Charles Babbage (1791-1871) is widely recognised as the great ancestral figure in the history of computing. He designed the first automatic calculating engines and failed to realise a complete design in physical form. This thesis argues that historical accounts of Babbage's work are based on a set of contemporary sources provided by Babbage himself and that the subsequent interpretation of the major movement to automate calculation and tabulation in the nineteenth century is dominated by Babbage's own account of events. George Bidden Airy (1801-1891), Astronomer Royal for forty five years, consistently rejected arguments advocating the utility of the engines. Airy had a defining influence on the fate of the engines yet his views barely feature in the canon. Using new archival sources, Airy's views are explored through a series of case studies: Babbage's Difference Engine No. 1 and the Swedish calculating engine used at the General Register Office for the 1864 Life Table. In addition, his views on the use of calculating aids are explored using instances in which he was petitioned by inventors, specifically by Thomas Fowler and William Bell. The thesis situates Airy's views in the context of manual methods used in the production of printed mathematical tables - techniques that automatic calculating engines were intended to replace - and in the context of contemporary expectations of the largely unbuilt engines. The treatment includes new work on the mathematical implications of the engines, specifically Babbage's speculations on computation as a systematic method of solution, and presents a revisionist view of tabular errors as the primary motive for the development of the engines.