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.

  • $\begingroup$ Bolay? do you mean Bolyai? $\endgroup$ – Gerry Myerson Dec 14 '17 at 11:43
  • 1
    $\begingroup$ I corrected the spelling of his name. $\endgroup$ – user2554 Dec 14 '17 at 14:53

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.

  • $\begingroup$ I voted your answer because it really helped me understand things about Gauss's formula (now i know it's form is similar to the one given by Schlafli, and that helps me a lot). Thanks!, but i still need further explanation. In particular, as far as i know, Gauss meant hyperbolic geometry when he talked about "non-euclidean geometry", but when you mentioned spherical triangle it made suspect that perhaps we are not talking about the same thing. $\endgroup$ – user2554 Dec 15 '17 at 7:55
  • $\begingroup$ In addition, since i don't understand well the rimannnian theory of 3-manifolds, i have a basic misunderstanding: in three-dimensional hyperbolic spaces with constant space curvature k, the Gauss-Bonnet theorem can be applied to figures with volume, but does it mean it can be applied to figures with finite area but zero volume? i ask because by analogy with the Gauss-Bonnet theorem for 2-manifolds, it cannot by applied to cross sections (sections of one dimension lower). $\endgroup$ – user2554 Dec 15 '17 at 8:04
  • $\begingroup$ @user2554 The link of a vertex of a polyhedron in any geometry is a spherical triangle. As for Gauss-Bonnet, in its original form (due to Gauss) it is stated for triangles. In any case, area as angle defect is a fundamental fact of hyperbolic geometry. $\endgroup$ – Igor Rivin Dec 15 '17 at 16:51
  • $\begingroup$ Sorry but i'm not a professional mathematician - i asked my question here simply because its the only place where there are experts to answer my question. As far as i know, "link" is a concept from knot theory, and i dont know the concept of a "link of a vertex of a polyhedron". What is it? $\endgroup$ – user2554 Dec 15 '17 at 17:31

For the completeness of the discussion, i must add a reconstruction of Gauss's procedure, which i discovered a few months ago. I already posted it on HSM stack exchange, so i simply copy it to here:

Gauss's procedure does imply Bolyai's result on the volume of orthoscheme tetrahedron, as i'll show here. However, Gauss's result is a little bit more limited than Bolyai, since in Gauss's tetrahedron 4 of the 12 face angles of the tetrahedron are right, while Bolyai refers to a slightly more general tetrahedron whose only 3 face angles ar right.

Preliminary discussion:

To see the connection between Schlafli formula and the first formula in Gauss's fragment, one needs to understand that Gauss thinks of the tetrahedron 1234 in such a way that the faces 124 and 134 are perpendicular and the edges 24 and 13 meet the intersection line 14 also at right angles. Therefore, the dihedral angles at sides 12 and 14 are constant right angles and don't contribute to the sum in Schlafli formula. In addition, Gauss defines the tetrahedron in such a way that the angles at vertex 3 are constant (so that an "observer" in hyperbolic space which is located at vertex 3 sees the rest of the vertexes at constant lines of sight). Since the three face angles at vertex 3 correspond to the length of sides of a spherical triangle, and the dihedral angles at sides 31,32,34 correspond to the angles of this spherical triangle, one gets that constancy of face angles at vertex 3 implies constancy of the dihedral angles 31,32,34.

Therefore, only the dihedral angle of the side 24 changes. The dihedral angle 24 is equal to face angle 341 since two face angles at vertex 4 are right so the third face angle 341 (which is one side of a spherical triangle) is equal to the opposite angle - which is dihedral angle 24. This leads directly to the first formula in Gauss's fragment (apart from a missing factor of $\frac {1}{2}$).

Derivation of explicit volume formula from Gauss's formula:

For the sake of consistency, we denote the angles 431, 234, and 214 as $\alpha$, $\beta$ and $\gamma$, respectively. Now lets look at the link of vertex 3 of the tetrahedron: it's a spherical triangle whose two edges lengths are $\alpha$ and $\beta$ and one angle is $\gamma$ (it is the dihedral angle of edge 31 and it's also equal to $\gamma$). In addition the sides $\alpha$$, \beta$ of this spherical triangle are orthogonal to each other. Therefore, by a combination of the spherical sine theorem and the spherical pythagoras theorem, we get:

$$ \frac{{sin(arccos(cos\alpha\cdot cos\beta))}}{{sin 90}} = \frac {{sin\beta}}{{sin\gamma}}$$, or:

$$sin\gamma = \frac {{sin\beta}}{{\sqrt{{1 - (cos\alpha \cdot cos\beta)^2}}}}$$

Now, Gauss's procedure for the calculation of the volume leads to the following integral:

$$\Delta = \frac {{tan\beta}}{{2 tan \gamma}}\int_{0}^{c}\frac {{x sinh(x) dx}}{{(cosh^2(x) - 1 + \frac {{cosh^2(x)}}{{sin^2\alpha cot^2\beta}})\sqrt{{\frac{{cosh^2x}}{{cos^2\beta}} - 1}} }}$$.

Now, the left factor of the denominator $cosh^2(x)(1 + \frac{{1}}{{sin^2\alpha \ cot^2\beta}})-1$, is exactly equal to $cosh^2(x)\cdot \frac{{1}}{{cos^2\gamma}}-1$, because subtitution of $cos\gamma = \sqrt {1 - \frac {{sin^2\beta}}{{1-(cos\alpha\cdot cos\beta)^2}}}$ in this expression gives the previous one.

Concluding remarks:

$$Vol(T) = \frac {{tan\beta}}{{2 tan \gamma}}\int_{0}^{c}\frac {{x sinh(x) dx}}{{(\frac {{cosh^2(x)}}{{cos^2\gamma}} - 1)\sqrt{{\frac{{cosh^2x}}{{cos^2\beta}} - 1}} }}$$

can be seen as treating a slightly more general case then the one treated by Gauss (note: the differences in notation between the Bolyai integral in the presentation and Gauss's integral are just due the different symbols of the angles 431, 234, and 214).

However, for the case treated by Gauss, his formulas are absolutely correct. He should also be given credit for the identification of the calculation of the orthoscheme tetrahedron as the basis for volume formulas of general tetrahedrons (without right angles). In one of his letters, he refered to those calculations of volumes as "die jungle" - i guess he refered to the extremely complicated integrals that arise in the attempts to the decompose the general tetrahedron into orthoscemes (this problem was only solved very recently).

  • Paul Stackel, the mathematician who edited Janos Bolyai's geometric works, had the following things to say about Bolyai's derivation of his integral formula:

It is most remarkable that the method that Gauss used for cubing the tetrahedron, is exactly the same as that of Johann. This is shown in a note from March 1832, from Gauss's estate, which is printed in the works (vol. VIII, p. 228); Gauss has exactly the same special tetrahedron (only 3142 instead $abc\delta$ means) and exactly the same decomposition by planes perpendicular to ab (31).

This quotation is taken from p. 113 of the book "Wolfgang und Johann Bolyai geometrische Untersuchungen" (here is a link: https://archive.org/details/wolfgangundjohan01stuoft/page/112/mode/2up), which was edited and translated to german by the Stackel.

Since it was the same Stackel who also pointed out that Bolyai discovered a form of Schlafli's formula, i think it confirms the view that Gauss really discovered a special case of Schlafli formula. This is a very significant point in Gauss's work on non-euclidean geometry that went unnoticed and was never commented on seriously by historians of mathematics. I think so because it's perhaps Gauss's only fragment dealing with calculations in hyperbolic space $H^3$, and not just in the hyperbolic plane.

  • It's still necessary to understand how Gauss arrived at the formula $\partial \Delta = -\frac{{1}}{{2}}(24)d(341)$ (he missed the factor $\frac {{1}}{{2}}$ at the first attempt); the second formula from his note can be derived with relative ease. I think dechipering Gauss's second fragment might serve as a clue for understanding Gauss's reasoning.

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