I apologize for the rather vague title. It's space-limited and I'm not sure how to concisely state what I want to know. Basically, I understand that the solutions to quadratic equations (and if I remember correctly cubic equations) often had surveying problems land surveying. However, quartic and quintic equations seem to have less to do with the physical realities of societies in which their solutions were discovered that linear, quadratic, and cubic equations, especially since the problems were most often stated geometrically. The final proof of the unsolvability of polynomials of degree greater than 4 by radicals, though, came in the 19th century when mathematicians were studying the zeros of polynomials primarily for the sake of testing the solvability of polynomials in general. So, I guess my question could be more accurately stated as: Were there any practical applications of finding the zeros of quartic and quintic equations at the time that the problems were solved?
Re: Historical question: Was the study of the polynomial zeros of practical applicati Not practical, but revolutionary. The ancients knew how to solve the quadratic but couldn't crack the cubic. For a couple of thousand years, mathematicians thought that their work consisted in understanding and perfecting the work of the ancients. When Cardano solved the cubic in 1539 (the date's approximate, and Cardano got the solution from Tartaglia) it was the very first time that mathematicians solved a problem the ancients couldn't. This gave the mathematicians of the middle ages a great psychic boost, realizing that they could finally surpass the ancients. This was one of the accomplishments marking the beginning of the Renaissance.
Re: Historical question: Was the study of the polynomial zeros of practical applicati yes! in fact, these problems still exist today: many economic situations are modelled by polynomial equations, and zeros of these polynomials represent "break-even" points, which separate "regions of profitability" from "regions of loss". another easy example is from "curve-fitting", we might know the value of a process (function) at only a few places of measurement, and we'd like to find a reasonable approximation by a smooth function good for any point. in general, this approach uses a n degree polynomial to approximate a function for which we only have n+1 data points of measurement. if we want to know when this approximation, p(x) = b, for some special value b, that is the same as solving (finding a root) the polynomial q(x) = 0, where q(x) = p(x) - b (subtract b from the constant term of p). if a general form for finding an exact solution for a polynomial of ANY degree were known, we could use this formula to solve q(x) = 0 for x, which would be VERY useful. unfortunately, for n > 4, there is no "general formula", although that doesn't mean that every polynomial can't be solved exactly, just that some of them can't be solved exactly...with ALGEBRA. it turns out there ARE method which give us "good approximations". in addition, when calculus started being developed, it turns out that polynomial functions could be made to be "quite good approximations" of other functions, provided the "other functions" were "nice enough". in particular, trigonometric functions like sine and cosine fall into the "nice enough" category, meaning polynomials could be used in some cases instead of the trigonometric functions, under certain circumstances (the classic example is using the polynomial p(x) = x instead of sin(x), for values of x "near enough" to 0 (such as low-angle trajectories)). certainly some "higher-order" polynomials were used less in the 16-th century than today (the average 16-th century person may not have even had much occasion to use numbers much), but as a whole, mankind was still ignorant of "what could be done", and as such, there was a certain sense of "exploring just for the sake of finding out". the same could still be said today, every time one mathematical question is answered, others appear in its wake. the motivation is usually NOT "how will this be useful?", but "where does this lead?". today's idle wondering may be tomorrow's vital fact. this has been shown to be true many times over. but even were it not so, we would probably still try to solve puzzles, for the sheer joy of it.
Re: Historical question: Was the study of the polynomial zeros of practical applicati I think it may be worthwhile to mention several hundred years ago, mathematicians of one court would challenge other courts' mathematician(s) with a problem (or set of problems). A typical problem involved finding the roots of a cubic, or quartic equation. IF I recall correctly, Tartaglia withheld his solution of the cubic as it gave him a definite advantage over his adversaries.
Re: Historical question: Was the study of the polynomial zeros of practical applicati Just to focus the discussion on that point, I'll guess No. Perhaps someone can show a practical problem in architecture where finding the height of a arch or something like that requires a cubic or quintic. That would still leave open the question of whether architects solved the problem that way or whether they used scale models. There is also the question of what is considered a practical application. For example, Cardano was (among other things) an astrologer. He no doubt considered an application of mathematics to astrology to be practical. Do we count it as such? I think the original post is asking whether the solution of polynomial equations was the focus of intense interest due to the need to solve such problems using the technology of the era when these solutions were developed. The history of math books I read (once upon a time) never mentioned any such pressing need as motivation - but perhaps those books were written by pure mathematicians.
Re: Historical question: Was the study of the polynomial zeros of practical applicati From a historical point of view, I believe special cases of the cubic equation cropped up naturally in celestial/astronomical issues. Many Arab mathematicians worked on those problems. And the "practical" application? A main, driving issue behind the financing of Islamic mathematicians was their ability to calculate the correct qibla, that is the local prayer direction towards Mecca.
Re: Historical question: Was the study of the polynomial zeros of practical applicati these are good points. i know for a fact that the grand cathedral at chartres was actually constructed through trial-and-error (apparantly it fell down more than once), some of which might have been avoided if more knowledge of the mathematics of arches was more widely known. and, of course, the "cutting edge" of mathematical research was (in the middle ages) often a closely guarded professional secret. even a widely-published mathematician such as Leonardo of Pisa did not reveal all of his "bag of tricks" in print, presumably because to a large extent court patronage (i.e. income, the middle age equivalent of today's research grant) was contingent upon being able to solve things nobody else could. it still makes me chuckle today to think about how Cardano and Tartaglia quarrelled about the cubic. it would appear, from the scant information history leaves behind, that a good deal of research into polynomials was based on professional pride, and that at the time various solutions to higher-degree polynomials were found, these were considered as some of the more pressing unsolved problems in existence. everyone knew that equations were powerful in their problem-solving potential, and it was no doubt hoped at one time that EVERY equation might one day be cracked open. it appears, that over time, a curious shift in priority occurred in mathematics. mathematics no doubt developed as a method to solve pragmatic problems: accounting (for), reckoning, estimating and predicting as-yet untranspired events (such as a harvest, or the building materials required for a bridge, etc.). at some point, the methods themselves took over: geometry was no longer about mensuration or surveying problems, and the techniques of al-jabr were not limited to solving astronomy questions, or business transactions, but became objects of study in their own right, divorced from any immediate application. so it may be impossible to give a completely accurate answer: the orginal impetus for some polynomials may well have been certain physical situations, but at some point, mathematical theory started growing faster than engineering practice. but this wasn't by any means a uniform development, like two cars running down a drag-strip. both fields grew by fits and starts, knowledge become lost, and re-discovered. for example: i came across a tid-bit that said that most of Euclid's book V was unusable in the 16th century, because of a bad translation. Tartaglia corrected this, making information available that had essentially been lost for centuries. by Galois' time, universities devoted solely to study of mathematics for mathematics sake were already a feature of most important european cities (Paris had more than one, one of the great set-backs of Galois' life was his failure to get into the school he wanted to go to). In Cardano's time, mathematics was mainly limited to the people who could afford to study it, and there was less of a line drawn between "applied and pure" mathematics (it was still not uncommon for problems to be posed verbally, rather than symbolically). Liber Abaci was still in wide-spread use as a text, which makes no differentiation between "pure" and "applied" mathematics (and which has several problems involving polynomials). mathematics appeared to be gaining the upper hand as a "pure science" by the beginning of the christian era, but no doubt the local politics of alexandria (where early christian sectarian conflict led to the burning of the great library), as well as the decline of the roman empire, gave "practical problems" a chance to catch up. were it not for the brilliance of islamic mathematicians during the "dark ages", it is doubtful that renaissance mathematics would have been so fecund.
Re: Historical question: Was the study of the polynomial zeros of practical applicati Thanks for all the responses. They were all very informative. Of course, now, I am even less sure of what I was trying to ask in the first place. I guess I was wondering if innovation in mathematics, specifically in the solution to the general polynomial equation (which led to group theory, an indispensable tool for modern physics), was driven by "practical" (read: physical/technological) applications or by the search for knowledge for knowledge's sake. I understand that the above statement is largely, if not completely, a false dichotomy, but I have been wondering about the "usefulness" of the search for knowledge lately, because I have been discussing the "usefulness"/"uselessness" dichotomy with people who claim that knowledge isn't "useful" unless it yields technological applications. I thought it might be helpful to point out that not every theory in science or theorem in mathematics was immediately applicable to technology. The search for analytic solutions to polynomial equations is therefore a fruitful example because it is both technologically applicable and theoretically deep.