The name of the wealthy family was Bronck (sometimes spelled Bronk).
The clan's patriarch, Jonas, settled on 500 acres north of the Harlem River in 1639 and promptly affixed his surname to
various features of the local geography, notably the Bronx River.

As one of his descendants explained, "The termination of `x' merely indicates the possessive case. Instead of writing Bronk's River or Bronk's farm, the Dutch took the phonetic short cut and made `x' do duty for the fusion of `k' and `s'; extremely simple, and a space saver too. Thus, when Jonas impressed his own family nomenclature on the region he settled, the Aquahung River became Bronk's River--the Bronx, as it remains today, correctly expressed in Dutch."

As far as Cecil can tell, the name "the Bronx" didn't signify the entire area now known as the Bronx until late in the 19th century.

In 1874 about 20 square miles of mainland Westchester county was annexed to New York City. This region was known thereafter as the Annexed District of the Bronx, in apparent reference to the Bronx River, then the district's eastern border.

In 1898 the Annexed District became part of the Borough of the Bronx--presumably still referring to the river. After a while, however, people forgot about the river and began casually referring to the entire borough as "the Bronx." The use of "the," in other words, is simply a historical accident.

Hope this helps John