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Certainly, Gauss later corrected the mistake, but this was apparently only after he became acquainted with Lobachevsky's work. According to Stackel's comments, the second formula in Gauss's second formula appears as formula 62 in Lobachevsky's work "On the Principles of Geometry" (1830), which can be found here. Immediately after that (in [43]-[49] of this work), Lobachevsky communicates his discovery of an expression for volume of hyperbolic orthoscheme tetrahedron as function of three dihedral angles.

A simple derivation of the formula for the element $dP$ of a pyramid was given by Lobachevsky in year 1830 in the treatise "on the principles of geometry". He writes: $$dP = \frac{1}{2}d\psi(r\mathbb{cos}\varphi-h)$$ In fact, in non-euclidean geometry $r\mathbb{cos}\varphi$ is greater than $h$ or $x$. On the other hand, he does not explicitly mention the formula for volume $ABCabc$, which Gauss seems to have had as early as 1832 (see p.229 of this volume). It also reads as: the volume ABCabc is equal to half the product of the base $BC$ and the inline of the infinitely small quadrilateral $ADda$.

Certainly, Gauss later corrected the mistake, but this was apparently only after he became acquainted with Lobachevsky's work. According to Stackel's comments, the second formula in Gauss's second fragment appears as formula 62 in Lobachevsky's work "On the Principles of Geometry" (1830), which can be found here. Immediately after that (in [43]-[49] of this work), Lobachevsky communicates his discovery of an expression for volume of hyperbolic orthoscheme tetrahedron as function of three dihedral angles.

A simple derivation of the formula for the element $dP$ of a pyramid was given by Lobachevsky in year 1830 in the treatise "on the principles of geometry". He writes: $$dP = \frac{1}{2}d\psi(r\mathbb{cos}\varphi-h)$$ In fact, in non-euclidean geometry $r\mathbb{cos}\varphi$ is greater than $h$ or $x$. On the other hand, he does not explicitly mention the formula for volume $ABCabc$, which Gauss seems to have had as early as 1832 (see p.229 of this volume). It also reads as: the volume ABCabc is equal to half the product of the base $BC$ and the area of the infinitely small quadrilateral $ADda$.

In the first part of this fragment, the (infinitesimal) volume element that Gauss gives a formula for is an hyperbolic wedge, and according to p.114-115 of the book "Geometry II: Spaces of Constant Curvature", this result is indeed correct. There, the author proves the result:

Regarding the second part of Gauss's fragment (about the pyramid $ABCD$) - the formula Gauss gives is consistent with Schlafli formula. To see this, note that if the dihedral angle $\theta$ is very small, than varying it changes only dihedral angles at sides $AB,AC,AD$ of the pyramid. The dihedral angles at sides $AC,AD$ are equal (we will denote it $\beta$), and if we increase $\theta$ than $\beta$ decreases by $\frac{1}{2}\mathbb{cos}\varphi d\theta$. To see why this statement is correct all we need is to consider a spherical triangle with two sides $\varphi$ that enclose an infinitesimal angle $\theta$. Than a simple calculation shows that $\beta-\frac{\pi}{2}$ - which is the change in dihedral angles $AC,AD$ - is $-\frac{1}{2}\theta\mathbb{cos}\varphi$. Now combine this with the fact that $$AB=x$$ $$AC=AD=r$$

$$Volumen = \frac{1}{2}(x-r\mathbb{cos}\varphi)\theta$$

In the first part of this fragment, the (infinitesimal) volume element that Gauss gives a formula for is an hyperbolic wedge, and according to p.114-115 of the book "Geometry II: Spaces of Constant Curvature", this result is indeed correct. There, the author (E.B. Vinberg) proves the result:

Regarding the second part of Gauss's fragment (about the pyramid $ABCD$) - the formula Gauss gives is consistent with Schlafli formula. To see this, note that if the dihedral angle $\theta$ is very small, than varying it changes only dihedral angles at sides $AB,AC,AD$ of the pyramid. The dihedral angles at sides $AC,AD$ are equal (we will denote it $\beta$), and if we increase $\theta$ than $\beta$ decreases by $\frac{1}{2}\mathbb{cos}\varphi d\theta$. To see why this statement is correct all we need is to consider the link of vertex $A$, which is a spherical triangle with two sides $\varphi$ that enclose an infinitesimal angle $\theta$. Than a simple calculation shows that $\beta-\frac{\pi}{2}$ - which is the change in dihedral angles $AC,AD$ - is $-\frac{1}{2}\theta\mathbb{cos}\varphi$. Now combine this with the fact that $$AB=x$$ $$AC=AD=r$$

$$Volumen = \frac{1}{2}(x-r\mathbb{cos}\varphi)\theta$$

Concluding remarks

According to one of the letters in the Bolyai-Gauss correspondence, after Gauss recommended Bolyai to determine the volume of hyperbolic tetrahedron, Bolyai's father replied that his son has already dealt with such volume calculations before. However, Since Bolyai's Appendix does not contain any volume calculation, Gauss's first fragment must be a result of original thought process of him. On the other hand, as noted by Stackel, his first fragment does not contain a factor of $\frac{1}{2}$, and this made me wonder if he was truly aware of Schlafli's formula in its full generality (at least in three dimensions). To be sure, his procedure shares some striking similarities with it, like the use of volume differential $\partial \Delta$ (his idea treats a tetrahedron as part of parametric family, instead of trying to calculate volume directly from length of sides) and of the dihedral angle at side $24$ (use which is somewhat hidden by the fact that it equals angle $341$).

Certainly, Gauss later corrected the mistake, but this was apparently only after he became acquainted with Lobachevsky's work. According to Stackel's comments, the second formula in Gauss's second formula appears as formula 62 in Lobachevsky's work "On the Principles of Geometry" (1830), which can be found here. Immediately after that (in [43]-[49] of this work), Lobachevsky communicates his discovery of an expression for volume of hyperbolic orthoscheme tetrahedron as function of three dihedral angles.

On the other hand, Stackel says the following:

A simple derivation of the formula for the element $dP$ of a pyramid was given by Lobachevsky in year 1830 in the treatise "on the principles of geometry". He writes: $$dP = \frac{1}{2}d\psi(r\mathbb{cos}\varphi-h)$$ In fact, in non-euclidean geometry $r\mathbb{cos}\varphi$ is greater than $h$ or $x$. On the other hand, he does not explicitly mention the formula for volume $ABCabc$, which Gauss seems to have had as early as 1832 (see p.229 of this volume). It also reads as: the volume ABCabc is equal to half the product of the base $BC$ and the inline of the infinitely small quadrilateral $ADda$.

In Vinberg's book mentioned before, he too does not mention Lobachevsky or Bolyai as the originators of the volume formula for hyperbolic wedge, and just remarks that Schlafli came close to it in 1853.

If Stackel and Vinberg did not confuse the historical facts, than I think the formula for volume of hyperbolic wedge is perhaps the only result in hyperbolic geometric that originated with Gauss; all his other unpublished results were covered and developed much more comprehensively by Bolyai and Lobachevsky. As such, it is an important fact.

(I know this post is very long, but I think this is the only place in Web where Gauss's work on hyperbolic volume is discussed comprehensively, so I had to collect all useful information here.)

(Here $\phi$ is the angle of face $ABC$ at corner $A$, as is clear from Gauss's drawing (not shown here).)

(Here $\varphi$ is the angle of face $ABC$ at corner $A$, as is clear from Gauss's drawing (not shown here).)