让-罗贝尔·阿尔冈Jean-Robert Argand

这本传记是关于让-罗贝尔·阿尔冈的，这个名字对于几乎所有通过复数的‘阿尔冈图’学习数学的人来说都是众所周知的。让我们在这本传记的开头就说明，名字“阿尔冈 Robert”以及上面给出的他的出生和死亡日期不太可能是正确的。它们指的是一个真实的人，但这个人不太可能是‘阿尔冈图’的作者。以下关于阿尔冈 Robert Argand的信息，可能错误地成为了发明‘阿尔冈图’的人传记的标准部分。

阿尔冈是巴黎的一名会计师和簿记员，只是一位业余数学家。关于他的背景和教育知之甚少。我们确实知道他的父亲是Jacques Argand，母亲是Eves Canac。除了他的出生日期，他受洗的日期也是已知的——1768年7月22日。关于他生平的少数其他已知事实中，有一点关于他孩子的信息。他的儿子出生在巴黎并继续住在那里，而他的女儿Jeanne-Françoise-Dorothée- Marie-Elizabeth 阿尔冈嫁给了Félix Bousquet，他们住在斯图加特。

如果这些信息不太可能是真的，那么此时了解它来自何处或许会有所帮助。Jules Hoüel出版了一部四卷本著作，题为Théorie Élémentaire des Quantités Complexes Ⓣ（复量的初等理论）。在Hoüel于1874年出版第4卷之前，他决定尝试寻找关于阿尔冈的生平信息。他知道Ami 阿尔冈（1750-1803）曾发明仪器并在巴黎住过一段时间，出生于日内瓦。这一定让Hoüel猜测阿尔冈图的发明者可能出生于日内瓦，于是他询问他在日内瓦的同事，能否找到阿尔冈的生平细节。我们上面呈现的关于阿尔冈的细节是Hoüel请求的结果，尽管提供信息的人曾表示怀疑他们找到的是正确的阿尔冈。尽管有这些怀疑，这些信息一直被视为确定的，直到20世纪90年代末，Gert Schubring的研究导致他声称[7]：-

……这些为数不多的已知数据似乎值得怀疑。

Schubring的论证主要基于这样一个事实：基本上没有证据表明阿尔冈的标准传记可能是正确的。他还有几条论据表明这个“标准传记”是错误的。其中之一是，阿德里安-马里·勒让德似乎见过阿尔冈，却将他描述为一个“年轻人”。如果阿尔冈是阿尔冈 Robert Argand，那么他见到阿德里安-马里·勒让德时将是38岁，不太可能配得上这种描述。另一件表明阿尔冈不是阿尔冈 Robert Argand的事情是，阿尔冈 Robert Argand是一名会计和簿记员，而从他的著作来看，阿尔冈表明他可能是钟表行业的专家技师。

阿尔冈因对复数给出几何解释而闻名，其中$i$被解释为旋转90°。复数的模的概念也归功于阿尔冈，但后来使用该术语的奥古斯丁·路易·柯西通常被认为是这一概念的创始者。阿尔冈图被教授给大多数学习数学的学生，阿尔冈的名字将通过这一重要概念在数学史上流传下去。然而，他的名字与复数的这种几何解释相关联，仅仅是一系列相当奇怪事件的结果。

第一个发表这种复数几何解释的是卡斯帕尔·韦塞尔。这一思想出现在卡斯帕尔·韦塞尔1787年的著作中，但直到卡斯帕尔·韦塞尔于1797年3月10日向丹麦皇家科学院的一次会议提交论文后才发表。该论文于1799年发表，但未引起数学界的注意。卡斯帕尔·韦塞尔的论文在1895年被重新发现，当时Christian Juel提请注意它，同年，索菲斯·李重新发表了卡斯帕尔·韦塞尔的论文。

这并不像乍一看那么令人惊讶，因为卡斯帕尔·韦塞尔是一名测量员。然而，阿尔冈也不是职业数学家，所以当他在1806年给出复数的几何解释时，是在一部他可能自费私下出版的回忆录中，但实际上没有证据表明它曾出版。唯一确定的是阿尔冈自己的陈述，他在1806年至1813年间的某个时候私下分发了极少数副本。无论是否出版都无关紧要，因为没有其出版的证据留存下来，人们会预期它比卡斯帕尔·韦塞尔的著作更不引人注意，毕竟后者是由丹麦皇家科学院出版的。也许更令人惊讶的是，阿尔冈的名字甚至没有出现在回忆录上，因此无法确定作者。

阿尔冈的著作为人所知的方式相当复杂。阿德里安-马里·勒让德从阿尔冈那里收到了一份著作Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques Ⓣ（论在几何构造中表示虚量的一种方式），并于1806年11月2日将其寄给François Français，尽管两人都不知道作者的身份。阿德里安-马里·勒让德在这封信中写道：-

有些人非常成功地从事科学研究，却不为人知，也不追求名声。最近我见到一个年轻人，他请我阅读他写的一部关于虚数的著作；他没有很好向我解释他的目的，但他让我明白，他认为所谓的虚量与其他量一样真实，并用线来表示它们。起初我向作者表示我非常怀疑，但我答应阅读他的论文。与我的预期相反，我发现了相当原创的思想，表述得非常好，有相当深厚的计算知识作为支撑，最终导向非常精确的结果，例如三角学的大部分公式、罗杰·柯特斯定理等。这里是这部著作的一个概要，你可能会感兴趣，它将使你能够判断其余部分。……我在这里只给出他思想的一小部分，但你会补足它，也许你会像我一样发现，它们足够原创，值得关注。至于其余部分，我只是把它留给你作为好奇的对象，我不会为自己辩护。

François Français于1810年去世后，他的兄弟Jacques Français整理他的论文，在其中发现了阿尔冈的小论文。1813年9月，Jacques Français发表了论文Nouveaux principes de Géométrie de position, et interprétation des symboles imaginaires Ⓣ（位置几何学的新原理，以及虚符号的解释），其中他基于阿尔冈的思想给出了复数的几何表示，并附有有趣的应用。Jacques Français本可以轻易地将这些思想据为己有，但他做的恰恰相反。他在论文结尾说，这个思想基于一位不知名数学家的著作，并请求这位数学家表明身份，以便他能够因其思想而获得荣誉：-

我必须……出于公正地声明，这些新思想的实质并不属于我。我是在阿德里安-马里·勒让德写给我已故兄弟François Joseph Français（1768-1810）的一封信中发现它们的，在这封信中，这位伟大的数学家与他分享了（作为已经传达给他的东西，以及作为纯粹好奇的对象）我的第2和第3个定义的实质、我的第1个定理，以及我的第2个定理的第3个推论[...]。我希望我对我所达到的结果的公开能够导致这些思想的第一作者为人所知，并揭示他自己在这个主题上所做的工作。

Jacques Français的文章出现在约瑟夫·热尔岗的期刊Annales de mathématiques上，阿尔冈回应了Jacques Français的请求，承认他是作者，并向Annales de mathématiques提交了他原始作品 Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques Ⓣ（《论在几何构造中表示虚量的一种方式》）的一个稍作修改的版本，附有一些新的应用。没有什么比一场争论更能引起世人对某事的注意，而这正是接下来发生的事情。Jacques Français、阿尔冈和弗朗索瓦-约瑟夫·塞尔瓦之间在约瑟夫·热尔岗的期刊的页面上进行了一场激烈的讨论。在这组通信中，Jacques Français和阿尔冈为几何表示的有效性辩护，而弗朗索瓦-约瑟夫·塞尔瓦则主张复数必须用纯代数来处理。

人们可能会以为阿尔冈对数学没有其他贡献。然而并非如此，尽管他将永远因阿尔冈图而被铭记，但他最好的工作是关于代数基本定理的，而为此他几乎没有得到什么荣誉。他在1806年的工作中给出了代数基本定理的一个漂亮证明（带有小漏洞），并在1813年于约瑟夫·热尔岗的期刊上发表其结果时再次给出。当然，阿尔冈是第一个在系数为复数的情况下陈述该定理的人。Petrova在[6]中讨论了该基本定理的早期证明，并指出阿尔冈给出了一个几乎现代形式的证明，而该证明在1813年第二次发表后被遗忘了。

1813年之后，阿尔冈确实在数学界获得了更高的知名度。他在1813年至1816年间又发表了八篇文章，全部发表在约瑟夫·热尔岗的期刊上。其中大多数要么基于他最初的论文，要么是对其他数学家发表的论文的评论。他的最后一篇出版物是关于组合的，其中他使用记号$(m, n)$表示从$m$个对象中选取$n$个对象的组合。

在[1]中，Jones将阿尔冈的工作总结如下：-

阿尔冈是一个背景不明、职业与数学无关、与他那个时代的文献接触不确定的人，他凭直觉发展了一个时机已成熟的关键思想。他自己利用了它。他工作的质量和意义得到了他那个时代一些天才的认可，但沟通的中断以及其他工作者类似发展的近似同时性，迫使历史学家无法将建立在他所耕耘的概念之上的成果完全归功于他。

在[7]中，Gert Schubring试图重建阿尔冈试图让阿德里安-马里·勒让德对他的几何解释产生兴趣的尝试：-

1806年秋天，阿德里安-马里·勒让德被阿尔冈找上门来，后者试图在直接交谈中向他概述自己手稿中的主要结果。阿德里安-马里·勒让德对方法及其应用表示怀疑。离开时，阿尔冈敦促阿德里安-马里·勒让德阅读他的手稿。阿德里安-马里·勒让德没有记住这个人的名字，并以为手稿会显示其作者的名字。当阿尔冈离开后，阿德里安-马里·勒让德意识到论文既没有标明作者的地址也没有标明作者的名字。在阅读《Éssai》时，阿德里安-马里·勒让德注意到了它的质量，他等待其作者的再次来访，但作者没有再出现。为了结束自己与这些概念的牵连，他在1806年11月2日的信中写给了François Français报告。由于阿德里安-马里·勒让德坚决要求不要用关于这篇论文的讨论来打扰他，无论是年长的还是后来年幼的Français都不敢向他询问这篇论文及其作者。另一方面，阿尔冈——显然是一个害羞的人——由于阿德里安-马里·勒让德不感兴趣和怀疑的反应，没有发表他的论文。只有通过Français兄弟对他思想的相当间接的接受，才促使阿尔冈组织了一次后来的印刷，他安排将其写作日期放在标题页上。

阿尔冈 1806年一定在巴黎，当时他遇到了 阿德里安-马里·勒让德，而且他1813年肯定在巴黎，因为他在那年发表的论文上给出了一个巴黎的地址。

我们必须为这部必然相当不令人满意的阿尔冈传记加上最后一条注记。他的信件和已发表著作都以阿尔冈之名出现，没有其他名字。这在我们看来更像是一个笔名，而非作者的真实姓名。当然，如果这是真的，那就意味着将来任何辨认阿尔冈的尝试都会变得更加困难（很可能不可能）。

This biography is about Argand, the man whose name is well-known to essentially everyone who has studied mathematics through the 'Argand diagram' for complex numbers. Let us state right at the beginning of this biography that the first names "Jean Robert" and the dates of his birth and death as given above are unlikely to be correct. They refer to a real person, but it is unlikely that this person is the author of the 'Argand diagram'. The following information about Jean Robert Argand has, probably incorrectly, become a standard part of the biography of the man who invented the 'Argand diagram'.

Jean-Robert Argand was an accountant and bookkeeper in Paris who was only an amateur mathematician. Little is known of his background and education. We do know that his father was Jacques Argand and his mother Eves Canac. In addition to his date of birth, the date on which he was baptized is known - 22 July 1768. Among the few other facts known of his life is a little information about his children. His son was born in Paris and continued to live there, while his daughter, Jeanne-Françoise-Dorothée- Marie-Elizabeth Argand, married Félix Bousquet and they lived in Stuttgart.

If this information is unlikely to be true, perhaps it would be useful at this point to understand where it comes from. Jules Hoüel published a four volume work entitled Théorie Élémentaire des Quantités Complexes Ⓣ. Before Hoüel published Volume 4 in 1874 he decided to try to find biographical information about Argand. He knew that Ami Argand (1750-1803), who had invented instruments and lived in Paris for a while, had been born in Geneva. This must have made Hoüel guess that the inventor of the Argand diagram might have been born in Geneva so he asked his colleagues in Geneva if they could find biographical details of Argand. The details about Jean-Robert Argand we have presented above are the result of Hoüel's request although those giving the information had expressed doubts that they had found the correct Argand. Despite the doubts, this information has been taken as definite until the late 1990s when Gert Schubring's research resulted in his claim that [7]:-

... these few known data seem to be doubtful.

Schubring's argument is based mainly on the fact that there is essentially no evidence to suggest that the standard biography of Argand might be correct. He also has a few arguments which suggest that this 'standard biography' is wrong. One is that Legendre, who appears to have met Argand, describes him as a 'young man'. If Argand was Jean Robert Argand he would be 38 years old when he met Legendre and unlikely to merit this description. Another thing which suggests that Argand is not Jean Robert Argand is that Jean Robert Argand is an accountant and bookkeeper while, from his writings, Argand shows he is probably an expert technician in the clock industry.

Argand is famed for his geometrical interpretation of the complex numbers where $i$ is interpreted as a rotation through 90°. The concept of the modulus of a complex number is also due to Argand but Cauchy, who used the term later, is usually credited as the originator this concept. The Argand diagram is taught to most school children who are studying mathematics and Argand's name will live on in the history of mathematics through this important concept. However, the fact that his name is associated with this geometrical interpretation of complex numbers is only as a result of a rather strange sequence of events.

The first to publish this geometrical interpretation of complex numbers was Caspar Wessel. The idea appears in Wessel's work in 1787 but it was not published until Wessel submitted a paper to a meeting of the Royal Danish Academy of Sciences on 10 March 1797. The paper was published in 1799 but not noticed by the mathematical community. Wessel's paper was rediscovered in 1895 when Christian Juel draw attention to it and, in the same year, Sophus Lie republished Wessel's paper.

This is not as surprising as it might seem at first glance since Wessel was a surveyor. However, Argand was not a professional mathematician either, so when he produced his geometrical interpretation of complex numbers in 1806 it was in a memoir which he may have published privately at his own expense but in fact there is no proof that it was published. All that is certain is Argand's own statement that he privately distributed a very small number of copies some time between 1806 and 1813. Whether it was published or not does not matter for, as no evidence survives of its publication, one would have expected it to be less noticeable than Wessel's work which after all was published by the Royal Danish Academy. Perhaps even more surprisingly, Argand's name did not even appear on the memoir so it was impossible to identify the author.

The way that Argand's work became known is rather complicated. Legendre received a copy of the work, Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques Ⓣ from Argand and he sent it to François Français on 2 November 1806 although neither knew the identity of the author. Legendre wrote in this letter:-

There are people who cultivate science with great success without being known and without looking for fame. Recently I saw a young man who asked me to read a work he had done on imaginary numbers; he did not explain his object to me very well, but he made me understand that he regarded the so-called imaginary quantities as real as the others, and represented them by lines. At first I showed the author I was very doubtful, but I promised to read his memoir. I found contrary to my expectation, quite original ideas, very well presented, supported by a rather deep knowledge of calculation, and finally that lead to very exact consequences such as most formulas of trigonometry, the theorem of Cotes, etc. Here is a sketch of this work that you may be interested in and that will allow you to judge the rest. ... I only give here a small part of his ideas, but you will make up for it, and perhaps you will find, like me, that they are original enough to deserve attention. For the rest I leave you simply as an object of curiosity and I will not defend myself.

After François Français's death in 1810 his brother Jacques Français worked on his papers and he discovered Argand's little memoir among them. In September 1813 Jacques Français published the paper Nouveaux principes de Géométrie de position, et interprétation des symboles imaginaires Ⓣ in which he gave a geometric representation of complex numbers, with interesting applications, based on Argand's ideas. Jacques Français might easily have claimed these ideas for himself, but he did quite the reverse. He ended his paper by saying that the idea was based on the work of an unknown mathematician and he asked that the mathematician should make himself known so that he might receive the credit for his ideas:-

I must ... out of justice declare that the substance of these new ideas does not belong to me. I found them in a letter from M Legendre to my late brother François Joseph Français, 1768-1810, in which this great mathematician shares with him (as a thing that has been communicated to him, and as an object of pure curiosity) the substance of my 2nd and 3rd definitions, of my 1st theorem, and the 3rd corollary of my 2nd theorem [...]. I hope that the publicity that I give to the results that I have reached can lead to the first author of these ideas being known, and to bring to light the work he has done himself on this subject.

The article by Jacques Français appeared in Gergonne's journal Annales de mathématiques and Argand responded to Jacques Français's request by acknowledging that he was the author and submitting a slightly modified version of his original work Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques Ⓣ, with some new applications, to the Annales de mathématiques. There is nothing like an argument to bring something to the attention of the world and this is exactly what happened next. A vigorous discussion between Jacques Français, Argand and Servois took place in the pages of Gergonne's Journal. In this correspondence Jacques Français and Argand argued in favour of the validity of the geometric representation, while Servois argued that complex numbers must be handled using pure algebra.

One might have expected that Argand would have made no other contributions to mathematics. However this is not so and, although he will always be remembered for the Argand diagram, his best work is on the fundamental theorem of algebra and for this he has received little credit. He gave a beautiful proof (with small gaps) of the fundamental theorem of algebra in his work of 1806, and again when he published his results in Gergonne's Journal in 1813. Certainly Argand was the first to state the theorem in the case where the coefficients were complex numbers. Petrova, in [6], discusses the early proofs of the fundamental theorem and remarks that Argand gave an almost modern form of the proof which was forgotten after its second publication in 1813.

After 1813 Argand did achieve a higher profile in the mathematical world. He published eight further articles, all in Gergonne's Journal, between 1813 and 1816. Most of these are based on either his original memoir, or they comment on papers published by other mathematicians. His final publication was on combinations where he used the notation $(m, n)$ for the combinations of $n$ objects selected from $m$ objects.

In [1] Jones sums up Argand's work as follows:-

Argand was a man with an unknown background, a nonmathematical occupation, and an uncertain contact with the literature of his time who intuitively developed a critical idea for which the time was right. He exploited it himself. The quality and significance of his work were recognised by some of the geniuses of his time, but breakdowns in communication and the approximate simultaneity of similar developments by other workers force a historian to deny him full credit for the fruits of the concept on which he laboured.

In [7] Gert Schubring attempts to give a reconstruction of Argand's attempts to interest Legendre in his geometrical interpretation:-

In the autumn of 1806, Legendre was approached by Argand, who tried to outline the main results in his manuscript to him in direct conversation. Legendre responded with scepticism as to the method and its applications. Upon leaving, Argand urged Legendre to read his manuscript. Legendre had not retained the name of this man and assumed that the manuscript would show the name of its author. When Argand had left, Legendre realised that the paper indicated neither the address nor the name of the author. Upon reading the 'Éssai', Legendre noticed its quality, he waited for a further visit from its author, but the author did not appear again. In order to end his own involvement with these conceptions he wrote the report to François Français in the letter of 2 November 1806. Since Legendre firmly asked not to be bothered with discussions on this paper, neither the elder nor later the younger Français dared to ask him about the paper and its author. On the other hand, Argand - apparently a shy man - abstained from publishing his paper, due to Legendre's uninterested and sceptical reaction. Only the quite indirect reception of his ideas via the brothers Français induced Argand to organise a later printing where he arranged for the date of its composition to be put on the title page.

Argand must have been in Paris in 1806 when he met Legendre and he was certainly in Paris in 1813 for he gives a Paris address on his paper published in that year.

We must add one last note to this, necessarily rather unsatisfactory, biography of Argand. His letters and published work all appear under the name Argand with no other names. This would look to us more like a non-de-plume that the actual name of the author. Of course, if this is true, it would mean that any attempt to identify Argand in future would be made even more difficult (probably impossible).