Ramble, Part 12: Italian Interlude
Jan. 24th, 2007 07:49 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowPrecedence quarrels - who came up with an idea first - are nothing new in mathematics; the Newton-Leibniz dispute over the discovery of calculus is probably the most famous. But the first such - at least, the first one I know of - has some interesting aspects that I'd like to discuss.
As I mentioned before, the mathematicians of the Muslim world, during the European Middle Ages, made some significant advances in the solution of cubic equations. The full solution, however, did not come until the early sixteenth century, and the story is a colorful one.
The story begins around 1515, when Scipione del Ferro, a professor of mathematics at the University of Bologna, came up with a method for solving cubic equations of the form x3+mx=n, where m and n are positive numbers. He did not publish his result, but passed it on to his student Antonio Fior. Around 1535, Nicolo Fontana of Brescia, better known as Tartaglia ("the stammerer"), announced that he had discovered the solution to equations of the form x3+px2=n, p and n being positive. Fior thought that Tartaglia was bluffing and challenged him to a public competition. A few days before the contest, Tartaglia discovered his own solution to the equation x3+mx=n. Thus armed with solutions to two types of equation, he was able to defeat Fior.
At this point Girolamo Cardano entered the picture. Cardano was, by profession, a doctor, but he was a talented mathematician as well. He was also a gambler (he wrote the first mathematical treatise on the subject), a bravo, and something of a bully. Somehow, he wheedled the secret from Tartaglia, under (so claimed Tartaglia later) a promise of confidentiality. In 1545, however, he published Ars magna, a masterful treatise on algebra which included, among other things, Tartaglia's methods. (In fact, it described the solution of the general cubic, not just the special forms del Ferro and Tartaglia had dealt with, and the general fourth-order equation as well.) Tartaglia protested vehemently, but was rebutted by Ludovico Ferrari, Cardano's student (who had come up with the solution of the fourth-order equation); according to Ferrari, Cardano had gotten the information from another of del Ferro's students. A public controversy ensued, and it may be that Tartaglia was lucky to escape with his life.
What is interesting from the historical point of view is the attitude towards mathematical discoveries it reveals. To del Ferro, Fior, and Tartaglia, such discoveries were to be kept secret, handed down from master to student, rather than publicized. This seems odd from today's perspective, where the importance of publication is a given ("publish or perish", as the saying goes), but there are, I think, reasons for it. (What follows is speculative, but it seems to me to fit the facts.)
The fundamental economic unit in the cities of medieval and Renaissance Italy was the guild; practitioners of any trade organized to safeguard their collective interests, by safeguarding business secrets from outsiders, passing them on only from master to apprentice. In many cities, this was enforced by the city government. (The case of the glassmakers in Venice, isolated on Murano Island for security reasons, is notorious in this regard.) In the days before a solid international regime on the protection of intellectual property by means of patents, copyright, and the like, this wasn't an unreasonable position to take. There was no mathematicians' guild, of course, but is it surprising that such an attitude would infect them as well? Add to this the fact that public competitions of the type Fior and Tartaglia engaged in were not uncommon and sometimes carried monetary prizes, and their stance is understandable.
It took a long time for the present attitude towards publication to take firm root. Even as late as the mid-nineteenth century, Karl Friedrich Gauss withheld much of his work from the public, although in his case the motive seems to have been aesthetic rather than financial. Nor have disputes over precedence disappeared; there is one going on right now, in connection with the proof of the Poincaré conjecture. The two are intertwined, of course; the first to publish is generally accorded the credit for a discovery. We'll address these problems again in later posts; the next one, however, will discuss some of the details of the work of del Ferro and the rest on cubic equations.
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Ramble Contents
As I mentioned before, the mathematicians of the Muslim world, during the European Middle Ages, made some significant advances in the solution of cubic equations. The full solution, however, did not come until the early sixteenth century, and the story is a colorful one.
The story begins around 1515, when Scipione del Ferro, a professor of mathematics at the University of Bologna, came up with a method for solving cubic equations of the form x3+mx=n, where m and n are positive numbers. He did not publish his result, but passed it on to his student Antonio Fior. Around 1535, Nicolo Fontana of Brescia, better known as Tartaglia ("the stammerer"), announced that he had discovered the solution to equations of the form x3+px2=n, p and n being positive. Fior thought that Tartaglia was bluffing and challenged him to a public competition. A few days before the contest, Tartaglia discovered his own solution to the equation x3+mx=n. Thus armed with solutions to two types of equation, he was able to defeat Fior.
At this point Girolamo Cardano entered the picture. Cardano was, by profession, a doctor, but he was a talented mathematician as well. He was also a gambler (he wrote the first mathematical treatise on the subject), a bravo, and something of a bully. Somehow, he wheedled the secret from Tartaglia, under (so claimed Tartaglia later) a promise of confidentiality. In 1545, however, he published Ars magna, a masterful treatise on algebra which included, among other things, Tartaglia's methods. (In fact, it described the solution of the general cubic, not just the special forms del Ferro and Tartaglia had dealt with, and the general fourth-order equation as well.) Tartaglia protested vehemently, but was rebutted by Ludovico Ferrari, Cardano's student (who had come up with the solution of the fourth-order equation); according to Ferrari, Cardano had gotten the information from another of del Ferro's students. A public controversy ensued, and it may be that Tartaglia was lucky to escape with his life.
What is interesting from the historical point of view is the attitude towards mathematical discoveries it reveals. To del Ferro, Fior, and Tartaglia, such discoveries were to be kept secret, handed down from master to student, rather than publicized. This seems odd from today's perspective, where the importance of publication is a given ("publish or perish", as the saying goes), but there are, I think, reasons for it. (What follows is speculative, but it seems to me to fit the facts.)
The fundamental economic unit in the cities of medieval and Renaissance Italy was the guild; practitioners of any trade organized to safeguard their collective interests, by safeguarding business secrets from outsiders, passing them on only from master to apprentice. In many cities, this was enforced by the city government. (The case of the glassmakers in Venice, isolated on Murano Island for security reasons, is notorious in this regard.) In the days before a solid international regime on the protection of intellectual property by means of patents, copyright, and the like, this wasn't an unreasonable position to take. There was no mathematicians' guild, of course, but is it surprising that such an attitude would infect them as well? Add to this the fact that public competitions of the type Fior and Tartaglia engaged in were not uncommon and sometimes carried monetary prizes, and their stance is understandable.
It took a long time for the present attitude towards publication to take firm root. Even as late as the mid-nineteenth century, Karl Friedrich Gauss withheld much of his work from the public, although in his case the motive seems to have been aesthetic rather than financial. Nor have disputes over precedence disappeared; there is one going on right now, in connection with the proof of the Poincaré conjecture. The two are intertwined, of course; the first to publish is generally accorded the credit for a discovery. We'll address these problems again in later posts; the next one, however, will discuss some of the details of the work of del Ferro and the rest on cubic equations.
Previous Next
Ramble Contents
