Gauss and Non-Euclidean Geometry

8. Conclusion

It becomes clear that a mathematician persuaded of the truth of non-Euclidean geometry and seeking to convince others is almost driven to start by looking for, or creating, non-Euclidean three-dimensional Space, and to derive a rich theory of non-Euclidean two-dimensional Space from it — as Bolyai and Lobachevskii did, but not Gauss. The only hint that he explored the non-Euclidean three-dimensional case is the remark by Wachter, but what Wachter said was not encouraging: “Now the inconvenience arises that the parts of this surface are merely symmetrical, not, as in the plane, congruent; or, that the radius on one side is infinite and on the other imaginary” and more of the same. This is a long way from saying, what enthusiasts for Gauss’s grasp of non-Euclidean geometry suggest, that this is the Lobachevskian horosphere, a surface in non-Euclidean three-dimensional Space on which the induced geometry is Euclidean. In particular, there is no three-dimensional differential geometry leading to an account of non-Euclidean space.

Gauss, by contrast, possessed a scientist’s conviction in the possibility of a non-Euclidean geometry which was no less, and no greater, than that of Schweikart or Bessel. The grounds for his conviction are greater, but still insubstantial, because he lacks almost entirely the substantial body of argument that gives Bolyai and Lobachevskii their genuine claim to be the discoverers of non-Euclidean geometry.