Etymology
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must (v.)

auxiliary of prediction, "be obliged, be necessarily impelled," from Old English moste, past tense of motan "have to, be able to," from Proto-Germanic *motanan (source also of Old Saxon motan "to be obliged to, have to," Old Frisian mota, Middle Low German moten, Dutch moeten, German müssen "to be obliged to," Gothic gamotan "to have room to, to be able to"), perhaps from PIE root *med- "take appropriate measures," but this old suggestion lately has been doubted. Used as present tense from c. 1300, eventually displacing motan, from the custom of using past subjunctive as a moderate or polite form of the present.

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must (n.1)
"new wine," Old English must, from Latin mustum (also source of Old High German, German most, Old French moust, Modern French moût, Spanish, Italian mosto), short for vinum mustum "fresh wine," neuter of mustus "fresh, new, newborn," perhaps literally "wet," and from PIE *mus-to-, from root *meus- "damp" (see moss).
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must (n.2)

"mold, moldiness," c. 1600, perhaps a back-formation of musty (q.v.).

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must (n.3)

"male elephant frenzy," 1878, from earlier adjective (1855), from Urdu mast "intoxicated, in rut," from Persian mast, literally "intoxicated," related to Sanskrit matta- "drunk, intoxicated," past participle of madati "boils, bubbles, gets drunk," from PIE root *mad- "wet, moist" (see mast (n.2)).

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must (n.4)

"that which has to be done, seen, or experienced," 1892, from must (v.). As an adjective, "obligatory, indispensable," by 1912, from the noun; must-read (n.) is from 1959.

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mustn't 
by mid-18c., contraction of must not; see must (v.).
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needs (adv.)

"of necessity, necessarily," late 14c. and surviving in archaic constructions involving must, from Middle English nede (see need), used as an adverb reinforcing must (v.), hence the genitive ending.

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moist (adj.)

late 14c., "slightly wet; well-irrigated, characterized by moistness," from Old French moiste "damp, wet, soaked" (13c., Modern French moite), which is of uncertain origin. Perhaps from Vulgar Latin *muscidus "moldy," also "wet," from Latin mucidus "slimy, moldy, musty," from mucus "slime" (see mucus). Alternative etymology [Diez] is from Latin musteus "fresh, green, new," literally "like new wine," from musteum "new wine" (see must (n.1)). If this wasn't the source, it influenced the form of the other word in Old French. Related: Moistly; moistness (mid-14c.).

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mustard (n.)

late 13c. (late 12c. as a surname), "seed of the mustard plant crushed and used as a condiment paste or for medicinal purposes," from Old French mostarde "mustard; mustard plant" (Modern French moutarde), from moust "must," from Latin mustum "new wine" (see must (n.1)); so called because it was originally prepared by adding must to the ground seeds of the plant to make a paste. As the name of the plant itself, by mid-14c. in English. As a color name, it is attested from 1848.

Mustard-pot is attested from early 15c. Mustard gas, World War I poison (first used by the Germans at Ypres, 1917), so called for its color and smell and burning effect on eyes and lungs; chemical name is dichlordiethyl sulfide; it contains no mustard and is an atomized liquid, not a gas. To cut the mustard (1907, usually in negative) is probably from slang mustard "genuine article, best thing" (1903) on notion of "that which enhances flavor."

I'm not headlined in the bills, but I'm the mustard in the salad dressing just the same. [O. Henry, "Cabbages and Kings," 1904]
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*med- 

Proto-Indo-European root meaning "take appropriate measures."

It forms all or part of: accommodate; accommodation; commode; commodious; commodity; empty; immoderate; immodest; Medea; medical; medicament; medicaster; medicate; medication; medicine; medico; medico-; meditate; meditation; Medusa; meet (adj.) "proper, fitting;" mete (v.) "to allot;" modal; mode; model; moderate; modern; modest; modicum; modify; modular; modulate; module; modulation; mold (n.1) "hollow shape;" mood (n.2) "grammatical form indicating the function of a verb;" must (v.); premeditate; premeditation; remedial; remediation; remedy.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit midiur "I judge, estimate;" Avestan vi-mad- "physician;" Greek mēdomai "be mindful of," medesthai "think about," medein "to rule," medon "ruler;" Latin meditari "think or reflect on, consider," modus "measure, manner," modestus "moderate," modernus "modern," mederi "to heal, give medical attention to, cure;" Irish miduir "judge;" Welsh meddwl "mind, thinking;" Gothic miton, Old English metan "to measure out."

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