Ancient Greek Astronomy
One of my astronomy classes this week focused on some of the highpoints of ancient thought about astronomy. It illustrated a fact often forgotten or never contemplated by modern people, that ancient peoples were just as intelligent, curious, and careful in their reasoning as modern thinkers.
It’s apparent from experience that our knowledge of the world must be built on knowledge handed to us by our predecessors. Further, to construct effective scientific models to explain and predict the behavior of nature, we must depend on the best available mathematical, computational, and technological tools. But ancient thinkers had only a meager inheritance of knowledge from their predecessors, along with geometry, and very crude computational tools. Today’s scientists have advantages incomprehensible to Aristotle, Aristarchus, and Eratosthenes: Calculus, differential equations, chaos theory, electronic calculators and personal computers, telescopes and other magical laboratory equipment. Nevertheless, ancient scholars were able to apply observations, logic, and geometry to tackle several very difficult astronomical questions, including the following:
(1) What is the relative size of the Moon and Earth?
(2) How large is the Moon’s orbit in comparison to the Earth’s radius?
(3) What is the relative size of the Sun and Moon?
(4) How far away from the Earth is the Sun, in units of the Earth’s radius?
(5) What is the radius of the Earth?
The questions about the Sun proved very difficult to answer. The available technology simply did not afford the ancients a reliable measurement of the angle between the Moon and the Sun during one of the Moon’s quarter phases, a critical datum needed for their calculations of the Sun’s radius and distance from the Earth. But on the other questions, they achieved remarkable results. The lesson for us is not merely astronomical facts (we can answer these questions to much a greater degree of accuracy anyway), but the fact that combining simple observations with logic and geometry can deliver new knowledge about the universe that we would probably never obtain any other way.
We should never assume that all modern people are smarter than all ancient people. The existence of the modern-day Flat Earth Society is sad proof that simply being contemporary does not guarantee a correct view of the universe. The Flat Earth Society appears at first glance to be a sophisticated, cleverly designed hoax, perhaps to serve as a parody of other groups that believe crazy things everyone else knows to be crazy. But the Flat Earth Society is deadly serious. It’s a small community advocating the belief that we live on a flat disk. They are not joking. To explain the mountain of observational evidence that the Earth is a round ball, they invent so many absurd explanations the rest of us are forced to question their sanity.
To be sure, on small distance scales, approximately 10 miles or so, the Earth is basically flat. So it makes sense that many ancient people assumed this kind of topology for the entire planet. But plenty of ancient thinkers knew the general shape was spherical, not because they wanted to be contrary, but because they made simple observations of the heavens and new enough geometry to interpret what they saw. We have no way of knowing how many individuals privately reached this conclusion, or who was the first to do so. But in the historical record, Aristotle (384-322 BCE) is the first to provide observational arguments for a spherical Earth. Aristotle noted the Earth always casts a curved shadow on the Moon during a lunar eclipse. If the Earth were a round, flat disk, the shadow would be curved, but the arc length would change with the orientation of the disk relative to the Sun and Moon. The arc length of the Earth’s shadow on the Moon is the same for every eclipse. Other flat shapes, such as a square, will not cast a curved shadow in any orientation. He also noted that as one travels south, new stars become visible, and that if one watches a ship sail directly away from the coast, first the hull, then the deck, and finally the mast disappears over the horizon. These facts are best explained by a spherical Earth.
Aristarchus (310-230 BCE) extended Aristotle’s argument in an ingenious calculation of the relative size of the Moon and Earth. The procedure is difficult to carry out without modern photography, but when carefully executed it yields a surprisingly accurate result. One needs to make an accurately-scaled drawing of the shadow cast by the Earth on the Moon during a lunar eclipse. The arc of the Earth’s shadow can then be extended into a circle whose diameter can be compared directly to the diameter of the Moon. Only one assumption is necessary, that the shadow cast by the Earth is the same size as the Earth itself. Since the Earth-Sun distance is large compared to the Earth-Moon distance, this turns out to be a good approximation.
Aristarchus also calculated the size of the Moon’s orbit relative to the Earth’s radius, the radius of the Sun relative to the Earth’s radius, and the distance between the Earth and Sun in units of the Earth’s radius. There are several important lessons we can draw from Aristarchus’ results.
First, even though Aristarchus’ geometrical arguments were sound, he needed good observational data to fill in the details. Unfortunately, a critical measurement needed for his calculation was the angle between the Sun and Moon during one of the Moon’s quarter phases. This measurement could not be made accurately in Aristarchus’ day, or even in our day. So his calculations of the Sun’s size and distance from the Earth are not as accurate as his other figures. But he was able to determine the Sun was larger than both the Earth and Moon, and far from the Earth compared to the Earth-Moon distance.
Second, Aristarchus reasoned that since the Sun is larger than the Earth, the Earth orbited the Sun, contrary to the popular belief that the Sun orbited the Earth. This suggestion was met with scorn, not only because it violated religious and philosophical dogma, but because it implied a corresponding observational fact that could not be verified. Aristarchus’ opponents pointed out that if the Earth went round the Sun, the Sun would be at the center of the Celestial Sphere. (To Aristarchus and his contemporaries, the Celestial Sphere was the reality, not a convenient fiction used to articulate our observations of the night sky.) If the Earth went round the Sun, our perspective on the stars of the Celestial Sphere would change, and the stars’ relative positions would change throughout the year. The effect is called parallax. But since this apparent shift in the stellar positions was not observed, the Earth could not be moving about the Sun.
This is a perfectly reasonable argument, with one critical assumption, that the stars are close to the Earth compared to the Earth-Sun distance. But that assumption turns out to be incorrect. The nearest star to the Earth (after the Sun) is over 270,000 times farther from the Earth than the Sun. The stars are so distant from the Earth that the small parallax shift created by the Earth’s motion about the Sun is very difficult to measure, even with modern technology. The first reliable measurement of stellar parallax was not made until 1838.
Third, Aristarchus was forced to determine his results in units of the Earth’s radius as opposed to stadiums (the common distant unit of the day), because he did not know the radius of the Earth.
Eventually, a librarian in Egypt figured out a way to measure the radius of the Earth using shadows cast by the Sun. Eratosthenes (276-194 BCE) was skilled in both geography and astronomy, and he brought both to bear on the problem of the Earth’s radius in a measurement of remarkable elegance. On the summer solstice, at noon in Syene, Egypt, the Sun was directly overhead so that its rays were directed at the center of the Earth. Evidence for this arrangement came from a water-well in Syene: the Sun cast no shadow in the well. At the same time in Alexandria, a short distance from Syene, the Sun cast shadows with a 7-degree angle. (Aristarchus knew the distance between Syene and Alexandria, so he could calculate the position of the Sun in the sky in Alexandria at the approximate moment that it was overhead in Syene.) So Eratosthenes reasoned that the distance between Syene and Alexandria was 7/360 of the circumference of the Earth, since there are 360 degrees in a circle. To determine the circumference, he needed only to multiply the distance by 360/7, or about 51. Given the circumference of the Earth, the radius is known from the formula C = 2πr. Using modern units, and assuming a conversion factor of 1 stadium = 0.15 kilometers, Eratosthenes calculated a radius of 6100 km, amazingly close to the accepted value of 6378 km.
The geometric concept that underlies all ancient and modern measurements of astronomical sizes and distances is angular size. The angular size of an object is calculated from two measurements. The first is the apparent size, which is determined simply by holding up a ruler next to an object as you look at it. For example, if you hold a ruler at arms’ length and measure the apparent size of the full Moon, you’ll get something between 5 and 8 mm, depending on the length of your arm. The second measurement is the length of your arm, which must be measured in the same units. The angular size of the moon is calculated from
(angular size in radians) = (apparent size) / (length of your arm)
From here, we calculate the diameter of an object from this formula:
(diameter) = (angular size in radians) x (distance)
The units of the diameter will match those of the distance, which must be measured independently. (In textbooks, these formulas often include factors of 2π and 360 degrees, and the angular size is measured in degrees. But the factors of 2π and 360 degrees effectively accomplish a conversion from degrees to radians, so the formulas above are equivalent, and simpler.)
