Mathematics and physics are by their nature very different disciplines indeed. In fact, mathematics possesses a methodology that is unique among all the sciences, since it is the only science in which deductive logic alone determines truth. It follows that mathematical theorems which were established thousands of years ago are every bit as true today as they were on the day that they were first proved. To many it may not seem surprising that mathematics has played an essential role in science, and particularly physics—after all, physics is a quantitative science. Nevertheless, mathematics and physics are by their nature very different disciplines indeed. If one factor has remained constant throughout the twists and turns of the history of progress in the physical sciences, it is the decisive role played by mathematics. It is the story of this role in many of its guises that the chapters of this book will attempt to portray.