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  3, 4, 5, 6, 8, 10, 12, 15, 16, 20.

  We now know that they cannot be constructed when the number of sides is

  7, 9, 11, 13, 14, 18, 19.

  which leaves one number in this range, 17, as yet unaccounted for. The story of the 17-gon will be told in its rightful place; it is important for more reasons than purely mathematical ones.

  In discussing geometry, there is no substitute for drawing on a sheet of paper with a real straightedge and real compass. It gives you a feel for how the subject fits together. I’m going to take you through my favorite construction, for the regular hexagon. I learned it from a book my uncle gave me in the late 1950s, called Man Must Measure, and it’s lovely:

  How to construct a regular hexagon.

  Fix the radius of the compass throughout, so all circles will be of the same size. (1) Draw a circle. (2) Choose a point on it and draw a circle centered at that point. This crosses the original circle in two new points. (3) Draw circles with these points as centers, to get two more crossings. (4) Draw circles with these points as centers; both pass through the same new crossing point. The six points can now be connected to form a regular hexagon. It is aesthetically pleasing (though mathematically unnecessary) to complete the picture with (5): Draw a circle centered on the sixth point. Then six circles meet at the center of the original one, forming a flower shape.

  Euclid used a very similar method, which is simpler but not quite so pretty, and he proved that it works: you can find it in Proposition 15 of Book IV.

  3

  THE PERSIAN POET

  W ake! For the Sun, who scattered into flight

  The Stars before him from the Field of Night,

  Drives Night along with them from Heav’n and strikes

  The Sultán’s Turret with a Shaft of Light.

  To most of us, the name of Omar Khayyám is indelibly associated with his long ironic poem, the Rubaiyat, and specifically with the elegant translation into English by Edward Fitzgerald. To historians of mathematics, however, Khayyám has a greater claim to fame. He was prominent among the Persian and Arab mathematicians who took up the torch that the Greeks had dropped, and continued the development of new mathematics after scholars in Western Europe descended into the dark ages and its scholars abandoned theorem-proving for theological disputation.

  Among Khayyám’s great achievements is the solution, by respectable methods of Greek geometry, of cubic equations. His techniques necessarily went beyond the straightedge and compass that tacitly limit Euclidean geometry, because these tools are simply not up to the job—a fact that the Greeks strongly suspected, but could not prove because they lacked the necessary point of view, which was not geometry but algebra. But Khayyám’s methods did not go much beyond straightedge and compass. He relied on special curves known as “conic sections” because they can be constructed by slicing a cone with a plane.

  The conventional wisdom in popular science writing is that every equation halves a book’s sales. If true, this is very bad news, because nobody would be able to understand some of the key themes of this book without being shown a few equations. The next chapter, for instance, is about Renaissance mathematicians’ discoveries of formulas that solve any cubic or quartic equation. I can get away without showing you what the quartic formula looks like, but we really will need to take a quick look at the formula for the cubic. Otherwise, all I can tell you is something like “multiply some numbers by some other numbers and add some numbers to that, and then take the square root, then add another number and take the cube root of the result; then do the same thing again with slightly different numbers; finally, add the two results together. Oh, and I forgot to mention—you have to do some dividing as well.”

  Some writers have challenged the conventional wisdom and even written books about equations. They seem to be following the old showbiz saying, “If you’ve got a wooden leg, wave it.” Now, there is a sense in which this book is about equations; but just as you can write a book about mountains without requiring your readers to climb one, you can write a book about equations without requiring your readers to solve one. Still, readers of a book about mountains probably won’t understand it if they have never seen a mountain, so it really will help us both a lot if I show you a few carefully selected equations.

  The ground rules, slanted heavily in your favor, are these: The word is “show.” I want to you to see the equation. You needn’t do anything with it. When necessary, I will pick the equation to pieces and explain which features matter for our story. I will never ask you to solve an equation or calculate with one. And I will do my utmost to avoid them whenever I can.

  When you get to know them, equations are actually rather friendly. They are clear, concise, sometimes even beautiful. The secret truth about equations is that they are a simple, clear language for describing certain “recipes” for calculating things. When I can tell you the recipe in words, or just give you enough feel for how it goes that the details don’t matter, I will. On rare occasions, though, it becomes so cumbersome to use words that I’ll have to use symbols.

  There are three kinds of important symbols for this book, and I’ll mention two of them now. One is our old friend x, “the unknown.” This symbol stands for a number that we do not yet know, but whose value we are desperately trying to find out.

  The second type of symbol is little raised numbers, like 2 or 3 or 4. They are instructions to multiply some other number by itself the appropriate number of times. So 53 means 5 × 5 × 5, which is 125, and x2 means x × x, where x is our symbol for an unknown number. They are read as “squared,” “cubed,” “raised to the fourth power,” etc., and collectively they are referred to as powers of the number concerned.

  I haven’t the foggiest idea why. They have to be called something.

  Either the Babylonian method for solving quadratic equations was passed on to the ancient Greeks, or they reinvented it. Heron, who lived in Alexandria somewhere between 100 BCE and 100, discussed a typical Babylonian-style problem in Greek terminology. Around the year 100, Nichomachus, probably an Arabian hailing from Judea, wrote a book called Introductio Arithmetica in which he abandoned the Greek tradition of representing numbers by geometrical quantities such as lengths or areas. To Nichomachus, numbers were quantities in their own right, not lengths of lines. Nichomachus was a Pythagorean, and his work shows it: he deals only in whole numbers and their ratios, and he uses no symbols. His book became the standard arithmetic text for the next millennium.

  Symbolism entered into algebra in the work of a Greek mathematician named Diophantus, sometime around 500. The only thing that we know about Diophantus is his age at death, and that comes to us via a route of dubious authenticity. A Greek collection of algebra problems contains one that reads like this: “Diophantus spent one-sixth of his life as a boy. His beard grew after a further one-twelfth. He married after another one-seventh, and his son was born five years later. The son lived to half his father’s age and the father died four years after the son. How old was Diophantus when he died?”

  Using that ancient algebraist’s own methods, or more modern ones, you can deduce that he must have been 84. It was a good age, assuming the algebra problem is based on fact, which is questionable.

  That’s all we know of his life. But we know quite a bit about his books, through later copies and references in other documents. He wrote one book on polygonal numbers, and part of it survives. It is arranged in Euclidean style, proves theorems using logical arguments, and has little mathematical significance. Far more significant were the 13 books of the Arithmetica. Six of them are still in existence, thanks to a thirteenth-century Greek copy of an earlier copy. Four more may have surfaced in a manuscript found in Iran, but not all scholars are convinced that it traces back to Diophantus.

  The Arithmetica is presented as a series of problems. In the preface, Diophantus says he wrote it as a book of exercises for one of his students. He used a special symbol for the unknown, and different symbols for
its square and cube that seem to be abbreviations of the words dynamis (power) and kybos (cube). The notation is not very structured. Diophantus adds symbols by putting them next to each other (as we now do for multiplication) but has a special symbol for subtraction. He even has a symbol for equality, though this may have been introduced by later copyists.

  Mostly, the Arithmetica is about solving equations. The first surviving book discusses linear equations; the other five treat various kinds of quadratic equations, often in several unknowns, and a few special cubic equations. A key feature is that the answers are always integers or rational numbers. Today we call an equation “Diophantine” if its solutions are restricted to integers or rational numbers. A typical example from the Arithmetica is, “Find three numbers such that their sum, and the sum of any two, is a perfect square.” Try it—it’s by no means easy. Diophantus’s answer is 41, 80, and 320. The sum of all three is 441 = 212. The sums of pairs are 41 + 80 = 121 = 112, 41 + 320 = 361 = 192, and 80 + 320 = 400 = 202. Clever stuff.

  Diophantine equations are central to modern number theory. A famous example is Fermat’s “last theorem,” which states that two perfect cubes, or higher powers, cannot add to form a similar power. With squares, this kind of thing is easy, and goes back to Pythagoras: 32 + 42 = 52 or 52 + 122 = 132. But you can’t do the same with cubes, fourth powers, fifth powers, or anything higher than the square. Pierre de Fermat scribbled this conjecture (without a proof; it wasn’t a theorem despite its name) in the margin of his personal copy of the Arithmetica around 1650. It took nearly 350 years before Andrew Wiles, a British-born number theorist living in America, proved that Fermat was right.

  The historical tradition in mathematics is sometimes very long.

  Algebra really arrived on the mathematical scene in 830, when the main action moved from the Greek world to the Arabic one. In that year the astronomer Mohamed ibn Musa al-Khwārizmī wrote a book called al-Jabr w’al Muqâbala, which translates roughly as “restoration and simplification.” The words refer to standard techniques for manipulating equations so as to put them into a better form for solution. From al-jabr comes our word “algebra.” The first Latin translation in the twelfth century bears the title Ludus Algebrae et Almucgrabalaeque.

  Al-Khwārizmī’s book contains hints of earlier influences, Babylonian and Greek, and also rests on ideas introduced in India by Brahmagupta around 600. It explains how to solve linear and quadratic equations. Al-Khwārizmī’s immediate successors worked out how to solve a few special kinds of cubic. Among them are Tâbit ibn Qorra, a doctor, astronomer, and philosopher who lived in Baghdad and was a pagan, and an Egyptian named al-Hasan ibn al-Haitham, generally referred to in later Western writings as Alhazen. But the most famous of them all is Omar Khayyám.

  Omar bore the full name Ghiyath al-Din Abu’l-Fath Umar ibn Ibrahim Al-Nisaburi al-Khayyámi. The word “al-Khayyámi” translates literally as “tent-maker,” which some scholars believe may have been the trade of his father Ibrahim. Omar was born in Persia in 1047, and spent most of his productive life at Naishapur. You can find it in an atlas as Neyshabur, a city near Masshad in the Khorosan province of northeastern Iran, close to the border with Turkmenistan.

  Legend has it that in his youth, Omar left home to study Islam and the Quran under the celebrated cleric Imam Mowaffak, who lived in Naishapur. There he struck up a friendship with two fellow students, Hasan Sabah and Nizam al-Mulk, and the three of them made a pact. If any of them became rich and famous—not unlikely for students of Mowaffak—that person would share his wealth and power with the other two.

  The students completed their studies and the years flew by; the pact remained in force. Nizam traveled to Kabul. Omar, politically less ambitious, spent some time as a tent-maker—another possible explanation of the name “Al-Khayyámi.” Science and mathematics became his passions, and he spent most of his spare time on them. Eventually Nizam returned, secured a position in the government, and became administrator of affairs to the sultan Alp Arslan, with an office in Naishapur.

  Since Nizam was now rich and famous, Omar and Hasan claimed their rights under the pact. Nizam asked the sultan for permission to assist his friends, and when it was granted he honored the agreement. Hasan received a well-paid government job, but Omar merely wished to continue his scientific studies in Naishapur, where he would pray for the health and well-being of Nizam. His old school friend arranged for Omar to be given a government salary, to free his time for study, and the deal was done.

  Hasan later tried to overthrow a senior official and lost his sinecure, but Omar continued serenely on and was appointed to a commission whose mandate was to reform the calendar. The Persian calendar was based on the movements of the sun, and the date of the first day of the new year was subject to change, which was confusing. It was just the job for a competent mathematician, and Omar applied his knowledge of mathematics and astronomy to calculate when New Year’s Day should fall in any given year.

  Around this time, he also penned the Rubaiyat, which loosely translates as “quatrains,” a poetic form. A rubai was a four-line verse with a rather specific rhyming pattern—more accurately, one of two possible patterns—and a rubaiyat was a collection of verses in this form. One verse makes a clear reference to his work on reforming the calendar:

  Ah, but my computations, People say,

  Reduced the Year to better Reckoning? Nay,

  ’Twas only striking from the Calendar

  Unborn To-morrow and dead Yesterday.

  Omar’s verses were distinctly nonreligious. Many of them praise wine and its effects. For instance:

  And lately, by the Tavern Door agape,

  Came shining through the Dusk an Angel Shape

  Bearing a Vessel on his Shoulder; and

  He bid me taste of it; and ’twas—the Grape!

  There are wry allegorical references to wine, as well:

  Whether at Naishápúr or Babylon,

  Whether the Cup with sweet or bitter run,

  The Wine of Life keeps oozing drop by drop,

  The Leaves of Life keep falling one by one.

  Other verses poke fun at religious beliefs. One wonders what the Sultan thought of the man he had put on retainer, and what the imam thought of the outcome of his teaching.

  Meanwhile, the disgraced Hasan, having been forced to leave Naishapur, fell in with a gang of bandits and made use of his superior education to become its leader. In the year 1090 those bandits, under Hasan’s command, captured Alamut castle in the Elburz Mountains, just south of the Caspian Sea. They terrorized the region, and Hasan became notorious as the Old Man of the Mountains. His followers, known as the Hashishiyun for their use of the drug hashish (a very potent form of cannabis), built six mountain fortifications, from which they would emerge to kill carefully selected religious and political figures. Their name was the origin of the word “assassin.” So Hasan managed to become rich and famous in his own right, as befitted a student of Mowaffak, though he was not, by this time, disposed to share his fortune with his old schoolmates.

  While Omar calculated astronomical tables and worked out how to solve the cubic, Nizam pursued his political career until, in a touch of exquisite irony, Hasan’s bandits assassinated him. Omar lived on to the age of 76, dying—so it is said—in 1123. Hasan died the following year, aged 84. The assassins continued to wreak political havoc until they were wiped out by the Mongols, who conquered Alamut in 1256.

  To return to Omar’s mathematics: Around 350 BCE the Greek mathematician Menaechmus discovered the special curves known as “conic sections,” which he used, scholars believe, to solve the problem of doubling the cube. Archimedes developed the theory of these curves, and Apollonius of Perga systematized and extended the subject in his book Conic Sections. What particularly interested Omar Khayyám was the Greek discovery that conic sections could be used to solve certain cubic equations.

  Conic sections are so named because they can be obtained by slicing a cone with a plane
. More properly, a double cone, like two ice-cream cones joined at their sharp ends. A single cone is formed by a collection of straight-line segments, all meeting at one point and passing through a suitable circle, the “base” of the cone. But in Greek geometry you can always extend straight-line segments as far as you wish, and the result is to create a double cone.

  The three main types of conic section are the ellipse, parabola, and hyperbola. An ellipse is a closed oval curve that arises when the cutting plane passes through only one half of the double cone. (A circle is a special kind of ellipse, created when the plane is exactly perpendicular to the cone’s axis.) A hyperbola consists of two symmetrically related open curves, which in principle extend to infinity, that arise when the cutting plane passes through both halves of the double cone. The parabola is a transitional form, a single open curve, and in this case the cutting plane must be parallel to one of the lines lying on the surface of the cone.

  Conic sections.

  At great distances from the tip of the cone, the curves of a hyperbola become ever closer to two straight lines, which are parallel to the lines where a parallel plane through the tip would cut the cone. These lines are called asymptotes.

  The Greek geometers’ extensive studies of conic sections constituted their most significant area of progress beyond the ideas codified by Euclid. These curves remain vitally important in today’s mathematics, but for quite different reasons from those that interested the Greeks. From the algebraic point of view, they are the next simplest curves after the straight line. They are also important in applied science. The orbits of planets in the solar system are ellipses, as Kepler deduced from Tycho Brahe’s observations of Mars. This elliptical orbit is one of the observations that led Newton to formulate his famous “inverse square law” of gravity. This in turn led to the realization that some aspects of the universe exhibit clear mathematical patterns. It opened up the whole of astronomy by making planetary phenomena computable.