**Giovanni Ceva**'s dates of birth and death have, until recently, been uncertain (for example they were unknown when [1] was written) but recently the correct information has been found in the parish records (see for example Paola Landra's work [2]). Ceva's father, Carlo Francesco Ceva (1610-1690), was a man who undertook many different activities such as buying and selling houses and land. He also worked for the duke of Milan in an official capacity, collecting excise duty. This meant that he was a wealthy man, able to provide a high quality lifestyle for his family. He married Paola Columbo, the daughter of Cristoforo Columbo and Elisabetta Caballina, on 20 September 1639. They had several children: Laura Maria Francesca Elisabetta Ceva (born 1640), Clara Giustina Bonaventura Ceva (born 1642), Iginio Nicolò Ceva (born 1644), Francesco Ceva (born 1645), Giovanni Benedetto Ceva (the subject of this biography, born 1647), Tommaso Ceva (who also has a biography in this archive, born 1648), Teresa Francesca Ceva (born 1650), and Cristoforo Vittore Ceva (born 1652). Laura, Iginio, Francesco, Tomasso, Teresa and Cristoforo all had ecclesiastical careers, with Francesco, Tomasso, Teresa and Cristoforo becoming Jesuits, joining the Society of Jesus.** **Giovanni Ceva's cousin, also named Carlo Francesco Ceva, was the Bishop of Tortona. We know no details of Ceva's youth, but he does make an intriguing comment that his youth was saddened by "many kinds of misfortune". He also suggests that his family opposed his academic pursuits.

Giovanni Ceva was educated at the Collegio di Brera, a Jesuit college in Milan, where he showed a particular aptitude for science, particularly for mathematics. After leaving the College, he followed the same path as his father, engaging in business activities and in administrative political roles in Milan, Genoa and Mantua. But he was also interested in undertaking scientific activities and, in addition to the financial dealing he was doing, he studied geometry and hydraulics. He entered the university of Pisa in 1670 and there he studied under Donato Rossetti (1633-1686), the professor of logic, who was a strong supporter of atomic theories. He also studied under Alessandro Marchetti (1633-1714), who succeeded his teacher Giovanni Alfonso Borelli (1608-1679) as professor of mathematics at Pisa in 1679. Marchetti:-

... was a man above prejudices, free to declare his sentiments, preferring experiment to authority, and reason to Aristotle.

While studying in Pisa, Ceva also met Michelangelo Ricci who, like Rossetti, was a member of the Mathematics-Physics Academy of Rome. It is highly likely, therefore, that Ceva spent some time at the Academy in Rome during his two years of residence in Pisa. The first mathematical problem he attacked was the classic problem of squaring the circle and he produced several incorrect solutions to it. He gave up this work, becoming somewhat discouraged. Following his stay in Pisa, Ceva carried on with his mathematical studies which he would publish in his work *De lineis rectis se invicem secantibus statica constructio* (1678), and, at the same time, he continued with similar activities to those of his father. He was at this time living in Mantua where he was appointed as Auditor and Commissioner to Ferdinando Carlo Gonzaga, the Duke of Mantua and Montferrat, taking over the role from his father. In this administrative position, Ceva was essentially responsible for the economy of Mantua and Montferrat. Major government roles did not stop Ceva finding time to pursue his scientific studies, however, for during this period he undertook the work that he wrote up in *Opuscula mathematica de potentiis obliquis, de pendulis, de vasis et de fluminibus *in 1682. In 1683 the Duke of Mantua granted Ceva citizenship of Mantua, showing how important the Duke considered Ceva's contributions to the state. Although he was very busy with his duties for the Duke, Ceva continued to undertake mathematical research. He corresponded with many of the leading scientists of the day which kept him in the forefront of mathematical and other scientific developments. He does comment, however, that his work was distracted by "serious cares and affairs of his friends and family."

On 15 January 1685 Ceva married Cecilia Vecchi; they had several children, Domitilla Francesca (born 1687), Carlo Francesco (born 1688 but died at the age of three), Massimo Cristoforo (born 1689), Pietro Antonio (born 1690), Maria Domitilla (born 1692 and died in the same year), Carlo Francesco (born 1693), and Costantino Innocenzo (born 1694). The Duke of Mantua extended the citizenship, which he had already given to Ceva, to his children. He was appointed Professor of mathematics at the University of Mantua in 1686, a post he held for the rest of his life. By the beginning of the 18^{th} century, however, the Gonzaga family were in crisis and the Habsburgs began to threaten Mantua. Ferdinando Carlo fled in 1701 leaving Anna Isabella Gonzaga as a regent. Ceva continued to undertake research on geometric problems and to work on economic issues, the results of his research being published in the two works of 1710 and 1711, respectively, mentioned below. In 1707 Austria annexed the duchy of Mantua and began to construct heavy fortifications. Giovanni Ceva quickly moved to support the new Austrian regime and in fact he was chosen by the people of a district of the city as their nominee to swear allegiance to their new political masters.

For most of his life Giovanni Ceva worked on geometry. He discovered one of the most important results on the synthetic geometry of the triangle between Greek times and the 19^{th} Century. The theorem states that lines from the vertices of a triangle to the opposite sides are concurrent precisely when the product of the ratio the sides are divided is 1. He published this in *De lineis rectis* (1678), which he dedicated to the Duke of Mantua, a book in two parts together with an important appendix. Ceva also rediscovered and published Menelaus's theorem in this work. That the book was not fully appreciated at the time of its publication is suggested by the fact that only one edition appeared and also by the fact that many of the results in the book were later rediscovered by others who were unaware of Ceva's work. The importance of Ceva's discoveries was only fully appreciated when pointed out by Michel Chasles in the 19^{th} century.

You can see Ceva's theorem and Menelaus's theorem.

In addition to his work on geometry, Ceva studied applications of mechanics and statics to geometric systems. Although he wrongly concluded that the periods of oscillation of two pendulums were in the same ratio as their lengths, he later corrected the error. We now look briefly at his other works.

As we mentioned above, Ceva published *Opuscula mathematica *in 1682. This work, dedicated to Michelangelo Ricci, is in four parts. It investigates questions of pure geometry as well as applications of mathematics, particularly to hydrodynamics. In *Geometria motus, opusculum geometricum in gratiam aquarum excogitatum* (1692) he, to some extent, anticipated the infinitesimal calculus in his study of curves such as parabolas and hyperbolas using infinitesimal methods of the type introduced by Bonaventura Cavalieri. It is clear from his correspondence with Vicenzo Viviani and Antonio Magliabechi (the librarian to the Duke of Tuscany) in 1685 that at this stage he was already undertaking research, motivated by his study of hydraulics, which eventually appeared in this 1692 publication. He dedicated *Geometria motus *to Ferdinando Carlo, the Duke of Mantua. He published a small work on his geometric researches, *Tria problemata geometris proposita* in 1710. In the following year he published *De re nummeraria quod fieri potuit, geometrice tractata*, one of the first works in mathematical economics; in it Ceva attempted to solve the conditions of equilibrium for the monetary system of a state like Mantua. As we have mentioned a number of times, Ceva did important work on hydraulics. On this topic he published *Opus hydrostaticum* (1728) which, other than *De lineis rectis*, is his most important contribution.

In his role in the government of Mantua he was able to use his knowledge of hydraulics and argue successfully against a project which proposed to divert the river Reno into the river Po. Ceva put forward his views in the pamphlet *Conseguenze del Reno, se coll'aderire al progetto de' Signori Bolognesi, si permettesse in Po grande* (1716) but these views were opposed by Eustachio Manfredi who replied with the pamphlet *Ragioni del Signor G. C. ... e del Signor D. M. Battaglia ... contra l'introduzione del Reno nel Po grande *(1716). The controversy continued with Ceva's *Replica ... in difesa delle sue dimostrazioni, e ragioni, per le quali non debbasi introdurre Reno in Po* (1717) to which Manfredi replied with *Osservationi alla replica di G. C. *(1717). The argument was finally settled in Ceva's favour with his final pamphlet on the topic *Risposta alle osservazioni del Sig. Dott. E. M. contro la di lui replica in proposito dell'immissione di Reno in Po grande, pretesa da' Signori Bolognesi* (1721).

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

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