Was Hawthorne Beach actually in New York? Greenwich,Ct. sits on New York border and Rye almost sits on the border. The below is an excerpt from a publication from the New York Central Railroad in 1897. I have seen in an old geographical directory a reference to Hawthorne Beach in Greenwich,Ct. 1890's. I have only listed it here because it showed in a listing of 'old amusement parks'.

HAWTHORN BEACH. RYE BEACH. MILTON POINT.
(from an old Timetable of the New York Central Railroad)
About six miles distant, on the Sound Shore, are Hawthorn Beach, Rye Beach, Milton Point, and other attractive resorts that afford excellent fishing and bathing. All of these points are reached by one or another of the beautiful drives for which Westchester County is noted. Proceeding northward from White Plains a charming landscape reveals itself in varied manifestations of wondrous beauty. Rare combinations of mountain, stream and foliage greet the eye in endless variety, the whole forming a panorama of rural scenery incomparable for beauty, picturesqueness and variety. A run of three miles brings us to KENSICO.