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  The majority of Omar’s extant mathematics is devoted to the theory of equations. He considered two kinds of solution. The first, following the lead of Diophantus, he called an “algebraic” solution in whole numbers; a better adjective would be “arithmetic.” The second kind of solution he called “geometric,” by which he meant that the solution could be constructed in terms of specific lengths, areas, or volumes by geometrical means.

  Making liberal use of conic sections, Omar developed geometric solutions for all cubic equations, and explained them in his Algebra, which he completed in 1079. Because negative numbers were not recognized in those days, equations were always arranged so that all terms were positive. This convention led to a huge number of case distinctions, which nowadays we would consider to be essentially the same except for the signs of the numbers. Omar distinguished fourteen different types of cubic, depending on which terms appear on each side of the equation. Omar’s classification of cubic equations went like this:

  cube = square + side + number

  cube = square + number

  cube = side + number

  cube = number

  cube + square = side + number

  cube + square = number

  cube + side = square + number

  cube + side = number

  cube + number = square + side

  cube + number = square

  cube + number = side

  cube + square + side = number

  cube + square + number = side

  cube + side + number = square

  Each listed term would have a positive numerical coefficient. You may be wondering why this list does not include cases like

  cube + square = side

  The reason is that in these cases we can divide both sides of the equation by the unknown, reducing it to a quadratic.

  Omar did not entirely invent his solutions but instead built on earlier Greek methods for solving various types of cubic equation using conic sections. He developed these ideas systematically, and solved all fourteen types of cubic by such methods. Previous mathematicians, he noted, had discovered solutions of various cases, but these methods were all very special and each case was tackled by a different construction; no one before him had worked out the whole extent of possible cases, let alone found solutions to them. “Me, on the contrary—I have never ceased to wish to make known, with exactitude, all of the possible cases, and to distinguish among each of the cases the possible and impossible ones.” By “impossible” he meant “having no positive solution.”

  To give a flavor of his work, here is how he solved “A cube, some sides, and some numbers are equal to some squares,” which we would write as

  x3 + bx + c = ax2.

  (Since we don’t care about positive versus negative, we would probably move the right-hand term to the other side and change a to –a as well: x3 – ax2 + bx + c = 0).

  Omar instructs his readers to carry out the following sequence of steps. (1) Draw three lines of lengths c/b, , and a, with a right angle. (2) Draw a semicircle whose diameter is the horizontal line. Extend the vertical line to cut it. If the solid vertical line has length d, make the solid horizontal line have length cd/. (3) Draw a hyperbola (solid line) whose asymptotes (those special straight lines that the curves approach) are the shaded lines, passing through the point just constructed. (4) Find where the hyperbola cuts the semicircle. Then the lengths of the two solid lines, marked x, are both (positive) solutions of the cubic.

  Omar Khayyám’s solution of a cubic equation.

  The details, as usual, matter much less than the overall style. Carry out various Euclidean constructions with ruler and compass, throw in a hyperbola, carry out some more Euclidean constructions—done.

  Omar gave similar constructions to solve each of his fourteen cases, and proved them correct. His analysis has a few gaps: the points required in his construction sometimes fail to exist when the sizes of the coefficients a, b, c are unsuitable. In the construction above, for example, the hyperbola may not meet the semicircle at all. But aside from these quibbles, he did an impressive and very systematic job.

  Some of the imagery in Omar’s poetry is mathematical and seems to allude to his own work, in the self-deprecatory tone that we find throughout:

  For “Is” and “Is-Not,” though with Rule and Line

  And “Up-And-Down” by logic I define,

  Of all that one should care to fathom, I

  Was never deep in anything but—Wine.

  One especially striking stanza reads:

  We are no other than a moving row

  Of Magic Shadow-shapes that come and go

  Round with the Sun-illumined Lantern held

  In midnight by the Master of the Show.

  This recalls Plato’s celebrated allegory of shadows on a cave wall. It serves equally well as a description of the symbolic manipulations of algebra, and the human condition. Omar was a gifted chronicler of both.

  4

  THE GAMBLING SCHOLAR

  “I swear to you, by God’s holy Gospels, and as a true man of honor, not only never to publish your discoveries, if you teach me them, but I also promise you, and I pledge my faith as a true Christian, to note them down in code, so that after my death no one will be able to understand them.”

  This solemn oath was—allegedly—sworn in 1539.

  Renaissance Italy was a hotbed of innovation, and mathematics was no exception. In the iconoclastic spirit of the age, Renaissance mathematicians were determined to overcome the limitations of classical mathematics. One of them had solved the mysterious cubic. Now he was accusing another of stealing his secret.

  The irate mathematician was Niccolo Fontana, nicknamed “Tartaglia,” the stammerer. The alleged thief of his intellectual property was a mathematician, a doctor, an incorrigible rogue, and an inveterate gambler. His name was Girolamo Cardano, aka Jerome Cardan. Around 1520, Girolamo, a true prodigal son, had worked his way through his father’s legacy. Broke, he turned to gambling as a source of finance, putting his mathematical abilities to effective use in assessing the chances of winning. He kept dubious company; once, when he suspected another player of cheating, he slashed the man’s face with a knife.

  They were hard times, and Girolamo was a hard man. He was also a highly original thinker, and he wrote one of the most famous and influential algebra texts in history.

  We know a lot about Girolamo because in 1575 he told us all about himself in The Book of My Life. It begins thus:

  This Book of My Life I am undertaking to write after the example of Antoninus the Philosopher, acclaimed the wisest and best of men, knowing well that no accomplishment of mortal man is perfect, much less safe from calumny; yet aware that none, of all ends which man may attain, seems more pleasing, none more worthy than recognition of the truth.

  No word, I am ready to affirm, has been added to give savor of vainglory, or for sake of mere embellishment; rather, as far as possible, mere experiences were collected, events of which my pupils . . . had some knowledge, or in which they took part. These brief cross-sections of my history were in turn written down by me in narrative form to become this my book.

  Like many mathematicians of the period, Girolamo practiced astrology, and he notes the astrological circumstances surrounding his birth:

  Although various abortive medicines—as I have heard—were tried in vain, I was normally born on the 24th day of September in the year 1500, when the first hour of the night was more than half run, but less than two thirds . . . Mars was casting an evil influence on each luminary because of the incompatibility of their positions, and its aspect was square to the moon.

  . . . I could easily have been a monster, except for the fact that the place of the preceding conjunction had been 29° in Virgo, over which Mercury is the ruler. And neither this planet nor the position of the moon or of the ascendant is the same, nor does it apply to the second decanate of Virgo; consequently I ought to have been a monster, and indeed was so near it that
I came forth literally torn from my mother’s womb.

  So I was born, or rather taken by violent means from my mother; I was almost dead. My hair was black and curly. I was revived in a bath of warm wine which might have been fatal to any other child. My mother had been in labor for three entire days, and yet I survived.

  One chapter of The Book of My Life lists the books Girolamo wrote, and the first on the list is The Great Art, one of three “treatises in mathematics” that he mentions. He also wrote on astronomy, physics, morality, gemstones, water, medicine, divination, and theology.

  Only The Great Art plays a part in our tale. Its subtitle, The Rules of Algebra, explains why. In it, Girolamo assembled methods for solving not just the quadratic equation, known to the Babylonians, but newly discovered solutions for cubic and quartic equations. Unlike Khayyám’s solutions, which depended on the geometry of conics, those in The Great Art are purely algebraic.

  Earlier, I mentioned two kinds of mathematical symbol, both of which we see in an expression such as x3, for the cube of the unknown. The first kind of symbol is the use of letters (x in this case) to stand for numbers—either unknown, or known but arbitrary. The second kind uses raised numbers to indicate powers—so the superscript 3 here indicates the cube x × x × x. Now we come to a third kind of symbol, the last that we will need.

  This third type of symbol is very pretty, and it looks like this:. This symbol means “square root.” For instance, , “the square root of nine,” means the number that when multiplied by itself gives the answer 9. Since 3 × 3 = 9, we see that = 3. It’s not always that easy, however. The most notorious square root, which according to a very unlikely legend caused the mathematician who drew attention to it, Hippasus of Metapontum, to be thrown overboard from a boat, is the square root of two, . An exact expression in decimals has to go on forever. It starts like this:

  1.4142135623730950488,

  but it can’t stop there, because the square of that number is actually

  1.99999999999999999999522356663907438144,

  which obviously is not quite the same as 2.

  This time we do know where the symbol came from. It is a distorted form of the letter “r,” standing for “radix,” the Latin for “root.” Mathematicians understand it that way and read as “root two.”

  Cube roots, fourth roots, fifth roots, and so on are shown by putting a small raised number in front of the “root” sign, like this:

  The cube root of a given number is the number that has the given number as its cube, and so on. So the cube root of 8 is 2, because 23 = 8. Again, the cube root of 2 can be expressed only approximately in decimal notation. It starts out like this:

  1.2599210498948731648

  and continues, if you have sufficient patience, forever.

  It is this number that turns up in the ancient problem of doubling the cube.

  By the year 400, Greek mathematics was no longer on the cutting edge. The action moved east, to Arabia, India, and China. Europe descended into the “Dark Ages,” and while these were not quite as dark as they have often been painted, they were dark enough. The spread of Christianity had the unfortunate side effect of concentrating learning and scholarship in the churches and monasteries. Many monks copied the works of mathematical greats like Euclid, but very few of them understood what they were copying. The Greeks could dig a tunnel through a mountain from both ends and make it meet in the middle; the early Anglo-Saxon method of surveying was to lay out a design, full scale, in a field. Even the notion of drawing to scale had been lost. If the Anglo-Saxons had wanted to make an accurate map of England, they would have made it the same size as England. They did make maps of conventional size, but not very accurate ones.

  By the end of the fifteenth century, the focus of mathematical activity was once again swinging back to Europe. As the Middle and Far East ran out of creative steam, Europe was getting its second wind, struggling free of the embrace of the Church of Rome and its fear of anything new. Ironically, the new center of intellectual activity was Italy, as Rome lost its grip on its own backyard.

  This sea change in European science and mathematics began with the publication, in 1202, of a book called the Liber Abbaci, written by Leonardo of Pisa, who much later was given the nickname Fibonacci—son of Bonaccio—and is now known by that name even though it was invented in the nineteenth century. Leonardo’s father, Guilielmo, was a customs officer in Bugia, now Algeria, and in his work must have come across people from many cultures. He taught his son the newfangled numerical symbols invented by the Hindus and the Arabs, the forerunners of our decimal digits 0 through 9. Leonardo later wrote that he “enjoyed so much the instruction that I continued to study mathematics while on business trips to Egypt, Syria, Greece, Sicily, and Provence, and there enjoyed disputations with the scholars of those places.”

  At first sight, the title of Leonardo’s book seems to indicate that it is about the abacus, a mechanical calculating device using beads that slide on wires, or pebbles in a groove in the sand. But just as the Latin word calculus, referring to one of those small pebbles, later acquired a different and more technical meaning, so the word abbaco, the counting frame, came to mean the art of computation. The Liber Abbaci was the first arithmetic text to bring the Hindu-Arabic symbols and methods to Europe. A large part of it is given over to the new arithmetic’s applications to practical subjects like currency exchange.

  One problem, about an idealized model of the growth of a population of rabbits, led to the remarkable sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on, where each number from 2 onward is the sum of the preceding two numbers. This “Fibonacci sequence” is Leonardo’s greatest claim to fame—not for its rabbit-breeding implications, which are nil, but for its remarkable mathematical patterns and its key role in the theory of irrational numbers. Leonardo could have had no idea that this little jeu d’esprit would come to eclipse the entire rest of his life’s work.

  Leonardo wrote several other books, and his Practica Geometriae of 1220 contained a large part of Euclid, plus some Greek trigonometry. Book X of Euclid’s Elements discusses irrational numbers composed of nested square roots, of the type . Leonardo proved that these irrationals are inadequate for solving cubic equations. This does not imply that the roots of the cubic cannot be constructed by ruler and compass, because other combinations of square roots might yield solutions. But it was the first hint that cubics might not be solvable using only Euclid’s tools.

  In 1494, Luca Pacioli pulled together much existing mathematical knowledge in a book on arithmetic, geometry, and proportion. It included the Hindu-Arabic numerals, commercial mathematics, a summary of Euclid, and Ptolemy’s trigonometry. A running theme was the element of design in nature, embodied in proportions—the human body, perspective in art, the theory of color.

  Pacioli continued the tradition of “rhetorical” algebra, using words rather than symbols. The unknown was “thing,” the Italian word cosa, and for a time, practitioners of algebra were known as “cossists.” He also employed some standard abbreviations, continuing (but failing to improve on) the approach pioneered by Diophantus. Morris Kline makes a telling point in his monumental Mathematical Thought from Ancient to Modern Times: “It is a significant commentary on the mathematical development of arithmetic and algebra between 1200 and 1500 that Pacioli’s [book] contained hardly anything more than Leonardo of Pisa’s Liber Abbaci. In fact, the arithmetic and algebra . . . were based on Leonardo’s book.”

  At the end of his book, Pacioli remarked that solving the cubic was no better understood than squaring the circle. But this would soon change. The first big breakthrough came about one-third of the way into the sixteenth century, in the city of Bologna. At first it passed unnoticed.

  Girolamo Cardano was the bastard son of a Milan lawyer, Fazio Cardano, and a young widow named Chiara Micheria, the mother of three children by her former marriage. He was born in Pavia, a town in the duchy of Milan, in 1501. When the plague came to
Milan, the pregnant Chiara was persuaded to move to the countryside, where she gave birth to Girolamo. Her three older children, who had remained behind, all died of the plague.

  According to Girolamo’s autobiography,

  my father went dressed in a purple cloak, a garment which was unusual in our community; he was never without a small black skullcap . . . From his fifty-fifth year on he lacked all of his teeth. He was well acquainted with the works of Euclid; indeed his shoulders were rounded from much study . . . My mother was easily provoked; she was quick of memory and wit, and a fat devout little woman. To be hasty-tempered was a trait common to both parents.

  Though a lawyer by profession, Fazio was skilled enough in mathematics to give advice about geometry to Leonardo da Vinci. He taught geometry at the University of Pavia and at the Piatti Foundation, a Milanese institution. And he taught mathematics and astrology to his illegitimate son, Girolamo:

  My father, in my earliest childhood, taught me the rudiments of arithmetic, and about that time made me acquainted with the arcane; whence he had come by this learning I do not know. Shortly after, he instructed me in the elements of the astrology of Arabia . . . After I was twelve years old he taught me the first six books of Euclid.

  The child had health problems; attempts to involve him in the family business were not successful. Girolamo managed to persuade his doubting father to let him study medicine at the University of Pavia. His father preferred law.

  In 1494, Charles VIII of France had invaded Italy, and the ensuing war continued sporadically for fifty years. An outbreak of hostilities closed the University of Pavia, and Girolamo moved to Padua to continue his studies. By all accounts he was a first-class student, and when Fazio died, Girolamo was campaigning to become the university’s rector. Although many people disliked him for speaking his mind, he was appointed by the margin of a single vote.