You may be right, Mack, when speaking of paid companies. But Engine 1 and Engine 2 were formed in 1731, when the first fire apparatus in the United States were purchased. Two hand-drawn pumpers were purchased. All citizens were required to respond whenever there was a fire, and they operated under the supervision of the Aldermen.
The city's first firehouse was built in 1736 in front of City Hall on Broad Street. A year later, on December 16, 1737, the colony's General Assembly created the Volunteer Fire Department of the City of New York, appointing 30 men who would remain on call in exchange for exemption from jury and militia duty. The city's first official firemen were required to be "able, discreet, and sober men who shall be known as Firemen of the City of New York, to be ready for service by night and by day and be diligent, industrious and vigilant." They used the two pumpers mentioned above.
Although the 1737 Act created the basis of the fire department, the actual legal entity was incorporated in the State of New York on March 20, 1798 under the name of "Fire Department, City of New York."
Source: Heroes of Ground Zero. FDNY A History". Public Broadcasting Service, May 3, 2015.