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: all 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. Even this situation was tenuous, as the first line of Kings would fail with the death of Siriondil, who left no direct heir. Eventually, the line of Kings would fail altogether with the disappearance of the last king, Eärnur. After this point, with there being no descendent of Elendil strong enough to make a serious claim and for fear of another Kin-strife, 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.
These complexities were resolved at the end of the Third Age with the enthronement of Aragorn Elessar, both a direct heir of Isildur and an heir of Anárion through his ancestor Firiel, as High King of the Reunited Kingdom of Arnor and Gondor.