### Cubic Skulduggery & Intrigue

Solution of a cubic equation, usually called Cardano’s formula.

Babylonian mathematicians knew how to solve simple polynomial equations, in which the unknown quantity that we like to call x enters in the form of powers, that is, x multiplied repeatedly by itself. When only x appears, we have a linear equation. If x-squared enters, we have a quadratic. The third power of x yields a cubic equation, the fourth power a quartic and so on [TM135 or search for “thatsmaths” at irishtimes.com].

Linear:    b x + c = 0

Quadratic:     a x2 + b x + c = 0

Cubic:     x3 + a x2 + b x + c = 0

Quartic:    x4 + a x3 + b x2 + c x + d = 0

Euclid discussed the solutions of linear and quadratic equations in geometric terms in his Elements. But nothing more happened in European mathematics for a thousand years. The Romans contributed precisely nothing to the subject.

Western civilization woke with a start after the slumbers of the medieval age. Universities in Bologna, Oxford, Salamanca, Paris and elsewhere began to flourish. In the early sixteenth century, several Italian Renaissance mathematicians were active in trying to solve polynomial equations of higher degree. The solution of the cubic and quartic equations was the first substantial advance in mathematics in Europe since the days of the ancient Greeks.

Precious and Secret Knowledge

In 1494 Luca Pacioli, an Italian Renaissance mathematician and friend of Leonardo da Vinci, published his Summa de Arithmetica, presenting methods of solving linear and quadratic equations. He stated that no general method of solving cubic equations was known and that probably no such method was possible. This acted as a spur to other mathematicians to find a solution.

The first successful attack on cubics was that of Scipione del Ferro (1465-1526). He held the chair of mathematics in the University of Bologna. He kept his method secret: such knowledge was valuable, as it could ensure victory in public challenges, which were issued from time to time. Del Ferro revealed his method to his student, Antonio Maria Fior. He also kept a notebook in which he described his discoveries and, after his death, this passed to his son-in-law, Hannibal dell Nave.

Stammerer Stumbles on Solution

Niccolo Fontana was a gifted mathematician born in Brescia. An injury in childhood left him with a speech impediment which resulted in his nick-name, Tartaglia – the Stammerer. Tartaglia, unaware of the solution method of del Ferro, found a solution method for cubics which have no linear term. But he was initially unable to solve “depressed cubics”, those without a quadratic term:

x3 + b x + c = 0

In 1535, Fior challenged Tartaglia to a mathematical contest. Each man was to submit thirty problems for the other to solve. After a hectic struggle, Tartaglia discovered the solution method for depressed cubics just eight days before the deadline and solved all of Fior’s thirty problems. The performance of Fior, who was of indifferent ability, was dismal, and he faded ignominiously from public view.

Left: Tartaglia. Right: Cardano.

A colourful character

At this point, Gerolamo Cardano, entered the fray. Born in Milan, he gained a reputation as a leading physician. He was a keen gambler and his understanding of the laws of chance enabled him to make a profit on this enterprise. His book Games of Chance is the earliest serious study of probability.

Cardano heard of Tartaglia’s great triumph and was determined to discover his method. He wrote repeatedly to Tartaglia, begging and cajoling him to reveal the secret of his method. In 1539 Tartaglia divulged the formula in poetic form, after Cardano had sworn a solemn oath never to reveal it. Cardano made rapid progress in solving the remaining species of cubic equations. He shared Tartaglia’s secret with his student Ludovico Ferrari and this brilliant protégé found a technique for solving the quartic equation, involving a polynomial of the fourth degree. But since Ferrari’s method reduced a quartic to a cubic of Tartaglia’s type, the oath that Cardano had sworn prevented him from publishing it.

Tartaglia’s poetic description of his method:

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.

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.

Cardano’s Oath:   I swear to you, by God’s holy Gospels, and as a true man of honour, not only never to publish your discoveries, if you teach me them, but I also promise you, and I pledge my faith as a true Christian, to note them down in code, so that after my death no one will be able to understand them.

However, in 1543 Cardano and Ferrari travelled to Bologna where they met Hannibal della Nave, Scipione del Ferro’s son-in-law, who gave then access to del Ferro’s notebook. Finding that Tartaglia’s solution method had been discovered earlier by del Ferro, Cardano felt relieved of his obligation to secrecy and in 1545 he published the methods for both cubic and quartic in his masterwork Ars Magna.

The culmination of the affair was another public contest, between Tartaglia and Ferrari. It was on the home turf of Cardano and Ferrari in Milan, and such was the hostile atmosphere that Tartaglia left Milan, lucky to escape physical abuse, with the contest unresolved. Later, it became evident that Ferrari had posed a problem that he himself could not solve. Presumably, he was fishing for information, behavior described by Tartaglia as “a very shameful thing”.