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This question is a continuation of my previously-posted question: Several questions about Gauss's contributions to electromagnetism. I wrote it after user vonbrand asked me to split my original question into several. I'm very interested to learn about Gauss's unpublished fragments on electromagnetism; especially I'm interested in Gauss's "Zur Electrodynamik" (Gauss's werke, volume 5, p. 601 - 630).

The first titles in this manuscript relate directly to electric circuits, and record, to my opinion, his discovery of Kirchoff's laws for branched electrical circuits. But in addition, it seems that he studies some interesting configurations for electrical circuits (see p.601-604 (titles 1-2) and p. 621-622 (titles 16-17)), which I didn't find comments about in the literature. The formulas related to these figures seem to be of a very interesting form (they involve uncommon functions in the basic theory of electric circuits). For these reasons, I refuse to believe that nobody knows what these figures describe, so if anyone knows something about these schematic figures, please explain.

It seems also that title 2 in this manuscript records his discovery of the "principle of minimum heat" (established by Kirchoff in 1848; it's related to minimum entropy principles), which Dunnington mentions in p.161 of his biography of Gauss - Gauss writes: "The basic principle is that $\sum ri^2$ is in minimum for the actual current distribution". Here, $r$ is the resistance of a resistor and $i$ the electric current (for explanation see this article: The Minimum Entropy Production Principle). So, did Gauss refer to this important variational principle in several of his letters (to Weber, perhaps?)?

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I'm quite not an electricity engineer, but it seems that the figure on p. 603 (title 2) - which actually represents an interesting electric circuit with a current source "a" (to use Gauss's designation) connected with a configuration of resistors that includes a "Y" shaped collection of 3 resistors (c,e,f in Gauss's notation) positioned in the centre of a circle of three other resistors (b,d,a in Gauss's notation) in a series (i.e, the resistors system is neither serial nor parallel) - is related to the $Y-\Delta$ transform invented in 1899 by Arthur Edwin Kennelly. Gauss gives in this note several formulas which describe the effective resistence of this configuration, as well as the intensity of current in each of the six resistors:

Gauss defines: $$(b+c)(d+e)+f(b+d+c+e) = p$$ $$bcde(\frac{{1}}{{b}}+\frac{{1}}{{c}}+\frac{{1}}{{d}}+\frac{{1}}{{e}}) + f(b+d)(c+e) = q$$

and then notes that the total resistence of the configuration without the piece "a" is: $q/p$.

These formulas seem to be very complicated so one can take it as an indication that he used later techniques from network analysis to derive them. Here is an image of the figure from Gauss's fragment.

Gauss's electrical circuit

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  • $\begingroup$ Maybe I'm misunderstanding the answer, but how could Gauss's works be related to something invented 50 years after his death? $\endgroup$
    – Cubic
    May 31, 2019 at 14:04
  • $\begingroup$ It's a common thing to find in Gauss's writings anticipations of tools and ideas developed much later. I ceased to be surprized of such findings very long ago, as there are many other such examples. Mathematicians and physicists in the past didn't have effective communication, so (for example) Maxwell independently rediscovered the Gauss's linking integral. Now to the point - you are invited to look at the figure p. 603 of volume 5 of Gauss's works - i'm quite sure there is no trivial way to reslove this configuration into parallel or serial resistors. $\endgroup$
    – user2554
    May 31, 2019 at 14:15
  • $\begingroup$ I'll say again - i'm not sure about the relation to the $Y-\Delta$ transform - but i'm sure the formulas given by Gauss in this fragment cannot be derived by non-special techniques. But - as i said - i'm not an electrical engineer - so my answer is only an educated guess that is intended to give idea for dechipering Gauss's note. $\endgroup$
    – user2554
    May 31, 2019 at 14:20
  • $\begingroup$ By the way, i searched in a certain university book on electromagnetism for information on $Y-\Delta$ transform, and accidently i found an exercise on electric circuits which deals with the same configuration as Gauss. I looked in the answers and saw the same formulas which Gauss gave in his note. So now one can assert that Gauss's formulas are correct and do constitute an anticipation of later techniques in network analysis (i didn't write nonesense). $\endgroup$
    – user2554
    May 31, 2019 at 22:58
  • $\begingroup$ Gauss doesn't give a formula for the resistance of the whole configuration, as far as I can see. Only for the configuration without the piece a. How can the current source put on a make a current flow through the whole? Don't there have to be two different points (on a, b, c, d, e, or f) on which one has to put a voltage difference and make a current flow? Which implies that the resistance of the whole is dependent on the two points you choose to put the difference in voltage on? $\endgroup$
    – Deschele Schilder
    Jun 8, 2019 at 13:45
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As for the figures on p.621, I think (but I'm not sure) that the problem which Gauss tries to solve in it is (in modern terms): "to find the self-inductance of a single circular loop". This is a very difficult problem in mathematical physics that is often omitted in textbooks on electromagnetism - who usually adhere to the simple problem of calculating the magnetic field and self-inductance of an infinite coil (that is made of an infinite number of nearly circular loops).

What made me make this conclusion is:

  • Of course, what Gauss himself writes in his verbal description of fragments 16-17.
  • The formula which he arrives involves the term $\log(\frac{8r}{ae^2}) = \log(\frac{8r}{a} )-2$, where $r$ is the radius of the circular coil and $a$ is the radius of the wire which the loop is made of. This is exactly the log term in the modern result for self-inductance of a single circular coil: see this link http://users.telenet.be/minnaert/blog/The_self_inductance_of_a_circular_loop.htm. What makes this thesis stronger is Gauss's words and figure in fragment 17, (although he switches there the notation for the radius of the wire from $a$ to $\rho$). In addition, the factor ''$R$'' (from the modern result) is absent from Gauss's formula because he refers to the induced electric field, not to the induced voltage (so Gauss's formula is derived from the modern result from dividing by $2\pi R$ ).

The only thing that is lacking for complete agreement is the difference between the numerical factors of the two formulas; a $\pi$ is missing from Gauss's. That might be the outcome of a different system of unit that Gauss used (maybe cgs?), but i'm not sure.

i've added here a figure of p.621 from Gauss's nachlass: Gauss's calculation of self-inductance of a single circular loop

Notice: to prevent misunderstandings, the fact that the self-inductance of a single circular loop can be infinite (in the limit $r>>a$) doesn't contradict the finite result for the self-inductance of an infinite coil ($L = n^2\mu_0\pi r^2 $), because the derivation of magnetic field inside an infinite solenoid is valid onle when the distances between consecutive loops are comparable to the radius of the wire from which the coil is made of. Otherwise the assumption that the magnetic field inside the solenoid is uniform is simply incorrect. In the case where $a$ tends to zero, $n$ (the density of the loops) goes to infinity, so one gets consistent results.

Some history:

According to this article - https://nvlpubs.nist.gov/nistpubs/bulletin/04/nbsbulletinv4n1p149_A2b.pdf ("on the self-inductance of circles"), the first to publish a formula for the self-inductance of circle is Kirchhoff in 1864; his formula was the first in a series of more accurate results by Maxwell, Max Wien and others. So Gauss's (unpublished) formula can be interpreted as a first result in this direction. Gauss's fragment 17 contains several additional formulas, one of which is identical to Kirchhoff's.

Update (26.07.2023)

An accidental search in Maxwell's "treatise on electricity and magnetism" led to a surprising find. In p.345-346 of volume 2 of his treatise, Maxwell refers to this calculation of Gauss in the context of the following problem:

To find the form of a coil for which the coefficient of self-induction is in a maximum, when the the total length and thickness of the wire are given.

, and states that the relevant result was first found by Gauss. Gauss's result states that for a coil with circular cross section, the maximal self-induction is obtained when the mean radius $a$ of the coil is $3.22$ times the radius $c$ of the cross section of the coil (the condition for maximality is $a=3.22c$). I guess this result was particulary beneficial for electrical circuits designers, since it allowed the introduction of coil inductors with optimised self-induction.

(note that $a$ is not the arithmetic mean of the radiuses of the toroidal coil, but actually a kind of geometric mean of the radiuses; I am not exactly sure of the details right now. Anyway, I believe the use of the geometric mean was one of Gauss's main innovations in this calculation.)

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