亚历山大-泰奥菲勒·范德蒙德Alexandre-Théophile Vandermonde

亚历山大-泰奥菲勒·范德蒙德的父亲是一名医生，原籍朗德里塞，但在东方度过了12年。他在巴黎开设了一家诊所，并在那里行医，他的儿子范德蒙德出生时他正在那里工作。他并不鼓励儿子从事医学职业，而是鼓励他投身音乐事业。当然，他年轻时对数学并不感兴趣。范德蒙德于1755年9月7日获得学士学位，1757年9月7日获得硕士学位。

他最初的热爱是音乐，他的乐器是小提琴。他追求音乐事业，直到35岁才转向数学。正是Fontaine des Bertins对数学的热情感染了范德蒙德。或许令人惊讶的是，他在1771年当选为Académie des Sciences院士，当时除了他的第一篇论文外，几乎没有证据表明他的数学天赋，尽管当时他还不是院士，但该论文于1770年11月在科学院宣读。然而，他在这篇论文以及1771年至1772年间提交给科学院的另外三篇论文中，确实对数学做出了相当显著的贡献。这四篇论文代表了他全部的数学成果，我们将在下面讨论其内容，以及一些数学史家对其贡献的看法。

范德蒙德当选为Académie des Sciences院士确实激励了他为科学院努力工作，并发表其他关于科学和音乐的著作。1777年，他发表了与艾蒂安·贝祖和化学家拉瓦锡一起进行的低温实验结果，特别是研究了1776年发生的一次非常严重的霜冻的影响。十年后，他发表了两篇关于钢铁制造的论文，这次是与加斯帕尔·蒙日和贝托莱合作。这项研究的目的是通过试验不同的铁和碳混合物来改进用于刺刀的钢材。他与加斯帕尔·蒙日的密切合作反映了一个事实：两人是非常亲密的朋友；事实上，如此亲密，以至于他被称为femme de Monge。

1778年，范德蒙德向Académie des Sciences提交了一部关于音乐理论的两部分著作的第一部分。第二部分于两年后提交。这部著作Système d'harmonie applicable à l'état actuel de la musique Ⓣ（适用于当前音乐状况的和声体系）并没有像人们可能期望的那样，由一位在两个领域都是专家的人提出一种数学的音乐理论。相反，这部著作的目的是提出这样一种观点：音乐家在评判音乐时应该忽略所有音乐理论，只依靠训练有素的耳朵。正如人们可能预料的那样，这被证明是一部有争议的著作，音乐家们对是否同意范德蒙德的观点产生了尖锐的分歧。尽管起初许多音乐家反对，但范德蒙德提出的观点多年来逐渐获得支持，到19世纪初，Académie des Sciences已将音乐从数学领域移至艺术领域。值得重复的是，一位最高级别的数学家竟然反对音乐作为一门数学艺术，而这一地位自古希腊时代以来一直保持，这很奇怪。

范德蒙德担任过的职位包括1782年国立工艺学院院长和1792年军队服装局局长。在1792年同年，他与约瑟夫·拉格朗日一起在Académie des Sciences的一个委员会中任职，该委员会必须检查新发明的乐器——和声小提琴。他参与了1794年10月成立的巴黎高等师范学校，并参与了政治经济学课程的设计团队。他的朋友加斯帕尔·蒙日也参与了巴黎高等师范学校，约瑟夫·拉格朗日和皮埃尔·西蒙·拉普拉斯也是如此。然而，该机构于1795年1月在自然历史博物馆开学后仅运作了六个月就被关闭了。

像加斯帕尔·蒙日一样，范德蒙德是法国大革命的坚定支持者，这场革命始于1789年7月14日攻占巴士底狱。早在这一事件之前，法国的革命政治对范德蒙德来说就如此激动人心，以至于使他偏离了可能更长的数学和科学职业生涯。然而，事实是他一生都健康状况不佳，若非如此，他很可能能够高度参与政治，同时继续从事数学和科学活动。

也许范德蒙德的名字今天最为人所知的是因为范德蒙德行列式。虽然他对行列式理论做出了重大贡献，这无疑是事实，但在他的四篇数学论文中，这个行列式却从未出现。因此，这个行列式以他的名字命名就显得相当奇怪，一些作者对此困惑了一段时间。昂利·勒贝格在[3]中的猜想（首次发表于1940年）认为，这是由于某人误读了范德蒙德的记号，从而相信这个行列式出现在他的工作中，这似乎是最有可能的。

范德蒙德的四篇数学论文，连同它们在Académie des Sciences的发表日期，分别是Mémoire sur la résolution des équationsⓉ（关于解方程的回忆录）（1771年）、Remarques sur des problèmes de situationⓉ（关于定位问题的注记）（1771年）、Mémoire sur des irrationnelles de différents ordres avec une application au cercleⓉ（关于各种无理阶及其在圆上的应用）（1772年）和Mémoire sur l'éliminationⓉ（关于消元的回忆录）（1772年）。

这四篇论文中的第一篇给出了方程根的$m$次幂之和的公式。它还给出了此类根的幂的对称函数之和的公式。这两者都不是新的，不久前已出现在爱德华·华林的工作中，但尽管范德蒙德知道这一点，他仍声称——在我看来[EFR]这是正确的——他的方法足够不同，使得第二次发表这些结果是值得的。这篇论文还表明，如果$n$是小于10的素数，则方程$x^{n} - 1 = 0$可以用根式求解。Jones在[1]中写道：-

……范德蒙德真正且未被承认的成名之处在于他的第一篇论文，在其中他通过研究在方程根的置换下不变函数，来处理代数方程可解性的一般问题。

利奥波德·克罗内克在1888年声称，现代代数的研究始于范德蒙德的这篇第一篇论文。奥古斯丁·路易·柯西相当明确地指出，范德蒙德比约瑟夫·拉格朗日更早拥有这一最终导致群论研究的非凡思想。

在他的第二篇论文中，范德蒙德考虑了棋盘上的骑士巡游问题。这篇论文是拓扑学思想研究的早期例子。范德蒙德考虑了移动的骑士所生成的曲线的交织，他在这一领域的工作标志着一些思想的开始，这些思想首先由卡尔·弗里德里希·高斯、然后由詹姆斯·克拉克·麦克斯韦在电路背景下加以扩展。

在第三篇论文中，范德蒙德研究了组合思想。他定义了符号

$[p]^{n} = p(p - 1)(p - 2)(p - 3) ... (p - n + 1)$

和

$[p]^{-n} = \Large\frac{1}{(p+1)(p+2)(p+3)...(p+n)}\normalsize .$

他给出了$[x + y]^{n}$展开的一个恒等式，并证明了

$\pi \pi = [\pi]^{\pi}. [-\pi]^{-\pi}$

有趣的是，当时还没有表示$n!$的记号，而范德蒙德用他的记号定义了更一般的东西。显然

$[n]^{n} = n!$

范德蒙德四篇论文中的最后一篇研究行列式理论。托马斯·穆尔[4]声称，由于这篇论文，范德蒙德是：

唯一有资格被视为行列式理论奠基人的人。

托马斯·穆尔之所以如此断言，是因为尽管像哥特弗里德·威廉·莱布尼茨这样的数学家在范德蒙德之前就研究过行列式，但所有早期工作都只是把行列式当作解线性方程组的工具。然而，范德蒙德把行列式视为一个函数，并给出了行列式函数的性质。他展示了交换两行和交换两列的效果。由此他推断出，具有两行相同或两列相同的行列式为零。最后，他给出了一种极为巧妙的行列式记号，但这种记号未能流传下来。

Alexandre-Theophile Vandermonde's father was a medical doctor who was originally from Landrices but had spent 12 years in the Orient. He had set up a medical practice in Paris and was working there as a doctor when his son Alexandre-Théophile was born. He did not encourage his son to follow a medical profession but rather encouraged him to take up a career in music. Certainly he was not interested in mathematics when he was young. Alexandre-Théophile was awarded his bachelier on 7 September 1755 and his licencie on 7 September 1757.

His first love was music and his instrument was the violin. He pursued a music career and he only turned to mathematics when he was 35 years old. It was Fontaine des Bertins whose enthusiasm for mathematics rubbed off on Vandermonde. Perhaps surprisingly he was elected to the Académie des Sciences in 1771 with little evidence of his mathematical genius other than his first paper which, although he was not a member at the time, was read to the Academy in November 1770. However, he did make quite a remarkable contribution to mathematics in this paper and three further papers which he presented to the Academy between 1771 and 1772. These four papers represent his total mathematical output and we will discuss their content below together with the views of a number of historians of mathematics on his contribution.

Vandermonde's election to the Académie des Sciences did motivate him to work hard for the Academy and to publish other works on science and music. In 1777 he published the results of experiments he had carried out with Bézout and the chemist Lavoisier on low temperatures, in particular investigating the effects of a very severe frost which had occurred in 1776. Ten years later he published two papers on manufacturing steel, this time joint work with Monge and Bertholet. The aim of this research was to improve the steel used for bayonets but experimenting with different mixtures of iron and carbon. That he work closely with Monge reflected the fact that the two were very close friends; in fact, so close, that he was known as femme de Monge.

In 1778 Vandermonde presented the first of a two part work on the theory of music to the Académie des Sciences. The second part was presented two years later. This work Système d'harmonie applicable à l'état actuel de la musique Ⓣ did not propose a mathematical theory of music as one might have expected from someone who was an expert in both fields. On the contrary the aim of the work was to put forward the idea that musicians should ignore all theory of music and rely solely on their trained ears when judging music. As one might expect this proved a controversial work with musicians being sharply divided as to whether they agreed with Vandermonde or not. Despite the opposition of many musicians at first, the ideas put forward by Vandermonde gained favour over the years and by the beginning of the nineteenth century the Académie des Sciences had moved music from the mathematical area to the arts area. It is worth repeating that it is strange that a mathematician of the highest rank should have argued against music as a mathematical art, a position it had held since the days of ancient Greece.

Positions which Vandermonde held include director of the Conservatoire des Arts et Métiers in 1782 and chief of the Bureau de l'Habillement des Armées in 1792. In the same year of 1792 he sat with Lagrange on a committee of the Académie des Sciences which had to examine the violon harmonique, a newly invented musical instrument. He was involved with the École Normale, which was founded in October 1794, and was on the team designing a course in political economy. His friend Monge was also involved with the École Normale as were Lagrange and Laplace. However the establishment only operated for six months after it opened in the Muséum d'Histoire Naturelle in January 1795 before being closed down.

Like Monge, Vandermonde was a strong supporter of the Revolution which began with the storming of the Bastille on 14 July 1789. The politics of Revolution in France long before this event had been so exciting for Vandermonde that it diverted him from a possible longer mathematical and scientific career. However the truth of the matter is that he suffered from poor health all his life and, but for this, he might well have been able to be highly involved in politics yet continue with mathematical and scientific activities.

Perhaps the name of Vandermonde is best known today for the Vandermonde determinant. While it is certainly true that he made a major contribution to the theory of determinants, yet nowhere in his four mathematical papers does this determinant appear. It is rather strange, therefore, that this determinant should be named after him and several authors have puzzled over the fact for some time. Lebesgue's conjecture in [3] (first published in 1940) that it resulted for someone misreading Vandermonde's notation, and therefore believing that this determinant was in his work, seems the most likely.

Vandermonde's four mathematical papers, with their dates of publication by the Académie des Sciences, were Mémoire sur la résolution des équations Ⓣ (1771), Remarques sur des problèmes de situation Ⓣ (1771), Mémoire sur des irrationnelles de différents ordres avec une application au cercle Ⓣ (1772), and Mémoire sur l'élimination Ⓣ (1772).

The first of these four papers presented a formula for the sum of the $m$th powers of the roots of an equation. It also presented a formula for the sum of the symmetric functions of the powers of such roots. Neither of these were new having appeared in Waring's work shortly before but, although he was aware of this Vandermonde claimed, rightly in my [EFR] opinion, that his approach was sufficiently different to make publication of these results for a second time worthwhile. The paper also shows that if $n$ is a prime less than 10 the equation $x^{n} - 1 = 0$ can be solved in radicals. Jones writes in [1]:-

... Vandermonde's real and unrecognised claim to fame was lodged in his first paper, in which he approached the general problem of the solubility of algebraic equations through a study of functions invariant under permutations of the roots of the equation.

Kronecker claimed in 1888 that the study of modern algebra began with this first paper of Vandermonde. Cauchy states quite clearly that Vandermonde had priority over Lagrange for this remarkable idea which eventually led to the study of group theory.

In his second paper Vandermonde considered the problem of the knight's tour on the chess board. This paper is an early example of the study of topological ideas. Vandermonde considers the intertwining of the curves generated by the moving knight and his work in this area marks the beginning of ideas which would be extended first by Gauss and then by Maxwell in the context of electrical circuits.

In his third paper Vandermonde studied combinatorial ideas. He defined the symbol

$[p]^{n} = p(p - 1)(p - 2)(p - 3) ... (p - n + 1)$

and

$[p]^{-n} = \Large\frac{1}{(p+1)(p+2)(p+3)...(p+n)}\normalsize .$

He gave an identity for the expansion of $[x + y]^{n}$ and also proved that

$\pi \pi = [\pi]^{\pi}. [-\pi]^{-\pi}$

It is interesting to note that at this time no notation existed for $n!$ yet with his notation Vandermonde had defined something more general. Clearly

$[n]^{n} = n!$

The final of Vandermonde's four papers studied the theory of determinants. Muir [4] claims that because of this paper Vandermonde was:-

The only one fit to be viewed as the founder of the theory of determinants.

The reason for this strong claim by Muir is that, although mathematicians such as Leibniz had studied determinants earlier than Vandermonde, all earlier work had simply used the determinant as a tool to solve linear equations. Vandermonde, however, thought of the determinant as a function and gave properties of the determinant function. He showed the effect of interchanging two rows and of interchanging two columns. From this he deduced that a determinant with two identical rows or two identical columns is zero. Finally he gave a remarkably clever notation for determinants which has not survived.