At the dawn of the Hellenistic Age, two Greek astronomers developed radical new cosmologies. Heraclides of Pontica proposed that the Earth rotated on its axis and that Mercury and Venus revolved around the Sun instead of the Earth. Aristarchus of Samos went further and proposed that all the planets, including the Earth, revolved around the Sun. In addition, Aristarchus made the first quantitative measurement of the distances to the Sun and the Moon, along with their sizes.

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.

In the last episode we examined Aristotle, a figure who really stands at the culmination of Classical Greek thought. In his works, Aristotle presented the ideas from all the great intellectuals ancient Greece had to offer, and examined them all critically. He adopted the ideas he thought made sense, and, more often than not, when he thought their ideas did not make sense, he explained why. And beyond just summing up the state of knowledge in Classical Greece, he then went on and synthesized it all into a remarkably intuitive and self-consistent description of the physical world. This physical framework was so compelling that it stood more or less unchallenged for centuries in Western Europe after his works were reintroduced from Arabia during the High Middle Ages.

Now, part of the reason for Aristotle’s towering reputation as the capstone of Classical Greek thought is due to unusually good fortune. Thanks to the vagaries of historical chance, much of Aristotle’s work happened to survive whereas we are left with bits and bobs from the works of various other philosophers of his era. So our understanding of his philosophy is unusually complete. Nevertheless, what he had to offer later cultures was compelling enough that they took him to be among the greatest authorities on the natural sciences.

Well, as we learned last episode, Aristotle died within a year of Alexander the Great, and Alexander’s death traditionally marks the transition from Classical Greece to Hellenistic Greece. Thanks to Alexander’s conquests to the east, Greek culture during this period began to receive a stronger influence from the various Mesopotamian cultures, in particular the Babylonians and Egyptians, and the Greeks, in turn, exerted more cultural influence in Egypt and the Near East. The center of gravity of intellectual life in the region started to shift away from Athens in Attica, to Alexandria in Egypt, and to a lesser degree, Antioch in Syria.

Now, it is something of an understatement to say that when Alexander the Great died, he did not leave much of a succession plan. The famous story, of course, is that as he was dying from whatever illness or poison it was that had befallen him, his generals asked him to whom he would leave his empire, if, heaven forbid, he should die. Alexander replied, “to the strongest.” A story as good as this means it is likely to be apocryphal, but, again, to many of the ancient historians, whether or not the events they were describing literally happened as they described them was of lesser importance than it is to us in the modern era — what was more important was whether or not they were accurately describing the spirit and moral lessons of those events. And, in this case, Alexander’s wish that his empire go “to the strongest,” did, in fact, accurately describe the events that unfolded afterwards. Now, some have speculated that this dying wish of Alexander the Great may have also been a willful mishearing of his true intention. In the Greek, Alexander’s statement “to the strongest” is rendered “toi kratistoi.” But Alexander also had a general named Craterus, at least in the Anglicized version of his name, and if Alexander had left his empire to him, the statement would have been “toi Krateroi,” not too far off from “toi kratistoi.” And according to other authors Alexander had, in fact, nominated another of his generals named Perdiccas to take over after his death.

But whatever Alexander’s wishes were, the extremely rapid expansion of the Macedonian Empire under his command led to its extremely rapid fragmentation after his death. There hadn’t been time to institute any sort of stable political or bureaucratic infrastructure to unify the sprawling empire. So immediately after Alexander’s death, his various generals began snatching whatever provinces of the empire they could, and fighting with each other for the rest in what became known as the Wars of the Diadochi, diadochi being the plural of “diadochus,” which means “successor.” These wars lasted more or less continuously for the next forty years.

So, now that we have officially left the Classical Age and entered into the Hellenistic Age, in this episode we’ll now turn to the astronomers of this new era. But before we really get into this in earnest and looking at the stars of this show, Heraclides and Aristarchus, it would be prudent for us to take a moment and see how our old friends, the Pythagoreans, have been doing this whole time. It’s been eight episodes since the episode on Pythagoras and the Pythagorean school now, and, of course, Pythagoras himself founded his school many centuries earlier, in the 6th century BC, in the middle of the Archaic Age. But his school, or cult, was highly influential and persisted for centuries. For astronomy, the most influential Pythagorean was certainly Philolaus who was more or less contemporaneous with Socrates. Most of what we learned about Pythagorean cosmology in Episode 11 came to us from Philolaus. But by the Hellenistic Age, Pythagoreanism as an intellectual movement really starts to peter out. You may recall from that episode that during the Classical Era, the Pythagoreans started to splinter into two traditions: one was the akousmatikoi, who were primarily interested in the religious rituals and the strict rules of life of Pythagoreanism. The other group was the mathematikoi, who were more interested in Pythagoras’s intellectual program of scholarship, learning, and debate in philosophy and mathematics. Now, Plato found much of Pythagoras’s philosophy compelling, and he was not ashamed to appropriate the elements of their philosophy that he liked, which turned out to be a lot of them. So after the establishment of Plato’s Academy, many of the mathematikoi, who, by and large, were not as interested in the religious trappings of their philosophical school, migrated into Platonism.

One other motivation for the collapse of the mathematikoi sect of Pythagoreanism around the transition to the Hellenistic Era is that Alexander the Great’s conquests to the east had posed a big problem for Pythagorean cosmology. Now you may remember that Pythagorean cosmology was a little unusual in that it rejected geocentrism. But it wasn’t a heliocentric cosmology either. Rather than placing the Earth or the Sun at the center of the universe, they, or at least Philolaus, placed an object he called the Central Fire, which is what provided light and life to all things. The Sun, in this cosmology, merely reflected the light of the Central Fire. The Pythagoreans had also posited the existence of another planet, called the Counter-Earth, which was on the other side of the Central Fire to balance out the Earth’s weight and keep the universe symmetric. But of course we don’t see the Central Fire. And we don’t see the Counter-Earth either. But this was no problem for the Pythagoreans. They just said that this is because the Earth rotates around the Central Fire in such a way that one direction always points towards it, and we happen to be on the side of the Earth facing outwards, away from the Central Fire. So we never see it, just like we never see the far side of the Moon. But as we learned in last month’s episode on Aristotle, by this point Greek astronomers had quite compelling evidence from a variety of observations that the Earth was spherical in shape, and, more importantly, that it wasn’t all that large. Alexander’s empire stretched all the way from Greece to India, which was a distance that covered a substantial fraction of the Earth’s known circumference. Yet even in far-off India the Central Fire of the Pythagoreans was not observed. So the collapse of the Pythagorean theory of the Central Fire in the wake of Alexander the Great’s conquests was probably another factor in the dissipation of the mathematikoi sect of the Pythagoreans.

A smaller contingent of the other sect, the akousmatikoi, ambled along for another several centuries, but by the close of the Classical Era, individuals drawn to a rigorous, ascetic way of life had other options to choose from besides the Pythagoreans, in particular the trendier Cynics. A few centuries later the philosopher Epictetus poked fun at those who thought they could just throw on an old cloak and grab a staff, the traditional garb of the Cynic, and then start calling themselves a philosopher, just as today wags might poke fun at young undergraduates who throw on a beret and start smoking cigarettes and call themselves Existentialists.

But, just before Pythagoreanism as an intellectual force is nearly extinguished, there are two, rather obscure Pythagoreans that had some influence on astronomy in the movement’s dying days. These two are Ecphantus and Hicetas. We don’t know much about either of these figures, but they were apparently both from Syracuse in Sicily and lived during the 4th century BC, so were roughly contemporaneous with Aristotle. Both were atomists, which already separated them to some extent from the dogma of Pythagoreanism, though their atomism differed from that of Democritus and their physics was different from Aristotle’s. They held that the atoms moved by the nous, that is, by the intellect of God. Aristotle, by contrast, though he wasn’t an atomist, held that movements in the universe were due to percussion — one thing hitting another. Recall that according to Aristotle, if I throw a ball, it travels through the air because the air continually pushes it forward. But to Ecphantus and Hicetas the motions of atoms are simply guided by nous. But the atomism of Ecphantus and Hicetas is really less interesting than their major innovation — they held that the Earth rotated. We appear to see the sphere of the fixed stars rotate above us once every day, but according to them, this is but an illusion. Really it is us who move on top of the Earth, and the fixed stars live up to their name and are truly fixed — they just appear to move by virtue of our own rotation on the Earth.

Now unfortunately there’s not much more to say about the views of these late Pythagoreans because hardly anything more about them survives. Ecphantus was referenced by Aëtius and St. Hippolytus, and Hicetas is referenced in a work of Cicero that ultimately draws from Theophrastus’s History of Physics. In fact, there is some debate as to whether or not Ecphantus even existed. Ecphantus’s views are in rather remarkable agreement with those of Heraclides, about whom I’ll have much more say in a moment. So some scholars have speculated that Heraclides may have fabricated Ecphantus as an authority he could point to when he himself was proposing that the Earth rotates, and thereby siphon off some of the aura of the Pythagoreans for his own idea. Hicetas’s claim to significance took even longer to bear fruit, but was certainly even more consequential for the history of astronomy. Some 2000 years later, Nicolaus Copernicus cited Hicetas as an ancient authority who claimed that the Earth rotated when Copernicus posited that the Earth rotated about the Sun. This citation of Copernicus points to an interesting way in which the science of today differs from the science of the early modern period. Today, the emphasis in science is on novelty — coming up with an idea that no one else has thought of. Of course, you’re expected to cite prior work, but the quality of your work is inextricably tied up in its originality. But Copernicus found it helpful to cite an ancient authority who held to a similar view in order to emphasize the continuity of his ideas with those of the past, and downplay its novelty. So what Ecphantus and Hicetas lacked in surviving works, they made up for in influence.

Well, I’ve already referenced one of the two astronomers who are the focus of this month’s episode — Heraclides of Pontica. Now I opened this episode with a whole soliloquy about how we are now transitioning from the astronomy of the Greek Classical Era to the Hellenistic Era. But I might have misled you a little bit, because Heraclides lived from around 388 to 315 BC, or so, so was more or less exactly contemporaneous with Aristotle, who, of course stands at the culmination of the Classical Era. But even though Heraclides’s life just barely peaks into the beginning of the Hellenistic Era, I think it’s fair to associate him more with the astronomy of this period than with the astronomy of the Classical Era. In particular, I’ve argued that Aristotle’s work acted as a sort of culmination of Greek philosophy of the Classical Era. But although Heraclides was contemporaneous with Aristotle and must have been known to him, Aristotle doesn’t have anything to say about Heraclides’s thought, at least in any surviving work. Why there is this omission is hard to say — maybe Aristotle didn’t feel he was familiar enough with Heraclides’s ideas, perhaps because they were so new, or maybe he just had a low opinion of them and didn’t think that they were worth discussing. Or maybe he did write about them, but those works just didn’t survive.

At any rate, Heraclides of Pontica was born in the unsurprising location of Heraclea Pontica, which was a Greek colony on the southern coast of the Black Sea. Like almost everyone else we’ve talked about in Greek astronomy, he was born into a wealthy and noble family. As a young man, probably in his early twenties, he travelled to Athens to learn at Plato’s Academy, and at least according to Diogenes Laërtius, he also studied with the Pythagoreans. Like Aristotle, he stayed in Plato’s Academy for quite a long time and rose to a position of some prominence in the school. You may recall from Episode 15 that towards the end of his life, Plato had a series of misadventures trying to tutor a couple of tyrants over in Sicily. Well, during Plato’s absence, Heraclides was chosen to run the Academy. After Plato died another philosopher named Speusippus took over, but after Speusippus’s death, Heraclides was apparently in the running to run the school once again. But, sadly for him, Xenocrates beat him out. You may remember him as Aristotle’s companion when he travelled to Assos. Whether out of sour grapes or whether he just felt it was the time to retire, after this defeat Heraclides then left Athens and returned to his native land where he lived another 23 years or so, dying around 315 BC.

His contemporaries, maybe not his best friends, described him has being fat, effeminate, and pompous, and apparently they tweaked his name from Heraclides Ponticus to Heraclides Pompicus. But he was evidently a crackerjack writer although only a few fragments of his works survive. He seems to have written a great deal of dialogs, but he had a more engaging style than Plato’s dialogs. Plato’s dialogs have a fairly cursory description of the setting and the individuals talking together, and the settings and characters are essentially an excuse for Plato to write out long philosophical soliloquies, maybe punctuated by a question or objection here or there. By contrast, Heraclides packed his dialogs with action and delighted in introducing all sorts of characters, and, in particular, in reviving long dead philosophers to toss their own opinions into the mix. And he frequently digressed into tangential stories. In that way I like to think of him as the stylistic godfather of this podcast.

Well, haters gonna hate, and some of his contemporaries were not at all enamored with his writing style. Epicurus, in particular, criticized his works for being “crammed with puerile stories.” Like many of the other philosophers of this era, we know from surviving summaries of his works that he wrote on a huge variety of subjects: geometry, music, rhetoric, history, ethics, in addition to what he is best known for today, namely his astronomy. It’s really unfortunate that his works were lost because beyond just having lost a few good stories, it’s around this time that Greek astronomy, and, in particular, Greek planetary models, start to change dramatically. Up until now the only serious planetary model had been Eudoxus’s theory of homocentric spheres, along with the variants on this model proposed by Callippus and Aristotle. But it’s around this time that new ideas with more staying power start to get introduced — ideas you may have heard of like epicycles, equants, and deferents. But unfortunately, due to the lack of surviving sources around this time, it’s very difficult to assign credit to say who developed what idea.

But, just as you fight with the army you have, not the army you wish you had, we have to make do with the sources that have survived not the sources that we wished survived. In the case of Heraclides, these largely come from quotations and summaries in works by Simplicius, Aëtius, and Proclus, although some of his ideas also make it into Vitruvius’s famous treatise on architecture. For those not so immersed in the classics, Vitruvius’s name persists in the popular consciousness more strongly thanks to Leonardo da Vinci’s drawing Vitruvian Man. I’m sure you’ve seen it, it’s the drawing of a man in two positions superimposed on top of each other: one standing with his arms straight out and then another with his arms slightly up and his legs spread out. Da Vinci then inscribed a square and a circle around the man. Da Vinci’s goal in the drawing was to sketch out the ideal proportions of the human body. The provenance of the name of the drawing, Vitruvian Man, is that the Roman architect Vitruvius had, in this same treatise that was purportedly on architecture, described the proportions of the ideal human body. He claimed that if you lay flat on your back with your hands and legs extended, then your hands and feet will fall at points on a circle centered on your belly button. Likewise if you stand up with your arms outstretched, your hands, feet, and head will now describe a square since the distance from head to toe will be equal to the distance between the two outstretched hands. Or at least, that’s how it would go for the ideal human body, according to Vitruvius. Incidentally, Da Vinci wasn’t the first artist to try to make a drawing to demonstrate this property, but his drawing was certainly the best. Now, why is it that a book on architecture has anything to tell us about the proportions of the human body, much less the astronomy of Heraclides? On the proportions of the human body the connection is clear. Architecture according to Vitruvius, is nothing more than the construction of forms in right proportion to each other, so a general understanding of correct proportion is necessary for the successful architect. Of course most of De Architectura is about architecture itself, not the proportions of the human body. But Vitruvius devotes the second to last book of his treatise to describing the other subjects that the successful architect must have knowledge of. The architect has to be able to draw for obvious reasons. Geometry is also a must since it is needed to construct the various forms in a building. But Vitruvius also explains that it is necessary for the architect to be familiar with music since certain war machines that an architect would be responsible for constructing, like catapults and ballistae, need to be tuned much like a musical instrument. And astronomy, of course, has deep connections to music, it being the study of the harmony of the stars, and it also has deep connections to geometry, optics, and vision. So, at least according to Vitruvius, the successful architect needs to have knowledge of all these subjects, and Vitruvius delivers his readers with what they need to know in this ninth book. At any rate, this is how one of the important sources about Heraclides’s ideas survives. And, I should probably confess a rather delicate detail after making this lengthy digression about Vitruvius — Vitruvius does not actually name Heraclides explicitly, though it is fairly clear that one of the passages is describing one of Heraclides’s ideas. All this is to say that when we’re relying on Roman books about architecture to tell us about Greek astronomers that they don’t even explicitly mention by name, you know we’re really in trouble.

Okay, well with that caveat, what can we say about Heraclides’s astronomy? Some of it was fairly conventional for the day. He held that the universe was a god, as are the Earth, Sun, and planets. He also seems to have been more or less in agreement with Plato on his doctrine of the transmigration of souls. After death, souls journey through the heavens, in particular passing somewhere through the Milky Way, before returning to Earth for another life.

But Heraclides had two genuinely novel contributions to astronomy. The first I have already mentioned — he held that the stars were fixed and that the Earth rotated once every day. Now, true, Heraclides was probably not the first to have this idea — I just talked about Ecphantus and Hicetas who also believed this. But Heraclides was of much greater renown as an astronomer, so the idea that the Earth rotated later came to be associated much more closely with him — and, not to put too fine a point on it, it’s not so obvious that Ecphantus and Hicetas actually existed.

But regardless, the idea that the Earth rotated every day was not a completely unknown hypothesis until Heraclides sauntered into the picture. Aristotle considered this idea, but rejected it. After all, you can’t have a single rotation explain the motions of everything you see in the heavens. Sure, the rotation of the Earth could explain the daily east-to-west motion of the fixed stars, but the planets still move independently of the stars. So some things move in the heavens regardless of what the Earth is doing. And if some things are moving about in the heavens independent of the Earth, why not all of them? To Aristotle, it made more sense to couple the motions of the planets to the rotating sphere of the fixed stars.

So why did Heraclides come to the opposite conclusion? Unfortunately, due to the paucity of sources, we don’t know exactly why Heraclides held that the Earth rotated — we just know that he did. And what is more, we know that in the end, Aristotle’s criticism of the idea won the day and few subsequent astronomers held to Heraclides’s view, at least until the early modern period.

Heraclides is also responsible for one other important development in the history of astronomy, though, unfortunately, what exactly it was is very hard to say and requires a lot of close reading of the primary sources. Historians of science will spend pages analyzing the meaning of individual Greek words in these sources and sometimes arguing that this or that word probably was copied incorrectly and if we substitute another similar word we get this meaning, etc.

I won’t get into all the minutiae of these scholarly arguments, but I do want to provide a big honking caveat here that starting with Heraclides and continuing for another few centuries we start to see the development of the theory of epicycles, but due to the poor quality of the sources it’s oftentimes hard to say who exactly developed what. So take any attributions here with a grain of salt.

Well, in the case of Heraclides, what we can probably say for certain is that in addition to believing that the Earth rotated, he also held that Mercury and Venus revolve around the Sun. Now Eudoxus’s model of homocentric spheres had always had problems with Venus. It either got the position way off or predicted too many retrograde motions. Callippus’s model improved on things, but I think it is fair to say that fundamentally this model, or any model where Venus and Mercury revolve around the Earth is aesthetically unsatisfactory. After all, we just never see them get all that far away from the Sun. Venus only gets around at most 47 or so degrees away from the Sun, and for Mercury it’s only 28 degrees. But in a model where Mercury and Venus both revolve independently around the Earth there’s no reason in principle why that has to be the case. It should be possible for them to appear anywhere in the zodiac independent of where the Sun is. For Eudoxus and Callippus, the orbital parameters of Mercury and Venus just happened to take on values that kept them close to the Sun. But there was no fundamental reason why they had to have these values, it’s just how things turned out. At least to modern scientists, seeing such a huge coincidence makes the hair on the backs of our necks stand up straight. This kind of fine tuning is what a model looks like when it is desperately crying to be put out of its misery. And Heraclides’s model could have been a nice coup de grâce for Eudoxus’s model. In Heraclides’s model the fact that Venus and Mercury are always seen close to the Sun is because they actually revolve around the Sun. It’s a direct consequence of the model itself — no fine tuning of the model’s parameters is needed.

This idea had a second bonus, which is that it explained why Mercury and Venus vary in brightness. This had been a big sticking point for Eudoxus’s model of homocentric spheres and it seems that several astronomers, including Autolycus, had brought it up as an issue. After all, in this model, the spheres were homocentric, they were all centered on the same place, the Earth, so the planets were always the same distance away from the Earth. Even if you said, well, it wasn’t the distance to the Earth that mattered, but the distance to the Sun, this still didn’t explain the fact that sometimes when you saw Venus at, say, 30 degrees away from the Sun it was brighter and at other times when it was the exact same angle from the Sun it was dimmer. But in Heraclides’s model, this variation in brightness was perfectly explained. Since Venus revolved around the Sun, sometimes it was closer to the Earth, so it was brighter, and other times it was on the other side of the Sun and so was dimmer.

And the benefits of this theory don’t end there. You may remember that Eudoxus’s original model did a terrible job at explaining the motions of Venus. His successor Callippus was able to do a better job at explaining the motions of Venus, but at the cost of adding in an extra sphere. So, to explain Venus’s motion Callippus had to calculate the superposition of the motions of five separate spheres. By contrast, Heraclides could get rid of one of the spheres right off the bat because the first one was there to explain the daily east-to-west rising and setting of the stars and for Heraclides this was just due to the rotation of the Earth. Then to reproduce the annual movement through the zodiac, this was just due to the fact that Venus on average followed the Sun’s motion because it was revolving around the Sun. And then by assuming that Venus revolves in a circle around the Sun, he could explain the major features of Venus’s motions — Venus bounces between a maximum eastern and western distance from the Sun, sometimes moving in the same direction as the Sun, and sometimes appearing to move backwards. And, what’s more, this easily explained why it takes longer for Venus to go from maximum western elongation to maximum eastern elongation than vice versa.

Now, having been presented with all this evidence, to us moderners who are used to the idea of planets revolving around the Sun, it seems like a slam dunk that this is the superior theory. But in the ancient world, the notion that some planets would revolve around the Earth and others around the Sun was strange. What was the difference between them? And, after all, the Earth was the center of the universe, so why would something revolve about some other point? Circular motion about any other point was impossible in Aristotelian physics. A few centuries later the astronomer Theon of Smyrna tried to rationalize the theory for his readers by an analogy. Theon pointed out that there are naturally different kinds of centers for an object. If we take the human body, for example, one natural center is the heart, which is the center of our vital force. But in a physical sense, the center of the body is the belly button. So it may be similar with the planets that the Earth remains the physical center of the universe, but the Sun acts as the center of their motions, at least for Mercury and Venus. At any rate, the fact that Theon felt the need, centuries later, to provide this intuition as to how it could be that the planets could revolve around a point which was not the physical center of the universe should be an indication that this was not an intuitive idea at the time.

Now, so far, what I’ve presented seems to be a fairly uncontroversial characterization of the viewpoints of Heraclides. But at this point the debates really start to begin in earnest. Some historians, the Italian astronomer Schiaparelli, whom we met in the episode about Eudoxus among them, believe that Heraclides went further than just saying that Mercury and Venus revolve around the Sun and claimed that all the planets except Earth revolved around the Sun. The evidence on this point seems to be ambiguous, but it’s not so hard to see how he may have made the leap from saying that Mercury and Venus revolve around the Sun to saying that all the planets revolve around the Sun. After all, one of the major pieces of evidence in favor of his theory was that it explained the variations in the brightness of Venus. But Mars also varies in brightness, and these variations aren’t exactly random. Mars is always at its brightest when it’s at opposition. Now, this isn’t really a problem for Eudoxus’s model in the same way that Venus’s variation in brightness is. The problem for Venus was that it would have different brightnesses at the same angle from the Sun, whereas this isn’t the case for Mars. But, if the variations in Venus’s brightness could be explained by making Venus revolve around the Sun, you can see how it’s not too hard to make the leap to saying that Mars’s variations in brightness are also due to it revolving around the Sun, even if this isn’t a problem for Eudoxus’s model per se.

However, there is a qualitative difference between the inferior planets, Venus and Mercury, and the superior planets. In the case of Venus and Mercury, the Earth is completely outside their orbit. But in the case of the superior planets Earth would be on the inside of their orbit around the Sun. This distinction of whether the Earth is inside or outside the planet’s orbit may seem rather trivial to us, but to the ancient Greeks it represented two competing planetary models: the eccentric hypothesis and the epicycle hypothesis. Ultimately the epicycle hypothesis came to win the day, so this idea is more familiar to us. The idea of epicycles is just that rather than revolving around the Earth directly, a planet revolves around some point in space in a small circle, and then this point revolves around the Earth. In the case of Mercury and Venus, this point they revolved around was the Sun. In the eccentric hypothesis, the planet itself revolved around the Earth, but in a circle that is somewhat offset from the Earth. This central point then revolved around the Earth. So in the epicycle hypothesis you have a large circle centered on the Earth and then the planet revolves around a smaller circle centered on a point on that larger circle. And in the eccentric hypothesis, it’s the opposite — you have a small circle centered on the Earth and then the planet revolves in a large circle around a point on that smaller circle. If you have some trouble distinguishing between these two models, that’s okay. It turns out that these two models are mathematically equivalent to each other and this observation of their equivalence was probably an important step towards the development of the theory of epicycles. This proof was due to a later astronomer by the name of Apollonius of Perga and I will have much more to say about his astronomy in next month’s episode. But although Apollonius is sometimes credited with originating epicycles, to some scholars, the way he writes about them suggests that his audience was probably already familiar with the idea, so the basic idea of epicycles may have come earlier, and, some have argued, gone all the way back to Heraclides of Pontica.

But Apollonius lived during the end of the 3rd century BC and we are still at the end of the 4th century and going into the 3rd century, so before we get to Apollonius, there is one other astronomer of towering importance that we have to spend some time with: Aristarchus of Samos. Aristarchus was strongly influenced by his teacher, Strato of Lampascus, so it would be no good unless I said a few words about him, too. Strato was born in 335 BC, so he would have been a young boy when Aristotle died. Nevertheless, he grew up to be the third head of Aristotle’s Peripatetic School after Aristotle’s student Theophrastus died. Strato’s astronomy was fairly conventional. His only notable views were that he believed that the stars reflected light from the Sun and that comets were stars in thick clouds, like a torch in the fog. But although there’s not much to say about his astronomy he is very notable for his physics. Despite heading up Aristotle’s philosophical school, Strato tossed aside many of the dogmas of Aristotle’s physics. At a fundamental level, he objected to the teleological nature of Aristotle’s physics. You may recall that in Aristotle’s physics the motions of objects are explained by resorting to some interior goal of the object. So rocks, composed of the element earth, tend to settle to the center of the universe simply because it’s in their nature to do that — nothing else has to cause a rock to fall. If the Earth were not present, a rock would simply drift naturally towards this special spot in the universe. Its motion is determined entirely by its interior disposition, so to speak. But Strato removed the teleology and instead placed at the foundation of his theory of motion the idea that heat provided the motive force for matter. Strato had a place for God in his universe, but his theology was also rather deistic. His idea of God was that it was a sort of blind, purposeless entity that causes the universe to move forward in time, almost as if he just promoted the idea of “the laws of physics” to be a god.

One of his more interesting positions was on the motion of falling bodies. Aristotle had held that the speed of a falling body is constant. But Strato rejected this tenet of Aristotle’s as well, and rather remarkably for the age, supported his own position that falling bodies accelerate by pointing to an observation. He noted that if you stand at a height and pour a thin stream of water from a jug, it will start off as a continuous stream, but at some point break up into droplets. Strato correctly inferred from this that this must mean that the water accelerates as it falls, since the distance between one droplet in the stream is increasing relative to neighboring parts of the stream. If bodies fell at a constant speed as Aristotle held, the stream would remain continuous since any droplet would remain at a constant distance relative to its neighbors. Given his relatively mechanistic worldview and his penchant for using observations of specific physical experiments to support his theories Strato is often recognized as the spiritual father of physics.

Well, Strato had another accomplishment of note, at least for our purposes, and that was being the teacher of Aristarchus, one of the greatest astronomers of ancient Greece. Aristarchus was about 25 years younger than Strato, born around 310 BC and lived until 230 BC. His teacher’s rejection of Aristotelian dogma may have encouraged his own independence of thought, and Aristarchus turned out to be a highly original astronomer. That said, although he is really only known for his astronomy today, during his own time he was renowned for much more. Vitruvius, in his book about architecture, said that there were just a few men who had achieved a broad mastery of learning and Aristarchus was one. Incidentally Vitruvius also singles out a few other figures we have met, namely Archytas and Philolaus, and also a few more that we will touch upon in the future: Eratosthenes, Archimedes, and Apollonius.

Well, only one work of Aristarchus survives, called On Sizes and Distances, but it is a true masterpiece of astronomical reasoning and I will walk you through it in just a bit. Unfortunately, seeing this great work survive makes us regret all the more the other works of his which were lost. We know that he wrote a book on optics, and there was at least one other work of his on astronomy which was also lost, but was quoted by Archimedes. This quotation by Archimedes appears in a rather delightful work called the Sand-Reckoner. It reads like something that could have come straight out of a modern order-of-magnitude physics class. The work poses the question “How many grains of sand could we fit in the universe?” To answer this question, Archimedes had to do two things. First, he had to figure out how large the universe was, and second, because the universe turns out to be pretty big relative to a grain of sand, he had to figure out a way of expressing very large numbers. I outlined the Greek numeral system back in Episode 10 about the Ionian School, but back then we were still focused on the Archaic Age, and by the Hellenistic Age the numbering system had seen a few improvements. You might recall that the basic strategy of the Greek numbering system was to use the letters of the alphabet as numbers. So, the first letter of the alphabet, alpha, represented the number 1, the second letter, beta, represented a 2, and so on up until you got to iota, which represented the number 10. Then the next letter, kappa, represented 20, lambda represented 30, and so on up until you got to rho, which represented 100. Then the numbers start incrementing by hundreds, so the next letter, sigma, represents 200, tau is 300, and so on. Now, those of you who are intimately familiar with the Greek alphabet and are doing some math in your head may realize that there’s a little problem here. I said that iota represents the number 10, but iota is only the ninth letter of the Greek alphabet. And, more to the point, the Greek alphabet has only 24 letters, but if you want to represent numbers all the way up to 1000, which the Greeks did, you would need 27, so the Greek alphabet would be three letters short. Their strategy to solve this was to retain a few obsolete characters that had originally existed in the Phoenician alphabet, but which the fallen out of use since the Greeks didn’t have any phonetic use for them. So the number six was not represented by the sixth letter of the Greek alphabet, zeta, but was instead represented by an older character called digamma, and then all the later characters got pushed back one, so zeta represented seven, eta eight, and so on. Then after pi, which represented 80, the Greeks used an old letter called koppa, not to be confused with kappa, to represent the number 90. And at the end of the sequence, after the last letter of the alphabet, omega, the Greeks used another obsolete character called san, or sampi.

So this system then allowed the Greeks to write numbers all the way up to 999. This worked well enough during the earlier parts of Greek history, but certainly by the Hellenistic Age people sometimes found it necessary to deal with numbers larger than 999. The first solution that evolved was to repeat the first nine letters but add a tick mark to the upper or lower left. So an alpha with a tick represents 1000, a beta with a tick represents 2000, and so on. Later on, once numbers bigger than 9,999 were needed, they switched to a different system. They repurposed the capital mu, which looks like a Latin M, to represent the number 10,000 or myriad. To get numbers beyond this, the Greeks would then write a lower case number above the M, which represented how many multiples of 10,000 there were. So a beta above the M would be 20,000, a gamma above the M would be 30,000, and a sequence of sigma lambda delta would be 234 myriads, or 2,340,000.

This system worked well enough for everyday uses, but it was not good enough for astronomy. Aristarchus, in particular, in his work On the Sizes and Distances, at one point had to represent the number 71,755,875. It was a bit cumbersome to fit all these digits above the M, so Aristarchus introduced an innovation where he wrote them to the left hand side. So this number was represented by zeta with a tick, rho, omicron epsilon, then the M, or really capital mu, epsilon with a tick, omega, omicron, and epsilon.

This was all well and good for Aristarchus, but it still wasn’t going to work for Archimedes, who was trying to estimate the number of grains of sand that would fit in the universe. A myriad is a large number, but a myriad grains of sand will still more or less fit in the palm of your hand. The whole universe will need a whole lot of myriads. And actually, Archimedes assumed an exceedingly small size for a grain of sand. He assumed that a myriad grains of sand would fit into a poppy seed. So, however many poppy seeds would fit in the universe, you would have to multiply that by 10,000 to get the number of grains of sand. At any rate, just writing numbers to the left of a capital M clearly wasn’t going to cut it, so Archimedes devised an ingenious technique to represent larger numbers. He basically took advantage of the fact that you could use Aristarchus’s technique to represent numbers up to, though not including, 10,000 squared. So the largest number in Aristarchus’s system was 99,999,999. Archimedes called these first order numbers. Archimedes then defined second order numbers to be ordinary numbers times a myriad squared. The second order numbers would then take you up to one less than $10^{16}$. Likewise he defined third order numbers to be ordinary numbers times $10^{16}$. Now for the purposes of estimating the number of grains of sand that can fit in the universe, this procedure is more or less sufficient. You need to go to a few higher orders, but you get the idea of what’s going on. In the end, Archimedes estimated that you needed an eighth order number to count the number of grains of sand in the universe, which works out to be around $10^{63}$. This is a vastly larger number than anyone had worked with in Greek mathematics up until this point. But having started a good thing going, Archimedes was not going to stop now, so he then went on to define the notion of a period. A period is the myriad order number, or 100 million to the power of 100 million. This is a stupendously large number, and really there’s no need for numbers this large for any physical quantities, but Archimedes wanted large numbers and that’s exactly what he got. So any number up to 100 million to the power of 100 million is a first period number. But then the numbers from there up to this number squared, or 100 million to the power of 200 million, are second period numbers, and then numbers up to 100 million to the power of 300 million are third period numbers, and so it goes. This gets you all the way up to the 100 millionth period, which works out to be 100 million raised to the power of $10^{16}$, or a 10 followed by 80 quadrillion zeros. You would be hard pressed to find a use for numbers any larger than this. In essence, Archimedes’s system is the modern positional number system restricted to three digits, but using a base 100 million numbering system rather than a base 10 numbering system like we use today.

Well, as I said, these mind-bogglingly large numbers were not actually necessary to solve the problem of estimating the grains of sand that could fit in the universe, it was just for fun. But because his ultimate goal in this text was to estimate the number of grains of sand that could fit in the universe, Archimedes also needed some estimate of the universe’s size. And for this he turned to Aristarchus. And it is here that we get the clearest description of Aristarchus’s cosmology. By far the most notable feature of this cosmology is that Aristarchus was the first astronomer to propose a heliocentric system. He claimed that the Sun, not the Earth, was at the center of the universe, and all of the planets, including the Earth, revolved around the Sun.

Now, why exactly Archimedes brought up Aristarchus’s heliocentric system to get an estimate for the size of the universe is actually a little unclear because Aristarchus never provided an estimate of its size, and anyway the fact that the system is heliocentric is not really relevant to getting a size. But, given that it is certainly the clearest description in the sources, and one which was written only a few decades after Aristarchus, we as historians of science can be glad that Archimedes recorded Aristarchus’s hypothesis anyway. What Archimedes basically did to get an estimate of the size of the universe was to misread Aristarchus’s cosmology, probably deliberately, and insert a crucial assumption that the ratio between the size of the Earth’s orbit to the Earth’s diameter is equal to the size of the sphere of the fixed stars to the size of the Earth’s orbit. Now, again, strictly speaking it’s not necessary to assume a heliocentric model to make this assumption, but it seems that in his original, lost text, Aristarchus had worded his description of the universe in such a way that Archimedes could sophistically read the words with an interpretation that suited him.

At any rate, what is not in question is that, independent of Archimedes’s motivations for citing him, cite him he did, and we can be sure that Aristarchus did in fact hold that the Earth revolved around the Sun. Archimedes’s paragraph in the Sand-Reckoner is a very clear description of Aristarchus’s heliocentric hypothesis, and it’s also very reliable since Archimedes knew Aristarchus well and was writing a short time after Aristarchus’s death, at most about 25 years, but it’s not even the only source. There are other sources which corroborate Aristarchus’s fantastical idea that the Earth we all stand on is somehow moving about the Sun.

Aristarchus also took up Heraclides’s idea that the Earth rotates on its axis, which meant that the fixed stars really were fixed in his theory. And given this, he also noted that the Earth’s axis of rotation was tilted relative to the axis of revolution around the Sun, and was able to correctly identify this as the mechanism of the seasons — why it’s hot in summer and cold in winter.

Unfortunately, because the original work in which Aristarchus put this idea forward has been lost, we don’t know why exactly he proposed a heliocentric theory. It’s certainly the case that a heliocentric system would have provided a better model of planetary motions than Eudoxus’s model of homocentric spheres. But as I mentioned earlier, Apollonius later effectively proved that the heliocentric model is mathematically equivalent to the theory of epicycles.

For whatever reason, though, Aristarchus’s heliocentric theory never caught on. For a long time many historians of science held that this may have been because Aristarchus’s idea that the Earth moved was seen as blasphemous. But more recent scholarship tends to reject this interpretation as an anachronism. Now, the idea that the Greeks considered Aristarchus’s hypothesis of heliocentrism to be blasphemous is not entirely implausible. After all, in the Platonic system, the Earth was the Hearth of the House of the Gods, represented by the goddess Hestia, and in a rightly ordered universe, all the gods would revolve about the hearth of their house. As everyone knows, of course, the fireplace in your house does not move. The philosopher Dercyllides wrote that “we must reject with abhorrence those who have brought to rest the things which move, and set in motion the thing which by their nature and position are unmoved.” There is another passage in particular, from Plutarch, which, in most renderings, claims that another philosopher who was contemporaneous with Aristarchus, named Cleanthes, had stated that Aristarchus ought to be charged with impiety for his claim that the Earth moved around the Sun.

However, historian and physicist Lucio Russo has argued that this statement that Cleanthes called for Aristarchus to be charged with impiety is not in the original manuscripts, and instead originates from a 17th century publication of Plutarch’s works. According to Russo, in the original manuscripts, the Parinisus B and E codices, the grammatical tense on Cleanthes and Aristarchus is switched so that in the original it is in fact Aristarchus who is accusing Cleanthes of being impious. In Russo’s reading this accusation was not serious — Plutarch’s character who is telling this story is laughing as he does so — but it fit the context because Cleanthes had regarded the Sun to be divine and the source of all life — in short, the Hearth of the Gods. But in a geocentric model, claiming that the Sun revolved around the Earth was, in Cleanthes’s theology, to claim that Hestia moved, which was blasphemous! So in this reading, Aristarchus seems to be cleverly pre-empting any claims that his heliocentric model is blasphemous because it caused the Earth to move, by pointing out that the logic of Cleanthes’s theology would lead to the same charge of impiety. But the subtlety of this reading seems to have been lost on the 17th century publishers who were responsible for widely disseminating Plutarch’s works. In the aftermath of the trial of Galileo Galilei the heliocentric theory was indelibly associated with heresy in their minds, so when they came upon this passage, they assumed that the scribe who wrote it had made a simple grammatical error and had accidentally switched who was the accuser and who was the accused. Trying to be helpful, the philologists corrected this supposed mistake, and later publishers have perpetuated this modification down through the centuries. This might sound a little bit conspiratorial, but it is often the case that philologists, editors, and publishers have to make educated guesses about mistakes or omissions in the original manuscripts to come up with something publishable.

At any rate, all this is to say that there is no conclusive evidence that later Greek philosophers rejected Aristarchus’s heliocentrism on the grounds that they considered it to be blasphemous. However, for whatever reason, his idea was still not especially popular. We only know of one other astronomer in the ancient world who accepted it, Seleucus of Seleucia, who was alive in the middle of the 2nd century BC, but that is it. It seems that the fact that the great astronomer Hipparchus rejected it about a century later dealt it a death blow, because Hipparchus’s geocentric system matched the data so well that later Greek astronomers had no real reason to reject his geocentric model whole cloth and only tweaked it here and there when the model’s predictions started to drift away from the observations. But with that said, had Hipparchus adopted Aristarchus’s heliocentrism, he likely could have produced a simpler model of equivalent accuracy. But unless some new document is unearthed in a ruin in the Mediterranean somewhere, we will probably never know why Aristarchus proposed this idea and why Hipparchus, along with almost everyone else, rejected it.

Well, I teased Aristarchus’s one surviving work, On the Sizes and Distances when I started talking about him and now it is time to give this great work the exposition it deserves. The basic goal of this treatise was not modest. It was nothing less than to establish mathematically the sizes of the Moon and Sun along with the distances to them. Aristarchus’s method was ingenious. He noted that when the Moon is at first quarter, that is, when exactly half of the Moon is illuminated by the Sun, this means that the angle from the Sun to the Moon, to the Earth, is exactly ninety degrees. So, if you draw lines between the Earth, Sun, and Moon, you have a right triangle, with the line between the Sun and the Earth being the hypotenuse. Now, if, at this moment, you measure the angle between the Sun and the Moon, this completely determines the ratio between the distance to the Sun to the distance to the Moon. And, in particular, Aristarchus took the angle between the Sun and the Moon at first quarter to be 87 degrees, which meant that the distance to the Sun was about 19 times larger than the distance to the Moon.

This was already an impressive derivation, but it still has the problem that it just gave you relative distances. You still didn’t know how far away the Moon was in any practical units like meters or miles, or less anachronistically, stades. Now, I discussed in the last episode how Greek mathematicians at the time had an estimate of the Earth’s size that wasn’t too far off from the truth, maybe about 50% too large. But if he could relate the size of the Earth to the size or distance to the Moon or Sun, he would be in business.

To get at that, Aristarchus looked to the appearance of a lunar eclipse. It had been well established by this point that lunar eclipses occurred when the Moon passed into the shadow cast by the Earth. By timing how long it takes the Moon to first enter into the shadow and move entirely into the shadow and then how long it took to get out of the shadow, Aristarchus had estimated that these two phases take an approximately equal amounts of time. In other words, the time it took for the light to be extinguished at the beginning of the eclipse plus the time it took for the light to reappear at the end of the eclipse was about the same as the time the Moon spent in its fully eclipsed phase. This then meant that at the Moon’s location, the width of the Earth’s shadow was about twice as wide as the Moon itself. By then combining this observation with a measurement of the angular diameter of the Moon and Sun, along with his earlier calculation that the Sun was 19 times further away than the Moon, Aristarchus could then set up a geometrical relationship between the positions and sizes of the Earth, Moon, and Sun, at this point in time and calculate the distance and size of the Moon and Sun in terms of the Earth’s radius. And, since he knew the size of the Earth’s radius in stades, this allowed him to make the first quantitative measurement of the distances and sizes of the Moon and Sun. Unfortunately the geometry of this step is a little more complicated than just a single right triangle, so the podcasting medium is not especially conducive to walking you through it, but if you have the inclination it is an enjoyable geometry problem to work it all out for yourself.

So, how did Aristarchus do? Ultimately he came up with the Sun’s radius as being about 6.7 times the radius of the Earth, and the radius of the Moon as being 35% the radius of the Earth. What are the correct values? Well, the Moon isn’t so far off, the correct value is 28% rather than 35%. But the Sun is way off, it’s really more like 109 times the size of the Earth rather than 6.7. And the distance to the Sun that he got was 380 Earth radii which compares really quite poorly to the true value of 23,500. So, even by the rather loose measurement standards of astronomy, the results are… not great.

So what went wrong? Well Aristarchus’s methodology was perfectly sound. The problem was that the numbers he used as his inputs were way off. Of course, we already know that the estimate of the Earth’s size at the time was about 50% too large. But that’s not the real issue here because these errors are large even when you’re just looking at the ratio between the sizes and distances to the Earth’s radius, so there are still big errors independent of whatever you take to be the size of the Earth. Instead, the main problem is that Aristarchus took the angle between the Sun and Moon at first quarter to be 87 degrees, whereas its true value is 89 degrees and 50 arcminutes. That might not sound like a big difference, but it’s somewhat deceptive, because in the calculation, as this angle approaches 90 degrees, the ratio between the Sun and Moon’s distance diverges to infinity. So those last few degrees make a big difference in the derived distance to the Sun.

Now, I was careful to say that Aristarchus took the angle to be 87 degrees rather than to say that Aristarchus measured the angle to be 87 degrees. We have no evidence that he actually measured this value. In fact, we can be fairly certain that he was well aware that the other number he used, the angular diameter of the Sun and Moon, was wrong. He used 2 degrees for his calculations when the correct value is half a degree. It’s a rather bizarre mistake to make that he used a value of 2 degrees because, first of all, unlike measuring the angle between the Sun and the Moon at first quarter, measuring the angular diameter of the Sun or Moon is really very easy to do. And, in point of fact, in Archimedes’s The Sand Reckoner, he explicitly says that Aristarchus had measured the angular diameter of the Sun to be half a degree. So what gives? Why did he use the wrong value here when another source indicates that he was well aware of the correct value? Why bother going through these detailed geometrical derivations just to use such inaccurate values in the end?

One theory is that there are just some typos. When Aristarchus says that the angular diameter of the Sun is 2 degrees, he doesn’t actually say, “2 degrees,” that’s just the modern rendition. What he literally says is that it is 1/15th of a sign of the zodiac. So, some have supposed that what he meant to say was that it subtends 1/50th of a sign of the zodiac, which is 0.6 degrees, pretty close to the true value of half a degree. But this still doesn’t explain why Aristarchus quotes him in a different context using the correct value. As for the angle between the Sun and the Moon at first quarter, it is certainly forgiveable if he measured 87 degrees since this is a very difficult measurement to make accurately.

But the general consensus about these inaccuracies is that Aristarchus just wasn’t all that interested in the specific numbers. If you read the original text, it doesn’t really read like an astronomical treatise. It basically reads like a geometrical text — it would fit right in with Euclid’s Elements. Aristarchus sets out some some assumptions and definitions and then goes through and derives a series of geometrical theorems. The thought is that Aristarchus wasn’t so interested in what we moderners would consider astronomy, he was more interested in solving a fun problem in geometry. The fact that this problem happened to correspond to a real physical system was almost incidental. So it was not so important to use accurate numbers in calculating the distances, because getting accurate distances wasn’t really the goal. The goal was to devise a method by which those distances could be obtained, in other words, to solve a particular problem in geometry, and Aristarchus succeeded at this.

The other thing to keep in mind when reading a work like this is that mathematics at the time of Aristarchus had no notion of decimals or irrational numbers or trigonometry. Once you’ve made the observation that the Earth, Sun, and Moon form a right triangle at first quarter, finding the ratio of the Sun’s distance to the Moon’s distance is easy with modern trigonometry — you just need to calculate the secant of 87 degrees, or 89 degrees 50 arcminutes if you’re using the correct value. But there was no secant or sine or cosine in Aristarchus’s day, and even if there was, he didn’t have a calculator he could just plug the angle into and get out the answer. So making the observation that the system formed a right triangle was just the first step for Aristarchus. Aristarchus then had to develop detailed geometrical arguments to figure out the ratio between the leg and hypotenuse of this triangle, and these geometrical arguments only worked for particular angles. Even then, he was not able to arrive at any exact numbers. Without a system of decimals, all he could do was provide an upper and lower bound on the various numbers he derived. So rather than saying that the ratio between the Sun and Moon’s distance was 19.11, the best he could do with the methods available to him was to say that it is somewhere between 18 and 20. And when he is calculating the ratio between the Sun’s radius and the Earth’s radius, he says that this ratio divided by this ratio minus one is greater than the rather implausibly precise ratio 71,755,875 to 61,735,500. So you can see why Aristarchus had to develop a new technique to represent large numbers.

Well, Aristarchus had a few other contributions to astronomy, he developed an improved sundial and attempted a more precise measurement of the length of the year. But these were relatively minor compared to his heliocentric theory and his method to measure the sizes and distances of the Sun and Moon. Had later astronomers taken up Heraclides’s and Aristarchus’s ideas with more enthusiasm it’s entirely plausible that the standard model of the Solar System going into the Middle Ages might have been a heliocentric model rather than a geocentric model. But it was not to be.

In the next episode we will look at two subsequent astronomers, Eratosthenes and Apollonius of Perga, one of whom continued Aristarchus’s project of combining geometry with measurement to estimate the size of astronomical bodies, though with more precision than Aristarchus; and the other, who repudiated Aristarchus’s heliocentrism and sent Greek astronomy, and thereby all of Western astronomy, down the path of the epicycle model for some 17 centuries. I hope you’ll join me then. Until the next full moon, good night and clear skies.

Additional references

• Heath, A History of Greek Mathematics
• Russo, Lucio, The Forgotten Revolution
• Zeller, Outlines of the History of Greek Philosophy