The Maritime Museum of San Diego is reconstructing the San Salvador, lead ship of the fleet of Juan Cabrillo, who arrived in San Diego in 1542, long before Jamestown! Apparently it's a more demanding task than expected, but I expect much progress before I next see it, when I return to California this summer. Sebastian Münster's trippy 1552 map "Die Nüw Welt" (no wonder Japan feels so close!) is part of their display.
