I am writing a story for young people about the history of the development of the cubic formula and complex numbers, partly because it has so much drama and partly because it's amusing that complex numbers were born of this almost farcical drama.

I've read the chapter on it in William Dunham's book, Journey Through Genius (chapter 6, Cardano and the solution of the cubic). I've also read a number of online accounts. (Two that seemed helpful: an MAA account, and information from the University of Saint Andrews.) I have a pretty good idea of what happened, but there are details that differ in different accounts. And it's hard to tell which sources are most accurate.

The question I have now is whether anyone uses the cubic formula nowadays? Is it used in the development of other mathematical ideas? My question seems to lie between math and math education.

Edited to add: My story is coming along. I would like, at some point, to consult with a math historian who can verify that I've told the story as accurately as possible. How would I find a math historian who knows about these developments and might be interested in checking out my manuscript?

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    $\begingroup$ I don't have any significant suggestions for what you're asking, and I agree with @user52817 who says the formula was never much used to solve cubic equations that arose outside of a theoretical math context. However, for additional history, see the references I gave in my answer to History of the theory of equations: John Colson. Of possible interest is the fact that although all cubics can be solved by radicals, some cubics with rational coefficients have real number solutions that (continued) $\endgroup$ – Dave L Renfro Sep 28 '19 at 19:13
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    $\begingroup$ cannot be expressed using a finite-length expression involving the four arithmetic operations and roots (of any positive integer order) in which every number under any of the radicals is a positive real number (i.e. the implicit appearance of non-real complex numbers CANNOT be avoided in any such expression of the solution). For references, see this 26 September 2005 sci.math post. Of course, there are alternative representations using trig. functions, hyperbolic functions, etc. in which complex numbers can be avoided. $\endgroup$ – Dave L Renfro Sep 28 '19 at 19:14
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    $\begingroup$ To concur with @user52817's answer, roots of cubic equations are ubiquitous in computer graphics, but they are not calculated via the cubic formula. See, e.g., Blinn, James F. "How to solve a cubic equation, Part 1: The shape of the discriminant." IEEE Computer Graphics and Applications 26, no. 3 (2006): 84-93. (PDF download.) $\endgroup$ – Joseph O'Rourke Sep 30 '19 at 12:52
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    $\begingroup$ A colleague of mine used the cubic equation to reverse engineer a number which has a particular Galois group. It can be useful for creating weird field extension problems if you know how to tinker... $\endgroup$ – James S. Cook Oct 1 '19 at 2:58
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    $\begingroup$ @SueVanHattum In case you didn't know about this, one place to consider asking about historical uses of math is the SE History of Science and Mathemetics site (but maybe not to find someone to read a manuscript). You may not wish to do that with the particular question issue, as your comment says you already had $4$ offers from math historians to look it over. Nonetheless, you should at least keep this in mind for any future such needs. $\endgroup$ – John Omielan Nov 3 '19 at 19:53

It seems unlikely that the Cardano formula has even been of serious analytic use, i.e., used to approximate roots of a cubic. At least since the inception of calculus, Newton's method can be used to do so.

In my thinking, the modern perspective of Cardano's formula is an algebraic perspective, not analytic, after Galois. Polynomials of degree 2, 3, 4 are solvable using rational operations and radicals precisely because the "symmetry group" of the roots, i.e., the Galois group of such a polynomial, has a certain algebraic structure called "solvability." The Galois group of a cubic is $S_3$ or $A_3$, i.e., the symmetric or alternating group. Both groups have the special algebraic structure of "solvability."

The Galois group of a polynomial of degree 5 or higher need not have this algebraic structure. Hence there cannot exist a universal "Cardano-like formula for fifth degree polynomials."

So the Cardano formula should be viewed as historical motivation for what eventually became the Abel–Ruffini theorem of the nineteenth century, which asserts that the general polynomial of degree 5 or higher cannot be solved by radicals, and which explains why polynomials of degree 2, 3, and 4 can be always solved by radicals.


Here's another computer-graphics example, which may or may not count as "nowadays". When I was in college 25 years ago I started a project to write a ray-trace renderer, which I then continued to expand after college. Ray-tracing is an elegant model that just recently has gained the hardware support to make it feasible in real time, e.g.: https://developer.nvidia.com/rtx/raytracing.

The ray-tracing model that I'm familiar with requires geometrically coding primitives like a cube, sphere, plane, etc., and then designers can put those pieces together to make more sophisticated compound objects in space. As part of my expansion, I also added the ability for my code to handle the primitive for a torus (thinking that would be useful for space-station animations and the like). Since a ray can intersect a torus in as many as 4 locations, geometrically this requires solving a 4th-degree equation, i.e., a quartic. The mechanism to do that would be, in the general case, reduce it to a cubic in $x^2$ and then call a function I wrote to solve that cubic -- i.e., an explicit implementation of the cubic formula.

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  1. (Tiny help, but maybe something.) This came up in a Google search. http://archives.math.utk.edu/ICTCM/VOL10/C003/paper.pdf The first page has a couple small motivating stories/quotes about why people are fascinated by cubics. You could trace back the two quotes to the original sources and see if there's anything else that's helpful above the quotes. But even as is, they give some mild motivation.

A. The Kac reference is from his autobiography. I don't have a copy of it, but you should be able to get it by interlibrary loan. This link gives a little more on the pull of the cubic for Kac: http://lab.rockefeller.edu/cohenje/assets/file/136KacEnigmasOfChanceAnnProb1986.pdf

Edit: I founf a copy of the Kac bio. It's pretty motivating, actually. He wrote an all new derivation of Cardano, and got it published, as a 16 year old.

[And I am no Kac, but I share/d a similar desire to at least forward derive the equation, as we do with completing the square for the quadratic. I didn't spend the summer doing it though...I guess that's why I'm not a mathematician. But surely the urge to do so is common. We had a recent discussion about repeated root solution to linear constant homo diffy qs...and I, with others, was bugged by how the common training/motivation is to "just try this solution, it might work" and then you prove it does. I checked several texts and found that going back to at least ~1900 (e.g. Granville), this is a common way to present the idea. However, it is not needed...Thomas/Finney derives the repeated root solution "forward".]

B. I have a copy of the Feynman book. There is really not much more on the cubic than within that quote. Although a page or two more on the general issue of "current" (early 80s) Greek education over-venerating the ancient (in art, science, mechanics).

  1. I don't see analytical solution of the cubic as a normal part of current science/engineering studies. And I have a pretty broad chemistry/physics/engineering training. A few cubic polynomials do occasionally occur in practical problems but even then people solve them by estimation. For example, my AP Chemistry text from the 1980s has an occasional cubic in the equilibrium chapter problems (just a few and at the end and harder). But the counsel was always to use estimation to find it, to within sig figs. In contrast analytical solution of quadratics is extremely common, to the extent that bsq-4ac is iconic (hard grooved into memory).
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    $\begingroup$ I wonder if you could sketch how Thomas and Finney derive the repeated root solution "forward". I was also annoyed by this, and in my own approach I take a limit of solutions to differential equations with distinct roots, but whose roots converge together to produce the double roots. $\endgroup$ – Steven Gubkin Sep 28 '19 at 14:46
  • $\begingroup$ I AFL (away from library) at the moment and I don't remember the exact algabra. Had something to do with taking the initial solution and somehow feeding it in and getting a second solution. Will put the details in when I get back. It was just some little kailedscope cover book (the blue AP text you see in the movie Stand and Deliver that Angel gets extra copies of). Was funny to see it better in there than other books I like (Granville, Spiegel, Kreyszig). $\endgroup$ – guest Sep 28 '19 at 15:51
  • $\begingroup$ See pp 621-622 of Elements of Calculus and Analytic Geometry, 1981, Thomas Finney (the AP text, last book Thomas was invovled in before becoming a brand). Soory my math is weak, but he does some sort of operator logic to create a separable equation of two factors and then solves the first order diffy Q by using an integrating factor. It directly gives the correct answer regardless of repeated root or not. [Not totally sorry for not describing the math well, since this is a segue from cubics. But even if I just transcribed it would need a longer answer space, not a comment.] $\endgroup$ – guest Sep 28 '19 at 16:58
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    $\begingroup$ @StevenGubkin you probably already know we can derive the repeated root solution via Laplace transform techniques. However, I think motivating a change of variable starting with solving $y''=0$ is the least high-brow approach. I asked about the history of this somewhere... I collect much here: math.stackexchange.com/q/206967/36530 $\endgroup$ – James S. Cook Oct 1 '19 at 2:55
  • $\begingroup$ @JamesS.Cook Thanks for compiling that list! The answer of Artem is what I "invented" to present to my students. $\endgroup$ – Steven Gubkin Oct 1 '19 at 11:00

(Second answer for follow-on question from Sue about contacting math historians.)

Sorry, to be so simplistic, Sue, but I would just reach out* to some general math historians of any stripe (at a university that has those) and just correspond. Either with whoever you get or by trying to work your way to one with the specific subject matter expertise you want. I would think any math historian (even if not an expert on this period) would know some right questions to ask, that would benefit you. And then he might be able to say, "talk to Jane Umptifratz" if you want someone especially strong on this exact period.

Also, you don't mention it, so apologies if this is also obvious, but a quick Amazon search shows several books on the derivation of the cubic/quintic, which after all is a fascinating story of human conflict as well as interesting mathematics:


https://www.amazon.com/Tales-Mathematicians-Physicists-Simon-Gindikin/dp/0387360263 (chapter within)


You could also try reaching out to some people who are math populizers: John Derbyshire or one of the fellows who wrote a Fermat Last Theorem book.

In addition, I suggest to look at the article on algebra in the Britannica as well as the articles on the protagonists. Often the person chosen to write a Britannica article has deep knowledge (much more so than the space of the article contains), so you could try to correspond with them.

Of course, with all these people be pleasant and don't expect them to help you and don't worry if some ignore you. Just move on to the next.

*In this day and age, it is relatively easy to Google someone's email or even phone or to find a way to contact them through their department or publisher or the like. (Even in the old days, I just got on the phone and called around.)

  • $\begingroup$ I have read a lot online, which is how I decide which version of the basic story I want to include in my story. But I hadn't thought of trying to contact authors of the online stuff. You have given me some good ideas. Thanks. (I'm near UC Berkeley, so maybe I'll try to find someone there. Maybe I can take them out to coffee. And that just gave me another idea.) $\endgroup$ – Sue VanHattum Oct 15 '19 at 4:54
  • $\begingroup$ @Sue VanHattum: There's also the MAA's History of Mathematics E-Mail List which, incidentally, I'm a member of but have only participated a handful of times in the last few years. $\endgroup$ – Dave L Renfro Oct 15 '19 at 5:36
  • $\begingroup$ That would be the perfect place to ask for a volunteer. If I wrote up my request (with my contact email), would you be willing to post it to the group? $\endgroup$ – Sue VanHattum Oct 15 '19 at 19:08
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    $\begingroup$ @Sue VanHattum: Sure, send me an email with the text of what you want posted. It doesn't have to be very detailed because I'll include the URL for your question posted here. $\endgroup$ – Dave L Renfro Oct 16 '19 at 4:31
  • $\begingroup$ Thank you! I will do that by this weekend. $\endgroup$ – Sue VanHattum Oct 16 '19 at 15:49

I've got an idiosyncratic addition to these answers:

(a) Regarding contemporary mathematical uses of the Cardano formula, I coauthored an article that "uses" it! The role it plays in the paper is essentially as inspiration: we noticed that the set of roots of a family of polynomials we cared about had the same form as the roots of a "depressed" cubic as given by the Cardano formula, and our search for why that was happening led us to a deeper understanding of our family of polynomials. (They turned out to be generalized Lucas polynomials!) As an addendum, we note that the same Cardano(/del Ferro/Tartaglia) solution method for the cubic can also be used to solve the polynomials in this family.

Aside: the article has been accepted for publication in the American Mathematical Monthly and will come out sometime next year. The review process led to a substantial revision that I believe vastly improves the article, but I can't post it because Taylor & Francis owns it, so we have to make do with the arXiv preprint till publication. That said, the treatment and use of the Cardano formula aren't really changed.

(b) Regarding historians of mathematics, do you know Michael Barany? He is a historian of mathematics active (and generally responsive) on Twitter, so he may be able to point you toward people with the relevant expertise.

(c) I am not a historian of math, but I have read Cardano's Ars Magna (in English translation), so if you need a fact-check on something that's in there, I might know the answer. Email me? Otoh, it was a library book and it was a decade ago, so maybe this is a dumb idea. And I'm unlikely to know anything about the del Ferro/Tartaglia/Cardano/Ferrari(/Bombelli??) saga that you don't already know.

(d) Addendum: this is of possible relevance, although it is not an answer to the question and you probably already know it. Picking up on Dave Renfro's comments on the OP, my understanding is that one of the Cardano formula's important historical roles was in getting mathematicians to take the complex numbers seriously. I learned about this from the first chapter in Tristan Needham's book Visual Complex Analysis, and Needham also points to John Stillwell's book Mathematics and Its History. Imaginaries are first discussed in print in Cardano's book but in a context unrelated to the cubics, and the discussion is highly speculative and Cardano is really unsure there is any point to what he is saying. (See the link above on Ars Magna which is a blog post I wrote that discusses this toward the end.) However, 30 years later, Rafael Bombelli discovered that certain real cubics with real roots, for which the Cardano(/del Ferro/Tartaglia) formula appears to "break" by requiring the extraction of a square root of a negative, can still be solved by the formula by going ahead and just treating the extraction of the root of a negative as something you're allowed to do: all the weird imaginary numbers you get that way cancel out of the final answer, and the final answer is the correct real root. According to Needham, this was the real impetus for mathematicians starting to take complex numbers seriously.

(I guess this is just registering a perhaps-lesser-known contribution that the cubic formula made to math history. I suppose while I'm brainstorming things of this type that you already know, I should mention the better-known fact that it [and Ferrari's quartic formula, also published in Ars Magna] are what set off the search for the solution to the quintic, which is what led to the development of Galois theory, therefore group theory, therefore abstract algebra as a whole, so between complex numbers and abstract algebra, I guess pretty much all of modern math is downstream from the Cardano formula somehow!)

  • $\begingroup$ Thank you, @benblumsmith! (I already have 4 offers from math historians to look it over.) I will definitely look at your paper when I get some time. (Good to hear from you!) $\endgroup$ – Sue VanHattum Oct 31 '19 at 4:38
  • $\begingroup$ I should've known you'd be all set on the historian-fact-check thing by now! Anyway, I look forward to reading the finished product in whatever form you make it available! $\endgroup$ – benblumsmith Oct 31 '19 at 12:51

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