An Old Nursery Rhyme

Monday's child is fair of face,
Tuesday's child is full of grace,
Wednesday's child is full of woe,
Thursday's child has far to go,
Friday's child is loving and giving,
Saturday's child has to work for its living,
But a child that's born on the Sabbath day
is fair and wise and good and gay.

Hopefully most people don't hold these to be true. In any case the Everything Perpetual Calendar helps find birth days of the week.

Sunday, the first day of the week, is named for the sun. Romans called this day dies solis, day of the sun. Anglo-Saxons of England called this day sunnandaeg, sun day. Early Christians made this their Sabbath, or day of worship and rest since Jesus was said to have risen from the dead on a Sunday.

Monday, the second day of the week, is named for the moon. The Romans called it dies lunae, day of the moon. Anglo-Saxons called it monandaeg, moon day.

Tuesday, the third day of the week, is named for Tyr, the Norse god of war. The Roman name for this day was dies Martis, Mar's day after their god of war. Anglo-Saxons called it Tiwesdaeg, Tiw's (Tyr's) day.

Wednesday, the fourth day of the week, is named for Woden (Odin), the supreme Norse god. Romans called this dies Mercurii, Mercury's day, for their messenger of the gods. Anglo-Saxons called this Wodnesdaeg, Woden's day.

Thursday, the fifth day of the week, is named for Thor, the Norse god of thunder. Romans called this day dies Jovis, Jove's day, after Jupiter, their ultimate god. Anglo-Saxons called this day Thuresdaeg, Thor's day.

Friday, the sixth day of the week, is named for Frigg, the Norse goddess of the heavens and married love and the wife of Odin. The Roman name for this day is dies Veneris, Venus's day, for their goddess of love. The Anglo-Saxon name for this day was Frigedaeg, Frigg's day.

Saturday, the seventh day of the week, is named for Saturn, the Roman god of farming. Romans called this day dies Saturni, Saturn's day. This became Saeterdaeg to Anglo-Saxons. Jews made this day their Sabbath because it is said that God had created the world in six days and rested on the seventh.

Michael Keith (the individual responsible for Keith Numbers, chessametics and the 3835 digit pi mnemonic, Cadaeic Cadenza) also brings us the world's most concise C function to calculate the day of the week (at 45 characters in length):1

d: day (d=1..31)
m: month (m=1..12)
Y: year
D: day of the week (0=Sunday, 1=Monday, ... 6=Saturday)

D = (d+=m<3?y--:y-2,23*m/9+d+4+y/4-y/100+y/400)%7;


1 Michael Keith points out his function is similar in spirit to Zeller's Congruence.
It is unknown how long days have been grouped into weeks of seven days, but some of the earliest records begin with the Hebrews. In Hebrew, the days are simply numbered one through seven, with seven being the Sabbath (Shabbat). Greek, Russian and Portuguese are similar, having religious names for Saturday and Sunday (though Russian has the Sabbath as day six, not seven).

Greek:
Deftera (second)
Tritê (third)
Tetartê (fourth)
Pemptê (fifth)
Paraskeuê (preparation)
Savvato (Sabbath)
Kyriakê (Lord's Day)

Portuguese:
Segunda-feira (second)
Terça-feira (third)
Quarta-feira (fourth)
Quinta-feira (fifth)
Sexta-feira (sixth)
Sábado (Sabbath)
Domingo (of God)

Russian:
Ponedelnik (after "do-nothing")
Vtornik (second)
Sreda (middle)
Chetverg (fourth)
Pyatnitsa (fifth)
Subbota (Sabbath)
Voskresenye (Resurrection)

Latin was a bit more creative, naming the days after the visible heavenly bodies, also the source for the names of the Roman gods. The European languages that don't follow the Hebrew numbering scheme usually use a variation of the Latin names.

Latin:
Lunae dies (Moon)
Martis dies (Mars)
Mercurii dies (Mercury)
Jovis dies (Jupiter)
Veneris dies (Venus)
Saturni dies (Saturn)
Solis dies (Sun)

French is closely related to Latin and, like the other Romance languages, retains similar names. The dies ("day") part has been contracted into a -di suffix and the last two days are derived from alternate Christian Latin names for Sabbath and God's day.

French:
Lundi
Mardi
Mercredi
Jeudi
Vendredi
Samedi (from Latin Sambati dies (Sabbath))
Dimanche (from Latin Dominius (of God))

English borrows from Latin and French, but it is primarily a Germanic language. Unlike French, it retains the Latin source for the "religious" days of Saturday and Sunday. But it localizes some of the Roman god names into their Anglo-Saxon/Nordic equivalents: Mars, Mercury, Jupiter, and Venus are replaced with Tiw (Tyr), Woden (Odin), Thor, and Friga (Freya).

English:
Monday
Tuesday (Tiw (aka Tyr))
Wednesday (Woden (aka Odin))
Thursday (Thor)
Friday (Friga (aka Freya))
Saturday
Sunday

German, like English, is Germanic with a heavy Latin influence. Dienstag, from "assembly day", replaces Mars. Mittwoch, or midweek, replaces Wednesday. Donnerstag means thunder day, a reference to Thor, god of thunder. The references to Roman or Norse gods were probably deliberately replaced as the Germans were Christianized.

German:
Montag
Dienstag (assembly day)
Mittwoch (Midweek, from mittauuechun, coined by Notkur, a monk, circa 1020 AD, to replace pagan reference to mercury)
Donnerstag (thunder day, reference to Thor, god of thunder)
Freitag (Freya, as in English)
Samstag/Sonnabend (Samstag is probably from French Samedi, Sonnabend is "Eve of Sunday")
Sonntag

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