Chapter Eight: How to Aim a Cannonball
Renaissance Europe, 1400–1600 CE
In the winter of 1512, French soldiers sacked the city of Brescia in northern Italy with a thoroughness that the inhabitants would not forget for generations. They burned, they looted, and they killed. A boy of about twelve — the son of a murdered mail rider, already poor, already largely self-taught — was caught in the slaughter and received a sword wound to his jaw and palate that shattered his teeth and left him with a permanent stammer. His mother, the account goes, nursed him back to health by licking his wounds when there was nothing else to hand.
The boy’s name was Niccolò Fontana. He would later take the nickname Tartaglia — “the stammerer” — and wear it, with characteristic stubbornness, as a badge rather than a wound. He became, against every obstacle his circumstances could throw at him, one of the leading mathematicians of the sixteenth century. And the experience of Brescia — the cannon fire, the chaos, the organised application of violence by well-equipped armies — left a mark on his intellectual life as permanent as the scar on his face.
Tartaglia spent his career thinking about war. Not because he glorified it, but because war was the most mathematically demanding enterprise his world contained, and Tartaglia went where the hard problems were.
The New Science of Violence
The fifteenth century changed warfare, and warfare changed mathematics.
Gunpowder had reached Europe from China via the Islamic world by the thirteenth century. By the fourteenth, cannon were appearing on European battlefields. By the fifteenth, they had become the decisive technology of military power — capable of reducing in hours the stone walls that had protected cities for centuries, capable of firing projectiles over distances that no previous weapon had approached, capable of killing at ranges where the attacker could not even be seen from the walls.
But cannon were, in an important technical sense, not understood. Gunners knew from experience roughly how to aim them — they adjusted elevation by eye, compensated for wind by feel, estimated range from years of practice. This was craft knowledge, passed from master to apprentice, unwritten, untheorised, and extremely variable in its results. A skilled gunner was an artist whose art died with him. There was no science of ballistics because no one had yet asked the question that a science requires: not how do you aim this particular cannon in this particular situation, but what are the general principles that govern where any projectile goes?
Tartaglia asked the question, and in 1537, twenty-five years after the sack of Brescia, he published the answer in a book called Nova Scientia — A New Science.
The title was deliberate and provocative. Tartaglia was claiming that the flight of a cannonball was a subject for mathematical analysis, not craft intuition. That it obeyed general principles that could be stated as propositions and proved by argument. That Euclid’s method — axioms, definitions, demonstrated theorems — could be applied not just to abstract geometric figures but to the messy, physical problem of a ball of iron hurtling through air.
The frontispiece of Nova Scientia is one of the most striking images in the history of mathematics. It shows a walled compound — the compound of knowledge — with a crowd gathered inside around a demonstration. The caption reads: “The Mathematical sciences speak: Who wishes to know the various causes of things, learn about us. The way is open to all.” At the single entrance to the compound stands Euclid, as gatekeeper. In the inner courtyard, visible only to those who pass through the outer one, stand Aristotle and Plato.
The message is clear: to understand the physical world, you must first pass through mathematics.
What Tartaglia Got Right, and What He Got Wrong
Tartaglia’s ballistics were partly right and importantly wrong, and the gap between his results and the correct theory is itself a story worth telling.
The dominant physical theory of his time was Aristotelian. According to Aristotle, a projectile’s motion had three stages: first, a straight line of “violent” motion in the direction it was fired, as the initial force of the powder drove it forward; then a curved transition as the violent force was exhausted; then a straight vertical drop as “natural” motion — the tendency of heavy objects to fall toward the earth’s centre — took over completely. This gave the trajectory a shape like a bent elbow: straight out, a curve, straight down.
Tartaglia’s observation — partly empirical, partly theoretical — was that the trajectory was continuously curved throughout, not bent in stages. This was correct. He also derived, by a combination of physical reasoning and mathematical argument, that the maximum range of a cannon was achieved when it was aimed at 45 degrees to the horizontal. This too was correct — it is a result that follows from the mathematics of projectile motion, and Tartaglia arrived at it without the calculus that would later make it straightforward to derive.
What he got wrong was the shape of the curve. Tartaglia described the middle section of the trajectory as a circular arc, which is a reasonable approximation but not exact. The exact shape is a parabola, and deriving that requires understanding how gravity acts continuously on a moving object — an understanding that required the conceptual machinery of calculus, which was still a century and a half away. It would take Galileo, working in the early seventeenth century and directly influenced by Tartaglia’s work, to arrive at the parabola.
But Tartaglia had done something more important than getting the exact answer. He had established that the question had an exact answer — that cannon fire was a mathematical problem with a mathematical solution, that the language of geometry and proof was the right language for understanding the physical world. This claim, which seems obvious to us now, was genuinely radical in 1537. It was a declaration that mathematics was not merely a tool for counting and measuring but a fundamental description of how nature behaves. Galileo would later say that the book of nature is written in the language of mathematics. Tartaglia helped write one of the early European chapters in that story.
The Stammerer and the Doctor
While Tartaglia was building his reputation in Venice as a mathematician of the practical and bellicose, a very different figure was making his own name in the same world.
Gerolamo Cardano was born in Pavia in 1501, the illegitimate son of a lawyer who was also, according to some accounts, a friend of Leonardo da Vinci. He became a physician, a gambler, an astrologer, a philosopher, and a mathematician — sometimes all in the same week. He was brilliant, erratic, vain, ruthless, intermittently destitute, and intermittently triumphant, and he was utterly unlike Tartaglia in almost every way except one: he was obsessed with cubic equations.
A cubic equation is one where the highest power of the unknown is three: something of the form x³ + bx² + cx + d = 0. The quadratic formula — known since Babylon, systematised by al-Khwarizmi — solved equations up to the second power. Nobody had a general method for the third power. The problem had been open for two thousand years.
In the early sixteenth century, unknown to most of the mathematical world, a professor at the University of Bologna named Scipione del Ferro had quietly solved a special case — the “depressed cubic,” where the x² term is missing, of the form x³ + px = q. Del Ferro told almost no one. He wrote his solution in a private notebook and shared it, near the end of his life, with a student named Antonio Maria Fior.
Fior, possessing what he believed was an unbeatable secret weapon, challenged Tartaglia to a public mathematical duel in 1535. Mathematical duels were serious events in Renaissance Italy — public contests with real stakes, witnessed by crowds, where each contestant proposed thirty problems for the other to solve within a fixed time. Winning enhanced your professional reputation; losing could end a career. Fior proposed thirty problems, all requiring the solution of depressed cubics. He expected Tartaglia to be helpless.
He had miscalculated. Tartaglia, alerted by the nature of Fior’s challenge that a solution to the depressed cubic must exist, worked furiously in the weeks before the contest and discovered his own method. On the day, he solved all thirty of Fior’s problems. Fior solved none of Tartaglia’s. The humiliation was total.
The Secret Written in Verse
News of Tartaglia’s triumph reached Cardano in Milan, and Cardano wanted the secret. He wanted it with the particular intensity of a man who understood exactly what it was worth — not just as a tool for winning duels, but as a step toward a general theory of equations that would transform algebra.
He wrote to Tartaglia. He flattered him, invited him to Milan, promised introductions to wealthy patrons. Tartaglia resisted. He intended to publish his method himself, and he had no intention of giving it away. The correspondence went back and forth for months, Cardano pressing and Tartaglia evading, until in March 1539 Tartaglia finally relented — on one condition.
Cardano swore an oath. He swore, in terms as solemn as he could devise, that he would never publish Tartaglia’s method: “I swear to you by the Sacred Gospel, and on my faith as a gentleman, not only never to publish your discoveries, if you tell them to me, but I also promise and pledge my faith as a true Christian to put them down in cipher so that after my death no one shall be able to understand them.”
Satisfied — or perhaps simply worn down — Tartaglia revealed his method. He gave it, characteristically, in verse: a twenty-five line poem that encoded the procedure for solving the depressed cubic in compressed, ambiguous language deliberately designed to prevent anyone who intercepted the letter from understanding it without guidance.
Cardano, who was not given the underlying proof, spent the following months reconstructing the mathematics from the method alone. He succeeded. More than that, he and his student Ludovico Ferrari extended the results: Cardano generalised to all forms of the cubic, and Ferrari used the cubic solution to crack the quartic — equations of the fourth degree, x⁴ and below — completing the solution of polynomial equations up to degree four in a single extraordinary burst of work.
And then Cardano found himself in an impossible position. He had sworn never to publish. He had results that were, collectively, the greatest advance in algebra since al-Khwarizmi. He could not publish them without breaking his oath.
The Broken Oath and Its Consequences
In 1543, Cardano travelled to Bologna with Ferrari, where they were shown the private notebooks of the late Scipione del Ferro. The notebooks contained del Ferro’s solution to the depressed cubic — the same result Tartaglia had given Cardano, but written down by del Ferro before Tartaglia had ever heard of the problem.
This changed everything, in Cardano’s mind. His oath had been sworn on the basis that Tartaglia was the original discoverer. Del Ferro’s notebook proved that Tartaglia was not — that the result had been found independently, earlier, by someone else. Cardano no longer felt bound.
In 1545 he published Ars Magna — The Great Art — the most important algebra book since al-Khwarizmi’s Al-Jabr. It contained the complete solution of the cubic in all its cases, Ferrari’s solution of the quartic, and a range of other algebraic results. Cardano acknowledged both del Ferro and Tartaglia in the text. He gave Tartaglia full credit for communicating the cubic solution to him. He explained the del Ferro discovery and why he felt released from his oath.
Tartaglia was not mollified. He was livid, and he remained livid for the rest of his life, pursuing Cardano through public pamphlets, accusations of theft, and eventually challenging him to a mathematical duel. Cardano, who considered the matter settled, declined and sent Ferrari in his place. The duel took place in Milan in August 1548. Ferrari, who had extended the cubic solution to the quartic and understood the Ars Magna more deeply than its original discoverer, dismantled Tartaglia’s arguments methodically before a hostile hometown crowd. Tartaglia, losing badly, left Milan overnight and never fully recovered his reputation.
The formula for solving the cubic bears both their names today: the Cardano-Tartaglia formula. Whether this is justice or another injustice depends on how you read the episode. The historical truth, which the sources support, is that del Ferro found it first, Tartaglia rediscovered it independently, Cardano generalized it and proved it and published it with attribution. Nobody emerges entirely clean. But the mathematics itself was correct, and it was in print, and it changed everything.
What the Cubic Formula Actually Says
Let us look at what Cardano published, because it is worth the effort.
A depressed cubic — one with no x² term — has the form:
x³ + px = qCardano’s formula gives the solution as:
x = ∛(q/2 + √(q²/4 + p³/27)) - ∛(-q/2 + √(q²/4 + p³/27))This formula is remarkable for several reasons. It involves cube roots, which no previous formula had required. It involves a square root nested inside a cube root — a compound radical, a thing that had no precedent in the algebra of the time. And in certain cases, it produces something deeply unsettling: the expression under the square root sign, q²/4 + p³/27, becomes negative.
A negative number under a square root sign. In Cardano’s time, this was not just unusual — it was, officially, impossible. Square roots of negative numbers did not exist. They made no sense geometrically and no sense arithmetically. When a quadratic equation had a negative discriminant, mathematicians simply said it had no solution. End of story.
But Cardano noticed something extraordinary. For a certain class of cubic equations — equations that everyone could see had real, genuine, positive solutions — the formula required you to compute square roots of negative numbers as intermediate steps, and then the negative parts cancelled out at the end, leaving a perfectly ordinary positive answer. The square root of a negative number appeared in the middle of the calculation and then disappeared.
He described this, in a phrase that has become famous in the history of mathematics, as involving “mental tortures” — “putting aside the mental tortures involved”, one could nevertheless proceed with the calculation and get the right answer. He knew something was there. He didn’t know what it was. He filed it away.
What Cardano had glimpsed was the complex numbers — numbers of the form a + b√(−1), where a and b are ordinary real numbers. They would not be properly understood for another two centuries, would not be given their modern name until the nineteenth century, and would not be placed on a rigorous footing until Gauss and Argand provided geometric interpretations of them in the early 1800s. But they were there, lurking in the Ars Magna, necessary for the cubic formula to work, impossible to wish away.
The cubic formula had forced mathematics to confront something it could not handle. That confrontation, stretched over two centuries, would eventually produce a completely new kind of number — and those numbers would turn out to be essential to quantum mechanics, to electrical engineering, to signal processing, to almost every branch of modern physics. All of it traceable to a formula for solving x³ + px = q, published in Nuremberg in 1545 by a physician who had broken a solemn oath.
Viète and the Revolution of Symbols
While Cardano and Tartaglia were fighting over cubic equations in Italy, a quieter revolution was under way that would ultimately matter more than the cubic formula itself.
The revolution was notational. And its central figure was a French lawyer.
François Viète was born in 1540 in the Vendée region of western France, studied law, and spent his professional life as a royal councillor and parliamentary advocate. Mathematics was, officially, a hobby — but it was a hobby to which he devoted an extraordinary amount of energy, and the results he produced in his spare time changed the face of algebra permanently.
The problem Viète identified was the one that had constrained algebra since al-Khwarizmi: every equation was stated in words, with specific numbers, addressing a specific problem. Even in the most advanced algebraic work of the Renaissance — Cardano’s Ars Magna included — you could write that “a cube and six things equal twenty” (meaning x³ + 6x = 20), but you could not write a general relationship that held for all values of the coefficients. There was no way to say “for any p and q, the depressed cubic x³ + px = q has the solution…” because there was no symbol for “any p” or “any q.” There were only specific numbers.
Viète’s innovation was to introduce letters as symbols not just for unknowns (that had been done before, sporadically) but for known quantities whose values were unspecified. He used vowels — A, E, I, O, U — for the unknowns, and consonants — B, C, D, F, G — for the given quantities. This meant you could write a relationship that was genuinely general: not “a cube and six things equal twenty” but “A cubed plus B times A equals D,” where B and D could be any numbers you chose. The equation described a whole family of problems, not just one.
This is the birth of modern symbolic algebra. It is such a natural idea, in retrospect, that it is almost impossible to appreciate how radical it was at the time. Before Viète, algebra was the art of solving specific equations. After Viète, algebra was the science of general relationships between quantities — a language for describing structure, not just a toolkit for calculating answers.
Viète’s notation was not quite our modern notation — his vowel-consonant system was awkward, and the symbols for operations like plus, minus, and equals were still being standardised across Europe. The equals sign, for instance, was introduced by the Welsh mathematician Robert Recorde in 1557, in a book that justified it by saying: “I will sette as I doe often in woorke use, a paire of paralleles, or Gemowe lines of one lengthe, thus: ═══, bicause noe .2. thynges, can be moare equalle.” Two parallel lines, because nothing is more equal. The equals sign, like so many mathematical innovations, arrived and then seemed so obviously right that within a generation no one could imagine writing without it.
The synthesis of Viète’s symbolic approach with the decimal number system (which was being standardised in Europe at roughly the same time, through the work of Simon Stevin in the Netherlands) and the algebraic results of Cardano and Ferrari gave European mathematics, by the end of the sixteenth century, something it had never had before: a powerful, flexible, general language for expressing mathematical relationships. The pieces were in place for the next leap.
The World That Warfare Built
Step back from the individual stories — Tartaglia and his cannonballs, Cardano and his oath, Viète and his vowels, Napier and his logarithms — and look at the shape of the century as a whole.
The sixteenth century in Europe was an era of profound disruption: the Protestant Reformation fracturing the religious unity of the continent, the printing press transforming the speed at which ideas spread, the voyages of exploration connecting Europe to the Americas and Asia and forcing a complete revision of humanity’s picture of the world, and gunpowder warfare making the old military and political order obsolete with terrifying speed. Armies were larger, more destructive, and more technically sophisticated than anything the medieval world had produced.
All of this disruption created demand for mathematics. Artillery required ballistics. Navigation required trigonometry and calculation. Commerce required algebra and accounting. Administration required the kind of systematic quantitative thinking that algebra had made newly possible. The printing press made mathematical books available, for the first time, to anyone who could afford them — and mathematical books sold, because their readers were merchants and engineers and navigators and military officers who needed what was in them.
This is the context in which the algebraic revolution happened: not in a university, not in a library funded by a generous caliph, but in the noisy marketplace of a Europe that was modernising itself through trade, warfare, and the restless circulation of printed ideas.
And the demand fed back into the mathematics. Tartaglia developed ballistics because a military commander asked him about cannon ranges. Napier developed logarithms because astronomers and navigators needed faster computation. Cardano generalised the cubic partly because the competitive mathematical culture of Renaissance Italy rewarded whoever could solve the hardest problems. None of them were working in a vacuum. All of them were responding to pressures from the world around them, and the mathematics they produced — even the most abstract results, even Cardano’s complex numbers — was shaped by those pressures in ways that are visible if you know where to look.
This is the book’s argument, arriving again in a different century. The Babylonian accountant, the Egyptian rope stretcher, the Kerala astronomer, the Baghdad judge, and now the Renaissance gunner and navigator: each one was doing mathematics because a problem had to be solved, and existing tools were not enough. The mathematics outlasted the immediate problem and became the foundation for the next generation’s work. Every generation inherits a toolkit built under pressure, and uses it to solve problems that were not imagined when the tools were made.
The Setup for Everything That Follows
By 1600, European mathematics had accumulated something extraordinary: a general symbolic language for expressing relationships between quantities, a systematic theory of equations up to the fourth degree, tables of logarithms for rapid computation, and a growing library of trigonometric results absorbed from the Islamic tradition. It had also accumulated, in Cardano’s complex numbers and in the unresolved problem of the quintic equation (degree five, which would wait until the nineteenth century and require entirely new mathematics to solve), several deep unsolved problems that pointed toward the next frontier.
The frontier was motion. Tartaglia had asked what path a cannonball follows. He had not been able to answer precisely, because answering precisely required understanding how velocity changes continuously under the influence of gravity — a mathematical concept that the algebra of his time could not handle. The geometry of the Greeks could describe static figures: shapes, areas, volumes. The algebra of the Renaissance could describe relationships between fixed quantities. But the world is not static and its quantities are not fixed. Things move. They accelerate. They change continuously.
Describing continuous change mathematically — finding a language adequate to the world in motion — was the problem that the next century would solve. It would be solved independently in England and in Germany, by two men who would spend the rest of their lives arguing about which of them had solved it first. It would draw on Tartaglia’s ballistics, Kepler’s planetary orbits, Napier’s logarithms, and the algebraic machinery that Viète had spent his legal career constructing.
And, in a broader historical sense, some of its component ideas had earlier precedents on the coast of the Arabian Sea, in work neither man knew.
The next chapter turns to the seventeenth century and the invention of calculus — or rather, to the collision of two independent inventions of calculus, and the furious priority dispute that followed. The mathematics of change was waiting to be discovered. Isaac Newton and Gottfried Leibniz were both heading toward it. Neither of their discoveries emerged in a vacuum, and some key series had earlier precedents in Kerala.