# What cubic problems did Tartaglia and Fior pose to each other?

I have been researching the history of finding roots to general polynomials and the story of solving for the roots of cubic polynomials ($ax^3+bx^2+cx+d=0$) lead me to find several sources describing a "mathematical duel" between the Italian mathematicians Tartaglia (Nicolo de Brescia) and Antonio Fior of 16th Century A.D..

Many sources, like this one and this one, all speak of Tartaglia and Fior posing 30 problems to each other before they competed in a public math contest and how Tartaglia was able to find a general solution to each of Fior's problems just before the contest.

My question, which might also be appropriate for the Math History SE (in Commit stage of Area 51 right now) is:

What were the 30 problems that Fior posed to Tartaglia and what are the 30 problems that Tartaglia posed to Fior? I am personally interested in seeing them and trying my hand at solving them based on what I've learned recently about cubic polynomials. I feel that the 60 collective problems are not only of historical interest in themselves but also I think they must have been designed challenging enough for each man to try to stump the other. I think those particular 60 problems would be much more instructive in their solution than just an arbitrary assortment of cubic polynomial problems given on worksheets across the Internet and particular math books.

Any information on where to find these 60 problems or a partial list of them is very appreciated!

• I believe Fior posed 30 cubics, while Tartagia posed 30 wide-ranging problems. This I got from one of William Dunham's books, I don't remember which one. He gives no further details on the problems. Apr 20, 2014 at 0:31
• The main issue is that Tartaglia possessed the "general solution" while other "algebraist" of his time can handle only some special case : see David Burton, The History of Mathematics. An Introduction (6th ed - 2006), page 317-321 : The Italian Algebraists: Pacioli, del Ferro, and Tartaglia. Apr 23, 2014 at 6:34
• It seems that some of Tartaglia's problems were from Euclidean geometry. May 31, 2019 at 2:19
• From my reading, it sounds like Tartaglia only solved the depressed cubic, and did not have the general solution. Sep 23, 2019 at 2:38

While these problems have great historical value, I'm not sure they are really that great as problems. It is not so easy to give a natural setting where you have to solve a cubic equation to solve a problem, so they will necessarily become somewhat contrived. (Even al-Khwarizmi's "practical" examples of quadratic equations suffer from this!)

Anyway, six of Fiore's problems are reprinted in The History of Mathematics: A Reader by John Fauvel and Jeremy Gray. Tartaglia's book Quesiti et Inventioni Diverse from 1546 is given as reference, but they do not say if more problems can be found there.

(Edit: Tartaglia's book can be found here, and Fiore's list begins on leaf 114r! It does not appear that Tartaglia's problems are included)

Fiore's problems 1,2,3,15,17,30:

• Find me a number such that when its cube root is added to it, the result is six, that is 6.
• Find me two numbers in double proportion [$x$ and $2x$] such that when the square of the larger number is multiplied by the smaller, and this product is added to the two original numbers, the result is forty, that is 40.
• Find me a number such that when it is cubed, and the said number is added to the cube, the result is five.
• A man sells a sapphire for 500 ducats, making a profit of the cube root of his capital. How much is this profit?
• There is a tree, 12 braccia high, which was broken into two parts at such a point that the height of the part which was left standing was the cube root of the length of the part that was cut away. What was the height of the part that was left standing?
• There are two bodies of 20 triangular faces [icosahedra] whose corporeal areas added together make 700 braccia, and the area of the smaller is the cube root of the larger. What is the smaller area?

Fauvel and Gray also have a section with seven problems together with discussions from the debate between Tartaglia and Ferrari in 1547. The problems are all posed by Ferrari, and show much more variety than the ones by Fiore. It appears that an account of these two debates were published by Ferrari, and has been reprinted by Arnaldo Masotti in 1974 (Cartelli di sfida matematica).

(Edit: This book can be found here.)

The ones related to cubic equations (15, 21, 23) are:

• Find me two numbers such that when they are added together, they make as much as the cube of the lesser added to the product of its triple with the square of the greater; and the cube of the greater added to its triple times the square of the lesser makes 64 more than the sum of these two numbers.
• Find me six quantities in continuous proportion starting with one, such that the double of the second with the triple of the third is equal to the root of the sixth.
• There is a cube such that its sides and its surfaces added together are equal to the proportional quantity between the said cube and one of its faces. What is the size of the cube?

The first fifteen of Fior's questions, a summary of the remaining fifteen, and four of Tartaglia's questions are in the article:

Martin Nordgaard, Sidelights on the Cardan-Tartaglia Controversy, 1937-8.

Here are a few more of Fior's:

• (5) Two men were in partnership, and between them they invested a capital of 900 ducats, the capital of the first being the cube root of the capital of the second. What is the part of each?
• (7) Find a number which added to twice its cube root gives 13.
• (8) Find a number which added to three times its cube root gives 15.
• (9) Find a number which added to four times its cube root gives 17.
• (12) A jeweler buys a diamond and a ruby for 2000 ducats. The price of the ruby is the cube root of the price of the diamond. Required the value of the ruby.
• (13) A Jew furnishes capital on the condition that at the end of the year he shall have as interest the cube root of the capital. At the end of the year the Jew receives 800 ducats, as capital and interest. What is the capital?
• (14) Divide thirteen into two parts such that the product of these parts shall equal the square of the smallest part multiplied by the same.

Here are the only four of Tartaglia's that Nordgaard knows of:

• Find an irrational quantity such that when it is multiplied by its square root augmented by 4, the result is a given rational number.
• Find an irrational quantity such that when it is multiplied by its square root diminished by 30, the result is a given rational number.
• Find an irrational quantity such that when to it is added four times its cube root, the result is thirteen.
• Find an irrational quantity such that when from it one subtracts its cube root, the result is 10.