**Marin Mersenne** was born into a working class family in the small town of Oizé in the province of Maine on 8 September 1588 and was baptised on the same day. From an early age he showed signs of devotion and eagerness to study. So, despite their financial situation, Marin's parents sent him to the Collège du Mans where he took grammar classes. Later, at the age of sixteen, Mersenne asked to go to the newly established Jesuit School in La Flèche which had been set up as a model school for the benefit of all children regardless of their parents' financial situation. It turns out that Descartes, who was eight years younger than Mersenne, was enrolled at the same school although they are not thought to have become friends until much later.

Mersenne's father wanted his son to have a career in the Church. Mersenne, however, was devoted to study, which he loved, and, showing that he was ready for responsibilities of the world, had decided to further his education in Paris. He left for Paris staying en route at a convent of the Minims. This experience so inspired Mersenne that he agreed to join their Order if one day he decided to lead a monastic life. After reaching Paris he studied at the Collège Royale du France, continuing there his education in philosophy and also attending classes in theology at the Sorbonne where he also obtained the degree of Magister Atrium in Philosophy. He finished his studies in 1611 and, having had a privileged education, realised that he was now ready for the calm and studious life of a monastery.

The Order of the Minims, having been set up by St Francis of Paula in 1436, was thriving at this time. They believed they were the least (minimi) of all the religions on earth, and devoted themselves to prayer, study, and scholarship. They wore a habit made of coarse black wool with broad sleeves and girded by a thin black cord (as seen in the portraits of Mersenne). Charles VIII introduced the Order into France and, due to their great simplicity, the monks were named 'les bons hommes'. After the French Revolution the Order dwindled considerably in number and today there exists only a few convents in Italy. Mersenne entered the Order on 16 July 1611, and was ordained a priest in Paris in July 1612 after a two and a half month probationary period in the monasteries at Nigeon and Meaux. His first posting was in 1614 to the monastery in Nevers where he taught philosophy and theology to the younger members of the community. In fact one of his students, Hilarion de Coste, later became his confidant and biographer. It was during this period of his life that he is thought to have discovered the cycloid - a geometric curve.

After two years teaching Mersenne was elected superior of the Place Royale monastery in Paris where he remained, except for brief journeys, until his death in 1648. It is believed that the Church supported him for most of his life, although in later years a fellow monk, Jacques Hallé, helped out with money and granted him access to his library. From the beginning of his time in Paris, mathematical problems played an important role in his life. Very early on he had links with important scholars in Paris whom he met often, exchanging ideas and discussing projects. The Minims realised that the biggest service he could give was through his books and they never asked any more of him.

In 1623 he published his first two papers consisting of studies against atheism and scepticism in France; *L'usage de la raison* and *L'analyse de la vie spirituelle*. Continuing his theological writing he had then wanted to disprove magic, however a fellow monk pointed out that it was not appropriate, leading to his publication of *Quaestiones celeberrime in genesim* that includes the disapproval of magicians in the Scriptures. This book contains 1900 columns of text from the Bible in its first six chapters. It was because of this publication that, in September 1624 when he returned to Paris, he met Gassendi who had been asked to comment on Mersenne's results, and later became his closest friend.

At this time France was going through a period of anti-witchcraft, expelling any sorcerers. *L'impiété des deistes*, in French, was aimed at the French public so that they might read and understand what was happening. It was during this time that Mersenne started to think about the theological criticism directed against Descartes and Galileo. In fact Mersenne's attitude to Galileo changed over a number of years as Garber points out in [16]:-

Marin Mersenne was central to the new mathematical approach to nature in Paris in the1630s and1640s. Intellectually, he was one of the most enthusiastic practitioners of that program, and published a number of influential books in those important decades. But Mersenne started his career in a rather different way. In the early1620s, Mersenne was known in Paris primarily as a writer on religious topics, and a staunch defender of Aristotle against attacks by those who would replace him by a new philosophy. ... In the early1620s, Mersenne listed Galileo among the innovators in natural philosophy whose views should be rejected. However, by the early1630s, less than a decade later, Mersenne had become one of Galileo's most ardent supporters.

Mersenne was beginning to realise that alongside religion it was science that really interested him. Mathematics was the area he studied in greatest depth, believing that without it no science was possible. He always had a philosophical approach to mathematics and believed that the cause of the sciences is the cause of God, see [5]. So, in *La vérité des sciences* he proved, via many great discoveries, the value of the human mind. It was around this time that Mersenne started to become a coordinator for all European scholars. From 1623 he began to make a careful selection of savants who met at his convent in Paris or corresponded with him from all across Europe and even from as far afield as Constantinople and Transylvania (present-day Hungary). His regular visitors, or correspondents, included Peiresc, Gassendi, Descartes, Roberval, Beeckman, J B van Helmont, Fermat, Hobbes, Étienne Pascal, and his son Blaise Pascal. He set up meetings of scholars from around Europe during which they would read and review scientific papers, both national and international, exchange contacts with other scholars and plan and discuss experiments and other work. This came to be known as the *Académie Parisiensis* and sometimes among friends as the *Académie Mersenne*. It was notably one of most resourceful centres of research at that time, meeting weekly at members' houses and later in Mersenne's cell due to his weakened health. The list of Mersenne's correspondents kept increasing and Mersenne himself did not hesitate to travel to meetings with scholars all around Europe.

Mersenne had a strong interest in music and spent a lot of time researching acoustics and the speed of sound. In 1627 he published one of his most famous works, *L'harmonie universelle*. In this work he was the first to publish the laws relating to the vibrating string: its frequency is proportional to the square root of the tension, and inversely proportional to the length, to the diameter and to the square root of the specific weight of the string, provided all other conditions remain the same when one of these quantities is altered. Mersenne had already started encouraging the talents of others and helped them to share their ideas and results with other scholars. When Roberval arrived in Paris, after joining Mersenne's circle of scholars, his talent was soon recognised by Mersenne who encouraged him to work on the cycloid.

The period between 1627 and 1634 was a transitional period in Mersenne's life. During this time he travelled to Holland for several months between 1629 and 1630. His main reason was to seek a cure for an illness with the help of spa water but he used the opportunity to visit scholars in the surrounding areas. The greater maturity in his writing in the seven years since his last publication became apparent when *Questions inouyes* and *Questions harmoniques* were printed in 1634. In October 1644 Mersenne travelled to Provence and Italy where he learnt of the barometer experiment from Torricelli. On his return to Paris, he reported this news to encourage French scholars to carry out the experiments too.

Throughout his lifetime Mersenne helped many potential scientists by steering them in the right direction and advising some on the next step to take. He became a role model for Huygens whom Mersenne took under his wing and through his encouraging letters inspired Huygens' *Theory of Music*. Huygens had intended to move to Paris in 1646 to be near Mersenne in order to enable them to contact each other more easily, however Huygens did not move until several years after Mersenne had died so they never met.

Galileo also has to be grateful to Mersenne for making his work known outside Italy. Mersenne insisted on publishing Galileo's work and without this Galileo's ideas might never have become as widely known. Continuing his travels into his old age, in 1646 Mersenne set off on a trip to Bordeaux. There he met Pierre Trichet whom he helped make his mark. The success of the scientific life over in Bordeaux and Guyenne, which later formed the Académie Royale des Sciences, was largely due to the advice and experience Mersenne was able to offer. He returned to Paris in 1647.

Mersenne fell ill after his visit to see Descartes in July 1648 and, unfortunately, his health never improved. He was advised to mix wine with his water to help him get better, however Minims do not drink wine. He had an abscess on the lung but the surgeon was unable to find it. Mersenne himself pointed out that the incision, which he asked for, had been attempted too low. Gassendi was there for Mersenne throughout his illness and remained with him until his death on 1 September 1648 in Paris, just 8 days from his 60^{th} birthday. He never gave up his life-long desire to advance science. He even asked, in his will, that his body be used for biological research.

After Mersenne's death, letters in his cell were found from 78 different correspondents including Fermat, Huygens, Pell, Galileo and Torricelli. Also several physics instruments were found in his cell and a lot of Mersenne's library was retrieved from which *L'optique et la catoptrique* was published in 1651. Inside this publication one of Roberval's texts was inserted. Later all the letters he sent and received from other scholars were accumulated and published in several volumes. These letters read like an international review of mechanics in the early 17^{th} century. Mersenne was aware of all the science that was going on, what all the scientists were doing, and only wanted for them all to work together in advancing science.

Mersenne studied the cycloid for several years quoting his research in *Quaestiones in Genesim* (1623), *Synopsis mathematica* (1626) and *Questions inouyes* (1634). He gave the definition of a cycloid as the locus of a point at distance *h* from the centre of a circle of radius *a*, that rolls along a straight line. He stated the obvious properties including the length of the base line equals the circumference of the rolling circle. We note that Mersenne referred to the cycloid as the 'roulette' but the term cycloid was adopted later. He attempted to find the area under the curve by integration but having failed, so he put the question to Roberval. In 1638 he announced that Roberval had indeed found the area under the cycloid.

Mersenne's name is best remembered today for Mersenne primes. He tried to find a formula that would represent all primes but, although he failed in this, his work on numbers of the form

2

^{p}- 1,pprime

has been of continuing interest in the investigation of large primes. It is easy to prove that if the number *n* = 2^{p} - 1 is prime then *p* must be a prime. In 1644 Mersenne claimed that *n* is prime if *p* = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 but composite for the other 44 primes *p* smaller than 257.

Over the years it has been found that Mersenne was wrong about 5 of the primes of the form 2^{p} - 1 where *p* is less than or equal to 257 (he claimed two that did not lead to a prime (67 and 257) and missed 3 that did: 61, 89, 107). Drake [13] has tried to both understand the source of Mersenne's work on these primes, and also to try to determine the rule that was being used. He suggests Frenicle de Bessy may be the source and also suggests that the errors might be misprints by the printer. Drake reconstructs Mersenne's rule for exponents as that they must differ by not more than one from a value of 2^{n} or by not more than three from a value of 2 to the power 2^{n}.

Mersenne undertook experiments to test Galileo's law of motion for falling bodies. In 1634 he presented the results that he had obtained when measuring the acceleration of falling bodies from heights of 147, 108 and 48 feet. These confirmed the time-squared law that Galileo had published in his *Dialogue on the two chief world systems* of 1632 but they also raised questions about the numerical data. One problem he tried to solve was whether acceleration was continuous as Galileo maintained or discontinuous as Descartes believed. Mersenne thought Galileo's assumption that a falling body passes through infinite degrees of speed was incompatible with a genuinely mechanistic explanation of acceleration. These ideas are discussed in detail in [24] and [25].

In some of his non-mathematical works Mersenne looks at permutations and combinations. He states practical rules for calculating the number of combinations or permutations, solving the problem of finding the number of permutations with or without repetitions and gives an example the making of anagrams. His main reason to study combinatorial analysis was, however, to optimise musical composition as he explains in *The book on the art of singing well* which is Book Six of *Harmonie universelle* (1636). In an unpublished manuscript preserved in the Bibliothèque Nationale at Paris he gave the 40320 permutations of 8 notes.

During the final four years of his life, Mersenne spent a lot of time investigating the barometer. Pascal had already proved that air was not weightless and it was Mersenne who found the density of air to be approximately ^{1}/_{19}^{th} that of water. He was informed of the barometer experiment, consisting of a glass tube about 3 feet long sealed at one end and filled with pure mercury, through several letters from De Verdus but it was not until October 1644, when he visited Torricelli in Italy, that he saw the experiment carried out. Torricelli used the pressure of the air to explain why the mercury moved up the glass tube. Mersenne was doubtful that the air pressure actually supported the mercury and on his return attempted to re-do the experiment but did not have the necessary equipment. Mersenne explained the problem to Étienne Pascal, his son Blaise Pascal, Petit, Roberval, and others in Paris. There is some confusion as to who, in 1647, initially suggested the experiments with the Torricellian tube and the mountain, later to be called the Puy de Dome experiments. Certainly Mersenne had briefed both Huygens and Le Tenneur but it was not until three weeks after Mersenne's death in 1648 that these experiments were carried out. They consisted of collecting results both at the foot of the Puy de Dome and at the summit. Tests were made as to whether the level of mercury in the column was lower when at the top of the mountain than it was at the bottom. If this had proved to be true, they realised that this would be due to the pressure of the air alone. Perier, who finally conducted the experiments, did indeed find that there was a significant difference in the level of the mercury hence drawing the correct conclusion that the air pressure was supporting it.

An interesting question is how Mersenne managed to pursue his scientific ideas freely at a time when the Church (of which he was a devoted member) moved to prevent such discussion. This topic is considered in detail in [18] where Hine writes:-

During the first half of the seventeenth century, debate over the Copernican hypothesis had spread beyond the ranks of astronomers and had stirred up so much controversy that the Church decided to intervene. In1616a theological examining body concluded that the idea of the earth's motion was philosophically false and in conflict with the Scriptures, and it suspended Copernicus's book until corrected. Historians have generally assumed that this decision and the subsequent condemnation of Galileo had such a devastating effect that scientific progress in Catholic countries was greatly retarded. However, the attitude of Mersenne, who was both a faithful member of a religious order and a central figure in the development of French science, does not support such a conclusion. An examination of Mersenne's reaction to Copernicanism indicates that no matter how disturbing the Church's decision, it was still possible, at least in France, to study Copernican ideas and to find them useful, despite some reservations. Mersenne was affected by such decisions of the Church, but less so than one might suppose.

**Article by:** *J J O'Connor* and *E F Robertson* based on an honours project by *Jessica Daniell* (University of St Andrews).

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