That Hinshaw who was apparently a partner with Nathan Thomas Beals was Amos S. Hinshaw (1839-1930), son of Stephen Hinshaw (and, although we don't know with certainty what his full middle name was, one could guess that it was probably Stephen):
Incidentally, I did perform a relationship search through my Rootsmagic database, and found that Nathan Thomas BEALS and Amos S. HINSHAW were indeed first cousins, once removed - through Charles and Hannah (PIGGOTT) DAVIS.Amos S. Hinshaw indeed did live in Westfield, and was a member of Westfield Monthly Meeting. And he was shown in the 1900 census with occupation "carriage builder", and in 1910 as "mechanic, carriage shop".
As to his relationship to Nathan Thomas Beals: inconsequential. Westfield was a small town in 1893, and there was a tight and close Quaker community - Amos and Nathan might have come together in partnership for reasons other than any genealogical relationship. However, probably coincidentally, Amos was 3rd cousin once removed to Solomon Hinshaw, who married Nathan's sister Elizabeth.
Regards,Jan HinshawThe Hinshaw Family Association
P.S.: I hope you don't mind, but I liked the "Beals & Hinshaw" scan you referenced, so I copied the image and added it to Amos Hinshaw's page (credited to you of course).