A Modern Vision of the Work of Cardano and Ferrari on Quartics - Final Considerations

Solving the corresponding cubic is usually the only time-consuming task in the whole process. A natural question to ask is whether there is a simpler method to solve quartics. Rene Descartes (1596-1650) developed a novel approach, which he published in 1637 in The Geometry ([2], p. 180), almost 90 years after the first edition of Ars Magna. Let us succintly present Descartes' method from a modern perspective. For this purpose we will analyze Problem IX, chapter 39, of Cardano's book: x⁴ + 4x + 8 = 10x². First we try to express x⁴ - 10x² + 4x + 8 as a product of two quadratic factors for certain undetermined coefficients a,b,c,d, say:

Let r₁, r₂ be the solutions of x² + ax + b = 0 and r₃, r₄ the solutions of x² + dx + c = 0. Then r₁ + r₂ = -a and r₃ + r₄ = -d. But r₁, r₂, r₃, r₄ are solutions of a quartic without cubic term, hence r₁ + r₂ + r₃ + r₄ = 0. Therefore d = -a. The original problem is thus reduced to finding a, b, c such that

Expanding terms we get c + b = a² -1 0, c - b = 4/a, and bc = 8. Adding and subtracting the first two equalities we obtain b = (1/2)(a² - 10 - 4/a), c=(1/2)(a² - 10 + 4/a). Then 32 = 4bc = (a² - 10)² - 16/a², which is equivalent to u³ - 20u² + 68u - 16 = 0 with u = a². By simple inspection we note that u = 4 is a solution, hence a = 2, c = -2, and b = -4. That is to say, x⁴ - 10x² + 4x + 8 = (x² + 2x - 4)(x² - 2x - 2). As expected, the solutions of these quadratics are 1 ± √3, -1 ± √5 respectively, which happen to be the solutions of the original quartic.

Ferrari's approach compares favorably with Descartes' method, as was illustrated in the solution of the preceding problem. Both require the solution of a corresponding cubic; as a matter of fact, Galois theory implies that any algebraic method for quartics using radicals will have to go through the solution of a cubic, except for some particular cases like biquadratics. For instance, Euler's method ([5], pp. 282-288) requires, in general, the solution of a cubic too.

There is a particular case that has all biquadratics as a special subcase: namely, quasisymmetric equations, meaning equations of the form a₀x⁴ + a₁x³ + a₂x² + a₁mx + a₀m² = 0. To solve such an equation, divide by x² and then define z = x + m/x. We obtain a₀z² + a₁z + (a₂ - 2a₀m) = 0, a quadratic equation in z. For each solution z we will have to solve the equation x² - zx + m = 0. Actually, two equations of the above-mentioned type appear in chapter 34 of Cardano's book (problems II and III); he solves them through a rather convoluted "Rule for a Mean". It is worth noting that the transformation z = x + m/x does not appear in Ars Magna; it belongs to a later period in the history of algebra.

From a pedagogical perspective one might ask: Which method is easier to learn and apply? Apparently it is a matter of taste. Among 20th century authors, Dickson ([3], pp. 51-52) chose a variation of Ferrari's method, and Feferman ([6], pp. 349-350) preferred Descartes' method, while Weisner ([9], pp. 140-143) presented Euler's method. In any case, probably it is best to acquaint students with at least two methods. Each corresponds to a different period in the development of algebra.

One of the authors (MH) regularly discusses Ferrari's and Descartes' method in a course on teaching of secondary mathematics intended for seniors planning to become high school mathematics teachers (syllabus). One class is devoted to cubics and another to quartics (80-minute classes). Both topics could very well be chosen as enrichment units for high school students taking either Algebra II or Pre-Calculus; not only will they learn new algebraic techniques, but their understanding of quadratics might be enhanced. At the college level, students of Abstract Algebra would surely benefit from a historical approach to the solution of cubics and quartics; the topic is interesting on its own and lies at the foundation of several mathematical developments of the 19th and 20th century.

Chapter 39 of Ars Magna can be seen as exemplifying three principles that should still hold in the classroom today; namely, the need to learn more than one approach to the solution of a problem whenever this is possible, the advisability to travel from the particular to the general, and the reliance on methods rather than rules.

One might marvel that as early as the 1540's Lodovico Ferrari, who started as a servant in the household of Cardano, could have solved an outstanding open problem in algebra, namely finding the real roots of any quartic equation. Our admiration goes also to Cardano for having written an epoch-making mathematical book.