What is the mathematical proof that the Earth is round?

The mathematical proof that the Earth is round comes from observations of the Earth's curvature and the way objects disappear over the horizon. One key piece of evidence is the fact that the Earth's shadow on the moon during a lunar eclipse is always round, which can only happen if the Earth is a sphere. Additionally, the way ships disappear hull-first over the horizon as they sail away is consistent with a curved surface. These observations, along with mathematical calculations, support the conclusion that the Earth is round.