In what century did ballet begin?
Ballet started in the 15th Century in the Renaissance court of Italy. The earliest references I can find to "Ballet" refer to Roman times when even then, it is thought to have originated with the Greek Tragedies. Like most other art forms, it was lost during the Middle Ages and revived again during the Renaissance period. I suspect that the question refers to "Classical Ballet". J S Bach (1685-1750) was a prolific composer, but did write anything labelled specifically as a "Ballet", although many modern ballet companies perform ballet to Bach's music, such as the Canata's which are essentially a Psalm or other religious text set to music. W A Mozart (1756-1791) wrote music for ballets specifically. My assumption being that "Classical Ballet" appeared between the death of Bach and the rise of Mozart. Somebody of superior subject knowledge may be able to correct or clarify my assumption.