c. 1600, "epicurean, having delicate tastes," from fine (adj.) + toothed "having teeth" (of a certain kind); see tooth (n.). By 1703 as "having fine teeth" (of a saw, file, comb, etc.); fine-tooth in this sense attested from 1804.
mid-13c., "unblemished, refined, pure, free of impurities," also "of high quality, choice," from Old French fin "perfected, of highest quality" (12c.), a back-formation from finire or else from Latin finis "that which divides, a boundary, limit, border, end" (see finish (v.)); hence "acme, peak, height," as in finis boni "the highest good." The English word is from c. 1300 as "rich, valuable, costly;" also in a moral sense "true, genuine; faithful, constant." From late 14c. as "expertly fashioned, well or skillfully made," also, of cloth, "delicately wrought." Of weapons or edges, "sharp" from c. 1400. In reference to quality of gold and silver, late 15c.
In French, the main meaning remains "delicate, intricately skillful;" in English since c. 1300 fine has been also a general broad expression of admiration or approval, the equivalent of French beau (as in fine arts, "those which appeal to the mind and the imagination," 1767, translating French beaux-arts). Related: Finer; finest. Fine print is from 1861 as "type small and close-set;" by 1934 in the extended sense "qualifications and limitations of a deal."
Old English toð (plural teð), from Proto-Germanic *tanthu- (source also of Old Saxon, Danish, Swedish, Dutch tand, Old Norse tönn, Old Frisian toth, Old High German zand, German Zahn, Gothic tunþus), from PIE root *dent- "tooth." Plural teeth is an instance of i-mutation.
The loss of -n- before spirants is regular in Old English, Old Frisian, and Old Saxon: compare goose (n.), five, mouth (n.). Also thought, from stem of think; couth from the stem of can (v.1); us from *uns.
Application to tooth-like parts of other objects (saws, combs, etc.) first recorded 1520s. Tooth and nail as weapons is from 1530s. The tooth-fairy is attested from 1964.
<a href="https://www.etymonline.com/word/fine-toothed">Etymology of fine-toothed by etymonline</a>
Harper, D. (n.d.). Etymology of fine-toothed. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/fine-toothed