Scipione
del Ferro is sometimes known as Ferreo, sometimes as Ferro, and
sometimes as dal Ferro. His role in the history of mathematics is an important
one and he deserves great credit for solving one of the outstanding ancient
problems of mathematics. In one sense he is well known, for his role in solving
cubic equations is explained in almost every general work on the history of
mathematics ever written, and yet, surprisingly, his name remains relatively
unknown.
Scipione
del Ferro's parents were Floriano and Filippa Ferro. Floriano Ferro was
employed in paper making which, because of the invention of printing in the
1450s, became an important trade at this time due naturally to a vastly
increased demand for paper. Of Scipione del Ferro's education little is known
but it is probable that it was at the University of Bologna which was founded
in the 11th century and so was a long established and famous university four
hundred years before del Ferro was born.
We
know that del Ferro was appointed as a lecturer in arithmetic and geometry at
the University of Bologna in 1496 and that he retained this post for the rest
of his life. However he was not only involved in academic activities for
records have survived which show that he was involved in business transactions
in the latter part of his life.
No
writings of del Ferro have survived. This must be due, at least in part, to his
reluctance to make his results widely known, preferring to communicate them
only to a few close friends and students. We do know however that he kept a
notebook in which he recorded his most important discoveries. This notebook
passed to del Ferro's son-in-law Hannibal Nave when del Ferro died in 1526. Now
Hannibal Nave was also a mathematician and he had married del Ferro's daughter
Filippa, who of course was named after del Ferro's mother. Hannibal Nave took
over del Ferro's lecturing duties at the University of Bologna in 1526 and also
his name since he adopted the name of dalla Nave alias dal Ferro. Nave still
had the notebook in 1543, for in that year Cardan and Ferrari travelled to
Bologna to see him and his father-in-law's notebook for Ferraris records this
in his writings. We quote the relevant passage from Ferrari below.
Now
the outstanding problem which del Ferro solved was to find a formula to solve a
cubic equation similar to the formula which had been known since the time of
the Babylonians for solving equations of second degree. Today we write
the solutions to ax2 + bx + c = 0 as
x = (-b + Ö b2-4ac))/2a and x = (-b -Ö (b2-4ac))/2a.
Now
in del Ferro's time, although such solutions were known, they were not known in
this form. Firstly, in the middle of the 16th century in Europe, zero was not
in use; secondly negative numbers were not in use; and thirdly there was no
understanding of a quadratic having two roots. Now mathematicians in the time
of del Ferro knew that the problem of solving the general cubic could be
reduced to solving the two cases x3 + mx = n and x3 = mx
+ n, where m and n are positive numbers. (The term in x2 can always
be removed by means of a suitable substitution.) Of course, if negative
coefficients had been in use then there would have been only one case.
There
has been much conjecture as to whether del Ferro came to work on the solution
of the cubic as a result of a visit which Pacioli made to Bologna. He
taught at the University of Bologna during 1501-02 and discussed mathematical
problems with del Ferro at that time. It is not known whether the two discussed
the algebraic solution of cubic equations, but certainly Pacioli had included
this topic in his famous treatise the Summa which he had published seven
years earlier. Some time after Pacioli 's visit to Bologna, del Ferro
solved one of the two cases of this classic problem (but as we mention below,
he may have solved both cases).
The
subsequent developments in the story of the solution of the cubic, namely the
contest in 1535 between Antonio Maria Fior (a student of del Ferro) and
Tartaglia , then the involvement of Cardan, are told in detail in the biographi
of Tartaglia and of Cardan . As far as this biography of del Ferro is concerned
we should stress that it was Cardan 's discovery that del Ferro had been
the first to solve the Cubic and not Tartaglia which made him feel that
he could honour his oath to Tartaglia not to divulge his method and
still publish the solution in Ars Magna for there Cardan considered he
is giving del Ferro's method, not that of Tartaglia. Ferrari, a student of
Cardan's wrote (on 1 April 1547) about their earlier trip to see Hannibal della
Nave :-
Four years ago when
Cardano was going to Florence and I accompanied him, we saw at Bologna Hannibal
della Nave, a clever and humane man who showed us a little book in the hand of
Scipione del Ferro, his father-in-law, written a long time ago, in which that
discovery [solution of ] was elegantly and learnedly presented.
In
Ars Magna Cardan writes with great respect for the achievements
of del Ferro (see for example [1]):-
Scipione Ferro of
Bologna, almost thirty years ago, discovered the solution of the cube and
things equal to a number [which in today's notation is the case x3 +
mx = n], a really beautiful and admirable accomplishment. In distinction this
discovery surpasses all mortal ingenuity, and all human subtlety. It is truly a
gift from heaven, although at the same time a proof of the power of reason, and
so illustrious that whoever attains it may believe himself capable of solving
any problem.
Now
the story that Fior was the only person to whom del Ferro divulged his solution
is common in most histories of mathematics, yet it is false. As we have seen
above the solution was written down by del Ferro and certainly was known to Nave.
Pompeo Bolognetti, who lectured at the University of Bologna on mathematics
from 1554 to 1568, also had access to the original solution by del Ferro as
well as the solution as given by Cardan in Ars Magna which had been
published by then. Bombelli , who published his Algebra in 1572, also
had access to details of del Ferro's work which no longer exists today. Bombelli,
like Cardan, expressed wonder at the genius of del Ferro and describes him as:-
... a man uniquely
gifted in this art [of algebra]...
Around
1925, Bortolotti examined sixteenth century manuscripts reproducing work by
Bolognetti, Cardan and Bombelli . One important manuscript is headed:-
Dal Ferro's rule for
the solution of cubic equations. From the Cavaliere Bolognetti, who had it from
the Bolognese master of former days, Scipione dal Ferro. On unknowns and cubes
equal to numbers.
The
manuscript gives a method of solution which is applied to the equation 3x3
+ 18x = 60. From research on this and the other manuscripts, Bortolotti concluded
that, contrary to the widely held belief that del Ferro only solved one case of
the cubic, that indeed he solved both cases. However Crossley believes that the
evidence from the Bolognetti manuscript adds weight to the belief that del
Ferro solved only one case.
Very
little is known about other work by del Ferro. He made an important
contribution to rationalising fractions, extending methods to
rationalise fractions which had square roots in the denominator (which
were know to Euclid) to fractions whose denominators were the sum of three cube
roots. We also know that del Ferro worked on another problem which was popular
in his time, namely examining which geometrical problems could be solved with a
compass set in a fixed position. Ferrari, in a letter to Tartaglia ,
states the del Ferro worked on such problems but he did not give any details of
del Ferro's results.
It
is sad that del Ferro's notebook has not survived. Indeed it is probable that
he would have attained considerably more fame had we been able to give details
of the problems which he solved and wrote down in his notebook.