Why are we even asking this question?
In Volume I of Wenzl Hause’s double-bass method, published around 1820-25, he gives these unusual instructions in the chapter about fingering (Fingersatz).

“If the first note to be gripped above the empty string has a ♭, you have to use the first (index) finger, but if it has a #, you use the second finger; but if the tone to be gripped is more than half or a whole tone away from the open string, you use the first, second or fourth finger. It is also done exactly so in the higher positions.”
Hause’s note seems to imply that if for example the first note you play on the G string is an A-flat, you have to play it with the first finger, but that if it’s a G-sharp, you have to play it with the second finger, and the A with the fourth finger. This seems to be consistent with Hause’s fingerings in the exercises of this first volume as well as those in Volume II of his method, the 90 Exercises.
He gives no reason for using these fingerings, he just says you have to do it that way. The question is, where does this even come from, and why must this way of playing be used?
Thanks to a tip from my esteemed colleague Joe Carver, it seems that the Parisian bassist and teacher Charles Labro (1810-1882) might have the explanation. This is from his Méthode de Contre-Basse, first published around 1860.

“As one can see in the above example, instead of making a minor third or a major third to the open string with the 4th finger of the left hand, as in the 1st and 2nd degrees of the 1st position, one can only make a minor second or a major second. Although very limited, this fingering is excellent and should preferably be chosen in the phrases of the rear position(*). The pressure of the strings near the nut being extremely troublesome, it is good to seek and use all the means that must facilitate the execution.”
(*): this is what we today call the “half position.”
The other interesting point is that strangely enough, Hause, who often goes to somewhat exaggerated lengths to explain the details of his instruction, says absolutely nothing about why one should use this way of fingering. At the time Hause published the second edition of his method, a G-sharp was likely still lower than an A-flat, as equal temperament was not yet the standard everywhere – that could possibly explain using the first finger for a note with a flat and the second finger for a note with a sharp. Labro, on the other hand, didn’t really care about whether the note had a sharp or a flat, as by then in France an E-flat was the same as a D-sharp.
At some point, František Simandl aka Franz Simandl (1840-1912) decided that enough was enough. He first published his own new method in 1874, still widely used wherever bass playing is taught, and that was the end of using the second and fourth fingers this way in the half position.
Ende gut, alles gut!
Written and added on February 3, 2025.
Updated on February 5, 2025.