## Friday 7 July 2023

### 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]

• 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]

1. Thank you! That is a fascinating topic
I am not sure that the lack of trigonometric functions would stop Eratosthenes from using gnomon readings. If you are capable of making a ring with degree markings, it is possible to use this instrument to construct a right triangle with specific angle, measure it's sides and there is your way to get a tan function. An inverse tan would imply a painstaking process of tweaking the angle until you get the correct sides ratio but thats no more time consuming than sailing the Nile with your transit instrument.
And speaking of Ptolemy, do you have an opinion there was he getting his coordinates with 1/12 degree precision? Was he digitizing existing maps, or using some sort of straightedge and compass construction based on periplous and itinerary data? The origin of Ptolemy's Geography is the most interesting question for me, given how large and influential this work is

1. It is possible, I agree, but (a) I very much doubt the precision of such an instrument; and (b) I think it's best to take Ptolemy at his word about the instruments in use: there's a second one in the same passage, which he says is more precise than the circular instrument.

Even the circular instrument looks far easier to engineer than a skaphe -- Ptolemy says it has precision better than a degree -- plus there's that 2nd century BCE graffito from Meroë pointing to its use.

Pliny speaks of skaphai as being for the use of travellers: for precision work, Ptolemy's instruments must be the way to go.

Ptolemy wasn't taking all his measurements himself of course. His coordinates are very precise -- for latitude, at least -- but not very accurate. Some are much more inaccurate than others -- I mentioned the ones in Tunisia for Carthage and Leptis Magna -- but are in broad agreement with other sources like Strabo, so that suggests measurements were taken at some point and never precisely checked.

I'm no expert on the origins of the Geography: you'll have to look elsewhere for that. The body of work written by Tupikova and Geus is certainly worth following up. There's a 2013 article by them that seems to suggest his coordinates imply the use of multiple meridians in the Geography: that would suggest a synthesis of multiple surveys. But as I say, Tupikova is the expert (even if I think she's wrong to focus on the gnomon and skaphe rather than the instruments Ptolemy himself describes).

2. I guess we really cant be sure how the Eratosthenes' computation went given the scarcity of sources. That is why Ptolemy's Geography is so interesting to me: it is really large, so there can't be many different ways to explain it all.

There is another detail that bothers me. Did they have to wait for equinox to estimate latitude? That may be not very practical, especially for travelers. Sun altitude on other dates can also be used, you just have to subtract Sun's declination on that date. Do you think it is possible that some of the erroneous latitudes can be explained by mixup of the dates of otherwise accurate measurements? Did they have some strange local calendar in Carthage?

3. Did they have to wait for equinox to estimate latitude? No, not necessarily: that just seems to be the standard time for reporting gnomon ratios, as in Pliny and Strabo. For other times of year you can do some fancy tricks with daylight hours and the angle subtended by the sun's path through the sky, though I'm not able to explain them fully with my current state of knowledge.

Ancient travellers might perhaps have had access to the calculation methods: the testimony in Pliny doesn't look very clear to me. But I doubt it. Given how erroneous Pliny's knowledge is about some pretty basic stuff, I suspect those methods must have been for experts only.

I have no compelling explanation for the error in Carthage's latitude. The latitude Ptolemy quotes is very close to being compatible with the gnomon ratio in Strabo, so my best guess is that erroneous measurements were taken on a single occasion, and were never checked. But how the error happened in the first place? -- no idea!

2. My new favorite blog! I've already caught some of the myths and half-truths I've been telling my students over the years JUST about Eratosthenes! What a resource, Peter, you are for us secondary Latin instructors!

1. Glad it's of some use! Future posts are likely to be less and less frequent, though -- it's become very hard to find any kind of audience this year. YouTube seems to be the only place people go to find out stuff now. Which is a pity, since making videos requires a much larger skillset and time commitment. (I am thinking of trying a video version of this, actually, but it's never going to be a high quality thing with animated bells and whistles.)

2. Just a single reader, but I would prefer if you'd continue to post here; for knowledge, I rather prefer text to video as a format.