I find it hard to imagine that this hoary tale is not in common parlance, but such is but one of the many delusions of the amateur mathematician in these dreary days1. It is a tale of booze soaked treachery and high stakes competetion from the days when mathematics was a royal sport and the exact solving of cubics was a high stakes game that only the most strong of mind and character dared attempt.
No tale can be said to have one true beginning, if I may be forgiven for utilizing this cliche2, but I shall choose as a suitable start to introduce the character who serves as the protagonist of this affair. Niccolo Fontana Tartaglia was born in Brescia, where he obtained his last name3 as a result of a French soldier stabbing him in the mouth, inflicting a life long speech impediment upon our protagonist. As a man, Tartaglia had obtained for himself a local reputation of being both well-endowed with skills of a mathematical nature and a boastful tongue, and as was the manner of the time, found himself challenged by one Antonio Fiore to a contest of mathematical duel, in which each contestant was to produce a list of 30 problems that their opponent must solve within several days. Tartaglia put together a wide variety of intricacies, but Fiore submitted only cubics, expecting to humble his oppugnant4. After many painstaking days and nights of formulaic manipulation, Tartaglia formulated an algorithm for solving all cubics, with which he then swept through the problems like an oiled scythe through autumn dried wheat, defeating a chastened Fiore. Keeping in with the style of the times, he kept his general solution a secret and used it only for challenging other mathematicians with solving seemingly impossible cubics.
Enter Gerolamo Cardano, a character who's life accomplishments included chopping off his son's ears, undergoing a prolonged prosecution for heresy for his calculating and publishing the horoscope of Jesus, compiling the first treatise on Probability theory that helpfully included detailed methods of cheating at games of chance, and his claims that deaf people were not idiots and could in fact use their minds much like that of those endowed with aural faculties. In short, a complicated man, and not to mention a literal bastard, who's potential in the ways of abstract reasoning was only exceeded by his capacity for devious and capricious actions. Cardano had heard of Tartaglia's victories over many, and grew convinced that his fellow Italian had obtained a general method of solving the cubic. For whatever initial reason5, Cardano felt driven to apprehend this solution for himself, and thus he extended an invitation to Tartaglia under the pretenses of introducing him to the Marchese of Milan at a soiree at Cardano's abode. Accounts differ as to whether said meeting actually occurred, but by Tartaglia's account, after being plied with liquors and panegyrics by the cunning Cardano, Tartaglia finally relented and shared his algorithm with Cardano, but only after forcing him to swear the following oath:
I swear to you by the sacred Gospel, and on my faith as a gentleman, not only never to publish your discoveries, if you tell them to me, but I also promise and pledge my faith as a true Christian to put them down in cipher so that after my death no one shall be able to understand them.
The following day Tartaglia left Milan in some hurry, perhaps sensing what history has show us to be true: that Cardano held neither faith as a Christian nor a Gentleman.
Time passed, as it is ever wont to do, and Cardano stayed true to his oath, holding Tartaglia's Method close to his bosom in a manner unto like that of a sympathetic woman and a small injured animal. And it is possible that in a another world, Cardano could have kept his oath, except for being that in this one he took in the prodigy Ferrari as apprentice. Together they worked on any number of the celebrated abstruse intricacies of the time, towards the interest of publishing a compendium of the pair's various techniques and transformations. But then it came to pass that Ferrari, through painstaking labors, secured a method by which one could solve a certain variety of quartic equation. Including this method in their proposed mathematical epic would secure their place as the finest mathematicians of the time, yet there was one miniscule fly trapped in the proverbial ointment: Ferrari's method reduced a problem of quartics to that of cubics.
Cardano, ever wily, realized that there might yet be a path by which he could side-step his oath. Coupling his knowledge of the duel between Fior and Tartagliga with his personal estimation of Fior as an uninspired man, Cardano theorized that not only did Fior have his own cubic method, but that it was almost certainly not of Fior's derivation, but that of his tutor, Del Ferro. Armed with suspicions, Cardano and Ferrari set off to Del Ferro's estate, where by some obfuscation they managed to gain access to the now deceased mathematician's papers. In which, they found Del Ferro's cubic method clearly outlined, and with that, a loophole by which oaths sworn could be forsaken. Sparing no time, the pair published their works in quartics, cubics, and much more in the Ars Magna6 to the great outrage of Tartaglia, who then publicly denounced the pair, revealing the Oath that Cardano transgressed, and then challenged Ferrari to a mathematical duel. Perhaps justice would have dictated that The Stammerer upset The Upstart, but no man observant of the ways of our world would imply that any form of justice is inherent in such workings. Tartaglia lost, and in the face of such disgrace and ignominy, retreated from the public eye, never to be heard from again.
A fact of note in this story is that several mathematicians seem to have independently derived solutions to the cubic but kept them as deeply held secrets. Earlier, I noted that the justification for this was simply a matter of the times, but it is likely that something more kept these mathematicians in a state of allowing the results to be known but the methods to be secret. It is that all general methods of solving the cubic make allowance for the appearance of the square root of negative 1 or i as it is now known, and at the time, most gentlemen of well breeding would have declared you to be a madman for even contemplating such profanity7. Indeed, a look at works of the time divulges a distressingly puritan squeamishness to even contemplating negative coefficients in polynomials. Thus we have that it took an audacious blasphemer such as Cardano to publish any such method, his justification being that it acheived results.
Nevertheless, what are we to make of the fact that if we approach history from the view point of a vectorist, we have that multiple mathematicians almost simultaneously and independently did work that would lead to the public acknowledgement of the existence of i, being at the very least, a necessary evil among the mathematician's lexicon. What if i wanted to be discovered? What if Someone (or Some Thing) wanted i to be discovered? Dwelling in some dark recess, unknowing of time, a blinking, throbbing, nameless urge flickered when brushed by human consciousness, like the hint of stirring movement spotted peripherally in the mirror of an unlit bathroom, and knew hunger. But for who's gain? Was Cardano the self-promoting inscrupulous bastard historians have painted him as, or another in a long list of patsies in some Archon's scheme, flicked about by unseen fingers (talons? unspeakable appendages?), smirking at his people's God while Some Thing, Some Where, twisted his desires, for devious purposes unknown to men?...perhaps they have already come to pass...perhaps they yet lay on the horizon...
1Indeed, I often have to be reminded that the concept of a mathematician held by most is that of a mere calculator, idly adding and subtracting large numbers for the supposed sheer thrill of it all, as opposed to the actual goal of discovering the only truths we can prove in this universe.
2but what better way to begin a story I have already described once as hoary?
3meaning "Stammerer."
4Fiore had learned method from Scipione Del Ferro which painstakingly avoided dealing with any negative coefficients and thus avoided explicitly dealing with any negative roots(but still they lurked, between lines of calculations, like some impossible beast waiting to strike). Ferro kept his method a secret to all but two, as it was common for natural philosophers of that day to hoard their discoveries as to benefit all the more from the rarity of their knowledge. See any of numerous works on the Calculus Wars between the continental acolytes of Liebniz and Newton's co-horts in the Royal Society for more on the evolution of that world into ours.
5glory, exploration, exploitation? or was this part of some more grand devious scheme in which the individual motivations of mere men were but pieces on some multidimensional game board?
6to Cardano's merit, he at no point claims either cubic method for himself, but gives credit where credit was due
7currently and historically, mass culture's view of i is sometimes strange to the rationalist, as why one would somehow single out this entity as being somehow more fantastic and strange than other such entities that we do not come across in the percievable physical world like pi(a transcendent), root of 2(an irrational), or a circle (show me an actual circle
8besides x^2 + y^2 = 1, smart ass
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