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.