The book is subtitled: An Alphabetical Journey Through the Great Proofs, Problems and Personalities. That is as good a one sentence summary as I could concoct. Consequently there are 25 chapters (not 26 as you would expect as X and Y are covered in one chapter). This structure creates difficulties for the author as logical progression is important in maths and it is unlikely a logical mathematical structure can be worked out in the order of the alphabet. On the whole Dunham manages this dilemma well, although there are some abrupt jumps - as he admits himself. The variation in topics is interesting in itself and the level of maths is not so difficult that each chapter needs a long introduction to preliminary ideas. It is not possible to discuss the maths in detail in a book only 295 pages long anyway. He is also able to concoct some logic to the order. For instance the first chapter is about arithmetic, which is a logical place to start mathematically and the first letter of the alphabet. He manages to discuss Newton in chapter K (Knighted Newton) and follow it with Leibniz, thus discussing their dispute over priority in calculus in consecutive chapters. Chapter X - Y on the Cartesian Plane precedes chapter Z on Complex numbers.
There are many ideas in this book that could be included in a review, but one particularly struck me while I was reading it. James Garfield was elected US President in March 1881. Sadly he was assassinated a few months later, but what struck me was that had produced a clever proof of Pythagoras Theorem. One wonders if the present incumbent of the White House knows what Pythagoras Theorem is. It is certain that he wouldn't have a clue about developing a proof, and what is even sadder he wouldn't have any curiosity the theorem or a proof.
For those with an interest in maths and familiarity with maths to a secondary school standard, this is an interesting and entertaining read.
The chapters are as follows:
Arithmetic - issues regarding whole numbers are more complex than might be supposed
Bernoulli Trials - primarily Jacob Bernoulli's discoveries regarding probability but includes conflict and rivalry with brother Jonann
Circle - circumference, area and particularly the calculation of pi
Differential Calculus - basic intro to the theory, with some applications such as calculating tangents and maximums, minimum and stationary points of functins
Euler - emphasis on the stunning breadth and depth of Euler's contribution to maths
Fermat - brief biography and description of some of his contributions, including Last Theorem
Greek Geometry - contributions of Ancient Greeks with particular emphasis on Euclid (of cause)
Hypotenuse - three proofs of Pythagoras Theorem
Isoperimetric Problem - to determine, from among all curves of the same perimeter, the one enclosing the largest area
Justification - general discussion of the idea of mathematical proof
Knighted Newton - Newton's strange and prickly personality as well as his enormous contributions to mathematics and physics
Lost Leibniz - Leibniz's contributions to maths especially calculus and his conflict with Newton regarding priority of its discovery
Mathematical Personality - what type of people are attracted to mathematics?
Natural Logarithm - e (ie 2.718281828459045 ... ) and natural logarithms; theory and practice
Origins - some very early mathematical landmarss; Egyptian, Mesopotamian, Chinese and Indian
Prime Number Theorem - the proportion of primes less than or equal to a number is roughly equal to the reciprocal of the natural logarithm of the number
Quotient - development of the number systems (natural, rational, irrational and real) from the basic arithmetic processes of addition, subtraction, multiplication, division and extraction of roots
Russell's Paradox - Bertram Russell's paradoxical personality and life and his mathematical paradox which involved the set of all those sets that are not members of themselves
Spherical Surface - how Archimedes determined the surface area of a sphere
Trisection - the impossibility of trisecting an angle using just a compass and unmarked straight edge
Utility - brief discussion of why mathematics and the natural world should mirror each other; followed by some examples of the usefulness of maths taken from trigonometry
Venn Diagram - a simple idea that was invented before John Venn, but he got the credit
Where are the Women? - why there haven't been many women mathematicians historically and how that has changed in recent times
X - Y Plane - the Cartesian Plane
Z - complex numbers