Byzantium/Constantinople was conquered in 1451, but the empire continued to decline starting around the 18th or 19th centuries. It officially died in WWI. As for how long it was a major threat to Europe, the empire's prime was the 16th century, under Suleiman the Magnificent.
John Adams was an ambassador to France during the American Revolution. Also served in Amsterdam securing a large loan that helped finance the war. He was also the first VP under George Washington, though Washington pretty much shut him out.
