W. H. J. Shaw seems to have been the first to publish a version of this effect in English, which he titled “The Flight Through Crystals”; see his Magic Up to Date, 1896, p. 19. In the same year, Carl Willmann published his handling in the German magazine Die Zauberwelt, Vol. 2 No. 6, June 1896, p. 88, “Das fliegende Tuch”.
Roughly a year later, Ellis Stanyon described the trick in Conjuring for Amateurs, 1897, p. 42. Stanyon states that “This is a very surprising trick, and a favourite with the most noted prestigitateurs.” One of those prestigitateurs was certainly David Devant, the actual details of whose method were eventually published in Prof. Hoffmann's Later Magic, 1904, p. 303, where Hoffmann declares the trick Devant's invention. Hoffmann's description provides details that Shaw and Stanyon lacked, although they got the basics right: a switch of a set of knotted silks, and a vanish of a red silk inside a clear glass tube, effected by a Pull. Peter Warlock stood behind the Devant attribution; see his column in The Stage, as referenced in Hugard's Magic Monthly, Vol. 11 No. 10, Mar. 1954, p. 117.
Two years before Devant's method was published, C. Lang Neil included Frank Kennard's “The Mysteriously-Joined Handkerchiefs” in Modern Conjurer, 1902, p. 231. Kennard's method avoided switching the handkerchiefs by concealing one inside the hem of another. Frank Ducrot marketed essentially the same concept (a pocket-silk) as “The 20th Century Handkerchief Trick,” c. 1903; see the advertisement in Mahatma, Vol. 7 No. 7, Jan. 1904, p. 80, which is presumably Ducrot's version.
This should not be confused with Hardin Burlingame's unrelated marketed trick, “U-Need-A Twentieth Century Handkerchief,” 1899; or Ellis Stanyon's marketed “20th Century Handkerchief Production,” 1903.