Very nice looking book, well-written. As is usual w/ books about math that actually include even a little math, it is slow going.

Detailed Review:

* Prologue: Gives the means of arriving at the definition, a science of patterns, and also emphasizes the importance of abstraction and notation. Remarks that computers have allowed visualization that was not possible before; Julia sets are older than our ability to easily see what they look like.

* Chapter 1: Counting
Covers the history of the development of the concept of number, as far as can be inferred from archaeological evidence. The Rhind papyrus has a description of how to calculate the volume of the frustum of a pyramid. The numbers used are concrete; but it it s clear that the reader is expected to consider them variables as necessary. A modern day person would use integration to derive a formula for the volume.

Moving on to the Greeks, who were really most interested in geometry. Discusses the Pythagorean theorem and the interpretation of numbers as lengths. The formula for (a + b)^2 and (a -b)^2 is presented as the Greeks would have understood it, geometrically. A proof of the irrationality of root 2 is shown; is that how the Greeks discovered it? A discussion of prime numbers; evidently the fundamental theorem of arithmetic is in Euclid's work, as well as the proof that there is an infinite number of primes.

The primes allow us to leap 1000 or so years forward, and discuss the prime density function, the ratio of the number of primes less than or equal to N to N. Wikipedia discusses the prime _counting_ function: https://en.wikipedia.org/wiki/Prime-counting_function. The proof that the number of primes are infinite is well presented; I've seen other place where the proof fails to properly cover the two possibilities, leaving the reader perplexed and uncertain.

Chebyshev discovers that between every number and its double there is at least one prime. And finally, an upper bound, on the function, 1/ln N, was proved.

The next topic, Finite Arithmetic, is a discussion of abstract algebra. It was then I wanted to find my Abstract Algebra textbook, only to discover that it was almost certainly in my currently inaccessible cubicle.

Chapter 4: Shape
About geometry and algebraic geometry. There is a solid discussion of Euclid and a sidebar on the golden ratio with some unexplained relationships. The regular solids and conic sections are discussed. Properties are described w/out much justification but with fine illustrations. Kepler's wierd obsession with the plantonic solids and the planetary orbits gets a mention. Then algebra is introduced, and there is a perplexing discussion of a general formula for a circle, that yields no insight to me (p.118). Old problems of compass and straightedge construction of various figures are discussed. These all rely on the fact that compass and straightedge constructions only yield a subset of mathematical operations; basic arithmetic and the taking of square roots, and that all the solutions require a calculation that uses more than these. This can't be covered in detail in a book like this, of course. The next step is non-Euclidean geometries: hyperbolic, spherical, Riemannian, projective.