[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.
[proverb, saying, maxim], Middle English saue, at first in a general sense, "what is said, talk, words," from Old English sagu "saying, discourse, speech, study, tradition, tale," from Proto-Germanic *saga-, *sagon- (source also of Middle Low German, Middle Dutch sage, zage, German Sage "legend, fable, saga, myth, tradition," Old Norse saga "story, tale, saga"), from PIE root *sek(w)- "to say, utter" (see say (v.)).
The surviving specific sense of "proverb, saying, maxim" is by late 13c. "[A] contemptuous term for an expression that is more common than wise" [Century Dictionary].