For a student studying Arabic at an American college in the 18th century, what kind of textbook would he have used, and what kind of text would he have read?

In the 18th century? As far as I'm aware the only college in America that taught Arabic in the 18th century was Harvard. America's other early universities didn't offer Arabic until the 19th century.

As far as I'm aware there is quite a bit more information about the study of Arabic in Europe at the time than there is about America. If that is of interest I'd be happy to sort of lay out the "state of the field" in the 18th century, but I don't have the right background to discuss what was specifically taught at Harvard at the time.