The celebration of the New Year did not occur on the first day of January until after the introduction of the Gregorian calendar system in 1582--even then only in France, the northern Italian city states, Portugal, and in the Spanish nations of Castile and Aragon. The new calendar was not accepted until 1600 in Scotland and 1752 in England.
From the earliest days of the Roman imperial calendar, the New Year was 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).
Regardless of when the beginning of the year was marked however, the day has traditionally been celebrated as a day of renewal throughout Christendom--for vows, vision, and vocation. It was on the first day of the new year, for instance, that Medieval 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 Lord’s great Epiphany.
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. But in Edinburgh, the purpose was not merely to have a grand excuse for a public party, but a way of celebrating the truth of Epiphany newness.