Etymology
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intermediate (v.)
c. 1600, "to interfere;" 1620s, "to mediate," from inter- "between" + mediate (v.). Related: Intermediated; intermediating.
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intermediate (adj.)
"being or occurring between" (two things), early 15c., from Medieval Latin intermediatus "lying between," from Latin intermedius "that which is between," from inter "between" (see inter-) + medius "in the middle" (from PIE root *medhyo- "middle").
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intermediacy (n.)
1713, from intermediate + abstract noun suffix -cy. Intermediateness is from 1826.
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intermezzo (n.)
1782, from Italian intermezzo "short dramatic performance (usually light and pleasing) between the acts of a play or opera," literally "that which is between," from Latin intermedius (see intermediate (adj.)).
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intermediary (adj.)

1757, "situated between two things;" 1818 as "serving as a mediator;" from French intermédiaire (17c.), from Latin intermedius "that which is between" (see intermediate). As a noun, "one who acts between others" from 1791 (Medieval Latin intermedium also was used as a noun). An earlier adjective was intermedial (1590s).

Intermediary, n., is, even with concrete sense of go-between or middleman or mediator, a word that should be viewed with suspicion & resorted to only when it is clear that every more ordinary word comes short of the need. [Fowler]
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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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middleweight (n.)

also middle-weight, "boxer or jockey of intermediate weight" (between a lightweight and a heavyweight), 1842, from middle (adj.) + weight (n.).

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middling (adj.)

"medium in rank, condition, or degree; intermediate," 1540s, from Middle English medlinge "intermediate between two things" (late 14c.), from middle (adj.) + present-participle suffix -ing (2). Used in trade to designate the second of three grades of goods. Hence "only medium, neither good nor bad" (1650s). As an adverb, "tolerably, passably," by 1719.

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non-stop (adj.)

also nonstop, "that does not stop," 1903, from non- + stop (n.); originally of railway trains not making intermediate stops. As an adverb by 1920.

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