Mersenne and Mixed Mathematics

Antoni Malet

Daniele Cozzoli Pompeu Fabra University

One of the most fascinating intellectual ªgures of the seventeenth century, Marin Mersenne (1588–1648) is well known for his relationships with many outstanding contemporary scholars as well as for his friendship with Descartes. Moreover, his own contributions to natural philosophy have an interest of their own. Mersenne worked on the main scientiªc questions debated in his time, such as the law of free fall, the principles of Galileo’s mechanics, the law of refraction, the propagation of light, the vacuum problem, the hydrostatic paradox, and the Copernican hypothesis. In his Traité de l’Harmonie Universelle (1627), Mersenne listed and de- scribed the mathematical disciplines: Geometry looks at continuous quantity, pure and deprived from matter and from everything which falls upon the senses; arithmetic contemplates discrete quantities, i.e. numbers; music concerns harmonic numbers, i.e. those numbers which are useful to the sound; cosmography contemplates the continuous quantity of the whole world; optics looks at it jointly with light rays; chronology talks about successive continuous quantity, i.e. past time; and mechanics concerns that quantity which is useful to machines, to the making of instruments and to anything that belongs to our works. Some also adds judiciary astrology. However, proofs of this discipline are

The papers collected here were presented at the Workshop, “Mersenne and Mixed Mathe- matics,” we organized at the Universitat Pompeu Fabra (Barcelona), 26 May 2006. We are grateful to the Spanish Ministry of Education and the Catalan Department of Universities for their ªnancial support through projects Hum2005-05107/FISO and 2005SGR-00929. We thank the editors of PoS for their kind welcome to our proposal to publish the papers, with our special gratitude to Roger Ariew for his initial support.

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borrowed either from astronomy (that I have comprised under cosmology) or from other sciences.1

Mersenne dealt with almost every one of the mixed and pure mathematical sciences listed above. He ªrst collected a number of treatises on geometry and mixed mathematics in the two editions of his Synopsis (Syn- opsis Mathematica 1626 and Universae geometriae synopsis 1644). He com- posed several treatises on music (Traité de l’harmonie universelle 1627; Ques- tions harmoniques 1634; Les preludes de l’harmonie universelle 1634; Harmonie universelle 1636; Harmonicorum libri XII 1648) and optics (De Natura lucis 1623; Opticae 1644; L’Optique et la catoptrique 1651). Later on he published further collections of essays concerning mechanics, pneumatics, hydro- statics, navigation, and the techniques for establishing weights and mea- sures (Cogitata physico-mathematica 1644; Novarum observationum physico- mathematicarum tomus III 1647). Moreover, Mersenne contributed to spread Galileo’s writings in France (Les nouvelles pensées de Galilée 1639; Les méch- aniques de Galilée 1634) as well as the ideas of thinkers such as Hobbes and Roberval. Mersenne discussed natural philosophical issues not only in his mathematical works but also in a number of philosophical essays, in his huge correspondence, and also in writings whose content is not directly related to science. These are notably the cases of the Quaestiones caeleber- rimae in Genesim, a huge commentary to St. Jerome’s Vulgate that contains contributions to optics and acoustics, of La vérité des sciences, the Harmonic- orum libri XII, and l’Harmonie Universelle, which contains contributions to arithmetic (Knobloch 1974). Many years ago, Robert Lenoble’s groundbreaking Mersenne ou la naissance du mécanisme portrayed the Minim friar as an early and inºuential advocate of the mechanization of the world picture (Lenoble 1943). Lenoble did much to establish Mersenne’s reputation as a Christian thinker who, out of his concern for the attacks the Christian faith was re- ceiving from many quarters, was willing to renounce some Aristotelian tenets in order to make room for philosophical innovations in mechanics, astronomy, music, and so on. In a move that was quite standard in his 1. “[. . .] la Géométrie considere la quantité continuë, pure et dénuée de la matière, et de tout qui tombe sous les sens: l’Arithmétique contemple la quantité discrette, à sçavoir les nombres. La Musique considere les nombres harmoniques, c’est à dire qui servent aux sons. La Cosmographie contemple la quantité continuë de tout le monde. L’Optique la considere jointe aux rayons de la lumière. La Chronologie parle de la quantité continuë qui est successive à sçavoir du temps passé: et la Méchanique parle de la quantité, qui sert aux Machines, aux instruments, et à tout ce qui appartient à nos ouvrages. Quelques-uns ajoustent l’Astrologie Judiciaire, mais cette partie n’a nulles demonstrations que celles qu’elle emprunte de l’Astronomie (que j’ai compris sous la Cosmographie) et qu’elle prend des autres sciences” (Mersenne 2003, pp. 38–9).

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days, Lenoble could easily construct Mersenne’s defense of the value of mathematics and advocacy of the mixed mathematical sciences as straight- forward support for a mechanistic understanding of nature. Richard Popkin substantially reªned Lenoble’s picture by placing Mersenne’s philosophical innovations within the emergence of modern skepticism and by characterizing them in terms of ‘mitigated skepticism’. According to Popkin, while Mersenne partially accepted the skeptics’ critique of the possibility of Aristotelian scientia, and consequently of the knowledge of the real nature of things, he advocated a kind of knowledge consisting “of information about appearances, and hypotheses and predictions about the connections of events and the future course of events” (Popkin 1979, p. 131). Along with the contributions of Lenoble and Popkin, the careful exegetical work of the curators of Mersenne’s correspondence brought to light new facets of Mersenne’s scientiªc work. This includes Mersenne’s view on the question of the motion of the Earth, or his evaluation of the ideas of a number of authors, such as Bruno, Campanella, the atomists and Galileo, whom he condemned in the early 1620s, but that later on evalu- ated more positively. Armand Beaulieu, one of the last editors of Mer- senne’s huge correspondence, has assembled a substantial amount of information about Mersenne’s biography and his works (Beaulieu 1995). In spite of Mersenne’s many works and his centrality in the seventeenth-century republic of letters, his writings have not generally received the close attention they deserve. The exceptions are perhaps mathematics proper (as opposed to “mixed” mathematics and to his philosophy of mathematics, on which see below) and music. Mathematics proper greatly interested Mersenne but his contributions to it are not central within his philosophical corpus (Coumet 1972; Knobloch 1974; Engelberg, Gertner 1981; Warusfel 1994). On the other hand, music was perhaps the science that most deeply and continuously interested Mersenne. Crombie has stressed that Mersenne combined a systematic program of measuring acoustical and optical quantities affecting perceptions with the investigation of the nature of light and sound. According to Crombie, the monochord was the instrument on which he based his investigations (Crombie 1994, vol. II, p. 822). There have been in recent years a number of substantial contributions to Mersenne’s theory of music, a facet of his thought with which we will not be concerned here (Bailhache 1993 and 1994; Crombie 1994; De Buzon 1994; Knobloch 1994; Fabbri 2003). Scholars have stressed the importance of music in Mersenne’s contribution to set up a post-Aristotelian world picture. Frédéric de Buzon has argued that Mersenne considered the principles of music rooted in those of arithmetic, geometry and physics, whereas Scholastics considered music a science sub- ordinated to arithmetic and Kepler grounded it on geometry. For Mer-

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senne, music showed the universal harmony of the order of creation; understanding this order needed both an experimental investigation and theological explanations (De Buzon 1994, pp. 123 and 127). In the Colloquium organized in 1988 to commemorate Mersenne’s fourth centenary, a number of substantial papers were read that illuminate points of Mersenne’s biography, the order of Minims to which he be- longed, and his intellectual connections to different philosophers (Con- stant, Fillon 1994). Interestingly, only Pierre Costabel’s essay on Mersenne and cosmology dealt with Mersenne’s scientiªc ideas as opposed to his role as a mathematical and philosophical intelligencer (Costabel 1994). Among recent contributions devoted to Mersenne Peter Dear’s mono- graph stands out (Dear 1988). Stressing Mersenne’s deliberate allegiance to some weak form of Aristotelianism, Dear has shown that Mersenne’s “metamathematics” (to use his word) is largely indebted to orthodox sources of Renaissance scholasticism, and in particular to discussions among Jesuits on the nature of mathematics, its objects, and its “scien- tiªc” status. Dear’s Mersenne stays distant to any sort of causal physics while granting new, central, unprecedented status to the mixed mathematical sciences. Recent historiography on Mersenne has been concerned with his reaction to Copernicanism and to Galileo’s work. Antonio Nardi has placed Mersenne’s contribution to hydrodynamics within the context of Galilean science (Torricelli, Castelli, and Galileo himself). According to Nardi, even if Mersenne’s work on hydrodynamics was not as cogent as Torri- celli’s, he provided sound and clear explanations of the main hydro- dynamical phenomena (Nardi 1994). William L. Hine’s review of Mer- senne’s Copernicanism reveals an important facet of his attitude toward science and religion (Hine 1973). Mersenne, who never thought the idea of the motion of the earth to be heretical, was not a Copernican, but he always freely discussed arguments in favor of Copernicanism. According to Hine, Mersenne did not see any reason to challenge the Church for he be- lieved that proofs for or against Copernicanism were not conclusive. Hine construed Mersenne’s interest in Copernicanism as part of his willingness to abandon Aristotelianism, a view not endorsed by other scholars. The recent scholarly interest in Mersenne’s reaction to Galileo’s ideas is not surprising given both the importance of Galilean mechanics within seventeenth-century science and the changing ways in which Mersenne under- stood Galileo’s contributions and methods. While in his Questiones in Genesim (1623) Mersenne took Galileo to be a heretic, by 1634 Mersenne did not consider the motion of the earth a heretical statement, although he still said that Galileo’s condemnation was justiªed by Galileo’s failing to

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keep his promise of 1616.2 Moreover, in 1634 Mersenne was publishing Galileo’s contributions to mechanics in a book titled Les méchaniques de Galilée, followed in 1639 by Les nouvelles pensées de Galilée, a free translation of the Discorsi. Perhaps more interestingly, Mersenne changed his views on the truth of Galileo’s law of free fall, and on the value of Galileo’s demon- stration of it. Domenico Bertoloni Meli has recently confronted the ways in which Galileo and Mersenne used numerical tables, including tables listing data on free fall. According to Bertoloni Meli, Mersenne, inspired by his belief in universal harmony, used numerical tables in order to show the symmetry and regularity of certain phenomena (Bertoloni Meli 2004). Thus, Mersenne’s numerical tables were not essentially intended to com- pare theoretical predictions with experimental data. Rather, Mersenne wanted numerical tables to illustrate a physical phenomenon in a clearer and nicer way, even when the experience could not be performed due to physical limitations. New light on Mersenne’s reception of Galileo’s free fall laws has been provided by Carla Rita Palmerino, who has linked it to Descartes’ inºuence on the Minim friar (Palmerino 1999; see below). Daniel Garber’s recent insightful paper has linked Mersenne’s understanding of Galileo to both the general status of Aristotelianism in Mersenne’s days and to its status within Mersenne’s natural philosophy (Garber 2004). According to Garber, Mersenne chose to understand Galileo as a “mixed mathematician working in the tradition of Aristotelian mechanics” rather than as a natural philosopher. This would be consistent with the high respect Mersenne always kept for Aristotelianism while promoting authors and approaches that effectively challenged it. As the foregoing suggests, many facets of both Mersenne’s thought and his involvement with contemporary issues and people still remain highly controversial. This special issue of PoS presents essays that explore connections between Mersenne’s contributions to mechanics, optics, cosmography, and music and their philosophical context. The outcome is a

2. On 8 February 1634 Mersenne wrote to Rivet: “Quant a Galilée, sa condamnation est particulierement appuyé sur sa promesse de l’an 1622 [really of 1616. Interestingly, in 1622 Campanella published his Apologia pro Galileo, which was one of Mersenne’s sources] de n’enseigner point cette doctrine, contre laquelle vous sçavez que l’escriture est claire: Terra in aeternum stat etenim ªrmavit orbem terrae qui non commovebitur, etc. 1 (Eccl. Cap. 1) Et c’est à l’eglise à prendre garde que chacun n’explique pas l’Escriture à sa phantasie. Je sais qu’on peut repondre qu’il y a d’autres passages qui la font mouvoir, comme Mota est terra a facie homini, etc. (Ps. 113) et qu’elle s’accomode a notre sens en parlant à nous, mais puisqu’il la faut souvent prendre dans la rigeur du sens, qui peut nous assurer qu’elle s’y doive prendre icy ou là, si ce n’est l’ensemble de ªdèles?” (Mersenne 1932–1988, vol. IV, pp. 37–8).

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sharper picture of Mersenne’s relations with Galileo, Descartes, Bruno, and also with Renaissance Scholasticism generally. The articles also offer a deeper understanding of Mersenne’s changing views on the relationship between philosophy, theology and mathematics through his involvement in discussions about the nature of light, free fall, and cosmology. Daniele Cozzoli’s paper reconstructs important facets of Mersenne’s optics. It shows Mersenne’s attempts to conciliate the new optical ideas of Kepler, Descartes, and Hobbes with Scholasticism. Moreover, Mersenne’s thoughts on the theory of light and optics generally serve to illuminate his philosophy of science and his Aristotelianism. The paper shows that Mersenne tried ªrst to accommodate novelties within the Aristotelian theory of light, and then, when he fully accepted new ideas on light, he always wanted to remain faithful to Aristotle, even though Aristotelianism no longer played an explanatory role. Miguel Angel Granada’s paper investigates Mersenne’s critique of Giordano Bruno’s ideas concerning the inªnity of the universe, the plural- ity of the worlds, and the soul of the world. Stressing the theological and metaphysical implications of these issues, Granada shows that in the early 1620s Mersenne attacked Bruno because of the theological implications of his cosmological ideas. Indeed, in L’impieté des déistes, athées et libertins de ces temps, Mersenne deªned Bruno “un des plus meschans hommes que la terre porta jamais” (Mersenne, 1624, I, p. 230). However, Mersenne did not at- tack Bruno’s atomism and, later on, in the 1630s, he defended Bruno’s view concerning the inªnity of the universe. According to Granada, this is not due to a changing attitude toward Bruno, but to the fact that Mer- senne rejected only the impious consequences of Bruno’s ideas. Carla Rita Palmerino’s article deals with the vexed issue of Mersenne’s reception of Galileo’s mechanics. It is well known that between 1634 and 1647 Mersenne switched from an initial enthusiastic acceptance of Gali- leo’s law of free fall to a skeptical judgment concerning its validity. In her paper, Palmerino argues that Mersenne’s change of mind was mainly due to Descartes’ inºuence generally, and in particular to Descartes’ criticism of a key principle in Galileo’s conceptualization of the motion of free fall, the principle that a body initially at rest should pass through inªnite degrees of speed. The paper shows that, after having searched for a mathematical science of motion, Mersenne considered Galileo’s theory of fall rather more compatible with an internal (to the body) or Aristotelian no- tion of gravity than with an external, mechanistic one. Finally, Carlos Calderón presents his two reconstructions of Mersenne’s monochord, one virtual and one material, both done by carefully and faithfully following Mersenne’s description in L’Harmonie universelle. The reconstructions shed light on the double nature the monochord had

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in Mersenne’s days, it being simultaneously both a musical instrument and a scientiªc one. The virtual reconstruction may be found at http:// www.calderon-online.com/mersenne/monochord_mersenne.htm.

References Bailhache, P. 1993. “Cordes vibrantes et consonances chez Beeckman, Mersenne et Galilée.” Pp. 73–91 in Musique et mathématiques. Nantes: Université de Nantes. Bailhache, Patrice. 1994. “L’Harmonie universelle: la musique entre les mathématique, la physique, la métaphysique et la religion.” Les études philosophique, 1–2: 13–24. Beaulieu, Armand. 1995. Mersenne. Le grand minime. Bruxelles: Fondation Nicolas-Claude Fabri de Peiresc. Bertoloni Meli, Domenico. 2004. “The Role of Numerical Tables in Gali- leo and Mersenne.” Perspectives on Science, 12: 212–237. Blay, Michel. 1994. “Mersenne expérimentateur : les études sur le mouvement des ºuides jusqu’en 1644.” Les études philosophiques, 1–2: 69–86. Constant, Jean-Marie & Fillon, Anne, eds. 1994. 1588–1988 Quatrième centenaire de la naisance de Marin Mersenne. Le Mans: Université du Maine. Costabel, Pierre. 1994. “Mersenne et la cosmologie.” Pp. 47–56 in 1588– 1988 Quatrième centenaire de la naisance de Marin Mersenne. Edited by Jean-Marie Constant & Anne Fillon. Le Mans: Université du Maine. Coumet, Ernest. 1972. “Mersenne: dénombrements, répertoires, numéro- tations de permutations.” Mathématiques Sciences Humaines, 38: 5–37. Crombie, Alistair C. 1994. Styles of Scientiªc Thinking in the European Tradi- tion, 3 vols. London: Duckworth. Dear, Peter. 1988. Mersenne and the Learning of the Schools. Ithaca: Cornell University Press. De Buzon, Frédéric. 1994. “Harmonie et métaphysique: Mersenne face à Kepler.” Les études philosophique, 1–2:119–28. Drake, Stillman. 1971. “The rule behind ‘Mersenne’s numbers’.” Physis– Rivista Internazionale di Storia della Scienza, 13 (4): 421–424. Engelberg, Don & Gertner, Michael. 1981. “A marginal note of Mersenne concerning the ‘Galilean Spiral’.” Historia Mathematica, 8 (1): 1–14. Fabbri, Natacha. 2003. Cosmologia e armonia in Kepler e Mersenne. Contrappunto a due voci sul tema dell’Harmonice Mundi. Firenze: Olschki. Garber, Daniel. 2004. “On the Frontlines of the Scientiªc Revolution: How Mersenne Learned to Love Galileo.” Perspectives on Science, 12: 135– 163.

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Hine, William L. 1973. “Mersenne and Copernicanism.” Isis, 64 (221): 18–32. Knobloch, Eberhard. 1994. “Desargues, Mersenne et Kircher : la musique et les mathématiques.” Pp. 111–124 in Desargues en son temps. Edited by Jean G. Dhombres & J. Sakarovitch. Paris: Librairie scientiªque. Knobloch, Eberhard. 1974. “Marin Mersennes Beiträge zur Kombina- torik.” Sudhoffs Archiv, 58: 356–379. Lenoble, Robert. 1943. Mersenne ou la naissance du mécanisme. Paris: Vrin. Mersenne, Marin. 1627. Traité de l’harmonie universelle, Paris: Fayard. Mersenne, Marin. [1636] 1963. Harmonie universelle contenant la theorie et la pratique de la musique...,3 vols. Paris: S. Cramoisy. Facsimile edition Paris: Centre national de la recherché scientiªque. Mersenne, Marin. 1932–1988. Correspondance du P. Marin Mersenne, religieux minime. Edited by P. Tannery, C. De Waard, M. B. Rochot and A. Beaulieu, 17 vols. Paris: Beauchesne-Editions du CNRS. Nardi, Antonio. 1994. “Théorème de Torricelli ou théorème de Mersenne?” Les études philosophique, 1–2: 87–118. Palmerino, Carla Rita. 1999. “Inªnite Degrees of Speed: Marin Mersenne and the Debate over Galileo’s Law of Free Fall.” Early Science and Medi- cine, 4: 269–328. Popkin, Richard H. 1979. A History of Scepticism from Erasmus to Spinoza. Berkeley-Los Angeles-London: University of California Press. Warusfel, André. 1994. “Deux textes mathématiques de Mersenne.” Les études philosophiques, 1–2: 41–51.

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