Measuring the Circumference of the Earth — Eratosthenes' Shadow Method

Around 240 BC the Greek scholar Eratosthenes, chief librarian at Alexandria, measured the size of the whole Earth with nothing but a vertical stick, its shadow, and a known distance. He knew that at noon on the summer solstice the Sun stood directly over Syene (modern Aswan) — sunlight reached the bottom of a deep well and cast no shadow. At Alexandria, far to the north, a vertical rod at the same moment cast a shadow of about 7.2°. That angle is one-fiftieth of a full circle, so the Earth's circumference had to be fifty times the distance between the two cities. This blueprint reproduces his measurement with a gnomon and a plumb line — and lands within a few percent of the true value, as he did.

中級者

手順

Understand the principle

The Sun is so far away that its rays reach Earth effectively parallel. If a vertical rod casts no shadow at one place and a shadow at another at the very same moment, the shadow angle equals the difference in latitude between the two places. Measure that angle and the ground distance, and simple proportion gives the whole circumference.

Build a vertical gnomon

Fix a straight dowel upright at the centre of a flat board — this is the gnomon. Hang a plumb line (a string with a small weight) beside it and adjust until the rod is exactly parallel to the string, so it is truly vertical. Any lean will spoil the angle.

このステップの材料:

Dowel Rod1 個

Red Alder Board1 個

必要な工具:

Cotton Kitchen String

Find true north and the noon line

Mark the north-south meridian line across the board so you can recognise local solar noon — the instant the shadow is shortest and points due north. A gnomon's shadow swings shortest exactly at midday.

必要な工具:

Chalk Line

Measure at solar noon

At local solar noon (ideally on the summer solstice, when the effect is largest and matches Eratosthenes' day), the shadow is at its shortest. This is the single moment when you take your reading.

Mark and measure the shadow

Mark the tip of the shadow with chalk. Measure the shadow's length from the base of the rod to that mark, and measure the height of the rod above the board. Record both lengths carefully in the same units.

必要な工具:

Chalk Line

Cotton Kitchen String

Work out the shadow angle

The angle of the Sun from vertical equals arctan(shadow length ÷ rod height). Eratosthenes found this angle to be about 7.2°. Because 360° ÷ 7.2° = 50, his shadow angle was exactly one-fiftieth of a full circle.

Take the north-south distance

You need the distance due north-south to a place where the Sun is overhead (no shadow) at that same moment. Eratosthenes used Syene, about 5,000 stadia — roughly 800 km — south of Alexandria. Use a known map distance between your site and that reference latitude.

必要な工具:

Cotton Kitchen String

Compute the circumference

Since the shadow angle is one-fiftieth of a circle, the full circumference is fifty times the distance: 50 × 800 km = 40,000 km. Eratosthenes' own figures gave 50 × 5,000 = 250,000 stadia. In general, circumference = distance × (360° ÷ shadow angle).

Check against the true value

The Earth's real circumference is about 40,075 km around the equator. A careful reading with a stick and its shadow lands within a few percent — the same accuracy Eratosthenes reached around 240 BC, proving the size of the planet can be found from a shadow and a length of string.