Gandalf himself gives the answer to this question in FotR, A Journey in the Dark, when the fellowship arrives at the gate:
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Those were happier days, when there was still close friendship at times between folk of different race, even between Dwarves and Elves.
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Part of the inscription on the door is:
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Celebrimbor of Hollin drew these signs.
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Celebrimbor, being an Elf, did not know the Dwarven language, which was kept quite secret by the Dwarves. It is only logical that he wrote in his own, the Elven, language.
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'Mercy!' cried Gandalf. 'If the giving of information is to be the cure of your inquisitiveness, I shall spend all the rest of my days in answering you. What more do you want to know?' 'The whole history of Middle-earth..
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