Etymology
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quarter day (n.)

mid-15c., "day that begins a quarter of the year," designated as days when rents were paid and contracts and leases began or expired, from quarter (n.1). They were, in England, Lady day (March 25), Midsummer day (June 24), Michaelmas day (Sept. 29), and Christmas day (Dec. 25); in Scotland, keeping closer to the pre-Christian Celtic calendar, they were Candlemas (Feb. 2), Whitsunday (May 15), Lammas (Aug. 1), and Martinmas (Nov. 11). Quarter in the sense "period of three months; one of the four divisions of a year" is recorded from late 14c. Related: Quarter days.

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Dorian (adj.)

"of Doris or Doria," c. 1600, first in reference to the mode of ancient Greek music, literally "of Doris," from Greek Doris, the small district in central Greece, traditionally named for Doros, legendary ancestor of the Dorians, whose name is probably related to dōron "gift" (from PIE root *do- "to give").

From 1620s as "native or inhabitant of Doris." Dorian was the name the ancient Greeks gave to one of their four great divisions (the others being the Aeolians, Ionians, and Achaeans). In addition to architecture and music, The Dorians had their own calendar and dialect (see Doric) and the Dorian states included Sparta, Argos, Megara, and the island of Rhodes.

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nones (n.)

late 14c. in reference to the Roman calendar, "ninth day (by inclusive reckoning) before the ides of each month" (7th of March, May, July, October, 5th of other months), from Old French nones and directly from Latin nonæ (accusative nonas), fem. plural of nonus "ninth," contracted from *novenos, from novem "nine" (see nine).

The ecclesiastical sense of "daily office said originally at the ninth hour of the day" is from 1709; originally fixed at ninth hour from sunrise, hence about 3 p.m. (now usually somewhat earlier), from Latin nona (hora) "ninth (hour)," from fem. plural of nonus "ninth." It was also used from c. 1300 in a general sense of "midday" (see noon). For for the nones, see nonce.

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indiction (n.)
late 14c., "period of fifteen years," a chronological unit of the Roman calendar that continued in use through the Middle Ages, from Latin indictionem (nominative indictio), literally "declaration, appointment," noun of action from past participle stem of indicere "to declare publicly, proclaim, announce," from in- "in" (from PIE root *en "in") + dicere "to speak, say, tell" (from PIE root *deik- "to show," also "pronounce solemnly").

Fixed by Constantine and reckoned from Sept. 1, 312. Originally for taxation purposes, it was "a common and convenient means for dating ordinary transactions" [Century Dictionary]. The name refers to the "proclamation," at the beginning of each period, of the valuation upon which real property would be taxed.
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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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Beltane (n.)

early 15c., from Lowland Scottish, from Gaelic bealltainn "May 1," important Celtic religious rite marking the start of summer, probably literally "blazing fire," from PIE root *bhel- (1) "to shine, flash, burn" + Old Irish ten "fire," from PIE *tepnos, related to Latin tepidus "warm," from PIE root *tep- "to be hot." But this derivation of the second element is hotly disputed by some on philological grounds, and fires were equally important in the other Celtic holidays. Also known as "Old May Day," because after the 1752 calendar reform it continued to be reckoned according to Old Style; it was one of the quarter-days of ancient Scotland.

The rubbish about Baal, Bel, Belus imported into the word from the Old Testament and classical antiquity, is outside the scope of scientific etymology. [OED]
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flora (n.)
c. 1500, "Roman goddess of flowers;" 1777, "the plant life of a region or epoch," from Latin Flora, "goddess of flowers," from flos (accusative florem, genitive floris) "flower," from *flo-s-, Italic suffixed form of PIE *bhle- "to blossom, flourish" (source also of Middle Irish blath, Welsh blawd "blossom, flower," Old English blowan "to flower, bloom"), extended form of root *bhel- (3) "to thrive, bloom."

Her festival, the Floralia, was April 28 to May 2 and featured "comic theatrical representations" and "excessive drinking" [Century Dictionary]. The French Revolutionary calendar had a month Floréal (April 20-May 20). Used as the title of systematically descriptive plant catalogues since 1640s, but popularized by Linnaeus in his landmark 1745 study of Swedish plants, "Flora Suecica."
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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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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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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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