Etymology
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extremism (n.)

"disposition to go to extremes in doctrine or practice," 1848, from extreme + -ism.

I would remind you that extremism in the defense of liberty is no vice! And let me remind you also that moderation in the pursuit of justice is no virtue. [Barry Goldwater (1909-1998), acceptance speech as Republican candidate for President, 1963]
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intemperance (n.)
early 15c., "lack of restraint, excess," also of weather, "inclemency, severity," from Old French intemperance (14c.) and directly from Latin intemperantia "intemperateness, immoderation, excess" (as in intemperantia vini "immoderate use of wine"), from in- "not, opposite of" (see in- (1)) + temperantia "moderation, sobriety, discretion, self-control," from temperans, present participle of temperare "to moderate" (see temper (v.)).
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moderate (v.)

early 15c., "to abate excessiveness, reduce the intensity of;" from Latin moderatus "within bounds, observing moderation;" figuratively "modest, restrained," past participle of moderari "to regulate, mitigate, restrain, temper, set a measure, keep (something) within measure," from PIE root *med- "take appropriate measures." Intransitive sense of "become less violent, severe, rigorous, etc." is from 1670s. Meaning "to preside over a debate" is first attested 1570s. Related: Moderated; moderating.

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temperature (n.)
mid-15c., "fact of being tempered, proper proportion;" 1530s, "character or nature of a substance," from Latin temperatura "a tempering, moderation," from temperatus, past participle of temperare "to mix in due proportion, modify, blend; restrain oneself" (see temper (v.)). Sense of "degree of heat or cold" first recorded 1670 (Boyle), from Latin temperatura, used in this sense by Galileo. Meaning "fever, high temperature" is attested from 1898.
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moderate (adj.)

"not excessive in amount, intensity, quality, etc.," late 14c., originally of weather and other physical conditions, from Latin moderatus "within bounds, observing moderation;" figuratively "modest, restrained," past participle of moderari "to regulate, mitigate, restrain, temper, set a measure, keep (something) within measure," from PIE *med-es-, from root *med- "take appropriate measures." The notion is "keeping within due measure." In English, of persons from early 15c., of opinions from 1640s, of prices from 1670s. Related: Moderateness.

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moderator (n.)

late 14c., moderatour, "that which regulates the movement of the celestial spheres," from Latin moderator "manager, ruler, director," literally "he who moderates," from moderatus "within bounds, observing moderation;" figuratively "modest, restrained," past participle of moderari "to regulate, mitigate, restrain, temper, set a measure, keep (something) within measure," from PIE root *med- "take appropriate measures."

Meaning "one who acts as an arbitrator, person who presides at a meeting or disputation" is from 1560s. Fem. form moderatrix attested from 1530s.

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mediocrity (n.)

c. 1400, mediocrite, "moderation; intermediate state or amount," from Latin mediocritatem (nominative mediocritas) "a middle state, middling condition, medium," from mediocris "of middling height or state, moderate, ordinary," figuratively "mediocre, mean, inferior," literally "halfway up a mountain" (see mediocre). Neutral at first; disparaging sense "quality of being moderate or middling in ability, accomplishment, etc." began to predominate from late 16c. The meaning "person of mediocre abilities or attainments" is from 1690s. Before the tinge of disparagement crept in, another name for the Golden Mean was golden mediocrity.

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measure (v.)

early 14c., mesuren, "to exercise moderation;" mid-14c., "to deal out or divide up by measure," also "to ascertain spatial dimensions, quantity, or capacity of by comparison with a standard;" from Old French mesurer "measure; moderate, curb" (12c.), from Late Latin mensurare "to measure," from Latin mensura "a measuring, a measurement; thing to measure by," from mensus, past participle of metiri "to measure," from PIE root *me- (2) "to measure." The native verb is mete. Intransitive sense of "to be of a (specified) measure" is from 1670s. To measure up "have the necessary abilities" is 1910, American English. Related: Measured; measuring.

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middle (adj.)

Old English middel, "equally distant from extremes or limits; intermediate," from Proto-West Germanic *midla- (source also of Old Frisian middel, Old Saxon middil, Middle Low German, Dutch middel, Old High German mittil, German mittel), from Proto-Germanic *medj, from PIE root *medhyo- "middle."

Middle finger "the third finger" (counting the thumb as the first)  so called from late Old English. Middle school is attested from 1838, originally "middle-class school, school for middle-class children;" the sense in reference to a school for grades between elementary and high school is from 1960. Middle management, the level below senior management, is by 1941.

Middle-of-the-road in the figurative sense is attested from 1894, originally political; edges of a dirt road can be washed out and thus are less safe, but the notion here probably is of the middle as "less exposed to ambush." Middle way in the figurative sense of "path of moderation" is from c. 1200. Middle ground as "place of moderation or compromise between extremes" is by 1961. Middle-sized "of medium size" is by 1620s.

In U.S. history, the Middle States (1784) were those between New England and the South (New York, New Jersey, Pennsylvania, Delaware). Middle America for "the 'silent majority,' the generally conservative middle class regarded as a homogeneous group" is by 1968.

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equity (n.)

early 14c., equite, "quality of being equal or fair, impartiality;" late 14c., "that which is equally right or just to all concerned," from Old French equite (13c.), from Latin aequitatem (nominative aequitas) "the uniform relation of one thing to others, equality, conformity, symmetry;" also "just or equitable conduct toward others," from aequus "even, just, equal" (see equal (adj.)).

In law, "fairness in the adjustment of conflicting interests; the settlement of controversies by the dictates of good conscience" (natural equity), late 14c., from Roman naturalis aequitas, the general principles of justice which corrected or supplemented the legal codes ("governed by benevolence, while justitia yields to another only what is strictly due," Lewis & Short).

Hence, in England and U.S., also "justice based on such principles, the system of jurisprudence as to what is fair and what is not," and "a court or jurisdiction in which these doctrines are applied" (1590s).

The Latin word also meant "a quiet, tranquil state of mind; moderation, evenness of temper."

The L. æquitas was somewhat influenced in meaning by being adopted as the ordinary rendering of Gr. ἐπιεικεια ...,which meant reasonableness and moderation in the exercise of one's rights, and the disposition to avoid insisting on them too rigorously. [OED]

From 1620s as "an equitable right, that to which one is justly entitled," especially a right recognized by courts of equity that is not provided for in the common or statute law (such as certain property rights of wives). Equities, "the ordinary shares of a limited company," carrying certain rights to assets and profits, is attested by 1904.

By 1980s it had taken on extended senses in sociology, e.g.: "allocating benefits in various policy fields in such a way as to provide groups, persons, and places with at least a minimum level of benefits so as to satisfy basic needs" [Stuart S. Nagel, "Equity as a Policy Goal," 1983].

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