### Gauss’s Great Triangle and the Shape of Space

In the 1820s Carl Friedrich Gauss carried out a surveying experiment to measure the sum of the three angles of a large triangle. Euclidean geometry tells us that this sum is always 180º or two right angles. But Gauss himself had discovered other geometries, which he called non-Euclidean. In these, the three angles of a triangle may add up to more than two right angles, or to less.

Gauss is considered by many to be the greatest mathematician who ever lived. Before the era of the euro, he was featured on the German 10 DM banknote. The front of the note shows Gauss and the normal probability curve, or bell curve, that he introduced and that is so fundamental.

The reverse side of the note shows a surveying sextant and a portion of the triangulation network in the vicinity of Hamburg that was constructed by Gauss in the course of his survey of the Kingdom of Hannover between 1821 and 1825.

The Great Triangle

Using his new invention, a surveying instrument called a heliotrope, Gauss took measurements from three mountains in Germany, Hohenhagen, near Göttingen, Brocken in the Harz Mountains and Inselberg in the Thüringer Wald to the south. In his survey of Hannover, Gauss had used these three peaks as “trig points”. The three lines joining them form a great triangle with sides of length 69km, 85km and 107km. The angle at Hohenhagen is close to a right angle, so the area of the triangle as close to half the product of the two short sides, or about 3000km².

Gauss assumed that light travels in a straight line. His sightings were along three lines in space. We should not confuse these with measurements along great circles on the curved surface of the Earth, which would form a spherical triangle. Gauss was considering a plane triangle.

In Euclidean geometry, the sum of the three angles of every triangle is equal to 180º. In hyperbolic geometry there is an angular deficit so that the sum of the three angles is less than 180º. In spherical or elliptic geometry there is an excess: the angles add up to more than 180º. The magnitude of the excess or deficit grows with the area of the triangle.

Curved Space

The geometry of physical space is a matter of measurement, and the character of space must be established by observations. 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 word “geometry” means measurement of the Earth.

Euclidean space is flat: the quantity that measures the curvature of space is called the Riemann tensor. For Euclidean geometry, all components of this tensor vanish identically. It was taken for granted that physical space is flat, but general relativity has changed all that. 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.

The observed total angle found by Gauss was 180º, within the limitations of observational errors. 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 amount

ε = ( GM/Rc² )*( A/R² )

due to the mass M of Earth. G is Newton’s constant, c the speed of light, R the radius of the Earth and A the area of the triangle. This correction comes to about 10^-13 radians [Hartle, 2003]. There would be further, smaller, contributions from the Sun and the other planets.

Was Gauss really observing the shape of space?

There is no clear documentary evidence that Gauss was actually seeking evidence of non-Euclidean geometry of physical space. Indeed, doubt has been cast by some experts on this idea: mathematician John Conway [MathForum, 1998] pointed out that a departure from Euclidean geometry large enough to be measurable on the scale of the Earth would result in massive distortions on an astronomical scale, and would have been evident long before Gauss made his measurements . Moreover, Bühler, in his biography of Gauss [Bühler, 1981], dismisses as a myth the idea that Gauss was measuring the curvature of space. He considered that the purpose of the great triangle was to act as a control to check the consistency of the measurements of the smaller triangles within it.

Of course, the Earth’s intrinsic (spherical) curvature means that when we combine several small triangles to make a large one, there is a discrepancy. Gauss was well aware of this and indeed he calculated the magnitude of this small spherical effect.

We now know that there is an angular discrepancy due to relativistic effects, but it is far too small to be directly observed. However, Gauss would not have had knowledge of the magnitude of this discrepancy. At the time of his survey, Gauss was immersed in the study of non-Euclidean geometry. He knew that the angular discrepancy grows with the area, so it is reasonable to suggest that he had spatial curvature in mind when he measured the great triangle, and that he might have thought it worthwhile to look for evidence of an angular discrepancy and a curved space.

Sources

Bühler, W. K., 1981: Gauss: A Biographical Study. Springer-Verlag, ISBN: 3540106626

MathForum, 1998: Conway, John H: Gauss and the really large triangle.
http://mathforum.org/kb/message.jspa?messageID=1381049

Hartle, J. B., 2003: Gravity: An Introduction to Einstein’s General Relativity. Addison-Wesley. ISBN: 9-780-80538-662-2