Cotillion or Quadrille

from the American Dancing Master, Elias Howe, 1862
 

Cotillion is an English word, and signifies any brisk and lively dance by two or more couples, thus a common Waltz or Polka, or any of the 
round dances, can properly be called Cotillions; all dances so called, were formerly in 6-8 or 3-4 time. The name was generally applied to dances 
formed like the first four in a quadrille. This kept the dancers nearly in continual motion.  Afterwards the side couples were added, which allows 
time to rest, and thus the English Cotillion and the French are now formed precisely alike. The proper name would seem to be Quadrille, as 
Cotillion could properly be applied to a large number of other dances. Cotillon is a French word, and is used by those that pattern after the 
French. 
 
Formerly it was fashionable to perform every change and movement with the proper and appropriate steps, but at present, in the common 
quadrille, little regard is paid to the steps; many simply walk through the figures with the notion that it is more dignified to do so. The change 
of "balance" is now generally made by the couple joining hands and chassaing or galloping forward and back, and the change of "forward two," 
is performed by both the lady and gentleman, being accompanied by their partners, thus in nearly or quite all of the changes of the quadrille 
the lady and gentleman join hands in their performance.