The Turbulent History
of the Cubic Equation:
From the Quanta Magazine website
A 490th Anniversary Celebration
Robert “Dr. Bob” Gardner
ETSU Abstract Algebra Club
Seminar on the History and Exploration of Math Problems (S.H.E.M.P.)
Spring 2025
Primary References
Simon & Schuster, 2005
Princeton University Press, 2020
Cardano, 1545
MIT Press 1968
Dover Publications, 1993
When and Where?
Nicolaus Copernicus (1473-1543) published On the Revolutions of the Celestial Spheres, in which he argued for a sun-centered universe.
The Years 1500 to 1550 (Eurocentric Version)
Exploration and exploitation of the Americas began by the French, English, Spanish, and Portuguese. Hernando De Soto (1500-1542) first explored the current southern U.S. (including this area).
Michelangelo (1475-1564) and Leonardo da Vinci (1452-1519) were active.
The reformation begins with Martin Luther (1483-1546).
The expedition of Ferdinand Magellan (1480-1521) circumnavigated the world.
The Italian renaissance ends with the Sack of Rome in 1527.
Source: Wikipedia page on the 16th Century.
From the Wikipedia page on the Mediterranean Sea (accessed 1/5/2025)
The Setting
From the Touropia.com website (accessed 1/5/2025)
Brescia
Pavia
Padua
The Quadratic Formula
The Babylonians
Completing the Square
The quadratic formula is now easily derived by completing the square. You might see this in a high school algebra class. We have that each of the following equations is equivalent:
The Quadratic Formula Today
al-Khwarizmi
From the Bayt al Fann website (accessed 1/5/2025).
al-Khwarizmi did not present equations symbolically, but instead expressed them verbally.
The Cubic Formula: Introduction and Prehistory
What Does it Mean to “Solve” a Cubic Equation?
In the early 16th century, things were very different. First, negative numbers were not yet recognized as numbers (nor were complex numbers). This affected the types of cubic equations and what could pass as a solution. The goal was to find a solution of a cubic with real coefficients.
Omar Khayyam
From the MacTutor biography of Omar Khayyam (accessed 1/6/2025).
Omar Khayyam (1048-1131), working in Persia (i.e., Iran), is best known as a philosopher and poet and for his collection of poems Rubaiyat. He studied cubic equations, showing that such an equation can have more than one solution, and giving geometric solutions of certain cubics in terms of the intersection of a
hyperbola and a circle. He seems to be the first
to consider a general theory of cubic equations.
The Cubic Formula:
dal Ferro and Company
Cubics Before Tartaglia
Image from Towards Data Science webpage (accessed 11/10/2024)
Scipione dal Ferro shared his result with the student Antonio Maria Fiore (dates uncertain) of Venice. After dal Ferro’s death, a manuscript describing the result was left to his son-in-law. This manuscript would end up with Annibale della Nave, a mathematician in Bologna. della Nave will appear in the story again later.
dal Ferro’s equation is derived in History of Mathematics (MATH 3040) Problem Study 8.14, and in Precalculus 1-Algebra (MATH 1040) in Section 4.5. The Real Zeros of a Polynomial Function.
The Cubic Formula: Niccolò Tartaglia
Niccolò Tartaglia
Image from MacTutor page on Tartaglia (accessed 11/10/2024)
Niccolò Tartaglia (circa 1500-1557) was born in Brescia in the Republic of Venice (northern Italy). At age 12 he was stabbed in the face by a French soldier during the occupation of Brescia, leading to a life-long stammer. “Tartaglia” means stammerer, the name he adopted. He taught math in Verona where from 1516 to 1518. In 1534
Niccolò Tartaglia
Image from MacTutor page on Tartaglia (accessed 11/10/2024)
and Tartaglia accepted. The contest was scheduled for February 22, 1535 in Venice.
The Cubic Formula: Tartaglia versus Fiore
From Stock.Adobe.Com (accessed 1/13/2025)
From MacTutor page on Tartaglia (accessed 11/10/2024)
Not Fiore! Just an impression of him…
Tartaglia versus Fiore
The contest was organized as follows:
Year of our Lord 1534, day 22, February in Venice*
These are the 30 proposed problems of Antonio Maria Fior for Maestro Nicolo Tartaglia
Record of the date: Questiti et invention diverse de Nicolo Tartaglia (1546/1554) [New Problems and Inventions]
*The Venetian new year was March 1, so this corresponds to our February 22, 1535
Fiore’s “Arsenal” and Questions
What did Tartaglia know and when did he know it?
Tartaglia’s “Arsenal”
Tartaglia knew how to solve four types of cubics:
*Tartaglia could develop equations of these types from given
irrational solutions (involving square roots), but could not solve
them in general.
Tartaglia’s Questions
Tartaglia included questions based on all four types of equations he could solve. For example [Toscano, page 50]:
Tartaglia’s Questions (continued)
I dunno!
You should.
Tartaglia 30, Fiore 0
Tartaglia was able to solve all of Fiore’s problems
in two hours.
Fiore was unable to solve any of Tartaglia’s problems.
The Cubic Formula: Tartaglia and Cardano
Image from Towards Data Science webpage (accessed 11/10/2024)
Gerolamo Cardano (1501–1576)
Gerolamo Cardano was a physician, astrologer, gambler, philosopher, and mathematician. He studied at the universities of Pavia and Padua (both in northern Italy).
He was working on his second math-related book when he learned of the contest between Tartaglia and Fiore. He wanted to include the Tartaglia’s cubic formula.
Cardano repeatedly invited Tartaglia to visit.
Come visit in Milan.
NO!
Tartaglia declined. Cardano’s invitations to visit.
I’ll set up a meeting…
…OK.
Cardano arranged an introduction to the commander in chief of Milan.
Tartaglia consented and went to Milan in March of 1539. Cardano tried to charm him and offer him credit for his solutions of cubics in a special chapter of Cardano’s
next mathematics book.
I promise…
Cardano pressured Tartaglia to reveal his work. Tartaglia eventually agreed, if Cardano would take an oath not to publish it.
According to Tartaglia, on March 25, 1539 Tartaglia agreed to reveal his work to Cardano, but on the condition that Cardano would not publish it.
Swear it!
Cardano’s secretary, Ludovico Ferrari, claims to have been present at the meeting and says that no such oath was taken.
Never happened!
What Follows is One of the Two Most Famous Controversies in the History of Mathematics
Gerolamo Cardano (1501–1576)
From MacTutor page on Tartaglia (accessed 11/10/2024)
Image from YouTube (accessed 11/10/2024)
Ludovico Ferrari (1522-1565)
From MacTutor page on Cardano (accessed 11/10/2024)
Niccolò Tartaglia (circa 1500-1557)
The Cubic Formula: Cardano and Ars Magna
The Rules of Algebra (Ars Magna) Girolamo Cardano, translated by Richard Witmer (MIT Press, 1968).
Reprinted by Dover Publications (1993, 2007).
Cardano’s The Practice of Arithmetic and Simple Mensuration appeared in May 1539 (only two months after Tartaglia’s visit) without the inclusion of Tartaglia’s cubic work, as agreed.
From MathScienceHistory.com website (accessed 1/16/2025)
Cardano’s 1539 Book
Ludovico Ferrari was born in Bologna and at the age of 14 started working for Cardano. Cardano recognized Ferrari’s intelligence and took responsibility for Ferrari’s education.
Ludovico Ferrari
After Tartaglia revealed his solution, Cardano was able to find a proof for it, and he started to look for solutions to other types of cubic equations (since negative numbers and zero are not recognized, there was not a single “cubic equation”).
In 1540 Ferrari found a solution to the quartic equation, but it relied on the solution of cubic equations and, since these were not public, Ferrari could not publish his work.
Ludovico Ferrari
In 1543, rumors were circulating that Scipione dal Ferro had left his solution of cubics to his son-in-law. Cardano and Ferrari traveled to Bologna and met with Annibale della Nave, who had possession of dal Ferro’s original work. After they convinced della Nave that they could solve the cubics which dal Ferro had solved, della Nave showed them dal Ferro’s papers. This was the evidence that Tartaglia was not the first to solve the cubic equations, but instead was preceded by dal Ferro by 20 years.
Scipione dal Ferro Gets Credit
Remember Antonio Fiorre?
Whether Tartaglia had Cardano take an oath or not, Cardano went ahead and published the work on the cubic, since it is work originally performed by dal Ferro. In 1545, “the book regarded by many mathematicians as marking the beginning of modern algebra,” Ars Magna (“The Great
Art”), was published by Cardano [Livio, page 71].
Cardano Publishes: Ars Magna (1545)
From Wikipedia page on Ars Magna (accessed 1/23/2025)
Cardano Credits Tartaglia, Ars Magna (pages 8 & 9)
Tartaglia was angered by Cardano’s publication of what Tartaglia considered his work. Within a year of the publication of Ars Magna, Tartaglia published “New Problems and Inventions” in which he accused Cardano of perjury. Cardano himself avoided the conflict.
Tartaglia is Unhappy!
You published MY work! PLAGARISM!
…dal Ferro’s work!
This is beneath me…
Cardano Solves 13 Types of Cubics
In ETSU’s History of Mathematics (MATH 3040) the textbook includes the following “Problem Study”:
The Depressed Case
Cardano’s Chapter XI of Ars Magna
Cardano’s derivation involves the dissection of a cube into smaller parallelepipeds. He proves geometrically:
Image of the first page of Cardano’s Chapter XI from the Archive.org webpage.
Cardano’s Chapter XI Computation
Cardano has shown:
Cardano’s Chapter XI Computation (cont.)
Cardano’s Chapter XI Computation (cont.)
Finally Cardano has:
The Quartic Formula: Cardano, Ferrari, and More Ars Magna
Ferrari’s Quartic Work
More on Ferrari, Cardano, Tartaglia, and the Quartic at a later date…
To solve a quartic, Ferrari introduces a particular cubic (the resolvent cubic) that can be solved. A solution to this then leads to a solution of the given quartic.
Quartics and Ars Magna
So in order to publish the quartic results, the cubic results have to be known!
This is work of dal Ferro, Ferrari, and me (with some credit to Tartaglia).
…AND SO: Ars Magna (1545)
Image from MacTutor page on Cardano (accessed 11/10/2024)
Recognition of the Mathematicians
From “Bologna – Scipione del Ferro Birthplace” video by Heidi Meyer Math:
https://www.youtube.com/watch?app=desktop&v=B2vxWbeDrgA (2:32 in length)
�
“Birthplace of Scipione dal Ferro, mathematician who first solved the cubic equation.”
Bologna, Italy (dal Ferro)
From https://www.flickr.com/photos/aldoaldoz/3236085544 (11/13/2024) (in “piazza Paolo VI”; accessed 11/13/2024)
“Here, having escaped the massacres of 1512, a poor boy wounded in the lips was then named after his undeserved speech. That name is glorious in the science of numbers.”
Brescia, Italy (Tartaglia)
From: https://www.shutterstock.com/image-photo/italy-brescia-monument-niccolo-fontana-tartaglia-1451618135
Brescia, Italy (Tartaglia)
“Dream of Gerolamo Cardano” J. Presno (date?).
From https://www.meisterdrucke.uk/fine-art-prints/J.-Presno/84540/Dream-of-Gerolamo-Cardano.html
Girolamo Cardano
Girolamo Cardano (continued)
“One of the most gifted and versatile men of his time, Cardano wrote a number of works on arithmetic, astronomy, physics, medicine and other subjects. His greatest work is his Ars Magna, the first great Latin treatise devoted solely to algebra.” [Howard Eves, An Introduction to the History of Mathematics 6th edition (1990), page 275. Emphasis added.]
“It has often been pointed out that three of the greatest masterpieces of science created during the [Italian Renaissance] appeared in print almost simultaneously: Copernicus, De Revolutionibus Orbium Coelesium (1543); Vesalius, De Febrica Humani Corporis (1543), and finally Girolamo Cardano, Artis Magnae Sive de Regulis Algebraicis (1545).
… To Cardano’s contemporaries it was a breakthrough in the field of mathematics, exhibiting publicly for the first time the principles for solving both cubic and biquadratic equations, giving the roots by expressions formed by radicals…” [Oystein Ore, Foreword in Richard Witmer’s translation of The Rules of Algebra (Ars Magna), Dover Publications (1993), page vii. Emphasis added.]
What Happens Next?
From the MacTutor biography of Bombelli (accessed 2/2/2025)
Rafael Bombelli (1525-1572)
Rafael Bombelli's Algebra (1572) gives a summary of late 16th century algebra. He gives the algebraic interaction of positive and negative numbers.
He also made contributions to the acceptance of complex numbers as numbers.
It was the theory of equations that (eventually) led to the acceptance of both negative numbers and complex numbers.
What Happens Next? CLASSICAL ALGEBRA!
François Viète was never a professional mathematician. In 1591 he published The Analytic Art. He is the one who introduced the use of letters to represent unknowns.
His book contained the theory of equations work of Cardano etc. He is the one to first use the term “coefficient.”
The stage was now set for classical algebra!
And then… MODERN ALGEBRA!
From the MacTutor biography of Abel (accessed 2/2/2025)
Niels Abel
(1802-1829)
From the MacTutor biography of Galois (accessed 2/2/2025)
Évariste Galois
(1811-1832)
References
Israel Kleiner's A History of Abstract Algebra, (Birkhäuser, 2007).
José Contreras, “An Episode of the Story of the Cubic Equation: The del Ferro-Tartaglia-Cardano’s Formulas,” Journal of Mathematical Sciences & Mathematics Education, 10(2), 24–37 (2015).
Howard Eves, An Introduction to the History of Mathematics 6th edition (Saunders College Publishing, 1990).
Fabio Toscano, The Secret Formula: How a Mathematical Duel Inflamed Renaissance Italy and Uncovered the Cubic Equation (Princeton University Press, 2020).
Mario Livio, The Equation That Couldn’t Be Solved: How Mathematical Genius Discovered the Language of Symmetry (Simon & Schuster, 2005).
Girolamo Cardano, The Rules of Algebra (Ars Magna), translated by T. Richard Witmer (MIT, 1968; reprinted by Dover Publications, 1993).
Questions?
THANK YOU!
“A Modern Vision of the Work of Cardano and Ferrari on Quartics” Convergence July 2009
Michel Helfgott, ETSU Department of Mathematics
2004-2016
Harald Helfgott,
University of Gottingen