In this paper the Lucasian Professor of Applied Mathematics in the University of Cambridge studies the role of ocean science in the service of mankind, and examines the interaction between the many disciplines which make up that science. The paper, which was written in May 1972 (and a version of which was published in the Bulletin of the Institute of Mathematics and its Applications for February 1973), was presented at a conference held in Greenwich on 12–14 September 1973 to celebrate the hundredth anniversary of the Royal Naval College. (Note: the National Institute of Oceanography referred to in the text is now known as the Institute of Oceanographic Sciences.)