I looked for Swale and his book in all the usual places, such as Google and public library databases (including the Boston Public Library’s), but couldn’t find a trace of either. Since every good genealogist knows not everything is online or online and publicly available, I contacted the BPL’s general reference department. Within a few minutes the librarian obtained Swale’s first name and the correct title.
The caption contained an error: John Henry Swale (1775-1837) wrote Geometrical Amusements in the early 19th century. By searching his name in Google Books, I found his book and several brief biographies, including an introduction to a volume written by T.T. Wilkinson, An Account of the Life and Writing of John Henry Swale (1858).
Wellington’s photo is a copy of an early 1800s sketch of Swale placed in a daguerreotype case from the 1850s or early 1860s—long after Swale’s death. It’s a curious mystery. Obviously someone in the family thought highly enough of Swale to have the copy made and placed in a case.
The only ways for Wellington to figure out if Swale is related to her is to either trace her own ancestry or look for his descendants. I’d start by trying to find Swale’s family information in Wilkinson’s book and by searching databases such as Ancestry.com and FamilySearch.org.
In the 1825 Directory of Lancaster (available on Ancestry.com), Swale appears as a professor of mathematics living at 12 Epworth St., Liverpool. These details give Wellington a few facts to start her search.