We measured the curvature – of which manifold? Gauss’s triangle saw the ground.
A story is widely repeated: that Gauss, in his geodetic survey, measured the angles of a triangle to test whether space is flat. This paper passes no judgement on historical intent. It judges numbers. NO NEW MATHEMATICS AND NO NEW PHYSICS IS CLAIMED. Gauss-Bonnet, spherical excess and the curvature of the Schwarzschild spatial slice are all standard.
First, the exact statement is Gauss-Bonnet. For a geodesic triangle, the angle sum minus pi equals the integral of the Gaussian curvature over the area. At constant curvature the right side is the curvature times the area, and on a sphere the area divided by the radius squared. On an octant of the unit sphere – three right angles, area four pi over eight – both sides come to pi over two and the difference is zero (checked here).
Second, the triangle does measure a curvature. With sides of sixty-nine, eighty-five and one hundred and seven kilometres, Heron’s formula gives an area of 2929.43 square kilometres; against an earth radius of 6371 kilometres the surface has Gaussian curvature 2.46368 times ten to the minus fourteen per square metre and the spherical excess is 14.8865 arcseconds (computed here). This is not a hidden small quantity. Fifteen arcseconds is a size triangulation dealt with routinely – surveys correct for spherical excess precisely because it is visible.
Third, the brake. That is the curvature of the surface, not of space. The earth’s mass times the gravitational constant divided by the speed of light squared is a length of 4.4350 millimetres, and the spatial slice around the earth has curvature of that value over the cube of the earth’s radius, namely 1.71503 times ten to the minus twenty-three per square metre. Over the same triangle the defect is only 1.03628 times ten to the minus eight arcseconds, a ratio of 1.43653 times ten to the ninth – nine orders of magnitude (computed here). Neither a nineteenth-century instrument nor a twenty-first-century one separates them with this triangle.
The confusion arises because the Theorema Egregium (Paper 91) is true. The surface’s curvature is intrinsic and can be measured from inside the surface alone. That is exactly why measured from the inside sounds like measured from inside space – but being intrinsic does not say whose intrinsic quantity it is.
Fourth, the cosmological curvature is smaller again. Taking a conservative radius of curvature of ten to the twenty-seventh metres, the defect over the same triangle is 6.042 times ten to the minus forty arcseconds, thirty-one orders below the previous case. Conversely, a null result at precision sigma bounds the curvature only below sigma divided by the area. At one arcsecond the radius of curvature is bounded above 3.858 earth radii, at a tenth of an arcsecond above 12.2, and even at a hundredth above only 38.58 (computed here). As a test of the geometry of space the triangle is far too small – and too small is a matter of area, not of instruments.
Fifth, the sharpest fence is not precision at all. A survey’s horizontal angles are reduced to the reference surface as part of the procedure. However good the instrument becomes, the angles of the triangle it outputs are quantities on that surface. So the object being measured is, by construction, a triangle on that surface, before any question about space is asked. This is a sharper fence than the orders of magnitude – those say too small to see, this one says a different object is being looked at.
The census. Surface curvature belongs to the earth’s surface, a two-dimensional manifold, at order ten to the minus fourteen, and over this triangle it was measured as fifteen arcseconds. Spatial curvature belongs to the slice around the earth, at order ten to the minus twenty-three, and at ten to the minus eight arcseconds it is not measurable. Cosmological curvature belongs to the cosmological slice, below ten to the minus fifty-four, and at ten to the minus forty arcseconds it is not measurable. Spacetime curvature is tidal and is not an angle at all, but another quantity. The separator is which manifold the triangle lies in – not difficulty and not precision. The same word, the same figure and the same angle sum name four different quantities.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem or physical law (there is none) / anything about what Gauss historically intended (this paper decides nothing there – reading the sources is the historians’ work). What is added: addresses and orders of magnitude for four curvatures, an actual computation of the bound a null result would give, and the observation that the sharpest fence is the reduction procedure rather than the precision. What remains: the sentence we measured the curvature does not yet name a quantity until the manifold is named. Gauss’s triangle did lie on something curved. It was the ground we stand on.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). “We Measured the Curvature” — of Which Manifold? [Paper 27]. Zenodo.
DOI: 10.5281/zenodo.22090333
ORCID: 0009-0006-8587-3450