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]
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."
"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.