Short answers, pulled from the story.
James Fox (fl. 1780-1830) was a British machine tool maker based in Derby. He became celebrated for his lathes, lace machinery, and is credited with inventing a screw-cutting machine, a wheel-teeth cutting engine, and a self-acting lathe, as well as reportedly making one of the first planing machines around 1814.
James Fox began as a butler in the service of the Reverend Thomas Gisborne at Foxhall Lodge in Staffordshire. Gisborne recognised Fox's interest in handicraft and enabled him to set up in business on his own account.
Fox supplied lace machinery in large quantities to the town of Nottingham and obtained considerable work from Arkwright and Strutt, the firms regarded as founders of the modern cotton manufacture. He also exported machinery to France, Russia, and Mauritius.
Fox reportedly made one of the first planing machines around 1814, but priority among Fox, Matthew Murray of Leeds, and Richard Roberts of Manchester has not been established. The secrecy maintained by all three makers has prevented historians from resolving the question.
An example of a Fox lathe is on display at Wortley Top Forge in South Yorkshire. Additional Fox machine tools survive as museum pieces in Birmingham and in Norway.
Fox's business at Derby was carried on by his sons and remained active into the final third of the 19th century. The later history of the firm is not known.