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This is the fourth volume in the introducing mathematics series to the lay person. It is highly desirable that the opening pages of a book should give a potential reader some indication of the scope and purpose of the book, its level of difficulty and the knowledge that it presupposes. First of all, it should be pointed out that, while this book follows Vision in Elementary Mathematics in time, it does not follow it in the development of the subject. Vision in Elementary Mathematics was aimed at the beginnings of education; it was intended to help the teacher or parent concerned with children between, say, five and thirteen years old; it did not assume any prior knowledge of mathematics apart from that minimum of arithmetic that most people have. This book does assume some background in mathematics. It supposes the reader to be fairly comfortable with the kind of topics covered in my earlier book Mathematician’s Delight. This does not mean that every chapter requires an understanding of calculus — far from it. If Vision in Elementary Mathematics was meant to help the teacher of children five to thirteen years old, this book may be helpful to a teacher who is revising the syllabus for pupils between eleven and eighteen years old. The discussion must make suggestions for the Mathematics and Science Sixth, but it must also concern itself with the eleven-year-olds, and with classes that are not being taught by a mathematics specialist. Chapters One and Three, for example, develop an approach originally published in the Scientific American. If this approach is criticized, it will probably be on the grounds that it is too childish. Again, a very considerable part of Chapter Nine has been tried out in schools, and found to be intelligible and entertaining to pupils who knew just a little algebra and Pythagoras’ Theorem. Wherever possible, an idea taken from modern mathematics has been explained in terms of quite elementary mathematics