Manhattan in 1628 by Dingman Versteeg, was published in 1904 (New York: Dodd Mead & Co.) and is available in digital format, free, at Google Books. The focus of the book is a letter, written from Manhattan Island in 1628, by Jonas Michaelius. Versteeg provides a transcript, a translation, and then his own historical sketch of Manhattan in 1628. My purposes for pointing it out is to bring attention to pages 29 to 33, where Versteeg's subject is persons with the surname Brouwer. Pages 29-33 (as wells as 133-134) have been segregated out and are available as a PDF online here. It is my belief that Manhattan in 1628 is the original source of much of the misinformation regarding Adam Brouwer's ancestry that has circulated for probably 100 years now, and can still be found online. The fact is, Adam Brouwer's ancestry, and even the identity of his parents, remains unknown.
On pages 29 through 33, you will find the names of various persons named Brouwer. They are (in order of appearance) Jan Jansz Brouwer; Jan Albertsz Brouwer, and his sons, Dirck Jansz Brouwer and Cornelis Jansz Brouwer; Jan, a son of the just mentioned Dirck Jansz Brouwer; Adam Brouwer Berkhoven (Versteeg associates Berkhoven with Berkhout); Jan Gerritsen Brouwer; Jan Brouwer, with a son, Jan, baptized at New Amsterdam; and then on page 134, Pieter Clementssen Brouwer. It is from this cast of completely unrelated individuals, who simply share a common surname, that many 20th century "researchers" (and I use the term loosely) found "evidence" to cobble together various ancestries for Adam Brouwer. Versteeg does caution, "It would be vain, however, unless additional evidence should come to light, to try to connect the descendants of Adam Brouwer with the Captain-Councillor of New Netherland..." (bottom page 32 to top of page 33). And although Versteeg does end with, "...though the two may have been distantly related," anyone reading this volume would be well advised to take Versteeg's cautionary words seriously.
Those interested in more, see "New Insight into the Origins of Adam Brouwer" (2008).