Why Beauty is Truth (13 page)

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.

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.