Monday, January 25, 2016

Tartaglia solving the cubic -- in verse

     Mathematical historians now credit both Cardano and Tartaglia with the formula to solve cubic equations, referring to it as the "Cardano-Tartaglia Formula." Tartaglia is known for reporting solutions of three different forms of the cubic equation in a poem (1534).  Below we offer Boston poet Kellie Gutman's English translation of Tartaglia's verse, followed by the original Italian.

When X Cubed    by Niccolò Tartaglia (1500–1557)       (Englished by Kellie Gutman)

When x cubed’s summed with m times x and then   
  Set equal to some number, a relation    
  Is found where r less s will equal n.

Now multiply these terms. This combination
  rs will equal m thirds to the third;
  This gives us a quadratic situation,    

Where r and s involve the same square surd.  
  Their cube roots must be taken; then subtracting
  Them gives you x; your answer’s been inferred.

The second case we’ll set about enacting
  Has x cubed on the left side all alone.    
  The same relationships, the same extracting:

Seek numbers r and s, where the unknown
  rs will equal m-on-3 cubed nicely,
  And summing r and s gives n, as shown.

Once more the cube roots must be found concisely
  Of our two newfound terms, both r and s,
  And when we add these roots, there’s x precisely.

The final case is easy to assess:
  Look closely at the second case I mention --
  It’s so alike that I shall not digress.

These things I’ve quickly found, they’re my invention,
  In this year fifteen hundred thirty-four,
  While working hard and paying close attention,   

Surrounded by canals that lap the shore.

And here is the original Italian version by Tartaglia:

Quando che’l cubo

Quando che’l cubo con le cose appresso
  Se agguaglia à qualche numero discreto
  Trovar dui altri differenti in esso.

Dapoi terrai questo per consueto
  Che'l lor produtto sempre sia eguale
  Al terzo cubo delle cose neto,

El residuo poi suo generale
  Delli lor lati cubi ben sottrati
  Varra la tua cosa principale.

In el secondo de cotesti atti
  Quando che’l cubo restasse lui solo
  Tu osservarai quest'altri contratti,

Del numer farai due tal part’à volo
  Che l’una in l’altra si produca schietto
  El terzo cubo delle cose in siolo

Delle qual poi, per commun precetto
  Torrai li lati cubi insieme gionti
  Et cotal somma sara il tuo concetto.

El terzo poi de questi nostri conti
  Se solve col secondo se ben guardi
  Che per natura son quasi congionti.

Questi trovai, e non con passi tardi
  Nel mille cinquecentè, quatro e trenta
  Con fondamenti ben sald’è gagliardi

Nella citta dal mar’ intorno centa.

     Kellie O. Gutman lives in Boston, Massachusetts. With her husband Richard J.S. Gutman, she has written and published two books. Gutman's translation of "Quando Che'l Cubo" first appeared in The Mathematical Intelligencer, 27 (1) 2005, 32-36.


  1. Thank you! I've always wanted to read this in English.

    1. I'm grateful to Kellie for the translation. Thanks for dropping by!