Trigonometric Delights

Eli Maor
Pages: 256
https://www.jstor.org/stable/j.ctt1r2f46

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. [Illustrations]
(pp. ix-x)
4. Preface
(pp. xi-2)
5. PROLOGUE: Ahmes the Scribe, 1650 B.C.
(pp. 3-10)

In 1858 a Scottish lawyer and antiquarian, A. Henry Rhind (1833-1863), on one of his trips to the Nile valley, purchased a document that had been found a few years earlier in the ruins of a small building in Thebes (near present-day Luxor) in Upper Egypt. The document, known since as the Rhind Papyrus, turned out to be a collection of 84 mathematical problems dealing with arithmetic, primitive algebra, and geometry.¹ After Rhind’s untimely death at the age of thirty, it came into the possession of the British Museum, where it is now permanently displayed. The papyrus as originally found...

6. Recreational Mathematics in Ancient Egypt
(pp. 11-14)

What is the meaning behind this cryptic verse? Clearly we have before us a geometric progression whose initial term and common ratio are both 7, and the scribe shows us how to find its sum. But as any good teacher would do to break the monotony of a routine math class, Ahmes embellishes the exercise with a little story which might be read like this: There are seven houses; in each house there are seven cats; each cat eats seven mice; each mouse eats seven ears of spelt; each ear of spelt produces seven hekat of grain. Find the total...

7. 1 Angles
(pp. 15-19)

Geometric entities are of two kinds: those of a strictly qualitative nature, such as a point, a line, and a plane, and those that can be assigned a numerical value, a measure. To this last group belong a line segment, whose measure is its length; a planar region, associated with its area; and a rotation, measured by its angle.

There is a certain ambiguity in the concept of angle, for it describes both the qualitative idea of “separation” between two intersecting lines, and the numerical value of this separation-the measure of the angle. (Note that this is not so with...

8. 2 Chords
(pp. 20-29)

When considered separately, line segments and angles behave in a simple manner: the combined length of two line segments placed end-to-end along the same line is the sum of the individual lengths, and the combined angular measure of two rotations about the same point in the plane is the sum of the individual rotations. It is only when we try to relate the two concepts that complications arise: the equally spaced rungs of a ladder, when viewed from a fixed point, do not form equal angles at the observer's eye (fig. 7), and conversely, equal angles, when projected onto a...

9. Plimpton 322: The Earliest Trigonometric Table?
(pp. 30-34)

Whereas the Egyptians wrote their records on papyrus and wood and the Chinese on bark and bamboo-all perishable materials-the Babylonians used clay tablets, a virtually indestructible medium. As a result, we are in possession of a far greater number of Babylonian texts than those of any other ancient civilization, and our knowledge of their history-their military campaigns, commercial transactions, and scientific achievements–is that much richer.

Among the estimated 500,000 tablets that have reached museums around the world, some 300 deal with mathematical issues. These are of two kinds: “table texts” and “problem texts,” the latter dealing with a variety...

10. 3 Six Functions Come of Age
(pp. 35-40)

An early Hindu work on astronomy, theSurya Siddhanta(ca. 400 A.D.), gives a table of half-chords based on Ptolemy's table (fig. 15). But the first work to refer explicitly to the sine as a function of an angle is theAryabhatiyaof Aryabhata (ca. 510), considered the earliest Hindu treatise on pure mathematics.¹ In this work Aryabhata (also known as Aryabhata the elder; born 475 or 476, died ca. 550)² uses the wordardha-jyafor the half-chord, which he sometimes turns around tojya–ardha(“chord–half”); in due time he shortens it tojyaorjiva.

Now begins...

11. Johann Müller, alias Regiomontanus
(pp. 41-49)

It is no coincidence that trigonometry up until the sixteenth century was developed mainly by astronomers. Aristarchus and Hipparchus, who founded trigonometry as a distinct branch of mathematics, were astronomers, as was Ptolemy, the author of theAlmagest.During the Middle Ages, Arab and Hindu astronomers, notably Abul-Wefa, al-Battani, Aryabhata, and Ulugh Beg of Samarkand (1393-1449), absorbed the Greek mathematical heritage and greatly expanded it, especially in spherical trigonometry. And when this combined heritage was passed on to Europe, it was again an astronomer who was at the forefront: Johann Müller (see fig. 17).

Müller was born in Unfinden, near...

12. 4 Trigonometry Becomes Analytic
(pp. 50-55)

With the great French mathematician Franςois Viète (also known by his Latin name Franciscus Vieta, 1540-1603), trigonometry began to assume its modern, analytic character. Two developments made this process possible: the rise of symbolic algebra–to which Viète was a major contributor–and the invention of analytic geometry by Fermat and Descartes in the first half of the seventeenth century. The gradual replacement of the cumbersome verbal algebra of medieval mathematics with concise, symbolic statements–a literal algebra–greatly facilitated the writing and reading of mathematical texts. Even more important, it enabled mathematicians to apply algebraic methods to problems that...

13. Franςois Viète
(pp. 56-62)

It is unfortunate that the names of so many of those who helped shape mathematics into its present form have largely vanished from today’s curriculum. Among them we may mention Regiomontanus, Napier, and Viète, all of whom made substantial contributions to algebra and trigonometry.

Franςois Viète was born in Fontenay le Comte, a small town in western France, in 1540 (the exact day is unknown). He first practiced law and later became involved in politics, serving as member of the parliament of Brittany. As was the practice of many learned men at the time, he latinized his name to Franciscus...

14. 5 Measuring Heaven and Earth
(pp. 63-79)

Since its earliest days, geometry has been applied to practical problems of measurement–whether to find the height of a pyramid, or the area of a field, or the size of the earth. Indeed, the very word “geometry” derives from the Greekgeo(earth) andmetron(to measure). But the ambition of the early Greek scientists went even farther: using simple geometry and later trigonometry, they attempted to estimate the size of the universe.

Aristarchus of Samos (ca. 310–230 B.C.) is considered the first great astronomer in history. Whereas most of his predecessors had derived their picture of the...

15. Abraham De Moivre
(pp. 80-86)

Abraham De Moivre was born in Vitry in the province of Champagne, France, on May 26, 1667, to a Protestant family.He showed an early interest in mathematics and studied it–secretly–at the various religious schools he was attending. In 1685 Louis XIV revoked the Edict of Nantes–a decree issued in 1598 granting religious freedom to French Protestants–and a period of repression followed. By one account De Moivre was imprisoned for two years before leaving for London, where he would spend the rest of his life. He studied mathematics on his own and became very proficient in it....

16. 6 Two Theorems from Geometry
(pp. 87-94)

In a circle the angle at the center is double of the angle at the circumference, when the angles have the same circumference as base.¹

In more common language, the proposition says that an angle inscribed in a circle (that is, an angle whose vertex lies on the circumference) is equal to half the central angle that subtends the same chord (fig. 28). Two corollaries from this theorem immediately follow: (1) In a given circle,allinscribed angles subtending the same chord are equal (this is Proposition 21 of Euclid; see fig. 29); and (2) All inscribed angles subtending the...

17. 7 Epicycloids and Hypocycloids
(pp. 95-107)

In the 1970s an intriguing educational toy appeared on the market and quickly became a hit: thespirograph.It consisted of a set of small, plastic–made wheels of varying sizes with teeth along their rims, and two large rings with teeth on their inside as well as outside rims (fig. 39). Small holes perforated each wheel at various distances from the center. You pinned down one of the rings onto a sheet of paper, placed a wheel in contact with it so the teeth would engage, and inserted a pen in one of the holes. As you moved the...

18. Maria Agnesi and Her “Witch”
(pp. 108-111)

Even today, women make up only about 10 percent of the total number of mathematicians in the United states;¹ worldwide their number is much smaller. But in past generations, social prejudices made it almost impossible for a woman to pursue a scientific career, and the total number of women mathematicians up to our century can be counted on two hands. Three names come to mind: Sonia Kovalevsky (1850–1891) of Russia, Emmy Noether (1882–1935), who was born in Germany but emigrated to the U.S., and Maria Agnesi of Italy.²

Maria Gaetana Agnesi (pronounced “Anyesi”) was born in Milan in 1718,...

19. 8 Variations on a Theme by Gauss
(pp. 112-116)

There is a story about the great German mathematician Carl Friedrich Gauss (1777–1855), who as a schoolboy was asked by his teacher to sum up the numbers from 1 to 100, and who almost immediately came up with the correct answer, 5,050. To the amazed teacher Gauss explained that he had merely noticed that by writing the numbers twice, first from 1 to 100 and then from 100 to 1, and adding the two sums vertically, each pair adds up to 101. Since there are one hundred such pairs, we get 100 x 101 = 10,100, and since this...

20. 9 Had Zeno Only Known This!
(pp. 117-128)

Can space be endlessly divided, or is there a smallest unit of space, a mathematical atom that cannot further be split? Is motion continuous, or is it but a succession of snapshots which, like the frames in an old motion picture, are themselves frozen in time? Questions such as these were hotly debated by the philosophers of ancient Greece, and they are still being debated today–witness the never-ending search for the ultimate elementary particle, that elusive building block from which all matter is supposedly made.

The Greek philosopher Zeno of Elea, who lived in the fifth century B.C., summarized...

21. 10 $\left( {\sin x} \right)/x$
(pp. 129-138)

Students of calculus encounter the function$\left( {\sin x} \right)/x$early in their study, when it is shown that$\mathop {\lim }\limits_{x \to 0} \left( {\sin x} \right)/x = 1;$this result is then used to establish the differentiation formulas${\left( {\sin x} \right)^'} = \cos x{\rm{ and }}{\left( {\cos x} \right)^'} = - \sin x$. Once this has been done, however, the function is soon forgotten, and the student rarely sees it again. This is unfortunate, for this simple-looking function not only has some remarkable properties, but it also shows up in many applications, sometimes quite unexpectedly.

We note, to begin with, that the function is defined for all values of$x$except 0; but we also know that as$x$gets smaller and smaller, the ratio...

22. 11 A Remarkable Formula
(pp. 139-144)

We are not quite done yet with the function$\left( {\sin x} \right)/x$. Browsing one day through a handbook of mathematical formulas, I came across the following equation:

$\frac{{\sin x}}{x} = \cos \frac{x}{4}.\cos \frac{x}{4}.\cos \frac{x}{8}...$.

As I had never seen this formula before, I expected the proof to be rather difficult. To my surprise, it turned out to be extremely simple:

$\sin x = 2\sin x/2{\rm{ }}\cdot{\rm{ }}\cos x/2$

$\begin{array}{l}= 4\sin x/4 \cdot \cos x/4 \cdot \cos x/2 \\ = 8\sin x/8 \cdot \cos x/8 \cdot \cos x/4 \cdot \cos x/2 \\ = ... \\\end{array}$

After repeating this process n times, we get

$\sin x = {2^n}\sin x/{2^n} \cdot \cos x/{2^n} \cdot ... \cdot \cos x/2.$

Let us multiply and divide the first term of this product by x (assuming,of course,that$x \ne 0$and rewrite it as$x \cdot [(\sin x/{2^n})/(x/{2^n})];$we then have

$\sin x = x \cdot \left[ {\frac{{\sin x/{2^n}}}{{x/{2^n}}}} \right] \cdot \cos x/2 \cdot \cos x/4 \cdot .......... \cdot \cos x/{2^n}$

Note that we have reversed the order of the...

23. Jules Lissajous and His Figures
(pp. 145-149)

Jules Antoine Lissajous (1822–1880) is not among the giants in the history of science, yet his name is known to physics students through the “Lissajous figures”–patterns formed when two vibrations along perpendicular lines are superimposed. Lissajous entered the École Normale Supérieure in 1841 and later became professor of physics at the Lycée Saint-Louis in Paris, where he studied vibrations and sound. In 1855 he devised a simple optical method for studying compound vibrations: he attached a small mirror to each of the vibrating objects (two tuning forks, for example) and aimed a beam of light at one of...

24. 12 $\tan x$
(pp. 150-164)

Of the numerous functions we encounter in elementary mathematics, perhaps the most remarkable is the tangent function. The basic facts are well known:$f\left( x \right) = \tan x$has its zeros at$x = n\pi \left( {n = 0, \pm 1, \pm 2,...} \right)$, has infinite discontinuities at$\left( {2n + 1} \right)\pi /2$, and has period$\pi$(a function f (x) is said to have a period P if P is the smallest number such that$f\left( {x + p} \right) = f\left( x \right)$for all x in the domain of the function). This last fact is quite remarkable: the functions$\sin x$and$\cos x$have the common period$2\pi$,yet their ratio,$\tan x$reduces the period to$\pi$. when it comes to periodicity, the ordinary rules of the algebra of...

(pp. 165-180)

From the sublime beauty of Euler’s formulas we now turn to a more mundane matter: the science of map making. It is common knowledge that one cannot press the peels of an orange against a table without tearing them apart: no matter how carefully one tries to do the job, some distortion is inevitable. Surprisingly, it was not until the middle of the eighteenth century that this fact was proved mathematically, and by none other than Euler: his theorem says that it is impossible to map a sphere onto a flat sheet of paper without distortion. Had the earth been...

26. 14 $\sin x = 2$: Imaginary Trigonometry
(pp. 181-191)

Imagine you have just bought a brand new hand-held calculator and find to your dismay that when you try to subtract 5 from 4, you get an error sign. Yet this is precisely the situation in which first-grade pupils would find themselves if their teacher asked them to take away five apples from four: “It cannot be done!” would be the class’s predictable response.

The history of mathematics is full of attempts to break the barrier of the “impossible.” Many of these attempts have ended in failure: for more than two thousand years mathematicians tried to find a construction, using...

27. Edmund Landau: The Master Rigorist
(pp. 192-197)

Edmund Yehezkel Landau was born in Berlin in 1877; his father was the well-known gynecologist Leopold Landau. He began his education at the French Gymnasium (high school) in Berlin and soon thereafter devoted himself entirely to mathematics. Among his teachers was Ferdinand Lindemann (1852–1939), who in 1882 proved the transcendence of$\pi$–the fact that$\pi$cannot be the root of a polynomial equation with integer coefficients–thereby settling the age-old problem of “squaring the circle” (see p. 181). From the beginning Landau was interested in analytic number theory–the application of analytic methods to the study of integers....

28. 15 Fourier’s Theorem
(pp. 198-210)

Trigonometry has come a long way since its inception more than two thousand years ago. But three developments, more than all others, stand out as having fundamentally changed the subject: Ptolemy's table of chords, which transformed trigonometry into a practical, computational science; De Moivre's theorem and Euler's formula${e^{ix}} = \cos x + i\sin x$, which merged trigonometry with algebra and analysis; and Fourier’s theorem, to which we devote this last chapter.

Jean Baptiste Joseph Fourier was born in Auxerre in north-central France on March 21, 1768. By the age of nine both his father and mother had died. Through the influence of some friends of...

29. Appendix 1 Let’s Revive an Old Idea
(pp. 213-217)
30. Appendix 2 Barrow’s Integration of $\sec \phi$
(pp. 218-219)
31. Appendix 3 Some Trigonometric Gems
(pp. 220-221)
32. Appendix 4 Some Special Values of sin α
(pp. 222-224)
33. Bibliography
(pp. 225-228)
34. Credits for Illustrations
(pp. 229-230)
35. Index
(pp. 231-240)