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The photo in this link is one of the correspondences between Gauss and Encke on prime numbers. Transcription of this letter can be found in Gauss' Werke, Band II, while the transcription of the first and the last paragraph is missing. Although I cannot read Gauss's handwriting very well, I tried to give my transcription:

(First paragraph)Vor allem statte ich Ihnen für die [gelegentliche?] Übersendung des Jahrbuchs von 1852 meinen verbindlichsten Dank ab.

(Last paragraph)Doch es ist Zeit abzubrechen. Ich sage noch meinen herzlichen Dank für Ihre Mitteilungen über der [Farbige der dorthin offenst?] Zustände, Noch sieht man Keinen Aus weg, aus dem Labyrinth in das uns die Nachäfferei der Franzosen gezerrt hat. Unter herzlichen Wünschen für Ihr Wohlbefinden

Could anybody check if it is correct or not? Thanks in advance.

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    Könnte es sein "über die Färbung der dortigen öffentl[ichen] Zustände"? – Christian Geiselmann Jul 14 '17 at 18:25
  • @ChristianGeiselmann I'd like to read the substantive as that, yes - But the article is definitely "den". – tofro Jul 15 '17 at 8:38
  • @tofro I think the first letter is a capital P with some color missing, not an F (compare with his other capital Fs, they are far more straight.) – sgf Jul 15 '17 at 13:27
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    @sgf No. don't think so. The P doesn't have a vertical bar and looks entirely different - See "Primzahl" – tofro Jul 15 '17 at 14:02
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    @CarstenS Ich mal einen Meta Thread eröffnet: german.meta.stackexchange.com/questions/1275/… – Thomas Jul 21 '17 at 11:42
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The first paragraph reads as follows (keeping the line breaks of the original)

Vor allem statte ich Ihnen für die gewogentliche Über-

sendung des Jahrbuchs von 1852 meinen verbindlichsten Dank ab.

gewogentlich is not in use in modern German. However, there is still the related adjective gewogen (see Duden).

As you can guess from the comments, the last paragraph is tricky.

Doch es ist Zeit abzubrechen. Jetz sage ich noch meinen herzlichen

Dank für Ihre Mittheilungen über [den Färbunge] der dortigen öffentl.

Zustände, [doch/Noch] sieht man keinen Ausweg aus dem Labirynth; in das uns

die Nachäfferei der Franzosen gezerrt hat. Unter herzlichen Wunschen

für Ihr Wohlbefinden

Both, the handwriting and the spelling are sloppy as if Gauß was in a hurry or just tired (e.g. the missing Umlaut-dots on Wünsche). In addition, there are old-fashioned spellings (Mittheilungen).

In modern spelling and without abbreviation I would transcribe it as

Doch es ist Zeit abzubrechen. Jetzt sage ich noch meinen herzlichen

Dank für Ihre Mitteilungen über [den Färbunge] der dortigen öffentlichen

Zustände, [doch/Noch] sieht man keinen Ausweg aus dem Labyrinth, in das uns

die Nachäfferei der Franzosen gezerrt hat. Unter herzlichen Wünschen

für Ihr Wohlbefinden

[den Färbunge] is really hard to decipher, due to the sloppiness.

The article could be den or der, I would exclude die, das, des, or dem.

I am pretty sure, the noun starts with F and has rbu in the middle. The letter in between could be a or ä. The ending includes a letter with a lower loop, and a common ending is ung. However, just based on the writing, it could be a q or y as well. And after that letter with the lower loop is something that looks like an e.

Putting it all together, we receive den Färbunge, but that does not make sense. As @CarstenS commented, it could be a an old fashioned word that is not in use anymore. It could also be a dialect word or a word of foreign origin. Taking into account the general sloppiness on the last page, I would go with the comment from @ChristianGeiselmann that Gauss meant to write über die Färbung or über die Färbungen and just misspelled the article (maybe he originally wanted to use another noun and kept the wrong article).

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    I think, "Färbungen" could be nowadays "Schattierungen". So the "öffentliche Zustände" (public conditions) would vary slightly and Encke described these variations in his letter. – IQV Jul 26 '17 at 5:56

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