Tartaglia versus Cardan

[This article consists of quotes, mainly from Tartaglia, concerning the 'cubic dispute'. For the full context of the quotations below, see the biographies of del Ferro, Cardan, Tartaglia, and Ferrari. Connecting comments are put in square brackets.]

Zuanne de Tonini da Coi: I have heard that some time ago you entered into a disputation with Master Antonio Maria Fior and that in the end you reached agreement whereby he was to propound thirty problems for you, each of a different kind, set down in writing and sealed, to be deposited with Master Per Iacomo di Zambelli, notary, and similarly you would propound thirty problems for him, each of a different kind also. This you both did, fixing a term of forty or fifty days for each of you to solve these problems and agreeing that whichever of you within that time should be adjudged to have solved the greater number of the questions you had been given would take the honours, together with some small reward you suggested for each problem. And it has been reported to me, and confirmed by Fina a Bressa, that you solved all thirty of his problems in the space of two hours. I find that hard to believe.

Tartaglia: All that you have been told or had reported to you is true. And the reason why I was able to solve his thirty problems in so short a time is that all thirty concerned work involving the algebra of unknowns and cubes equalling numbers. He did this believing that I would be unable to solve any of them, because Brother Luca asserts in his treatise that it is impossible to solve such a problem by any general rule. However, by good fortune, only eight days before the time fixed for collecting the two sets of thirty sealed problems, I had discovered the general rule for such expressions.

All the problems I propounded for him were indeed each of a different kind. I did this in order to show him my versatility, and that my grounding lay not merely in one or two, or even three, private discoveries of mine, or in secrets, although I had kept them to myself for greater safety. Moreover, I could have set him another thousand, not just thirty; instead, as agreed, I propounded all thirty each of a different kind, to show that I thought little of him and had no cause whatever to fear him.

[Zuan Antonio de Bassano was a bookseller who frequently travelled between Milan and Venice. He took messages between Cardano and Tartaglia. The following exchange took place on 2 January 1539.]

Zuan Antonio: Master Niccolo Tartaglia, I have been directed to you by a worthy man, a physician of Milan, named Master Girolamo Cardano, who is a very great mathematician. And because he has understood that you have been engaged in disputation with Master Fior, putting to him for a wager certain questions that could only be answered by knowing the general rule for resolving the case of the cubic, which general rule you had found by your own discovery. Therefore his excellency prays you that you will kindly make known to him the rule discovered by you, and if you think fit will make it public under your name in his present work, but if you do not think fit that it should be published he will keep it secret.

Tartaglia: Tell his excellency that he must pardon me: when I propose to publish my invention, I will publish it in a work of my own, and not in the work of another man, so that his excellency must hold me excused.

Zuan Antonio: If you object to make known to him your discovery, his excellency has bidden me to pray that you will, however, give him the said thirty questions that were proposed to you, with your resolution of them, and at the same time the thirty questions that were proposed by yourself.

Tartaglia: I cannot do that, because as soon as he shall have one of the said cases with its solution, his excellency will at once understand the rule discovered by me, with which many other rules may perhaps be found, based on the same material.

Zuan Antonio: His excellency has given me eight questions to give to you, praying that you will resolve them for him. The questions are these:
1. Divide me ten into four parts in continued proportion, of which the first shall be two.

2. Divide me ten into four parts in continued proportion, of which the second shall be two.

[Problems 3, 4 and 5 omitted.]

6. Find me four quantities in continued proportion, of which the second shall be two, and the first and fourth added together shall make ten.

7. Make me of ten three parts in continued proportion, of which the first multiplied by the second will make eight.

8. Find me a number which when multiplied by its root plus three will make twenty-one.
Tartaglia: Those are questions put by Master Zuanne da Coi, and by no-one else. I know them by the two last, because a similar one to the seventh he sent me two years ago, and I made him confess that he did not understand the same, and a similar one to that last (which induces an operation of the square and cube equal to a number) I gave him out of courtesy solved, not a year ago, and for that solution I found a rule specially bearing upon such problems.

Zuan Antonio: I know well that these questions were given to me by his said excellency, Master Girolamo Cardano, and no other.

Tartaglia: The said Master Zuanne da Coi must have been to Milan and proposed them to his excellency, and he, being unable to resolve them, has sent them to be worked out by me, and this I hold for certain, because the said Master Zuanne promised me a year ago that he would come here to Venice, but for all that he has never been, and I think he has repented of his purpose and given its turn to Milan.

Zuan Antonio: Do not think that his excellency would have sent you these problems If he had not understood them and known how to solve them, or that they should proceed from another person, for his excellency is one of the most learned men in Milan, and the Marquis dal Vasto has given him a great provision for his competency.

[Later in the conversation Zuan Antonio repeats his request for the thirty questions]

Tartaglia: Of those - though I can ill spare the time - I will make a copy. But his excellency, whatever his competence, will be unable to resolve them, for to do that would mean his excellency had a wit like to my own, which he has not.

Zuan Antonio: Be pleased, then, to give me his.

Tartaglia: They are these precisely as he wrote them:-

Glory to God, 1534, and the 22nd day of February, in Venice. These are the thirty problems proposed by me Antonio Maria Fior to you Master Niccolo Tartaglia.
1. Find me a number that when its cube root is added to it, the result is six, that is 6.

2. Find me two numbers in double proportion 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.

3. Find me a number such that when it is cubed, and the said number is added to this cube, the result is five.

[We only give a selection.]

15. A man sells a sapphire for 500 ducats, making a profit of the cube root of his capital. How much is the profit?

17. There is a tree, 12 braccia high, which was broken into 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?

30. There are two bodies of twenty triangular faces the areas of which when added together make 700 braccia, and the area of the smaller is the cube root of the larger. What is the smaller area?
Cardano to Tartaglia (12 February 1539): I wonder much, dear Master Niccolo, at the unhandsome reply you have made to one Zuan Antonio de Bassano, bookseller ... I would pluck you out of this conceit, as I plucked out lately Master Zuanne da Coi, that is to say, the conceit of being the first man in the world, wherefore he left Milan in despair; I would write to you lovingly and drag you out of the conceit of thinking you are so great - would cause you to understand from kindly admonition, out of your own words, that you are nearer to the valley than the mountain-top. In other things you may be more skilled and clever than you have shown yourself to be in your reply; and so I must in the first place state that I have held you in good esteem, and as soon as your book on Artillery appeared, I bought two copies, the only ones Zuan Antonio brought, of which I gave one to Signore the Marquis ...

The third point is, that you told the said bookseller that if one of my questions were solved all would be solved, which is most false, and it is a covert insult to say that while thinking to send you six problems, I had sent but one, which would argue in me a great confusion of understanding; and certainly, if I were cunning, I would wager a hundred scudi upon that matter; that is to say, that they could not be reduced either into one, or into two, or into three questions. And, indeed, if you will bet them, I will not refuse you, and will come at an appointed time to Venice, and will give bank security here if you will come here, or will give it to you there in Venice if I go thither. This is not mere profession, for you have to do with people who will keep their word ...

I send you two questions with their solutions, but the solutions shall be separate from the questions, and the messenger will take them with him; and if you cannot solve the questions he will place the solutions in your hand. You shall have them each to each, that you may not suppose I have sent rather to get than to give them; but return first your own, that you may not lead me to believe that you have solved the questions, when you have not.
In addition to this, be pleased to send me the propositions offered by you to Master Antonio Maria Fior, and if you will not send me the solutions, keep them by you, they are not so very precious. And if it should please you, in receiving the solutions of my said questions - should you yourself be unable to solve them, after you have satisfied yourself that my first six questions are different in kind - to send me the solution of any one of them, rather for friendship's sake, a for a test of your great skill, than for any other purpose, you will do me a very singular pleasure.
1. Make me of ten four quantities in continued proportion whose squares added shall make sixty.

2. Two persons were in company, and possessed I know not how many ducats. They gained the cube of the tenth part of their capital, and if they had gained three less than they did gain, they would have gained an amount equal to their capital. How many ducats had they?
[Friendlier correspondence passed between them. Cardano tried to tempt Tartaglia to Milan to meet his patron, Alphonso d'Avalos, Marquis del Guasto. This might result in Trataglia being offered a better job.]

Tartaglia (comment written on a letter from Cardan): I am reduced by this fellow to a strange pass, because if I do not go to Milan the lord marquis may take offence, and such offence might do me mischief, I go thither unwillingly: however, I will go.

[On 25 March 1539 Tartaglia visited Cardano in Milan. Alphonso d'Avalos, Marquis del Guasto, was not in Milan at the time. The following conversation, reported by Tartaglia, took place in Cardano's house.]

Cardano: And I also wrote to you that if you were not content that I should publish them, I would keep them secret.

Tartaglia: Enough that on that head I was not willing to believe you.

Cardano: I swear to you by the sacred Gospel, and on the faith of a gentleman, not only never to publish your discoveries, if you will tell them to me, but also I promise and pledge my faith as a true Christian to put them down in cipher, so that after my death nobody shall be able to understand them. If you will believe me, do; if not, let us have done.

Tartaglia: If I could not put faith in so many oaths I should certainly deserve to be regarded as a man with no faith in him; but since I have made up my mind now to ride to Vigevano to the lord marquis, because I have been here already three days, and am tired of awaiting him so long, when I am returned I promise to show you the whole.

Cardano: Since you have made up your mind at any rate to ride at once to Vigevano to the lord marquis, I will give you a letter to take to his excellency, in order that he may know who you are; but before you go I should wish you to show me the rule for those cases of your, as you have promised.

Tartaglia: I am willing ...

[Tartaglia then gave Cardano his rule in a poem he had written.]
When the cube and things together
Are equal to some discreet number,
Find two other numbers differing in this one.
Then you will keep this as a habit
That their product should always be equal
Exactly to the cube of a third of the things.
The remainder then as a general rule
Of their cube roots subtracted
Will be equal to your principal thing
In the second of these acts,
When the cube remains alone,
You will observe these other agreements:
You will at once divide the number into two parts
So that the one times the other produces clearly
The cube of the third of the things exactly.
Then of these two parts, as a habitual rule,
You will take the cube roots added together,
And this sum will be your thought.
The third of these calculations of ours
Is solved with the second if you take good care,
As in their nature they are almost matched.
These things I found, and not with sluggish steps,
In the year one thousand five hundred, four and thirty.
With foundations strong and sturdy
In the city girdled by the sea.
This verse speaks so clearly that, without any other example, I believe that your Excellency will understand everything.

Cardano: How well I understand it, and I have almost understood it at the present. Go if you wish, and when you have returned, I will show you then if I have understood it.

[Cardano, however, in his attempts to understand the rule came across the problem of the irreducible case. He wrote asking Tartaglia's help. The following extract is from Tartaglia's reply.]

Tartaglia to Cardano (August 1539): Master Girolamo, I have received a letter of yours, in which you write that you understand the rule; but that when the cube of one-third of the coefficient of the unknown is greater in value than the square of one-half of the number you cannot resolve the equation by following the rule, and therefore you request me to give you the solution of this equation "One cube equal to nine unknowns plus ten". To which I reply, and say, that you have not used the good method for resolving such a case; also I say that such your proceeding is entirely false. And as to resolving you the equation you have sent, I must say that I am very sorry that I have given you already so much as I have done, for I have been informed, by person worthy of faith, that you are about to publish another algebraic work, and that you have gone boasting through Milan of having discovered some new rules in Algebra. But take notice, that if you break your faith with me, I certainly shall not break promise with you (for it is my custom); nay, even undertake to visit you with more than I have promised. ...

Tartaglia (note to himself): I propose to see whether I can perhaps alter the data he possesses, that is, turn him away from the right track and make him take some other ...

[Later Ferrari entered the dispute and began writing to Tartaglia instead of Cardano.]

Ferrari to Tartaglia: You have the infamy to say that Cardano is ignorant in mathematics, and you call him uncultured and simple-minded, a man of low standing and coarse talk and other similar offending words too tedious to repeat. Since his excellency is prevented by the rank he holds, and because this matter concerns me personally since I am his creature, I have taken it upon myself to make known publicly your deceit and malice.

Ferrari to Tartaglia: As for the twenty-second problem in the disputation. You at first say that it is not a question for a mathematician. To which I reply, that, if by a mathematician you mean someone like you, that is, someone who spends the whole time on roots, fifth powers, cubes and other trifles, then you are quite right. I promise you that if it were up to me to reward you, taking example from the custom of Alexander, I would load you up so much with roots and radishes that you would never eat anything else in your life.

[Eventually Tartaglia agreed to enter a public debate with Ferrari. It was held in the Church of Santa Maria del Giardino, Milan, on 10 August 1548. Don Ferrante di Gonzaga, the Governor of Milan, was referee. Each posed sixty-two problems on a wide range of scientific and mathematical topics. We give five of the problems Ferrari set Tartaglia.]
15. 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 make 64 more than the sum of these two numbers.

17. Divide eight into two parts such that their product multiplied by their difference comes to as much as possible, proving everything.

21. Find me six quantities in continuous proportion starting with one, such that the double of the second and the triple of the third is equal to the root of the sixth.

23. 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?

27. There is a right-angled triangle, such that when the perpendicular is drawn, one of the sides with the opposite part of the base makes 30, and the other side with the other part makes 28. What is the length of one of the sides?
[Ferrari won the contest. Ferrari couldn't solve Problem 27 above.]

Tartaglia to Ferrari (following the contest): ... and you ask the length of one of the sides. I reply to you, that you have proposed this to me in order that I should elucidate to you what you do not understand, and it is true, in your Ars Magna, Question 14, on page 71 you propose a similar problem, namely, you want one of the sides with the opposite part to make 29, and the other with the other to make 31. In which triangle the sides fixed by you are rational, that is, one would be 20, the other 15, the base 25, the perpendicular 12, the greater part of the base 16, and the lesser 9. And in the end you did not know how to solve such a problem by a general rule. It is a very shameful thing, to put forward such a question in public, and not to know how to solve it by a general rule. I have the same opinion of your Problems 26 and 19, but I reserved to myself my reply to you on them, in front of the referees ...

Written by J J O'Connor and E F Robertson
Last Update September 2005