There is no clear documentary evidence that Gauss was actually seeking evidence of non-Euclidean geometry of physical space. Was Gauss really observing the shape of space? There would be further, smaller, contributions from the Sun and the other planets. This correction comes to about 10^-13 radians. G is Newton’s constant, c the speed of light, R the radius of the Earth and A the area of the triangle. If Gauss had been able to take measurements of sufficient accuracy, he might have found that the sum of the three angles of his great triangle differed from two right angles by an amountĭue to the mass M of Earth. The observed total angle found by Gauss was 180º, within the limitations of observational errors. Thanks to Einstein, we know that physical matter distorts the space around it the Riemann tensor is non-zero and, near the planets and stars, space is curved. It was taken for granted that physical space is flat, but general relativity has changed all that. For Euclidean geometry, all components of this tensor vanish identically. The word “geometry” means measurement of the Earth.Įuclidean space is flat: the quantity that measures the curvature of space is called the Riemann tensor. For thousands of years it was assumed, on the basis of such observations, that Euclid’s geometry is a faithful and precise representation of physical space. The geometry of physical space is a matter of measurement, and the character of space must be established by observations. The magnitude of the excess or deficit grows with the area of the triangle.
In spherical or elliptic geometry there is an excess: the angles add up to more than 180º. In hyperbolic geometry there is an angular deficit so that the sum of the three angles is less than 180º. In Euclidean geometry, the sum of the three angles of every triangle is equal to 180º. We should not confuse these with measurements along great circles on the curved surface of the Earth, which would form a spherical triangle. His sightings were along three lines in space.
Gauss assumed that light travels in a straight line.
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