Etymology
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impeach (v.)

formerly also empeach, late 14c., empechen, "to impede, hinder, prevent;" early 15c., "cause to be stuck, run (a ship) aground," also "prevent (from doing something)," from Anglo-French empecher, Old French empeechier "to hinder, stop, impede; capture, trap, ensnare" (12c., Modern French empêcher), from Late Latin impedicare "to fetter, catch, entangle," from assimilated form of in- "into, in" (from PIE root *en "in") + Latin pedica "a shackle, fetter," from pes (genitive pedis) "foot" (from PIE root *ped- "foot").

In law, at first in a broad sense, "to accuse, bring charges against" from late 14c.; more specifically, of the king or the House of Commons, "to bring formal accusation of treason or other high crime against (someone)" from mid-15c.  The sense of "accuse a public officer of misconduct" had emerged from this by 1560s. The sense shift is perhaps via Medieval Latin confusion of impedicare with Latin impetere "attack, accuse" (see impetus), which is from the Latin verb petere "aim for, rush at" (from PIE root *pet- "to rush, to fly").

The Middle English verb apechen, probably from an Anglo-French variant of the source of impeach, was used from early 14c. in the sense "to accuse (someone), to charge (someone with an offense)." Related: Impeached; impeaching.

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impeachable (adj.)

early 15c., empechable, "liable to impeachment," from Old French; see impeach + -able. Related: impeachably; impeachability.

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appeach (v.)
obsolete variant form of impeach. Related: Appeached; appeaching.
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impeachment (n.)
late 14c., enpechement "accusation, charge," from Old French empeechement "difficulty, hindrance; (legal) impeachment," from empeechier "to hinder, impede" (see impeach). As a judicial proceeding on charges of maladministration against a public official, from 1640s.
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peach (v.)

"to inform against, betray one's accomplices," 1560s (earlier pechen, "to accuse, indict, bring to trial," c. 1400), a shortening of appeach, empeach, obsolete variants of impeach. For form, compare peal (v.), also Middle English pelour "an accuser," from appellour. Related: Peached; peaching; peacher.

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depeche (n.)

"a dispatch," 1520s, from French dépêche (15c.), from dépêcher "to dispatch," from Old French despeechier, from des- (see des-) + stem of empeechier "to hinder" (see impeach). Not directly related to dispatch. Sometimes nativized as depeach.

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dispatch (v.)

1510s, "to send off, send to a destination," usually implying urgent importance or haste, from Spanish despachar "expedite, hasten" or cognate Italian dispacciare "to dispatch." For first element, see dis-.

The second element apparently has been confused or corrupted, and its exact source and meaning is uncertain. One proposal is that it is Vulgar Latin *pactare "to fasten, fix" or *pactiare. Another says it is Latin -pedicare "to entrap" (from Latin pedica "shackle;" see impeach), and the Spanish and Italian words seem to be related to (perhaps opposites of) Old Provençal empachar "impede." See OED for full discussion.

Meaning "get rid of promptly by killing" is attested from 1520s; that of "attend to, finish, bring to an end, accomplish" is from 1530s. Related: Dispatched; dispatching.

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*ped- 

Proto-Indo-European root meaning "foot."

It forms all or part of: antipodes; apodal; Arthropoda; babouche; biped; brachiopod; cap-a-pie; centipede; cephalopod; cheliped; chiropodist; expedite; expedition; foot; foosball; fetch (v.); fetter; fetlock; gastropod; hexapod; impair; impede; impediment; impeach; impeccable; isopod; millipede; octopus; Oedipus; ornithopod; pajamas; pawn (n.2) "lowly chess piece;" peccadillo; peccant; peccavi; pedal; pedestrian; pedicel; pedicle; pedicure; pedigree; pedology; pedometer; peduncle; pejoration; pejorative; peon; pessimism; petiole; pew; Piedmont; piepowder; pilot; pinniped; pioneer; platypus; podiatry; podium; polyp; pseudopod; quadruped; sesquipedalian; stapes; talipes; tetrapod; Theropoda; trapezium; trapezoid; tripod; trivet; vamp (n.1) "upper part of a shoe or boot;" velocipede.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit pad-, accusative padam "foot;" Avestan pad-; Greek pos, Attic pous, genitive podos; Latin pes, genitive pedis "foot;" Lithuanian padas "sole," pėda "footstep;" Old English fot, German Fuß, Gothic fotus "foot."

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