Elias AlIen (c.1588-1653) was known as the best mathematical instrument maker of his day. He lived and worked in London, creating a thriving business - he was the first English instrument maker to support himself solely through the production of instruments - and teaching his skills to many apprentices who became the core of the trade during the latter part of the century. My thesis provides a full biography of Allen, set within the framework of the community of people who were in some way connected to mathematics. The second part of the dissertation is devoted to the most important of Allen's instruments: the Gunter quadrant aqd Gilnter sector, and various of William Oughtred's designs - the circles of proportion; the horizontal instrument and double horizontal dial, and the universal equinoctial ring dial. These are described, their uses explained, and used as the base for discussions of the ways in which instruments influenced and were influenced by the development of mathematics. This section concludes with a catalogue of all the Allen instruments in British museums. As well as a comprehensive literature survey of the mathematical texts printed in England during Allen's lifetime, I have given considerable time to a 'reading' of the instruments themselves - through study of the originals, through production of my own versions, through reconstructions of the methods,of use, and, in the case of the sector, through computer analysis of the accuracy in use. The conclusion of my thesis is that the mathematical culture of seventeenth century England was far broader than that which is normally portrayed in histories of mathematics, involving a wide range of people with very different backgrounds and very different approaches to and understandings of mathematics. Above all, it is shown to be rooted strongly in a geometrical interpretation of mathematics and one which is inherently practical. In such a culture the role of instruments is fundamental and thus instrument makers like Elias Alien have a place at the heart of the mathematical community.