The celebration of the New Year did not actually occur on the first day of January until after the adoption of the Gregorian calendar in 1582--and even then only in France, the northern Italian city states, Portugal, and the Spanish nations of Castile and Aragon. The new calendar and thus, the change in the celebration of the New Year, was not adopted in Scotland until 1600 and in England and its American colonies until 1752.
From the earliest days of the Roman imperial calendar, the New Year was instead celebrated on March 25--which is why September, October, November, and December are derived from the Latin words septem (seven), octo (eight), novem (nine), and decem (ten). This also explains why presidential inaugurations were once constitutionally scheduled to be held in March--and so it was from 1789 until the 1933 ratification of the Twentieth Amendment.
Throughout Christendom, the beginning of the New Year, regardless of when it was celebrated, has been set apart as a day of renewal--for vows, vision, and vocation. It was on this day that guild members took their annual pledge, that husbands and wives renewed their marriage promises, and that young believers reasserted their resolution to walk in the grace of the Gospel.
In Edinburgh beginning in the seventeenth century, revelers would gather at the Tron Church to watch the great clock tower mark their entrance into the New Year--which was the inspiration behind the relatively recent Times Square ceremony in New York. In Edinburgh though, the purpose was not merely to have a grand excuse for a public party, but rather to celebrate the truth of Epiphany newness.