希皮奥内·德尔·费罗Scipione del Ferro

希皮奥内·德尔·费罗有时被称为Ferreo，有时被称为费罗，有时被称为dal 费罗。他在数学史上的角色很重要，他因解决了数学中一个杰出的古老问题而值得高度赞扬。从某种意义上说，他很有名，因为他在解决三次方程中的作用在几乎所有已写成的数学史通论著作中都有解释，然而，令人惊讶的是，他的名字仍然相对不为人知。

费罗的父母是Floriano和Filippa费罗。Floriano费罗从事造纸业，由于15世纪50年代印刷术的发明，当时对纸张的需求大幅增加，造纸业自然成为一项重要的行业。关于费罗的教育情况所知甚少，但很可能是在博洛尼亚大学接受的，该大学建于11世纪，因此在费罗出生前四百年就已是一所历史悠久且著名的大学。

我们知道费罗于1496年被任命为博洛尼亚大学的算术和几何讲师，并且他终生保留了这个职位。然而，他不仅参与学术活动，因为保存下来的记录显示他在晚年还参与了商业交易。

费罗的著作没有留存下来。这至少部分是由于他不愿将自己的成果广泛传播，宁愿只告诉少数亲密的朋友和学生。然而我们确实知道他保留了一本笔记本，其中记录了他最重要的发现。1526年费罗去世时，这本笔记本传给了费罗的女婿Hannibal Nave。Hannibal Nave也是一位数学家，他娶了费罗的女儿Filippa，她自然是以费罗的母亲命名的。Hannibal Nave于1526年接替费罗在博洛尼亚大学的讲课职责，也继承了他的名字，因为他采用了dalla Nave别名dal 费罗的名字。1543年Nave仍然持有这本笔记本，因为那一年吉罗拉莫·卡尔达诺和洛多维科·费拉里前往博洛尼亚拜访他并查看他岳父的笔记本，洛多维科·费拉里在其著作中记录了此事。我们在下面引用洛多维科·费拉里中的相关段落。

费罗解决的突出问题是找到一个求解三次方程的公式，类似于自巴比伦时代以来已知的求解二次方程的公式。今天我们将$ax^{2} + bx + c = 0$的解写成

$x = \Large \frac{-b + \sqrt{b^{2} - 4ac}}{2a}$和$x = \Large \frac{-b - \sqrt{b^{2} - 4ac}}{2a}$。

在费罗的时代，尽管这样的解已知，但并非以这种形式为人所知。首先，在16世纪中叶的欧洲，零尚未被使用；其次，负数尚未被使用；第三，人们还不理解二次方程有两个根。费罗时代的数学家知道，求解一般三次方程的问题可以归结为求解$x^{3} + mx = n$和$x^{3} = mx + n$两种情况，其中$m$和$n$是正数。（$x^{2}$中的项总是可以通过适当的代换消去。）当然，如果当时已经使用负系数，那么就只有一种情况了。

关于费罗是否是因为卢卡·帕西奥利访问博洛尼亚而开始研究三次方程的求解，有很多猜测。卢卡·帕西奥利在1501-02年间在博洛尼亚大学任教，并在那时与费罗讨论数学问题。不知道两人是否讨论过三次方程的代数解法，但可以肯定的是卢卡·帕西奥利已将这一主题纳入他七年前出版的著名论著Summa中。在卢卡·帕西奥利访问博洛尼亚之后某时，费罗解决了这个经典问题两种情况中的一种（但正如我们下面提到的，他可能两种都解决了）。

三次方程求解故事的后续发展，即1535年安东尼奥·马里亚·菲奥尔（费罗的学生）与尼科洛·塔尔塔利亚之间的竞赛，然后是吉罗拉莫·卡尔达诺的介入，在我们的尼科洛·塔尔塔利亚和吉罗拉莫·卡尔达诺传记中有详细叙述。就这本费罗的传记而言，我们应当强调，正是吉罗拉莫·卡尔达诺发现费罗是第一个求解三次方程的人，而不是尼科洛·塔尔塔利亚，这使他感到自己可以遵守对尼科洛·塔尔塔利亚不发誓不泄露其方法的誓言，同时仍在Ars MagnaⓉ（《大衍术》）中发表该解法，因为在那里吉罗拉莫·卡尔达诺认为他给出的是费罗的方法，而不是尼科洛·塔尔塔利亚的方法。吉罗拉莫·卡尔达诺的学生洛多维科·费拉里（于1547年4月1日）写到了他们早先拜访汉尼拔·德拉·纳韦的旅行（例如见[3]）：-

四年前，当吉罗拉莫·卡尔达诺前往佛罗伦萨而我陪同他时，我们在博洛尼亚见到了Hannibal della Nave，一个聪明而人道的人，他给我们看了一本小书，是他岳父费罗手写的，很久以前写的，其中那个发现[三次方程的解法]被优雅而博学地呈现出来。

在Ars Magna Ⓣ（伟大的艺术）中，吉罗拉莫·卡尔达诺对费罗的成就表示了极大的尊重（例如见[1]）：-

博洛尼亚的费罗，大约三十年前，发现了立方与等于数的东西的解法[用今天的记号就是$x^{3} + mx = n$的情况]，一个真正美丽而令人钦佩的成就。在区分上，这个发现超越了所有凡人的才智和人类的精妙。它确实是来自天堂的礼物，尽管同时也是理性力量的证明，如此辉煌，以至于任何达到它的人都可以相信自己能够解决任何问题。

关于Fior是唯一被费罗透露其解法的人的故事在大多数数学史中很常见，但它是假的。正如我们上面所见，解法是由费罗写下的，并且肯定为Nave所知。Pompeo Bolognetti，从1554年到1568年在博洛尼亚大学讲授数学，也可以接触到费罗的原始解法以及吉罗拉莫·卡尔达诺在Ars Magna Ⓣ（伟大的艺术）中给出的解法，后者当时已经出版。拉法耶尔·蓬贝利，他在1572年出版了他的Algebra，也可以接触到费罗工作的细节，这些细节今天已不复存在。拉法耶尔·蓬贝利，像吉罗拉莫·卡尔达诺一样，对费罗的天才表示惊叹，并描述他为：-

...一个在这门[代数]艺术上独一无二的天才...

大约在1925年，Bortolotti（见[2]）检查了十六世纪的手稿，复制了Bolognetti、吉罗拉莫·卡尔达诺和拉法耶尔·蓬贝利的工作。一份重要的手稿标题为：-

费罗解三次方程的法则。来自Cavaliere Bolognetti，他从昔日的博洛尼亚大师费罗 dal 费罗那里得到它。论未知数与立方等于数。

手稿给出了一种解法，应用于方程$3x^{3} + 18x = 60$。通过对这份手稿和其他手稿的研究，Bortolotti得出结论：与普遍认为费罗只解了三次方程的一种情形的看法相反，他实际上解了两种情形。然而Crossley在[3]中认为，来自Bolognetti手稿的证据增加了费罗只解了一种情形的看法的分量。

我们对费罗的其他工作略知一二。他对rationalising分数做出了重要贡献，将有理化分母中含有平方根的分数（欧几里得已知的）的方法扩展到分母是三个立方根之和的分数。我们还知道费罗研究了当时另一个流行的问题，即考察哪些几何问题可以用固定位置的圆规解决。洛多维科·费拉里在给尼科洛·塔尔塔利亚的一封信中陈述费罗研究过这类问题，但他没有给出费罗结果的任何细节。

遗憾的是费罗的笔记本没有保存下来。事实上，如果我们能够给出他解决并写在笔记本中的问题的细节，他很可能会获得相当大的名声。

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 $ax^{2} + bx + c = 0$ as

$x = \Large \frac{-b + \sqrt{b^{2} - 4ac}}{2a}$ and $x = \Large \frac{-b - \sqrt{b^{2} - 4ac}}{2a}$.

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 $x^{3} + mx = n$ and $x^{3} = mx + n$, where $m$ and $n$ are positive numbers. (The term in $x^{2}$ 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 $x^{3} + mx = n$], 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 $3x^{3} + 18x = 60$. 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 known 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.