保罗·鲁菲尼Paolo Ruffini

保罗·鲁菲尼的父亲Basilio 鲁菲尼是Valentano的一名医生。鲁菲尼年幼时[4]：-

……具有神秘的气质，似乎注定要成为神职人员……

当鲁菲尼十几岁时，全家搬到了意大利北部艾米利亚-罗马涅大区摩德纳附近的雷焦。他于1783年进入摩德纳大学，在那里学习数学、医学、哲学和文学。在摩德纳，他的数学老师中有Luigi Fantini，教鲁菲尼几何，以及鲁菲尼 Cassiani，教他微积分。

埃斯特家族统治着摩德纳，1787年，Cassiani被任命为埃斯特家族地产的顾问。Cassiani在摩德纳开设的分析基础课程于1787-88年由鲁菲尼接手，尽管当时他仍是学生。1788年6月9日，鲁菲尼毕业，获得哲学、医学和外科学学位。此后不久，他又获得了数学学位。

鲁菲尼从Cassiani那里接手的分析基础课程一定教得很好，因为1788年10月15日，他被任命为分析基础教授。Fantini曾在鲁菲尼读本科时教他几何学，他发现自己的视力恶化，1791年不得不辞去摩德纳的职位。鲁菲尼于1791年被任命填补数学基础教授的空缺。然而，鲁菲尼不仅仅是一位数学家。他受过医学训练，同样在1791年，摩德纳医学院法庭授予他行医执照。

这是法国大革命后战乱频仍的时期。到1795年初，法国已在各条战线上取得胜利。在意大利北部，法国军队威胁着奥地利-撒丁的阵地，但其指挥官未能采取主动。1796年3月，他被拿破仑·波拿巴取代，后者执行了一场精彩的机动战役。4月12日发起攻势，接连击败并分割了奥地利和撒丁军队，然后向都灵进军。撒丁国王请求停战，尼斯和萨伏依被并入法国。波拿巴继续对奥地利人作战，占领了米兰，但在曼图亚受阻。在曼图亚落入其军队之前，他与帕尔马公爵和摩德纳公爵签署了停战协定。拿破仑的军队占领了摩德纳，鲁菲尼极不情愿地发现自己身处政治动荡之中。

拿破仑建立了由伦巴第、艾米利亚、摩德纳和博洛尼亚组成的内阿尔卑斯共和国。尽管不想卷入其中，鲁菲尼却发现自己被任命为内阿尔卑斯共和国初级委员会的代表。然而，他很快离开了这个职位，并于1798年初回到摩德纳大学从事科学工作。他被要求宣誓效忠共和国，而鲁菲尼发现出于宗教原因他无法做到。由于未能宣誓，他失去了教授职位并被禁止教学。

鲁菲尼似乎并未因失去讲席而受到太大困扰，事实上他是一个非常冷静的人，对周围的一切戏剧性事件都泰然处之。他不能教数学这一事实意味着他有更多时间行医，从而帮助他极为关爱的病人。另一方面，这给了他机会从事最原创的项目之一，即证明五次方程无法用根式解决。

用根式解多项式方程意味着找到其根用系数表示的公式，使得该公式只涉及加、减、乘、除和开方运算。二次方程（二次的）从巴比伦时代起就为根式可解所知。三次方程已由del 希皮奥内·德尔·费罗、尼科洛·塔尔塔利亚和吉罗拉莫·卡尔达诺用根式解出。洛多维科·费拉里在1540年用根式解出了quartic，因此尽管许多数学家尝试过，250年来无人能用根式解出五次方程。在认真尝试理解这个问题的人中有埃伦弗里德·瓦尔特·冯·切恩豪斯、莱昂哈德·欧拉、艾蒂安·贝祖、亚历山大-泰奥菲勒·范德蒙德、爱德华·华林和约瑟夫·拉格朗日。

看来在鲁菲尼之前，没有人真正相信五次方程不能用根式求解。当然，没有数学家发表过这样的主张，甚至约瑟夫·拉格朗日在其著名论文Reflections on the resolution of algebraic equations中也说他会回到五次方程解的问题上，显然，他仍然希望通过根式求解它。1799年，鲁菲尼出版了一本关于方程理论的书，其标题表明了他关于五次方程不能用根式求解的主张：General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than four is impossible。该书的引言开头写道：-

次数大于四的一般方程的代数解总是不可行的。看，一个非常重要的定理，我相信我能够断言（如果我没有弄错的话）：给出它的证明是出版本卷的主要原因。不朽的约瑟夫·拉格朗日以其崇高的思考，为我的证明提供了基础。

鲁菲尼在他的工作中使用了group theory，但他不得不为自己发明这个主题。约瑟夫·拉格朗日曾使用过排列，人们可以争辩说群出现在约瑟夫·拉格朗日的工作中，但由于约瑟夫·拉格朗日从未组合过置换，我们现在是在事后才看到他的论文中群论的萌芽。鲁菲尼是第一个引入元素的阶、共轭、置换群中元素的循环分解以及本原和非本原概念的人。他证明了一些显著的定理（当然不是用下面引用的现代术语）：-

一个置换的阶是分解为不相交循环后各循环长度的最小公倍数。

$S_{5}$中一个5阶元素是一个5-循环。

如果$G < S_{5}$的阶能被5整除，那么$G$有一个5阶元素。

$S_{5}$没有指数为3、4或8的子群。

这是一项出色的工作，除了一处空缺外，证明了鲁菲尼所声称的定理。证明以现代符号给出在[4]中。然而，数学家们对鲁菲尼的工作反应出奇地少。1801年，鲁菲尼将他的书寄了一本给约瑟夫·拉格朗日。他没有收到回复，于是他又寄了第二本并附上一封信[4]：-

由于不确定您是否收到了我的书，我再寄给您一本。如果我在任何证明中有误，或者如果我说了一些我认为是新的、而实际上并不新的东西，最后如果我写了一本无用的书，我恳请您真诚地指出来。

鲁菲尼没有收到回复，他在1802年又写道：-

没有人比我更有资格……收到我冒昧寄给您的这本书。……在写这本书时，我主要想证明高于四次的方程不可能求解。

一些数学家接受了鲁菲尼的证明，尽管不得不说，比萨的教授Pietro Paoli在1799年写道[4]时是受了爱国动机的影响：-

我非常愉快地读了您的书……并极力推荐那个最重要的定理，它排除了求解高于四次方程的可能性。我与您以及我们的意大利一同感到无比欣喜，因为一个理论在这里诞生并完善，而其他国家对此贡献甚少……

要理解这段引文，必须认识到约瑟夫·拉格朗日出生在都灵，当时是意大利的一部分。撇开这种爱国反应不谈，数学界似乎几乎忽略了鲁菲尼的伟大成果。那么鲁菲尼是如何反应的呢？他在1803年发表了第二个证明，希望它可能更容易理解，在引言中写道：-

在本论文中，我将尝试证明同一命题[五次方程不可解]，希望推理不那么深奥，并且完全严格。

至少鲁菲尼收到了Malfatti关于这篇论文的评论，但不幸的是Malfatti没有理解鲁菲尼的论证，并提出了一个错误的反对意见。鲁菲尼在1808年和1813年发表了进一步的证明。关于这最后一个证明，Ayoub在[4]中写道：-

还有什么能比这更优雅呢？这个证明本质上就是后来所称的皮埃尔·汪策尔对尼尔斯·阿贝尔证明的修改，并于1845年发表。它类似于鲁菲尼的证明并不奇怪，因为皮埃尔·汪策尔在他的论文中说……“利用尼尔斯·阿贝尔和鲁菲尼的工作……”。

鲁菲尼并没有停止努力使自己的工作得到数学界的承认。当让·巴蒂斯特·约瑟夫·德朗布尔在一份关于1789年以来数学状况的报告中写道：-

鲁菲尼提议证明这是不可能的……，

鲁菲尼回复道：-

……我不仅提议证明，而且实际上证明了……。

人们不得不为鲁菲尼感到极度惋惜。如果某位数学家写信给他，指出证明中有错误甚至漏洞，那么至少鲁菲尼会有机会改正它。然而，似乎没有人真正想知道五次方程不能用根式求解。鲁菲尼请求巴黎科学院对他的证明的正确性作出判断，约瑟夫·拉格朗日、阿德里安-马里·勒让德和西尔维斯特·佛朗索瓦·拉克鲁瓦被任命审查它。他们再次提出了一份就鲁菲尼而言极不令人满意的报告：-

……如果一件事不重要，就无人注意，而约瑟夫·拉格朗日本人“以他的冷静”发现其中几乎没有值得注意的东西。

Royal Society 也被要求对正确性发表意见，鲁菲尼 收到了稍微客气一些的回复，其中说，尽管他们没有对具体工作表示认可，但他们相当确信它证明了所声称的内容。唯一确实承认其重要性和正确性的人是 奥古斯丁·路易·柯西。这更加令人惊讶，因为 奥古斯丁·路易·柯西 是所有数学家中最不善于归功于他人的人之一。他在 1821 年写信给 鲁菲尼，距 鲁菲尼 去世不到一年 [1]：-

……你关于方程一般解的论文是一项在我看来一直值得数学家关注的工作，并且据我判断，它完全证明了高于四次的方程不可能用代数方法求解。

事实上，奥古斯丁·路易·柯西 在 1813 年至 1815 年间写了一部关于置换群的重要著作，并在其中推广了 鲁菲尼 的一些结果。他肯定深受 鲁菲尼 思想的影响。通过 奥古斯丁·路易·柯西 的这种影响，也许是 鲁菲尼 的工作对数学发展产生影响的唯一途径。

我们在鲁菲尼的职业生涯中停在了1799年左右，那时他开始发表关于五次方程的文章。他离开摩德纳大学，在摩德纳的军事学校教了7年应用数学。他继续行医，为从社会最贫穷到最富有的人看病。拿破仑倒台后，鲁菲尼于1814年成为摩德纳大学校长。政治局势仍然极其复杂，尽管他个人能力出众、备受尊敬、以诚实著称，他担任校长的时期一定非常艰难。

除了校长职务，鲁菲尼还在摩德纳大学担任应用数学讲席、实用医学讲席和临床医学讲席。1817年爆发了斑疹伤寒疫情，鲁菲尼继续治疗病人，直到他自己也染上了这种病。虽然他部分康复，但从未完全恢复健康，1819年他放弃了临床医学讲席。然而，他并没有放弃科学工作，1820年他根据自己患病的经历发表了一篇关于斑疹伤寒的科学文章。

鲁菲尼的工作还有另外一些方面值得提及。他写了几部哲学著作，其中一部反驳了皮埃尔·西蒙·拉普拉斯的一些哲学思想。他还写了关于probability以及概率在法庭案件证据中的应用的文章。

鉴于本文中关于五次方程不可解性的信息，有理由问为什么 尼尔斯·阿贝尔 被归功于证明了该定理，而 鲁菲尼 却没有。Ayoub 提出 [4]：-

……数学界还没有准备好接受如此革命性的想法：多项式不能用根式求解。此外，置换方法太奇特，而且必须承认，鲁菲尼 早期的论述不易理解。……比如说在 1800 年至 1820 年间，数学界的心态……从试图求解五次方程转变为证明其不可能性……

Paolo Ruffini's father, Basilio Ruffini, was a medical doctor in Valentano. As a young child Paolo was [4]:-

... of a mystical temperament and appeared to be destined for the priesthood...

The family moved to Reggio, near Modena in the Emilia-Romagna region of northern Italy, when Paolo was a teenager. He entered the University of Modena in 1783 where he studied mathematics, medicine, philosophy and literature. Among his teachers of mathematics at Modena were Luigi Fantini, who taught Ruffini geometry, and Paolo Cassiani, who taught him calculus.

The Este family ruled Modena and, in 1787, Cassiani was appointed as a councillor for the Este estates. Cassiani's course at Modena on the foundations of analysis was taken over by Ruffini in 1787-88 although he was still a student at this time. On 9 June 1788 Ruffini graduated with a degree in philosophy, medicine and surgery. Soon after this he graduated with a mathematics degree.

Ruffini must have made a good job of the foundations of analysis course he took over from Cassiani for, on 15 October 1788, he was appointed professor of the foundations of analysis. Fantini, who had taught Ruffini geometry when he was an undergraduate, found his eyesight deteriorating and in 1791 he had to resign his post at Modena. Ruffini was appointed to fill the position of Professor of the Elements of Mathematics in 1791. However, Ruffini was not only a mathematician. He had trained in medicine and, also in 1791, he was granted a licence to practise medicine by the Collegiate Medical Court of Modena.

This was a time of wars following the French Revolution. By early 1795 France had won victories on every front. In northern Italy the French army threatened Austrian-Sardinian positions, but its commander failed to take the initiative. In March 1796 he was replaced by Napoleon Bonaparte who executed a brilliant campaign of manoeuvres. Taking the offensive on 12 April and successively defeated and separated the Austrian and the Sardinian armies and then marched on Turin. The King of Sardinia asked for an armistice and Nice and Savoy were annexed to France. Bonaparte continued the war against the Austrians and occupied Milan but was held up at Mantua. Before Mantua fell to his armies he signed armistices with the duke of Parma and the duke of Modena. Napoleon's troops occupied Modena and, much against his wishes, Ruffini found himself in the middle of the political upheaval.

Napoleon set up the Cisalpine Republic consisting of Lombardy, Emilia, Modena and Bologna. Although not wishing to get involved, Ruffini found himself appointed as a representative to the Junior Council of the Cisalpine Republic. However, he soon left this position and, in early 1798, he returned to his scientific work at the University of Modena. He was required to swear an oath of allegiance to the republic and this Ruffini found he could not bring himself to do on religious grounds. By failing to swear the oath he lost his professorship and was barred from teaching.

Ruffini did not seem greatly disturbed by the loss of his chair, in fact he was a very calm man who took all the dramatic events around him in his stride. The fact that he could not teach mathematics meant that he had more time to practise medicine and therefore help his patients to whom he was extremely devoted. On the other hand it gave him the chance to work on what was one of the most original of projects, namely to prove that the quintic equation cannot be solved by radicals.

To solve a polynomial equation by radicals meant finding a formula for its roots in terms of the coefficients so that the formula only involves the operations of addition, subtraction, multiplication, division and taking roots. Quadratic equations (of degree 2) had been known to be soluble by radicals from the time of the Babylonians. The cubic equation had been solved by radicals by del Ferro, Tartaglia and Cardan. Ferrari had solved the quartic by radicals in 1540 and so 250 years had passed without anyone being able to solve the quintic by radicals despite the attempts of many mathematicians. Among those who had made serious attempts to understand the problem were Tschirnhaus, Euler, Bézout, Vandermonde, Waring and Lagrange.

It appears that nobody before Ruffini really believed that the quintic could not be solved by radicals. Certainly no mathematician has published such a claim and even Lagrange in his famous paper Reflections on the resolution of algebraic equations says he will return to the question of the solution of the quintic and, clearly, he still hoped to solve it by radicals. In 1799 Ruffini published a book on the theory of equations with his claim that quintics could not be solved by radicals as the title shows: General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than four is impossible. The introduction to the book begins:-

The algebraic solution of general equations of degree greater than four is always impossible. Behold a very important theorem which I believe I am able to assert (if I do not err): to present the proof of it is the main reason for publishing this volume. The immortal Lagrange, with his sublime reflections, has provided the basis of my proof.

Ruffini used group theory in his work but he had to invent the subject for himself. Lagrange had used permutations and one can argue that groups appear in Lagrange's work but since Lagrange never composed permutations it is rather with hindsight that we now see the beginnings of group theory in his paper. Ruffini is the first to introduce the notion of the order of an element, conjugacy, the cycle decomposition of elements of permutation groups and the notions of primitive and imprimitive. He proved some remarkable theorems (not of course with the modern terminology quoted below):-

The order of a permutation is the least common multiple of the lengths in the decomposition into disjoint cycles.

An element of $S_{5}$ of order 5 is a 5-cycle.

If $G < S_{5}$ has order divisible by 5 then $G$ has an element of order 5.

$S_{5}$ has no subgroups of index 3, 4 or 8.

It is remarkable work and, except for one gap, proves the theorem as Ruffini claimed. The proof is given in modern notation in [4]. However there was a strange lack of response to Ruffini's work from mathematicians. In 1801 Ruffini sent a copy of his book to Lagrange. He received no response and so he sent a second copy with a covering letter [4]:-

Because of the uncertainty that you may have received my book, I send you another copy. If I have erred in any proof, or if I have said something which I believed new, and which is in reality not new, finally if I have written a useless book, I pray you point it out to me sincerely.

Again Ruffini received no reply and he wrote yet again in 1802:-

No one has more right ... to receive the book which I take the liberty of sending you. ... In writing this book, I had principally in mind to give a proof of the impossibility of solving equations of degree higher than four.

Some mathematicians accepted Ruffini's proof although one would have to say that Pietro Paoli, the professor at Pisa, was influenced by patriotic motives when he wrote in 1799 [4]:-

I read with much pleasure your book ... and recommend greatly the most important theorem which excludes the possibility of solving equations of degree greater than four. I rejoice exceedingly with you and with our Italy, which has seen a theory born and perfected and to which other nations have contributed little...

To understand this quotation one has to realise that Lagrange was born in Turin which was part of Italy at the time. This patriotic reaction apart, the world of mathematics seemed to almost ignore Ruffini's great result. So how did Ruffini react? He published a second proof in 1803 which he hoped might be more easily understood, writing in the introduction:-

In the present memoir, I shall try to prove the same proposition [insolubility of the quintic] with, I hope, less abstruse reasoning and with complete rigour.

At least Ruffini received comments from Malfatti concerning this paper, but unfortunately Malfatti had not understood Ruffini's arguments and raised a fallacious objection. Ruffini published further proofs in 1808 and 1813. Of this last proof Ayoub writes in [4]:-

Can anything be more elegant? This proof is essentially what was later called the Wantzel modification of Abel's proof and was published in 1845. It is no surprise that it should resemble Ruffini's proof, since Wantzel says in his paper ..."using works of Abel and Ruffini...".

Ruffini did not stop trying to have his work recognised by the mathematical community. When Delambre wrote in a report on the state of mathematics since 1789:-

Ruffini proposes to prove that it is impossible ...,

Ruffini replied:-

... I not only proposed to prove but in reality did prove ... .

One has to feel desperately sorry for Ruffini. If some mathematician had written to him showing him there was an error or even a gap in the proof, then at least Ruffini would have had the chance to correct it. However, it seemed that nobody really wanted to know that quintics could not be solved by radicals. Ruffini asked the Institute in Paris to pronounce on the correctness of his proof and Lagrange, Legendre and Lacroix were appointed to examine it. Again they produced a report which was highly unsatisfactory as far as Ruffini was concerned:-

... if a thing is not of importance, no notice is taken of it and Lagrange himself, "with his coolness" found little in it worthy of attention.

The Royal Society were also asked to pronounce on the correctness and Ruffini received a somewhat kinder reply which said that although they did not give approval of particular pieces of work they were quite sure that it proved what was claimed. The one person who did acknowledge the importance and correctness was Cauchy. This is all the more surprising since Cauchy was one of the worst of all mathematicians at giving credit to others. He wrote to Ruffini in 1821, less than a year before Ruffini's death [1]:-

... your memoir on the general resolution of equations is a work which has always seemed to me worthy of the attention of mathematicians and which, in my judgement, proves completely the impossibility of solving algebraically equations of higher than the fourth degree.

In fact Cauchy had written a major work on permutation groups between 1813 and 1815 and in it he generalised some of Ruffini's results. He had certainly been greatly influenced by Ruffini's ideas. This influence through Cauchy is perhaps the only way in which Ruffini's work was to make an impact on the development of mathematics.

We left the story of Ruffini's career around 1799 when he began his publications on the quintic. He left the University of Modena to spend 7 years teaching applied mathematics in the military school in Modena. He continued to practise medicine and tend to patients from the poorest to the richest in society. After the fall of Napoleon, Ruffini became rector of the University of Modena in 1814. The political situation was still extremely complex and despite his personal skills, the great respect in which he was held, and his reputation for honesty, his time as rector must have been a very difficult one.

As well as the rectorship, Ruffini held a chair of applied mathematics, a chair of practical medicine and a chair of clinical medicine in the University of Modena. In 1817 there was a typhus epidemic and Ruffini continued to treat his patients until he caught the disease himself. Although he made a partial recovery, he never fully regained his health and in 1819 he gave up his chair of clinical medicine. He did not give up his scientific work, however, and in 1820 he published a scientific article on typhus based on his own experience with the disease.

There are further aspects of Ruffini's work which should be mentioned. He wrote several works on philosophy, one of which argues against some of Laplace's philosophical ideas. He also wrote on probability and the application of probability to evidence in court cases.

Given the information in this article about the insolubility of the quintic, it is reasonable to ask why Abel has been credited with proving the theorem while Ruffini has not. Ayoub suggests that [4]:-

... the mathematical community was not ready to accept so revolutionary an idea: that a polynomial could not be solved in radicals. Then, too, the method of permutations was too exotic and, it must be conceeded, Ruffini's early account is not easy to follow. ... between 1800 and 1820 say, the mood of the mathematical community ... changed from one attempting to solve the quintic to one proving its impossibility...