Etymology
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require (v.)

late 14c., requeren, "to ask (someone) a question, inquire," a sense now obsolete, from Old French requerre, requerir "seek, procure; beg, ask, petition; demand," from Vulgar Latin *requaerere, from Latin requirere "seek to know, ask, ask for (something needed)," from re-, here perhaps meaning "repeatedly" (see re-), + quaerere "ask, seek" (see query (v.)). In some later English senses probably directly from Latin.

Still in 16c.-17c. commonly "to ask or request (to have or do something)," but this original sense of the word has been taken over by request (v.).

Also from late 14c. as "to stand in need of, want; to need for some end or purpose." The sense of "demand that (someone) do (something)" is from 1751, via the notion of "to ask for imperatively, or as a right" (late 14c.). The meaning "demand as necessary or essential on general principles" is from early 15c. Related: Required; requiring.

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required (adj.)

"that must be done as a condition," c. 1600, past-participle adjective from require (v.). Required reading, that which must be read to attain an understanding of a subject, is attested from 1881.

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requisite (adj.)

"needed, necessary, required by circumstances or the nature of things, so needful that it cannot be dispensed with," mid-15c., from Latin requisitus, past participle of requirere "seek to know, ask, ask for" (see require). As a noun, "that which is necessary, something indispensable," from c. 1600. Related: Requisiteness.

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requisition (n.)

c. 1400, requisicioun, "a request, an act of requesting or demanding," from Old French requisicion (12c.) and directly from Medieval Latin requisitionem (nominative requisitio) "examination, a searching," noun of action from past-participle stem of Latin requirere (past participle requisitus) "seek to know, ask, ask for" (see require).

The meaning "action of formally calling upon someone to perform some action, etc." is by 1550s, originally legal. The sense of "action of requiring a certain amount of something to be furnished" is by 1776.

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request (n.)

mid-14c., requeste, "act of asking for a favor, service, etc.; expression of desire for something to be granted or done," from Old French requeste (Modern French requête) "a request," from Vulgar Latin *requaesita, from Latin requisita (res) "(a thing) asked for," from fem. of requisitus "requested, demanded," past participle of requirere "seek to know, ask, ask for" (see require).

From late 14c. as "that which one asks for." By 1928 as "a letter, telephone call, etc., asking for a particular song to be played on a radio program, often accompanied by a personal message or dedication."

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requirement (n.)

1520s, "request, requisition" (a sense now obsolete), from require + -ment. Meaning "things required, a need, something necessary" is from 1660s. Meaning "that which must be accomplished, necessary condition" is by 1841. Related: Requirements. Fowler points out that requirement "means properly a need" and requisite "a needed thing," though the distinction is a fine one.

That which is required by the nature of the case, or is only indirectly thought of as required by a person, is called a requisite ; that which is viewed as required directly by a person or persons is called a requirement ...; a requisite is more often material than a requirement ; a requisite may be a possession or something that may be viewed as a possession, but a requirement is a thing to be done or learned. [Century Dictionary, 1895]
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re- 

word-forming element meaning "back, back from, back to the original place;" also "again, anew, once more," also conveying the notion of "undoing" or "backward," etc. (see sense evolution below), c. 1200, from Old French re- and directly from Latin re- an inseparable prefix meaning "again; back; anew, against."

Watkins (2000) describes this as a "Latin combining form conceivably from Indo-European *wret-, metathetical variant of *wert- "to turn." De Vaan says the "only acceptable etymology" for it is a 2004 explanation which reconstructs a root in PIE *ure "back."

In earliest Latin the prefix became red- before vowels and h-, a form preserved in redact, redeem, redolent, redundant, redintegrate, and, in disguise, render (v.). In some English words from French and Italian re- appears as ra- and the  following consonant is often doubled (see rally (v.1)).

The many meanings in the notion of "back" give re- its broad sense-range: "a turning back; opposition; restoration to a former state; "transition to an opposite state." From the extended senses in "again," re- becomes "repetition of an action," and in this sense it is extremely common as a formative element in English, applicable to any verb. OED writes that it is "impossible to attempt a complete record of all the forms resulting from its use," and adds that "The number of these is practically infinite ...."   

Often merely intensive, and in many of the older borrowings from French and Latin the precise sense of re- is forgotten, lost in secondary senses, or weakened beyond recognition, so that it has no apparent semantic content (receive, recommend, recover, reduce, recreate, refer, religion, remain, request, require). There seem to have been more such words in Middle English than after, e.g. recomfort (v.) "to comfort, console; encourage;" recourse (n.) "a process, way, course." Recover in Middle English also could mean "obtain, win" (happiness, a kingdom, etc.) with no notion of getting something back, also "gain the upper hand, overcome; arrive at;" also consider the legal sense of recovery as "obtain (property) by judgment or legal proceedings." 

And, due to sound changes and accent shifts, re- sometimes entirely loses its identity as a prefix (rebel, relic, remnant, restive, rest (n.2) "remainder," rally (v.1) "bring together"). In a few words it is reduced to r-, as in ransom (a doublet of redemption), rampart, etc.

It was used from Middle English in forming words from Germanic as well as Latin elements (rebuild, refill, reset, rewrite), and was used so even in Old French (regret, regard, reward, etc.).

Prefixed to a word beginning with e, re- is separated by a hyphen, as re-establish, re-estate, re-edify, etc. ; or else the second e has a dieresis over it: as, reëstablish, reëmbark, etc. The hyphen is also sometimes used to bring out emphatically the sense of repetition or iteration : as, sung and re-sung. The dieresis is not used over other vowels than e when re is prefixed : thus, reinforce, reunite, reabolish. [Century Dictionary, 1895]
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microsurgery (n.)

"surgery so delicate as to require the use of a microscope," 1912, from micro- + surgery. Related: Microsurgical.

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non-invasive (adj.)

also noninvasive, "not tending to spread; not require the introduction of instruments into the body," by 1850, from non- + invasive.

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deskill (v.)

also de-skill, "alter a workplace so as no longer to require skilled workers" (usually through technology), 1941, from de- + skill. Related: Deskilled.

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