Earth curvature table sparks debate between mathematicians and flat-earthers

The Earth curvature table that debunks (or doesn't) flat-earth theory

A calculation using elementary trigonometry, the same taught in high school, has become a battleground between those who defend the Earth's sphericity and those who claim the horizon doesn't deceive. The debate starts with a three-column table measuring the vertical drop of the Earth's surface at various distances. Critics argue it is flawed because it confuses an object's height with its actual visibility. The data fueling the controversy: from sea level, some testimonies claim islands have been sighted from more than 200 kilometers away, something classical geometry considers impossible. The original table calculates curvature based on an Earth radius of 6,373 kilometers and horizontal displacement, applying formulas for angle and vertical drop. Experts point out that this drop does not equal the apparent occlusion of an object, as atmospheric refraction and observer height completely alter the result. That is where the trouble begins.

What exactly does the Earth curvature table measure?

The table measures the vertical distance between the spherical surface and the horizontal tangent at a point, assuming a perfect sphere. For short distances, the drop is centimeters; for hundreds of kilometers, the drop skyrockets. But defenders of the spherical model insist that this drop does not imply an object disappears from view, because observer height and refraction allow seeing beyond the geometric horizon. The confusion between drop and occlusion is the core of the misunderstanding. Some argue the table is wrong because it measures the drop from a tangent that does not correspond to the real line of sight. Counterarguments state that at oceanic distances, accumulated curvature is undeniable and explains why ships disappear hull-first rather than mast-first. Trigonometry isn't subjective, but its interpretation is.

Impossible sightings: Mallorca from Barcelona and Montgó from Valencia

The debate intensifies with specific cases: from the Barcelona coast, at sea level, people claim to have photographed Mallorca; from Valencia, the profile of Montgó is confused with the island's. The distance between Barcelona and Mallorca exceeds 200 kilometers, which, according to curvature calculations, would make seeing the island impossible without an observation height of thousands of meters. Graphic testimonies, however, show mountain silhouettes that some identify as Mallorca and others as Montgó or simple pareidolia. Atmospheric refraction, which can bend light and allow seeing objects beyond the geometric horizon, is cited by sphere defenders as an explanation. Flat-earthers counter that such refractions are exceptional and do not explain repeated sightings. The debate becomes a clash of interpretations.

The Polaris argument and the Sun and Moon appearing the same size

Another flat-earth argument points out that Polaris remains fixed in the sky, which would be incompatible with a rotating and tilted Earth. It is also claimed that the Sun and Moon appear the same size even though the former is 400 times farther away and 400 times larger, a coincidence presented as suspicious. The scientific community responds that Polaris stays fixed because it aligns with Earth's rotation axis, and the match in apparent sizes is a well-documented numerical coincidence. Flat-earthers argue these explanations are ad hoc and that the official model requires too many coincidences. Occam's razor is invoked from both sides.

The UN logo and the 'human cattle' conspiracy

A recurring argument is that the UN logo represents a flat Earth, interpreted as a sign that the global elite hides the truth. The UN logo is an azimuthal equidistant projection, a standard way to represent a sphere on a plane to show all continents, not a declaration of flat-earth theory. Those defending this theory claim that if Earth were flat with a dome, like ancient cosmologies, humanity would be confined to a farm. Skeptics respond that Earth's shape depends on repeatable measurements, not a logo or a conspiracy. The discussion shifts toward the sociology of belief.

Why doesn't the curvature table settle the debate?

The table doesn't settle the debate because it measures a geometric magnitude that doesn't directly correspond to visual experience. For small distances, the drop is imperceptible; for large distances, refraction, observer height, and the definition of horizon come into play. No table replaces direct measurement, and direct measurements, such as those by satellites and geodesy, confirm sphericity. Flat-earthers demand tests accessible to ordinary citizens, and that is where the debate stalls: science offers models, not personal experiences. The gap between model and perception is the true battlefield.

The analysis hits a wall: there is no way for a trigonometry table, however correct, to convince someone who distrusts the authority producing it. Conversely, no photographic sighting, however spectacular, overturns a model backed by centuries of geodesy. The discussion remains open, with the same questions and increasingly refined answers.