Inspired by this question, how might one determine the shape of the world given a cosmology with no cycles? The moon, sun, stars, and planets are all fixed. The Earth does not rotate. You can't simply walk around it because it is essentially tidally locked; one side is a barren, hellish desert, the other is a frozen wasteland.
Astronomy will still help, even if there is no motion. Imagine for a moment that there's no moon, no other planets, and no distant stars: just the planet you live on and its sun, and that the planet is tidally locked.
From one specific point (in the dead center of your planet's "hot side") you will see the sun directly overhead, all the time. At every other point on that hemisphere, you'll see the sun somewhere other than directly overhead. As an extreme example, at the very furthest extent of the hot side, you'll see the sun on the horizon.
If you can measure the angle between the sun and horizon (and your people should really develop this ability, since it's far and away the best way to navigate the hot side of such a planet), and you plot out the angles on a piece of paper, you'll eventually notice that they describe the behavior of a curved, not flat, surface. In fact you can measure the changes in angle to work out how curved the surface is, and comparing that angle to known distances will let you work out how big your planet is.
This was the method Eratosthenes of Cyrene used to determine Earth's circumference ca. the 3rd century BC. He had to make his measurements on one particular day in order to account for the apparent movement of the Sun, but your people wouldn't have to; they could run this sort of experiment any time they liked. They wouldn't even have to travel particularly far; the cities he used were only about 800km apart, and if you just wanted a proof of sphericity, not an accurate measurement, they could be closer.
Now you might point out that Eratosthenes's method only makes sense if you already think the Earth is spherical, and that may be true. The idea of a spherical Earth appears to be about 300 years older, and was one of several theories floated at the time, more or less because the Greeks thought it was suitably elegant and geometrical. What Eratosthenes and other astronomers of his era did was provide evidence to support an idea that was already popular.