The earth is like a point compared to the sky
The earth, when compared with the distance to the stars, has the size of a single point. This sounds strange but it follows from simple observations.
If you and another observer stand at very different places on earth and look at the same star on the same night, the star appears in exactly the same direction and at the same distance. No matter how far apart the two observers are, their measurements of a star's position agree perfectly. Two travellers separated by hundreds of miles see the sky as though they were standing together.
A second piece of evidence comes from the instruments astronomers use. A gnomon (a vertical rod that casts a shadow) works equally well anywhere on earth. The angle of the shadow at noon on a given day depends on the observer's latitude, but the geometry that governs the shadow assumes the rod is at the centre of the earth. In practice, this assumption always works. Observers on the surface of the earth get accurate results from their instruments, as though they really were standing at the centre of the universe.
A third indication is the behaviour of the horizon itself. At any point on earth, the horizon appears to bisect the sky (that is, to cut it exactly in half). If you look around, the dome above you matches the invisible dome below you: precisely half the celestial sphere is visible at any moment. If the earth had any noticeable size compared with the distance to the stars, this would not be possible. Standing on the surface of a large body, an observer's horizon would cut off more than half the sky below and reveal less than half above. Only if the earth is so small compared with the sky that you are effectively at its centre can the horizon bisect the sky evenly.
Figure 3 — If the earth had significant size (top), an observer on its surface would see less than half the sky. Because the earth is effectively a point (bottom), the horizon always passes through the centre and bisects the sphere exactly. Use the slider to change the observer's latitude and watch how the angle of the horizon changes while always dividing the sky into two equal halves.
How much is half of the sky?
Ancient observers could verify this empirically, because they were thoroughly familiar with the whole sky. They knew that the zodiac (the band of twelve constellations through which the sun, moon, and planets move) contains exactly twelve equal segments. At any moment of the night, they could look up and see exactly six of them at once. The other six were below the horizon, waiting to rise. This simple count, repeated on any clear night from anywhere on earth, confirms that the horizon divides the sky into two equal halves. A second confirmation comes from watching a single star through a full day and night: the time it spends above the horizon, from rising to setting, equals the time it spends below the horizon before rising again. Each star's night-time arc is mirrored by its invisible daytime arc. Half the sphere above, half below, always.
As you move around the surface of the earth, what changes is not whether your horizon bisects the sky (it always does) but the angle at which it does. An observer at the equator has a horizon plane tilted one way; an observer near the North Pole has a horizon plane tilted almost perpendicular to the first. Each observer sees exactly half the celestial sphere, but a different half. The difference between latitudes is entirely in the tilt of the horizon, not in the portion of sky it reveals.
Star distances
The practical conclusion is that, for the purposes of astronomy, the earth may be treated as a single point at the centre of a vast sphere of stars. The distances between observers on earth, however large they seem from the ground, are negligible compared with the distances to the heavens. The ancients had no way to measure stellar distances, but they had already grasped that the distances must be vast beyond comprehension.