Etymology
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fiche (n.)

1949, "slip of paper, form," especially "the form filled in by foreign guests in French hotels" [OED], from French fiche "card, index card, slip, form" (15c.), verbal noun from Old French fichier "to attach, stick into, pin on" (12c.), from Vulgar Latin *figicare, from Latin figere "to fix, fasten" (from PIE root *dheigw- "to stick, fix"). Sense of "card, strip of film" is a shortening of microfiche (1950).

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microfiche (n.)

"flat piece of film containing micrographs of the pages of a book, etc.," 1950, from French microfiche, from micro- + French fiche "slip of paper" (see fiche).

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*dheigw- 

Proto-Indo-European root meaning "to stick, fix." 

It forms all or part of: affix; crucifix; crucify; dig; dike; ditch; fibula; fiche; fichu; fix; fixate; fixation; fixity; fixture; infibulate; infibulation; microfiche; prefix; suffix; transfix.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit dehi- "wall;" Old Persian dida "wall, stronghold, fortress," Persian diz; Latin figere "to fix, fasten, drive, thrust in; pierce through, transfix;" Lithuanian dygstu, dygti "germinate;" Old Irish dingid "presses, thrusts down;" Old English dic "trench, ditch," Dutch dijk "dam." 

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