I'm taking a History of Science class right now that focuses mainly on the Scientific Revolution era. It's really interesting to me to see all of our primary sources were originally written in Latin because some of them are from about 200 years ago. Which, to me, seems like a very short time for a language to completely die off. So how did this happen? Latin was once the most common language in the world and now it's gone?
Suppose we take classical Latin, rather than Old Latin, as our starting point, then Latin has an active history of 100 BC through to the 16th/17th century, when it started to lose its place even in Academia. So 1800 years is not bad for a language, especially one that lost its native speaking population.
More specifically on your question, Latin remained the lingua academica throughout Europe through the middle ages, renaissance, and early modern periods. However this was its last real bastion of active usage. The vernacular languages of Europe had long since replaced Latin in everyday usage, and gradually took over the domains of literature, diplomacy, and eventually scientific and academic use. In the 18th century it rapidly diminished, and by the start of the 19th it was an oddity rather than the ordinary.
The real question is how it managed to survive so long.