Etymology
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may (v.1)

Old English mæg "am able" (infinitive magan, past tense meahte, mihte), from Proto-Germanic root *mag-, infinitive *maganan (Old Frisian mei/muga/machte "have power, may;" Old Saxon mag/mugan/mahte; Middle Dutch mach/moghen/mohte; Dutch mag/mogen/mocht; Old High German mag/magan/mahta; German mag/mögen/mochte; Old Norse ma/mega/matte; Gothic mag/magan/mahte "to be able"), from PIE root *magh- "to be able, have power." A present-preterit verb (with can, shall, etc.). Also used in Old English as a "auxiliary of prediction."

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May 

fifth month of the modern calendar, early 12c., Mai, from Old French mai and directly from Latin Majus, Maius mensis "month of May," possibly from Maja, Maia, a Roman earth goddess (wife of Vulcan) whose name is of unknown origin; possibly from PIE *mag-ya "she who is great," fem. suffixed form of root *meg- "great" (cognate with Latin magnus).

"[R]eckoned on the continent of Europe and in America as the last month of spring, but in Great Britain as the first of summer" [Century Dictionary, 1897]. Replaced Old English þrimilce, month in which cows can be milked three times a day. May marriages have been considered unlucky at least since Ovid's day. May-apple, perennial herb native to North America, so called for its time of blooming and its yellowish fruit, is attested from 1733, American English.

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may (v.2)

"to celebrate May Day, to take part in May Day festivities," late 15c., from May. Maying as "the observance of May Day with all its sports and games" is attested from late 14c. (maiing).

And as a vapour, or a drop of raine
Once lost, can ne'r be found againe:
                     So when or you or I are made
                     A fable, song, or fleeting shade;
                     All love, all liking, all delight
                     Lies drown'd with us in endlesse night.
Then while time serves, and we are but decaying;
Come, my Corinna, come, let's goe a Maying.
[Robert Herrick, "Corinna's Going a-Maying," 1648] 
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may-fly (n.)

also mayfly, popular name of ephemeral insects of early spring, 1650s, from May + fly (n.).

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May Day 

"first of May," on which the opening of the season of flowers and fruit formerly was celebrated throughout Europe, mid-13c.; see May + day (n.). May Queen "girl or young woman crowned with flowers and honored as queen at the games held on May Day," seems to be a Victorian re-invented tradition; the phrase Queen of Maij is attested from c. 1500.

May Day's association with communism (and socialism and anarchism) dates to 1890. A U.S. general strike for an eight-hour workday began May 1, 1886, and culminated in the Haymarket bombing affair in Chicago on May 4. By 1890 strikes, protests, and rallies were being held in Europe by socialist and labor organizations on May 1, at first in support of the eight-hour day, more or less in commemoration of the 1886 strike.

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mayflower (n.)

"a flower that appears in May," c. 1600s; from May + flower (n.). Used of the hawthorn and locally for the lady's smock, the marsh marigold, and other plants that bloom in May. A popular ship name in early 17c.

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devil-may-care (adj.)

"reckless, careless," 1828 (but suggested in other forms by 1793). As an oath or expression by 1815.

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maybe (adv.)

"perhaps, possibly," early 15c., from (it) may be; see may (v.1) + be (v.). In early 19c. still sometimes written as two words. As a noun, "something that may be or happen," 1580s.

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might (v.)

Old English mihte, meahte, originally the past tense of may (Old English magen "to be able"), thus "*may-ed." The noun might-have-been "something that might have happened but did not," also "someone that might have been greater but wasn't," is by 1848.

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*meg- 
Proto-Indo-European root meaning "great."

It forms all or part of: acromegaly; Almagest; Charlemagne; maestro; magisterial; magistral; magistrate; Magna Carta; magnate; magnitude; magnum; magnanimity; magnanimous; magni-; Magnificat; magnificence; magnificent; magnify; magniloquence; magniloquent; Magnus; maharajah; maharishi; mahatma; Mahayana; Maia; majesty; major; major-domo; majority; majuscule; master; maxim; maximum; may (v.2) "to take part in May Day festivities;" May; mayor; mega-; megalo-; mickle; Mister; mistral; mistress; much; omega.

It is the hypothetical source of/evidence for its existence is provided by: Armenian mets "great;" Sanskrit mahat- "great, mazah- "greatness;" Avestan mazant- "great;" Hittite mekkish "great, large;" Greek megas "great, large;" Latin magnus "great, large, much, abundant," major "greater," maximus "greatest;" Middle Irish mag, maignech "great, large;" Middle Welsh meith "long, great."
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