April. Not only is it the cruelest month (or, in the notation of first-order logic, ∀m (a≤_{C}m), but it’s also Mathematics Awareness Month and National Poetry Month. For the past few years, I’ve been taking Stephen Ornes’ suggestion and making it Math Poetry Month.

This month, I stumbled on an early example of mathematical poetry in the solution to the cubic equation. A cubic equation is a polynomial with a 3 as the largest exponent. The more familiar quadratic equation has the form ax^{2}+bx+c=0, while a cubic equation generally has the form ax^{3}+bx^{2}+cx+d=0.

Many of us memorized the quadratic formula in middle or high school (perhaps to the tune of “Pop goes the weasel”). But did you ever learn the cubic formula? Probably not—it’s quite complicated and not nearly as easy to derive. While versions of the quadratic formula were known to the ancient Babylonians and medieval Islamic mathematicians, the cubic equation stubbornly resisted a general solution for many more years.

The eventual solution of the cubic equation is one of the more colorful stories in math history. You can read expanded (sometimes embellished) accounts of it elsewhere, but the part that concerns us is that Niccolò Tartaglia (ca. 1500-1557) had discovered a way to solve certain kinds of cubic equations. Another mathematician, Girolamo Cardano (1501-1576), wanted to learn the formula and promised not to publish it. Tartaglia shared the formula with Cardano as a poem, and Cardano ended up publishing it. (You can read more about their dispute here.)

Convergence, a magazine published by the Mathematical Association of America, is a great resource for learning about mathematics through its history. I went to a Convergence article, “How Tartaglia solved the cubic equation” by Friedrich Katscher, to learn more about the poem Tartaglia sent to Cardano.

In case you doubted that everything sounds more romantic in Italian, here is Tartaglia’s original Italian poem.

Quando chel cubo con le cose appresso

Se agguaglia à qualche numero discreto

Trouan dui altri differenti in esso.

Dapoi terrai questo per consueto

Che'llor produtto sempre sia eguale

Alterzo cubo delle cose neto,

El residuo poi suo generale

Delli lor lati cubi ben sottratti

Varra la tua cosa principale.

In el secondo de cotestiatti

Quando che'l cubo restasse lui solo

Tu osseruarai quest'altri contratti,

Del numer farai due tal part'à uolo

Che l'una in l'altra si produca schietto

El terzo cubo delle cose in stolo

Delle qual poi, per communprecetto

Torrai li lati cubi insieme gionti

Et cotal somma sara il tuo concetto.

El terzo poi de questi nostri conti

Se solue col secondo se ben guardi

Che per natura son quasi congionti.

Questi trouai, & non con paßi tardi

Nel mille cinquecentè, quatroe trenta

Con fondamenti ben sald'è gagliardi

Nella citta dal mar'intorno centa.

Incidentally, the rhyme scheme is terza rima (aba bcb cdc etc.), which first appeared in Dante's *Inferno*. Perhaps Tartaglia was subliminally urging Cardano to abandon all hope of solving the cubic equation.

This is the beginning of Katscher’s translation of the poem into English:

When the cube with the cose beside it

Equates itself to some other whole number,

Find two others, of which it is the difference.

Hereafter you will consider this customarily

That their product always will be equal

To the third of the cube of the cose net.

Its general remainder then

Of their cube sides, well subtracted,

Will be the value of your principal unknown.

See Katscher's article for a complete translation of the poem into both English and symbolic notation. He even points out an error in Tartaglia’s poem. (The third of the cube is not the same as the cube of the third.)

As a side note, one of my favorite things about the history of the cubic equation is its relationship to complex numbers, which involve the square roots of negative numbers. I’ll leave you with this teaser: the equation x^{3}=15x+4 has three real solutions, but if you use the cubic formula to solve it, you’ll end up making an excursion to the complex plane before you end up with real numbers again. Rafael Bombelli's attempts to deal with the square roots of negative numbers that arose this way ended up laying a foundation for complex analysis.

For more on the solution to the cubic equation, check out another Convergence article, "Solving the cubic with Cardano" by William B. Branson. While today we favor algebraic symbol manipulation, Branson shows how Cardano conceptualized the cubic equation as an equation involving real, 3-dimensional cubes.

More resources for Math Poetry Month:

How to Celebrate Math Poetry Month

Intersections—Poetry with Mathematics

The Poetry of Calculus (The Calculus of Poetry?)

Measure Yourself by the Standard of the Capybara

You Were on the Moon: Astropoetry from Tychogirl

Setting Mathematics in Verse

In Praise of Fractals and Poetry

Quiz: Can You Tell Mathematics from Poetry?