The son of a weaver from Market Bosworth in Leicestershire, Simpson was largely self-educated. His mathematical interests were aroused when a peddler gave him a copy of the popular textbook, Cocker's Arithmetic. He left home early, for by 1724 he was reported to be in nearby Nuneaton, practicing as an astrologer. By 1735 he had arrived in London where he worked initially as a weaver but also as a part-time teacher of mathematics. He soon became well known through a series of popular textbooks among which were A New Treatise on Fluxions (1737) and his Treatise of Algebra (1745). In 1743 he was appointed to the Royal Military Academy at Woolwich. He also served (1754–60) as editor of the Ladies Diary – a journal that sought to interest the “fair sex” in “Mathematicks and Philosophical Knowledge.”

In mathematics he is best known for his formulation in 1743 of what has since been known as Simpson's rule, allowing the area under a curve to be approximated by using parabolic arcs.