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]
"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.
"occupying a middle or intermediate place;" mid-14c., of persons, "of middle rank" (but this is possibly from, or mixed with, mean (adj.1)); from Anglo-French meines (plural), Old French meien, variant of moiien "mid-, medium, common, middle-class" (12c., Modern French moyen), from Late Latin medianus "of the middle," from Latin medius "in the middle" (from PIE root *medhyo- "middle").
From late 14c. as "in a middle state, between two extremes." Meaning "intermediate in time, coming between two events or points in time" is from mid-15c. (the sense in meanwhile, meantime). The mathematical sense "intermediate in a number of greater or lesser values, quantities, or amounts" is from late 14c.