雅各布·黎卡提Jacopo Riccati

雅各布·黎卡提的父亲是Conte Montino 黎卡提，而他的母亲来自Colonna家族，这是罗马贵族家族之一。尽管几个世纪以来Colonna家族一直与教皇冲突，但到17世纪，这种冲突已经结束，家族成员在教会、国家和军队中担任高级职务。Conte Montino 黎卡提在他儿子黎卡提十岁时去世，尽管他的母亲没有再婚，但这个男孩由他的母亲和他父亲的一个兄弟给予了良好的教养。年轻的黎卡提的这位叔叔看到了他是一个多么有才华的男孩，并建议他的母亲让他在布雷西亚的耶稣会学院接受良好教育，该学院被公认为“贵族学校”。黎卡提于1687年十一岁时进入该学院，在接下来的六年里，他在那里接受了极好的教育。显然他非常重视自己的教育，因为后来他把自己所有的儿子都送到了布雷西亚的同一所学院。家族传统意味着黎卡提在学院毕业后应继续深造，攻读法律学位，而这正是他在1693年进入帕多瓦大学时所遵循的道路。

尽管黎卡提进入帕多瓦是为了学习法律，但他当然对科学感兴趣，尤其是天文学，因此，除了法律课程外，他还参加了由斯特凡诺‧德力‧安杰利教授的天文学课程。他很快与他的讲师斯特凡诺‧德力‧安杰利成为朋友，后者此时已相当年迈。斯特凡诺‧德力‧安杰利曾受教于博纳文图拉·卡瓦列里，并拥护博纳文图拉·卡瓦列里引入的不可分方法。当斯特凡诺‧德力‧安杰利读到艾萨克·牛顿的Philosophiae Naturalis Principia Mathematica Ⓣ（自然哲学的数学原理）时，他意识到这对他一生研究的某些无穷小方法呈现了一种新的、令人兴奋的发展。大约在1695年，他把自己的艾萨克·牛顿的Principia副本给了黎卡提，并解释了他对这部作品的迷恋。这次与Principia的相遇鼓励黎卡提学习数学，但他完成了法律学位，于1696年6月7日从帕多瓦大学毕业。

黎卡提在毕业不久后于1696年10月15日结婚。他的妻子是Elisabetta dei Conti d'Onigo，他们育有十八个孩子，其中九个在童年夭折，九个活到成年。这些孩子中最著名的两个是Vincenzo Riccati（生于1707年），他对数学物理做出了重要贡献，在本档案中有传记；以及Giordano Riccati（生于1709年），他在学习法律后，对音乐律制和建筑做出了显著贡献，也有MacTutor传记。黎卡提有独立的经济来源，并在Castelfranco Veneto拥有一个大庄园，这是一个位于帕多瓦以北约30公里、威尼斯西北约40公里的小镇。在那里，他照顾家庭，并在城镇管理中发挥了重要作用，在1698年至1729年间担任了九年市长。建筑师Francesco Maria Preti（1701-1774）出生于Castelfranco Veneto，在黎卡提担任市长期间，他设计了镇上的建筑，包括1723年完工的大教堂。作为数学家，黎卡提基本上是自学成才，研究当时主要数学家的著作，并与他们中的许多人通信。他的阅读材料包括当时的科学期刊，特别是Commentari dell' Accademia delle Scienze di Bologna、Acta Eruditorum Lipsiae和位于圣彼得堡的帝国科学院的Proceedings。两本在威尼斯印刷的本地期刊也引起了他的兴趣，即Galleria di Minerva和Giornale de' Letterati d' Italia。这些期刊发表诗歌、哲学著作、短篇小说和一些数学。黎卡提 certainly not someone who worked on mathematics to the exclusion of other topics. 相反，正如Sergio Bittanti在[6]中指出的，他对所有学术主题都感兴趣：-

黎卡提有着广泛的兴趣，从数学到诗歌，从物理到宗教，他的著作和丰富的藏书就是见证。

我们引用黎卡提本人的话：-

我并不想声称每个主题都应被深入探究。依照自己的天赋和兴趣，一个人至少应选择一个主题，并深入研究它。在其他主题上，则应效仿蜜蜂从每朵花中吸取一滴花蜜……

黎卡提除了对数学和物理学做出下文所述的重要贡献外，还发表了哲学著作（在[14]中讨论）和文学作品（在[19]中讨论）。事实上，他最初作为数学家崭露头角，是因为解决了Giornale de' Letterati d'Italia中提出的一个困难数学问题。黎卡提自学了最新的数学进展，正是他对这些新方法的透彻了解，使他能在1710年解决这个问题。这标志着他数学成果的开端，他很快便发表了大量重要论文。

他很快获得了名声，并拒绝了彼得大帝约在1725年邀请他担任圣彼得堡科学院院长的提议，以及帕多瓦大学数学讲席的邀请，还有其他诱人的提议，如维也纳宫廷顾问。然而，他无需薪水，乐于留在意大利与他的大家庭在一起，在那里他可以完全按照自己希望的方式从事研究。Sergio Bittanti写道[6]：-

黎卡提是一位不张扬、善良的人，他更喜欢自己的家而非科学院和大学。他的生活方式非常简单，很少旅行。大概他唯一长时间离家是在1719年夏天，当时遵照医生的建议，他搬到Val di Sole，以利用那个山谷有益健康的水。

在Val di Sole期间，黎卡提遇到了尼古拉·伯努利二世，他们就解决微分方程进行了数学讨论。此时我们应该看看黎卡提与其他数学家和科学家的互动。这些包括：Giovanni Rizzetti（1675-1751），以批评艾萨克·牛顿的光理论而闻名；Gabriele Manfredi，博洛尼亚大学数学教授和校长，著名数学家和天文学家Eustachio Manfredi的兄弟；Giovanni Poleni，帕多瓦大学教授；Antonio Vallisneri（1661-1730），在帕多瓦大学担任实用医学和理论医学讲席，并且是Giornale de' Letterati d'Italia的编辑；Ramiro Rampinelli，一位在罗马和博洛尼亚任教授的数学家；以及Bernardino Zendrini（1679-1747），为威尼斯共和国工作的科学家。在这些科学家中，Manfredi对黎卡提的数学方法影响最大，特别是通过他的书De constructione aequationum differentialium primi gradusⓉ（论一阶微分方程的构造），1707年在博洛尼亚印刷。黎卡提终生热衷于研究使用分离变量法解微分方程的方法，正是通过阅读这本书而来。黎卡提与Gabriele Manfredi互动的细节由Sandra Giuntini在[10]中详细考虑。A C Garibaldi在评论[10]时写道：-

这次科学交流的主题，首先是黎卡提在某些微分方程中分离未知量的方法，然后是1724年Suzzi——黎卡提的一位年轻弟子——与丹尼尔·伯努利之间争论的关于可化为等积的月牙形的问题。

这段引文中提到的Suzzi是Giuseppe Suzzi。他和Ludovico da Riva是黎卡提的私人学生，在1722年和1723年期间跟随他学习数学。事实上，黎卡提为教授Suzzi和da Riva编写了详细的讲义（共154页），这些讲义后来以Delia separazione delle indeterminate nelle equazioni differenziali di prima e di secondo grado, e della riduzione delle equazioni differenziali del secondo grado e d'altri gradi ulteriori Ⓣ（《论一阶和二阶微分方程中的变量分离，以及二阶及更高阶微分方程的化简》）为名出版。Suzzi和da Riva是极为出色的学生，后来分别成为帕多瓦大学的数学教授和天文学教授。

除了上面提到的科学家之外，黎卡提还与雅各布·赫尔曼、尼古拉·伯努利二世和玛利亚·阿涅西等顶尖数学家通信。玛利亚·阿涅西的著名教材Instituzioni analitiche ad uso della gioventù italiana Ⓣ（《供意大利青年使用的分析教程》）（第一卷于1748年出版，第二卷于1749年出版）就是在她就其内容与黎卡提通信期间写成的。她在序言中写道：-

在第二卷中，当处理积分学时，读者会发现一种关于多项式的新方法；这归功于著名的黎卡提伯爵，他在所有科学领域都具有独特的价值，为知识界所熟知。他如此善意地将这样一份礼物赐予我，而我并不配得到它，现在我对他和公众做出了应有的公正对待。

1749年，黎卡提的妻子Elisabetta去世，此时他从Castelfranco Veneto搬到Treviso，家族在那里还有另一处住所。事实上，在Elisabetta去世前的几年里，黎卡提家的孩子们每年大部分时间都住在Treviso的家中。正是在Treviso的这个家族住所中，黎卡提去世了，他被安葬在Treviso的大教堂中，黎卡提家族在那里有一座小教堂。

他在水力学方面的工作对威尼斯城很有用，他帮助沿运河修建了堤坝[1]：-

他经常被威尼斯元老院咨询，特别是关于沿河流和运河修建堤坝的问题，在这个和其他问题上，他的专业知识都受到尊重。

然而，他最著名的是在解微分方程方面的工作。在微分方程的研究中，他降低方程阶数和分离变量的方法很重要。他考虑了许多一般类型的微分方程，并找到了被广泛采用的解法。他主要以黎卡提微分方程而闻名，对此他进行了详尽的研究，并给出了某些特殊情况的解。尽管他可能早在1715年就开始研究这个方程，但关于该微分方程的第一个书面记录似乎出现在他1720年写给Giovanni Rizzetti的一封信中。该方程由黎卡提在我们上面提到的1722-23年讲义中讨论过[1]：-

在阐述已知的一阶微分方程积分方法时，黎卡提研究了那些可以通过适当的代数变换进行积分的方程，然后才考虑那些需要变量替换的方程。接着，他讨论了约翰·伯努利提出的一些方法，并阐述了Gabriele Manfredi用于积分齐次方程的方法。他进一步指出，为了确定具有给定性质的曲线，有时将其与通常坐标以外的某些坐标相关联可能是有用的。随后，黎卡提通过许多例子讨论了他自己设计的积分方法。其中，一种方法涉及将方程化为齐次方程，而另一种更有趣的方法是黎卡提所称的“半分分离”法。半分分离技术包括三个操作。首先，将整个方程乘以或除以未知函数的适当函数，使其可积；其次，在完成此积分后，将结果视为一个新的未知量，从而消去其中一个原始变量；最后，对结果应用前两个步骤，直到获得所需的新结果。

他的工作对丹尼尔·伯努利等领先数学家产生了广泛影响，后者在其Exercitationes quaedam mathematicae Ⓣ（数学练习）中研究了该方程，而莱昂哈德·欧拉则将黎卡提的思想扩展到任意阶非齐次线性微分方程的积分。黎卡提还研究了摆线摆、流体中的阻力定律以及微分几何。

Bittanti描述了黎卡提生命的终结[6]：-

黎卡提是一个坚强而勤奋的人，在他的一生中始终保持着活跃而富有创造力的头脑。1754年4月2日，他突然发烧，两周后的4月15日，他去世了。

黎卡提的Opere Ⓣ（著作集）于1765年出版，共四卷，由其子Giordano Riccati编辑。

Jacopo Riccati's father was Conte Montino Riccati while his mother came from the Colonna family, one of the noble Roman families. Although for centuries the Colonna family had been in conflict with the Pope, by the 17th century this conflict was over and members of the family held high offices in the church, state and military. Conte Montino Riccati died when his son Jacopo was ten years old and, although his mother did not remarry, the boy was given a good upbringing by his mother and one of his father's brothers. This uncle of the young Jacopo saw what a talented boy he was and advised his mother to have him well educated at the Jesuit college in Brescia which was recognised as the "school for the nobility". Jacopo entered the college in 1687 when he was eleven years old and there he was given an excellent education during the following six years. Clearly he valued his education highly, since later in life he sent all of his own sons to the same college in Brescia. Family tradition meant that Riccati would be expected to continue his studies after college by taking a law degree and this is precisely the course he followed when he enrolled at the University of Padua in 1693.

Although he entered Padua to read law, Jacopo Riccati was certainly interested in the sciences, particularly in astronomy, so, as well as courses in law, he attended an astronomy course taught by Stephano degli Angeli. He soon became friends with his lecturer Angeli who was by this time quite an old man. Angeli had been taught by Bonaventura Cavalieri and had championed the method of indivisibles which Cavalieri had introduced. When Angeli read Isaac Newton's Philosophiae Naturalis Principia Mathematica Ⓣ he realised that this presented a new and exciting development of certain methods with infinitesimals which he had worked on all his life. Around 1695 he gave his copy of Newton's Principia to Riccati and explained his fascination with the work. This encounter with the Principia encouraged Riccati to study mathematics but he completed his law degree, graduating from the University of Padua on 7 June 1696.

Riccati married soon after graduating, on 15 October 1696. His wife was Elisabetta dei Conti d'Onigo and they had eighteen children, nine of whom died in childhood and nine survived to adulthood. The two most famous of these children were Vincenzo Riccati (born 1707) who made important contributions to mathematical physics and has a biography in this archive, and Giordano Riccati (born 1709) who, after studying law, made notable contributions to musical temperament and to architecture, and also has a MacTutor biography. Riccati was of independent means and had a large estate at Castelfranco Veneto, a small town about 30 km north of Padua and about 40 km north west of Venice. There he cared for his family and also played a major role in the administration of the town, being mayor for nine years between 1698 and 1729. The architect Francesco Maria Preti (1701-1774) was born in Castelfranco Veneto and, during the period that Riccati was mayor, he designed buildings in the town including the cathedral completed in 1723. As a mathematician Riccati was, essentially, self-educated studying the works of the leading mathematicians of the day and corresponding with many of them. Among his reading material was the scientific journals of the day, in particular the Commentari dell' Accademia delle Scienze di Bologna, the Acta Eruditorum Lipsiae, and the Proceedings of the Imperial Academy of Sciences based in St Petersburg. Two local journals which were printed in Venice also interested him, namely the Galleria di Minerva and the Giornale de' Letterati d' Italia. These journals published poetry, philosophical writings, short stories and some mathematics. Riccati was certainly not someone who worked on mathematics to the exclusion of other topics. Rather the reverse, he was interested in all scholarly subjects as Sergio Bittanti points out in [6]:-

Riccati had far-reaching interests, ranging from mathematics to poetry, from physics to religion, as witnessed by his works and his rich library.

We quote from Riccati himself:-

I do not want to claim that every topic should be probed in detail. Following one's own talent and inclination, one should select at least one topic, and study it in depth. In the others, one should follow the example of the bee which sucks a drop of nectar from each flower ...

Indeed Riccati, in addition to the important contributions to mathematics and physics which we describe below, also published philosophical works (discussed in [14]) and literary writings (discussed in [19]). In fact he first made his mark as a mathematician by solving a difficult mathematical problem which appeared in the Giornale de' Letterati d'Italia. Riccati had studied on his own the latest mathematical advances and it was his thorough knowledge of these new methods which enabled him to solve this problem in 1710. This marked the beginning of his mathematical output and he was soon publishing numerous significant papers.

He soon attained fame and turned down an offer from Peter the Great to become President of the St Petersburg Academy of Science in around 1725, an offer of the chair of mathematics at the University of Padua, as well as other tempting offers such as Advisor to the Court in Vienna. However, he had no need of a salary and was pleased to remain in Italy with his large family where he could pursue his own studies in exactly the manner that he wished. Sergio Bittanti writes [6]:-

Riccati was an undemonstrative, kind man who preferred his home to academies and universities. His way of life was a very simple one, and he travelled very little. Probably, the only extended period he spent away from home was the summer of 1719, when, following the recommendation of his physician, he moved to Val di Sole to take advantage of the healthy water of that valley.

While in the Val di Sole Riccati met with Nicolaus(II) Bernoulli and they had mathematical discussions regarding solving differential equations. At this point we should look at Riccati's interactions with other mathematicians and scientists. These include: Giovanni Rizzetti (1675-1751), famed as a critic of Newton's theory of light; Gabriele Manfredi, professor of mathematics and chancellor of the University of Bologna, and the brother of eminent mathematician and astronomer Eustachio Manfredi; Giovanni Poleni, who was a professor at the University of Padua; Antonio Vallisneri (1661-1730), who held the chairs of Practical Medicine and Theoretical Medicine at the University of Padua and was an editor of the Giornale de' Letterati d'Italia; Ramiro Rampinelli, a mathematician who was a professor at Rome and at Bologna; and Bernardino Zendrini (1679-1747), a scientist working for the Republic of Venice. Of these scientist, it was Manfredi who had the greatest influence on Riccati's approach to mathematics, particularly through his book De constructione aequationum differentialium primi gradus Ⓣ, printed in Bologna in 1707. Riccati's life-long passion for studying methods of solving differential equations using separation of variables came through his reading of this book. Details of Riccati's interactions with Gabriele Manfredi are considered in detail by Sandra Giuntini in [10]. A C Garibaldi, in a review of [10], writes:-

The subject of this scientific exchange is, first, a method of Riccati for separating the indeterminates in some differential equations, and then a question on the lunules quarrables which was disputed between Suzzi, one of Riccati's young disciples, and Daniel Bernoulli in 1724.

Suzzi, mentioned in this quote, is Giuseppe Suzzi. He and Ludovico da Riva were Riccati's private pupils studying mathematics with him during 1722 and 1723. In fact Riccati produced detailed lecture notes (consisting of 154 pages) for teaching Suzzi and da Riva which were subsequently published as Delia separazione delle indeterminate nelle equazioni differenziali di prima e di secondo grado, e della riduzione delle equazioni differenziali del secondo grado e d'altri gradi ulteriori Ⓣ. Suzzi and da Riva were students of exceptional quality, becoming professors of mathematics and astronomy, respectively, at the University of Padua.

In addition to the scientists mentioned above, Riccati also corresponded with the leading mathematicians such as Jacob Hermann, Nicolaus(II) Bernoulli and Maria Gaetana Agnesi. Agnesi's famous text Instituzioni analitiche ad uso della gioventù italiana Ⓣ (first volume published 1748, second volume 1749) was written while she was corresponding with Riccati about its content. She wrote in the Preface:-

In the second volume, when dealing with Integral Calculus, the reader will find a new method for polynomials; this is due to the famous Count Jacopo Riccati, a personality of unique merit in all sciences, and well known to the literate world. He was so kind as to favour me with such a gift, which I did not deserve, and I now do justice to him, and to the public, as it should be.

In 1749 Elisabetta, Riccati's wife, died and at this time he moved from Castelfranco Veneto to Treviso where the family had another home. In fact for a few years before Elisabetta's death the Riccati children had lived in the Treviso home for a large part of each year. It was in this family home in Treviso that Riccati died and he was buried in the Cathedral in Treviso where the Riccati family had a chapel.

His work in hydraulics was useful to the city of Venice and he helped construct dikes along the canals [1]:-

He was often consulted by the senate of Venice, particularly on the construction of dikes along rivers and canals, and his expertise was deferred to on this and other topics.

However, he is best known for his work on solving differential equations. In the study of differential equations his methods of lowering the order of an equation and separating variables were important. He considered many general classes of differential equations and found methods of solution which were widely adopted. He is chiefly known for the Riccati differential equation of which he made elaborate study and gave solutions for certain special cases. Although he probably began studying the equation in 1715, the first written record of the differential equation seems to be in a letter he wrote to Giovanni Rizzetti in 1720. The equation was discussed by Riccati in the 1722-23 lecture notes we mentioned above [1]:-

In expounding the known methods of integration of first-order differential equations, Riccati studied those equations that may be integrated with appropriate algebraic transformation before considering those that require a change of variable. He then discussed certain devices suggested by Johann Bernoulli and expounded the method used by Gabriele Manfredi to integrate homogeneous equations. He further pointed out that in order to determine a curve endowed with an assigned property, it may at times be useful to relate it to some coordinates other than the usual ones. Riccati then discussed, with many examples, the integration methods that he himself had devised. Of these, one involves the reduction of the equation to a homogeneous one, while another more interesting method is that of "halved separation," as Riccati called it. The technique of halved separation comprises three obrations. In the first, the entire equation is multiplied or divided by an appropriate function of the unknown so that it becomes integrable; second, after this integration has been carried out, the result is considered to be equal to a new unknown, and one of the original variables is thus eliminated; and finally, the first two procedures are applied to the result until a new and desired result is attained.

His work had a wide influence on leading mathematicians such as Daniel Bernoulli, who studied the equation in his Exercitationes quaedam mathematicae Ⓣ, and Leonard Euler who extended Riccati's ideas to integration of non-homogeneous linear differential equations of any order. Riccati also worked on cycloidal pendulums, the laws of resistance in a fluid and differential geometry.

Bittanti describes the end of Riccati's life [6]:-

Count Riccati was a strong and hard-working person, with an active and fertile mind throughout the years of his life. On 2 April 1754, he had a sudden bout of fever and a fortnight later, on 15 April, he passed away.

Riccati's Opere Ⓣ was published in four volumes in 1765, edited by his son Giordano Riccati.