Measuring How Far the Sun Is Compared to the Moon — Aristarchus' Half-Moon Method

Around 270 BC Aristarchus of Samos worked out that the Sun is far more distant than the Moon — and did it with pure geometry. His key idea: when the Moon is exactly half-lit, the Sun, Moon and Earth form a right angle at the Moon. Measure the angle in the sky between the half-Moon and the Sun, and trigonometry gives the ratio of their distances. This blueprint reproduces his method with a simple sighting instrument. It also shows, honestly, why his answer of '19 times' fell short of the true 390 — the geometry is perfect, but the angle is almost impossible to measure by eye. From it Aristarchus argued the Earth goes round the Sun, eighteen centuries before Copernicus.

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Understand the geometry

When the Moon shows exactly half its disc lit, the sunlight is hitting it side-on — which means the angle Sun–Moon–Earth is a perfect right angle. The Earth, Moon and Sun then form a right triangle, with the Earth–Moon line and Earth–Sun line as two sides. The angle between them at the Earth unlocks the distance ratio.

Catch a precise half-Moon

Watch for the exact moment of first-quarter or last-quarter Moon, when the terminator (the line between light and dark) is dead straight and half the disc is lit. This is when the right angle at the Moon holds — and judging it precisely is the hardest part.

必要な工具:

Dowel Rod

Measure the Sun–Moon angle

With both the Sun and the half-Moon in the sky, measure the angle between them using a cross-staff or a graduated sighting board. NEVER look at the Sun directly — sight it by the shadow its pinhole or a rod casts on the board while you line the other end up with the Moon.

必要な工具:

Red Alder Board

Cotton Kitchen String

Note Aristarchus' reading

Aristarchus measured this angle as 87°, just short of a right angle. The true value is about 89.85° — so close to 90° that the naked eye cannot tell the difference. Record your own measured angle for the calculation.

Do the trigonometry

In the right triangle, the ratio of the Sun's distance to the Moon's distance is 1 ÷ cosine(your angle). For 87°, cos 87° ≈ 0.0523, so the Sun works out about 1 ÷ 0.0523 ≈ 19 times farther than the Moon.

Compare honestly with the truth

The real Sun is about 390 times farther than the Moon, not 19. The gap is not a flaw in the reasoning — it is because the true angle (89.85°) is so near 90° that being off by even 3° throws the answer out enormously. The method is exact; only the instrument's precision limits it.

See why it changed history

Even the modest '19 times' meant the Sun must be vastly bigger than the Earth. Aristarchus reasoned that the small Earth would sooner orbit the giant Sun than the reverse — proposing a Sun-centred cosmos around 270 BC, roughly 1,800 years before Copernicus revived the idea.

Push for better precision

Repeat on several half-Moons and average your readings, and use the finest angle scale you can build. Every fraction of a degree you gain nearer the true 89.85° pushes your Sun-distance ratio up toward the real value — a direct lesson in how measurement precision drives discovery.