*The Mathematical Universe*was written by William Dunham, a Professor of Mathematics at Muhlenberg College, Allentown, Pennsylvania, United States.

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

*- primarily Jacob Bernoulli's discoveries regarding probability but includes conflict and rivalry with brother Jonann*

**B**ernoulli Trials*- circumference, area and particularly the calculation of pi*

**C**ircle*- basic intro to the theory, with some applications such as calculating tangents and maximums, minimum and stationary points of functins*

**D**ifferential Calculus*- emphasis on the stunning breadth and depth of Euler's contribution to maths*

**E**uler*- brief biography and description of some of his contributions, including*

**F**ermat*Last Theorem*

*- contributions of Ancient Greeks with particular emphasis on Euclid (of cause)*

**G**reek Geometry*- three proofs of Pythagoras Theorem*

**H**ypotenuse*- to determine, from among all curves of the same perimeter, the one enclosing the largest area*

**I**soperimetric Problem*- general discussion of the idea of mathematical proof*

**J**ustification*- Newton's strange and prickly personality as well as his enormous contributions to mathematics and physics*

**K**nighted Newton*- Leibniz's contributions to maths especially calculus and his conflict with Newton regarding priority of its discovery*

**L**ost Leibniz*- what type of people are attracted to mathematics?*

**M**athematical Personality*- e (ie 2.718281828459045 ... ) and natural logarithms; theory and practice*

**N**atural Logarithm*- some very early mathematical landmarss; Egyptian, Mesopotamian, Chinese and Indian*

**O**rigins*- the proportion of primes less than or equal to a number is roughly equal to the reciprocal of the natural logarithm of the number*

**P**rime Number Theorem*- development of the number systems (natural, rational, irrational and real) from the basic arithmetic processes of addition, subtraction, multiplication, division and extraction of roots*

**Q**uotient*- Bertram Russell's paradoxical personality and life and his mathematical paradox which involved*

**R**ussell's Paradox*the set of all those sets that are not members of themselves*

*- how Archimedes determined the surface area of a sphere*

**S**pherical Surface*- the impossibility of trisecting an*

**T**risection*angle*using just a compass and unmarked straight edge

*- 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*

**U**tility*- a simple idea that was invented before John Venn, but he got the credit*

**V**enn Diagram*- why there haven't been many women mathematicians historically and how that has changed in recent times*

**W**here are the Women?*- the Cartesian Plane*

**X - Y**Plane*- complex numbers*

**Z**
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