I’ve just returned from a science fiction convention, and in a panel about mechanical computing, someone put out the speculation (I have no idea if this can be supported from historical evidence) that as the Greeks knew all about conic sections, they might have known that the planets moved in elliptical orbits. The epicycles (it was suggested) come about because if you’re making machinery to forecast the heavens, such as the Antikythera mechanism, it’s easier to make circular wheels than elliptical ones. So you model the elliptical orbits as circles, then correct them with smaller circles moving on the circles. Thus, not bad science, but good engineering.
That’s reasonable, but most likely the Greeks didn’t know the orbits were ellipses, but just knew them as solutions to Newton’s laws and used calculus to approximate them as deviations from circles.
Obviously I disagree. How can you boldly state that they didn’t have things? Maybe there’s no written record, but someone writing about the Antikythera Mechanism should know the limits of that! Indeed, the Method of Mechanical Theorems was discovered after even the Mechanism. But there is a written record. There’s a lot more calculus in Archimedes than the method of exhaustion. That’s barely calculus and it’s interesting more because it’s rigorous than because it’s a forerunner to calculus. But the Method is a pretty big chunk of integral calculus.
The missing part is differential calculus; Newton’s laws do not appear in the written record. But Vitruvius appears to discuss how the planetary orbits are the result of inertia and centripetal force, so it is really not a big stretch to posit further elaboration.
I think using ellipses only really gets you good mileage once you have the planets moving around the sun. If, like Aristotle, you have the planets moving around the earth, then epicycles are just being a very general way of representing periodic motion phenomenologically.
Apollonius introduced epicycles and proved a theorem of the commutativity of addition: a small epicycle about a large orbit appears the same as a large epicycle about a small orbit. This makes it seems like he was using them to represent heliocentrism, not just fitting data.
The Ptolemaic model has an epicycle and an equant for each planet. One of them corresponds to heliocentrism and the other to the non-circularity of an orbit. (That’s very vague because the correspondence is different for the inner planets vs the outer planets. The role of epicycle vs deferant gets switched and (thus) which orbit, the planet or the earth’s has its non-circularity approximated differs.)
I’ve just returned from a science fiction convention, and in a panel about mechanical computing, someone put out the speculation (I have no idea if this can be supported from historical evidence) that as the Greeks knew all about conic sections, they might have known that the planets moved in elliptical orbits. The epicycles (it was suggested) come about because if you’re making machinery to forecast the heavens, such as the Antikythera mechanism, it’s easier to make circular wheels than elliptical ones. So you model the elliptical orbits as circles, then correct them with smaller circles moving on the circles. Thus, not bad science, but good engineering.
That’s reasonable, but most likely the Greeks didn’t know the orbits were ellipses, but just knew them as solutions to Newton’s laws and used calculus to approximate them as deviations from circles.
The Greeks didn’t have Newton’s laws, or calculus except for the method of exhaustion for calculating certain areas.
Obviously I disagree. How can you boldly state that they didn’t have things? Maybe there’s no written record, but someone writing about the Antikythera Mechanism should know the limits of that! Indeed, the Method of Mechanical Theorems was discovered after even the Mechanism. But there is a written record. There’s a lot more calculus in Archimedes than the method of exhaustion. That’s barely calculus and it’s interesting more because it’s rigorous than because it’s a forerunner to calculus. But the Method is a pretty big chunk of integral calculus.
The missing part is differential calculus; Newton’s laws do not appear in the written record. But Vitruvius appears to discuss how the planetary orbits are the result of inertia and centripetal force, so it is really not a big stretch to posit further elaboration.
I think using ellipses only really gets you good mileage once you have the planets moving around the sun. If, like Aristotle, you have the planets moving around the earth, then epicycles are just being a very general way of representing periodic motion phenomenologically.
Apollonius introduced epicycles and proved a theorem of the commutativity of addition: a small epicycle about a large orbit appears the same as a large epicycle about a small orbit. This makes it seems like he was using them to represent heliocentrism, not just fitting data.
The Ptolemaic model has an epicycle and an equant for each planet. One of them corresponds to heliocentrism and the other to the non-circularity of an orbit. (That’s very vague because the correspondence is different for the inner planets vs the outer planets. The role of epicycle vs deferant gets switched and (thus) which orbit, the planet or the earth’s has its non-circularity approximated differs.)
Heliocentrism was around then as well, e.g. Aristarchus of Samos.