Etymology
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Words related to solve

idiom (n.)

1580s, "form of speech peculiar to a people or place;" meaning "phrase or expression peculiar to a language" is from 1620s; from French idiome (16c.) and directly from Late Latin idioma "a peculiarity in language," from Greek idioma "peculiarity, peculiar phraseology" (Fowler writes that "A manifestation of the peculiar" is "the closest possible translation of the Greek word"), from idioumai "to appropriate to oneself," from idios "personal, private," properly "particular to oneself."

This is from PIE *swed-yo-, suffixed form of root *s(w)e-, pronoun of the third person and reflexive (referring back to the subject of a sentence), also used in forms denoting the speaker's social group, "(we our-)selves" (source also of Sanskrit svah, Avestan hva-, Old Persian huva "one's own," khva-data "lord," literally "created from oneself;" Greek hos "he, she, it;" Latin suescere "to accustom, get accustomed," sodalis "companion;" Old Church Slavonic svoji "his, her, its," svojaku "relative, kinsman;" Gothic swes "one's own;" Old Norse sik "oneself;" German Sein; Old Irish fein "self, himself").

[G]rammar & idiom are independent categories; being applicable to the same material, they sometimes agree & sometimes disagree about particular specimens of it; the most can be said is that what is idiomatic is far more often grammatical than ungrammatical, but that is worth saying, because grammar & idiom are sometimes treated as incompatibles .... [Fowler]
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*leu- 
Proto-Indo-European root meaning "to loosen, divide, cut apart."

It forms all or part of: absolute; absolution; absolve; analysis; analytic; catalysis; catalyst; catalytic; dialysis; dissolve; electrolysis; electrolyte; forlorn; Hippolytus; hydrolysis; -less; loess; loose; lorn; lose; loss; Lysander; lysergic; lysis; -lysis; lyso-; lysol; lytic; -lytic; palsy; paralysis; pyrolusite; resolute; resolution; resolve; soluble; solute; solution; solve; solvent.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit lunati "cuts, cuts off," lavitram "sickle;" Greek lyein "to loosen, untie, slacken," lysus "a loosening;" Latin luere "to loose, release, atone for, expiate;" Old Norse lauss "loose, free, unencumbered; vacant; dissolute;" Old English losian "be lost, perish."
solution (n.)
late 14c., "a solving or being solved," from Old French solucion "division, dissolving; explanation; payment" or directly from Latin solutionem (nominative solutio) "a loosening or unfastening," noun of action from past participle stem of solvere "to loosen, untie, dissolve," from PIE *se-lu-, from reflexive pronoun *s(w)e- (see idiom) + root *leu- "to loosen, divide, cut apart." Meaning "liquid containing a dissolved substance" is first recorded 1590s.
unsolved (adj.)
1660s, from un- (1) "not" + past participle of solve (v.).