What song does Amaryllis sing in The Music Man?

The first song Amaryllis plays is source music: the incomplete first phrase of "Rustic Dance" by C.R. Howell. I have found vintage sheet music published in dates ranging from 1924 to 1944, plus it is in many standard piano lesson books (including the one I learned from as a kid in the 1970s.) I found a listing for undated vintage sheet music whose seller suggested it was circa 1890. C.R. Howell was born in 1850, so given shorter life spans of the time, it would be likely that the song was written before the 1912 setting of the show Music Man. The second song Amaryllis plays, the "cross-hand piece," was composed by Meredith Willson for the show, and was designed to be a partner song with "Goodnight My Someone."