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According to the Greek mythographer Apollodorus (Lib. 3.14.3), Adonis' genealogy was known in three conflicting accounts:
The last of these genealogies seems to have been the most popular of the three. It is also the most relevant to our study since the other two presume a purely natural and ordinary birth.1) Hesiod claimed that Adonis was the son of Phoenix and Alphesiboea.
2) Some people held that Adonis was the son of Cinyras, founder and king of Paphos in Cyprus, and his wife Metharme, daughter of Pygmalion, king of Cyprus.
3) Panyasis claimed that Adonis was the son of Thias, king of Assyria and his daughter Smyrna.
As the story goes, Aphrodite was outraged with Smyrna, either: 1) "because she did not honor the goddess" (Apollodorus) or 2) because her mother boasted that her daughter was more beautiful than the goddess herself (Hyginus). Aphrodite therefore gave Smyrna an incestuous passion for her father Panyasis:
… with the complicity of her nurse she shared her father's bed without his knowledge for twelve nights (agnoounti tō patri nuktas dōdeka suneunastē). But when he was aware of it, he drew his sword and pursued her, and being overtaken she prayed to the gods that she might be invisible; so the gods in compassion turned her into the tree which they call Smyrna (myrrh). Ten months afterwards the tree burst and Adonis, as he is called, was born … (Apollodorus, Lib. 3.14.4).Ovid too represents Smyrna as engaging in incest with her father and thereby conceiving Adonis:
Plena patris thalamis excedit et inpia diro semina fert utero conceptaque crimina portat.
Forth from the chamber she went out, full of her father, with crime conceived within her womb. (Met. 10.469-70)
The other mythographers followed this same story (Plutarch, Parallela 22; Antoninus Liberalis, Transform. 34; Fulgentius, Mythologies 3.8; Lactantius Placidus, Narrat. Fab. 10.9; Scholiast on Theocritus 1.107).
This myth was etiological: it was meant to explain why myrrh trees oozed with drops of sap. These drops were the sad teardrops of Smyrna, from whom the myrrh tree had taken its name. The Greek word smyrna comes from a Semitic word, akin to the Arabic word murr. The Latin cognate is myrrha, from which the English word "myrrh" is derived. There is no etymological relationship between any of these words and the name of Jesus’ mother, however. Jesus is said in the New Testament to have been born of ‘Mary’ (Mat 1:16; Mk 6:3; Lk 1:30-31; Acts 1:14). In Greek, this name is Maria or Mariam, a very popular Jewish name in first century, it being the Greek transliteration of the Hebrew name Miriam. Its popularity was of course not attributable to the myth of Adonis’ birth but to the fact that Moses’ sister was named Miriam. Miriam had led the women in the prophetic song of triumph after Pharaoh’s army had been drowned and the Israelites had crossed through the Red Sea (Deut 15:20-21). While the etymology for Miriam is uncertain, it possibly came from the Egyptian mr ("love") and Yah (short for Yahweh). If so, the name would have meant, ‘loved of Yahweh’. Another possibility is that the name was derived from the root mr, which means "bitterness" (cf. Ruth 1:20).
Nothing is said or implied in the Adonis myth about his having been born of a virgin. Smyrna was impregnated by her father. This is expressly stated not only by Apollodorus, but by the other mythographers as well.
Nor do we find anywhere that Adonis was born "at Bethlehem, in the same sacred cave that Christians later claimed as the birthplace of Jesus", as we read in the above quote. The quotation marks within this quote were apparently lifted from a secondary source, but the source is making a bogus claim. According to the extant primary sources (e.g. Apollodorus, Hyginus, Theocritus, etc.) we can only assume that Adonis was born either in Paphos (a city of Cyprus) or somewhere in Assyria.
The New Testament accounts of Jesus’ virgin birth bear no resemblance to this myth and should be deemed independent of it.