One of the most disconcerting theories in the history of science was proposed for the first time in the Alexandria of the third century BC. The theory was so humbling in its implications for humanity’s place in the cosmos that even the man who came up with it may have been tempted to disown it, so at odds with religious doctrine that to repeat it would come to be regarded by later generations as blasphemy of the blackest stripe, so repellent to all of our natural human sensibilities that it would take the world as a whole almost 2000 years to fully accept it, long after the evidence that it was in fact the case had grown into a veritable mountain. Such was the theory of heliocentrism.
To call the astronomy that existed prior to the appearance of this bold new theory “primitive” would be wildly unfair. The fact is that the average person living earlier than the third century BC knew far more about the movements of the sky than most of us do today — for, living closer to the rhythms of nature as they did, they rightly considered such knowledge essential to their survival. As early as 5000 BC, the Mesopotamians were devising star maps and instruments for studying the sky in the service of an eminently practical cause: the cycles of changes they saw occurring above their heads could, they had learned, tell them when to plant and when to harvest their crops to the best effect, thereby aiding them immeasurably in their never-ending struggle to grow enough food to support their burgeoning civilization, the first of its kind in the world. Gazing up at the heavenly bodies whose perambulations across the sky dictated so much of their lives, the Mesopotamians extrapolated, reasonably enough, that the heavens must be in motion above a stationary platform on which they stood and observed them. It was equally self-evident to them that the gods deigned to pass signals to mortals via the movements of those mysterious illuminated spheres.
As with the world’s first civilization, so with its second. Egyptian religion too came to a large extent to be an exercise in star-gazing. Indeed, the priests of Egypt became the most accomplished of all astronomers of early ancient times. Already by the time they built the Pyramids of Giza around 2500 BC, the Egyptians had devised a sophisticated calendar by watching the heavens. Each new year began with the first appearance of the star we call Sirius, the brightest of them all. (This event occurs in the Mediterranean region during what we now consider to be the middle of summer.) The year that followed consisted of the familiar 365 days, divided into the familiar twelve months; the Egyptians even understood the need to add an extra day to every fourth year to keep their calendar in sync with Sirius. In time, the Egyptians may have become the first people to realize that the Earth is round, just like most of the other heavenly bodies; this would have constituted the first step toward cosmic egalitarianism. As such, it may have been almost as controversial in its time as the later theory of heliocentrism.
We believe that Pythagoras, the Greek mathematician and philosopher whose work in geometry would later be codified and expanded upon by Euclid, visited Egypt at some time around 540 BC and talked with the land’s star-gazing priests. Then he returned to his city of Croton, an outpost of Greek culture near the toe of the boot that is modern Italy, to argue in writing for a spherical Earth. As proof, he noted how the round shadow of the Earth could be seen on the face of the Moon during a lunar eclipse, and how one could see terrestrial features that were farther away from a higher perch on the Earth’s surface than one could from a lower one, implying that it was the curvature of said surface rather than the range of the human eye itself that was the limiting factor. By the time of Socrates, Plato, and Aristotle, the fact that the Earth was a globe was taken as a given.
But none of these Three Wise Men of classical Greek philosophy had any inkling that the Earth wasn’t the locus around which everything else in the cosmos spun. Their model of the universe had the Moon orbiting the Earth every 27 days, the Sun every 365 and one quarter days, and the five planets that can be observed with the naked eye following more eccentric paths that still ultimately centered on the Earth. Meanwhile the stars were a distant, stationary backdrop to it all, shining down from the very edges of a cosmos that had the Earth at its very center.
The philosophers were willing to accept that the Earth must be rotating; how else to explain the apparent march of the Sun across the sky every day, or the similar marches of the Moon and the stars every night? But they could not imagine that the Earth might be moving laterally as well as spinning; this seemed incompatible with the fact that the stars occupied the same positions in relation to one another night after night. If the Earth really was moving, the stars ought to appear to shift about, thanks to the optical phenomenon known as parallax.
For an example of parallax, think about the gauges in a car. They give accurate readings when one looks at them from the driver’s seat, but if one moves over to the passenger seat, their needles no longer align as they should with the markings behind them. The Greeks were very interested in optics from a very early date, and were well aware of this phenomenon. Yet they saw no sign of it in the night sky, leading them to conclude that the Earth must be fixed in space, the linchpin of what we now call the solar system and, indeed, the cosmos.
And then along came the Alexandrian astronomer Aristarchus to upend this sanguine formulation.
As with too many of the figures we meet in these pages, we know distressingly little about this first person in recorded history to seriously propose that the Earth does not lie at the center of the universe. Ancient sources tell us that he came to Alexandria from his birthplace on the Greek island of Samos, and a bit of conjecturing from what scant information we have — including the known orientation of the heavens at the time when he made some of his observations — allows us to conclude that he probably lived from about 310 to 230 BC. We know as well that he studied at the Lyceum in Athens under a natural philosopher, astronomer, and mathematician named Strato before he made his way to Alexandria.
In classical times, most astronomers tended to double as mathematicians because attempts to track the heavens tended to depend so heavily upon the latest mathematical techniques. Aristarchus was no exception to this rule. We can imagine him as a colleague and perhaps friend of Euclid in Alexandria, perhaps a little later of Archimedes. As we’ll see, Archimedes was at the very least thoroughly familiar with Aristarchus’s writings.
The one complete treatise of Aristarchus that has survived — sadly, not the one where he broached the theory of heliocentrism — is entitled On the Sizes and Distances of the Sun and Moon. In it, he attempts to determine those things by observing the heavenly bodies in question, both under normal conditions and during a lunar eclipse, and then applying trigonometry to the angles and ratios he has collected. He gives his solutions not as fixed distances but as ratios to the radius of the Earth, the latter being itself an unknown quantity at this juncture in history.
On the surface, it isn’t a terribly impressive piece of work. He does come surprisingly close to the real size of the Moon in offering up a figure of .35 Earth-radii; the correct figure is .29 Earth-radii. Yet he believes the Sun to have a radius about 6.7 times that of the Earth, rather than the 109 Earth-radii we now know to be the case. He is also well off the mark when it comes to the distances of both heavenly bodies from the Earth.
He says nothing in this particular paper about whether he believes the Sun to revolve around the Earth or vice versa. He may simply not have considered the question relevant to his present goals — or it may have been this very treatise, flawed though it is, that moved his thinking in the direction of the great truth about our status in the universe which he would later reveal to an unwelcoming world. For, in marked contrast to the extensive calculations that went into On the Sizes and Distances of the Sun and Moon, his heliocentrism is directly supported by no mathematics or even empirical observations that have come down to us. It may instead have arisen, at least initially, strictly from a discomfort with the fundamental incongruity which he had described in that other paper: the idea of a Sun with a diameter 6.7 times that of the Earth orbiting our little planet rather than vice versa.
As it is, though, we learn about Aristarchus’s earthshaking insight only secondhand. Archimedes mentions it in a digression within a treatise of his own, one which we’ve met before: in fact, the very same treatise in which he employs his new system of writing extremely large numbers to estimate how many grains of sand would fit into the universe.
The “universe” is the name given by most astronomers to the sphere, the center of which is the Earth. This is the common account. But Aristarchus brought out a book consisting of certain hypotheses… His hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun in the circumference of a circle, the Sun lying in the middle of the orbit.
Archimedes’s description of the theory of heliocentrism as turning up in a “book of hypotheses” might be taken to suggest that Aristarchus was doing something that scientists often do today: faced with a discontinuity in his research — namely, that of a Sun that was much larger than the Earth orbiting it like a subordinate body — he was presenting a number of hypotheses for himself or others to follow up on at some future date. Not having his original work, we cannot know how deeply he actually delved into the idea and its ramifications, nor how much mathematics, if any, he brought to bear upon it.
It’s equally difficult to say for sure what Archimedes thought of the idea. He delivers no ringing endorsement of it when he first brings it up in his own treatise on the size of the universe, but, when he presents his solution to the puzzle he has posed for himself, he nevertheless relies upon it in ways which I’ll soon describe.
Before I do so, however, it’s important to note that Aristarchus and Archimedes both persisted in seeing the stars as fixed lights shining down from the border of a perfect sphere that held the Earth, the Sun, the Moon, and all of the other heavenly bodies near its center; this held true for them regardless of the exact arrangement of said bodies in relation to one another. Thus the lack of stellar parallax in the night sky would seem to be as strong an argument as ever against heliocentrism.
But rather than taking this lack as incontrovertible proof that the Earth must occupy a fixed, unchanging point in space, Archimedes, quite possibly following the lead of Aristarchus before him, argued that it could instead mean that the stars were simply so far away that the Earth’s movement around the Sun was insignificant by comparison — so insignificant that no change in the apparent positions of the stars could be observed from the Earth. Whether it originated with Aristarchus or Archimedes, this was a brilliant stroke of insight. We now know that the stars do in fact “move” in the sky as the Earth ambles along in its orbit — in other words, the phenomenon known as stellar parallax does exist — but the shifts are too tiny to be observed with the naked eye alone.
And now we can return to Archimedes’s puzzle. He saw that he could determine the volume of his spherical universe easily enough if he could just determine the distance between its center and its border-demarcating ring of stars. And he now understood as well that this distance must be vastly greater than anyone had ever dared to imagine before if it was to be compatible with both a heliocentric universe and the lack of stellar parallax. Yet the reality was that he had absolutely no way of finding it with the tools available to him. And so, after coming this far in his treatise, he suddenly veers wildly off the rails of any recognizable logic: he decides, apparently arbitrarily, that the ratio between the diameter of the universe and the distance of the Earth from the Sun must be the same as that between the latter distance and the diameter of the Earth. The conclusion he finally offers up posits that the diameter of the universe is about two light years in modern measurements, its volume enough to hold, so he calculates, 1063 grains of sand.
Archimedes’s reasoning at this last stage is so patently flawed — not a quality for which the man is generally known — that we have to question how seriously he really meant for anyone to take his conclusion. The treatise would seem to have been written primarily as a proof of concept of sorts, in order to demonstrate how his new system of writing very large numbers might be used for just about the largest number anyone in his world would ever need to write. In his mind, the rest may have been no more than an exercise in mathematical fun and games — a temptation to which the great mathematician and proto-scientist was anything but immune, as we’ve already seen.
While Archimedes provided the most detailed account of Aristarchus’s theory that has reached us today, he was not the only ancient writer to mention it. The other texts that remain to us mostly describe the controversy that surrounded the introduction of the theory rather than the theory itself, making it clear that Aristarchus faced an immediate backlash from many of his peers. There is some debate among modern historians as to what extent the reaction may have had a religious component. A piece of seemingly affirmative evidence is provided by a passage in Plutarch which states that another Alexandrian astronomer “thought to lay an action for impiety against Aristarchus.” And adding a bit of circumstantial fuel to this fire is the fact that Aristarchus’s mentor Strato was well-known for his advocacy of a strict atheistic materialism; he was one of the first philosophers to argue for a universe composed only of matter and energy, with no room for gods.
Still, we must also allow that a pagan religious milieu which actually worshiped the Sun as one of the greatest of all the gods would not necessarily have taken heliocentrism as quite so dangerous a challenge to its dogma as Christianity later did during the time of Nicolas Copernicus and Galileo Galilei. Likewise, we must allow that there were in fact legitimate empirical objections to be made from the data that was to hand in ancient Alexandria. For anyone who hewed to the notion that the simplest explanation for a phenomenon is most likely to be the correct one, the apparent absence of stellar parallax which Archimedes had used to conclude, whether facetiously or in earnest, that the stars were really, really far away could still be taken as prima-facie evidence that the Earth didn’t move at all.
Indeed, we don’t even know how seriously Aristarchus took his own theory, much less whether he seriously attempted to defend it from the attacks of his learned peers. All told, he makes for a sadly ambiguous sort of pioneer. Heliocentrism would appear to have been a leap too far for even the bold intellectual milieu of Alexandria, perhaps even for the man himself.
So, let us turn to another extraordinary achievement by another of the city’s mathematician-astronomers.
The man named Eratosthenes was both of those things and much, much more, being if anything an even more prodigiously diverse intellect than Archimedes. He is believed to have been born in 276 BC in the city of Cyrene, a vassal of the Ptolemaic Empire that lay to the west of Egypt proper. Like Aristarchus, he made a name for himself at the Lyceum in Athens before coming to Alexandria, reportedly to become the tutor to the eventual King Ptolemy IV, who was still a young boy at the time. Eratosthenes got his hand in at just about everything in Alexandria: history, philosophy, poetry, fiction, geography, and of course mathematics and astronomy. He was apparently also good at currying royal favor: in 245 BC, likely not long at all after his arrival, Ptolemy III made him Head Librarian.
We’ve already learned that, for all that it was widely accepted by that time that the Earth was a sphere, no one could yet say how big that sphere might be in a concrete unit of measure. Eratosthenes now set out to answer that question.
At solar noon on the summer solstice — i.e., the longest day of the year — the Sun shined down on Alexandria from a position slightly to the south, as evidenced by the shadows that extended northward from the objects its light struck. But Eratosthenes learned, either through personal observations or from the reports of travelers, that the further south of Alexandria that one journeyed, the closer the Sun came to being directly overhead at this instant. When one ventured as far as the city of Syene (modern-day Aswan) at the traditional southern boundary of Egypt, one found that the Sun’s rays struck the surface of the Earth from a perfectly vertical angle. At locations even further to the south than Syene, by contrast, the Sun shined down from the north. Eratosthenes realized that these facts provided him with some of the tools he would need to calculate the circumference of the Earth.
Next, he needed to know the exact distance between Syene and Alexandria. It’s tempting to imagine that he employed Archimedes’s freshly invented odometer for this purpose, but we have no way of knowing for sure; instead of that instrument, he may have borrowed one of his king’s “pacers,” a man trained to measure distances by taking careful, even steps as he walked and keeping a precise count of them. By whatever means, he was able to conclude that the two cities lay 497 miles (800 kilometers) apart, a very credible estimate indeed.
Now came the masterstroke. At noon on the next summer solstice in Alexandria, Eratosthenes measured the length of the shadow of an object of a known height, then did some basic trigonometry to determine that the Sun was exactly 7.2 degrees off the vertical. Knowing as he did that the Sun stood directly overhead at the same moment in Syene, he could conclude that the angular distance from Syene to Alexandria was 2 percent of the way around the entire planet — this because 7.2 degrees is 2 percent of a full 360 degrees. Thus he needed only multiply the distance of 497 miles (800 kilometers) by 50 to find the circumference of the Earth. He came up with 24,850 miles (40,000 kilometers), an astonishingly accurate figure. We now know the exact value to be 24,901 miles (40,074 kilometers); Eratosthenes came within one quarter of a percentage point of perfection. Luck certainly played a role in his success; many of his estimates had to break just the right way for him to come so close. Still, his work would remain remarkable even if it was off by an order of magnitude more.
In addition to this, his most famous feat, Eratosthenes made many other contributions to astronomy. He was the first to realize that the changing orientation of the Sun in the sky at different times of year indicated that it must be tilted in relation to the Earth, even as he still hewed, like most in Alexandria, to the traditional geocentric model of the universe. He calculated the angle of this tilt to be 23.85 degrees; the correct value is 23.77 degrees. He understood that it explained the existence of seasons, and explained as well why variations in seasonal temperatures and the length of each day grew more extreme as one traveled north from Alexandria, less so as one traveled south.
Virtually everything we know about Eratosthenes’s achievements comes to us second-hand; only scattered fragments of his writings have survived. Nevertheless, we have learned from the hints and commentaries of others that he proposed a system for locating places on the surface of the Earth that was shockingly similar to our modern system of latitude and longitude; he naturally placed Alexandria at 0 degrees longitude, thus making it the Greenwich of its age. Noting the existence of marine fossils in landlocked regions, he speculated that the continents of the Earth were not fixed eternally in place, but rather moved about and changed their shape over the course of eons; in doing so, he anticipated the theory of plate tectonics. He noted how the cycles of the ocean’s tides were linked to the cycles of the Moon far above, and pondered what mysterious force might cause that to be the case; in doing this, he anticipated the theory of gravity. And in the realm of pure mathematics, he was the inventor of the Sieve of Eratosthenes, a foolproof if labor-intensive way of identifying the set of prime numbers up to a theoretically infinite value.
Eratosthenes had an exceptionally long life by the standards of his time: he reached more than 80 years of age. According to legend, he ended his own life on his own terms. In 194 BC, he began to lose his eyesight, a horrifying prospect for one whose days revolved around reading, writing, and observing. So, he elected to commit suicide by starving himself to death. At his funeral, a fitting epitaph was read: “Thou didst fall asleep in the slumber to which all must come, O Eratosthenes, after pondering over high matters.”
The work of Aristarchus, Eratosthenes, and their peers in the Alexandria of the third century BC became essential building blocks of later ancient astronomy. With, that is, one notable exception: that most paradigm-shattering idea of all, Aristarchus’s theory of heliocentrism.
The most important astronomer of the second century BC was named Hipparchus. Although he was most identified with the Greek island of Rhodes, he too may very well have spent time in Alexandria. Regardless, his record of achievement is long and impressive: he charted the positions and magnitudes of 1080 separate stars, inventing as part of his efforts a scale for classifying stars by their brightness that is still used in an only slightly modified form today; he charted the apparent movements of the Sun, the Moon, and the other planets around the Earth with equally admirable precision; he regularized and otherwise improved upon Eratosthenes’s system of latitude and longitude; he determined the length of a solar year to within four minutes of the figure we use, and the length of a lunar month down to less than a second’s precision. Most famously of all, he compared his star chart with one drawn by another astronomer 166 years earlier, and discovered thereby what we call today the precession of the equinoxes: an extremely slow drift in the position of the stars in the sky, on the order of one degree every 72 years, caused, we now know, by a slight wobble in the Earth’s rotation. And, because mathematics and astronomy were so intimately connected during ancient times, he gave to trigonometry its first table of sines in the course of doing all these things. Yet for all his obvious genius, Hipparchus rejected out-of-hand the notion of heliocentrism.
In fact, there came only one more astronomer in ancient times who was willing to take seriously Aristarchus’s theory. Seleucus, a contemporary of Hipparchus named for the Seleucid Empire of the East in which he lived, became perhaps a much more strident advocate for the idea than Aristarchus himself had been: Plutarch tells us that Aristarchus “supposed it only, but Seleucus affirmed it to be true.” But none of Seleucus’s writings have survived, and, apart from a passing mention by the later geographer Strabo that he studied the tides as well, we know nothing else about this man who in an alternate version of history might have become the Galileo to Aristarchus’s Copernicus.
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14 Comments for "Chapter 6: Alexandrian Astronomy"
Another super interesting article Jimmy!
Typo: “He was the first the realize that the changing orientation of the Sun”.
He was the first to realize
exactly 7.2. degrees
-> exactly 7.2 degrees
Knowing that the distance (D) between Alexandria (A) and Syrene
This really explains the genius of these folks in an easily-understandable way. Makes their insights all the more impressive, and I can’t help but wonder why ancient astronomical science retreated to formalistic traditionalism, or how different history would have been if that decline could have been avoided.
Great article, as always! I was just a bit perplexed by the ending. The quote from Dryer you give seems to suggest the idea that late Greek astronomers were instrumentalists. To use scholarly terminology, they wanted to only to ”save the phenomena”, but did not make any ontological commitments – or in more layman terms, their astronomical systems were just tools for calculating the future positions of planets, but not any theories or even hypotheses about what moves them or even how they really move.
Now, instrumentalism is a quite respectable, even if contested interpretation of the work of astronomers like Ptolemy, but I find it hard to reconcile with your statement that their work was ”replete with metaphysical import that had no basis in empiricism”. If one goes with the instrumentalist reading, then by definition there won’t be any metaphysics in their work – it’s just a calculatory tool for ordering known empirical data and predicting future observations.
The connection to astrology that you made seemed also a bit of a leap. It’s one thing to construct geocentric models of planetary movement, and completely another thing to use planetary positions as a supposed source for divination – the former requires intricate mathematics, the latter just observing the skies and making up wild tales about the constellations, comets etc. True, we have the example of Ptolemy who wrote a book on both topics, but this connection seems just a happenstance.
I think astrology is a little bit more complex than you give it credit for — if “credit” is the right word here. It’s much more than just “making up a wild tales.” Doing it properly requires a fair amount of real mathematics and, yes, real astronomy, albeit of the geocentric rather than the heliocentric stripe. Astrological horoscopes, for example, are not particularly simple to create. I say this even as I’m sure we can both agree that the whole thing is just an elaborate castle in the air with no basis in reality. My position here is that, once astronomers lost the desire to find the “physically true system of the world” if that truth didn’t correspond to their own, often metaphysical or religious understanding of how things *ought* to be, the doorway to astrology opened wide.
I do understand the connection to instrumentalism that is easy to read into the Dreyer quote in particular, but I think that may be a slightly anachronistic reading. Instrumentalism is a relatively recent philosophical invention, a product of nineteenth- and twentieth-century epistemology, and its adherents choose to practice it *consciously*. It is in many ways the ultimate rejection of metaphysics, given its determination to focus on physical phenomena only. Ancient astronomers of the Ptolemy school, on the other hand, merely became less interested in physical truths than metaphysical ones.
I think that you may be reading Ptolemy as well a bit anachronistically, although you’re certainly not the only one ever to have done so. Ptolemy is often described as someone who developed an ingenious if hugely mistaken physical model of the universe from the information available to him at the time — and “Oh, yes, he also wrote this weird astrology book.” It’s pretty clear, however, that he actually saw his astronomy book and his astrology book as part of the same broad study: the astronomy book set up the model of the universe, then the astrology book explained how to read its metaphysical import. But I’ll have more to say about Ptolmy in a future chapter. (To any readers who don’t know: the Ptolemy we are speaking is a famous Alexandrian astronomer and astrologer from the second century AD, not any of the Ptolemaic monarchs…)
My point was not so much about the particular case of Ptolemy, but more about the conceptual difference between the mathematics of geocentric model of solar system and the astrology. I admit I might have been a bit harsh about the latter, but I’d still say that in its most simplest form all it needs is just looking at the night sky, when a baby is born, and making connections with the planets and other lights you see and the supposed fate of the child. Adding the mathematical model makes astrology more complex – and creates a deceptive resemblance to science – but the choice of the model seems completely arbitrary. I am not expert on the cultural history of astrology, but I do know that many of the ancient civilisations beyond Greek had their own astrologies, probably with their own underlying astronomical models. I don’t see any reason why even heliocentric models of astronomy couldn’t be connected to astrological explanations – and probably someone has done that very thing at some point in history.
As for Ptolemy himself, I actually agree with you that astrology and astronomy were most likely just two interconnected sides of the same coin in his mind. Instrumentalist reading of his work was a big thing at least since Pierre Duhem (and judging just by the quote, I thought Dreyer might have been part of that tradition). Still, this interpretation hinges on ignoring pretty much everything beyond Almagest, and I think all evidence seems to show that Ptolemy believed in his epicycles. I am interested to find out what you’ll have to say about him!
The Alexandrian astronomers were in need of a breakthrough that hadn’t occurred yet–the concept of inertia and inertial motion. Once that came (over a thousand years later), heliocentrism quickly won out.
When making an extraordinary claim like “The intuitive model of the cosmos that has been accepted for all of human history is entirely wrong”, there’s nothing really weird about being confronted first with the question, “Are you sure that the measurements you made by just kinda eyeballing the position of the planets at night?”
It wasn’t really until Tycho Brahe that anyone could respond to that question with a properly authoritative, “No, I’m pretty confident in my measurements”. Heck, Galileo got in trouble much less for claiming the Earth went ’round the sun and much more for responding to “Are you sure about your numbers” by writing a short pamphlet wherein he called the pope an idiot.
I suppose lack of telescopes was also an important factor impeding the progress toward heliocentric model. Once you start to see that other planets have their own satellites, it becomes that much more difficult to believe that Earth is somehow more special.
Come to think of it, Greek astronomers had a lot of things going against them:
* Lack of proper equipment. No telescopes. Also, no mechanical clocks for precise measurement of times.
* Lack of mathematical innovations. Calculating without zero and expressing your ideas without algebraic notation is just pain.
* Unfounded philosophical assumptions. Ellipses, you say? We accept only perfect circles in heavens.
* Lack of mass communication. No printing press, no cheap paper, not even books in codex form. Chances are, even if you do make an innovation, only few close buddies will hear about it and no one will remember it in couple of generations.
* Lack of institutional support. It’s definitely easier to do research, when you have several universities, some learned societies and a number of petty monarchs to choose from – even Catholic church might provide for your living, if you just give lip service to their dogmas. Oh, you are from ancient Greek or Rome? Well, better hope the current king or emperor is more into reading philosophical books than using them to heat his bath.
Typo: I think the italian city is Crotone
Great article! 🙂
It seems that the ancient Greek name was actually Croton or Kroton. Thanks!