## Monday 17 April 2023

### How Eratosthenes measured the earth. Part 1

 1. The spherical earth | 2. Eratosthenes’ method | 3. Distance | 4. Angle of the sun

In the 200s BCE, a Libyan mathematician and geographer by the name of Eratosthenes calculated the circumference of the earth. He did it with pretty basic equipment, and his result was off by only 16%.

It’s an impressive story. But to tell it properly, we have to start a few centuries earlier.

Before Eratosthenes could measure the earth, he had to know that it’s spherical. And contrary to what some famous people have claimed, he isn’t the one who worked that out. That discovery happened about 150 years before his time.

 Earth, view centred on Alexandria. The polar circumference is 40,008 km; Eratosthenes measured it as 252,000 stadia, or around 46,620 km plus or minus 1900 km.
 Note. I’ve previously written about the measurement of the earth and the discovery of its shape in antiquity (1 2): those pieces were geared towards dispelling widely believed myths. Here I’ll focus on telling the true story so far as it’s understood.

## Part 1. The spherical earth

Ancient northern Africans and southern Europeans knew the earth was spherical well over 2000 years ago. Its shape was never forgotten, and many ancient writers talk about it, from Plato and Aristotle, to the Romans, to Christian church fathers, to mediaeval philosophers and poets. Ovid, Augustine, Bede, Roger Bacon, Dante all knew the earth is round.

That said, a handful of flat-earthers did exist, driven by philosophical or religious preconceptions. Ancient Epicureans thought that the universe comes in layers, which seems that it should imply a universal up and down. Around 500 CE, some Syrian Christian leaders taught that the universe has the shape of the Ark of the Covenant. These are isolated cases. Even more importantly, they didn’t engage in evidence-based debate: these ideas had no impact outside their own circles. Beyond those circles, the earth’s spherical shape was common knowledge.

 An ancient flat-earther: the shape of the cosmos as depicted by Kosmas, a 6th century CE Christian ‘traveller’ (who didn’t travel much, and certainly not to India as his nickname ‘Indikopleustes’ would suggest). Left: an illumination from a manuscript of Kosmas depicting what he believed was the form of the cosmos. Right: a schematic drawing by Johannes Zellinger, here mirror-flipped to match the illumination.

You may think: your average Jackie wasn’t reading Aristotle or Bede, therefore they must have believed the earth is flat. Well, that’s possible. But they weren’t reading flat-earthers like John Chrysostom or Theodore of Mopsuestia either. Don’t make assumptions about what ‘average people’ thought. If every mediaeval source tells us that the earth is round, and no one treats it as a matter of debate, we should take that as our starting point.

Still, at some point, there had been a time when everyone genuinely was a flat-earther. There has to have been a first time that the earth’s shape was discovered.

It so happens that it was in Greece, in the late 400s BCE. We don’t have an explicit record of the discovery. What we do know is that before 400 BCE, absolutely everyone on record who talks about the earth’s shape says that it’s flat, without exception. After 400 BCE, virtually everyone knows for a fact that it’s spherical. When Plato talks about it, around 360, he’s already taking the spherical shape for granted, as common knowledge.

 Note. Flat-earthers before 400 BCE; references are to the edition of Diels and Kranz. Anaximander, 12.A.10, 12.A.11, 12.A.21, 12.A.25, 12.B.5; Anaximenes, 13.A.6, 13.A.7§4, 13.A.20; Anaxagoras, 59.A.1§8, 59.A.42§3, 59.A.47, 59.A.87; Archelaos, 60.A.4§4; Empedokles, 31.A.50, 31.A.56; Leukippos, 67.A.1, 67.A.26; Diogenes of Apollonia, 64.A.1; Demokritos, 68.A.94, 68.B.15§2. Plato taking the spherical earth for granted: Phaedo 108e–109a.

## In the beginning

In Greek mythological thought, the cosmos came in layers: the thin, fiery aether in the upper reaches of the cosmos; beneath that the dense misty air that humans breathe; then the surface of the earth and sea; then the dark underworld; and at the very bottom, the bottomless void of Tartaros, a netherworldly counterpart to the sky. That’s the picture we get in Homer and Hesiod, in the first part of the 600s BCE.

 A diagram of the cosmos imagined by Homer. This diagram appears in the margin of an 11th century CE manuscript of the Iliad, cod. Marciana 453 (or ‘Venetus B’) fol. 103r. From top to bottom the layers are: aithér ‘aether’; aér ‘mist’, that is breathable air; háides ‘Hades, unseen’, that is the underworld; and tártaros ‘void’. Already in Homer, the earth’s surface is imagined as the centre of the cosmos.
 Note. Diogenes Laertios 8.48 claims that a round earth was known before the 400s by Pythagoras; Aëtios, Opinions of the philosophers 3.10.1, claims it was known by Thales. These are both false, without the slightest doubt. (1) Diogenes Laertios also ascribes round-earthism to Parmenides and Hesiod, and those are certainly untrue. (2) More reliable sources tell us that Pythagoras thought the earth and all the planets are attached to cosmic spheres, with the earth’s surface facing away from the ‘central fire’ (58.B.37, 44.A.16 Diels-Kranz); Thales thought the earth was like a piece of wood floating in water (11.A.14 Diels-Kranz). (3) Pre-Sokratic philosophers show an overwhelming consensus that the earth is flat, and shaped like either a pillar section, a drum, or a table. If any of them were round-earthers, a well-informed writer like Aristotle when talking about earlier beliefs about the earth’s shape would certainly have highlighted the fact.

From this starting point we can trace Greek thinkers coming up with new ideas, and making incremental advances. Their ideas weren’t always grounded in empirical evidence. Most of them are dead wrong. Even so, they paved the way for the discovery of the round earth in the 400s.

Pre-Sokratic thinkers made three key advances that made the round-earth model possible. Their advances stood the test of time up to the early modern era.

1. The idea that the earth is suspended in space, in the centre of a spherical cosmos.
2. The realisation that astronomical observations of the sky have correspondences to the geometry of the earth.
3. The concept of universal centripetal and centrifugal motion, playing roles analogous to gravity and buoyancy.

These points haven’t survived into present-day science — or at least points 1 and 3 haven’t — but they were central to how people understood the cosmos up until Copernicus and Newton came along. And they were the basis for Eratosthenes’ work.

## Space, the celestial sphere, and proto-gravity

There are no surviving books written by any of the so-called ‘pre-Sokratic’ philosophers, the natural philosophers who lived before Sokrates’ time. That’s a pity, because, wrong as they were about most things, they did essential groundwork. We have to rely on later reports of what they said.

The first key advances came from the Ionian philosopher Anaximander, or Anaximandros, in the early 500s BCE. Like all the pre-Sokratics he thought the earth was flat. He taught that it has the shape of a cylinder, and that we live on the top side of it.

So far, not great. But Anaximander also taught that this cylinder is suspended in space, in the centre of the cosmos. In his cosmology the planets, including the sun and moon, are attached to invisible rings that rotate around the earth — and these rings don’t stop at the horizon: they go all the way round. In other words Anaximander’s sky isn’t the ceiling of a flat cosmos. It’s a celestial sphere, with the earth at its centre, held in place by the force of sheer isotropy.

The cylindrical earth was a dead end. But Anaximander’s celestial sphere, and the earth’s suspension at its centre, were essential steps towards developing a concept of a spherical earth.

 Note. Earth as cylinder: 12.A.10 Diels-Kranz; similarly 12.A.11.3, 12.A.25, 12.B.5. Celestial bodies making a full circle around the earth: 12.A.11.4, 12.A.18, 12.A.21, 12.A.22. Earth suspended in space: 12.A.11.3. For discussion see Couprie 2011: 99–114.

Over time, as Greek traders and colonists watched the stars and the seasons in Greek colonies and trading centres, in Ukraine to the north and Egypt to the south, they realised Anaximander’s model needed tweaking. They observed that the days are different lengths, depending on how far north or south you are. That the climate is colder in the north, and warmer in the south.

Most of the tweaks they came up with are wrong. Really, really wrong. Every one of them continued to assume that the earth is flat. But their efforts show that they realised there were problems, and they were working on solutions.

Anaximenes and Anaxagoras, who also lived in western Anatolia, disliked the idea that the earth is suspended purely by isotropy. So they suggested that it’s held up in the centre of the cosmos by air pressure from underneath, like when a boiling pot makes the pot lid bounce up. Leukippos and Demokritos, the atomists, from Abdera in northern Greece, taught that the earth was flat but tilted. The reasoning was that mountain tops are cold; therefore, things that are high up are cold; therefore, places like Ukraine must be ‘higher up’ than Egypt and Ethiopia, and the earth slopes downwards to the south. And Archelaos of Athens taught that the earth isn’t perfectly flat but concave, and this is why sunrise and sunset happen at different hours in different parts of the world. If these ideas seem daft, remember that failed theories are still part of the process.

 Left: schematic drawing of part of Anaximander’s cosmos, with a cylindrical earth hovering in the centre of the rotating sphere of fixed stars. Right: the tilted earth of Leukippos and Demokritos, held up by air pressure. (Not to scale.)
 Note. Air pressure supporting the earth: 13.A.20 Diels-Kranz. Earth tilted downwards to the south: 67.A.1§33. Earth’s surface as concave disc: 60.A.4§4.

Archelaos was wrong about the earth being a concave disc, but another of his ideas turned out to be much more significant. He rejected Anaximenes’ notion that the earth is held in place by air pressure. Instead, he taught that the sun and stars are hot fiery bodies in the sky; liquid water flows into the centre, and the centre loses its heat by boiling off the water as aér (‘mist, breathable air’), which is in turn burned off in the form of the sun and stars. As a result earth, in the centre, is where you find cold things like rock and liquid water, and the celestial sphere is where you find fiery heavenly bodies.

That is: it seems to be implied in Archelaos’ picture of things that there’s a cosmic force driving cold matter towards the centre of the cosmos, while heat moves towards the heavens.

In a very limited sense, Archelaos invented the idea of cosmic centripetal and centrifugal forces.

 Note. Archelaos 60.A.4§2–3 Diels-Kranz: ‘The cause of movement is the distinguishing of heat and cold from one another. Heat is set in motion, while the cold is static. In its liquid state water flows into the centre, where it is boiled and becomes aér (mist, breathable air) and earth, and the one is borne upward, the other settles downward. The earth is static and becomes (cold) for the following reason: it lies in the centre, making up a zero-sized portion, so to speak, of the universe; (and aér) is released from the conflagration. This is how the stars first ignited, of which the biggest is the sun, then the moon, and then the others, some smaller, some bigger.’

This isn’t gravity. Maybe it’s a stretch even to call it proto-gravity. But it’s useful as groundwork. A century later, when Aristotle talks about the spherical earth, gravity, and natural motion, he describes it in very similar terms: cold matter falls ‘downwards’ towards the centre of the cosmos, fiery matter moves ‘upwards’, that is, outwards.

 Note. For Aristotle’s doctrine of ‘heavy’ and ‘light’ materials having natural centripetal and centrifugal motion, see On the sky 311a–313b. Earlier in On the sky, at 297b, he cites the centripetal motion of ‘heavy’ materials (‘the nature of weight to be borne towards the centre’) as the cause of the earth’s spherical shape.

So, thanks to Anaximander and Archelaos, we’ve got an earth that’s suspended in space; and we’ve got a primitive version of proto-gravity. Without these ideas, I doubt it would have been possible even to imagine a spherical earth.

## Spherical cosmos to spherical earth

We don’t know who it was that first argued that the earth is a sphere. We don’t know for sure what reasoning they used. But we can be certain that it was based on astronomy, not on observations relating to the surface of the earth.

One scholar, Dirk Couprie, suggests that the main credit should go to the astronomer Oinopídes of Chios (Couprie 2011: 169). We don’t have anything written by Oinopídes, but ancient reports claim that he’s the one who discovered that the ecliptic, the path that the sun and planets follow in a circle around the sky, is slanted at an angle from the celestial equator (41.7 Diels-Kranz).

 The celestial sphere, spherical earth, and the ecliptic. For an observer on the earth’s surface, the celestial sphere of fixed stars appears to rotate around the celestial poles. The celestial equator, the horizontal circle in blue, bisects the celestial sphere at the halfway point. The ecliptic, in lilac, is another circle around the sky, which represents the path taken by the sun, moon, and planets. (For a heliocentric observer, the ecliptic becomes the plane in which the planets orbit the sun.) Eratosthenes measured the angle between these two circles — the obliquity of the ecliptic — as 24°. The true figure was 23.8°. Today it is 23.4°: the angle wobbles slowly over the millennia.

I think we have a glimpse of the discovery of the earth’s shape in the work of Kleomedes, an astronomer who lived more than 500 years later, in the Roman era. Kleomedes was writing in a time when he could take round-earthism for granted. The first part of his work Meteora, ‘the heavens’, describes the ecliptic; the zones of the celestial sphere, and their corresponding zones on earth; then he goes on to talk about specific evidence that the earth is spherical.

 Note. For a good English translation of Kleomedes see Bowen and Todd 2004. I do not know of any freely available online translations.

Kleomedes highlights two key points. First, circular movements in the heavens — the rotation of the stars, and the movement of the planets along the ecliptic — imply that celestial geometry is spherical. Second, the axis of the celestial sphere changes depending on how far north or south you are.

Based on these, Kleomedes gets to the most direct piece of reasoning: for every point in the celestial sphere, there is a point on earth that is directly beneath it. The line of reasoning seems to be that if you take these points together, they imply that the earth, too, has spherical geometry.

That’s the general pattern of the logic that Kleomedes gives us. We can’t be certain, but I’d say it’s a decent bet that the original discovery followed a similar pattern.

Only after he’s laid out these principles does Kleomedes get to discussing specific evidence for the earth’s shape (Meteora 1.5 = pp. 72-86 ed. Ziegler).

• The length of time between sunrise and sunset is different in different places.
• Lunar eclipses are observed at different hours depending on how far east or west you are.
• The celestial pole has a different azimuth depending on how far north or south you are.
• Different stars appear in the sky depending on how far north or south you are.
• When you sail towards islands, mountains appear to rise up out of the sea gradually.

But he puts these points in a secondary position. He doesn’t treat them as ways of working out the earth’s shape, they’re there as corroboration. He uses them to make the spherical shape more tangible, believable, and digestible.

It didn’t take long for people to start estimating the size of the spherical earth. When Aristotle discusses evidence for the earth’s shape, he also brings up its size. The available estimates at the time were much too high, but even so, Aristotle is impressed at how small the earth must be.

Moreover, the appearance of the stars makes it apparent not just that the earth is round, but also that its size is not great. Just a small change of position to the south or north causes the horizon to appear distinctly different, and causes a significant change in the stars overhead ... Also, those mathematicians who have tried to calculate the size of its circumference reckon it as 400,000 stadia [≈ 74,000 km or 45,980 miles].
Aristotle, On the sky 297b.30–298a.17 (my translation)

This figure is nearly double the true size of the earth, and even so, Aristotle found it startlingly small. Over the following century the figure would drop further, thanks to a lower estimate by Archimédes, and a data-based calculation by Eratosthenes.

In parts 2 to 4 we‘ll look at how Eratosthenes actually arrived at his calculation. As here, the focus won’t be on rebutting popular accounts which are packed with inaccuracies, but on telling the true story.