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The name Jan Thomaszen was a patronomic name, meaning he was the son of Thomas. I don't think that he actually used Eckerson, as his name has also been found as Jan Eckens, and several other variations, but his children adopted the name Eckerson out of the need to establish surnames. According to most records that I have seen, Jan Thomaszen's father was called Thomas Hugesson-from Hugh probably.
Johannes de la Montagne was the first physician in New Amsterdam, he was also appointed a memeber of the council under Governor Kieft and Governor Stuyvesant in New Amsterdam. According to what I have read, some of the old records were in his hand writing, and maybe that is why some of his descendants are translating some of the old Dutch records. I for one am very much in their debt, as they have done an outstanding job. Incidently, a grandson of Johannes de la Montagne married a daughter of Jan Thomaszen, and according to the records of Domine Henricus Selyns made in 1686, Jan Thomaszen and Johannes were neighbors in New York.