# Marin Mersenne

Marin Mersenne, Marin Mersennus or le Père Mersenne (∗ 1588-09-08; † 1648-09-01) was a French monk who also studied or worked in philosophy, mathematics, and music theory.

## Life

Marin Mersenne (pronounced Mehr-SENN) was born of peasant parents near Oizè, 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 received his full holy orders in 1613.

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, with other kindred spirits such as René Descartes, Étienne Pascal, Gilles de Roberval and Nicolas-Claude Fabri de Peiresc, he studied mathematics and music. He corresponded with Giovanni Doni, Constantijn Huygens and other scholars in Italy, England and Holland. He was a staunch defender of Galileo Galilei, assisting him in translations of some of his mechanical works. For four years, Mersenne devoted himself entirely to philosophic and theological writing, and published Quaestiones celeberrimae in Genesim (1623); L'Impieté des déistes (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. 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 René 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. Among his correspondents was Josh, Dekar, Galilei, Roberval, Pascal, Bekman and another scientists. Peter L. Bernstein in his book Against the Gods: the Remarkable story of risk writes: "The Académie des Sciences in Paris and the Royal Society in London, which were founded about twenty years after Mensenne's death, were direct descendants of Mersenne's activities." (Bernstein 1996, p. 59).

He died through complications arising from a lung abscess.

## Works

### Quaestiones celeberrimae in Genesim (1623)

It 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 may appear 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 diatribe against magical and divinatory arts, cabalism, 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 to angelology. He also criticises Pico della Mirandola, Cornelius Agrippa and Francesco Giorgio with Robert Fludd as his main target. Fludd responded with Sophia cum moria certamen (1626), wherein Fludd 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 (1637)

This book contains the famous Mersenne Laws which describe the frequency of oscillation of a stretched string. This frequency is:

a) Inverse proportional to the length of the string (this was actually known to the ancients, and is usually credited to Pythagoras himself).

b) Proportional to the square root of the stretching force, and

c) Inverse proportional to the square root of the mass per unit length.

The exact formula for the lowest frequency is

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

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

In this book, Mersenne, a disciple of René Descartes, also introduced several innovating concepts that can be considered as the basis of modern reflecting telescopes:

- Instead of using an eyepiece, as did Galilei who had to tilt the mirror to have easy access to the image, Mersenne introduced the revolutionary idea of a second mirror that would reflect the light coming from the first mirror. This allows one to focus the image behind the primary mirror in which a hole is drilled at the centre to unblock the rays.

- Mersenne invented the afocal telescope and the beam compressor that is useful in many multiple-mirrors telescope designs.

- Mersenne recognized also that he could correct the spherical aberration of the telescope by using nonspherical mirrors and that in the particular case of the afocal arrangement he could do this correction by using two parabolic mirrors.

- Much earlier than Laurent Cassegrain, he found the fundamental arrangement of the two-mirrors telescope combination, a concave primary mirror associated with a convex secondary mirror and discovered the telephoto effect that is critical in reflecting telescopes, although it is obvious that he was far from having understood all the implications of that discovery.

Unfortunately, because of the harsh criticism that he encountered, especially that of RenÃ© Descartes, he made no attempt to build a telescope of his own invention.

### Other

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

However, he 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 center of a network for exchange of information. These include such notables as: Pierre de Fermat, Pascal, Gassendi, Roberval, Beaugrand, Huygens, Pell, Galileo and Torricelli.

His 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

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

as the ratio for an equally-tempered semitone ($\sqrt[12]{2}$). 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.

## Music

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

## Mersenne primes

In 1644 he conjectured that the only Mersenne primes less than or equal to M(257) were:
M(2), M(3), M(5), M(7), M(13), M(17), M(19), M(31), M(67), M(127), M(257).
The correct list is:
M(2), M(3), M(5), M(7), M(13), M(17), M(19), M(31), M(61), M(89), M(107), M(127).

Those in italics were incorrect inclusions, those in bold were omissions.