How did Ptolemy prove in the following excerpt that the Earth is not cylindrical?

Question

I have trouble interpreting the following excerpt:

“Nor could it [the Earth] be cylindrical, with the curved surface in the east-west direction, and the flat sides towards the poles of the universe, which some might suppose more plausible. This is clear from the following: for those living on the curved surface none of the stars would be ever-visible, but either all stars would rise and set for all observers, or the same stars, for an equal [celestial] distance from each of the poles, would always be invisible for all observers.”

I understand that if the Sun rotated around the curved side of the cylinder, no one living on the curved side would see any ever-visible stars because their view of the celestial views is blocked. However Ptolemy states that “all stars would rise and set for all observers,” which I believe would be wrong because the two poles and the curved surface would be oriented in different directions on a cylinder. He also states that the other possibility is that “the same stars, for an equal [celestial] distance from each of the poles, would always be invisible for all observers.” However, I don’t understand how his explanation in the excerpt directly above would prove that “the same stars… would always be invisible for all observers.”

Here’s the link to my source (page 41, chapter 5:) https://isidore.co/calibre/get/pdf/Ptolemy%26%2339%3Bs%20Almagest%20-%20Ptolemy%2C%20Claudius%20%26amp%3B%20Toomer%2C%20G.%20J__5114.pdf

Answer 1

If the Earth were a cylinder, and stationary in the centre of the universe, then the apparent daily motion of the stars must be due to the stars' actual motion in the heavens. The stars he thought to be on a sphere that encircles the Earth, and there is a pole to this sphere, at the Northern pole there is a bright star.

If the Earth were a cylinder, and the stars moved in circles around the axis of the cylinder; the axis of the cylinder pointing directly at the poles of the celestial sphere; then for any observer on the curved surface the pole star would be on the Northern horizon. The Southern pole would be on the southern horizon an all the stars would rise and set. That is clearly not the case.

Ptolemy then considers the possibility of a cylindrical Earth with the axis of the cylinder not pointing towards the pole of the sphere. Then some stars would rise and set and others would remain always below the horizon. Ptolemy (in Egypt) would see the Pole Star at about 30 degrees above the horizon. But he knew that the position of the pole star was not the same in the Northern part of the Empire. There were stars that were sometimes visible in Alexandria that remained always below the Horizon in Rome. In Rome the Pole Star was 40 degrees above the Horizon. This is not consistent with a cylindrical Earth.

Answer 2

As I wrote in answer to another one of your questions, when I can’t understand a written passage, I try to picture it. So, here’s an image of an hypothetical cylindrical Earth in the celestial sphere from Ptolemy’s cosmology. The Earth is the blue cylinder in the middle, with an hypothetical observer located at the black dot on its side. As you can see, light from a star at the celestial north pole (yellow ray) can’t reach that observer, because the CylindEarth is in the way; such a star would be below the horizon and forever hidden to the BlackDotObserver.

QED.