Etymology
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Words related to gain

virile (adj.)

late 15c., "characteristic of a man; marked by manly force," from Old French viril (14c.) and directly from Latin virilis "of a man, manly, worthy of a man," from vir "a man, a hero," from PIE root *wi-ro- "man." Virile member for "penis" is recorded from 1540s.

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gainful (adj.)
"producing profit or advantage," 1550s, from gain (n.) + -ful. Phrase gainfully employed attested from 1796. Related: Gainfully (1540s).
gainly (adj.)
"well-formed and agile," 1886, probably a back-formation from ungainly. Earlier "ready, prompt" (1620s), from gain (n.).
gainer (n.)
"one who gains or profits," 1530s, agent noun from gain (v.). As "one who (deliberately) gains weight" by 2000s.
invitation (n.)
mid-15c., "act of inviting, solicitation," from Latin invitationem (nominative invitatio) "an invitation, incitement, challenge," noun of action from past participle stem of invitare "invite, treat, entertain," originally "be pleasant toward," from in- "toward" (from PIE root *en "in").

The second element is obscure. Watkins suggests a suffixed form of the PIE root *weie- "to go after something, pursue with vigor" (see gain (v.)); de Vaan also traces it to a PIE form meaning "pursued." Meaning "the spoken or written form in which a person is invited" is from 1610s.
regain (v.)

1540s, "gain again, recover," as what has escaped or been lost, from French regaigner (Modern French regagner), from re- "again" (see re-) + gaginer, from Old French gaaignier "to earn, gain; trade; capture, win" (see gain (v.)). Meaning "arrive at again, return to" is from 1630s. Related: Regained; regaining.

vim (n.)
1843, usually said to be from Latin vim, accusative of vis "strength, force, power, vigor, energy," from Proto-Italic *wis-, traditionally from PIE root *weie- "to go after, pursue with vigor or desire," with noun derivatives indicating "force, power" (see gain (v.)) and related to the root of virile. But de Vaan seems to have doubts ("more easily explained from an original root noun"), and based on the early uses OED suggests the possibility that the English word is of "a purely inventive or interjectional origin."
violation (n.)
c. 1400, from Old French violacion and directly from Latin violationem (nominative violatio) "an injury, irreverence, profanation," from past participle stem of violare "to treat with violence, outrage, dishonor," perhaps an irregular derivative of vis "strength, force, power, energy," from PIE root *weie- "to go after, pursue with vigor or desire" (see gain (v.)).