After learning surgery from Henry Bracken of Lancaster, he worked as a surgeon in Sedbergh for a year, then went to study medicine at Edinburgh, walking 150 miles there with his savings stitched into his coat.

After a rudimentary education at the Revd Charles Udal's school in Garsdale, Dawson worked until he was about twenty as a shepherd on his father's freehold, developing an interest in mathematics in his spare time with the aid of books that he bought with the profits from stocking knitting or borrowed from his elder brother, who had become an excise officer.

His stay in the capital was brief, but he gained experience in the London hospitals, attended surgical and medical lectures, and made a contact, with Edward Waring, the Lucasian professor of mathematics at Cambridge, that was to be important for his future work as a mathematician.

For a fee of about 5 shillings a week for unlimited tuition, in addition to the cost of accommodation and food, sometimes in Dawson's house but more commonly in a local inn, pupils were taught in a characteristic peripatetic fashion.

Pupils who went on to Cambridge and did not achieve the rank of senior wrangler included, in addition to Richard and Adam Sedgwick and Haygarth, the lord chief justice Sir Nicholas Conyngham Tindal, the mathematician Miles Bland, who was at Sedbergh School, and several bishops.

An anonymous correspondent writing from Trinity College, Cambridge, in the European Magazine urged the university to recognize his status as the first mathematician of England by awarding him an honorary degree.

Described by Adam Sedgwick as ‘a firm believer and a good sober practical Christian of the old school’, Dawson abhorred the doctrines of David Hume and applauded James Beattie's attack on Humean scepticism.

His 24-page pamphlet outlining his views on the damaging consequences and unsure foundations of an acceptance of determinism, The Doctrine of Philosophical Necessity Briefly Invalidated (1781), elicited a dismissive, unsigned rejoinder in the Monthly Review (65, 1781, pp.

The original of one of the portraits, painted by Joseph Allen in 1809 and showing Dawson teaching a seated pupil, had already been lost by the mid-nineteenth century, but it survived in the form of a copy by the vicar of Sedbergh, the Revd D. M. Peacock, and an engraving by W. W. Barney.

He died, on 19 September 1820, and a monument high in the nave of St Andrew's Church in Sedbergh was erected, in the form of a bust of him by Robert William Sievier, with an inscription, dated August 1825, by his former pupil John Bell.