Measuring Hannover, Gauss thought a lot about how to make maps of a curved Earth. In 1827 published the result he called Theorema Egregium, the “remarkable theorem”: the curvature of a surface can be measured by those who live inside it, without going out to look from the outside.

A consequence: in a plane, the angles of a triangle add up to 180°. In a sphere, they add up to more. Take the triangle with one vertex at the north pole and the other two at the equator:

The two angles at the equator are right angles. With 90° at the pole, the triangle has three right angles: 270°.

The two angles at the equator measure 90°, because the meridians cut the equator at right angles. Adding the angle $\alpha$ of the pole:

$$90° + 90° + \alpha = 180° + \alpha.$$

Anything above 180° is proportional to the area of ​​the triangle and shows that the surface is curved. That’s why no flat map of the Earth is perfect: there is always distortion of distances, angles or areas.

Gauss also realized that there could be geometries in which Euclid’s parallel postulate did not exist. worth it. But he did not publish anything: in a letter from 1829, he said that he feared “the shouting of the Boeotians”, those who did not would understand. Non-Euclidean geometries were eventually published by Lobachevsky and János Bolyai.

All topics by Carl Friedrich Gauss · ← The normal curve, the Gaussian bell · Systems of equations: the determinant and Gaussian elimination →

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