The optical illusion of concealing a break in a ring by gliding the fingers around its circumference while appearing to turn the ring to display its entirety can be found in The Art of Legerdemain Discovered by Hocus Pocus Junior, 1634, “How to seeme to put a ring through ones cheek”.
The same optical illusion is used to conceal the gap in a key ring in the Linking Rings; see the Tarbell System Incorporated Magic, Lesson 47, 1926, p. 9. David P. Abbott reported having learned this technique from Han Ping Chien; see Abbott's Book of Mysteries, 1977, p. 85. Whether it was Chien's invention or simply a technique used in China is not clear.
In “Sandsational Rope Trick”, marketed as a booklet in 1949, George Sands applied the idea to rope magic, to make a length of rope appear to be a continuous loop without ends.