I have checked US Patents for around this date.
These chairs are called "collapsible chairs" or "folding chairs". Most likely the patent is for the folding mechanism.
The problem is that the patent dates displayed on many items ARE NOT for the issue date of the patent (which is how you find the patent/item)! Individuals/companies many times put in the date that the patents WAS REQUESTED which could have been months/year before the patent was actually issued (due to time to process and updates/improvements that were made to the request). Many times it is like finding a needle in a haystack.
In addition, the actual design of the chair that was submitted can be altered IF the patent was actually for the folding mechanism only...then it will very hard to find a match on the patent diagrams.
Please look at the folding mechanism and see if there are any numbers, patent information, identifiers to help in finding this.
Just FYI, here is an example of patents issued in that time frame
https://www.google.com/?tbm=pts&gws_rd=ssl#tbm=pts&q=Collapsible+chair+Oct+18+1875