Who doesn't like to contemplate the beauty of a lunar eclipse like what happened in the early hours of last Friday, November 19th? But did you know that more than 2.200 years ago, a Greek calculated the distance between the Earth and the Moon from observing a lunar eclipse and using only mathematics, a candle and a coin?

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Aristarchus of Samos was a astronomer and Greek mathematician who lived between 310 BC and 230 BC He had important contributions in Mathematics, and was the first scientist to propose that the Earth should revolve around the Salt and of its own axis. But his great work was “On the Sizes and Distances between the Sun and the Moon”.

Aristarchus' calculations in the XNUMXrd century BC of the relative sizes (left) of the Sun, Earth, and Moon, from a XNUMXth century Greek copy
Aristarchus' calculations in the XNUMXrd century BC of the relative sizes (left) of the Sun, Earth, and Moon, from a XNUMXth century Greek copy

It couldn't have been an easy life for a scientist at that time. Even before the invention of the telescope, the astrolabe, the internet and the coffee, Aristarchus was committed to calculating the sizes and distances between the Earth, the Moon and the Sun. For this, he used all the “technology” available, some mathematics and a lot of creativity.

With a coin in his hand, Aristarchus pointed it towards the Moon and positioned it so that it completely covered the star. Then he measured the distance between the coin and his eye, which was equivalent to 108 times the diameter of the coin. By the principle of similarity between triangles, he concluded that the Moon would be at a distance of 108 times its diameter. But what would the moon's diameter be?

By similarity to triangles, Aristarchus concluded that the distance to the Moon was 108 times its diameter (D)
By akin to triangles, Aristarchus concluded that the distance to the Moon was 108 times its diameter (D). Image: Marcelo Zurita

To measure the size of the Moon, Aristarchus observed an eclipse and calculated the diameter of the Earth's shadow cast on the Moon. This calculation he made by measuring the time it took the Moon to enter and exit the Earth's shadow. The time it takes to enter Earth's shadow is the time the Moon travels its own diameter. And the time until it completely leaves the shadow, is the time the Moon travels through the diameter of that shadow.

So Aristarchus calculated that the Earth's shadow was about twice the diameter of the Moon. Great. But until then, we didn't know the diameter of the Moon.

Engraving excerpt showing Aristarchus of Samos (310 BC - 230 BC)
Engraving excerpt showing Aristarchus of Samos (310 BC – 230 BC). Source: Atlas of Adreas Cellarius (1646)

There, we have a coincidence of nature. Sun and Moon are approximately the same apparent size in the sky. The Greeks already knew this because during total eclipses of the Sun, the Moon completely obscured the Sun for a few moments. So Aristarchus can use that same ratio of 108 times the diameter of the object and consider that the Earth's shadow cone projects to a distance 108 times the diameter of the planet. Knowing this and knowing the size of the Earth's shadow, Aristarchus used, once again, the principle of similarity between triangles and calculated that the diameter of the Moon would be about a third of the diameter of the Earth. Thus, the distance between Earth and Moon would be equivalent to 36 Earth diameters (which is a third of 108 times the Earth's diameter).

Actual distance between Earth and Moon in an OSIRIS-REX Probe photo
Actual distance between Earth and Moon in an OSIRIS-REX Probe photo. Credits: NASA/Goddard

Obviously you expected Aristarchus to give you a distance in kilometers. But look, not even the kilometer existed at that time (the decimal metric system only came into force in the XNUMXth century), and humanity would still wait a few years before discovering the Earth's diameter. Still, it remains to be said where the candle comes into this story.

So it is. At that time there were no clocks or hourglasses either. Then, to measure the durations of the eclipse, Aristarchus lit a candle and measured how much it burned during each phase. Since the candle burned steadily, it could convert the measure of the burnt length of the candle into a measure of time.

A candle clock, like the one used by Aristarchus to measure the durations of an eclipse
A candle clock, like the one used by Aristarchus to measure the durations of an eclipse. Credits: Benutzer:Flyout/wikimedia.org

The difficulty to obtain precise measurements with that “technology” is clear. Because of this, Aristarchus' calculations were not so accurate. The Moon is actually 3,7 times smaller than the Earth and is at a distance of 30 Earth diameters. But see that, even with such precariousness, Aristarchus came very close to real values. Not bad, considering the previous estimate was that the Moon's diameter would be 72 times smaller than Earth's.

And speaking of Earth, a few years later, Eratosthenes accurately calculated the diameter of our planet, and this gave more tangible values ​​to the calculations of Aristarchus of Samos.

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