(This question was posted before in History of Science and Mathematics stackexchange, but since I recieved no comments after 6 days, I decided to reask it here.)
P. 220-221 of volume 12 of Gauss's werke contain a complete list of the prize problems which Gauss suggested to the Goettingen university in the years 1830, 1834, 1842 and 1849. Those prize problems can shed some light on his intentions for further studies of several mathematical questions whose solution he did not complete. Here is a Google translation (from Latin) of some of the problems in the list:
1830, May 6. [1] Derive the general criterion for the solvability of the differential trinomial $pdx^2+2qdxdy+rdy^2$ into two factors, each of which is a complete differential. [2] Determine the line joining two given points which, when rotated about a given axis, produces a surface of smallest area. [3] To explain the character of the curve, in which the radius of curvature is everywhere reciprocally proportional to the length of the curve. [4] To progress the current state of our knowledge about the periodically variable stars. [5] Demonstrate the equality between the infinite series $$1+(\frac{1}{2})^3x+(\frac{1\cdot3}{2\cdot4})^3x^2+(\frac{1\cdot3\cdot5}{2\cdot4\cdot6})^3x^3+etc$$ and the square of the infinite series $$1+(\frac{1}{4})^2x+(\frac{1\cdot5}{4\cdot8})^2x^2+(\frac{1\cdot5\cdot9}{4\cdot8\cdot12})^2x^3+etc$$ 1834, May 21. [1] Determine the moment of inertia of the five Platonic solids with respect to any axis through the center.[2] To explain the various methods of solving Kepler's problem, especially by means of infinite series, as well as addressing the degree of convergence of these developments. 1842, May 23. [1] To understand the methods for finding any number of right angled spherical triangles whose sides and angles have rational sines and cosines.
Note that I have omitted a few problems for which the translation was not good enough.
Comments:
- Problem [2] from 1830 concerns "minimal surfaces" and was solved by Gauss's pupil Benjamin Goldschmidt. Since the solution of this problem (the catenary) was already well known during the 18th century (L. Euler discovered it), I guess Gauss's real intention was to make the methods related to minimal surfaces more rigorous.
- Problem [3] from 1830 concerns the so-called "Euler spiral".
- Problem [4] from 1830 concerns, according to other sources I read ,the construction of a reliable "photometer" (this is the subject of another question I posted long ago on hsm stackexchange).
- Problem [5] from 1830 was not set as a prize problem by the Goettingen university (it remained merely a suggestion). However, Gauss apparently solved it by himself in p.191-193 of volume 10 of his werke; he apparently shows that when $x=1$ than the first series and the square of the second series are both equal to $2\frac{\varpi^2}{\pi^2}$. It seems that Gauss's identity is a case of Clausen's formula (Clausen deduces the same identity in his article); this raises the question why Gauss suggested to set this as a prize problem if Thomas Clausen has already proved it in 1828.
- Problem [1] from 1834 was solved by the german mathematician Feodor Deahna in his prize awarded essay from 1835 "Momenta inertiae singulorum quinque corporum regularium". Apart from the fact that the Platonic solids are spherical tops, I find this problem to be also very interesting.
What is most interesting to me is problem [1] from 1830 - it deals with a differential form (such as the first and second fundamental forms), therefore it seems that it is concerned with a certain problem of differential geometry. Since it was only 2 years after Gauss's groundbreaking memoir on differential geometry, I guess it is reasonable to believe it might be related to some differential-geometric problem that occupied Gauss's mind during this time.
My questions
- Since the only problem whose meaning I did not recognize yet is problem [1] from 1830, my main question is about the meaning of it, and how it merges into the context of Gauss's differential geometric ideas.
- Regarding the other problems, since it is not possible to cover the whole context for all these problems in a single post, I mainly aim to make a "brain storming" on Gauss's list of problems, so any interesting/useful comments on the other problems will be blessed!
Several clues regarding problem [1]: Based on several clues in Gauss's Nachlass, I have a very "unfounded intuition" that problem [1] is related to a problem in differential geometry which occupied him during this time. In p.446 of volume 8 of Gauss's werke, Gauss states the following:
The conditional equation that $$pdx^2+2qdxdy+rdy^2$$ is a pure product of two total differentials is the following: $$0 = 2(q^2-pr)(\frac{\partial^2p}{\partial y^2}-2\frac{\partial^2q}{\partial x\partial y}+\frac{\partial^2r}{\partial x^2})+p(\frac{\partial p}{\partial y}\frac{\partial r}{\partial y}-2\frac{\partial q}{\partial x}\frac{\partial r}{\partial y}+(\frac{\partial r}{\partial x})^2)+q(\frac{\partial p}{\partial x}\frac{\partial r}{\partial y}-\frac{\partial p}{\partial y}\frac{\partial r}{\partial x}+4\frac{\partial q}{\partial x}\frac{\partial q}{\partial y}-2\frac{\partial q}{\partial x}\frac{\partial r}{\partial x}-2\frac{\partial p}{\partial y}\frac{\partial q}{\partial y})+r(\frac{\partial p}{\partial x}\frac{\partial r}{\partial x}-2\frac{\partial p}{\partial x}\frac{\partial q}{\partial y}+(\frac{\partial p}{\partial y})^2)$$
In his comments on this note, Stackel remarks that in the same notebook in which Gauss wrote this statement, immediately after it appears a short note entitled "general solution to the problem of development of a surface", which appears in p.447-448 of the same volume. The mathematician Julius Weingarten comments on Gauss's approach in this "general solution", and says several things which I was unable to understand but concludes that Gauss's solution is incomplete.
Rephrasing Gauss's problem in modern terms, I guess what he attempted to solve is: "given the surface metric, find all possible isometric embeddings of it in $\mathbb{R^3}$" (I interpreted "development" as "isometric embedding"). This problem in an extension of the Theorema Egregium, and since it is perhaps the only aspect that Gauss left uncomplete in his memoir of differential geometry, I suspect there might be a connection. Many mathematicians - among them Ferdinand Minding and Weingarten himself - have contributed to its solution.
Since I don't have enough knowledge in differential geometry to understand what is going on there, I stress that this supposed connection between Gauss's prize problem from 1830 and the problem of "finding all isometric embeddings" is entirely a conjecture of mine based on "crossing the loose ends". I have no idea how the problem of factoring a differential form into two total differentials is related to isometric embedding of a given metric.