Early depictions of the Quadrille

Dancing the quadrille. Detail from a series of illustrations from about 1821 attributed to William Hawkes Smith of Birmingham satirising the craze for the quadrille in the late Georgian period.

The quadrille was introduced in France around 1760: originally it was a form of cotillion in which only two couples were used, but two more couples were eventually added to form the sides of a square. The "quadrille des contredanses" was now a lively dance with four couples, arranged in the shape of a square, each couple facing the center. One pair was called the "head" couple, the adjacent pairs the "side" couples. A dance figure was often performed first by the head couple and then repeated by the side couples.

Reaching English high society around in the early part of the nineteenth century the quadrille became a craze. As it became ever more popular it evolved into forms that used elements of the waltz, including The Caledonians and The Lancers. 

Writing in 1815 the London Dancing Master Thomas Wilson says: "Quadrilles are of that Species of Dancing that at present claim a high precedence in Fashionable Circles ' and from their partaking greatly bf the style of Cotillions in their Composition may notwithstanding their more fashionable appellation and their more short and less complex Figures be properly considered as petit (sic) or short Cotillions."
By the time of these illustrations although there was an enormous amount of quadrille music the form of the dance had settled down as consisting five parts named:

1. Le Pantalon ("Trousers")
2. L’été ("Summer")
3. La Poule (The Hen")
4. La Pastourelle ("The Shepherd Girl")
5. Finale

Sometimes La Pastourelle was replaced by another figure; La Trénis or occasionally  this was added as a 6th part.
For the most part, the figures and steps of each of the parts remained the same whatever music was being used. However, some publishers, notably the Whites of Bath used dancing teachers to create new choreography which they offered as an alternative to the standard set.