Etymology
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Words related to *mei-

administer (v.)
Origin and meaning of administer

late 14c., aministren, later administren, "to manage as a steward, control or regulate on behalf of others," from Old French aministrer "help, aid, be of service to" (12c., Modern French administrer), and directly from Latin administrare "to help, assist; manage, control, guide, superintend; rule, direct," from ad "to" (see ad-) + ministrare "to serve, attend, wait upon," from minister "inferior, servant, priest's assistant" (see minister (n.)).

The -d- was restored 14c.-16c. in French and after 15c. in English. In reference to punishment, justice, etc., "to dispense, bring into operation" (especially as an officer), from mid-15c. In reference to medicines, medical treatment, etc., "to give," from 1540s. Related: Administered; administering.

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administration (n.)

mid-14c., "act of giving or dispensing;" late 14c., "management (of a business, property, etc.), act of administering," from Latin administrationem (nominative administratio) "aid, help, cooperation; direction, management," noun of action from past-participle stem of administrare "to help, assist; manage, control, guide, superintend; rule, direct," from ad "to" (see ad-) + ministrare "to serve, attend, wait upon," from minister "inferior, servant, priest's assistant" (see minister (n.)).

Early 15c. as "management of a deceased person's estate under a commission from authority." Meaning "management of public affairs" is from 1680s; hence, "executive power in a government" (1731), though later in Britain government was used in this sense. Meaning "a U.S. president's period in office" is first recorded 1796 in writings of George Washington.

The administration of government, in its largest sense, comprehends all the operations of the body politic, whether legislative, executive, or judiciary; but in its most usual, and perhaps in its most precise, signification, it is limited to executive details, and falls peculiarly within the province of the executive department. ["The Federalist," No. 72 (Hamilton)]
comminute (v.)

"make (solids) small or fine, reduce to a fine powder," 1620s, from Latin comminutus, past participle of comminuere "to lessen, break into smaller parts," from com "with, together" (see com-) + minuere "to make smaller" (from PIE root *mei- (2) "small"). Related: Comminuted; comminuting; comminution (1570s, from Latin comminutionem).

diminish (v.)

early 15c., diminishen, "to lessen, make or seem to make smaller," from merger of two obsolete verbs, diminue and minish

Diminue (late 14c.) is from Old French diminuer "make small," from Latin diminuere "break into small pieces," variant of deminuere "lessen, diminish," from de- "completely" (see de-) + minuere "make small" (from PIE root *mei- (2) "small"). Minish (mid-14c.) is from Old French menuisier, from Latin minuere.

 Meaning "to lower in power, importance, or estimation" is from mid-15c. Intransitive sense of "to grow less, become or appear smaller" is from 1510s. Related: Diminishes; diminishing

meiosis (n.)

in biology, "division of a cell nucleus," 1905, from Greek meiosis "a lessening," from meioun "to lessen," from meion "less," from PIE root *mei- (2) "small."

Earlier (1580s) it was a rhetorical term, a figure of speech "weak or negative expression used for a positive and forcible one, so that it may be made all the more emphatic," as when one says "not bad" meaning "very good" or "don't mind if I do" meaning "I really would like to," or this example from "Mark Twain":

"YOUNG AUTHOR." — Yes Agassiz does recommend authors to eat fish, because the phosphorus in it makes brains. So far you are correct. But I cannot help you to a decision about the amount you need to eat,—at least, not with certainty. If the specimen composition you send is about your fair usual average, I should judge that perhaps a couple of whales would be all you would want for the present. Not the largest kind, but simply good, middling-sized whales.

 Related: meiotic; meiotically.

Menshevik (adj.)

1907, from Russian men'shevik, from men'she "lesser" (comparative of malo "little," from PIE root *mei- (2) "small") + -evik "one that is." So called by Lenin because they were a minority in the party. The Russian word was used earlier in reference to the minority faction of the Social-Democratic Party when it split in 1903. See Bolshevik. As a noun from 1917. Russian plural mensheviki occasionally was used in English. Related: Menshevism; Menshevist.

menu (n.)

1837, "detailed list of dishes to be served at a banquet or meal," from French menu de repas "list of what is served at a meal," from French menu (adj.) "small, detailed" (11c.), from Latin minutus "small," literally "made smaller," past participle of minuere "to diminish," from root of minus "to diminish" (from PIE root *mei- (2) "small"). Computer sense of "list of options displayed on a screen" is by 1967, from the expanded sense of "any detailed list," which is attested by 1889.

metier (n.)

"one's skill, talent, or calling," 1792, from French métier "trade, profession," from Old French mestier "task, affair, service, function, duty," from Gallo-Roman *misterium, from Latin ministerium "office, service," from minister "servant" (see minister (n.)).

mince (v.)

late 14c., mincen, "to chop (meat, herbs, onions, etc.) in little pieces," from Old French mincier "make into small pieces," from Vulgar Latin *minutiare "make small," from Late Latin minutiæ "small bits," from Latin minutus "small" (from PIE root *mei- (2) "small").

From 1540s in reference to speech, "to utter primly or in a half-spoken way as affected delicacy, clip affectedly in imitation of elegance," of words or language, "to restrain in the interest of decorum," 1590s. The meaning "walk with short or precise steps or with affected nicety" is from 1560s. The etymological sense is "to make less, make small." Related: Minced; mincing.

minestrone (n.)
Italian vegetable soup, 1871, from Italian minestrone, with augmentative suffix -one + minestra "soup, pottage," literally "that which is served," from minestrare "to serve, to prepare (soup, etc.)," from Latin ministrare "to serve, attend, wait upon," from minister "inferior, servant, priest's assistant" (see minister (n.)).