The comparative study of different cultural traditions of navigation and encounter is both fascinating and frustrating. Bridging the techniques of navigation between European and indigenous societies was a much celebrated feature of Enlightenment voyages of discovery, particularly in oceanic theatres of empire. While navigators and their readers have been captivated by stories of cartographic exchanges on islands and ice floes, historians of science have never quite been persuaded that these episodes bridge the great divide between seemingly rival sources of authority: practices of precision measurement inscribed in the divided circle, and narrative practices of description emerging out of ethnographic observation and exchange. Rather than seeing these as being incommensurable, it is proposed instead that each way of knowing presents a different aspect of a more general set of problems about the nature of science and empire: namely, the relationships between orientation, autonomy and reciprocity.