The Universe in Zero Words

by Mackenzie, Dana

  Tartaglia, of course, felt that Cardano had betrayed him. He fired off a volley of insulting open letters, as well as a book of his own. Cardano, however, remained above the fray and allowed Ferrari to do the answering for him—a task that Ferrari took to with great zest, and ultimately with the successful outcome recounted at the beginning of this chapter.

  It may seem unjust that the formula for solving the cubic is now known as Cardano’s formula—not del Ferro’s, or Tartaglia’s, or even Ferrari’s. But as has already been stated in the last chapter, mathematics thrives when it is communicated openly. It is not enough merely to discover America—you must make the discovery known to the rest of the world. Cardano alone took that final step, and reaped the glory.

  CARDANO’S FORMULA had a lasting impact that far exceeded the importance of the problem it solved. For example, it provided one of the first motivations for the use of imaginary numbers and complex numbers in mathematics. Imaginary numbers are numbers whose square is negative (a property no real number has). Using imaginary numbers, we can say that –1 has two square roots, which are denoted by i and –i. Without imaginary numbers, we would have to say that –1 has no square roots. Once we have imaginary numbers, we can define complex numbers as numbers that have both a real and an imaginary part, such as 1 + 2i.

  Not only modern-day mathematics but also modern-day physics would be unthinkable without imaginary numbers. In quantum mechanics, for instance, elementary particles such as photons are defined to be “wave functions.” The wave function for a photon at some point will in general have complex-number values, such as 0.2 + 0.3i. The imaginary part of the wave function accounts for the wavelike properties, or “phase,” of the photon; for instance, it explains why a light beam that shines through two slits forms a diffraction pattern on the other side, rather than two bright bars (see Young’s experiment, on page 143). Thus, imaginary numbers seem to be woven in a very real way into the fabric of the universe.

  Above A title page vignette illustrated in Cardano’s Opera omnia, 1663, depicting Ptolemy and Euclid.

  Prior to Cardano, it did not occur to anybody to assert that –1 has two square roots but they are imaginary. You might compare it to asking a child how many imaginary friends she is inviting to her birthday party. But in the case of cubic polynomials, these “imaginary friends” actually left behind some real birthday presents! In 1572, Rafael Bombelli presented an example of this phenomenon. The equation x3 = 15x + 4 has a real solution, x = 4, that can be verified by substitution. However, Cardano’s formula gives:

  Cardano had no interest in such nonsense: “So progresses arithmetic subtlety, the end of which, as is said, is as refined as it is useless,” he once wrote. But Bombelli realized that these expressions have meaning. The first cube root is equal to 2 + i and the second is equal to 2 – i, and therefore x = (2 + i) + (2 – i) = 4. In the final solution the imaginary quantities have disappeared, but we could not have gotten to the solution without them.

  Thankfully, today’s students are no longer expected to learn Cardano’s formula. Nevertheless, nineteenth-century students were expected to know how to solve cubics. Albert Einstein, in his university exams, correctly solved a problem with Cardano’s formula—in contrast to the surprisingly persistent legend that he was a poor mathematics student.

  Another long-delayed ramification of Cardano’s formula involved the solution of higher-degree equations. After the cubic and quartic had been tamed, one might have expected the solution of fifth-degree polynomials, or quintics, to follow shortly thereafter. But strangely, another 250 years went by with very little progress. Some quintic equations can be solved. But no universal solution, applicable to all quintics, was ever found.

  In 1824, a Norwegian mathematician named Niels Henrik Abel finally showed that there could not be any Cardano-like formula for the solutions to a fifth-degree equation. (“Cardano-like formula” refers to any formula that involves square roots, cube roots, fourth roots, etc., possibly nested inside one another. Mathematicians call this a “solution by radicals.”) Abel’s theorem may have closed one chapter of mathematics, but it opened another. His proof led mathematicians to a deeper understanding of the concept of symmetry, a topic to be discussed in chapter 14.

  * * *

  * An exception is the Persian poet and mathematician Omar Khayyam (1050–1130), who showed how to solve a large class of cubic equations by geometrical constructions (e.g., by finding the intersection point of a parabola and a circle). However, the results are not readily convertible into numerical form, and are not equivalent to the later work of Tartaglia and Cardano.

  8

  order in the heavens kepler’s laws of planetary motion

  Another great battle of sixteenth- and seventeenth-century science was fought over a “revolutionary” theory that actually was not a revolution. In 1543, while on his deathbed, Nicolaus Copernicus published a book called De revolutionibus orbium coelestium, which placed the Sun, not Earth, at the center of the solar system. Although Copernicus’ theory was at odds with the ecclesiastical understanding of the cosmos, it was definitely not a new idea. Aristarchus of Samos, a Greek philosopher, had already discussed a heliocentric model of the universe in the fourth century BC.

  In the early years of the 1600s, two events thrust the “Copernican” (but really Aristarchan) theory into the center of a storm of controversy. The first was the invention of the telescope in 1608. Secondly, using one of these new instruments, Galileo Galilei discovered four small moons orbiting Jupiter. For Galileo, and for anyone else who took the trouble to look through the telescope, here was direct visual evidence of objects in the universe that did not orbit Earth. Galileo’s discovery sounded the death knell for the dogma that Earth was the center of the universe.

  It is easy to paint Galileo as the great champion of the Copernican theory, and indeed his story is full of drama and martyrdom. Brought to trial by the Inquisition in 1633 on a charge of heresy for advocating the view that the Earth was not the center of the universe, Galileo was convicted, forced to recant, and confined to house arrest for the rest of his life. However, there is a second hero of the story who is not quite as well known, yet perhaps deserves equal credit: Johannes Kepler.

  The function r(θ) represents the distance of a planet from the Sun when its location on the zodiac is θ degrees. The angular position θ(t) is itself a function of time, t. The total time it takes the planet to go around the Sun is T. The constants p and R describe (roughly) the width and length of the orbit. The eccentricity, ε, describes how far the orbit deviates from a perfect circle. C1 and C2 are two empirical constants.

  Although he is primarily considered an astronomer, Kepler had a real gift for mathematics and for bold conjecture. He was able to spot patterns where no one else had before—and sometimes where none existed. For example, when Galileo announced his discovery that Jupiter had four moons, Kepler conjectured that Mars must have two moons and Saturn eight, in order to make a geometrical progression: 1 (Earth), 2 (Mars), 4 (Jupiter), 8 (Saturn). Amazingly, he was right about Mars, but the “pattern” was an utter coincidence. Jupiter has 63 moons that we know about, and Saturn has 62.

  It is easy to understand why Kepler, with his speculative temperament, was one of the first scientists to wholeheartedly embrace Copernicus’ theory. It is a bit more of a surprise to see him chastising Galileo for not doing the same. In 1597, when Galileo wrote to him that he agreed with Copernicus but dared not publish his opinion, Kepler wrote back: “I would have wished, however, that you, possessed of such an excellent mind, took up a different position … Have faith, Galilei, and come forward!” Nevertheless, Galileo remained publicly silent for thirteen more years, until his discovery of the Galilean moons of Jupiter gave him the evidence he needed to “come forward.”

  NOWADAYS, KEPLER’S FAME rests on three mathematical laws that he discovered by analyzing the painstaking observations of the orbits of the planets that had been taken by his former employer, t
he Danish astronomer Tycho Brahe. His laws form a bridge from the old style of astronomy, which was concerned with describing the cosmos, to a new style that explains the motions of the planets and other celestial bodies. They are still descriptive laws, but they are so precise that they virtually beg for a mathematical proof. Isaac Newton provided the proof in 1686, roughly three-quarters of a century later.

  Kepler’s first law states that planets orbit the Sun in ellipses, not circles, with the Sun at one focus. To express this in equation form, we could write it as follows:

  Here r(θ) represents the distance from the planet to the Sun when it is θ degrees away from aphelion (its greatest distance from the Sun). The number p represents the distance when the planet is 90 degrees away from aphelion, and the number ε represents the eccentricity, or departure from circularity, of the planet’s orbit. Notice that if the eccentricity ε is zero, the equation becomes r(θ) = p: in other words, the distance from the planet to the Sun is a constant. In this case, and only in this case, the orbit is a circle.

  Ironically, Kepler’s first law was actually a departure from strict Copernicanism. The Sun is not at the center of Earth’s orbit, but slightly displaced from it. At its closest approach to the Sun, called perihelion, Earth is about 91.3 million miles (147 million kilometers) away. At its greatest distance, Earth is 94.5 million miles (152 million kilometers) away. More importantly, Kepler’s law made a clean break with centuries of tradition that tried to describe planetary orbits either as circles, or as complicated combinations of circular motions. Aristotle had considered circles to be the most perfect curves, and therefore the only ones that could describe the motion of the heavenly bodies. In the third century, Ptolemy had refined Aristotle’s system with a complicated, Rube Goldberg-esque arrangement of circular motions superimposed on other circular motions—but even so he could not predict the planets’ motions very accurately. Kepler’s law (to modern eyes) is simple, economical, and far more beautiful than Plutarch’s theory. To determine the shape of any planet’s orbit, you need to know only two numbers: p and ε.

  Left Kepler’s Second Law: The line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

  But Kepler wasn’t finished. He discovered two more patterns in Brahe’s planetary data that have since been elevated to the status of “laws.” Kepler’s second law states that planets speed up when they get closer to the Sun, and they do so in a precisely quantifiable way. The area that a planet sweeps out in any given, fixed-time interval is the same no matter where the planet is in its orbit. Because Earth is closer to the Sun at perihelion, it must sweep out a fatter triangle in one day at perihelion than it does when it is at aphelion, as shown above.

  Earth’s orbit is so nearly circular (ε = 0.0167), that most of us are not aware of these subtle differences. Nevertheless, they do affect us, more profoundly than most people realize. At present, Earth’s closest approach to the Sun falls during the northern hemisphere’s winter. This means that the northern hemisphere has somewhat shorter (and milder) winters than it would if the perihelion came in summertime. However, this will not always be the case. In about 13,000 years, the situation will be reversed, and we will experience longer and more severe winters. The existence of orbital variations like these has been proposed as a contributor to ice ages.

  While Kepler’s first two laws were the culmination of an eight-year struggle to understand Brahe’s data on the orbit of Mars, Kepler’s third law seems to have occurred to him quite suddenly—on March 8, 1618, as he was putting the finishing touches to a book called Harmony of the World. Unlike the first two laws, which describe the motion of an individual planet, the third law provides a basis for comparison between planets. It says that the length of a planet’s year is proportionate to the 3/2 power of its distance to the Sun. (Another way of saying this is that the square of the orbital period is proportional to the cube of the mean distance from the Sun.) For instance, Pluto is 39.5 times farther from the Sun than Earth is. Thus it takes (39.5)3/2 = 39.5 √39.5 = 248 years to orbit the Sun.

  Opposite The orbit of Mars, an engraving from Astronomia nova by Kepler (1571–1630).

  Kepler’s third law is actually more useful in reverse. It is easy to measure the orbital time of a planet around the Sun, or a moon around a planet, or two stars around each other. The hard part is measuring the distance between them. Kepler’s law gives us an immediate way of converting orbital periods to distances. Later improvements, using Newton’s law of gravitation, enable us to infer the mass of moons, planets, or stars from their orbital periods. Such calculations are essential, for instance, in the search for extrasolar planets (i.e., planets in other solar systems) that might be capable of supporting life. If we didn’t know how big and how far away from its sun a planet is, we would not know whether it is habitable. If, one day, we do find evidence of life on a distant planet, we will owe it to Kepler and his third law.

  9

  writing for eternity fermat’s last theorem

  Pierre de Fermat was not a practical joker. The son of a wealthy leather merchant in southern France, he earned a law degree at the University of Orleans in 1631, bought a seat in the parlement (supreme court) at Toulouse, and became a member of the nobility. From the evidence of his letters, he was a shy, taciturn man who disliked controversy.

  But Fermat had one unusual characteristic: He loved mathematics. In an era when mathematicians were starting to reach across national boundaries and turn their subject into an international enterprise, he achieved worldwide fame that lasted long after his death. By a curious twist of fate, his most lasting legacy was a problem that he almost certainly did not solve. That problem, called Fermat’s Last Theorem, unintentionally became the greatest practical joke in mathematical history—a deceptively simple statement that defied all efforts at proof for more than 350 years.

  To the best of our knowledge, Fermat was self-taught. However, during his student days he formed friendships with a small circle of people who were interested in mathematics, and this apparently stimulated him to start doing his own research. One of his friends moved to Paris in 1636 to work in the royal library, and brought the work of this previously unknown provincial mathematician, Fermat, to the attention of Father Marin Mersenne.

  The numbers x, y, z, and n are positive integers, and n is greater than 2. In contrast to the previous equations I have discussed, Fermat’s Last Theorem states that this equation has no solutions.

  In an era before scientific academies and scientific journals, when most universities did not even have a professor of mathematics, Mersenne was the focal point of mathematics in France. He held regular meetings at his convent and kept in touch with nearly every mathematician in Europe. If you wanted to publicize a new discovery, you would send it to Mersenne. The rest of the world would find out soon enough.

  Fermat himself never visited Paris, never ventured out of the south of France, and met Mersenne only one time, in 1646. He adamantly refused to have anything published under his own name. Nevertheless, his results became known everywhere, thanks to Mersenne, and other mathematicians in France and abroad avidly desired to learn his methods.

  Yet Fermat was very close-lipped. His normal modus operandi was to send his discoveries as problems to other mathematicians, often artfully concealed so that the true nature of his discovery would not be apparent to the recipient unless they had been working on similar problems themselves. In this way Fermat could ascertain whether he had found something new, without giving away what it was.

  Of course this sort of challenge both tantalized and annoyed other mathematicians. René Descartes called Fermat a “braggart,” and Bernard Frénicle de Bessy accused him of posing impossible problems. Fermat seems to have been torn between his desire for recognition and a nearly pathological fear of revealing too many of his secrets. On the one hand, he was fond of quoting a motto of Sir Francis Bacon: Multi pertranseant ut augeatur scientia (“Many must pass by in order that kno
wledge may grow”). On the other hand, by his reluctance to publish, Fermat made many parts of his own work inaccessible to “passersby.”

  FERMAT WAS THE FIRST modern European mathematician to take an active interest in number theory, the study of equations with integer solutions. An ongoing theme in his work was Pythagorean triples: in other words, whole numbers a, b, and c such that a2 + b2 = c2. As Fermat knew from studying a book that had been recently translated from the ancient Greek, Diophantus’ Arithmetica, the Greeks had a general method for solving this equation. Fermat came up with innumerable variations on the theme: finding two Pythagorean triples with the same hypotenuse c; Pythagorean triples whose areas were square numbers, or twice a square, or such that the sum of the legs a + b was square. He was able to resolve all of these problems to his satisfaction, even on occasion proving that there was no solution. (For example, no Pythagorean triangle has an area that is a perfect square.)

  One day, probably between 1636 and 1640, he came up with another variation: Could a cube be written as a sum of two cubes? More generally, did the equation xn + yn = zn ever have whole-number solutions if the exponent n was greater than 2? In the margin of his personal copy of Diophantus, Fermat wrote: “No cube can be split into two cubes, nor any biquadrate [fourth power] into biquadrates, nor generally any power beyond the second into two of the same kind. For this I have discovered a truly wonderful proof, but the margin is too small to contain it.” This handwritten note, which Fermat never intended anyone to see, became one of the most famous quotes in mathematical history. As number theorist André Weil has written, “How could he have guessed that he was writing for eternity?”