Heir of Elendil
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Heir of Elendil was a title granted to those descended from Elendil, the founder of the Realms in Exile in the late years of the Second Age. In the North-kingdom of Arnor, whose rulers were descended through Elendil's elder son Isildur, its definition was simple: their rulers came from Elendil's direct male line, unbroken down to Aragorn himself.
The political situation in the South-kingdom of Gondor was more complicated, because its first independent King, Meneldil, was not a direct heir of Elendil in the strictest sense, being the son of Isildur's younger brother Anárion. In Gondor, therefore, the title was taken to refer to any descendants of Elendil who ruled in their own stead, a convenience that allowed Gondor's Kings to represent themselves as true Heirs of Elendil. This situation was complicated still further with the loss of Gondor's last King, Eärnur. After this point, the Stewards who took up the rule of Gondor acted in place of the heirs of Elendil, and took on the powers associated with the title.