With these words the young Hungarian mathematical prodigy János Bolyai, reputedly the best swordsman and dancer in the Austrian Imperial Army, wrote home about his discovery of non-Euclidean geometry in 1823. Bolyai's discovery indeed marked a turning point in history, and as the century progressed mathematics finally freed itself from the lingering sense that it must describe only the patterns in the ‘real’ world. Some of the doors which these discoveries flung open led directly to new worlds whose full exploration has only become possible with the advent of high speed computing in the last twenty years.

Paralleling the industrial revolution, mathematics grew explosively in the nineteenth century. As yet, there was no real separation between pure and applied mathematics. One of the main themes was the discovery and exploration of the many special functions (sines, cosines, Bessel functions and so on) with which one could describe physical phenomena like waves, heat and electricity. Not only were these functions useful, but viewed as more abstract entities they took on a life of their own, displaying patterns whose study intrigued many people. Much of this had to do with understanding what happened when ordinary ‘real’ numbers were replaced by ‘complex’ ones, to be described in Chapter 2.