洛必达Guillaume de l'Hôpital

要给出洛必达的全名需要整整一段，所以我们只给出一个大大简化的版本：洛必达-François-路易·安托万 Marquis 洛必达，Sainte-Mesme侯爵，Entremont伯爵和Ouques-la-Chaise领主。这个家族在法国世代显赫，可追溯至大约12世纪。Hôpital这个名字有多种拼法，较早的版本是l'Hospital或Lhospital，而洛必达是这个名字相对现代的拼法。他的父亲是Anne-Alexandre 洛必达，国王军队的中将；他是Sainte-Mesme伯爵和奥尔良公爵。洛必达的母亲是Elisabeth Gobelin，Claude Gobelin的女儿，Claude Gobelin是国王军队的总监和国家参事。Sturdy写道[5]：-

Anne-Alexandre 洛必达 与奥尔良家族的显赫关系以及加斯东·奥尔良对他的信任，为洛皮塔尔家族的声誉增添了耀眼的光彩；这也为他们提供了仅次于国王本人的保护。

小时候，洛必达 对拉丁语之类的科目没有天赋，但他发展出了很强的数学能力，并对这门学科怀有真正的热情。贝尔纳·勒布耶·德·丰特奈尔 在 [7] 中叙述说，当 洛必达 十五岁时，有一次他正与罗阿内公爵和一位阿尔诺先生讨论数学。他们告诉他 布莱兹·帕斯卡 提出的一个关于摆线的非常困难的问题。几天后，洛必达 就解决了这个问题。鉴于他的家庭背景，洛必达 走上军旅生涯并担任骑兵团上尉并不令人意外。他当然没有放弃对数学的兴趣，正如 贝尔纳·勒布耶·德·丰特奈尔 所解释的 [7]：-

他投身军旅，但并未放弃他最珍爱的热情。他甚至在帐篷里也研究几何学。他退到那里不仅是为了研究，也是为了隐藏自己致力于研究这件事。因为必须承认，法兰西民族虽然和其他民族一样彬彬有礼，却仍处于那样一种野蛮状态：它会怀疑科学一旦达到某种程度是否与贵族身份不相容，以及一无所知是否更为高贵。……我本人就见过一些与他同时服役的人，他们大为惊讶：一个像他们一样生活的人，竟然是欧洲最杰出的数学家之一。

然而他因近视——十步之外便看不清——而从军队辞职。朱利安·罗威尔·柯立芝 写道 [3]：-

人们倾向于相信，是 洛必达 对数学的热爱，而非他视力的缺陷，促使他放弃军旅生涯而转向科学生涯。

当然，从那时起他就把注意力转向了数学。几乎可以肯定，如果不是 1691 年末他与 约翰·伯努利 的一次偶然相遇，洛必达 在今天的数学界会完全默默无闻。约翰·伯努利 当时 24 岁，他刚刚在讲授了数学的最新发展——即 哥特弗里德·威廉·莱布尼茨 的微分学——之后抵达巴黎。洛必达 当时是 尼古拉斯‧马勒伯朗士 在奥拉托利会聚会圈中的成员，该圈子汇集了巴黎最杰出的数学家和科学家。对 约翰·伯努利 来说，去那里结识法国最杰出的数学家是显而易见的选择，而他很快发现 洛必达 是最热情的一位。洛必达 则对见到 约翰·伯努利 很感兴趣，因为他很快就清楚地意识到，在无穷小方法的新发展方面，约翰·伯努利 比巴黎其他任何人都懂得多得多。约翰·伯努利 告诉 洛必达 以及 尼古拉斯‧马勒伯朗士 圈子中的其他人，他知道曲线曲率半径的一般公式。尽管 克里斯蒂安·惠更斯、哥特弗里德·威廉·莱布尼茨 和 艾萨克·牛顿 等人也知道这一点，但在巴黎它被认为是一个重要的未决问题，因此 洛必达 尽管可能是法国最优秀的数学家之一，也意识到自己可以从 约翰·伯努利 那里学到很多。约翰·伯努利 同意每周给 尼古拉斯‧马勒伯朗士 的圈子讲四次课，并在接下来的六个月里一直这样做。洛必达 听了这些课，但随后从巴黎搬到了他在乌克的庄园，在那里他聘请 约翰·伯努利 给他上私人课程。到 1692 年 11 月，约翰·伯努利 已离开乌克，回到巴塞尔，并从那里与 洛必达 通信。

洛必达曾发表过几篇简短的数学札记，但在1692年，当约翰·伯努利在乌克给他上课时，洛必达把de Beaune问题的解答寄给了克里斯蒂安·惠更斯。Florimond de Beaune曾征求一条切线截距为定长的曲线，而约翰·伯努利已把这个解答收入他给洛必达讲授的课程中。洛必达并未声称他寄给克里斯蒂安·惠更斯的解答是他自己的，但克里斯蒂安·惠更斯作出了合理的推测，认为那是他的。此后不久，洛必达用笔名发表了这一解答。等到约翰·伯努利看到发表的解答时，他已回到巴塞尔，自然极为不快。大约有六个月，洛必达与伯努利之间的通信中断了；随后，在1694年3月17日，洛必达给约翰·伯努利寄了一封信，其中提出了一个引人注目的提议[11]：-

我很乐意给你300镑的聘金，从今年1月1日开始。……我保证不久会增加这笔聘金，我知道它非常微薄，只要我的事务稍有起色。……我不会不讲道理到要求你为此付出全部时间，但我会请你每隔一段时间给我一些时间，按我的要求工作，并把你的一些发现告诉我，同时请你不要把它们中的任何一项透露给别人。我甚至要求你不要把我这里寄给皮埃尔·伐里农先生或其他人，也不要寄去你留在我这里的任何文稿副本；如果它们被发表，我绝不会高兴。请就这一切答复我……

虽然没有找到约翰·伯努利回信的副本，但从洛必达的下一封信可知，约翰·伯努利很快接受了这一提议。通信继续下去，在1695年约翰·伯努利寄给洛必达的一封信中，他作出了如下引人注目的承诺[11]：-

你只需让我知道你的明确意愿，哪怕我此生不再发表任何东西，因为我将严格遵从它们，我写的任何东西都不会再被人看到。

1695年9月1日，约翰·伯努利离开巴塞尔，去格罗宁根就任数学教授的新职。1696年，洛必达的名著Analyse des infiniment petits pour l'intelligence des lignes courbesⓉ（《用于理解曲线的无穷小分析》）出版；这是第一部关于微分学的教科书。在引言中，洛必达承认自己受惠于哥特弗里德·威廉·莱布尼茨、雅各布·伯努利和约翰·伯努利，但洛必达认为由他提供的基础是他自己的想法。他写道：-

我必须承认，我非常感激伯努利先生们的劳作，尤其是现任格罗宁根教授的那些劳作。我自由地使用了他们的发现，也使用了哥特弗里德·威廉·莱布尼茨先生的发现，因此我坦率地把他们愿意认作自己的一切归还给他们。

他也知道艾萨克·牛顿的贡献，写道：-

我必须在此公正地承认（正如哥特弗里德·威廉·莱布尼茨先生本人在1694年8月的'Journal des Sçavans'中所做的那样），博学的艾萨克·牛顿爵士同样发现了类似微分学的东西……但哥特弗里德·威廉·莱布尼茨先生的方法要容易和迅速得多，这是由于他所使用的记号，更不用说它在许多场合所提供的奇妙帮助。

这本书以两个定义开始：

定义1。变量是那些连续增加或减少的量，而常量则在其他量变化时保持不变。

定义2。变量连续增加或减少的无限小部分称为该量的微分。

接下来是两个公理：

公理1。承认两个量，如果它们的差是一个无穷小量，那么它们可以不加区分地互相代替（或使用）；或者（同样的事情）一个量如果只增加或减少一个无穷小量，可以认为它保持不变。

公理2。承认一条曲线可以被视为无穷多条无穷小直线的集合；或者（同样的事情）视为一个有无穷多条边的多边形，每条边的长度都是无穷小，使得相邻直线之间的夹角决定曲线的曲率。

在著作的第二章中，洛必达继续确定曲线的切线。鉴于他将曲线定义为有无穷多条边、每条边长度无穷小的多边形，他可以将曲线上一点的切线定义为该点处无穷小直线所生成的直线。他给出了许多计算切线的例子以及一个一般方法。在第三章中，他考虑极大和极小问题，给出了来自力学和地理学的例子。在后面的章节中，他继续考虑拐点、尖点、曲率、evolutes、involutes和高阶导数。在第9章中，可以找到现在称为洛必达法则的规则，用于求一个以分数形式给出的函数在一点处分子和分母都趋于零时的极限。

这本书是一个极其重要的贡献。它被使用了很长时间，直到1781年还在出版新版本，它也是下一代微积分书籍的典范。但是，当然，我们必须考察这部著作在多大程度上依赖于约翰·伯努利。嗯，当这本书出现时，他并没有过于强烈地抱怨，只有在洛必达去世后，他才更加强烈地说这本书本质上是他的。约翰·伯努利，尽管作为数学家享有盛誉，却卷入了各种优先权争议，特别是与他的兄弟雅各布·伯努利的争议，他的主张没有被太认真对待。另一方面，洛必达的个性和对概念的深刻理解使同事们站在他一边。例如，Robinson在[1]中写道：-

洛必达拥有非常吸引人的个性，除其他外，他谦虚而慷慨，这两种品质在他那个时代的数学家中并不普遍。

直到1921年，约翰·伯努利给洛必达所授课的手稿副本才被发现，人们看到这本书多么紧密地遵循了课程笔记。此外，当两人之间的协议被看到时，对事件有了更多的理解。事实上，由于他们之间的协议，约翰·伯努利在洛必达的书出版时并没有处于可以抱怨的位置。在[11]中，Truesdell试图正确看待这一安排：-

我们不应过于严厉地评判洛必达的做法。虽然起初或许是经济上的需要迫使约翰·伯努利接受了这一安排，但在他于1695年就任格罗宁根讲席之后，这种安排仍在继续。洛必达身为贵族，习惯于为他人提供的服务付费，如果约翰·伯努利是政治家、律师，甚至可能是建筑师，他所做的事就不会被认为是错的。当然，这没有什么值得洛必达骄傲的。仔细考察洛必达向哥特弗里德·威廉·莱布尼茨和克里斯蒂安·惠更斯报告其数学进展的那些信件可以看出，除了一两处可能的例外，洛必达并没有撒谎，而是以居高临下的口吻提到约翰·伯努利，完全不承认对他有任何亏欠，并且在出处问题上，其写法暗示了某种东西却没有真正断言。

那么，为什么约翰·伯努利如此迅速地同意了洛必达的提议？尽管正如Truesdell所暗示的，金钱可能起了一定作用，但这似乎并非全部原因。当然，在当时，社会地位远比现代世界重要得多，约翰·伯努利在很大程度上会感到不得不顺从一位贵族。在晚年，约翰·伯努利夸耀他从洛必达那里得到的钱，夸大了他所收到的数额。但也可能对年轻的约翰·伯努利来说，他的数学发现广为人知，比世人知道他是作者更为重要。

但我们必须赞扬洛必达迅速理解了呈现给他的新颖数学。他是一位非常能干的数学家，当然能够以极大的清晰度写作。然而，确实看起来他自己很少或没有做出数学发现，他对最速降线问题的解决方案可能不是他自己的。这个问题由艾萨克·牛顿、哥特弗里德·威廉·莱布尼茨和雅各布·伯努利独立解决，如果解决方案确实归功于他，这将使洛必达处于非常好的伙伴中。除了数学家解决的问题外，往往他们提出的问题的质量使他们脱颖而出。在这方面，值得注意的是洛必达在思考分数阶导数的概念是否有意义。有一封他写给哥特弗里德·威廉·莱布尼茨的信，日期为1695年，其中他问$d^{n}y/dx^{n}$对$n = \large\frac{1}{2}\normalsize$是否有意义。

人们可能认为，作为贵族，洛必达会很容易被选入Académie des Sciences。然而，他的贵族身份使得通过正常程序选举成为不可能，但在1699年科学院重组后，他被授予荣誉地位。

洛必达考虑出版一部关于积分学的著作，但在得知哥特弗里德·威廉·莱布尼茨将出版一部关于该主题的著作后，他放弃了他的计划。他去世后发现了一本书的手稿，该书于1707年以Traité anlytique des sections coniques Ⓣ（圆锥曲线的分析论著）为标题出版。该文本的第二版于1720年出现。

洛必达与Marie-Charlotte de Romilley de La Chesnelaye结婚；他们有一个儿子和三个女儿。

最后让我们给出朱利安·罗威尔·柯立芝对洛必达的评价[3]：-

他是一个社会地位极高之人的光辉典范，对学问的热爱驱使他将短暂一生的大部分时间投入到科学写作中。

To give Guillaume de l'Hôpital's full name would take a whole paragraph so we give just a much shortened version: Guillaume-François-Antoine Marquis de l'Hôpital, Marquis de Sainte-Mesme, Comte d'Entremont and Seigneur d'Ouques-la-Chaise. The family had been a prominent one in France over many generations going back to around the 12th century. There are various spellings of the name Hôpital, the earlier versions being l'Hospital or Lhospital with l'Hôpital being a relatively modern form of the name. His father was Anne-Alexandre de l'Hôpital, a Lieutenant-general in the King's Army; he was Comte de Sainte-Mesme and Duc d'Orléans. Guillaume's mother was Elisabeth Gobelin, the daughter of Claude Gobelin who was an Intendant in the King's Army and a Councillor of State. Sturdy writes [5]:-

The prominent association which Anne-Alexandre de l'Hôpital had with the house of Orléans and the trust which Gaston d'Orléans placed in him conferred a brilliant lustre on the reputation of the l'Hôpitals; it also afforded them a protection second only to that of the king himself.

As a child, l'Hôpital had no talent for subjects like Latin, but he developed strong mathematical abilities and a real passion for the subject. Bernard de Fontenelle, in [7], recounts that when l'Hôpital was fifteen years old he was, on one occasion, discussing mathematics with the Duke of Roannès and a Mr Arnaud. They told him about a very difficult problem on the cycloid that Blaise Pascal had proposed. A few days later l'Hôpital had solved the problem. Given his family background it is not surprising that Guillaume de l'Hôpital followed a military career and served as a captain in a cavalry regiment. He certainly did not give up his interest in mathematics, as Fontenelle explains [7]:-

He entered into the service, but without giving up his dearest passion. He studied geometry even in his tent. It is not just that he retired there to study, it was also to hide his application to study. For it must be admitted that the French nation, although as well mannered as any other, is still in that sort of barbarism by which it wonders whether the sciences, taken to a certain point, are incompatible with nobility, and whether it is not more noble to know nothing. ... I have personally seen some of those who served at the same time, greatly astonished that a man who lived like them was one of the leading mathematicians in Europe.

However he resigned from the army because of nearsightedness being unable to see beyond ten paces. Julian Coolidge writes [3]:-

One is inclined to believe that it was l'Hôpital's love of mathematics rather than the imperfection of his vision that led him to abandon a military career in favour of a scientific one.

Certainly from that time on he directed his attention to mathematics. It is almost certain that l'Hôpital would be totally unknown in the world of mathematics today but for a chance meeting between him and Johann Bernoulli towards the end of 1691. Bernoulli at this time was 24 years old and he had just arrived in Paris after giving lectures on the latest development in mathematics, namely Leibniz's differential calculus. l'Hôpital was at the time a member of Nicolas Malebranche's circle at the Congregation of the Oratory which contained the leading mathematicians and scientists of Paris. It was an obvious place for Bernoulli to go to meet the leading French mathematicians and he soon discovered that l'Hôpital was the most enthusiastic. l'Hôpital, for his part, was intrigued to meet Bernoulli, for it quickly became clear to him that he was much more knowledgeable about the new developments in infinitesimal methods than anyone else in Paris. Bernoulli told l'Hôpital, and others in Malebranche's circle, that he knew the general formula for the radius of curvature of a curve. Although others such as Huygens, Leibniz and Newton knew this, it was thought in Paris to be an important open question so l'Hôpital, although probably one of the best mathematicians in France, realised he could learn much from Bernoulli. Bernoulli agreed to give four lectures a week to Malebranche's circle and he did so for the next six months. l'Hôpital attended these lectures but then moved from Paris to his estate at Ouques where he employed Bernoulli to give him private lessons. By November 1692 Bernoulli had left Ouques and returned to Basel from where he carried out a correspondence with l'Hôpital.

L'Hôpital had published a few brief mathematical notes, but in 1692, while Bernoulli was giving him lessons at Ouques, l'Hôpital sent a solution of de Beaune's problem to Huygens. Florimond de Beaune had asked for a curve for which the subtangent had a fixed length and Bernoulli had included the solution in the course he had given l'Hôpital. L'Hôpital did not claim that the solution he sent Huygens was his own but Huygens made the reasonable assumption that it was. Shortly after this l'Hôpital published the solution under a pseudonym. By the time Bernoulli saw the published solution he was back in Basel and, naturally enough, he was highly displeased. For about six months the correspondence between l'Hôpital and Bernoulli ceased then, on 17 March 1694, l'Hôpital sent a letter to Bernoulli with a remarkable proposition [11]:-

I will be happy to give you a retainer of 300 pounds, beginning with the first of January of this year. ... I promise shortly to increase this retainer, which I know is very modest, as soon as my affairs are somewhat straightened out. ... I am not so unreasonable as to demand in return all of your time, but I will ask you to give me at intervals some hours of your time to work on what I request and also to communicate to me your discoveries, at the same time asking you not to disclose any of them to others. I ask you even not to send here to Mr Varignon or to others any copies of the writings you have left with me; if they are published, I will not be at all pleased. Answer me regarding all this ...

Although no copy of Johann Bernoulli's reply has been found, we know from l'Hôpital's next letter that Bernoulli rapidly accepted the proposition. The correspondence continued and in one of the letters Bernoulli sent to l'Hôpital in 1695 he made the following remarkable promise [11]:-

You have only to let me know your definite wishes, if I am to publish nothing more in my life, for I will follow them precisely and nothing more by me will be seen.

On 1 September 1695 Johann Bernoulli left Basel to take up a new appointment as professor of mathematics at Groningen. In 1696 L'Hôpital's famous book Analyse des infiniment petits pour l'intelligence des lignes courbes Ⓣ was published; it was the first text-book to be written on the differential calculus. In the introduction l'Hôpital acknowledges his indebtedness to Leibniz, Jacob Bernoulli and Johann Bernoulli but l'Hôpital regarded the foundations provided by him as his own ideas. He wrote:-

I must own myself very much obliged to the labours of Messieurs Bernoulli, but particularly to those of the present Professor at Groningen. I have made free use of their discoveries, as well as those of Mr Leibniz, so that I frankly return to them whatever they please to claim as their own.

He was also aware of Newton's contributions, writing:-

I must here in justice own (as Mr Leibniz himself has done in 'Journal des Sçavans' for August 1694) that the learned Sir Isaac Newton likewise discovered something like the Calculus Differentialis ... Bit the method of Mr Leibniz is much more easy and expeditious, on account of the notation he uses, not to mention the wonderful assistance it affords on many occasions.

The book begins with two definitions:

Definition 1. Variable quantities are those that increase or decrease continuously while a constant quantity remains the same while other vary.

Definition 2. The infinitely small part by which a variable quantity increases or decreases continuously is called the differential of that quantity.

There follow two axioms:

Axiom 1. Grant that two quantities whose difference is an infinitely small quantity may be taken (or used) indifferently for each other; or (what 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.

Axiom 2. Grant that a curved line may be considered as the assemblage of an infinite number of infinitely small straight lines; or (what is the same thing) as a polygon with an infinite number of sides, each of infinitely small length such that the angle between adjacent lines determines the curvature of the curve.

In the second chapter of the work L'Hôpital went on to determine tangents to a curve. Given his definition of a curve as a polygon with an infinite number of sides each of infinitely small length, he can define the tangent at a point on the curve as being the straight line produced from the infinitely small straight line at that point. He gives many examples of computing tangents as well as a general method. In the third chapter he considers maximum and minimum problems giving examples from mechanics and geography. In later chapters he goes on to consider points of inflection, cusps, curvature, evolutes, involutes, and higher order derivatives. In Chapter 9 is found the rule, now known as L'Hôpital's rule, for finding the limit of a function giiven as a fraction whose numerator and denominator tend to zero at a point.

This book was an extremely important contribution. It was used for a long time, with new editions produced until 1781, and it was also a model for the next generation of calculus books. But, of course, we have to examine the question of how dependent the work was on Johann Bernoulli. Well, he did not complain too vigorously when the book appeared and only after L'Hôpital's death did he become more forceful in saying that the book was essentially his. Johann Bernoulli, although having an outstanding reputation as a mathematician, had been involved in various priority disputes, particularly with his brother Jacob Bernoulli, and his claims were not taken too seriously. On the other hand, L'Hôpital's personality and deep understanding of the concepts led colleagues to take his side. For example, Robinson writes in [1]:-

L'Hôpital possessed a very attractive personality, being, among other things, modest and generous, two qualities which were not widespread among mathematicians of his time.

Only in 1921 did a manuscript copy of the course given by Johann Bernoulli to L'Hôpital come to light and it was seen how closely the book followed the course notes. Also when the agreement between the two men was seen, more understanding of the events became possible. In fact Bernoulli had not been in a position to complain when L'Hôpital's book was published because of the agreement between them. In [11] Truesdell tries to put the arrangement into perspective:-

We should not judge L'Hôpital's procedure too harshly. While perhaps financial necessity compelled Bernoulli to accept the arrangement initially, it continued after he had settled in his professorship at Groningen in 1695. L'Hôpital, being a noblemen, was accustomed to pay for the services of others, and what he did would not have been considered wrong had Bernoulli been a politician, a lawyer, perhaps even an architect. Certainly it was nothing for L'Hôpital to be proud of. Careful examination of the letters in which L'Hôpital reported his mathematical progress to Leibniz and Huygens shows that with one or two possible exceptions L'Hôpital did not lie, but rather referred to Bernoulli in a condescending tone without acknowledging any debt whatever to him and in matters of provenance wrote in such a way as to suggest without actually asserting.

So why did Johann Bernoulli agree so quickly to L'Hôpital's proposal? Although as Truesdell suggests, money may have played a part, it does not seem to be the whole reason. Certainly social status was far more significant at this time than in the modern world and Bernoulli would, to quite a large extent, feel obliged to be subservient to a nobleman. Towards the end of his life Bernoulli boasted of the money he had received from L'Hôpital, exaggerating the amount he had received. But it is also possible that it may have been more important to the youthful Bernoulli that his mathematical discoveries became well known than that the world knew him to be the author.

But we have to give L'Hôpital credit for understanding quickly the novel mathematics that was being presented to him. He was a very competent mathematician and certainly was able to write with great clarity. It does appear, however, that he made few if any mathematical discoveries of his own and his solution of the brachistochrone problem was probably not his own. The fact that this problem was solved independently by Newton, Leibniz and Jacob Bernoulli would put l'Hôpital in very good company indeed if the solution was indeed due to him. As well as the problems a mathematician solves, it is often the quality of the questions they ask which single them out for greatness. In this respect it is worth noting the fact that l'Hôpital was wondering whether the notion of a fractional derivative makes sense. There is a letter from him to Leibniz dated 1695 in which he asks whether $d^{n}y/dx^{n}$ has a meaning for $n = \large\frac{1}{2}\normalsize$.

It might be thought that being from the nobility, L'Hôpital would have found it easy to be elected to the Académie des Sciences. However, his nobility made election under the normal processes impossible but after the reorganisation of the Academy in 1699 he was given honorary status.

L'Hôpital considered publishing a work on integration but, on learning that Leibniz was going to publish a work on the topic, he dropped his plans. A manuscript of a book was discovered following his death and this was published under the title Traité anlytique des sections coniques Ⓣ in 1707. A second edition of this text appeared in 1720.

L'Hôpital married Marie-Charlotte de Romilley de La Chesnelaye; they had one son and three daughters.

Let us end by giving Coolidge's assessment of L'Hôpital [3]:-

He was a shining example of a man of the highest social distinction whose love of learning drove him to devote much of his short life to scientific writing.