Etymology
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Words related to assume

ad- 

word-forming element expressing direction toward or in addition to, from Latin ad "to, toward" in space or time; "with regard to, in relation to," as a prefix, sometimes merely emphatic, from PIE root *ad- "to, near, at."

Simplified to a- before sc-, sp- and st-; modified to ac- before many consonants and then re-spelled af-, ag-, al-, etc., in conformity with the following consonant (as in affection, aggression). Also compare ap- (1).

In Old French, reduced to a- in all cases (an evolution already underway in Merovingian Latin), but written forms in French were refashioned after Latin in 14c. and English did likewise 15c. in words it had picked up from Old French. In many cases pronunciation followed the shift. Over-correction at the end of the Middle Ages in French and then English "restored" the -d- or a doubled consonant to some words that never had it (accursed, afford). The process went further in England than in France, where the vernacular sometimes resisted the pedantic, resulting in English adjourn, advance, address, advertisement (Modern French ajourner, avancer, adresser, avertissement). In modern word-formation sometimes ad- and ab- are regarded as opposites, but this was not in classical Latin.

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sub- 
word-forming element meaning "under, beneath; behind; from under; resulting from further division," from Latin preposition sub "under, below, beneath, at the foot of," also "close to, up to, towards;" of time, "within, during;" figuratively "subject to, in the power of;" also "a little, somewhat" (as in sub-horridus "somewhat rough"), from PIE *(s)up- (perhaps representing *ex-upo-), a variant form of the root *upo "under," also "up from under." The Latin word also was used as a prefix and in various combinations.

In Latin assimilated to following -c-, -f-, -g-, -p-, and often -r- and -m-. In Old French the prefix appears in the full Latin form only "in learned adoptions of old Latin compounds" [OED], and in popular use it was represented by sous-, sou-; as in French souvenir from Latin subvenire, souscrire (Old French souzescrire) from subscribere, etc.

The original meaning is now obscured in many words from Latin (suggest, suspect, subject, etc.). The prefix is active in Modern English, sometimes meaning "subordinate" (as in subcontractor); "inferior" (17c., as in subhuman); "smaller" (18c.); "a part or division of" (c. 1800, as in subcontinent).
*em- 

Proto-Indo-European root meaning "to take, distribute." 

It forms all or part of: assume; consume; emption; example; exemplar; exemplary; exemplify; exempt; exemption; impromptu; peremptory; pre-emption; premium; presume; presumption; prompt; pronto; ransom; redeem; redemption; resume; sample; sejm; subsume; sumptuary; sumptuous; vintage.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit yamati "holds, subdues;" Latin emere "buy," originally "take," sumere "to take, obtain, buy;" Old Church Slavonic imo "to take;" Lithuanian imu, imti "to take."

For the sense shift from "take" to "buy" in the Latin verbs, compare Old English sellan "to give," source of Modern English sell "to give in exchange for money;" Hebrew laqah "he bought," originally "he took;" and colloquial English I'll take it for "I'll buy it." 

presume (v.)

late 14c., presumen, "to take upon oneself, to take liberty," also "to take for granted, believe or accept upon probable evidence, presuppose," especially overconfidently, from Old French presumer (12c.) and directly from Latin praesumere "anticipate," in Late Latin, "assume," from prae "before" (see pre-) + sumere "to take, obtain, buy," from sus‑, variant of sub‑ "up from under" + emere "to take" (from PIE root *em- "to take, distribute").

To presume is to base a tentative or provisional opinion on such knowledge as one has, to be held until it is modified or overthrown by further information. [Century Dictionary]

The intransitive sense of "to venture beyond the limits of ordinary license or propriety" and that of "to press forward presumptuously" are from early 15c. Related: Presumed; presumedly; presuming; presumingly.

assuasive (adj.)
"mitigating, soothing," 1708, probably from assume on model of persuasive, etc.
assumable (adj.)
"capable of being assumed," 1780 (re-assumable is from 1724), from assume + -able. Related: Assumably; assumability.
assumpsit (n.)
"legal action for recovery of damages through breach of contract," legal Latin, literally "he has taken upon himself," perfect indicative of assumere "to take up, take to oneself" (see assume). The word embodies the allegation that the defendant promised or undertook to perform the specified act.
assumption (n.)

c. 1300, "the reception, uncorrupted, of the Virgin Mary into Heaven" (also the Aug. 15 Church festival commemorating this, Feast of the Assumption), from Old French assumpcion, asumpsion (13c.) and directly from Latin assumptionem (nominative assumptio) "a taking up, receiving, acceptance, adoption," noun of action from past-participle stem of assumere "take up, take to oneself" (see assume).

Meaning "minor premise of a syllogism" is late 14c. Meaning "appropriation of a right or possession" is mid-15c. in English, from a Latin use (Cicero). Meaning "action of taking for oneself" is recorded from 1580s; that of "something taken for granted" is from 1620s.

assumptive (adj.)
early 15c., from Mwdieval Latin assumptivus, from assumpt-, past participle stem of assumere/adsumere "take up, take to oneself" (see assume) + -ive. Oldest sense in English is medical, of bloodletting, "withdrawing humours from opposite parts of the body."
unassuming (adj.)
"unpretentious, modest," 1726, from un- (1) "not" + present participle of assume (v.). Related: Unassumingly.