1. MATH 110-02 -- Algebra through History November 8, 2019
Nicolo Fontana (“Tartaglia”), Girolamo Cardano, Lodovico Ferrari, and the Solution of Cubics and Quartics
MATH Algebra through History
November 8, 2019

2. Nicolo Fontana Born in 1499 or 1500 in Brescia (in northern Italy)
His father (a mail delivery rider) was killed by robbers in 1506
During an attack on his native city by French soldiers in the War of the League of Cambrai in 1512, he took refuge in the city’s cathedral with his mother and siblings
But breaking into the church, a soldier wounded him severely in the jaw and palate with a saber
He recovered, but had a severe speech impediment for the rest of his life as a result – acquired the nickname “Tartaglia” = stammerer

3. A contemporary portrait

4. Another “Renaissance man”
He contributed significant advances to mathematics, as we will see, but he was also:
A military engineer
A land surveyor
A bookkeeper
Spent most of his life in the Republic of Venice – a major Mediterranean power in this period with extensive territories around the Adriatic Sea

5. Mathematical works
He published a translation of Euclid into Italian in 1543 that corrected a number of errors in earlier versions that had been transmitted by way of Arabic
He also translated Archimedes into Italian
These were the first translations into a modern European language
Best-known book was a General Trattato di Numeri e Misure, published in Venice, 1556 – 1560
Discusses arithmetic, practical applications, etc.

6. The saga of cubic equations
As we have seen, a number of other Italian mathematicians had been interested in extending the methods of Al-Khwarizmi to cubic equations: 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 +𝑑=0
Negative numbers were still not understood or used, though, so for the Renaissance mathematicians, these came in different ``flavors’’
``cubes and roots equal to numbers:” 𝑎𝑥 3 +𝑏𝑥=𝑐,
``cubes and squares equal to numbers:” 𝑎𝑥 3 + 𝑏𝑥 2 =𝑐,
And so forth … eight (or even more, in some presentations) different possibilities where all the coefficients are written as positive numbers

7. Cubic equations
First person to solve some of these algebraically was Scipione del Ferro (1465 – 1526 CE)
But he did not reveal his solution
There was a certain tradition at this time of algebraic contests and solution techniques for particular types of equations could be valuable knowledge
The contests often had associated monetary prizes, …
Del Ferro, on his deathbed, did finally reveal his method for the ``cubes and roots equal to numbers” case to his student Antonio Maria Fior

8. Enter Tartaglia
Tartaglia had also discovered a different method for the “cubes and squares equal to numbers case”
In 1535, Fior challenged him to an ``algebraic duel’’ – a cubic-solving contest
Each of them was to submit 30 questions for the other one to solve(!)
Tartaglia’s questions included a number of different types of problems to really test his opponent
Fior’s problems, on the other hand, were 30 different ``cubes and roots equal to numbers” cubic equations to solve

9. Tartaglia is inspired; Fior is humiliated
Fior was confident that only he had the secret for that kind of cubic
But Tartaglia was a good enough mathematician that, building on what he already knew, he was able to figure out the method for that other case as well on Feb. 13, 1535 (he recorded the date!)
He solved all of Fior’s problems in a matter of a couple of hours
Fior couldn’t make headway on a number of Tartaglia’s questions (he wasn’t especially insightful or knowledgeable)
Tartaglia was satisfied with having discovered something new; he did not even claim the prize for having won the contest(!)

10. Now Tartaglia wants to keep his secret
Tartaglia, probably wanting to keep his method “proprietary” for use in other contests, never publishes his method
At this point, Girolamo Cardano (1501 – 1576 CE) enters the story!
Cardano had become interested in solving cubic equations too and knowing Tartaglia had been successful in his contest with Fior, he contacts Tartaglia to ask him what he had done and ask for permission to include the formulas (with credit to Tartaglia) in a book he planned to publish later
Tartaglia refuses!

11. Cardano – another polymath

12. Cardano is persistent
Cardan eventually wheedled the formula out of Tartaglia in 1539 in Milan (Tartaglia had traveled there to meet with Cardano because Cardano promised to introduce him to possible aristocratic patrons)
The terms:
He agreed not to publish it, and moreover he said
He would only write it down in a coded form (from a poem that Tartaglia had composed), so that no one could discover the formula after Cardano’s death by reading the papers he left behind

13. Cardano and Ferrari
Cardano had a very able student/assistant named Lodovico Ferrari (1522 – 1565 CE)
Almost as soon as Tartaglia left to return to Venice, Cardano and Ferrari set to work to understand and generalize Tartaglia’s method
Ferrari eventually used some of the same ideas to solve general quartic equations too
In 1543, Cardano learned that Scipione del Ferro had actually solved cubic equations before Tartaglia, so he apparently no longer felt he was bound by the terms of his “deal” with Tartaglia

14. The Ars Magna
In 1545, Cardano published a very famous and influential book called Artis magnae, sive regulis algebraicis liber unus
It contained solutions to cubic and quartic equations, credited del Ferro and Tartaglia properly (they ``dotted the I’s and crossed the t’s” they would not be accused of plagiarism)
But Tartaglia was furious that Cardano had not kept the terms of the “deal”
He challenged Cardano and/or Ferrari to another equation-solving contest in 1548

15. Ferrari takes up the challenge
Even though Tartaglia had a lot of experience at these contests by this time,
Ferrari understood the algebra (cubic and quartic equations in all cases) better and won this contest “hands down”
Tartaglia apparently never really recovered from this and his reputation fell to the point that he lost jobs, stipends, etc.
Died in 1557
Next time, we’ll see what Tartaglia and Cardan actually did to solve cubics(!)

16. Ferrari’s later life
He turned down a request from the Holy Roman Emperor of the time to tutor his son in mathematics
Went to work as a tax assessor for the governor of Milan, Ferrando Gonzaga; became extremely rich
Took at least some priestly vows after this (evidence: a portrait)
Retired to Bologna, where he lived with his widowed sister
She poisoned him(!) and he died in 1565
After taking his money, she remarried, whereupon her new husband left her and took all the money(!) leaving her penniless