On things that should never be forgotten

And some things that should not have been forgotten were lost. History became legend. Legend became myth. And for two and a half thousand years, the ring passed out of all knowledge. — JRR Tolkien, Lord of the Rings

There’s a wonderful scene the 2009 movie Agora, starring Rachel Weisz, where the mathematician Hypatia ruminates about the nature of the heavenly motions. Hypatia was the head of the Platonic Academy in Alexandria in the 4th century, and a brilliant mathematician. In the movie she is shown as dissatisfied with all those Ptolemaic circles within circles. At one point in the clip, an older scholar reminds the assembled group that there is another way to understand the way things move in the sky, a heliocentric theory proposed by Aristarchus. At that point, Aristarchus had been dead for almost six hundred years, and his theory had fallen into obscurity.

Aristarchus’ theory is one of history’s great missed opportunities, and the forgetting of the first heliocentric model is not a historical oddity, but a tragedy. His theory had been a great advance. Aristarchus used it to explain eclipses, and to estimate the distances to the Moon and Sun. How did he do this? And why was his theory forgotten?

The ‘how’ question is easier to answer, but it requires us to talk about how we measure distances in everyday life, and why those methods fail when you attempt to measure the distance to astronomical objects. The ‘why’ question is more subtle.

How do we measure distances in everyday life?

There are a variety of ways we judge distances as we navigate the world. Here is a short list (from Wikipedia):

Parallax (binocular vision)
Motion parallax (following moving objects across our field of view)
Depth from motion (change in apparent size)
Perspective (using knowledge of shapes)
Relative size (compared to a known object)
Luminance contrast (distant things appear hazy)
Occulomotor cues (you perceive how much your eye needs to distort to focus on the object)
Occlusion (near things occlude far things)
Texture gradients (details are discerned if close)

None of these will work for measuring the distances to astronomical objects, except for parallax, but even then the parallax effect is so small we have to use artificial means to measure it. Therefore: We simply cannot judge distances to objects like the Moon or Sun using the naked eye alone. Most of us have imbibed this message from such a young age that we can find it hard to understand what it must have been like not to know it.

The physicist Erwin Shrödinger, in his wonderful essays collected in Nature and the Greeks and Science and Humanism relates how the philosopher Epicurus [341-270 BCE], whose life overlapped that of Aristarchus [ca. 310-230 BCE], said that the “…Sun is just as large as it appears.” What does that even mean? At arms length, my thumb easily covers the Sun and Moon. Does that mean they are both smaller than my thumb? A rigorous understanding of the apparent sizes of objects requires a ray theory of light, and the first inklings of such a theory appear in Euclid’s Optics, written around 300 BCE. Aristarchus’ writings are full of geometric figures, and his methods for measuring the distances to the Moon and Sun rely heavily on a theory of light and shadow, i.e. ray theory. But ray theory requires a source of illumination, and even here the Greeks were a bit fuzzy on things. The Pythagoreans believed that the Moon, Sun, and planets were illuminated by a central fire. Aristarchus believed the source of illumination was the Sun, that the Earth was also a celestial body, and this theory allowed him to argue that bodies like the Earth and Moon could cast shadows on one another. This is how he developed the first correct theory of solar and lunar eclipses. But more than this, it’s what allowed him to measure the distance to the Moon.

How did Aristarchus estimate the distance to the Moon?

By measuring the time it took for the Moon to cross the center of the Earth’s shadow during an eclipse, we can estimate the distance. This is surprisingly simple, once you get the right idea in mind, and the geometry. (We’ll use round numbers below for simplicity.) The eclipse shown in the figure at the top of the page is only a partial one, but the duration of the eclipse would be similar to the one shown in this conceptual figure:

[Figure by Sagredo – Geometry of a Lunar Eclipse, Public domain, via Wikimedia Commons]

The longest lunar eclipses take about t = 3 hours = 1/8 day.
To keep things simple, assume the Earth’s shadow is a cylinder with the same diameter as the Earth, d=2 r. (While this doesn’t look like a good assumption given the figure above, keep in mind that the figure is not to scale, and it is a reasonable approximation if the Sun is much farther away than the Moon. This is where we require the assumption that the Moon shines by reflected light from the Sun.)
The Moon travels around the circumference of its orbit in about one month, T = 30 days. Assume the speed along the orbit is constant. Therefore, using ‘speed = distance traveled/time’ we have (2π R)/T = 2r/t.
Solve for R/r = T/(π t) = 30/(3/8) = 80. (Note: I have used π = 3, which in technical mathematical terms is a ‘brutal’ approximation, keeping only the first significant digit.)

This means that the Moon is about 80 Earth radii away from us. The true value is closer to 60, but the order of magnitude is correct.

Why was Aristarchus’ theory forgotten?

Naked eye observations of lunar eclipses easily reveal that the Earth’s shadow is curved, not a straight line, as can be seen in the photo at the top of the page. Lunar eclipses occur several times a year, so they are fairly common. Why wasn’t it obvious from simple observations like this that Earth was a sphere? Many ancient astronomers, like Aristarchus, understood what they were looking at, and were convinced the Earth was a sphere, but some later cultures did not. Discoveries can be forgotten.

Psychologists and cognitive scientists have found that when a sensory stimulation, an observation, or other piece of evidence, violates our intuition we sometimes don’t see it, or we don’t process it. We can memorize a 200 word poem easier than 200 random words. Why? It has something to do with the poem having structures and patterns that help us to understand it (e.g. rhythm, rhyme, as well as imagery, and other aspects that help it to make ‘sense’ to us), where a random set of words does not. Memorizing chess pieces on a board is similar: if the pattern arises from a game, a chess master will do far better than an ordinary mortal at memorization. If the pattern of pieces does not arise from a game, they do no better than anyone else.

We are bombarded all day long with sensory input. If we did not organize this somehow into an internal theory of the world, we would not be able to function. Most of this becomes second nature to us as we grow up, so we stop thinking about it. This internal ‘world theory’ becomes a kind of filter on our experience. It allows us to function and thrive, yet it can also keep us from seeing things that are right in front of us.

How can learn to see in new ways?

The ancients were not stupid, or even ignorant, in some ways, but they did have an intuition about how the world worked, and this intuition could make it difficult for them to see things right overhead that violated that intuition. This is not uncommon today. Psychologists have a term for this difficulty in integrating new information that runs counter to our intuition: cognitive dissonance. What types of things are right under our nose that we overlook because they don’t fit our preconceived notions of how the world works? The challenge is always to try and see the world anew, to be astonished by it, like a child. The best scientists are often a bit childlike, because they try to preserve this sense of wonder intentionally.

I wrote about this at greater length in my Aeon essay Behold: Science as seeing . One goal of education should always be to reawaken our sense of newness, to keep us alive to the fact that the way we see the world is not the only ‘right’ way, and to rekindle our sense of wonder, so we stay open to new ideas and new ways of doing things. Aristarchus drew upon evidence that was open and available to everybody. It was above their heads, and available to anybody who was able to see with an open mind.

Image at top: Lunar eclipse, 21 January, 2019, by Roberto Mettifogo, CC BY 4.0 https://creativecommons.org/licenses/by/4.0, via Wikimedia Commons.

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