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关于拉格朗日中值定理的几点注记
拉格朗日中值定理（Lagrange's Mean Value Theorem）是17大数学家拉格朗日（Joseph-Louis Lagrange）发现的数学定理，它可以用来求函数在任意一段区间内的极值，从而简化数学计算。
这个定理在很多地方都有应用，比如微积分学和统计学等科学领域，都有它的身影。
首先，拉格朗日中值定理的本质是，对于在定义域上连续不断的函数，可以证明它在任意一段区间内（a，b）中必定存在某一数c，使得这个函数在区间内的极值点等于函数f （c）,即f（c）=f（a）+f（b）。
这里，c一般可以在[a,b]之间任取，而它说明了极值点所在的位置以及函数值。
其次，拉格朗日中值定理的证明方式主要是利用微积分中的泰勒公式。
这一公式利用平均值定理和微分的性质联系起来，从而可以证明拉格朗日中值定理的有效性。
此外，拉格朗日中值定理也可以用来求函数积分和导数，即
积分：∫f（x） dx=f（b）-f（a）
导数：f'（c）=f（b）-f（a）
最后，拉格朗日中值定理在统计学上也有应用，它可以用来解决统计推断的问题，比如：一个函数是一个随机变量的函数，那么一个函数满足拉格朗日中值定理就意味着它的期望可以根据拉格朗日函数的值来求出来。
总而言之，拉格朗日中值定理是一种伟大的数学发现，它极大地丰富了数学的理论，它也在许多领域有很广泛的应用，比如可以用来求函数极值、积分和求导等，它也被广泛应用于各种统计问题之中，它可以为这些问题提供有力的解决方案。
法国大革命下的科学家群体拉格朗日和天体力学约瑟夫-路易斯·拉格朗日，意大利数学家、天文学家，在分析学、数论、经典力学和天体力学等领域都做出了重大贡献。
“I cannot say whether I will still be doing geometry ten years from now. It also seems to me that the mine has maybe already become too deep and unless one finds new veins it might have to be abandoned. Physics and chemistry now offer a much more glowing richness and much easier exploitation. Also, the general taste has turned entirely in this direction, and it is not impossible that the place of Geometry in the Academies will someday become what the role of the Chairs of Arabic at the universities is now.”– Joseph Louis Lagrange, Letter to d’Alembert (1781)“我不能说十年后我是否还在做几何学。
在我看来，矿井可能已经变得太深了，除非有人找到新的矿脉，否则它可能不得不被放弃。
物理和化学现在提供了更丰富和更容易的开发，此外，一般的品味已经完全转向了这个方向。
几何在学院的地位有一天可能成为今天阿拉伯语的椅子的角色。
”–约瑟夫·路易斯·拉格朗日，给达朗贝尔的信（1781年）约瑟夫-路易斯-拉格朗日Joseph-Louis Lagrange（1736-1813）自学成才的数学家1736年1月25日，拉格朗日（Lagrange）出生在都灵，原名朱塞佩·卢多维科·拉格朗日，以前是萨沃伊公国的首都，但在1720年成为撒丁王国的首都。
数学家名人故事：拉格朗日拉格朗日（—），法国著名的数学家、力学家、天文学家，变分法的开拓者和分析力学的奠基人。
他曾获得过世纪欧洲最大之希望、欧洲最伟大的数学家的赞誉。
拉格朗日出生在意大利的都灵。
由于是长子，父亲一心想让他学习法律，然而，拉格朗日对法律毫无兴趣，偏偏喜爱上文学。
直到岁时，拉格朗日仍十分偏爱文学，对数学尚未产生兴趣。
岁那年，他偶然读到一篇介绍牛顿微积分的文章《论分析方法的优点》，使他对牛顿产生了无限崇拜和敬仰之情，于是，他下决心要成为牛顿式的数学家。
在进入都灵皇家炮兵学院学习后，拉格朗日开始有计划地自学数学。
由于勤奋刻苦，他的进步很快，尚未毕业就担任了该校的数学教学工作。
岁时就被正式聘任为该校的数学副教授。
从这一年起，拉格朗日开始研究极大和极小的问题。
他采用的是纯分析的方法。
年月，他把自己的研究方法写信告诉了欧拉，欧拉对此给予了极高的评价。
从此，两位大师开始频繁通信，就在这一来一往中，诞生了数学的一个新的分支——变分法。
年，在欧拉的推荐下，拉格朗日被提名为柏林科学院的通讯院士。
接着，他又当选为该院的外国院士。
年，法国科学院悬赏征解有关月球何以自转，以及自转时总是以同一面对着地球的难题。
拉格朗日写出一篇出色的论文，成功地解决了这一问题，并获得了科学院的大奖。
拉格朗日的名字因此传遍了整个欧洲，引起世人的瞩目。
两年之后，法国科学院又提出了木星的个卫星和太阳之间的摄动问题的所谓六体问题。
面对这一难题，拉格朗日毫不畏惧，经过数个不眠之夜，他终于用近似解法找到了答案，从而再度获奖。
这次获奖，使他赢得了世界性的声誉。
年，拉格朗日接替欧拉担任柏林科学院物理数学所所长。
在担任所长的年中，拉格朗日发表了许多论文，并多次获得法国科学院的大奖：年，其论文《论三体问题》获奖；年，其论文《论月球的长期方程》再次获奖；年，拉格朗日又因论文《由行星活动的试验来研究彗星的摄动理论》而获得双倍奖金。
在柏林科学院工作期间，拉格朗日对代数、数论、微分方程、变分法和力学等方面进行了广泛而深入的研究。
拉格朗日——数学世界里一座高耸的金字塔Joseph-Louis Lagrange(1736~1813)拉格朗日(Lagrange, 1736—1813)是18世纪伟大的数学家、力学家和天文学家, 1736年生于意大利都灵. 青年时代, 在数学家雷维里(F. A. Revelli)指导下学习几何学后, 萌发了他的数学天才. 17岁开始专攻当时迅速发展的数学分析. 19岁时, 拉格朗日写出了用纯分析方法求变分极值的论文, 对变分法的创立做出了贡献, 此成果使他在都灵出名. 当年, 他被聘为都灵皇家炮兵学校教授. 1763年, 拉格朗日完成的关于“月球天平动研究”的论文因较好地解释了月球自转和公转的角速度的差异, 获得巴黎科学院1764年度奖, 此后他还四次获得巴黎科学院征奖课题研究的年度奖. 1766年, 在达朗贝尔和欧拉的推荐下, 普鲁斯国王腓特烈大帝写信给拉格朗日说: 欧洲最大之王希望欧洲最大之数学家来他的宫廷工作. 拉格朗日决定接受邀请, 于当年的8月21日离开都灵前往柏林科学院, 并担任了柏林科学院数学部主任一职, 一直到1787年才移居巴黎.拉格朗日的学术生涯主要在18世纪后半期. 当时数学、物理学和天文学是自然科学的主体. 数学的主流是由微积分发展起来的数学分析, 以欧洲大陆为中心; 物理学的主流是力学; 天文学的主流是天体力学. 数学分析的发展使力学和天体力学深化, 而力学和天体力学的课题又成为数学分析发展的动力. 拉格朗日在数学、力学和天文学三个学科中都有重大历史性的贡献, 但他主要是数学家, 研究力学和天文学的目的是表明数学分析的威力. 他的全部著作、论文、学术报告记录、学术通讯超过500篇. 几乎在当时所有的数学领域中, 拉格朗日都做出了重要贡献, 其中最突出的贡献是在把数学分析的基础脱离几何与力学方面起了决定性的作用. 使得数学的独立性更为清楚, 而不仅是其他学科的工具.他的工作总结了18世纪的数学成果, 同时又开辟了19世纪数学研究的道路.拉格朗日在使天文学力学化、力学分析化方面也起了决定性作用, 促使力学和天文学更深入发展. 他最精心之作当推《天体力学》，在其中他投注了37年的心血, 用数学把宇宙描绘成一个优美和谐的力学体系, 被哈密顿(Hamilton)誉为“科学诗”.拉格朗日科学的思想方法,也给后人留下了深远的影响. 拉格朗日常数变易法, 其实质就是矛盾转化法.他在探索微分方程求解的过程中, 巧妙地运用了高阶与低阶、常量与变量、线性与非线性、齐次与非齐次等各种转化. 拉格朗日解决数学问题的精妙之处, 就在于他能洞察到数学对象之间的深层次联系, 从而创造有利条件, 使问题迎刃而解.拉格朗日是欧洲最伟大的数学家之一, 拿破仑曾称赞他“是一座高耸在数学世界的金字塔”.。
数学家拉格朗日的小故事
今天给你讲个拉格朗日的小故事。
拉格朗日呀，那可是个超级厉害的数学家。
他年轻的时候就特别痴迷数学。
据说有一次，他陷入了一个超级复杂的数学难题当中。
他就像着了魔一样，整天把自己关在房间里，对着那些密密麻麻的公式和草稿纸发呆。
他的家人可担心了，想着这孩子是不是魔怔了。
但是拉格朗日不管啊，他满脑子都是怎么解开这个难题。
有一天，他突然灵感乍现，就像脑袋里有个小灯泡突然亮了起来。
他兴奋得在房间里跳了起来，大喊着：“我找到啦！”这一嗓子可把他的家人吓了一跳，还以为出了什么事呢。
拉格朗日对数学的热爱可不仅仅是解决难题。
他在很多数学领域都做出了巨大的贡献。
他研究的东西很深奥，但是又非常有用。
比如说在力学方面，他的理论就像是给科学家们开了天眼一样。
还有啊，拉格朗日在学术交流的时候也特别有趣。
他和其他数学家讨论问题的时候，就像武林高手过招。
他说话不紧不慢的，但是每一句话都像一颗重磅炸弹，让那些同行们又佩服又觉得压力山大。
他的一生都奉献给了数学，就像一个不知疲倦的探险家，在数学的神秘大陆上不断地开拓新的疆土。
他的故事也激励着后来的数学家们，让大家都想成为像他一样厉害的人物呢。
拉格朗日的故事
拉格朗日，即约瑟夫·路易斯·拉格朗日，是一位杰出的法国数学家和物理学家。
他于1736年出生在巴黎一个平凡的家庭中。
拉格朗日年轻时表现出极高的数学才华。
他从小就对数学产生了浓厚的兴趣，并在学校里展现出了卓越的成绩。
他的天赋和努力使他很快在法国的数学界崭露头角。
拉格朗日的成就主要集中在许多领域，但最重要的还是在变分法和微积分方面。
他对拉格朗日力学的贡献尤其引人注目。
他发展了一套优雅而强大的数学方法，用于解决刚体在力学系统中的运动问题。
拉格朗日是一位勤奋且谦虚的学者。
他不仅独立开创了许多重要的数学理论，而且还与其他杰出的数学家合作，推动了数学的发展。
他的学术著作广泛流传，对后来的研究产生了深远的影响。
尽管他在学术界享有很高的声誉，但拉格朗日的生活并不一帆风顺。
他经历了许多困难和挫折，包括家庭的不幸和健康问题。
然而，他坚韧不拔地迎接挑战，继续研究和探索数学的奥秘。
拉格朗日最终在1783年因病去世，享年47岁。
他被公认为一位卓越的数学家和物理学家，并为世界的数学领域做出了巨大的贡献。
他的才华和成就将永远在数学史上闪耀。
拉格朗日的故事给了我们许多启发。
他的坚韧精神和对知识的追求不仅是一位伟大科学家的品质，更是一种人生态度。
他不畏困难，不放弃追求，为我们树立了榜样。
欧洲最大之数学家——拉格朗日拉格朗日约瑟夫•拉格朗日（Joseph Louis Lagrange，1736—1813），法国数学家、力学家和天文学家，原籍意大利。
普鲁士国王腓特烈大帝尊称他为“欧洲最大之数学家”，他在数学、力学和天文学三个学科领域中都有历史性的贡献，其中尤以数学方面的成就最为突出。
拉格朗日生于意大利西北部的都灵，父亲是法国陆军骑兵里的一名军官，后由于经商破产，家道中落。
据拉格朗日本人回忆，如果幼年时家境富裕，他也就不会作数学研究了，因为父亲一心想把他培养成为一名律师。
拉格朗日个人却对法律毫无兴趣。
到青年时代，在数学家雷维里的教导下，拉格朗日喜爱上了几何学。
17岁时，他读了英国天文学家哈雷的介绍牛顿微积分成就的短文《论分析方法的优点》后，感觉到“分析才是自己最热爱的学科”，从此他迷上了数学分析，开始专攻当时迅速发展的数学分析。
18岁时，拉格朗日用意大利语写了第一篇论文，是用牛顿二项式定理处理两函数乘积的高阶微商，他又将论文用拉丁语写出寄给了当时在柏林科学院任职的数学家欧拉。
不久后，他获知这一成果早在半个世纪前就被莱布尼兹取得了。
这个并不幸运的开端并未使拉格朗日灰心，相反，更坚定了他投身数学分析领域的信心。
19岁这年是拉格朗日一生中最值得自豪的时期，这一年，他取得了数学上的巨大成就。
他以欧拉的思路和结果为依据，用纯分析的方法求变分极值。
第一篇论文“极大和极小的方法研究”，拉开了他研究变分法的序幕。
1755年8月，拉格朗日在给欧拉的信中，论述他的研究成果。
欧拉充分肯定了他的新方法，并在自己的论文中将此方法命名为“变分法”，至今已发展成为一个数学分支。
变分法的创立，使拉格朗日在都灵声名大震，并使他在19岁时就当上了都灵皇家炮兵学校的教授，从此走向数学研究的道路，逐步成为当时第一流的科学家。
1756年，受欧拉的举荐，拉格朗日被任命为普鲁士科学院通讯院士。
1766年，拉格朗日接受普鲁士国王腓特烈的邀请，前往柏林任普鲁士科学院数学部主任，从此开始了他一生科学研究的鼎盛时期。
欧拉和拉格朗日的故事
欧拉（Leonhard Euler，1707年-1783年）是18世纪欧洲最杰出的
数学家之一，他的杰出成就涵盖了数学、物理、机械学、天文学和哲学。
他被认为是数学史上最伟大的数学家之一，也是瑞士和俄罗斯数学家最重
要的代表。
拉格朗日（Joseph Louis Lagrange，1736年-1813年）是18世纪末
和19世纪初欧洲最杰出的数学家之一，他的成就也遍及各个领域，包括
数学、物理和天文学。
拉格朗日是欧拉及其学派的主要继承人之一，他的
工作为现代数学和物理学奠定了基础，被认为是史上最杰出的数学家之一。
欧拉和拉格朗日在数学历史上有许多共同点。
他们二人都出生于欧洲，欧拉是瑞士人，而拉格朗日是意大利人。
他们都在年轻时就表现出了超凡
的才华，分别成为了当时欧洲数学界的核心人物。
两人在数学上有许多交
流和协作，使得他们的理论成果更为复杂和准确。
在科学领域，欧拉和拉
格朗日是数学和物理学的领袖人物，同时也是理论物理学和天文学的巨匠。
欧拉和拉格朗日的逝世时间也有些相似之处，他们均活到了很高的年龄，同时在晚年遭遇疾病的困扰，最终不幸逝世。
欧拉逝世的时候，他的
大部分成就已经被世人广泛认可，而拉格朗日的成就则多数被认为是在其
逝世后才被充分认识和重视的。
欧拉和拉格朗日并肩成为数学史上最伟大的数学家之一，他们给后世
留下了无穷无尽的贡献，他们的理论成果被广泛应用于现代数学、物理学、天文学和理论物理学的发展当中。
拉格朗日英文作文简介English:Lagrange, whose full name is Joseph-Louis Lagrange, was an influential mathematician and astronomer in the 18th century. He was born in Turin, Italy in 1736 and later moved to Berlin, Prussia where he spent most of his career. Lagrange made significant contributions to various fields of mathematics, including number theory, calculus, and mechanics. He is best known for his work in the calculus of variations and celestial mechanics, where he developed the method of Lagrange multipliers and the concept of Lagrangian points in the study of planetary motion. His work laid the foundation for many important mathematical and scientific discoveries, and he is remembered as one of the leading mathematicians of his time.中文翻译:拉格朗日，全名为约瑟夫-路易斯·拉格朗日，是18世纪影响深远的数学家和天文学家。
他于1736年出生在意大利都灵，并后来移居普鲁士柏林，在那里度过了他的大部分职业生涯。
18世纪欧洲最伟大的数学家拉格朗日的故事导读：约瑟夫·拉格朗日(Joseph-Louis Lagrange，1736~1813)全名为约瑟夫·路易斯·拉格朗日，法国著名家、物理学家。
他在数学、力学和天文学三个学科领域中都有历史性的贡献，其中尤以数学方面的成就最为突出。
拉格朗日(1736—1813)，法国著名的数学家、力学家、天文学家，变分法的开拓者和分析力学的奠基人。
他曾获得过18世纪“欧洲最大之希望、欧洲最伟大的数学家”的赞誉。
拉格朗日出生在意大利的都灵。
由于是长子，父亲一心想让他法律，然而，拉格朗日对法律毫无兴趣，偏偏喜爱上文学。
直到16岁时，拉格朗日仍十分偏爱文学，对数学尚未产生兴趣。
16岁那年，他偶然读到一篇介绍牛顿微积分的文章《论分析方法的优点》，使他对牛顿产生了无限崇拜和敬仰之情，于是，他下决心要成为牛顿式的数学家。
在进入都灵皇家炮兵学院学习后，拉格朗日开始有计划地自学数学。
由于刻苦，他的进步很快，尚未毕业就担任了该校的数学教学工作。
20岁时就被正式聘任为该校的数学副教授。
从这一年起，拉格朗日开始研究“极大和极小”的问题。
他采用的是纯分析的方法。
1758年8月，他把自己的研究方法写信告诉了欧拉，欧拉对此给予了极高的评价。
从此，两位大师开始频繁通信，就在这一来一往中，诞生了数学的一个新的分支——变分法。
18世纪欧洲最伟大的数学家拉格朗日的故事1759年，在欧拉的推荐下，拉格朗日被提名为柏林科学院的通讯院士。
接着，他又当选为该院的外国院士。
1762年，法国科学院悬赏征解有关月球何以自转，以及自转时总是以同一面对着地球的难题。
拉格朗日写出一篇出色的论文，地解决了这一问题，并获得了科学院的大奖。
拉格朗日的名字因此传遍了整个欧洲，引起世人的瞩目。
两年之后，法国科学院又提出了木星的4个卫星和太阳之间的摄动问题的所谓“六体问题”。
面对这一难题，拉格朗日毫不畏惧，经过数个不眠之夜，他终于用近似解法找到了答案，从而再度获奖。
Joseph-Louis LagrangeBorn: 25 Jan 1736 in Turin, Sardinia-Piedmont (now Italy) Died: 10 April 1813 in Paris, FranceArticle by:J J O'Connor and E F RobertsonJanuary 1999Edited by: XiaJingbo , mail to : xjb@Joseph-Louis Lagrange is usually considered to be a French mathematician, but the Italian Encyclopaedia [40] refers to him as an Italian mathematician. They certainly have some justification in this claim since Lagrange was born in Turin and baptised in the name of Giuseppe Lodovico Lagrangia. Lagrange's father was Giuseppe Francesco Lodovico Lagrangia who was Treasurer of the Office of Public Works and Fortifications in Turin, while his mother Teresa Grosso was the only daughter of a medical doctor from Cambiano near Turin. Lagrange was the eldest of their 11 children but one of only two to live to adulthood.Turin had been the capital of the duchy of Savoy, but became the capital of the kingdom of Sardinia in 1720, sixteen years before Lagrange's birth. Lagrange's family had French connections on his father's side, his great-grandfather being a French cavalry captain who left France to work for the Duke of Savoy. Lagrange always leant towards his French ancestry, for as a youth he would sign himself Lodovico LaGrange or Luigi Lagrange, using the French form of his family name.Despite the fact that Lagrange's father held a position of some importance in the service of the king of Sardinia, the family were not wealthy since Lagrange's father had lost large sums of money in unsuccessful financial speculation. A career as a lawyer was planned out for Lagrange by his father, and certainly Lagrange seems to have accepted this willingly. He studied at the College of Turin and his favourite subject was classical Latin. At first he had no great enthusiasm for mathematics, finding Greek geometry rather dull.Lagrange's interest in mathematics began when he read a copy of Halley's 1693 work on the use of algebra in optics. He was also attracted to physics by the excellent teaching of Beccaria at the College of Turin and he decided to make a career for himself in mathematics. Perhaps the world of mathematics has to thank Lagrange's father for his unsound financial speculation, for Lagrange later claimed:-If I had been rich, I probably would not have devoted myself to mathematics.He certainly did devote himself to mathematics, but largely he was self taught and did not have the benefit of studying with leading mathematicians. On 23 July 1754 he published his first mathematical work which took the form of a letter written in Italian to Giulio Fagnano. Perhaps most surprising was the name under which Lagrange wrote this paper, namely Luigi De la Grange Tournier. This work was no masterpiece and showed to some extent the fact that Lagrange was working alone without the advice of a mathematical supervisor. The paper draws an analogy between the binomial theorem and the successive derivatives of the product of functions.Before writing the paper in Italian for publication, Lagrange had sent the results to Euler, who at this time was working in Berlin, in a letter written in Latin. The month after the paper was published, however, Lagrange found that the results appeared in correspondence between Johann Bernoulli and Leibniz. Lagrange was greatly upset by this discovery since he feared being branded a cheat who copied the results of others. However this less than outstanding beginning did nothing more than make Lagrange redouble his efforts to produce results of real merit in mathematics. He began working on the tautochrone, the curve on which a weighted particle will always arrive at a fixed point in the same time independent of its initial position. By the end of 1754 he had made someimportant discoveries on the tautochrone which would contribute substantially to the new subject of the calculus of variations (which mathematicians were beginning to study but which did not receive the name 'calculus of variations' before Euler called it that in 1766).Lagrange sent Euler his results on the tautochrone containing his method of maxima and minima. His letter was written on 12 August 1755 and Euler replied on 6 September saying how impressed he was with Lagrange's new ideas. Although he was still only 19 years old, Lagrange was appointed professor of mathematics at the Royal Artillery School in Turin on 28 September 1755. It was well deserved for the young man had already shown the world of mathematics the originality of his thinking and the depth of his great talents.In 1756 Lagrange sent Euler results that he had obtained on applying the calculus of variations to mechanics. These results generalised results which Euler had himself obtained and Euler consulted Maupertuis, the president of the Berlin Academy, about this remarkable young mathematician. Not only was Lagrange an outstanding mathematician but he was also a strong advocate for the principle of least action so Maupertuis had no hesitation but to try to entice Lagrange to a position in Prussia. He arranged with Euler that he would let Lagrange know that the new position would be considerably more prestigious than the one he held in Turin. However, Lagrange did not seek greatness, he only wanted to be able to devote his time to mathematics, and so he shyly but politely refused the position.Euler also proposed Lagrange for election to the Berlin Academy and he was duly elected on 2 September 1756. The following year Lagrange was a founding member of a scientific society in Turin, which was to become the Royal Academy of Sciences of Turin. One of the major roles of this new Society was to publish a scientific journal the Mélanges de Turin which published articles in French or Latin. Lagrange was a major contributor to the first volumes of the Mélanges de Turin volume 1 of which appeared in 1759, volume 2 in 1762 and volume 3 in 1766.The papers by Lagrange which appear in these transactions cover a variety of topics. He published his beautiful results on the calculus of variations, and a short work on the calculus of probabilities. In a work on the foundations of dynamics, Lagrange based his development on the principle of least action and on kinetic energy.In the Mélanges de Turin Lagrange also made a major study on the propagation of sound, making important contributions to the theory of vibrating strings. He had read extensively on this topic and he clearly had thought deeply on the works of Newton, Daniel Bernoulli, Taylor, Euler and d'Alembert. Lagrange used a discrete mass model for his vibrating string, which he took to consist of n masses joined by weightless strings. He solved the resulting system of n+1 differential equations, then let n tend to infinity to obtain the same functional solution as Euler had done. His different route to the solution, however, shows that he was looking for different methods than those of Euler, for whom Lagrange had the greatest respect.In papers which were published in the third volume, Lagrange studied the integration of differential equations and made various applications to topics such as fluid mechanics (where he introduced the Lagrangian function). Also contained are methods to solve systems of lineardifferential equations which used the characteristic value of a linear substitution for the first time. Another problem to which he applied his methods was the study the orbits of Jupiter and Saturn.The Académie des Sciences in Paris announced its prize competition for 1764 in 1762. The topic was on the libration of the Moon, that is the motion of the Moon which causes the face that it presents to the Earth to oscillate causing small changes in the position of the lunar features. Lagrange entered the competition, sending his entry to Paris in 1763 which arrived there not long before Lagrange himself. In November of that year he left Turin to make his first long journey, accompanying the Marquis Caraccioli, an ambassador from Naples who was moving from a post in Turin to one in London. Lagrange arrived in Paris shortly after his entry had been received but took ill while there and did not proceed to London with the ambassador. D'Alembert was upset that a mathematician as fine as Lagrange did not receive more honour. He wrote on his behalf [1]:- Monsieur de la Grange, a young geometer from Turin, has been here for six weeks. He has become quite seriously ill and he needs, not financial aid, for the Marquis de Caraccioli directed upon leaving for England that he should not lack for anything, but rather some signs of interest on the part of his native country ... In him Turin possesses a treasure whose worth it perhaps does not know.Returning to Turin in early 1765, Lagrange entered, later that year, for the Académie des Sciences prize of 1766 on the orbits of the moons of Jupiter. D'Alembert, who had visited the Berlin Academy and was friendly with Frederick II of Prussia, arranged for Lagrange to be offered a position in the Berlin Academy. Despite no improvement in Lagrange's position in Turin, he again turned the offer down writing:-It seems to me that Berlin would not be at all suitable for me while M Euler is there.By March 1766 d'Alembert knew that Euler was returning to St Petersburg and wrote again to Lagrange to encourage him to accept a post in Berlin. Full details of the generous offer were sent to him by Frederick II in April, and Lagrange finally accepted. Leaving Turin in August, he visited d'Alembert in Paris, then Caraccioli in London before arriving in Berlin in October. Lagrange succeeded Euler as Director of Mathematics at the Berlin Academy on 6 November 1766.Lagrange was greeted warmly by most members of the Academy and he soon became close friends with Lambert and Johann(III) Bernoulli. However, not everyone was pleased to see this young man in such a prestigious position, particularly Castillon who was 32 years older than Lagrange and considered that he should have been appointed as Director of Mathematics. Just under a year from the time he arrived in Berlin, Lagrange married his cousin Vittoria Conti. He wrote to d'Alembert:-My wife, who is one of my cousins and who even lived for a long time with my family, is a very good housewife and has no pretensions at all.They had no children, in fact Lagrange had told d'Alembert in this letter that he did not wish to have children.Turin always regretted losing Lagrange and from time to time his return there was suggested,for example in 1774. However, for 20 years Lagrange worked at Berlin, producing a steady stream oftop quality papers and regularly winning the prize from the Académie des Sciences of Paris. Heshared the 1772 prize on the three body problem with Euler, won the prize for 1774, another one onthe motion of the moon, and he won the 1780 prize on perturbations of the orbits of comets by theplanets.His work in Berlin covered many topics: astronomy, the stability of the solar system, mechanics,dynamics, fluid mechanics, probability, and the foundations of the calculus. He also worked onnumber theory proving in 1770 that every positive integer is the sum of four squares. In 1771 heproved Wilson's theorem (first stated without proof by Waring) that n is prime if and only if (n -1)! + 1 is divisible by n. In 1770 he also presented his important work Réflexions sur la résolution algébrique des équations which made a fundamental investigation of why equations of degrees up to 4 could be solved by radicals. The paper is the first to consider the roots of an equation as abstractquantities rather than having numerical values. He studied permutations of the roots and, althoughhe does not compose permutations in the paper, it can be considered as a first step in thedevelopment of group theory continued by Ruffini, Galois and Cauchy.Although Lagrange had made numerous major contributions to mechanics, he had not produced acomprehensive work. He decided to write a definitive work incorporating his contributions andwrote to Laplace on 15 September 1782:-I have almost completed a Traité de mécanique analytique, based uniquely on the principle of virtual velocities; but, as I do not yet know when or where I shall be able to have it printed, I am not rushing to put the finishing touches to it.Caraccioli, who was by now in Sicily, would have liked to see Lagrange return to Italy and hearranged for an offer to be made to him by the court of Naples in 1781. Offered the post ofDirector of Philosophy of the Naples Academy, Lagrange turned it down for he only wanted peace todo mathematics and the position in Berlin offered him the ideal conditions. During his years in Berlinhis health was rather poor on many occasions, and that of his wife was even worse. She died in 1783after years of illness and Lagrange was very depressed. Three years later Frederick II died andLagrange's position in Berlin became a less happy one. Many Italian States saw their chance andattempts were made to entice him back to Italy.The offer which was most attractive to Lagrange, however, came not from Italy but from Parisand included a clause which meant that Lagrange had no teaching. On 18 May 1787 he left Berlin tobecome a member of the Académie des Sciences in Paris, where he remained for the rest of hiscareer. Lagrange survived the French Revolution while others did not and this may to some extentbe due to his attitude which he had expressed many years before when he wrote:-I believe that, in general, one of the first principles of every wise man is to conform strictly to the laws of the country in which he is living, even when they are unreasonable.The Mécanique analytique which Lagrange had written in Berlin, was published in 1788. It had been approved for publication by a committee of the Académie des Sciences comprising of Laplace, Cousin, Legendre and Condorcet. Legendre acted as an editor for the work doing proof reading and other tasks. The Mécanique analytique summarised all the work done in the field of mechanics since the time of Newton and is notable for its use of the theory of differential equations. With this work Lagrange transformed mechanics into a branch of mathematical analysis. He wrote in the Preface:- One will not find figures in this work. The methods that I expound require neither constructions, nor geometrical or mechanical arguments, but only algebraic operations, subject to a regular and uniform course.Lagrange was made a member of the committee of the Académie des Sciences to standardise weights and measures in May 1790. They worked on the metric system and advocated a decimal base. Lagrange married for a second time in 1792, his wife being Renée-Françoise-Adélaide Le Monnier the daughter of one of his astronomer colleagues at the Académie des Sciences. He was certainly not unaffected by the political events. In 1793 the Reign of Terror commenced and the Académie des Sciences, along with the other learned societies, was suppressed on 8 August. The weights and measures commission was the only one allowed to continue and Lagrange became its chairman when others such as the chemist Lavoisier, Borda, Laplace, Coulomb, Brisson and Delambre were thrown off the commission.In September 1793 a law was passed ordering the arrest of all foreigners born in enemy countries and all their property to be confiscated. Lavoisier intervened on behalf of Lagrange, who certainly fell under the terms of the law, and he was granted an exception. On 8 May 1794, after a trial that lasted less than a day, a revolutionary tribunal condemned Lavoisier, who had saved Lagrange from arrest, and 27 others to death. Lagrange said on the death of Lavoisier, who was guillotined on the afternoon of the day of his trial:-It took only a moment to cause this head to fall and a hundred years will not suffice to produce its like.The École Polytechnique was founded on 11 March 1794 and opened in December 1794 (although it was called the École Centrale des Travaux Publics for the first year of its existence). Lagrange was its first professor of analysis, appointed for the opening in 1794. In 1795 the École Normale was founded with the aim of training school teachers. Lagrange taught courses on elementary mathematics there. We mentioned above that Lagrange had a 'no teaching' clause written into his contract but the Revolution changed things and Lagrange was required to teach. However, he was not a good lecturer as Fourier, who attended his lectures at the École Normale in 1795 wrote:- His voice is very feeble, at least in that he does not become heated; he has a very pronounced Italian accent and pronounces the s like z ... The students, of whom the majority are incapable of appreciating him, give him little welcome, but the professors make amends for it.Similarly Bugge who attended his lectures at the École Polytechnique in 1799 wrote:-... whatever this great man says, deserves the highest degree of consideration, but he is too abstract for youth.Lagrange published two volumes of his calculus lectures. In 1797 he published the first theory of functions of a real variable with Théorie des fonctions analytiques although he failed to give enough attention to matters of convergence. He states that the aim of the work is to give:- ... the principles of the differential calculus, freed from all consideration of the infinitely small or vanishing quantities, of limits or fluxions, and reduced to the algebraic analysis of finite quantities.Also he states:-The ordinary operations of algebra suffice to resolve problems in the theory of curves.Not everyone found Lagrange's approach to the calculus the best however, for example de Prony wrote in 1835:-Lagrange's foundations of the calculus is assuredly a very interesting part of what one might call purely philosophical study: but when it is a case of making transcendental analysis an instrument of exploration for questions presented by astronomy, marine engineering, geodesy, and the different branches of science of the engineer, the consideration of the infinitely small leads to the aim in a manner which is more felicitous, more prompt, and more immediately adapted to the nature of the questions, and that is why the Leibnizian method has, in general, prevailed in French schools.The second work of Lagrange on this topic Leçons sur le calcul des fonctions appeared in 1800.Napoleon named Lagrange to the Legion of Honour and Count of the Empire in 1808. On 3 April 1813 he was awarded the Grand Croix of the Ordre Impérial de la Réunion. He died a week later.References for Joseph-LouisLagrange1.J Itard, Biography in Dictionary of Scientific Biography(New York1970-1990).2.Biography in Encyclopaedia Britannica. [Available on the Web]Books:3.W Barroso Filho, La mécanique de Lagrange, Principes et méthodes(Paris, 1994).4.M T Borgato and L Pepe, Lagrange : Appunti per una biografiascientifica (Torino, 1990).5.P Cossali, Elogio di L Lagrange (Padua, 1813).6.J B J Delambre, Notice sur la vie et les ouvages de M le Comte JL Lagrange, Mémoires de la class des sciences matématique del'institut 1812 (Paris, 1816).7. D Herrmann, Joseph Louis Lagrange (1736-1813) (Berlin-Treptow,1963).8.O Stamfort, Lagrange, in H Wussing and W Arnold, Biographienbedeutender Mathematiker (Berlin, 1983).Articles:9. C Baltus, Continued fractions and the Pell equations : The work ofEuler and Lagrange, Comm. Anal. Theory Contin. Fractions3(1994), 4-31.10.W Barroso Filho and C Comte, La formalisation de la dynamique parLagrange (1736-1813), in Sciences à l'époque de la RévolutionFrançaise (Paris, 1988), 329-348.11.G Beaujouan, Documents nouveaux concernant Lagrange, Rev. Hist. Sci.Appl.3 (1950), 110-132.12.E Bellone, Boltzmann and Lagrange : 'classical' quanta and beliefsabout irreversibility (Italian), in Science and philosophy(Milan, 1985), 517-532.13.M T Borgato and L Pepe, The family letters of Joseph-Louis Lagrange(Italian), Boll. Storia Sci. Mat.9 (2) (1989), 192-318.14.M T Borgato and L Pepe, An unpublished memoir of Lagrange on thetheory of parallels (Italian), Boll. Storia Sci. Mat.8(2) (1988), 307-335.15.M T Borgato and L Pepe, Lagrange in Turin (1750-1759) and hisunpublished lectures at the Royal School of Artillery (Italian), Boll. Storia Sci. Mat.7 (2) (1987), 3-43.16.B Buraux-Bourgeois, L'analyse diophantienne chez Lagrange, inAnalyse diophantienne et géométrie algébrique (Paris, 1993),13-23.17.I Chobanov, Lagrange and mechanics : myth and reality (Bulgarian),Annuaire Univ. Sofia Fac. Math. Méc.78 (2) (1984), 3-46.18.C Comte, Joseph-Louis Lagrange poète scientifique et citoyeneuropeen, La Recherche208 (1989), 394-396.19.P Costabel, Lagrange et l'art analytique, Comptes rendus del'Académie des sciences. Série génerale, La vie des sciences(1989), 167-177.20.P Delsedime, La disputa delle corde vibranti ed una lettera ineditadi Lagrange a Daniel Bernoulli, Physis - Riv. Internaz. Storia Sci.13 (2) (1971), 117-146.21.P Dugac, La théorie des fonctions analytiques de Lagrange et lanotion d'infini, in Konzepte des mathematisch Unendlichen im 19.Jahrhundert (Göttingen, 1990), 34-46.22.S B Engelsman, Lagrange's early contributions to the theory offirst-order partial differential equations, Historia Math.7 (1) (1980), 7-23.23.C G Fraser, Isoperimetric problems in the variational calculus ofEuler and Lagrange, Historia Math.19 (1) (1992), 4-23.24.C G Fraser, Lagrange's analytical mathematics, its Cartesianorigins and reception in Comte's positive philosophy, Stud. Hist.Philos. Sci.21 (2) (1990), 243-256.25.C G Fraser, Joseph Louis Lagrange's algebraic vision of the calculus,Historia Math.14 (1) (1987), 38-53.26.C G Fraser, J L Lagrange's changing approach to the foundations ofthe calculus of variations, Arch. Hist. Exact Sci.32 (2) (1985), 151-191.27.C G Fraser, J L Lagrange's early contributions to the principlesand methods of mechanics, Arch. Hist. Exact Sci.28 (3) (1983), 197-241.28.C G Fraser, Isoperimetric problems in the calculus of variationsof Euler and Lagrange (Spanish), Mathesis8 (1) (1992), 31-53.29.M Galuzzi, Lagrange's essay 'Recherches sur la manière de formerdes tables des planètes d'après les seules observations', Rev.Histoire Math.1 (2) (1995), 201-233.30.J V Grabiner, The calculus as algebra, the calculus as geometry :Lagrange, Maclaurin, and their legacy, in Vita mathematica(Washington, DC, 1996), 131-143.31.I Grattan-Guinness, A Paris curiosity, 1814 : Delambre's obituaryof Lagrange, and its 'supplement', in Science and philosophy(Milan, 1985), 664-677.32.A T Grigor'yan, Lagrange's works on mechanics (Russian), VoprosyIstor. Estestvoznan. i Tehn. Vyp.20 (1966), 46-51.33.A T Grigor'yan, Lagrange's work on mechanics. On the occasion ofthe bicentennial of the publication of 'Mécanique analytique' (Russian), Studies in the history of physics and mechanics 1989 'Nauka' (Moscow, 1989), 157-170.34.R R Hamburg, The theory of equations in the 18th century : the workof Joseph Lagrange, Arch. History Exact Sci.16 (1) (1976/77), 17-36.35.Y Hirano, Quelques remarques sur les travaux de Lagrange - quiconcernent la théorie des équations algébriques et la notionpréliminaire de groupes, Historia Sci. (2) 5 (1) (1995), 75-84.36.H N Jahnke, A structuralist view of Lagrange's algebraic analysisand the German combinatorial school, in The space of mathematics (Berlin, 1992), 280-295.37.G Julia, La vie et l'oeuvre de J.-L. Lagrange, Enseignement Math.39 (1942-1950), 9-21.38.T Koetsier, Joseph Louis Lagrange (1736-1813) : his life, his workand his personality, Nieuw Arch. Wisk. (4) 4 (3) (1986), 191-205.39.A Korn, Joseph Louis Lagrange, Mathematische GeschichteSitzungsberichte12 (1913), 90-94.grange (o Lagrangia), Giuseppe Luigi, Enciclopedia Italiana XX(Rome, 1933), 380-381.41.G Loria, Essai d'une bibliographie de Lagrange, Isis40 (1949),112-117.42.V A Ogannisjan, Joseph Louis Lagrange (Russian), Armjan. Gos. Ped.Inst. Sb. Naucn. Trud. Ser. Fiz.-Mat. Vyp.3 (1966), 99-110. 43.M Panza, Eliminating time : Newton, Lagrange and the inverse problemof resisting motion (Italian), Conference on the History ofMathematics (Rende, 1991), 487-537.44.M Panza, The analytical foundation of mechanics of discrete systemsin Lagrange's 'Théorie des fonctions analytiques', compared with Lagrange's earlier treatments of this topic II, Historia Sci. (2)1 (3) (1992), 181-212.45.M Panza, The analytical foundation of mechanics of discrete systemsin Lagrange's 'Théorie des fonctions analytiques', compared with Lagrange's earlier treatments of this topic I, Historia Sci. (2)1 (2) (1991), 87-132.46.L Pepe, Supplement to the bibliography of Lagrange : the 'rapports'to the first class of the Institute (Italian), Boll. Storia Sci.Mat.12 (2) (1992), 279-301.47.L Pepe, Three 'first editions' and an unpublished introduction toLagrange's 'Théorie des fonctions analytiques' (Italian), Boll.Storia Sci. Mat.6 (1) (1986), 17-44.48.L Pepe, Lagrange and his treatises on mathematical analysis(Italian), in Symposia mathematica XXVII(London-New York, 1986), 69-99.49.R Roth, The origin of the theory of groups : The theorem of Lagrange(1771) (Spanish), Bol. Mat.3 (1969), 137-141.50.G Sarton, Lagrange's personality (1736-1813), Proc. Amer. Philos.Soc.88 (1944), 457-496.51.A L Shields, Lagrange and the 'Mecanique analytique', TheMathematical intelligencer10 (4) (1988), 7-10.52.Z Siki'c, Joseph Louis Lagrange (on the 250th anniversary of hisbirth), Matematika (Zagreb)15 (4) (1986), 47-50.53.J-M Souriau, La structure symplectique de la mécanique décrite parLagrange en 1811, Math. Sci. Humaines94 (1986), 45-54.54.A S Sumbatov, Developments of some of Lagrange's ideas in the worksof Russian and Soviet mechanicians, La 'Mécanique analytique' de Lagrange et son héritage, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat.Natur.126 (1992), suppl. 2, 169-200.55.V Szebehely, Lagrange and the three-body problem, La 'Mécaniqueanalytique' de Lagrange et son héritage, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur.126 (1992), suppl. 2, 201-213.56.R Taton, Lagrange et la Révolution française (juillet 1789 -novembre 1795), La 'Mécanique analytique' de Lagrange et sonhéritage, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur.126 (1992), suppl. 2, 215-255.57.R Taton, Sur quelques pièces de la correspondance de Lagrange pourles années 1756-1758, Boll. Storia Sci. Mat.8 (1) (1988), 3-19.58.R Taton, Le départ de Lagrange de Berlin et son installation à Parisen 1787, Rev. Histoire Sci.41 (1) (1988), 39-74.59.R Taton, Les débuts de la carrière mathématique de Lagrange : lapériode turinoise (1736-1766), in Symposia mathematica XXVII(London-New York, 1986), 123-145.60.R Taton, Inventaire chronologique de l'oeuvre de Lagrange, Rev.Hist. Sci.27 (1974), 3-36.61.F Verhulst, Perturbation theory from Lagrange to van der Pol, NieuwArch. Wisk. (4) 2 (3) (1984), 428-438.。
