Etymology
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Minerva 

in ancient Roman mythology, one of the three chief divinities (with Jupiter and Juno), a virgin goddess of arts, crafts, and sciences; wisdom, sense, and reflection (later identified with Greek Athene), late 14c., Mynerfe, minerve, from Latin Minerva, from Old Latin Menerva, from Proto-Italic *menes-wo- "intelligent, understanding," from PIE root *men- (1) "to think, remember," with derivatives referring to qualities of mind or states of thought (see mind (n.)). Compare Sanskrit Manasvini, name of the mother of the Moon, manasvin "full of mind or sense." Related: Minerval.

Minerva Press, a printing-press formerly in Leadenhall Street, London; also a class of ultra-sentimental novels, remarkable for their intricate plots, published from about 1790 to 1810 at this press, and other productions of similar character. [Century Dictionary]
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invita Minerva 
Latin adverbial phrase, used with reference to literary or artistic creation, "without inspiration," literally "Minerva unwilling;" i.e. "without inspiration from the goddess of wisdom;" ablative fem. of invitus "against the will, unwilling, reluctant," according to de Vaan from PIE compound *n-uih-to- "not turned to, not pursuing," related to the source of invitation. With Minervā, ablative absolute of Minerva.
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*men- (1)

Proto-Indo-European root meaning "to think," with derivatives referring to qualities and states of mind or thought.

It forms all or part of: admonish; Ahura Mazda; ament; amentia; amnesia; amnesty; anamnesis; anamnestic; automatic; automaton; balletomane; comment; compos mentis; dement; demonstrate; Eumenides; idiomatic; maenad; -mancy; mandarin; mania; maniac; manic; mantic; mantis; mantra; memento; mens rea; mental; mention; mentor; mind; Minerva; minnesinger; mnemonic; Mnemosyne; money; monition; monitor; monster; monument; mosaic; Muse; museum; music; muster; premonition; reminiscence; reminiscent; summon.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit manas- "mind, spirit," matih "thought," munih "sage, seer;" Avestan manah- "mind, spirit;" Greek memona "I yearn," mania "madness," mantis "one who divines, prophet, seer;" Latin mens "mind, understanding, reason," memini "I remember," mentio "remembrance;" Lithuanian mintis "thought, idea," Old Church Slavonic mineti "to believe, think," Russian pamjat "memory;" Gothic gamunds, Old English gemynd "memory, remembrance; conscious mind, intellect."

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Athena 

Greek goddess of wisdom, skill in the arts, righteous warfare, etc., from Latin Athena, from Greek Athē, name of a common Greek goddess, dating to Minoan times, depicted with a snake and protecting the palace. "Like the goddess itself, the name is pre-Greek" [Beekes]. Identified by the Romans with their Minerva.

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Pluto (n.)

Roman god of the underworld, early 14c., from Latin Pluto, Pluton, from Greek Ploutōn "god of wealth," from ploutos "wealth, riches," probably originally "overflowing," from PIE root *pleu- "to flow." The alternative Greek name or epithet of Hades in his function as the god of wealth (precious metals and gems, coming from beneath the earth, form part of his realm). The planet (since downgraded) was discovered 1930 by U.S. astronomer Clyde W. Tombaugh; Minerva also was suggested as a name for it. The cartoon dog first appeared in Walt Disney's "Moose Hunt," released April 1931.

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hag (n.)
early 13c., "repulsive old woman" (rare before 16c.), probably from Old English hægtes, hægtesse "witch, sorceress, enchantress, fury," shortened on the assumption that -tes was a suffix. The Old English word is from Proto-Germanic *hagatusjon, which is of unknown origin. Dutch heks, German Hexe "witch" are similarly shortened from cognate Middle Dutch haghetisse, Old High German hagzusa.

The first element probably is cognate with Old English haga "enclosure, portion of woodland marked off for cutting" (see hedge (n.)). Old Norse had tunriða and Old High German zunritha, both literally "hedge-rider," used of witches and ghosts. The second element in the prehistoric compound may be connected with Norwegian tysja "fairy; crippled woman," Gaulish dusius "demon," Lithuanian dvasia "spirit," from PIE *dhewes- "to fly about, smoke, be scattered, vanish."

One of the magic words for which there is no male form, suggesting its original meaning was close to "diviner, soothsayer," which were always female in northern European paganism, and hægtesse seem at one time to have meant "woman of prophetic and oracular powers" (Ælfric uses it to render the Greek "pythoness," the voice of the Delphic oracle), a figure greatly feared and respected. Later, the word was used of village wise women.

Haga is also the haw- in hawthorn, which is an important tree in northern European pagan religion. There may be several layers of folk etymology here. Confusion or blending with heathenish is suggested by Middle English hæhtis, hægtis "hag, witch, fury, etc.," and haetnesse "goddess," used of Minerva and Diana.

If the hægtesse once was a powerful supernatural woman (in Norse it is an alternative word for Norn, any of the three weird sisters, the equivalent of the Fates), it might originally have carried the hawthorn sense. Later, when the pagan magic was reduced to local scatterings, it might have had the sense of "hedge-rider," or "she who straddles the hedge," because the hedge was the boundary between the civilized world of the village and the wild world beyond. The hægtesse would have a foot in each reality. Even later, when it meant the local healer and root collector, living in the open and moving from village to village, it may have had the mildly pejorative Middle English sense of hedge- (hedge-priest, etc.), suggesting an itinerant sleeping under bushes. The same word could have contained all three senses before being reduced to its modern one.
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