**Giovanni Ceva, The Mathematician With Many Hats**

*By Ashley Langham*

Giovanni Ceva was a mathematician that specialized in geometry. He was also a physicist, hydraulics engineer, politician, professor, and tax collector. He is most known for the geometrical theorem named after him that concerned straight lines intersecting at a common point emerging from the triangle’s vertices, or angular points. He is most known for his 1678 publication *De Lineis Rectis Se Invicem Secantibus Sciatica Constructio *(Concerning Straight Lines) which contains his famous theorem. Unfortunately, at the time he published it, it wasn’t very popular and only managed to have one edition. It wasn’t until the 19th century, when Michel Chasles, French mathematician, recognized that other mathematicians’ geometrical theorems had actually all derived from Ceva’s own work, from his 1678 publication.

Ceva had a very long life and accomplished many things. He made important contributions to physics in the study of motion and hydraulics. He also was involved in the business and political activities of the town Mantua and Montferrat. Read on to learn more about his work and life!

# Giovanni Ceva (1647–1734): His Early Life and Theorem

Ceva was born in Milan to a wealthy family. His father, Carlo Francesco Ceva bought and sold real estate and worked in politics as a tax collector for the Duke of Milan. His mother was Paola Columbo. Together they had seven other children, along with Giovanni. Giovanni Ceva, later followed in his father’s footsteps, and pursued a life in politics, along with his other pursuits. There is not much information about his childhood but in his own words, he described it as very “sad and one full of misfortunes”, where his family didn’t approve of his academic choices.

His earliest known education was when he attended Collegio di Brera, at a Jesuit College in Milan and showed early talent for mathematics and science. After he graduated, he continued his studies at the University of Pisa in 1670. He followed in the immense footsteps of Galileo, who also studied and became a professor there, almost a century prior. It was during his studies he fell into his first mathematical ”rut”. He couldn’t figure out how to square a circle while constructing a square with the same area as the given circle, using straight edges and a pair of compasses. And, in fact, in 1882, the Lindemann-Weierstrass theorem proved that his “problem” was actually impossible to solve. So, when Ceva gave up on trying to solve this problem, he actually saved himself a lot of time and effort.

It was around this time, he followed his father’s lead into politics and was appointed to his father’s position. He was appointed by the Duke of Mantua as the town’s auditor and commissioner. He became responsible for the economy of two states, (Mantua and Montferrat). After securing quite a significant post, he felt he was finally able and ready to get married. He married Cecilia Vecchi in 1685 and had seven children, five of which survived and also became citizens of the state of Mantua.

This didn’t distract him from his mathematical pursuits. By this time, he had already published his first book, *Concerning Straight Lines,* and took a separate career in the study of geometry. In his first publication he mentioned his biconditional (a solution where the converse is true), theorem concerning:

*The three lines of a triangle containing the vertices of that triangle intersect opposite sides at three points and are concurrent.*

In this same piece of work, Ceva was able to enrich Greek mathematician Medelaus of Alexandria’s theory, established in 100 AD, that uses a more rudimentary understanding of geometrical relationships. In Menelaus’s theorem he establishes a biconditional relationship that states:

*If three sides of a triangle are crossed by a straight line then the product of the three of nonadjacent line segments formed is equal to the product of three other line segments (**Britannica**)*

Unfortunately, his discoveries in geometry did not make him famous for his time. In fact, it would take another century until he was even credited properly for his work. He did, however, make enough waves in the mathematical field that he was appointed as the professor of mathematics at the University of Mantua and maintained that post for the rest of his life. He even kept his post during the occupation and protection of Austrian forces, in 1707. It was during this time, he actually gained and held full support from the new Austrian regime to continue his duties as professor and politician. This was as a result of his swift and strong allegiance to the new regime.

Despite his book not picking up more relevance, he established some notoriety in the well-known mathematician circles. He kept several close relationships, through correspondence, with other, arguably more famous, mathematicians.

# Giovanni Ceva’s Other Works

Along with his many jobs and contributions to mathematics, he also established himself in the study of motion. He wrote *Opuscula Mathematica *(Mathematical Essays) which was an investigation of forces and questions around geometry and hydraulics, i.e. liquid mechanics. In this collection of essays he discussed parallelogram of forces, a method for applying two forces to one object, pendulum motion, and the behavior of bodies in flowing water. (Britannica) As a part of his political duties, he was also appointed superintendent of waters in Mantua in 1684. It would be his works in hydraulics that helped him in this post. He was able to successfully argue against a potential project that would divert the river Reno away from the town of Mantua, cutting off an essential water source.

In 1692, he also applied the questions of geometry to his study of motion in his book *Geometria Motus *(The Geometry of Motion). In this work, he continued broadening upon his theories on the study of motion and hydraulics. He also dedicated the work to the Duke of Mantua. Well into his eighties, he wrote *Opus Hydrostaticum *(Hydrostatics). He also wrote one of the first mathematical economics books examining the conditions for balance in a monetary system in a relatively small town, like Mantua, in *De Re Nummaria* (Concerning Money Matters).

He died in Mantua on May 13, 1734. His work would not only influence other mathematicians in the field of geometry, he would also make significant contributions to the study of physics, particularly in hydraulics.

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