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This trick is most often credited to the British professional, Oswald Williams, and that is fine, as far as it goes. But the trick is also frequently credited to, or said to have been popularized by, Oswald Rae, another British pro, twelve years the junior of Mr. Williams. That attribution is also correct in general. Here is why.
Probably sometime in the late 1920s, Oswald Williams originated a trick called “Watch Your Watch”. While recounting a story of having his wristwatch stolen on the street, the magician removes his watch from his arm, makes it disappear in his (Devil's) handkerchief and shows it back on his wrist (duplicate). This trick was an item much commented on in Williams's repertoire.
Oswald Rae built on the patter premise and watch trick, adding to the robber's booty Rae's finger ring and a roll of bills from his pocket. All three items disappeared from his handkerchief and reappeared in their original locations: wrist, finger and vest pocket. When Rae came to Canada and the U.S. to perform at both annual conventions for the SAM and IBM in May-June of 1930, this trick was in his performances and quickly became associated with him in U.S. publications. (An effect description of a 1930 performance can be read in Max Holden's Programmes of Famous Magicians, 1937, p. 39.) The holdup presentation and watch effect were Oswald Williams's, and it is true that Rae popularized the trick; but Rae also added the simultaneous finger-ring and bill vanishes and reproductions, which is the form in which the trick has endured.
Decades later, Tommy Wonder, Alan Shaxon, Rich Marotta and Jim Steinmeyer developed, published and marketed notable versions of the Holdup Trick.
Sid Lorraine suggested that Oswald Williams was probably inspired in his creation of “Watch Your Watch” by Wright and Larsen's “The Ghost Watch” in The Sphinx, Vol. 26 No. 5, Jul. 1927, p. 167. The effects do have strong points of resemblance, particularly in their use of a holdup presentation. The Wright and Larsen trick uses a pocket watch, which disappears from a paper cone and reappears back on its watch chain.