**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 17^{th} 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. 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 of1719, 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 in1724.

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* (On the separation of variables in differential equations of first and second order, and on the reduction of differential equations of second order and higher orders). 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 giovent 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. On2April1754, he had a sudden bout of fever and a fortnight later, on15April, he passed away.

Riccati's *Opere* was published in four volumes in 1765, edited by his son Giordano Riccati.

**Article by:** *J J O'Connor* and *E F Robertson*

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