displayed as an axiomatic system. This is in marked contrast with the pattern established by Euclid and his contemporaries, which has for many years been the geometry taught in the main school course in this country.

First ideas about axiomatic structures have arisen from comparison of algebras, leading to the definition of an abstract group. We claim that the difficult philosophical concept of mathematical proof is more easily grasped in algebra, where the sets of axioms are less extensive than in geometry and one is less likely to make implicit assumptions (e.g. those arising from the way a figure is drawn). One object of this book is to lead up to an axiomatic treatment of transformation geometry, and this is reached in Chapter S. On the way, further

algebraic structures are studied, notably vector spaces, and these are also used to set up other geometries in a similar fashion. At each stage there are relevant and interesting results about configurations, but the selection of material in

geometry has been made primarily to bring out the nature of proof, the variety of geometries which can he invented, and the relatl.ons between different geometries.

The algebra chapters need less introduction, and could stand on their own; previous experience of vectors, matrices, linear transformations and linear equations is woven into a single, more rigorous account, culminating in the chapter on eigenvalues and eigenvectors. It has always been the S.M.P. view,

however, that links between algebra and geometry are mutually beneficia!. ,Algebraic methods certainly assist the development of geometry, though the extent to which they are used must remain a matter of taste, and it is equally true that most. People find geometrical illustration illuminates formal algebr~ic processes and theorems. In these ways this book is conceived as the inevitable seque! to earlier S.M.P. writing on algebra and geometry. With its preoccupation with mathematical proof and formal development of consistent theories, it is hoped that it will act also as a bridge to university mathematics.