Woodhouse attended school in North Walsham, 1790 at Caius College, Cambridge. Late period, he became a professor of mathematics, astronomy and experimental philosophy. In 1802 he became a member of the Royal Society. From 1820 to 1822 he was professor of mathematics at Cambridge Lucasischer.
Woodhouse was interested in the theoretical foundations of the calculus, the notation of the differential calculus, the nature of imaginary numbers and related topics. He wrote several works on calculus and reached there the " Continental " mathematics.