Old English tapur, taper "candle, lamp-wick," not found outside English, possibly a borrowing and dissimilation of Latin papyrus (see papyrus), which was used in Medieval Latin and some Romance languages for "wick of a candle" (such as Italian papijo "wick"), because these often were made from the pith of papyrus. Compare also German kerze "candle," from Old High German charza, from Latin charta, from Greek khartēs "papyrus, roll made from papyrus, wick made from pith of papyrus."
1580s, "shoot up like a flame or spire," via an obsolete adjective taper, from taper (n.), on the notion of the converging form of the flame of a candle. Sense of "become slender, gradually grow less in size, force, etc." first recorded c. 1600. Transitive sense from 1670s. Related: Tapered; tapering.