What was the optical problem discussed and solved by Gauss in his 1831 letter to Brandes?

Question

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). An almost complete english translation of Gauss's letter is the following:

"The circumstances to which you briefly allude at the end of your letter has caused me to devote a spare hour to the essay which you mention. The proper meaning of my remarks there doesn't appear to have been comprehended by all; but the essay itself requires some modification. I now find, as a result of a more laborious investigation, that the indistictness which depends upon the value of the fourth power of the distance of the incindent rays from the axis in the expression for the longitudinal aberration has the least possible influence when the objective is so constructed that those rays, which are incident infinitely near the axis, and also those incident at a distance = $R\sqrt{\frac{6}{5}}$ (where $R$ is the radius of the objective) are combined in one point A, and the eyepiece will be so placed as to observe that point in the axis where the rays combine whose distances = $(\frac{3}{5} - \frac{\sqrt{6}}{10})R$ and $(\frac{3}{5} + \frac{\sqrt{6}}{10})R$ from the axis. If you imagine a plane, perpendicular to the axis, and cutting this at this point, the image upon it would be the more indistinct, the greater the circle around A, which is filled by rays from the object through the object glass, and the intensity of the rays upon each part of the circle must also be noticed. It may seem rather empirical; but i think the most advisable course to go upon the same principles as lie the root of least squares. If $dx$ be an element of this circle, $y$ the distance of this element from A, and $z$ the intensity of the ray, I assume that $\int zy^2dx$ will be the value of the total indistinctness and make this a minimum. I arrived at the following results:

• If the object-glass be constructed in such a manner that the term in the longuitudinal aberration which depends upon the square of the distance from the axis = 0, and the eyepiece be so placed that A falls at that point where these rays cut it, the value of this integral = $E$.

• If, with the same arrangement, the eyepiece be placed in such a manner as to make this integral as small as possible (when A becomes the point of union of rays whose distance is $R(\frac{1}{2})^{\frac{1}{4}}$), the integral is $\frac{1}{4}E$.

• On the other hand, with the above arrangement and the best position for the eyepiece, the integral = $\frac{1}{100}E$, the absolute minimum.

The above result, namely, that the purely imaginary image produced by those rays which lie infinitely near the axis, should be combined at the point of union with rays of a greater distance from the axis than the semi-diameter of the aperture appears at the first very surprising and paradoxical; but the reason is soon seen with a little study. The so-called principal image (by rays lying near the axis) is of the least importance, as it were, on acount of its feeble intensity, and its of more consequence that those rays from the parts which lie nearer the borders of the glass, should be kept near together among each other, so to speak, which is best done in the above manner. I am sorry the the limits of the letter will not allow me to enlarge upon this subject: an accurate calculation is needed - in loose talk, an important point is easily overlooked; but these hints will suffice for those who understand. I generally find that with the best position for the eyepiece, the integral is always equal to $\frac{1}{4}E(1-\frac{8}{5}\mu^2 + \frac{2}{3}\mu^4)$, when the object-glass is so constructed that rays at a distance $\mu R$ are united in one point with the so-called principal image. This is a minimum, for $\mu = \sqrt{\frac{6}{5}}$ and it then $\frac{1}{100}E$. If $\mu = 1$ it would only equal to $\frac{1}{60}E$, and for $\mu$ infinitely small = $\frac{1}{4}E$. Not only is Bohnenberger wrong in this, but I was also wrong myself then, because I have not differed from him as much as I ought to have done. I had then only considered the whole magnitude of the indistinct image, without having sufficiently noticed the intensity of each separate part."

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"?)

Answer 1

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!