Etymology
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quotient (n.)

in mathematics, "the result of the process of division, quantity resulting from the division of one number by another, number of times one quantity is contained in another," mid-15c., quocient, from Latin quotiens "how often? how many times?; as often as," pronominal adverb of time, from quot "how many?" (from PIE root *kwo-, stem of relative and interrogative pronouns). The Latin adverb quotiens was mistaken in Middle English for a present participle of quot in -ens.

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gradient (n.)
"steep slope of a road or railroad," 1835, principally in American English, probably from grade (n.) by analogy of quotient, etc. [OED]. It was used 17c. as an adjective, of animals, "characterized by walking;" in that case it is probably from Latin gradientem, present participle of gradi "to walk."
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*kwo- 
also *kwi-, Proto-Indo-European root, stem of relative and interrogative pronouns.

It forms all or part of: cheese (n.2) "a big thing;" cue (n.1) "stage direction;" either; hidalgo; how; kickshaw; neither; neuter; qua; quality; quandary; quantity; quasar; quasi; quasi-; query; quib; quibble; quiddity; quidnunc; quip; quodlibet; quondam; quorum; quote; quotidian; quotient; ubi; ubiquity; what; when; whence; where; whether; which; whither; who; whoever; whom; whose; why.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit kah "who, which;" Avestan ko, Hittite kuish "who;" Latin quis/quid "in what respect, to what extent; how, why," qua "where, which way," qui/quae/quod "who, which;" Lithuanian kas "who;" Old Church Slavonic kuto, Russian kto "who;" Old Irish ce, Welsh pwy "who;" Old English hwa, hwæt, hwær, etc.
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I.Q. 

1922, abbreviation of intelligence quotient, a 1921 translation of German Intelligenz-quotient, coined 1912 by German psychologist William L. Stern.

Intelligence is a general capacity of an individual consciously to adjust his thinking to new requirements: it is general mental adaptability to new problems and conditions of life. [Stern, "The Psychological Methods of Testing Intelligence," 1914]

Earlier, i.q. was an abbreviation of Latin idem quod "the same as."

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intelligence (n.)
Origin and meaning of intelligence
late 14c., "the highest faculty of the mind, capacity for comprehending general truths;" c. 1400, "faculty of understanding, comprehension," from Old French intelligence (12c.) and directly from Latin intelligentia, intellegentia "understanding, knowledge, power of discerning; art, skill, taste," from intelligentem (nominative intelligens) "discerning, appreciative," present participle of intelligere "to understand, comprehend, come to know," from assimilated form of inter "between" (see inter-) + legere "choose, pick out, read," from PIE root *leg- (1) "to collect, gather," with derivatives meaning "to speak (to 'pick out words')."

Meaning "superior understanding, sagacity, quality of being intelligent" is from early 15c. Sense of "information received or imparted, news" first recorded mid-15c., especially "secret information from spies" (1580s). Meaning "a being endowed with understanding or intelligence" is late 14c. Intelligence quotient first recorded 1921 (see I.Q.).
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