Is it believed that Aristotle had sufficiently precise data to conclude that the arcs were round circular, and implied a geometric argument as above? Could he? Or should we take the word "curved" at its face value and rather view the whole argument as one of philosophical flavour rather than a mathematical one?
I think the conclusive remark of Aristotle about the shape of the earth was not only from the curved line of the shadow but from all the other observations he made or people made during those times.
As the general environs of interpretations and conjectures contribute to a final outcome or belief. see the notes-
By around 500 B.C., most ancient Greeks believed that Earth was round, not flat. But they had no idea how big the planet is until about 240 B.C.(**much before the Aristotle observation) when **Eratosthenes devised a clever method of estimating its circumference.
It was around 500 B.C. that Pythagoras first proposed a spherical Earth, mainly on aesthetic grounds rather than on any physical evidence.
Like many Greeks, he believed the sphere was the most perfect shape.
Possibly the first to propose a spherical Earth based on actual physical evidence was Aristotle (384-322 B.C.), who listed several arguments for a spherical Earth: ships disappear hull first when they sail over the horizon, Earth casts a round shadow on the moon during a lunar eclipse, and different constellations are visible at different latitudes.(italics mine)
Greek philosopher and scientist Aristotle (384-322 BC) proved that the Earth is round by carefully observing lunar eclipses. The shadow that the Earth casts on the Moon during a lunar eclipse, Aristotle noticed, was always curved, something only a sphere will do. If the Earth were flat, it would not consistently cast a curved shadow; after observing a number of lunar eclipses with the same result, Aristotle became convinced – correctly – that the Earth is round.
In about 350 BC, Aristotle wrote, “the earth is spherical…in eclipses the outline is always curved: and, since it is the interposition of the earth that makes the eclipse, the form of this line will be caused by the form of the earth’s surface, which is therefore spherical.” one can try Aristotle’s revolutionary experiment for yourself – from any place on Earth a lunar eclipse will always cast a rounded shadow.
Around 350 BC, the great Aristotle declared that the Earth was a sphere (based on observations he made about which constellations you could see in the sky as you travelled further and further away from the equator) and during the next hundred years or so, Aristarchus and Eratosthenes actually measured the size of the Earth!
Around this time Greek philosophers had begun to believe the world could be explained by natural processes rather than invoking the gods, and early astronomers began making physical measurements, in part to better predict the seasons. The first person to determine the size of Earth was Eratosthenes of Cyrene, who produced a surprisingly good measurement using a simple scheme that combined geometrical calculations with physical observations.
It has actually been known that the Earth was round since the time of the ancient Greeks. It was Pythagoras who first proposed that the Earth was round sometime around 500 B.C. As I recall, he based his idea on the fact that he showed the Moon must be round by observing the shape of the terminator (the line between the part of the Moon in light and the part of the Moon in the dark) as it moved through its orbital cycle. Pythagoras reasoned that if the Moon was round, then the Earth must be round as well. After that, sometime between 500 B.C. and 430 B.C., a fellow called Anaxagoras determined the true cause of solar and lunar eclipses - and then the shape of the Earth's shadow on the Moon during a lunar eclipse was also used as evidence that the Earth was round.
during lunar eclipses, one only sees a small arc of the Earth (at most about 1/12 of the full circle). Is it really possible to tell apart a circular arc from an elliptic one with a naked eye given such a short sample?
Regarding the above as earth is spinning as well as rotating about the Sun, during different observations of the eclipses the same edge of the earth is not necessarily repeated, therefore if there had been major changes in the curvature the lines may have been disturbed.
On the whole, taking the stages of measurements of those days one can say that His observation was correct(based on mathematical findings rather than philosophical), though we now know that Earth is not a perfect sphere.