Gorham's early education was obtained at Exeter, New Hampshire, and his academic training at Harvard, from which he received the degrees of B. A. in 1801, M. B. in 1804, and M. D. in 1811.
Soon after gaining his first medical degree, he went to London and took private lessons in experimental chemistry with Friedrich Accum, who at that time was the most noted teacher of chemical manipulation in Europe.
Later, he studied chemistry with Thomas Hope at the University of Edinburgh.
In London, he became acquainted with Benjamin Silliman. Each man on his return to the United States taught chemistry with conspicuous success, Gorham at Harvard and Silliman at Yale. Gorham continued his medical studies at Harvard, one of his teachers being Dr. John Warren, whose daughter, Mary, he married.
In 1809, Gorham was appointed an adjunct professor of chemistry and materia medica, and when Aaron Dexter resigned as Erving Professor of Chemistry and Mineralogy in 1816, Gorham succeeded him.
The latter appointment must have been regarded as a significant scientific event because Gorham delivered a formal inaugural address, and soon afterward John Adams, Ex-President of the United States, wrote a long letter to Gorham setting forth in grandiloquent language the opportunities for chemists to make discoveries which would benefit humanity.
Realizing the necessity for an adequate textbook adapted to the needs of college students, Gorham wrote The Elements of Chemical Science, dedicated to Aaron Dexter, his teacher and former colleague. It was one of the first systematic text-books on chemistry written by an American and published in this country and was a standard work for many years.
Although Gorham’s regular duties as a teacher and a physician were onerous, he found time to prepare and publish several original papers in chemistry, the more important of which are: “Analysis of Sulphate of Barytes from Hatfield, Mass”, “Indigogene”, “Chemical Examination of a Quantity of Sugar Supposed to Have Been Intentionally Poisoned”, and “Chemical Analysis of Indian Corn”.
In 1814 - 18, Gorham served as a librarian in the Massachusetts Medical Society and in 1823-26, as recording secretary.
His work as Erving Professor of Chemistry ceased in 1827 when he was succeeded by John W. Webster.
Gorham was a Fellow of the American Academy of Arts and Sciences and a member of the Massachusetts Medical Society.
Gorham was an accomplished teacher and was especially helpful to his students, often assisting them personally in their studies.
On June 2, 1808, Gorham married Mary Warren.