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In the Hesiodic Catalogue of Women, Graecus is presented as the son of Zeus and Pandora II, sister of Hellen the patriarch of Hellenes. Hellen was the son of Deucalion who ruled around Phthia in central Greece. The Parian Chronicle mentions that when Deucalion became king of Phthia, the previously called Graikoi were named Hellenes. Aristotle notes that the Hellenes were related with Grai/Greeks (Meteorologica I.xiv) a native name of a Dorian tribe in Epirus which was used by the Illyrians. He also claims that the great deluge must have occurred in the region around Dodona, where the Selloi dwelt. However, according to the Greek tradition it is more possible that the homeland of the Greeks was originally in central Greece. A modern theory derives the name Greek (Latin Graeci) from Graikos, "inhabitant of Graia/Graea," a town on the coast of Boeotia. Greek colonists from Graia helped to found Cumae (900 BC) in Italy, where they were called Graeces. When the Romans encountered them they used this name for the colonists and then for all Greeks (Graeci.) The word γραῖα graia "old woman" comes from the PIE root *ǵerh2-/*ǵreh2-, "to grow old" via Proto-Greek *gera-/grau-iu; the same root later gave γέρας geras (/keras/), "gift of honour" in Mycenean Greek. The Germanic languages borrowed the word Greeks with an initial "k" sound which probably was their initial sound closest to the Latin "g" at the time (Goth. Kreks). The area out of ancient Attica including Boeotia was called Graïke and is connected with the older deluge of Ogyges, the mythological ruler of Boeotia. The region was originally occupied by the Minyans who were autochthonous or Proto-Greek speaking people. In ancient Greek the name Ogygios came to mean "from earliest days".
Where do modern theorist believe the name of Greek originated from ?
A modern theory derives the name Greek (Latin Graeci) from Graikos, "inhabitant of Graia/Graea," a town on the coast of Boeotia.