This book is intended for pupils in the top classes in high schools and for students in mathematics departments of universi- ties and teachers' colleges. It may also be useful in the work of mathematical societies and may be of interest to teachers of mathematics in junior high and high schools.The subject matter is concerned with both algebra and geom- etry. There are many useful connections between these two disciplines. Many applications of algebra to geometry and of geometry to algebra were known in antiquity; nearer to our time there appeared the important subject of analytical geometry, which led to algebraic geometry, a vast and rapidly developing science, concerned equally with algebra and geometry. Algebraic methods are now used in projective geometry, so that it is uncertain whether projective geometry should be called a branch of geometry or algebra. In the same way the study of complex numbers, which arises primarily within the bounds of algebra, proved to be very closely connected with geometry; this can beseen if only from the fact that geometers, perhaps, made a greater contribution to the development of the theory than algebraists.The book is intended for quite a wide circle of readers. The early sections of each chapter may be used in mathematical classes in secondary schools, and the later sections are obviously intended for more advanced students (this has necessitated a rather complicated system of notation to distinguish the various parts of the book).