But the young canon remained cautious. Only in his later years would he set down his definitive conclusions in a work entitled De revolutionibus orbium coelestium (On the Revolution of the Heavenly Spheres), which firmly placed the sun at the centre of its own planetary system. However, not until he was lying on his deathbed in 1543 would Copernicus allow his revolutionary work to be published. This would literally change the world for ever, and would go on to spark great controversy within the Catholic Church. Even the Protestant leader Martin Luther poured scorn on the idea: ‘This fool wishes to overturn the whole science of astronomy. Does not the Holy Bible tell us that Joshua commanded the sun to stand still, and not the earth?’ Just under half a century later Copernicus’ work would be read by Galileo, who would become a convinced advocate of heliocentric astronomy, before being appointed professor of mathematics at Padua, thus bringing Copernicus’ idea full circle.
Just six years after Copernicus left Padua in 1503, the city would be overrun by the forces of the Holy Roman Emperor Maximilian I and the League of Cambrai, causing the university to be shut down. Though the imperial troops only occupied the city for a few weeks, teaching would not resume here until 1517. Amongst the first influx of new students was a young man known as Tartaglia, who would later take up residence in Venice, where he would make an algebraic discovery that would involve him in the greatest mathematical controversy of his age.
The man known to history as Tartaglia was born Niccolò Fontana in 1500 at Brescia, fifty miles east of Milan (though this was part of Venetian territory at the time). His father was a despatch rider, travelling on horseback across country to neighbouring towns to deliver mail; but in 1506 he was murdered by robbers, plunging his already-poor family into near-destitution. Worse was to follow in 1512 when Brescia was sacked and pillaged by the invading troops of the French king Louis XII, who had arrived in Italy to support the League of Cambrai against Venice. During the sack more than 45,000 citizens of Brescia were put to the sword by the rioting foreign soldiery. In the course of this mayhem the twelve-year-old Tartaglia, along with his mother and brothers, managed to take sanctuary in the local cathedral, but to no avail. A French soldier slashed Tartaglia across the head with his sword, leaving him for dead. In fact, the blow had only severed his jaw and palate. His mother found him alive, and over the coming month nursed him back to health. But the shock and the wound ensured that the young boy would never fully recover the power of speech – hence his nickname Tartaglia, which means ‘the stammerer’. After this he would never shave, growing a beard to camouflage his frightful scars.
As a result of this distressing setback Tartaglia would spend most of his time at home, where he developed an interest in mathematics. Consequently his mother sought out and found a patron for her exceptional son, who took him to study in Padua and then Venice. But as Tartaglia’s talent blossomed, and he began outstripping all those around him, this had a detrimental effect on his character, causing him to overcompensate for his lowly origins and ugly appearance by becoming unbearably proud and arrogant. On his return to Brescia he soon fell out with his patron, and around the age of seventeen he left home to teach mathematics in Verona. He is known to have been desperately poor during the ensuing decade or so; despite this, he seems to have got married and had a family sometime in his early thirties. Then in 1534 he moved to Venice, where he quickly began attracting a reputation for brilliance by defeating a number of renowned mathematicians in the public contests that were becoming so popular. These were a development from the philosophical disputations held by medieval theologians, the intellectual equivalent of the jousting tournaments between knights. One of the basic rules was that no contestant should submit to his opponent a problem that he could not himself solve. Victory was a means to advancement for the exceptionally skilled in a hierarchical society; even so, the work Tartaglia gained as a mathematics teacher and tutor to the sons of noble families brought him only a modest income.
Tartaglia was soon making significant contributions to his field. One of his first achievements was to translate Euclid’s Elements into Italian. Prior to this, Euclid’s work had only been available in Latin translations taken from badly corrupted Arabic sources, rendering parts of the text incomprehensible. Tartaglia’s work revived the study of this formative text, which would prove an inspiring influence on Galileo, leading him to conclude that ‘the book of the world is written in the language of mathematics’. Tartaglia was also the first to understand that cannonballs travel in a trajectory when fired from a gun. (Previously, in accord with Aristotle’s thinking, it had been assumed that the cannonball travelled in a straight line, and then simply dropped straight down out of the sky when its momentum was spent.) This enabled Tartaglia to calculate precisely the angle at which to fire a cannonball, if it was to follow a certain trajectory and hit a particular spot. He would publish a book giving tables of angles and target distances, thus hugely improving the accuracy of cannon fire.
However, his major achievement was to discover a general formula for solving cubic algebraic equations (that is, those that contain an unknown to the power of three, or x3 as we would write it*). The solution of the cubic had become the major mathematical challenge of the age: many sought it, while others simply despaired. When Luca Pacioli had published his masterwork Summa in 1494, a work that was said to contain all the mathematical knowledge known to that date, he had offered the opinion that no one would ever find a general solution for cubic equations. Unlike equations involving a simple unknown such as x, or even x2, cubic equations involving x3 were simply too complex for there to be a general formula that would give a solution. Yet soon after moving to Venice, Tartaglia managed to prove Pacioli wrong and came up with a solution to the cubic. However, news soon reached him that a young mathematician by the name of Antonio Fior had also discovered a method for solving cubic equations. In the prime of his manhood, Fior was even more arrogant than Tartaglia had been, and he reckoned that the uneducated Tartaglia was merely bluffing in an attempt to enhance his reputation. With the aim of achieving an even greater reputation for himself, he challenged Tartaglia to a mathematical duel.
In fact, Fior had not himself discovered a method for solving the cubic. This had been passed on to him sometime earlier by his teacher, Scipione del Ferro, who had decided on his deathbed that he did not wish the secret he had discovered to die with him. During this period mathematicians were not in the habit of publishing their original discoveries. On the contrary, they were in the habit of keeping them to themselves. These were the methods they could use to overcome their opponents at public mathematical contests, thus gaining them prestige, and on occasion leading to academic posts. However, the method passed on by del Ferro to his eager pupil, Fior, did in fact only solve a certain type of cubic equation. As it happens, we now know that Tartaglia too had only discovered a method for solving one type of cubic equation – and this was the very same method that had previously been discovered by del Ferro.
Preparations were soon under way for the great mathematical contest between Tartaglia and Fior. This was to be held on 20 February 1535 at the University of Bologna, where it was expected to attract a large crowd of mathematicians and aficionados to see this great problem resolved once and for all. Each contestant was to submit a list of thirty cubic equations for his opponent to solve using his own method; according to the rules of such competitions, no one could set a problem that he himself was unable to solve.
It was estimated that each contestant would take at least forty days to work his way through the list he had been given, during which time the contest was liable to seesaw either way in an exciting fashion. At the end of the allotted time, the contestant who had correctly solved the most problems would be declared the winner. The prize for winning the contest was characteristically medieval – the loser would have to pay for a feast to be enjoyed by the winner and thirty of his friends. The real prize would of course be the renown accruing to the winner, who would become fa
mous in universities and courts all over Europe for having discovered the finest solution to this problem, which had defeated all-comers.
However, as the day of the contest approached, Tartaglia became more and more nervous, suspecting that perhaps Fior had in his possession methods for solving every type of cubic equation. Racking his brains, Tartaglia set to work night and day in an effort to discover a method for solving all cubic equations. Just before dawn on 13 February, seven days before the contest was due to begin in Ferrara, Tartaglia at last discovered the answer. By manipulating the different types of cubic equation and making a number of ingenious substitutions, he was able to transform the different types of cubic equation into the very type for which he had a solution.
The day of the contest arrived, and the two contestants duly presented themselves at Bologna University before the assembled authorities and onlookers, including many members of Venetian society who had travelled to see honour done to their city. Each contestant in turn presented his list of thirty problems to the authorities. Fior’s list included a number of difficult problems – one involving a transaction with a Jewish moneylender, another concerned with the price of a sapphire – each of which could be reduced to a cubic equation. Unfortunately all the equations set by Fior were of the one type of cubic equation that he knew how to solve. Tartaglia’s list, on the other hand, contained individual problems involving all types of cubic equation, the large majority of which Fior had no idea how to solve. The result was a crushing victory for Tartaglia, who managed to solve all thirty of the equations put to him in just two hours, leaving Fior humiliated and struggling to come up with a single answer. Tartaglia graciously declined to accept the thirty meals that were his due, and set off in triumph back to Venice.
But, unbeknown to Tartaglia, this was just the beginning. Word spread through Italy of his great triumph, soon reaching Milan and the ear of Girolamo Cardano, who was possibly the most brilliant and certainly the most unscrupulous mathematician of the age. Cardano would later write to Tartaglia asking permission to include his solution of the cubic in a book he was writing on methods of calculation. He assured Tartaglia that he would credit him as the sole discoverer of this new method, which Tartaglia had ensured still remained known only to himself. But Tartaglia refused to be drawn, replying that he intended to publish his secret in a book of his own.
Girolamo Cardano had been born at Pavia, just outside Milan, in 1501. He was the illegitimate son of a young widow and a distinguished lawyer in Milan called Fazio Cardano, who was sufficiently adept at mathematics to have been consulted by Leonardo da Vinci on difficulties that he was having with certain geometric problems. Even in his youth, Cardano seems to have been a difficult character, and it was with some reluctance that his father sent him abroad to the Venetian Republic to study medicine at the University of Padua. Here his father’s misgivings were confirmed: Cardano proved an exceptional student, yet his arrogance knew no bounds. As a student he even put himself up for election as the university rector, winning the post because no one else could afford the expensive entertaining that was part of the job. He expected to finance this by gambling, but to no avail. As if this were not enough, his ensuing behaviour proved so obnoxious that, despite his evident brilliance, the authorities were only persuaded to grant their ex-rector his doctorate after a third vote. His reputation preceded him and, when he arrived back in Milan, the College of Physicians refused to grant him permission to practise locally. In the end he had to return to a village outside Padua, where he practised for five years as a lowly country doctor. Despite such setbacks, Cardano would go on to achieve an international reputation, to the point where he even treated European royalty. Even more astonishing was the fact that he simultaneously gained a supreme facility in mathematics. Yet for Cardano this was to prove no abstract pursuit, and he was soon using his mathematical abilities to great advantage in gambling, a pastime to which he became addicted, to the point where he boasted in his autobiography, ‘Not a day went past on which I did not gamble.’ His pioneering understanding of probability theory, which was a century ahead of its time, enabled him to know in advance when the odds were in his favour and place his bets accordingly – though this did not stop him from cheating, when he deemed it necessary. On top of these vices he had many others, which may be gleaned from his disarmingly frank autobiography. Despite such revelations, he still claimed in this work: ‘I have but one ingrained and outstanding fault – the habit I have of saying things which I know will upset people. I am fully aware of the effect of this, but persist in it regardless of all the enemies it creates for me.’
Besides being a boastful liar, he was also devoid of conscience. In 1534 his father managed to secure for him a post as a mathematics lecturer in Milan. Even so, this did not prevent him from gambling away all his possessions, including his wife’s jewellery, so that he ended up in the poorhouse the following year – the very year of Tartaglia’s triumph in Bologna.
Cardano bided his time, and it was not until three years later that he wrote to Tartaglia, offering to publish his solution in the work that he was writing. This was intended as a rival to Pacioli’s Summa, no less, and included an entire chapter listing Pacioli’s mistakes. Despite Tartaglia’s persistent rejections, Cardano would not be put off and continued to pester him. In 1539 he even went so far as to send the bookseller Zuan Antonio da Bassano as an intermediary, suggesting to him that if he could not wheedle the secret out of Tartaglia, he should ask him for the list of thirty questions that Fior had submitted to him at the contest in Bologna, as well as Tartaglia’s thirty correct answers. Cardano suspected that he might be able to gain a clue from these questions and answers, which would enable him to solve the problem of the cubic for himself. Tartaglia was well aware of this, and remained adamant. Finally, Cardano appeared to admit defeat and issued a friendly invitation to Tartaglia to visit him in Milan, suggesting that he might be able to advance Tartaglia’s career. Although Tartaglia was well respected as a mathematician in Venice, his earnings as a teacher of this subject remained meagre; as a result he accepted Cardano’s offer with alacrity, and in 1539 travelled to Milan to meet his colleague.
Here Cardano soon revealed his hand: if Tartaglia was willing to confide to him the secret of the cubic, he was willing to introduce him to Alfonso d’Avalos, the governor of Milan, who happened to be a friend of his. D’Avalos would certainly be interested in the discoveries that Tartaglia had made on how to increase the accuracy of cannon fire, and was liable to offer him a well-paid post as a military adviser. After giving some thought to the matter, Tartaglia warily agreed to tell him the secret of the cubic, but only after he had made Cardano take the following solemn oath:
I swear to you on the Sacred Gospel and on my faith as a man of honour not only never to publish your discovery, if you reveal it to me, but also promise on my faith as a Christian only to note this down in code, so that even after my death no one will be able to understand it.
Cardano duly swore the oath, and Tartaglia wrote down for him the twenty-four-line poem that he had composed to memorise the secret of how to solve the three types of cubic equation. The words unfolded in all their enigmatic glory, like some magic spell:
When the lonely cube on one side you have found,
With the other terms being together bound:
… two numbers multiplied, swift as a bird,
Reveal the simple answer of one third …
Ending triumphantly:
These things I did discover before all others
In fifteen hundred and thirty-four
In the city girt by the Adriatic shore.
Cardano duly penned a letter of introduction to d’Avalos. At this point it becomes unclear precisely what happened. The two main contemporary sources, both written down many years after the event, were each heavily biased: one was written by Tartaglia himself, the other by Cardano’s pupil Ludovico Ferrari, who claimed to have been present (he almost certainly was not). Tartagli
a knew that d’Avalos was not in Milan, but was visiting the fortifications at Vigevano, fifteen miles away on the banks of the River Ticino; so he set off with Cardano’s letter of recommendation. However, on the way he seems to have come to the conclusion that he had somehow been tricked by Cardano, and thereupon turned his horse back to Venice. His only comfort was that Cardano was sworn to secrecy.
But Cardano was not a man so easily bound. He began pondering: how had a distinctly mediocre mathematician such as Fior discovered how to solve the cubic, a problem that had defeated the finest minds of the age? The obvious answer appeared to be that he had learned it from his talented master, Scipione del Ferro. In 1543 Cardano travelled to Bologna, where he visited the man who had been entrusted with del Ferro’s notebooks and mathematical papers. Amongst these Cardano soon found the formula for solving the cubic. Evidently del Ferro had discovered this first. Cardano returned to Milan and included the formula in his book Ars Magna, which he published the following year.
Venetians Page 23