The earliest known description of the illusion of having a ring tumble ring by ring down a linked chain seems to be J. T. Ferguson's “The Chinese Rings: A Novelty in Working” in The Magic Wand, Vol. 7 No. 3, Nov. 1916, p. 47. Ferguson describes the sequence done with the chain held horizontally, rather than in the usual vertical position, but the mechanics are the same. It is possible he is claiming credit for the management of the sequence in this position, but since he makes no mention of a preceding version and no earlier description of the tumbling ring illusion has been discovered, this seems the first record of it, and J. T. Ferguson its inventor.
While Ferguson may have discovered the principle of the tumbling ring illusion independently, there were two toys that might have aided that discovery: the tumbling ring and Jacob's Ladder blocks. While some sources claim these novelties are ancient, published references and descriptions of them seem to exist no earlier than the last few decades of the nineteenth century. The tumbling ring puzzle is done with an even number of rings joined in an interlinked double chain. The tumbling ring illusion described by Ferguson seems to be a simplification of the double-chain toy. The earliest description of the toy so far discovered is in the November 1888 issue of La Nature, p. 584. The author implies that the toy first appeared for sale on the Champ de Mar near the site where the Eiffel Tower was being built (1887-8). The story of the toy was immediately translated for the English Mechanic and World of Science, Nov. 23, 1888, p. 251, and The Popular Science News, Jan. 1889, Vol. 23 No. 1, p. 1. These sources imply that the toy was invented in France around 1888 or shortly before. However, W. W. Durbin recalled buying a set of the tumbling rings in a New York magic shop in 1879 (see Werner C. Dornfield's Trix and Chatter, 1921, p. 224). Assuming his recollection of the date was correct, this would place the toy roughly nine years earlier in the U.S. Further documentation is needed to support this.
The Jacob's Ladder block novelty is made of flat wooden blocks strung together by a system of parallel ribbons. A clear description of it can be found in Magic: Stage Illusions and Scientific Diversions Including Trick Photography by Albert A. Hopkins, 1897, p. 399; and a responsible history of the toy and the piece of conjuring apparatus on which it is based is given in “The Yawning Mouth” by Volker Huber, Gibecière, Vol. 1 No. 1, Winter 2005, p. 37.