Episode 16: The Homocentric Spheres of Eudoxus

April 18, 2022

In working on the problem of doubling the cube, Plato's friend Archytas devised an ingenious solution that involved a three dimensional curve determined by the intersection of a torus with a cylinder. Archytas's student Eudoxus then seems to have been inspired by this solution to develop the first serious model of planetary motions in ancient Greece, his theory of homocentric spheres.


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.

We left off last month on yet another cliffhanger. Plato, after disparaging the use of grubby data for astronomy in his earlier works, had, by the end of his life, seemed to recognize the importance of developing a model for the planetary motions. This may have been partly for quasi-theological reasons. Recall that the Athenian Stranger in the Laws had argued that the word “planet” was blasphemous, because its literal meaning, “wanderer”, implied that the gods were just moving about willy-nilly on the sky. In Plato’s philosophy, such arbitrariness was anathema. Plato felt that these motions had to have some deeper logic behind them, even if this logic eluded the casual observer. Now, Plato’s model of the Solar System could explain some of these motions. He could explain the daily rising and setting of the stars and planets, and, by placing the planets on additional spheres with a much slower rotation in the opposite direction from the stars, he could explain their slow eastward drift relative to the background stars.

But Plato seems to have recognized that his model was fairly limited. His spheres rotated at a constant speed, so he could not explain the fact that their motion was not observed to be constant. Even worse, sometimes the planets appeared to stop their usual west-to-east drift relative to the background stars and go backwards, in what is called retrograde motion. So Plato knew that there had to be more to the story and called upon his students to see if anyone would rise to the challenge of explaining the planetary motions. And one of them did — a man by the name of Eudoxus.

So that’s the cliffhanger we left off on from last episode, but I’m afraid you’re going to get another bait and switch at this point, because before we can talk about Eudoxus we need to talk about another philosopher who taught Eudoxus — the philosopher Archytas.

Now, if you’ve been paying close attention, you may actually recognize the name Archytas because I briefly mentioned him in last month’s episode on Plato. He was the fellow who generously bought Plato out of slavery when things went pear shaped for him over in Syracuse. But Archytas was really one of the more remarkable figures in ancient Greece and deserves a more thorough treatment than just being the guy who bailed out Plato, not least for his influence on Eudoxus.

Archytas was contemporaneous with Plato, so probably born around 435 BC, though possibly as much as 25 years later. Archytas was born over in Magna Graecia in the city of Tarentum, which is today known as Taranto in southern Italy. At the time, Tarentum was one of the most important cities in Magna Graecia. It had been founded centuries earlier as a Spartan colony and had an excellent natural port, so it became a crucial commercial hub for trade between Magna Graecia and the rest of Greece. However, we hear from two sources, Herodotus and Aristotle, that in 473 BC, about 50 years before Archytas was born, the Tarantines were routed in a battle with the Iapygian tribes to the north. The two had been neighbors for centuries and had been enemies the entire time, periodically skirmishing. As befitted a thalassocratic civilization like Greece, the Greeks in the area basically only controlled narrow strips of land along the coast. If you imagine Italy as a boot, Tarentum basically sits right where the inner part of the heel of the boot comes up and attaches to the sole, nestled in that corner. The heel itself is called the Salentine peninsula, or Salento today, and was prime real estate. Naturally the Greeks attempted to expand from Tarentum down into this peninsula and ran into the Iapygians as they did so.

At any rate, their centuries old conflict erupted once again in 473 BC during the Battle of Kailia. I mention this because Herodotus notes that this was the “greatest slaughter of Greeks ever known.” Apparently nearly the entire aristocracy of Tarentum was wiped out in this one battle. This ended up having dramatic political implications, because while there had been a popular movement to establish democracy for some time, the democrats could never achieve much success. Tarentum was, after all, a Spartan colony, and the Spartans had always been suspicious of democracy. But with the aristocracy all dead, the commoners could finally take control of the reins of power and establish democratic rule over the city. So by the time of Archytas, Tarentum had been a democracy for about 50 years.

I go into all this detail about the broader political atmosphere of Archytas’s hometown because, rather unusually for a man of his intellectual stature, having made major contributions to geometry, mechanics, and philosophy, Archytas was also a prominent political figure in Tarentum. His role in the political life of Tarentum has been compared by many historians to be similar to that of Pericles in Athens. The two were both indispensable men for their respective city-states and played an outsized role in shaping their city-states in their images.

In Archytas’s case, we have records that he was elected to the position of strategos, which was essentially the commander-in-chief, seven years in a row. Not only was his seven year command apparently unprecedented, it was unconstitutional as well. The law of Tarentum forbade any man from holding the office of strategos two years in a row, so the Tarentines had have held Archytas in extremely high regard to willingly depart from this prohibition six times in a row. Beyond this, it seems that later on he had acquired even more power for himself and become an autokrator. As this is where we derive the modern word autocrat, you might correctly surmise that it involves a single individual holding quite a lot of power. In ancient Greece an autokrator would not merely command the military, but also be entrusted with the general power to engage in war and diplomacy without needing to consult the assembly. An autokrator did not have unlimited power, and it was generally understood that he would need to get the approval of the assembly after the fact, in contrast to the more powerful role of dictator who could more or less do what he wanted for a specified period of time, usually a year.

Nevertheless, all this is to say that Archytas was a big man. In fact, given that Plato not only knew Archytas but was friends with him, it has been speculated that Plato’s fixation on the idea of the philosopher-king may have been inspired by Archytas’s own life. Here was the ideal philosopher-king in the flesh, a man who could make advances in geometry and use the wisdom gained thereof to rule his state justly. Plato seems to have met Archytas during his travels around the Mediterranean in his 40s after the death of Socrates. During this time Plato had been exposed to the Pythagoreanism that was present throughout Magna Graecia and generally liked what he saw. Archytas was typically classified by later writers as a Pythagorean himself, but because we don’t know a great deal about his philosophy, we don’t really have much independent evidence that he was a true Pythagorean. It also seems to be the case that later authors in antiquity just assumed that any philosopher from Italy was a Pythagorean.

Well, we don’t know much more about his life, but it is speculated that Archytas died in a shipwreck. The only evidence for this, however, is somewhat oblique. In Horace’s Odes, the poet writes:

The sea, the earth, the innumerable sand,
Archytas, thou couldst measure; now, alas!
A little dust on Matine shore has spann’d
That soaring spirit; vain it was to pass
The gates of heaven, and send thy soul in quest
O’er air’s wide realms; for thou hadst yet to die.

The poem in full describes a sailor coming upon the shipwrecked body of Archytas on the shore of Matine and three times sprinkling sand on the corpse to fulfill the requirement that the body be buried, for without a proper burial, a soul would be condemned to wander the Earth for a hundred years before being permitted to enter Hades. But, this is a poem written by a poet some three centuries later and it is not obvious that it is based on a literal event in the philosopher Archytas’s life. But it does speak to his stature in the ancient world that a Roman poet would feature him in one of his odes centuries later.

Well, during his remarkable life Archytas wrote some works, but we don’t actually know how many. To further impress on you his status in the ancient world, the main problem we have with Archytas is that too many works are attributed to him, and almost all of them are spurious. Today there are only four fragments that are generally regarded as authentic, though, as always in the classics, you can make a name for yourself by disputing the received wisdom and arguing that this or that work is, in fact, counterfeit.

Two of the fragments come from a work called Harmonics which dealt with the geometric theory of proportions and apparently an application of this theory to acoustics. Another comes from a work called On Sciences. The fragment we have is somewhat more philosophical in character and muses on the value of mathematics and the sciences and tries to determine the role they should play in a just society. The last fragment comes from a work called the Discourses and argues that logic is the superior form of mathematics. Now, again, as a reminder from the last episode, the word “logic” in this context is not the logic we think of today, but something closer to arithmetic. Really I should have used the more literal translation “logistic” to distinguish the two, but it is too late for the last episode so we’re just going to be stuck with it now. His rationale for this argument was that there were certain proofs which could be made using logistic or numbers more broadly, which could not be made with geometry. But overall it is rather difficult to assess the strength of this argument for the plain reason that much of it just doesn’t survive. When he says that logistic is the highest of the sciences, we unfortunately don’t know what exactly he meant by that. Mostly we have to turn to Plato’s writings and assume that Plato is using the term in the same way that Archytas was.

The general premise of the argument, though, that the philosopher should consider the various forms of knowledge and determine which of them is highest, has some similarities to the passage in Plato’s Republic that we talked about in the last episode. And in fact, Plato may have been inspired to write that passage in response to Archytas and to demonstrate that the highest form of knowledge was not logistic, but astronomy. And the overall set of fields of knowledge that Archytas identified in his argument here: logistic or arithmetic, geometry, astronomy, and music, were by the Middle Ages codified as the “Quadrivium,” the fields of knowledge that any man would need to master before he could begin to study the highest forms of knowledge: philosophy and theology.

Well, as I have alluded to, in addition to his public accomplishments, in the intellectual life, Archytas was renowned principally for his advancements in mechanics and geometry. He was apparently the first Greek philosopher to make a serious attempt at formulating a theory of optics and mechanics, but unfortunately, what exactly he did in these fields has not survived.

But his achievements in geometry have survived, or at least enough of them have to justify his reputation as a great geometer. He seems to have been largely responsible for the results in Book VIII of Euclid’s Elements. These deal with the theory of proportions. For example, from Proposition 22, we have that if three numbers are in continued proportion and the first is a square, then the third will be a square as well. So, as an example, if we start with the number 9, which is a perfect square, and then we double it, we get 18, and then if we double that again, we get 36, which is again, indeed, a perfect square.

Perhaps his greatest mathematical triumph, though, was the so-called Delian problem. The Delian problem goes like this. Suppose I give you a cube. How can you construct another cube that has twice the volume of the cube I have given you. The name comes from a little tale about how the inhabitants of the island of Delos, who were suffering from a plague and went to the Oracle of Delphi. Evidently the Oracle pitied them because unlike the usual cryptic responses, the response given by the Oracle in this case was apparently rather straightforward. The Oracle told them that the problem was that their altar was too small and needed to be twice as large. If they built this new altar, the plague would go away.

Now, the altar was a large stone cube, so the first idea the Delians had was to construct a second cube identical to the first and then just stack the second on top of the first or next to it. But this was clearly no good as no matter which place they put it the plague did not go away. Evidently they needed to maintain the cubic shape. So on their second try, they constructed a stone cube whose sides were twice the length of the original. But the plague persisted and so they realized that this solution was no good because even though the sides were twice as long, the volume was eight times as much. Clearly the Oracle was looking for a cube with twice the volume. At this, the Delians were stumped. They did not know how to construct a cube that was twice the volume of the original and went to Plato for help, who posed the problem to Archytas among others and gave it the name the “Delian problem.”

Well, much later, in the 19th century AD, the French mathematician Pierre Wantzel proved that this problem is, in fact, impossible to solve using the traditional tools of geometry, the straightedge and the compass, that had been codified by our old friend Oenopides. Incidentally Wantzel in the same paper also proved the impossibility of solving another ancient problem in geometry that had bedeviled mathematicians for centuries: that of trisecting an angle. And he published these results at the age of 23, which is fortunate because he died a mere ten years later at the age of 33. The last classic open problem from geometry, how to construct a square with the same area as a given circle, was not proved impossible for another 45 years.

So the Delian problem as it came to Archytas was an impossible problem, at least by the most rigorous standards of the day where the solution had to be limited to the use of a straightedge and a compass. But Archytas did not know that and these limitations did not stop him from finding a solution. No doubt he tried to find a straightedge and compass solution at first, but he ultimately abandoned this approach and was able to come up with an astonishing solution to the problem using solid geometry. Now, solutions using a straightedge and compass are known as planar solutions because the only operations you can do are in a plane with a straightedge and a compass. Now you might object that a cube is inherently a three dimensional object, so how could you possibly find a planar method to double the volume of a cube. But in this case the goal is simply to produce a line whose length is equal to the length of a side of a cube of the desired size. So, in principle, not knowing about Wantzel’s impossibility proof, one might suppose that it would be possible to find a planar method to double the volume of a cube.

But, if you relax these constraints and allow yourself to operate in three dimensions, it does in fact become possible to find ways to double the volume of a cube. Now, I won’t go into the details of Archytas’s solution except to say that commentators ever since have been almost uniformly been astounded at its genius. It involves rotating several figures in three dimensional space and determining the intersection of these curved surfaces and seems to have come down from the gods without any precedent. In fact it is such a singular feat of mathematical imagination that more modern historians of mathematics have variously tried to either reverse engineer how a mathematician of his day could have worked this out or argued that this was evidence that some later mathematician developed the solution and had just attributed to Archytas after the fact.

Regardless, the thing that is most important to us for the history of astronomy, and the main subject of this show, Eudoxus, in particular, is that one of these three dimensional curves that Archytas produced in his solution to the Delian problem is a curve which is today known, appropriately as the Archytas curve, and this curve was critical not only to doubling the cube, but to Eudoxus’s theory of planetary motion as well. Now given that this a podcast and I can’t just show you the curve, you’ll have to maybe close your eyes at this point and do a little visualization to imagine what it looks like. If you’re driving, maybe at this point, pull over to the side of the road. The idea is to imagine a torus with an inner radius of zero. So think of a donut that is so fat that there is essentially no hole in the middle — the inner part of one side of the donut just touches the inner part of the opposite side. Now take a cylinder with the same radius as the circle that makes up the torus and stick it through the torus, like you were putting the donut on a stick. But now, because we’re working with abstract mathematical objects rather than literal donuts on sticks, imagine that instead of sticking the cylinder through the axis of the torus, we offset it, so that one edge of the cylinder is going through the center of the torus, and the opposite edge is going through the outer edge of the torus. Then the line where the torus intersects the cylinder traces out a three dimensional curve which is known today as the Archytas curve.

Well, as I said I won’t go into how exactly you can use this curve to double the cube, but just keep this image of intersecting cylinders and tori in the back of your mind as we talk about Eudoxus.

Now I had mentioned that Archytas’s principal intellectual achievements were in geometry, mechanics, and optics, rather than astronomy. Almost none of his cosmology, if he had one, has survived. But before we move on to the main act of Eudoxus, I would be remiss if I didn’t mention one cosmological idea of his that survived and turned out to be extremely influential. This was his argument that the universe was infinite in size. His argument was quite simple. He said, suppose the universe was finite in size. What would happen if I went out, right up to the edge, and extended my staff? Surely my staff would not stop because there is nothing there to stop it. So it would extend out further than the edge of the universe. But that would mean that the new edge of the universe is where the end of my staff is. So I could move over there and repeat the process. And since I can do this indefinitely, the universe can have no edge.

Now, from a modern perspective, this is not regarded as a very compelling argument. It bakes in a number of questionable assumptions. We could, for example, imagine that there is an unbreakable wall at the edge of the universe. So the thought experiment of extending your staff beyond it then becomes impossible. Alternatively, how do we know that the staff could survive being pushed past the edge of the universe. Maybe that part of the staff is just destroyed. And, from a more modern perspective it presupposes a view of space that modern physics abandoned about a century ago. In modern physics it is perfectly possible to have a finite volume of space without any edges because space is allowed to be curved, much like the Earth has a finite surface area but no edges anywhere that you can fall off of.

So, it is possible to find issues with Archytas’s argument for an infinite universe either from the standpoint of modern physics or from a more simplistic argument that the ancients would have accepted. Nevertheless, although this argument was not universally accepted in its day — Aristotle, as you might imagine, wasn’t convinced — but it was extremely influential for centuries to come and many, many respectable astronomers did find it compelling in its simplicity, perhaps Isaac Newton chief among them.

Okay, so that is the story of Archytas and now it is time to turn to one of his and Plato’s greatest students: Eudoxus of Cnidos. Being the student of these two figures, he was, as you might expect, born about a generation after Archytas and Plato, perhaps in 408 BC. He lived to be 53 years old, so that would mean that he died around 355 BC and we are told that he flourished in the early 360s BC. As his name suggests, he was born in the city of Cnidos which was in the Doric Hexapolis in Anatolia. Over the last several episodes I’ve made much of the region of Ionia, which was along the western coast of Anatolia. In the northern portion of this coast were the Aeolians, and in the southern portion were the Dorians. In particular, there were six major city-states which had formed a federation known as the Doric Hexapolis, of which Cnidos was a member.

The details of his life are somewhat sketchy, but it seems that sometime in his youth he traveled from Cnidos to Tarentum where he was able to learn from Archytas. Then at the age of 23 he traveled to Athens to learn from Plato. Now, I’ve said on a few occasions that philosophers in ancient Greece were, as a rule, aristocrats as only aristocrats had the leisure time to spend contemplating geometry and the nature of the good life. But Eudoxus may have been the rare exception to this rule. I have not been able to find any evidence as to whether or not Eudoxus was noble, but the sources all make a point of saying that he was poor. In fact he was so poor that he could not afford to live in Athens proper and had to live in the nearby town of Piraeus instead and had to walk seven miles to Plato’s Academy every day. In the snow and uphill both ways no doubt.

He likely had expressed an interest in astronomy and his friends raised the money for him to travel to Egypt to learn their astronomy over there. We hear that he managed to acquire a letter of introduction from Agesilaus, the King of Sparta, to the Pharaoh, and was able to study with the priests for 16 months in Egypt. While he was there he seems to have learned some observational techniques and apparently made observations at Egyptian observatory near the holy city of Heliopolis.

After his time in Egypt he made his way to the city of Cyzicus in northwestern Anatolia and there began to collect a following, and at some point he and his school moved to Athens. Towards the end of his life, he returned to his native Cnidos and there entered politics. But even then he evidently could continue to make time for his astronomy as he built an observatory for himself in his hometown and the sources make a point of saying that he observed the star Canopus. Why they mention this star in particular we do not know for certain, but we can guess at a few ideas. Canopus is somewhat special in that it is very bright, the second brightest star in the night sky in fact, and at the latitude of Greece was just barely be visible on the southern horizon at certain times of the year. In fact, due to the precession of the equinoxes it’s no longer visible from Greece, but in the ancient world the fact that the star could have easily been seen in Egypt but could not be seen in the northern parts of Greece was invoked as evidence for a spherical shape of the Earth.

We are aware of four works that he wrote. Two of these, the Mirror and Phaenomena, were later combined and set to verse by the poet Aratus. This treatment was something of a blessing and a curse because, on the one hand, Aratus was evidently quite a good poet and these works became extremely popular in the ancient world. In fact, a line from Eudoxus, filtered through Aratus’s verse even makes its way into the Bible in the Acts of the Apostles. In chapter 17 of the Acts, St. Paul, during his travels throughout the known world, arrives in Athens and is distressed to find that the city is full of idols, so he makes an effort to convert the pagan Athenians and tell them of the good news of the resurrection of Jesus Christ. He is taken to the Areopagus which is the public square for debate, and then takes a rather clever tack in his evangelization. Rather than upbraiding the Athenians and preaching to them a message of fire and brimstone, telling them that they will burn in hell if they do not repent of their idolatry and believe in the true God, he begins with flattery and tells them that he admires them for being such a religious people since everywhere you go in the city there are idols and temples. But he points out to them that they themselves know that their religion is not complete because they have an altar dedicated to “an unknown God.” So he tells them that this unknown God that they have worshiped among their many gods is none other than the one true God who created the heavens and the Earth. And during this sermon he says,

The God who made the world and everything in it, being Lord of heaven and earth, does not live in shrines made by man, nor is he served by human hands, as though he needed anything, since he himself gives to all men life and breath and everything. And he made from one every nation of men to live on all the face of the earth, having determined allotted periods and the boundaries of their habitation, that they should seek God, in the hope that they might feel after him and find him. Yet he is not far from each one of us, for ‘In him we live and move and have our being’; as even some of your poets have said, ‘For we are indeed his offspring.’

This last line here, “For we are indeed his offspring” is a direct quote from Aratus’s poetic rendition of Eudoxus’s text. That this work was known centuries later to a Pharisee born all the way over in Tarsus speaks to the influence that this text had. And St. Paul’s quotation of the work in turn speaks to his unique sensitivity to the cultures he was preaching to such that he could gain converts to his bizarre foreign religion in dozens of cities across the ancient Mediterranean.

Now when I got into this I said that Aratus’s repackaging of Eudoxus was both a blessing and a curse. It was a blessing because, as I just described, its popularity ensured its survival. But the curse that came along with this was that Aratus was a poet, not an astronomer, so some of the ideas seem to have gotten a little garbled in his telling, even if it is a pleasant sounding garble. Fortunately, the later astronomer Hipparchus wrote a commentary on the poem and helped to set the record straight where Aratus took some creative liberties.

Well, in addition to the Mirror and Phaenomena, Eudoxus also may have written a text called Sphaerics which had to do with spherical geometry, which of course is an essential piece of mathematics when doing astronomy, particularly the positional astronomy that was the focus of the field from antiquity up until really the 19th century. The work doesn’t survive, but, assuming that it did exist, it would have formed the basis for a later text that was also called Sphaerics written by Theodosius of Bithynia in the 2nd century BC. We’ll get into this surviving version of Sphaerics in a later episode because it was critical for the transmission of Greek mathematics to Islamic civilization during the medieval era and then to Western Europe in the High Middle Ages after the Crusades. But that is certainly getting ahead of ourselves.

The last work of Eudoxus was also lost and is generally called On Speeds and apparently contained the main body of his theory of planetary motions. But fortunately for us, the ideas in the work survive in two other texts. One is in Aristotle’s Metaphysics which is a sadly brief treatment which gets lumped together with the later model of Callippus which we’ll talk about in the next episode, along with Aristotle’s own ideas because, well, you should know by now that he just couldn’t help himself to tell you what he thought about everything and everyone. But Eudoxus’s ideas in On Speeds also survive in a commentary on Aristotle’s De Caelo by the author Simplicius, who, as he was writing this commentary, was drawing from an earlier work describing Eudoxus’s ideas by Sosigenes, who, in turn was drawing from an even earlier author, Eudemus, who had written a text called History of Astronomy, which would have been fantastic if it had survived because it would have made my job here a lot easier, but, sadly it didn’t. So, as I said back in Episode 9, when trying to understand the ideas of the ancient Greek astronomers, we are too often reading summaries of commentaries on quotations of the original work, so we’re looking through a glass darkly.

In fact, because of this difficulty in piecing together from these various fragments what Eudoxus’s model actually was, it was unjustly neglected for many centuries. Eudoxus’s model was soon superseded by the planetary models of later Greek astronomers like Hipparchus and Ptolemy and so his original model fell into obscurity. When it was invoked it was generally in a derogatory way and commentators would criticize it on account of the large number of spheres it proposed. So it wasn’t until the 1820s AD that a German astronomer by the name of Christian Ludwig Ideler revisited the theory and recognized that it had been unjustly denigrated and that it deserved more consideration than it had been given in the past. But even then, piecing together what the theory actually was took another 50 years when the Italian astronomer Giovanni Schiaparelli fully worked out the details in the 1870s. Incidentally, this won’t be the last we’ll hear of Giovanni Schiaparelli. One day, maybe a few decades from now, when we at long last get to the astronomy of the late 19th and early 20th centuries, we will return to Schiaparelli for the pivotal role he played in the study of Mars, and in particular a feature that he discovered which he called canali, which means channels, but got rather mistranslated into English as “canals” and set off a flurry of research about the nature of these features and whether or not they had been constructed by a Martian civilization. But we will have to wait until sometime in the 2040s to hear the full story and for the time being will have to turn our attention back to Eudoxus in classical Greece.

Before jumping into his planetary model I would be remiss if I didn’t give some attention to his geometry because he was renowned for his advances in geometry as well. In fact, ancient sources regarded Eudoxus as one of the all time greats in geometry, second to none other than Archimedes. It seems that he was largely responsible for Book V of Euclid’s Elements which deals with the theory of proportions and magnitudes. This might sound a bit anodyne, but it was something of a breakthrough at the time. Really since the time of Pythagoras when Pythagorean mathematicians had discovered that the square root of two was irrational, mathematicians had been at a loss for how to deal with this troublesome numbers. The Pythagorean approach had always been to treat the integers as the fundamental units of mathematics and build the entire edifice of mathematics from combinations of integers, whether that be sums, products, or ratios. But here was the square root of two which seemed to stand outside that entire system. It could not be produced by any sum, product, or ratio of integers. Eudoxus’s insight was that this hierarchy could be inverted. Rather than taking the integers as the fundamental building blocks of mathematics, he took geometrical objects as the fundamental building blocks. With his theory of proportions, line segments of arbitrary lengths could be combined together in sums, products, or ratios. In this system the square root of two posed no particular problem at all. It could be represented with a line of a particular length just like any other, and this resolved the centuries old conundrum that had bedeviled mathematicians of the day as to what to do with these roots.

Okay, now that we have payed our homage to Eudoxus’s geometry we can at long last move on to his astronomy. Now, quite unlike Plato, it’s clear that his experience learning from the priests in Egypt had taught him the value of careful observation of the skies, so he had a much more empirical bent than any of the astronomers we’ve discussed so far with the exception of Oenopides. In addition to building his observatory in Cnidos, he also built several instruments. It’s likely that he used an instrument called a dioptra in his observatory, which is essentially a disc with a tube attached that could freely rotate, rather like the needle of a compass. By looking through the tube at a star, you could then measure the angle made with some other object. Now, the dioptra had been around for many centuries in use as a surveying instrument, and Eudoxus probably wasn’t even the first to use the tool for astronomy. In fact, if you’ll recall the episode on Pythagoras, I had mentioned that he had come from the island of Samos which had been renowned for its engineering marvel, the Eupalinian aqueduct which was a tunnel more than a kilometer long excavated from a mountain. The engineers who built the aqueduct likely used the dioptra to achieve the accuracies necessary for their work. And in astronomy, we’ll hear a little bit in the next episode about two Athenian astronomers, Meton and Euctemon, who built an observatory and probably used a dioptra for their measurements of the heavens.

But in addition to his use of the dioptra, Eudoxus apparently invented a new instrument of his own which was called the “arachne.” Unfortunately only the name survives, so we’re not totally sure what it was, but based on the name alone, which literally translates to “spider”, it may have been something called a planispheric astrolabe. This is a device that can be used to measure time based on the position of the sun. It looks like a metal disk that has a web of lines traced onto it to represent different solar latitudes at different times of the year. Then there is a dial in the center that can be turned to an appropriate date or latitude and it looks rather like a spider sitting in a web.

Okay so now we have arrived at the main show, what history has remembered Eudoxus for — his model of planetary motion. Eudoxus was the first Greek astronomer to develop a serious model of the planets. This model today is known as the model of homocentric spheres, or for those unusual scholars who have a predilection for Latin over Greek, the model of concentric spheres. The basic model is fairly straightforward and is best introduced by its model of the Moon’s motion. We know that the Moon rises in the east and sets in the west every day. But, over the course of the month, the Moon’s position relative to the background stars also changes. It slowly drifts east relative to the background stars, moving through the zodiac, until it returns to the same position in the zodiac a month later. Eudoxus modeled this motion by imagining that the Moon is attached to the equator of a sphere. This sphere rotated west to east with a rotational period of one month. But the poles of this sphere were in turn attached to another sphere, and this second sphere rotated faster, at a rate of once per day, from east to west. The combination of the motions of these two spheres then produced lunar motion that was east to west on a daily basis, but slowly, over the course of the month, moved the Moon through the zodiac and its various phases. Because these two spheres were both centered on Earth they got the name “homocentric spheres”.

But Eudoxus did not stop there. The problem with using only two spheres to describe the Moon’s motion was that the Moon does not pass straight through the ecliptic. Its orbit is somewhat inclined, so sometimes it is slightly above the ecliptic, and sometimes it is slightly below it by up to five degrees. So sometimes it is in the northern part of a zodiacal constellation, and other times its in the southern part. Now, you might think that this is no problem for Eudoxus’s model, because this second sphere describing the monthly motion could just be at a slightly different inclination relative to the first sphere that caused the daily motion. But the problem is that the Moon does not always reach its highest ecliptic latitude in the same spot every month. The constellation where this happens gradually drifts westward with a period of just over 18 and a half years. This kind of motion cannot be explained by superimposing the rotations of two spheres. So Eudoxus added a third. This third sphere has an inclination very close to that of the second sphere describing the monthly motion and has very nearly the same period. But because the period is not exactly the same, there ends up being this long term periodicity, sort of like how you can produce beat frequencies by simultaneously playing two tones of very similar frequencies.

Now, this was a crucial point that Schiaparelli had to reproduce after the fact, because the later authors who described Eudoxus’s theory actually got it wrong. Simplicius, in particular, said that the third sphere had a period of this 18 and a half years. But this is very obviously wrong, because it implies that the Moon would cross the ecliptic only every 9 years or so, when in fact it’s easy to see that it crosses the ecliptic twice a month.

So this model of three spheres for the Moon, where one corresponds to the Moon’s daily rotation, one corresponds to its monthly rotation, and one corresponds to its latitudinal variation, is the basic idea behind Eudoxus’s theory. And it actually works fairly well for the Moon. Eudoxus had exactly the same model of the motion of the Sun. Three spheres, one corresponding to its daily motion, one corresponding now to its yearly motion through the zodiac, and one corresponding to the latitudinal variation about the ecliptic. Now, if you’re only half listening to this, you might just be nodding along and thinking, “yep, that makes sense.” But actually, this last point is a bit strange. What is the latitudinal variation of the Sun about the ecliptic? The ecliptic is, by definition, the path that the Sun takes in the sky. So how can the Sun vary about its own path? We know today, that this path is caused by the Earth’s revolution around the Sun, and there is essentially no variation in this path, at least none that would have been measurable in Eudoxus’s day. So what is going on here?

To tell the truth, we don’t really know why Eudoxus believed that this third sphere was necessary for the Sun. It’s much harder to measure the Sun’s position in the zodiac than it is for the Moon or planets because you can’t actually see the zodiac when the Sun is up. Since the Moon and planets all display some variation in latitude through the ecliptic, perhaps it was natural to assume that the Sun did as well. After all, the Sun is just another light in the sky, brighter than the rest, to be sure, but surely it moves in a similar manner. Now, these later sources justified Eudoxus’s assumption that the Sun’s ecliptic latitude varies on the basis that the location where the Sun rises on the horizon during the solstices changes from year to year. This is a real effect due to the precession of the equinoxes, but it’s a very, very small effect with a period of about 26,000 years and was almost certainly outside of the observational capacities of the time. So we’re left with the somewhat unsatisfying explanation that Eudoxus probably included this third sphere in analogy with the motions he saw of the Moon and planets.

So there is Eudoxus’s model of the lunar and solar motions. His model of the motions of the planets is just a wee bit more complicated because for their motions he found that he needed to use four spheres instead of three. So, just like in the case of the Moon, we have one sphere for the daily rising and setting of each of the planets. Then there is one sphere for their sidereal period, the time it takes the planet to return to the same constellation in the zodiac. For Mars, this is a bit shy of two years, and for Saturn it’s close to 30 years. So, so far it’s just the same as with the Moon and Sun. Then Eudoxus put the poles of the third sphere somewhere on the zodiac, and then added a fourth sphere at a small-ish angle relative to the third, and fixed the planet to the equator of this fourth sphere. Eudoxus is clear that these two spheres rotate at the same rate and in the same direction. Now, part of the reason that Eudoxus’s theory of planetary motions was more or less forgotten for millennia is that it’s not exactly obvious what the motion of the planets will be due to the simultaneous rotation of these two spheres, and it wasn’t until the late 1890s when Giovanni Schiaparelli once again worked it out. Now, with modern mathematics this isn’t really too hard a problem to do and if you work it out it turns out that the planet traces a figure-eight shape called a “hippopede”, which literally translates to “horse fetter” apparently because it resembles the path that Greek horse riders would take to exercise their horses.

But before getting into what exactly that means for the planetary motion, one of the things that Giovanni Schiaparelli had to figure out was whether or not Eudoxus could have figured out that this was the path that the planet would take. Because it’s one thing to derive this curve using our modern knowledge of algebra. But Eudoxus didn’t know about algebra. So could he have figured it out using the mathematics known to him at the time? Schiaparelli was able to show that Eudoxus could have found this shape using solid geometry. In particular, this curve is given by the intersection of a sphere with a cylinder if the curved surface of the cylinder is just tangent to the surface of the sphere. This, of course, is not too different than the curve of Archytas, which is the intersection of a cylinder and a torus that we talked about earlier in the episode.

Okay, so to see what Eudoxus’s model of planetary motion is, we can imagine a planet that rises and sets on a daily basis, and also moves uniformly through the zodiac with a period of a year for Mercury and Venus and about 30 years for Saturn, with the rest of the planets being somewhere in between. Then superimposed on this uniform motion through the zodiac, each planet traces out a figure eight. So this means that the planet will have some variation in its ecliptic latitude, sometimes appearing in the northern parts of the zodiacal constellations, and sometimes appearing in the southern parts. This effect is just like we saw for the Moon and Sun. But in addition to this, there is also the other direction of the figure eight, an extra motion along the direction of the ecliptic. What this does is make the planet sometimes move faster than average along the ecliptic, and sometimes slower than average. And, in fact, if the dimensions of the figure eight are just right, not only will the planet sometimes move slower than average along the ecliptic, it will sometimes move backwards. So, at least qualitatively, Eudoxus’s model is able to explain one of the most puzzling features of planetary orbits — retrograde motion.

But how does the model do quantitatively? Now here we get a little bit stuck because we need to know three things: first, the sidereal period of the planet, how long it takes it to return to the same constellation in the zodiac; second, the synodic period, how long it takes to go from opposition to opposition; and thirdly then this angle between the third and fourth spheres in Eudoxus’s model. Now, of course, we can measure the sidereal and synodic periods ourselves, but Eudoxus never recorded the angles he used between his third and fourth spheres, or if he did, it didn’t survive. So what we have to do is basically ask, how well could his theory do, assuming we picked the best possible values for these angles. In a sense all we can do is figure out how good Eudoxus’s model could have possibly been in the best case scenario rather than knowing how good it actually was.

Well, if we go through this exercise as Schiaparelli did, it turns out that the model actually works remarkably well for Jupiter and Saturn. The maximum error for Jupiter’s motion is 44 arcminutes. For comparison, the width of your pinky finger held at arm’s length subtends an angle of about a degree, so 44 arcminutes is about three-quarters that width. Another way to think about it is that the width of the Moon on the sky is about 30 arcminutes, so the error in Jupiter’s position would have been 50% more than the width of the Moon. So, potentially measurable, but by the standards of ancient Greece, really not very far off at all. And it does even better for Saturn, being off by at most 9 arcminutes. For reference, 9 arcminutes is about the width of a penny at a distance of 25 feet, so this error would not have been measurable at all in Eudoxus’s day.

Unfortunately the theory starts to run into trouble with Mars. The best the model can do produces a trajectory for Mars that takes it 30 degrees away from the ecliptic, which is a huge amount. Even worse, no possible choice of angle between the third and fourth sphere can produce retrograde motion for Mars. But there’s another wrinkle when it comes to explaining Mars’s motion. I mentioned a while back that Eudoxus’s theory comes to us via the author Simplicius, and Simplicius separately records measurements from Eudoxus of the synodic and sidereal periods of the planets. Now, by and large, these periods are pretty close to their modern day values. But for Mars, Simplicius records that its synodic period is about 8 and a half months, whereas its true value is more like two years. So this is way, way off and any competent astronomer of the time would have known it. Nevertheless, if we play along and just stick in this super wrong synodic period into the model, we actually do get a trajectory of Mars that is pretty close to the real thing. We get retrograde motion, and the retrograde arc is predicted to be 16 degrees which is not too far off from its true value of 15 and a half on average. And the maximum deviation in latitude away from the ecliptic is predicted to be about 5 degrees, which is also about right. So Eudoxus’s model can reasonably account for Mars’s motion after all, just at the cost of using a totally incorrect synodic period. And lest you get too excited about that, there is another wrinkle, too, which is that if you use this bad synodic period, Eudoxus’s model does correctly predict how long Mars goes retrograde for, but it predicts that it goes retrograde too often. You would see three retrograde motions every cycle instead of just one.

So, even bending over backwards for Eudoxus here we have a hard time saying that his model works to explain Mars’s motion. Now, we shouldn’t be too harsh with Eudoxus for failing to explain Mars. As we will see, Mars will turn out to be a trouble child for the next 2000 years to come, and explaining its bizarre motion will be perhaps the central problem of astronomy. In modern terms, the motion of Mars was such a difficult problem because its orbit is relatively eccentric, and it gets quite close to Earth, so our nearby vantage point accentuates the anomalies produced by its eccentricity.

But, there are still two planets left, how did Eudoxus do with Mercury and Venus? In this case, because on average these planets have the same sidereal motion as the Sun, the angle between the third and fourth sphere ends up being set just by the maximum deviation these planets make away from the ecliptic. In the case of Mercury this is about 2 and a quarter degrees. This then predicts that the retrograde arc of Mercury is 6 degrees, which is a fair amount lower than its true value, but, to be fair to Eudoxus, this happens as Mercury is very close to the Sun, so it’s a very difficult measurement to make. So Eudoxus might not have known that his prediction here was off.

Venus is another story, though. Like Mars, Venus has the problem that it’s just not possible to produce retrograde motion, and this is true no matter what angle you assume between the third and fourth spheres. Even worse, from a qualitative point of view, the motion of the planet along this figure eight is uniform. This means that it’s perfectly symmetric and takes the same amount of time going from the easternmost point on the figure eight to the westernmost point as it does going from the westernmost point to the easternmost point. But this is not at all observed for Venus. It takes 441 days to go from the easternmost point to the westernmost point, but only 143 days to do the opposite. Eudoxus’s theory just cannot explain this behavior. Plus, Eudoxus’s theory would predict that Venus crosses the ecliptic four times every sidereal period, but in fact it only does it twice.

So, in sum, given the observations available to him in his day, Eudoxus’s theory does quite a good job predicting the motions of Mercury, Jupiter, and Saturn. But it fails completely for Venus and Mars. But, for the first serious attempt at explaining planetary motions, this is a pretty good start, especially considering that it will take astronomers another 2000 years to figure it out, so it is clearly not a trivial problem.

Now, as I mentioned earlier, Eudoxus’s model was discarded by later Greek astronomers who developed more sophisticated theories. But not only did they discard his theory, they also seem to have disparaged it because of its proliferation of spheres. After all, the Moon and Sun both had three spheres apiece, and the five planets had four spheres apiece, plus there was the sphere of the fixed stars, so there were 27 spheres in all in Eudoxus’s model. But I don’t think that this was a fair criticism at all. Eudoxus’s theory was really quite elegant. And it wasn’t just elegant, it was also parsimonious. Eudoxus’s model essentially only had three real parameters that had to be fit to data: the sidereal period, the synodic period, and the inclination between the third and fourth spheres. Modern astronomy by contrast requires five parameters to fit an orbit: the semi-major axis, eccentricity, inclination, longitude of ascending node, and argument of perihelion. So Eudoxus’s model is actually more constrained than that of modern astronomy and nevertheless does a decent job with the Sun, Moon, and three out of the five planets.

Now, one question that we unfortunately don’t have an answer to is whether or not Eudoxus believed his model of planetary motion to be a literal, physical model of the universe. That is, we don’t know whether he literally believed that each planet was actually attached to the equator of some sphere, whose poles were attached to another sphere, whose poles were in turn attached to a third sphere, whose poles were then attached to a fourth sphere. It’s not clear whether or not he believed those spheres to actually exist, or whether it was a just convenient mathematical model to predict where the planets would go in the sky and explain with a simple device their apparently complicated motions.

Of course, we may ask this question really of any scientific model — do we believe the model to be literally true, or just a convenient mathematical procedure for predicting the outcomes of certain kind of experiments. And really, this is not so much a question of physics or astronomy, but metaphysics. Metaphysical questions around how literally we should interpret scientific models had gone rather dormant after the development of Newtonian physics in the 18th and 19th centuries. But these questions started to get raised again in the early 20th century with the development of relativistic physics and especially quantum physics, where the models that started getting proposed were extraordinarily accurate, but had conceptual implications were frankly bizarre. Is the electron really a wave even though sometimes you can model it as a particle, or is it really a particle that sometimes you can model as a wave? Or is it really neither of those things?

In thinking about these ideas, philosophers of science repurposed a term that actually went all the way back to Plato: “saving the phenomena.” This phrase will crop up again and again, so it’s worth dwelling on it to unpack the various meanings it takes. In 20th century philosophy of science, this term came to be used as a shorthand for a more formal attitude towards scientific models. They were simply methods to predict the outcomes of experiments and told you nothing more about the fundamental reality of nature. But the original meaning of “saving the phenomena” in ancient Greek science comes from a line of Plato’s that got written down by Eudemus in his History of Astronomy, and then borrowed by Sosigenes and then commented on by Simplicius. When Plato had issued his famous challenge to his students to explain the motions of the planets, he had apparently also stipulated a constraint — that the fundamental components of the model should be uniform circular motion. The goal then, was to produce a model using this assumption that “saved the phenomena”, or with a more accurate though less literal translation from the Greek, “saved the appearances” or “kept the appearances.” Which is to say that the model would reproduce the observed motions of the planets. The model would maintain the appearances of the planets on the sky.

Well, this particular hypothesis, that the fundamental components of the model be uniform circular motions, has been fairly vigorously denigrated over the last 100 or 200 years as an irrational predilection that Greek astronomers had for the supposed perfection of the circle. The heavens were supposed to be perfect, or at least as close to perfection as could be seen in this life, and therefore had to be represented by the most perfect geometrical objects, and, the criticism goes, this dogma then blinded Greek astronomers, along with later medieval astronomers, to the obvious deficiencies of their models. Now, I think in the case of Plato himself, these criticisms may be fair. He certainly had a huge bias for what kinds of mathematical objects he deemed philosophically suitable for the heavens. But I think the criticism may be somewhat less fair for later Greek astronomers, and I want to perhaps make an effort to defend their systems of uniformly rotating circles and spheres superimposed on each other. Firstly, uniform circular motion has one huge advantage over other models — it is extremely simple to use in calculations. If modern astronomers had, in some freak catastrophe, forgotten how to model planetary motions, the first thing they would probably think of is to fit them with circular motion. And I think they would be right to do so. Any deviation from circular motion complicates the math tremendously. In fact, for elliptical motion, there is actually no way to represent in closed form the position of a planet at any given time. Secondly, models built on uniform circular motion are really surprisingly accurate. True, we know today that the orbits of the planets are elliptical, but they are really not all that elliptical. If I showed you a drawing of Mars’s orbit, whose extreme eccentricity plagued astronomers for centuries, you’d still have a hard time distinguishing it from a circle. And what’s more, we now know thanks to the work of Joseph Fourier in the late 18th and early 19th centuries that elliptical motion can be decomposed into a superposition of circular motions. So although these models made an assumption which was fundamentally wrong, in practice the models had tractable calculations and really weren’t all that far off from the truth.

All this is to say that Eudoxus was the first astronomer to develop a model of planetary motions built on top of the idea that the planets moved in circles at uniform speeds. But he would certainly not be the last. This would be the mode by which astronomers would model planetary motion for two millennia to come.

Well, Eudoxus got Greek astronomy off to a decent start by more or less describing the motions of three out of the five planets, along with the Sun and Moon to boot. But it was clear that there were deficiencies and so a student of his by the name of Callippus of Cyzicus tried to rectify these, as did another student of Plato, Aristotle. But it is late and the story of their efforts will have to wait until the next episode. I hope you’ll join me then. Until the next full moon, good night and clear skies.