In his memoirs 'Quesiti et inventiones diversi' of 1546, Tartaglia relates
that in 1530 a fellow countryman from Brescia, Zuanne de Tononi da Coi,
challenged him to solve the equation x^3 + 3x^2 = 5. After working for
some time on such equations, Tartaglia claims in the 'Quesiti' that he had
found a general method of solving equations of the form x^3 + ax^2 = b. But
he does not reveal his method.
Some 30 years before, del Ferro had solved cubics of the form x^3 + px = q
and divulged the solution to his student Fiore who challenged Tartaglia to
a public disputation in 1535. Tartaglia relates with some glee that after
much concentrated thought he discovered the solution of x^3 + px = q and
roundly defeated Fiore, who could only solve cubics of the one type. In
1539, Tartaglia explained his solution of the latter equation to Cardano,
who published it in 1545 after having learnt of del Ferro's solution.
The mystery remains: what was Tartaglia's solution of da Coi's problem?
Clearly it could not have been the well-known reduction to the form x^3 +
px = q, because Tartaglia required so much time and effort to learn Fiore's
solution.
Furthermore, the algebraic expression for the solution of x^3 + ax^2 = b
in terms of the coefficients is longer and more complicated than that of
x^3 + px = q. A further mystery is why Tartaglia never subsequently
referred to his solution even though he engaged in a long and bitter
priority dispute with Cardano.
In an article published in Vol 11, No 4 of The Australian Mathematical
Society Gazette in 1984, I argued that his solution may have been of the
form 'the shortest line segment from the intersection of the parabola y =
x^2 and the hyperbola (x + a)y = b to its asymptote'. (Of course expressed
in the mathematical language of Tartaglia).
My reason for this conjecture is that this is exactly how Archimedes
expresses the solution of the problem of cutting a sphere by a plane so
that the volumes of the pieces are to one another in a given ratio. This
result appears in the book 'On the sphere and the cylinder'. Actually, the
situation is a little more complicated than this. Archimedes states the
solution without proof, but his commentator Eutocius gives a complete
solution, including a 'diorismos', i.e. necessary and sufficient conditions
for the existence of a solution. The connection with the solution of the
cubic of da Coi's type is pointed out in Heath's edition of Archimedes.
Could Tartaglia have known Archimedes' work? He could and he did. In his
'General trattato di numero et misure', published in six installments from
1556 to 1560, Tartaglia relates that he acquired a Latin edition of 'On the
sphere and the cylinder' in 1531 and proceeds to give an Italian
translation.
Some mysteries remain. Did Tartaglia's book contain Eutocius' work? Did he
succeed in finding an algebraic solution? Why did he never reveal the
connection?
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Associate Professor Phill Schultz
Director of Postgraduate Studies,
Department of Mathematics,
The University of Western Australia,
Nedlands, 6907, Australia
Phone:(08)9380-3381 Fax:(08)9380-1028
e-mail schultz@maths.uwa.edu.au
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