late 14c., "first, original, first in order of time," from Old French prime and directly from Latin primus "first, the first, first part," figuratively "chief, principal; excellent, distinguished, noble" (source also of Italian and Spanish primo), from Proto-Italic *prismos, superlative of PIE *preis- "before," from root *per- (1) "forward," hence "in front of, before, first, chief."

The meaning "of fine quality, of the first excellence" is from c. 1400. The meaning "first in rank, degree, or importance" is from 1610s in English. Arithmetical sense (as in prime number, one indivisible without a remainder except by 1) is from 1560s; prime meridian "the meridian of the earth from which longitude is measured, that of Greenwich, England," is from 1878. Prime time originally (c. 1500) meant "spring time;" the broadcasting sense of "peak tuning-in period" is attested by 1961.

c. 1300, "main, principal, chief, dominant, largest, greatest, most important;" also "great, large," from Old French principal "main, most important," of persons, "princely, high-ranking" (11c.) and directly from Latin principalis "first in importance; original, primitive," from princeps (genitive principis) "first man, chief leader; ruler, sovereign," noun use of adjective meaning "that takes first," from primus "first" (see prime (adj.)) + root of capere "to take" (from PIE root *kap- "to grasp"). 

Latin, from petitio "petition" (see petition (n.)) + genitive of principium (see principle (n.)). It translates Greek to en arkhē aiteisthai "an assumption at the outset."

In logic, the assumption of that which in the beginning was set forth to be proved; begging the question: a fallacy or fault of reasoning belonging to argumentations whose conclusions really follow from their premises, either necessarily or with the degree of probability pretended, the fault consisting in the assumption of a premise which no person holding the antagonistic views will admit. [Century Dictionary]

"fundamental principles," c. 1600, plural of Latin principium "a beginning, origin" (see principle (n.)). Especially as the short form of the title of Newton's book (published 1687). Hence principles "earliest or elementary parts of a subject" (1530s).

"honorable, moral, having (good or right) principles," 1690s, from principle, which was used as a verb 17c.-18c. meaning "to ground (someone) in principles" (whether good or ill). In later instances from the noun.

Proto-Indo-European root meaning "to grasp."

It forms all or part of: accept; anticipate; anticipation; behave; behoof; behoove; cable; cacciatore; caitiff; capable; capacious; capacity; capias; capiche; capstan; caption; captious; captivate; captive; captor; capture; case (n.2) "receptacle;" catch; catchpoll; cater; chase (n.1) "a hunt;" chase (v.) "to run after, hunt;" chasse; chasseur; conceive; cop (v.) "to seize, catch;" copper (n.2) "policeman;" deceive; emancipate; except; forceps; gaffe; haft; have; hawk (n.); heave; heavy; heft; incapacity; inception; incipient; intercept; intussusception; manciple; municipal; occupy; participation; perceive; precept; prince; purchase; receive; recipe; recover; recuperate; sashay; susceptible.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit kapati "two handfuls;" Greek kaptein "to swallow, gulp down," kope "oar, handle;" Latin capax "able to hold much, broad," capistrum "halter," capere "to grasp, lay hold; be large enough for; comprehend;" Lettish kampiu "seize;" Old Irish cacht "servant-girl," literally "captive;" Welsh caeth "captive, slave;" Gothic haban "have, hold;" Old English hæft "handle," habban "to have, hold."

Proto-Indo-European root forming prepositions, etc., meaning "forward," and, by extension, "in front of, before, first, chief, toward, near, against," etc.

It forms all or part of: afford; approach; appropriate; approve; approximate; barbican; before; deprive; expropriate; far; first; for; for-; fore; fore-; forefather; foremost; former (adj.); forth; frame; frau; fret; Freya; fro; froward; from; furnish; furniture; further; galore; hysteron-proteron; impervious; improbity; impromptu; improve; palfrey; par (prep.); para- (1) "alongside, beyond; altered; contrary; irregular, abnormal;" paradise; pardon; paramount; paramour; parvenu; pellucid; per; per-; percent; percussion; perennial; perestroika; perfect; perfidy; perform; perfume; perfunctory; perhaps; peri-; perish; perjury; permanent; permeate; permit; pernicious; perpendicular; perpetual; perplex; persecute; persevere; perspective; perspire; persuasion; pertain; peruse; pervade; pervert; pierce; portray; postprandial; prae-; Prakrit; pre-; premier; presbyter; Presbyterian; preterite; pride; priest; primal; primary; primate; primavera; prime; primeval; primitive; primo; primogenitor; primogeniture; primordial; primus; prince; principal; principle; prior; pristine; private; privilege; privy; pro (n.2) "a consideration or argument in favor;" pro-; probably; probe; probity; problem; proceed; proclaim; prodigal; produce; profane; profess; profile; profit; profound; profuse; project; promise; prompt; prone; proof; proper; property; propinquity; prophet; prose; prostate; prosthesis; protagonist; Protean; protect; protein; Proterozoic; protest; proto-; protocol; proton; protoplasm; Protozoa; proud; prove; proverb; provide; provoke; prow; prowess; proximate; Purana; purchase; purdah; reciprocal; rapprochement; reproach; reprove; veneer.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit pari "around, about, through," parah "farther, remote, ulterior," pura "formerly, before," pra- "before, forward, forth;" Avestan pairi- "around," paro "before;" Hittite para "outside of," Greek peri "around, about, near, beyond," pera "across, beyond," paros "before," para "from beside, beyond," pro "before;" Latin pro "before, for, on behalf of, instead of," porro "forward," prae "before," per "through;" Old Church Slavonic pra-dedu "great-grandfather;" Russian pere- "through;" Lithuanian per "through;" Old Irish ire "farther," roar "enough;" Gothic faura "before," Old English fore (prep.) "before, in front of," (adv.) "before, previously," fram "forward, from," feor "to a great distance, long ago;" German vor "before, in front of;" Old Irish air- Gothic fair-, German ver-, Old English fer-, intensive prefixes.