艾蒂安·贝祖Étienne Bézout

艾蒂安·贝祖的父亲是Pierre Bézout，他是内穆尔镇的一名地方法官。人们可能会以为贝祖会从事同样的职业，因为他的祖父也曾在内穆尔担任地方法官。贝祖的母亲是Hélène-Jeanne Filz。

正如我们已经指出的，家族传统几乎要求贝祖追随他父亲和祖父的脚步。然而，莱昂哈德·欧拉非凡的数学比父母的意愿更为强大，因为贝祖一旦读了莱昂哈德·欧拉的著作，就希望献身于数学。1756年，他发表了一篇论文Dynamique。次年，他发表了Quantités différentielles，1758年又发表了Rectification des courbes。后两篇论文是对积分的研究。

1758年，贝祖被任命为Académie des Sciences的力学副研究员，同年又被任命为皇家审查官。1763年，他被任命为海军卫队考试官，这一职位由舒瓦瑟尔公爵提供给他。他在这一角色中被赋予的一项重要任务，是编写一本专为向学生教授数学而设计的教科书。

贝祖 因这项任务所产生的教科书而闻名。第一部是 Cours de mathématiques à l'usage des Gardes du Pavillon et de la Marine Ⓣ（《供“帕维永”与海军警卫队使用的数学》），一部四卷本著作，于 1764-67 年出版。

1768年，夏尔·艾蒂安·路易·加缪的炮兵考官去世了。贝祖被任命接替他，成为炮兵部队的考官。他开始编写另一本数学教科书，结果写出了Cours complet de mathématiques à l'usage de la marine et de l'artillerieⓉ（供海军和炮兵使用的完整数学），这是一部六卷本著作，于1770年至1782年间出版。这是一本非常成功的教科书，多年来一直是希望进入巴黎综合理工学院的学生所学习的书。Grabiner在[1]中写道：-

教授非数学家的经验塑造了这些著作的风格：贝祖先讲几何再讲代数，他观察到初学者对数学推理还不够熟悉，无法理解代数论证的力量，尽管他们确实能欣赏几何中的证明。他避开了令人恐惧的术语“公理”、“定理”、“附注”，并试图避免过于紧密和详细的论证。

考虑到这种方法针对的是贝祖为其编写教材的读者群，他的书因缺乏严谨性而受到了一定程度的批评，这是可以预料的。然而，尽管如此，这些书对于那些需要使用数学的人来说是可以理解的，因此非常受欢迎并被广泛使用。它们的使用范围超出了法国，因为它们被翻译成英文并在北美使用。特别是哈佛大学将它们采用为微积分教科书。

回到关于 贝祖 职业生涯的更多信息，我们应当注意到，他于 1768 年在 Académie des Sciences 被提升为力学部的通讯院士，随后于 1770 年进一步晋升为领薪院士。

正如我们已经指出的，贝祖 以教科书作者而闻名，但他也以其在代数方面的工作而闻名，特别是在方程方面。1763 年任职之后，他忙于教学职责，并且确实非常认真地对待这些职责。因此，他能用于研究的时间相对较少，并且他有意识地决定限制自己工作的范围，以便能在狭窄的领域内做出有价值的成果。

贝祖 进行研究的方式很有意思，因为即使在今天，这仍是一种获得成果的好方法。他研究相当一般的问题，但由于通常无法用当时可用的数学知识来解决，他便研究一般问题中他能够求解的特殊情形。这种方法往往缓慢地导致对一般情形越来越多的理解，而一般情形最终可能变得可解。贝祖 对这种方法有一个名称，即“简化假设法”。

他关于方程理论的第一篇论文Sur plusieurs classes d'équations de tous les degrés qui admettent une solution algébriqueⓉ（论几类可代数求解的任意次方程）考察了如何通过将一个单未知数的方程写成两个双未知数的方程来攻克它。他在这篇论文中写道：-

众所周知，一个确定方程总是可以被视为两个双未知数方程的结果，当其中一个未知数被消去时。

当然，从表面上看，这无助于解方程，但贝祖做了一个简化假设，即两个方程之一是特别简单的形式。例如，他考虑了其中一个方程只有两项的情况，即$n$次项和一个常数项。这篇论文已经引入了贝祖将做出最重要贡献的主题，即从一组联立方程中通过消元法产生一个关于其中一个未知数的单一结果方程的方法。

他还在使用行列式解方程方面做了重要工作。这出现在他于1764年发表的一篇论文Sur le degré des équations résultantes de l'évanouissement des inconnues Ⓣ（消去未知量后所得方程的次数）中。由于这篇论文中关于解联立方程组的思想，詹姆斯·约瑟夫·西尔维斯特在1853年将方程系数矩阵的行列式称为Bézoutiant。

贝祖在方程理论方面发表的这些以及更多论文被收集在Théorie générale des équations algébraiquesⓉ（代数方程的一般理论）中，该书于1779年出版。这部著作包含一个被称为贝祖定理的结果：-

由任意数量的、未知数个数与之相同的完整方程，且次数任意，所得最终方程的次数等于各方程次数的乘积。

贝祖所说的完整方程，是指由一个多项式定义的方程，该多项式包含未知数的所有可能的乘积项，且这些项的次数不超过该多项式的次数。人们必须理解贝祖所面临的问题，因为他没有我们这种简单的下标记号用$x_{1}, x_{2}, x_{3}, ...$来表示未知数，甚至也不能用下标记号来标记他的方程。尽管如此，贝祖，这位准备进行冗长而困难的代数运算的人，仅仅对一个归纳论证稍加含糊其辞，就证明了他的定理。

在这部著作中，贝祖还首次给出了科林·麦克劳林关于两条代数曲线相交结果的一个令人满意的证明。

Grabiner [1]告诉我们：-

[贝祖]结婚早且幸福；尽管他在社交场合拘谨且有些忧郁，但认识他的人都谈到他极其善良和热心。到[1763年]，贝祖已经成为了父亲……

他于1783年去世后，在他的出生地Nemours镇竖立了一座雕像，以纪念他的伟大成就。

Étienne Bézout's father was Pierre Bézout who was a magistrate in the town of Nemours. One might have expected Étienne to follow the same career, for his grandfather had also been a magistrate in Nemours. Étienne's mother was Hélène-Jeanne Filz.

As we have already indicated the family tradition almost demanded that Étienne follow in his father and grandfather's footsteps. However the remarkable mathematics of Leonard Euler proved stronger than his parents wishes, for once Bézout had read Euler's works he wished to devote himself to mathematics. In 1756 he published a memoir Dynamique. In the following year he published Quantités différentielles and in 1758 Rectification des courbes. These latter two papers were investigations of integration.

In 1758 Bézout was appointed an adjoint in mechanics of the Académie des Sciences and, in the same year, as royal censor. He was appointed examiner of the Gardes de la Marine in 1763, the post being offered to him by the Duke of Choiseul. One important task that he was given in this role was to compose a textbook specially designed for teaching mathematics to the students.

Bézout is famed for the texbooks which came out of this assignment. The first was Cours de mathématiques à l'usage des Gardes du Pavillon et de la Marine Ⓣ, a four volume work which appeared in 1764-67.

In 1768 Camus, who was the examiner for the artillery, died. Bézout was appointed to succeed him becoming examiner of the Corps d'Artillerie. He began work on another mathematics textbook and as a result he produced Cours complet de mathématiques à l'usage de la marine et de l'artillerie Ⓣ, a six volume work which appeared between 1770 and 1782. This was a very successful textbook and for many years it was the book which students hoping to enter the École Polytechnique studied. Grabiner writes in [1]:-

The experience of teaching non-mathematicians shaped the style of the works: Bézout treated geometry before algebra, observing that beginners were not yet familiar enough with mathematical reasoning to understand the force of algebraic demonstrations, although they did appreciate proofs in geometry. He eschewed the frightening terms "axiom", "theorem", "scholium", and tried to avoid arguments that were too close and detailed.

As might be expected given this approach aimong at the readership for whom Bézout intended his texts, his books came in for a certain amount of criticism for lacking rigour. However, despite this they were books which could be understood by those who needed to use mathematics and as a result were very popular and widely used. Their use spread beyond France for they were translated into English and used in North America. In particular Harvard University adopted them as calculus textbooks.

Returning to give more information about Bézout's career, we should note that he was promoted to associé in mechanics at the Académie des Sciences in 1768 and then further promoted to pensionnaire in 1770.

As we have indicated Bézout is famed for being a writer of textbooks but he is famed also for his work on algebra, in particular on equations. He was much occupied with his teaching duties after his 1763 appointments and he took these very seriously indeed. As a consequence he could devote relatively little time to research and he made a conscience decision to restrict the range of his work so that he could produce worthwhile results in a narrow order.

The way Bézout went about his research is interesting since still today it is a good approach for obtaining results. He attacked quite general problems, but since an attack was usually beyond what could be achieved with the mathematical knowledge then available, he attacked special cases of the general problems which he could solve. This approach often leads slowly to more and more understanding of the general case which may eventually become soluble. Bézout had a name for this approach to mathematics, namely the "method of simplifying assumptions".

His first paper on the theory of equations Sur plusieurs classes d'équations de tous les degrés qui admettent une solution algébrique Ⓣ examined how a single equation in a single unknown could be attacked by writing it as two equations in two unknowns. He wrote in this paper:-

It is known that a determinate equation can always be viewed as the result of two equations in two unknowns, when one of the unknowns is eliminated.

Of course on the face of it this does not help solve the equation but Bézout made the simplifying assumption that one of the two equations was of a particularly simple form. For example he considered the case when one of the two equations had only two terms, the term of degree $n$ and a constant term. Already this paper had introduced the topic to which Bézout would make his most important contributions, namely methods of elimination to produce from a set of simultaneous equations, a single resultant equation in one of the unknowns.

He also did important work on the use of determinants in solving equations. This appears in a paper Sur le degré des équations résultantes de l'évanouissement des inconnues Ⓣ which he published in 1764. As a result of the ideas in this paper for solving systems of simultaneous equations, Sylvester, in 1853, called the determinant of the matrix of coefficients of the equations the Bézoutiant.

These and further papers published by Bézout in the theory of equations were gathered together in Théorie générale des équations algébraiques Ⓣ which was published in 1779. This work includes a result known as Bézout's theorem:-

The degree of the final equation resulting from any number of complete equations in the same number of unknowns, and of any degrees, is equal to the product of the degrees of the equations.

By a complete equation Bézout meant one defined by a polynomial which contains terms of all possible products of the unknowns whose degree does not exceed that of the polynomial. One has to understand the problems that faced Bézout for he did not have our simple suffix notation to denote the unknowns by $x_{1}, x_{2}, x_{3}, ...$ nor could he even label his equations with a suffix notation. Despite this Bézout, who was prepared to enter long and difficult algebraic manipulations, proved his theorem with just a little hand waving over an inductive argument.

In this work Bézout also gave the first satisfactory proof of a result of Maclaurin on the intersection of two algebraic curves.

Grabiner [1] tells us that:-

[Bézout] married early and happily; although he was reserved and somewhat sombre in society, those who knew him spoke of his great kindness and warm heart. By [1763] Bézout had become a father ...

After his death in 1783 a statue was erected in Nemours, the town of his birth, to commemorate his great achievements.