Other people have explained how hieroglyphs were deciphered so I’m going to focus on Cuneiform.
It began to be successfully deciphered after German philogists began looking for and find the proper names of ancient Persian shahanshahs in the cuneiform inscriptions at Persepolis that were known from ancient Greek, Hebrew, and later Sassanid Persian sources. (I’m talking about men like Cyrus, Cambyses, Darius, and Xerxes, BTW…)
Georg Friedrich Grotefen then suggested that since that the Sassanid shahanshahs’ standard method of address in inscriptions was “(Name), great king, king of kings, son of (Father’s Name)” then perhaps the Achaemenid shahanshahs had used that form as well. As it turns out, he was right, though his work was not immediately accepted and in some respects, he did make mistakes in his attempted deciphering.
The later French scholar Eugène Burnouf used a similar method to decipher a list of Darius the Great’s satrapies.
Meanwhile, in 1835, a man named Henry Rawlinson rediscovered the Behistun Inscription near Kermanshah that had three identical texts carved into it in Old Persian, Babylonian and Elamite, which was to cuneiform what the Rosetta Stone was to hieroglyphs.
It certainly helped, however, that Akkadian and Old Persian both belong to very well understood language families, the Afro-Asiatic and Indo-European families. Transcribing Elamite or Sumerian (both of which are long-extinct language isolates) would have been very difficult, if not impossible, had there not also been cuneiform used to write Akkadian or Old Persian.
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