It works much the same these days (at least where I live): if a man and woman are married, he is legally presumed to be the father. If parents are unmarried, mom can write whatever she wants on the birth certificate, but it doesn't become legally binding until he either signs a voluntary acknowledgement of paternity, or until court-ordered paternity testing shows him to be the father. (And if a man and woman are married, but some other man is the father and they all want to acknowledge it, the married man also has to sign a relinquishment of paternity.)