Voyagers to the New World.
William Morrow: 1979.
Here is a book that should be updated every decade or two. This is a thorough review of the main theories of precolumbian contacts between the Americas and the rest of the world.
Davies covers everything from the colonists’ theories, to the Mormons, to scientific hypotheses, and even UFO contacts. His most interesting writing involves the very earliest ideas.
By reviewing the scientific literature on plants, arts, histories, ships, etc., the author is pretty convincing that there was probably scant contact between the Old and New worlds, and if there were, such contact had little, if any, impact on American culture. Not entirely convincing, and I personally suspect there may have been more contact than Davies thinks.
The author’s final chapter deals with alternative theories on the striking similarities between cultures: these are cognitive, archetypal, and dream/hallucinogenic theories which, he urges, need further research.