# A “couple” of questions on Gauss's mathematical diary

Throughout my upbringing, I encountered the following annotations on Gauss's diary in several so-called accounts of the history of mathematics:

"... A few of the entries indicate that the diary was a strictly private affair of its author's (sic). Thus for July 10, 1796, there is the entry

ΕΥΡΗΚΑ! num = Δ + Δ + Δ.

Translated , this echoes Archimedes' exultant "Eureka!" and states that every positive integer is the sum of three triangular numbers—such a number is one of the sequence 0, 1, 3, 6, 10, 15, ... where each (after 0) is of the form $\frac{1}{2}n(n+1)$, $n$ being a positive integer. Another way of saying the same thing is that every number of the form $8n+3$ is a sum of three odd squares... It is not easy to prove this from scratch.

Less intelligible is the cryptic entry for October 11, 1796, "Vicimus GEGAN." What dragon had Gauss conquered this time? Or what giant had he overcome on April 8, 1799, when he boxes REV. GALEN up in a neat rectangle? Although the meaning of these is lost forever the remaining 144 are for the most part clear enough." "

The preceding paragraphs have been quoted verbatim from J. Newman's The World of MATHEMATICS (Vol. I, pages 304-305) and the questions that I pose today were motivated from my recent spotting of [2]:

• Why is there no mention whatsoever to the REV. GALEN inscription in either Klein's or Gray's work?

• What is the reason that E. T. Bell expressed that Gauss had written the Vicimus GEGAN entry on October 11, 1796? According to Klein, Gray, and (even) the Wikipedians it was written on October 21, 1796. As far as I understand, Klein and Gray are just reporting the dates that appear on the original manuscript. Did Bell actually go over it?

• Last but not least, is there a compendium out there of all known potential explanations to the Vicimus GEGAN enigma? The only ones whereof I have notice can be found on page 112 of [1]:

"... Following a suggestion of Schlesinger [Gauss, Werke, X.1, part 2, 29], Biermann ... proposed that GA stood for Geometricas, Arithmeticas, so reading GEGAN in reverse as Vicimus N[exum] A[rithmetico] G[eometrici cum] E[xspectationibus] G[eneralibus]. Schumann has since proposed other variants; including, for GA, (La) G(rangianae) A(nalysis)..."

References

1. J. J. Gray. " A commentary on Gauss's mathematical diary, 1796-1814, with an English translation". Expo. Math. 2 (1984), 97-130.
2. F. Klein. "Gauß' wissenschaftliches Tagebuch 1796–1814". Math. Ann. 57 (1903), 1–34.
3. M. Perero. Historia e Historias de Matemáticas. Grupo Editorial Iberoamérica, 1994, pág. 40.
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Speaking of reading in reverse, does J.J. Gray's article really start on page 197 and end on page 130? –  Barry Cipra Jan 25 '12 at 21:35
Good catch, Barry! –  J. H. S. Jan 25 '12 at 22:17
J.H.S., perhaps Bell's "October 11" was similarly a typo. A good copy editor will catch page citations that don't make sense, and will certainly question a date like "October 41," but it's easy to imagine a date that makes sense escaping notice. –  Barry Cipra Jan 25 '12 at 22:38

I think part of the answer may be found by consulting Volume X of Gauss's Werke. "REV. GALEN" doesn't actually appear in the Tagebuch itself, a facsimile of which appears following page 482. It was jotted down by Gauss elsewhere, as explained on page 539, in the commentary (which runs for nearly three pages) on the Tagebuch entry dated April 8, 1799.

Just above the excerpted paragraphs from Men of Mathematics, Bell writes, "A facsimile reproduction [of Gauss's diary] was published in 1917 in the tenth volume (part 1) of Gauss' [sic] collected works, together with an exhaustive analysis of its contents by several expert editors." I think it's safe to assume that Bell actually looked at this 1917 publication (and I think it's reasonable to assume that the 1973 edition I'm looking at right now is not substantially different), and I think it's fair to conjecture that Bell paid more attention -- but maybe not enough! -- to the transcription and commentary than he did to the facsimile.

As for the misdating of "Vicimus GEGAN," the correct date is clear enough in both the facsimile and in the transcription on page 507. For one thing, it appears immediately below an entry dated October 18. My guess is that either Bell or the typesetter made a simple mistake.

Finally, a useful reference, especially for "GEGAN" (and a related notation, "WAEGEGAN") is Mathematisches Tagebuch : 1796-1814. Unfortunately, my command of German is insufficient to give a good synopsis of what's to be found there. I hope an actual historian will weigh in here.

Added Feb. 21: It turns out there is a 2005 edition of Mathematisches Tagebuch 1796-1814 (the copy I found earlier was a 1985 edition) which has an update referring to a 1997 paper by Kurt Biermann. Here is a relevant Zentralblatt review of that paper:

Zbl 0888.01025 Biermann, Kurt-R. Vicimus NAGEG. Confirmation of a hypothesis. (Vicimus NAGEG. Bestätigung einer Hypothese.) (German) [J] Mitt., Gauss-Ges. Gött. 34, 31-34 (1997).

The author, a well-known expert on Carl Friedrich Gauss, reports on a Gauss-manuscript, which was found recently in the Göttingen astronomical observatory by H. Grosser and which confirms a hypothesis by Biermann from 1963. At that time Biermann read the frequent code GEGAN in Gauss' diary and manuscripts in inverse order as standing for (vicimus) N[exum medii] A[rithmetico-] G[eometricum] E[xpectationibus] G[eneralibus]. This in turn was alluding (in Biermann's opinion) to Gauss' discovery of the connections between the arithmetic geometric mean and the general theory of elliptic functions. The recently found Gauss-manuscript shows, for the first time, the code NAGEG, and, on the same sheet (which is reproduced in the article), the well-known GEGAN alongside with the picture (by Gauss' hand) of a lemniscate. Thus a remarkable historical hypothesis has been essentially solved after more than three decades. [R.Siegmund-Schultze (Berlin)]

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Incidentally, Jeremy Gray's commentary is reprinted in a new (2004) edition of Carl Friedrich Gauss: Titan of Science by G. Waldo Dunnington, published by the MAA. (Unfortunately, the libraries here only have the original, 1955 edition, so I'm only going by what I came across online.) The 1903 article by Felix Klein can be found online if you're persistent. –  Barry Cipra Jan 26 '12 at 23:09
Maybe I shouldn't be so coy about Klein: Go to de.wikisource.org/wiki/Tagebücher and search for Carl Friedrich Gauß (under 19. Jahrhundert), click on the link, and then click on the pdf icon. –  Barry Cipra Jan 26 '12 at 23:17

I believe I have a reference for Schuhmann:

Schuhmann E. 1976 Vicimus GEGAN, Interpretationsvarianten zu einer Tagebuchnotiz von C.F. Gauss, Naturwiss. Tech. Medezin. 13.2, 17-20.

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