Etymology
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medial (adj.)

1560s, "pertaining to a mathematical mean," from Late Latin medialis "of the middle," from Latin medius "in the middle, between; from the middle," as a noun (medium) "the middle;" from PIE root *medhyo- "middle." Meaning "occupying a middle position, existing between two extremities or extremes" is attested from 1721.

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medial (n.)
"a medial letter," 1776, from medial (adj.).
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medially (adv.)

"in a medial or central position," 1804, from medial (adj.) + -ly (2).

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in medias res 

Latin, literally "in the midst of things," from medias, accusative fem. plural of medius "middle" (see medial (adj.)) + accusative plural of res "a thing" (see re). From Horace, in reference to narrative technique:

Semper ad eventum festinat, et in medias res,
Non secus ac notas auditorem rapit (etc.)
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Midi 

"southern France," 1883, from French midi "south," literally "midday" (12c.), from mi "middle" (from Latin medius "middle;" see medial (adj.)) + di "day" (from Latin dies, from PIE root *dyeu- "to shine"). At midday in the northern hemisphere the sun is in the south of the sky. Compare Latin meridianus "of midday, of noon;" also "southerly, to the south" (see meridian), and Middle English mid-dai in its secondary sense "south, to the south" (late 14c.).

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mitten (n.)

late 14c., mitain (from mid-13c. in surnames) "a glove, a covering for the hand," especially "a covering for the hand, differing from a glove in not having a separate covering for each finger, the thumb only being separated," from Old French mitaine "mitten, half-glove" (12c.) and from Medieval Latin mitta, both of uncertain origin; both perhaps from Middle High German mittemo, Old High German mittamo "middle, midmost" (reflecting the notion of "half-glove"), or from Vulgar Latin *medietana "divided in the middle," from Latin medius (see medial (adj.)).

From 1755 as "lace or knitted silk glove for women covering the forearm, the wrist, and part of the hand," worn fashionably by women in the early 19c. and revived towards the end of it. Hence get the mitten (1825), of men, "be refused or dismissed as a lover" (colloquial), from the notion of receiving the mitten instead of the "hand."

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*medhyo- 
Proto-Indo-European root meaning "middle." Perhaps related to PIE root *me- (2) "to measure."

It forms all or part of: amid; intermediate; mean (adj.2) "occupying a middle or intermediate place;" medal; medial; median; mediate; medieval; mediocre; Mediterranean; medium; meridian; mesic; mesial; meso-; meson; Mesopotamia; Mesozoic; mezzanine; mezzo; mezzotint; mid (prep., adj.); middle; Midgard; midriff; midst; midwife; milieu; minge; mizzen; moiety; mullion.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit madhyah, Avestan madiya- "middle," Greek mesos, Latin medius "in the middle, between; from the middle," Gothic midjis, Old English midd "middle," Old Church Slavonic medzu "between," Armenian mej "middle."
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stave (n.)
"piece of a barrel," 1750, back-formation from staves (late 14c.), plural of staff, with the usual change of medial -f- to -v- (compare leaves/leaf). The plural form possibly was in Old English but not recorded there.
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meson (n.)

subatomic particle, 1939, from Greek mesos "middle" (from PIE root *medhyo- "middle") + subatomic particle suffix -on. Earlier mesotron (1938). So called for being intermediate in mass between protons and electrons. An earlier use of the word, from the Greek noun meson "center," meant "the medial plane which divides the body into two equal and symmetrical parts" (by 1883).

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rh- 

consonantal digraph, an initial sequence used in Latin (and thus in English words from Latin) to represent Greek initial aspirated r-. The medial Greek form of it usually is represented by -rrh-, as in catarrh, diarrhea, hemorrhage, myrrh, Pyrrhic. As it was pronounced as simply "r" in Middle English (as in Old French and Spanish), the -h- tended to be dropped in spelling but was restored in early Modern English with the classical revival.

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