As scotandrsn’s post explained, the entire concept of a composer, as we know it, is relatively modern. Asking how many composers there were in the 17th or 18th Centuries is like asking how many painters or sculptors there were at the same time. Impossible to answer. A composer was just another adornment to any noble’s court. He was a servant in the household, and composed only what his master wanted to hear. If the master was throwing a party, for example, he might request that the composer provide a suitable fanfare to be played as the courtiers arrive, followed by an hour’s worth of dance music, a couple of salon songs, and finally background music for the feast. The music would be composed, rehearsed, played, then burned or filed away and most likely never heard again. Only pieces that were especially favored were singled out for preservation.
Of course, not everybody followed this model – there are always exceptions, such as in the Church or among the commoners. But I think this account serves as an accurate depiction of the way that “artistic” music was generally brought into being from the post-Renaissance era through the Enlightenment.
The movie Amadeus actually provides an excellent glimpse into this world. It portrays Mozart as a man unable to accept the constricted role of a mere court servant. Though he could be making a comfortable living through the patronage of nobility, his self-centered, rebellious, impulsive manner of behaving closes the doors of opportunity. As an outcast of the court, he soon finds himself unable to make a living and spirals downward into self-destruction.
What makes Mozart’s death at such a young age particularly poignant is that the role of the composer in society was changing. As the revolutionary era swept through Europe, many of the aspects of polite society once reserved for the aristocracy (including artistic music) were embraced by the emerging merchant class. Had he lived but a few years longer, Mozart might have been able to enjoy the public acclaim he so desired.
But back on the subject of the court composer, there were other elements of the patronage system that contrived to keep the composer out of the limelight. One of these was the promotion of individual performers. It was not until the 19th century that an opera singer or instrumental virtuoso was expected to confine his or her performance to what was written on the page. Soloists were allowed to ad lib to their hearts’ desire, using the work of the composer and/or librettist as a mere vehicle for the display of their performing talents.