The reception by British and Irish mathematicians of late-eighteenth-century French mathematical analysis, and its applications to astronomy, is here re-examined. The main early British participants were John Robison, John Playfair, Robert Woodhouse, John Toplis, John Brinkley, John West, William Spence, John Herapath, James Ivory, William Wallace, and Mary Somerville. Their activities and publications, and those of some others, are outlined. The reviews of John Playfair and Robert Woodhouse, many little-known, are highlighted. Finally, we discuss why, despite the work of these individuals, reform of British mathematics was at first so slow ; and why, in contrast, the eventual modernisation of the Cambridge Mathematical Tripos examinations, following the efforts of Charles Babbage, John Herschel, George Peacock and others, led to rapid improvement.