Dance cards appear to have originated in the 18th century, but their use first became widespread in 19th-century Vienna, especially at the large balls during Fasching before Lent.
From the 19th century until World War I, dance cards for the elite of Austria-Hungary were often very elaborate, with some even incorporating precious metals and jewels.
Conversely, claiming one's "dance card is full" implies that even though they may be interested, they have no time for another person.
The "dance card" contains administrative data about the mission, aircraft, and aircrew as well as a list of the maneuvers to be flown.
The card serves as a table of contents for the mission and a quick reference for the aircrew's use during the flight.