This tract provides an introduction to four finite geometrical systems and to the theory of projective planes. Of the four geometries one is based on a nine-element field and the other three can be constructed from the nine-element 'miniquaternion algebra' a simple system which has many though not all the properties of a field. The three systems based on the miniquaternion algebra have widely differing properties; none of them has the homogeneity of structure which characterizes geometry over a field. While these four geometries are the main subject of this book many of the ideas developed are of much more general significance. The authors have assumed a knowledge of the simpler properties of groups fields matrices and transformations (mappings) such as is contained in a first course in abstract algebra. Development of the nine-element field and the miniquaternion system from a prescribed set of properties of the operations of addition and multiplication are covered in an introductory chapter. Exercises of varying difficulty are integrated with the text. Language: English