PDF Ebook Hidden Harmonies: The Lives and Times of the Pythagorean Theorem, by Robert Kaplan, Ellen Kaplan
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Hidden Harmonies: The Lives and Times of the Pythagorean Theorem, by Robert Kaplan, Ellen Kaplan
PDF Ebook Hidden Harmonies: The Lives and Times of the Pythagorean Theorem, by Robert Kaplan, Ellen Kaplan
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A squared plus b squared equals c squared. It sounds simple, doesn't it? Yet this familiar expression opens a gateway into the riotous garden of mathematics, and sends us on a journey of exploration in the company of two inspired guides, 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. As in the authors' best-selling The Nothing That Is and Chances Are..., the excitement of mathematics leaps from the pages of Hidden Harmonies.
- Sales Rank: #127425 in Audible
- Published on: 2013-02-27
- Format: Unabridged
- Original language: English
- Running time: 425 minutes
Most helpful customer reviews
10 of 11 people found the following review helpful.
Mathematics as a rich, human endeavor
By Sanjoy Mahajan
This book is a joy. In it, the Kaplans poetically describe how they see mathematics: "For all the statement's concision [on Dijkstra's proof of the Pythagorean theorem], it has been torn from the history, associations, intuitions, and implications of the Pythagorean Theorem, and made into a cell in cyberspace that discloses nothing" (pp. 155-156). Mathematics, in this view, is one of the richest human creations ("made by minds for minds" [p. xii])---and filled with intuition and history.
I am inspired by that implicit image throughout their book, and by how the Kaplans use the Pythagorean Theorem to illustrate these themes. I am often find myself so interested in following up an idea in the text that I wish the footnotes had been placed right at the bottom of the page rather than at the end of the book.
The book is inviting in many ways. As a small example, the diagrams have hand-drawn labels, making the diagrams feel more like friends in a process of discovery rather than like finished products that mean the end of the story. Furthermore, the book's language is welcoming: The zeroth chapter ("An Outlook on Insights") ends with, "Come and see" (p. xii). And I want to.
Who should read this book? Anyone who wonders what mathematics is, and why people found and find it fascinating. In much school mathematics, the dessicated presentations of results---without the intuitions, associations, and history---kill the spirit of mathematics. This book is a wonderful antidote. It was what I had hoped, for the Kaplans co-authored perhaps my favorite book about teaching mathematics: Out of the Labyrinth: Setting Mathematics Free.
12 of 15 people found the following review helpful.
Proofs Positive
By Rob Hardy
If someone says something is true, well, maybe it is or maybe it isn't. If something is demonstrated by mathematics as true, then there is a higher probability that it really is true. But if something is mathematically proven, that cinches it. A formal mathematical proof, once one accepts some basic conditions, is the strongest version of certainty we have. The Pythagorean theorem was proved long ago, and once a mathematical statement is proved, it needs no further proof. There is an allure to this particular theorem, however, that has led people to re-prove it hundreds of times. In _Hidden Harmonies: The Lives and Times of the Pythagorean Theorem_ (Bloomsbury Press), husband and wife mathematicians Robert and Ellen Kaplan have shown plenty of the proofs made in a surprising variety of styles. The theorem is not "more true" because it is "more proved," but the proofs as laid out here have something to tell us not just about right triangles, but about the concepts involved in proving something mathematically, and about larger topics in mathematics as well.
So, you take any triangle that has a corner that is a right angle; the theorem tells you that it has two short legs labeled a and b, and a longest leg (hypotenuse) labeled c, and then that a^2 + b^2 = c^2. The formula has a genuine appeal because it is so simple. Right triangles have an appeal because they are everywhere: a ladder leaning against a wall, a diagonal support on a door, a short cut across a city block, and so on. The Babylonians, Egyptians, Greeks, and of course Pythagoras himself play roles in the theorem's history, but the best part of the Kaplans' book is a sampling of the many proofs that have come down to us from the ancients and the moderns. The most famous one is the one in Euclid, and of course it is here, but it is far from the simplest or most elegant of proofs. There is a lovely proof involving two identical right triangles within a trapezoid. The author of this proof was none other than James A. Garfield, before he was elected president in 1880. The authors say of it (and it is true of so many other proofs in these pages), "Obvious once you see it - like so much in math; but (again, like so much in math) not at all obvious before." Leonardo da Vinci had a proof (is there anything into which that astonishing mind did not extend itself?) which the authors call "cryptic," but which is really very pretty, based on a couple of irregular hexagons which contain irregular quadrilaterals and also contain the triangle and squares on all three sides. Many of these diverse proofs were gathered by Elisha Scott Loomis, born in a log cabin in 1852, who collected Pythagorean proofs like others collect stamps. Some were his own; on 1 August 1900 he came up with five proofs. Of the proofs he collected, the authors say, some "are torturous, some teasing; some pert, some monstrous." There's a proof here based on building a gadget in the shape of a right triangle and showing that, since there is no such thing as perpetual motion, it could not move, and therefore the theorem is true. One proof was invented by Einstein (although it had been made by others before him). Another relies on filling the infinite Euclidean plane with square tiles and a diagonal grid upon them.
The famous number theorist G. H. Hardy wrote, "If you are feeling particularly hostile toward the whole universe and you want to do something evil, show this argument to your calculus students and tell them they need to learn differential equations to understand how to prove the Pythagorean theorem." He calls this "Pythagoras Made Difficult," and yes, calculus can be used in a proof. The authors joke afterwards, "`There now, wasn't that easy?' as the contortionist said to the Fat Lady." Not easy, no, but the authors are not joking when concerning a different proof they invite us to "recall from your days of trigonometry that cos (r - s) = cos r x cos s + sin r x sin s." I don't know about you, but my days of trigonometry are gone beyond recall, and plenty of the latter chapters here will be forever beyond me. That's OK; everyone hits a limit of understanding in mathematics. The geometric and algebraic proofs, however, are often surprising and lovely, and the authors display good humor throughout. They made it fun for me to trace out those shapes and equations within my comprehension, while increasing my respect for the proofs and provers I will never understand.
0 of 0 people found the following review helpful.
"...giddy sort of sensation..."
By Kindle Customer
I bought this book because of a reference in another book. I bought a hard copy because of the Kindle review for the product. I'm not a mathematician but I do enjoy reading about math. While I understood only about 11 1/4 of the math in Hidden Harmonies, I think that the book is fantastic. It is well written and usually this is not true of math books and at times for books in general.
The concepts were presented with a great deal of clarity. There are a number of quotations that would demonstrate this but, I will share only one--"...let's linger a moment to relish a proof that depends on the impossibility of perpetual motion." "Linger" and "relish" what fantastic mathematical words.
Hidden Harmonies is a book for those who are just inquisitive about math and for those with more of an appetite for the intricacies of mathematical concepts.
I do plan to use some of the drawings of proofs in my art work so for me the math book is also an art book.
Enjoy it.
See all 13 customer reviews...
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