As the authors of The Nothing That Is and The Art of the Infinite, Kaplan and Kaplan have a well-earned reputation for writing fascinating expositions of mathematics and its history which are accessible to non-experts. Hidden Harmonies is endlessly fascinating, but not as accessible as their earlier work. That being said, grab yourself a good history of mathematics text to serve as a back-up and dive right in — Hidden Harmonies will reward your efforts with some wonderful history, numerous interesting proofs of the title theorem, and an often amusing look into the world of mathematics.
The book opens with a story about a 6-year-old English boy giving a proof of the Pythagorean Theorem (PT) by drawing in the sand. We learn that the boy never grew to be the great mathematician he seemed destined to become. Instead, he threw himself to his death when well-meaning grownups took away his Euclid. A true story, we wonder? No, say the Kaplans, the tale is from a short story by Aldous Huxley entitled “Young Archimedes.” Its purpose in the present volume is to remind us that the myth of mathematicians as somehow superhuman is just that: a myth. Mathematics is done by human beings. Aldous Huxley and mathematics — who knew?
The remainder of the book provides lively and entertaining glimpses into the lives and life’s work of a large cast of interesting characters (all of them human). In addition (and this is where this volume really stands out from the many volumes devoted to PT), the authors discuss how we came to know what we know about them and how certain we are as to the truth of what we think we know. It is this attention to the historiography of mathematics which makes this book a wonderful choice as a secondary text for a history of mathematics course as well as a great way to recruit undergraduates to work on research in the history of mathematics. For mathematicians and historians of mathematics it is a wonderful journey through well-trod paths that will likely provide new insights even to the most jaded of Pythagorean junkies. I will provide a brief glimpse into several chapters — to do more would “give away the plot” of a book which is truly unique in its approach and style.
The tale begins with the Babylonians who, we are told, “did things with numbers and shapes of a finesse and intricacy that will take your mind’s breath away [even though they] stood two-thirds as tall and lived half as long as we do.” What we know about Babylonian mathematics we have inferred from a small set of ancient tablets, few of them even close to complete. As a result, there is lively — even heated — debate among those working in this area as to just how much the Babylonians knew and when and how they knew it (a sort of Babylon-gate debate if you will). To give you an idea of the fun, here’s a sample of one scholar describing the work of another: “The pretentious and polemical attempt … to find an alternative explanation of the table on Plimpton 322 is so confused and misleading that it should be completely disregarded…” (quoted on p. 8). It’s probably not as much fun for the scholars involved!
We are treated to a lively discussion of Babylonian mathematics and the clever means by which historians of mathematics attempt to reconstruct it. It’s here that many readers may need to consult another text. (There are also references to historians of mathematics which are much more interesting if you know the work or the person.) The key tablets are shown in photos which are then translated just below (page 24 is a good example). The translations involve converting Babylonian numerals (base 60) into something which make sense to modern readers. The convention is to use a sort of hybrid system in which base 60 places are separated by colons. Thus 11:5 represents 11×60 + 5. The transition from the integer part to the fractional part (the “sexagesimal point,” so to speak) is signaled by a semicolon, so that 12;15 indicates 12 + 15/60.
The chapter provides an exciting introduction to the wonderful mathematics practiced by these ancient people. This includes an extended discussion of the various means by which they might have computed square roots to the sort of accuracy presented in the tablets. Since there is (to date) no direct evidence as to the methods they employed, historians compare their values for a particular root with those obtained by later scholars (Plato, Heron, et al) and attempt to argue, based on the accuracy of the reported answers, that they might have had one or more of these techniques at hand.
Moving forward four chapters and several thousand years we encounter Elisha Scott Loomis (born 1852) and author of The Pythagorean Proposition: It Proofs Analyzed and Classified and Bibliography of Sources for Data of the Four Kinds of Proofs. We are treated to a wonderful collection of proofs (correct, neat, tidy and none of the above) as well as a fascinating bit of information about the Harvard Library copy of this book. On the inside front cover is a printed label:
This book was stolen from the Harvard College Library. It was later recovered. The thief was sentenced to two years at hard labor. 1932. (p. 82).
The quote is wonderful in itself, but also serves as a window into the Kaplans’ approach. The authors amaze, amuse, inform and bemuse — sometimes in a single paragraph.
In Chapter 9 we are treated to a discussion of the parallel postulate and non-Euclidean geometries. In particular, a clever sequence of theorems due to Legendre and Scott E. Brodie prove that PT is equivalent to the parallel postulate. The authors cite Brodie’s page at the wondrous Cut-The-Knot web site. I did a little sleuthing of my own and discovered that Brodie has several other nice contributions to Cut-The-Knot. A bit more sleuthing revealed that Brodie is an ophthalmologist practicing in New York. (My thanks to Dr. Brodie for returning my email and granting permission to reveal his profession.) The chapter concludes with a tour of geometry and trigonometry on various surfaces — a tour which is likely to fascinate undergraduates as well as bright high school students.
On the off chance that I haven’t convinced you to buy a copy for yourself and recommend your school library do the same, consider Chapter 6 in which we meet the Sierpinski Gasket, computer scientist Edsger Dijkstra, and the law of cosines. Of Dijkstra we learn of his seminal work in computer science and also that “arrogance in computer science is apparently measured in nano-Dijkstras” (p. 154). Among other contributions, Dijkstra proved the following neat result. Let sgn(a) be 1 if a > 0, –1 if a< 0, and 0 if a = 0). Then
sgn(a2 + b2 – c2) = sgn(α + β – γ),
where a, b, c are the lengths of the sides opposite the three angles α, β, γ. The authors comment on this formula: “There’s our whole book so far, freeze-dried to half a line of squiggles.” (page 153) Amusing, but false. Reducing Hidden Harmonies to a half line of squiggles would be equivalent to passing out the score to Beethoven’s 9th in lieu of a performance by the Chicago Symphony.
I hope my selected glimpses into its contents demonstrate that Hidden Harmonies is a book which rewards careful reading on many levels. In its attention to detail and clever word play it reminds me of Douglas Hofstadter’s Gödel, Escher, Bach. Amidst all the fun lurk numerous proofs of PT — some of which belong in Erdős’ famous book, while others most certainly do not. I recommend it for college libraries and as a secondary text for courses in the history of mathematics.
At one point the authors tell that us “The shortest distance in proof space is from insight to confirmation, but the richest voyages include those detours to sudden views that open up speculation on other and deeper vistas” (p. 207). Detours such as those are everywhere in this wonderful book — wander down a few and enjoy the view!
Richard Wilders is Marie and Bernice Gantzert Professor in the Liberal Arts and Sciences at North Central College in Naperville, IL. He teaches courses in the history of mathematics and science and is eagerly awaiting the next Kaplan book.