Oddbean

First, Euclidean geometry is a mathematical model that applies perfectly to flat surfaces, and it’s useful for small-scale calculations. However, when we deal with large scales—like the Earth—we need to use non-Euclidean geometry (specifically spherical geometry) because the Earth is a three-dimensional object. The concept of curvature you’re referring to doesn’t contradict Euclidean geometry; instead, it shows the limits of Euclidean geometry when applied to large, spherical objects like planets. Think of how map projections distort continents because a flat map can’t perfectly represent a spherical surface. In fact, the Earth's curvature has been measured countless times through experiments like: The Eratosthenes experiment (250 BC), which measured the Earth's circumference using the angle of shadows at two distant locations. Satellites orbiting the Earth, which require precise knowledge of the Earth’s curvature to function properly. Airplane flight paths, which use great circle routes, demonstrating how spherical geometry explains the shortest distance between two points on Earth, not Euclidean flat distances. To claim the Earth is flat by using Euclidean geometry would be like insisting that because a triangle has 180 degrees in Euclidean geometry, it must also have 180 degrees on a spherical surface—which isn’t true. In spherical geometry, triangles can have more than 180 degrees, and that’s been empirically verified. So, the issue isn’t reconciling curvature with Euclidean geometry—it’s that Euclidean geometry isn’t the right tool for the job on a planetary scale. Using it to argue the Earth is flat is like using a ruler to measure the volume of a sphere—it’s the wrong tool for the task.

No special pleading here—just basic physics and geometry. Water does conform to the shape of its container, and on Earth, that container is the curved surface of a sphere. At large scales, water follows the curvature of the Earth due to gravity. You can observe this curvature over large bodies of water with experiments like: • Ship hulls disappearing first as they move away from shore, which wouldn’t happen if the Earth were flat. • Lighthouses being visible from farther away the higher they are, another clear sign of curvature. • The Bedford Level Experiment, which initially seemed to support flat Earth but, when repeated correctly with proper controls, confirmed Earth’s curvature. If water didn’t follow Earth’s curvature, satellite imagery, aircraft navigation, and even basic GPS wouldn’t work. The fact that we use curvature in these technologies—and they function perfectly—shows this isn’t about preference or special pleading; it’s about observable, measurable reality.