chain-saw n.

c. 1300, "connected series of links of metal or other material," from Old French chaeine "chain" (12c., Modern French chane), from Latin catena "chain" (source also of Spanish cadena, Italian catena), which is of unknown origin, perhaps from a PIE root *kat- "to twist, twine" (source also of Latin cassis "hunting net, snare").

As a type of ornament worn about the neck, from late 14c. As a linear measure ("a chain's length") from 1660s. From 1590s as "any series of things linked together." The meaning "series of stores controlled by one owner or firm" is American English, 1846. The figurative use "that which binds or confines" is from c. 1600.

Chain-reaction is from 1916 in physics; the specific nuclear physics sense is from 1938. Chain-mail armor is from 1795, from mail (n.2). Before that, mail alone sufficed. Chain letter is recorded from 1892; at first usually to raise money; decried from the start as a nuisance.

Nine out of every ten givers are reluctant and unwilling, and are coerced into giving through the awful fear of "breaking the chain," so that the spirit of charity is woefully absent. [St. Nicholas magazine, vol. xxvi, April 1899] 

Chain of command is from 1915. Chain-lightning, visible as jagged or broken lines, is from 1834. Chain-smoker, one who smokes one after another, lighting the next from the stump of the last, is attested from 1885, originally of Bismarck (who smoked cigars), thus probably a loan-translation of German Kettenraucher. Chain-smoking (n.) is from 1895.

[toothed cutting tool] Middle English saue, from Old English sagu, from Proto-Germanic *sago "a cutting tool" (source also of Old English seax "knife," Old Norse sög, Norwegian sag, Danish sav, Swedish såg, Middle Dutch saghe, Dutch zaag, Old High German saga, German Säge "saw"), from PIE root *sek- "to cut" (source also of Latin secare "to cut").

In reference to its use as a musical instrument, by 1905. Saw-grass, the long, toothed grass found in the Southern U.S., is attested by 1822. The saw-fly (1773), destructive to plants, is so called for the construction of the insect's egg-depositing organ.