Algebraic Solution of Cubic and Quartic Equations

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Algebraic Solution of Cubic and Quartic Equations

Overview

The solution of the cubic and quartic equations was one of the major achievements of Renaissance algebra. The publication of the results in Girolamo Cardano's book The Great Art brought charges that Cardano had broken his promise to Tartaglia, who claimed he had made the major discovery in the cubic case. Attempts to identify all solutions of the cubic, quartic, and higher order equations would require the invention of complex numbers and would lead to the discovery of the theory of groups, one of the most important ideas in modern abstract algebra.

Background

Our word "algebra" is derived from the Arabic Kitab al-jabr w'almuqabala, a book by the Arabic Mathematician Muhammad ibn-Musa al-Khwarizmi (c. 780-c. 850) which described an art of "restoration and reduction," that is, finding the value of unknown quantities in an equality by rearranging terms. The book was translated into Latin in 1145 as the Liber algebrae et almucabala by Robert of Chester (fl. 1145), an English scholar living in Muslim Spain. In Christian Western Europe mathematics was in a far less advanced state than in countries under Muslim control. The Greek scholarly tradition had continued in the Byzantine Empire, which, however, was under frequent attack by its neighbors. In 1543 Turkish forces overran its capital, Constantinople. Byzantine scholars found refuge in Italy, where rich and powerful families like the Medici added scholars to their entourage and manuscripts to their libraries. The appearance of Gutenberg's printing press made mathematical ideas far more widely available. Over 200 new books on mathematics appeared in Italy before 1500.

In 1545 a book entitled Ars Magna, or The Great Art, by the Italian mathematician Girolamo Cardano (1501-1576), appeared. This work incorporated significant new results—the solution of the cubic and quartic equations. In modern treatments of algebra, a quadratic equation is any equation of the form

where A is a number other than zero, and B and C are constants that can be positive, negative, or zero. The letter x, of course, is the unknown to be found. Until quite recently, however, mathematicians did not have the tools available to deal with the case in which A, B, and C are all positive numbers. Further, there was a tendency to avoid the appearance of negative numbers. A method of solution for some quadratic equations had been developed by the ancient Babylonians based on a process now taught as "completing the square." The al-jabr discusses the six possible variations of the quadratic equation that can be written without negative numbers or a zero, for example, Ax² = Bx, or Ax² + Bx = C. In the latter case there are two possible solutions given by the quadratic formula, which involves the common operations of arithmetic, addition, subtraction, multiplication, and division as well as taking a square root.

A cubic equation has the modern form

With B set to zero, this is known as the reduced cubic equation. The quartic or biquadratic equation has the form

As with the earlier solution of the quadratic, in treating these equations Cardano had to consider many special cases to avoid negative quantities and the need to take the square root of a negative number. As with the quadratic, Cardano's solution to the cubic and quartic involved adding subtracting, multiplying, dividing, and taking roots, in this case including cube roots as well as square roots.

Cardano freely admitted that the solutions he presented were not his original discovery. The solution to the reduced cubic equation x³ + Ax = B had been found in 1515 by Scipio del Ferro (1465-1526), a professor at the University of Bologna. He communicated this solution to his student Antonio Fior some 20 years later. The Italian mathematician Niccolò Fontana, better known as Tartaglia (1500?-1557), then announced that he had discovered the solution to the cubic equation lacking a first order term, x³ + Ax² = B, as well as the solution to del Ferro's case. Fior doubted that Tartaglia could have found such a solution and arranged a contest in which he and Tartaglia exchanged sets of 30 problems. At the end of the agreed-upon time Tartaglia had solved all of Fior's problems while Fior could solve none of Tartaglia's. Cardano invited Tartaglia to his home, hinting that he might be able to introduce him to a possible patron. There, Tartaglia disclosed his method to Cardano in return for assurances that it would not be published.

The solution to the quartic was obtained by the Italian Ludovico Ferrari (1522-1565), who had been Cardano's secretary and would become his son-in-law. Cardano wrote that Ferrari had developed it at his request. The essence of the solution was to define a new variable, related to the unknown, in such a way that the quartic could be written as a cubic, which could then be solved.

Tartaglia responded to Cardano's book by publishing one of his own, describing his own research on the cubic equation and attacking Cardano's integrity for breaking his promise. The meeting between Cardano and Tartaglia had taken place in 1539. Cardano learned of del Ferro's solution in 1542, however, and felt he was no longer bound by his promise, as Tartaglia's results had been in good measure anticipated by del Ferro. Ferrari rose to the defense of Cardano, and issued a public mathematical challenge to Tartaglia, to which Tartaglia responded. After six such exchanges Ferraro and Tartaglia engaged in a public oral debate in a church in Milan in 1548.

Despite the argument over whether Cardano had acted properly, there are important discoveries that are undoubtedly Cardano's. It was Cardano who discovered a systematic method to get the general cubic equation into reduced form so that del Ferro's solution could be used. Cardano was also the first to show that the cubic equation could have three real solutions. He was also among the first mathematicians to use imaginary numbers in expressing the solutions of algebraic equations, although a full understanding of their properties would not come for nearly three centuries.

Impact

The extreme competitiveness of the mathematicians involved in solving the cubic and quartic equations is consistent with the aggressive individualism of the Renaissance. Cardano was more flamboyant than most. The illegitimate son of a lawyer, he played dice and chess to gain income. He received a medical degree from the University of Padua in 1526, but was prevented until 1534 from practicing in his native Milan because of his illegitimacy. Although his patients once included the Pope, he went to jail for heresy in 1570 for having cast a horoscope of Jesus, only to become the papal astrologer a year later. Tartaglia too had engaged in questionable practices. He published a translation of Archimedes by the Belgian scholar William of Moerbeke (c. 1220 -1286) in a manner suggesting that it was his own work. Ferrari was probably involved in intrigues as well. It is reported that he was poisoned by a relative.

A major step forward in algebra would occur with the work of the French lawyer and writer François Viète (1540-1620). While the above discussion has followed modern practice in using letters to represent both known and known numbers, it is only in the work of Viète that this is accomplished. The al-jabr used no mathematical symbols at all, while Cardano's work used letters for known quantities but not the unknown. With Viète's new notation it became easier to think of solving an algebraic equation as finding the values of x for which a definite function of the variable x would equal zero. This set the stage for the study of functions themselves and the study of transformations of functions caused by introducing new variables, ideas important in modern algebra, trigonometry, and calculus.

The solution of equations involving powers of unknown quantities has repeatedly served to inspire new developments in mathematics. The Greek mathematician Diophantus (fl. third century a.d.) raised the question of the existence of whole number solutions for equations involving whole number powers of different unknowns. The French mathematician Pierre de Fermat (1601-1665) conjectured that there would be no whole number solutions, A, B, C, to equations of the form

When n is greater than two. The search for a proof of Fermat's conjecture would occupy mathematicians for centuries.

The introduction of complex numbers, that is numbers of the form, a + ib, where the "imaginary" unit i has the property that i² = -1, was motivated in part by the study of algebraic equations. If complex numbers are allowed as solutions, then every quadratic equation has two solutions, every cubic three, and every quartic four. Understanding the nature of complex numbers was a source of many new ideas in nineteenth-century mathematics.

With the solutions of the cubic and quartic equations known, it might seem that the solution of the quintic equation, which included the fifth power of the unknown, and even higher order equations would be achieved eventually. Despite several centuries of effort, these solutions were not found. That such exact solutions, involving roots and arithmetic operations only, are not possible was demonstrated in 1824 by the Norwegian mathematician Niels Henrik Abel (1802-1829). Abel's results were generalized to all equations involving powers of the unknown higher than fourth by the French mathematician Evariste Galois (1811-1832). The study of algebraic transformations by these two mathematicians would lead to a general theory of transformations now known as group theory, which is considered one of the major areas of both abstract and useful mathematics by modern mathematicians and scientists.

DONALD R. FRANCESCHETTI