Martin Gardner was very fond of vanishing area puzzles and devoted two chapters to them in his first book. There are actually two distinct types. Sam Loyd's Vanishing Chinaman and similar puzzles have pictures which are reassembled so a part of the picture appears to disappear. But the physical area remains fixed. The second type cuts up an area and reassembles it to produce more or less area, as in the classic chessboard dissection which converts the 8×8 square into a 5×13 rectangle. Gardner had managed to trace such puzzles back to Hooper in 1774. In 1989, I was visiting Leipzig and reading Schwenter which referred to an error of Serlio, in his book of 1535. Serlio hadn't realised that his dissection and reassembly gained area, but it is clear and this seems to be the origin of the idea. I will describe the history and some other versions of the idea.