Working with Cubic Equations

In order for us to trace the history of the solution of the cubic equation, we must look back to ancient Egypt. The first solutions to cubic equations at this time were geometric, which is fitting since ancient Egyptians and Greeks new very little to nothing about algebra. The original cubic equations resulted from problems in land measuring and especially from the duplication of the cube.

Hippocrates of Chios was probably the first person to show that the duplication of a cube could be reduced to finding two mean proportionals between a given line and one twice its length ( Gardner, "A Brief History of Equations"). Hippocrates and others (Menaechmus and Archimedes) were believed to have come close to solving the problem of doubling the cube using intersecting conic sections. Later in the 7th century, it was a Tang dynasty mathematician Wang Xiaotong that solved 25 cubic equations of the form

But Xiaotong could not solve the cubic for the general case, for all values of p and q.

Then, in the 11th century, Persian mathematician Omar Khayyanm made giant strides in solving cubic equations. He discovered that a cubic equation can have more than one solution and stated that it cannot be solved using just a compass and straightedge construction (which was a common methodology used to prove propositions during that time and previously). Omar did find a geometric solution to the cubic equation 

Here is a specific example of Khayyam's geometric solution of a cubic equation.

In the 12th century, another Persian mathematician, Sharaf al-Din al-Tusi wrote the "Treatise on Equations", which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He used what became known as the "Ruffini-Horner" method to approximate the root of a cubic equation. It was also during this time that the importance of the discriminant of the cubic equation to find algebraic solutions to certain types of cubic equations was realized by Sharaf.

In the early 16th century, the Italian mathematician Scipione del Ferro (1465 - 1526) found a method for solving a class of cubic equations, those of the form 

.

Unfortunately during this time, negative numbers were not known to del Ferro, or else he probably would have realized that all cubic equations can be reduced to the above form if we were to allow m and n to be negative.

In 1530, Niccolo Tartaglia was able to work out a general method to solve cubics of the above form, and was persuaded by Gerolamo Cardano to reveal his secret method for solving cubic equations. As the story goes, Cardano promised that he would not publish Tartaglia's work. A couple of years later, Cardano was discovered the solution of the general cubic equation

However he was unable to publish his findings because the solution was largely dependent on Tartaglia's solution of the depressed cubic. Lucky for Cardano, he found a solution to the depressed cubic in his pupil's, Ludovico Ferrari, own hand writing, and was then able to publish his work. This is when, in 1545, Cardano published his book "Ars Magna", the "Great Art." In this book he gave credit to del Ferro with the original solution to the depressed cubic.

Even though many today might believe that it was Cardano who, on his own, found the solution to solving the general cubic equation, it is evident that work on cubic equations began hundreds of years before Cardano's time.