My question is a direct continuation of my already posted question https://hsm.stackexchange.com/questions/6772/did-gausss-expression-for-the-differential-of-the-hyperbolic-volume-of-the-tetr (asked on "history of science and mathematics" stackexchange). I simply didn't find any sources that say that Gauss's result was a nonesense; 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 fron his expression for the volume differential), and in particular, the 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)". I ask again a very similar question simply because i didn't get an answer to the title question in my previous post, and since questions that have answers are likely to "go the graveyard", i preffered to ask it again.

I really 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, so i can't succeed in this task without help. 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:

$$\alpha^2\cdot cotg^2341 - \beta^2\cdot tanh^2(l_{24}) = 1$$

when: $$\alpha = cotg431,\quad \beta = cotg 234.$$

Literally speaking, Gauss's formula says the length of the side 24 is an inverse hyperbolic function of a trigonometric function of the angle 341. 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 expreesion for the differential:

$$\partial \Delta = - \frac{{1}}{{2}} l_{24}\cdot \partial A_{341},$$

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 appreciating well 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'll exhaust all of my options.