Why Beauty is Truth Page 7
This is when he frittered away his inheritance and turned to gambling, which became an addiction for the rest of his turbulent life. And not only that:
At a very early period in my life, I began to apply myself seriously to the practice of swordsmanship of every class, until, by persistent training, I had acquired some standing even among the most daring . . . By night, even contrary to the decrees of the Duke, I armed myself and went prowling about the cities in which I dwelt . . . I wore a black woolen hood to conceal my features, and put on shoes of sheep-pelt . . . often I wandered abroad throughout the night until day broke, dripping with perspiration from the exertion of serenading on my musical instruments.
It scarcely bears thinking about.
Awarded his medical degree in 1525, Girolamo tried to enter the College of Physicians in Milan and was rejected—nominally for illegitimacy but in fact, largely because of his notorious lack of tact. So instead of joining the prestigious college, Girolamo set himself up as a doctor in the nearby village of Sacco. This provided a small income, but the business limped along. He married Lucia Bandarini, the daughter of a captain in the militia, and moved closer to Milan, hoping to earn more money to provide for his family, but again the college turned him down. Unable to pursue a legitimate medical career, he reverted to gambling, but even his mathematical expertise failed to restore his fortunes:
Peradventure in no respect can I be deemed worthy of praise; for so surely as I was inordinately addicted to the chess-board and the dicing table, I know that I must rather be considered deserving of the severest censure. I gambled at both for many years, at chess more than forty years, at dice about twenty-five; and not only every year, but—I say it with shame—every day, and with the loss at once of thought, of substance, and of time.
The entire family ended up in the poor house, having long ago pawned their furniture and Lucia’s jewelry. “I entered upon a long and honorable career. But away with honors and gain, together with vain displays and unseasonable delights! I ruined myself! I perished!”
Their first child arrived:
After having twice miscarried and borne two males of four months, so that I . . . at times suspected some malefic influence, my wife brought forth my first born son . . . He was deaf in his right ear . . . Two toes on his left foot . . . were joined by one membrane. His back was slightly hunched but not to the extent of a deformity. The boy led a tranquil existence up to his twenty-third year. After that, he fell in love . . . and married a dowerless wife, Brandonia di Seroni.
Now Girolamo’s late father came to their rescue, in a rather indirect manner. Fazio’s lecturing post at the university was still open, and Girolamo got the job. He also did a bit of doctoring on the side, despite being unlicensed. A number of miraculous cures—probably luck, given the state of medicine in that period—gave him a high reputation. Even some members of the college took their medical problems to him, and for a while it looked as though he might finally gain entrance to that esteemed institution. But once again, Girolamo’s tendency to speak his mind scuppered that; he published a vitriolic attack on the abilities and character of the college’s membership. Girolamo was aware of his lack of tact but apparently did not see it as a fault: “As a lecturer and debater, I was much more earnest and accurate than in exercising prudence.” In 1537 his lack of prudence caused his latest application to be turned down.
But his reputation was becoming so great that the college eventually had no real choice, and he was made a member two years later. Things were looking up; all the more so when he published two books about mathematics. Girolamo’s career was advancing on several fronts.
Around this time, Tartaglia made a major breakthrough—a solution to a broad class of cubic equations. After some persuasion, and with reluctance, he confided his epic discovery to Cardano. It is hardly surprising that six years later, when he received a copy of Cardano’s algebra text The Great Art, or on the Rules of Algebra and found a complete exposition of his secret, Tartaglia was incensed.
Cardano had not stolen the credit, for he gave full acknowledgment to Tartaglia:
In our own days Scipione del Ferro of Bologna has solved the case of the cube and first power equal to a constant, a very elegant and admirable accomplishment . . . In emulation of him, my friend Niccolò Tartaglia of Brescia . . . solved the same case when he got into a contest with his [del Ferro’s] pupil Antonio Maria Fior, and moved by my many entreaties, gave it to me.
Nonetheless, it was galling for Tartaglia to see his precious secret given away to the world, and even more galling to recognize that many more people would remember the author of the book than the erstwhile secret’s true discoverer.
That, at least, was Tartaglia’s view of the affair, which constitutes almost all of the existing evidence. As Richard Witmer points out in his translation of The Great Art, “We are dependent almost exclusively on Tartaglia’s printed accounts, which by no stretch of the imagination can be regarded as objective.” One of Cardano’s servants, Lodovico Ferrari, later claimed to have been present at the meeting and said that there had been no agreement to keep the method secret. Ferrari later became Cardano’s student, and he solved—or helped to solve—the quartic, so he cannot be considered a more objective witness than Tartaglia.
What made matters worse for poor Tartaglia was that it wasn’t just a case of lost credit. In Renaissance Europe, mathematical secrets could be translated into hard cash. Not just through gambling, Cardano’s preferred route, but through public competitions.
It is often said that mathematics is not a spectator sport, but that was not true in the 1500s. Mathematicians made reasonable livings by challenging each other to public contests, in which each would set his opponent a series of problems, and whoever got the most answers right was the winner. These spectacles were less thrilling than bare-hands fighting or swordplay, but onlookers could place wagers and find out which contestant won, even if they had no idea how he did it. In addition to the prize money, winners attracted pupils, who would pay for tuition, so the public competitions were doubly lucrative.
Tartaglia was not the first to find an algebraic solution to a cubic equation. The Bolognese professor Scipione del Ferro discovered his solution of some types of cubic somewhere around 1515. He died in 1526, and both his papers and his professorship were inherited by his son-in-law, Annibale del Nave. We can be sure of this because the papers themselves came to light in the University of Bologna library around 1970, thanks to the efforts of E. Bartolotti. According to Bartolotti, del Ferro probably knew how to solve three types of cubic, but he passed on the method for solving only one type: cube plus thing equals number.
Knowledge of this solution was preserved by del Nave and by del Ferro’s student Antonio Maria Fior. And it was Fior, determined to set himself up in business as a mathematics teacher, who came up with an effective marketing technique. In 1535 he challenged Tartaglia to a public cubic-solving contest.
There were rumors that a method for solving cubics had been found, and nothing encourages a mathematician more than the knowledge that a problem has a solution. The risk of wasting time on an unsolvable problem is ruled out; the main danger is that you may not be clever enough to find an answer you know must exist. All you need is lots of confidence, which mathematicians seldom lack—even if it turns out to be misplaced.
Tartaglia had rediscovered del Ferro’s method, but he suspected that Fior also knew how to solve other types of cubic and would thus have a huge advantage. Tartaglia tells us how much this prospect worried him, and how he finally cracked the remaining case shortly before the contest. Now Tartaglia had the advantage, and he promptly wiped out the unlucky Fior.
Word of the defeat spread; Cardano heard of it in Milan. At that time he was working on his algebra text. Like any true author, Cardano was determined to include the latest discoveries, for without them his book would be obsolete before it was even published. So Cardano approached Tartaglia, hoping to wheedle the secret out of him
and put it in The Great Art. Tartaglia refused, saying that he intended to write his own book.
Eventually, however, Cardano’s persistence paid off, and Tartaglia divulged the secret. Did he really make Cardano swear to keep it hidden, knowing that a textbook was in the offing? Or did he succumb to Cardano’s blandishments and then regret it?
There is no doubt that he was extremely angry when The Great Art appeared. Within a year he had published a book, Diverse Questions and Inventions, which laid into Cardano in no uncertain terms. He included all of the correspondence between them, supposedly exactly as written.
In 1574, Ferrari came to the support of his master by issuing a cartello—a challenge to a learned dispute on any topic Tartaglia cared to name. He even offered a prize of 200 scudi for the winner. And he made his opinions very clear: “This I have proposed to make known, that you have written things which falsely and unworthily slande . . . Signor Girolamo, compared to whom you are hardly worth mentioning.”
Ferrari sent copies of the cartello to numerous Italian scholars and public figures. Within nine days, Tartaglia responded with his own statement of facts, and the two mathematicians ended up by exchanging twelve cartelli between them over a period of eighteen months. The dispute seems to have followed the standard rules for a genuine duel. Tartaglia, who had been insulted by Ferrari, was allowed the choice of weapons—the selection of topics to be debated. But he kept asking to debate Cardano rather than his challenger, Ferrari.
Ferrari kept his temper under control and pointed out that in any case it had been del Ferro, not Tartaglia, who had solved the cubic to begin with. Since del Ferro had made no fuss about Tartaglia’s unjustified claim of credit, why wasn’t Tartaglia willing to behave likewise? It was a good point, and Tartaglia may have recognized that, because he considered withdrawing from the contest. However, he didn’t, and one possible reason was the city fathers of Brescia, his hometown. Tartaglia was after a lectureship there, and the local dignitaries may have wanted to see how he acquitted himself.
At any rate, Tartaglia agreed to the debate, which was held in a Milanese church before large crowds in August 1548. No record of the proceedings is known, save for a few indications by Tartaglia, who said that the meeting ceased when suppertime approached. This hints that the debate may not have been especially gripping. It seems, though, that Ferrari won handily, because afterward he was offered some plum positions, accepted the post of tax assessor to the governor of Milan, and soon became very rich. Tartaglia, on the other hand, never claimed to have won the debate, failed to get the Brescia job, and descended into bitter recrimination.
Unknown to Tartaglia, Cardano and Ferrari had an entirely different line of defense, for they had visited Bologna and inspected del Ferro’s papers. These included the first genuine solution of the cubic, and in later years they both insisted that the source of the material included in The Great Art was del Ferro’s original writings, not Tartaglia’s confidence to Cardano. The reference to Tartaglia was included merely to record how Cardano himself heard of del Ferro’s work.
There is a final twist to the tale. Soon after the second edition of The Great Art was published, in 1570, Cardano was imprisoned by the Inquisition. The reason may have been something that had previously seemed entirely innocent: not the content of the book, but its dedication. Cardano had chosen to dedicate it to the relatively obscure intellectual Andreas Osiander, a minor figure in the Reformation but one strongly suspected of being the author of an anonymous preface to Nicolaus Copernicus’s On the Revolutions of the Heavenly Spheres, the first book to propose that the planets go around the Sun, not the Earth. The Church considered this view heretical, and in 1600 it burned Giordano Bruno alive for maintaining it—hanging him upside down from a stake, naked and gagged, in a Roman market square. In 1616, and again in 1633, it gave Galileo a lot of grief, for the same reason, but by then the Inquisition was content to put him under house arrest.
To appreciate what Girolamo and his compatriots achieved, we must revisit the Babylonian tablet that explains how to solve quadratics. If we follow the recipe and express the calculation steps in modern symbolism, we see that in effect the Babylonian scribe was saying that the solution to a quadratic equation x2 – ax = b is
This is equivalent to the formula that every school student used to learn by heart, and that nowadays is found in every formula book.
The Renaissance solution of the cubic equation is similar but more elaborate. In modern symbols it looks like this: Suppose that x3 + ax = b. Then
As formulas go, this one is relatively simple (believe me!), but you need to develop a lot of algebraic ideas before it can be so described. It is by far the most complicated formula we will look at, and it uses all three types of symbol that I have introduced: letters, raised numbers, and the sign, as well as both square roots and cube roots. You don’t need to understand this formula, and you certainly don’t need to calculate it. But you need to appreciate its general shape. First, some terminology that will prove very useful as we proceed.
An algebraic expression like 2x4 – 7x3 – 4x2 + 9 is called a polynomial, which means “many terms.” Such expressions are formed by adding various powers of the unknown. The numbers 2, –7, –4, and 9, which multiply the powers, are called coefficients. The highest power of the unknown that occurs is called the degree of the polynomial, so this polynomial has degree 4. There are special names for polynomials of low degree (1 through 6): linear, quadratic, cubic, quartic, quintic, and sextic. The solutions of the associated equation 2x4 – 7x3 – 4x2 + 9 = 0 are called the roots of the polynomial.
Now we can dissect Cardano’s formula. It is built from the coefficients a and b, employing some additions, subtractions, multiplications, and divisions (but only by certain integers, namely 2, 4, and 27). The esoteric aspects are of two kinds: There is a square root—in fact the same square root occurs in two places, but one is added while the other is subtracted. Finally, there are two cube roots, and these are the cube roots of quantities that involve the square roots. So aside from harmless operations of algebra (by which I mean those that merely shuffle the terms around), the skeleton of the solution is, “Take a square root, then a cube root; do this again; add the two.”
That’s all we will need. But I don’t think we can get away with less.
What the Renaissance mathematicians initially failed to grasp, but later generations soon realized, is that this formula is not just a solution to one type of cubic. It is a complete solution to all types, give or take some straightforward algebra. For a start, if the cube term is, say, 5x3 instead of x3, you can just divide the entire equation by 5—the Renaissance mathematicians were certainly smart enough to spot that. A more subtle idea, which required a quiet revolution in how we think of numbers, is that by allowing the coefficients a and b to be negative if necessary, we can avoid fruitless distinctions among cases. Finally, there is a purely algebraic trick: if the equation involves the square of the unknown, you can always get rid of it—you replace x by x plus a carefully chosen constant, and if you do it right, the square term miraculously disappears. Again, it helps here if you stop worrying whether numbers are positive or negative. Finally, the Renaissance mathematicians worried about terms that were entirely missing, but to modern eyes the remedy is obvious: such a term is not actually missing, it just has coefficient zero. The same formula works.
Problem solved?
Not quite. I lied.
Here is where I lied. I said that Cardano’s formula solves all cubics. There is a sense in which that’s not true, and it turned out to be important. I didn’t tell a very bad lie, though, because it all depends on what you mean by “solve.”
Cardano himself spotted the difficulty, which says a lot for his attention to detail. Cubics typically have either three solutions (fewer if we exclude negative numbers) or one. Cardano noticed that when there are three solutions—say 1, 2, and 3—the formula does not seem to yield those solutions in any sensible way. Instead,
the square root in the formula contains a negative number.
Specifically, Cardano noted that the cubic x3 = 15x + 4 has the obvious solution x = 4. But when he tried out Tartaglia’s formula, it led to the “answer”
which seemed meaningless.
Among European mathematicians in those days, few brave souls were willing to contemplate negative numbers. Their Eastern counterparts had come to terms with negative quantities much earlier. In India, the Jains developed a rudimentary concept of negative quantities as early as 400, and in 1200 the Chinese system of “counting rods” used red rods for positive numbers and black rods for negative ones—though only in certain limited contexts.
If negative numbers were a puzzle, their square roots had to be even more baffling. The difficulty is that the square of either a positive or a negative number is always positive—I won’t explain why here, but it is the only choice that makes the laws of algebra work consistently. So even if you are happy using negative numbers, it seems that you have to accept that they cannot have sensible square roots. Any algebraic expression involving the square root of a negative quantity must therefore be nonsense.
And yet Tartaglia’s formula led Cardano to just such an expression. It was worrying in the extreme that in cases in which you knew the solution by some other route, the formula seemed not to produce it.
In 1539 a worried Cardano raised the matter with Tartaglia:
I have sent to enquire after the solution to various problems for which you have given me no answer, one of which concerns the cube equal to an unknown plus a number. I have certainly grasped this rule, but when the cube of one-third of the coefficient of the unknown is greater in value than the square of one-half of the number, then, it appears, I cannot make it fit into the equation.