**Gilles Roberval**'s parents were Pierre Personne and Jeanne Le Dru. The obvious question one might ask at this point is why Gilles Roberval's father was not named Roberval. The reason is that Gilles Roberval was named Gilles Personne and this was the name under which he was known for at least the first twenty-five years of his life, only adding the "de Roberval" after 1628. Pierre and Jeanne Personne were of humble origins, living in the village of Roberval, about 50 km north (and a bit east) of Paris. They worked on the land but probably were well enough off to lead a comfortable life. Recent research (in 2003) has shown that Gilles was born in a field at Noël-Saint-Martin, Villeneuve-sur-Verberie, close to Roberval, where his mother was bringing in the harvest. He was baptised the next day and the record of this has been found in the archives of the Oise region. We know that Gilles had a large extended family, had brothers and sisters but only one sister, Marie, outlived him.

Gilles began to study mathematics at the age of 14 years when the parish priest of Rhuis, a village just 2 km to the north of Roberval, realised that young Gilles was highly intelligent and began to give him lessons. The parish priest was actually the chaplain to the queen, Marie de Médici, and he not only instructed Gilles Personne in mathematics but also in Latin and probably Greek. At some stage (no record exists to indicate exactly when) he left his home district and travelled widely visiting many places in France. At this time he earned his living teaching mathematics while he discussed advanced topics with university teachers in the towns he visited. Auger writes [2] that he rode from town to town with an ink bottle strapped to the saddle of his horse. On his travels he went to Bordeaux where he met Pierre de Fermat. In September 1627 he was at La Rochelle when King Louis XIII of France, who had declared war on the Huguenots, besieged the town which was the most important of the Huguenot cities of France, and the centre of Huguenot resistance. Gilles Personne, as he was still known at the time, studied both practical and theoretical aspects of the problems of fortifications and ballistics which resulted from this siege.

He arrived in Paris in 1628 and made contact with Marin Mersenne's circle, particularly with Claude Hardy, Claude Mydorge, Étienne Pascal and Blaise Pascal. He became a member of the circle; in fact he later become the only professional mathematician in the group. At around this time, having received permission from the head of the village of Roberval, he added "de Roberval" to his name becoming Gilles Personne de Roberval. We will refer to him as Roberval from this point on. He spent the years from 1628 to 1632 building up his knowledge and skills in mathematics with the aim of getting a position as a professional mathematician. He achieved this in 1632 when he was appointed professor of philosophy in the Collège Gervais in Paris. This was a small institution attached to the university of Paris. It had been founded in the 14^{th} century for students from Bayeux but by the time Roberval was appointed it no longer had that restriction. After he was appointed to the Collège Gervais, Roberval rented accommodation next to the College [5]:-

Before we consider briefly what Roberval's life in these rooms was like, we need to understand a little of his character. Sturdy writes that [5]:-Roberval moved into rooms there where he resided until the end of his life; he never became a property owner in Paris. He had second floor accommodation comprising simply two rooms which looked into the college courtyard in one direction and the rue Boutebrie in the other. While teaching at the college he prepared himself as a candidate for one of the most prestigious mathematical positions in Paris: the Ramus Chair at the College Royal which fell vacant in1634and was to be filled through public competition.

More relevant to his life style was his reputation for being mean [5]:-... he was of notoriously difficult and irascible temper.

So how did he furnish his two rooms? They had only beds, tables and chairs, and these items were of poor quality. No pictures hung on the walls, and there were no ornaments. However, he did have some books in his rooms by authors such as Euclid, Archimedes, Viète, Torricelli, Gassendi, Descartes, Mersenne, Kepler, Vitruvius, Herodotus, Cicero and Quintillian. He had several dictionaries, books on grammar, and (despite not being a religious man) a Latin Bible. When he died not even a bottle of wine was found in his rooms but a large amount of cash was found, something of the order of eight times his annual salary.His apparent reluctance to spend money earned him a reputation as an inveterate miser whose avarice was a by-word among his associates and colleagues. This is not entirely fair, for he was ready to contribute generously to the dowries of his niece and great-niece. On the other hand the items devoted to personal convenience and comfort ...[were]few in number and do convey the impression of somebody whose lifestyle was exceedingly austere.

On 24 June 1634 he was declared the winner of the competition for the Ramus Chair, and was appointed to that chair of mathematics in the Collège Royale. This was a competitive appointment and Roberval had to compete for reappointment every three years. In 1655 he was appointed to Gassendi's chair of mathematics at the Collège Royale, in addition to the Ramus chair, and he held both chairs for the rest of his life. The years between 1648 and 1653 were difficult ones for anyone living in Paris with civil wars known as the Frondes. In August 1648 there was insurrection in Paris and the people barricaded the streets. After a siege by the army, the rebellion faded away by the spring of 1649 but civil war erupted again with a battle being fought around Paris in the summer of 1652. By 1653 the civil war ended although a continuing war with the Spanish was still going on. By 1655 Roberval felt that France had achieved peace between its own people and felt confident enough to make a major purchase of land [5]:-

Sturdy carries out a calculation in [5] which suggests that Roberval paid a higher price for the farm than one would expect. Perhaps this particular property meant a lot to him and certainly by the end of his life many members of his extended family were living in the area. The farm was mixed, set up for grain production, and had a dairy herd of cattle as well as a beef herd. Roberval brought in extra income by leasing out small plots on his farm to individuals. He employed two managers to run the farm together with a number of labourers. His nephew Antoine Personne lived on the farm and managed it for a number of years. Menerval is, like Roberval's birthplace, north of Paris, but it is further to the west about 80 km from Paris. It was well situated for its produce to be sold in Paris, and also near enough for Roberval to be able to make trips to the farm.In1655, when the Frondes had died down, he bought from Pierre de Brûlart, sieur de Coullet, a farm situated at Menerval.

Let us now look at Roberval's contributions to mathematics. It is rather difficult to assess his importance and influence for, although he made outstanding contributions of fundamental importance, he only published two works during his lifetime. These were *Traité de mécanique des poids soutenus par des puissances sur des plans inclinés à l'horizontale* (1636) and *Le système du monde d'après Aristarque de Samos* (1644). One might reasonably ask why, given that he produced such a large quantity of innovative mathematics, he published so little. We do not know the answer to this question for certain, but one likely theory is that he wanted to keep his discoveries out of the public domain so that he could use the material in the three-yearly competitions for the Ramus Chair [3]:-

Whatever the reason for the lack of publications during his lifetime, quite a lot of his work was published inCandidates were required not only to lecture, but also to demonstrate theorems and solve problems put to them by all comers; as result, the practice grew up of the incumbent trying to ensure his re-appointment by proposing problems which only he could solve.

*Divers ouvrages de mathématique et de la physique par messieurs de l'Académie Royale des Sciences*in 1693. Most of Roberval's material in this 1693 publication was at least fifty years old, so it did not have the impact that it would have done had it been published shortly after being written. Other texts have been published much later, for example

*Eléments de géométrie*in 1996, and other material has not yet been published (and perhaps never will be published) such as his courses on astronomy, surveying, architecture and physical geography.

Roberval developed powerful methods in an early study of integration, writing *Traité des indivisibles* which he claimed was based on Archimedes and not Cavalieri. It was published as part of *Divers ouvrages de mathématique et de la physique par messieurs de l'Académie Royale des Sciences* (1693). In it he computed the definite integral of a rational power of *x* and of sin *x*. Computing the integral of sin *x* allowed him to solve the problem:

He was (rightly) very proud of having solved this problem and, using similar techniques, he was the first to square the surface off the oblique cone in 1644. He worked on the cycloid computing its quadrature before 1636 and he also computed the cubature of the solid it generates by rotating it about its base. He compared the lengths of curves, a topic not considered since the times of the ancient Greeks, equating the spiral and parabola in their ordinary forms. Before August 1648 he had discovered the equality of the length of the generalised cycloid and the ellipse. This means that he solved the problem before Blaise Pascal, who receives the credit for achieving this first in 1659. He computed the arc of the cycloid before 1640 by reducing the problem to the integration of the sine. He therefore solved this problem before Torricelli who found a solution after 1644. He also computed the arc length of a spiral.Trace on a right cylinder, with a single motion of the compass, a surface equal to that of a given square.

In addition to his discoveries on plane curves, Roberval is important for his method of drawing the tangent to a curve, already suggested by Torricelli. This method of drawing tangents makes Roberval the founder of kinematic geometry. He writes:-

He developed this method of computing tangents while working on the cycloid some time before 1636. At first he kept this discovery a secret, but he taught the method between 1639 and 1644. Roberval wrote a treatise on algebra and one of analytic geometry which appeared in his posthumous 1693 publication. He certainly introduced algebraic methods into solving geometric problems before René Descartes did, but although he deserves credit for this nevertheless he did not produce Cartesian geometry.By means of the specific properties of the curved line, examine the various movements made by the point which describes it at the location where you wish to draw the tangent: from all these movements compose a single one; draw the line of direction of the composed movement, and you will have the tangent of the curved line.

However, some of his most important contributions were in the area of mechanics. Here is most impressive work was the discovery of the law of composition of forces in 1636. He discovered this general principle while studying a body suspended by two strings. In 1647 he wrote to Torricelli about his discoveries in mechanics (see for example [1]):-

He then went on to give Torricelli an overview of an intended new eight-volume work on mechanics. The content was planned as follows: Book I, On the centre of action of forces in general; Book II, On the balance, Book III, On the centre of action of particular forces; Book IV, On the chord; Book V, On instruments and machines; Book VI, On the forces which act within certain media; Book VII, On compound movements; Book VIII, On the centre of percussion of moving forces. The treatise has never been found and probably was never written, but parts of each of the eight books exists in Roberval's manuscripts.We have constructed mechanics which is new from its foundation to its roof, having rejected, save for a small number, the ancient stones with which it had been built.

We mentioned above that *Le système du monde d'après Aristarque de Samos* (1644) was one of only two publications by Roberval in his lifetime. In this work he praises Aristarchus's heliocentric system but he did not totally reject Ptolemy's earth centered system with the sun and planets circling the earth, or Tycho Brahe's system which has the earth at the centre, but has the planets circling a sun which circles the earth. Roberval writes in the Preface:-

It is interesting to see that Roberval believes in universal attraction well before it was proposed by Isaac Newton:-Perhaps all three of these systems are false and the true one unknown. Still, that of Aristarchus seemed to me to be the simplest and the best adapted to the laws of nature.

In 1666 Roberval was one of a group of scientists making astronomical observations from Jean-Baptiste Colbert's Paris residence. In addition to Roberval the others involved were Christiaan Huygens, Pierre de Carcavi, Adrien Auzout, Bernard Frénicle de Bessy and Jacques Buot. In many ways this can be seen as a meeting of the Académie Royale des Sciences before its official foundation. Colbert, who was the French Minister of Finance, chose the small group who met in the King's Library on 22 December 1666, which was the founding meeting of the Académie Royale des Sciences. Roberval was one of these founding members and went on to play an important role as an enthusiastic and energetic member of the Academy in is early years [5]:-In all this worldly matter, and in each of its parts, resides a certain property by the force of which this matter contracts into a single continuous body.

In 1669 he invented the 'Roberval balance' which is now almost universally used for weighing scales of the balance type. He presented details to the Academy on 21 August of that year. He also worked with Jean Picard in cartography and wrote on mapping France. He worked on one of the big questions of the day, whether a vacuum could exist, and designed apparatus which was used by Blaise Pascal in his experiments. In a report written on 20 September 1647 he confirmed Pascal's experiments on the vacuum above an inverted column of mercury in a tube. He explained the suspension of mercury in the tube by the pressure of air on the exterior mercury.He read papers on various subjects four times in1667, nineteen times in1668and twice in1669; nine other contributions are recorder during these years. Again, when a German inventor approached Colbert in1668with the offer of a secret machine which would solve the problem of calculating the longitude, Roberval was one of a small committee appointed by Colbert to examine the claim(the others were Auzout, Carcavi, Huygens, Picard and a senior naval officer). As a senior member of the Académie des Sciences he made an outstanding contribution to its early activities, bringing to its deliberations an enthusiasm and forcefulness ...

Jacqueline Pascal, the sister of Blaise Pascal, wrote a letter on 25 September 1647 in which she describes a meeting between Roberval and Descartes. The latter did not believe in the existence of a vacuum [4]:-

We see from this letter the friction between Roberval and Descartes. Not only did they argue over scientific issues, but they also showed a strong mutual dislike of each other. As well as meeting with other scientists in Paris, Roberval also corresponded regularly with Pierre de Fermat and with Evangelista Torricelli until his death in 1647. He had some close friends such as Abbé Gallois who was the editor of... they began to discuss the problem of the vacuum. Monsieur Descartes became particularly serious on the subject. The others explained a recent experiment to him and asked him what he thought entered into the space of the emptied tube. He said that it was his "subtle matter". My brother[Blaise Pascal]responded to this theory as best he could. Believing that my brother was having some difficulty expressing himself, Monsieur de Roberval took on Monsieur Descartes with not a little passion - although he remained civil. Monsieur Descartes responded rather bitterly that he could speak to my brother as long as he desired because my brother spoke reasonably but that he wouldn't continue to talk with Monsieur de Roberval, because the latter spoke out of too many prejudices. With that, he glanced at his watch and saw that it was noon. He stood up, because he had a dinner date in the Faubourg Saint-Germain. Monsieur de Roberval also had a date in the same neighbourhood. So Monsieur Descartes led him over to a carriage where the two of them were alone. They appeared to be joking with each other, but there was a bit of an edge to their humour, as Monsieur de Roberval confirmed after he returned from dinner ...

*Le Journal des Sçavans*from 1665 to 1674, secretary of the Academy in 1668 and 1669 and later professor of Mathematics and Greek at the Collège de France. Other close friends included Pierre de Carcavi and Desnoyers, the Queen Mother's Secretary.

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