may seem like no big thing, but the Arabic numeral system that we use does quite a lot of the heavy lifting for us. Figuring out how to do this in a numbering system that lacks a zero is not quite as trivial.

Well as far as his astronomy goes, Apollonius is generally credited with developing the system of deferents and epicycles, though as I mentioned in the last episode, the evidence is somewhat ambiguous and a case can be made that the model had been developed earlier and was simply refined by Apollonius. Nevertheless since Apollonius is the figure most closely associated with epicycles, now is as good a time as any to go into a little more detail about how this model worked.

The theory of epicycles is famously a geocentric model of the Solar System. So the Earth is at the center of the universe, and the Sun, Moon, and all five of the classical planets all revolve around the Earth. Now, the simplest model you can make is to suppose that the planets revolve around the Earth in perfect circles moving at a constant speed. This more or less works since over time you see the planets drifting through the constellations of the zodiac, some, like Mars, doing so relatively rapidly, taking a bit less than two years to move through a full 360 degrees, and others like Saturn taking their time at a rate of nearly 30 years to do so. But if you look more carefully, this simple model of perfect circular motion cannot explain the observed motions of the planets. The planets don’t move at a constant speed through the zodiac. Sometimes they move faster and sometimes they move slower. And, in fact, sometimes they even slow to a stop and start moving in the other direction.

Now, Eudoxus had tried to model this phenomenon as a superposition of the motions of two spheres centered on the Earth. This worked to produce the qualitative behavior of retrograde motion, and really worked quite well for the more distant planets like Jupiter and Saturn whose retrograde motion is somewhat minimal, but for the closer planets with more extreme retrograde motion it did not do a very good job and his successor Callippus had to fix this by superimposing the motions of even more spheres, which produced an extremely complicated motion.

The model of deferents and epicycles tries to explain the variable motion of the planets another way. Rather than directly revolving in a circle around the Earth, a planet revolves in a little circle around an invisible point in space. This little circle is called an epicycle and its center is called the deferent. The deferent, in turn, revolves in a circle around the Earth. The motion along the epicycle is in the same sense as the motion of the deferent. So if one is counterclockwise, the other will be counterclockwise as well. The period of revolution of the deferent is just the sidereal period of the planet, the time it takes to journey 360 degrees through the zodiac. So, for Mars that’s a bit less than two years and for Saturn it’s close to 30 years. Simultaneously, the planet also travels around its epicycle once per year.

Or, at least, that’s how we’d describe it today. In modern terms we would say that the planet rotates around its epicycle once per year. But the ancient Greeks defined one revolution a little differently. The way we would do it is with respect to absolute space. So, how long does it take the planet to move 360 degrees in its epicycle around the deferent. But the Greeks defined a single revolution to be when the planet goes from its most distant point from the Earth back to its most distant point from the Earth. Since the center of the epicycle, the deferent, is moving as the planet revolves around its epicycle, it has to travel a little farther than it otherwise would to get back to its most distant point from Earth. So according to the Greeks, the period of a planet’s motion around the epicycle is what is called the synodic period, which is the time from opposition to opposition. For the more distant planets, this isn’t really a very big difference from a year because the deferent moves so slowly that the planet doesn’t have to move much farther to return to its most distant point. For Saturn this period is 1 year and 13 days. But for Mars the difference is fairly substantial, the time it took to go around its epicycle according to the Greeks was a bit more than two years.

Now, one advantage the Greeks had in defining a revolution in this way is that it worked for all of the planets. Every planet, superior or inferior, revolved around its epicycle once per synodic period, and its deferent revolved around the Earth once per sidereal period. If we use the modern definition of a revolution as going through 360 degrees, the aesthetics of the model are not quite as nice because we have to treat the inferior planets differently from the superior planets. With this definition, for the inferior planets, it’s the deferent that revolves around the Earth once per year, not the epicycle, and the period of the motion of the epicycle is the synodic period. So the Greeks seem to have had it right in defining their system the way they did because everything was much cleaner.

So that was how they determined the periods of the two motions, the period of the deferent about the Earth and the period of the planet’s motion about the deferent. But what about the sizes of the deferent and the epicycle? Now, as with any planetary model, there’s really no way to get the absolute scale of the distances to the planets in something like meters unless you go and do parallax measurements from the Earth. But you can say something about the relative size of the epicycle to the deferent. This is just set by the distance that the planet travels in its retrograde motion. With a smaller epicycle relative to the deferent the planet travels a shorter distance in retrograde and with a larger epicycle it travels a longer distance.

Now, the main advantage of the theory of epicycles is that it can produce that puzzling retrograde motion that we see. But it doesn’t have to. The Sun and Moon both have variable motion across the sky, but the motion isn’t so variable that we see either one go backwards in retrograde motion. So, what Apollonius did, or maybe another Greek astronomer before him, was to reverse the sense of revolution in the epicycle for the Moon and Sun. Then the epicycle no longer produces a retrograde motion, but it does produce a variable motion, sometimes faster, sometimes slower, that more or less corresponds to the observed motion of the Moon and Sun.

In last month’s episode I also brought up a competing theory to the theory of epicycles, the theory of eccentric circles. The idea of this theory is that each planet revolves in a circle around the Earth, but this circle is a little bit offset from the Earth. Then, in turn, this offset revolves around the Earth as well. So in the case of epicycles you have a point on big circle which revolves around the Earth, and then the planet revolves in a little circle around that point on the big circle. Whereas in the theory of eccentric circles, it’s the reverse. The planet revolves in a big circle around a point which revolves in a little circle around the Earth.

One of Apollonius’s major contributions was to show that these two models were absolutely identical. Now, Apollonius did this using the tools he had available to him at the time, which were the tools of geometry, and showing this equivalence geometrically is not completely trivial. It’s not the most challenging problem, but geometrically, the fact that these two models are mathematically identical is not obvious from the outset. With modern vector arithmetic, though, this is a pretty trivial thing to show. The position of the planet can be treated as a vector, and you can decompose the vector into the sum of two pieces. In the case of the epicycle model, you have a vector which is big and which points to the location of the deferent, and then you have a smaller vector which goes from the deferent to the planet on the epicycle. In the eccentric circle model, the center of the planet’s orbit is given by a small vector, and then the planet’s location is a big vector from that center to its location on the orbit. The fact that these two models are identical is just a consequence of the fact that it doesn’t matter in which order you add up these two vectors.

In mathematical terms, we say that vector addition commutes. A lot of important operations in mathematics are commutative. Addition and multiplication of ordinary numbers both are. It doesn’t matter if I add 11 + 7 or 7 + 11, either way I get 18, and similarly 5 times 9 is the same as 9 times five. But not all operations are. Division is one of the simpler operations which is not commutative. 8 divided by 4 is 2, but 4 divided by 8 is one half, not 2. But, vectors do happen to add commutatively, so it doesn’t matter if you add the big vector to the small vector like you do in the eccentric circle model or if you add the small vector to the big vector like you do in the epicycle theory.

Now, given that epicycles and eccentric circles are equivalent models and Apollonius knew it, this then raises the question of why it is that epicycles won out. A big reason is that identifying the important points in a planet’s orbit was more intuitive in the theory of epicycles. It is very easy to see when the planet is at its first stationary point in the epicycle model, whereas it’s less obvious in the eccentric circle model. But this wasn’t the only reason. Another reason that epicycles came to dominate Greek astronomy and Western astronomy thereafter is that it fit in well with physical models of the Solar System. Now in the case of Apollonius and later astronomers like Hipparchus and Ptolemy, it is not clear whether or not they believed that their models were literal physical models of the universe or just mathematical devices to predict planetary motions, as was the case with Eudoxus and his homocentric spheres a century earlier.

But regardless of what they thought, other astronomers did want to see a literal, physical model of the Solar System. We saw that with Aristotle who envisioned the spheres of the heavens as being interlocking crystalline spheres. With the epicycle model, later astronomers could envision the epicycles as being like a ball bearing constrained between two spheres so that it would roll in a circle in the space between these two spheres. But in the eccentric circle model, the centers of the planetary orbits were all enclosed in each other so you couldn’t really separate out the planetary motions into distinct physical layers. So from a physical perspective, the epicycle model made a lot more sense to later astronomers than the eccentric circle model did.

The development of the theory of epicycles marked an important milestone in the history of astronomy in ancient Greece where astronomy definitively started to diverge from the tradition of pure philosophy. In the Archaic Age and through most of the Classical Age, astronomy was just a branch of natural philosophy, which was just a branch of pure philosophy. Philosophers would speculate on all sorts of matters, ethics, politics, rhetoric, and the physical world was one of the things they felt they could speculate on. Some, like those in the Ionian School, had more of an emphasis on natural philosophy than the philosophers in other schools, but philosophers like Plato and Aristotle had no qualms about opining on the nature of the physical world. Over the course of the Classical Era, astronomy started to become more mathematical in nature. Understanding Eudoxus’s model required some mathematics, and understanding Callippus’s variant on it required even more. And, of course, the use of mathematics for astronomy was celebrated by many philosophers like Plato, who saw it as no less than the highest form of mathematics.

But when we get to Apollonius, now well into the Hellenistic Age, the theory of epicycles was a mathematically more complicated kind of model, and wasn’t easy for more casual observers to deal with. So the kinds of people who could work with these models were, in short, mathematicians. The mathematics required to work with the model started to outstrip the mathematical ability of the philosophers. So we start to see that the philosophers of the Hellenistic Age and beyond, largely stuck to the more primitive Aristotelian model of nested spheres which was conceptually easier to understand. The model produced less accurate results than the epicycle model, but these were philosophers, not astronomers, they weren’t out every night carefully observing the positions of the planets to check.

So, when a new philosophical movement arose during the Hellenistic Age called Stoicism, it was primarily interested in ethics, metaphysics, and epistemology, and the Stoics more or less ignored the model of epicycles and held to Aristotle’s system of 54 interlocking, nested spheres. In other respects the astronomy of the Stoics was even more primitive. Some Stoics held that the celestial bodies were nourished by exhalations from the Earth and that they move around the sky to be above the points on the Earth where they will receive the most nourishment. This idea in effect rejected the idea that a theory of planetary motion was even possible at all since according to them the planets are just moving wherever they want so that they can get the most nourishment from the Earth.

At any rate, it’s around now in the Hellenistic Age that Greek astronomy had become sophisticated enough that its development really had to be left to specialists. No longer can we turn to philosophers to learn about astronomy, just as we should never turn to astronomers to learn about matters of ethics or ontology or metaphysics, and Apollonius of Perga is a useful marker for when this division became definite.

Well the other major astronomer of the late 3rd century was Eratosthenes. Eratosthenes was born around thirty years earlier than Apollonius in the year 276 BC, but lived a longer life and died around the same time as Apollonius did in 194 BC. He was born in Cyrene which was a city on the coast of North Africa, today on the eastern end of Libya. In what is by now a familiar story, as a young man he moved to Athens to study. He seems to have picked and chosen his intellectual diet there. He studied at a number of schools, including the Stoics, the Cynics, and the Platonists and in his time in Athens, some ten years or so, he developed a scholarly reputation and wrote works on a wide variety of subjects including Platonic philosophy, poetry, history, and sport. Now, I say he developed a scholarly reputation rather than saying exactly what kind of a scholarly reputation it was because he acquired a few nicknames for himself: the All-Rounder and Beta, because he supposedly had the talent to become a second-rate scholar in any field he put his mind to.

Well this assessment was maybe more coming from his intellectual enemies because Eratosthenes was respected enough and had sufficient breadth of scholarship to be tapped by the Pharaoh of Egypt at the time, Ptolemy III, to be the head librarian of the Library of Alexandria, which by this point in time had grown to become the greatest library in the world. So in 245 BC, at the age of 30 or so, he moved to Alexandria where he remained for the rest of his life.

Now, the previous head librarian was Apollonius of Rhodes, not to be confused with the Apollonius of Perga I was talking about at the beginning of this episode. There is a tale, probably apocryphal, that Apollonius of Rhodes was forced out of his position after showing a draft of his epic poem Argonautica, which tells the story of Jason and the Argonauts. According to the tale, the reception to his poem was so poor that he was forced to resign from his plum position. Now, at the time it seems that Argonautica did receive something of a mixed reception when it first came out. But there doesn’t seem to be any real evidence that he was forced out of his position as head librarian as a consequence. Given that the replacement occurred not long after Ptolemy III ascended to the throne it seems to have been more part of the general cabinet shuffles that take place any time a new leader assumes office.

And really it’s no surprise that Ptolemy III would take an interest into who was head librarian of the Library of Alexandria. At the time it was a major source of what we might call today soft power, and enlarging the library to ensure that it was the best in the world was an important project for the Egyptian pharaohs. The library itself was located in the palace complex where the pharaoh lived, so it was close to the seat of power. The pharaohs of the time were serious about supporting this project to the extent that there are records that the pharaohs would demand regular updates from the head librarian about the number of books added to the collection and whether or not this was meeting their targets. And there were standing orders that any ship arriving in Alexandria would be searched for books. Any books found had to be surrendered to the library so that scribes could copy the book and return the original to the owner.

The ultimate goal, of course, was to collect a copy of every book in the known world. But they also were very interested in ancient manuscripts and had an entire section devoted to the oldest copies of the works of Homer that they could find. Now Greek paganism didn’t have a set of holy texts like Judaism or Hinduism does. The closest thing the Greeks had to scripture was the works of Homer. Everyone read Homer, had memorized passages from Homer, and quoted Homer to find solace in the same way that a Jew or a Christian might quote one of the psalms. So the librarians were intensely interested in finding manuscripts of Homer that were as close to the original as possible and wrote extensive commentaries on the works in the same way that we see Medieval monks writing extensive commentaries on the Gospels more than a millennium later.

Now, running an organization on the scale of the Library of Alexandria was not easy in the best of times, but the demands of the position of head librarian seem to have become even more stressful towards the later part of Eratosthenes’s tenure. In 220 BC, Eumenes II, who was the king of a small state called Pergamon on the western coast of Anatolia decided to found a library that would rival the Library of Alexandria, naturally called the Library of Pergamon. And rival it did. The two libraries increasingly competed with each other for the largest collection, the oldest books, and the most learned scholars. The tremendous amount of money poured into these two rival projects stimulated a robust market in forgeries. Since neither library wanted to let any manuscript get away from them in case it ended up in the hands of their rival, their standards decreased to nothing and they would take absolutely any shred of papyrus with ink on it, even if they knew it was likely fake. But fearing the consequences of missing an original text of Homer or Hesiod, they would pay for it, label it, potentially as dubious, and store it.

The rivalry between the Library of Alexandria and the Library of Pergamon does not seem to have been a friendly one. The two libraries would poach each other’s scholars and slander each other. In fact, slander from the Library of Pergamon seems to be the origin of the story that the Library of Alexandria stole the original copies of the works of the three great tragedians, Aeschylus, Sophocles, and Euripedes. The story goes that the Library requested to borrow the original copies from the state library of Athens. Athens, in turn, required that the library post as collateral 15 talents of gold, something on the order of maybe 100 million dollars in today’s money. The library then made high quality copies of the works, but returned the copies to Athens and kept the originals, and told Athens that they could keep the collateral.

Another devious tactic that the Library of Alexandria employed to stay a step ahead of Pergamon, at least according to Pliny the Elder, was to throttle shipments of papyrus to the Library of Pergamon. At the time, papyrus was the dominant medium of written material in the Mediterranean. Since it came from the papyrus plant which grew much more easily in Egypt than in Greece or other parts of the Mediterranean, Egypt more or less had a monopoly on the export of the material. The preeminence of the Library of Alexandria over the Library of Pergamon was such a matter of national pride that exports of papyrus to the rival library were halted, or at least severely curtailed to prevent the Library of Pergamon from copying books at too high a rate. To circumvent this embargo, the Library of Pergamon began a project to find alternative writing material to papyrus and they hit upon parchment. Now it is sometimes stated that the Library of Pergamon invented parchment and this is not true. But it does seem to be the case that out of necessity they had to make big improvements in the techniques to produce parchment. And they were successful at this to such an extent that etymologically, the word parchment in English ultimately derives from the name of this town Pergamon.

So getting one over the librarians at Pergamon was what occupied Eratosthenes in his day job, at least for the last 25 years or so of his life. In 195 BC, around the age of 80 he became blind, and being loath to live without being able to read, he starved himself to death a year later.

Now it is maybe a little bit of a stretch to discuss Eratosthenes in a history of astronomy. Really he was more of a geographer, at least among his many scholarly interests. The one contribution to true astronomy that we do know about is that he invented a device called an armillary sphere. An armillary sphere is a really pretty thing, it’s a small sphere surrounded by a series of rotating rings which represent different features of the celestial sphere like the celestial equator, the tropics, and the ecliptic. By rotating the rings appropriately you can determine things like where on the horizon the Sun will rise on a particular day of the year, or what it’s altitude will be at noon.

But the contribution that Eratosthenes is best known for today is his measurement of the size of the Earth, which is maybe technically geography rather than astronomy, but I think is close enough to let slide. After all, you do need to know distances on the Earth to measure distances to the heavenly bodies, or at least you did up until the development of radar measurements in the mid 20th century.

I’ve mentioned before that the size of the Earth was at this point not totally unknown. Aristotle had simply stated that according to “the mathematicians,” whoever they were, the Earth’s circumference was 400,000 stades. Archimedes, too, had just stated that the circumference of the Earth was 300,000 stades, but didn’t even have the common courtesy that Aristotle did to vaguely cite some group of people. Very probably these results from Archimedes came about from measuring how the altitudes of stars changed in different cities. A strong candidate at least for Archimedes’s claim that the Earth had a circumference of 300,000 stades is that it came from a geographer named Dicaearchus of Messana who was in the generation after Aristotle and studied from the great philosopher. As with Eratosthenes, reducing him to be just “a geographer” is selling him a little short because his interests and work was wide and varied. But one of his more celebrated results was to measure the heights of mountains, in particular Mt. Pelion, by means of an instrument called a dioptra or diopter. I mentioned the diopter back in Episode 16 because Eudoxus used it, though probably didn’t invent it, but the basic idea was that it was a circular plane with notches at the different angles and a tube attached that you could look through. The tube could then rotate around so that you measure the angle between different objects. By measuring the angle that the top of a mountain appeared at a certain distance, and then moving towards or away from the mountain by some measured amount and seeing how that angle changed, he calculated the height of various mountains and found that they were not as tall as had generally been assumed. After all, one objection to the theory that the Earth was spherical was mountains. They’re right in front of us, you can see them with your own eyes and they’re big, so clearly the Earth is not a sphere. QED. And, strictly speaking, of course, this objection is correct, the Earth is not a perfect sphere, but by measuring the heights of mountains Dicaearchus was able to show that they were not very large relative to the size of the Earth’s circumference, so these mountains, big as they might seem to us on the ground, were not substantial deviations from the sphericity of the Earth.

At any rate, all this is to say that by the time of Eratosthenes there had been a couple of estimates of the Earth’s size and they were in the ballpark of being correct, but from our perspective they had two main problems: One, they were only sort of correct, and two, we don’t really know how they were actually made.

Eratosthenes then is notable for making the first measurement of the Earth’s size that doesn’t have these two problems. We know what technique he used and his result was really pretty good, maybe even too good.

So how did Eratosthenes do it? What he did was that he had heard reports that at noon on the summer solstice a gnomon in the town of Syene cast no shadow. In the more poetic accounts the story is that the town was known for the fact that on the summer solstice at noon the Sun would be visible from the bottom of a well. This meant that in this town at noon on the summer solstice, the Sun was directly overhead. Eratosthenes then measured the elevation of the Sun at noon on the summer solstice in Alexandria and found that it was 1/50th the circumference of a circle away from the zenith, which works out to be 7.2 degrees away from the zenith.

Now the town of Syene is far down in southern Egypt and both were on the Nile, which goes more or less almost exactly north-south. The pharaohs had been good about surveying their domain, and according to the author Martianus Capella, Ptolemy III had employed a team of professional pacers to measure the distances between all the cities in his kingdom.

So, since Eratosthenes knew how the angle of the solar elevation changed between two cities which were north and south of each other, and he knew the distance between them, 5000 stades, with some basic geometry he could work out the circumference of the Earth to be 250,000 stades. Later, though, it seems that some sources, or maybe even Eratosthenes himself, bumped up this number from 250,000 stades to 252,000 stades, probably so that it divided into 360 more nicely since 252 / 360 is 7/10ths, whereas 250 / 360 is 25 / 36ths.

So how good is this number? As always it’s a little tricky to say for sure since the length of a stade varied, but fortunately we’re on much more solid ground with Eratosthenes than we were with Aristotle. The sources that describe Eratosthenes’s measurement, in particular Pliny the Elder, say that Eratosthenes used a stade such that one schoenus equaled 40 stades. What is a schoenus you ask? It was defined to be 12,000 royal cubits, and once we get to cubits we’re on pretty solid footing since a cubit, at least this cubit, is known to be about 52.5 centimeters. So the stade that Eratosthenes used was 157.5 meters in today’s units.

Eratosthenes’s measurement of a circumference of 252,000 stades therefore works out to a radius of the Earth of 6320 km, which is remarkably only 50 km off from the true value of 6370. So Eratosthenes was only 0.8% off from the correct value!

Now I should mention that this is the consensus view of Eratosthenes’s result, but there is a minority opinion that Eratosthenes was not this close and we are using the wrong length of a stade. After all, generally speaking, the stade in the ancient Greek world was much larger, closer to 185 meters rather than 157 meters. In the early 19th century the French archaeologist Jean-Antoine Letronne compared contemporary measurements of the distances between cities to the distances recorded by the ancient Greek geographer Strabo and found a result that was, on average, consistent with this shorter value of around 157 meters. The problem, though, is that it’s not entirely clear what distances the ancient geographers used. The roads or rivers between cities were not straight lines. Were they measuring the distance as the crow flies, or the actual distance it takes to travel from one place to the next? If it was the travel distance, then when we compare to modern straight line measurements we would get an anomalously small length for the stade.

Regardless, even if the consensus view holds and the stade Eratosthenes used really was about 157 meters, it does seem to be the case that Eratosthenes did have a healthy dose of good luck to get as accurate a measurement as he did. After all, he did not actually travel to Syene to measure the elevation of the Sun on the summer solstice. As far as we know he just accepted as truth the story that the Sun was directly overhead at that location on that day. That would imply that the city of Syene is directly on the tropic of Cancer. But, in fact, it is not. Now, you might want to just look up the latitude of Syene to see this, but we have to be a little careful with these sorts of things because the tropics are not stationary, they move a little bit over time. Due to a phenomenon called nutation, the Earth’s axis of rotation wobbles a little bit. One component of this is due to the precession of the Moon’s orbit and is fairly rapid with a period of about 18 years. But there are longer term components that can be 1000s of years long. So the axis of the Earth’s rotation was a little different in Eratosthenes’s day than it is today. Back then it was 23.72 degrees and today it is more like 23.43 degrees, and on average it is continuing to decrease. So this means that the city of Syene was somewhat closer to the Tropic of Cancer than it is today, but it still wasn’t on it. So Eratosthenes’s assumption that the Sun was directly overhead on the summer solstice was not exactly correct. But when he made his measurement of the elevation of the Sun in Alexandria, he expressed the result as a fraction of a circle: 1/50th the circumference of a circle, which works out to be 7.2 degrees, whereas the correct value is closer to 7.5 degrees. Now, luckily for him, this underestimate partially compensated for his incorrect assumption that Syene was on the Tropic of Cancer. Then by bumping his result up to 252,000 stades to get a number that was more nicely divisible by 360, he happened to further cancel out this error and just so happened to end up remarkably close to the modern value.

Now, the tale that Eratosthenes leaned on a report that the Sun was visible from the bottom of a well in Syene makes for a good story, and it did mean that his investigation into the size of the Earth did not require any travel from him. But strictly speaking, it’s not necessary that Eratosthenes made these measurements on the summer solstice specifically. Later on the astronomer Cleomedes mentions that you can also measure the elevation of the Sun at noon in these two cities on the winter solstice and you will still find that there is a difference of 1/50th of a circle.

Well the last thing to say about Eratosthenes as far as his astronomy goes is that he is credited with making the first measurement of the obliquity of the ecliptic, that is, the angle of the ecliptic relative to the celestial equator, or in more modern terms, the angle of the axis of the Earth’s rotation with respect to its orbit around the Sun.

Now, in principle he could have done this by incorrectly noting once again that Syene is on the Tropic of Cancer and measuring its latitude by measuring the altitude of the north star. This would have been a little more tricky than it seems because the north star at the time was not our familiar north star of Polaris. In fact, there wasn’t really a north star at the time, the celestial north pole was about halfway between the stars Kochab and Pherkad, which are the two stars at the end of the bucket of the Little Dipper. So, without any star to look at directly, you’re just looking between two stars and this could have been a difficult measurement to make. And furthermore, of course, as far as we know Eratosthenes never visited Syene. So instead what he did to measure the obliquity of the ecliptic was a much better technique that allowed him to stay right at home. He just measured the elevation of the Sun at noon on the summer solstice and the elevation of the Sun at noon on the winter solstice. One half the difference between these two angles was the obliquity of the ecliptic. He ended up getting 23.85 degrees which is very close to the value it had at the time of 23.72 degrees, and this value was noticeably smaller than the value that the ancient Greek astronomers had previously assumed it to take of exactly 24 degrees.

So, with that, there is just one more astronomer of note of the Hellenistic Era, who comes in at the tail end of this period, and that is Hipparchus. But he is certainly a strong figure to end the era on, as Hipparchus was not just the greatest astronomer of his day, but perhaps the greatest astronomer in all of antiquity. As such, he deserves no less than to have an entire episode devoted to him, so we will have to save Hipparchus for next month. I hope you’ll join me then. Until the next full moon, good night and clear skies.

## Additional references

- Evans, History and Practice of Ancient Astronomy
- Fried & Unguru, Apollonius of Perga’s Conica Text, Context, Subtext
- Rawlins, Eratosthenes’ Geodesy Unraveled: Was There a High-Accuracy Hellenistic Astronomy?
- Fischer, Another Look at Eratosthenes’ and Posidonius’ Determinations of the Earth’s Circumference