Trigonometric Delights

You may, as I did, find the juxtaposition of the two title words a bit hard to swallow. The word trigonometry brought to my mind cold analytic facts about right triangles, tedious derivations of half- and double-angle formulae and Laws of Sines and Cosines, and dull, lifeless stories about ladders and flagpoles. When first studying trigonometry, I would not have used the word "delight."

Trigonometric Delights, by Eli Maor, has changed my mind. Professor Maor takes his readers back to trigonometry in a way that is, well, delightful. Students, teachers, professional and recreational mathematicians will enjoy reading this book.

Maor writes Trigonometric Delights from an historical perspective, but it is not a history book. It contains many theorems and results of trigonometry, but it is not a textbook. Rather, Maor achieves a satisfying blend of mathematics and history, creating a work that informs, teaches, and stimulates thought, while underscoring that mathematics is a human endeavor, not a stale collection of facts that exist in a vacuum. His book is the labor of a missionary whose aim is to deepen our appreciation of ideas and the people who developed them, ideas about which we have heard, but have not fully enjoyed. It is evident throughout that Maor is devoted to his subject. His love for trigonometry is contagious. He writes enthusiastically and engagingly.

Maor takes us from Ahmes the Scribe and the Rhind Papyrus (circa 1650 BC), to Fourier and his theory of trigonometric series. In between there are intriguing discussions of many well-known and some not so well-known historical figures and their roles in the development of trigonometry. For example, we learn of the inclusion in Ptolemy’s Almagest of a table of chord lengths of a certain circle. This is the first known table of trigonometric function values, which are actually half-angle values for the sine function. (Interestingly, Ptolemy discussed neither right triangles nor the more modern definitions of sine or cosine functions as coordinates of points on the unit circle). Another example: Maor writes of the great geodetic survey undertaken for the government of France by Abbé Jean Picard, and, later, several generations of Cassinis. This survey eventually led to the determination that the earth is an oblate spheroid, a fact predicted by Newton, but disputed by many of his contemporaries.

Mathematical gems fill the pages. Maor elegantly derives the Law of Sines and the double-angle and half-angle formulae for sine and cosine from Euclid’s simple theorem about circles: The measure of an angle inscribed in a circle is double the central angle of the arc it subtends. He goes on to argue convincingly that "the Law of Sines is really a theorem about circles," since "each side of a triangle inscribed in a unit circle is equal to the sine of the opposite angle." Another lovely example is his description of the relationship of the function sin(x)/x to the ratio of circumference to diameter of a circle on a sphere. Yet more delightful is a surprising geometric construction (via straightedge and compass) of the sum of any convergent infinite geometric series. Maor accomplishes this by rewriting the series

as

(assuming the angle a=arc cos(x) is given). I am tempted to describe more, but these examples suffice.

The book is accessible to motivated, mature students with a background merely consisting of high school algebra, geometry, and, in some parts, a bit of calculus. Teachers will find it useful, too. Twice in the past several weeks I have had occasion to use what I learned in this book. The first was in a calculus class when a student asked how the sine function got its name. The second was in a Modern Geometries course. The question arose whether there is a device for producing straight lines in a fashion analogous to the way we use compasses to construct circles. That is, when we use a compass we do not copy models of circles, but rather use a property of circles to build them. Is there such a device for creating straight lines? Maor’s wonderful etymology of sine handled the first question, his discussion of hypocycloids the second. (Do you remember spirographs?)

Most of the information in this book is not new. Much of it I had heard or read about before, but only in bits and pieces, here and there. It is a pleasure to find it all assembled here so skillfully. Though any of the chapters or biographical vignettes can be read independently, Maor has succeeded in weaving them together in a way that makes delightful reading from cover to cover. Trigonometric Delights is a welcome addition to my bookshelf. I recommend it for yours and for your school's library.

Sean Bradley ( sbradley@clarke.edu) is assistant professor of Mathematics at Clarke College in Dubuque, Iowa.