Mathematicians have developed the language of topology; but the conventional development leaves until last the discussion and ideas about shapes, so those who have not mastered the initial ideas of analysis and general topology are barred from participation by a rigid formalism. This book, first published in 1976, adopts a different approach, developing a language close to that of ordinary discourse. It is intended to encourage readers of varying backgrounds, but especially students, to think spatially. The text is well illustrated, with toned drawings creating three-dimensional effects where appropriate, and there are abundant exercises. Useful appendices accompany the text, providing hints and solutions to these exercises and also a sketch of how the treatment can be modelled within a conventional topology course for more advanced students. The book can be regarded as an example of the emerging discipline of mathematics education, as well as being about surfaces.

arithmetic of the integers, linear algebra, an introduction to group theory, the theory of polynomial functions and polynomial equations, and some Boolean algebra. It could be supplemented, of course, by material from other chapters. Again, Course 5 (Calculus) aiscusses the differential and integral calculus more or less from the beginnings of these theories, and proceeds through functions of several real variables, functions of a complex variable, and topics of real analysis such as the implicit function theorem. We would, however, like to make a further point with regard to the appropriateness of our text in course work. We emphasized in the Introduction to the original edition that, in the main, we had in mind the reader who had already met the topics once and wished to review them in the light of his (or her) increased knowledge and mathematical maturity. We therefore believe that our book could form a suitable basis for American graduate courses in the mathematical sciences, especially those prerequisites for a Master's degree.