Customer Reviews

The Pythagorean Theorem: A 4,000-Year History

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Eli Maor is a fine mathematician who has produced some wonderful books on math topics for a general--well, let me say, educated--readership. His book, Trigonometric Delights, is my favorite. It is very interesting and engaging. I want to say "for an educated reader" again, though it seems rather redundant. Why would anyone who didn't know anything about trig and have an interest in the subject even bother to pick up the book? Still, as someone who spent more than ten years in high school math classrooms, I also found his work useful to interest and inspire my students (and myself).

Since the class I taught most often was geometry, I was very happy to see this book on the Pythagorean theorem. I have to admit, as an avid reader on the subject, I was familiar with much of what's here; particularly, the historical development and the more "Euclidean" applications of the theorem. On the other hand, he developed some proofs and problems I hadn't seen before which I found quite interesting.

Overall, however, I didn't find this book quite as engaging as some of his other work. I got the feeling he started off wanted to write a book that would have more universal appeal than some of his other titles. I mean, after all, nearly everyone knows what the Pythagorean theorem is, or has at least heard of it. But there wasn't nearly enough of the "simple" stuff and the last half of the book really goes quite far afield into mathematics without which someone without a pretty decent background in the subject will have a difficult time; particularly since the development is rather sparse in what feels like an aborted effort to keep things simple. Even some of the earlier demonstrations and proofs are a bit difficult if you don't have the background in Greek mathematics which, unfortunately, is often lacking these days.

Still, as someone who loves geometry and has a pretty good background in it, I found much here to like. Any reader who feels confident in their mathematical ability will probably find much here to like too.

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"To this day, the theorem of [Greek mathematician] Pythagoras [which states that the square of a right-angled triangle's longest side or hypotenuse is equal to the sum of the squares of the other two sides, written in the language of mathematics as (c^2 = a^2 + b^2) or, more commonly, (a^2 + b^2 = c^2)] remains the most important single theorem in the whole of mathematics. That seems like a bold and extraordinary thing to say, yet it is not extravagant; because what Pythagoras established is a fundamental characterization of the space in which we move, and it is the first time that it is translated to numbers...In fact, the numbers that compose right-angled triangles [called Pythagorean Triples such as (3,4,5), (28, 45, 53) and (65, 72, 97)] have been proposed as messages which we might send out to planets in other star systems a test for the existence of rational life there."

The above quotation is found in this fascinating book authored by history of mathematics professor and author Eli Maor. (Note that the above quotation was not said by Maor.) It catches the importance of this deceptively simple theorem, a theorem children's writer Lewis Carroll (who was also a mathematician) called "dazzlingly beautiful."

What did I learn from this book? Answer: there's a lot more to the Pythagorean theorem than (a^2 + b^2 = c^2)!! Maor may be the first author who has examined all the mathematics, history of mathematics, and physics books and collected just the material directly and indirectly related to the Pythagorean theorem.

The result is that Maor has brought the long history of the Pythagorean theorem back to life. Sometime around 570 BCE Pythagoras proved (notice I said "proved" and not "discovered") a theorem about right triangles that made his name immortal. He also pondered the workings of the universe and tried to relate its workings to the laws of musical harmony. In the subsequent centuries, this theorem was used and developed by others such that it has become central to almost every branch of science, pure or applied. After twenty-five centuries, this theorem was expanded and thrust into four-dimensional space-time by Albert Einstein to formulate his own picture of the universe.

Yes, there is simple mathematics in this book. To understand it, all you will need is some high school algebra and geometry and a bit of elementary calculus.

Do you have to follow the mathematics found in this book? NO. Personally, I found that you could skim, even skip the mathematical parts and still not lose the essential flow of the main narrative. (Actually, the more difficult mathematics is relegated to the book's appendices.)

Throughout the book are diagrams and even some pictures to enhance its main narrative. As well, there are eight pages of colour photographs found near the book's center.

A feature of this book is that it contains "sidebars." These are brief sections (there are ten) found at the end of some chapters that usually focus on some aspect of the Pythagorean theorem. My two favourites have the following titles: "The Pythagorean Theorem in Art, Poetry, and Prose" and "Four Pythagorean Brainteasers." You don't have to read each sidebar.

Another feature of this book is its chronology. It more or less summarizes the main events in this book in chronological order. This chronology begins in the year 1800 BCE and ends in the year 1996.

Finally, a note on the book's cover picture (displayed above by Amazon). It shows the detail or "zooming in" of a beautiful larger 1649 picture called "Allegory of Geometry" by artist Laurent de la Hyre (displayed on this book's inside back flap). The book's cover picture zooms in on several geometric figures, the one on the top left showing Euclid's proof of the Pythagorean theorem.

In conclusion, this book is essential for anyone that wants to be familiar with the four thousand year history of the Pythagorean theorem. I leave you with some actual lines from Gilbert and Sullivan's "Pirates of Penzance:"

"I'm very well acquainted, too, with matters mathematical,
I understand equations, both simple and quadratic,
About Binomial Theorem I'm teeming with a lot o'news,
With many cheerful facts about the square of the hypotenuse."

(first published 2007; list of colour plates; preface; prologue; 16 chapters; epilogue; main narrative 215 pages; 8 appendixes; chronology; bibliography; illustrations credits; index)

<<Stephen Pletko, London, Ontario, Canada>>

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I loved e: the story of a number, both the story and the mathematics in it. But for some reason this book does not catch the same spirit. It doesn't have the exciting thread of a story that makes you want to turn to the next page, and the many different proofs make it feel like it's a patchwork of items forcing itself to support the topic rather than a natural inspiring thread that helps you see the growth in the mathematics. I found it disappointing.

There are many books popularizing Mathematics. In order to fascinate and impress the layman
they mostly take results and theorems from number theory, Mersenne primes, Fermat numbers or Theorem,
formulas like that of even perfect numbers...
As such the book would get three stars, as one of the multitude of mediocre books out there.
Another way to gain attention is to choose something easily recognizable (almost everybody has heard of the Pythagorean theorem) and a sensation!
Pythagoras theorem is not due to Pythagoras!

But the author misses several important points! He misleads. He does not popularize!

1) There more than 400 different proofs of the Pythagorean Theorem.
Yet none of these 400 proofs can be attributed to Babylonians or to Egyptians.
The first to formulate and PROVE the theorem was, just that, Pythagoras of Samos.

2) Babylonians were superb calculators but lousy geometers.
The very crude value 3 for pi, appearing even in the Bible, was used by the Babylonians.
Any child, who learns how to draw a circle on the sand, can easily establish that pi must be greater than 3. And after 2 cuttings in the middle of the radius that pi>3+1/8=3.125
And yet the author calls Babylonians superb geometers!!

3) On the other hand the Egyptians had a remarkable approximation for pi, (16/9)^2=3.1605.
It appears at problem 50 of Rhind papyrus. But they did not leave us any hint of how they arrived at this value. For them it was irrelevant how you reason.
A very plausible answer of how they did it , appeared in 1977, in a wonderful paper of Hermann Engels "Quadrature of the Circle in Ancient Egypt". The paper illustrates the practical character of the Egyptian Geometry. AND it contains an unexpected side result. The Egyptians did NOT know the relation a^2+b^2=c^2 for the sides of a right triangle.
And yet Eli Maor does not refer to Engel in the text under the title "Did the Egyptians know it ?"

4) The author shows a clear agenda in the last paragraph of the first chapter. It reads:
"Plimpton 322 thus shows that the Babylonians were not only familiar with
the Pythagorean theorem, but that they knew the rudiments of number theory
and had the computational skills to put the theory into practice--quite remarkable
for a civilization that lived a thousand years before the Greeks produced
their first great mathematician."

Eli Maor has obviously not understood
what a giant intellectual leap the Greeks took when they passed from the particular to the general, from
observing to proving.
Or even from 7 Babylonian kush, 7 Egyptian khet or 7 horses to the number 7.
What are the properties of the number 7? What is its role in the multiplicative structure of the integers ?
The author of Plimpton 322 would stand agape in a classroom where a Greek tutor
explained that 7 is a prime and proved that its square root is irrational.
Concepts like prime, irrational, theorem, proof were outside his mental capacity.
He would consider such concepts as sheer Greek gibberish.
But for us moderns Mathematics is exactly that Greek "gibberish". Axioms, theorems, proofs,...
To put it bluntly, for us moderns, Pythagoras is a Mathematician, the author of Plimpton 322 is not!
Maor's claims about Babylonians, Theorems, Geometry, and Number theory are simply wishful thinking.

Allow me to conclude by using the words of T. E. Rihll, a real expert on the History of Science, when she writes about the matter of Babylonians vs Pythagoras, in her Oxford new survey of "Greek Science", page 18.

"...herein lies the difference between Greek activities, which we are calling Science, and what went before or elsewhere,which we do not call science. The Babylonians observed a mathematical regularity, and compiled or calculated tables of similar regularities. The Greeks, or rather (as later tradition asserts) a Greek called Pythagoras, observed this mathematical regularity and proved geometrically that it holds for all particular cases.
The search for the general, the abstract, and the process of arguing rationally about the case: these are the hallmarks of science, which are absent from non-scientific knoweledge and understanding."

The theorem is still Pythagoras and only 2500 years old!

The Pythagorean Theorem could rightfully be called the 'Crown Jewel of Mathematics'. For from its truths and intellectual spawn come all the wonders of our modern word--high rises, automobiles, cell phones, interplanetary probes, you name it! Unfortunately, the last serious book on this subject was written over 80 years ago by an Ohio school teacher, Elisha Loomis. Enter Dr. Eli Maor! He has written an absolutely marvelous book about 'The Crown Jewel' that will captivate anyone with a good high school mathematics background. Read it and behold a wonder!

Reading such book it reminds me time when Issac Assimov was manufacturing inteligence. Reading books is easier than write them. On page 142 A case of overuse, NY Times rectangle, on it I was surprised by author either ignorance or lack of visual practice. While he correctly reasoned that without right angle puzzle is difficult to solve, everyone can see that hypoteuse is identical with base of rectangle. And even when we do not know parameters of height it is.. , everyone should know from 5 year of school that area of triangle is 1/2 of area of its rectangle . I see that author is twisting Pythagoras elbows to make some science out of it.But things are perhaps not so over complicated as book presents. Any one can enjoy it in more simple way. I for example use the same model on demonstrating current social - economical bust in local pub. Just look on whole reactangle as available material of our universe , and on triangle as faculty of productive mankind. While marketable ideology shifts point E once to left and next to right the productive cappacity or absorbtion of wealth really does not change, as well as so called growth. Of course somes at upper reaches of distributive structure can get out of picture. I am some times surprised what can be done from recycled knowledge and who has on it copyright.

As a social history book,you will probably be overwhelmed by the math formulas.As a mathmetical history book,you will be impressed by the story of this classical theorem.I never realised that so many of the isolated ancient cultures,developed their own version of the theorem.Much like the story of the 'Great Flood',it became a part of the universal folk-history of mankind.The noted scholars of yesteryear elucidated a profound numerical 'law of Nature',if not the most important.This arithmetical model has forever changed the way humankind has approached and proposed number problems and numerical functions.Einstein once said,'God does not play with dice.'.In other words,the cyclical laws of Nature are univerisal and steadfast,even if against the desires of humankind.The abstract rhythms of physical Nature can be measured and demonstrated in numerical constructions.Not too many books around,that deal with the sole topic of the encyclical global story of the Pythagorean theorem.I generally disagree with giving someone's name to an object,locus or idea.For example ,Mount McKinley or Pike's Peak,to name a couple.A better name would be ,'the Universal Theorem'.Yet,a lombard Greek has his surname attached,forever assuming his sole creative ownership over this profound theorem is correct.We still think so and his 'locus classicus' remains to this day.

I purchased this book for my husband as a gift and he enjoyed the read. It is always difficult to find exactly the correct book for him ... and this time I did it! with the help of Amazon.com. Thank you Amazon.com