NOTICE--Application will be made to the Legislature of New York, at the next session, for a charter to construct a Railroad from Brooklyn to Coney Island, to be called the "Brooklyn, Fort Hamilton and Bath Railroad Company." Brooklyn, November 28, 1845. n28 1aw3*
--Brooklyn Eagle, Friday, November 28, 1845
Interestingly, in 1836, an Act was passed to provide for this construction.
But these documents from 1913 state "No record of any construction. In the state engineer's report for 1882 the Company is described as extinct."
In 1845, you got to Coney by stage, cart, or steamboat.
NOTICE is hereby given that an application will be made at the next session of the Legislature of the State, for the incorporation of a company to be called the New York, Gowanus, Fort Hamilton and Coney Island Ferry Company, with a capital of one hundred thousand dollars. New York, Dec. 19, 1845. ja13 1aw6w*(and intermittently through March 2, 1846).
--Brooklyn Eagle, Tuesday, January 13, 1846
It appears to have passed the Legislature about Wednesday, March 11, 1846.