finish v.

late 14c., "set (one's eyes or mind) on something" (a figurative use), probably from Old French verb *fixer, from fixe "fixed," from Latin fixus "fixed, fast, immovable; established, settled," past-participle adjective from figere "to fix, fasten, drive, thrust in; pierce through, transfix," also figurative, from PIE root *dheigw- "to pierce, stick in;" hence "to fix, fasten."

Sense of "fasten, attach" is c. 1400; that of "to make (colors, etc.) fast or permanent" is from 1660s. The meaning "settle, assign" evolved into "adjust, arrange" (1660s), then "to repair" (1737), but this sometimes was objected to (see below). Sense of "tamper with" (a fight, a jury, etc.) is from 1790. As euphemism for "castrate a pet" it dates from 1930. Related: Fixed; fixing.

To fix is to make fast, or permanent; to set immoveably, &c.: hence, to fix a watch, is to stop it, or prevent it from 'going;' which, it must be admitted, is a very unsatisfactory mode of repairing that article. [Seth T. Hurd, "A Grammatical Corrector; or, Vocabulary of the Common Errors of Speech," 1847]

c. 1300, "relation by marriage" (as opposed to consanguinity), from Old French afinite "relationship, kinship; neighborhood, vicinity" (12c., Modern French affinité), from Latin affinitatem (nominative affinitas) "relationship by marriage; neighborhood," noun of state from affinis "adjoining, adjacent," also "kin by marriage," literally "bordering on," from ad "to" (see ad-) + finis "a border, a boundary" (see finish (v.)).

The spelling was re-Latinized in early Modern English. It has been used figuratively in English since c. 1600 of structural relationships in chemistry, philology, geometry, etc. The meaning "natural liking or attraction, a relationship as close as family between persons not related by blood" is from 1610s.