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, 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).
