As I understand it (and I make no guarantees about this), mid 19th century, the Italians and French were playing three stringers tuned in 4ths and 5ths. It was the Germans and Austrians, responding to writing of Beethoven etc that went to the four string bass. The advantage being a wider range, but, according to some, a loss of tonal quality.
I think it is not appropriate to say the 3 string bass is a forerunner of the 4 string bass.
E.g. have a look at Gary Karr's Amati bass built in 1611. This seems to be a 4 stringer by its origine like most other basses made since.
The 3 stringer is a "fashion" instrument in the "classic" time periode in southern Europe - Italy, Spain, maybe (?) France. In other European countries they never were very common.
Thomas