Eric was the second son of Sir Alfred Pearce Gould, KCVO, and his wife, Florence. He was born in London on 6 May 1887. His father was an eminent surgeon and the family lived at 10 Queen Anne Street, Marylebone
He was educated at Charterhouse from 1899 until 1904 and came up to Christ Church with a Scholarship in Natural Science. He took First Class Honours. He trained at the Middlesex Hospital where he was awarded the Senior Broderip Scholarship and the Lyell Medal. He qualified M.B, B.Ch.in 1911. He became a Fellow of the Royal College of Surgeons in 1912 and proceeded to M.Ch. in 1913, the year in which he won a Radcliffe Travelling Fellowship. This enabled him to study surgery in the United States and pathology in Berlin. He gained his M.D. in 1918.
In the First World War, he served as a temporary surgeon in Gibraltar from 1914 until 1917 when he was invalided out. He suffered from severe asthma and, frequently, needed injections of adrenaline. Returning to the Middlesex in 1920, he was Dean of the Medical School from 1925 until 1929. He was President of the Medical Defence Union and a member of the Court of Examiners of the Royal College of Surgeons.
In 1916, he married Audrey Mitchell Jackson at Eastry in Kent.
At the Middlesex, he followed in the footsteps of his father, occupying the same wards and, for a period, having the co-operation of the same ward sister. He was said to have had a great ability to impart knowledge, inherited from his father, as well as oratorical gifts which made him one of the best medical speakers of his generation.
He was called up at the outbreak of the war and died in the Royal Naval Hospital, Devonport on 1 August 1940.
He is buried in Ford Park Cemetery, Plymouth: Church section T.9.30.
He left £39,717-15-4.
His youngest brother, Alfred Leslie, also a doctor, was killed on 19 May 1918 and is commemorated on the Christ Church Memorial.