Mathematics is relentlessly seeping into the very foundations of the civilisation in which we live and the community of which we are part, but I wonder how many of my readers—teachers of mathematics in grammar schools or in the universities—realise the magnitude and understand the nature of this seepage. It is permeating commerce, sociology, economics, medicine, linguistics, …, as well as more traditional science and technology. In this article I shall confine myself mainly to engineering, although I believe that the principles I shall develop will apply generally to any ancillary mathematics.