Chapter 7

Nicolo Tartaglia’s self regard is dramatically inflated after his victory over Fior. He is aware he now possesses unique knowledge that will provide him with an income for years to come. He guards it jealously and to good effect. His enhanced reputation brings him various teaching posts, including a public lectureship in Milan as well as his Venetian teaching post. This gives him the money and time to turn his mind to subjects no one else is thinking about.

One of those subjects is artillery. No one has yet put mathematics to work in the service of kings and generals at war and Tartaglia finds himself well equipped. However, he is filled with a pacifist’s remorse. For all his vain self-aggrandising nature, he remains a devout man of high moral standards. The creation and refinement of killing machines does not sit well with him, as this passage from his writings makes clear:

One day I fell to thinking it a blameworthy thing, to be condemned — cruel and deserving of no small punishment by God — to study and improve such a damnable exercise, destroyer of the human species, and especially Christians in their continual wars … I destroyed and burned all my calculations and writings that bore on the subject. I much regretted and blushed over the time I had spent on this.

That attitude only softened when Suleiman the Magnificent, Sultan of the Ottoman Empire, began to slaughter Christians in the crusades. ‘It no longer appears permissible to me at present to keep these things hidden,’ Tartaglia says. ‘I have hence resolved to publish them partly in writing and partly by word of mouth, to every faithful Christian, so that each may be better fitted in offence as well as defence.’ In 1537, convinced he can help the Pope’s cause, he self-publishes his treatise on the trajectory of cannonballs, The New Science of Artillery.

Good as the book is, it misses its target. Tartaglia simply doesn’t have the reach, the connections, to attract the attention of military minds. So when Jerome gets in touch two years after its publication and offers to help, Tartaglia takes the bait.

Jerome is desperate to know Tartaglia’s secrets. Having heard a great deal about them, he is suffering a base professional jealousy. Jerome sees algebra as his specialist subject and yet he is painfully aware that he doesn’t know all there is to know about it.

It is a particular problem because he has decided to write a revolutionary book on arithmetic. This is to be no high-minded endeavour to impress his academic rivals, but a maths book for the common people. It will be sixty-eight chapters long, with an introduction that lays out the journey the reader is to take.

That decision was made fourteen years ago and, in the years since, he has already completed much of the book. So far, The Practice of Arithmetic has dealt with basic operations, such as multiplying and dividing. It has taught integers and fractions, and even the supernatural properties of numbers. Jerome has reached the more advanced chapters now. The problem is that Tartaglia’s confidante, the hollow-eyed Brescian schoolteacher Zuanne da Coi, has come to Milan. He is talking loudly about Tartaglia’s new solutions to the cubic equations. Jerome knows these new solutions must be published in his book. But how to get them?

Fortunately, Jerome has trained a student, Lodovico Ferrari, whose talent will help achieve this tricky task. Lodovico came to Jerome’s household as a fourteen-year-old servant on 30 November 1536. Initially, he was a replacement for his cousin, who had run away from the notion of working hard for a living. Unlike his cousin, Jerome noted, Lodovico could read and write, so he put the boy in charge of the household secretarial work. When Lodovico proved able and clever, Jerome began to teach him mathematics. Lodovico may well have been the guinea pig on whom Jerome tested the efficacy of his blossoming maths text.

Five years later, in 1541, Jerome had become convinced of Lodovico’s gift and was determined to give it space to flower. He resigned one of his teaching posts, as a lecturer at the Piatti Foundation in Milan, knowing this would give Lodovico an opportunity to break into academia.

There were only two candidates for the newly vacant position: Lodovico and the teacher Zuanne da Coi. The choice was resolved in a public maths duel — of course — and Lodovico won easily. The twenty-year-old servant is now also a lecturer in geometry, feels a great debt to Jerome, and is fiercely loyal to his cause.

Jerome and Lodovico work well together, researching new algebraic solutions to difficult problems. But still they cannot crack Tartaglia’s secret. Eventually, they decide on a more direct approach. They will ask Tartaglia to share his knowledge for the common good.

Aware of Tartaglia’s prickly reputation, Jerome and Lodovico enlist a local bookseller named Zuan Antonio as a go between. He will convey Jerome’s carefully phrased question to Tartaglia and bring back the reply, with all communication directed to Antonio as a third party.

It starts off politely enough:

Master Nicolo Tartaglia, I have been directed to you by a worthy man, a physician of Milan, named Master Girolamo Cardano, who is a very great mathematician. And because he has understood that you have been engaged in disputation with Master Fior, putting to him for a wager certain questions that could only be answered by knowing the general rule for resolving the case of the cubic, which general rule you had found by your own discovery. Therefore his excellency prays you that you will kindly make known to him the rule discovered by you, and if you think fit will make it public under your name in his present work, but if you do not think fit that it should be published he will keep it secret.

The reply, predictably, came in the negative:

Tell his excellency that he must pardon me: when I propose to publish my invention, I will publish it in a work of my own, and not in the work of another man, so that his excellency must hold me excused.

The full back-and-forth becomes a little tedious — most particularly for those involved. However, Jerome does not give up. He even offers never to publish the solutions, if Tartaglia will only reveal them. Tartaglia refuses.

Jerome then asks for the list of questions which Fior had set him in the competition for Tartaglia’s post. These Tartaglia reluctantly supplies, commenting that, ‘his excellency, whatever his competence, will be unable to resolve them, for to do that would mean his excellency had a wit like to my own, which he has not’.

Chiding his correspondent for such an ‘unhandsome reply’, Jerome remarks that whatever Tartaglia might think, he is himself ‘nearer to the valley than the mountain-top.’ It is then that he drops into the conversation the fact that he has passed a copy of Tartaglia’s book on artillery to a friend. A friend who happens to be Alphonso d’Avalos, Marquis del Guasto.

The Marquis, as Tartaglia knows — and Jerome knows that he knows — is a powerful military man. Jerome now dangles the chance of an offer of employment before his rival. He invites Tartaglia to come to Milan to meet the Marquis. Jerome’s masterful move puts Tartaglia in a bind. In the margins of a letter from Jerome, Tartaglia scrawls out his frustration: ‘I am reduced by this fellow to a strange pass, because if I do not go to Milan the lord marquis may take offence, and such offence might do me mischief, I go thither unwillingly: however, I will go.’

And go he does. When he arrives, however, the Marquis is out of town. Jerome invites Tartaglia to visit his house instead. There he repeats his promise that, if he could just learn the secrets of the cubic equations, he will refrain from publishing them. Tartaglia brazenly says he is not willing to take Jerome at his word. Jerome swears several oaths and says he’ll even write the solutions down only in code, so that no one can discover them, even after his death.

And it is then — for reasons that remain unclear — that The Stammerer wavers. He says he will ride off in search of the Marquis; on his return, he will show Jerome the solutions. Seizing the moment, Jerome says no — tell me now. And, using a poem as a cover, Tartaglia does so:

When the cube and things together

Are equal to some discreet number,

Find two other numbers differing in this one.

Then you will keep this as a habit

That their product should always be equal

Exactly to the cube of a third of the things.

The remainder then as a general rule

Of their cube roots subtracted

Will be equal to your principal thing

In the second of these acts,

When the cube remains alone,

You will observe these other agreements:

You will at once divide the number into two parts

So that the one times the other produces clearly

The cube of the third of the things exactly.

Then of these two parts, as a habitual rule,

You will take the cube roots added together,

And this sum will be your thought.

The third of these calculations of ours

Is solved with the second if you take good care,

As in their nature they are almost matched.

These things I found, and not with sluggish steps,

In the year one thousand five hundred, four and thirty.

With foundations strong and sturdy

In the city girdled by the sea.

Yes, it seems unbelievably, painfully contrived. But this is Renaissance Italy; this is how they are. Indeed, Tartaglia is by no means the first algebraist to summarise his insight in poetic form. Decades earlier, for example, the mathematician Luca do Borgo had expressed his algebraic rules in Latin quatrains.

Tartaglia is proud of his rhyme: it ‘speaks so clearly’, he tells Jerome, ‘that, without other example, I think your excellency will be able to understand the whole’. Jerome, in reply, assures Tartaglia that he has understood almost everything about the solution on his first reading of the poem. Remember your promise not to publish, is Tartaglia’s response. And then there is silence. Five months pass. Tartaglia, regretting his weakness and hearing disturbing rumours, writes to Jerome from Venice:

I am very sorry that I have given you already so much as I have done, for I have been informed, by a person worthy of faith, that you are about to publish another algebraic work, and that you have gone boasting through Milan of having discovered some new rules in Algebra. But take notice, that if you break your faith with me, I certainly shall not break promise with you (for it is my custom); nay, even undertake to visit you with more than I have promised.

Jerome holds his nerve. He has indeed published a book containing algebraic rules, The Practice of Arithmetic. But, no, he hasn’t broken his promise. He sends Tartaglia a copy, so that he can verify this for himself. Tartaglia’s response is cruel and petty. He writes to Jerome that the book contains nothing new. It is simply a synthesis of what we already knew. What’s more, he adds, the book is confused and full of mistakes: ‘The whole of this work of yours is ridiculous and inaccurate, a performance which makes me tremble for your good name.’ He points to one particular howler: ‘your excellency has made such a gross mistake that I am amazed thereat, forasmuch as any man with half an eye must have seen it — indeed if you had not gone on to repeat it in divers examples, I should have set it down to a mistake of the printer’.

Nor has he finished. So numerous are the errors, Tartaglia says, and sometimes so rudimentary, that Jerome is clearly incapable of innovative work. The idea that Jerome could have worked out Tartaglia’s solution to the cubic equation for himself, he writes, ‘sets me off laughing’. Enter Lodovico Ferrari, Jerome’s brilliant pupil, spoiling for a fight. Ferrari clearly believes it is below his beloved master’s dignity to pursue the quarrel. But Ferrari has no trouble with brawls himself, having lost two fingers to a rival’s dagger in a tavern fight. Now, seeming to relish taking on Tartaglia, he writes on his master’s behalf:

You have the infamy to say that Cardano is ignorant in mathematics, and you call him uncultured and simple-minded, a man of low standing and coarse talk and other similar offending words too tedious to repeat. Since his excellency is prevented by the rank he holds, and because this matter concerns me personally since I am his creature, I have taken it upon myself to make known publicly your deceit and malice.

He even makes a mathematical joke. With all Tartaglia’s talk of square roots, cube roots, and more, he says, ‘I promise you that if it were up to me to reward you, taking example from the custom of Alexander, I would load you up so much with roots and radishes that you would never eat anything else in your life.’

You probably had to be there. But the reality is that very many people were, for the correspondence between Tartaglia and Ferrari is sold in the streets like newspapers. Indeed, it circulates around Europe as other academics take it upon themselves to explain to everyone what the dispute is about. The row has become famous, as have its protagonists. The unexpected side effect is that the public is learning mathematics in order to follow the soap opera. To be precise, the public is learning about cubic equations and conic sections.

That might seem implausible now, but in Renaissance Italy the public is still aware that these subjects belong to a revered intellectual lineage. For most of history, conic sections — the points where a cone meets a surface — were just a mathematical challenge, the geometrical Sudoku of the professional mathematician. The Greeks, for instance, loved playing around with these puzzles and, a couple of hundred years before the birth of Christ, a geometer and astronomer called Apollonius of Perga wrote an eight volume treatise on the mathematics of the conic section. More than a thousand years later, Omar Khayyam, the Persian mathematician and poet, wrote his Treatise on Demonstration of Problems of Algebra, in which he showed that the intersection of a hyperbola with a circle provides a geometric method for solving cubic equations. This, it is worth saying, is not a pointless pursuit. Conic sections trace out an ellipse or parabola and can be used to calculate the trajectories of planets through the sky, or artillery through the air.

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‘I simply had to have them in the book.’ Jerome’s tone is defensive, almost sly. ‘Students of algebra needed to know these solutions. Do you understand?’

I nod, but slowly and with my head inclined; I can see his point, but I can also see Tartaglia’s. ‘I suppose so,’ I concede eventually. ‘It’s funny: we teach our children to solve these equations, but I don’t think they ever know why they are useful.’

Jerome frowns at this. ‘Why not?’

‘Most of our children give up studying mathematics as soon as they are able.’

‘And come back to it when they are older?’

I shake my head. ‘No. They almost never come back. They may go through their lives knowing the formula for solving a quadratic equation, but never once apply it to anything.’

‘So there are bakers on the streets of sixteenth-century Milan now who feel more in tune with mathematics than the educated children of your city?’

‘More in tune than the educated adults, if I’m honest.’

You know it’s true. Remember this? x equals minus b, plus or minus the square root of b squared minus four ac, all over two a. It is a phrase that means something to people educated to secondary level across the world. Few of us realise that the Babylonians knew it too, as the formula to solve a quadratic equation of the form (as we would write it now): ax2 + bx + c = 0.

Jerome and Tartaglia and all their peers certainly knew this trivial piece of mathematics. What hadn’t been resolved was how far you could go. Could there be solutions for all equations that involved the cubic: x3? And what about the quartic: x4? The quintic: x5?

Years earlier, Scipio Ferreus had gone some of the way with cubic solutions and Tartaglia had gone further still. Jerome and Lodovico had derived two more solutions to algebraic problems. Jerome was eager to publish them, but he was hamstrung. His work relied on Tartaglia’s solution, revealed in the poem, and he had promised not to publish that. And then he found a way to sidestep The Stammerer.

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In 1543, someone — we don’t know who — whispers in Jerome’s ear that it was Scipio Ferreus, not Tartaglia, who came up with the first solution. If that is true, why not go straight to the source? Or, given that the source is dead, the source’s son-in-law?

Here, fortune smiles on Jerome. It turns out that the son-in-law, Annibale della Nave, is still very much alive, and living in Bologna. Jerome and Lodovico pitch up at his door. Annibale shows them Ferreus’s papers. In an ‘aha’ moment worthy of any great detective story, they see the solution — and see immediately that it is the solution that Tartaglia hinted at in his poem. Yes, Tartaglia might have worked out a solution to the cubic equation, but Ferreus had worked it out first. That means the solution is not the exclusive property of Tartaglia. And so, Jerome reasons, it could be published in his next work, a book on the ‘Great Art’ of algebra.

‘My heart was skipping,’ Jerome says. His eyes glint. ‘After that, the solutions poured out.’

‘The quartic? And all the various cubic solutions?’

He nods. ‘And now I was free to publish them. My oath to Tartaglia was irrelevant’

‘Because you could publish his solution as Ferreus’s work.’

He grins: a wide, ugly leer. ‘Exactly,’ he says.

In his new book, The Great Art (Ars Magna), Jerome gives Tartaglia and Ferrari all credit for their contributions, which, to his credit, he lays out with great skill. Scipio Ferreus also receives credit where it is due.

To Tartaglia, however, a credit is not enough. The Stammerer is incensed.

Jerome doesn’t understand why, but then there are many things to do with these equations and their solutions that he doesn’t understand. The negative solutions, for example, which he calls ‘fictitious’. And there are worse cases. There are, for instance, the solutions that involve the square roots of negative numbers — the famed imaginary numbers.

Jerome’s Ars Magna is the first published acknowledgement that mathematical procedures can produce the square roots of negative numbers. In it, they appear as the solution to a relatively simple problem: ‘divide 10 into two parts, one of which multiplied into the other shall produce 40’. The only solutions, Jerome showed, were (5 + √-15) and (5 – √-15).

Jerome had seen that solving cubic equations often produces square roots of negative numbers along the way. That is because his formula for solving an equation of the form x3 = 3px + 2q was = 3√(q + √(q2 – p3)) + 3√(q – √(q2 – p3)).

Once you start plugging numbers into that formula, square roots of negative numbers can appear very quickly. You can start with very ordinary numbers and sometimes progress through to solutions that contain very ordinary numbers, but you must be prepared to encounter strange and scary beasts along the way. And Jerome was.

Not that he has received the credit he deserved for such boldness. Nowadays, mathematicians tend to credit Descartes with recognising the importance of these ‘imaginary’ numbers. In his 1637 book La Géométrie, Descartes says, ‘Neither the true nor the false roots are always real; sometimes they are imaginary.’

By ‘false’, Descartes means negative numbers, which were themselves considered suspicious and problematic. How much more troublesome the imaginary numbers seemed. Newton called them ‘impossible’ in his Universal Arithmetic of 1707. Newton’s arch rival, Gottfried Leibniz, was more positive. In 1702, Leibniz, who was a great admirer of Jerome’s work, spoke of the imaginary number as ‘a fine and wonderful resource of the human spirit, almost an amphibian between being and not being’.

In the end, it was the Swiss mathematician, Leonhard Euler, who brought imaginary numbers into the mainstream. In the eighteenth century, he followed Descartes’ idea and named them imaginary numbers, denoting the square root of -1 as i. He connected i to the real world by showing that a mathematical constant known as e — Euler’s number — is connected to pi via i: eiπ = -1.

That e ties together with i and pi is one of the great mysteries of the mathematical universe. e was being used in myriad calculations — from calculations of compound interest, to the power of cannons — and so it wasn’t long before Euler’s work turned Jerome’s ‘useless’ numbers into an essential component of a mathematician’s toolkit. By the end of the eighteenth century, they were required everywhere.

If I want to take something that varies over time, for instance, and calculate its exact value at a certain time, I need the imaginary numbers. That is because they exist in the formula and, as soon as the formula involves squaring a number, they become real. For all their ‘imaginary’ nature, if you don’t put them in, you get the wrong answer.

So, when are the imaginary numbers required in a formula? The answer is when there is more than one dimension to a problem — which, in the real world, is always the case. Say I wanted to calculate how fast a team of oxen could plough a field. It is not just about how much power they apply to the plough — there is also the issue of the soil’s resistance to their movement. And the amount of resistance changes depending on how fast the plough is moving. It is a complex problem and it requires complex numbers. And ‘complex numbers’ is the name we give to the combination of real and imaginary numbers.

How does ploughing a field relate to the solutions for cubic equations? Well, I can plot a graph of the speed of the oxen versus the resistance of the soil to the plough. Because the soil’s frictional resistance to the ploughshare depends on the speed of the oxen, it wouldn’t be a straight line; it would be a curve. And if I wanted to add in a third factor — the change in resistance as the spring sunshine dries out the soil, say — then I would have not just a curve, but a three dimensional curve. That is a solid object, essentially, something like a curvy cone. If I want to know where this curvy cone intersects with another factor — the availability of labourers during the day, say, so that I can work out the most efficient time at which to start ploughing — then I am trying to find solutions to where the curvy cone meets another, related curve. I am, in fact, looking for the solutions to a cubic — possibly quartic — equation. We are back in the territory of Omar Khayyam and Apollonius of Perga.

No one wanting to plough a field in Jerome’s day would think about consulting a mathematician. However, people calculating the interest owed on loans certainly did. Bankers and loan sharks lent money with property as security — your house, perhaps, or the contents of a grain-storage barn. With the value of the property changing on a weekly or monthly basis, depending on the economic climate, and accepted interest rates varying on a daily basis, those calculations involved solving what were, at the time, rather complex equations. The bankers didn’t need to think in terms of conic sections because the educated men they hired as in-house mathematicians generally had plug-and-play formulae at their disposal. Except, of course, where they didn’t. In those cases, a new solution method would, quite literally, put money in the bank.

Things are no different now. When I was doing my PhD I worked with a colleague, Daniel, who eventually pursued a career in finance. His abilities in solving ‘differential equations’ — the name given to equations that involve a cohort of smoothly varying factors, all of which can change the outcome — have earned him a fortune. His variables are not mud and ploughs, sun and workmen, but commodity prices and shipping times, along with the minutiae of supply and demand. This is why, for decades now, the world’s banks have been mopping up some of the best universities’ mathematicians and physicists. All of these people know the value of i. To them, it is priceless.

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‘And why didn’t you work for the bankers?’ Jerome looks genuinely puzzled. ‘You know quantum theory, too. You can solve these equations. You could have made your fortune.’

I raise my eyebrows and smile. ‘I could ask you the same thing, surely?’

‘I never cared about money,’ Jerome says with a shrug.

‘Me neither. Otherwise I wouldn’t be a writer, would I?’

Jerome acknowledges the joke with a grin, but says nothing.

‘And I would never have discovered all of this.’ I tap my stone on the psi. ‘I would have had all the knowledge and none of the wisdom. Daniel can solve equations for money, but he doesn’t understand the nature of reality.’

‘And you do?’ He knows the answer.

‘Not yet,’ I reply. ‘But I’m ever hopeful.’

I can see the smile twitching at the corner of his mouth. ‘And is it the Copenhagenists that give you such optimism?’

He is being sardonic, and enjoying it. Jerome had very little time for the Danes. He was once asked by King Christian III of Denmark to travel to Copenhagen and become the royal physician for a salary that was twice what he was earning at Pavia — plus a house, servants, and horses. Jerome politely declined. The Danes, to his mind, were uncultured barbarians and their climate would be the end of him. What, he later wrote, is the use of riches and comfort when the cold and the damp are ‘an entrance to death’s caverns’?

‘I told the King of Denmark that I couldn’t help him because I couldn’t be in two places at the same time,’ Jerome says. He looks thoughtful. ‘I said I was a widower with children and that I had to oversee their education. I couldn’t be in the north when my heart was in the south.’

‘That was an expedient excuse, though, wasn’t it? Your children’s education didn’t stop you going to Scotland, after all.’

Jerome laughs, a low chuckle that makes his shoulders shake for a moment. ‘No,’ he says. ‘But Archbishop Hamilton needed my help more than King Christian.’

‘And now you need his.’

Jerome picks up his pen. ‘Yes,’ he says. ‘And now I need his.’