William Horner was educated at Kingswood School Bristol. At the almost unbelievable age of 14 he became an assistant master at Kingswood school in 1800 and headmaster 4 years later. He left Bristol and founded his own school in 1809 in Bath.
Horner's only significant contribution to mathematics was Horner's method for solving algebraic equations. It was submitted to the Royal Society on 1 July 1819 and was published in the same year in the Philosophical Transactions of the Royal Society.
Some years earlier Ruffini had described a similar method which had won him the gold medal offered by the Italian Mathematical Society for Science who had asked for improved methods for numerical solutions to equations. However neither Ruffini nor Horner was the first to discover this method as it was known to Chu Shih-Chieh 500 years earlier.
In the 19th and early 20th centuries Horner's method has had a prominent place in English and American textbooks on algebra. It is not unreasonable to ask why that should be. The answer lies simply with De Morgan who gave Horner's name and method wide coverage in many articles which he wrote.
After Horner died in 1837 his son, also called William, carried on running the school in Bath.