The History of Mathematics

Mathematics After the Middle Ages

What was the scandal between mathematicians working on cubic and quartic equations?

The early work on cubic equations was a tale of telling secrets—all taking place in Italy. Antonio Maria Fiore (c. 1526-?)—considered a mediocre mathematician by scholars—received the secret of solving the cubic equation from his teacher Scipione del Ferro (1465–1526), who was the one who actually discovered the formula and told Fiore just before his death. It didn’t take long for Fiore to spread the rumor of its solution. A self-taught Italian mathematical genius known as Niccoló Tartaglia (1500-c. 1557; nicknamed “the stutterer”) was already discovering how to solve many kinds of cubic equations. Not to be outdone, Tartaglia pushed himself to solve the equation x3 + mx2 = n—bragging about it when he had accomplished the task.

Fiore was outraged—a fortuitous event for the study of cubic (and eventually quartic) equations. Demanding a public contest between himself and Tartaglia, the mathematicians were to give each other 30 problems with 40 to 50 days in which to solve them. Each problem solved earned a small prize, but the winner would be the one to solve the most problems. In the space of 2 hours, Tartaglia solved all Fiore’s problems—all of which were based on x3 + mx2 = n. Eight days before the end of the contest, Tartaglia had found the general method for solving all types of cubic equations—while Fiore had solved none of Tartaglia’s problems.

The story did not end there: Around 1539, Italian physician and mathematician Girolamo Cardano (1501-1576; in English, known as Jerome Cardan) stepped into the picture. Impressed with Tartaglia’s abilities, Cardano asked him to visit. He also convinced Tartaglia to divulge his secret solution of the cubic equation, with Cardano promising not to tell until Tartaglia published his results.

Apparently, keeping secrets was not practiced in Italy at this time, and Cardano eventually beat Tartaglia to publication. Cardano eventually encouraged his student Luigi (Ludovico) Ferrari (1522-?) to work on solving the quartic equation (or the general polynomial equation of the fourth degree). Ferrari did just that, and in 1545, Cardano published his Latin treatise on algebra, Ars Magna (The Great Art)—which included a combination of Tartaglia’s and Ferrari’s works in cubic and quartic equations.


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