米歇尔·罗尔Michel Rolle

米歇尔·罗尔的父亲是一个店主。罗尔几乎没有受过正规教育，在接受了一些初等教育后主要是自学。他先是为一名公证人做抄写员，然后在他家乡Ambert周边地区担任几位律师的助手。1675年，可能是为了寻求更好的生活，他去了巴黎，在那里担任抄写员和算术专家。然而，到达巴黎后不久他就结婚了，孩子也很快出生。他的收入不足以养活不断增长的家庭，但他一直在自学高等数学，正是他在这门学科中培养的技能提供了突破。

1682年，他通过解决雅克·奥扎南公开提出的一个问题而获得了一定的名声。法国国王路易十四的财政总监兼海军国务秘书Jean-Baptiste Colbert因这一成就奖励了罗尔。Colbert为罗尔安排了一笔养老金，这使他走上了财务安全的道路，但还有其他同样重要的后果[6]：-

与罗尔的奖金所带来的经济利益同等重要的是，这件事让卢瓦侯爵注意到了他，而卢瓦侯爵恰好正在寻找一个人来教他的一个儿子数学。罗尔被卢瓦侯爵雇用，后者对他的教学和数学才能印象极为深刻。从1783年起，卢瓦侯爵有能力给予他的门徒更多的科学赞助，并在1685年让罗尔成为科学院院士时这样做了。

罗尔解决的问题是由雅克·奥扎南于1682年8月31日在Journal des sçavans上提出的；问题如下：——

求四个数，其中任意两个的差是一个完全平方数，此外前三个数之和也是一个完全平方数。

雅克·奥扎南指出，具有这些性质的四个数中最小的一个至少有50位数字，但罗尔找到了满足条件的四个数，每个数只有七位数字。他通过在 Journal des sçavans上发表他的解答而使其为人所知。上面引文中提到的卢瓦侯爵是弗朗索瓦·罗尔·勒泰利耶，卢瓦侯爵，法国战争国务秘书。他雇用罗尔来辅导他的第四个儿子卡米耶·勒泰利耶（1675-1718）。卢瓦侯爵安排罗尔在战争部担任行政职务，但罗尔不喜欢这项工作，很快就辞职了。罗尔于1685年当选为Académie Royale des Sciences院士，他在当选后所做出的令人印象深刻的数学工作充分证明了卢瓦侯爵对他的信任是正确的。

在继续讨论罗尔所做的有趣数学贡献之前，让我们再提供一些关于他生平的事实。他于1699年成为Académie des Sciences的几何学家津贴领取者。1708年，罗尔中风。他恢复得相当好，但心智能力下降，此后未再做出数学贡献。首次中风后他又活了十一年，但在1719年第二次中风，这次致命。

现在让我们来看看罗尔的重要数学贡献。他研究Diophantine analysis、代数（使用克劳德-加斯帕·巴歇·德·梅齐里亚克的方法，涉及欧几里得算法的使用），并在较小程度上研究几何。他于1690年出版了他最重要的著作Traité d'algèbreⓉ（代数论），内容是关于方程理论的。在这部论著中，他发明了$\sqrt[n]{x}$的记号来表示$x$的$n$次根，因此它成为标准记号；今天仍在使用。在Traité d'algèbreⓉ（代数论）中，罗尔使用欧几里得算法来求两个多项式的最大公因数。他还用它来解丢番图线性方程。然而，这部著作中最重要的部分也许是他引入“级联”概念的地方。让我们看看这个想法是如何运作的：如果$P(x) = 0$是一个给定的多项式方程，其实根为$a$和$b$，那么他构造一个多项式$P'(x)$，他称之为“第一级联”，使得$P'(b) = (b - a)Q(b)$，其中$Q(x)$是一个次数较低的多项式。当然，用我们的术语来说，$P'(x)$是$P(x)$的一阶导数。罗尔然后构造“第二级联”，即二阶导数，并以此方式继续。在$P(x)$的任意两个相邻根之间有一个$P'(x)$的根，在$P'(x)$的任意两个相邻根之间有一个$P''(x)$的根，等等。他的方法是：从一个给定的多项式开始，做一个线性变换以得到一个所有根都是正的多项式（他从未证明他的变换总是有效，但它确实有效），然后继续构造多项式的级联，直到得到一个线性多项式。然后可以沿着级联向上回溯，近似地求出每个多项式的根。朱利叶斯·沙因写道：——

级联方法具有重要的历史意义。微积分和方程理论的一些基本原理可以明确地追溯到它们作为该方法的附带命题的起源。它扩充了方程根的极限的概念，提供了科林·麦克劳林推导出他的公式所依据的基础，开创了确定根的现代级数方法，并讨论了方程中虚根及其导数之间的关系。罗尔的定理，作为微积分的一个重要命题，也起源于该方法。

事实上，罗尔最为人所铭记的是“罗尔定理”，该定理于1691年发表在Démonstration d'une Méthode pour resoudre les Egalitez de tous les degrezⓉ（《解所有次数方程的方法的论证》）中。这部著作是为了给某些方法（特别是级联法）提供证明而写的，他在Traité d'algèbreⓉ（《代数论》）中给出这些方法时没有给出理由。他的证明基于Johann van Waveren Hudde引入的方法。熟悉的罗尔定理表述为：

如果$f (a) = f (b) = 0$，那么对于某个满足$a ≤ x ≤ b$的$x$，有$f '(x) = 0$。

“罗尔定理”这个名称是Giusto Bellavitis在1846年给这个基本结果起的。在他1691年的工作中，罗尔采用了这样的观念：如果$a > b$，那么$-b > -a$。今天看来很奇怪的是，这并不是当时的通行做法，而是与勒内·笛卡儿等人使用的实数排序相反。罗尔在他1691年的工作中使用的另一个符号是等号“=”。这个符号不是罗尔发明的，而是罗伯特·雷科德发明的，但勒内·笛卡儿使用了不同的符号，等号=并不常用。罗尔在1699年发表了另一部关于不定方程解的重要著作，Méthode pour résoudre les équations indéterminées de l'algèbreⓉ（《解不定代数方程的方法》）。

从我们刚才关于罗尔工作的叙述中，可能会以为他正在发展无穷小分析。这将是一个严重的错误，因为罗尔将无穷小分析描述为一堆巧妙的谬误，并且他认为这些方法可能导致错误。他的回忆录Du nouveau systême de l'infiniⓉ（《论新的无穷系统》）（1703年）开头如下（例如见[4]）：-

几何学一直被认为是一门精确的科学，而且确实是数学其他部分中广泛存在的精确性的来源。在其原理中，人们只能找到真实的公理，所有提出的定理和问题要么得到了可靠的证明，要么能够被可靠地证明。如果有任何虚假或不确定的命题混入其中，它们会立即被逐出这门科学。但自从无穷小量的新系统被混入几何学以来，几何学的这种精确性特征似乎不再占主导地位。我看不出这个系统为真理产生了任何东西，在我看来，它常常掩盖错误。

现在我们应该注意到，这篇回忆录虽然直到1703年才由科学院发表，但实际上包含的材料曾在1700-01年科学院中进行的一场激烈辩论的早期阶段被使用过。这场激烈的分歧发生在罗尔和皮埃尔·伐里农之间，最终以喧闹收场（有人说罗尔缺乏教养表现在他的恶劣行为上）。Academy成立了一个委员会来决定这两位数学家中谁是正确的，但未能得出明确结论。罗尔 Blay [2]写道：-

从1700年起，这种反对极为活跃，由罗尔领导。他的批评主要基于两个论点：一个强调新微积分的基本概念和原理不充分且缺乏逻辑严密性；另一个则试图表明（借助精心挑选的例子），新微积分会导致错误，因为它不能得出与皮埃尔·德·费马，尤其是Hudde的经典代数启发方法相同的结果。

特别是，罗尔指出，与洛必达在Analyse des infiniment petits pour l'intelligence des lignes courbesⓉ（《无穷小分析，用于理解曲线》）（1696年）中给出的以下公理相关的困难：-

承认两个量，其差为无穷小量，可以不加区分地相互替代（或使用）；或者（同样地）一个量仅增加或减少一个无穷小量，可以视为保持不变。

例如，罗尔在1701年3月12日和3月16日向科学院致辞。他让听众考虑曲线

$y - b = \Large\frac{(x^2 - 2ax + a^2 - b^2 )^{2/3}} {a^{1/3} }$。

然后他应用了新的无穷小方法，发现

$dy = \Large\frac{4(x - a)dx}{3(ax^2 - 2a^2 x + a^3 - ab^2 )^{1/3}}$

因此，这种方法表明曲线在$x = a$处有一个转折点。罗尔随后应用Hudde的方法表明曲线在三个点$x = a, x = a - b$和$x = a + b$处有转折点。这是一个巧妙构造的例子，但皮埃尔·伐里农能够看出罗尔分析中的微妙错误。他后来回复[2]：-

首先，令 dy = 0，将得到 x = a；其次，令 dy 相对于 dx 为无穷大或 dx = 0，将得到 x = a - b 或 x = a + b。当人们看到，如果所求曲线有某个极大值或极小值对应于一条平行于[x轴]的切线，那么这只能出现在 x = a 的端点处；而如果它有某个极大值或极小值对应于垂直于该轴的[切线]，那么这只能出现在 x = a - b 和 x = a + b 的端点处。

在科学院决定不再对此主题进行进一步讨论之后，罗尔在Journal des sçavans的页面上继续了这场争论，而Joseph Saurin对此表示反对。值得永远称赞的是，罗尔最终承认自己错了。他向皮埃尔·伐里农、贝尔纳·勒布耶·德·丰特奈尔和尼古拉斯‧马勒伯朗士承认了这一点。Jean Itard 以如下评价结束了他的文章[1]：-

罗尔是一位技艺高超的代数学家，他打破了笛卡尔式的技巧；而他对无穷小方法的反对，归根结底是有益的。

Michel Rolle's father was a shopkeeper. Michel had little formal education being largely self-educated after receiving some elementary schooling. He worked as a transcriber for a notary and then as an assistant to several attorneys in the district around his home town of Ambert. In 1675, probably seeking a better life, he went to Paris where he worked as a scribe and arithmetical expert. However, quite soon after he arrived in Paris he married and children quickly followed. His income was not sufficient to support his growing family but he had been studying higher mathematics on his own and it was the skill that he had developed in this discipline which provided the breakthrough.

In 1682 he achieved a certain fame by solving a problem which had been publicly posed by Jacques Ozanam. Jean-Baptiste Colbert, the controller general of finance and secretary of state for the navy under King Louis XIV of France, rewarded Rolle for this achievement. Colbert arranged a pension for Rolle which started him on the road to financial security, but there were other equally important consequences [6]:-

Of equal importance with the financial benefits to which Rolle's prize led, was the fact that it brought him to the notice of Louvois who happened to be looking for someone to teach mathematics to one of his sons. Rolle was hired by Louvois, who came to be greatly impressed by his pedagogic and mathematical skills. From 1783 onwards Louvois was in a position to confer further scientific patronage on his protégé and did so when he made Rolle a member of the Académie in 1685.

The problem which Rolle solved was posed in the Journal des sçavans on 31 August 1682 by Jacques Ozanam; it was the following:-

Find four numbers the difference of any two being a perfect square, in addition the sum of the first three numbers being a perfect square.

Ozanam stated that the smallest of the four numbers with these properties would have at least 50 digits, but Rolle found four numbers satisfying the conditions with each number having seven digits. He made his solution known through publishing it in the Journal des sçavans. Louvois, who is referred to in the above quote, was François Michel le Tellier, Marquis de Louvois, the French Secretary of State for War. He employed Rolle to tutor his fourth son, Camille le Tellier (1675-1718). The Marquis de Louvois arranged for Rolle to have an administrative post in the Ministry of War, but Rolle disliked the work and soon resigned. Rolle was elected to the Académie Royale des Sciences in 1685 and the impressive mathematical work he produced following his election fully justified the Marquis de Louvois' faith in him.

Before going on to discuss the interesting mathematical contributions Rolle made, let us give a couple of further facts about his life. He became a Pensionnaire Géometre of the Académie des Sciences in 1699. In 1708 Rolle suffered a stroke. He recovered his health fairly well but his mental capacity was diminished and he made no further mathematical contributions after this stroke. He survived for eleven years after this first stroke but in 1719 he suffered a second stroke which proved fatal.

Let us now look at Rolle's important mathematical contributions. He worked on Diophantine analysis, algebra (using methods of Claude Gaspar Bachet de Méziriac involving the use of the Euclidean algorithm) and, to a lesser extent, on geometry. He published his most important work Traité d'algèbre Ⓣ in 1690 on the theory of equations. In this treatise, he invented the notation $\sqrt[n]{x}$ for the $n$th root of $x$ and, as a consequence, it became the standard notation; it is used today. In Traité d'algèbre Ⓣ Rolle used the Euclidean algorithm to find the greatest common divisor of two polynomials. He also used it to solve Diophantine linear equations. Perhaps the most significant part of the work, however, is where he introduces the notion of 'cascades'. Let us see how this idea worked: If $P(x) = 0$ is a given polynomial equation with real roots $a$ and $b$ then he constructs a polynomial $P'(x)$, which he called the 'first cascade,' so that $P'(b) = (b - a)Q(b)$ where $Q(x)$ is a polynomial of lower degree. Of course in our terminology $P'(x)$ is the first derivative of $P(x)$. Rolle then constructs the 'second cascade' which is the second derivative, and continues in this fashion. Between any two consecutive roots of $P(x)$ there is a root of $P'(x)$, between any two consecutive roots of $P'(x)$ there is a root of $P''(x)$, etc. His method is to start with a given polynomial, make a linear transformation to obtain a polynomial all of whose roots are positive (he never proves that his transformation always works but it does), then to continue to construct the cascade of polynomials until a linear polynomial is reached. One can then move back up the cascade, finding approximately the roots of each polynomial. Julius Shain writes:-

The method of cascades has an important historical significance. Some basic principles of the calculus and the theory of equations can definitely be traced to their origin as incidental propositions of the method. It amplified the concepts of limits of roots of equations, provided the fundamentals from which Maclaurin derived his formula, began modern methods of series for determining roots, and discussed the relationship of imaginary roots in equations and their derivatives. Rolle's theorem, an important proposition of the calculus, also owes its origin to the method.

In fact Rolle is best remembered for 'Rolle's Theorem' which was published in Démonstration d'une Méthode pour resoudre les Egalitez de tous les degrez Ⓣ in 1691. This work was written to provide proofs of certain methods (in particular the method of cascades) which he gave without justification in Traité d'algèbre Ⓣ. His proofs were based on methods introduced by Johann van Waveren Hudde. The familiar Rolle's Theorem states:

If $f (a) = f (b) = 0$ then $f '(x) = 0$ for some $x$ with $a ≤ x ≤ b$.

The name 'Rolle's Theorem' was given to this basic result by Giusto Bellavitis in 1846. In his 1691 work Rolle adopted the notion that if $a > b$ then $-b > -a$. It seems strange today to realise that this was not the current practice at the time but was in opposition to the ordering of the real numbers used by Descartes and others. Other notation Rolle used in his 1691 work was the equals sign '='. This notation was not invented by Rolle, rather it was invented by Robert Recorde, but Descartes had used a different notation and the sign = was not in common use. Rolle published another important work on solutions of indeterminate equations in 1699, Méthode pour résoudre les équations indéterminées de l'algèbre Ⓣ.

It might be assumed from what we have just written about Rolle's work that he was developing the infinitesimal calculus. This would be a serious error, for Rolle described the infinitesimal calculus as a collection of ingenious fallacies and he believed that the methods could lead to errors. He begins his memoir Du nouveau systême de l'infini Ⓣ (1703) as follows (see for example [4]):-

Geometry has always been considered as an exact science, and indeed as the source of the exactness which is widespread among other parts of mathematics. Among its principles one could only find true axioms and all the theorems and problems posed were either soundly demonstrated or capable of sound demonstration. And if any false or uncertain propositions were slipped into it they would immediately be banned from this science. But it seems that this feature of exactness doe not reign anymore in geometry since the new system of infinitely small quantities has been mixed to it. I do not see that this system has produced anything for the truth and it would seem to me that it often conceals mistakes.

Now we should note that this memoir, although only published by the Academy of Sciences in 1703, actually contained material which had been used in the early stages of a vigorous debate which took place in the Academy of Sciences in 1700-01. This vigorous disagreement was between Rolle and Pierre Varignon and it ended in uproar (some say that Rolle's lack of breeding showed in his bad behaviour). The Academy set up a commission to decide which of the two mathematicians was correct but it failed to come to a definite conclusion. Michel Blay [2] writes:-

This opposition, extremely active from 1700 on, was led by Michel Rolle. The burden of his critique rested on two arguments, one stressing the inadequacy and the lack of logical rigour of the fundamental concepts and principles of the new calculus, the other pretending to show (with the aid of cleverly selected examples) that the new calculus led to error, insofar as it did not yield the same results obtained in the classical, algebraically inspired methods of Fermat and, more especially, Hudde.

In particular Rolle suggested that there were difficulties associated with the following axiom given by de l'Hôpital in Analyse des infiniment petits pour l'intelligence des lignes courbes Ⓣ (1696):-

Grant that two quantities whose difference is an infinitely small quantity may be taken (or used) indifferently for each other; or (which is the same thing) that a quantity which is increased or decreased only by an infinitesimally small quantity may be considered as remaining the same.

For example, Rolle addressed the Academy on 12 March and 16 March 1701. He asked his audience to consider the curve

$y - b = \Large\frac{(x^2 - 2ax + a^2 - b^2 )^{2/3}} {a^{1/3} }$.

He then applied the new method of the infinitesimally small and found that

$dy = \Large\frac{4(x - a)dx}{3(ax^2 - 2a^2 x + a^3 - ab^2 )^{1/3}}$

so that this method showed that the curve has a turning point at $x = a$. Rolle then applied Hudde's method to show that the curve has turning points at three points, $x = a, x = a - b$ and $x = a + b$. This is a cleverly constructed example, but Varignon was able to see the subtle error in Rolle's analysis. He later replied [2]:-

One will obtain first x = a by setting dy = 0; and second, x = a - b or x = a + b by making dy infinite in relation to dx or dx = 0. When one sees that if the desired curve has some maximum or minimum that meets a tangent parallel to the [x-axis], this can only be at the extremity of x = a; and that if it has one that meets ... [tangents] perpendicular to this axis, this can only be at the extremity of x = a - b and of x = a + b.

After the Academy decided that no further discussion of this topic was take place, Rolle continued the argument in the pages of the Journal des sçavans opposed by Joseph Saurin. To his eternal credit, Rolle eventually conceded that he was wrong. He acknowledged this to Varignon, Fontenelle and Malebranche. Jean Itard ends his article [1] with the following assessment:-

Rolle was a skillful algebraist who broke with Cartesian techniques; and his opposition to infinitesimal methods, in the final analysis, was beneficial.