# Trigonometric Delights

In the first two sentences of the preface, Maor writes

This book is neither a textbook of trigonometry —of which there are many— nor a comprehensive history of the subject, of which there are almost none. It is an attempt to present selected topics in trigonometry from a historic point of view and to show their relevance to other sciences.

I could not think of a better characterisation of the book than this. All I can add to this description is to give an idea of which kind of topics were selected and what kind of applications have benefited from these developments.

The successive chapters are organised more or less chronologically, starting with a prologue about the Egyptian Rhind papyrus from around 16-17th century B.C. and ending with Fourier series (18th century). There is of course much attention for the history, but what strikes me in particular, is how much attention is given to the etymology of the mathematical terminology. The origin of the words algorithm and algebra is described in several publications as originating from the Arab author al-Khwarizmi and from al-jabr, which is part of the title of his book, but what is the origin of words such as sine, secant, and many other common mathematical words? Maor carefully pays attention to this. He also shows how trigonometry, which originally was about angles like in pyramid building problems in Egypt, were somewhat made more abstract, in a geometric context of triangles by the Greek, but later, it became more and more part of analysis. The sine and cosine were not only tabulated for computational purposes, but they became functions so that now we see x in sin x as a real or even a complex number, not necessarily corresponding to a physical or geometric angle. The original idea of an angle in degrees or radians in the goniometric unit circle has become somewhat obsolete.

But of course it all starts with angles and chords in planar circles for the Greek, and even earlier in astronomy, which is essentially a three dimensional spherical discipline as practised by Babylonians and almost any civilisation of antiquity. This is the subject of the first two chapters. Then appeared tables of goniometric values of what became our basic goniometric functions. This opens the possibility to introduce algebra (goniometric identities) and gradually also analysis (involving series) into the discipline. This helped considerably to discover (actually re-discover) the heliocentric interpretation of our solar system and to measure our own planet by triangulation and those practical problems in turn stimulated the development of associated trigonometric identities in triangles. But before the heliocentric model, the trajectories of the planets required also more general curves than ther circle like epi- and epo-circles which allow an easy description in terms of trigonometric formulas. Then Maor ventures into a period of proper analysis with the Sine integral and many other relations and series expansions, not in the least for the fascinating number π. These were obtained by master minds such as Gauss and Euler. As complex numbers entered the picture, with Euler's fabulous formula, we are fully involved in complex analysis, conformal maps and ultimately Fourier analysis.

This marvelous survey by Maor of some episodes in the historical evolution of mathematics also allows to sketch some biographies of important mathematicians. There are the "usual suspects" from Greek antiquity (including Zeno whose paradoxes are discussed when infinitesimals from analysis are introduced). Also Regiomontanus (15th C.), François Viète (16th C.), De Moivre (17th C.), Maria Agnesi and her "witch" (18th C.), Jules Lissajous (19th C.), Edmund Landau (20th C.) are discussed in somewhat more detail. aot only the history and mathematicians, also the applications are well documented: astronomy, cartography, spirographs, periodic oscillation, music; and there are detailed mathematical derivations of several trigonometric and other mathematical identities, conformal maps, series converging to π, the solution of the Basel problem by Euler, how Gauss showed that any trigonometric summation formula can be represented geometrically, etc.

All these items are treated requiring only some elementary trigonometric formulas. Some of the standard identities are collected in appendices. In another appendix we find Maor's plea to re-introduce the unit circle and the geometric definitions of the trigonometric functions like cos and sin being projections of the circular point on x- and y-axis, etc. instead of the "New Math" approach. Also Barrow's integration of sec x is moved to an appendix. All the chapters are completed with a section containing notes and references to the sources used. There are many useful mathematical graphs and some grayscale images.

This is an interesting mixture of mathematical history, illustrating the evolution and the usefulness of trigonometry throughout the centuries, and on top of that, it gives some mathematical training by deriving formulas and identities that are easily accessible with only some elementary mathematics knowledge. The book appeared originally in 1998 and is here reprinted in its original form as a volume in the Princeton Science Library. So this is a fortunate occasion to bring this great book back under the attention of a broad audience.

**Submitted by Adhemar Bultheel |

**16 / May / 2020