Why Beauty is Truth Page 8
Here Cardano is describing exactly the condition for the square root to be that of a negative number. It is clear that he had an excellent grasp of the whole business and had spotted a snag. It is less clear whether Tartaglia had a comparable level of understanding of his own formula, because his response was, “you have not mastered the true way of solving problems of this kind . . . Your methods are totally false.”
Possibly Tartaglia was merely being deliberately unhelpful. Or possibly he did not see what Cardano was getting at. At any rate, Cardano had put his finger on a puzzle that would exercise the combined intellects of the world’s mathematicians for the next 250 years.
Even in Renaissance times, there were hints that something important might be going on. The same issue arose in another problem discussed in The Great Art, to find two numbers whose sum is 10 and whose product is 40. This led to the “solution” 5 + and 5 – . Cardano noticed that if you ignored the question of what the square root of minus fifteen meant, and just pretended it worked like any other square root, then you could check that these “numbers” actually fit the equation. When you added them, the square roots canceled out, and the two 5’s added to 10, as required. When you multiplied them, you got 52 – ()2, which equals 25 + 15, which is 40. Cardano did not know what to make of this strange calculation. “So,” he wrote, “progresses arithmetic subtlety, the end of which is as refined as it is useless.”
In his Algebra of 1572, Rafaele Bombelli, the son of a Bolognese wool merchant, noticed that similar calculations, manipulating the “imaginary” roots as if they were genuine numbers, could convert the weird formula for Cardano’s puzzling cubic into the correct answer, x = 4. He wrote the book to fill in some spare time while he was reclaiming marshland for the Apostolic Camera, the Pope’s legal and financial department. Bombelli noticed that
and
so the sum of the two strange cube roots becomes
which equals 4. The meaningless root was somehow meaningful, and it gave the right answer. Bombelli was probably the first mathematician to realize that you could carry out algebraic manipulations with square roots of negative numbers and get usable results. This was a big hint that such numbers had a sensible interpretation, but it didn’t seem to indicate what that interpretation was.
The mathematical high point of Cardano’s book was not the cubic but the quartic. His student Ferrari managed to extend Tartaglia and del Ferro’s methods to equations that contain the fourth power of the unknown. Ferrari’s formula involves only square roots and cube roots—a fourth root is just the square root of a square root, so those are not needed.
The Great Art does not include a solution of the quintic equation, in which the unknown appears to the fifth power. But as the degree of the equation increases, the method for solving it gets more and more complicated, so few doubted that with enough ingenuity, the quintic too could be solved—you probably had to use fifth roots, and any formula would be really messy.
Cardano did not spend time seeking such a solution. After 1539 he returned to his numerous other activities, especially medicine. And now his family life fell apart in the most horrific manner: “My [youngest] son, between the day of his marriage and the day of his doom, had been accused of attempting to poison his wife while she was still in the weakness attendant upon childbirth. On the 17th day of February he was apprehended, and fifty-three days after, on April 13th, he was beheaded in prison.” While Cardano was still trying to come to terms with that tragedy, the horror got worse. “One house—mine—witnessed within the space of a few days, three funerals, that of my son, of my little granddaughter, Diaregina, and of the baby’s nurse; nor was the infant grandson far from dying.”
For all that, Cardano was incurably optimistic about the human condition: “Nevertheless, I still have so many blessings, that if they were another’s he would count himself lucky.”
5
THE CUNNING FOX
Which road to take? Which subject to study? He loved them both, but he must choose between them. It was a terrible dilemma. The year was 1796, and a brilliant 19-year-old youth was faced with a decision that would affect the rest of his life. He had to decide on a career. Although he came from an ordinary family, Carl Friedrich Gauss knew that he could rise to greatness. Everyone recognized his ability, including the duke of Brunswick, in whose domain Gauss had been born and where his family lived. His problem was that he had too much ability, and he was forced to choose between his two great loves: mathematics and linguistics.
On 30 March, however, the decision was taken out of his hands by a curious, remarkable, and totally unprecedented discovery. On that day, Gauss found a Euclidean construction for a regular polygon with seventeen sides.
This may sound esoteric, but there was not even a hint of it in Euclid. You could find methods for constructing regular polygons with three sides, or four, or five, or six. You could combine the constructions for three and five sides to get fifteen, and repeated bisections would double the number of sides, leading to eight, ten, twelve, sixteen, twenty, . . .
But seventeen was crazy. It was also true, and Gauss knew full well why it was true. It all boiled down to two simple properties of the number 17. It is a prime number—its only exact divisors are itself and 1. And it is one greater than a power of two: 17 = 16 + 1 = 24 + 1.
If you were a genius like Gauss, you could see why those two unassuming statements implied the existence of a construction, using straightedge and compass, of the regular seventeen-sided polygon. If you were any of the other great mathematicians who had lived between 500 BCE and 1796, you would not even have got a sniff of any connection. We know this because they didn’t.
If Gauss had needed confirmation of his mathematical talent, he certainly had it now. He resolved to become a mathematician.
The Gauss family had moved to Brunswick in 1740 when Carl’s grandfather took a job there as a gardener. One of his three sons, Gebhard Dietrich Gauss, also became a gardener, occasionally working at other laboring jobs such as laying bricks and tending canals; at other times he was a “master of waterworks,” a merchant’s assistant, and the treasurer of a small insurance fund. The more profitable trades were all controlled by guilds, and newcomers—even second-generation newcomers—were denied access to them. Gebhard married his second wife, Dorothea Benze, a stonemason’s daughter working as a maid, in 1776. Their son Johann Friederich Carl (who later always called himself Carl Friedrich) was born in 1777.
Gebhard was honest but obstinate, ill-mannered, and not very bright. Dorothea was intelligent and self-assertive, traits that worked to Carl’s advantage. By the time the boy was two, his mother knew she had a prodigy on her hands, and she set her heart on ensuring that he received an education that would allow his talents to flourish. Gebhard would have been happier if Carl had become a bricklayer. Thanks to his mother, Carl rose to fulfill the prediction that his friend, the geometer Wolfgang Bolyai, made to Dorothea when her son was 19, saying that Carl would become “the greatest mathematician in Europe.” She was so overjoyed that she burst into tears.
The boy responded to his mother’s devotion, and for the last two decades of her life she lived with him, her eyesight failing until she became totally blind. The eminent mathematician insisted on looking after her himself, and he nursed her until 1839, when she died.
Gauss showed his talents early. At the age of three, he was watching his father, at that point a foreman in charge of a gang of laborers, handing out the weekly wages. Noticing a mistake in the arithmetic, the boy pointed it out to the amazed Gebhard. No one had taught the child numbers. He had taught himself.
A few years later, a schoolmaster named J. G. Büttner set Gauss’s class a task that was intended to occupy them for a good few hours, giving the teacher a well-earned rest. We don’t know the exact question, but it was something very similar to this: add up all of the numbers from 1 to 100. Most likely, the numbers were not as nice as that, but there was a hidden pattern to them: they formed a
n arithmetic progression, meaning that the difference between any two consecutive numbers was always the same. There is a simple but not particularly obvious trick for adding the numbers in an arithmetic progression, but the class had not been taught it, so they had to laboriously add the numbers one at a time.
At least, that’s what Büttner expected. He instructed his pupils that as soon as they had finished the assignment, they should place their slate, with the answer, on his desk. While his fellow students sat scribbling things like
1 + 2 = 3
3 + 3 = 6
6 + 4 = 10
with the inevitable mistake
10 + 5 = 14
and running out of space to write in, Gauss thought for a moment, chalked one number on his slate, walked up to the teacher, and slapped the slate face down on the desk.
“There it lies,” he said, went back to his desk, and sat down.
At the end of the lesson, when the teacher collected all the slates, precisely one had the correct answer: Gauss’s.
Again, we don’t know exactly how Gauss’s mind worked, but we can come up with a plausible reconstruction. In all likelihood, Gauss had already thought about sums of that kind and spotted a useful trick. (If not, he was entirely capable of inventing one on the spot.) An easy way to find the answer is to group the numbers in pairs: 1 and 100, 2 and 99, 3 and 98, and so on, all the way to 50 and 51. Every number from 1 to 100 occurs exactly once in some pair, so the sum of all those numbers is the sum of all the pairs. But each pair adds up to 101. And there are 50 pairs. So the total is 50 × 101 = 5050. This (or some equivalent) is what he chalked on his slate.
The point of this tale is not that Gauss was unusually good at arithmetic, though he was; in his later astronomical work he routinely carried out enormous calculations to many decimal places, working with the speed of an idiot savant. But lighting calculation was not his sole talent. What he possessed in abundance was a gift for spotting cryptic patterns in mathematical problems, and using them to find solutions.
Büttner was so astonished that Gauss had seen through his clever ploy that, to his credit, he gave the boy the best arithmetic textbook that money could buy. Within a week, Gauss had gone beyond anything his teacher could handle.
It so happened that Büttner had a 17-year-old assistant, Johann Bartels, whose official duties were to cut quills for writing and to help the boys use them. Unofficially, Bartels had a fascination for mathematics. He was drawn to the brilliant ten-year old, and the two became lifelong friends. They worked on mathematics together, each encouraging the other.
Bartels was on familiar terms with some of the leading lights of Brunswick, and they soon learned that there was an unsung genius in their midst, whose family lived on the brink of poverty. One of these gentlemen, councilor and professor E. A. W. Zimmerman, introduced Gauss to the duke of Brunswick, Carl Wilhelm Ferdinand, in 1791. The duke, charmed and impressed, took it upon himself to pay for Gauss’s education, as he occasionally did for the talented sons of the poor.
Mathematics was not the boy’s sole talent. By the age of 15 he had become proficient in classical languages, so the duke financed studies in classics at the local gymnasium. (In the old German educational system, a gymnasium was a type of school that prepared its pupils for university entrance. It translates roughly as “high school,” but only paying students were admitted.) Many of Gauss’s best works were later written in Latin. In 1792, he entered the Collegium Carolinium in Brunswick, again at the duke’s expense.
By the age of 17 he had already discovered an astonishing theorem known as the “law of quadratic reciprocity” in the theory of numbers. It is a basic but rather esoteric regularity in divisibility properties of perfect squares. The pattern had already been noticed by Leonhard Euler, but Gauss was unaware of this and made the discovery entirely on his own. Very few people would even have thought of asking the question. And the boy was thinking very deeply about the theory of equations. In fact, that was what led him to his construction of the regular 17-gon and thus set him on the road to mathematical immortality.
Between 1795 and 1798, Gauss studied for a degree at the University of Göttingen, once more paid for by Ferdinand. He made few friends, but the friendships he did strike up were deep and long-lasting. It was at Göttingen that Gauss met Bolyai, an accomplished geometer in the Euclidean tradition.
Mathematical ideas came so thick and fast to Gauss that sometimes they seemed to overwhelm him. He would suddenly cease whatever he was doing and stare blankly into the middle distance as a new thought struck him. At one point he worked out some of the theorems that would hold “if Euclidean geometry were not the true one.” At the forefront of his thoughts was a major work that he was composing, the Disquisitiones Arithmeticae, and by 1798 it was pretty much finished. But Gauss wanted to make certain that he had given due credit to his predecessors, so he visited the University of Helmstedt, which had a high-quality mathematics library overseen by Johann Pfaff, the best-known mathematician in Germany.
In 1801, after a frustrating delay at the printer’s, the Disquisitiones Arithmeticae was published, with an effusive and no doubt heartfelt dedication to Duke Ferdinand. The duke’s generosity did not end when Carl left the university. Ferdinand paid for his doctoral thesis, which he presented at the University of Helmstedt, to be printed as the regulations required. And when Carl started to worry about how to support himself when he left university, the duke gave him an allowance so that he could continue his researches without having to be bothered about money.
A noteworthy feature of the Disquisitiones Arithmeticae is its uncompromising style. The proofs are careful and logical, but the writing makes no concessions to the reader and gives no clue about the intuition behind the theorems. Later, he justified this attitude, which continued throughout his career, on the grounds that “When one has constructed a fine building, the scaffolding should no longer be visible.” Which is all very well if all you want people to do is admire the building. It’s not so helpful if you want to teach them how to build their own. Carl Gustav Jacob Jacobi, whose work in complex analysis built on Gauss’s ideas, said of his illustrious predecessor, “He is like the fox, who erases his tracks in the sand with his tail.”
Around this time, mathematicians were gradually coming to realize that although “complex” numbers seemed artificial and their meaning incomprehensible, they made algebra much neater by providing solutions to equations in a uniform way. Elegance and simplicity are the touchstones of mathematics, and novel concepts, however strange they appear at first, tend to win out in the long run if they help to keep the subject elegant and simple.
If you work purely with traditional “real” numbers, equations can be annoyingly erratic. The equation x2 – 2 = 0 has two solutions, plus or minus the square root of two, but the very similar equation x2 + 1 = 0 has none at all. However, this equation has two solutions in complex numbers: i and – i. The symbol i for was introduced by Euler in 1777 but not published until 1794. A theory couched solely in terms of “real” equations is littered with exceptions and pedantic distinctions. The analogous theory of complex equations avoids all of these complications by swallowing wholesale one big complication at the very outset: to allow complex numbers as well as real ones.
By 1750, the circle of ideas initiated by the mathematicians of Renaissance Italy had matured and closed. Their methods for solving cubic and quartic equations were seen as natural extensions of the Babylonian solution of quadratics. The connection between radicals and complex numbers had been worked out in some detail, and it was known that in this extension of the usual number system, a number had not one cube root but three; not one fourth root but four; not one fifth root but five. The key to understanding where these new roots came from was a beautiful property of “roots of unity,” that is, nth roots of the number 1. These roots form the vertices of a regular n-sided polygon in the complex plane, with one vertex at 1. The remaining roots of unity space themselves out equally arou
nd a circle of radius 1, centered at 0. For instance, the left-hand figure (next page) shows the locations of the fifth roots of unity.
More generally, from any particular fifth root of some number it is possible to obtain four more, by multiplying it by q, q2, q3, and q4. These numbers are also spaced equally around a circle centered at 0. For example, the five fifth roots of 2 are shown in the right-hand figure.
(Left) The fifth roots of unity in the complex plane. (Right) The fifth roots of two.
This was all very pretty, but it suggested something much deeper. The fifth roots of 2 can be viewed as the solutions of the equation x5 = 2. This equation is of the fifth degree, and it has five complex solutions, only one of which is real. Similarly, the equation x4 = 2 for fourth roots of 2 has four solutions, the equation for 17th roots of 2 has 17 solutions, and so on. You don’t have to be a genius to spot the pattern: the number of solutions is equal to the degree of the equation.
The same seemed to apply not just to the equations for nth roots, but to any algebraic equation whatsoever. Mathematicians became convinced that within the realm of complex numbers, every equation has exactly the same number of solutions as its degree. (Technically, this statement is true only when solutions are counted according to their “multiplicity.” If this convention is not used, then the number of solutions is less than or equal to the degree.) Euler proved this property for equations of degree 2, 3, and 4, and claimed that similar methods would work in general. His ideas were plausible, but filling in the gaps turned out to be almost impossible, and even today it takes a major effort to push Euler’s method to a conclusion. Nevertheless, mathematicians assumed that if they were solving an equation of some degree, they should expect to find precisely that many solutions.