索菲·热尔曼Sophie Germain

索菲·热尔曼是安布鲁瓦兹-弗朗索瓦索菲·热尔曼（1726-1821）的次女，他是一位富有的商人、金匠和珠宝商，后来成为丝绸商人，母亲是玛丽-玛德莱娜·格吕格吕（?-1823），她是金匠让·格吕格吕的女儿，让·格吕格吕是哲学家和政治经济学家的朋友。安布鲁瓦兹-弗朗索瓦于1789年当选为国民议会代表，后来成为法国银行的董事。索菲·热尔曼的姐姐是玛丽-玛德莱娜索菲·热尔曼（生于1770年5月29日），妹妹是安热莉克-安布鲁瓦兹索菲·热尔曼（生于1779年）。玛丽-玛德莱娜索菲·热尔曼于1790年与公证人夏尔·莱尔贝特（1752-1836）结婚，他们有一个儿子阿芒-雅克·莱尔贝特（1791-1864），后来成为律师、运动员和政治家。安热莉克-安布鲁瓦兹索菲·热尔曼于1809年与医生勒内-克洛德·热弗鲁瓦（1767-1831）结婚，在他1831年去世后，她于1833年与另一位医生若阿尚-亨利·迪托谢（1776-1847）结婚，他是一位领先的植物学家和生理学家。我们给出了索菲·热尔曼姐妹的一些细节，因为她没有结婚，因此她的姐妹及其家庭在她的生活中扮演了重要角色。

索菲·热尔曼出生在圣但尼街336号的家中，这里是那些对自由改革感兴趣的人的聚会场所，她在早年就接触到了政治和哲学讨论。1789年法国大革命爆发，同年5月法国成为君主立宪制国家，随后7月14日攻占巴士底狱。索菲·热尔曼的父亲当选为巴黎第三等级的代表，然后是国民议会，并深入参与了政治事件。索菲·热尔曼当时十三岁，在此之前她一直在家接受教育。随着巴黎陷入混乱，正常生活暂停，她决定去她父亲藏书丰富的图书馆，通过读书来打发时间。

索菲·热尔曼十三岁时大革命爆发，她翻阅父亲的书，偶然看到J.É. 蒙蒂克拉的Histoire des MathématiquesⓉ（数学史）。她读到J.É. 蒙蒂克拉记述阿基米德对数学的热情：-

[阿基米德]会忘记吃喝。他的仆人不得不替他记住用餐时间，几乎不得不强迫他满足这些人的需求。

她读到阿基米德因不愿停止做数学而死于罗马士兵之手。她被这个故事打动，决定自己也必须成为数学家。这里有某种东西能让她如此投入，以至于逃离周围正在进行的大革命的混乱。她告诉父母自己想成为数学家，但他们完全反对她的想法，告诉她这不是女孩该从事的职业。

然而，索菲·热尔曼在她父亲的图书馆里继续学习，从阅读艾蒂安·贝祖的Cours complet de mathématiques à l'usage de la marine et de l'artillerieⓉ（海军和炮兵用数学完整教程）（1770）第一卷开始，该卷涵盖算术。她偶然看到Jacques Antoine-Joseph Cousin（1739-1800）关于微积分的一本书，即Leçons de calcul différentiel et de calcul intégralⓉ（微分和积分学教程）（1777），她觉得这本书引人入胜。为了继续阅读更高级的微积分文本，她自学了拉丁语和希腊语。她在夜里裹着毯子阅读艾萨克·牛顿和莱昂哈德·欧拉，而她的父母在睡觉——他们拿走了她的火、她的灯和她的衣服，试图迫使她远离书本。最终，她的父母减少了对她学习的反对，因为有一天早上醒来，看到索菲·热尔曼不在床上，发现她在图书馆里睡着了，那里冷得连墨水瓶里的墨水都冻成了固体。尽管索菲·热尔曼既没有结婚也没有获得专业职位，她的父亲在她一生中继续在经济上支持她。在接下来的几年里，她继续学习微分和积分学，而1793-94年恐怖统治时期周围的可怕事件反而帮助她专注于学习。

公共工程中央学校于1794年由拉扎尔·卡诺和加斯帕尔·蒙日创办。次年它更名为巴黎综合工科学校。索菲·热尔曼在学校开办时十八岁，正好是开始大学教育的年龄。然而，女孩无论如何都不可能成为学生。这并没有阻止索菲·热尔曼，她从巴黎综合工科学校获得了许多课程的讲义，包括路易·安托万-François Fourcroy的化学课程和约瑟夫·拉格朗日的分析课程。在约瑟夫·拉格朗日的讲座课程结束时，他邀请学生把自己的书面观察寄给他。索菲·热尔曼用笔名M LeBlanc提交了一篇论文，其独创性和洞察力使约瑟夫·拉格朗日寻找作者。我们注意到，笔名“M LeBlanc”并非编造的名字，而是曾就读于该校的学生安托万-August LeBlanc的名字。他在数学上并不很有天赋，很快就放弃了在巴黎综合工科学校的学业。

当约瑟夫·拉格朗日发现向他投稿的“M LeBlanc”是一位女性时，他对她工作的尊重依然如故，并成为她的赞助人和数学顾问。然而，她的教育杂乱无章、毫无系统，她从未接受过她所渴望的专业训练。约瑟夫·拉格朗日确实让他的同事们认识到索菲·热尔曼是一位有数学天赋的女孩，其中几位还给她写了信。例如，加斯帕尔·蒙日写信给她，讨论当重物位置发生无穷小变化时与杠杆相关的问题。其他人则与她讨论某些数学悖论。然而，尽管Jacques Antoine-Joseph Cousin确实提出与她见面，却没有人尝试提供系统的学习。值得注意的是，由于索菲·热尔曼是一位未婚女性，她与男性会面存在社交上的困难。

并非每个人都以她认为自己应得的尊重对待她。一个例子是拉朗德，他于1797年拜访了索菲·热尔曼。她开始与他谈论皮埃尔·西蒙·拉普拉斯的Exposition du système du mondeⓉ（《宇宙体系解说》），该书前一年才出版。拉朗德告诉她，她不应该读这样的著作，而应该读他那本书的第二版Astronomie des damesⓉ（《女士天文学》）（1795年）。这本“女士天文学”连一个数学方程都没有，索菲·热尔曼对他的建议感到受辱。拉朗德于1797年11月4日给她寄了一封道歉信，但她从未原谅他。

另一个冒犯她的人是希腊学家Anase de Villoisson。他这样做是因为他在自己写的一首诗中赞美了她，而我们从Villoisson于1802年7月14日写给她的道歉信中知道她有多么不快（见[35]）：-

小姐：
我冒昧地随信附上我那部不幸作品新版本的样本，其中有我曾告诉过您的更正和增补。小姐，在我怀疑真理的敬意会冒犯您的谦逊之前，M Paugens已将其插入他《法国文库》的第三期；您的谦逊与您的才华一样罕见。我以我的名誉重复我的歉意以及深切而永恒的遗憾：我本不该允许自己以书面形式谈到您，小姐，我的钦佩将永远沉默，并被求得原谅一个错误或一个非自愿过失的愿望所束缚，也被我向您母亲和姐姐立下的深切敬意所束缚。我有幸成为，小姐，
您谦卑而顺从的仆人，
Anase de Villoisson。

索菲·热尔曼就他1798年Essai sur le Théorie des NombresⓉ（《数论初探》）中提出的问题写信给阿德里安-马里·勒让德，此后两人之间的通信实际上成了一种合作。阿德里安-马里·勒让德将她的一些发现收入了Théorie第二版的附录中。她的一些信件后来发表在她的Oeuvres Philosophique de Sophie GermainⓉ（《索菲·热尔曼哲学著作集》）中。

索菲·热尔曼最著名的通信对象是卡尔·弗里德里希·高斯。她对他1801年Disquisitiones ArithmeticaeⓉ（《算术研究》）中提出的方法有了透彻的理解。1804年至1809年间，她给他写了十几封信，起初再次采用“M LeBlanc”这个假名，因为她担心自己身为女性会被忽视。收到她的第一封信后，卡尔·弗里德里希·高斯写信给天文学家Heinrich Wilhelm Matthias Olbers（1758-1840）（例如见[7]）：-

我最近有幸收到巴黎年轻数学家LeBlanc的来信，他热情地熟悉了高等数学，并显示出他对我的《算术研究》钻研得多么深入。

在他们通信期间，卡尔·弗里德里希·高斯对她的数论证明给予了高度赞扬，他在给同事的信中也重复了这一评价。直到1806年法国占领她的家乡布伦瑞克之后，索菲·热尔曼的真实身份才向卡尔·弗里德里希·高斯透露。她回想起阿基米德的命运，担心卡尔·弗里德里希·高斯的安全，便联系了一位与她家相识的法国将军。卡尔·弗里德里希·高斯既不认识这位将军，也不认识索菲·热尔曼，在他打听之后，她被迫透露了自己的身份，写信给他说[4]：-

我对你来说并不像你可能以为的那样完全陌生，但因为害怕女性科学家所遭受的嘲笑，我之前在与你通信时用了M LeBlanc这个名字。……我希望我今天向你透露的信息不会使我失去你在一个借来的名字下给予我的荣誉……

当卡尔·弗里德里希·高斯收到这封信时，他对她更加赞扬。他写道：-

但是，我该如何向你描述我的钦佩与惊愕呢？当我看到我尊敬的通信者M LeBlanc先生变身为这位杰出的索菲·热尔曼，他给出了如此辉煌的榜样，让我难以相信。对抽象科学总体的品味，尤其是对数的奥秘的品味，极为罕见；人们对此感到惊讶；这门崇高科学的迷人魅力只向那些有勇气深入其中的人展现……事实上，没有什么能比你对这门科学的偏爱更令我感到如此荣幸且毫不含糊地证明，这门科学——它以其众多的欢乐丰富了我的生活——的吸引力并非虚幻。

我们在上文提到，索菲·热尔曼的妹妹Angélique-Ambroise 索菲·热尔曼于1809年嫁给了医生Rene-Claude Geoffroy。他在巴黎Braque街4号有一栋漂亮的联排别墅，婚后一段时间，整个索菲·热尔曼家族搬进了那所房子。我们还要注意到，索菲·热尔曼与她的外甥，即她姐姐的儿子Amand-Jacques Lherbette关系密切。她指定Lherbette为她的文学遗嘱执行人，事实上，在她去世后，他出版了她的一些未完成作品。

1808年，德国物理学家Ernst Florens Friedrich Chladni（1756-1827）访问了巴黎，在那里他进行了振动板的实验，展示了所谓的Chladni图形。
你可以在THIS LINK看到一些Chladni图形。

法兰西学院设立了一项奖金竞赛，挑战如下：-

……构建一个弹性表面的数学理论，并指出它与经验证据的吻合程度。

所有参赛作品的截止日期定为两年。

大多数数学家没有尝试解决这个问题，因为约瑟夫·拉格朗日曾说过可用的数学方法不足以解决它。然而，索菲·热尔曼在接下来的十年里尝试推导弹性理论，与一些最杰出的数学家和物理学家竞争和合作。

事实上，索菲·热尔曼是1811年竞赛中唯一的参赛者，但她的工作没有获奖。她没有从物理学原理推导出她的假设，当时她也不可能这样做，因为她没有接受过分析和变分法的训练。然而，她的工作确实激发了新的见解。约瑟夫·拉格朗日，竞赛的评委之一，纠正了索菲·热尔曼计算中的错误，并提出了一个他认为可能描述Chladni图案的方程。

竞赛截止日期延长了两年，索菲·热尔曼再次提交了唯一的参赛作品。她证明了约瑟夫·拉格朗日的方程在几种情况下确实产生了Chladni图案，但无法从物理原理给出约瑟夫·拉格朗日方程的令人满意的推导。由于这项工作，她获得了荣誉奖。

索菲·热尔曼在1815年重新开放的竞赛中的第三次尝试被认为值得获得一公斤黄金奖章的奖励，尽管其数学严谨性仍有不足。令公众失望的是，她并未如预期般出现在颁奖典礼上。尽管这是她科学生涯的高峰，但有人提出[4]：-

……她认为评委们没有充分认识到她的工作……

并且

……科学界没有给予她似乎应得的尊重。

当然，西莫恩·德尼·泊松，她在弹性问题上的主要竞争对手，也是比赛的评委之一，对她的工作只发来一封简短而正式的确认信，避免与她进行任何严肃的讨论，并在公开场合无视她。

正如一位传记作者所说：-

尽管是索菲·热尔曼首先尝试解决一个难题，但当其他受过更多训练、能力更强、人脉更广的人在她的工作基础上继续推进，弹性成为重要的科学主题时，她却被排除在外。女性根本不被认真对待。

索菲·热尔曼试图扩展她的研究，在1825年提交给法兰西研究院一个委员会的一篇论文中，该委员会成员包括西莫恩·德尼·泊松、Gaspard de Prony和皮埃尔·西蒙·拉普拉斯。这项工作存在若干缺陷，但委员会没有向作者报告这些缺陷，而是干脆忽略了这篇论文。它后来从de Prony的论文中被发现，并于1880年发表。

在这一时期所做的工作中，有关于费马大定理的工作以及一个后来被称为索菲·热尔曼定理的定理。从1738年到1840年恩斯特·爱德华·库默尔的贡献之前，这一直是与皮埃尔·德·费马大定理相关的最重要结果。她于1819年5月12日写信给卡尔·弗里德里希·高斯[4]：-

尽管我在振动曲面理论（如果我能对我所想的圆柱曲面做一些实验并感到满意的话，我还有很多要补充的）上努力了一段时间，但我从未停止思考数论。……早在我们的科学院提出将证明皮埃尔·德·费马方程的不可能性作为奖项主题之前很久，这个挑战……就常常折磨着我。

Larry Riddle在[30]中写道：-

在这封信中，她提出了证明皮埃尔·德·费马大定理的宏伟计划。她的目标是证明对于每个奇素数指数$p$，存在无穷多个形式为$2Np + 1$的辅助素数，使得非零$p$次幂剩余$x^{p} \mod (2Np + 1)$的集合不包含任何连续整数。如果$x^{p} + y^{p} = z^{p}$有解，那么索菲·热尔曼观察到任何这样的辅助素数都必须整除数$x, y$或$z$之一。她的信件和在各图书馆发现的手稿显示了她对小于100的素数$p$以及$N$从1到10的辅助素数的分析。……然而，正如索菲·热尔曼向卡尔·弗里德里希·高斯承认的那样，她甚至对于单个素数指数也无法确立无穷多个辅助素数的存在。事实上，索菲·热尔曼的宏伟计划注定要失败，因为后来表明对于每个奇素数$p$，只有有限个辅助素数满足非连续$p$次幂剩余条件。索菲·热尔曼本人最终在给阿德里安-马里·勒让德的信中证明，对于$p = 3$，辅助素数7和13是唯一有效的。尽管如此，这是第一次有人设计出证明皮埃尔·德·费马大定理对无穷多个素数指数成立而非逐例处理的计划。

Andrea Del Centina在[18]中重新评估了索菲·热尔曼关于皮埃尔·德·费马大定理的工作，并得出结论：-

有必要重新评价索菲·热尔曼在皮埃尔·德·费马最后定理方面的工作。她不仅独立于阿德里安-马里·勒让德发展出了归于她名下的定理，事实上她还完成了通常归功于他的部分额外工作，而且她还证明（或几乎证明）了多年后被重新发现的结果。近200年后，她的思想仍然处于核心地位。1985年，利用她方法的一个推广，伦纳德·阿德曼和托马斯·利特尔·希思-Brown [皮埃尔·德·费马最后定理的第一种情形，1985]以及独立的Fouvry [Brun-爱德华·查尔斯·蒂奇马什定理：在皮埃尔·德·费马Ⓣ定理中的应用（Brun-蒂奇马什定理：在费马定理中的应用），1985]，证明了皮埃尔·德·费马最后定理的第一种情形对无穷多个指数成立。

索菲·热尔曼继续从事数学和哲学工作，直至去世。去世前，她草拟了一篇哲学论文，该文在她去世后以Considérations générales sur l'état des sciences et des lettresⓉ（关于科学与文学状况的一般评论）为题发表在Oeuvres philosophiques上。她的论文受到August Comte的高度赞扬。Amy Dahan Dalmédico在[16]中写道：-

在这篇论文中，她试图找出所有人类活动中的智力过程。她相信智力宇宙充满了类比。人类精神识别出这些类比，进而导致对自然现象和宇宙规律的发现。我们应当认识到索菲·热尔曼的生活与我们自己的生活之间的类比，它们应当引导我们在偏见面前追求卓越。

在[21]中，Jesse A Fernandez Martinez指出了索菲·热尔曼如何预见了August Comte：-

Comte对尼古拉·德·孔多塞和Saint-Simon的亏欠经常被提及。直到最近才发现，在他的体系的主要特征上，索菲·热尔曼对他的预见有多么明显。Dühring在其《从开端到现在的哲学批判史》（第3版，莱比锡，1878年）中，在给出她工作的完整摘要后说：“从上文可以看出，在索菲·热尔曼的著作中无需使用这个词就能找到的实证主义，包含了迄今为止与Auguste Comte的名字联系在一起的那些本质特征。”

她于1829年罹患乳腺癌，但未被此病和1830年革命的战斗所吓倒，她完成了关于数论和曲面曲率（1831年）的论文。

索菲·热尔曼于1831年6月在巴黎萨瓦街13号去世，该处至今仍在，并有一块纪念牌。她的死亡证明上列出的身份不是数学家或科学家，而是“食利者”（有产者）。在[35]中，作者引用了H J Monans，Woman in Science（1913年）的话：-

然而，说来奇怪，当国家官员前来为这位最杰出的法国科学院成员的显赫同事和合作者开具死亡证明时，却将她标注为食利年金领取者——而非数学家。这还不是全部。当埃菲尔铁塔建成时，工程师们不得不特别关注所用材料的弹性，在这座高耸的建筑上刻有七十二位学者的名字。但人们不会在这份名单中找到那位天才之女的名字，她的研究为建立金属弹性理论做出了巨大贡献，即索菲·热尔曼。她是否因为与玛利亚·阿涅西在法兰西学院中无资格相同的理由——因为她是女性——而被排除在这份名单之外？看来如此。如果情况确实如此，那么对于那些对一位如此有功于科学、并以其成就赢得了名人堂中令人羡慕地位的人如此忘恩负义的人，就更应感到羞耻。

索菲·热尔曼被安葬在拉雪兹神父公墓。
你可以在THIS LINK看到她的坟墓照片。

Marie-Sophie Germain was the middle daughter of Ambroise-François Germain (1726-1821), a prosperous merchant, goldsmith and jeweller who later became a silk-merchant, and Marie-Madeleine Gruguelu (?-1823) the daughter of the goldsmith Jean Gruguelu who was a friend of philosophers and political economists. Ambroise-François was elected a deputy to the National Assembly in 1789 and later became a director of the Bank of France. Sophie's elder sister was Marie-Madeleine Germain (born 29 May 1770) and her younger sister was Angélique-Ambroise Germain (born 1779). Marie-Madeleine Germain married the notary Charles Lherbette (1752-1836) in 1790 and they had one son Amand-Jacques Lherbette (1791-1864) who became a lawyer, sportsman and politician. Angélique-Ambroise Germain married the doctor Rene-Claude Geoffroy (1767-1831) in 1809 and, after his death in 1831, she married another medical man Joachim-Henri Dutochet (1776-1847), a leading botanist and physiologist, in 1833. We have given some details of Sophie's sisters because she did not marry and, as a consequence, her sisters and their families played a large part in her life.

Sophie's home at 336 rue St Denis, where she was born, was a meeting place for those interested in liberal reforms and she was exposed to political and philosophical discussions during her early years. The year 1789 saw the outbreak of the French Revolution when in May of that year France became a constitutional monarchy followed by the storming of the Bastille on 14 July. Sophie's father was elected a deputy of the Third Estate for the city of Paris, then the National Assembly, and became very involved in the political events. Sophie was thirteen years old at this time and up until then she had been educated at home. With Paris in chaos and normal life suspended, she decided that she would go to her father's library, which was extensive, and pass the time reading books.

Sophie, thirteen years old when the Revolution broke out, looked through her father's books, coming across Étienne Montucla's Histoire des Mathématiques Ⓣ. She read Montucla's account of Archimedes' passion for mathematics:-

[Archimedes] would forget to eat and drink. His servant would have to remember his meal times for him and would almost have to force him to satisfy these human needs.

She read about Archimedes' death at the hands of a Roman soldier because he would not stop doing mathematics. She was moved by this story and decided that she too must become a mathematician. Here was something which would let her become so involved that she would escape from the chaos of the Revolution going on around her. She told her parents she wanted to become a mathematician but they were totally opposed to her ideas, telling her that it was no occupation for a girl.

Sophie pursued her studies in her father's library, however, beginning with reading the first volume of Étienne Bézout's Cours complet de mathématiques à l'usage de la marine et de l'artillerie Ⓣ (1770) which covered arithmetic. She came across a book by Jacques Antoine-Joseph Cousin (1739-1800) on the calculus, namely Leçons de calcul différentiel et de calcul intégral Ⓣ (1777), which she found fascinating. In order to move on to reading more advanced texts on calculus, she taught herself Latin and Greek. She read Newton and Euler at night while wrapped in blankets as her parents slept - they had taken away her fire, her light and her clothes in an attempt to force her away from her books. Eventually her parents lessened their opposition to her studies, after waking up one morning, seeing Sophie was not in her bed, and finding her asleep in the library which was so cold that the ink had frozen solid in the ink well. Although Germain neither married nor obtained a professional position, her father continued to support her financially throughout her life. She continued to study the differential and integral calculus over the following years with the horrific events around her during the Reign of Terror in 1793-94 only helping her concentrate on her studies.

The École Centrale des Travaux Publics was founded in 1794 by Lazare Carnot and Gaspard Monge. In the following year it was renamed the École Polytechnique. Sophie Germain was eighteen years old when the École opened and at exactly the right age to begin a university education. There was no way, however, a girl could become a student. This did not stop Germain who obtained lecture notes for many courses from École Polytechnique including Antoine-François Fourcroy's chemistry course and Joseph-Louis Lagrange's analysis course. At the end of Lagrange's lecture course he invited his students to send him their written observations. Using the pseudonym M LeBlanc, Germain submitted a paper whose originality and insight made Lagrange look for its author. We note that the pseudonym "M LeBlanc" was not a made-up name but rather it was the name of the student Antoine-August LeBlanc who had attended the École. He was not very gifted mathematically and had quickly given up his studies at the École Polytechnique.

When Lagrange discovered "M LeBlanc" who had submitted a paper to him was a woman, his respect for her work remained and he became her sponsor and mathematical counsellor. Her education was, however, disorganised and haphazard and she never received the professional training which she wanted. Lagrange certainly made his colleagues aware that Germain was a girl with mathematical talent and several of them wrote to her. Monge, for example, wrote to her about problems associated with a lever when infinitesimal changes in the position of a weight occur. Others discussed certain mathematical paradoxes with her. There was, however, no attempt to provide structured learning although Jacques Antoine-Joseph Cousin did offer to meet with her. It is worth noting that, since Germain was an unmarried woman, there were social difficulties in her meeting with men.

Not everyone treated her with the respect she felt she deserved. One case was Jérôme Lalande who visited Germain in 1797. She started to talk to him about Laplace's Exposition du système du monde Ⓣ which had only been published in the previous year. Lalande told her that she should not be reading such works, rather she should be reading the second edition of his book Astronomie des dames Ⓣ (1795). This "astronomy for ladies" does not contain a single mathematical equation and Germain felt insulted by his suggestion. Lalande sent her a letter of apology on 4 November 1797 but she never forgave him.

Another who offended her was the Hellenist, Anase de Villoisson. He did this by praising her in a poem he had written and we know how displeased she was from Villoisson's letter of apology written to her on 14 July 1802 (see [35]):-

Mademoiselle:
I dare to take the liberty to offer you, adjoined to this, a sample of the new edition of my unfortunate work with corrections and additions which I have told you about. M Paugens, Mademoiselle, had inserted it in the third issue of his 'Bibliothèque Française' before I became suspicious that the homage of the truth would shock your modesty, which is as rare as your talent. I repeat with my excuses and lively and eternal regret on my word of honour that I should never have permitted myself to speak of you, Mademoiselle, in writing, and that my admiration will always be silent and enchained by the desire to obtain pardon for an error, or for an involuntary fault, and by the deep respect which I have vowed to your mother and sister. I have the honour to be, Mademoiselle,
Your humble and obedient servant,
Anase de Villoisson.

Germain wrote to Legendre about problems suggested by his 1798 Essai sur le Théorie des Nombres Ⓣ, and the subsequent correspondence between the two became virtually a collaboration. Legendre included some of her discoveries in a supplement to the second edition of the Théorie. Several of her letters were later published in her Oeuvres Philosophique de Sophie Germain Ⓣ.

However, Germain's most famous correspondence was with Gauss. She had developed a thorough understanding of the methods presented in his 1801 Disquisitiones Arithmeticae Ⓣ. Between 1804 and 1809 she wrote a dozen letters to him, initially adopting again the pseudonym "M LeBlanc" because she feared being ignored because she was a woman. After receiving her first letter, Gauss wrote to the astronomer Heinrich Wilhelm Matthias Olbers (1758-1840) (see, for example, [7]):-

I recently had the pleasure of receiving a letter from LeBlanc, a young mathematician in Paris, who made himself enthusiastically familiar with higher mathematics and showed how deeply he penetrated into my 'Disquisitiones Arithmeticae'.

During their correspondence, Gauss gave her number theory proofs high praise, an evaluation he repeated in letters to his colleagues. Germain's true identity was revealed to Gauss only after the 1806 French occupation of his hometown of Braunschweig. Recalling Archimedes' fate and fearing for Gauss's safety, she contacted a French general who was a friend of her family. Gauss knew neither the general nor Sophie Germain and, after he made enquiries, she was forced to reveal her identity, writing to him [4]:-

I am not as completely unknown to you as you might believe, but that fearing the ridicule attached to a female scientist, I have previously taken the name of M LeBlanc in communicating to you. ... I hope that the information that I have today confided to you will not deprive me of the honour you have accorded me under a borrowed name ...

When Gauss received this letter, he gave her even more praise. He writes:-

But how to describe to you my admiration and astonishment at seeing my esteemed correspondent M LeBlanc metamorphose himself into this illustrious personage [Sophie Germain] who gives such a brilliant example of what I would find it difficult to believe. A taste for the abstract sciences in general and above all the mysteries of numbers is excessively rare; one is astonished at it; the enchanting charms of this sublime science reveal themselves only to those who have the courage to go deeply into it ... Indeed nothing could prove to me in so flattering and less equivocal manner that the attractions of this science, which has enriched my life with so many joys, are not chimerical, as the predilection with which you have honoured it.

We mentioned above that Sophie's younger sister Angélique-Ambroise Germain married the doctor Rene-Claude Geoffroy in 1809. He had a fine town house at 4 Rue du Braque in Paris and, some time after the marriage, the whole Germain family moved into that home. Let us also note that Sophie became close to her nephew, Amand-Jacques Lherbette, the son of her elder sister. She named Lherbette as her literary executor, and indeed after her death he would publish some of her unfinished work.

In 1808, the German physicist Ernst Florens Friedrich Chladni (1756-1827) had visited Paris where he had conducted experiments on vibrating plates, exhibiting the so-called Chladni figures.
You can see some Chladni figures at THIS LINK.

The Institut de France set a prize competition with the following challenge:-

... formulate a mathematical theory of elastic surfaces and indicate just how it agrees with empirical evidence.

A deadline of two years for all entries was set.

Most mathematicians did not attempt to solve the problem, because Lagrange had said that the mathematical methods available were inadequate to solve it. Germain, however, spent the next decade attempting to derive a theory of elasticity, competing and collaborating with some of the most eminent mathematicians and physicists.

In fact, Germain was the only entrant in the contest in 1811, but her work did not win the award. She had not derived her hypothesis from principles of physics, nor could she have done so at the time because she had not had training in analysis and the calculus of variations. Her work did spark new insights, however. Lagrange, who was one of the judges in the contest, corrected the errors in Germain's calculations and came up with an equation that he believed might describe Chladni's patterns.

The contest deadline was extended by two years, and again Germain submitted the only entry. She demonstrated that Lagrange's equation did yield Chladni's patterns in several cases, but could not give a satisfactory derivation of Lagrange's equation from physical principles. For this work she received an honourable mention.

Germain's third attempt in the re-opened contest of 1815 was deemed worthy of the prize of a medal of one kilogram of gold, although deficiencies in its mathematical rigour remained. To public disappointment, she did not appear as anticipated at the award ceremony. Though this was the high point in her scientific career, it has been suggested that [4]:-

... she thought the judges did not fully appreciate her work ...

and that

... the scientific community did not show the respect that seemed due to her.

Certainly Poisson, her chief rival on the subject of elasticity and also a judge of the contest, sent a laconic and formal acknowledgement of her work, avoided any serious discussion with her and ignored her in public.

As one biographer phrases it:-

Although it was Germain who first attempted to solve a difficult problem, when others of more training, ability and contact built upon her work, and elasticity became an important scientific topic, she was closed out. Women were simply not taken seriously.

Germain attempted to extend her research, in a paper submitted in 1825 to a commission of the Institut de France, whose members included Poisson, Gaspard de Prony and Laplace. The work suffered from a number of deficiencies, but rather than reporting them to the author, the commission simply ignored the paper. It was recovered from de Prony's papers and published in 1880.

Among her work done during this period is work on Fermat's Last Theorem and a theorem which has become known as Germain's Theorem. This was to remain the most important result related to Fermat's Last Theorem from 1738 until the contributions of Kummer in 1840. She wrote to Gauss on 12 May 1819 [4]:-

Although I have laboured for some time on the theory of vibrating surfaces (to which I have much to add if I had the satisfaction of making some experiments on cylindrical surfaces I have in mind), I have never ceased to think of the theory of numbers. ... A long time before our Academy proposed as the subject of a prize the proof of the impossibility of Fermat's equation, this challenge ... has often tormented me.

Larry Riddle writes in [30]:-

In this letter she laid out her grand plan to prove Fermat's Last Theorem. Her goal was to prove that for each odd prime exponent $p$, there are an infinite number of auxiliary primes of the form $2Np + 1$ such that the set of non-zero $p$th power residues $x^{p} \mod (2Np + 1)$ does not contain any consecutive integers. If there were a solution to $x^{p} + y^{p} = z^{p}$, then Germain observes that any such auxiliary prime would have to necessarily divide one of the numbers $x, y$, or $z$. Her letter and manuscripts found in various libraries showed her analysis for the primes $p$ less than 100 and for auxiliary primes with $N$ from 1 to 10. ... However, as Germain admitted to Gauss, she was unable to establish the existence of an infinite number of auxiliary primes even for a single prime exponent. Indeed, Germain's grand plan was doomed to failure as it was later shown that for each odd prime $p$ there are only a finite number of auxiliary primes that satisfy the non-consecutive $p$th power residue condition. Germain, herself, eventually proved in a letter to Adrien-Marie Legendre that for $p = 3$, the auxiliary primes 7 and 13 were the only ones that worked. Nevertheless, this was the first time anyone had devised a plan to prove Fermat's Last Theorem for infinitely many prime exponents rather than on a case by case basis.

Andrea Del Centina re-evaluates Germain's work on Fermat's Last Theorem in [18] and concludes:-

A revaluation of Germain's work on Fermat's Last Theorem is necessary. Not only did she develop the theorem attributed to her independently from Legendre, in fact she did part of the additional work commonly credited to him, but she also proved (or nearly proved) results that were rediscovered many years later. After almost 200 years, her ideas were still central. In 1985, using a generalisation of her method, Adleman and Heath-Brown [The first case of Fermat's last theorem, 1985] and, independently, Fouvry [Théorème de Brun-Titchmarsh. Application au théorème de Fermat Ⓣ, 1985], proved that the first case of Fermat's Last Theorem holds for infinitely many exponents.

Germain continued to work in mathematics and philosophy until her death. Before her death, she outlined a philosophical essay which was published posthumously as Considérations générales sur l'état des sciences et des lettres Ⓣ in the Oeuvres philosophiques. Her paper was highly praised by August Comte. Amy Dahan Dalmédico writes in [16]:-

In the essay, she tried to identify the intellectual process in all human activities. She believed the intellectual universe is filled with analogies. The human spirit recognises these analogies, which then leads to the discovery of natural phenomena and the laws of the universe. We should recognise the analogies between the life of Sophie Germain and our own, and they should lead us to strive for excellence in the face of prejudice.

In [21] Jesse A Fernandez Martinez notes how Germain anticipated August Comte:-

Comte's indebtedness to Condorcet and to Saint-Simon has frequently been mentioned. It is only recently that it has been discovered how distinctly he was anticipated in the main features of his system by Sophie Germain. Dühring, in his Critical History of Philosophy from Its Beginnings to the Present Time (3rd ed., Leipzig, 1878), says, after giving a full abstract of her work: "One sees from the above that the Positivism which, without the use of the word, one finds in the writings of Sophie Germain, contains the essential features of that which has hitherto been associated with the name of Auguste Comte."

She was stricken with breast cancer in 1829 but, undeterred by that and the fighting of the 1830 revolution, she completed papers on number theory and on the curvature of surfaces (1831).

Germain died in June 1831 at 13 rue de Savoie in Paris, which still stands today and has a commemorative plaque. Her death certificate listed her not as mathematician or scientist, but 'rentier' (property holder). In [35] the author quotes from H J Monans, Woman in Science (1913):-

And yet, strange as it may seem, when the state official came to make out the death certificate of this eminent associate and co-worker of the most illustrious members of the French Academy of Science, he designated her as a rentiere-annuitant - not as a mathematicienne. Nor is this all. When the Eiffel Tower was erected in which the engineers were obliged to give special attention to the elasticity of the materials used, there were inscribed on this lofty structure the names of seventy-two savants. But one will not find in this list the name of that daughter of genius, whose researches contributed so much toward establishing the theory of the elasticity of metals, Sophie Germain. Was she excluded from this list for the same reason that Agnesi was ineligible in the French Academy - because she was a woman? It would seem so. If such, indeed, was the case, more is the shame for those who were responsible for such ingratitude toward one who had deserved so well of science, and who by her achievements had won the enviable place in the hall of fame.

Germain was buried at the Cimetière du Père Lachaise.
You can see a picture of her grave at THIS LINK.