A question on Gauss' "Vicimus GEGAN"

Question

The 43rd entry(Oct. 1796) of Gauss' mathematical diary "Vicimus GEGAN" remained a mystery for a long time. K. R. Biermann found evidence that GEGAN is related the famous arithmetic-geometric mean(AGM) of Gauss. He also referred this entry to the link between the AGM and general elliptic functions. Since I do not have access to Biermann's article, I am still confused about Biermann's conclusion.

--The 51st entry(Jan. 1797) of the diary shows that Gauss started his research on lemniscate;

--The 92nd entry(July. 1798) of the diary suggests that Gauss found the connection between Jacobi theta-functions and lemniscate functions;

--The 98th entry(May. 1799) of the diary shows that Gauss numerically computed the AGM between $1$ and $\sqrt{2}$ and verified the AGM is equal to a certain lemniscatic integral to 11 digital places;

--The 102nd entry(Dec. 1799) shows that Gauss proved his conjecture.

As is indicated in these later entries, Gauss was likely to be unaware of the link between elliptic functions and the AGM before 1799(i.e., he did not know as much of elliptic functions in 1796 as in 1799). What is the exact argument of Biermann in his 1997 paper?

Answer 1

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There are two parts to Biermann's hypothesis on the meaning of "Vicimus GEGAN":

i) GEGAN should be read in inverse order as standing for (vicimus) N[exum medii] A[rithmetico-] G[eometricum] E[xpectationibus] G[eneralibus];

ii) With "Vicimus GEGAN" the great Gauß was alluding to his discoveries on the connection between the arithmetic-geometric means and the general theory of elliptic functions.

Biermann's paper in the Mitt. Gauß-Ges. Göttt. does confirm the first part of the hypothesis or at least the portion of it wherein he asserts that GEGAN should be read backwards. To be more precise, Biermann reported on a manuscript (by Gauß) found in the Göttingen astronomical observatory by Herr Dr. Hartmut Grosser sometime in 1997; on a certain sheet of the manuscript (there is a copy of it in the penultimate page of the paper!), we can find the abbreviation NAGEG at least three times, the well-known GEGAN, and even a drawing of something which resembles a lemniscate.

Despite the fact that Biermann wrote in "das Fazit" of his paper something like "it remains to determine exactly what the E stands for", he wrapped it up expressing his belief that the sketch of the lemniscate just below the "GEGAN" on that page of the manuscript also validated the second part of his conjecture:

Was nun aber der Fund so bedeutsam macht, sind die Majuskeln G E G A N mit der Skizze einer Lemniskate darunter. Das Wort GEGAN begegnet uns auf der Seite noch mehrfach, teils kunstvoll kalligraphisch verschnörkelt, auch in der Form gegan. Was nun aber allem die Krone aufsetzt, ist die Begegnung mit dem so lange vergeblich gesuchten Schlüsselwort NAGEG, auch in der Form nageg.

Welche Schlußfolgerungen können nun aus diesem Sachverhalt gezogen werden?

1. Das gemeinsame Auftreten des Schlüsselworts GEGAN und einer Lemniskate beweist, daß die Hypothese ... , wonach GEGAN mit dem agM in Verbindung steht, begründet war.

2. Das nunmehr bei Gauß nachgewiesene Schlüsselwort NAGEG stützt entscheidend die obige Hypothese ..., wonach die Gaußschen Schlüsselwörter rückwärts zu lesen sind.

Fazit: Die vom Verfasser bei seinen Dechiffrierungsversuchen Gaußscher Schlüsselwörter und Buchstabengruppen erzielten Resultate werden durch das von Herrn Dr. Grosser gefundene und freundlicherweise als Kopie zur Verfügung gestellte Gaußsche Notizblatt bestätigt. Mag unsicher bleiben, ob etwa E zu Exspectatio bzw. zu Expressio bzw. zu Explicatio zu ergänzen sei usw. – der Sinn der zunächst rätselhaften Schlüsselwörter dürfte nunmehr unstrittig sein: Die Chiffren bezogen sich auf die große Gaußsche Entdeckung des Zusammenhangs zwischen lemniskatischen Funktionen, agM und Potenzreihen und der daraus resultierenden Theorie der Modulfunktion und der elliptischen Transzendenten. Teils haben sie retrospektiven ("vicimus", "demonstravimus"), teils programmatischen ("revolve!") Charakter. Dergestalt ermöglicht uns das Studium der Gaußschen Verschlüsselungen einen Blick in die geistige Werkstatt eines Genies.

Translation:

What makes this discovery so significant are the capital letters G E G A N with the sketch of a lemniscate below them. We encounter the word GEGAN several additional times on this page, sometimes in ornate calligraphic flourishes, and also in the variant gegan. What tops it all is the encounter with the keyword NAGEG, also in the variant nageg, the search for which had so long proved elusive.

What conclusions can now be draw from this factual situation?

1. The joint appearance of the keyword GEGAN and a lemniscate proves that hypothesis ..., according to which GEGAN is linked to the AGM, was well founded.

2. The keyword NAGEG now documented in Gauss's writings decisively supports hypothesis ..., according to which Gaussian keywords are to be read in reverse.

In conclusion: The results achieved by the author in his attempts to decipher Gaussian keywords and letter groups are being confirmed by the sheet of scratch paper found and kindly supplied in copy by Dr. Grosser. While it may remain uncertain whether, for example, E should be expanded into Exspectatio, or Expressio, or Explicatio etc, the sense of the initially enigmatic keywords should now be beyond dispute:

These ciphers referred to Gauss's major discovery of the connection between lemniscatic functions, the AGM, and power series and the resulting theory of the modular form and elliptic transcendentals. Their character is in part retrospective ("vicimus", "demonstravimus"), in part programmatic ("revolve!"). In this way the study of Gaussian cipherings allows for a peek into the mental workshop of a genius.

If after reading Biermann's paper through and through, you still find hard to reconcile the second part of his hypothesis with the entries in Gauß's diary which you listed in your question, I think you can do no worse than to take a look at this paper:

D. A. Cox, The arithmetic-geometric mean of Gauß. L'Enseignement Mathématique, vol. 30, 1984, pp. 275-330.

In particular, on page 321 of Cox's paper you will find this:

... how much did [Gauß] know about the agM [in May, 1799]? Unfortunately, this a very difficult question to answer. Only a few scattered fragments dealing with the agM can be dated before May 30, 1799 (see [12, X.1, pp. 172-173 and 260]). As for the date 1791 of his discovery of the agM, it comes from a letter he wrote in 1816 (see [12, X.1, p. 247]), and Gauß is known to have been wrong in his recollections of dates. The only other knowledge we have about the agM in this period is an oral tradition which holds that Gauß knew the relation between theta functions and the agM in 1794 (see [12, III, 493]). We will soon see that this claim is not as outrageous as one might suspect.

Interestingly enough, Cox did not mention Biermann's hypothesis on the "Vicimus GEGAN" entry at all in his article...

Answer 2

@ José Hdz. Stgo.:

Thank you for this great answer. I think I should add a few comments on the related materials I collected from different books. (Since I am neither a native speaker of German language nor well-versed in it, sometimes I need google translation or deepL translation for a better understanding on all these mysteries)

I believe Biermann used some hypotheses originated from L. Schlesinger on Gauss' work of AGM and elliptic functions(see Fragmente zur Theorie des arithmetrisch-geometrischen Mittels aus den Jahren 1797-1799, L. Schlesinger, 1911). Schlesinger made a great effort on dating some fragments on elliptic functions in Gauss' Leistenotizen(in which most of the records are dated no later than 1798), which is extremely important for understanding the development of Gauss' theory on elliptic functions. Schlesinger pointed out that the fragments in the Leistenotizen containing AGM are consisted of:

a)AGM and the rectification of ellipses;

b)AGM and the series whose exponents are quadratic functions(i.e., theta functions);

c)Series expansion of elliptic integrals as well as linear differential equation for elliptic integral of the first kind and the second kind.

In 1911 Schlesinger dated these materials no later than 1798 as well as other records in the Leistenotizen. It is very likely that this hypothesis contradicts what Gauss wrote in his dairy on May, 30th 1799(which implies that Gauss did not know the relation between AGM and elliptic integral before 1799), so later Schlesinger changed it to the summer of 1799(Gauss's Werke, X-1, pp. 273).

In the same year(Nov. 1799), Gauss started his Scheda Ac, another important record on Gauss' own development of elliptic function theory. In the Scheda Ac one can see the encrypted word GALEN(Gauss's Werke, X-1, pp. 273). I trust Biermann's interpretation of the GALEN that Gauss realized the importance of the reciprocal of AGM(which is equal to some elliptic integral of the first kind), and I do not reject Biermann's hypothesis that "Vicimus GEGAN" is a discovery on AGM. but I do not think "Vicimus GEGAN" is a discovery on the AGM and elliptic integrals/thetafunctions:

I)The date of the record "Vicimus GEGAN" is Oct. 21, 1796. Gauss started his research on lemniscate months after that(Jan. 1797). It is highly improbable that Gauss had already known the relation between the general AGM and the elliptic integral of the first kind when he had not even started his research on a special case(the lemniscatic integral).

II)I doubt whether the oral tradition that Schering recorded is reliable. Gauss already know that

$$\theta_4(e^{-\pi})=1-e^{-\pi}+e^{-4\pi}-\cdots=\sqrt{\frac{\varpi}{\pi}}$$

$$\theta_2(e^{-\pi})=2e^{-\pi/4}+e^{-9\pi/4}+\cdots=\sqrt{\frac{\varpi}{\pi}}$$

$$\theta_3(e^{-\pi})=1+e^{-\pi}+e^{-4\pi}+\cdots$$

$$\theta_3^4=\theta_4^4+\theta_2^4$$

in 1798(Scheda Aa, see Gauss's Werke, III, pp. 418), where $\varpi$ is the lemniscatic constant

$$\varpi=2\int_{0}^1\frac{\mathrm{d}x}{\sqrt{1-x^4}}.$$

If he did know the relation between AGM and the theta functions in 1794, he can prove that $$AGM((\theta_3(q))^2,(\theta_4(q))^2)=1$$ without any effort, which would definitely answer Gauss' question on May 30th, 1799. Pfaff's letter(Gauss's Werke, X-1, pp. 273, which is quoted in Hidden Harmony—Geometric Fantasies: The Rise of Complex Function Theory by Jeremy Gray) suggests that the proof was elusive to Gauss even in Nov. 1799.

III)If Gauss had discovered anything about AGM in 1796, it might be something else, e.g. the asymptotic formula for AGM in Scheda Ac(Gauss's Werke, X-1, pp. 186, Gauss also said he found that AGM can be written as the quotient of two transcendental functions long ago in the 101st entry of his diary). But it is likely that we can never know what Gauss discovered on Oct. 21, 1796.