柯瓦列夫斯卡娅Sofia Kovalevskaya

柯瓦列夫斯卡娅是瓦西里·瓦西里耶维奇·科尔文-克鲁科夫斯基（1801-1879）和伊丽莎白·舒伯特之女，瓦西里是炮兵将军，伊丽莎白是俄罗斯贵族中受过良好教育的成员。柯瓦列夫斯卡娅的名字有不同的写法，我们在此应稍作说明。忽略从俄语转写的问题，她父亲的名字是克鲁科夫斯基，但他向纹章局申请，要求被认定为贵族成员。这一申请被拒绝，直到1858年他退休时，申请才获准，他成为科尔文-克鲁科夫斯基，科尔文这一名字来自匈牙利国王马蒂亚斯·科尔维努斯。瓦西里在1843年继承了两处大地产，一处在帕利比诺，另一处在莫希诺。大约在他继承这些地产的时候，他娶了伊丽莎白·费奥多罗夫娜·舒伯特（1820-1879），她是费奥多尔·费奥多罗维奇·冯·舒伯特和索菲·冯·舒伯特之女。伊丽莎白生于1820年12月15日，比丈夫大约年轻二十岁。我们在此指出，关于瓦西里·科尔文-克鲁科夫斯基的出生年份，资料来源有所不同，日期在1800年至1803年之间。还值得注意的是，伊丽莎白·舒伯特的祖父是天文学家和制图学家特奥多尔·弗里德里希·舒伯特（1789-1865），水星上有一座以他命名的环形山。柯瓦列夫斯卡娅有一个姐姐阿纽塔·瓦西里耶夫娜·科尔文-克鲁科夫斯卡娅（1843-1887），还有一个弟弟费奥多尔·瓦西里耶维奇·科尔文-克鲁科夫斯基（1855-1919）。费奥多尔后来在圣彼得堡大学物理-数学系学习，之后在一个政府部门工作。

关于柯瓦列夫斯卡娅的名字，也需要稍作说明。她出生时的名字是柯瓦列夫斯卡娅·瓦西里耶夫娜·克鲁科夫斯卡娅，直到1858年她父亲的贵族申请获批后，才改姓科尔温-克鲁科夫斯基。她常被称为索菲或索尼娅，前者是柯瓦列夫斯卡娅的英语化形式，后者是昵称，她成年后朋友们这样称呼她。柯瓦列夫斯卡娅是她丈夫姓氏科瓦列夫斯基的阴性形式，该姓氏常转写为Kovalevskaia，偶尔转写为Kovalevskaja。她也以索尼娅·科瓦列夫斯基之名著称，用的是她姓氏的阳性形式，她本人有时也使用这种形式。在本传记中，我们将统一使用柯瓦列夫斯卡娅这一形式。

1855年，柯瓦列夫斯卡娅的父亲克鲁科夫斯基将军被派往卡卢加，该地位于莫斯科西南约70公里处，全家在那里住到1858年。家里为阿纽塔请了保姆和家庭女教师，搬到卡卢加时阿纽塔十二岁。

1858年父亲退休后，柯瓦列夫斯卡娅住在克鲁科夫斯基家的乡村庄园帕利比诺，由家庭教师和女教师教育。帕利比诺靠近立陶宛边境，是一座大庄园，有羊和牛，有养鱼的湖泊，有可供打猎的森林，还有菜园。柯瓦列夫斯卡娅的父亲全力管理庄园，全家在宅邸里生活舒适。

柯瓦列夫斯卡娅在很小的时候就被数学吸引。她的叔叔彼得·瓦西里耶维奇·克鲁科夫斯基，即她父亲的哥哥，经常访问帕拉比诺。她在[62]中写到他：-

尽管他是我们家族中最年长的成员，本应是家族之首，但事实是，任何想指挥他的人都可以指挥他，整个家族都把他当作一个年长的孩子。他长期以来一直有怪人和梦想家的名声。他的妻子几年前去世了；他把自己的全部、相当可观的地产交给了唯一的儿子，只给自己留下非常微薄的月养老金。就这样没有任何明确的事务要处理，他常常来帕利比诺，和我们连续住上几周。他的到来总是被视为节日，当他和我们在一起时，家里的气氛不知怎的变得更活跃、更舒适。

彼得·瓦西里耶维奇叔叔非常尊重数学，经常谈论这门学科。柯瓦列夫斯卡娅在她的自传[62]中写道：-

这些概念的含义我自然还无法领会，但它们作用于我的想象，使我对数学产生了一种敬畏之情，视其为一种崇高而神秘的科学，向入门者开启一个凡人无法企及的奇迹新世界。

当柯瓦列夫斯卡娅11岁时，她育儿室的墙壁上贴满了米哈伊尔·奥斯特罗格拉德斯基关于微分与积分分析的讲义。这需要一些解释！在全家从卡卢加搬到帕利比诺之前，他们对整座庄园宅邸进行了重新装修。墙纸是从圣彼得堡订购的，但他们在计算所需卷数时犯了一个小错误，结果少了一卷。他们没有再订购一卷，而是决定用旧纸张来裱糊育儿室，并在阁楼里寻找了一些。米哈伊尔·奥斯特罗格拉德斯基关于微分与积分分析的讲义就在那里，因为柯瓦列夫斯卡娅的父亲瓦西里·瓦西里耶维奇在接受军事训练时曾听过他的课。她注意到纸上有一些东西她曾听叔叔提到过。研究墙纸成了柯瓦列夫斯卡娅接触微积分的入门。

正是在家庭教师约西夫·伊格纳季耶维奇·马列维奇（1813-1898）的指导下，柯瓦列夫斯卡娅开始了她第一次正规的数学学习，从算术开始（她觉得枯燥），然后转向初等几何和代数。她写道，正是作为他的学生，[62]：-

我开始对数学产生如此强烈的吸引力，以至于我开始忽视其他学科。

柯瓦列夫斯卡娅的父亲决定停止她的数学课，但她借了一本布尔东的Algebra Course，在夜里全家人都睡着时阅读[62]：-

由于我整天都处于家庭女教师的严格监视之下，我在这件事上不得不耍些小聪明。就寝时，我常把书塞在枕头底下，等大家都睡着后，我便借着圣像灯或夜灯的微光通宵阅读。在这种情况下，我当然不敢梦想继续系统地学习我心爱的科目。如果不是下面这件事促使我父亲在一定程度上重新考虑他对我的教育的看法，我的数学知识很可能长期局限于⟦E1⟧的内容。

她提到的这件事发生在一年后，当时邻居、海军学院物理学教授尼古拉·蒂尔托夫送给她家一本他写的物理教科书，柯瓦列夫斯卡娅试图阅读它。她看不懂光学章节中遇到的三角公式，便尝试自己解释它们。蒂尔托夫意识到，她在处理正弦概念时，使用了历史上发展出这一概念的同样方法。蒂尔托夫与柯瓦列夫斯卡娅的父亲争论，认为应该鼓励她进一步学习数学，但直到两年后，他才允许柯瓦列夫斯卡娅跟随亚历山大·斯特兰诺柳布斯基（1839-1903）上私人课程，后者曾是蒂尔托夫的学生。这些关于解析几何、微分和积分的课程是在全家人在圣彼得堡时进行的，他们每年都会在那里待一段时间，探望舒伯特家的姑母们。在那里，柯瓦列夫斯卡娅加入了她家的社交圈，其中包括作家费奥多尔·米哈伊洛维奇·陀思妥耶夫斯基。

我们现在需要稍微谈谈柯瓦列夫斯卡娅的妹妹Anyuta，因为她对年轻的柯瓦列夫斯卡娅有很强的影响。大约在1863年，Anyuta对她的一位朋友，当地牧师的儿子，在大学学习期间假期回到Palibino时告诉她的激进思想变得热情起来。Anyuta想去圣彼得堡生活，甚至提议住在一个年轻人没有仆人一起生活的公社里。她的父亲不允许这种行为，Anyuta不得不留在Palibino。她的反应是秘密写作，并以男性笔名发表了两篇文章。嗯，这是除了柯瓦列夫斯卡娅之外对所有人秘密进行的，因为两姐妹彼此之间没有秘密。1865年2月28日，全家去了圣彼得堡。Dostoevsky来拜访，Anyuta和柯瓦列夫斯卡娅都会见他。通常他们不被允许与他单独相处，但有一次他们的父母出去了，三人能够一起聊天。柯瓦列夫斯卡娅写了一些诗，展示给Dostoevsky看，他称赞了它。然而，随着会面的进行，Anyuta和Dostoevsky在她的虚无主义观点上产生了分歧。Dostoevsky开始越来越多地将注意力转向柯瓦列夫斯卡娅，而她也练习钢琴演奏来娱乐他。尽管Dostoevsky表示希望娶Anyuta，但尽管她钦佩他并认为他是个了不起的人，她觉得他们并不合适。1865年4月，全家回到了Palibino。

到1868年，阿纽塔在圣彼得堡时已与一个持虚无主义观点的激进团体有了接触。她认为，自己旅行和接受教育的唯一途径就是结婚，通过这个激进团体，她和一位朋友被介绍给了弗拉基米尔·奥努弗里耶维奇·科瓦列夫斯基（1842-1883），想法是如果他娶了她们中的一个，她们两人就能“逃脱”，一个作为妻子，另一个由一对已婚夫妇陪伴。此时柯瓦列夫斯卡娅已经十八岁，有一次她和妹妹一起来见弗拉基米尔，那次见面后，他坚持要娶柯瓦列夫斯卡娅，而不是阿纽塔或她的朋友。原本只是名义上的婚姻，当弗拉基米尔任由感情主导时，情况发生了变化。他写信给柯瓦列夫斯卡娅：-

遇见你让我相信灵魂的亲和力，我们两人如此迅速……而真诚地走到一起，至少对我来说，成为了朋友……。现在我不禁想象我们共同的未来中许多欢乐和美好的事情。的确，运用最冷静的判断，不带幼稚的热情，几乎可以肯定地说，柯瓦列夫斯卡娅·瓦西里耶夫娜将成为自然科学的某个分支中出色的医生或学者……

柯瓦列夫斯卡娅意识到，嫁给弗拉基米尔，她就能实现她妹妹的愿望，出国接受高等教育。如果未婚，她父亲不会允许她离家去上大学，而俄罗斯的女性未经父亲或丈夫的书面许可，不能与家人分开生活。因此，十八岁时，她于1868年9月嫁给了年轻的古生物学家弗拉基米尔·科瓦列夫斯基，此前她极其艰难地获得了父亲的许可。起初，柯瓦列夫斯卡娅住在圣彼得堡并在那里的大学听课，一切顺利。她写信给回到帕利比诺的妹妹：-

谢切诺夫的讲座明天开始；所以我的真正生活从上午9点开始……。弗拉基米尔·奥努弗里耶维奇和朋友们将郑重地护送我走后面的楼梯，这样就有希望躲过校方和好奇的目光……

伊万·米哈伊洛维奇·谢切诺夫（1829-1905）是圣彼得堡的生理学教授。柯瓦列夫斯卡娅再次写信给安尤塔：-

我忘了告诉你，梅奇尼科夫答应让我听他的课，并为我争取到听物理课的许可……我正在学习生理学，尤其是解剖学；我们从彼得·伊万诺维奇·博科夫那里弄到一副骨架，兄弟此刻正在摆弄它……

关于这段引文有两点说明。伊利亚·伊里奇·梅奇尼科夫是一位动物学家，当时是圣彼得堡的讲师（Docent）。她提到的“兄弟”实际上是她的丈夫，她一直称他为“兄弟”，这清楚地表明了当时他们之间的关系。

尽管她的婚姻此时让柯瓦列夫斯卡娅得到了她渴望的教育，但它很快就给她带来了问题，在其十五年间，它一直是间歇性悲伤、恼怒和紧张的来源，她的专注常被与丈夫的频繁争吵和误解打断。虽然她现在被允许听大学讲座，但她真正的爱好是数学，她觉得为此她必须去德国。

1869年春天，柯瓦列夫斯卡娅与丈夫前往海德堡学习数学和自然科学，却发现女性不能在大学注册入学。最终她说服大学当局允许她非正式地听课，条件是她必须获得每位授课教师的许可。柯瓦列夫斯卡娅在那里成功地学习了三个学期，听了古斯塔夫·基尔霍夫、赫尔曼·冯·亥姆霍兹、莱奥·柯尼希斯贝格尔和Paul Du Bois-Reymond的课，据一位同学的回忆录记载，她：-

……以其非凡的数学才能立即引起了老师们的注意。莱奥·柯尼希斯贝格尔教授、著名化学家古斯塔夫·基尔霍夫……以及其他所有教授都为这位天赋异禀的学生而欣喜若狂，称她为非凡的现象。

遵循莱奥·柯尼希斯贝格尔的建议，1871年柯瓦列夫斯卡娅搬到柏林，师从卡尔·魏尔斯特拉斯——莱奥·柯尼希斯贝格尔以前的老师。她到达后，卡尔·魏尔斯特拉斯给了她一些题目来测试她的数学能力，当她在一周内解出这些题目时，他立即确信了她的才华。尽管卡尔·魏尔斯特拉斯及其同事做出了努力，参议院仍拒绝允许她参加大学的课程。具有讽刺意味的是，这实际上帮助了她，因为在接下来的四年里，卡尔·魏尔斯特拉斯私下辅导她。到达柏林后不久，她听说安尤塔在巴黎公社——一个激进的社会主义团体，于1871年3月接管了巴黎的统治。柯瓦列夫斯卡娅赶到巴黎，当时公社正遭到法国军队的镇压。许多人受伤和被杀，但两姐妹安然无恙地逃脱了。自然，这一事件给柯瓦列夫斯卡娅留下了深刻的印象。

到1874年春天，柯瓦列夫斯卡娅已经完成了三篇论文。卡尔·魏尔斯特拉斯认为每一篇都值得获得博士学位。这三篇论文分别关于Partial differential equations、Abelian integrals和Saturn's Rings。其中第一篇是一项非凡的贡献，于1875年发表在奥古斯都·利奥波德·克雷勒的Journal上。关于将abelian integrals归约为更简单的elliptic integrals的论文重要性较小，但它包含了一系列熟练的操作，显示了她对卡尔·魏尔斯特拉斯理论的完全掌握。

1874年，在收到卡尔·魏尔斯特拉斯、Paul Du Bois-Reymond和拉扎勒斯·福克斯非常肯定的报告后，柯瓦列夫斯卡娅被哥廷根大学授予博士学位，成绩为最优等。她未出席而被授予学位，并写信给院长解释她的处境：-

尊敬的院长将欣然允许我在申请数学系哲学博士学位入学的信中补充一些内容……这仅仅是为了满足我最亲爱的朋友们的一个愿望……我希望向他们提供一个无可争辩的证明：我献身于数学研究，是遵循我天性的坚定倾向，而且，这项研究并非没有成果。同时，我希望尊敬的院长不会误解我，如果我坦率承认，我不知道自己是否有足够的镇定来参加一场严格的考试，我担心这种不寻常的处境，以及不得不面对面地回答那些我完全不认识的人，会让我慌乱，尽管我知道考官们会尽力为我着想。除此之外，我的德语说得很差。

Pozefsky在[83]中写道：-

1874年，这对夫妇带着外国的博士学位回到圣彼得堡，并开始了一项乌托邦式的计划。他们相信自己的科学知识将使他们能够进行万无一失的投资，并期望他们的新资本能让他们在余生中平静地研究科学。但计划以灾难告终。他们挥霍了继承的遗产，深陷债务……

Vladimir从未从那次事件中恢复过来，并继续推行他那些“快速致富”的计划，但均未成功。柯瓦列夫斯卡娅很快恢复过来，想继续她的学术生涯，但此时在圣彼得堡，尽管拥有博士学位和卡尔·魏尔斯特拉斯的强力推荐信，她却无法获得学术职位。这是多种原因共同造成的，但她的性别是一个主要障碍，她对虚无主义哲学的热情也是如此。她的虚无主义信仰有多极端尚不清楚，因为各传记给出的看法有些不同。例如，在[48]中，Kennedy说她是：-

……首先是一位明确反对革命的贵族……[并且]是一位精英主义者，认为一个有价值的社会需要一个精英知识分子阶层。

在51中，Koblitz将柯瓦列夫斯卡娅描绘成一个激进分子（[28]，[29]）：-

在青春期早期接触到进步思想后，她热切地接受了‘六十年代’的理念，即使在她因社会主义同情而职业受损时也毫不掩饰，虽然她从未直接投身其中，但她羡慕她的激进朋友们参与积极的政治工作，把护照借给他们，接收密码信件，并以一些带有风险的小方式帮助他们。

另见Ann Hibner Koblitz在THIS LINK所写的柯瓦列夫斯卡娅传记。

然而，她的拒绝导致了一段长达六年的时期，在此期间她既没有进行研究，也没有回复卡尔·魏尔斯特拉斯的来信。她苦涩地发现，她能得到的最好的工作就是给小学女生班级教算术，并说道：-

不幸的是，我在乘法表方面很薄弱。

由于无法找到工作，柯瓦列夫斯卡娅和她的丈夫决定以夫妻而非兄妹的身份生活。1878年10月，她生下了一个女儿，柯瓦列夫斯卡娅 Vladimirovna（被称为Fufa），但从1880年起，她越来越多地回归数学研究。这是由巴夫尼提·列波维奇·切比雪夫邀请她在1880年于圣彼得堡第六届自然科学家大会上发表演讲所促成的。她提交了那篇作为她博士论文一部分而写的关于阿贝尔积分的未发表论文。1882年，她开始研究光的折射，并就此主题写了三篇文章。1891年，维多·沃尔泰拉发现柯瓦列夫斯卡娅犯了与加布里尔·拉梅相同的错误，而这些论文正是基于后者的工作，尽管她指出了他在问题表述中所犯的其他几个错误。然而，这三篇文章中的第一篇仍然是一篇有价值的论文，因为它包含了对卡尔·魏尔斯特拉斯关于积分某些偏微分方程的理论的阐述。

此处一段未译出，以下为英文原文In the spring of 1883, Vladimir, from whom Sofia had been separated for two years, committed suicide after he was again in financial difficulties which led to a stock scandal. After the initial shock, Kovalevskaya immersed herself in mathematical work in an attempt to rid herself of feelings of guilt. Mittag-Leffler managed to overcome opposition to Kovalevskaya in Stockholm, and obtained for her a position as privatdocent. To prove her competence, she had to lecture for a year without pay and without an official position. Having no other option, she reluctantly agreed. She began to lecture there in early 1884, giving a course in German on partial differential equations. Her lectures were well received and she was appointed to a five year extraordinary professorship in Higher Analysis in June of that year on condition that she lecture in Swedish, which she did. Not everyone was happy, for example August Strindberg (1849-1912), the famous author, wrote in a local paper [76]:-

一位女教授是有害且令人不快的现象——甚至可以说，是一个怪物。

柯瓦列夫斯卡娅把女儿留在了莫斯科，只在1884年夏天见过她，但在1885年，当Fufa八岁时，她把她带到了斯德哥尔摩。1889年6月，她成为自物理学家劳拉·巴斯和玛利亚·阿涅西以来第一位在欧洲大学担任讲席的女性。

柯瓦列夫斯卡娅在斯德哥尔摩的岁月里，她完成了许多人认为她最重要的研究。她讲授分析领域最新课题的课程，并成为新期刊Acta Mathematica的编辑。她承担了与巴黎和柏林数学家联络的任务，并参与了国际会议的组织工作。她的地位引起了社会的关注，她重新开始写回忆录和戏剧，这是她年轻时喜欢做的事情。Karen Rappaport写道[84]：-

柯瓦列夫斯卡娅曾在约斯塔·米塔格-莱弗勒家中住过一段时间，在那里她结识了他的妹妹Anna Leffler并与之建立了友谊。Anna Leffler是一位著名的女权倡导者和作家，她鼓励了柯瓦列夫斯卡娅的文学倾向。1887年，她们合作创作了一部名为《为幸福而斗争》的戏剧。该剧基于柯瓦列夫斯卡娅在她临终妹妹床边想到的一个想法。1887年秋天Anyuta去世后，柯瓦列夫斯卡娅感到孤独和沮丧。姐妹俩一直很亲密，柯瓦列夫斯卡娅深深地感受到了这一损失。

French Academy of Sciences的Bordin奖主题于1886年公布。参赛作品必须是对刚体研究问题的重要贡献。柯瓦列夫斯卡娅参赛，并于1886年凭借其论文Mémoire sur un cas particulier du problème de le rotation d'un corps pesant autour d'un point fixe, où l'intégration s'effectue à l'aide des fonctions ultraelliptiques du temps Ⓣ（关于重体绕固定点旋转问题的一个特殊情形的回忆录，其中积分使用超椭圆函数进行）获得了Bordin奖。共有十五篇论文参赛。为表彰这项工作的卓越性，奖金从3,000法郎提高到5,000法郎。参赛作品本应是匿名的，许多柯瓦列夫斯卡娅的传记声称她获奖只是因为评委不知道参赛者是一位女性。然而，这完全是不实的，因为该奖项的主题是专门为契合她的研究专长而选择的。这并非只为她而做，因为奖项主题常常被选来鼓励杰出的年轻数学家。她亲自接受了该奖项，Academy of Sciences院长在祝贺演讲中说[76]：-

我们的评委发现，她的工作不仅见证了渊博而广阔的知识，还见证了一个极具创造力的头脑。

柯瓦列夫斯卡娅在这一课题上的进一步研究于1889年赢得了瑞典科学院颁发的一项奖，同年，在巴夫尼提·列波维奇·切比雪夫的倡议下，柯瓦列夫斯卡娅当选为帝国科学院的通讯院士。尽管沙皇政府曾多次拒绝让她在自己国家担任大学职位，但Imperial Academy的规则被修改，允许选举女性。

柯瓦列夫斯卡娅最后发表的作品是一篇短文Sur un théorème de M Bruns Ⓣ（关于M 海因里希·布伦斯的一个定理），她在文中给出了海因里希·布伦斯关于均匀物体势函数性质定理的一个新的、更简单的证明。1891年初，正值她数学能力和声誉的巅峰时期，柯瓦列夫斯卡娅死于流感并发肺炎。她曾前往法国，回到斯德哥尔摩后不久便病倒了。医生对她的病情做出了错误诊断，等他们意识到她患的是肺炎时，已经来不及挽救她了。她被安葬在斯德哥尔摩以北的索尔纳北方公墓。

卡尔·魏尔斯特拉斯说柯瓦列夫斯卡娅可以[51]：-

……通过行动表明，由于偏见，女性一直被排斥在人类最高追求之外。

利奥波德·克罗内克称她为“最罕见的研究者之一”，而约斯塔·米塔格-莱弗勒在谈到作为教师的柯瓦列夫斯卡娅时说：-

我们深知她以何等鼓舞人心的热忱阐释[她的]思想……又是何等乐意地奉献出她知识的财富。

Koblitz 写道 [51]：-

在她去世时，Kovalevskaia 确实被认为与她同代的任何人比肩。这包括 儒勒·昂利·庞加莱、埃米尔·皮卡 和 约斯塔·米塔格-莱弗勒。

让我们以引用 1895 年对 A Russian childhood 1895 年出版的评论 [3] 来结束，以展示 柯瓦列夫斯卡娅 所遭受的偏见：-

尽管有所有这些冒险和成功，她的生活却是一场忧郁的失败，而她知道它是如此。甚至她的科学成就也不过是对她从老师那里继承来的思想的大师级发挥。很难想象她如何能对父母和孩子更加残忍和漠不关心。她强烈的激情渴望是巴尔扎克毕生努力追求的两件事——成名和被爱。她像他一样，达到了两者的极致。但她的生活，如同开始时一样，在悲惨中结束；而他的生活则高尚地得到了满足。男人抛弃了自我；女人则将任性之魔紧紧束缚在自己的要害之处。

Sofia Kovalevskaya was the daughter of Vasily Vasilievich Korvin-Krukovsky (1801-1879), an artillery general, and Elizaveta Shubert, both well-educated members of the Russian nobility. Sofia's name is written in different forms and we should say a little about that here. Ignoring the problems of transliteration from Russian, her father's name was Krukovsky but he made applications to the Department of Heraldry to allow him to be considered a member of the nobility. This was refused until he retired in 1858 when his application was allowed and he became Korvin-Krukovsky, the name Korvin coming from Matthias Corvinus, the King of Hungary. Vasily had inherited two large estates in 1843, one at Palibino and the other at Moshino. Around the time he inherited these estates, he had married Elizaveta Fedorovna Shubert (1820-1879), the daughter of Fyodor Fyodorovich von Shubert and Sophie von Shubert. Elizaveta had been born on 15 December 1820, and was about twenty years younger than her husband. Let us note at this point that sources differ in giving Vasily Korvin-Krukovsky's year of birth, with dates ranging between 1800 and 1803. It is also worth noting that Elizaveta Shubert's grandfather was the astronomer and cartographer Theodor Friedrich Schubert (1789-1865) who has a crater on Mercury named after him. Sofia had an older sister Anyuta Vasilievna Korvin-Krukovskaya (1843-1887), and a younger brother Fyodor Vasilievich Korvin-Krukovsky (1855-1919). Fyodor went on to study in the Physics-Mathematics Faculty of St Petersburg University and then worked in a government ministry.

We need to also say a little about Sofia Kovalevskaya's name. She was given the name Sofia Vasilievna Krukovsky, only adopting the name Korvin-Krukovsky after her father's application for nobility was accepted in 1858. She is often called Sophie or Sonya, the first being an anglicised version of Sofia, the second being a familiar version by which she was known by her friends after she became an adult. Kovalevskaya is the female version of her husband's name Kovalevsky which is often transliterated as Kovalevskaia, and infrequently as Kovalevskaja. She is also known as Sonya Kovalevsky, using a masculine version of her surname, a form she sometimes used herself. We will use the form Sofia Kovalevskaya throughout this biography.

In 1855 Sofia's father, General Krukovsky, was posted to Kaluga, about 70 km south west of Moscow, and the family lived there until 1858. The family had a nanny and a governess for Anyuta who was twelve years old when they moved to Kaluga.

Sofia lived at Palibino, the Krukovsky country estate, from 1858 when he father retired, and was educated by tutors and governesses. Palibino was near the Lithuanian border and was a large estate with sheep and cattle, lakes stocked with fish, forests with game, and vegetable gardens. Sofia's father was fully occupied managing the estate and the family lived comfortably in the manor house.

It was at a very young age that Sofia was attracted to mathematics. Her uncle Pyotr Vasilievich Krukovsky, her father's elder brother, often visited Palabino. She writes about him in [62]:-

Although he was the oldest member of our family and should have been its head, the truth was that he was ordered about by anyone who felt like it, and the whole family treated him like an elderly child. He had long held the reputation of an eccentric and a dreamer. His wife had died some years before; he had handed over his entire and rather good-sized estate to his only son, leaving for himself only a very small monthly pension. Left thus without any definite business affairs to attend to, he used to come to Palibino often and stayed with us for weeks on end. His arrival was always regarded as a holiday, and the atmosphere at home became somehow livelier and cosier when he was with us.

Uncle Pyotr Vasilievich had a great respect for mathematics and often spoke about the subject. Sofia wrote in her autobiography [62]:-

The meaning of these concepts I naturally could not yet grasp, but they acted on my imagination, instilling in me a reverence for mathematics as an exalted and mysterious science which opens up to its initiates a new world of wonders, inaccessible to ordinary mortals.

When Sofia was 11 years old, the walls of her nursery were papered with pages of Ostrogradski's lecture notes on differential and integral analysis. This requires some explanation! Before the family moved from Kaluga to Palibino, they had the whole manor house redecorated. Wallpaper was ordered from St Petersburg but they had made a small error in working out how many rolls would be required and ended up one short. Rather than order one roll, they decided to paper the nursery with old sheets of paper and looked for some in their attic. Ostrogradski's lecture notes on differential and integral analysis were there because Vasily Vasilievich, Sofia's father, had attended his course when undergoing his military training. She noticed that there were certain things on the sheets she had heard mentioned by her uncle. Studying the wallpaper was Sofia's introduction to calculus.

It was under the family's tutor, Yosif Ignatievich Malevich (1813-1898), that Sofia undertook her first proper study of mathematics, beginning with arithmetic (which she found boring) and then moving on to elementary geometry and algebra. She writes that it was as his pupil that [62]:-

I began to feel an attraction for my mathematics so intense that I started to neglect my other studies.

Sofia's father decided to put a stop to her mathematics lessons but she borrowed a copy of Bourdon's Algebra Course which she read at night when the rest of the household was asleep [62]:-

Since I was under my governess's strict surveillance all day long, I was forced to practice some cunning in this matter. At bedtime I used to put the book under my pillow and then, when everyone was asleep, I would read the night through under the dim light of the icon-lamp or the night lamp. Under such circumstances, of course, I did not dare dream of continuing the systematic study of my favourite subject. My mathematical knowledge would likely have remained confined for a long time, to the contents of Bourdon's 'Algebra' if I had not been aided by the following incident, which motivated my father to reassess his views on my education to some degree.

The incident she refers to happened a year later when a neighbour, Nikolai Tyrtov, Professor of Physics at the Naval Academy, presented her family with a physics textbook which he had written, and Sofia attempted to read it. She did not understand the trigonometric formulae that she came across in the chapter on optics and attempted to explain them herself. Tyrtov realised that in her working with the concept of sine, she had used the same method by which it had been developed historically. Tyrtov argued with Sofia's father that she should be encouraged to study mathematics further but it was two years later before he permitted Sofia to take private lessons with Aleksandr Strannoliubskii (1839-1903) who had been a student of Tyrtov. These lessons on analytic geometry, and differential and integral calculus, took place when the family were in St Petersburg, where they spent some time each year visiting the Shubert aunts. There Sofia joined her family's social circle which included the author Fyodor Mikhailovich Dostoevsky.

We now need to say a little about Sofia's sister Anyuta at this point, because she was a strong influence on the young Sofia. In around 1863 Anyuta had become enthusiastic about the radical ideas that one of her friends, the son of the local priest, told her about when he was home in Palibino from his university studies during the vacation. Anyuta wanted to go to St Petersburg to live, and even proposed living in a commune where young people were living together without servants. Her father was not going to allow such behaviour and Anyuta had to remain at Palibino. She reacted by secretly writing and getting two pieces published under a male pseudonym. Well, it was done secretly from everyone except Sofia, for the two sisters kept nothing from each other. On 28 February 1865 the family went to St Petersburg. Dostoevsky came to visit and both Anyuta and Sofia would meet him. Usually they were not allowed to be alone with him but on one occasion their parents were out and the three were able to chat together. Sofia had written some poetry which was shown to Dostoevsky and he praised it. As the meetings progressed, however, Anyuta and Dostoevsky disagreed over her Nihilist views. Dostoevsky began to turn his attention more and more to Sofia and in turn she practised her piano playing to entertain him. Although Dostoevsky expressed a wish to marry Anyuta, as much as she admired him and thought he was a wonderful person, she felt they were not suited. The family returned to Palibino in April 1865.

By 1868, Anyuta had made contact with a radical group with Nihilist views when in St Petersburg. She believed her only way to travel and get an education was to marry and through the radical group, she and a friend were introduced to Vladimir Onufrievich Kovalevsky (1842-1883), with the idea that if he married one or other of them they could both "escape", one as a wife and the other chaperoned by a married couple. By this time Sofia was eighteen years old and on one occasion she came with her sister to meet Vladimir who, after that meeting insisted he would marry Sofia rather than Anyuta or her friend. What had been intended only as a nominal marriage changed when Vladimir allowed his feelings to take over. He wrote to Sofia:-

Meeting you makes me believe in the affinity of souls, so swiftly ... and genuinely did the two of us come together and, on my part at least, become friends .... Now I cannot keep from picturing much that is joyous and good in our common future. Indeed, employing the coolest possible judgment, without childish enthusiasm, one can state almost positively that Sofia Vasilievna will become a splendid doctor or scholar in some branch of the natural sciences ...

Sofia realised that by marrying Vladimir she could do what her sister had wanted and go abroad to enter higher education. If unmarried, her father would not allow her to leave home to study at a university, and women in Russia could not live apart from their families without the written permission of their father or husband. So, at the age of eighteen, she married the young palaeontologist Vladimir Kovalevsky in September 1868 having, with the greatest difficulty, received her father's permission. At first things went well for Sofia living in St Petersburg and attending the University there. She wrote to her sister back in Palibino:-

Sechenov's lectures begin tomorrow; and so my real life begins at 9 a.m. .... Vladimir Onufrievich and friends will solemnly escort me by way of the back stairs so that there is hope of hiding from the administration and from curious stares ...

Ivan Mikhailovich Sechenov (1829-1905) was Professor of Physiology at St Petersburg. Again Sofia writes to Anyuta:-

I forgot to tell you that Mechnikov promised to admit me to his lectures and get permission for me to attend the physics lectures .... I'm studying physiology and particularly anatomy; we got a skeleton from Pyotr lvanovich Bokov and brother is poking it at this moment ....

Two comments about this quote. Ilya Ilyich Mechnikov was a zoologist who at this time was a docent at St Petersburg. The "brother" she refers to is in fact her husband, whom she always referred to as "brother", a clear indication of their relationship at this time.

Although her marriage was allowing Kovalevskaya to get the education she craved at this time, it soon caused her problems and, throughout its fifteen years, it was a source of intermittent sorrow, exasperation and tension and her concentration was broken by her frequent quarrels and misunderstandings with her husband. Although she was now allowed to attend university lectures, her real love was mathematics and she felt that for this she had to go to Germany.

In the spring of 1869 Kovalevskaya travelled with her husband to Heidelberg to study mathematics and the natural sciences, only to discover that women could not matriculate at the university. Eventually she persuaded the university authorities to allow her to attend lectures unofficially, provided that she obtain the permission of each of her lecturers. Sofia studied there successfully for three semesters attending lectures by Gustav Kirchhoff, Hermann Helmholtz, Leo Königsberger, and Paul Du Bois-Reymond and, according to the memoirs of a fellow student, she:-

... immediately attracted the attention of her teachers with her uncommon mathematical ability. Professor Königsberger, the eminent chemist Kirchhoff, ... and all of the other professors were ecstatic over their gifted student and spoke about her as an extraordinary phenomenon.

Following Leo Königsberger's advice, in 1871 Kovalevskaya moved to Berlin to study with Karl Weierstrass, Königsberger's former teacher. When she arrived, Weierstrass gave her some problems to test her mathematical skill and when she solved these in a week he was immediately convinced of her brilliance. Despite the efforts of Weierstrass and his colleagues the senate refused to permit her to attend courses at the university. Ironically this actually helped her since over the next four years Weierstrass tutored her privately. Soon after arriving in Berlin, she heard that Anyuta was in the Paris Commune, a radical socialist group that took over running Paris in March 1871. Sofia rushed to Paris where the Commune was being suppressed by the French army. Many were wounded and killed but the two sisters escaped uninjured. Naturally, this episode made a big impression on Sofia.

By the spring of 1874, Kovalevskaya had completed three papers. Weierstrass deemed each of these worthy of a doctorate. The three papers were on Partial differential equations, Abelian integrals and Saturn's Rings. The first of these is a remarkable contribution which was published in Crelle's Journal in 1875. The paper on the reduction of abelian integrals to simpler elliptic integrals is of less importance but it consisted of a skilled series of manipulations which showed her complete command of Weierstrass's theory.

In 1874 Kovalevskaya was granted her doctorate, summa cum laude, from Göttingen University after it received very positive reports from Karl Weierstrass, Paul Du Bois-Reymond, and Lazarus Fuchs. She was awarded the degree in absentia, writing to the Dean to explain her position:-

The very reverend Dean will graciously permit me to add something to the letter in which I present myself for admission to the degree of Doctor Phil, in the mathematical faculty ... It is only a wish to satisfy my dearest friends ... I wish to give them an incontestable proof that, in devoting myself to the study of mathematics, I am following the determined bent of my nature, and that, moreover, this study is not without result. At the same time I hope the very reverend Dean will not misconstrue me if I acknowledge openly that I do not know whether I have sufficient aplomb to undergo an examen rigorosum, and I fear that the unusual position, and having to answer, face to face, men with whom I am altogether unacquainted, would confuse me, although I know the examiners would do all they could for me. In addition to this, I speak German very badly.

Pozefsky writes in [83]:-

In 1874 the couple returned to St Petersburg, foreign doctorates in hand, and embarked on a utopian scheme. They believed that their knowledge of the sciences would allow them to engage in fail-safe investing and expected their new capital to permit them to study science in tranquillity for the rest of their lives. But the plans ended in disaster. They squandered their inheritances, fell deeply into debt ...

Vladimir never recovered from his episode and continued, unsuccessfully, with his "get-rich-quick" schemes. Sofia quickly recovered and wanted to continue her academic career but, now in St Petersburg, despite having a doctorate and letters of strong recommendation from Weierstrass, she was unable to obtain an academic position. This was for a combination of reasons, but her sex was a major handicap as was her enthusiasm for the nihilist philosophy. It is unclear how extreme her nihilist beliefs were for the biographies give somewhat different views. For example in [48] Kennedy says she was:-

... above all an aristocrat who definitely opposed revolution ... [and] an elitist who believed that a worthwhile society needs an elite intellectual class.

In [51] Koblitz paints a picture of Kovalevskaya as a radical ([28], [29]):-

Exposed to progressive thought in early adolescence, she embraced eagerly the ideas of the 'sixties, made no secret of her socialist sympathies even when they harmed her professionally, and while she never committed herself directly, she envied her radical friends their involvement in active political work, gave them her passport, accepted coded letters, and helped them out in small ways that entailed some risk.

See also Ann Hibner Koblitz's biography of Kovalevskaya at THIS LINK.

Her rejections, however, resulted in a six year period during which time she neither undertook research nor replied to Weierstrass's letters. She was bitter to discover that the best job she was offered was teaching arithmetic to elementary classes of school girls, and remarked:-

I was unfortunately weak in the multiplication table.

Unable to get a job, Kovalevskaya and her husband decided to live as man and wife rather than brother and sister. In October 1878, she gave birth to a daughter, Sofia Vladimirovna (who was called Fufa), but from 1880 increasingly returned to her study of mathematics. This was prompted by an invitation from Pafnuty Chebyshev to deliver an address to the Sixth Congress of Natural Scientists in St Petersburg in 1880. She delivered the unpublished paper on abelian integrals which she had written as part of her doctoral thesis. In 1882 she began work on the refraction of light, and wrote three articles on the topic. In 1891, Volterra discovered that Kovalevskaya had made the same mistake as Lamé, on whose work these papers were based, though she had pointed out several others which he had made in his presentation of the problem. The first of these three articles was still a valuable paper however, because it contained an exposition of Weierstrass's theory for integrating certain partial differential equations.

In the spring of 1883, Vladimir, from whom Sofia had been separated for two years, committed suicide after he was again in financial difficulties which led to a stock scandal. After the initial shock, Kovalevskaya immersed herself in mathematical work in an attempt to rid herself of feelings of guilt. Mittag-Leffler managed to overcome opposition to Kovalevskaya in Stockholm, and obtained for her a position as privatdocent. To prove her competence, she had to lecture for a year without pay and without an official position. Having no other option, she reluctantly agreed. She began to lecture there in early 1884, giving a course in German on partial differential equations. Her lectures were well received and she was appointed to a five year extraordinary professorship in Higher Analysis in June of that year on condition that she lecture in Swedish, which she did. Not everyone was happy, for example August Strindberg (1849-1912), the famous author, wrote in a local paper [76]:-

A female professor is a pernicious and unpleasant phenomenon - even, one might say, a monstrosity.

Kovalevskaya had left her daughter in Moscow and only saw her during the summer of 1884, but in 1885, when Fufa was eight years old, she brought her to Stockholm. In June 1889 she became the first woman since the physicist Laura Bassi and Maria Gaetana Agnesi to hold a chair at a European university.

During Kovalevskaya's years at Stockholm, she carried out what many consider her most important research She taught courses on the latest topics in analysis and became an editor of the new journal Acta Mathematica. She took over the task of liaison with the mathematicians of Paris and Berlin and took part in the organisation of international conferences. Her status brought her attention from society, and she began again to write reminiscences and dramas that she had enjoyed doing when young. Karen Rappaport writes [84]:-

Sofia lived for a time in Mittag-Leffler's home, where she met and developed a friendship with his sister, Anna Leffler. Anna Leffler, a well-known advocate of women's rights and a writer, encouraged Sofia's literary leanings. In 1887 they collaborated on a play entitled 'The Struggle for Happiness'. It was based on an idea that had occurred to Sofia while she sat at the bedside of her dying sister. After Anyuta died in the fall of 1887, Sofia felt lonely and despondent. The sisters had been close, and Sofia felt the loss deeply.

The topic of the Prix Bordin of the French Academy of Sciences was announced in 1886. Entries were to be significant contributions to the problem of the study of a rigid body. Kovalevskaya entered and, in 1886, was awarded the Prix Bordin for her paper Mémoire sur un cas particulier du problème de le rotation d'un corps pesant autour d'un point fixe, où l'intégration s'effectue à l'aide des fonctions ultraelliptiques du temps Ⓣ. There had been fifteen papers entered for the prize. In recognition of the brilliance of this work the prize money was raised from 3,000 to 5,000 francs. Entries for the prize were supposed to be anonymous and many biographies of Kovalevskaya claim that she only won because the judges did not know the entry was from a woman. This, however, is totally false for the topic of the prize was especially chosen to fit in with her research expertise. This was not done just for her, for often the prize topic was chosen to encourage an outstanding young mathematician. She accepted the prize in person and the President of the Academy of Sciences, in his speech of congratulation, said [76]:-

Our judges have found that her work bears witness not only to profound and broad knowledge, but to a mind of great inventiveness.

Kovalevskaya's further research on this subject won a prize from the Swedish Academy of Sciences in 1889, and in the same year, on the initiative of Chebyshev, Kovalevskaya was elected a corresponding member of the Imperial Academy of Sciences. Although the Tsarist government had repeatedly refused her a university position in her own country, the rules at the Imperial Academy were changed to allow the election of a woman.

Kovalevskaya's last published work was a short article Sur un théorème de M Bruns Ⓣ in which she gave a new, simpler proof of Bruns's theorem on a property of the potential function of a homogeneous body. In early 1891, at the height of her mathematical powers and reputation, Kovalevskaya died of influenza complicated by pneumonia. She had travelled to France and had fallen ill shortly after returning to Stockholm. The doctors made an incorrect diagnosis of her illness and by the time they realised she was suffering from pneumonia, it was too late to save her. She was buried in the Northern Cemetery of Solna, just to the north of Stockholm.

Weierstrass said that Kovalevskaya could [51]:-

... show through deeds that women have been alienated from the highest strivings of mankind because of prejudice.

Kronecker described her as "one of the rarest investigators" while Mittag-Leffler, speaking of Kovalevskaya as a teacher, said:-

We know with what inspiring zeal she explained [her] ideas ... and how willingly she gave the riches of her knowledge.

Koblitz writes [51]:-

At the time of her death, Kovalevskaia was indeed considered the equal of anyone of her generation. This included Poincaré, Picard, and Mittag-Leffler.

Let us end by showing something of the prejudice that Kovalevskaya suffered by quoting from the 1895 review of 1895 publication of A Russian childhood [3]:-

With all these adventures and successes, her life was a melancholy failure, and she knew it to be such. Even her scientific achievements were but the masterly working out of ideas derived from her teachers. It is difficult to conceive how she could have been more cruel and unregardful of her parents and of her child. Her intense passionate desire was for two things which Balzac strove for all his laborious years - to be famous, and to be loved. She attained both, as he did, to the uttermost. But her life ended, as it began, in wretchedness; while his was nobly satisfied. The man had cast out selfhood; the woman fastened the demon of self-will in her very vitals.