Introduction -- The nine mathematicians whose works are represented in the following pages are among the most famous in the whole history of mathematics. Each of them made a significant contribution to the science -- a contribution which changed the succeeding course of the development of mathematics. That is why we have called this book Breakthroughs in Mathematics. Just as surely as there are technological breakthroughs which change our way of living, so are there breakthroughs in the pure sciences which have such an impact that they affect all succeeding thought.