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My question is a direct continuation of my already posted question Did Gauss's expression for the differential of the hyperbolic volume of the tetrahedron agree with later results?.

I simply didn't find any sources that say that Gauss's result was a nonsense; in his commentary on Gauss's relevant note, Stackel doesn't say Gauss was mistaken (apart from a factor of $1/2$ that was missing from his expression for the volume differential), and in particular, the book "Mathematics and Its History", p379, 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 - Cubirung der Tetraeder))".

I want to place Gauss's result in historical context, but the conceptual framework of hyperbolic geometry is as new to me as it was to Gauss's contemporaries. My previous post helped me understand the Gauss's formula for the orthoscheme tetrahedron; it connects the length of the side 24 with the angle 341 by the formula:

$$\alpha^2\cdot \cot^2(341) - \beta^2\cdot \tanh^2(l_{24}) = 1$$

when: $$\alpha = \cot(431),\quad \beta = \cot(234).$$

Now, I understand the method of exhaustion is universal and doesn't depend on type of geometry; whether the geometry is hyperbolic, elliptic or euclidean, one can find volumes by dividing it into slices and then integrate them. But I don't know how to move from the Gauss's expression for the differential:

$$(1) \partial \Delta = - \tfrac12 l_{24}\cdot \partial A_{341},$$

to the volume function; in particular, I lack an expression for the area of the face 341 as a function of the length 24. If anybody can help me with that, then I believe I'll be able to derive an expression for the volume function.

Perhaps I'm not completely appreciating the difficulty of the subject (I know three-dimensional hyperbolic geometry is a pretty advanced topic) and there are very few people who can answer my question, but I won't give up until I've exhausted all of my options.

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  • $\begingroup$ Gauss's formula covers a very special case, why do you expect it to imply the general formula of Lobachevski? For the background on the volume of hyperbolic tetrahedron, read the paper of Milnor that I recommended in the answer on your previous question (it is free on Internet). $\endgroup$ – Alexandre Eremenko Dec 14 '17 at 5:54
  • $\begingroup$ Lobachevsky's formula also covers a very special case - the case of orthoscheme tetrahedron - exactly the case treated by Gauss. Only recently the problem of the general tetrahedron was solved by a number of authors. Even if Gauss's result is a special case of Lobachevsky's formula, i still want to see the resulting volume function from Gauss's result - and to see how this integral is a special case of a more general integral. $\endgroup$ – user2554 Dec 14 '17 at 7:56
  • $\begingroup$ Alexandre Eremenko - woudn't be better if i'll ask this question on mathoverflow rather then here? $\endgroup$ – user2554 Dec 14 '17 at 9:52

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