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Old English tapur, taper "candle," not found outside English, possibly a dissimilated borrowing from Latin papyrus (see papyrus), which was used in Medieval Latin and some Romance languages for "wick of a candle" (e.g. Italian papijo "wick"), because these often were made from the pith of papyrus. Cf. also German kerze "candle," from Old High German charza, from Latin charta, from Greek khartes "papyrus, roll made from papyrus, wick made from pith of papyrus."
"shoot up like a flame or spire," 1580s, from taper (n.). Sense of "gradually decrease in size, force, etc." first recorded c.1600. Related: Tapered; tapering.