Brothers Richard and David Garfinkle have composed a book which presents mathematics from both the Mathematician’s as well as the User’s point of view. The book provides a study of mathematics from its origins of counting through the development of arithmetic, algebra, geometry, analysis, calculus, probability, and statistics. The studies of these disciplines are presented in a historical style which details how one branch of mathematics evolves into another.

The underlying theme of this book is that mathematics, which governs so much of our lives, is a common inheritance of everyone. But somehow this inheritance gets lost by many as they pass through their respective educational systems. A primary purpose of this book is to help people recover their rightful inheritance of using and appreciating mathematics.

The Garfinkles’ book is written in a light-hearted manner without sacrificing mathematical rigor. The authors begin by taking us on a journey of discovering the roots of mathematics and then delving into what is described as its three primary processes, which they playfully dub as abstraction, mistrust, and laziness:

- abstraction being the mental process of precise focusing and generalizing on a small set of details,
- mistrust which requires one to prove a statement based upon known truths, and
- laziness which insists one works as efficiently as possible much like a carpenter who develops or invents a tool to make his job less labor intensive and more accurate. Mathematicians often refer to this as ‘elegance’.

The depth of the mathematics and physics developed throughout the book are substantial. For example, the chapter on calculus details the development of differentiation and integration of both simple and complex functions. In the section on differential calculus, the authors present a historical perspective on the developments of derivatives, limits, and second derivatives. They carry this topic through to an explanation of the Taylor Series and vectors. The discussion on integral calculus begins with the Reimann Sum and concludes with the integration of logarithmic and exponential functions. The applications presented in this section include such diverse examples as the growth of infectious diseases used in medicine to Carbon-12 and Carbon-14 dating used in geology and archeology.

The range of topics covered in X Marks the Spot is remarkable. Examples of just two of the topics included are comprehensive explanations of set theory as well as non-Euclidean geometries. There is even a section on computers that digs deep into both hardware and software, covering topics ranging from Boolean Algebra and binary arithmetic to both assembly language and compiler language software.

The section of the book on physics demonstrates the use of mathematics in Galilean and Newtonian physics as well as Relativity and Quantum Mechanics.

The last chapter is devoted to math education and includes recommendations for how to teach math as well as how math should fit into the overall curriculum.

X Marks the Spot is a book for anyone who teaches mathematics at any level. This book will be helpful for any course, particularly in developing the introduction portion of a course. The professor or teacher who is looking for practical examples to help answer the eternal question of “When is this stuff ever used?” will find a multitude of examples to help answer such questions. And those interested in the history and philosophy of mathematics would find this a useful text and a welcome addition to their library. The book is well documented and contains over 200 drawings and illustrations within its over 450 pages.

Tom French has a B.S. and M.S. degree in Mathematics from Minnesota State University, Mankato. He has 35 years of engineering and business experience with Lockheed Martin. He was part of the design team that first implemented medium-scale and large-scale integrated circuits into computers. Tom has taught mathematics and computers in numerous professional academies, colleges, universities, and high schools.