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Volume 5 of Gauss's works contains a section which includes "Essays on various objects in mathematical physics". Just to emphasize the importance of this section, i'll mention that it includes, for example, his theory of falling objects on the rotating earth (precursor to the theory of Coriolis force), as well as his treatise about the achromatic double lenses, which was the subject of a previous unanswered post (What was Gauss's theoretical work related to his invention of the "Double-Gauss lens"?). The same section includes a 2-pages "letter to Brandes on the same subject" (the subject of achromatic double lenses). This letter is of great interest to me, because it includes uncommon and interesting formulas in optical theory - it involves, for example, certain integrals (so this letter doesn't seem to deal with the traditional aspects of optics).

I searched in Clemens Schaefer's treatise for information about this, and in article 84, it states that his letter to Brandes is a "result of a new calculation, which takes the intensity of light into account". Another useful piece of information which i inffered from Schaefer's treatise is that the calculation Gauss made to arrive at his results is reproduced in J.C.E Schmidt's "textbook on analytical optics" (published in 1834).

Therefore:

  • What was the specific problem which Gauss solved in this letter (Gauss only gives final results)? unfortunately, i was not able to understand much from the section on optics in Schaefer's treatise, as it immediately throws the reader into the details of Gauss's occupation with optics and doesn't give much background and supporting motivation.

Update:

While my posted answer clarified a little bit the context of writing of Gauss's letter, it still doesn't give a direct answer to the title question. Therefore, in order to futher eludicate the answer, i'll focus now more on the details of Gauss's letter.

Gauss writes in his letter:

I now find through a more in-depth investigation that the indistinctness, which depends on the fourth power of the distance of the striking rays in the expression for the length deviation, has the smallest possible total influence, if one constructs the objective in such a way...

There, the main riddle to dechiper now is:

  • what is the meaning of the term "indistinctness" which Gauss uses? why does it depend on the fourth power of the distance of the striking rays in the expression for the length deviation? (and what is "the expression for the length deviation"?)
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  • $\begingroup$ I found the following sentence from the article "is lens design legal?" - in 1831 Gauss balanced forst-order and second-order spherical aberrations in the presence of defocusing. I'm quite sure this article refers to "Gauss's letter to brandes", but unfortunately i don't have access to this article. $\endgroup$ – user2554 Jul 4 at 15:14
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The article "Is lens design legal?" gives some details about Gauss's letter to Brandes; although it doesn't explain much about Gauss's formulas and calculations, it describes very well the historical significance and meaning of this letter. The article discusses "figures of merit" of optical systems; a kind of quantitive measure of the quality of an image formed by a lens system. The following passage is taken from this article:

In 1831 Gauss balanced first-order and second-order spherical aberrations in the presence of defocusing. In his published "letter to Brandes" Gauss wrote: "Based on your letter with regard to my recent paper on lens design I decided to spend one more hour on this topic, which apparently has been misunderstood by some readers." And: "Originally I minimized the total size of the blurred image without weighting its parts" - Apparently Gauss had used Chebyshev's worst case rule to optimize transversal aberrations - "But that is somewhat arbitrary. I consider it now most appropriate to apply the principle of least mean squares, which i call total-undeutlichkeit (total blur) in this context". Obviously, Gauss used the second Gauss moment of transversal aberrations as the figure of merit for lens design. The concluding sentence reads: "The limits of a letter forbid a detailed proof, but my hints will suffice for the expert." (This was written nearly 160 years ago.)

The principle used by Gauss to quantify "blurriness" (the principle of least mean squares) is at the foundation of current theories related to image proccesing (like algorithms for removing blurriness of an image), and this letter suggests a possible reason why "Gaussian blur" is called by this name (although i'm not quite sure about the naming of "Gaussian blur"). The article explains more about the consequences of Gauss's calculation but unfortunately i'm not familiar enough with optical theory for understanding it.

More detailed explanation:

In Gauss's letter he gives the following definition for the "total blur" of an image:

$$\int i \rho^2 ds$$,

where $i$ is the intensity of light falling on a point of the image plane, $ds$ is the area element, and $\rho$ is the distance of that point from a certain point $A$ which Gauss defines (i didn't understand the definition of $A$). Gauss definition is therefore coinciding with the algorithm of "weighted least-squares" - here the weights are the intensities of Light. Note that according to this definition, even the image of a point source might have non-zero blur - exactly because of the spherical aberration! More generally, the quantification of the quality of a lens system is done by substracting the blur of the source from the blur of it's image (that definition is relevant for extended sources).

Since the modern approach to identify high-contrast images (images with low blur) is to subject it to a Fourier transform and see if there are significant high-frequency components, one can state affirmatively that Gauss's integral formula captures the essence of the modern approach. Gauss gives in his letter several formulas that describe the conditions needed to minimize the total blur.

Note: this is a very partial answer that articulates only one aspect of Gauss's letter, and a further examination of it is needed. Any informative and usefull comments will be blessed!

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