Fact-check: Eratosthenes (IBM, 1961) (video)

Video review/fact-check of the IBM short film ‘Eratosthenes’ (1961).

How Eratosthenes measured the earth. Part 4

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

(e) Angle of the sun

Eratosthenes didn’t use gnomons to measure the sun’s angle.

Here once again I have to digress from telling the true story, and dispel a myth. It’s a myth that even many specialists take for granted. I’m just as guilty: in the past I’ve repeated the popular wisdom that Eratosthenes’ measurement was based on gnomon readings. Well, I was wrong.

The sun at midday (source: Stellarium)

The gnomon is the most basic instrument for measuring the sun’s motion. It’s a vertical rod casting a shadow on a horizontal surface. Egyptian observers had been thoroughly familiar with gnomons for many centuries before the Ptolemaic era. Gnomons had several uses:

1. a way of expressing latitude
2. determining exact time of midday, dates of equinoxes and solstices
3. orientation (determining the direction of due north, south, etc.)

The first of these was a comparatively recent innovation: it doesn’t make sense to measure latitude until you know the earth is spherical. The idea is that latitude is expressed as the ratio between a gnomon and the length of its shadow, at midday at the equinox. This ratio is an indirect expression of the sun’s angle.

A gnomon reading measures the ratio of the gnomon to its shadow at midday. This diagram shows a reading taken on the equinox, when the sun’s rays are parallel to the earth’s equator. Using trigonometry, you can use this ratio to calculate the angle θ, since tan θ = o/a.

And this is exactly what Pliny the Elder does (1st century CE). He quotes gnomon readings taken at the equinox.

So in Egypt at midday on the equinox, the shadow of an umbilicus — what they call a ‘gnomon’ — measured a little over half of the gnomon. In the city of Rome, the shadow is one ninth longer than the gnomon. In the town of Ancona, it’s 1/32 more than that. And in the region of Italy called Venetia, the shadow is the same length as the gnomon.

Pliny, Natural history 2.182

Strabo gives the latitudes of Alexandria and Carthage in the same way (2.5.38). Now, remember from Part 2 that at midday on the equinox, the sun’s angle from the vertical is equal to your latitude. This means we can convert Pliny’s and Strabo’s figures to the modern way of expressing latitude — degrees away from the equator — by plugging them into a calculator and using the ‘inverse tan’ function.

	Gnomon reading	Calculated latitude	Actual latitude
Egypt	a little over 0.5	over 27°	24.1° (Aswan) to 31.2° (Alexandria)
Alexandria	0.600 (3/5)	31.0°	31.2°
Carthage	0.636 (7/11)	32.5°	36.9°
Rome	0.889 (8/9)	41.6°	41.9°
Ancona	0.920 (8/9 + 1/32)	42.6°	43.4°
Venice	1.00	45.0°	45.4°

These are reasonably accurate, apart from Carthage.

Note. Ptolemy also puts Carthage too far south, at 32.7°: Geography 4.3.7. And no, it isn’t because Strabo and Ptolemy are confusing Carthage with Leptis Magna — one might imagine that, since the Greek name of Leptis Magna, Neapolis ‘new town’, means the same as Carthage’s Punic name, Qrt ḥdšt ‘new town’. But Ptolemy lists Carthage and ‘Neapolis’ separately: see 4.3.13.

Our earliest evidence for the practice of using gnomon readings as a measure of latitude dates to Pytheas of Massalía in the 300s BCE. Pytheas’ book apparently started out by reporting the latitude of Massalía (modern Marseille).

Hipparchos says that in Byzantion the ratio of a gnomon to its shadow is the same as what Pytheas reports for Massalía ...

Pytheas fr. 6c Mette (Strabo 2.5.8; also fr. 6a, i.e. Strabo 1.4.4)

Elsewhere Strabo reports (2.5.41) that Hípparchos’ summer solstice gnomon ratio for Byzantíon is 120 : 41.8, that is, 2.871. That puts the sun’s angle from the vertical at 19.2°. Taking the earth’s inclination to the ecliptic as 23.8° in that era, that implies a latitude of 43.0° N. The real latitudes of Marseille and Byzantíon are 43.3° N and 41.0° N, respectively.

Pytheas’ book doesn’t survive. We aren’t actually told that he carried on taking gnomon readings on his trip into the North Sea. But it’s fairly strongly implied: we do hear about him reporting on the behaviour of the summer tropic at the Arctic Circle (Strabo 2.5.8).

People wrote books reporting gnomon readings in Africa too. In the early 200s BCE Philon, a Ptolemaic diplomat, reported gnomon readings and other observations of the sun in Meroë.

(Hipparchos states that) Phílon described the parallel of latitude at Meroë, and related it in his Voyage to Aithiopia. He stated that 45 days before the summer solstice the sun is directly overhead; and he reports the ratios of gnomons to their shadows at the solstices and equinoxes. Eratosthenes’ (figures) agree very closely with Philon.

Philon, FGrHist 670 F 2 (Strabo 2.1.20)

‘Agree very closely’ implies their figures weren’t exactly identical: that is, Eratosthenes had another source in addition to Philon. Eratosthenes was well equipped with gnomon readings, it seems.

The second use of gnomons — determining solstices, equinoxes, and the moment of midday — was of much longer standing in Egypt. The Egyptian economy depended on the flooding of the Nile, so the Egyptians were highly motivated to keep track of the solar year. Martin Isler points to reliefs dating back to the 20th century BCE which he explains as gnomons with a forked tip and held vertical by struts. These gnomons are taller than a person — much taller, in the second image below — and the forked tip is designed to improve precision.

Left and centre: reliefs depicting gnomons used in ritual contexts, from a chapel of Senusret I (1900s BCE, left) and the Pylon of Ramesses II at Luxor (1200s BCE, centre). Right: diagram illustrating the movement of the gnomon’s shadow. The view is from overhead, with the gnomon at the bottom of the diagram, casting a shadow upward. The fork on the gnomon is oriented east-west: as the sun transits, the fork improves precision in determining the exact moment of midday. (Source: Isler 1991)

The third use of the gnomon, determining true north, may be even older, though the merkhet was better suited to this purpose. The great pyramid complex at Giza, built in the 2500s BCE, is famously oriented so that the main buildings are aligned with the cardinal directions to a precision of around 0.08°. Gnomons probably weren’t the primary tool for achieving that, but they must certainly have been an important piece of ancillary equipment.

Note. Nell and Ruggles 2014 survey possible orientation methods used at Giza, arguing that the primary method was alignment with circumpolar stars; they allow that gnomons and merkhets served ancillary roles. A stellar method seems inevitable given that some pyramid complexes have slightly different orientations, and precession seems to be responsible for that. They find no suitable pairs of circumpolar stars, however. I suggest Phecta and Megrez, in the Great Bear (Gamma and Delta Ursae Majoris, the two stars on the left side of the ‘scoop’ of the ‘big dipper’). According to Stellarium they were at their closest to being in a north-south alignment in the year –2586, with right ascensions differing by just 1.28 s. That is, in 2587 BCE, when one of these stars was due north, the other would be just 0.005° east or west.

Isler goes on to describe further Egyptian refinements to the gnomon, including (a) the use of plumb bobs to ensure the gnomon is exactly vertical; (b) designs for handheld gnomons to be used by travellers; (c) in southern Egypt, the technique of leaning the gnomon towards due north at a fixed angle to compensate for the sun being directly overhead in summer.

By Eratosthenes’ time, then, the gnomon was fully developed. The idea of using it as a measure of latitude was new-ish, but it worked well — and to judge from Pytheas’ measurement at Massalia, under ideal conditions it could be quite accurate.

But there’s a problem. A gnomon measurement is a ratio between two lengths. How do you convert a ratio to an angle?

You could use the inverse tan function, as I mentioned. Just one snag: trigonometric functions are a modern thing. Aristarchos had started to dabble in trigonometric inequalities, but he was a long way from developing the Taylor series, which is what modern calculators use to compute trigonometric functions. Hipparchos apparently computed a table of chord lengths in a circle, which implicitly draws on the sine function: that sounds good. But Hipparchos himself states that he reckoned the circle in 24 portions, that is, in steps of 7.5°. When Ptolemy quotes latitudes, they’re precise to 1/12 of a degree! Hipparchos’ trigonometric table absolutely didn’t have that kind of precision.

And that’s just one angle. Comparing two angles compounds the inaccuracies.

Note. Hipparchos using steps of 7.5°: commentary on Aratos, 148,26–150,3 Manitius. See also Neugebauer 1975: 299–300, with a reconstruction of Hipparchos’ chord table at 1132 (Table 8).

Moreover, if Eratosthenes had quoted gnomon readings for Alexandria, Syene, and Meroë, then it’s a little remarkable that none of the figures are quoted in any source. It’s not like they’d be hard to express. The real latitudes are 31.2°, 24.1°, and 16.9°. If ancient sources quoted equinoctial ratios of 3/5, 4/9, and 3/10, those would be close enough: they’d translate to latitudes of 31.0°, 24.0°, and 16.7°. But only the first ratio appears in any ancient source, and it isn’t in connection with Eratosthenes (Strabo 2.5.38).

The only numerical figure we get that is actually related to Eratosthenes’ angular measurements is this one.

So the distance from Syene to Alexandria must be 1/50 of a great circle of the earth.

Eratosthenes, Measurement M6 Roller (Kleomedes 1.7, 100,19–21 Ziegler)

Eratosthenes jumped directly to the angular measurement, as a proportion of a circle. He didn’t stop and invent trigonometry on the way. Gnomons must have served only an ancillary role, just as they did at Giza 2300 years earlier. Eratosthenes’ measurement wasn’t done with gnomon readings.

Irina Tupikova (2018) suggests that the instrument he used instead was the skáphē, a bowl with a gnomon sticking up out of the centre and gauge markings along the side of the bowl indicating the number of degrees from the vertical. I’ll accept that that’s possible, but I doubt ancient skáphai were capable of the necessary precision. (The precision of the skáphē isn’t attested or well studied.)

A much likelier candidate is a device described by Ptolemy in the 100s CE. This device has the advantage that there’s evidence of it being used at Meroë in the 200s BCE.

We make a bronze ring of a suitable size ... We use this as a meridian circle [i.e. oriented north-south], by dividing it into the normal 360° of a great circle, and subdividing each degree into as many parts as [the size] allows. Then we take a smaller ring, and fit it inside the first ... the smaller ring can rotate freely inside the larger, with a north-south motion, in the same plane. At two diametrically opposite points on one lateral face of the smaller ring we fix little plates, of equal size, pointing towards each other and the centre of the rings ... [W]e observed the sun’s movement towards the north and south by turning the inner ring at noon until the lower plate was completely enshadowed by the upper one. When this was the case, the tips of the pointers indicated to us the distance of the sun from the zenith in degrees, measured along the meridian.

Ptolemy, Almagest 1.12 (64,12–66,4 Heiberg, tr. Toomer)

Left: the instrument described by Ptolemy (base image: Toomer 1984: 61). The instrument is lined up north-to-south, with the meridian; at midday the inner ring is turned so that A casts its shadow on B, then the angle of the sun is read on the outer ring. Right: a 2nd century BCE graffito from Meroë depicting a seated observer using a similar device, with the sun at top left (source: Garstang 1914, Plate VI No. 1). Garstang (1914: 4) and Depuydt (1998: 173–176) interpret the device as a ‘transit instrument with circle’, i.e. Ptolemy’s device, combined with ‘an azimuth instrument’, i.e. a gnomon, represented by the vertical line extending upward from the ring and touching a sunray.

In 1914 John Garstang described a building in Meróë, dating to the 2nd century BCE, whose outside wall contained graffiti with astronomical calculations and observations. One graffito (above, right) showed something close to Ptolemy’s ring transit device.

Gnomons still had their uses. You’d need one to pinpoint the direction of the meridian, and the exact moment of midday/ And they weren’t made obsolete by Ptolemy’s device: a gnomon doesn’t require as much precision in its engineering.

But when Kleomedes reports that Eratosthenes measured the distance from Syene to Alexandria as 1/50 of a great circle of the earth, we should take it that he means exactly what he says. Eratosthenes didn’t do trigonometry on two gnomon ratios in order to compare them. He directly compared two angular measurements.

(f) The upshot

Most people, if they know anything about Eratosthenes’ measurement, learned about it from Carl Sagan in his 1980 TV series Cosmos.

Eratosthenes’ only tools were sticks, eyes, feet, and brains — plus a zest for experiment. With those tools he correctly deduced the circumference of the earth to high precision, with an error of only a few percent.

Carl Sagan, Cosmos episode 1 (first broadcast 28 Sep. 1980)

This is mostly false. Sagan both diminished and exaggerated the accomplishment.

Eratosthenes was standing on the shoulders of giants. Sagan’s version diminishes the huge amount of work done by previous explorers, researchers, astronomers, and writers; the political infrastructure, the rich publication history, the high precision instrumentation. He glosses over the reasoning and the tools that Eratosthenes himself devised: the most sophisticated geographer of his time, the inventor of lines of latitude and longitude, and the principle that observations at the same meridian can be directly compared.

At the same time, by looking only at Hultsch’s selectively reported result, he also glosses over the blunders and the inaccuracies. The Ptolemaic understanding of the route of the Nile leaves a lot to be desired; the distance measurements are pretty poor; and it turns out that while shadow lengths can tell you your latitude accurately under ideal conditions, they’re not very reliable.

In light of all that, it’s lucky for Eratosthenes’ modern reputation that his measurement was only 16% high. It remained the most accurate measurement of the earth’s size until the modern era. But there was no way at the time for anyone to be sure of that — no one knew Eratosthenes’ measurement was the most accurate one, until modern measurements improved on him!

It’s only in hindsight that we know the techniques and technology he used were the very best that were available. The principle of his measurement, however, was impeccable. Eratosthenes should be given credit for his methodology, more than for his lucky result.

References

• Borchardt 1921. ‘Ein weiterer Versuch zur Längenbestimmung der ägyptische Meilen (itr-w).’ In: Regling, K.; Reich, H. (eds.) Festschrift zu C. F. Lehmann-Haupts sechzigstem Geburtstage. Wien/Leipzig. 119–123.
• Bowen, A. C.; Todd, R. B. 2004. Cleomedes’ lectures on astronomy. Berkeley/Los Angeles.
• Bruins, E. M. 1964. Codex Constantinopolitanus Palatii veteris no. 1, 3 vols. Leiden.
• Carlos Carman, C.; Evans, J. 2015. ‘The two earths of Eratosthenes.’ Isis 106: 1–16. [JSTOR]
• Couprie, D. L. 2011. Heaven and earth in ancient Greek cosmology. New York.
• Depuydt, L. 1998. ‘Gnomons at Meroë and early trigonometry.’ Journal of Egyptian archaeology 84: 171–180. [JSTOR]
• Diels, H.; Kranz, W. 1960. Die Fragmente der Vorsokratiker, 9th ed. Berlin. [Internet Archive: vol. 1, vol. 2] (Note: see Kirk and Raven 1957 for a selection of the fragments, with English translations in the footnotes, and cross-referenced to Diels & Kranz)
• Duncan-Jones, R. P. 1980. ‘Length-units in Roman town planning: the pes monetalis and the pes drusianus.’ Britannia 11: 127-33. [JSTOR]
• Garstang, J. 1914. ‘Fifth interim report on the excavations at Meroë in Ethiopia. Part I. General results.’ Annals of archaeology and anthropology (Liverpool) 7: 1–10. [Google Books]
• Hultsch, F. 1882. Griechische und römische Metrologie, 2nd ed. Berlin. [Internet Archive]
• Isler, M. 1991. ‘The gnomon in Egyptian antiquity.’ Journal of the American Research Center in Egypt 28: 155–185. [JSTOR]
• Kirk, G. S.; Raven, J. E. 1957. The presocratic philosophers. A critical history with a selection of texts. Cambridge. [Internet Archive] (cf. Diels and Kranz 1960)
• Loret, V. 1903. ‘L’átour et la Dodècaschène.’ Sphinx: revue critique embrassant le domaine entier de l’égyptologie 7: 1–24. [Persée]
• Mette, H. J. 1952. Pytheas von Massalia. Berlin.
• Nell, E.; Ruggles, C. 2014. ‘The orientations of the Giza pyramids and associated structures.’ Journal for the history of astronomy 45.3: 304–360. [DOI]
• Neugebauer, O. 1975. A history of mathematical astronomy. Berlin/Heidelberg.
• Priskin, G. 2004. ‘Reconstructing the length and subdivision of the iteru from late Egyptian and Graeco-Roman texts.’ Discussions in Egyptology 60: 57-71. [Academia.edu preprint]
• Roller, D. W. 2010. Eratosthenes’ Geography. Princeton.
• Tupikova, I. 2018. ‘Eratosthenes’ measurements of the earth: astronomical and geographical solutions.’ Orbis terrarum 16: 221–254. [Academia.edu]
• —— 2022. ‘A common-sense approach to the problem of the itinerary stadion.’ Archive for the history of exact sciences 76: 319–361. [DOI]

Further scholarschip on the stadion:

• Engels, D. 1985. ‘The length of Eratosthenes’ stade.’ American journal of philology 106: 298–311. [JSTOR]
• Gulbekián, E. 1987. ‘The origin and value of the stadion unit used by Eratosthenes in the third century B.C.E.’ Archive for history of exact sciences 37: 359–363. [JSTOR]
• Oxé, A. 1963. ‘Die Masstafel des Julianus von Askalon.’ Rheinisches Museum 106: 264–286. [Universität zu Köln | JSTOR]
• Pothecary, S. 1995. ‘Strabo, Polybius, and the stade.’ Phoenix 49: 49–67. [JSTOR]
• Priskin, G. 2004. ‘Herodotus on the extent of Egypt.’ Göttinger Miszellen 201: 63–67. [Academia.edu preprint]

Further scholarschip on the measurement of the earth:

• Diler, A. 1949. ‘The ancient measurements of the earth.’ Isis 40: 6–9. [JSTOR]
• Drabkin, I. E. 1943. ‘Posidonius and the circumference of the earth.’ Isis 34: 509–512. [JSTOR]
• Dutka, J. 1993. ‘Eratosthenes’ measurement of the earth reconsidered.’ Archive for the history of exact sciences 46: 55–66. [JSTOR]
• Nissen, H. 1903. ‘Die Erdmessung des Eratosthenes.’ Rheinisches Museum 58: 231–245. [Universität zu Köln | JSTOR]
• Priskin, G. 2006. ‘The Egyptian heritage in the ancient measurements of the earth.’ Göttinger Miszellen 208: 75–88. [Academia.edu preprint]
• Rawlins, D. 1982. ‘Eratosthenes’ geodesy unraveled: was there a high-accuracy Hellenistic astronomy?’ Isis 73: 259–265. [JSTOR]
• Russo, L. 2013. ‘Ptolemy’s longitudes and Eratosthenes’ measurement of the earth.’ Mathematics and mechanics of complex systems 1.1. 67–79. [DOI]
• Taisbak, C. M. 1974. ‘Posidonius vindicated at all costs? Modern scholarship versus the Stoic earth measurer.’ Centaurus 18: 253–269. [DOI]

How Eratosthenes measured the earth. Part 3

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

Distances along stretches of the Nile as Eratosthenes described them in his Geography, fr. 98 Roller (see below for details). The ‘5300 stadia’ from Alexandria to Syene may include a 300 stadia region around Syene where the sun was overhead at the summer solstice; alternatively the extra 300 stadia could be explicable by Priskin’s theory that the 5300 stadia represent a conversion from 106 Egyptian schoinoi or itrw at the rate of 50 : 1.

(d) Distance

Distances quoted by Eratosthenes

Eratosthenes’ chosen sites were Meroë, Syene, and Alexandria. He reckoned the straight-line distance between each pair as 5000 stadia.

[T]he meridian through Syene is drawn roughly along the course of the Nile from Meroë to Alexandria, and the distance is about 10,000 stadia. And Syene is situated halfway between them, that is, 5000 stadia from Meroë.

Eratosthenes, Geography fr. 34 Roller (Strabo 2.5.7)

So the distance from Syene to Alexandria must be one fiftieth of a great circle of the earth: and this is 5000 stadia. Therefore, the entire circle works out to be 250,000 stadia. And this is how Eratosthenes’ calculation goes.

Eratosthenes, Measurement M6 Roller (Kleomedes 1.7, 100,19–23 Ziegler)

From Syene to Meroë, Timosthenes the commander of Philadelphos’ fleet gave no measurement but said it was a journey of 60 days; Eratosthenes gave it as 625 [Roman] miles; Artemidoros, 600 miles.

Pliny, Natural history 5.59

Note. The third passage doesn’t appear in Roller’s edition of the Eratosthenes fragments, for some reason. Pliny’s figure of 625 miles represents 5000 stadia: a Roman mile was 5000 Roman feet. At 2.85 and 12.53 Pliny quotes the rate of 1 stadion = 625 Roman feet = 1/8 of a Roman mile.

How did he obtain this distance? The most popular answer is based on Martianus Capella, writing in the 5th century CE:

In fact Eratosthenes was informed by Ptolemy’s royal surveyors [mensores regios] about the number of stadia from Syene to Meroë ...

Martianus Capella 6.598

At this point we could go on a long tangent. Were these mensores a special category of surveyor known as ‘bematists’? What does that job title entail, exactly? It’d take half an hour. Let’s cut to the chase. Martianus’ mensores weren’t military survey technicians, counting footsteps or wheeling a hodometer through hundreds of kilometres of trackless desert. Or if that is what Martianus imagined, then he’s just wrong.

Some of the scenery on the direct route between Syene and Meroë. (Base image: Google Earth)

Eratosthenes used reports published in books that were readily available. And these books described the course of the Nile, not the straight-line desert route between Syene and Meroë.

First, in his Geography he said that he simply repeated the distances he found quoted in these reports, without verifying them.

Eratosthenes states that, for more remote regions, he reports distances that have been handed down, and that he has not verified them but repeated them as he has received them. But sometimes he adds, ‘in a more or less straight line.’

Eratosthenes, Geography fr. 131 Roller (Stabo 2.1.41)

Second, we have independent testimony about Greek-speaking writers describing routes and distances between Syene and Meroë. Several Ptolemaic figures travelled to Meroë and beyond, and wrote books about their observations: Dalion, Aristokreon, Bion, Simonides, Philon, Timosthenes, Basilis — and these are just the ones we know about. Pliny tells us that four of these writers explicitly reported distances south of Syene (though he doesn’t report their figures), and that some of them reckoned the total distance from Meroë to the Mediterranean coast as 1250 Roman miles (10,000 stadia).

These people were explorers, admirals, diplomats. They weren’t trundling hodometers along the riverbank, and they certainly weren’t counting paces.

Note. Pliny, Natural history 6.183. Dalion sailed further up the Nile beyond Meroë and described the wildlife (FGrHist 666). Eratosthenes did not draw on Aristokreon’s account (Aristokreon quoted the distance from Syene to the Mediterranean as 750 Roman miles, i.e. 6000 stadia; FGrHist 667 F 1 = Pliny, NH 5.59). Bion wrote a description of the ethnography and geography of Aithiopia and Nubia (FGrHist 668). Simonides wrote an account of his five years living in Meroë (FGrHist 669). Philon was an agent for the Ptolemaic government who wrote a Voyage into Aithiopia (FGrHist 670). Timosthenes, the naval commander, wrote a substantial account of his voyages (FGrHist 2051) in which he quoted a 60 day travel time up the Nile from Syene to Meroë.

Put these two points together, and you’ve got Eratosthenes describing the course of the Nile as follows.

For Eratosthenes says the Nile flows northward from Meroë for 2700 stadia; then it turns back southward, in the direction of the winter sunset, for 3700 stadia; after almost reaching the latitude of Meroë, and projecting a long distance into Libya, it makes its second turn northward for 5300 stadia, until it gets to the great cataract, turning slightly eastward; then 1200 stadia to the smaller cataract at Syene; then 5300 stadia further to the sea.

Eratosthenes, Geography fr. 98 Roller (Strabo 17.1.2)

Notice the nice round numbers. They aren’t pure guesswork: these people were experienced professionals, after all. But they weren’t whipping out a theodolite at every bend in the river.

We can only conjecture how Eratosthenes reckoned this meandering route as 5000 stadia in a straight line. In the second stretch, at least, his figures imply a bearing directly southwest. Further north, it isn’t clear where exactly he imagined the river ‘turning slightly eastward’. The 1200 stadia stretch from the great cataract to Syene surely represents the Triakontaschoinos, or ‘land of 30 schoinoi’: this is probably the origin of Pliny’s claim that Eratosthenes reckoned the schoinos as 40 stadia (see below). Here are some speculative reconstructions.

The Nile with two hypothetical reconstructions of Eratosthenes’ account of its route. Distances are in stadia. Left: the Nile with real locations marked. Centre: my suggestion for how Eratosthenes might have envisaged the upper Nile. Right: reconstruction by Rawlins 1982, overlaid on the real geography. Eratosthenes’ stated figures are in bold, other figures are calculated (by myself or Rawlins). My reconstruction preserves a distance of 5000 stadia between Syene and Meroë in a north-south line; Rawlins puts them at different longitudes, has an unattested 5300 stadia for the north-south distance between them, and doesn’t link the 1200 stadia section with the length of the Triakontaschoinos. Left and right base images are from Google Earth, with spherical geometry centred on the Syene meridian; centre image is on flat geometry.

Whatever the details, a lot of approximation was involved. It takes a lot of massaging to make Eratosthenes’ measures look like the real course of the Nile. And we certainly shouldn’t be imagining some poor sap counting footsteps through hundreds of kilometres of desert along the direct route.

Eratosthenes’ distances were travellers’ educated estimates. They weren’t exact measures carefully surveyed with high-precision equipment.

Note. Priskin 2004 argues that these distances, and the 5300 stadia from Syene to Alexandria in the same passage, are derived from native Egyptian reckonings. The idea is that Eratosthenes drew on a Greek-language report in which distances in Egyptian schoinoi were converted to stadia at the rate of 50 stadia to the schoinos. I used to find this argument compelling, but it doesn’t stack up in light of the 3rd century BCE geographical literature mentioned above. Eratosthenes’ use of Timosthenes is directly attested (FGrHist 2051 T5–T8), and we know his figures had strong similarities to Philon’s (Geography fr. 40 Roller = FGrHist 670 F 2).

The length of the stadion

So, we’ve got distances of 5000 stadia. What’s that in modern units?

Our evidence for the length of a stadion varies greatly in its character and usefulness. Here’s the executive summary: the stadion as a metrological unit was anywhere between 177 m and 192 m, with the most traditional figure being an eighth of a Roman mile, that is, 183–185 m.

It’s also been suggested that stadia could also act as an itinerary unit when measuring long distances. You can think of this distinction as scientific versus practical. In metrological units, the distance from New York to Los Angeles is 3940 km (2450 miles) as the crow flies. In itinerary units, it’s six or seven days’ drive.

We have three types of evidence for the stadion.

1. Physical ancient racetracks, the literal meaning of stadion. These range from 177.3 m (Delphi) to 192.3 m (Olympia).
2. Ancient sources report a variety of conversion rates to other reckonings, namely the Roman mile and the Egyptian schoinos.
3. Reports of long distance measurements may indicate that the stadion could also be an itinerary measure.

The first two categories are basically metrological. Ancient units aren’t as precise as modern units, but the Roman mile was relatively stable and precise. Modern scholarship has a habit of treating it as 1480 m, thanks to a 19th century study; a more recent study puts it anywhere in the range 1465 to 1481 m. One eighth of that puts the stadion at 183 to 185 m, in good agreement with the racetracks. We do hear of some variations — 7.5 stadia to the mile, or 8.25, or what have you — but nothing too serious.

Note. Duncan-Jones 1980: 127 reports on evidence for the centuria which implies a Roman foot of 292.92–296.25 mm, and therefore a mile of 1464.60–1481.25 m (1 Roman mile = 5000 Roman feet). The 19th century study is that of Hultsch 1882: 88–90, who reports a range for the foot of 295.5–297.0 mm. The 8 : 1 rate goes back at least to Polybios (2nd cent. BCE; 3.39.8).

As for the schoinos — that’s the Greek name; the Egyptian word was itrw, literally ‘river’ — modern scholars like to quote a prototypical length of 10.5 km. This figure was devised in the early 1900s from some sample distance measurements, in conjunction with the fact that 10.5 km happens to be a nice round multiple of Egyptian royal cubits: 20,000 royal cubits of 0.525 m = 10.5 km. (There is no documentary basis for the 20,000 cubit figure.)

Note. For the origin of the 10.5 km and 20,000 cubit figures, see Tupikova 2022: 322–323, citing Loret 1903 and Borchardt 1921.

Now, here’s what Pliny says about converting between the stadion, the Roman mile, and the schoinos.

By Eratosthenes’ reckoning, a schoinos is 40 stadia, that is 5 miles; some have given a rate of 32 stadia to the schoinos.

Pliny, Natural history 12.53

Three things to notice here.

• Pliny repeats the usual 8 : 1 conversion rate for the mile, implying a stadion of 185 m.
• If we instead started from a schoinos at 10.5 km, his figures would imply a mile of 2630 m and a stadion of 263 m or 328 m.
• The 40 : 1 rate probably comes from the distance of 1200 stadia that Eratosthenes quoted for the Triakontaschoinos or ‘land of 30 schoinoi’, from the great cataract northward to Syene.

The key to resolving this apparent inconsistency is that the schoinos was not normally a metrological unit. It was an itinerary unit: specifically, a chunk of a boat journey upriver (recall that itrw means ‘river’). It may have described a stretch of the Nile covered by one towing team (New Pauly s.v. ‘schoinos’).

So you’d expect its physical length would vary depending on terrain. And that’s exactly what ancient sources tell us.

[From Alexandria to the top of the Delta] Artemidoros states that the journey upriver is 28 schoinoi, which is 840 stadia, reckoning 30 stadia to the schoinos. But when I made the trip, they gave the distances using different reckonings in different places, so that they agreed on 40 stadia or even more to the schoinos ... Artemidoros himself [says] ... that from Memphis to the Thebaid the conversion is 120 stadia to the schoinos, and from the Thebaid to Syene it is 60 ...

Strabo 17.1.24

Pliny and Strabo make it crystal clear that they’re talking about variations in the length of the schoinos, not the stadion. They’re using stadia to express that variation. Pliny makes similar statements about the Persian schoinos (Natural history 6.125).

Lengths for the stadion based on racetracks and ancient conversion rates, treating them as if the mile and schoinos were metrological units of 1480 m and 10.5 km respectively. The ‘sane range’ for the stadion, in green, includes the physical racetracks, and all conversions with the Roman mile except for two late figures from Dío Cassius and Julian of Ascalon. A full tabulation of the sources and their conversion rates is given below in Appendix A.

Some modern reports are cautious enough about the schoinos to call the 10.5 km figure a convenient ‘average’. Even that’s doing it a kindness.

The upshot is that the schoinos is utterly useless for reconstructing other ancient units.

Note. In this series I wanted to tell the true story without distractions, but there’s no dodging some widely repeated falsehoods. The following claims come from Friedrich Hultsch (1882: 60–62), and they are all false. For fuller explanations see Appendix B below. Egyptian measurements were rigidly regulated by the royal cubit (0.525 m; ‘dem Ptolemäischen Maße ... welches nach dem genauen und beständigen Maßstabe der alten ägyptischen Königselle geregelt war’). The schoinos was ‘firmly determined’ (fest bestimmt) as 12,000 cubits. Different conversion rates reflect variations in the length of the stadion, not the schoinos. The length of the stadion is to be determined exclusively from (a) conversion rates with the schoinos in the pseudo-Heronic Geometrika (ii.6 = iii.5 Bruins) — though not the conversion rate that the Geometrika quotes for the mile! — and (b) Pliny’s statement that Eratosthenes used a 40 : 1 rate with the schoinos. The length of Eratosthenes’ stadion was 157.5 m. Eratosthenes commissioned ‘bematists’ to pace out distances.

Our useful metrological evidence boils down to the racetracks and the Roman mile. And as we’ve seen, they’re broadly in agreement, if not very precise: they produce a metrological stadion of 177–192 m, or let us say, 185 m ± 8 m.

It remains to consider the possibility of reckoning in itinerary stadia. A useful 2022 article by Irina Tupikova goes over this, looking at ancient sources generally, not just Eratosthenes.

Where distances quoted by ancient sources can be compared with the real distances, her analysis indicates that distances under 1000 stadia (in metrological terms, 185 km = 115 miles) tend to be significantly overstated. Distances over 1000 stadia are very variable, sometimes overstated, sometimes understated.

You can explain this in two basic ways:

1. Ancient surveying methods weren’t very accurate.
2. Distances between cities are a representation of journey times.

Take a modern analogy. Let’s say you’re aware that it’s six or seven days’ drive from New York to Los Angeles. If you accept a common reckoning that you can drive 800 km (500 miles) in a day, you can estimate the total distance as 4800–5600 km (3000–3500 miles). The real distance by road is 4500 km (2800 miles). (Why the discrepancy? All sorts of reasons: fatigue, terrain, supplies, accidents, etc.)

So we could say that the distance is 4800–5600 km in ‘itinerary kilometres’, which are about 7%–17% shorter than actual kilometres. I’m not saying this would be wise. But you could.

This is the kind of conclusion Tupikova reaches about stadia. For land distances over 125 Roman miles, Pliny the Elder reports distances which could be taken to suggest an ‘itinerary mile’ which is shorter than the 1480 m mile by 7% to 23%; for sea distances under 125 miles, his miles tend to be around 36% short. When Strabo reports distances from Eratosthenes’ Geography, they could represent an itinerary stadion around 13% to 31% short of the 185 m stadion.

If you take the ‘itinerary stadion’ at face value, Eratosthenes’ measurement turns out to be highly accurate. Tupikova reckons that the measure of 157.5 m represents Eratosthenes’ ‘itinerary stadion’: using that, his measurement of the earth’s circumference at 252,000 stadia is impressively precise: it comes to 30,690 km, just 0.79% shy of the true value.

Except that’s total nonsense. It’s circular. The argument goes: if we use distance units based on the figures that Eratosthenes gives for the distances between Alexandria–Syene–Meroë, then his measurement of the earth, which is also based on those same figures, turns out to be accurate too.

When you put it that way, the 0.79% precision isn’t quite so amazing.

That isn’t what the ‘itinerary kilometre’ and the ‘itinerary stadion’ are for. They don’t work like that. If someone travelling across America converted ‘six days’ drive’ to kilometres at the rate of 800 : 1, and you confronted them with the true metrological distance, they wouldn’t say: ‘Well actually I was working in itinerary kilometres, and in those, my measurement is perfectly precise.’ No: they’d admit that it was an estimate.

Eratosthenes would say the same. In fact he did say the same. Remember, he openly stated that he didn’t verify his figures.

Eratosthenes didn’t quote the distance from Syene to Meroë as 5000 stadia because he was using ‘Eratosthenean stadia’, and Eratosthenean stadia happen to be 1/5000 of that distance. He did so because he relied on imprecise measurements.

The circumference of the earth

We’re left with the metrological stadion, of 177–192 m. Using that standard, Eratosthenes’ figure of 5000 stadia for the distances Alexandria–Syene and Syene–Meroë is too high. 5000 stadia is between 890 and 960 km; the real distances, as the crow flies, are 840 km and 800 km respectively; the north-south separation, ignoring the different longitudes, is 790 km for both distances.

That puts Eratosthenes’ figure for the earth’s circumference between 44,680 km (stadia of 177.3 m) and 48,460 km (stadia of 192.3 m). The traditional reckoning of 8 stadia = 1 Roman mile (1480 m) gives 46,620 km. The earth’s true polar circumference is 40,008 km. Eratosthenes’ result is between 11.7% and 21.1% too high; the traditional reckoning is 16.5% high.

There’s no need to make excuses for that. This is still impressive for the 3rd century BCE. It doesn’t need to be accurate within 0.79% to be a terrific achievement. And it has to be said, given that Eratosthenes didn’t verify his data, he got lucky. The estimates he used could easily have been much wilder.

Next week, the finale: part 4, the problem of measuring the sun’s angle.

Appendix A. Table of ancient sources

Here I give the lengths of physical racetracks and ancient conversion rates between the stadion, schoinos, and Roman mile, treating them as if all three were metrological units.

Source	Conversion rate	Resulting length of stadion
Ancient racetracks (Gulbekian 1987: 360; New Pauly s.v. ‘stadion’)	—	177.3–177.5 m (Delphi) 181.3 m (Epidauros) 184.3–185.0 m (Athens) 191.4 m (Miletos) 192.25–192.30 m (Olympia)
Herodotos 2.6, 2.9	60 stadia = 1 Egyptian schoinos	175.0 m (schoinos = 10.5 km)
Herodotos 2.149; pap. Heidelberg 1289	1 stadion = 400 Egyptian cubits	210.0 m (royal cubit = 0.525 m)
Eratosthenes, reported by Pliny, Natural history 12.53	40 stadia = 1 schoinos	262.5 m (schoinos = 10.5 km)
‘others’, reported by Pliny, Natural history 12.53	32 stadia = 1 schoinos	328.1 m
Artemidoros, reported by Strabo 17.1.24	30, 60, or 120 stadia = 1 schoinos	350.0 m, 175.0 m, or 87.5 m
Strabo 17.1.24	30 or 40 stadia = 1 schoinos	350.0 m or 262.5 m
Geometrika (pseudo-Heron) ii.6, iii.5 Bruins	30 stadia = 1 schoinos 45 stadia = 1 ‘barbarian’ schoinos 60 stadia = 1 ‘Persian’ schoinos	(these illustrate that the schoinos is an itinerary measure, not metrological; also claims that the Roman mile = 5400 Roman feet)
Eratosthenes, reported by Pliny, Natural history 5.59, 2.85, 12.53	40 stadia = 5 Roman miles; Syene to Meroë (5000 stadia) = 625 Roman miles	185 m (mile = 1480 m)
Eratosthenes and Strabo, reported by Julian of Ascalon (i.201 Hultsch)	8¼ stadia = 1 mile	179.4 m (mile = 1480 m)
Polybios 3.39.8	8 stadia = 1 Roman mile	185 m (mile = 1480 m)
Polybios, reported by Strabo 7.7.4	8⅓ stadia = 1 Roman mile	177.6 m (mile = 1480 m)
‘most people’, according to Strabo 7.7.4	8 stadia = 1 Roman mile	185 m (mile = 1480 m)
Plutarch, Gaius Gracchus 7.2	a little under 8 stadia = 1 Roman mile	a little over 185 m (mile = 1480 m)
Censorinus 13.2	1 stadion = 625 Roman feet (⅛ mile)	185 m (mile = 1480 m)
Dio Cassius 38.17.7, 39.50.2, 46.44.4, etc.	7½ stadia = 1 Roman mile	196.1 m (assuming post-200 CE mile ≈ 1471 m: Hultsch 1882: 97)
Julian of Ascalon i.201 Hultsch	7½ stadia = 1 Roman mile	196.1 m (post-200 CE mile ≈ 1471 m)
Geometrika 22.1 Heiberg	7½ stadia = 1 mile = 4500 royal feet = 5400 ‘Italian’ feet	(meaning obscure; the ancient Roman mile = 5000 Roman feet)

Appendix B. Problems with Hultsch’s account of ‘Eratosthenes’ stadion’

• If the schoinos really were ‘firmly determined’, then the ‘average’ schoinos of 10.5 km would mean (a) at the 40 : 1 and 32 : 1 rates that Pliny quotes, the stadion was 263 m or 328 m; (b) Artemidoros’ reckoning would produce a stadion anywhere from 88 m to 350 m. Hultsch’s reckoning of 157.5 m doesn’t fit in here anywhere.
• All scholars in the last century have quoted a rate of 20,000 cubits in the ‘average’ schoinos, not 12,000 (see e.g. Priskin 2004; Tupikova 2022: 323). As we’ve seen, (a) even that’s just a byproduct of the 10.5 km reckoning, and (b) it’s far too precise, given that the schoinos is itinerary while the cubit is metrological.
• The only basis for the 12,000 cubit figure is the pseudo-Heronic Geometrika, which comes with its own problems.
• It’s idiosyncratic: e.g. it quotes the rates 1 ‘mile’ = 5400 ‘Italian’ feet = 4500 phileteric feet, rather than the 5000 foot standard.
• The Geometrika gives a rate of 30 stadia to the schoinos.
• The Geometrika treats all these units as metrological: this wasn’t the case in antiquity.
• The vocabulary of the Geometrika implies a Byzantine date. It refers to ‘Italian’ feet and miles rather than ‘Roman’, surely on the grounds that by the time it was written, ‘Roman’ included all of the units it lists. It was only around the 5th century CE that ‘Roman’ started to be used for the people of the eastern empire. The Geometrika should be grouped with Julian of Ascalon among sources too late to be relevant.
• Hultsch selectively disregards the 8 : 1 rate with the mile which Pliny attributes to Eratosthenes in the same sentence; the 8¼ : 1 rate with the mile which Julian of Ascalon attributes to Eratosthenes; the 30 : 1 rate with the schoinos in the Geometrika; and all of Polybios’, Artemidoros’, Strabo’s, and Plutarch’s testimony (among others).
• These selections are transparently aimed at trying to make Eratosthenes’ measurement work out to be as accurate as possible.
• The only basis for the notion of ‘pacing out distances’ is the etymology of the rare word βηματίζω ‘measure’. The primary meaning comes from the distance unit the βῆμα (‘pace’, usually 2.5 or 3 feet). Ultimately both are derived from βη- ‘walk’, but etymology isn’t a reliable guide to meaning. Prior to the Byzantine period βηματίζω appears only in contexts where literal measurement by pacing would be impossible (Dionysios Chalkous fr. 3.5 West) or where we know other measuring methods were used (Polybios 3.39.8).
• Although Hultsch’s account has long been rejected, much of his misinformation lingers. Popular accounts still accept without question that the stadion and the schoinos were 157.5 m and 12,000 cubits. Even Tupikova’s 2022 article gives airtime to the 157.5 m stadion, and the notion of surveyors pacing out distances.

How Eratosthenes measured the earth. Part 2

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

Around 240–230 BCE Eratosthenes measured the circumference of the earth as 252,000 stadia. This translates to 46,620 km, plus or minus 1900 km. That’s 12% to 21% higher than the true distance, which is 40,008 km, but it’s still a great illustration of using down-to-earth practical observations to measure something on a colossal scale.

The Alexandria meridian according to Eratosthenes. The sites on the meridian are recorded in Eratosthenes, Geography fr. 34, fr. 60 ed. Roller. See Roller 2010: 25–26 for discussion. (Base image: Google Earth; spherical projection, centred on the Alexandria meridian)

It didn’t come out of the blue. Astronomers in Greece 150 years earlier had discovered that the earth is spherical, as we saw last time. Eratosthenes also drew on books published by Ptolemaic explorers reporting distances along the Nile, and observations of the sun using astronomical instruments. Not just sticks in the ground! But the simplest instrument, the sundial stick or gnomon, had been in use for this purpose and carefully refined by Egyptian astronomers for two thousand years.

We know of two earlier estimates of the earth’s circumference: Aristotle reports an estimate of 400,000 stadia (74,000 km), Archimedes reports one of 300,000 stadia (55,500 km). But where they quoted guesstimates, Eratosthenes used empirical observation.

Note. Aristotle, On the sky 297b.30–298a.17; Archimedes, Sand-reckoner 8 (ii.246 Heiberg, 222 tr. Heath).

(a) Eratosthenes

Eratosthenes was born around 280 BCE in Cyrene, a city founded in Libya some 350 years earlier by Greek colonists from Thera. (‘Libya’ works both in modern English and ancient Greek, by the way. The difference is that ancient Greek Libýe refers to the whole of the Maghreb, not just modern Libya.)

Note. Eratosthenes’ birth date is uncertain. The Souda lexicon puts it in 276–273 BCE, but another source, Strabo 1.2.2, states that he studied under Zeno of Kition, the founder of Stoicism, who died in 262/1 BCE.

He studied in Cyrene, then at the Academy in Athens. In Athens he was headhunted by Ptolemy III. He moved to Egypt and became the director of the Mouseion or ‘shrine of the Muses’ — more famously known as the ‘library of Alexandria’.

Eratosthenes made important contributions in many areas. He was the first to devise the concept of lines of latitude and longitude, and he used them to make the first attempt at a comprehensive atlas of the known world. His studies of music theory and tuning were influential in ancient music. Modern mathematicians know him for the ‘sieve of Eratosthenes’, an early method for generating prime numbers. He wrote epic and elegiac poetry, and wrote extensive scholarly commentaries on older literature, especially comedy. And he produced the most influential chronography of his era, synthesising multiple histories with different calendar systems into a single united timeline.

All of his writings are lost, except one, the Katasterismoi, an account of the mythical stories behind the constellations. Even that book is heavily contaminated by later alterations.

His calculation of the earth’s size, published in a book called On the measurement of the earth, was as famous in antiquity as it is today. Only fragments and indirect reports survive: they’re collected in Roller’s edition of Eratosthenes’ fragmentary Geography (Roller 2010: 263–267).

A hieroglyphic symbol for the island of Elephantine (Aswan, Egypt), devised to honour Eratosthenes’ measurement of the earth, in the form of a try square and plumb bob, tools used to align a gnomon.

(b) The methodology

Conceptually, the procedure was straightforward.

1. Choose two sites and measure the distance between them in a straight line along the earth’s surface. This represents a fraction of a great circle going all the way around the earth.
2. Determine that fraction by measuring the sun’s angle at both sites at the same time.
3. Multiply to find the circumference of the whole circle.

The sun’s rays at sites 1 and 2 are practically parallel, since the sun is very distant. Some simple geometry (omitted here) then shows that the difference in the sun’s observed angle at sites 1 and 2 is equal to the sites’ angular separation along the great circle around the earth. (It doesn’t matter whether you’re measuring the sun’s angle from the horizontal, or vertical, or some other reference point: the difference in angle is the important thing.)

Eratosthenes’ chosen sites were 5000 stadia apart, where a stadion is roughly 185 metres. (We’ll come back to this in part 3.) And the difference in angle turned out to be 1/50 of a full circle. Therefore, the circumference of the earth came to 250,000 stadia.

According to most reports, Eratosthenes adopted a ‘rounded’ figure of 252,000 stadia, around 46,620 km. That’s probably because it gives a tidy figure of 700 stadia per degree of latitude: 252,000 = 700 × 360. It also gave tidy figures for the distances between the pole and the Arctic Circle, and between the tropic and the equator.

Note. 250,000 stadia: Measurement M6 Roller (Kleomedes 1.7 = 100,19–102,5 Ziegler). 252,000 stadia: Measurement M1 Roller (Geminos 16.6); M2 (Macrobius, On the dream of Scipio 2.6.3); M3 (Vitruvius 1.6.9); M4 (Pliny, Natural history 2.247); M5 (Censorinus 13.2); M7 (Martianus Capella 6.596); Geography frs. 30, 34 Roller (Strabo 2.5.6–7). (Carlos Carman and Evans 2015 argue that these two different figures are the result of using upper and lower bounds for the distance of the sun. I’m not persuaded. For one thing, Kleomedes explicitly states that the angular separation was specifically ‘1/50 of a great circle’, not 7.129° as the authors calculate for the lower bound. For another, there’s good reason to infer that Eratosthenes used instruments that measured the sun’s angle directly, without trigonometry: see part 4.) A third figure, 259,200 stadia (i.e. 720 stadia per degree of latitude), appears in Geography fr. 29 Roller (Markianos of Herakleia, Períplous of the outer sea 1.4). Roller 2010: 144 suggests that this may have come from an alternate calculation by Dionysodoros of Melos.

That’s the result. There were major technical hurdles to obtaining the measurements, though.

1. Simultaneous measurements. Sites 1 and 2 need to be hundreds of kilometres apart. Using ancient techniques, how do you ensure that the measurements at each site are taken at the same time?
2. Distance. How do you tell how far apart the sites are?
3. Angle of the sun. How do you measure the difference in the sun’s angle? Come to that, how do you make even one measurement of its angle?

Some popular accounts of the story give a good explanation of the first point, but misrepresent the second and third. Perhaps those points seem trivial to a modern audience. But Eratosthenes didn’t have ordinance survey maps or trigonometry. Here and in parts 3 and 4 we’ll look at each problem, and the solutions that he found. Or at least the kind of solution he found.

(c) Simultaneous measurements

The simultaneity problem required a geographer with a good conceptual command of the earth’s spherical geometry and of the idea of meridians. Eratosthenes was ideally placed: he was the geographer who invented the idea of lines of longitude and latitude.

Time zones were the solution. Ancient astronomers knew that observers at different longitudes see an astronomical event, like a lunar eclipse, at different hours of night depending on how far east or west they are. As we saw in Part 1, this is one of the points that Kleomedes cites to corroborate the earth’s shape.

But the reverse is also true. Observers at the same longitude see an astronomical event at the same hour. So if observers at the same longitude were to measure the sun’s angle at the same hour of the same day — say, midday on the equinox — then their measurements would be simultaneous.

In his Geography Eratosthenes had already plotted out the Alexandria meridian. To the north it ran through Rhodes (Greece), Lysimacheia (Gallipoli, Türkiye), and Olbia (50 km west of modern Kherson, Ukraine); to the south it went through Syene (modern Aswan, Egypt) and the Kushite capital Meroë (Al Bagrawiya, Sudan).

His data weren’t perfect. These cities aren’t actually at the same longitude: in modern notation, they range from 26.9° E (Lysimacheia) to 33.7° E (Meroë). Longitude was tremendously difficult to determine accurately until the 1700s CE.

The sites used for Eratosthenes’ measurement. (Base image: Google Earth; spherical projection)

For his measurement of the earth, Eratosthenes chose the sites along the Nile: Alexandria, Syene, and Meroë. Some sources report that he used readings taken at Syene and Meroë; others, readings at Alexandria and Syene. The simplest interpretation is that he used both pairs. He certainly reckoned each pair as being the same distance apart. Their placement, with Syene on the Tropic of Cancer, and Alexandria and Meroë equidistant to north and south, made them ideal for calibrating each other.

Note. Syene was of intrinsic interest for two reasons. First, because it was regarded as the southern border of Egypt. Second, because it lay on the tropic, meaning that at midsummer the sun was directly overhead at midday. Earlier Egyptian astronomers were well aware of this, and had devised refinements to the gnomon to compensate for it (Isler 1991: 164–167).

According to Eratosthenes, the equator is 252,000 stadia. ... From the equator to the summer tropic is 4 sixtieths (of the total circumference); and this is the parallel drawn through Syene. The tropic passes through Syene because there, at the summer solstice, a gnomon is shadowless at midday. And the meridian through Syene is drawn roughly along the course of the Nile from Meroë to Alexandria, and the distance is about 10,000 stadia. And Syene is situated halfway between them, that is, 5000 stadia from Meroë.

Eratosthenes, Geography fr. 34 Roller (Strabo 2.5.7)

The time selected for the measurements was midday on either of the equinoxes. Midday transits are the obvious choice because that’s a unique moment in the day, when the sun is at its highest in the sky. As we’ll see in part 4, ancient astronomers used devices designed specifically to measure the angle of the sun’s shadow at midday; the very name ‘meridian’ means ‘midday’ (Latin meridies = Greek mesembria, which means both ‘midday’ and ‘due south’).

He may perhaps have used solstice readings too, but the evidence doesn’t actually say that. The midsummer solstice comes up in this connection simply because Syene is located on the tropic, and at the tropic, the midday sun is directly overhead at the summer solstice. In any case equinoctial readings are more useful: equinoxes happen twice as often. When Pliny and Strabo express latitudes in terms of gnomon readings, they cite only equinoctial readings.

Note. Eratosthenes using transit readings from Syene and Meroë: Measurement M7 Roller (Martianus Capella 6.598). Alexandria and Syene: M6 Roller (Kleomedes 1.7 = 96–103 Ziegler). Readings taken at the equinoxes: M3 Roller (Vitruvius 1.6.9), M7 Roller (Martianus Capella). Reference to Eratosthenes’ measurement in connection with solstices: Geography 34 Roller (Strabo); Measurement M6 Roller (Kleomedes). Strabo relates that the explorer Phílon published equinox and solstice gnomon readings from Meroë, then adds that ‘Eratosthenes agrees very closely with Phílon’ (Geography fr. 40 Roller = Strabo 2.1.20 = FGrHist 670 F 2). Equinoctial gnomon readings: Pliny, Natural history 2.182; Strabo 2.5.38.

As it happens, equinoctial readings also have the benefit that at the equinoxes, and only at the equinoxes, the sun’s rays are parallel to the earth’s equator. As a result, at midday the sun’s angle from the vertical is equal to your latitude. Alexandria is at 31.2° N, and at midday on the equinox, the sun’s angle from the vertical is 31.2°. (I haven’t found any indication that Eratosthenes was aware of this point.)

At midday on the equinox, the sun’s angle to the vertical is equal to the latitude where the reading is taken.

Next week, part 3: solving the problem of distance.

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.