# Marin Mersenne

Last updated

Marin Mersenne
Born8 September 1588
Died1 September 1648 (aged 59)
Paris
NationalityFrench
Known for Acoustics, Mersenne primes

Marin Mersenne, Marin Mersennus or le PèreMersenne (French: ; 8 September 1588 – 1 September 1648) was a French polymath, whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for Mersenne prime numbers, those which can be written in the form Mn = 2n − 1 for some integer n. He also developed Mersenne's laws, which describe the harmonics of a vibrating string (such as may be found on guitars and pianos), and his seminal work on music theory, Harmonie universelle , for which he is referred to as the "father of acoustics". [1] [2] Mersenne, an ordained priest, had many contacts in the scientific world and has been called "the center of the world of science and mathematics during the first half of the 1600s" [3] and, because of his ability to make connections between people and ideas, "the post-box of Europe". [4] He was also a member of the Minim religious order and wrote and lectured on theology and philosophy.

A polymath is an individual whose knowledge spans a significant number of subjects, known to draw on complex bodies of knowledge to solve specific problems. The term entered the lexicon in the 20th century and has now been applied to great thinkers living before and after the Renaissance.

In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century.

An integer is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 1/2, and 2 are not.

## Life

Mersenne was born of peasant parents near Oizé, Maine (present-day Sarthe, France). He was educated at Le Mans and at the Jesuit College of La Flèche. On 17 July 1611, he joined the Minim Friars and, after studying theology and Hebrew in Paris, was ordained a priest in 1613.

Oizé is a commune in the Sarthe department in the region of Pays-de-la-Loire in north-western France.

Sarthe is a department of Pays de la Loire situated in the Grand-Ouest of the country. It is named after the River Sarthe, which flows from east of Le Mans to just north of Angers.

Le Mans is a city in France on the Sarthe River. Traditionally the capital of the province of Maine, it is now the capital of the Sarthe department and the seat of the Roman Catholic diocese of Le Mans. Le Mans is a part of the Pays de la Loire region.

Between 1614 and 1618, he taught theology and philosophy at Nevers, but he returned to Paris and settled at the convent of L'Annonciade in 1620. There he studied mathematics and music and met with other kindred spirits such as René Descartes, Étienne Pascal, Pierre Petit, Gilles de Roberval, Thomas Hobbes, and Nicolas-Claude Fabri de Peiresc. He corresponded with Giovanni Doni, Jacques Alexandre Le Tenneur, Constantijn Huygens, Galileo Galilei, and other scholars in Italy, England and the Dutch Republic. He was a staunch defender of Galileo, assisting him in translations of some of his mechanical works.

Nevers is the prefecture of the Nièvre department in the Bourgogne-Franche-Comté region in central France. It was the principal city of the former province of Nivernais. It is 260 km (160 mi) south-southeast of Paris.

The Annunciation, also referred to as the Annunciation to the Blessed Virgin Mary, the Annunciation of Our Lady, or the Annunciation of the Lord, is the Catholic and Eastern Orthodox celebration of the announcement by the Archangel Gabriel to the Blessed Virgin Mary that she would conceive and become the mother of Jesus, the Jewish messiah and Son of God, marking His Incarnation. Gabriel told Mary to name her son Yeshua, meaning "YHWH is salvation".

René Descartes was a French philosopher, mathematician, and scientist. A native of the Kingdom of France, he spent about 20 years (1629–1649) of his life in the Dutch Republic after serving for a while in the Dutch States Army of Maurice of Nassau, Prince of Orange and the Stadtholder of the United Provinces. One of the most notable intellectual figures of the Dutch Golden Age, Descartes is also widely regarded as one of the founders of modern philosophy.

For four years, Mersenne devoted himself entirely to philosophic and theological writing, and published Quaestiones celeberrimae in Genesim (Celebrated Questions on the Book of Genesis) (1623); L'Impieté des déistes (The Impiety of the Deists ) (1624); La Vérité des sciences (Truth of the Sciences Against the Sceptics, 1624). It is sometimes incorrectly stated that he was a Jesuit. He was educated by Jesuits, but he never joined the Society of Jesus. He taught theology and philosophy at Nevers and Paris.

The Society of Jesus is a religious order of the Catholic Church headquartered in Rome. It was founded by Ignatius of Loyola with the approval of Pope Paul III in 1540. The members are called Jesuits. The society is engaged in evangelization and apostolic ministry in 112 nations. Jesuits work in education, research, and cultural pursuits. Jesuits also give retreats, minister in hospitals and parishes, sponsor direct social ministries, and promote ecumenical dialogue.

In 1635 he set up the informal Académie Parisienne (Academia Parisiensis), which had nearly 140 correspondents, including astronomers and philosophers as well as mathematicians, and was the precursor of the Académie des sciences established by Jean-Baptiste Colbert in 1666. He was not afraid to cause disputes among his learned friends in order to compare their views, notable among which were disputes between Descartes and Pierre de Fermat and Jean de Beaugrand. [5] Peter L. Bernstein, in his book Against the Gods: The Remarkable Story of Risk, wrote, "The Académie des Sciences in Paris and the Royal Society in London, which were founded about twenty years after Mersenne's death, were direct descendants of Mersenne's activities." [6]

Jean-Baptiste Colbert was a French politician who served as the Minister of Finances of France from 1661 to 1683 under the rule of King Louis XIV.

Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' Arithmetica.

Jean de Beaugrand was the foremost French lineographer of the seventeenth century. Though born in Mulhouse, de Beaugrand moved to Paris in 1581. He also worked as a mathematician and published works on geostatics. He is credited with naming the cycloid. He lived and worked in Paris as an artist until his death in 1640.

In 1635 Mersenne met with Tommaso Campanella but concluded that he could "teach nothing in the sciences ... but still he has a good memory and a fertile imagination." Mersenne asked if Descartes wanted Campanella to come to Holland to meet him, but Descartes declined. He visited Italy fifteen times, in 1640, 1641 and 1645. In 1643–1644 Mersenne also corresponded with the German Socinian Marcin Ruar concerning the Copernican ideas of Pierre Gassendi, finding Ruar already a supporter of Gassendi's position. [7] Among his correspondents were Descartes, Galileo, Roberval, Pascal, Beeckman and other scientists.

Tommaso Campanella, baptized Giovanni Domenico Campanella, was a Dominican friar, Italian philosopher, theologian, astrologer, and poet.

Marcin Ruar, pseudonym Aretius Crispinus) (1589–1657) was a Socinian writer, advocate of religious tolerance, and rector of the Racovian Academy from 1620 to 1622.

Pierre Gassendi was a French philosopher, priest, astronomer, and mathematician. While he held a church position in south-east France, he also spent much time in Paris, where he was a leader of a group of free-thinking intellectuals. He was also an active observational scientist, publishing the first data on the transit of Mercury in 1631. The lunar crater Gassendi is named after him.

He died September 1 through complications arising from a lung abscess.

## Works

### Quaestiones celeberrimae in Genesim (1623)

Quaestiones celeberrimae in Genesim was written as a commentary on the Book of Genesis and comprises uneven sections headed by verses from the first three chapters of that book. At first sight the book appears to be a collection of treatises on various miscellaneous topics. However Robert Lenoble has shown that the principle of unity in the work is a polemic against magical and divinatory arts, cabalism, and animistic and pantheistic philosophies. He mentions Martin Del Rio's Investigations into Magic and criticises Marsilio Ficino for claiming power for images and characters. He condemns astral magic and astrology and the anima mundi , a concept popular amongst Renaissance neo-platonists. Whilst allowing for a mystical interpretation of the Cabala, he wholeheartedly condemned its magical application, particularly angelology. He also criticises Pico della Mirandola, Cornelius Agrippa, Francesco Giorgio and Robert Fludd, his main target. Fludd responded with Sophia cum moria certamen (1626), wherein he admits his involvement with the Rosicrucians. The anonymous Summum bonum (1629), another critique of Mersenne, is an openly Rosicrucian text. The cabalist Jacques Gaffarel joined Fludd's side, while Pierre Gassendi defended Mersenne.

### L'Harmonie universelle (1636)

L'Harmonie universelle is perhaps Mersenne's most influential work. It is one of the earliest comprehensive works on music theory, touching on a wide range of musical concepts, and especially the mathematical relationships involved in music. The work contains the earliest formulation of what has become known as Mersenne's laws, which describe the frequency of oscillation of a stretched string. This frequency is:

1. Inversely proportional to the length of the string (this was known to the ancients; it is usually credited to Pythagoras)
2. Proportional to the square root of the stretching force, and
3. Inversely proportional to the square root of the mass per unit length.

The formula for the lowest frequency is

${\displaystyle f={\frac {1}{2L}}{\sqrt {\frac {F}{\mu }}},}$

where f is the frequency, L is the length, F is the force and μ is the mass per unit length.

In this book, Mersenne also introduced several innovative concepts that can be considered the basis of modern reflecting telescopes:

• Much earlier than Laurent Cassegrain, he found the fundamental arrangement of the two-mirror telescope combination, a concave primary mirror associated with a convex secondary mirror, and discovered the telephoto effect that is critical in reflecting telescopes, although he was far from having understood all the implications of that discovery.
• Mersenne invented the afocal telescope and the beam compressor that is useful in many multiple-mirror telescope designs. [8]
• He recognized also that he could correct the spherical aberration of the telescope by using aspherical mirrors and that in the particular case of the afocal arrangement he could do this correction by using two parabolic mirrors, though a hyperboloid is required. [9]

Because of criticism that he encountered, especially from Descartes, Mersenne made no attempt to build a telescope of his own.

### Other

Mersenne is also remembered today thanks to his association with the Mersenne primes. The Mersenne Twister, named for Mersenne primes, is frequently used in computer engineering and in related fields such as cryptography.

However, Mersenne was not primarily a mathematician; he wrote about music theory and other subjects. He edited works of Euclid, Apollonius, Archimedes, and other Greek mathematicians. But perhaps his most important contribution to the advance of learning was his extensive correspondence (in Latin) with mathematicians and other scientists in many countries. At a time when the scientific journal had not yet come into being, Mersenne was the centre of a network for exchange of information.

It has been argued that Mersenne used his lack of mathematical specialty, his ties to the print world, his legal acumen, and his friendship with the French mathematician and philosopher René Descartes (1596–1650) to manifest his international network of mathematicians. [10]

Mersenne's philosophical works are characterized by wide scholarship and the narrowest theological orthodoxy. His greatest service to philosophy was his enthusiastic defence of Descartes, whose agent he was in Paris and whom he visited in exile in the Netherlands. He submitted to various eminent Parisian thinkers a manuscript copy of the Meditations on First Philosophy , and defended its orthodoxy against numerous clerical critics.

In later life, he gave up speculative thought and turned to scientific research, especially in mathematics, physics and astronomy. In this connection, his best known work is Traité de l'harmonie universelle (also referred to as Harmonie universelle) of 1636, dealing with the theory of music and musical instruments. It is regarded as a source of information on 17th-century music, especially French music and musicians, to rival even the works of Pietro Cerone.

One of his many contributions to musical tuning theory was the suggestion of

${\displaystyle {\sqrt[{4}]{\frac {2}{3-{\sqrt {2}}}}}}$

as the ratio for an equally-tempered semitone (${\displaystyle {\sqrt[{12}]{2}}}$). It was more accurate (0.44 cents sharp) than Vincenzo Galilei's 18/17 (1.05 cents flat), and could be constructed using straightedge and compass. Mersenne's description in the 1636 Harmonie universelle of the first absolute determination of the frequency of an audible tone (at 84 Hz) implies that he had already demonstrated that the absolute-frequency ratio of two vibrating strings, radiating a musical tone and its octave, is 1 : 2. The perceived harmony (consonance) of two such notes would be explained if the ratio of the air oscillation frequencies is also 1 : 2, which in turn is consistent with the source-air-motion-frequency-equivalence hypothesis.

He also performed extensive experiments to determine the acceleration of falling objects by comparing them with the swing of pendulums, reported in his Cogitata Physico-Mathematica in 1644. He was the first to measure the length of the seconds pendulum, that is a pendulum whose swing takes one second, and the first to observe that a pendulum's swings are not isochronous as Galileo thought, but that large swings take longer than small swings. [11]

## Battles with occult and mystical thinkers

Two German books which circulated around Europe in 1614-15, Fama fraternitatis and Confessio Fraternitatis, claimed to be manifestos of a highly select, secret society of alchemists and sages called the Brotherhood of Rosicrucians. The books were hoaxes, but were obviously written by a small group who were reasonably knowledgeable about the sciences of the day, and their main theme was to promote educational reform (they were anti-Aristotelian). However, these books also promoted an occult view of science containing elements of Paracelsian philosophy, neo-Platonism, Cabalism and Hermeticism. In effect, they sought to establish a new form of scientific religion with some pre-Christian elements.

Mersenne led the fight against acceptance of these ideas, particularly those of Rosicrucian promoter Robert Fludd, who had a lifelong battle of words with Johannes Kepler. The Rosicrucian ideas were defended by many prominent men of learning, and some members of the European scholarly community boosted their own prestige by claiming to be among the selected members of the Brotherhood. However, it is now generally agreed among historians that there is no evidence that the order of Rosicrucians ever existed. [12]

During the mid-1630s Mersenne gave up the search for physical causes in the Aristotelian sense (rejecting the idea of “essences”, which were still favoured by the scholastic philosophers) and taught that true physics could only be a descriptive science of motions (Mécanisme), which was the direction set by Galileo Galilei. Mersenne had been a regular correspondent with Galileo and had extended the work on vibrating strings originally developed by his father, Vincenzo Galilei. [13]

## Music

An air attributed to Mersenne was used by Ottorino Respighi in his second suite of Ancient Airs and Dances .

## List of works

• Euclidis elementorum libri, etc. (Paris, 1626)
• Les Mécaniques de Galilée (Paris, 1634)
• Questions inouies ou récréation des savants (1634)
• Questions théologiques, physiques, etc. (1634)
• Harmonie universelle First edition on line from Gallica (Paris, 1636–1637). Translation to English by Roger E. Chapman (The Hague, 1957)
• Nouvelles découvertes de Galilée (1639)
• Cogitata physico-mathematica (1644)
• Universae geometriae synopsis (1644)

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Mersenne's laws are laws describing the frequency of oscillation of a stretched string or monochord, useful in musical tuning and musical instrument construction. The equation was first proposed by French mathematician and music theorist Marin Mersenne in his 1637 work Traité de l'harmonie universelle. Mersenne's laws govern the construction and operation of string instruments, such as pianos and harps, which must accommodate the total tension force required to keep the strings at the proper pitch. Lower strings are thicker, thus having a greater mass per unit length. They typically have lower tension. Guitars are a familiar exception to this - string tensions are similar, for playability, so lower string pitch is largely achieved with increased mass per length. Higher-pitched strings typically are thinner, have higher tension, and may be shorter. "This result does not differ substantially from Galileo's, yet it is rightly known as Mersenne's law," because Mersenne physically proved their truth through experiments. "Mersenne investigated and refined these relationships by experiment but did not himself originate them". Though his theories are correct, his measurements are not very exact, and his calculations were greatly improved by Joseph Sauveur (1653–1716) through the use of acoustic beats and metronomes.

Henri Louis Habert de Montmor was a French scholar and man of letters.

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Harmonie universelle is the work of Marin Mersenne, published in Paris in 1636. It represented the sum of musical knowledge during his lifetime.

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