Scipione del Ferro

Scipione del Ferro (born February 6, 1465 in Bologna , † November 5, 1526 there ) was an Italian mathematician . From 1496 he was professor of arithmetic and geometry at the University of Bologna .

Luca Pacioli (around 1445–1514 / 17), who was at the University of Bologna from 1501–1502, took the view in his Summa de Arithmetica (1494) that there is no general solution method for equations of the 3rd degree and higher. This premonition turned out to be false for equations of degree 3 and degree 4, but was later proved to be correct for equations of degree greater than 4 using Abel-Ruffini's Theorem . Pacioli's view must have caught del Ferro's interest. He managed to find a solution to the reduced cubic equation . The knowledge was never published by him; only on his deathbed did he pass it on to his students Hannibal del Nave (his son-in-law and successor, around 1500 to 1558) and Antonio Maria Fior . ${\displaystyle x^{3}+ax+b=0}$

History of the solution

The vestibule of the Basilica di Santa Maria dei Servi in Bologna was the place for public mathematical challenges.

Nicolo Tartaglia , who from 1535 (when Fiore challenged him with the question of the solutions to 30 equations that can be expressed in the above form) used del Ferro's solution in public competitions to make money and possibly found this solution independent, at least as looked at his private secret knowledge, passed it on in claused form to Gerolamo Cardano , who had to swear to keep the solution to himself. After Cardano learned that del Ferro had found the solution long before Tartaglia's use of the same, he no longer felt bound by the oath and published the general solution, which went far beyond del Ferro's or Tartaglia's special case, in his work Ars magna de Regulis Algebraicis by 1545 . This also contains the general solution for equations of the fourth degree, which Cardano expressly ascribed to his student Lodovico Ferrari . Tartaglia then accused Cardano of intellectual theft, but was sentenced by a court in Milan to publicly retract that allegation.

Solution from del Ferro

${\displaystyle x^{3}+px+q=0}$	${\displaystyle x^{3}+6x-20=0}$
${\displaystyle D=\left({\frac {p}{3}}\right)^{3}+\left({\frac {q}{2}}\right)^{2}}$	${\displaystyle D=2^{3}+(-10)^{2}=108}$
For the case one can now calculate: ${\displaystyle D>0}$
${\displaystyle u={\sqrt[{3}]{-{\frac {q}{2}}+{\sqrt {D}}}}}$	${\displaystyle u={\sqrt[{3}]{10+{\sqrt {108}}}}=1+{\sqrt {3}}}$
${\displaystyle v={\sqrt[{3}]{-{\frac {q}{2}}-{\sqrt {D}}}}}$	${\displaystyle v={\sqrt[{3}]{10-{\sqrt {108}}}}=1-{\sqrt {3}}}$
Then: ${\displaystyle x=u+v}$	${\displaystyle \quad x=1+{\sqrt {3}}+1-{\sqrt {3}}=2}$
	Sample: ${\displaystyle 2^{3}+6\cdot 2-20=0}$

The solution found is the only real one, the Cardan's formula in its modern version provides two further complex solutions.

Web links

• Scipione del Ferro , English biography in the McTutor project