It is the age of exploration. Intrepid European colonists have come to settle the Americas amidst a storm of interest over the wondrous materials that may be found in the New World. Unfortunately for them, the indigenous have spent thousands of years cultivating their magic, distinguishing themselves in appearance, thought, language, and most severely: power. By the time of Christopher Columbus, the gulf has become so wide that the indigenous cannot even recognize people from the Old World as rational beings - regarding them instead as curiosities and pests beneath consideration.
These unfortunate circumstances have stymied European expansion into the continent, and unable to meaningfully combat the indigenous, the colonial powers have instead sought to broker communication between their peoples in the hopes that the use of familiar language will elevate themselves in their eyes. This is no easy task, given the usual methods for learning learning language through immersion are infeasible; intrepid linguists must observe from a distance lest they be discovered and potentially killed.
Altogether, the question stands: is this possible? Can a dedicated force of 16th century linguists learn an alien language through observation alone, given any sort of contact is likely to result in death? On what sort of timescale can this be achieved? I will accept any answer giving a historical example of any language having been successfully studied (grammar, lexicon, etc. to fluency) this way at any point in history.
Misc.:
- Capturing indigenous is impossible
- It is possible to remain hidden within earshot of the indigenous
- Infiltrating indigenous camps, towns, and other kinds of settlement or buildings is very dangerous
- Acquiring indigenous artifacts is incredibly difficult, even if they have been abandoned
- For simplicity's sake, we can imagine there's only one language
- The indigenous language is not too far removed from real ones
- The colonists are fielding technology roughly in line with what their real counterparts had
- The colonists are allowed to have anachronistic culture or knowledge, however
- Europe is desperate for successful colonies; the wondrous materials native to the continent are as valuable to them as modern technology would be to Rome