Cardano's Method
Introduction
Cardano's Method, also known as the method of solving cubic equations, is a significant mathematical technique developed by the Italian mathematician Gerolamo Cardano in the 16th century. This method represents a pivotal moment in the history of algebra, as it provided the first systematic solution to cubic equations, which are polynomial equations of the third degree. The method is detailed in Cardano's seminal work, "Ars Magna" (The Great Art), published in 1545.
Historical Context
Gerolamo Cardano (1501-1576) was a polymath who made substantial contributions to various fields, including mathematics, medicine, and philosophy. During the Renaissance, the study of algebra was undergoing significant transformations, and mathematicians were striving to solve polynomial equations of higher degrees. Before Cardano's time, quadratic equations had been solved by ancient mathematicians, but cubic equations remained an unsolved challenge.
The breakthrough in solving cubic equations was initiated by Scipione del Ferro, who discovered a method to solve a specific type of cubic equation. However, del Ferro's work was not published, and it was passed on to his student Antonio Maria Fior. Fior's challenge to Niccolò Tartaglia, another mathematician, led to Tartaglia discovering a general solution for cubic equations. Tartaglia shared his method with Cardano under the promise of secrecy. Cardano later published the method in "Ars Magna," crediting both Tartaglia and del Ferro.
The Method
Cardano's Method involves reducing a general cubic equation to a simpler form and then solving it using a series of algebraic manipulations. The general form of a cubic equation is:
\[ ax^3 + bx^2 + cx + d = 0 \]
Cardano's approach focuses on the depressed cubic equation, which lacks the quadratic term. By making a suitable substitution, the original cubic equation can be transformed into a depressed cubic equation of the form:
\[ t^3 + pt + q = 0 \]
- Reduction to Depressed Cubic
To reduce the general cubic equation to the depressed form, Cardano used the substitution:
\[ x = t - \frac{b}{3a} \]
This substitution eliminates the quadratic term, resulting in the depressed cubic equation. The coefficients \( p \) and \( q \) are derived from the original coefficients \( a \), \( b \), \( c \), and \( d \).
- Solving the Depressed Cubic
Cardano's Method then proceeds to solve the depressed cubic equation using the following steps:
1. **Introduce Auxiliary Variables:**
Cardano introduced two auxiliary variables \( u \) and \( v \) such that:
\[ t = u + v \]
2. **Formulate a System of Equations:**
By substituting \( t = u + v \) into the depressed cubic equation, Cardano derived a system of equations:
\[ u^3 + v^3 + (3uv)(u + v) = -q \] \[ 3uv = -p \]
3. **Solve for \( u \) and \( v \):**
The system of equations can be solved by expressing \( u^3 \) and \( v^3 \) in terms of \( p \) and \( q \):
\[ u^3, v^3 = \frac{-q \pm \sqrt{q^2 + 4p^3}}{2} \]
4. **Find the Roots:**
Once \( u \) and \( v \) are determined, the roots of the original cubic equation can be found by back-substitution.
Example
Consider the cubic equation:
\[ x^3 - 6x^2 + 11x - 6 = 0 \]
- Step 1: Reduce to Depressed Cubic
Using the substitution \( x = t + 2 \):
\[ (t + 2)^3 - 6(t + 2)^2 + 11(t + 2) - 6 = 0 \]
Expanding and simplifying:
\[ t^3 - t - 2 = 0 \]
- Step 2: Solve the Depressed Cubic
Introduce \( t = u + v \) and solve the system:
\[ u^3 + v^3 = 2 \] \[ 3uv = 1 \]
Solving for \( u \) and \( v \):
\[ u^3, v^3 = 1 \pm \sqrt{1 + 8} = 1 \pm 3 \]
Thus, \( u^3 = 4 \) and \( v^3 = -2 \).
- Step 3: Find the Roots
The roots are:
\[ u = \sqrt[3]{4}, \quad v = \sqrt[3]{-2} \]
Thus, \( t = \sqrt[3]{4} + \sqrt[3]{-2} \).
Substituting back, we get the roots of the original equation.
Impact and Legacy
Cardano's Method had a profound impact on the development of algebra. It not only provided a solution to cubic equations but also paved the way for the solution of quartic equations by Cardano's student, Lodovico Ferrari. The publication of "Ars Magna" marked the beginning of modern algebra and influenced subsequent mathematicians, including René Descartes and Isaac Newton.
See Also
References
- Cardano, Gerolamo. "Ars Magna." 1545.
- Katz, Victor J. "A History of Mathematics: An Introduction." 1998.
- Boyer, Carl B. "A History of Mathematics." 1968.