Born on 5 February 1817, he was the eldest son of John Augustus Thrupp (1785–1844) of Spanish Place, Manchester Square, London, himself the eldest son of Joseph Thrupp of Paddington Green, by his first wife, Mary Burgon.
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After his father died and left him money, Thrupp spent time on archæology and chess, through which he knew Henry Thomas Buckle.
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By his first wife, Mary Burgon, Joseph was father of Dorothea Ann Thrupp the hymn-writer, and of John Augustus Thrupp (1785–1814), the father of John Thrupp the historian; also of Charles Joseph Thrupp, the father of Admiral Arthur Thomas Thrupp.