The torn and restored plot was described in a trick with the translated title of “The Letter” in R.P's Ein Spiel Karten, 1853, p. 27 of the Pieper translation. The pieces of card in R.P's trick restored all at once. The earliest piece-by-piece restoration appears to be C.F. Waite's “The Triangle Card Trick”, published in The Magic Wand, Vol. 3 No. 27, Nov. 1912, p. 425.
The double-card fold-and-tear method was invented by Charles Jordan, who contributed it, in the context of a Torn and Restored Tissue Paper to Floyd Thayer's Magical Bulletin, Vol. 3 No. 8, Aug. 1915, n.p. The method was later applied to playing cards, as indicated by Karl Fulves in Charles Jordan's Best Card Tricks, 1992, p. 18. The idea of using a pair of pseudo-duplicates like the Nine and Ten of Hearts as a convincing touch in a torn-and-restored-card effect is also Jordan's. See his “Incomprehensible Destroyed Card,” 1919, and included in Fulves's two Jordan compilations, Charles T. Jordan: Collected Tricks, 1975, p. 46; and Charles Jordan's Best Card Tricks, ibid., p. 15.
For Dr. Jacob Daley's handling, see “Impromptu Torn and Restored Card Effect” in The Sphinx, Vol. 37 No. 3, May 1938, p. 61. Lu Brent also had a double-card fold-and-tear handling in Novel Magic, 1932, p.27. While R. W. Hull is frequently credited with the principle, he came late to the game with his handling in More "Eye-Openers", 1933, p. 22. In his description he cites Lu Brent as well as Joe Berg's “Berg's Paper Fold” as the sources for the fold-and-tear principle. All three parties seem unaware of Jordan's 1919 trick. Hull claims only the touch of tearing one corner from the signed selection, so that it can be handed to the person who did the signing, while three other pieces of the second card are handed out to three other people. Those three pieces are then gathered and the restoration effected. Only three-quarters of the card is restored, and the fourth quarter is fitted into place as proof the card is the same one.