Hidden Harmonies: The Lives and Times of the Pythagorean Theorem

by Ellen Kaplan, Ellen Kaplan

A squared plus b squared equals c squared. It sounds simple, doesn't it? Yet this familiar expression is a gateway into the riotous garden of mathematics, and sends us on a journey of exploration in the company of two inspired guides, acclaimed authors Robert and Ellen Kaplan. With wit, verve, and clarity, they trace the life of the Pythagorean theorem, from ancient Babylon to the present, visiting along the way Leonardo da Vinci, Albert Einstein, President James Garfield, and the Freemasons-not to mention the elusive Pythagoras himself, who almost certainly did not make the statement that bears his name.
How can a theorem have more than one proof? Why does this one have more than two hundred-or is it four thousand? The Pythagorean theorem has even more applications than Ancient Egyptians used it for surveying property lines, and today astronomers call on it to measure the distance between stars. Its generalizations are stunning-the theorem works even with shapes on the sides that aren't squares, and not just in two dimensions, but any number you like, up to infinity. And perhaps its most intriguing feature of all, this tidy expression opened the door to the world of irrational numbers, an untidy discovery that deeply troubled Pythagoras's disciples.
Like the authors' bestselling The Nothing That Is and Chances Are . . . -hailed as "erudite and witty," "magnificent," and "exhilarating"- Hidden Harmonies makes the excitement of mathematics palpable.

Genres: Mathematics, History, Nonfiction, Science

304 pages, Hardcover

First published January 4, 2011

Reviews

[Kyle · Rating 4 out of 5]

This was a very interesting book. You might think it would be hard to have an entire book on the Pythagorean theorem, but the Kaplan's do make it engaging. The history surrounding its discovery and use is quite interesting. I learned that Neugebauer probably was wrong about Plimpton 322, which I was unaware of before. Instead of being Pythagorean it seems more likely it was creating problem sets for scribe students. You also learn several proofs of the Pythagorean theorem, with the proofs well explained (there was only one proof that left me mystified in the book: the one "proving" the Pythagorean theorem with a torus, with further google searches not helping me see how the proof presented works). And you learn of the many ways of extending the definition for multiple dimensions, different spaces, and just a good amount of generalizations that touch upon the Pythagorean theorem.

There is a certain style to the writing that may put off some people in that it makes many allusions and has a semi-mysterious air to some paragraphs. That is it sort of makes grand pronouncements that I'm not sure as grand as they are proclaimed. I found that sometimes it was a little bit too much for my taste, but I still found the vast majority of the book to be interesting. The math is well written with good figures.

I would not recommend the Kindle edition because of formatting issues, though. Flipping back and forth to see figures is one thing, but the footnotes are in a strange order, and there are some strange spacings in some of the words like "in e quality" that while not incomprehensible, detract from the readability.

One other comment is that they do not dumb down proofs, so if you have little mathematical sophistication, you may find the entire thing unreadable and the generalizations not understandable without putting in a good deal of work. I think they present things well, but it would still be rather daunting without mathematical training. All in all, a fun read, and I will probably try some of the Kaplan's other books.

[Jim · Rating 3 out of 5]

Heard Robert Kaplan speak on NPR and just had to check out this book. Fascinating read, but recommended for math weanies only!

[Ryan Boyce · Rating 2 out of 5]

For a book about math, there was a lot more math than I was expecting. Hidden Harmonies is exactly what it advertises, a biography of the Pythagorean Theorem. The authors take readers on a journey through different times and societies encountering the Theorem and it's many proofs and consequences, adding insights about what the proofs tell us about mathematics and about the societies that created them along the way.

My biggest criticism of the book is that I'm not sure who it was written for. Most of the actual content of the book is decently technical proofs. Since I have a background in mathematics, nothing was particularly difficult, I just had to decide if it was worth the effort to take myself fully through the logic that the authors presented. This is certainly not a leisurely read at all.

Between proofs, the authors like to wax poetic with big words like "tautology" and philosophical terms with capital letters like "Being" and "Becoming". I found most of these discussions irritating, although occasionally they would hit on something interesting. It's also possible that I would appreciate the philosophical musings more if I had paid more attention to the specifics of the proofs.

With all that being said, I did take away some insight from the book. I have complained for a long time about physicists who rely on abstraction instead of understanding the world around them, but that's exactly what I've done with mathematics. Geometric proofs have much more to say in terms of spatial reasoning and logic, and yet for the most part I rely on algebra and higher abstractions to inform my view of the world. All my fancy mathematics tends to protect me from having to deal with the raw logic that it encodes, which is exactly what I hate about formalism in physics. Abstraction is the two edge sword wielded by understanding.

[Ton van Gessel · Rating 4 out of 5]

I do like this book. This is really a book about mathematics. There are quite a lot of different proofs and implications of the Pythagorean theorem in this book. To me it was a great pleasure trying to really understand these proofs. It taught me a lot about mathematics, geometry and proofs. This is by no means an easy read. Especially if you don't have a formal math education like me.

[Tom]

Do NOT try to read this book 47 years after your last Geometry or math class. It will simply not make any sense. Great book, I'm sure, for the right person. A long time ago I was that person but not now.

[Aiman Adlawan · Rating 5 out of 5]

I never knew how Pythagoreon theorem can be applied to our daily lives. Great story about how the theory was developed.

[William]

Whoa...proofs and I think I'm pooped... Started reading this one the other night before bed, thought my confusion was fatigue only to pick it up again today realizing I might be a bit ditzy for this one. Maybe it's because trying to read with my contacts in.

I want to love math and am not sure if this book lies within my zone of comprehension. Wouldn't be the first time.




...Update, just returned this one to the library...I need more background info to understand the historical and math references. I'll be back.

[Jack Vinson · Rating 4 out of 5]

I heard the authors speak and couldn't stop myself from buying. The book is loaded with proofs of the Pythagorean Theorem (a^2 + b^2 = c^2) of right triangles.

Great for math geeks and the curious about the development of thought and thinking.

[Jason · Rating 2 out of 5]

Interesting in places, but I just couldn't stand the writing style. I ended up skipping around because I was interested in the math, but didn't read it all.