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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 affeet all succeeding thought.
The mathematicians whose works we examine bridge a span of more than 2200 years, from Euclid, who lived and worked in Alexandria around 300 B.C., to Bertrand Russell, whose major mathematical work was accomplished in the fist years of the twentieth century. These nine chapters survey the major parts of mathematics a great many of its branches are touched on. We shall have occasion to deal with geometry, both Euclidean and non-Euclidean, with arithmetic, algebra, analytic geometry, the theory of irrationals, set-theory, calculus of probability, and mathematical logic. Also, though this is a matter of accident, the authors whose works we study come from almost every important country in the West: from ancient Greece and Hellenistic Rome, Egypt, France, Germany, Great Britain, and Russia.
Introduction
Geometry
Euclid—The Beginnings of Geometry
Lobachevski—Non-Euclidean Geometry
Descartes— Geometry and Algebra Joined
Arithmetic
Archimedes—Numbers and Counting
Dedekind—Irrational Numbers
Russell—The Definition of Number
Advanced topics
Euler—A New Branch of Mathematics: Topology
Laplace—The Theory of Probability
Boole—Algebra and Logic Joined
Suggestions for further reading