I've responded separately to Maryanne, but here's the little bit I know for anyone else who might be interested:
FIRST GENERATION: Charles & Catherine Dux, born Hesse-Darmstadt (probably) circa 1822. Emigrated to NYC circa 1855. Lived on West 39th and West 50th streets, perhaps others. Lived at 457 W. 39th Street during a time period when the Beckers (Joseph and Anna Dux Becker, described in my earlier message) lived across the street at 438 W. 39th. The first few children listed below seem to have been born in H.D. before they emigrated to New York.
SECOND GENERATION: Anna (this is pure conjecture), b. ca. 1840. Jacob [see 1 below], b. ca. 1851. August [see 2 below], b. 1854. Joseph, b. ca. 1856. [no further information] Philip [see 3 below], b. 1859.
(1) Jacob Dux, born ca 1851. A printer. M'd Emma (b. ca. 1853, California). Children inc. Lena, August [see 4 below], George & David. In 1880, lived at 220 W. 40th Street, NYC.
(2) August Dux, born 1854 in Hesse-Darmstadt. A butcher. M'd Barbara [Kay] ca. 1878. Children inc. Regina (1879), Emma (1881), Philip G. (1886), and August (1892). In 1900, lived on 4th Avenue in the Bronx.
(3) Philip Dux, born 1859. M'd Mary ca. 1886. Proprietor of a printing shop, and living at 768 Ninth Avenue, in 1910. Children unknown.
THIRD GENERATION: (4) August A. Dux, b. 1877, son of Jacob Dux. M'd Gertrude Kuhn. In 1900 census, listed as a carpenter living on 153rd or 158th street in New York.