The governance of empires and the claims of rival European states to exercise sovereignty over disparate territories have long posed interpretive problems for historians. Traditionally, they tended to focus on the empire of one or another European state, though pioneering works concerned with the Age of Exploration as a whole had to take a more comparative approach. Not surprisingly, both the comparative and the single-empire approaches lent themselves to the mapping of global spaces, color-coded to delimit the claimed possessions of the empire or empires discussed.

A key issue in the history of Europe’s global empire building was the contested nature of virtually all of the extra-European territories claimed, just as similar contentions and border disputes marked relations among states within Europe. No map can capture the complexity of those disputes any more than a current map of the United States with “red”...

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