My question is mainly about getting explanation to Gauss's third note (note III) in his writing "Zur Astralgeometrie", (see Cubierung der Tetraeder, pages 228-229), and about placing it in the right historical context. In this note Gauss is writing the following:
"In the tetrahedron 1234 (1,2,3,4 are it's vertexes), whose faces 124 and 134 are orthogonal, denote the volume as $$\Delta,$$ then it holds that: $$\partial \Delta = -24.\partial341,$$ and if the face angles at vertex 3 are constant, then the following also holds:
$$\alpha\alpha\cdot cotg^2341 - \beta\beta\cdot (tgi.24/i)^2 = 1$$
when: $$\alpha = cotg431,\quad \beta = cotg 234.$$
Two things i didn't understand about Gauss's notation immediately came to my mind:
- What does the notation 24 point "something" mean? does the point mean product? or something else?
- What is the "i" that appears in his formulas? how is it defined?
The questions i asked are of great interest to me since I think the problem of determining the volume of the non-euclidean tetrahedron really represents the peak of Gauss's work on hyperbolic geometry, and I really want to get a formal statement of his result, and to understand how is it compared with later results of Lobachevsky and Schlafli.
Update: recently i became aware of a certain interconnection between this fragment (CUBIRUNG DER TETRAEDER) and a second fragment of this kind - "Volumenbestimmungen in der Nichteuklidischen Geometrie" (Gauss's werke, volume 8, p.232-233). This fragment was found among the pages of Gauss's copy of Lobachevsky's 1840 publication (these two fragments are the only ones that deal with the probelm of hyperbolic tetrahedron). As a matter of fact, Stackel mentions, in his commentary of Gauss's "cubirung der tetraeder", that Gauss used the results of the second fragment implicitly when he derived his first displayed formula from p. 228 ($\partial \Delta = -\frac{{1}}{{2}}(24)d(341)$) - he used the two formulas:
- $$d(341) = -sinh(14)\cdot d(13)$$
- $$\partial \Delta = \frac{{1}}{{2}}d(13)\cdot (24)\cdot sinh(14)$$
to set up this relation ($\partial \Delta = -\frac{{1}}{{2}}(24)d(341)$). The second of these formulas was also mentioned in his later fragment, and though this fragment was written later (Gauss wrote this since the publication of Lobachevsky captured his attention), it seems that he was aware of it earlier.
I made this update because maybe dechipering Gauss's second fragment might serve as a breakthrough in understanding what Gauss knew on hyperbolic tetrahedrons.