On the Eighth Day of Christmas
The song goes "Eight maids a-milking." A traditional view [allegedly debunked by Snopes (Mikkelson, 2000)] is that this relates to the Eight Beatitudes. Mikkelson makes a fairly convincing argument, but not absolutely so. There is still room to accept that Catholics, under persecution from King Henry VIII's Church of England, had to hide the fact they were using the song to help remember key Catholic tenets. Most notably, the 7 Sacraments (for the 7th Day) are not the same for Anglicans and Catholics, other than that, the other tenets do not differ - still - if Catholics were using this song as a tool of remembrance, and were caught doing so, it could have meant imprisonment or even death. Regardless, Snopes is not our ultimate authority and even if the tradition is ancient or not, today we can use this as a reminder of the Eight Beatitudes.
Circumcision of Our Lord - January 1 - Within the Octave of Christmas
Circumcision was part of the ancient Law, starting with Abraham. Jesus fulfilled the old Law, but when He was only 8 days old, that fulfillment had not yet taken place. The Holy Family, still under the old Law complied and brought Jesus to the Temple to fulfill this part of the Law.
The various Christian denominations that observe the feast emphasize different aspects of the events surrounding Jesus' circumcision. In 1969 the (modern) Catholic Church changed the name of the observance to the "Solemnity of Mary," a name that reflects their emphasis on Mary's role as mother of the Savior (Traditionalists still hold to the traditional lectionary). Orthodox Christians continue to observe the day as the "Feast of the Circumcision of Our Lord." Episcopalians call the festival the "Feast of the Holy Name of Our Lord Jesus Christ." They emphasize the significance of Jesus' name, given to Mary by Gabriel - the angel of the Annunciation - which means literally "God saves" or "God helps." Lutherans compromise by calling the festival the "Feast of the Circumcision and the Name of Jesus." (Feast of the Circumcision, n.d.).
Feast of the Circumcision. (n.d.) Encyclopedia of Christmas and New Year\'s Celebrations, 2nd ed.. (2003). Retrieved January 1 2022 from https://encyclopedia2.thefreedictionary.com/Feast+of+the+Circumcision
Mikkelson, D. (2000). The Twelve Days of Christmas: Was the song 'The Twelve Days of Christmas' created as a secret code by persecuted Catholics? Snopes. https://www.snopes.com/fact-check/twelve-days-christmas/