Foote was born in Lee, Massachusetts and was educated at the Albany Institute. He studied law with Judge Daniel Cady in Johnstown, New York. After being admitted to the bar, he settled in western New York, and was district attorney and then judge of the court of common pleas of Seneca County, New York. His specialty was patent law, and he made several valuable inventions. In 1864 he was appointed to the board of appeals at the U. S. Patent Office. On July 28, 1868, he was appointed the eleventh Commissioner of Patents. 
Foote was one of the signatories of the 1848 Declaration of Sentiments. He was also the author of several books and papers on mathematics. He died in St. Louis, Missouri on October 29, 1883. On August 12, 1841, he married Eunice Newton (born July 17, 1819).  Eunice was a member of the editorial committee for the 1848 Seneca Falls Convention, an early and influential meeting of the women's rights movement. Elisha and Eunice were the parents of Mary Foote Henderson.
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- Goodwin, Nathaniel, "The Foote family: or, The descendants of Nathaniel Foote, one of the first ... " Hartford, Press of Case, Tiffany and company, 1849. p. 159
- Goodwin, p. 159