# Gauss-Bonnet Theorem: Neither Gauss nor Bonnet [closed]

Tristan Needham says (p.174),*

"While Gauss and Bonnet certainly paved the road to [the Gauss-Bonnet Theorem], neither one of them was even aware of this extraordinary result, let alone stated it!"

Needham assigns the honor to Leopold Kronecker and Walther von Dyck.

(Added). By "the Gauss-Bonnet Theorem," Neeham means $$\mathcal{K}(S_g) = 4 \pi (1-g) = 2 \pi \chi(S_g)$$ where $$S_g$$ is a closed, orientable surface of genus $$g$$, $$\mathcal{K}(S_g)$$ is its total curvature, and $$\chi(S_g)$$ is its Euler characteristic.

My question is:

Q. Is Needham's recounting historically accurate?

* Needham, Tristan. Visual Differential Geometry and Forms. Princeton University Press, 2021.

• Surely better for hsm.stackexchange.com Nov 21 at 1:11
• The "classical" form of the Gauss-Bonnet theorem talks about the sum of angles of a geodesic triangle in terms of the integral of the curvature over it; this is a statement in differential geometry. However, in modern differential topology books it is usually stated as a relation between the Euler characteristic of a compact surface and the integral of the curvature over it. Perhaps this distinction is at play. Nov 21 at 6:41
• Perhaps Needham's statement stems from a relatively narrow reading of what the Gauss-Bonnet theorem is stating. At its most broad, you could view Gauss-Bonnet as a statement of what the integral of the curvature over a 2-dimensional manifold (perhaps with corners) is, in terms of the curvature of the boundary curves (including jump angles) and the euler characteristic of the 2-dimensional manifold. Gauss's version assuming the region is a geodesic triangle is one instance of the theorem, but it of course can be stated in greater generality. Nov 21 at 22:21
• A belated addition to explain what Needham means by "the Gauss-Bonnet theorem." Nov 23 at 0:32

Shiing-Shen Chern's Historical remarks on Gauss-Bonnet seems an authoritative source. The formula goes back to Gauss (1827), Bonnet (1848), and Binet (unpublished). Gauss considered a triangle, Bonnet and Binet generalized it to smooth closed curves, where the sum of the angles is replaced by the integral of the geodesic curvature.

The equation for compact surfaces of arbitrary genus, referred to by Needham, was written up later by von Dyck (1888) (relying on earlier developments by Kronecker). So yes, the question in the OP "is Needham accurate" can be answered in the affirmative.

Daniel Gottlieb has examined the "sociology of mathematics" which governs the naming of theorems. In the context of the Gauss-Bonnet theorem he writes: Part of this story shows that the name of a theorem is not really for an attribution. It is very convenient to have a name for important theorems, and the main point is that people should know approximately what theorem is meant by the name rather than who gets the credit. Still, one can reflect that Bonnet's name is famous and Dyck's is virtually unknown these days.

Because of Binet's independent work, some authors speak of the Gauss-Binet-Bonnet theorem, here is one example.

And here is the footnote by Bonnet, in which he credits Binet. After having completed this paper, I saw a note by Binet, appended to a paper by Olinde Rodrigues in the "Correspondence of the École Polytechnique". In that note Binet derived Gauss's theorem in a similar way as I did.

• Bonnet and Binet? That’s almost as confusing as Monet and Manet! Nov 21 at 15:14
• @SamHopkins: Don't forget Brunn and Debrunner, of Borromean link fame. Nov 21 at 22:27
• @RyanBudney: Or Leonhard Euler, of St Petersburg, Russia, and Lenny Euler, the owner of Euler's Liquor Lounge in St. Petersburg, Florida. Nov 22 at 8:04