In the 1880 Federal Census, this family is living on Broad St., District 50, in Providence, RI.
William's family Bible is in the possession of one of his descendants, and copies of some of the family pages were given to me by Joyce (Remington) Gallup. On one page is written: "The Property of William R. Remington presented to him by his Mother Sarah Remington as a New Years gift Jan 1851." He would have been 23 years old.
He may have been the William R. Remington who was a Justice of the Peace at Warwick. (source: Acts and Resolves of the General Assembly of the State of Rhode Island and Providence Plantations Passed at the May Session, 1858)