Ridgeway was born 6 August 1853, at Ballydermot, King's County, Ireland, the son of Rev. John Henry Ridgeway and Marianne Ridgeway. He was a direct descendant of one of Cromwell's settlers in Ireland. He was educated at Portarlington School and Trinity College, Dublin, after which he studied at Peterhouse, Cambridge then Gonville and Caius College, Cambridge, completing the Classical tripos there in 1880.