Thursday 23 January 2020

Detecting the earth’s curvature

How did ancient observers work out that the earth is (very nearly) spherical?

It’s pretty well-known these days that the spherical shape of the earth was discovered by ancient observers. As I wrote in an older post, the turning point seems to be a little before 400 BCE in Greece. Before that date, all reports have the earth as flat; after 400, round-earthers pop up quickly, and there are only a handful of flat-earthers. Flat-earthism has never been anything more than a fringe opinion since then, in any place that had access to their findings.

What’s obscure, though, is this: how exactly did the Greeks discover it? What was the key piece of evidence?

A ship receding over the horizon. (NB: This is not how the ancients discovered the shape of the earth.) Notice the distortion caused by refraction. Source: ‘Mathias Kp’, preview image for ‘Ship sailing into the horizon’, YouTube, Feb. 2016.

Ancient sources don’t tell us directly. Let’s jump straight to academic opinions. Here are some lecture notes from a reputable astronomy professor. This is what Ohio State University astronomy students have learned since at least 2004:

Ancient Greek philosophers argued earth was a sphere, on several grounds:
  • Sphere a "perfect" shape.
  • Ships disappear over horizon.
  • Positions of constellation above horizon change as one goes north or south.
  • Earth casts round shadow on moon during a lunar eclipse.
Prof. David Weinberg, Ohio State, A161 lecture notes

Weinberg’s first point is completely imaginary. Ancient writers who discuss evidence for the earth’s shape talk about matter falling towards local minima in the earth’s surface, not about ‘perfection’. The second point, about ships going over the horizon, is the biggest myth here: it doesn’t appear in any ancient Greek source. It’s a second-hand misrepresentation of a Roman source, and as we’ll see below, the Roman writer himself suffered from some severe misunderstandings. The third point isn’t precisely what the ancient sources say, but close enough. The fourth point is the most accurate: it does appear in surviving ancient accounts, Aristotle and Ptolemy.

Note: The lecture notes have some other errors too, especially in the bit about Eratosthenes. Most of them are copied from Carl Sagan’s inaccurate treatment in Cosmos (1980): I’ve dealt with that in an earlier post. The thing about the well is untrue. Two more specific points: the angular distance between Eratosthenes’ cities was 7.2°, not 7.5°; and his calculated circumference was ca. 46,600 km, not the 39,300 that Weinberg states. The standard Greek stadion was 185 m, plus or minus a metre. It’s true there’s some confusion over the length of the stadion, thanks to some variations, and some misreporting in the early 20th century, but the 185 m standard really is unproblematic: see here for more discussion. Anyway, Eratosthenes’ high figure comes from the fact that he didn’t have great figures for the distances between cities. His data seem to have been based on traditional measures of Egypt dating back to well over a thousand years before his lifetime.

Ships going over the horizon get brought up very, very frequently when people think about how ancient people detected the earth’s shape. You’d think it’s something people observe every day. I wonder how many people have actually seen it. Of them, I wonder how many saw it without a telescope or a really good camera.

The problem is that it doesn’t actually work very well. Not because it’s false! Ships do indeed descend past the horizon.

It’s because human eyesight isn’t good enough. Sure, with a really good zoom, or a telescope, it’s possible to observe the phenomenon. But most people’s eyesight can’t resolve details that fine.

A Nikon P900 can see this, but your eyes might not be up to the task: the schooner Denis Sullivan, photographed from Frankfort, Michigan, 2 July 2016. The ship is apparently about 18 km offshore, judging from how much of it is concealed. The height is 29 m. I assume that about a third of it is concealed by the horizon, and camera height at 2.5 m above sea level. At that distance, the angular size of what’s visible here would be about 0.06°, less than an eighth the diameter of the moon. According to Wikipedia, someone with 6/6 vision (20/20, for American readers) can discern contours 1.75 mm apart at a distance of 6 m. That’s an angular size of 0.0167°. The sails appear 3.6 times larger than that, so the feat is possible. But only about a third of people have 6/6 vision. Calculations are based on Walter Bislin’s Advanced Earth Curvature Calculator, and account for atmospheric refraction. Photo source:


When Aristotle discusses empirical evidence for the earth’s shape, in On the sky 297a-298a, the evidence that he actually mentions is as follows:

  • Gravity -- or as Aristotle puts it, ‘the nature of mass to be borne towards the centre’ (τὸ φύσιν ἔχειν φέρεσθαι τὸ βάρος ἔχον πρὸς τὸ μέσον) -- ensures that all parts of the earth come to rest at a local minimum, so the resulting shape must be roughly spherical.
  • The earth’s shadow on the moon during a lunar eclipse is always circular, and only a sphere has a shadow that is invariably circular.
  • Even a relatively short journey to north or south changes which stars are visible.

By ‘short journey’, he may mean as little as a twelfth of a degree of the earth’s curvature, since that’s the precision in latitude we find reported in Ptolemy’s Geography. That’s a bit over 9 km -- a couple of hours’ walk.

From all this, then, it is clear that not only is the earth’s shape curved, but also that it is not a huge sphere. Otherwise people would not be able to see it so quickly, when they move only a short distance. ... All the mathematicians who try to calculate the size of its circumference say that it is about 400,000 (stadia, i.e. 74,000 km).
Aristotle, On the sky 298a.6-8, 15-17

Even that figure is too high, obviously, and a century later Eratosthenes came closer.


Ptolemy’s evidence for the earth’s shape, Almagest i.1.14-16 (ch. i.4), is a bit different:

  • Lunar eclipses take place at the same time for all observers, but they are reported at an earlier hour by observers further east, and at a later hour by observers further west; and the difference in hour is proportional to the east-west distance separating the observers.
  • Alternative shapes for the earth -- concave, plane, polyhedral, cylindrical -- are ruled out by various astronomical observations (omitted here).
  • Travelling north or south changes which stars are visible in the sky, and the change is proportional to the north-south distance travelled.
  • Observers on a ship moving towards a mountain see the mountain gradually rising up out of the sea as they approach.

The last point comes within spitting distance of the ships-going-over-the-horizon trope, but it’s far more realistic than the popular idea. A trireme 20 km away may be too small to make out properly, but mountains are a much bigger target.

Samothraki seen from Thasos, 75 km away. Photo by Borislav Angelov. Source: Google Maps.

Here’s an example from the Greek world: Mt Fengari, on the island of Samothraki, seen from the shore of Thasos, 76 km away. Fengari is the highest peak in the Aegean Sea, at 1611 m.

The photo is taken from the edge of the shore, so let’s assume eye height at 2 m. A basic geometrical calculation would have it that the bottom 394 metres of the island are concealed by the earth’s curvature. However, we also need to account for atmospheric refraction. The vertical distortion from refraction actually reduces the effect of the earth’s curvature, so that distant objects are more visible.

Diagram illustrating the relationship between a distant object’s actual location, and the place where it appears as a result of refraction. Source: Walter Bislin’s Calculator.

With the standard refraction figures used by the Advanced Earth Curvature Calculator, created by Walter Bislin, a Swiss engineer, it turns out that the actual portion of Samothraki that is concealed by the horizon is the first 322 metres. That’s 20% of the mountain’s height. The angular size of the visible part of the mountain is just under 1°, double the apparent diameter of the moon, so the effect ought to be noticeable for someone who knows the shape of the island well.

Now, when I said mountains, you probably thought of Mt Olympus, the highest peak in Greece at 2917 m. Actually, Olympus doesn’t work well for this. But let’s do the calculation anyway.

Mt Olympus seen from Sani, Halkidiki, 80 km away. Source: Sani Resort website.

Using Bislin’s calculator again, this time assuming 3 m eye height, it turns out that the first 349 metres of its height are concealed: more than a tenth of the mountain’s height. But the effect is going to be harder to see. The land at Olympus’ base is higher than 349 metres, so the skyline is still above the horizon. The apparent shape of the land wouldn’t be very different from how it looks up close.

Basically, for best results, sail towards islands.


There’s just one ancient writer who mentions the trope of ships going over the horizon: dear old Pliny the Elder.

It is the same reason why land is not seen from ships, but is visible from ships’ masts. Also, when a ship is sailing far away, if a shining light is attached to the top of the mast, it appears to go down gradually and is finally concealed.
Pliny, Natural history 2.164

So, this line is the ultimate source of the myth. It isn’t hard to imagine that this is an experiment that someone might actually have tried.

But notice the difference. In the popular myth, you’re supposed to discern the contours of a distant ship, unaided. In Pliny’s version, it’s the light that you’re supposed to observe descending into the sea. That’s much easier to believe: a light-emitting source is way, way more visible than distant contours.

Still, I’m pretty sure Pliny isn’t the source that professors teaching the history of astronomy are getting it from. (If they were reading ancient sources, they’d know Aristotle doesn’t talk about spherical ‘perfection’.) I’m betting the modern myth is filtered through a much more recent source: Copernicus.

It is understood by sailors that waters also press down into the same shape (a sphere): for land which is not visible from (the deck of) a ship is regularly seen from the top of the mast. Conversely, if something shining is placed at the top of the mast as the ship is moved away, it seems to people remaining on shore to go down gradually, until at last it is hidden as if setting.

Weirdly, Copernicus bases the structure of his introduction on Ptolemy, but his arguments are inspired by Pliny -- the worst possible choice, out of the three ancient sources we’ve looked at so far.

Because Pliny is not a good source of evidence for the shape of the earth. OK, yes, he does say it’s a sphere. But most of his ‘evidence’ is ridiculous. He thinks mountains in the Alps are over 50 Roman miles high, that is 74 km (NH 2.162); he thinks the earth’s shape is demonstrated by the shape of drops of water; that a convex meniscus on a liquid surface is a consequence of the earth’s curvature; that heavy objects placed in a cup of liquid don’t cause it to overflow, because the surface acqures a convex curve (NH 2.163).

If you’re looking for empirical evidence for the earth’s shape, Pliny should not be your main resource. Copernicus, I’m afraid, gets C+ for treatment of textual evidence.


Cleomedes’ discussion of the earth’s shape is similar to Ptolemy’s, in that he spends time rejecting alternative shapes, then at the end he tacks on the appearance of mountains when approaching them by sea. This is slightly odd given that he never cites Ptolemy, but let’s not get into that. His exact arguments are (Circular motions of heavenly bodies 1.5, = pp. 72-86 Ziegler):

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

However, this is already some way into his treatise: he invokes a number of arguments for sphericity earlier on in his book, too. More about that in a moment.

We still haven’t got to the root of the question. How did the person who worked out the earth’s shape do it?

One thing we can be absolutely sure of is this: they didn’t work it out by looking at ships or mountains. They were looking at the sky. All of the evidence cited by Aristotle, Ptolemy, and Cleomedes is based on astronomy, not geography. Ptolemy and Cleomedes only tack on the thing about mountains as an afterthought, to make it easier for readers to accept.

Here are two theories. First, Otto Neugebauer:

... [I]t seems plausible that it was the experience of travellers that suggested such an explanation for the variation in the observable altitude of the pole and the change in the area of circumpolar stars, a variation which is quite drastic between Greek settlements, e.g., in the Nile Delta and in the Crimea.
Neugebauer 1975: 576 (more generally see 575-578)
And second, Dirk Couprie:

Several sources ascribe the discovery of the ecliptic (or the Zodiac) to Oenopides, who lived about one century after Anaximander and was a younger contemporary of Anaxagoras (DK 41A7). This makes Oenopides a serious candidate for the discovery of the sphericity of the earth as well, as the ecliptic must be thought of as inclined to the celestial equator, which is the projection of the equator of a spherical earth on the celestial sphere.
Couprie 2011: 169 and 201-202
Couprie’s theory about Oenopides and the ecliptic may take a little explaining.

Left: the ecliptic plotted on a celestial sphere. Right: the ecliptic plotted on a rectilinear map of the stars as seen from earth.

The ecliptic is a path against the fixed stars, which the sun, moon, and planets stick close to at all times. On a rectilinear map it looks like an S-shape, but plotted onto a spherical sky it is a circle at a fixed angle to the celestial equator. That angle is 23.5°, but it wobbles slowly: in Eratosthenes’ time it was closer to 23.9°. The Greeks called it hēliakos ‘the sun’s (path)’, ekleiptikos ‘(the path) of eclipses’, or zōidiakos ‘belt-like’, since it is a circle around the earth. That of course is where we get the name for in reference to the constellations along the ecliptic, the zodiac.

Note, added later. As a respondent points out in a comment below, I committed an elementary blunder here: ζῳδιακός has nothing to do with ζώνη ‘belt’.

Now, that’s the geocentric point of view. In reality, the ecliptic is the plane in which the earth and other planets revolve around the sun. The earth’s equator is at an angle to that plane, and that’s what produces the phenomenon.

Couprie’s idea is this. The astronomer Oenopides is said to have discovered the ecliptic, but we know that’s not true. It was well known to Babylonian astronomers a millennium earlier. However, the ecliptic implies a spherical geometry for the sky. So, Couprie thinks, what Oenopides really discovered is that that somehow implies a spherical geometry for the earth too.

It doesn’t imply that all by itself, mind. It does tend to imply that it’s the earth that’s rotating, not the stars -- but ancient testimony is pretty hostile to that theory (Aristotle On the sky 296a.26-27; Ptolemy Almagest i.1.24-25 = ch. i.7).

However, if you take it in conjunction with Neugebauer’s point about Greek colonists in Ukraine and Libya noticing different astronomical phenomena, then you get a line of reasoning that looks very similar to the opening chapters of Cleomedes’ work. Cleomedes doesn’t present the earth’s shape as a premise. He works his way up to it.

Cleomedes starts off by establishing the spherical geometry of the sky; he describes the celestial equator, tropics, and arctic and antarctic circles, and how these have corresponding zones on earth; then he moves on to the planets and their motion relative to the ecliptic, and how the ecliptic is at an angle to the celestial equator; and then he gets to the key point that

The Earth is spherical in shape, and thus [located] downwards from every part of the heavens; as a result its latitudes do not have an identical position relative to the zodiac, but different ones are located below different parts of the heavens.
Cleomedes 1.3 = p. 36.21-26 Ziegler (tr. Bowen and Todd)

This is basically the conjunction of the spherical cosmology, the ecliptic, and Neugebauer’s point about the angle of the celestial sphere being different depending on how far north or south you are. Cleomedes carries on in exactly this way, talking about how ‘the heavens slope’. Only later on does he get into explicit arguments to support the earth’s sphericity.

It’s pretty likely that his manner of exposition is very close to the original reasoning. It isn’t a certainty. The ecliptic doesn’t come up in Aristotle’s or Ptolemy’s discussions of evidence for the earth’s shape. But I think the beginnings of the idea must have followed something like Cleomedes’ reasoning.

I want to add, as a postscript, that though Greek thinkers prior to 400 were all flat-earthers, including beloved names like Thales and Democritus, their work wasn’t a waste of time. Anaximander, in particular, can be credited with the important realisation that the earth isn’t the base of the cosmos, but is suspended in space. He was wrong about why it is suspended -- pre-Socratic philosophers thought it must be held up by air pressure -- but it was a crucial step. Without that notion, I doubt the spherical earth could have been discovered until many centuries later.


  • Bowen, A. C.; Todd, R. B. 2004. Cleomedes’ lectures on astronomy. University of California Press.
  • Copernicus, N. 1543. De revolutionibus orbium coelestium. Ioh. Petreius (Nürnberg).
  • Couprie, D. L. 2011. Heaven and earth in ancient Greek cosmology. Springer.
  • Neugebauer, O. 1975. A history of ancient mathematical astronomy (2 vols). Springer.