"through, by means of," 1580s (earlier in various Latin and French phrases, in the latter often par), from Latin per "through, during, by means of, on account of, as in," from PIE root *per- (1) "forward," hence "through, in front of, before, first, chief, toward, near, around, against."

1680s, Latin, "by families, by stocks;" in legal use, for inheritances, etc., opposed to per capita. See per- + stirpes.

[A]pplied to succession when divided so as to give the representatives belonging to one branch the share only that their head or ancestor would have taken had he survived. Thus, in a gift to A and the children of B, if they are to take per capita, each child will have a share equal to that of A; but if they are to take per stirpes, A will take one half and the other half will be divided among the children of B. [Century Dictionary]

c. 1300, performen, "carry into effect, fulfill, discharge, carry out what is demanded or required," via Anglo-French performer, performir, altered (by influence of Old French forme "form") from Old French parfornir "to do, carry out, finish, accomplish," from par- "completely" (see per-) + fornir "to provide" (see furnish). Church Latin had a compound performo "to form thoroughly, to form."

Theatrical/musical senses of "act or represent on or as on a stage; sing or render on a musical instrument" are from c. 1600. The verb was used with wider senses in Middle English than now, including "to make, construct; produce, bring about;" also "come true" (of dreams), and to performen muche time was "to live long." Related: Performed; performing; performable.

Middle English thoro, thorow, "perfect, complete," mid-13c., a two-syllable stressed form of thurgh "passing or cutting all the way through," which is an adverb (represented by modern through, "from end to end, from side to side") used as an adjective. The notion in thorough is "going all the way through."

Þurh-, thurgh- was an active word-forming element and prefix in Old English and Middle English, often in making transitive verbs of motion (thurghcomen, thurghgon, thurghfallen, thurghserchen, and compare thoroughfare) or intensive adjectives (thurgh-fin, thurgh-hot, thurgh-stif, thurgh-wet, and compare thoroughgoing). It also often translates Latin per-.

The stressed form of through began to develop in the adverb in late Old English. The stress and spelling change seems to not directly track with the sense shift. For the form, compare borough from Old English burh, furrow from furh. Related: Thoroughly; thoroughness.

The Old English adverb is attested as þurh, þurg, þuruh, þorh, þorch, þerh. It became through, the common modern form, by transposition. 

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.

word-forming element meaning "before," from Old French pre- and Medieval Latin pre-, both from Latin prae (adverb and preposition) "before in time or place," from PIE *peri- (source also of Oscan prai, Umbrian pre, Sanskrit pare "thereupon," Greek parai "at," Gaulish are- "at, before," Lithuanian prie "at," Old Church Slavonic pri "at," Gothic faura, Old English fore "before"), extended form of root *per- (1) "forward," hence "beyond, in front of, before."

The Latin word was active in forming verbs. Also see prae-. Sometimes in Middle English muddled with words in pro- or per-.