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He was educated at Charterhouse School and the University of Cambridge. In 1843, at the age of 16, he joined the bank of Deacon, Williams and Co. He then in 1845 joined Dawson Turner Turner and Gurney in Yarmouth, the banking firm of his grandfather Dawson Turner. He was also a director of Barclay and Co. and a clerk of the House of Commons.
In 1877 he became financial editor of The Economist and became editor-in-chief on the death of Walter Bagehot the same year, a position he held until 1883. He produced the three volume Palgrave's Dictionary of Political Economy (1894, 1896 and 1899) and also edited the collected historical works of his father, Sir Francis Palgrave. He also edited The Banking Almanac until his death, and for a time was editor of The Bankers’ Magazine.
He was the son of Francis Palgrave and his wife Elizabeth Turner, daughter of the banker Dawson Turner. His brothers were Francis Turner Palgrave, William Gifford Palgrave and Sir Reginald Palgrave. He married in 1859 Sarah Maria Brightwen, daughter of George Brightwen of Saffron Walden.
- Murray Milgate, 1987 . "Palgrave, Robert Harry Inglis," The New Palgrave: A Dictionary of Economics, v. 3, pp. 789–91.
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