This one-step sequence appears in the second edition of F. Leslie Clendenen's 1914 compilation, Dance Mad. It is titled simply "One Step" with the attribution "by E. Clarence Rothard". Rothard is listed in the thank-you credits in the back of the second edition, but his location is not given and a cursory search has not turned up any specific information about him. An "Edna Rothard Passapae" was teaching dance in Newark, New Jersey, by at least 1918 and might have been the daughter of E. Clarence, since it was not unusual for daughters to follow their parents as dance teachers. But even if she was, that doesn't necessarily mean that E. Clarence Rothard was also based in New Jersey.
Rothard's one-step is quite a lengthy sequence. There are eight separate sections of the dance, some of which are repeated. The overall pattern of the dance segments is ABCCDEFGHDH, each letter representing four bars of 2/4 music (eight beats) for a 44-bar (88-beat) dance.
The specified tune is "Hungarian Rag" (left; click to enlarge), a popular 1913 composition by German-American composer Julius Lenzberg adapted from Liszt's Hungarian Rhapsodies. The piece is still well-known; period recordings of it may be found here and here, and a modern version here. The sheet music is available online at the Charles Templeton Sheet Music Collection of Mississippi State University.
The sheet music is notated as 16b (repeated), 16b (repeated), 20b, 16b, and 16b. How exactly this is supposed match up with a 44b dance is an interesting question, but since the dance is broken into four-bar units, one can make it fit by adding another repeat of one of the 16b strains for a total of 132 bars, which allows for three times through the dance.