April. Not only is it the cruelest month (or, in the notation of first-order logic, ∀m (a≤Cm), 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 ax2+bx+c=0, while a cubic equation generally has the form ax3+bx2+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 x3=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?