Admittedly that would make much more sense to pre-1752 New Yorkers. Until then England and its colonies still used the Julian calendar, by which the New Year began on March 25th, rather than January 1st.
“The Orrery”, The Universal Magazine of Knowledge and Pleasure, 1749.
Because the formula used by the Julian calendar did not accurately measure a true year (i.e., the time it takes the earth to orbit the sun), in 1582 Pope Gregory XIII adjusted the 365-day Julian calendar to align more closely with the actual time required for the earth’s orbit around the sun. Perhaps for more terrestrial reasons, he also restored the beginning of the New Year to January 1st.
While most Roman Catholic countries adopted this Gregorian or “New Style” calendar, protestant countries which did not recognize the authority of the Pope, such as England, took considerably longer. In consequence, for quite some time two calendars were in use in Europe. Moreover, years until 1752 actually started twice in domains under English rule: the “legal” year began on March 25th, but use of the Gregorian calendar elsewhere and lingering memory of the traditional celebration that had itself been supplanted in the Middle Ages, led to January 1st being regarded as New Year’s Day.
To avoid misunderstanding (if not confusion!), documents created in the period between the new New Year (January 1) and the old New Year (March 25th) were often dated with the both the “Old Style” and “New Style” years – a system known as “double dating.” Such dates are usually identified with a slash mark between the “Old Style” and “New Style” year, as depicted here.
John de Witt to Lewis Morris, January 2, 1733/4. (Rutherfurd Collection, MS 531)
Needless to say, this continues to cause puzzlement among researchers and even more so when authors, such as Benjamin Franklin below, neglected to double date.
Benjamin Franklin to Cadwallader Colden, January 27, 1747/8. (Benjamin Franklin Papers, MS 1348)
Finally, in 1752, an Act of Parliament instituted the Gregorian calendar. However, the discrepancy between the two calendars amounted to 11 calendar days, so the calendar year in the year of adoption went from September 3rd to September 14th – a change reflected in a letter Benjamin Franklin wrote on the very day the “New Style” calendar was adopted (note the “NS.” above the year).
Benjamin Franklin to Cadwallader Colden, September 14, 1752. (Benjamin Franklin Papers, MS 1348)
For more details about the two calendars, you may want to read this slightly more comprehensive explanation of how they work.