Ancient Astronomy: Eratosthenes, Aristarchus and Precession

Measuring the Earth: Eratosthenes

Around 240 BCE, the Greek scholar Eratosthenes devised one of the most elegant measurements in the history of science. He knew that at noon on the summer solstice, sunlight shone directly down a deep well at Syene (modern Aswan, Egypt) — meaning the Sun was at the zenith there. At the same moment in Alexandria, roughly 800 km to the north, he measured the angle of a shadow cast by a vertical stick and found it to be about 7.2°.

Eratosthenes reasoned that if the Sun’s rays are essentially parallel (the Sun being so distant), then the angular difference between the two shadow angles must equal the angle subtended at Earth’s centre between the two cities. Since 7.2° is approximately 1/50th of 360°, the full circumference of the Earth must be 50 times the distance between the two cities: 50 × 800 km = 40 000 km. This gives a diameter of about 40 000 ÷ π ≈ 12 700 km — remarkably close to the modern value. The geometry involved is straightforward: if you know the arc length (the north–south distance between the cities) and the central angle (the shadow angle difference), you can calculate the full circumference.

Aristarchus: Moon and Sun Measurements

Aristarchus of Samos is famous for proposing a heliocentric model of the Solar System nearly 1,800 years before Copernicus. But he also devised clever geometric methods to estimate the sizes and distances of the Moon and Sun.

Diameter of the Moon: Aristarchus used lunar eclipses. During a total lunar eclipse, Earth casts a shadow across the Moon. By timing how long the Moon took to cross Earth’s umbral shadow and comparing that to the Moon’s own travel time across its full diameter, he estimated the ratio of Earth’s shadow width (approximately equal to Earth’s diameter at that distance) to the Moon’s diameter. He found Earth’s shadow to be about 2–3 times the Moon’s diameter, which makes the Moon roughly one third the width of Earth. His figure was less accurate than Eratosthenes’ Earth measurement.

Distance to the Moon: Aristarchus used the angular diameter of the Moon (about 0.5°) and a geometry argument based on the Moon’s apparent size. Knowing or estimating Earth’s diameter, if the Moon’s angular size is 0.5°, simple trigonometry gives its distance as approximately 60 Earth radii — close to the modern value of ~60.3 Earth radii.

Distance to the Sun: Aristarchus observed the Moon at exactly half-phase (first or last quarter), when the angle at the Moon, Earth–Moon–Sun, must be exactly 90°. He then measured the angle at Earth, Moon–Earth–Sun — the gap between the Moon and the Sun in the sky — as approximately 87°. Using trigonometry, he concluded the Sun was about 18–20 times further than the Moon. The true ratio is about 400 — Aristarchus’ measurement was inaccurate because the half-phase angle is extremely sensitive to small errors — but his method was entirely sound.

Diameter of the Sun: Because the Sun and Moon have nearly equal angular diameters (~0.5°), the Sun’s diameter must be in the same ratio to the Sun’s distance as the Moon’s diameter is to the Moon’s distance. If the Sun is 400 times further away and has the same angular size, it must be 400 times larger in diameter.

Axial Precession: The Wobbling Earth

Earth’s axis is tilted at about 23.5° to the plane of its orbit. But this tilt does not point in a fixed direction forever. Like a spinning top that slowly wobbles, Earth’s axis slowly traces out a cone in space — a motion called axial precession. The period of this wobble is approximately 26 000 years.

At present, Earth’s north pole points roughly toward Polaris (the Pole Star) in the constellation Ursa Minor. But this has not always been the case — and it will not always be. Around 3000 BCE, the star Thuban (in Draco) served as the pole star for ancient Egyptians. In about 12 000 years’ time, the brilliant star Vega (in Lyra) will lie near the celestial north pole. The cycle then continues, and Polaris will be the pole star once more in roughly 26 000 years.

Precession also shifts the date on which Earth is at any particular position in its orbit relative to the stars. The equinoxes precess (move slowly westward along the ecliptic) by about 1° every 72 years. This is why the astronomical constellations of the zodiac no longer match the Sun’s position at the dates associated with astrology’s zodiacal signs — the sky has shifted significantly since those signs were defined in antiquity.