## De lineis rectis se invicem secantibus statica construction [Staticæ constructionis liber primus [-secundus]].

Milan: Lodovico Monza, 1678.

First edition, rare, of Ceva’s first and most important book. “Ceva’s most important mathematical work is De lineis rectis (Milan, 1678), dedicated to Ferdinando Carlo. Chasles mentions it with praise in his Aperçu historique. In this work Ceva used the properties of the center of gravity of a system of points to obtain the relation of the segments which are produced by straight lines drawn through their intersections. He further utilized these properties in many theorems of the theory of transverse lines—for example, in placing at the points of intersection of the straight lines weights that are inversely proportional to the segments. From the relations of the weights, which are determined by the law of the lever, the relation of the segments is then derived. Ceva first applied his method, which is a combination of geometry and mechanics, to five basic figures, which he called ‘elements.’ He then used these in special problems, in which given relations are used to calculate others. The theorem of Menelaus concerning the segments produced by a transverse line of a triangle is proved, as is the transversal theorem concerning the concurrency of the transverse lines through the vertices of a triangle, which is named after Ceva. This theorem was established again by Johann I Bernoulli. Ceva worked with proportions and proved their expansion; he calculated many examples in detail and for all possible cases. (Occasionally he treated examples in a purely geometrical manner to demonstrate the advantage of his method.) In the second book of the De lineis, Ceva went on to more complex examples and applied his method to cylindrical sections, ellipses in the triangle, and conic sections and their tangents. In a geometrical supplement, not related to either of the first two books, Ceva dealt with classical geometrical theorems. He solved problems on plane figures bounded by arcs of circles and then calculated the volumes and centers of gravity of solid bodies, such as the paraboloid and the two hyperboloids of rotation. Cavalieri’s indivisibles are used successfully in this case” (DSB). Only one other copy on ABPC/RBH since 1961.

“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 [including] Giovanni Benedetto Ceva (born 1647) and Tommaso Ceva (born 1648, also a mathematician) … 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 …

“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 … 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 18th 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 19th Century. The theorem states that lines from the vertices of a triangle to the opposite sides are concurrent precisely when the product of the ratios in which 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 19th century” (MacTutor). Ceva died at Mantua in 1734.

It is now known that Ceva’s theorem was proved earlier by Yūsuf al-Muʾtamin, the third king of the Banu Hud dynasty who ruled over the Taifa of Zaragoza from 1081 to 1085. Al-Mu'taman was a scholar and patron of science, philosophy and the arts. He was well versed in astrology and mathematics, a discipline in which he wrote the most important treatise from al-Andalus, the Kitab al-Istikmal (‘Book of Perfection’).

Riccardi I, 342.

4to (239 x 173mm.), pp. [viii], 83, [1], with 10 folding engraved plates. Woodcut vignette on title-page, woodcut printer's device on final verso, woodcut initials, head- and tailpieces. Contemporary carta rustica, manuscript tile on spine. Very fine.

Item #5193

Price: \$9,500.00