>
> One of the earliest and probably the most acrimonious, was that between
> Tartaglia (c. 1499-1557) and Cardano (1501-1576) over the algebraic
> solution of the cubic. Archimedes had produced a geometric solution in
> his work 'On the sphere and cylinder' and there is little doubt that
> Tartaglia was familiar with this work. But Tartaglia certainly
> independently found an algebraic solution about 1535.
>
> Unfortunately for him, he was not the first. Scipio del Ferro, Professor
> of Mathematics at Bologna had found the same solution 20 years before and
> passed it on to his pupils. Cardano came later on the scene. He delayed
> publication of his 'Ars Magna' in 1545 so as to include a rather more
> complete solution. He acknowledges the contribution of Tartaglia, and
> that the latter had sworn him to secrecy, but he claims that his oath
> was superseded by his learning of the prior claim of Del Ferro.
>
> Tartaglia was furious, not only for moral and intellectual reasons, but
> also because there were material gains associated with being the sole
> possessor of mathematical knowledge. He published diatribes about
> Cardano's perfidy, and tried to challenge him to mathematical contests.
> Cardano remained above the fray, sending his pupil Ferrari instead. The
> latter defeated Tartaglia, who died an embittered man.
>
This is not quite what the sources tell. First of all, there was no dispute
on priority. In his Ars magna, Cardano gives full credit to Tartaglia. He gives
the history of the solutions of algebraic equations, starting with al-Hwarizmi,
Fibonacci, Luca Pacioli for second degree equations and then Scipione del
Ferro, who passed his knowledge to Antonio Maria Fiore who challenged
Tartaglia with this knowledge. Tartaglia found finally the solution of the
30 problems Fiore had proposed. They were all of the same kind (except one)
leading to cubic equations of the form
x^3 + px = q.
Tartaglia found furthermore the solution of the equations of the form
x^3 + q = px
x^3 = px + q.
He could not solve the general equation, as is clear from his writings. He did
not know to transform the general equation in such a way that the term x^2
disappears. He gave the solutions to Cardano. This is acknowledged by Cardano
in his Ars Magna. So the priority is clear. Tartaglia wanted to publish his
result by his own. Occupied with the translation of Euclid's Elements into
Italian, he postponed this publication.
TARTAGLIA says that Cardano had sworn not to publish the result. This is in
his Quesiti. But Lodovico Ferrari, apostrophied by Tartaglia as "il suo creato",
says in his second Cartello di sfida that he was also present in the house of
Cardano in Milan when Cardano and Tartaglia were talking to each other and
that Tartaglia were telling lies. So it is not clear whether Cardano ever
swore an oath. Ferrari also mentions in this cartello that he and Cardano
went to Bologna where they saw a note book of Ferro's with the solution of
the equation of the first type. And, after all, the result is one that must be
made public, it has to be known by all mathematicians. Cardano has no right to
withhold it. Moreover, Cardano had the solution for all types of cubic
equations, no exceptions.
So far, I have found only one qualifying remark in Cardano's papers on his
struggle with Tartaglia. This is in his autobiography. He says that he wrote up
the ars magna in 1545 and that he had worked on it during the time he struggled
with Giovanni Colla and Niccolo Tartaglia (Giovanni Colla had also found the
solution of the cubic in the meantime). He had picked up a few things of
Tartaglia in his book, but he (Tartaglia) prefered to have him (Cardano) as
rival, as superior rival, of course, instead of a friend who owes him thanks.
That is all he said.
Did Cardano swear an oath? We shall never know, I think.
Heinz Lueneburg
PS.: Cardano has published the ars magna in Nuremberg. Nunes, about 60 years
earlier, has published his libro de algebra in Anvers. How did the mail work
at that time?