Syene — the modern Egyptian city of Aswan, on the eastern bank of the Nile approximately 900 km south of Alexandria — sits at latitude 24°05’N. This puts it almost exactly on the Tropic of Cancer, the line on the Earth’s surface at which the Sun stands precisely overhead at noon on the summer solstice. At every other latitude, the noon Sun stands either north or south of vertical; at the Tropic of Cancer, once a year, it stands directly above.
The practical consequence is that a vertical pole at Syene casts no shadow at noon on the summer solstice. A deep vertical well at Syene is illuminated all the way to the bottom by direct sunlight. The local Hellenistic-Egyptian population had been observing this phenomenon for centuries before Eratosthenes — the Greek polymath who served as chief librarian of the Library of Alexandria from approximately 245 to 195 BC — used it as the empirical basis for a famous geometric computation.
The measurement
Eratosthenes’s reasoning, as preserved in the 1st-century AD treatise of Cleomedes (the original Eratosthenes treatise On the Measurement of the Earth is lost), worked as follows. At noon on the summer solstice at Syene, the Sun was directly overhead. At the same moment at Alexandria, a vertical gnomon (a measuring pole) cast a shadow at an angle of approximately 7.2° from vertical — about one-fiftieth of a full circle of 360°.
The geometric reasoning: if the Sun is so distant from Earth that its rays arrive essentially parallel, then the angular difference between the two noon observations equals the angular separation along the Earth’s surface between the two cities. The Alexandria–Syene line therefore subtends approximately 1/50 of the Earth’s circumference. If the surface distance between the two cities is known, the total circumference is fifty times that distance.
Eratosthenes used the standard Hellenistic Egyptian estimate of the Alexandria–Syene distance: approximately 5,000 stadia. The resulting Earth circumference was approximately 250,000 stadia.
How accurate
The accuracy of Eratosthenes’s figure depends entirely on which ‘stadion’ he used (the Greek and Egyptian world had several incompatible stadia of different lengths). The Greek ‘Olympic’ stadion of approximately 185 metres gives a circumference of about 46,000 km — about 15% too large. The Egyptian-Ptolemaic stadion of approximately 157 metres gives about 39,000 km — about 2% short of the modern figure of 40,075 km. Modern scholarship is divided on which unit Eratosthenes intended; either way, the figure is impressively close to correct.
The two observational assumptions — that Syene sits exactly on the Tropic of Cancer and exactly due south of Alexandria — are both very slightly wrong. Syene is approximately 55 km north of the actual Tropic, and Alexandria is approximately 3° west of Syene’s meridian. The errors partly cancel and partly compound; the substantive Eratosthenes figure is within the margin produced by his standing assumptions.
What the Tropic does today
The Tropic of Cancer is not a fixed line. The Earth’s axial tilt oscillates slowly between approximately 22.1° and 24.5° on a 41,000-year cycle (the Milankovitch obliquity cycle); the Tropic of Cancer correspondingly moves slowly south, currently at a rate of approximately 15 metres per year. The Tropic is now approximately 55 km south of Aswan; in Eratosthenes’s day it was substantially closer to the city than it is today.
The summer-solstice noon Sun no longer reaches exactly vertical above Aswan. It is still close enough that the substantive shadow-free phenomenon Eratosthenes used remains observable each 21 June.