Etymology
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New Year's Eve 

"evening before the first day of the new year," c. 1300; "þer þay dronken & dalten ... on nwe gerez euen." The Julian calendar began on January 1, but the Christian Church frowned on pagan celebrations of this event and chose the Feast of the Annunciation (March 25) as its New Year's Day. The civic year in England continued to begin January 1 until late 12c., and even though legal documents then shifted to March 25, popular calendars and almanacs continued to begin on January 1. The calendar reform of 1751 restored the Julian New Year in England. New Year's was the main midwinter festival in Scotland from 17c., when Protestant authorities banned Christmas, and continued so after England reverted to Christmas, hence the Scottish flavor ("Auld Lang Syne," etc.). New Year's gathering in public places began 1878 in London, after new bells were installed in St. Paul's.

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legem pone (n.)
"payment of money, cash down," 1570s, old slang, from the title in the Anglican prayer-book of the psalm appointed for Matins on the 25th of the month; it was consequently associated especially with March 25, the new year of the old calendar and a quarter day, when payments and debts came due and money changed hands generally. The title is from the first two words of the fifth division of Psalm cxix: Legem pone mihi, Domine, viam justificationum tuarum "Teach me, O Lord, the way of thy statutes."
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leap year (n.)
"year containing 366 days," late 14c., lepe gere (not in Old English), from leap (v.) + year. Probably so called from its causing fixed festival days, which normally advance one weekday per year, to "leap" ahead one day in the week. Compare Medieval Latin saltus lunae (Old English monan hlyp) "omission of one day in the lunar calendar every 19 years."

Dutch schrikkeljaar "leap year" is from Middle Dutch schricken "leap forward," literally "be startled, be in fear." The 29th of February is schrikkeldag. Danish skudaar, Swedish skottår are literally "shoot-year;" German schaltjahr is from schalten "insert, intercalate." The Late Latin phrase was annus bissextilis, source of the Romanic words; compare bissextile.
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blue moon (n.)

"a long time," 1821, often in phrases indicating something rarely occurring. Compare at the Greek calends (from an ancient Roman phrase alluding to the fact that the Greeks had nothing corresponding to the Roman calends), and the native in the reign of Queen Dick and Saint Geoffrey's Day "Never, there being no saint of that name," reported in Grose (1788). Nevermass "date which never comes" is from 1540s. Blue moon is suggested earliest in this couplet from 1528:

Yf they say the mone is blewe,
We must beleve that it is true.

Though this might refer to calendrical calculations by the Church. Thus the general "rareness" sense of the term is difficult to disentangle from the specific calendrical one (commonly misinterpreted as "second full moon in a calendar month," but actually a quarterly calculation). In either case, the sense of blue here is obscure. Literal blue moons do sometimes occur under extreme atmospheric conditions.

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dog days (n.)

"period of dry, hot weather at the height of summer," 1530s, from Latin dies caniculares, the idea, though not the phrase, from Greek; so called because they occur around the time of the heliacal rising of Sirius, the Dog Star (kyōn seirios). Noted as the hottest and most unwholesome time of the year; often reckoned as July 3 to August 11, but variously calculated, depending on latitude and on whether the greater Dog-star (Sirius) or the lesser one (Procyon) is reckoned.

The heliacal rising of Sirius has shifted down the calendar with the precession of the equinoxes; in ancient Egypt c. 3000 B.C.E. it coincided with the summer solstice, which also was the new year and the beginning of the inundation of the Nile. The "dog" association apparently began here (the star's hieroglyph was a dog), but the reasons for it are now obscure.

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