Etymology
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ratification (n.)

"act by which a competent authority gives sanction and validity to something done by another," mid-15c., ratificacion, from Old French ratification (14c.) and directly from Medieval Latin ratificationem (nominative ratificatio), noun of action from past-participle stem of ratificare "to confirm, approve" (see ratify).

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ratify (v.)

mid-14c., ratifien, "confirm, approve, sanction, validate by formal act of approval," from Old French ratifier (13c.), from Medieval Latin ratificare "confirm, approve," literally "fix by reckoning," from Latin ratus "fixed by calculation; determined; approved; certain, sure; valid" (past-participle adjective from reri "to reckon, think;" from PIE root *re- "to reason, count") + combining form of facere "to make" (from PIE root *dhe- "to set, put"). Related: Ratified; ratifying.

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rating (n.1)

1530s, "a fixing of rates, proportionate distribution of charge or compensation," verbal noun from rate (v.2). Meaning "a classification according to grade or rank" is from 1764.

Ratings "statistical estimate of the size of an audience for a particular broadcast," originally radio programs, began in 1930 in U.S. under a system set up by pollster and market researcher Archibald M. Crossley (1896-1985), and were called Crossley ratings or Crossleys until ratings began to be preferred c. 1947.

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rating (n.2)

"a reproving," 1570s, verbal noun from rate (v.1).

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ratio (n.)

1630s, in theological writing, "reason, rationale," from Latin ratio "a reckoning, account, a numbering, calculation," hence also "a business affair; course, conduct, procedure," also in a transferred sense, of mental action, "reason, reasoning, judgment, understanding, that faculty of the mind which forms the basis of computation and calculation." This is from rat-, past-participle stem of reri "to reckon, calculate," also "to think, believe" (from PIE root *re- "to think, reason, count").

Latin ratio often was used to represent or translate Greek logos ("computation, account, esteem, reason") in works of philosophy, though the range of senses in the two do not overlap (ratio lacks the key "speech, word, statement" meaning in the Greek word; see Logos).

The mathematical sense of "relation between two similar magnitudes in respect to quantity," measured by the number of times one contains the other, is attested in English from 1650s (it also was a sense in Greek logos). The general or extended sense of "corresponding relationship between things not precisely measurable" is by 1808.

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ratiocinate (v.)

"to reason, from two judgments to infer a third," 1640s, from Latin ratiocinatus, past participle of ratiocinari "to reckon, compute, calculate; to deliberate, meditate; to reason, argue, infer" (see ratiocination). "Now rare in serious use" [OED]. Related: Ratiocinant; ratiocinative; ratiocinatory.

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ratiocination (n.)

"process of reasoning, mental process of passing from the cognition of premises to the cognition of the conclusion," 1520s, from Latin ratiocinationem (nominative ratiocinatio) "a reasoning, calm reasoning," noun of action from past-participle stem of ratiocinari "to reckon, compute, calculate; to deliberate, meditate; to reason, argue, infer." This is a compound of ratio "reckoning, calculation," also "judgment, reason" (see ratio) + -cinari, which probably is related to conari "to endeavor, to try," from PIE *kona-, from root *ken- "to hasten, set oneself in motion" (see deacon).

Most writers make ratiocination synonymous with reasoning. J.S. Mill and others hold that the word is usually limited to necessary reasoning. [Century Dictionary]
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ration (v.)

"put (someone) on a fixed allowance," 1859, from ration (n.); sense of "divide into rations, apportion in fixed amounts" is from 1870. Related: Rationed; rationing. Middle English racionable (late 15c.) meant "reasonable."

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ration (n.)

1550, "the mental process of reasoning," later, "the relation of one number to another" (1660s), then "fixed allowance of food or other means of subsistence for a fixed period of time" (1702, often rations, from French ration in this sense).

All are from Latin rationem (nominative ratio) "reckoning, numbering, calculation; business affair, procedure," also "reason, reasoning, judgment, understanding," in Medieval Latin "a computed share or allowance of food." This is from rat-, past participle stem of reri "to reckon, calculate," also "think" (from PIE root *re- "to reason, count").

The first sense listed for the English word is obsolete; the second has gone with ratio. The military pronunciation (rhymes with fashion) took over in English from the preferred civilian pronunciation (rhymes with nation) during World War I. That war also gave the word a specific sense of "officially limited allowance for civilians in times of war or dearth" (by 1917).

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rational (adj.)

late 14c., racional, "pertaining to or springing from reason;" mid-15c., of persons, "endowed with reason, having the power of reasoning," from Old French racionel and directly from Latin rationalis "of or belonging to reason, reasonable," from ratio (genitive rationis) "reckoning, calculation, reason" (see ratio).

In arithmetic, "expressible in finite terms," 1560s. Meaning "conformable to the precepts of practical reason" is from 1630s. Related: Rationally. It is from the same source as ratio and ration; the sense in rational is aligned with that in related reason (n.), which got deformed in French.

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