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Most students learn that Copernicus put the Sun at the centre of the solar system in 1543. Fewer know that a Greek astronomer proposed the same idea around 1,800 years earlier, or that the same man tried to measure how far away the Sun is using nothing but a half moon. His answer was badly wrong, and the reason it was wrong is one of the clearest lessons in measurement you will find. This post covers both ideas, with the trigonometry worked through.
Who was Aristarchus?
According to the MacTutor biography of Aristarchus from the University of St Andrews, Aristarchus of Samos lived from about 310 BC to about 230 BC. He probably studied in Alexandria under Strato of Lampsacus. The Greeks called him "Aristarchus the mathematician", and the Roman architect Vitruvius credited him with a bowl-shaped sundial.
Only one of his works survives: On the Sizes and Distances of the Sun and Moon. It does not mention the Sun-centred idea at all.
How do we know he proposed a Sun-centred universe?
Through Archimedes. In his book The Sand-Reckoner, Archimedes summarised a lost book of hypotheses by Aristarchus. In the translation quoted by MacTutor, the hypotheses are "that the fixed stars and the sun remain unmoved, that the earth revolves about the sun on the circumference of a circle, the sun lying in the middle of the orbit". MacTutor adds that Plutarch reports Aristarchus also explained the daily turning of the stars by the Earth spinning on its axis.
Aristarchus also had to say the stars were extremely far away. If the Earth moves round the Sun, nearby stars should seem to shift position over a year. MacTutor notes he needed some such assumption, "for otherwise parallax effects would be visible". The Greeks did not accept the theory. Our post on Copernicus and heliocentrism tells how the idea returned.
How did he use a half moon to measure the Sun's distance?
This is the part you can work through. The Moon shines by reflected sunlight. At the exact moment it looks half lit, the angle at the Moon between the directions to the Earth and the Sun is 90°. So Earth, Moon and Sun form a right triangle with the right angle at the Moon.
If you measure the angle θ at the Earth between the Moon and the Sun, trigonometry gives:
Earth–Moon distance ÷ Earth–Sun distance = cos θ
Worked example with Aristarchus' value. MacTutor reports that he took θ = 87°.
cos 87° = sin 3° ≈ 0.0523.
So the Sun is 1 ÷ 0.0523 ≈ 19.1 times farther away than the Moon.
Without trigonometry, Aristarchus proved that sin 3° lies between 1/20 and 1/18, so the ratio lies between 18 and 20. That matches.
Now with modern values. The NASA Moon fact sheet gives the Moon's mean distance as 0.3844 million km, and the NASA Earth fact sheet gives the Earth's mean distance from the Sun as 149.598 million km.
Ratio = 149.598 ÷ 0.3844 ≈ 389.
cos θ = 1 ÷ 389 ≈ 0.00257, so θ ≈ 89.85°, about 89°51′.
You can see the sensitivity directly. MacTutor gives the true angle as 89°50′. Using that, cos θ = sin 10′ ≈ 0.00291, and the ratio comes out near 344. At 89°51′ it is about 382. A single arcminute, a sixtieth of a degree, shifts the answer by almost 40. Judging the exact moment of half moon by eye is far less precise than that.
What else did he get right and wrong?
MacTutor reports that Aristarchus used 2° for the apparent width of the Sun and Moon in the sky, about four times too large. NASA's Moon fact sheet gives the Moon's apparent diameter as 1,896 seconds of arc, which is about 0.53°. Archimedes later credited Aristarchus with a value of ½°, much closer, so he may have improved it later. He was right that the Sun and Moon appear the same size, which is why the Moon can just cover the Sun in a total eclipse.
The historian Otto Neugebauer, quoted by MacTutor, argued that Aristarchus was showing that such measurements could be made in principle rather than chasing accurate data. On that reading, the method was the achievement.
Common mistakes to avoid:
Putting the right angle at the Earth. At half moon the right angle is at the Moon. Draw the triangle before writing any trigonometry.
Mixing up sine and cosine. The Earth–Moon side is adjacent to θ, and the Earth–Sun line is the hypotenuse, so the ratio is cos θ.
Trusting one significant figure near 90°. If your angle is close to a right angle, ask how much the answer moves for a 1° change.
What can a student take from Aristarchus?
A correct method can give a wrong answer. Check the method and the data separately.
Do a sensitivity check. Change your input slightly and see how far the output moves. This habit catches errors in physics practicals.
Unpopular ideas can be right. The Sun-centred model was rejected for centuries before the evidence came in.
This week, do three trigonometry problems where the angle is close to 90° and compare answers for angles 1° apart. If you cannot see why the Earth–Moon–Sun triangle has its right angle at the Moon, the Fermi Simulation Lab has interactive simulations for ideas easier to see than read.
Next step: redo the half-moon calculation with θ = 89° and θ = 89.5°, and write one sentence on what the two answers tell you.
