## Wednesday, July 30, 2008

### Review of ‘The Pythagorean Theorem’ by Eli Maor

This is an excellent book on the history of the Pythagorean Theorem. I learnt a great deal of history of mathematics in relation to Pythagoras’s Theorem. This book is suitable to any student who has basic knowledge of calculus but the layperson will also find it interesting.
The book starts with the assertion that the Babylonians knew Pythagoras’s Theorem 1000 years before Pythagoras but it was the Greeks who proved the result.
There are a number of gems in the book which are not that well known in the mathematics community:
Hypotenuse is derived from the Greek words hypo meaning ‘under’ or ‘down’ and teinen meaning ‘to stretch’. Maor points out the reason for this is that the hypotenuse of a right triangle in Euclid’s Elements was always on the bottom. (I did not know this).
There are over 400 proofs of Pythagoras’s Theorem.
It was the French lawyer ‘Francois Viete’ who first converted verbal algebra into symbolic algebra.
Many more of these gems crop up throughout the book.
Maor does give a number of different proofs of Pythagoras’s Theorem.
More importantly the author does not shy away from producing mathematical expressions and symbols in a popular book like this. Here are a few examples:
1. Every even perfect number is of the form 2^(n-1)*(2^n-1).
2. Viete’s Identity product which expresses 2/π in terms of √2.
3. Shows how the area of one arch of the cycloid is 3 times the area of the circle generating it.
4. Gives an excellent brief description of Hilbert Spaces and non Euclidean geometry.
5. Explains why Pythagoras’s Theorem is not valid in non-Euclidean geometry.
There are many more fantastic mathematical examples. The more serious mathematics is left for the appendices.
Additionally Maor has provided an excellent general history of mathematics such as:
The first woman mathematician was Hypatia (370 to 415).
The University of Gottingen was world renown for mathematics up until the Second World War.
How Edmund Landau (1877 to 1938) shunned all references to geometry. Maor points out that Landau wrote a 372 page book ‘Differential and Integral Calculus’ and it does not contain a single illustration.
How Euler discovered differential geometry but its modern form is due to Riemann and Gauss.
There are also non-mathematical examples of history in the book such as the first European University was Bologna founded in 1088 and why the Christians burned the Library of Alexandria.
You will learn a lot from this book because it has been thoroughly researched and shows the different fields where Pythagoras’s Theorem is used.
The author has also made excellent use of illustrations so the layperson can understand without learning all the details.
Maor has an exceptional method of writing very technical mathematics in a seamlessly way.