Read this line from the text: They aim to kindle interest and to direct the activity of the awakened student along sound lines. In using the phrase sound lines, the author wants to suggest the professor's arguments are

In using the phrase sound lines, the author wants to suggest that the professor's arguments are logical and responsible. "Along sound lines" means done by the book, in a responsible way and with full consideration. The professor seeks to direct the student in a way that is responsible and conscientious.