Monday, 30 November 2015

Eratosthenes and the well

Eratosthenes is one of the most famous individuals of the Greco-Roman world -- and justly so: he was a leading expert in cartography, philology, mythography, ethnography, geometry, and astronomy. In an age of nepotism, his raw talent and hard work got him headhunted by Ptolemy III while he was still living in Athens.

Nowadays, he is most famous for making a reasonably accurate calculation of the circumference of the earth, using 3rd century BCE data and methods. This was a celebrated feat in his own time too. The island of Elephantine, at Syene (modern Aswan), had a new hieroglyphic symbol for its name created, apparently in honour of his calculation, in the form of a plumb bob and try square.
Hieroglyph
for Elephantine
(one of many variants)

No myths so far: this story is all true. By calculating the relative angle of elevation of the midday sun in three cities on the same meridian and at the same season -- Alexandria, Syene (modern Aswan), and Meroë (the chief city of what the Greeks called Aithiopia; modern Bagrawiya, Sudan) -- with figures for the distances between these cities, and with the assumption that the earth's curvature was spherical -- he came up with a figure variously reported as either 250,000 or 252,000 stadia.

In terms of angular precision, the calculation was very exact. In terms of absolute distances, not so much. Eratosthenes' own writings on the subject do not survive. It's most likely that he calculated 250,000 stadia, but that the figure got "rounded" to 252,000 so as to give a tidy figure of 700 stadia per degree of the earth's circumference (360° × 700 = 252,000). More importantly, Ptolemaic methods for surveying long distances were very inexact -- not nearly as good as Roman measures, for example -- and, to boot, there were many variants of the stadion ranging from ca. 157 metres to ca. 262 metres. The problem of Eratosthenes' units is hair-pullingly complicated. (Maybe we'll revisit the subject one day. But then again, maybe not: it really is a messy topic.)

But how he calculated the earth's circumference -- that's where the myths come in. Here's an extract from an especially popular and influential account:
One day, while reading a papyrus book in the library, he came upon a curious account. Far to the south, he read, at the frontier outpost of Syene, something notable could be seen on the longest day of the year. On June 21st, the shadows of a temple column or a vertical stick would grow shorter as noon approached. And as the hours crept towards midday, the sun's rays would slither down the sides of a deep well, which on other days would remain in shadow. And then precisely at noon columns would cast no shadows, and the sun would shine directly down into the water of the well. At that moment the sun was exactly overhead. It was an observation that someone else might easily have ignored -- sticks, shadows, reflections in wells, the position of the sun: simple everyday matters. Of what possible importance might they be?
-- Carl Sagan, Cosmos (1980; episode 1, "The shores of the cosmic ocean")
NOT how Eratosthenes calculated the earth's circumference
The story of the well is repeated in a lesson at the Khan Academy, in an even more colourful (and fictional) account in Julia Diggins' 1965 book String, Straightedge, and Shadow, and in many more places.

And it is mostly false. There was a well like this at Syene: but it certainly had nothing to do with Eratosthenes, and nothing at all to do with his calculation.

Here's Pliny the Elder's account of the well (Natural History 2.183 [§75]):
Similarly they say that in the town of Syene, 5000 stadia south of Alexandria, no shadow is cast at noon on the solstice. A well was made to test this, and it was entirely illuminated. This showed that the sun was directly overhead at that time. Onesicritus states that the same thing happens at that season in India south of the river Hyphasis [i.e. the modern Beas].
(The source Pliny was using is vague: "south of the Hyphasis" sounds like it should indicate Punjab, but the tropic runs considerably south of Punjab. It does run close to Pataliputra, though, further to the south-east, which was the capital of the Maurya Empire until the early 2nd century BCE: that is probably what Pliny's Hellenistic source was talking about.) Strabo also describes the well (17.1.48), and his phrasing tends to suggest that it was moderately famous, though unlike Pliny he doesn't say that its unique feature was the very reason it was built.

No ancient source, at all, anywhere, connects this well to Eratosthenes. In the surrounding context of these passages, however, both Pliny and Strabo do mention the actual method that Ptolemaic surveyors used for measuring latitude: they used an instrument called the gnomon (Pliny NH 2.182; Strabo 17.1.48).

Gnomons were originally for determining the date of the solstice. This practice goes back many thousands of years, in many different civilisations. Meton used a gnomon in Athens in 432 BCE for this purpose; the Shang people in ancient China were using them to measure solstices in the 13th century BCE. In Egypt, one of the reliefs in the jubilee chapel of Senusret I depicts a 9-metre-high gnomon in connection with the festival of Min in the 20th century BCE.

In its simplest form, the gnomon was a vertical rod that cast a shadow. Measuring the ratio between the shadow and the length of the rod would constitute a gnomon reading. Old Egyptian gnomons typically had a bifurcated tip to make the shadow more defined (as in the Senusret relief). The Egyptians could use hand-held ones to mark the passage of time at night. Travellers in the Greco-Roman world had portable gnomons or sundials, in the form of a round vessel with a stylus embedded in the centre and markings on the sides: this set-up was called a skaphion (Greek) or umbilicus (Latin). The Egyptians were well aware that near the tropic it was difficult to measure the solstice with a vertical gnomon, because the shadows are so short at midsummer, and so they adopted the practice of tilting the gnomon to the north and supporting it with struts. (Further reading, for those with JSTOR access.) In later gnomons, plumb bobs were used to ensure it was exactly vertical: to judge from the hieroglyphic for Elephantine mentioned above, it looks like this was the case with the Ptolemaic gnomon. (And incidentally, remember that the symbol for Elephantine included a try square? It so happens that gnomon was also the Greek word for a try square.)

Ancient travellers regularly took gnomon readings as a way of recording their latitude. Pliny, just before the passage quoted above, tells us
Portable timepieces are not used the same way everywhere, because the sun's shadows change every 300 stadia, or at most 500 stadia [i.e. about 0.5° latitude]. So in Egypt at the equinox the shadow is only a little more than half the length of the umbilicus -- what they call a gnomon. In the city of Rome the shadow is 8/9 the length of the gnomon at the same season; in the town of Ancona it is 1/35 longer (i.e. 8/9 + 1/35, or 0.917); and in the region of Italy called Venetia the shadow is the same length.
Vitruvius gives another list of equinoctial gnomon readings in various cities. Ptolemy has an extended account of gnomons and latitude in book 2 of his Almagest. And Martianus Capella indicates that Eratosthenes' measurement was based on gnomon readings taken at the equinox (De nuptiis 6.597).

The practice of taking gnomon readings as a geographical measurement goes back at least to Pytheas of Massalia, a famed traveller of the early 4th century BCE. According to Martianus Capella (De nuptiis 6.595), Pytheas took gnomon readings all the way from southern France to "Thoule" (either Iceland, or one of the island groups in the North Sea, or perhaps Scandinavia). Also before Eratosthenes' time, Philon, a surveyor for Ptolemy II, reported gnomon readings at Meroë in his book the Aithiopika or Voyage to Aithiopia. The book itself does not survive, but this report does:
Philon discusses the latitude (κλῖμα) of Meroë in the Voyage to Aithiopia that he wrote. He says the sun is directly overhead 45 days before the summer solstice, and he reports his readings of the ratio between the gnomon and its shadow at the solstices and equinoxes. Eratosthenes agrees very closely (συμφωνεῖν ἔγγιστα) with Philon.
Philon, New Jacoby 670 F 2 (=Strabo 2.1.20)
So, to recapitulate:
  • Eratosthenes didn't use a well to measure anything: the instrument of choice was the gnomon.
  • Taking latitude readings from a gnomon was standard practice for Ptolemaic surveyors decades before Eratosthenes came along.
  • The summer solstice barely came into it; gnomon readings were taken year-round, but it looks like equinoctial readings were the basis of Eratosthenes' calculation.
  • Eratosthenes didn't take the readings himself: he probably used Philon's published work as the basis of his calculation.
The well at Syene was just a striking, large-scale visualisation of the Tropic of Cancer. Pliny's wording suggests it may not even been built until after Eratosthenes' calculation anyway. And native Egyptians had known for centuries that Syene was on the Tropic of Cancer. (In fact Syene was slightly north of the tropic: it's at 24.09° N, and the tropic was at 23.72° N in Eratosthenes' time. The plane of the ecliptic shifts slightly over the millennia: currently it's at 23.44°.)

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