Woodhouse was made a fellow of the Royal Society on 16 December 1802. He was appointed Lucasian professor of mathematics in 1820 but this chair provided such a small income that he was happy to resign in 1822 so that he might accept the better paid position as Plumian professor of astronomy and experimental philosophy. He also became the first director of the Cambridge University observatory which had just been newly built. He held this latter professorship, and was head of the Observatory, until his death in 1827. Fellows of a Cambridge College were not allowed to be married so Woodhouse had to resign his fellowship in order that he might marry Harriet Wilken on 20 February 1823. In fact Woodhouse would have married sooner but his income was, until he became Plumian professor, insufficient to support a wife. Harriet, who came from Norwich, was the daughter of an architect. Her brother William Wilkens was even better known as an architect than was their father. Robert and Harriot had one son.
Woodhouse was interested in the theoretical foundations of the calculus, the importance of notation, the nature of imaginary numbers and other similar topics. He wrote an three papers in the Philosophical Transactions of the Royal Society in 1801, 1802 and an important book Principles of Analytic Calculation in 1803 which attempted to put the calculus on a rigorous algebraic foundation using a formal series expansions method similar to that developed by Lagrange :-
He demanded that analysis in general and the calculus specifically be placed upon a purely algebraic footing free of geometric and physical encumbrances such as limits or infinitesimals.In essence Woodhouse was dealing with Taylor series of a function, from which he could directly read off the first, second, third etc. derivatives from the coefficients of the terms of the series without involving any limiting process.
His support of Continental methods was aimed at his fellow professors at Cambridge who were still using Newton's method of fluxions, but it had little effect; people would not readily move away from their traditional approaches. Woodhouse, therefore, failed to have much immediate influence as a reformer in mathematical studies at Cambridge. If the methods of Lagrange, rather than those of Cauchy, had become the accepted methods then Woodhouse would have a much more prominent role in the history of mathematics today. However he did begin the Cambridge move towards Continental style mathematics which was taken further by Herschel, Peacock and Babbage. Peacock, in particular, considered Woodhouse's work to be of major importance but in general people did not appreciate his writing which, one would have to say, was rather difficult to understand and rather uninspiring.
Another attempt by Woodhouse to bring mathematics at Cambridge up-to-date was in 1804 when he published a paper on elliptic integrals in the Philosophical Transactions of the Royal Society. He fully realised the significance of the topic which had received little attention at Cambridge. He wrote several widely used elementary texts which, during his lifetime, brought him more fame. Woodhouse's other works include A Treatise on Plane and Spherical Trigonometry (1809), A Treatise on Isoperimetrical Problems and the Calculus of Variations (1810), Treatise on Astronomy (1812) and a work on gravitation published in 1818.
Article by: J J O'Connor and E F Robertson