Eratosthenes

What this page shows

Each stick is vertical. At local noon the shadow length and the stick height give the sun’s zenith angle, the angle between the vertical and the sun. With the rays taken as parallel, the difference of the two zenith angles is the angle between the two verticals. The radius is the distance between the places divided by that angle in radians.

The preset is the historical account of Syene and Alexandria at the June solstice. In that account the sun is overhead at Syene, so the first stick has no shadow, and the zenith angle at Alexandria is about 7.2°, one fiftieth of a circle. The second shadow is the length that gives 7.2° for the stick height you see.

The places are given modern coordinates: Alexandria at 31°12′ N, 29°55′ E, and Aswan, which is ancient Syene, at 24°05′20″ N, 32°53′59″ E. They are not on the same meridian. The distance filled in is 843,429 m, the great-circle distance between those coordinates on the GRS80 mean sphere, to the nearest metre. The account in stadia is history. This page computes in metres and does not convert a stadion.

With the historical angle and that modern distance, the radius that comes back is not the GRS80 mean radius. The 7.2° is a round figure from the account, and the cities do not lie on one meridian. The page does not adjust either fact. It divides the distance you enter by the angle the shadows give. The GRS80 mean radius, …, is there for the comparison.

Assumptions

• The sticks are vertical and of equal height. Each shadow is the one at local noon, with the sun on the meridian.
• The zenith angle is atan(shadow / height), in radians when the radius is formed. The shadow length is unsigned. The page does not know which way the shadow points.
• The radius is the distance divided by the absolute difference of the two zenith angles. Equal lengths, including two sticks with no shadow, give no angle. The radius is then not determined. A plane gives equal shadows, and so do two places with the same noon zenith angle.
• The June solstice declination in the note is +23.44°, and the December figure is −23.44°. The note is the noon zenith angle at the two preset latitudes, the difference of latitude and declination. It does not change the sticks, it is not a second radius, and it is not a midnight-sun calculation.
• In the historical account the sun is overhead at Syene. On a sphere with declination +23.44°, Aswan’s noon sun is about 0.65° from the zenith, not zero. The preset follows the account.
• No refraction. The drawing is a schematic. The cards are the result.
• The comparison radius is the GRS80 mean radius from the shared geometry module, (2a + b) / 3.

Formulas

Zenith angle z = atan(shadow / height). With z in radians, and d the distance between the places, the radius is d / |z₁ − z₂|. If z₁ = z₂, that division is not done.

The solstice note uses a declination of ±23.44°. At latitude φ the noon zenith angle on a sphere is |φ − declination|, and only while that is within 90°.

Sources

• Cleomedes, On the Heavens I.7, reports the account from Eratosthenes. At the summer solstice the sun was overhead at Syene, and at Alexandria it stood one fiftieth of a circle from the zenith. The distance in that account is 5,000 stadia. See Bowen, A. C. and Todd, R. B. (2004). Cleomedes’ Lectures on Astronomy. University of California Press.
• Moritz, H. (1980). Geodetic Reference System 1980. Bulletin Géodésique, 54, 395–405. The comparison uses the mean radius from that system, taken from the shared geometry module.
• The mean obliquity at J2000.0 is 23° 26′ 21.4″. This page rounds it to 23.44°. Jet Propulsion Laboratory, Solar System Dynamics, astrodynamic parameters. ssd.jpl.nasa.gov/astro_par.html
• The preset coordinates are Alexandria 31°12′ N, 29°55′ E and Aswan 24°05′20″ N, 32°53′59″ E. The 843,429 m is the great-circle arc between them on the GRS80 mean sphere, to the nearest metre.