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: What were the scientific principles of Gauss's photometer?).
- 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.
- I guess the meaning of problem [1] from 1842 is: "to deduce the analogous rule for generating euclidean Pythagorean triples in the case of spherical geometry". I think so because if Gauss's problem was asked for the case of euclidean planar geometry, than rational sines and cosines of the angles of right angled triangle means that $\mathbb{sin\alpha}=\frac{a}{c}, \mathbb{cos\alpha}=\frac{b}{c}$ (where $a,b,c$ are integers) and by the Pythagorean theorem we get $a^2+b^2=c^2$. However ,in spherical geometry the angles of a triangle don't sum up to a constant (which is $\pi$ in euclidean geometry), so for the problem to be well defined Gauss adds the condition that the sides of the spherical triangle should also have rational sines and cosines.
Regarding problem [1] from 1842, I also suspect that Gauss's motivation for searching such a rule was related to his brief interest in crystallography - according to p.167 of Dunnington's biography, "Gauss was interested in the question of rationality or irrationality of the rations of crystallographic coefficients", and since this has something to do with spherical geometry, there might be a connection. But the connection to crystallography is entirely an unfounded speculation of mine.
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.
So my main question is about the meaning of problem [1] from 1830, although any interesting/useful comments on any of the other problems will be blessed!
