玛利亚·阿涅西Maria Gaetana Agnesi

玛利亚·阿涅西是Pietro Agnesi的女儿，Pietro Agnesi来自一个靠丝绸发家的富裕家庭。Pietro Agnesi与他的三个妻子有二十一个孩子，而阿涅西是孩子中最年长的。正如Truesdell在[20]中所写，Pietro Agnesi：-

……属于介于贵族和仅仅富有者之间的一个阶层。这样的资产者可以拥有适合领主的家宅，举止像骑士，与一些贵族自由交往，从事生活中更精致的事物，成为有才能之人的赞助人。[Pietro Agnesi]正是这样做的……

一些关于阿涅西的记述把她的父亲描述为博洛尼亚的数学教授。[16]中清楚地表明这完全错误，但不幸的是，这个错误被沿袭到[1]中，并且还将在许多其他地方见到。

阿涅西可以为阿涅西提供高质量的导师，事实上他确实为她提供了最好的导师，他们都是来自教会的年轻学者。她展现出非凡的才能，在幼年就掌握了拉丁语、希腊语和希伯来语等多种语言。9岁时，她发表了一篇拉丁语演讲，为女性接受高等教育辩护。这并非如某些人所声称的那样是阿涅西的作品，而是一篇由她的一位导师用意大利语写的文章，她将其翻译并[20]：-

……她凭记忆将其背诵给由她父亲在花园中安排的一次学术聚会……

1738年，她出版了Propositiones Philosophicae，这是一系列关于哲学和自然科学的论文集。该卷包含191篇哲学论题，阿涅西将在与特别邀请的重要国际和国内人士的辩论中为之辩护，这些人都是她父亲邀请到家里的。在[4]中，德布罗斯描述了1739年7月16日发生的一个这样的夜晚：-

……我被带进一个宽敞精美的房间，在那里我看到大约三十个来自欧洲各国的人围成一圈，而阿涅西小姐独自和她的妹妹坐在沙发上。她是一个大约二十岁的女孩，既不丑也不美，举止非常朴素而甜美。……带我来的人贝洛尼伯爵想搞一场公开表演。他先用拉丁语向这个年轻女孩发表了一篇精彩的演讲，以便所有人都能听懂。她很好地回答了他，之后他们用同一种语言就喷泉的起源以及在某些喷泉中看到的类似海洋潮汐的涨落原因展开了辩论。她在这个话题上说得像天使一样，我从未听过如此令人愉悦的事情。……

她非常推崇艾萨克·牛顿的哲学，看到像她这个年纪的人如此精通这些抽象主题，真是奇妙。然而，尽管我对她的学识感到惊叹，但听到她说拉丁语如此纯正、流畅和准确，我或许更加惊叹……

这看起来可能像是一件极其令人反感的事情，阿涅西的父亲像马戏团表演一样炫耀女儿的才华。在某种程度上，情况必定如此，但公平地说，这种表演在当时相对常见。当然，尽管阿涅西总是完全顺从父亲的意愿，但她对他安排的这种表演并不十分满意。我们再次引用[4]，德布罗斯写道：-

她告诉我，她非常遗憾这次访问采取了学位论文答辩的形式，而且她不喜欢公开谈论这类事情，因为在这种场合，一个人觉得有趣，二十个人却厌烦得要命。……听到有人说她想进修道院，我很恼火，而且这并非出于需要，因为她非常富有。

在20中，Truesdell进一步解释了她想成为修女的愿望：——

她确实请求父亲允许她成为修女。父亲想到最心爱的孩子竟想离开自己，惊恐不已，恳求她改变主意。她同意继续住在他家里照顾他，但有三个条件：她可以随时去教堂；她穿着朴素谦卑；她完全放弃舞会、剧院和世俗娱乐。

阿涅西集中精力研究宗教书籍和学习数学。大约在这个时候，她写了一篇关于洛必达的Traité analytique des section coniques的评注，但从未出版。没有正规指导学习数学几乎是不可能完成的任务，只有少数人曾以这种方式取得巨大成就。然而，阿涅西在努力学习数学方面很幸运，因为一位修士拉米罗Rampinelli——一位曾在罗马和博洛尼亚任教授的数学家——来到米兰，并成为阿涅西家的常客。在Rampinelli的帮助下，阿涅西学习了Reyneau的微积分教材Analyse démontrée（1708年）。阿涅西明白她欠Rampinelli的情，在她著名著作Instituzioni analitiche ad uso della gioventù italiana的序言中写道：——

我以对数学最强烈的倾向为支撑，强迫自己独自投入所有学习，若不是[Rampinelli的]可靠指导和明智引领使我走出困境，我本会完全陷入难以逾越的巨大迷宫之中……；我深深感激他，我微薄的才能所足以取得的一切进展（无论它们是什么），都归功于他。

Rampinelli鼓励阿涅西写一本关于微分学的书。她用意大利语写了这本书作为教学文本，根据序言，该书试图呈现这些材料：——

……以恰当的清晰与简洁……它按照那种自然的顺序展开，这种顺序也许提供了最好的教导和最大的启发。

阿涅西用她父亲的钱，得以安排在自己家中私人印刷这本书，这样她就能亲自监督整个印刷过程。然而，她希望从一流数学家那里获得更多意见，因此在1745年7月20日，她写信给雅各布·黎卡提。正是Rampinelli建议雅各布·黎卡提或许可以向阿涅西提供建议，而他显然联系了雅各布·黎卡提——后者曾是他自己的老师之一——并且雅各布·黎卡提同意阅读阿涅西这本书的最终稿并提出建议。

雅各布·黎卡提迅速回复了阿涅西的第一封信，并承诺将文本转交给他的两个儿子Vincenzo Riccati和Giordano Riccati，以便他们也能对作品发表评论。一旦阿涅西收到雅各布·黎卡提对文本第一部分的评论，她就组织印刷该部分，而后面部分也寄给雅各布·黎卡提供他评论。到1747年，阿涅西将书的后面部分寄给雅各布·黎卡提，并向他解释前面部分的印刷已接近完成。

雅各布·黎卡提于1747年2月1日写信给Rampinelli，向阿涅西提供他自己早期关于积分的一些工作，供她收入书中。阿涅西收录了这项工作，并适当致谢雅各布·黎卡提。在她的信中，阿涅西试图让雅各布·黎卡提更快回复他对她草稿的注释，因为印刷商还有别的工作要做，她写信给雅各布·黎卡提说：——

……如果必须再次暂停印刷，我不知道何时才能重新开始，因为即使现在，继续印刷第一部分（很快就要完成）对我来说也极为困难。

阿涅西著名的两卷本著作Instituzioni analitiche ad uso della gioventù italianaⓉ（《供意大利青年使用的分析方法》）的第一卷于1748年出版，而阿涅西继续就第二卷的材料与雅各布·黎卡提通信，第二卷于次年出版。这部著作给她带来了很大的名声。巴黎Académie des Sciences的一个委员会就这部著作所作的报告指出：——

像作者所做的那样，把这些散见于现代数学家著作中、且往往以彼此极为不同的方法呈现的发现，归结为几乎统一的方法，需要很高的技巧和睿智。秩序、清晰和精确贯穿于这部著作的所有部分。……我们认为它是最完整、最优秀的论著。

教宗本笃十四世写信给阿涅西，说他年轻时曾研习数学，能看出她的工作将为意大利和博洛尼亚科学院带来荣誉。此后不久，他任命阿涅西为博洛尼亚大学的名誉读者。随后，博洛尼亚科学院院长和另外三位教授找到阿涅西，邀请她接受博洛尼亚大学的数学讲席。事实上，此后不久，阿涅西收到了教宗本笃十四世写于1750年9月26日的一封信：-

我们曾设想，您应被授予著名的数学讲席，由此自然可知，您不应感谢我们，而是我们感谢您……

很可能阿涅西既未接受也未拒绝这一提议。正如Truesdell在[20]中所写：-

十月，[阿涅西]收到一份教宗敕书，确认了对她的任命。她早已献身于神圣的隐退生活；尽管她的名字留在大学名册上长达四十五年，她从未前往博洛尼亚。

这确实解释了在许多记述中出现的关于阿涅西是否曾拥有数学讲席的混淆。弗里西是阿涅西一位兄弟的校友，在她的书出版之后曾造访阿涅西家。他在[6]中描述了她父亲如何对她施加严厉约束，她选择住在宅中远离其他家人居住的地方，并在那里帮助生病的老妇人。然而[6]：-

……她立即毫无明显困难地顺从了父亲的意愿……以如此的优雅与敏锐参与他家中的常规学术聚会，提出或回答各种问题、难题和科学疑问……

1752年父亲去世后，阿涅西全身心投入慈善工作。她[20]：-

……重新开始研习天主教教义，并对穷人、受苦者、绝症患者和痴呆者做出代价高昂的虔诚善举。她先是在已故父亲的房子里，后来又在其他地方，为年老体弱的妇女建立了一所收容所。

阿涅西将所有的钱都花在了这项慈善工作上，最终在她曾担任院长的济贫院中一贫如洗地去世。

论著Instituzioni analitiche ad uso della gioventù italiana Ⓣ（《供意大利青年使用的分析方法》）中不包含阿涅西的原创数学。相反，书中包含许多精心挑选以阐明思想的例子；一篇评论称其为：-

……一部通过例子而非理论来阐述的著作。

书中讨论了如今被称为“阿涅西的女巫”的三次曲线。关于这条曲线为何被称为“女巫”，一直有许多争论。皮埃尔·德·费马讨论过这条曲线，1703年，格兰迪给出了该曲线的一种作图法。1718年，格兰迪给它起了拉丁名“versoria”，意为“转动帆的绳索”，他因其形状而如此命名。格兰迪将拉丁语“versoria”译为意大利语“versiera”，而阿涅西在她的书中非常正确地指出，这条曲线被称为“la versiera”。

约翰·科尔森曾将艾萨克·牛顿的De Methodis Serierum et Fluxionum Ⓣ（级数与流数的方法）从拉丁文译成英文，于1736年出版；他还在1760年（科尔森去世之年）之前将阿涅西的Instituzioni analitiche ad uso della gioventù italiana Ⓣ（供意大利青年使用的分析方法）译成英文，尽管他的英译本直到1801年才出版。科尔森把“la versiera”误认为“l'aversiera”，后者意为“女巫”或“女魔鬼”。关于这条曲线如何以“阿涅西的女巫”之名而为人所知，详见[21]。

关于这条曲线的更多信息见THIS LINK。

Maria Gaetana Agnesi was the daughter of Pietro Agnesi who came from a wealthy family who had made their money from silk. Pietro Agnesi had twenty-one children with his three wives and Maria was the eldest of the children. As Truesdell writes in [20], Pietro Agnesi:-

... belonged to a class intermediate between the patricians and the merely rich. Such a bourgeois could have a household fit for a lord, comport himself like a knight, mingle freely with some nobles, occupy himself with the finer things of life, be a patron of men of talent. [Pietro Agnesi] did just that...

Some accounts of Maria Agnesi describe her father as being a professor of mathematics at Bologna. It is shown clearly in [16] that this is entirely incorrect, but the error is unfortunately carried forward to [1] and will also be seen in a number of other places.

Pietro Agnesi could provide high quality tutors for Maria Agnesi and indeed he did provide her with the best available tutors who were all young men of learning from the Church. She showed remarkable talents and mastered many languages such as Latin, Greek and Hebrew at an early age. At the age of 9 she published a Latin discourse in defence of higher education for women. It was not Agnesi's composition, as has been claimed by some, but rather it was an article written in Italian by one of her tutors which she translated and [20]:-

... she delivered it from memory to an academic gathering arranged by her father in the garden...

In 1738 she published Propositiones Philosophicae a series of essays on philosophy and natural science. The volume contained 191 philosophical theses which Agnesi would defend in disputes with specially invited audiences of important international and national people who her father would invite to his house. In [4] de Brosses describes one such evening which took place on 16 July 1739:-

... I was brought into a large fine room, where I found about thirty people from all countries of Europe, arranged in a circle and Mlle Agnesi, all alone with her little sister, seated on a sofa. She is a girl of about twenty years of age, neither ugly nor pretty, with a very simple and very sweet manner. ... Count Belloni, who took me, wanted to make a public show. He began with a fine discourse in Latin to this young girl, that it might be understood by all. She answered him well, after which they entered into a dispute, in the same language, on the origin of fountains and on the causes of the ebb and flow which is seen in some of them, similar to tides at sea. She spoke like an angel on this topic, I have never heard anything so pleasurable. ...

She is much attached to the philosophy of Newton, and it is marvellous to see a person of her age so conversant with such abstract subjects. Yet however much I was amazed at her learning, I was perhaps more amazed to hear her speak Latin with such purity, ease and accuracy...

It might look as if this were an extremely distasteful affair, with Agnesi's father showing off his daughter's talents like a circus act. To some extent this must be the case, but it is fair to say that shows of this type were relatively common at the time. Certainly, although Agnesi always acted in total obedience to her father's wishes, she was not very happy with the spectacle that he put on. Again we quote [4] where de Brosses wrote:-

She told me that she was very sorry that the visit had taken the form of a thesis defence, and that she did not like to speak publicly of such things, where for every one that was amused, twenty were bored to death. ... I was much annoyed to hear it said that she wished to enter a convent, and it was not through need, for she is very rich.

In [20] Truesdell explains further about her wishing to become a nun:-

She did ask her father's permission to become a nun. Horrified that his dearest child should desire to leave him, he begged her to change her mind. She agreed to continue living in his house and caring for him on three conditions: that she go to church whenever she wished, that she dress simply and humbly, that she abandon altogether balls, theatres, and profane amusements.

Agnesi concentrated her efforts on studying religious books and learning mathematics. Around this time she wrote a commentary on de L'Hôpital's Traité analytique des section coniques but it has never been published. Learning mathematics without proper instruction is an almost impossible task and only a few have ever achieved great things in this way. Agnesi was fortunate, however, in her bid to learn mathematics for a monk, Ramiro Rampinelli, a mathematician who had been a professor at both Rome and Bologna, arrived in Milan and became a frequent visitor to the Agnesi house. With Rampinelli's help Agnesi studied Reyneau's calculus text Analyse démontrée (1708). Agnesi understood the debt she owed to Rampinelli and in the preface to her famous book Instituzioni analitiche ad uso della gioventù italiana she wrote:-

With all the study, sustained by the strongest inclination towards mathematics, that I forced myself to devote to it on my own, I should have become altogether tangled in the great labyrinth of insuperable difficulty, had not [Rampinelli's] secure guidance and wise direction led me forth from it ...; to him I owe deeply all advances (whatever they might be) that my small talent has sufficed to make.

Rampinelli encouraged Agnesi to write a book on differential calculus. She wrote the book in Italian as a teaching text which, according to the preface, attempted to present the material:-

... endowed with proper clarity and simplicity..., which proceeds with that natural order which provides, perhaps, the best instruction and the greatest light.

Agnesi, with her father's money, was able to arrange for the private printing of the book in her own home where she could supervise the whole operation herself. However, she wished to obtain more input from leading mathematicians so, on the 20 July 1745, she wrote to Riccati. It was Rampinelli who suggested that Riccati might offer Agnesi advice and he had clearly contacted Riccati, who had been one of his own teachers, and Riccati had agreed to read the final draft of Agnesi's book and make suggestions.

Riccati replied quickly to Agnesi's first letter and promised to pass the text to his two sons, Vincenzo Riccati and Giordano Riccati, so that they could also comment on the work. Once Agnesi received Riccati's comments on the first part of the text she organised printing of that part while later parts were sent to Riccati also for him to comment on. By 1747 Agnesi was sending Riccati later parts of the book and explaining to him that printing of the earlier parts was nearly complete.

Riccati wrote to Rampinelli on 1 February 1747, offering Agnesi his some of his earlier work on integration for inclusion in her book. Agnesi included the work with proper acknowledgement to Riccati. In her letters Agnesi tried to get Riccati to reply more quickly giving his notes on her draft since the printer had other work to undertake, and she wrote to Riccati saying that:-

... if it becomes necessary to suspend the printing again, I do not know when I could start it anew, because even now it has been extremely difficult for me to continue with printing the first part (which soon will be finished).

The first volume of Agnesi's famous two volume work Instituzioni analitiche ad uso della gioventù italiana Ⓣ was published in 1748 while Agnesi continued corresponding with Riccati over the material for the second volume which was published the following year. The work was to bring her much fame. A report on it made by a committee of the Académie des Sciences in Paris states:-

It took much skill and sagacity to reduce, as the author has done, to almost uniform methods these discoveries scattered among the works of modern mathematicians and often presented by methods very different from each other. Order, clarity and precision reign in all parts of this work. ... We regard it as the most complete and best made treatise.

Pope Benedict XIV wrote to Agnesi saying that he had studied mathematics when he was young and could see that her work would bring credit to Italy and to the Academy of Bologna. Soon after this he appointed Agnesi to the position of honorary reader at the University of Bologna. Then Agnesi was approached by the president of the Academy of Bologna and three other professors of the Academy and invited to accept the chair of mathematics at the University of Bologna. Indeed, shortly after this, Agnesi received a letter from Pope Benedict XIV written on 26 September 1750:-

We have had the idea that you should be awarded the well known chair of mathematics, by which it comes of itself that you should not thank us by we you...

It is probable that Agnesi neither accepted nor rejected this offer. As Truesdell writes in [20]:-

In October [Agnesi] received a papal rescript confirming her appointment. She had already devoted herself to a holy, retired life; while her name remained on the rolls of the university for forty-five years, she never went to Bologna.

This does explain the confusion which appears in many accounts as to whether Agnesi ever held a chair of mathematics. Frisi, who was a school friend of one of Agnesi's brothers, visited the Agnesi house after the time that her book was published. He describes in [6] how her father imposed severe constraints on her, and she chose to inhabit rooms of the house away from where the rest of the family lived and there she helped old women who were ill. However [6]:-

... she immediately, with no apparent difficulty, gave way to her father's wishes ..., taking part in the usual academies in his house with such grace and penetration, propounding or answering questions, problems, and scientific doubts...

After the death of her father in 1752, Agnesi devoted herself entirely to charitable work. She [20]:-

... resumed her study of Catholic doctrine and her costly acts of piety towards the poor and suffering, the hopelessly ill and the demented. First in her late father's house and afterwards in other places she established a hospice for old infirm women.

Agnesi spent all her money on this charitable work and she died in total poverty in the poorhouse of which she had been the director.

The treatise Instituzioni analitiche ad uso della gioventù italiana Ⓣ contains no original mathematics by Agnesi. Rather the book contains many examples which were carefully selected to illustrate the ideas; one review calls it an:-

... exposition by examples rather than by theory.

The book includes a discussion of the cubic curve now known as the 'witch of Agnesi'. There has been much argument over the reason why the curve is called a 'witch'. The curve was discussed by Fermat and, in 1703, a construction for the curve was given by Grandi. In 1718 Grandi gave it the Latin name 'versoria' which means 'rope that turns a sail' and he so named it because of its shape. Grandi gave the Italian 'versiera' for the Latin 'versoria' and indeed Agnesi quite correctly states in her book that the curve was called 'la versiera'.

John Colson, who had translated Newton's De Methodis Serierum et Fluxionum Ⓣ from Latin to English for publication in 1736, translated Agnesi's Instituzioni analitiche ad uso della gioventù italiana Ⓣ into English before 1760 (the year of Colson's death) although his English translation was not published until 1801. Colson mistook 'la versiera' for 'l'aversiera' which means 'the witch' or 'the she-devil'. See [21] for a detailed description of how the curve has become known as the 'Witch of Agnesi'.

More information about this curve is at THIS LINK.