The title of my lecture is rather misleading. When I was asked to deliver it I thought I could profitably indicate how very definitely mathematics is used in a particular activity, such as gunnery, the application being neither complicated nor varied. If one was asked to write a paper on the utility of mathematics in engineering or in wireless electricity, the problem would be much more diffuse. The man who makes or fires a gun is only, after all, making one particular machine or doing one particular job. Hence it is possible briefly to state what mathematical difficulties he encounters.