Scipione del Ferro

Born: 6 February 1465 in Bologna (now Italy)
Died: 5 November 1526 in Bologna, Papal States (now Italy)

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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. 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 Ferrari records this in his writings. We quote the relevant passage from Ferrari below.

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 quadratic equations. Today we write the solutions to ax2 + bx + c = 0 as

x = [-b + √(b2 - 4ac)]/2a and x = [-b - √(b2 - 4ac)]/2a.

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. 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 to cubic equations as a result of a visit which Pacioli made to Bologna. Pacioli 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 our biographies 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 (see for example [3]):-

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 cubic equations] 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.

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 (see [2]) 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 in [3] believes that the evidence from the Bolognetti manuscript adds weight to the belief that del Ferro solved only one case.

We know a little 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.

Article by: J J O'Connor and E F Robertson

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  1. History Topics: Quadratic, cubic and quartic equations
  2. History Topics: Tartaglia versus Cardan
  3. Chronology: 1500 to 1600

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