Interesting question. I don't have a theory about the pedals, but I wonder what the five tuning forks are doing on the right-hand side?

There were many different kinds of pianos in the 19th century, some with 2 and some with 4 or even more pedals. About these fours pedals, I assume that one pedal is for resonance-to hold the dampers up like today, one would be una corda (also like today, where the piano plays just one string), one could be celesta- even softer than una corda, and one could be the imitation of fagot- a different sound, much louder.

The pedals could be compared to the different registers of the organ in that they completely change the colour of the instrument. They were therefore used differently, not like we do today-which is all the time using a bit of both when we want more piano or more resonance, but more by sections. For example in variations pieces, each variation could be characterized by a different pedaling, offering another sound and atmosphere.

I have no idea about the tuning fork, but it could also be to show the multiplicity of tunings. Today all pianos are tuned at 440 (or 442?) Herz,, while at that time there were a few possibilities.

I'm back to this question of why 4 pedals on a piano, after having practiced Friday morning at the RCM - where I teach - on a wonderful Viennese pianoforte from the beginning of 19th century. This one had 5 pedals of which one damper, one fagot and no less than three pedals whose function was to make the sound softer. It seems that a lot of research at that time concentrated on making the pianos play all shades of pianissimi (soft sounds). Today it is a well known fact that piano manufactures are concentrating on making the instruments sound as loud as they can.

The reason for these different kinds of developments is the close connection between the composers, interpreters and their influence on the manufacturing companies. Thus Chopin had a great collaboration with Pleyel, many of Liszt's pieces absolutely required the double escape mechanism invented by Erard, and Schubert's intimate music was written exactly for those pianoforte's with the multiple softening pedals.
For French speakers, here is a funny video on the subject: