# In the scholastic challenges of renaissance Italy, what restrictions were considered appropriate regarding the incumbent's choice of subject?

EDIT

Following Mauro's comment, I have altered my question to ask only about any restrictions that may have been considered concerning the suitability of the incumbent's choice of questions for the challenger.

Renaissance mathematicians were well known for keeping their discoveries secret. According to William Dunham in Journey Through Genius :

To understand this behaviour - almost incomprehensible in the “publish or peril” world of today - we must consider the nature of the Renaissance university. There, academic appointments were by no means secure. Along with patronage and political influence, continued service depended on the ability to prevail in public challenges that could be issued by any quarter at any time. … a public humiliation could be disastrous to one’s career.

Dunham then gives a colourful account of one such challenge. It starts with Scipione del Ferro’s discovery of a method for solving a certain class of “depressed” cubic equations; those which lack a term of the second degree - i.e., those of the form $ax^3 + cx + d = 0$. Ferro, on his deathbed, revealed the formula to his student, Antonio Fior. Fior then (rashly) issued a challenge to Tartaglia (Niccolo Fontana). Tartaglia had boasted that he could solve depressed cubics missing a linear term - i.e., those of the form $ax^3 + bx^2 + c = 0$. Tartaglia won the challenge, having hastily worked hard in the allotted time to obtain a method for solving the depressed cubics of the form presented by Fior, while Fior lacked the ability to meet Tartaglia’s challenge.

One assumes that a challenge is born of an “I know something you don’t know” claim. Those conventions governing the number of problems and the time limit seem easy enough to decide. Dunham tells that each of the participants issued their opponent with a list of 30 problems requiring solution. Apparently the challenge lasted over a period of some days since Tartaglia spent some days working to solve the alternative diminished cubic before providing his solutions. Curiously, Dunham states that even if the challenger was stumped by the incumbent’s problems, he could be confident that his secret knowledge, supposedly known only to himself, would guarantee the failure of his opponent, resulting in a stalemate.

What is less clear is how to fairly judge the incumbent's subject choice. Presumably the incumbent must present a set of problems on the same general subject. If the incumbent assumes that the challenger has some “secret” knowledge, then obviously there is no point in asking the same type of problems as the challenger is posing. In this example, there is no point in Tartaglia asking questions about cubics with no second degree term, so he asks questions about cubics with no linear term. Since the challenger is making the challenge, obviously he cannot choose the type of problem that the incumbent may ask since that would give the challenger an unfair advantage.

However, what if Fior possessed Cardano’s method for solving the general cubic. What then would be Tartaglia’s options for subject choice. He could not be required to also pose questions requiring the solution of cubics since that would give Firo an unfair advantage.

Q: What rules governed the scholastic challenges of the Renaissance? What rules determine the incumbent’s choice of questions?

Q : What restrictions were considered appropriate regarding the incumbent's choice of subject, and who judged the fairness of the questions posed?

• Frankly speaking, I think that there was no "codified rules" at all... – Mauro ALLEGRANZA Jul 25 '16 at 19:38
• @MauroALLEGRANZA Yes, that makes sense. Perhaps I should rephrase my question to focus on the restrictions on the incumbent's subject choice. – Nick Jul 25 '16 at 19:48
• Or, we could concentrate on whether or not any social rules were ever written out or described. If a commentator wrote, "At University X, it was the gentleman's rule that an incumbent could only counter with a challenge of the same branch of mathematics - geometry for geometry, discrete for discrete, matrix for matrix, etc. It is reported in the Chronicle of X 1525 that an one whose name has been blotted from history for antisocial behavior answered a graph traversal challenge with an infinite series summation and was promptly run out of town with pitchforks.", that could help immensely. – Columbia says Reinstate Monica Jul 20 at 1:12
• @RobertColumbia If one must be run out of town, then do it in style with a red hot poker up your jacksie. – Nick Jul 20 at 4:49