In his biography of Gauss, G. Waldo Dunnington describes the Gauss-Bolyai episode and their correspondence. In particular, he describes the contents of one letter from Gauss to Janos-Bolyai:



> 
> In the above mentioned letter Gauss gave as a sample of his own research a proof that in non-euclidean geometry the area of a triangle is proportional to the deviation of the sum of the angles from 180 degrees... In the same letter Gauss urged Johann to busy himself with the corresponding problem for space, namely, to determine the cubic content of the tetrahedron (space bounded by four planes). The papers of Johann contain several processes which can serve as a solution, among them the method which Gauss had in mind and which he indicated in one of his notebooks at the time of sending his letter to Wolfgang on March 6, 1832.
> 
> 
> 


In addition, Stillwell's book "Mathematics and Its History" (p.379) emphasizes that Gauss "did have many of the results of non-euclidean geometry by this time, **including** the answer to the volume problem he raised to test his young rival (see Gauss (1832 - [Cubierung der Tetraeder](http://gdz.sub.uni-goettingen.de/pdfcache/PPN236010751/PPN236010751___LOG_0038.pdf))". Looking at the same letter, I also saw that Gauss wrote that unlike the case of 2-dimensional content of simplexes (triangles), where the area is proportional to the angular deficit, in the case of 3-dimensional content of tetrahedrons such a simple formula does not exist.


A translation of Gauss's note is:



> 
> In the tetrahedron $1234$, whose faces $124$ and $134$ are orthogonal. Its volume is $$\Delta,$$ then it holds that:
> $$\partial \Delta = -24\cdot\partial341,$$ and if the face angles at vertex 3 are constant, then the following also holds: $$\alpha^2\cdot \mathbb{cotg}^2341 - \beta^2\cdot \mathbb{tanh}^2(l\_{24}) = 1$$ where $$\alpha = \mathbb{cotg}431,\quad 
> \beta = \mathbb{cotg}234.$$
> 
> 
> 


For the tetrahedron under the conditions described in Gauss's note, Gauss's formula therefore connects the length of the side $24$ with the angle $341$; literally speaking, it says the length of the side $24$ is an inverse hyperbolic function of a trigonometric function of the angle $341$. In his commentary on Gauss's note, Paul Stackel says Gauss missed a factor of $1/2$ in his expression for the volume differential.


As this is Gauss's only result dealing with calculations of volume in hyperbolic space $H^3$ (and not only in the hyperbolic plane), this is a very significant point in his work on non-euclidean geometry that has not recieved enough attention by historians of mathematics (except Stackel's and Stillwell's remarks).


Therefore, my questions are:


* What is the meaning of Gauss's results? how to derive them?
* How to derive a volume function $\Delta(l\_{24})$ for the tetrahedron satisfying Gauss's conditions by the analytic procedure outlined by Gauss?



How to derive from Gauss's results on the volume of hyperbolic orthoscheme tetrahedron the formula of Bolyai?
I am not sure of the notation, but I assume this can be derived from the Schlafli formula for the volume of a tetrahedron (so this seems to indicate that Gauss knew Schlafli's formula three quarters of a century prior to Schlafli):


$$
dV = -\frac12 \sum\_{ij}l\_{ij} d \alpha\_{ij},$$ where $l$ is the length of the edge, and $\alpha$ is the dihedral angle of the edge. Notice, in particular, that if the dihedral angle at an edge does not change, then that edge does not contribute to the sum. Further note that if you look at the link of the vertex $1$ (wlog), this is a spherical triangle, whose angles are the dihedral angles $12, 13, 14$ while its sides are the face angles of the three adjacent faces. By the Gauss-Bonnet theorem (note the first author), the variation of the area of the face is equal to minus the sum of the variations of the angles.


Put this all together, and you should get Gauss' formula. As for Schlafli's formula, there are many nice proofs, a simple geometric one by Vinberg (which appeared in the Geometry of Spaces of Constant Curvature survey), and a very pretty analytic one by Hellmuth Kneser, which appeared in Deutsche Mathematik, and thus is hard to find, but there is a more recent exposition a paper of Feng Luo <https://arxiv.org/abs/math/0412208>.


