On January 1 each year, the Church commemorates both the feast of the Circumcision of Christ as a child, and Saint Basil the Great, Archbishop of Caesarea in Cappadocia, who is attributed with the long poetic Eucharistic Prayer, or Anaphora, that we use on some major feast days and the Sundays during Lent.
Since many are recovering from celebrating the New Year from the night before, attendance at Liturgy on January 1st is usually pretty sparse in most parishes. The two feasts seem to fall on the same day from mere coincidence, and not for any particular theological reason.
It is the 8th day after Christ’s Nativity, and so according to the Convenant of Abraham, male children are circumcised 8 days after their birth. Jesus was no different from us, and therefore like us, he was subjected to Jewish law at the time. This simply reminds us of the connection between the old covenant and the new, the Old Testament and the New Testament, the teachings of the prophets that are fulfilled in Jesus Christ.
St. Basil is commemorated this same day as a remembrance of the date of his death in the year 379. Basil was a noted theologian, writing many books and treatises, his most famous, On the Holy Spirit, expresses the doctrine of the Trinity. He was a noted episcopal leader, overseeing the province of Pontus as archbishop, which was half of Asia Minor (present-day Turkey). He was also known for his great influence on monasticism, both throughout the East, and in the West through Saint Benedict. Numerous religious orders of men and women, east and west, bear the name of Saint Basil. In particular, he emphasized monastic communities rather than isolated ascetics or the solitude of hermits.
Finally, Basil atttended the Council of Constantinople in 360, and his teachings on the Holy Trinity and divinity of the Holy Spirit are seen in the final Profession of Faith (Creed) approved by that council.