# The Nature and Power of Mathematics

## by Donald M. Davis

This book was published by Princeton University Press in December 1993.
In paperback, its ISBN was 0-691-02562, but Princeton sold all their
copies.
It was republished by Dover Publications
in December, 2004.
The new ISBN is 0-486-43896-1, and it is priced at $19.95.
New and used copies are available at amazon.com.
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The publisher's blurb reads as follows:

In this engaging book, Donald Davis explains some of the most fascinating
ideas in mathematics to the nonspecialist, highlighting their philosophical and
historical interest, their often surprising applicability, and their beauty.
The three main topics discussed are non-Euclidean geometry, with its
application to the theory of relativity; number theory, with its application to
cryptography; and fractals, which are an application in art, among other areas,
of early mathematical work on iteration. Other topics include the influence of
Greek mathematics on Kepler's laws of planetary motion, and the theoretical
work that led to the development of computers.

Assuming the reader has some background in basic algebra and geometry,
Davis relies on exercises to develop some of the important concepts. These
exercises are designed to improve the reader's ability in logic, and enable him
or her actually to experience mathematics at increasingly advanced levels.

### Table of Contents

- 1. Some Greek mathematics
- 1.1 pi and irrational numbers: an introduction to Greek mathematics
- 1.2 Euclidean geometry
- 1.3 Greek mathematics and Kepler

- 2. Non-Euclidean geometry
- 2.1 Formal axiom systems
- 2.2 Precursors to non-Euclidean geometry
- 2.3 Hyperbolic geometry
- 2.4 Spherical geometry
- 2.5 Models of hyperbolic geometry
- 2.6 The geometry of the universe

- 3. Number theory
- 3.1 Prime numbers
- 3.2 The Euclidean algorithm
- 3.3 Congruence arithmetic
- 3.4 The Little Fermat Theorem

- 4. Cryptography
- 4.1 Some basic methods of cryptography
- 4.2 Public key cryptography

- 5. Fractals
- 5.1 Fractal dimension
- 5.2 Iteration and computers
- 5.3 Mandelbrot and Julia sets