Emerson was born at Hurworth where his father Dudley was a schoolmaster. Educated at Newcastle and York, William was unable to follow his father's profession, being of too 'warm' a temper, and so devoted the rest of his life to studious retirement. He published a great many mathematical works including The Doctrine of Fluxions (1748), and took the utmost care when advancing propositions or publishing inventions, for most of which he provided working models. In The Principles of Mechanics (1754) he shows a wind-powered vehicle in which the vertically-mounted propeller provides direct power to the front wheels via a system of cogs. A Treatise of Navigation followed in 1755 and The Elements of Optics in four books (1768). Emerson's Newtonian analysis was controverted by the great surgeon-teacher John Dawson at Sedbergh.
Emerson lived and died in Hurworth and is buried in the churchyard, where his tomb bears epitaphs in Latin and Hebrew. Emerson has been described as eccentric, even clownish, but at the same time possessing great intellectual energy and independence of mind. His clothes were old and his manner extremely uncouth. He wore his shirt back to front, all his coat-buttons undone and his legs wrapped in sacking so as not to scorch them as he sat over the fire. He declined an offer to become FRS because it would cost too much after all the expense of farthing candles he had been put to in the course of his life of study. Emerson rode regularly into Darlington on a horse like Don Quixote's, led by a hired small boy. In old age, plagued by the stone, he would alternately pray and curse, wishing his soul 'could shake off the rags of mortality without such a clitter-me-clatter.'