Etymology
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Demeter 

in Greek religion, the Olympian goddess of agriculture and useful vegetation, protectress of the social order and of marriage, mother of Persephone, from Greek Dēmētēr; the second element generally given as māter (see mother (n.1)); the first element possibly from da, Doric form of Greek "earth" (see Gaia), but Liddell & Scott find this "improbable" and Beekes writes, "there is no indication that [da] means 'earth', although it has also been assumed in the name of Poseidon." The Latin masc. proper name Demetrius means "son of Demeter."

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Ceres 

Roman goddess of agriculture (identified with Greek Demeter), also the name given to the first-found and largest asteroid (discovered 1801 by Piazzi at Palermo), from PIE *ker-es-, from root *ker- (2) "to grow." Her festival, Cerealia, was April 10.

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Kore 

in Greek mythology, the daughter of Zeus and Demeter, also called Persephone in her aspect as Hades's wife, from Greek korē "maiden," from PIE *korwo- "growing" (hence "adolescent"), from suffixed form of root *ker- (2) "to grow."

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Persephone 

wife of Hades, queen of the netherworld, identified with Kore, daughter of Zeus and Demeter, from Greek Persephone. De Vaan writes that "The name was always considered obscure" until a thorough investigation published in 2006 reported that the original form was persophatta, "as found in eight attestations, seven of which are on 5th c. BC Attic vases (by seven different painters)." He analyzes it as *perso-, cognate with Sanskrit parsa- "sheaf of corn," + a second element from the PIE root *gwhen- "to hit, strike" (see bane) thus "a female thresher of corn."

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Eleusinian (adj.)

1640s, "pertaining to Eleusis," town outside Athens, site of the mystery associated with the cult of Demeter, goddess of harvests, and her daughter.

The name is literally "arrival" (eleusis), from eleusomai "to go, come," from PIE *elu-to-, from root *leudh- "to grow up, come out" (see liberal (adj.)). This is also the root of Greek eleutheros "freedom."

The best agreement (semantically and formally) to this old ablauting verb is found in Celtic, with the OIr. preterite lod, luid  "I, he went" ... formally as good, but semantically less convincing, is the further comparison with Skt. ró(d)hati, Go. liudan "to grow, rise" (whence the old word for "people", OHG liut, etc. ...). [Beekes]
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