In mathematics, ‘advanced” ideas are apt to become ‘elementary’ with the passage of time. Progress of this kind is not always genuine: mathematical teaching (in universities as well as elsewhere) has often debased itself by superficial treatment of matters that cannot be properly understood without deep and careful thought. But in recent years great changes (representing, for the most part, genuine progress) have taken place in university mathematics, and corresponding changes at other levels are now urgently needed. Our original purpose in compiling this book was to offer some practical help to those who are trying to narrow the gap (or, if they are students, to bridge the gap) between ’school mathematics’ and ‘university mathematics’ as these terms are now generally understood in England and Wales. In countries where the standard period of undergraduate study is rather short, it is necessary for educational efficiency, as well as being desirable on other grounds, that a student entering a university to study mathematical subjects should have had a sound and fairly up-to-date introduction to pure mathematics: he should have acquired some understanding of the logical structure and the conceptual basis of this subject, and a habit of careful and critical thinking, as well as a modicum of technical skill of the traditional kind. He cannot be expected to do this without some rigorous training; but if he has some talent for the subject he is likely to enjoy such training, and to benefit from it even if he does not become a mathematical specialist.