The Venetians: A New History: from Marco Polo to Casanova (31 page)

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). Tartaglia 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.

When Tartaglia discovered what had happened, he was furious. He angrily accused Cardano of betraying his oath, but Cardano pointed out that his oath only concerned Tartaglia’s discovery, which he had now learned was not entirely original; del Ferro’s discovery pre-dated his by some years, and this was the formula that he had included in
Ars Magna
. Tartaglia in fact had only himself to blame – if only he had published his method of solving the cubic, instead of keeping it to himself, none of this would have occurred. As it was, Cardano had triumphed over him, albeit in a most underhand manner. Tartaglia would never really get over this defeat, and would die an embittered impecunious old man in Venice in 1557.

*
I have used the simplified modern notation here, which Tartaglia would not have used. At the time, even those who studied algebra had a different name, calling themselves ‘cossists’ – after the Italian word
cosa
, meaning ‘thing’, this being the name they gave to the unknown quantity (which we would call x).

12

The Loss of Cyprus

M
EANWHILE
V
ENICE ALSO
found itself facing some difficulty where money was concerned – a crisis that was only avoided by some swift and astute financial manipulation. The Republic’s monetary policy had continued to rely heavily upon silver, whilst maintaining a Gold Standard to keep its currency in line with other European currencies. This fixed the gold ducat as worth 124 silver
soldi di piccoli
(literally ‘little coins’). The soldi were considerably smaller than ducats and, in terms of actual precious metal, silver in fact remained the more valuable of the two – a situation reinforced by the fact that gold was more plentiful throughout Europe. This suited the Republic’s finances while there was more gold on the markets, and while Venice still had access to silver from the East. All this began to change after the Spanish conquest of Mexico in 1521, when silver began to flow into Europe from the New World. This flow became a torrent with the discovery by the Spanish in 1545 of the Potosi ‘Silver Mountain’ in Peru, which would soon be accounting for more than half the silver mined throughout the world. Consequently the price of silver plummeted in Europe against that of gold. As a result, the Venetian authorities took the drastic step of transferring the Republic’s finances to a Silver Standard – though still at the rate of 124 silver soldi to the gold ducat. This bold move brought a great reward for the authorities, enabling them to pay off the public debt with the official currency, which was now the vastly cheaper silver soldi. Bond-holders, and others who had bought quantities of the public debt, had little choice but to accept this payment. The government had benefited at the expense of its many investors; but, as ever, the interests of the Republic always overrode those of individual citizens.

And never was this more so than when the Republic was confronted with the prospect of war. Venice still faced a potential threat from the powers of Europe as well as from the expanding Ottoman Empire. Caught between a rock and a hard place, the Republic persisted in its policy of appeasement towards the Turkish sultan, Suleiman the Magnificent, while at the same time irritating the European powers by refusing to support any aggressive moves against the Turks. This situation was epitomised in 1535 when Venice antagonised the Holy Roman Emperor, Charles V, by its unwillingness to support his imperial fleet, under the command of the Genoese admiral Giovanni Andrea Doria (great-nephew of Andrea Doria), when it embarked to seize Tunis, which stood in the path of Ottoman expansion along the North African coastline.

However, the policy of appeasing the Turks was to prove terribly flawed when, two years later, Suleiman the Magnificent decided it was time to declare war on Venice: he had simply been biding his time. In August 1537 the Ottoman fleet appeared off the Venetian island of Corfu, under the command of the formidable Khaireddin Barbarossa, a Greek-born former pirate. Corfu guarded the entrance to the Adriatic, and this move directly threatened not only Venice, but also the kingdom of Naples, just a hundred miles across the water, which was part of the European empire ruled by Charles V. Venice appealed to Charles V and the European powers for help, but received no reply. Even more ominously, the imperial fleet under Giovanni Andrea Doria, which happened to be sailing through nearby waters, simply turned tail and set course for Genoa. Doria explained that he could not possibly put into battle without direct orders from Charles V. It was evident that he had not forgotten Venice’s lack of support two years previously, and that the Genoese still saw Venice as their real enemy.