Marin Mersenne: Educator of Scientists

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Marin Mersenne: Educator of scientists

Boria, Vittorio, Ph.D. The American University, 1989

Copyright ©1989 by Boria, Vittorio. All rights reserved.

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MARIN MERSENNE:

EDUCATOR OF SCIENTISTS

by

Vittorio Boria ofm cap.

submitted to the

College of Arts

in Partial Fulfillment of

the Requirements for the Degree of

Doctor of Philosophy

in

Mathematics Education

Signature of Committee

Chair: ^cL.oC

Dean of t|he College

6 October 1989______Date

1989

The American University

Washington, D.C.

a n AMERICAN UNiVERSITY LIBRARY 0 COPYRIGHT

by VITTORIO BORIA

1989 ALL RIGHTS RESERVED Pro Pace et Prosperitate MARIN MERSENNE;

EDUCATOR OF SCIENTISTS

BY

Vittorio Boria

ABSTRACT

The teaching of mathematics in the sixteenth and early seventeenth century France was reduced to a bare minimum.

Lack of enthusiasm in the teachers, lack of motivation in the students, and absence of creativity in researchers relegated mathematics to an insignificant role in education.

Marin Mersenne (1588-1648), a French monk, spent about forty years promoting a greater interest in science. A scientist himself, he made contributions in music, in number theory, the study of functions, and physics. He corresponded with many scholars of his time, and motivated them to search for the solution of many specific problems. He had correspondents in France, Spain, England, Germany, Holland,

Belgium, Czechoslovakia, Poland, Italy, Egypt, and Turkey.

Thanks to his mediation, many scholars could communicate among themselves, sharing the results of their studies.

He was able to bring scholars together at regular meetings, where scientific papers were read, national and international publications were reviewed, correspondence with

ii and from the most remote corners of Europe were exchanged,

experiments were designed and discussed, and quarrels among

scientists were settled. Eventually such regular meetings

gave birth to the French Academy of Sciences. From Italy he

brought to France Torricelli's barometric experiment, which

led to the discovery of the atmospheric pressure, against the

established Aristotelian philosophy. As a consequence, long

established principles were brought under closer scrutiny: modern science was born.

Mersenne was especially able to motivate young gifted students to pursue mathematical careers. He did this by proposing to them some problems in line with their interests and their level of preparation, suggesting relevant reading material, introducing them to well known scientists, involving them in his experiments, and sharing the credit with them.

By the third decade of the seventeenth century the curriculum of the schools in France was beginning to be challenged, the teaching of mathematics became a noble profession, and the sciences started to attract the attention of every educated person. For his efforts in popularizing science and involving great scholars in scientific research and publication, Mersenne today can be a model for mentors.

Ill ACKNOWLEDGEMENTS

It is with a heart full of gratitude to the

Father, the source of all knowledge, to the Son, the Alfa and Omega, and to the Holy Spirit, the Paraclete, that I present this work.

I am happy to acknowledge the contribution of my

Franciscan Capuchin confreres of the St. Francis Friary, in

Washington, D.C., and of the beloved Province of St.

Francis, and the MISEREOR Bischoflischen E. V. for their fraternal and financial support during the past three years.

I am indebted to the faculty of the Mathematics and Statistics and the Education Departments of The American

University.

I am very grateful to Dr. Mary Gray and Dr. David

Sadker, members of my committee, for their advice, patience, and consent to serve. Dr. Steven Schot, the chair of the committee, besides being a valuable teacher in the classroom, inspired and directed the research with necessary advice. I am specially indebted to him.

"Rétribuât vobis Dominus retributionem suam!"

IV TABLE OF CONTENTS

ABSTRACT ...... Ü

ACKNOWLEDGEMENTS ...... iv

CHAPTER ONE. INTRODUCTION ...... 1

Relevance of the Research...... 5 Biographical Notes ...... 8 Description of C o n t e n t ...... 17 Limitations ...... 18

CHAPTER TWO. STATUS OF MATHEMATICS AND PHYSICS IN THE XVIth AND XVIIth CENTURIES AND MERSENNE'S EDUCATION .... 20

The Curriculum of Mathematics at La Flèche . . 33 Education in the Order of the Minims .... 46

CHAPTER THREE. MERSENNE AND HIS CONTACTS WITH OTHER SCHOLARS ...... 53

Mersenne's Editorial Activity ...... 58 Correspondence ...... 91 Academies ...... 123

CHAPTER FOUR. MERSENNE, THE EDUCATOR AND THE M E N T O R ...... 143

Mersenne's Scientific Methodology ...... 150 Christiaan Huygens ...... 168 Mersenne and his Young Correspondents: an A n a l y s i s ...... 184 CHAPTER FIVE. SOME MATHEMATICAL AND GEOMETRICAL RESULTS BY M E R S E N N E ...... 194

Mersenne's Philosophical Approach to Mathematics ...... 195 Number T h e o r y ...... 206 Multiply Perfect Numbers ...... 221 The Cycloid ...... 226 Mersenne and the C y c l o i d ...... 232

CONCLUSION ...... 256

SELECTED BIBLIOGRAPHY...... 273

VI CHAPTER ONE

INTRODUCTION

Today the teaching of mathematics in many of the technically most advanced countries is the topic of much research. Although various studies differ on the aims and goals and the suggested recommendations regarding the problem of numeracy, all of them agree with the findings: the learning of mathematics is dangerously inefficient.

Analyzing the situation in the United States, Izaak Wirszup, of the University of Chicago, says that this is "one of the earnest challenges" in the history of the country.' One of the commonly voiced complaints is that

The vast majority of our high school graduates have not studied physics, chemistry, geography, or a foreign language ... they cannot even apply basic mathematics and science to simple jobs. This results in a personal tragedy of shattered hopes for countless of young Americans. It is also a national tragedy.:

The situation becomes even more critical when one compares the results of tests administered to sample groups

'I. Wirzup. "Education and National Survival," Educational Leadership 41 (January 1984): 4-11.

'Ibid., 7; see B. Vobedja, "Student Improve Basic Skills, but Few Can Apply Them Well," Washington Post. 58, (February 1 1989), 2. 2 in various countries of the world. Students from American and Canadian schools often ranked much lower not only than those from Japan and from Eastern European countries, but even from some of the emerging nations such as Korea and

Taiwan.: Students' and parents' attitudes about the academic achievement as well as the difference between the intended and the implemented curricula were also analyzed.

It was found that on the average, students in the Western

Hemisphere were less motivated than students in the other above-mentioned countries.

Some of the factors that have been blamed for the present crisis are:

I. The lack of competency, heavy work load, low salary,

and stressful conditions of the teachers, coupled with

the public's indifference to the quality and status of

educators;

II. The quality of the curriculum. A recent article in the

Washington Post quotes a study by an "ad hoc" committee

on the present situation:

Math education is an enterprise rooted in antiquity, with some of today's curricula

^Besides the above mentioned study by I. Wirszup, see also Hiroshi Kida, David W. Shwalb and Barbara J. Shwalb, Evaluation in Education: An International Review Series. (London, Pergamon Press, 1985); Kenneth J. Travers and Curtis C. McKnight, "Mathematics Achievement in U. S. Schools: Preliminary Findings from the Second lEA Mathematics Study," Phi Delta Kaooan 66 (February 1985): 407-413; H. W. Travers and J. K. McKnight, "Mathematics Achievement of Chinese, Japanese and American Children," Science 231 (February 14 1986): 693-99. matching very closely educational patterns of five hundred years ago. Yet ... mathematics education has entered a period of significant change, certain to last into the next century;'

III. The style, content, number, and cost of textbooks.

In the history of mathematics, however, the present situation is not at all a new problem. The teaching of mathematics has given rise to controversies throughout the centuries. A quick overview shows that educators have always differed in their opinion whether mathematics enhances logical thinking and creativity. They have also been unable to agree on the type and amount of mathematics that should be taught in the classroom. Examples of these differences of opinion will be presented in Chapter Two of this study.

This paper focuses on the mathematical education in

France during the second half of the sixteenth and the first half of the seventeenth century. In particular, it will concentrate on the contributions made to mathematics education by the monk Marin Mersenne. We will show in the following chapters that the teaching of mathematics went through various stages during this period. It went from a subject that was held in low esteem by philosophers, and one that was primarily used by merchants, to the point of such respectability that university chairs were eventually

'Eleanor W. Orr, "Trouble W t h Numbers," Washington Post 26 February 1989: Book World 6. 4 appointed in mathematics. By the first decades of the seventeenth century mathematics began to attract the attention of the most gifted minds, and many of the significant contributions already made by the ancient Greeks were being rediscovered, reformulated, and elaborated upon.

This rejuvenation of old ideas also led French scholars to reexamine the role of mathematics in education. The improvements made are evident from the fact that although in

1550 there were hardly any good textbooks available in

French schools, a century later France was a haven of scientific learning. By this time the French had established an Academy of Sciences, held public debates, and exchanged information on new mathematical discoveries. The details of this turn-about are the topic of this dissertation. Because of the broad nature of the topic, this paper will limit itself to studying the contributions of Marin Mersenne, a Minim monk. Fr. Mersenne devoted most of the sixty years of his life to the study of mathematics and fostered scientific communication among the savants of seventeenth-century France and Europe. In addition, he published a very large amount of his own and other scientists' research in many fields of knowledge.

The assertion of this paper is that, even though

Mersenne never taught mathematics or any other natural science courses in a classroom, his contacts with the well- known scientists of the century, his services to mathematics 5

through his own discoveries and the results he was able to

extract from others made him a leading figure in spreading

mathematical education in France and throughout Europe. His

ability in quantifying old problems and proposing new ones

was unmatched in his lifetime. The insistence with which he

encouraged the most learned to produce their best results,

even though sometimes he was not able to solve the problems

he proposed, led to the discovery of several important laws

of nature and mathematical formulas.

Relevance of the Research

Educational psychologists have always acknowledged

the role model that teachers play in the classroom. In

recent times the need of mentors for novice teachers has

also been recognized. The problem is deciding what

qualities mentors should possess to be effective. Probably

no single figure can completely satisfy the description of

an ideal mentor, but the analysis of the characteristics of

known effective educators can help identify the qualities desirable in a mentor. Besides, it is also an accepted fact that the role of the mentor is not relegated to the walls of a classroom; but many aspects of daily life merge in developing the ideal mentor.

Communication skills, ability in keeping the student

interested in the problem being studied, adaptation to the needs and interests of the student, updating his or her 6 knowledge, are some of the characteristics of an effective teacher.®

The assertion of this study is that Mersenne can be proposed as an ideal mentor. This assertion will be supported by the analysis of his scientific activities. He will be portrayed as:

I. coordinating the work of Descartes, Huygens, Beekmans

in Holland; Fermat, Roberval, Desargues in France;

Cavendish, Pell, Der Haak in England, Niceron, Kirch and

Maignan in Rome; Hevelius, Komensky and Magni in Poland;

- just to name a few of the numerous of his numerous

correspondents - to find satisfactory answers to

emerging scientific questions. No one else before him

was able to attract the attention of so many European

scholars to one problem and concentrate their efforts to

solve it. The role of Mersenne was so outstanding,

unique and led to so mucjh cooperation among the

international community of scientists that Mersenne can

truly be called an educator of scientists;

II. grooming potential scientists at home and abroad.

Christiaan Huygens is the most prominent example; others

include Blaise Pascal, Francois Des Verdus, Thibaud, and

Spinula. Mersenne kept these budding scientists among

his correspondents and informed them of up-to-date

®Anita E. Woolfolk, Educational Psvchology. 3rd ed., (Englewood, N.J.: Prentice Hall Inc., 1987), 425. 7

developments in their fields of interest. He also

assigned to them some experiments in which he was

personally involved, and got them in touch with

prominent scientists who could help them with the

solution of their problems;

III. publishing the results of his own research and that of

other prominent scientists. This encouraged others to

publish their results as well and to make public their

views. Thanks to Mersenne's monumental contributions to

science, French mathematicians came to be known

throughout the rest of Europe. Melanchton's remark that

the French would seize any old book of mathematics from

other countries, because of the paucity of their own,®

would soon be obsolete and leave room for Maggiotti's

admission that scientists in Paris had an unusual number

of publications on original problems.

The role of the mentor outside the classroom in the growth of the whole person can be a significant part in the solution of the critical situation in which mathematics education in the United States finds itself at present.

Mersenne's above mentioned contributions make this message relevant to both learners and educators in our society.

®See J. Fontes, Memoirs de 1 'Académie Scientifigue de Touluse 9th series, vol. 7 (1894), reported by Margolin J. C., L'Enseignment de Mathematioue en France 11540-1570). in French Renaissance Studies edited by P. Sharatt, (Edinburg: Edinburg University Press, 1976), 547. 8

Much research has recently been conducted on Mersenne's activities as a scientist and mathematician, but nothing has been written on his role as educator and as the inspiring mind behind many of the interests of researchers and discoverers of the seventeenth century. Another of the aims of this paper is to fill this gap, not only in order to offer a better understanding of this secretaire de 1*Europe savante of the seventeenth century, but also to propose him as a role model in non-formal education.

Biographical Notes

Marin Mersenne was born in Le Maine, the province of

Oize, France, on September 8, 1588. The limited resources of his parents did not allow him to further his education beyond the ordinary level attainable in his hometown. But his natural intellectual gifts and his eagerness to study were impressive enough for the sixteen year old young man to be granted a stipend at the newly founded Jesuit Collège du

Mains of La Flèche. Two years later, Rene' Descartes, eight years younger than Mersenne, joined the same school. They might have known each other at this school, but Mersenne was at a more advanced stage in his education. Descartes later 9

defined his alma mater I'une des plus celebre ecoles de

1 'Europe.:

According to Mersenne's most renowned biographer,

Robert Lenoble, Mersenne, during his stay at La Flèche, probably came in contact with the literature of the agnostic philosophy circulating secretly among the college's

students. He probably started to sketch a strategy to fight

against it. That may explain why he chose to pursue his

education in philosophy in Paris at the Collège Royale du

France, while also attending classes of theology at the

Sorbonne.

At the age of twenty three he obtained the degree of

Maoister Artium in philosophy and joined the Order of the

Minims. After completing the theological courses, he was ordained a priest in 1614.

Compared with his contemporaries, Mersenne's education was definitely a privileged one. Such an outstanding background enabled him to read and discuss any

^Descartes' first biographer, the Abbe' Andre' Baillet IVie de Mens. Descartes. 2 vols., Paris: Daniel Horthemels, 1691; reprint in A Garland Series. N. Y.: 1987) attributes to these years the life-long friendship between these two great representative of the seventeenth century. But the lack of any other testimony of communication between them until 1623 and the silence about it by Mersenne's first biographer and confident Hilarion de Coste,o. min., or any other of their contemporaries, induced the historian Lenoble who first did a complete analysis of Mersenne's life, to postpone the date of the close friendship of the two until 1623. At that time Mersenne was just gaining recognition for his publications while the young Descartes was still struggling in the midst of a career crisis. 10 topic with a variety of people, as his books and his correspondence testify.

His first assignment in the Order was to teach philosophy and theology to the younger members of the

Community at Nevers. This was his only teaching experience in a formal classroom. However, it must have given him much satisfaction, for he always kept in touch with his best students. Among them was Fr. Jean Francois Niceron, o. min., who later on taught mathematics at the house of studies of the Minims in Rome at the Trinita' dei Monti.

Because of Fr. Niceron's position in Rome and his continuing travels he had the opportunity to come to know the celebrities of Rome and Italy, like Torricelli, Cavallieri,

Galileo, and Kircher, S.J., of the Collegio Romano (the future Universita' Gregoriana). He used this opportunity to introduce the scientists of his native France to the

Italians and sometimes to deliver their correspondence.®

Another of his students was the above mentioned Fr. Hilarion

De Coste, o. min., who afterwards became his superior and confidant. They spent many years in the same monastery at

La Place Royale in Paris. De Coste was of great help to

' At his deathbed (he died at 34), Niceron left a manuscript on optics entitled La perspective curieuse. Mersenne enriched the booklet with a study of his own and prepared its publication, but it was his friend Roberval, from the College du France, who published it posthumously with the title Optics and Catoptrics. 11

Mersenne by copying old manuscripts for him and by helping him with the correspondence. At the death of his mentor,

Fr. Hilarion carefully recorded the list of Mersenne's correspondents.’ In 1651, he published the first biography of his friend, Mersenne. Today it is still the primary source of information on Mersenne and his activities.

After teaching for two years, the Minim monk was elected correcteur or superior of the monastery in Paris.

But it soon became obvious to him and to his confreres that his talents would be put to better use if he had more time for private study. Writing apologetical books in defense of the scholastic philosophy and fostering better understanding between theologians and philosophers on the one hand, and the scientists on the other, became his main occupation. A dialogue between the long accepted scholastic philosophy and the emerging new scientific revolution was often not possible. Sometimes, in fact, there was not any dialogue.

Reciprocal attacks and condemnations were the preferred

’ The work is incomplete. In fact it did not contain all the names of Mersenne's correspondents because, as we will see, some of ersenne's his correspondence was lent by the author or receiver to other interested researchers, and some had already been disposed by Roberval. The document also does not provide enough information about the number of letters from the same correspondent, the dates, and the content of the letters. Finally, it is clearly biased in favor of the high ranking officials and clergy. A closer look at the Correspondance instead shows that Mersenne had many correspondents also among the middle class intelligentsia. However, it is the most valuable piece of information about Mersenne's keen interest in his correspondents. 12 approach. This was in fact the initial strategy followed also by Mersenne.

To reach his goal more directly he realized that he should "attack" adversaries, like the skeptic school of

Pyrrho, the alchemists, the liberals and atheists, using their own arguments and methodology. So he committed himself to master and deepen his knowledge of the natural sciences, especially mechanics, the other branches of physics, and the mathematical sciences.

Except for the very first two booklets, L'usage de la raison and a prayer book, all his other publications with a religious theme had always a scientific background to support his theological or philosophical theses. This is the case of the Ouaestiones celeberrimae in Genesim (Paris,

1623). When arguing about the apparitions of angels, for example, he took 40 columns to describe the optical laws of light.

Starting from 1624, his religious publications became rarer but his scientific production kept increasing more and more. In fact, besides the above mentioned books, he wrote also the Questions rares et curieuses, theoloaioues. naturelles, morales, politiques et de controverse, résolues par raison de la Philosophie et de la

Théologie. (1630), a reprint of the first volume of

L*Impiété* des Deistes. 1634, and the Questions

Theoloaioues. PhVsioues. Morales et Mathématiques. Paris, 13

1634. Both books seem to have been prepared as a source­

book for preachers and teachers of the Catholic faith.

Actually, the theological content of the book was limited to

the title only, because the content dealt uniquely with

problems of natural science. All the other works have a

purely scientific aim. The ultimate goal of all his

editorial productivity can be said to have been the

principle of proving that religion and scientific progress

are not necessarily contradictory concepts. After the

initial burst of the first books against the adversaries of

religion, science stopped being an instrument to fight the

adversaries and became a means of starting a dialogue and

cooperation sometimes alongside and sometimes outside the

field of religion.

The next chapter will show the monumental amount of

editorial production that resulted from the long hours spent

in the cell of the Couvent de la Place Royale. Even today,

with all the technological facilities available, his

editorial output would be very considerable; but considering

the difficulties of the seventeenth century it borders on

the unbelievable. In the course of this dissertation,

special evidence will be provided to show the role that

Mersenne played as a channel of communication among

scholars, sometimes even those of opposite scientific positions. Here we limit ourselves to a brief hint of his

open mindedness in matters of religious differences. 14

To his friend and correspondent, the Protestant theologian Andre Rivet of Leyden, in Holland, he wrote:

May God grant you that before you leave this world you may have the consolation to see your holy and greatly desired union of the Christian accomplished, together with the Jewish religion in one faith, understanding and charity. This kind of arguments [of religious differences] displeases me to death. We have to acknowledge that the Jews are so tenacious in their beliefs because of their zeal to serve God and therefore they try to please Him by obeying His Law. The same is true about the other heretics as well.'"

Many of his correspondents were not Catholics, but he offered his services and asked collaboration from anyone whom he considered to be capable of cooperation, independently of his religious affiliation." After the first fiery outbursts of his early years - late twenties and early thirties - he adopted the attitude that the cause of the sciences was also the cause of God.

’“C . de Waard and others, eds., Correspondance du R. P. Marin Mersenne. vol. 16, 615, henceforth referred as Correspondance. A statement that makes Mersenne a forerunner of the Second Vatican Council, see De Libertate Reliaiosa. 6 and De Oecumenismo. 4. Even though he fought strongly against the Unitarian Socinianists, he was very worried that in Poland they were persecuted to death. He did not hesitate to host in his monastery those young members of the sect on their way to England, Holland or elsewhere.

"Bernard Rochot, one of the editors of the Correspondance. quoting Lenoble, about this point wrote: "The fact is that from a hammer of heretics, Mersenne later became, as his natural tendencies pushed him to, a man of a tolerance a donner le vertige. It seems that at this point he set himself to prove that it was quite possible to be a good Christian and a true scientist, and that one could enjoy a perfect freedom of spirit, without ever having to reject any of the dogmas of the faith." La Correspondence Scientifique du Pere Mersenne. (Paris: Palais de la Descouverte, 1966), 7. 15

Moreover, he fostered weekly meetings of the intelligentsia of Paris where scientific papers were read, national and international publications were reviewed, correspondence with and from the most remote corners in

Europe were exchanged, experiments were planned and discussed, and quarrels among scientists were settled. The

Academia Parisiensis survived its founder and about thirty years later was formally established through a royal decree as the Académie Française du Sciences.

Furthermore, Mersenne loved to have personal encounters with individual philosophers or scientists located in Paris or passing through Paris. He was the main reason why Gassendi, Fermat, Campanella, Descartes and others would travel to Paris and meet with him personally, hoping to be invited to attend the Thursday afternoon meetings of the Academia. More than that, he did not hesitate to travel inside France and in the rest of Europe to meet with the greatest minds ;of thé cities he was visiting. There he performed experiments with them or for them. Back home he would report the status of the sciences in the countries he had visited, inspiring the members of the Academia to a healthy emulation.

The seriousness of his scientific mind is reflected in the sophisticated machinery that he was able to assemble in his laboratory, where long and tiring experiments were performed. Claims of other scientists were accurately 16

checked and the results were shared with the rest of the world. If to all of this we add the amount of readings that he subjected himself to — of which his correspondence gives witness — one can rightly appreciate the claim of this paper that Mersenne's contribution to scientific progress is not measured only by the numbers of theorems he may have proved, or the natural laws he might have discovered. His role model as an educator, interested in the progress of science and of the scientist, is his greatest contribution.

However such a hyperactive lifestyle could not last for long. His biographers tell us that he never had been of strong health. In the summer of 1647, against the advice of his doctors and friends, he undertook an enervating trip to the South of France to visit his friends Fermat and Auzout.

He suffered a stroke and was forced to undergo surgery. An inexperienced young doctor inadvertently cut an artery in his upper arm. He never recovered from this injury. But even in his weakness he did not stop writing to his friends, designing experiments and encouraging others to perform them, taking part in the meetings which now took place in his cell to facilitate his participation. On Sundays, his great friend Gassendi would come to the monastery, help him with the devotions in the morning and entertain him with some topic of scientific interest in the afternoons. A number of other admirers would join them in the latter sessions. A week before his sixtieth birthday he died 17

attented by his friends Gassendi, Roberval and Hobbes.

Descartes had left Paris just few days earlier to his

retreat in Holland to avoid the hustle and bustle of the

scientific circle in Paris. A life spent in the promotion

of science had one more last service to render: In his will

Mersenne arranged that doctors could make an autopsy of his

body and so verify their diagnosis. His death was reported

in the Gazette Renaudot with the notice:

On the first of this month died in the monastery of the Minims of the Place Royale Fr. Marin Mersenne, aged 60. He was renown for the numerous beautiful works that he wrote on theology, philosophy and mathematics."

Description of Content

The role of Mersenne in the progress of sciences and

his contribution to foster wholehearted dedication to the

spreading of scientific information and education will be

developed in the following five chapters. The present

chapter is the first one and aims at introducing the subject

of the research, as well as the bibliography on which the

research is based. The next chapter will introduce the

scientific education of a typical young French student in an

institution of higher learning of the seventeenth century, with special attention to the private schools that Mersenne

attended.

""Necrologe de la Semaine," Gazette Renaudot. 3 (1648), reported in Correspondance vol. 16, 658. 18

The third chapter will deal with the role of

Mersenne in fostering cooperation among scientists in every field. In this Mersenne is a unique figure in the history of science, probably still unsurpassed. This dedication to the cause of science started giving tangible results right away. The fourth chapter will deal with the influence of

Mersenne on the younger generation of scientists. The final chapter will discuss how Mersenne's study of number theory was the fruit of his cooperation with Frenicle, Fermat, and

Descartes; and how the discovery of the properties of the cycloid and the plane and solid figures that it generates was due to an international concentration of effort and cooperation.

Limitations

Even though Mersenne is not an unknown person in the history of science, his works and his correspondence were known, until recently, mainly through the works of his correspondents. In 1985 his Correspondance was published in sixteen volumes. It would certainly be a great contribution to researchers to complete the work with a final volume containing an analytic index by the subject and the names of correspondents.

Mersenne was scrupulously careful to keep the mail he received, but he was not always treated with equal care by others; nor did he keep copies of his own writings. His 19 thought, therefore, is sometimes known to us only through the responses he received. This will certainly be one difficulty Mersenne-scholars will always face.

One would also expect that, through his exceptional amount of mail, Mersenne would sometimes jot down some personal feelings about his own claims of priority over those of other contestants. However, nothing of the sort is evident from his correspondence. He was too conscious of the futility of such claims and the danger of an abrupt end to a free flow of communication that this action could cause. Mersenne's personal discoveries are known only by the credit that others gave him for his work. The historical validity of such attributions has been often argued by historians because of the lack of better evidence.

This research also will be affected by this limitation. In case of controversy this study will generally take the position followed by the editors of Mersenne's correspondence, which is the most detailed and the most accurate study of Mersenne. CHAPTER TWO

STATUS OF MATHEMATICS AND PHYSICS

IN THE XVIth AND XVIIth CENTURIES

AND MERSENNE*S EDUCATION

In one of his Dialogues on Education (1528)

Erasmus of Rotterdam' (1466 -1536) expressed in one single line all he had to say about the learning of mathematics on the part of the young students; arithmeticen. musicam et astroloaiam deaustasse sat erit (it will be enough for him to be given a taste of arithmetic, music and astrology). In two other books he published, the De Pueris Instruendis

(1529) and the De Rations Studiis (1511-12), mathematics stands out by its absence, because when it is mentioned it is not to be praised: in De Pueris he refers to the boys with a natural inclination to mathematics, music or astronomy as if they were bizarre individuals.^

'Erasmus was called "the greatest man we come across in the history of education." (L. Bolgar, 1954, reported by James McComica, "The fate of Erasmian Humanism," in Universities. Society and the Future. S. Stphmison ed., (London: Pergamon Press, 216), 37) .

^Reported by Jean-Claude Margolin, L*Enseignement de Mathématiques en France fl540-15701 in French Renaissance Studies edited by P. Sharatt, (Edinburg: Edinburg University Press, 1976).

20 21

If this was the general situation in Europe, it was even worse in France during the sixteenth century and at the beginning of the seventeenth century. While the teaching of mathematics was flourishing in Italy with Luca

Paccioli (1445-1514), Nicolo' Tartaglia (1500-1557),

Girolamo Cardano' (1501-1576), and Rafaele Bombelli (1527-

1572); in Germany with Johannes Warmer (1468-1528),

Christoff Rudolff (1499-1550); in Austria with Michael

Stiffel (1496-1567), Georg Joachim von Lauchen [called also

Rheticus] (1514-1576); and in Switzerland with Joseph

Scaliger (1510-1609), just to name a few, France had no one except Francois Viète (1540-1603). Mathematicians in French universities were left isolated and in obscurity with few students to teach, whereas in Italy Tartaglia, Cardano and others were working on the cossist mathematics' to solve equations of third degree, and Rheticus had a number of arithmeticians working for him for nearly twelve years. In

Italy mathematicians were challenging each other to solve and propose new problems in public debates, and Rudolff in

Germany was writing books on algebra in German, while the few French mathematicians, like Charles De Bouvelles, were still writing on Arithmetic:

They standardized all sorts of artisan or trade practices, because one finds there effectively rules

'Cardano was rector of the University of Padua.

'"Cossist mathematics" from the Italian cosa was used to mean the unknown value in an algebraic equation. 22

relative to the money exchange, conversion of units of measures used in different countries and other types of calculations. . . . One can easily realize that such books were more in the hands of merchants, financiers, treasurers, bank tellers, and metal casters, than in those of scholars.'

And even worse:

And one could also very easily count all the books of mathematics which were produced after Charles de Bouvelles, which, after all, were just replicas of each other.®

Margolin lists six authors covering the time

between 1554 and 1588. Only in 1596 do the records of the

Jesuit Collèges in France signal the presence of a

professeur de matematiaues / while the University of Padua

in Italy and in Germany's universities students were

receiving instructions in algebra and some advanced

geometry.® When educators like Jan Louis Vives (1492-1540)

'J. C. Margolin, L*Enseignement de Mathématiques en France (in P. Sharatt ed. French Renaissance Studies. [Edinburg: Edinburg University Press, 1976], 129).

®Ibid.

^ • C. Margolin, L'Enseignement de Mathématiques en France. 130. It is worth mentioning that Viete was known outside France only after King Henry IV, when challenged by the Dutch to present even one French mathematician able to solve an equation of 45 degrees, proudly presented him. Viete found 22 positive solutions, which he thought had meaningful applications, and discarded the negative solutions. See "Viete" in Dictionary of Scientific Bibliography. 16 vols. C. C. Gillespie ed., (New York: New York, Scribner, 1970-1980; vol. 11).

®M. S. Mahoney, The Mathematical Career of Pierre Fermat. 1601 -1665. (Princeton, N.J.: Princeton University Press, 1973), 130. 23 and Pierre de La Ramees [Latinized as Ramus] (1515-1572) pleaded for a revision of the universities' curricula so that mathematics would be given a more prominent place and more independent chairs, they did not elicit much sympathy.

In fact the level of mathematics expected of graduating students was so low that some were getting their Master of

Arts Degree without any knowledge of mathematics and with only two years of philosophy and one of physics [introduced in 1466], which was a purely descriptive physics.*

Schwarzerd Philip Melanchton (1497-1560), writing in 1549 to a friend, the printer Johannes Petreius of Nuremberg, said that mathematics was "little studied in France, and it was therefore very easy for some German scholars and for many of his students, to earn their living in France giving lectures in mathematics: they did not need to be afraid of competition.

However, in 1531, King Francois I, following the advice of many scholars around him [some of whom were

’In Venice the School of Rinaldo already had a syllabus based on logic, philosophy and mathematics in 1408. See P. L. Rose, The Italian Renaissance of Mathematics. (Librairie Droz, Geneva, 1975). In Prague, for an M.A. Degree it was required to study the six books of Euclid, the sphere, and aliquid in musica et aritmetica (something in music and in arithmetic). Reported by The Universities in the Middle Ages, vol. I, (Oxford: Oxford University Press, 1951), 449.

'“J. Fontes, Memoirs de la Académie des Sciences de Toulouse t. VI, (1894). Reported by Margolin, L'Enseignement de Mathématiques en France. 112. 24 mathematicians] created the Collège du France, affiliated with the University of Paris, with one chair in mathematics." Oronce Fine (1494-1555), Peter Ramus" and

Jacques Pelletier (1517-1582), better known for their innovative views in mathematical education of the youth than for their mathematical achievements, occupied these chairs.

In particular, these men were instrumental in popularizing mathematics and bringing to contemporary students the contributions of the ancient Greeks" whose works they translated into Latin, thus making known their most important ideas. A growing awareness of the usefulness and

" Mersenne studied in this College after graduating from the College at La Fleche, which was run by the Jesuits. The initial reaction of the University of Paris to the new institution was strong and bitter. In fact the lecteur revaux was not given the title of University lecturer, but was simply called professeur a' Paris. See S. D'Irsay, Histoire des Universités Françaises. (Paris: A. Picard, 1933), 162.

" To foster the study of mathematics, Ramus converted his chair of rhetorics at the College Royal du France into a chair of mathematics, thus giving the College a second chair in mathematics. By Ramus' will the occupant of the chair had to win a public dispute among the contestants every three years. Roberval was able to occupy the chair from 1634 to 1675, the year of his death. No other university in France followed Ramus' example.

"Federigo Commandino (1509-1575) had already prepared the latin translation of Euclid, Apollonius, Archimedes, Aristarchus, Autolicus, Hero, Pappus, Ptolomey, and Serenus with rich commentaries and precise details. Wilhelm Holtmann (Basel 1575) published the latin version of Diophantes' Arithmetics which was reworked by Claude Bachet de Mezirist (1621) and was the version that Fermat used. Mersenne himself contributed to this movement by translating into French several of the above mentioned Greek authors in the Svnopsis Mathematics (1626). 25 need for mathematics in science appeared, for example, in

1586 in the Jesuits' Ratio Studiorum where one finds statements like the following:

It [mathematics] teaches poets about the rising and the setting of the stars . , . not to mention the services rendered by the mathematicians to the State, to medicine, to navigation and to farming. We need then to double our efforts so that mathematics flourishes in our colleges as the other subject do."

The foundation for the new generation of mathematicians who would dominate the international scene during the major part of the XVII century was prepared.

Royal chairs of mathematics were established at Marseilles,

Nantes, Toulon, Brest and at Cahors. At La Flèche, the college opened by King Henry IV to be a model school and where Mersenne and Descartes would study their philosophy courses, had a chair in mathematics beginning in 1608."

B. Rochot, after listing a few of the most relevant names of French mathematicians of this period, concludes:

All these names sum up an important epoch in the history of ideas, when the modern sciences were really born, after the rather confused initial effervescence of the western Renaissance. France was in a good position to benefit from the trend and to promote it. The politics of Richelieu and the first literary works, called classical nowadays, put us in the front line of the other nations. Spain is in decline; Italy, fractioned into principalities cannot recognize the universal value of Galileo; England passes from one

" Section De Matematicis in Ratio Studiorum. (Rome: Society of Jesus, 1586), 198 ff.

"F. Dainville, L'Education des Jesuites. XVI-XVII Siecles. (Paris: Edition de Mimuit, 1978), 9. 26

crisis to another towards the "kingdom" of Cromwell. Only Holland is favored wits its "golden century".

Despite this awakened interest in mathematics at the beginning of the XVII century, the universities, in

France as in the rest of the Europe, were not pioneers in fostering and spreading the results of the most advanced research in mathematics." W.T. Costello, a scholar in the history of science, reports that in 1640 Seth Ward could not find among his teachers in Cambridge anyone who could explain to him the first rudiments of trigonometry." In

Italy, Galileo learned mathematics, not at the University of

Pisa, where he studied philosophy, but from a private tutor,

Ostillo Ricci.'* The same is true of other major figures of the century. Descartes advanced beyond the rudiments acquired at La Flèche by studying with Isaac Beechman and

Johann Faulhaber in Holland," Franz van Schooten, the teacher of Christiaan Huygens, studied with Descartes; and

Leibniz, the cofounder of calculus, learned mathematics, not

'®G. Cosentino, "L'Insegnamentop delle Matematiche nei Collegi Gesuitici," Phvsis 13 (1971): 209.

"'W. T. Costello, The Scholastic Curriculum at Earlv Seventeenth-Centurv Cambridge (Cambridge University Press, 1958), 102-103.

'*P. L. Rose, The Italian Renaissance of Mathematics (Geneva: Librairie Droz, 1975), 281.

"s. Sirven, Les Annes d*Apprentissage de Desartes. 73 and Andre' Baillet, La Vie de Mons. Descartes, vol. 1, 39 say that Descartes retired to the town of St. Geminian for two years to attend, above everything else, to the study of mathematics. 27 at the University of Leipzig, his alma mater, but in Paris from Christiaan Huygens in 1672. Pierre Fermat,'® John

Wallis" and Isaac Newton” were practically self-taught, despite Barrow's influence on Newton.

Even when the universities began offering courses in mathematics, the number of students attending them was extremely small. Le Catalogue des Classes dans la Province de Paris pour 1627 of the Jesuit schools in France, in the region of Paris, reports that from a student population of

12,565 attending the 14 colleges, only 64 were registered for the mathematics courses offered in the two schools” [La

Flèche, being one]. The above mentioned John Wallis writing about the study of mathematics at the University of

Cambridge in the early 1630's at Immanuel College, said:

"Amongst more than 200 students in our college, I do not know of any two (perhaps not any) who had more Mathematics than I, (if so much), which was then but little . . . .

'“Se e M. Mahoney, The Mathematical Career of Pierre Fermat. 21.

"See W. T. Costello, The Scholastic Curriculum of 1600. 55; J. F. Scott, The Mathematical Works of John Wallis (London: Taylor and Francis, 1938), 5.

''See M. Mahoney, The Mathematical Carreer of Pierre Fermat. 21.

"Reported by F. Dainville, "L'Enseignment des Mathématiques en France Mediterenee du XVIe Siecle," 15.

" The Work of Thomas Hume, vol. Ill, reported by W. T. Costello, The Scholastic Curriculum of 1600. 27. 28

Even among the teachers there was little enthusiasm, if any, for mathematics. As Mahoney points out,

"Perhaps only at the time of Fermat's death in 1665, if

indeed then, can one find the elements of an emerging profession of mathematics."" When the Jesuits made it mandatory to have a chair of mathematics, "... mathematicians often added to their teaching assignment other activities, sometimes very absorbing."'® From the records of the Jesuit schools, it appears that at times the teachers were students who had just taken the course and were now pursuing their degrees in theology. As soon as they received their degrees in theology they would discontinue their teaching of mathematics. Despite the regulation of the Ratio Studiorum of the Society, teachers of philosophy or other subjects taught mathematics." The first time the University of Geneva did not have a theologian or someone with training in humanities to teach

M. Mahoney, The Mathematical Career of Pierre Fermat. 2 .

'®F. Dainsville, "L'Enseignment des Mathématiques en France Mediterenee du XVIe Siecle," 14.

" Ibid.. 12; S. Sirven, in Les Annes d 'Apprentissage de Descartes says that Descartes was taught mathematics by the theology student, Jean Francois, who, after graduating in theology continued in the role of professor of mathematics during the years 1609-12. Francois returned to resume the same post in 1621; see also C. de Rochemonteix, Un Collège des Jesuites aux XVIIe et XVCIII Siecles; Le Collège Henrv IV a La Flèche 4 vols. (Le Mains; Society of Jesus, 1889), 110, note 2. 29

mathematics was when the physician Esaie Colladon (1562-

1611) was appointed to a chair of philosophy. He also

taught science." Even afterward, mathematics in Geneva was

taught in private lessons only and the teachers were given

neither an official title nor a salary from the school. Not

until 1724 was a regular professorship of mathematics

definitely established and assigned to Jean Louis Calendrini

and Gabriel Kramer.” Other teachers of mathematics were at

the same time discharging administrative or disciplinary

duties, like treasurers, deans of students, principals of the school, which certainly did not help to make mathematics the first choice of the students.

The main reason for the apathy surrounding mathematics and the exact sciences in general was greatly due to the Aristotelian approach to education, even in the

" Revue d*Histoire des Sciences 6 (1953): 233. Even in the Protestant circles, sometimes in the absence of trained clergymen, the teaching of mathematics was withheld. The status of mathematics and other natural sciences in Geneva was much worse than in any other part of Europe. The educator Jean Robert Chouet struggled for over twenty years to see the first chair of mathematics formally established in the public academy of the town in 1724. It was assigned to Louis Calendrini and Gabriel Kramer, whose works in mathematics were renowned throughout Europe. However a non­ paid professorship in mathematics was already held by Pastor Etienne Jalaber in 1704. See M. Heyd, Between Orthodoxy and the Enlightenment (The Hague: M. Nijhoff Publ., 1982), 212- 228.

'*M. B. Heyd, Between Orthodoxy and the Enlightenment. 35. 30

Protestant schools and state universities.” Sandwiched between logic and ethics (first year) and metaphysics (third year), natural philosophy (physical sciences) and mathematics courses seemed out of place and of lesser

importance. The above mentioned Cristoff Clavius, S.J., who was entrusted by his Order with the task of preparing the textbook in mathematics for the Jesuit schools, blamed the philosophers who would denigrate the natural sciences and, in particular, mathematics in their classes. These subjects, they claimed, "are not sciences, they in fact do

”See G. Cosentino, "L'Insegnamento delle Matematiche nei Collegi Gesuitici," 125; P. L. Rose, The Italian Renaissance of Mathematics. 32. It is interesting to note the explanation of the aversion to mathematics at Cambridge given by W. T. Costello, The Scholastic Curriculum of 1600. 103; "While there is no avowed hostility towards Italian mathematics, . . . one may suspect that Galileo's aversion to Aristotle may have put off Aristotelian Cambridge. Galileo had fathered the new mechanics of freely falling bodies, and in the Discorsi had made a mathematical investigation of motion, that is of the relationship among distance, velocity and acceleration. It is possible that the Cambridge physicists felt this to be an intrusion on Aristotelian physics, which, we recall, felt itself alone to be the proper study of being crua in motion. Further, Galileo through the persona of Salviati attacked scholastic physics (defended by Simplicio) in the matter of actual infinity." G. Cosentino also explains the discrepancy between mathematicians and philosophers: "The difference between the [new] mathematical sciences and the fundamentally Aristotelian complex of philosophical and mathematical learning was quite evident. If nothing else, it was very difficult to find room for [the new] mathematics in the system of sciences of Aristotelian derivation. Aristotelian, in fact, attributed to mathematics a more practical than theoretical value. They had also a tendency to see in mathematics an explicative, rather than a cognitive value." (Ibid.. 206). 31

not have demonstrations and do not make abstraction from the

ens and the b o n u m . It might have been due to Clavius'

influence on the Superior General of the Society of Jesus

that caused the Jesuit chapter to ask the philosophy teachers to stop depreciating mathematics in class and,

finally, to forbid the same person to teach philosophy and mathematics. According to him, "Natural philosophy without mathematics is truncated and imperfect."” In the following section it will be pointed out how influential Clavius was in Mersenne's mathematical thinking.

During the second half of the century, this negative attitude towards mathematics subsided. For all the objections philosophers had to mathematics, they had to admit that the proofs of mathematical theorems were providing the highest level of certainty attainable by human abstraction, and the new discoveries in science could be rationally explained only by a comprehensive mathematical presentation. This reversal of attitudes to mathematics in the seventeenth century is described by Francois De Denville regarding the schools run by the Jesuits:

We witness an evolution after the 1660s towards a longer tenure of the instructors and towards their concentration in the teaching assignment of the exact

"C. Clavius, S.J., Algebra, in Mathematicorum Tomus Primus. (Mainz: 1611), 342.

”C. Clavius, Modo quo disciplinas mathematicae in Scholae Societatis Jesu possent oromoveri. in Monumenta Paedaaoqicae S.J.. reported by G. Cosentino, Phvsis 12 (1965): 207. 32

sciences. Career professors are now holding their chairs for a long time, sometimes for decades, and in some instances up to their old age. Compared with the recent past, mathematicians are now the seniors on the teaching staff of the schools.”

We have seen how the same situation is true also

for the chairs in the Royal Collèges; e.g., with Roberval,

who held the chair of mathematics for 41 years."

”F. Dainville, L'Education des Jesuites. 45.

"One can notice that the situation of Physics at the beginning of the XVII century was not any better than that of mathematics. P. Duhem fL'Origine de la Statique vol. 2 [Paris; J. Vrin, 1906], 132) summarizes it as follows: "At the end of the sixteenth century and the beginning of the seventeenth the study of Static flourished in the Netherlands and in Italy with Galileo; but the first third of the seventeenth century passes without any noteworthy publication in this field of science being published in French." The French readers willing to get an introduction in static or hydrostatics did not have at their disposal anything except the books of Jerome Cardano De la Subtilité et les Subtiles Inventions. The first books on physics published by a French author, Solomon de Caus, did not appear until 1615, when the physics of Stevin were introduced. They, however, lacked any originality. Duhem has this to say about the work: "Solomon de Caus had to refer to Cardan's notions of hydrostatics and static when in his book he passed to describe the machines so far invented or brought to perfection. De Caus' contribution was to formulate with order and clarity what the geometer-astrologer had announced in a confused way in his strange book." (Duhem, Ibid.. 136) In 1634 three books of physics appeared simultaneously in French: Simon Stevin's Oeuvres Mathematicrues in Leyde, Les mechaniques de Mr Galilée by Mersenne, and P. Herigone's Cours Mathématique in Paris. This eventful date was the signal for a strong movement which started attracting the attention of the French qeometres to the laws regulating the equilibrium of weights. The school of physics was born in France also; and the first maitres are the two rivals, Descartes and Roberval, who despised each other but, who had the same advisor and confident: P. Mersenne. 33

The Curriculum of Mathematics at La Flèche

Because of the lack of interest in mathematics and

science in France during this period, it is helpful and practical to look carefully and in greater detail at the

curriculum of mathematics at La Flèche that Mersenne

followed during his education at the Collège. Mahoney writes that "the nature and development of the scientific curriculum in the French schools and universities during the seventeenth century is a critical lacuna in historical scholarship."” One cause of such a lacuna is certainly due to the disastrous fate of the private libraries in France during the French Revolution. Precious manuscripts collected in some schools and monasteries were claimed by public libraries and entrusted to untrained personnel, who, at times, did not have a proper appreciation of their

Les Mechaniques de Galilée is practically Mersenne's French adaptation of Galileo's Le Meccaniche. which was circulating in various manuscript copies throughout Europe among the most learned scholars. Mersenne, here as in other situations, took the liberty to change, add and modify Galileo's notes and present them to the learned bonnes hommes — an euphemism to denote the ordinary amateur scientists — who would have found it difficult to understand the original paper in Italian.

”M. Mahoney, The Mathematical Career of Pierre Fermat. 48, note 53; see also 12, note 33. One would in fact look in vain for a description of the course content in mathematics, for example, in Stephen d'Irsay's History of Mathematics or in other sources dealing with education during the Renaissance. 34

content and often lost them or shelved them without

carefully recording them.”

However, the task of tracing back the curriculum

content in this period is made easier when examining the

Jesuit schools. Even though they had regional and

relatively independent administrations, their educational

goals, methods and curricula were discussed and evaluated by

the Jesuit educators at the General Chapters of the Society.

The results of the debates were codified in the already

”Franz van Schooten wrote that at La Fleche, for example, there was the tradition that the graduating student would sign and donate his book notes to the school. Unfortunately, none of these notes are still preserved. See C. Adams and A. Tannery, Oeuvres de Mr. Rene' Descartes vol.10, (Paris: J. Vrin, 1885), 646. The library of the Minims, at the Palace Royale, was one of the richest libraries in the country and specialized in mathematical and scientific publications. Mersenne, together with his confreres Robert Regnault and Francois de la Noue - who later became librarians of the Cardinals Rochelieu and Mazzarello respectively -was put in charge of it for one year. This library, started in 1614, had 8000 books by 1626. Most of them were donations from aged scholars and patrons. It is worth noting that this library was probably the first monastic library to adopt Gabriel Naude's method of book classification by the subject content, rather than by the author's name. Naude was a close friend of Mersenne and his study on magnetism was included in De Hvdraulicis. Whitmore says about the libraries of the Minims: "They were the supreme expression of all that was best in the intellectual activities in the Order in the XVII century." See P. J. S. Whitmore, The Order of the Minims in France. (The Hague: M. Nijkoff, 1967), 5. However cfr. Costabel, P., "L'initiation mathématique de Descartes" in Archives de Philosophie 46 (1983); 638 for a different opinion about the possible help that the recovery of the lost papers of La Flèche could give to reconstruct the curriculum of mathematics at La Fleche. 35 mentioned Ratio Studiorum." binding all the schools

conducted by the Society. Besides, the Superiors of every house, school or any activity of which they were in charge, periodically had to submit quite detailed reports, which were kept in the Archives of the Roman Curia fArchivium

Romanum Societatis lesu = ARSI). The aim of the Archives was expressed in a circular sent by the General in 1595: servire descriotionem rerum qestarum. sed et historiae.

Propterea. rem ob rationem. usum documentorum. exceotis centum annorum elapsis. ab A.R.P. Generali concedi solet.”

These annual reports had to include the names of the members of the community and their assignments. In this way, it is still possible to trace back to the beginning the names of the teachers at the Collège at La Flèche and, at least in part, the curricula in vogue around this period. These reports then would be a source of information about the amount of mathematics to which Mersenne was exposed when attending the school.

"The Jesuits' Ratio Studiorum. together with Melanchton's doctrines on education and the local enterprise of thousands of schoolmasters and city officials throughout Europe, completely transformed the education of the young in this period.

” "To preserve the description of the accomplishments, but also for history. Therefore for this reason, the use of the documents, except those older than a hundred years, is usually allowed by the V.R. Father General". Reported by G. Cosentino, "L'Insegnamento delle Matematiche nei Collegi Gesuitici," 231. 36

The initial cycle of education was run in the

chambres abecedaires. which, besides teaching the first

rudiments of writing and reading, introduced the young

pupils to counting and the first mathematical operations.

The teachers at this level were not, in general, members of

the Jesuit community. When Mersenne was accepted at the

college, he had already passed this level of studies.

The second cycle lasted three years and was

exclusively a course in the humanities. The student was

introduced to the grammar of Latin and Greek, the classical

books of literature of both languages, and the rhetoric (the

trivium).

Finally, the third cycle, again over three years, concentrated mainly on the introduction to philosophy.

During the first year, courses in logic and ethics were taught. Physics or natural philosophy - mathematics was later added starting in 1608 - was taught in the second year; finally, metaphysics was taught in the third year.

The whole curriculum was designed around the

Aristotelian-scholastic approach by a regulation of the 37

Constitutions of the Society of Jesus.” The curriculum of

physics included the "eight books of Physica. the four books

De Coelo et de Mundo. and the first book De Generations by

Aristotle."”

The entire curriculum in mathematics was divided

into four general fields: arithmetic, geometry, music and

astronomy," known as the medieval quadrivium. As in other

”"In the teaching of the logic, natural philosophy and metaphysics, the instructor will follow the philosophy of Aristotle." Constitutiones Societatis Jesu cum earum declarationibus (Rome: Society of Jesus, 1583), part IV, ch. XIV, par. 3, reported by G. Cosentino, "L'Insegnamento delle Matematiche nei Collegi Gesuitici," 207. The Ratio of the Minims also established that Aristotelian philosophy and Thomist theology were mandatory in the houses of study of the Order.

”C. de Rochemonteix, Un Collège des Jesuites aux XVIIe et XVIIIe Siecles. Le Collège Henrv IV a La Fleche. vol. 4, 32. Comenius in his The Great Didactic. [English translation by L. Keatinge, 2 vols. London: A and C. Black, 1921], 25-28 makes the following remark on the way physics was taught in schools: "The method of all arts shows that the schools contrive to teach pupils to judge by the eyes of others, and be wise by proxy. They do not teach pupils to discover springs and thence to lead off various streams; they show them streams drawn from authors and bid them return to the well-heard by following these secondary channels... Hardly anyone teaches physics by ocular demonstrations and experiments, but all by recitation of Aristotle or someone else. It comes in short to this: Man ought to be taught as far as possible, to be wise; not out of the books, but from the heavens and earth, from oaks and beeches, i.e., to know and scrutinize things themselves, not the observations and testimonies of others concerning things." As it will be pointed out later, Mersenne was in correspondence with Comenius, whose books on education he greatly admired.

"C. de Rochemonteix, Un Collège des Jesuites aux XVIIe et XVIIIe Siecles. 37-49. 38

Jesuit colleges, the textbook adopted for arithmetic was the

Epitome Arithmeticae Praticae and the Euclides Elementorum

both by Christof Clavius.” Clavius was the first to

introduce the division between the mathematics speculative. which Mersenne developed extensively in La Vérité des

Science, and the mathematics applicata. The first included arithmetic, algebra and geometry; applied mathematics

included astronomy, perspective, geodesic, music, the

"science of calculations" or applied arithmetic, and mechanics.” At La Flèche mathematics was mainly oriented towards les arts mechaniques." The Ratio Studiorum of the

Jesuits in 1599 made provision for students who showed more ability and dedication to the subject, by urging the instructors to give them more attention and more assignments.” Besides attending the morning classes.

”C. Clavius, Epitome Arithmeticae Praticae. (Basel: ex typographiae D. Basae, 1583). The book had several editions in Latin and was printed in several parts of Europe. In 1686 it was translated into Italian by Lorenzo Castellano and approved by the author. This translation dominated the scene in the Italian schools for over a century; Euclides Elementorum. (Rome: Grassini, 1574). The first six books were translated into Chinese under the supervision of Clavius' former student, Matteo Ricci, who had gone as a missionary to China.

"See C. Clavius, Opera Mathematics, vol.l, 3-4.

"See R. Descartes, in Discourse on Method and Meditations. English translation by L. Lafleur, (Indianapolis: The Liberal Arts Press Inc., 1968), 15.

”G. Cosentino, "L'Insegnamento delle Matematiche nei Collegi Gesuitici," 208. If these students were members of the Society of Jesus, they were later sent to Rome at the Colleaio Romano to study under Fr. Clavius, and after his 39

however, each evening the students participated in special

excercitationes and were assisted by tutors, who often were

students in the upper grades."

As mentioned previously, the Collège at La Flèche

began teaching mathematics in the academic year 1608-1609.

In that year, Mersenne was in the third year of the third

cycle and was concentrating on metaphysics. In the second

year, he most certainly took courses in physics; but did he

study mathematics at the school?

A regulation in 1665 from the General of the

Jesuits established that students in the third year of the

third cycle, who for any reason had not had physics and/or

mathematics the previous year, had to study these subjects

during the third year." One is inclined, though, to ask whether the regulation was issued to put an end to an abuse whereby students might have been graduated without the

death, under Fr. A. Kircher, S. J., or others. They were being prepared to hold the chairs of mathematics in their native countries. This fact in itself provided greater uniformity and established the standards for the schools of the Jesuits. If, instead, they did not belong to the Jesuits, then they were given a chance to have special classes in civil and military engineering. Mersenne's interest in several experiments about the range of the ball shot by a canon at different inclinations, his books on ballistics, and his description of a submarine that could be used also for military purposes may have originated in these classes.

"Presiding over such exercitationes or disputationes was a requirement for graduation during the Middle Ages, but by this time the requirement was dropped.

"See F. Dainville, L'Education des Jesuites. 328. 40

proper qualifications in mathematics and physics, and

whether the ordinance applied to the Jesuit students only or

everybody in the school as well. This regulation came from

Rome seventeen years after Mersenne's death and therefore

cannot help determine whether or not Mersenne had courses in

Mathematics at La Flèche. Mersenne's biographers do not give any hint about this, nor does the De Rochemonteix's study of the college. Mersenne himself is of no help either, because of his reluctance to talk about himself.

One thing is certain: if he participated in the regular course of mathematics, it was at the age of 21, in comparison to Descartes at 15, Fermat at 16, Huygens at 17,

Pascal at 12: a bit too old for an introduction to mathematics.

But if the first book in mathematics that he produced. La Vérité des Sciences, is any indication of his mathematical background, it is evident that Mersenne had certainly mastered the arithmetic in Clavius' book. In fact, the content of the applied arithmetic in La Vérité des

Sciences is an enrichment of his school-year's textbook."

"it is rather difficult to draw a parallel between the two books, because the aims and the methods of both were quite different. Clavius' book is a textbook for an introduction to arithmetic to young students in the classroom, and therefore the method he followed was a direct approach with many examples and proposed numerical cruastiunculae. Mersenne's book instead is an apologetical work directed at an older group which is expected to have a vast background in philosophy, theology, the classics, and the other scientific subjects as well as a wide scope of interests. The approach is less direct. In fact, he used the 41

The difficulty in determining the content of Mersenne's

mathematical education arises especially when one wants to

decide whether Mersenne had an introduction to algebra or

cossist mathematics. The following three points shed light

on the question:

I. It is clear from the many references to the authors in

his book that by the time Mersenne was writing La

Vérité he had read the books by Viète, by Anderson

Alexander (1582-1625), Giambattista Benedetti (1530-

1590),and the Algebra" by Clavius.However, even

though in several parts of the book Mersenne

introduced concepts from algebra, in the last chapter

of the third book in the above mentioned La Vérité.

which was supposed to deal with the subject in greater

depth, he wrote:

About the numbers that are less than zero, i. e., less than nothing, you could think of them by what I have told you about the irrational numbers, because just as we imagine the existence of the square or cubic roots of ten, even though there is dyslogia where the agnostic and the independent are given room and time to voice their difficulties and restrictions. La Vérité is rich in references to ancient and contemporary mathematical texts while at the same time providing enough mathematical examples to clarify the points it makes. However, it may be illuminating to list the headings of the chapters of the two books side by side, so as to make evident the similarities and the differences. See Appendix One.

"C. Clavius' Algebra was first published in 1608 in Rome, therefore too late for Mersenne to have had a course in it.

’“See, for example. La Vérité des Sciences, 711. 42

nothing of the sort, in the same way the algebraists use the numbers which are the numbers less than nothing, and which they denote by the symbol 0 - 8 to mean a number 8 times less than zero.”

And a little later he added:

I should explain the rule of algebra and its various equations before entertaining you about the content of geometry, but it would take us too much time, because it is very obscure and intricate until someone makes it easier for u s . ”

He does not go beyond the solution of a linear

equation.

II. Most of the mathematicians needed more time to study

under the supervision of more experienced tutors and

with more advanced textbooks in order to supplement the

math they learned in college. Was Mersenne formally

introduced into higher studies in mathematics? The

previous chapter indicated that in 1609-1611 Mersenne

was taking graduate courses at the Collège de France;

the following three years he was in the formation

program with the Minims; in 1614-18 he was involved in

teaching philosophy and theology; the next year he was

correcteur at Nevers. All these were full time jobs

that certainly did not leave him too much time for

private study. This is particularly true if one

considers that in 1623 he started his literary

’’Ib i d . . 560.

” Ibid.. 563 43

production with an average of 250 printed pages per

year for the first three years, in addition to an equal

amount of unpublished pages. One cannot see how he

could possibly have taken two years off as Descartes,”

”See note on page 23 about Descartes' enrollment at the University of Leyden. It has also been mentioned how Descartes retreated to St. Geminian for two years to devote himself to mathematics. In fact he wrote: "As soon as I finished my studies which usually admit one to the ranks of the learned, I changed my opinion completely. For I found myself saddled with so many doubts and errors that I seemed to have gained nothing in trying to educate myself unless it was to discover more and more fully how ignorant I was" (Discourse on Method. 5). Descartes was particularly fortunate because it was during these years that he met Beechman, who opened him to a new version of applied mathematics. P. Costabel ("L'Initiation Mathématique de Descartes," Archives de Philosophie 46 (1983): 637-46) asserts that, according to the Journal of Beechman, the real active initiation of Descartes must be placed from 1619 to 1628. Such an initiation was driven by the search to probe what, in mathematics, could help ad omnem coanitionem humanarum. During this initiation the assimilation of information coming from various sources and the originality of the reflexions of the young scholar came to a happy marriage. During this time Descartes also met Johann Faulhaber (1580 - 1635) from whom he, most likely, learned the cossist notation. 44

Fermat,” and others” did and still produce so much in

such a short period of time.

III. Finally, Mersenne never worked with algebra in any of

his subsequent publications. One has reasons,

therefore, to question if the lack of time mentioned in

La Vérité is only a rhetorical remark and if he might

be hiding a lack of confidence in his knowledge and

ability in algebra.

Based on this evidence, it is most likely that

Mersenne was not from the cossist mathematicians of his time.” All these facts apparently explain why Mersenne was never a professional mathematician, but needed in many

”Fermat also, before going to the law school, retreated to Bourdeaux, where he studied the books of algebra by F. Viete, probably at the school of Jacques Beaugrand. His analytical tendencies often put him at odds with Roberval, Frenicle and other mathematicians in Paris. But this helped him develop his theory of maxima and minima, tangents, etc., which he used in the study of the cycloid to rescue Roberval from embarrassment, as will be explained in the next chapter.

”For example, Marino Ghetaldi, who studied in Rome with C. Clavius before meeting Viete in Paris and becoming one of his devoted students.

“Among other concepts in algebra that Mersenne found contradictory, he listed in the La Vérité des Sciences the fact that a negative number from which another quantity is subtracted yielded a larger amount than the previous one, e.g. -10 - 2 = -12 > -10, while if to the first term a positive quantity is added, the result is smaller, i.e. -10 + 2 = - 8 < - 101 However, note also the reaction of Mersenne to Pierre Herigone's Cursus Mathematicus. About geometry Michelangelo Ricci wrote to Torricelli that his impression was that Mersenne was not well versed in it but that he had made many observations about it (Correspondance. vol. 13, 341). 45

instances to depend on the calculations that others performed for him. This also justifies, in part, the repeated changes of data that are encountered in some of his publications. Actually, those who pursue research on

Mersenne can rightly wonder how, in spite of the little training he had in mathematics, his very busy schedule, and his vast range of interests, he still found the energy and the will to educate himself and keep himself in touch with the latest scientific discoveries.

In geometry, it is quite possible that the Collège de la Flèche might have adopted the Euclides Elementorum by

Clavius, because the Geometria Practice by the same author was published only in 1604, too late for it to be adopted in

France during the time Mersenne was there. In any case

Mersenne in La Vérité des Sciences shows great familiarity with Euclid's geometry and its various commentaries, especially those by the ancient Theodose, Apollonius, and

Eratosthenes, and those by his contemporaries Ramus,

Maurilicus, and Jan Baptiste Benoit. In fact on the chapter: "What is necessary to be a good geometer?",

Mersenne included:

a. Very good familiarity with Euclid;

b. Theodose on the sphere;

c. Apollonius for the sections on the space, on the

plane and about the conics;

d. Erastotenes' books on median proportionalities; 46

e. Trigonometry and logarithms.

Because of his philosophical background, Mersenne

most likely enjoyed more geometry more than arithmetic.

Again in La Vérité des Sciences he wrote;

If there is anything self evident, clear and certain in the world, that is geometry. The proof for this is that in the past one thousand years nothing wrong has been found with the fifteen books of Euclid's Elementa or in the section on the conics by Paergeus.”

The book of clavius included considerations on the

circle (its divisions and properties), including the classical problem of the quadrature of the circle, the planar and spherical triangle, which led into trigonometric functions, as well as the concepts of the planar and spacial curves. He focused particularly on the conical sections

(the circle, parabola, ellipse, hyperbola and their respective areas), regular and irregular planar polygons

(their perimeters and areas), and solids (surface areas and volume).

Education in the Order of the Minims

In order to understand the personality of Mersenne and, consequently, the philosophy of science that led him in his scientific research, it will be useful to look into the educational environment of the Minims, with whom Mersenne spent about fifty years of his life.

^M. Mersenne, La Vérité des Sciences, book 2, 423. 47

Mersenne's decision to pursue a degree in theology

in Paris and afterwards to join the austere Order of the

Minims came as a great surprise to his parents. They had

hoped he would pursue a career in law; but when he chose to

study theology, they expected him to join the secular clergy

so that, with his education, he could aspire to a high ecclesiastical rank. His choice of the Minims, who, by law, cannot accept ecclesiastical titles or offices such as the episcopate, was instead a clear message that he had no ambition to such positions of power and authority. He never explained the reasons for his selection of the religious life nor his preference for the Minims to the more education-oriented Jesuits, the Franciscans, or the

Dominicans, whom he certainly met frequently at the Sorbonne in Paris. However, one can legitimately ask, to what extent did the religious life help Mersenne reach his educational goals, if he had any?

According to B. Rochot;

Let's recall the advantages that religious life was offering in those times to a spirit with strong tendencies to research and, at the same time, incapable of freeing himself from the conventions of his times: without the concerns for daily and personal needs, an absolutely free time, except for prayer. Moreover, the internationality of the Order offered him the advantage of free communication throughout Europe even in times of war.”

”P. Rochot, La Correspondance Scientifique du Pere Mersenne. 7. 48

However, Mersenne's field of research was

certainly not in line with the Order's goals. In general,

the study of mathematics for religious reasons was often

objected to, not only by the Minims but even by the Jesuits

and all the clergy, as being a field not conducive to piety

and the pastoral care.

The Minims formalized their attitudes towards

education in their Ratio Studiorum. For example, the Ratio

of 1639 listed six points tjiat were considered weaknesses of

human knowledge: vagueness of human sciences, difficulty (if

not impossibility) to attain them, insipidness and unpleasantness of their possession, the temptation of pride

and superiority above others, the ease with which human knowledge is lost, and the short span of time a person can make use of it. A Minim, therefore, was advised to

"indulge" himself in such amenities only to avoid idleness.

And the Ratio went on to advise the monk:

It is allowed to study, but without ostentation, without jealousy, without ambition, always remembering that you are the least one, from the Order of the Minims, whose members are warned against accepting the doctoral insignia, or soliciting any degree . . . ,”

”M. Lesguiller, Ratio studiorum (Paris: 1639), reported by P. J. S. Whitmore, The Order of the Minims in the XVII Centurv France. 261. This negative attitude toward education was the reason why, in the monastery there were monks who could not read or write and therefore could not be assigned to some offices. On the other hand, the young students of the Order were allowed to get a degree in theology or philosophy, but only from the Universities of Salamanca or Alcala'. 49

and concluded with the negative remark about books; "The big

books that generate so many great problems!"” The timing of

this Ratio suggests that at the time Mersenne was at the

peak of his scientific productivity, the community of the

monks may have been discussing the role of education in

their style of life. In fact the "golden age" of the Minims

faded out with the death of Mersenne's correspondent and mathematician P. Maignan, o. min., and his students. One

can also add the difficulties that Mersenne faced in the community when, for example, he came back from Holland, where, according to the civil authority and some of his confreres, he had met some "heretics in the disguise of scientists." This incidence created a situation of mistrust for Mersenne that probably lingered for a long period.

Besides, the number of visitors coming to see him in the monastery, and who were accommodated in the library or in his room, often disrupted the quiet of the monastery; because of his frequent exits from the monastery, his disruptions of the timetable of the day were quite frequent.

The fact that, during his visit to Rome, the Convento della

Santa Trinita' ai Monti initially denied him hospitality.

"Certainly Mersenne readers would have wished Mersenne had given more weight to this latter remark and shortened his treatises! In fact in certain circles he was criticized for the size of his volumes; Descartes, in all probability, never read thoroughly any of them. 50

may be a clue to the uneasy situation that his unusual type

of activity was creating around him.

On the other hand, even though the Minims were not

specially known for their educational standards, one has to

consider the fact that the above mentioned monastery in Rome

was erected with the same purposes of the Colleaio Romano of

the Jesuits. Some Minims teaching there were

internationally renowned for their scientific collections

and publications; the libraries of the Minims, as was pointed out earlier, were among the best monastic collections of that time;” and quite a few friars had published works of considerable value. During the famous barometric experiment conducted by Pierre Petit at the top of the Dome du Puvs two weeks after Mersenne's death, the

Provincial Superior of the Minims was invited to participate as an authoritative witness. The monastery at the foot of the mountain was used as the headquarters of the expedition and the local Superior was assigned to monitor the height of the barometer installed in the monastery during the experiment.

“'There is evidence to show that these men were allowed to possess books of their own, a modification to the rule of poverty. The description of Louis X I V s visit to Maignan at Toulouse and of Martin Lister's to Charles Plumier, o. min., (1646-1704) at the Place Royale make their rooms appear more like laboratories and libraries than traditional cells (P. J. S. Whitmore, The Order of the Minims in the XVII Centurv France. 212). 51

In spite of the above mentioned attitudes towards

education, it is important to note that Mersenne owed to the

Order his excellent preparation in Hebrew at the school of his confrere Jean Bruno. A second confrere had provided him with details of Arabic music and instruments, as well as a general introduction to Coptic (Egyptian) script. The original permission from the Order to publish was granted him in reference to his philosophical and theological training with the purpose of defending the Catholic Church, which was then challenged by the incipient liberalism of the

Renaissance. However, when Mersenne's interests shifted more towards scientific publications, his superiors always granted him the nihil obstat to publish his studies." As a matter of fact, he was even exempted from the major assignments in the monastery that could have deprived him of the time and the concentration he needed for his studies.

Finally, the fact that Mersenne made frequent enquiries about other confreres interested in sciences and

“*The publication of his books Ouaestiones celeberrimae in Genesim and Observationes et Emendationes ad Francisci Georaii Problemata was authorized by the General Superior of the Order P. Francisco de Mayada, and the cooitata Phvsico- Mathematica was dedicated to and approved by P. Lorenzo da Spezzano, who in 1647 had taken over the office of de Mayada. In general, the permission was granted on condition that two theologians of the Order would examine and approve the manuscript. Often his co-workers Robert Regnault, Francois de La Noue, or Francois Niceron were appointed censors, and Mersenne did not have any problem getting their approval. 52

mathematics even outside the limits of his Province, and

that he felt free to criticize in La Vérité des Sciences

those clergy who did not approve of monks dedicating

themselves to scientific research,” are signs that the Order

allowed him to continue to pursue his scientific research.

The remarks of the Ratio were probably more a philosophical position than a behavioral attitude inside the

administration of the Order. In fact, the reminder of pursuing education and research "without ostentation, envy or ambition" might explain the spirit of service towards science and scientists that motivated Mersenne, as well as his persistent reluctance to use the opportunity to talk about himself or his own accomplishments.”

““M. Mersenne, La Vérité des Sciences. In the general introduction (non numbered pages) Mersenne wrote; "I find it disgraceful that some people of little spirit and short discernment think the truth of sciences is useless and of little help to piety and religion. I am sure that if they will understand the content of this volume they will be quick to acknowledge their mistake; there is nothing in science which is not of the greatest use for the mind and the love of God."

”One can therefore advance the hypothesis that the decision of Mersenne to join the humble Order of the Minims was the result of a dissatisfaction with, or even mistrust in, his education. In fact, his biographers say that Mersenne plunged himself into the publishing activity only at the insistence of some of his older confreres. Otherwise, maybe Mersenne would have spent his years in retirement in the monastery and away from the circle of the French "intelligentsia." Descartes himself had experienced the same dissatisfaction, but he overcame the crisis by retiring in St. Geminian for a period of discernment and of independent study. CHAPTER THREE

MERSENNE AND HIS CONTACTS

WITH OTHER SCHOLARS

Mersenne lived in a period when the scientific outlook in Europe was undergoing a critical phase of transition. The discovery of the magnetic compass had led to the finding of the Americas and new passages to the Far

East, in particular the Oceania; gun powder was extending the power of the European nations outside the old continent, with an economical boost for the colonizing powers; and most of all, the newly discovered printing press was changing forever the history of the world in every field. The result was the gradual emergence of the dominance of the human- centered and experience-oriented type of knowledge over that of philosophical and theological speculation. Mersenne was well aware of the potential for this dramatic development of science, if the opportunities were properly used. For this purpose, the scholars needed to concentrate their interests on the essentials: the enthusiasm for keeping alive new discoveries and the desire for constantly feeding more challenges to the young generation of scientists. The

53 54 transitional period of science required clear definitions of the realms of science', nature, and religion and the

'when Galileo's books were condemned in Rome, Mersenne wrote to his friend and correspondent, the Protestant theologian Rivet of Leyden, asking his opinion about the condemned theories. In reply to Rivet's letter Mersenne wrote back: "You are right not to look at the theory about the earth's movement as a heresy, because the Holy Scriptures is not intended to teach us philosophy or theology" (Correspondance. vol. 8, 675). This was exactly the point that Galileo had already expressed in 1615 in a letter written to the Gran Duchessa of Tuscany: "Scriptures tell us how to get to heaven, not how the heavens are regulated1" Gassendi later accepted this view, which eventually became the official position of the Church. On another occasion, indicating how religion and science needed different approaches of understanding, Mersenne wrote: "Some want demonstrations in religion as in mathematics, others imagine that everything is false, or at least that we cannot be sure of anything, because, they say, everything has two sides; others again claim that everything is God . . ." fCorrespondance. vol.6, 228-29). At a later date he wrote to Rivet about the inevitability of having a dangerous confrontation between science and religion if the boundaries of both were not drawn in time (ibid., May 1644). And to the theologians he had this serious warning: "I warn the theologians not to rush to think or to state that anything is against the teachings of the faith or goes against the divine Scriptures. First they have to take a serious analysis, because it happens often that we are prevented by some preconceived ideas which are not always certain in subjects like physics and astrology and almost every field in mathematics. Which is not surprising, since God has left the whole world to our discussions." (Ouaestiones in Genesim. col. 1510) The same warning goes to the censors of books (La Vérité des Sciences, book 2, 354). To avoid the pitfalls, Mersenne suggested to those who intend to study theology: "I want them to master (optime versatos est! all the sciences, especially physics, medicine and mathematics." (L'Impieté des Deistes. book 1, 220-221). About three hundred fifty years afterwards the Vatican II decree Op tatam Totius (On Priestly Formation, Art. 5, 13, 1962) made this the official position of the Church: "Before Seminarians commence their specifically ecclesiastical studies, they should already have received that literary and scientific education which is a prerequisite for higher studies in their country." 54

explicit statement of their mutual reinforcing values and

their limitations. Mersenne accepted the challenge and set

himself to the task.

His main contribution to the intellectual

development in the seventeenth century consisted undoubtedly

in the strategy he devised in order to unlock the secrets of

science and his ability to share his plans with other

scholars and amateurs in their fields. In Les Preludes de

l'Harmonie Universelle, he so described his vision of

science in this way:

The sciences have sworn among themselves an inviolable partnership; it is almost impossible to separate them; for they would rather suffer than be torn apart; and if anyone insists in doing so, he gets for his trouble only imperfect and confused fragments. Yet, they do not arrive all together, but hold each other by the hand, so that they follow each other in a natural order which it is dangerous to change, because they refuse to enter in any other way where they are called.^

The only way the scientists could enter into the

secret realm of nature was to work together. Unfortunately,

Mersenne realized, that was not the way scientists generally

work,® and that was why they were still debating old

®M. Mersenne, Les Preludes de l'Harmonie Universelle (Paris, 1634), question 5, 17.

®Descartes, for example, wrote in the Discourse on the Method. (Part Two, 10): "... Frequently there is less perfection in a work produced by several persons than in one produced by a single hand . . . e.g., when we observe how the various parts of a city are arranged: here is a large unit, there a small one, and the streets, how crooked and uneven they are, one would rather suppose that chance and not the decision of rational men had so arranged them." Therefore, he concluded: "Faced with divergence of opinions, I could not accept the testimony of the majority. 55

problems, as in the past, particularly about philosophy, which, for Mersenne, included also natural philosophy, i.e., physics. He wrote:

Philosophy would long ago have reached a high level if our predecessors and fathers had put this into practice, and we would not waste our time on the primary difficulties, which appear as severe as in the first centuries when they noticed them. We would have the experience of the phenomena which would serve as principles for a solid reasoning; truth would not be deeply sunken, nature would have taken off most of her envelopes, one would see the marvels she contains in all her individuals/"

Part of the reason for this isolation of the scholars was due to the lack of public and private support.

Mersenne decried the fact that mathematicians and scientists were neglected, while poets and writers were welcome in the

for I thought it worthless as a proof of anything somewhat difficult to discover, since it is much more likely that a single man will have discovered it than a group of people. Nor, on the other hand, could I select anyone whose opinion seemed to me to be preferable to those of others, and I was thus constrained to embark on the investigation for myself." (Ibid.. 14) .

Mersenne, Les Preludes de l'Harmonie Universelle. 21. Cooperation is asked also among the drafters of a project and its executioners. Talking from experience he wrote in the Les Nouvelles Pensées de Galilée: "Before starting a project which needs machines, engineers and artisans, the project and the models should be exposed to the public and in particular to expert geometers, who know all the secrets of machines and can foresee the inconveniences that the atmospheric or water conditions can cause. On a fixed day the architects should bring their projects to be examined by the most knowledgeable engineers, mathematicians and carpenters, to select the best projects. In fact, theory and practice should go hand in hand, not only in the execution of the projects but even before at the drafting stage." (Pp. 16-17 in the original and 20-21 in the reprinted edition.) 56

king's court or among the nobility of France.® To

compensate for this, he made himself responsible for

actively promoting the growth of a self-confident scientific

community in Europe, with broad agreement on an acceptable

conception of nature, of workable methods of investigating

it, and of what could be known about it for certain.®

He took communication and cooperation to be the

means with which to reach the goal. Unlike his friends

Descartes, Fermat and Desargues, who prided themselves on

their very limited interest in reading in order to avoid

being influenced by the discoveries of others, Mersenne was

an avid reader.^ He was only too glad to be able to use the

libraries of his friends and patrons, in particular the rich

collection of Jacques Halle'. Whenever he wrote to his

friends he always mentioned the latest editorial news of

“see M. Mersenne, La Vérité des Sciences book 3, 751- 752 where Mersenne cited the case of the disciple of Viète, the British Alexander Anderson, who could have reached the heights of Archimedes and Apollonius were he given some financial and moral support. He therefore pleaded that at least after Anderson's death his merits be recognized and his manuscripts printed.

“See A. Crombie, "Problem of Scientific Acceptability," Phvsis 17 (1975): 188.

^Digby's description of Mersenne's cell in Paris made it look more like a library and a scientific laboratory than a traditional monastic cell. His correspondence shows him buying books from all the countries in Europe; during his visit to Italy, Ricci, a young scholar, described him to Torricelli as visiting all places in Rome looking for new books and ancient manuscripts (Correspondance vol. 13, 279). 57

Paris or of places sometimes as distant as Poland, England,

Holland or Italy. He frequently requested, as a favor from

his correspondents, information about publications. In

fact, some publishers regularly sent him newsletters

concerning their major publications.®

Once he was introduced to the techniques of the printing process, he became a non-stop publisher: in twenty

five years he produced twenty five books of various lengths and topics comprising a total of about 8,000 pages. The topics ranged from speculative theology, apologetics and scriptural dissertations to mathematics, music and physics.

A detailed description of Mersenne's publications can be found in Lenoble's Mersenne. ou La Naissance du Mechanism.® and a very rich bibliography in Robert Dear's Mersenne and the Learning in the School. In this chapter the editorial productivity of Mersenne, his commitment to contribute to the advancement of sciences through his correspondence, and the regular meetings of scientists of the Academia

®See for example the correspondence with the Polish publisher Jan Howelcke (Latinized Hovelius) of Gdansk (vol. 10, 605) or Cavalieri's letter of November 23, 1641 (Correspondance. vol 10, 792).

“However, the book does not report two more books: L'Usaae de 1 'Horologe Universelle, and Du Mouvement et de la Chute des Corps Pesants (see Bibliography for details). Besides, Lenoble was not able to prove Mersenne's authorship of La Vie Spirituelle except by quoting the monk's first biographer, Hilaire De Coste. However, in 1986, A. Crombie was able to identify the booklet in the Vatican Library. 58

Parisiensis will be analyzed together with the resulting

effects.

Mersenne*s Editorial Activity

Mersenne's works are generally of encyclopedic

nature. Often one volume will include several books bound

together, where he tries to find room for almost every bit

of information he was able to gather in the interval of time

from publication to publication. One could look upon his books as the forerunners of our modern scientific magazines.

But they were not a confused collection of information, as they have sometimes been taken to be even by some modern historians. For example Lynn Thorndike wrote: "His books tended to be catch-alls, and consequently their titles may not exactly or completely describe their content."'® The introverted Descartes did not bother reading many of these books. In fact, he rarely mentioned them in his own books or in his correspondence with Mersenne or others. This may be because the two corresponded so often that Descartes

'“Martha Ornstein calls La Vérité des Sciences "a little book" (The Role of Scientific Societies in the XVIIth Centurv. Chicago, Chicago Unversity Press, 1975, 254). A book of 1080 pages is not really a little book in volume. And as for the content, the book is where the philosophy of science of Mersenne is spelled out. 59 thought he knew beforehand the content of the books." Even

Lenoble, the most authoritative expert on Mersenne, said about his books:

In his chaotic works, which at times appear to be like a garbage dump, the ancient and the new intermingle in a picturesque disorder. But one has to understand and choose.

But a careful analysis of his works reveals well- conceived plans, where the apparent digressions from the central topic can be often justified. The problem was that the printing technique of his time could not express his imaginative and original creativity: nowadays, writers benefit from the use of parenthesis, footnotes, appendices, and from other techniques to insert a section with no apparent connection to the main content of the book, without, at the same time, breaking up the train of thought of the main topic. In lieu of these modern printing aids

Mersenne, and for that matter all of his contemporaries, used Scholia. Corollaries, and Monita to achieve the same goals. Mersenne needed an editor who could have tailored the material to the taste of the readers more suitably than

"Writing to Constantin Huygens on February 25, 1637, Descartes said: "I know him better as a person (a very good one, indeed) than as a writer, on whom I never spent more than half an hour." (Correspondance. vol. 6, 209).

'“R. Lenoble, Mersenne ou la Naissance du Mechanisme. 4. 60

the author himself.'® A superficial reader is sometimes

tempted to judge his books the way Lenoble and other readers

of Mersenne did. However, justice has to be done to

Mersenne. When I. Beeckmann saw the Somaire des Seize

Livres de la Musique — the plan of the Harmonie Universelle

— he wrote to Mersenne:

Great and certainly wonderful! In fact what you promise to include in your books of music is even philosophical! If you will be able to carry out your project in a decent way, you will take away (praeripueris) from us the burden of meditating on philosophical matters ! '"

'®This is clearly demonstrated in the modern reprints of Mersenne works, in particular the reprint of the Questions Inouves. Questions Harmoniques. Questions Theoloaioues. Les Mechanicfues de Galilée. Les Preludes de l'Harmonie Universelles edited in the Corpus des Quevres de Philosophie en Langue Française. A. Fayard, published in cooperation with the Centre National des Lettres. Andre Pessel editor, 1985. Each question starts a new page and the answer does not take more than two pages. The books are collections of current problems of science to which Mersenne gives short and appropriate solutions. He had in mind to continue the series and collect one thousands problems, but he could not reach his goal. Having editors publishing someone's manuscripts was a common practice in Mersenne's times. When, for example, Galileo wrote The Assaver, in opposition to the Jesuit Father Qrazio Grassi's Libra. quite a few members of the Lyncean Academy in Rome were all engaged in revising it, from the philosophical, theological, scientific and stylistic points of view. "In all, there were six authoritative 'censors,' from the Academy of the Lynceans and of different background, who could not have failed to observe any theological or philosophical aspect... ." (Redondi, Pietro, Galileo Heretic. Princeton, Princeton University Press, 1987, 45-46)

"Mersenne, Correspondance. vol. 2, 284. 61

Popkins" has shown how Mersenne in La Vérité des

Sciences was following a very carefully planned scheme, so

that the criticism of Thorndike against the book is not valid.'® William Hine did the same, in part, about the

Ouaestiones celeberrimae in Genesim. showing how the position of Mersenne on Copernicanism throughout his life was consistent and did not keep changing, as P. Boutroux and others had supposed." P. Costabel and P. Lerner in their critical edition of Mersenne's book Les Nouvelles Pensees de

Galilée also traced the origins, development, and spread of the book, showing in detail its similarities and differences in comparison with Galileo's Dialoghi. The present study attempts to show the plan behind Mersenne's De Hvdraulico-

Pneumaticis phaenomena and is based on the examination of the copy of the book kept at the Library of Congress.

'“Richard Popkin, "Father Mersenne's War Against Pyrrhonism," The Modern Schoolman 34 (1956-57), 61-78. The complaint about the negligence of Mersenne in the history of modern philosophy that Popkins laments has since been repaired by Peter Dear's Mersenne and the Learning of the School. which is the only book in English so far published on Mersenne.

'“when La Vérité des Sciences appeared in 1625 Mersenne was accused of plagiarism because it was thought that the book was a translation from the Greek of Herbert de Cherbury's De Veritate. But when the book was indeed translated into French in 1638, Descartes was the first to point out the difference between the two books (Baillet, La Vie de Mons. Descartes vol 2, 14-15).

'^William L. Hine, "Mersenne and Copernicanism" in Isis. 64 (1973): 18-32. 62

The work is contained in the Coaitata Phvsico-

Mathematica" with six other works. The fourth folio of the volume, containing the titles of the books included in the volume, reads:

TRACTATUS isto volumine continenti:

I. De Mensuris. Ponderibus et Numéris Hebraicis. Graecis

et Romanis ad Galliam Redactis.

II. De Hvdraulico-Pneumatico Phoenomenis.

III. De Arte Nautica. seu Histometria et Hvdrostatica.

IV. De Musica Theories Practice.

V. De Mechanicis Phaenomenis.

VI. De Ballisticis. seu Acoustimolocicis Phaenomenis.

After the title page of the first book, a letter of dedication to James Halle', an adviser to the King, and an introduction to the book on measures and weights follow.

[This latter section appears to have been duplicated and a copy of the reprint inserted in the book at a later time.

That seem to be a plausible explanation of the repetition of this section in the original copy preserved at Library of the Congress.] A general introduction to the whole work

"in the catalogues of the British Museum it appears that some copies of the Hvdraulica et Pneumatics Phenomena were bound separately under the title Hvdraulica Phenomena. Arscfue naviaandi. Harmonia Theories. Practica et Mechanics Phenomena. It was not possible to examine the copy; therefore I cannot say if the other books mentioned in the title are also bound together or if the title is only the heading of the Hvdraulica et Pneumatics Phoenomena. which appears on the book bound in the Coaitata. 63

follows, containing twenty seven points on the content of

the book, and finally, a Praefatio Praefationum. Two

subtitles on ratios and proportions follow immediately.

Again the title page, the dedication and the introduction to

the first book are repeated. None of these pages are

numbered. The text itself consists of forty pages.

The book on fluids is entitled; Hvdraulica

Pneumatica Arscme Naviaandi. Harmonia Theories practice et

Mechanics phaenomena. autore M. Mersenno M . . Parisiis.

Sumptibus Anthonii Bertier. via lacobaea M DC XLIV; Cum

Privilégie Reoii.^ it is dedicated to the esquire loanni

Marchioni D'Estrampes-Valencay. The dedicatory letter is dated March 6, 1644. The introduction, which is in fourteen pages long, contains fourteen points. The first thing that impresses the reader is the realization that the work is the result of long years of persistent research®® enriched by

’®This is a third different title of the book. This fault is probably due to the printers and proof readers, who were entrusted with the task of publishing the book. As appears in the introduction of La Vérité (page not numbered), once the book was in the hand of the publisher, Mersenne would intervene only when the book was already printed, so that he was left with the only option of pointing out the most evident mistakes on the last pages, (see Correspondance vol. 1, 81 for Bredeau Celement's comment on the work of the printers: "who often follow their own opinion rather than the manuscript . . .").

^Already in 1626 Mersenne showed his interest in the topic as his letter to Descartes on January of the same year reveals (see Correspondance. vol.l, letters nos. 45 and 192). In the same year he published a booklet of 22 pages: De Hvdrostatica et de iis ouae ad Aouam Pertinent, which was practically a summary of Stevin's Hvdrostatica. Nor did his interest fade with the publication of these books. A 64

many references to previously published books,®’ detailed

description of experiments®® and constant consultations with

other contemporary scientists.®® In 1625 he had expressed

classical example is the insistence, during his visit to Italy, with which he enquired from Torricelli about his experiment of a paraboloid vase full of water where the level of water descended at a constant rate. Mersenne wanted to make of it a clock with a precision up to the seconds. Again he asked his correspondent Columbi, S. J., to perform more experiments with the syphon, declaring that he would not hesitate to change everything that he had printed that far, if need be.

®’Among the many authors that Mersenne has consulted one could mention Macrobius, Summum Scinionis; Dounot Didier (1574-1640) Confutation de 1'Hydrostatic ou Balance en 1'Eau. (Paris, 1615); Bettinus Aoiarius; R.P. Fournier Hvdroaraphia Française. Paris, 1637; Froidmont Metereologicorum Libri. Antwerpen 1627, Galileo Dialoahi. Bessones lacobus, Pallisius, Lydiaticum, Vitruvius, Barbarus, Huygens, his contemporary and friend Claude Giraud Cercles oui se Descrivent dans l'Eau. However, it is possible, as Hine states about Mersenne's quotations in the Ouaestiones Celeberrimae in Genesim. that sometimes the quotations are from secondary sources, without his having seen the original works (Hine, "Mersenne and Copernicanism," ISIS. 64 (1973): 20.

®®The following statement demonstrates how much Mersenne wanted to base his work only on seriously performed experiments, regardless of what other authorities may have found: "About the sea tides see Galileo and his controversy, since I had in mind to report only what I could prove with well established experiments." (Prop. 54, Monitum, p. 223). Among the experiments Mersenne performed on hydrostatics one can mention the eliopile to study the principle of perpetual motion, and the ripples in a water reservoir to study waves.

®®See Mersenne ' s frequent correspondence with Descartes about Mersenne's intention to consult other scientists on the various topics he was working on re.a.. about the one on raising water in pipes by means of pistons which were creating a space which needed to be filled because in nature non datur vacuum (Correspondance. vol. 9, p. 595)]; Huygens, about the same topic wrote to him: "It has been a long time since we have formulated a theory and practiced it." (Correspondance. vol. 9, 26 Aug. 1639); Jean Baptiste Van 65

in La Vérité his amazement at the energy that water can

produce if put to the service of man: "Hydraulics and

pneumatics together with static produce such prodigious

effects that it seems that nature can imitate the most wonderful works of God.

The text starts defining the technical terms most

frequently used. The definitions are short and clear. At the end of the section the reader is told that if other technical terms are used they will be first explained so that the reader will not be inconvenienced by the technicality of the terminology. Six postulates

Helmont from Brussels about the pressure at various levels of depth in liquid said: "Water seems to be heavier at a deeper level because of the pressure exerted on it perpendicularly by the amount of liquid above it" (see Correspondance vol. 3, 81; This position was contrary to that of Aristotle, who believed that elements do not weigh in proprio loco (in their own milieu). But the lateral pressure was ignored by all during this period, even by Galileo and Benedetti); Jean Rey from Bugue, about the status of water vapor when Mersenne seemed to be oscillating between the two proposed solutions: that vapor is water in gaseous form or that water is air (see Correspondance vol. 3, 189, 287). Der Haak in London was kept informed about Mersenne's experiments with water pumps; Daguin, a little known artisan, wrote to Mersenne asking him to calculate the amount of water pumped by Daguin's machine per minute if the daily amount is 2400x268.28 liters]. The interest of Mersenne in the topic may have resulted from his visit to Beeckman in Holland in 1630 and the unsuccessful efforts to improve the effects of the fountains of Paris, where his friends Descartes, Desargues, and Roberval were involved.

Mersenne, La Vérité des Sciences, introductory pages, near the end. 66

(Suppositions vel Postulatal spell out the assumptions used

in the book. They are:

I. The water is so fluid that it naturally tends to

gravitate to the center of the earth. (In Ars Naviaandi this

postulate read: The centers of the earth and of the water

coincide.)

II. The fluidity of the water forces it to spread in

eguipotential layers with respect to the center of

gravity* (donee illius partes undeouaoue ad punctum R

seu centrum aecmaliter accédant). (See the figure from

Mersenne's original book).

III. The two connected vases BK and FC cannot fill with

water unless it has come to a height greater than D;

*Robert Boyle, (1627 - 1684) in his Paradoxa Hvdrostatica refined this postulate as: All assignable equal portions of the upper layers will be equally pressed by the water perpendicularly incumbent thereon. (Postulate 2, 1). 67

IV. The waters of the biblical flood could not reach the

top of the mounts of Armenia or other mountain tops

without first covering the land below;

V. Water flows from higher pressure to lower areas, so

that in the graph the water in QT and VI are at the

same level, but if IV is kept always filled up to IV,

it can exert a force on QT because of the extra height

IS.*

The text is then developed in 54 propositions.

Right from the beginning the variables determining the study

of fluid dynamics (e.g., the mass, time, velocity and height of the water level in the pipes, pressure) as well as their mutual relationships are clearly identified. In fact, the work begins by showing that the mass of water flowing

from a pipe is proportional to the height of the water column,^ the cross section of the orifice, and the square of

“other works of Mersenne, like the Ars Naviaandi and Les Mechanicmes de Galilée also follow the same initial pattern: definition of terms and starting axioms. Mersenne was well aware how non-professional people were turned off by the technical terms. In the introduction to La Vérité he showed how reluctantly he had to use the appropriate terms but that they, in the long run, convey much better the concepts much better than a circumlocution, (see La Vérité introductory pages, towards the end.)

Castelli (Della Misura delle Acque Correnti Roma Stamperia Camerale, 1628) had said that the velocity of the water at the orifice was proportional to the height of the fluid in the pipe. In the Les Preludes de L'Harmonie Universelle Mersenne in 1636 described an experiment he had performed to link the time of exhaustion of a cylindrical container which he had previously marked with eight equally distant points. He used the oscillations of a pendulum to measure time. This experiment led him to conclude that the 68

the time; besides the relation between height and velocity

of flow is experimentally fixed as;

VoC i/h

as described by the graph.*

height of the water in the pipes is proportional not to the quantity of water but to the square of the quantity of drained water.

“The same result was reached independently by I. Beeckman in 1615 as described in Journal de Savants (see vol.1,58-59, Les Hague, 1939) and E. Torricelli in 1644 and announced in Le Qpere Geometriche. Mersenne's independence from Beeckman can be shown by his lack of knowledge of Flemish, in which the Journal was written, even though it is certain that Mersenne did see the notes of the Dutch scholar. Besides, given Mersenne's habit of rushing to publish whatever had the appearance of novelty, one cannot explain why he had to wait until 1644 to make it public, if it came from Beeckman's Journal. Instead, a letter from Descartes dated January 28, 1639, in which he thanked Mersenne for showing him the results of his experiment "that a pipe four times taller gives only twice as much water" (Correspondance. vol. 8, 300), shows the originality of Mersenne's experiment. [Later Descartes tried to repeat the experiment, but his ineptitude, as he himself acknowledged, prevented his reaching any satisfactory result (Correspondance vol. 13, 58-59).] Jan Baptiste Baliani from Genoa was also informed of the law by Mersenne himself four years before Torricelli published it. When Mersenne was able to obtain Torricelli's book, he wrote to him expressing his satisfaction at seeing the result that he had described in his book confirmed in Torricelli's work. However, he inquired from Torricelli whether he could prove that the droplet dropping in the pipe did in fact fall at the same speed that it did as it dropped freely outside the pipe: "It seems, in fact, that the upper water in A slows the flow of the rising water in B, while, the water in B seems to be forced at a faster speed than the initial point A whatever the inclination of the pipe. This is because of the flight from vacuum or because of the continuity [of the flow]. . . . I ask you the law by which one could find how much faster the droplet would flow through the length of the pipe than if the single droplet would fall in the free space, fibid. vol. 13, 334) However, Edme Mariotte's posthumous work Traite' du Movement des Eaux (1686) gives credit to Torricelli for the discovery of this law and does not mention Mersenne, with 69

(r

Mersenne goes on to study the relation between the height of the vertical column in the reservoir and the vertical and horizontal ranges of the jet of water flowing out from a pipe/* Later he studied the curve described by

whose work Mariotte was certainly familiar. Torricelli himself never disputed Mersenne's claim of independence of the discovery. Knowing that Torricelli had rushed to publish the book to claim priority over Roberval on the study of the cycloid, and his feelings against Mersenne personally, it is hard to believe that Torricelli would have not challenged an unjustified claim on the part of Mersenne.

“while working on the book, in 1634, Mersenne posed to Gassendi the following five questions, most probably to compare results rather than to look for a solution: I. Why the vertical thrust of water reaches 3/4 of the height of the pipe, whether the pipe is four feet high or one foot high only? II. Why the horizontal range does not keep the same proportion, but rather the horizontal range from a pipe one foot high is exactly half the distance of the horizontal range from a four feet high pipe, if the taps of both pipes are kept at the same level? III. Let the pipe be 24 ft and the orifice 1 line, the vertical thrust will be 18 ft due to the weight of 1 line of water, which weighs a few ounces. To reach the same height by the compression of air, as one does in the artificial water springs, one needs all the force of a man, which is incomparably larger than the force of the above mentioned weight. Why? 70

a droplet of water from the jet and its relation with

respect to time.* This gives him reason to introduce the

Archimedean screw, and to generalize his results to the

general motion of a projectile. Practical applications of

IV. Why the horizontal range of the same spring caused by the same force as in the above question goes much farther than the horizontal range caused by the height of a pipe 24 feet high? (Correspondance. vol. 5, 197).

“This section on the curves described by the droplet seems to be the real motivation behind the whole work, because he claimed that it was an original contribution, never attempted by anyone before him. The terms defined at the beginning of the book are used mainly in this section. In fact, they are explained again; the various steps are described in great detail; the book studies the different initial directions of the droplet as it leaves the orifice, and it considers the effect of the air resistance, both for the vertical and the horizontal components of the motion. The reader is given hints on how to repeat the experiments on a smaller scale and the suggestion to record the results on tables is made, giving several examples. Carefully drawn graphs of experiments performed by the author help the reader to understand more easily the description. All curves are drawn on a quadrant of the plane which is clearly explained. Mersenne, as Descartes had suggested, found that, neglecting air resistance, the curve described by the droplet leaving the orifice at 45 degrees above the horizontal, was a perfect parabola. If the initial direction is horizontal the curve is a half parabola. To prove it he followed exactly the same principle he had described in the introduction to Les Nouvelles Pensees; namely, the horizontal speed of the particle is kept constant in vacuum, while gravity acts on the vertical component of the velocity. If air resistance is considered, the curves can be drawn experimentally by finding the horizontal range at various heights. In any case, the droplet leaving the orifice at a vertical direction will never reach the level of the water in the pipe, due to air resistance. Dechamps de Bergerasc more than once objected to Mersenne's solution saying that the curve of the droplet with initial velocity at 45 degrees with the horizontal was a hyperbola and the one with horizontal initial velocity an ellipse (see Correspondance vol.12 and 13). 71

the theory are to natural and artificial water springs, to

pneumatic machines,®’ (in particular the syphon,“) and to the

principle of perpetual motion (which he discards as

impossible)

®’The 37th proposition of the Hvdraulica reads: To search the reason of the rise of water through a syphon or any other pneumatic instrument. And the solution is: "The parts of the whole are so coherent that they cannot fall off at a point without other parts immediately replacing theem; so that there cannot be empty spots, unless the surrounding area forces them in because there are no other places [where such empty spots] can take refuge, except in such spots. He who understands this process correctly, will be able to solve many difficult problems in the future." Therefore, Mersenne is applying the principle to all the hydraulic machines just as Pascal did later.

“of interest here is Mersenne's description of the reason why water rises in syphons, filters and in other pneumatic machines. Here, as in Questions Inouve. he explained it by invoking the homogeneity of the fluid: The parts of the mass are so attracted among themselves by the cohesive forces that if one of the parts were to fail to follow the next, the other parts would immediately take over, so that there is no room for air bubbles unless the surrounding air forces them in because of greater air pressure. If one understands this point well, he will solve many difficulties which otherwise would be impossible to solve. In his experiments he applied this principle to all the pneumatic machines, as Pascal did. Boyle in Paradoxa Hvdrostatica (Arnold Leers, Armsterdam, 1670, 162) followed the same explanation without recourse to fuaa vacui. which Descartes had claimed.

“Mersenne was aware of the experiments which were performed in many parts of Europe to study the problem of perpetual motion. In particular he himself experimented with the eliopile and in a letter to Theodore de Haak he described the studies and failure of an Italian in Paris who was toiling on the project for forty years (Correspondance. vol. 9, 306). Huygens, Descartes, and Comenius also reported about a machine in Amsterdam that was supposed to set a pendulum in perpetual oscillation. In the Questions theoloaioues (question 15) therefore he wrote that it would be better for the experimenters to spend their time on more profitable goals than wasting time on trying to create the perpetual motion. The reason is that one cannot expect a 72

Continuing the examination of theoretical

hydrodynamics, Mersenne describes the similarities and

differences between the flow of water in pipes and in rivers, the resultant motion of two different currents joining at a point, and Archimedes' buoyancy principle.'

Practical uses of these laws were applied to the construction of submarines,“ pneumatic air tubes and

machine to react with the same intensity as the first applied impulse, unless it is supported by an additional force. However, if the system on which a machine operates is taken to include the natural forces of a windmill at the sea shore, for example, or the tides of the sea then, he argued, perpetual motion is indeed possible. And he suggested the use of the tides to put machines in motion.

“Proposition 49 describes the "reason why a human body plunged in water at any depth does not feel the weight of the water." Mersenne, following Descartes' explanation, claimed that the water removed by the human body presses against the bottom surface of the sea, which reacts by rejecting the water against the body so that this upward force balances the pressure of the weight of the cylinder of water above the body. Boyle (Works of Robert Bovle. Appendix 2, 214) later corrected the mistake, but referred to Mersenne as "a well known writer, and, for what I know, the last one who dealt with Hydrostatics, except for Pascal (Blaise)."

“Because of Mersenne's unawareness of the weight of water, he neglects its effects on divers, while Boyle (Work of Robert Bovles. 423) quoting descriptions of travellers in the Americas, where the Spanish were forcing the Indians to fish for pearls for them, was well aware of the problems. However Fournier in his Hvdraqraohie Françoise wrote quoting from Mersenne's Phaenomena Hvdraulica on the submarine: "One could experiment and see if the air in the submarine could be renewed with the heliopile. This suggestion, like many others, was made by the Rev. Marin Mersenne, Minim, a man of pious conversation, and who is of such a rare experience that I hardly know anyone else like him. But his special characteristic is that he communicates his interests with frankness and clarity with anybody." (Reported in Correspondance. vol. 9, 55-56). 73

mattresses, instruments to measure the relative densities*

of liquids and the humidity in the atmosphere,®^ devices to

measure the depth of the oceans and lakes, ways of finding

water springs and water wells, and practical suggestions on

the construction of ponds and water reservoirs. The last

proposition investigates how to find the amount of

rainfall,®® from which Mersenne tried to calculate how long

it must rain for the earth to be submerged, e.g., in

reference to the biblical flood.

Some corollaries to the preceding proposition, and

some short monita help break the monotony of the presentation.®® The author also uses the monita* to justify

“For Mersenne, following Aristotle in De Coelo. book 4, ch. 4, "Water has no weight in water, just as air has no weight in air" (Nouvelles Pensees de Galilée. 69). He held this idea in spite of the fact that Gianbattista Benedetti and Galileo were of different opinion, and that his friend. Van Helmont from Bruxelles, had written to him: "Water seem to be heavier at lower depths because it supports the water perpendicularly above it. Therefore holes in a ship are more dangerous the deeper they are." (Correspondance. vol 3, 81).

“Boyle in the Paradoxa Hydrostatica later developed further the description of the instrument which he will call statical hvdroscope.

“Edme Mariotte (Traité du Mouvement des Eaux) made use of Mersenne's design of the hydrometer.

“some of the main corollaries and monita are the following: the Jus Acruae (how the waters of the lakes, and castles' reservoirs should be distributed to the civilian population); the different shapes of containers gives him a reason to recall briefly the properties of the conic; mixture of metals and how to find the proportions of the parts; eulogy to Galileo; perpetual motion; sea tides. 74

his digressions from the main dissertation, to draw the

reader's attention to particular situations (such as a

reference book). In this way he can introduce examples that

the reader can verify himself, or some historical notes on

the water clock of the Egyptians/’ For example, Mersenne

feels that he needs to introduce a section on the conics

because, first of all, not all the pipes and containers are

necessarily of cylindrical shape, but also because in the

*In Les Mechanicmes de Galilée the digressions are introduced as Additions and they are numbered (10 additions).

^’Mersenne was fascinated by the device, and even after the publication of the Cogitata he still tried to enhance its precision (see his letters to Torricelli during his visit to Italy Correspondance vol. 13, 548), using a paraboloid container devised by Torricelli, so that the upper level of the fluid will decrease at a constant rate (ibidem. 289, 310). It is also worth mentioning that Mersenne wrote an anonymous L'Usage de 1'Horologe Phvsicrue. whose authorship he revealed in several letters. He also made use of the oscillations of a pendulum to determine time; in fact he inspired Christiaan Huygens to devise the isochronous clock (see Correspondance. vol 2, nos. 206, 256;). When Perre Petit in a letter dated 1658 mentioned to Christiaan Huygens the law of the oscillations of the pendulum, he added: "as we know from the experiments that I performed several times with the late P. Mersenne and Gassendi." He was probably referring to the observations Mersenne had done around 1643 while working on the Cogitata (see Ouevres de Christiaan Huvgens. 1889, vol. 2, 255, 272). Finally it is worth mentioning that Theodore de Haak from London wrote to Mersenne: "If possible, I will try to learn how to construct this clock with a clog-wheel. If I succeed, I will certainly notify you." fCorrespondance. vol.15, 354-55). It is not clear at all what clock he was talking about, whether one similar to that Mersenne had constructed or something different. Unfortunately this is also the last letter between the two friends so we do not have any more elaboration about it. 75

pages that follow he intends to describe the shape of the

curve traced by a droplet flowing from a pipe kept at a

constant level of water.* The reader not interested in this

section could skip it without missing anything from the main

topic, but if he cared to read the details, he would find

the book easy to follow and even enjoyable. To make it

easier for the reader to follow his train of thought without

being forced to turn back and forth to the diagrams,

Mersenne repeated some diagrams several times. A minor

problem was presented by the overabundance of details on the

graphs, which could have been drawn separately, leaving the

main curves easier to understand. Also, in the description

of the graphs, sometimes it was not immediately clear to

which graph the author was referring. Otherwise, the book

was very easy to read and understand. Already a

contemporary, Godfroid de Brusselles, wrote to Gassendi:

"The books in the Cogitata Mathematics are excellent, pure

and proud results of the human mind."* Later, Robert Boyle*

^Writing about the parabola, Mersenne took the opportunity to correct some of his statements on parabolic mirrors and introduced some new concepts on the topic. He realized how the topic was out of place and excused himself to the reader for it. Also in Les Nouvelle Pensees he used the same tactic to supplement the information he missed in L'Harmonie Universelle.

*Mersenne, Correspondance. vol. 13, 215.

*The historians R. Rouse and R. Ince in their Historv of Hvdraulics (Iowa Institute of Hydraulic Research, SIU, 1957), 231, state that the world owes to Boyle the introduction of the word "hydraulics" into the scientific literature. We have seen, however, how Mersenne had used the 76

was one of those most interested in Mersenne's books. In

his works Boyle quoted from Les Nouvelles Pensees and

referred several times to the Harmonicorum Libri and the

Hvdraulica Phaenomena. Cavendish from Hamburg asked to

obtain a copy of the Cogitata. Torricelli was presented with a copy of the Cogitata after Mersenne's visit to Italy;

Cavallieri mentioned to Torricelli that he had read the section on fluid mechanics. In fact, Cavallieri was interested in Mersenne's books because the idea of the speculum ustorium (the burning reflector) that he had taken from Giambattista Porta and Nicolo' Zucchi and developed in his Specchio Ustorio was further improved and the possibility of its application to the telescope revised in the conic section of the Hvdraulica Phaenomena. While

Cavallieri discussed the device simply from an abstract and

term long before Boyle. A quick inspection to the general index of R. Boyle's Works by Thomas Birch shows how often Boyle referred to Mersenne's experiments, especially those referring to the weighing of air and of liquid. Here is a quote from a letter from Boyle to his sister that shows how his appreciation went as far as covering something that he saw as a flaw in one of Mersenne's book: "And as to that industrious benefactor to the experimental knowledge, the learned and pious Mersennus, his charity made him much more fearful to neglect the doing what good he could do to others, than to venture to lessen his reputation by an indecorum, that in a mathematical book, and in a chapter of arithmetical combinations, he brings in not only a remedy against the erysipelas, but even a medicine for corns, where he tells us, that they may be taken away, by applying and daily renewing for ten days, or for a fortnight, the middle stalk, the blade and the root (for that I suppose he means by the unusual word thallum) of garlic, bruised." (Vol. 2, 200) 77

theoretical point of view and was quite skeptical about its

possible realization,* Mersenne believed in it so much that

he discussed it in Harmonie Universelle. Hvdraulica

Phaenomena. the Harmonicorum Libri. and in his

correspondence with Descartes, Fabri de Peiresc and others.

Even though he was not able to complete his project, in a

marginal note on Harmonie Universelle he designed a frame

for the project which shows his conviction that it was

possible to produce it. His insistence paid off at long

last in the work of his countryman Cassegrain, the German

Herschler, the Scott James Gregory of Edinburgh (1639-1675),

[who described it in Qptica Promote (London 1663)], and

Isaac Newton of Cambridge [who at last succeeded in

constructing the first reflecting mirror].* Even though

Newton did not explicitly admit it, Mersenne, who was not an

unknown figure to Newton if he could write his name side by

*In the Specchio Ustorio. 77 he wrote: "I have taken the opportunity to mention this [idea] but only as something whimsical, to give satisfaction, in other words, to those frivolous people who crave for cake rather than for bread. For in my view they will never match the excellence of the refracting telescope either by combination of mirrors or by the addition of lenses as anyone who wishes to try will, I believe, find out." Reported by Piero E. Ariotti's "Bonaventura Cavallieri, Marin Mersenne, and the Reflecting Telescope," ISIS. 8 (1972): 303-321.

*It is worthwhile to recall that while Mersenne was soon forgotten in continental Europe after his death, in England, where there was a reaction against Galileo, Mersenne's presentation of the ideas of Galileo in the Les Mechanicmes de Galilée* and Les Nouvelles Pensees de Galilée were read and discussed in the public meetings of the British Royal Society. 78

side with Kepler,* was one of those giants "on whose

shoulders he stood and who enabled him to look further."*

“Writing to Hamington once (May 20, 1698, Correspondence of Isaac Newton, vol. 4, 474) Newton said: "... I presume that you have consulted Kepler, Mersenne and other writers on the construction of figures. ..."

*It is remarkable that the first two papers presented by Newton to the Royal Society in London are linked to topics that Mersenne had written about extensively. The first, presented in 1671, was on the reflecting telescope, mentioned above. The other paper dealt with the famous discovery on the composition of the white light. Communicating the discovery to the Society in 1672 Newton wrote: "The oddest, if not the most considerable detection, which had hitherto been made in the operation of nature." [Newton, Correspondence. vol.3, 234.] But the phenomenon was already described by Mersenne in his Questions theoloaioues of 1634 [295-28]: "Is light visible and distinct of the colors?" The answer was: "Colors are certainly distinct from light, whose rays take the color of the various surfaces, when they penetrate the object through the pores and are reflected back differently. One could say that surfaces do not have a color of their own. Their colors are the effect of the different reflexions and refractions of the rays that they absorb and emit: even though it is difficult to tell the number of rays or the intensity of light necessary to give a surface a white, red or any other color; or the number of times that a ray needs to be refracted or reflected back to obtain the color that one needs. This secret presupposes a very subtle and very exact speculation of the surface, as well as the size of the each pore on the surface of the body. One could in fact imagine the pores like small concave mirrors, each of different shape from the others. Now, the prism, or the crystal triangle, or the glasses cut in various shape could help to overcome this difficulty, because their refractions, which produce the blue, green, yellow and many other colors could be measured: One could thus find how much of the light is lost by the various rays in a diaphane object, even though it would not still be difficult to apply this to find the amount of light absorbed in the pores of an opaque body." (Emphasis supplied). The same experience is stressed again in the question 42: "Is white better than the other colors?" to which he answered: " White could be said to be a dim light, or light at its start because it is closer to it than to any other color. In fact its impression on the eye is dazzled by its intensity and because it is composed of all the other 79

Notwithstanding the many editorial problems, his

books were generally well accepted by the readers and were

in great demand. Some of the books, in fact, were looked

colors. Take a ray of sun, for example, which represents the principal. When it transverses a triangle, or the prism of crystal or of glass or when it is refracted (literally bended) in a diaphane body." Yet in both cases Mersenne is not mentioned neither by Newton nor by Newton's contestants, in particular Mersenne's pupil Huygens. That the white light refracted through a prism gives origin to the spectrum of color was well known from antiquity. Kuhn recorded the observation, (see Thomas S. Kuhn "Newton Optical Papers" in Selected Papers and Letters of Newton, edited by Birch, T.) "Before Newton began his experiments at least four natural philosophers, Descartes, Marcus Marci, Boyle, and Grimaldi had discussed in optical treatises the colored iris produced by a prism, and Hooke had based much of his theory of light upon the colors generated by a single refraction of sunlight at an air-water interface. What made Newton's experiment stand out from the previous ones is that he performed the experiment in a dark room, where a ray of sunlight was made to pass through two holes on the parallel surfaces of a cylinder before it was refracted by the prism. In this way the spectrum could be projected on a screen placed at any distance from the prism, or the white light could be recovered through a convergent lens. Actually, Mersenne did not describe how he performed his experiment and there is no reason to give him more credit than he deserves. However it is clear that he knew that the white light was the composition of all the colors of the iris, and not just the two primary colors red and blue (see Kuhn, Ibid.) and that colored light was the effect of the different refractions of the white light, and in opaque bodies, the effect of absorption and reflection of the white light. Mersenne's contributions to astronomy are described by Pierre Humbert's article "Mersenne et les astronomes de son temps," Revue d'Histoire des Sciences et Leur Applications 2 (1948): 29-32, which is an issue totally dedicated to Mersenne in celebration of the 400th anniversary of his birth. Even though Mersenne's personal discoveries in astronomy are very few, he is behind many discoveries made by the French during this period. Above all, he was always one of the first to be informed about any discovery or improvements of the telescope. However the article fails to mention our Minim's project of the reflected telescope and his book on astronomy: Cosmographie Celestial. Paris: 1644. 80

upon as precious treasures. His confrere Thibaut once wrote

to him:

If there is anything I have learned in mathematics, I have learned most of it from those books of yours which I could obtain. Besides, Your Paternity has taken the pain to satisfy all my requests, for which I hereby expresses my gratitude.*

Fermat bound Mersenne's books together to make them easier to manage.* Fr. Kircher, S.J., who was teaching mathematics at the Colleoio Romano and was preparing his own book of music, read Mersenne's L'Harmonie Universelle in four days - which was not a small task given the size of the

*Mersenne, Correspondance. vol. XV, 75. The medical doctor De Villers is even more emphatic in his appreciation. About the Preludes de l'Harmonie Universelle he wrote: "You sent me the first parts of your book to hear my criticism. But I cannot find anything that cannot satisfy the most critical minds in this subject. There is nothing else for me to do but praise your enterprise for working on a subject that very few are able to appreciate, in fact many hate the practice it requires as well as the theory, which is subject to the humiliations of those who practice it. Your effort is even more praiseworthy because you have researched the hard core of harmony; you have given recognition and respectability to a little known and misinterpreted subject. In brief, of a servant of servants you created a bride's honor girl, if not a princess itself." (Correspondance. vol. 4, 120). De Villers is not afraid to give Mersenne practical advice and express his divergent opinions. His appreciation, therefore, is more reliable. Viviani kept two personal copies of Les Nouvelles Pensees which he annotated in several places.

*See Correspondance. vol. 6, 191-197. 81

book - and was greatly impressed. I. Beeckman often quoted

from Traite» de l'Harmonie Universelle, as did Fermat.“

There are many reasons for this appeal to the

reader. Unlike F. Bacon or other contemporary writers, who

did not have confidence in the literary strength of their mother tongue, Mersenne often published in French; his books were often addressed to scientists as well as to priests, artisans, philosophers, etc. This secured him a wider range of readers. The style was direct, sometimes in the form of a dialogue, and always simple.* At times, when he had to

“See Correspondance vol. 9, 133; vol. 11, 135; Reineri testified that Mersenne's La Vérité des Sciences was completely sold out in Leyden, and about the books on music and the miscellaneae ouaestiones he wrote to Mersenne hie in oretio sunt (are in high demand here) Correspondance. vol. 7, 116; see also letter of Huygens to Renieri of 16 and 26 September, 1637, in De Briefwisselino van C. Huvoens. (Worp ed., vol. 2 1913), 306, 316.

“Lenoble himself asserted about Mersenne's reports on his experiments: "Some pages of the Cogitata and Tomus III are a model of an experimentalist consciousness, and deserve a place in the anthologies besides the most well-known pages of Lavoisier or Pasteur." (Lenoble, Mersenne. ou la Naissance du Mécanisme. 389-390) A. Beaulieu, the editor of Volumes 14, 15 and 16 of Mersenne's correspondence, describes Mersenne's style of writing as follows: "Merenne's spelling, which is the XVIIth century spelling, of course, is correct (except for the proper names). This is not true about his correspondents, who are rather fancy in their spelling. This is the case, for example of some Minime, probably of very modest level of education, who seem to ignore the very elementary rules of the French grammar. Among the lay people, Pierre Desnoyers, secretary of Queen Marie Louise in Warsaw, is better, but seems to neglect all the rules of the times. In comparison with others, Mersenne's French is correct, lively and colorful; his Latin, as well as that of his correspondents', is excellent, at times maybe too fancy; the Greek is used only when the word, or group of words, seems more forceful than the Latin and the French; he uses only the biblical Hebrew for a more 82 use a more technical vocabulary he apologized to the readers. He tried to explain the advantages of an appropriate vocabulary, but always defined clearly the new terms before using them, as in the case of the above mentioned Hvdraulica. Finally, his ability to appreciate the contribution of the skilled artisan as well as the professional theoretician helped to captivate the sympathy of the readers.

When writing, Mersenne never seems to forget his main goals: to serve the sciences* and to instruct the reader while amusing him.“ Here and there he inserted a humorous line, like the following about the alchemists' pretense to be able to change everything into anything they wanted: "Oh the wonderful alchemy1 It can produce anything from nothing when the alchemists are able to produce nothing from everything!

exact translation of a word or a quotation." It seems only reasonable to apply these observations to his printed works also.

“see introduction to Les Nouvelles Penses de Galilée, non-numbered pages.

“e .o .. the subtitle to the Questions Theoloaiques reads: ou' chacun trovera' du contentement et de l'exercise.

“m . Mersenne, Ouaestiones coeleberrimae in Genesim. col. 1483. In La Vérité des Sciences. 78, he again said: "We then see the blowers, who after having glued, calcinated, burned and incinerated every element, become so poor that are forced to beg for their bread, if it were not for some dupe, whom they are able to convince that in a short time they will show him wonders. But only after having relieved him of all his belongings!" 83

The readers soon found that the best way to read

Mersenne's books was first to make the necessary corrections

in the text using the list in the Errata - Corrige, or, as

Mersenne called them, in the Menda. It was important never

to skip the introductory pages,* where the author gave

further details of the changes incurred during the printing

of the book. In fact Mersenne used to send copies of parts

of the book to some of his closest friends. He then tried

to include the reaction of the readers in the introduction,

or should we say, introductions, which was written only when

the printing of the book was completed. This is how Claude

Richard, S.J., (1588 -1664), from Madrid, would start

reading Mersenne's books as he explained in his letter to

him;

I treasure very much your books, but before I start reading them I first do all the corrections that are listed in the errata-corrige. but there are several others which the index does not indicate, and some of them quite noticeable, not only for the misprints but in the content itself. Verdussen attributes all the faults to the proof-readers.*

How much Mersenne was appreciated by many scholars

is demonstrated also by the strong support he was given when the British Robert Fludd attacked him in his writings.

“This is Mersenne's advice (e.g. see Praefatio Praefationum to his Cogitata); because this was the last part he sent to the publisher, he used it to alert the reader to the corrections that escaped during the proof­ reading and to the new information he had gathered since the inception of the printing of the book.

57Correspondance. vol. 13, 120. 84

Gassendi, Francis La Noue, and Gabriel Naude wrote several tracts in his defense.

Mersenne was also quoted in many contemporary books. His biographer. De La Coste, counted about forty books where Mersenne was named or quoted. Among them

Hevelius, the publisher and astronomer of Dantzig, in

Selenoqraphia wrote; As much as the convex mirror is smaller and the object is farther, so much the image is smaller

(First theorem of the catoptric in Synopsis, part II, p.505). At the same time Mersenne is quoted about ordinary objects, as well as about shade.* Kircher, in his Maanes reported, with reservations, the experiment that Mersenne refers to in his Ballistica about the magnet dropped in cold water. Dochuz assured Mersenne that it would loose its magnetic properties that, he claimed, it would recover if the water was warmed up.*

P. Paolo Casati (1617-1707) in Terra Machinis

Mota, has three interlocutors; one is Galileo, another is

Mersenne, and the third is Guldi.* Fr. Grégoire de St.

Vincent, S.J., asked him to write a presentation of his new

“ibid.. vol. 15, 306.

“a . Kircher, Maanes. sive de Arte Magnetica (Rome; 1641), 115; other quotations can be found on pp. 444, 469, 457.

*P. Duhem, L'Origines de la Statioue. vol. 2, 225. 85

publication De Quadratura Circuli et Hyperbole;®’ the

theologian of the Sorbonne, Morin, had shown to Mersenne and

Gassendi a sketch of his forthcoming publication. Both

scholars wrote to him encouraging him to pursue his work.

When the book was ready, he asked them for permission to

include their letters in the introduction of the book.

Realizing that the positions of Morin were not

scientifically tenable and theologically sound, both

scientists denied him the request, which Morin ignored. In

1767 Blasius Ugolini included Mersenne's Ouaestiones

Celeberrimae in Genesim in his Thesaurus

Anticfuitatum. *

The educational value of Mersenne's works was of

decisive importance: Mersenne's example and encouragement

helped scholars who were reluctant to send their manuscripts

“it is to be remembered that Mersenne in the Universae Geometriae Svnopsis was the first to state correctly that the quadrature of the circle is impossible.

*B. Ugolini, Thesaurus Antiauitatum Sacrarum Complectens Selectissima Clarissimorum Virorum Qpuscula. in cfuibus Veterum Hebraeorum Mores. Instituta. Ritus Sacri et Civiles Illustrantur O p u s . Venice: Herts-Colletti, 1767. Mersenne's work is included in vol. 32, 497-554 with the title Ouaestiones et Commentarium in Genesim (497-554) et De Musica Haebreorum (515-554). 86

to the press.* After meeting Mersenne in Rome, Maggiotti

wrote to Torricelli: "In France, mathematicians are well

organized, meet often, exchange ideas and have new publications appear every day.What a difference from the grim description by Melanchton of the French situation reported in the previous chapter1 Mersenne's role in this transformation was of capital importance.

Mersenne's experience gave confidence to perspective authors to seek his advice. Many would notify him about their intentions to publish some results of their research; others would share their fears of the possible reactions of the readers.* He was generous in giving

“To induce scholars to send their manuscripts to the press Mersenne used some stratagems such as announcing in his own books or letters the writer's intentions, quoting some results from their manuscripts, asking others to solicit their friends to publish, and helping the authors get the necessary privileoium from the King and the theologians of the Sorbonne.

“Mersenne's impression of the cooperation among the Italians was quite different. Writing from Rome to Isaac Bouliard, he said: "The Italians are so reserved that it is impossible to see anything of their work. They are not like you, the honor of the astronomers, that communicate your findings before and after the publications of your work!" (Correspondance vol. 13, 315).

“Doni, sending to Mersenne a complimentary copy of his recently published book on music, wrote: "I will consider that the troubles I had to go through in preparing this booklet which I am sending will have been worthwhile, if it will deserve your acceptance. Your sharp judgement, supported by your very rare education in this field, will weigh for me more than the statements of the other learned people. . . . Please accept it . . . and defend me with the charity proper of your Institute, against the malignity of those who assert that my findings are pure imaginations and fruitless work, or not to the level I had when I was there. 87

advice about the content, style, or editorial problems to

anyone who requested help. To Rivet, his Protestant friend

and lecturer at Leyden - who was about to give a written

response to some attacks he had received from a Catholic writer - he suggested: "If your book against Grotius is not yet published, try to avoid any offensive language, and do everything with as much kindness and moderation as you can.

You will get much more profit and honor."* On the other hand, to the minister Daillet from Paris he suggested:

I expect that you and Mr. des Marrets will not let the calumny and injuries of this minister of Satan pass without punishment. He does not deserve anything better, and you have better give him what he deserves without damaging him.*

His advice was appreciated and considered seriously. The Polish educator and reformer, Komensky, in a letter to Hartlib wrote: "In fact our wise Mersenne advised that he really doubts whether it was worth putting so much effort to write in Latin when we could present more easily the same work in a new language, with even better results."*

When Theodore der Haak told him about the project in Pell of setting up a library of mathematics, Mersenne foresaw the difficulties of the project's success and suggested a more

. ." (Correspondance. vol. 15, 329)

“Mersenne, Correspondance. vol. 15, 193.

*Ibid.. vol 11, 162.

“Reported in Correspondance. vol. 10, 322. 88

practical solution with more immediate results. He wrote:

"His plans are praiseworthy. But rather than putting

together a great collection of all those who wrote about

mathematics, as he is proposing, it would be more valuable

to publish the works of a dozen of the best in each section

of mathematics."®® He went on to suggest that after having

reproduced the ancient authors whose textbooks we have, such

as Euclid, Apollonius, Archimedes, Theodore, Pappus,

Ptolemy, etc., together with their not-yet published

manuscripts, (Golius has some in Leyden, and others are in

Rome), the best of the moderns could be added, e.g., Viète

for analysis, Clavius for his four or five books, and

Herigone, who has just finished his five volumes in Latin

and French. The same suggestion is made to Komensky.

Finally, Mersenne reviewed every type of book he could put

his hands on.

More than giving simple advice, he also went so

far as to edit the works of his contemporaries: Mydorge's

De Conicis. Desargues's Un Methods Aisee pour Apprendre et

Enseigner a* Lire et Escrire la Musicnie. Roberval's

Aristarcus Sami/* Hobbes' Gives. G. Naude's De Magnate, and

®®Ibid. vol. 8, 580.

^This book was printed twice in one year, 1644: once at the printery of Gulliem, and then, perhaps after Roberval insisted on introducing some changes in the work by the Bertier's. In 1647 the book was already out of print, so Mersenne printed it again in his Tomus III after Roberval had a chance to incorporate more changes. To fully appreciate Mersenne's efforts to edit the best works of 89

La Mothe Le Vayer's Discours Sceptique sur la Musique.

Morever, he offered to edit the works of several others. He

wrote to Galileo about his Discorso soora il flusso e

riflusso del mare (which was later incorporated into

Galileo's masterpiece Dialpgo):

We have come to know that you have achieved (described) a new system about the movement of the earth, which, however, you are not able to publish because of the prohibition of the Inquisition; if you want to leave it to us, and send a copy of it to us by any means you choose, we would take the risk of publishing it, to the extent that you wish.^'

mathematics, it is worth describing the difficulties he had to face to reach his goal. The books of Viete present one such example.

"Mersenne, Correspondance. vol 2, 173-76. Had Galileo heeded Mersenne's offer, he probably might have been spared the bitter consequences that the publication of the Dialoqo brought to him. However, as soon as Mersenne was able to obtain a copy of the book through the mediation of Gassendi he incorporated the laws of Galileo into his Traite* des mouvements et de la chute de corps pesans and in the Hvdraulvcis et Pneumaticis Phaenomena. In the Introduction to the Les Mechaniques de Galilée, he pleaded with scholars who corresponded with Galileo to prod him to publish all his notes. Unfortunately the Italian school of Galileo misinterpreted his interest in publishing Galileo's papers. They thought that Mersenne was using Galileo's fame to his own advantage, especially because Mersenne's rendition of Galileo was never a simple translation but was always accompanied by comments, corrections and additions. Raffaello Magiotti on 25 April 1637 wrote to Galileo: "This friar [Mersenne] publishes thick and many books, trying to ruin the reputation of others in order to create one for himself. And he might succeed with the mob. The works of his that I have been able to see are mainly in French. I am sorry I do not own them; otherwise I would have sent them to you, so that at the appropriate time and place you could reach him with a whip . . . "(Correspondance vol. 6, 241.) And he wrote again in the same terms to a mathematician in Florence, Famiano Michelini, the very same day. A month later he again mentioned Mersenne to Galileo in the same 90

The same offer is repeatedly presented to

Torricelli, who could not afford to publish his studies; to

the son of Kepler about his father's tables; and to Fermat.

Morever, he translated several works of ancient Greek and medieval mathematicians. For these services, Mersenne can truly be called an educator of scientists.

way. By 1647 this animosity against Mersenne in Italy was not yet overcome. Oratio Grassi, S.J., wrote to Baliani about the experiment of the barometer, telling him how already seven years earlier his confrere Honore Fabri had performed the experiment. Grassi concluded: "I thought you would like to know this... and you can see by yourself how the good P. Mersenne takes advantage of others' discoveries..." (Correspondance. vol. 15, 594). They tried, therefore, to hide from Mersenne all the manuscripts or information about Galileo. But Mersenne had contacts and friends everywhere! The publisher of Amsterdam Elzevir, on his way from Venice, where he had just signed a contract with Galileo to print his Discorso intorno a le Nuove Scienze. spent the winter 1636-1637 in Paris. Mersenne had all this time to copy the manuscript and start its translation! (see also Maarten van den Hove's [Hortensius in Latin] apologies to Diodati, dutifully reported to Galileo, for inadvertently letting news about Galileo's publications filter to Mersenne through Beeckman in Correspondance. vol. 6, 326.) This suspicious atmosphere, coupled with Mersenne's difficult handwriting, explains the silence of Galileo in response to Mersenne's letters, even when Diodati from Paris informed him about Mersenne's intentions to defend Galileo in writing, and when Mersenne sent him copies of Descartes' Discourse de la Methods. The same cool, even if interested and always polite, relations continued with Torricelli and his students Ricci and Maggiotti during Mersenne's visit to Italy. However, if one realizes that when publishing Les Nouvelles Pensees. to avoid problems with the Inquisition Mersenne had to hide his own name, it becomes evident that the Italians were ignoring the fact that Mersenne was doing a service to Galileo at his own risk. And we can even say that probably very few in Europe, and certainly no one in France did more than Mersenne in defense of Galileo, and still managed to avoid irritating the Roman Inquisition, at the same time. 91

Correspondence

If Mersenne's books gave an impetus to the development of science in the XVIIth century, his correspondence has played a much more important role up to the present. In fact one could safely say that the details of the history of scientific thought in almost every field of the seventeenth century cannot be fully grasped without taking into account the correspondence of the monk of the

Palace Royale.”

The new era introduced by the printing press had not - nor has yet - superseded the service rendered by the exchange of information through correspondence. Mersenne's monastic room was therefore converted into a workshop where the key books and the new ideas of French thought of this century were sometimes conceived, often developed, and almost always analyzed and reviewed in the worldwide correspondence he received or sent.” This is true of

”See R. Taton, "Role et Importance de Correspondances de XVII siecle," Revue de Svnthese 81-82 (1976) vol.96: 137- 45.

^Letter writing was taught as part of rhetoric already in the schools of Greece and Rome, and it continued throughout the centuries in Europe until the seventeenth century. The style changed throughout the centuries. Erasmus of Rotterdam, rejecting the principles of the Middle Ages, laid down the method which dominated during the Renaissance period, which was the style that Mersenne followed, (see W. Voise', "L'Art Epistolaire, son Passe', Son Avenoir" in Revue de Svnthese) . Hilaion de Coste described Mersenne's room as a "whirpool of all the studies" (La Vie du R. P. Marin Mersenne. 59); Thomas Hobbes in his autobiography 92

Fermat's geometrical” and combinatorial” theorems;

Desargues' principles of projective geometry;” Descartes'

analytical geometry;” Roberval's, Fermat's and Descartes's

calculus of the infinitesimals; Gregory of St. Vincent's

quadrature of the circle;” and of the Menelaus' and

Archimedes' helices.” This list limits itself to a few points in mathematics, excluding the whole development of number theory and the study of the cycloid which will be the topic of the next chapter.

wrote: "If anyone discovered an important porism it was to Mersenne he brought it, and in his own clear style, stripped of all figures of rhetoric, sententiousness, show or subterfuge, he discussed with the learned the problems they brought to him so that they could weigh his answers on the spot or carry them home with them for further thought. Out of the many discoveries he would publish the best, marking each with its author's name. Mersenne was the Pole around which resolved every star in the world of science." Autobiography. 141. According to Anthony Woods, Hobbes, during his stay in Paris, communicated with Mersenne about once or twice a week fAthenae Oxonienses vol.3: 1207).

”See for example Correspondance. vol. 9, 145-151; vol. 10, 653-661.

”See ibid.. vol 6, 195; vol. 4, 197.

”See ibid.. vol. 9, 249; vol. 10, 436, 520; R. Taton, Desarcfues (Paris: Centre Nationale de Recherche Scientifique, 1985), 26.

”See Correspondance. vol. 4, 436, 478; vol. 6, 325, 345, 553.

”See, for example, vol. 6, 295-298; vol 7, 408, 417, 454; vol. 10, 633, 705.

79 Ibid.. vol.9, 93; vol. 10, 662-671. 93

It is not so much the contributions of the best scholars of the century but that of the army of lower caliber amateur scientists that gives relevance to

Mersenne's correspondence. Their reactions, sometimes even resistance to and rejection of new ideas, forced these researchers to rework their proofs and reword their concepts in clearer and more exact terms before they committed their studies to the press and even afterward.®® This painstaking process helped prepare the scientific background for the qualitative leaps that the ideas of Copernicus, Kepler,

Galileo, and later of Newton, represent in the philosophy of science. Since at that time a mathematician was also a theologian and philosopher, as well as a physicist and chemist, the gains in one field of science were immediately reflected in these other fields.

It is instructive, for example, to follow the development of the experiment, proposed by Descartes and performed by a friend of Mersenne,®’ which should have answered the question of what happens to a bullet fired by a

®°A typical example is presented by the objections that Mersenne passed on to Descartes in reaction to his Discourse on the Methode. or the Dioptriaue and Descartes' replies [see Correspondance vol. 6, 247-262, 316-319; vol. 10, 427, 431 note, 434, 487, 522, 541, 568 ff.]

®’Most probably the military engineer Perrier, who is known to have worked many experiments, including the famous barometric experiments, with Mersenne. 94

rifle in a perfectly vertical direction.®® Actually the

question was not a new one; Descartes found it described in

the Recreations des Savants and Beeckman in 1630 had noted

in his private journal;

It is said that balls fired from a cannon directed perpendicularly upward, have never been retrieved. If this is true, then the attractive action of the earth is not a reality, and what is initially set in motion, moves on for ever.”

The results would have solved the questions of the

attraction and the rotation of the earth, which were the

8 2See, Correspondance. vol. 3, 98.

“l. Beeckman, Journal. vol. 3, 171-172. This is how Mersenne described the experience to his close friend Gassendi: "An arquebuse and a stone bow were tied to a perpendicular pole, 30 or 40 men were dispersed in a large area to find where the projectiles of lead would fall. After firing the pieces of artillery though, even though we had a net on the larger ditches of the castle, no one could find the returning bullet. One must necessarily conclude that the winds of the middle region have swept them away, or that they melt on the air, or finally that they float in the atmosphere. I rather prefer the first alternative, otherwise, in the other cases I should come up with an explanation." I Correspondance. vol.4, 315). A modern experimenter might be tempted to smile at Mersenne's result; however more than a century later, Moustier repeated the experiment with the same result as at Mersenne's first attempt. At a second attempt the bullet fell 585 meters away from the place where it was launched and 715 meters away in the third attempt, rather than the 20 meters away from where it should have fallen, according to Bezenberg's calculations. D'Alembert, ignoring the later successful experiment of Mersenne, concluded: "Either Mersenne and M. Petit did not search carefully enough the whole area for the bullet, which could have dug a hole on the ground and was buried in it, or - which is equally possible - they might have searched it too close to the place of launch, in the mistaken belief that it should fall within a given limit." 95

outstanding problems of science at that time. Descartes,“

Rey“ and Van Helmont” did not admit any gravitational force

from the center of the earth and Mersenne was skeptical

about the Copernican and Galilean notion of the rotation of

the earth.

When Mersenne reported that he had performed the

experiment with an arquebuse, and had failed to find the

fired bullet, Descartes, in a letter dated May 15, 1634, objected that the experiment should have been performed with a 30 or 40 pounds heavier cannon ball. If the disappearance of the bullet was because it had melted in the atmosphere, the larger mass would require higher temperature to melt; but if it did come back, the larger volume would make it easier to locate.” Gassendi and Cornu were involved in the topic, each coming up with his own interpretation of the results. Of particular interest was Fabri de Peiresc's suggestion to tie to the tail of the ball a flaming branch so that the trace of smoke could help one follow the trajectory described. Repeated experiments allowed Mersenne

“see Correspondance. vol. 3, 212.

“See ibid. vol. 3, 238-39.

“ibid, 32: "The falling bodies are not attracted by virtue of the center... In the same way, our hands tend to fall on their own, otherwise we would feel the attraction."

”lbid.. vol. 3, 141-42. 96

finally, on May 31, 1636,“ to retrieve the bullets at some

distance from the spot where they were fired. The

experiment must have laid a basis for a legend of scientific

credulity, because in 1690, more than forty years after

Mersenne's death, Varignon, on the cover of his Nouvelles

conjectures de la Pesanteur. [Paris: J. Vrin, 1690, 135]

printed the scene of a French military engineer operating a

still-smoking cannon pointed vertically upward, with a monk

on the opposite side of the cannon. Both men were watching

a bullet flying rapidly skyward. The caption read:

Retournet-t-il ? Below the cartoon Descartes' letter of 13

July 1638, in which the experiment is mentioned, was quoted.

The cartoon was again used in an 1890 publication.”

Actually the experiment attracted the attention of very

prominent scientists even decades after the original

correspondents died. The French physicists d'Alembert in

1774, Laplace in 1778, Poisson in 1734, and Benzenberg in

1804 studied the problem in detail” .

86See Correspondance. vol. 5, 586, letter to Doni.

“ Camille Flammarion, Astronomie Populaire (Paris: 1890), 76-77 and the caption reads: "Since the bullet did not fall on their heads, they thought they could assume that the bullet was lost in the atmosphere, where certainly it did remain for a long time."

“For more information on the literature about the problem of a projectile shot vertically see, for example. Correspondance. vol. 4, Appendix 4, 618-22. 97

But the experiment suggested to the physician

Villiers, when he read Mersenne*s letter on the problem, to

perform a related experiment to test the rotation of the

earth. He threw stones with a stone-bow toward the four

cardinal points and measured the distances. He expected

that if the earth rotated towards the east, the distance

travelled in the western direction would have been shorter than the one toward the east. To his amazement he found that the distance was independent of the direction.

Furthermore, since he performed the experiment from a boat, he found that the boat reacted always in a direction opposite to that of the thrown piece. He wondered what the reaction of the boat would be if he pointed the bow perpendicularly upward.

About this time Mersenne was at work on the

Harmonie Universelle, his most important work, together with the Latin version of it. In the second book of the work, about motion, Mersenne proposed: "Determine whether a bullet fired horizontally from the height of a tower reaches the ground at the same time as a second bullet falling directly downward from the same height."®’ Descartes and Mersenne expected that the bullet shot horizontally would take more time to drop to the ground. Their reasons were: a stone.

®’M. Mersenne, Harmonie Universelle, book 2, prop. 22, 155-156; the same question is posed also in the Harmonicorum Libri XII. towards the end of the introductory pages. 98

thrown perpendicularly upward by someone riding a horse or

on a postal cart, always falls on the hand of the rider no matter what his speed, his direction of motion, or the height of the stone;” and it takes exactly the same amount of time that it would take if the experiment were performed by a person at rest. Hence the time needed for the stone in the upward motion is the same as in the downward motion, and therefore the two.stones in the experiment should reach the ground at the same time. Repeating the experiment with arrows shot, using different intensity of forces, Mersenne found that the ones that fell further away from the foot of the tower generally took a longer time than the ones falling in a perpendicular direction. Mersenne, therefore, advanced the conjecture that the arrow or stone shot horizontally started falling only after it has gone an initial horizontal distance, thus explaining the longer time it takes to reach the ground.”

Villier, Cornu, Debeaume, and many other scholars living in small towns in France depended almost exclusively on Mersenne's letters to keep them updated on the latest scientific discoveries and topics of investigation by the

Academia Parisiensis or by individual scientists. It is informative to read in their letters how much they expected

”M. Mersenne, Les Nouvelles Pensees de Galilée. Introduction, non-numbered page.

”lbid.. 156. 99

mail from Mersenne. Mersenne, in his letters, would suggest

useful reading material, propose new problems that he

himself or others had thought about, give step by step

information on how to repeat some experiments, and assign to

his correspondents a role in some aspects of the researches

that were conducted in Paris.”

On his part, he was equally eager to learn from

every one of his correspondents. He was not embarrassed to

ask questions of anybody. Often his letters were simply a

long list of question after question. This characterized

him as; "The most curious person and of undigested knowledge."” At times the correspondents let their annoyed

feelings be known. Descartes would complain if Mersenne let more than two or three weeks pass without writing. Even though he would reply regularly in one letter - sometimes

“The case of Giovanni Spinula from Bologna may illustrate the point. In a letter he wrote to the French Minim, he pleaded: "I plead you Paternity to tell me which book in algebra you could find there (in Paris) for me. The book should be very simple and easy, even if in French, which I understand well enough. In particular I would appreciate if you could point out who has so far written on the "algebra speciosa" or the algebra of Viete in a clear and well programmed method. . . Furthermore, please propose to me a syllabus or a set of characteristic problems on each section of algebra. The problems should be challenging enough and whose solutions are not yet found, but at the same time they should be useful and interesting." (Correspondance vol. 15, 1647).

“Mersenne, Correspondance vol. 8, 486. 100

thirty or more pages long - to two or three letters from

Mersenne - he would request Mersenne to write regularly.”

Mersenne's letters have always represented a very

important point of reference for historians of the seventeenth century. During his lifetime they were used already in scientific circles to claim the priority of certain observations or demonstrations,” or to show the positions of various scholars on certain topics; they

“After Mersenne's return from Italy, he brought to France the great discovery by Torricelli of the vacuum created by a column of mercury in a tube. Mersenne had written about it often but he never had mentioned it to Descartes. However, about four years afterward, he asked Descartes to write about the theory behind the experiment. Descartes, who had just recently come to know about the phenomenon, wrote back; "You ask me to write about the experiment of the mercury, when you never mentioned it before, just as if I was supposed to guess it. This puts me in a very awkward position, because if my writing fails to explain the real reason of the phenomenon, I will be ridiculed as if I had done the experiment and failed to obtain the correct result; and if I do not write about it I will lose the respect of the scholars. So, please, tell me in writing all you have observed in your experiments and if I will make use of it I will feel obliged to give you the credit that you deserve. . . . " Then before closing the letter, he once again expressed his bitterness with Mersenne's silence: "I am amazed that you and Mr. Pascal have kept the experiment hidden from me for four years!" I Correspondance. vol. 15, 132) .

®^The scholar Giovanni Battista Baliani, governor of Saverno, introduced the second edition of the De Motu Gravium Solidorum with the letter where Mersenne was giving him full credit for his priority on the theory on the falling bodies, even before Galileo (Correspondance vol 14, 504-505). Similarly, the letter of Cavallieri to Mersenne in March 1646 about the Pell's Law was printed in Controversia de Vera Circuli Mensura Anno DCXLIV Exortae inter Christianum Severini Loaomontanum Cinbum. Pars Prima. Amsterdolami. Apud lohannem Blove. MDCXLII. 101

represented valuable sources of authoritative information.

In fact, scientists sometimes requested copies of his

correspondence; at other times Mersenne offered them to some

scientist to provide him more information on a certain

topic. This was the case, for example, when Rivet was preparing the Geooraphia Sacra; Mersenne sent him a "batch

of letters," which unfortunately were never returned and were apparently lost. The same happened with the letters sent to Hovelius to help with his study on the

Selenographia.

In 1653 Claude Clerselier (1614 - 1684), at the suggestion of Pierre Chanut who had inherited Descartes' papers, published the correspondence of Descartes. The letters to Mersenne represented the most important and most numerous set in the publication. R. Boyle in London was aware of this correspondence and quoted from it in his books. A. Beaulieu, the editor of the last three volumes of

Mersenne's correspondence, counted at least 35 sources where some of the letters to or from Mersenne were reprinted during the centuries, and these did not include quotations from the correspondence. In fact, it is ironic that

Mersenne's correspondence helped in the study of the intellectual activity of the correspondents before it was used to learn about Mersenne himself. For example,

Descartes' first biography by Baillet depended heavily on 102 the exchange of letters between the two friends and

correspondents.

The genesis of the publication of the

Correspondance du P. Marin Mersenne itself is of great interest. When Paul Tannery, together with Charles Henry and Charles Adam, worked on the Collected Works of Descartes and Fermat, he suggested, in 1885, that the publication of

Mersenne's correspondence would be of great importance to the history of science. Unfortunately, he died before he was able to start the work. But his wife, assisted by the

Dutch historian C. de Waard, who had already helped Dr.

Tannery finish the fourth volume on Fermat and had just finished his publication of the Journal de I. Beeckmann. planned to carry on her husband's project. The first volume appeared in 1932 and the third volume in 1936. At that time the University of Paris took over the project and entrusted the edition to Mr. Pitard, who had joined the project previously in 1934 under De Waard. The Second World War slowed the work for a while, but then R. Lenoble, B. Rochot and A. Beaulieu spelled each other off in directing the work, which was accomplished exactly 100 years after the first proposal by Tannery had appeared in Revue

Philosophique. Variété'. 1886 (t. 22, 296). Under the auspices of the Center of Alexandre Koyres, Paris, a general index for the first ten volumes was published in 1975. A proposal for a seventeenth volume including a general index 103

by name, as well as by subject was often advanced by the

editors themselves. Unfortunately nothing has so far

appeared.”

The correspondence of Mersenne with the above- mentioned Italian scientist and politician Baliani” (1585 -

“For greater details on the genesis and progress of the publication of the Correspondance see Paul Tannery "Une lettre inédite de Descartes" in Revue Philosophique. Variantes, 22 (1886), 296; Heiber et Zeuthen Memoirs Scientifique de Paul Tannery, vol 6, no. 4; Charles Adam, Le Pere Mersenne et ses Correspondance. (Paris, Privately printed, 1896); C. De Waard, "A la Recherche de la Correspondance de Mersenne" Revue d'Histoire des Sciences 2 (1948) is an extremely important description of the history of the collection of the letters collected by Mersenne, even if now a bit outdated; B. Rochot, La Correspondance Scientifique du Pere Mersenne. Palais de la Découverte, 1966 and "Le Pere Mersenne et les Relàtiones Intellectuelles dans l'Europe du XVII siecle" Cahiers d'Histoire Mondiale. 10 (1966) 55-73; A. Beaulieu, "La Correspondance de Mersenne" in Revue de Svnthese. 97 (1976), 71-76. [This issue of the magazine is very important because it is completely dedicated to the publication of the correspondences of some important historical figures and the problems connected with the project. In particular, it is worth singling out the article by P. Dibon "Les exchanges epistolaire dans l'Europe savante du XVIIe siecle", where Mersenne is given a prominent place.] A Beaulieu, "Voies ardues pour l'Edition d'une Correspondance" Nouvelles de la République des Lettres. 1 (1981), 41-63; idem. "Problems d'Edition de la Correspondence d'un Homme Prodigieux: Marin Mersenne" in Trevor H. Levere éd., Editinq Texts in the History of Sciences and Medicine, (edited by Trevor H. Lèvere, New York: Garland Publications, Inc., 1982). This study complements, and in parts corrects, the above mentioned study by de Waard, especially on the role, claimed by de Waard himself, of Libri, the chief librarian of the Bibliotegue Nationale of Paris in the eighteenth century.

“Giovanni Battista Baliani is an interesting figure in the history of science. He dedicated to study all the time he could spare from his gubernatorial duties. (He was, in fact. Governor of Savona, Italy.). He wrote a study on falling bodies,contained in the De Motu Gravium Solidorum (Genoa, 1638) and De Motu Gravium Solidorum et Liquidorum. (Genoa, 1646). He also published a treatise on pestilence: 104

1666) present a case study in the correspondence of

Mersenne. Baliani, who already in 1614 was in

Trattato della Pestilenza (Genoa, 1647) and Ooere Diverse (Genoa, 1666, the publication was completed only after Baliani's death) introduced by a letter by Mersenne acknowledging Baliani's priority over Galileo on the study of falling bodies. In the same book Balliani printed also his letter to Mersenne on November 28, 1647, dealing with his theory on the pressure of air and the equilibrium of the column of mercury with such a pressure. In fact, he had printed this reply right after forwarding the letter to Mersenne, but had second thoughts after reading Descartes' Meditations. in which Descartes ridiculed the idea of vacuum. Some of Baliani's achievements are: his design of using the heat energy produced by the friction of a plate of iron rotating over a second one, his proof that ice was less dense than water (Correspondance. vol.l, 524; vol.2, 71), and the principle of inertia. He was a member of the Academia dei Lincei of Florence after Galileo proposed his nomination. S. Drake (Galileo at Work. University of Chicago Press, 1981, 424-425) shows how Baliani's work marks progress in the understanding of the concepts of mass, and the analysis of acceleration. However, Baliani's relations with Galileo are cautious. He did not like the idea of being identified as a disciple of Galileo, even though he admired the scholar of Pisa (Correspondence. vol.15, 465: me eius admiratoris . non eius sectatoris For imitatoris. as Moscovici. L'Experience du Mouvement. 22 -23. has it! ut alioui existimant. nomen mereri). Moscovici characterizes Baliani as the "the most accurate and most constant analyst of Galileo's works" (Ibid.. 85). His claim of priority over Galileo on the law of the falling bodies, as well as a different approach, created such a rift between him and the school of Galileo, that Galileo had to intervene and give credit to Baliani, even though Galileo stressed he had reached his conclusion independently from him. As a consequence Baliani was cut off from the epistolary exchange with many of Galileo friends. He often learned about the events in Pisa or Bologna and Firenze only through his friends in Paris: the barometric experiment, the deaths of Torricelli and Cavallieri being only the most remarkable examples. His relations with Mersenne are much more relaxed. They met in Genoa twice on Mersenne's way to and from Rome. There they performed some experiments together, and Baliani introduced Mersenne to the intelligentsia of the town. Baliani's correspondence reveals the respect and admiration both had for each other. 105

correspondence with Galileo, came to know Mersenne through

his books, and their mutual friend Niceron. On his frequent

trips to and from Rome, Niceron used to stop in Savone or

Genoa to pay Baliani a visit. A frequent topic among the

scholars at that time was an exchange of information about

scientists working in their home towns and their

intellectual activities. Mersenne was the teacher and the

mentor of the Minim Niceron. Once Baliani and Mersenne

started corresponding, a lifelong bond of friendiship linked

them for the rest of their lives. From 1639 to 1648

Mersenne saved 21 letters of his friend. Unfortunately none

of Mersenne's letters to Baliani, probably more than forty,

were saved, other than the excerpt that Baliani printed in

the introduction to his second edition of the De Motu

Gravium Solidorum et Liouidorum (1646).

Mersenne's first letter to Baliani was to tell him

how much he appreciated the book. Baliani either through

Niceron or from Mersenne's letter, learned of Debaume's

criticism that he had postulated statements that needed a

solid proof.’” Mersenne, himself, had serious doubts about

the second postulate: "The oscillations of two pendula of

equal length are isochronous, independently of their

’”See Correspondance. vol. 8, p. 350. On March 5, 1639 Debaume wrote to Mersenne: "To come to the Italian [Baliani], I find that he puts in his postulates what he should have put in his propositions, which is what I intend to do." 106

amplitude." He had already objected to a similar claim by

Galileo’®’ questioning whether Galileo had really performed the experiment; therefore he rejected Baliani's assertion also. Baliani claimed that experience was the only proof he could invoke in his support. In fact he said that if two pendula AD and AC of equal arm length, were released simultaneously, one from D and the other from C, the sound at their impact with a barrier at AB would reach the experimenter's ears at exactly the same time. (See diagram)

4 c

B

’°’ln Les Nouvelles Pensees de Galileo. 1638, book 1, Article XX, [84]-[92], [197]-[199] the title reads: The proportion that should be given to pendula so that their oscillations be isochrones as one wishes. See also Mechaniques de Galilée. 70; Harmonie Universelle, book 3, proposition 23, 221; Harmonicorum Libri. 785. What is surprising about Mersenne is that he linked the oscillations of a simple pendulum to the oscillations of the vibrating string of a lute, thus showing that the same law regulates the oscillations of a string with one free end to move on a given path (see letter by the Academy of Paris to the Academia dei Lincei of Florence). 107

Baliani could justify his reasoning the same way as

Galileo did. The ball attached at the string AC has an

initial potential energy which as the ball descends towards

B is converted in kinetic energy. When it reaches the point

D the ball has a certain speed which helps it cover the

distance DB much faster than the pendulum AD. In this way the pendulum AC compensates for the larger distance it has to cover and it will reach the point B in the same time as

AD. If a fraction of a second between the two sounds could be perceived, chances were that they were released at some slightly different instants. Morever, Baliani added,

Mersenne himself had claimed that if the pendulum was free to oscillate beyond B, the ratio between the amplitudes of consecutive oscillations was likely to be 40/41. Even though Baliani had done the experiment many times and with great care, he had not noticed any decrease in the amplitude and period of the oscillations. However, Baliani was willing to retract his assertion if someone could prove him wrong.

Furthermore, Mersenne had difficulty accepting the fourth postulate in Baliani's book:

The momentum of the sphere sliding on a inclined plane’” is to the weight of the body, like the length of the inclined plane to the perpendicular distance of the inclined plane from the horizontal, if they are drawn starting at the same point; and furthermore in

’“Moscovici explains that here momentum means the component of the weight along the path of the inclined plane. 108

such a case the proportion of the gravity to the momentum is reciprocal to the proportion of the same lines on which the body is sliding.’”

In modern notation:

mg:M = AC:AB

It is really difficult to imagine how Mersenne, a

skilled experimenter, had difficulty with the statement. If one is to believe Baliani's reply to his correspondent's objections, Mersenne went so far as to doubt that the velocity of a body sliding on an inclined plane would depend on the angle of inclination’”. On the contrary, he does not

’“Moscovici, L'Experience du Mouvement. 35.

’“Moscovici, L'Experience du Mouvement. 36. Mersenne had done some research on this topic, also. For example in Les nouvelles Pensees de Galilée, book 4, [185] he reported Galileo's statement in Discorsi (Leyden, Elzevir, 1638, 166) of the third day: "We have therefore to agree as if it were a fundamental maxim, that the speed acquired by a body descending on planes of different inclinations, are equal, if their height is equal (emphasis by Mersenne): for example if a sphere slides along an inclined plane of four toises, and the height is of one toise, the final speed will be equal to the one sliding one toise on the perpendicular and four on the inclined, and so on." P. Constabel, the editor of the reprint of Mersenne's book, comments on this page by simply reporting Descartes' severe comment: "He does not prove the point neither it is true . . . It seems that he has written this third dialogue only to justify with a long circumlocution that the amplitudes of a vibrating string are the same. 109 react to Baliani's statement that the ratio of the distance travelled by the object and the time were proportional to the inverse of the final speed of the sliding object.’”

Again Baliani's reply was that experience was on his side.

In a second letter,’” Baliani made a remark on

Mersenne's L*Usage de Quadrant, ou de 1 'Horologe Phvsicme. p.12, regarding the length of a pendulum which determines a certain unit of time with a precision up to a second.

Baliani suggested that the correct length should be three feet and two inches (royal measure) rather than three and a half feet, as Mersenne had stated. Morever, this measure had to include the distance to the center of the sphere appended to the pendulum. In reply Mersenne sent him a string with a noose at one end. It was supposed to be the exact measure of the length of a pendulum which measured exactly one second per oscillation.’®’^ Baliani conferred which he fails to do; but he concludes that the weight descends faster by sliding through the arc of a circle than through its chord, but still he was not able to derive it from his own propositions." FLes Nouvelles Pensees de Galilée, critical edition by P. Constabel, (Paris; Centre Nationale de Recherche Scientifique, 1973), 21.]

’°®However, Mersenne's letters to Baliani were lost, so we do not know exactly what Mersenne's objections were. Mersenne might have been objecting to the fact that the ratios of the average speeds in the two cases were equal.

106Correspondance., vol. 10, 128.

’“Experimenting with this pendulum, Baliani realized that the oscillations were retarded because of the friction of the noose with the finger where the noose was oscillating. He, therefore, suggested that it should be suspended on a sharp edge to minimize the friction 110

with Galileo about Mersenne's value and Galileo answered

giving him the "golden rule" to fix the length of the pendulum by counting the number of oscillations per second.

The Jesuit Cabeo of Ferrara was asked to perform the experiment. He reported that he had found that the length should be 233 millimeters, probably because he thought that the time for one oscillation was proportional to the length of the pendulum and not to its square root. On a marginal note to his Harmonie Universelle Mersenne wrote:

Someone, in particular Mr. Cornu, is said to have experimented that the cord should not be longer than two feet and ten inches (=0.9164 m) to obtain 3600 oscillations in one hour. Anyone is free to make adjustments as he thinks best, but three feet should be enough.’”

Again in Novarum Observationum Tomus Tertius

Mersenne criticised Cabeo*s value and fixed the length to three feet. Mersenne noted that the period of oscillations varied slightly with the change of the temperature. Baliani explained the phenomenon by saying that wet dust was the cause of the slower oscillations during the winter.

Baliani's correspondence with Mersenne for the first two years centered mainly on the pendulum. Since

Baliani became familiar with Mersenne's handwriting and

(Correspondance. vol. 14, 306-307).

’“in Phaenomena Ballistica. 38-44, Mersenne wrote that he had performed the experiment with strings three feet and three and a half feet long. The three feet string seemed to give more accurate oscillations. Ill

style, Cavallieri sometimes sent him Mersenne's letters for

translation and rewriting. One such letter is of great

importance for the later development of the pendulum clock

by Huygens. Unfortunately, this letter was lost, but from

Cavallieri's letter to Baliani we can understand part of its

content.

The points of interest are:

I. Balls of different substances and weights, hanging on

strings of equal length are isochronous "to use the

words of the Pere Mersenne";

II. The pendulum ab (see explanatory graph reproduced by

Moscovici in the French translation) can oscillate with

the same period as the rod cd. Cavallieri explained

that this was possible because the whole mass of the

rod could be considered as a ball around the center of

oscillation e."* The ratio of the two lengths was 3/4.

tO

109In. Moscovici's L*Experience du Mouvement, the original letter in Italian, the center of oscillation "e" is labelled "c", but in the French translation and in the graph the point is, correctly, labelled "e". The author does not specify if the difference is due to Cavallieri or is a misprint due to Moscovici's publisher. The latter seems to be the most probable explanation. 112

III. One could ask whether any rod of any length, thickness

and shape could be isochronous with the pendulum.

Cavallieri answered that the surface could cause more

friction with air and therefore slow down the

oscillations. His suggestion was to examine the

inclinations at which each part impacts the air and its

effect on the period of oscillation. IV. Finally, Cavallieri added the following explanation to an inquiry by Mersenne which is not clearly

understood from the answer, but which seems to refer to

possible constraints to be imposed on the amplitude of

the oscillations of the pendulum to obtain isochronous

swings:

In triangles of 60, 90 and 120 degrees and of equal height, it seemed to me that isochronous strings are almost in the proportion of the secants of those triangles. If ac, the height, is the full sine, then ab, the secant of 30 degrees is 115470. Likewise, ab, the secant of 45 degrees is 141421. Just as ab, the secant of 60 degrees is 200000, which are like the given strings ae, ae, ae", which are related among themselves as 3, 6 and 9 following the observations of the priest. But, the conclusions from these observations are not clear, because the rest does not follow the proportion of the other plane figures. For this reason I told him that I did not find anything useful in this. I am mentioning this to you so that you could tell him that I spent some time reflecting on his question, which seemed to be very promising. However I am afraid that 113

concentrating too much on it could be harmful to my health and therefore I let it go.”®

# e.'' Given the innovative content of the letter, it is

surprising that it is not included in Mersenne's

Correspondance. particularly in view of the fact that the

editors of the collections published not only his direct

correspondence, but also many of his contemporaries'

references to him.

In a later letter Mersenne suggested that Baliani

perform an experiment to measure the speed of sound by

having a departing ship reach a constant speed, and fire a

cannon when it was three, six and twelve miles away.

Recording exactly the instants when the cannons were fired

and the sound was heard at the port, one would have the

distance and time necessary to calculate the speed. Even

though the method was still very primitive, Baliani wrote

back saying that it seemed to him that the experiments postulated the knowledge of the speed of sound. Mersenne

110See, Correspondance vol. 14, 351. 114

again tried to explain his design, but again Baliani found

it unrealistic and declined to perform the experiment.

From the summer of 1647 until Mersenne's death the

experiments with the barometer were almost exclusively the

topic of their letters. Baliani's correspondence in this

regard is important in explaining the difficulty that an amateur scholar encountered with the repetition of an experiment. Mersenne himself succeeded, after repeated failures, only through the assistance of his good friends

Petit and Blaise Pascal, after repeated failures. Mersenne had to explain to Baliani in two different letters the small precautions necessary to succeed and to answer the common objections several times before Baliani was able to succeed in the experiment after five months.

On the other hand though, Baliani, who was more a theoretician than an experimentalist, found it relatively easy to explain the phenomenon by means of the pressure of the column of air above the bowl of mercury outside the barometer, so that the weight of the mercury's column in the tube balances the weight of the air outside. He evidently was proud to state that without any experimental evidence he had already told Galileo that a vacuum could be created in space. Subsequently, he answered correctly some doubts that

Mersenne had expressed. One such problem was: Why does the thumb that closes the opening of the tube not feel the weight of the air above it, if a vacuum is present between 115 the mercury level and the thumb? Baliani answered that the force was in fact what Mersenne had described as se fortiter attrahi a vacuo (the thumb was strongly attracted by the vacuum). Suction is not a force of attraction by the empty space, but it is the lack of resistance to the great push of the air above the thumb. The fact that the force is not crushed by the huge weight of the column of air above could be explained by the relatively small pressure on the thumb, due to the large area on which the force is exerted.

Besides, one must realize that the pressure is not only from above but from every direction, so that an equilibrium can be established.

The empiricist Mersenne, who felt more confident with results obtained experimentally than from theoretical abstractions proposed to Baliani, as he did to Huygens,

Descartes and Magni, the following observation of Roberval.

A swimming bladder, deflated of all the air and with its opening tightly closed so no air could penetrate it, was introduced into a tube. After a vacuum was formed the bladder appeared inflated. The theory of the vacuum that was just starting to be accepted was seriously threatened.

Mersenne wrote repeatedly to his correspondents with requests for an explanation. Baliani received three such letters in less than two months. Only after Mersenne's death was Baliani able to come up with the explanation the young Huygens had already advanced: The internal part of the 116

bladder with little air entrapped, in the vacuum of the tube

is at a higher pressure and enables the bladder to expand.

But if air is introduced in the tube, the air pressure

reduces the bladder to its original shape.

Baliani's stature as a scientist was greatly

enhanced by his correspondence with Mersenne. Theirs is an

example of how fruitful and how enmeshing the correspondence

with Mersenne could become. Baliani's insight was able to

explain the principles behind the great topics of the epoch.

Mersenne, on his side, is to be credited for the atmosphere

of friendship and interest he was able to create with his

correspondent, so that the cooperation could become fruitful

and enjoyable for both. His readiness to communicate

whatever was new and his ability to overcome the

nationalistic boundaries in favor of an international

community of researchers, rightly place him among the

contributors in the seventeenth century who did most to

promote the progress of science. The work that such an

amount of correspondence required from him was a way to

achieve the goal he had set for himself; to ask from the

most privileged all they could produce for the benefit of

knowledge.

Mersenne's poor health at the end of his life

interrupted the exchange of correspondence between the two

savants, but not before Baliani tried to express to Mersenne his sympathy for the misfortune that had happened to him. 117

due to the inexperience of a young surgeon. Baliani, in

fact, even tried to advise Mersenne about some medications

that could relieve him from the pains of his arm.

The correspondence between Descartes and Mersenne

has been often quoted in the previous pages and more will be

reported in the following pages. It is therefore desirable

to focus for a moment on the relations between them in more

detail.

Around 1636, Descartes realized that the cultural

circles of Paris were not conducive for him to concentrate

on his studies, partly because of his difficult temperament.

He therefore decided to leave Paris for an undisclosed

destination in Holland. Mersenne was the only one in France

who was informed of the philosopher's address. They had

agreed that Mersenne would keep him informed about

developments in France, and that Mersenne would become the

channel through whom Descartes would communicate with any

other scientist in France. Mersenne carried out this

service faithfully for over twenty eight years, writing to

Descartes on the average of a letter every two weeks. As a

consequence, if Descartes' first biographer is to be believed, French scholars

. . . crowded the monastery bringing their questions for Descartes to Mersenne or requesting the answers to their problems. This occupation kept this priest very busy, but at the same time he was so kind that he never complained. In fact, he not only asked Descartes to answer all the questions presented to him in the packet he was sending, but he went so far as soliciting him to 118

propose on his part some questions to the others, with the pledge of sending back to him their solutions.’”

The large number of letters exchanged between them

is proof of the faithfulness with which Mersenne carried out

his pledge. In fact no one shared his time and attention as

much as Descartes did. When Descartes was involved in a

bitter confrontation with Grotius at the University of

Utrecht, and sanctions were levelled against him, Mersenne

as an act of support dedicated one of his books to him.

Besides, he often quoted him in his publications referring to him as the Illustris Vir. but never by name, probably at the request of Descartes, to avoid confrontations with

Descartes' many detractors. The role that Mersenne played in the publication and dissemination of Descartes' books has been already mentioned in the previous section. On his part, Descartes did not hesitate to ask for all kinds of services from the monk. In fact, he used Mersenne more than anyone else. He incorporated in his correspondence with

Mersenne what is now considered to be the best commentary on his own printed works. So much so, that the earlier mentioned Baillet ended up defining Mersenne as the first

Cartesian”®. But already Leibniz, the Christiaan Huygens'

”’André Baillet, La Vie de Mons. Descartes, vol. 1, 168.

”®Lenoble, Mersenne. ou la Naissance du Mécanisme. 448- 449, writes about Mersenne's intellectual relations with Descartes: "Descartes is not the preferred teacher nor the best friend of Mersenne. Among the great mechanicists who enjoyed his favors, he will prefer those who are the most 119

disciple, fought against such a judgement. In The Nouvelles

Lettres et Opuscules inédits de Leibniz Precedes d'une

Introduction he wrote:

I do not believe that Père Mersenne can be counted among the followers of Descartes. I can easily see that he did not follow his (philosophical) system, even though he was very close to him. But then this is true also of Mr. Hobbes, Gassendi and Roberval.”®

Lenoble, the authority on Mersenne, has proved

once and for all that Baillet's statement was definitely biased and aimed at proving the absolute superiority of

Descartes over all of his contemporaries. But in reality,

few people were as able as Mersenne to appreciate the

disengaged from any philosophical system: To the metaphysics of Descartes he prefers the sensualism of Hobbes, to the physics of Descartes that of Galileo, of the Pascals and of Roberval. But his intimate friend was certainly Gassendi. Not because he had in common with him the atomistic view (on this point he is closer to Descartes), but he is the comrade of his first attempts, the cautious empiricist who, like him, was convinced that there could not be any 'demonstrative' science, a generous heart with whom he feels confident. Between themselves they talk about Descartes as everybody in the 'New School' does: the Illustris Vir. of whom one admires the genius but at the same time does not accept the boastful metaphysical positions and the ill- tempered answers. Ruarus hit right to the point when he wrote in 1643: 'If it is useful I will gladly see the new things that your Gassendi has to say in philosophy; I have only glimpsed at the Cartesian philosophy, but could not pay sufficient attention to it.'"

’"Leibniz, , p. 33. Again in a letter to Remond de Monmort (Opera Philosophica. Berlin, ed. Erdmann, 1840, 704) he wrote: "Fr. Mersenne was not as Cartesian as one would expect. This priest mixed with Roberval, Fermat, Gassendi, Descartes, Hobbes. He did not particularly care to go into their dogmas and their confrontations, but was respectful towards everybody and was particularly effective in encouraging them all." 120

Descartes' original contributions in science and

mathematics, while at the same time maintaining their

independence from his philosophical system. But even in

science, Mersenne, while encouraging Descartes to pursue his

own thoughts, did not always agree with him. Even though an

Aristotelian himself, Mersenne had long before rejected the principle of authority.’” This is true in particular

regarding the barometric experiments. Descartes firmly believed that the space above the column of mercury in the barometer was filled with the "spirit of mercury," or a

"refined material" that could penetrate even glass, because

"nature abhors a vacuum." The vacuum theory, sponsored by

Baliani, Pascal and others, did not entirely resolve all the questions that were troubling Mersenne, but it was a theory that could be tested in the laboratory. Mersenne searched until the end of his life for the definite proof that would sweep away all his doubts. For this reason, he thought of measuring the height of the barometric column in various places of the same city and at various heights above sea level. Unfortunately, his poor health did not allow him to perform the experiment himself. The person of whom he requested this service could not see the use of it.

”^For example, in Les Preludes de l'Harmonie Universelle, quest. 1, cor.2, Mersenne wrote; "I have started talking about the main difficulties in music by means of the reason rather than by the authority of men, which I do not reject if it is accompanied by the demonstration of their assertion." 121

Mersenne has the merit of not giving in to Descartes'

pressure of denying the possibility of a vacuum in space.

Only two weeks after Mersenne's death, his friend Petit

performed the experiment at the Dome de Point for the young

Pascal proving definitely that pressure was the factor

determining the height of the column.

But exactly how did Descartes feel about Mersenne?

Perhaps a letter written from Bannius to Constantin Huygens, the father of Christiaan Huygens, gives an exact account:

Your letter was such an accurate description of the man Mersenne that anyone who would read it, especially the heroic Descartes, would immediately identify him. In fact he (Descartes) tells me frequently, that the man (Mersenne) is of a vast but undigested erudition; in other words, his knowledge is long and wide but lacks depth. However it is more profitable to appease him rather than to fight against him.”®

These last lines correctly express the real attitude of Descartes towards Mersenne: he used him as long as he thought he could profit from his services, but his language clearly showed that he did not hold Mersenne in great esteem on an intellectual level. For this reason he sometimes ignored for years problems presented to him by the monk, when he did not feel his academic superiority directly threatened, as we will see in the following chapter.”® His

’"Mersenne. Correspondance, vol. 15. 324.

”®0n his part Mersenne is responsible for withholding from Descartes important information, like the barometric experiment mentioned above, delaying some of the mail, diverting Descartes' correspondence to people who were not intended to receive it, like Fermat and Roberval. However, Descartes also profited from such liberties taken by 122

selfish behavior became even more evident when, during one

of his rare visits to Paris, Mersenne's health took a turn

for the worse. Descartes did not think twice about leaving

Paris immediately. Mersenne died only few days later.

However, Descartes never again found another person who

replaced the services of Mersenne with equal devotion and

attention.

On the other hand, the length, the curiosity and

the frequency of Mersenne's correspondence at times annoyed

some addressees. Even though, out of politeness, they would

not complain to Mersenne directly, they would answer only

few of the many questions Mersenne posed to them or they would ignore his letters. Quite often Mersenne was forced to ask the same questions more than once. A common complaint from his correspondents centered about his terrible handwriting and arbitrary notations. It was already pointed out how Galileo used this as an excuse to avoid answering Mersenne's letters. Torricelli, who was not any better disposed towards him, once commented to Rocca about Mersenne's handwriting saying: "This illegible scrawl which Plato would have attributed not to chickens but to pigs!"’" Mersenne was well aware of the problem with his handwriting. At times he tried a better quality of ink or

Mersenne in dealing with his correspondence.

’"Correpondance. vol. 12, 187. 123

suggested that the receiver should get the help of others to

read his scribbling. He also wondered about the big fuss that Galileo was making about it, claiming that his handwriting was read all over Europe!

Baliani (the devout correspondent), Descartes (the interested interlocutor), and Galileo (who ignored the correspondence of Mersenne), are the catalysts of people's responses to Mersenne's efforts to exchange information with the whole of Europe. His merits were recognized by

Rochemonteix who called him Le Secretaire de l'Europe

Savante.”®

Academies

It has already been pointed out that the universities of the seventeenth century did not play the role of centers of research in mathematics. However, in the previous century scholars throughout Europe felt the need for such centers, which they called academies. In fact, they sprang up all over Europe, sometimes as the result of the interest of wealthy supporters, as in Italy, and at times in response to the efforts of concerned citizens, as in France and England.”® The role played by Mersenne,

”®C. de Rochemonteix, Un College des Jesuites aux XVIIe et XVIIIe siecles vol.4., 127.

”®For further information about the relation of scientific research and the universities see also M. Ornstein, The Role of Scientific Societies in the XVIIth Century, especially the conclusion of the book 257-263. 124

together with other scientists, as a successful instigator

and organizer of the first cells which eventually resulted

in full fledged academies, transcended the borders of his native country. Even though he did not see his plans fully realized during his lifetime, he deserves credit, also, for the historical records he left the community of the scholars concerning the development of the idea of academies. His interest in the establishment of these centers is amply documented in his publications and in his correspondence which will be examined in this section.

In his very first publication, the Ouaestiones celeberrimae in Genesim. writing on the three kinds of music of antiquity, he mentioned that the French Academy of Music in 1572 had denied that music had to follow necessarily the tunes of the three accepted types, but that a new melody could be formed from the combination of the three. This presented him with the opportunity to describe in detail the formation and successes of the academy of music which enjoyed the support of King Charles IX's decree of 1570.

Mersenne was so interested in the plan that he did meticulous research, as he wrote; Quae omnia accurate perleai: ideooue ne tam honest! conatu oblivione seoelientur. paucis illos aperlo. [I read carefully all

(about such academy); therefore, I will disclose some of my findings so that they may not be buried in the tomb of 125

oblivion.]’®® The founders of this academy, Jean Antoine de

Baise and Jean Thebault de Corenville, created the academy

as a means to educate the youth in France to morality and honesty. The academy recruited the best scholars available

at the time in every field. It was presided over by a megalodidascalos or prefect (dean, in our contemporary terminology). The academy provided classes in natural sciences, languages, music, poetry, geography, mathematics, painting, military sciences and gymnastics. The school even had a staff attending the kitchen, gardening, tailor-shop and financial offices of the school.

In its short lifetime, the academy recorded many achievements, which Mersenne was careful to note. It added momentum to the French language through compositions of religious and secular poetry; it attempted to find general principles behind the practice of music; and it proposed topics of research such as musical interpretation in order

"to find whether any musician in all of the Christian Empire since Guidone Aretino to the present time has been able to reach the perfection of music found in antiquity."’®’ Despite its initial success and the encouragement, support and frequent inquiries of the king about its progress, it failed

’®®M. Mersenne, Ouaestiones celeberrimae in Genesim. col. 1683-87.

121Ibid. . 126

to attain its goals "because of the envy of some people."’®®

However, Mersenne did not believe that this was sufficient

reason to despair. Mersenne took the initiative in pleading

for the entire community of the French intelligentsia to

work together:

Far better academies can still be founded if only the best educated people would pool together their abilities. One caveat: only let envy not spoil such great efforts, as it usually happens in France!

I wish this Academy of Music could be started, so that it would flourish and produce fruit. Its effects would be an uninterrupted glory to God, a deepening of our knowledge of music, the education of the musicians, whom I greet with all my heart . . .’®®

This appeal is only an echo of the one already

voiced in the introductory pages. In fact, there he had

drawn up a plan describing what the academies should be

like: first of all, the academies appeared to him to be the

safest way to restore the sciences. To insure them a widely representative body and invest them with authority and privileges, he envisaged them under the protection of the

Pope of Rome and of the Christian kings, at least in Europe.

At this stage, an international academy seemed to be utopian dream. The first step would be to establish national academies, where scientists would spare no experience, no efforts, no expenses so that in a brief period of time sciences would flourish in their full vigor. Hundreds or

’®®Ibid..

’®®Ibid. 127

more people could be involved, some of them attending to

theological studies, others to philosophy, still others to

mathematics, medicine, chemistry, jurisprudence or any other

discipline. Their goal should be to find the causes,

reasons, scopes and goals of all phenomena. In this

grandiose project, there was room for academic dissent in

every field, for the benefit of science. In this way, participants could produce an encyclopedia for the

advancement of science."*

Actually, the project was not new. In Italy there were many such cultural centers. Sprat states that "of these, the first arose in Italy, where they have since so much abounded, that there was scarcely any big city without one of them."’®® The best known was the Academia dei Lincei founded in 1603 with the generous support and encouragement of Count Frederick Cesi. He offered his own house as a

’®*In the Harmonie Universelle (book 7, On Instruments of Percussion, proposition 31, p.61) Mersenne suggested that the academy could produce a Dictionary of Music containing the history, the rules, and the biography of the most relevant musicians. He himself set the example writing brief notes about the best musicians "so that the names of those who have excelled in any part of this science would not be forgotten..." (ibidem, preface to the reader, non numbered page; Traite des Instruments, books 1 and 2, proposition 12, 92). One of his confreres had already published a universal catalogue: Idea Bibliotecae Universalis cruam meditatur F. Petrus Blanchet. ex Ord. Min.. Paris, S. Cramoisy, 1631. Mersenne had encouraged the work and had written about it to Descartes, who also was enthusiastic about the project; [see Lenoble, Mersenne. ou La Naissance du Mechanisme. 435.]

’®®T. Sprat, History of the Roval Society (St. Louis: Washington University Press, 1958), 28. 128

center, which he had furnished with a library, a botanical

garden, a collection of natural attractions, a laboratory

for all fields, and printing facilities. Galileo,

Torricelli, Viviani, and Baliani were the greatest

representatives of this academy. It produced the Gesta

Lynceorum which are the earliest recorded publications of

scientific endeavor by a society. It even printed Galileo's

the Saqqiatore and On the Sun's Spots. However, with the

death of the founder, the society lost its main support and

by 1657 it had already stopped operating. One of Mersenne's

letters in 1643 on behalf of the Academia Parisiensis to the

Academia dei Lincei. testifies as to the active exchange of

information between the two centers.’®®

In Paris itself there were informal gatherings of

scholars, mainly in the private homes of the scholars, where

scientists would present papers or discuss current ideas.

One, in particular, remained famous because it gave rise to

the Académie Français in 1629. Cardinal Richelieu took the

academy under his tutelage and chartered it in 1634.

Mersenne took part in its weekly meetings, but it is not

certain that he was a formal member. However, it roused his

’®®See Correspondance. vol. 7, 427. The same year another great Academy, the Academia del Cimente, was started by Della Porta. In its ten years of activity (1657-1667) the society gained renown throughout Europe for its scientific achievements. 129 hopes because on August 2, 1634, he wrote to Nicholas Claude

Fabri de Peiresc about this group:

We have now a French Academy which convenes at Mr. Le Garde. Its members are Messrs Servianm Botru, Balzac and some twenty others. They are trying to standardize the rules of the language, will write a grammar and a dictionary to formalize the dictions characteristic of comedy, drama and for speeches. . . . If it will last we will gain much from it."^

Other gatherings of scholars were held at

Descartes' home (1625-1626)."® Mersenne was also a member of the group that gathered at the Depuy brothers' and he knew about the society at Renaudot's.

The problem with all these Parisian groups was that they needed organization, clear goals and serious commitment from the organizers and the members. Mersenne, therefore, started fulfilling these needs with a new group of the best mathematicians in Paris. On May 23, 1635, he wrote to Peiresc: "If M. Gassendi will come to Paris he will see the noblest academy of the world which has just started in this city. Its members are all mathematicians.""®

"^Correspondance. vol. 4, 281. The projected Dictionary was accomplished in 1694.

"®It is most likely that Mersenne and Descartes came to know each other, at least formally, during these meetings. They may have been at that time introduced by their mutual friend, Gassendi, rather than in their youth at the College of La Flèche.

’^Correspondance. vol. 5, 210. It is curious to note that all initial information about this academy is obtained almost exclusively from Mersenne's correspondence with Peiresc. One explanation may be that while his letters to Peiresc were still preserved, most of Mersenne's letters to 130

With pride and expectation Mersenne started

referring to this group as the Academia Parisiensis. while

Descartes and others called it Mersenne's Academy."® Its

founding members were Etienne Pascal (president of Aides at

Clermont in Auvergne), Mydorge, Hardy, Roberval, Desargues, the Abbe Chambron, and others."’ Later the Lyonnese

Gassendi, the Englishman Hobbes, Blaise Pascal (in the company of his father),’®® and others joined the club.

Descartes, Baliani, etc. were lost, and therefore we cannot tell if he mentioned the formation of "his" academy to the others. Another reason might be that Mersenne knew that Peiresc was interested in this kind of scientific gathering, himself having been a member of the Academia dei Lincei of Rome and a patron of minor scholars like Gassendi and Mersenne, and he had a group of scientist amateurs of Lyon, his hometown, gathering in his home.

’®®See Correspondance vol. 7, 364, Descartes'letter to Mersenne.

’®’See Correspondance. vol. 5, 371. It is frequently believed that Descartes, Fermat and Christiaan Huygens were members of this Academy (see M. Ornstein's The Role of Scientific Societies in the XVII Century; Sergescu, "Mersenne l'Animateur," Revue d'Histoire des Sciences 2 (1948): 10). But at this time Descartes was in self-imposed seclusion in Holland, and he took part in the meetings of the academy only in 1647, during one of his last visits to Paris; Fermat was never in Paris, and therefore could not have been a member of the society. Christiaan Huygens was living with his father, Constantin, in The Hague. Mersenne was already in correspondence with Constantin, but not yet with Christiaan. It was only when the Académie Rovale des Sciences was formally instituted that participation at the meetings was not required from members who were out of town.

’®® About Blaise Pascal's participation and of the meetings of the academy his sister Gilberte left the following account: "Staring at the participants he (Blaise) was following their reasoning and was progressing so much that he could not miss the meetings that were held every week. Here all the learned people of Paris gathered to bring their papers to the consultation of the others. . . . Very 131

The meetings were held at regular intervals (every

week, according to Gilette, Blaise Pascal's sister), at the homes of the members,’®® as the following quotation from a

letter to Fermat from Roberval (April 2, 1637) shows:

At the assembly of our mathematicians, which on that day convened at M. de Motholon's, counsellor, your letter was received and read with great admiration by the participants.’®*

In general, at the meetings there would be one or two presentations of papers by some previously designated members, followed by an interval of discussion on the topics. New ideas were aired and discussed; experiments, if necessary and possible, were performed. At the end, some of the letters from outsiders to Mersenne on mathematical or scientific topics were read, replies were formulated and someone would be assigned the task of answering them in the order proposed. Most of the time, Mersenne would take the assignment; other times the most interested of the members would take the job, as the controversy of the cycloid will show in the next chapter.

Mersenne's personal input in mathematics among such distinguished scholars was probably quite low, as was

often some propositions coming from Italy, Germany or other foreign countries were presented for the consideration of the academy." (B. Pascal, Ouevres Completes. 1342).

’®®It was already pointed out that during Mersenne's last month of life, when he was forced to lie in bed, the meetings were held in his room, to allow him to take part.

’®*Ouevres de Fermat, vol. 2 (1894), 103. 132

already pointed out in the second chapter of this paper. But

he certainly had an acute perception in distinguishing the

originality of the ideas and formulating them in new ways,

and in suggesting new experiments and alternative

explanations of the observed phenomena. No one described

better than the young Pascal, years later, the contributions

of Mersenne at these meetings:

He was particularly gifted in formulating wonderful questions; probably no one equalled him in this: and even though he was not lucky enough to solve them, which is what constitutes the real honor of a scientist, one has to admit what we owe to him, that he was instrumental in many discoveries, which perhaps would have never been reached if he had not initiated them.’®®

Morever, he was able to enshrine the outcome of

the discussions in these meetings both in his correspondence

and in his books.’®® This proved how fruitful and well

organized such assemblies were. One has to admit that the

Paris Academy of Mersenne was one of the strongest and most

resourceful centers of research of the seventeenth

’®®B. Pascal, Histoire de la Roulette, in Ouevres Completes. 194.

’®®One should be careful, however, not to assume from Hobbes' and other admirers' statements Mersenne's books and correspondence were uniquely the result of the academy's views. In fact the tracts entitled Questions, for example, date from before the formation of the academy. Besides, in music Mersenne was an uncontested authority, as he himself acknowledges in the above mentioned introduction to the Ouaestiones celeberrimae. 133

century."^ Even though the individual members claimed

credit for their personal efforts, nevertheless the

"’^All the almost-contemporary documents of the Academy Rovale des Sciences seem to agree with this assertion. For example. Fontanelle in Histoire de 1 'Académie Rovale des Sciences (Paris, G. Martin, J.B. Coignard, H.L. Guerin, 1733), 4-5 wrote: "It has been more than fifty years now that those who were in Paris used to see one another at Fr. Mersenne's. He used his familiarity with the most competent people in Europe to be a liaison in their communication. Messrs. Gassendi, Descartes, Hobbes, Roberval, the two Pascals, father and son, Blondel and some others used to meet at his place. He would propose them some problem of mathematics, or would ask them to perform some experiments with respect to certain aspects, and never before had anyone cared more for the new science born from the union of geometry with physics. More regular assemblies were held at Mr. de Mon(t)mor, Master or Reauetes and afterwards at Mr. Thevenot. (Here) they would examine the experiments and the new discoveries, (their) uses or the (possible) consequences that could be deduced." The same interpretation is again expressed by Giovanni Battista Cassini, later to be nominated to the Academy, in his De L'Origine et du Progress de l'Astronomie et de son Usage dans la Géographie et dan la Navigation. (Bologne, 1652), later reprinted in the above mentioned Memoirs. vol. 8, (Compagnie des Libraire, 1730), 31-32: "Many years before this academy was established, there were many assemblies of physics and mathematics in Paris. In 1638 Mersenne started such assemblies, which were afterwards continued by Mr. de Montmor and by Mr. Thevenor. Many scholars enjoyed coming together to converse about the observations on astronomy, on analytical problems, on experiments in physics and of the new discoveries in anatomy, in chemistry and in botanies. One would often see Messrs. Gassendi, Descartes, Fermat, Desargues, Hobbes, de Roberval, Bouillaud, Frenicle, Petit, Pacquet, Auzout, Blondel, the Pascals, father and son, and many others renowned for their works, of whom there would be too many to name them all. Many foreigners also join them." Even though the above documents reveal many inaccurate points, (e.g. dates, names) both single out the Mersenne group as the most prominent among the groups that preceded the formal national academy. If nothing else, the above statements could be interpreted to mean that the members of the Mersenne's academy who joined the Académie des Sciences were the most influential. 134

continuous and readily available consultation with the other

academicians made it possible for them to feel confident

about their results. Unlike Descartes and Fermat, who

preferred to work alone, they had all the academicians to

support them and defend them in their claims."® Arnold

Rogow, describing the solitude that Hobbes experienced

during his controversies at home after the publication of

his Leviathan. commented:

He was not without friends but almost no one was ready to come to his defense in writing or even to assist him in efforts to get his books published. Perhaps then, more than once in the decades following his return to England in 1651, Hobbes remembered, with nostalgia, the years in France and Saint Germain when his adversaries (Descartes among them) had been insignificant in comparison with his friends. Perhaps he even wished, at particularly low points in his later life, that he had a way to return to the salon of Montmor, Mersenne's successor, and resume his discussion...but such a venture was not possible."®

The highlights of the discussions of the Academy

were the centers of percussion or oscillation, the conics

and the cycloid.’*® But what dominated its activities for

’®®This was the spirit that animated Blaise Pascal, for example, in writing the Histoire de la Roulette: to defend the claims of the colleague Roberval against the claims of Torricelli.

’®®A. A. Rogow, Thomas Hobbes - a Radical in the Service of Reaction. New York: W. W. Norton, 1986), 177.

’*®One of the remarks that Purver Margery makes regarding the Mersenne group is that they failed to recognize the emergence of biology with its varied branches as a new science (The Roval Societv: Concept and Creation (The MIT Press, Cambridge, Massachussetts, 1967), 174). But they seem to forget that Mersenne's plan was to have an academy toute mathématicienne, even though he had in mind a wider academy where all the sciences could be hosted. 135

the last four years of Mersenne's life was the problem of

the barometer. Papers were presented with great care,

statements were tested with experiments performed

repeatedly, and meetings were attended regularly.

Mersenne's vast communication system was used as a network

of information with the scholars of the rest of the world

about the activities of the center. This commanded the

respect and admiration of outsiders both at home and

abroad”’. Mersenne, more than anyone in the group, deserves

the credit for this. However, this first cell of the future

Académie Française was not yet fulfilling the aspirations of

its founder. One reason was that the center was formed by

the members on a voluntary basis, without any legal status,

and most of all without financial support to sustain the

group and to publish their works. They could not afford to

have a fixed center, to furnish a respectable laboratory for

their experiments, or to open a library for consultation at

leisure. It is true that the library of the Palace Royale

of the Minims, contrary to the ordinary regulations of the

Order, was open to the public, as were other collections of

some rich or noble families or ecclesiastical authority open

Morever, Mersenne had several botanists, anatomists, etc. as friends. Harvey, who discovered blood circulation, was his correspondent.

’^Recall the admission by Magiotti to Torricelli: In France mathematicians are well organized, meet frequently and published many books every day. Correspondance vol. 10, 345. 136

to the public. Yet, not even the collection of them was not

able to meet the needs of every scholar.”®

To overcome this difficulty, Mersenne suggested,

as a first step, that everybody should pool his resources

together:

Those who have specialized laboratories or rare collections should enlist their possessions according to the field of specialization and explain the use and utility of their instruments: for example we should need to locate the collection of the rarest shells which are very precious; the same thing about the rare flowers; tubers of tulips or other plants, because, for

”®Within each group a persisting set of weaknesses forced consideration of alternate forms of associations. Most annoying, although certainly not most serious, was the unpredictable and often ephemeral nature of the voluntary association. Le Pailleur's Académie expired in 1654 (after about 29 years) with the patron's death; the existence of Bourdelot's Academy was interrupted by its founders hurried departure to Queen Christine's court in 1652; Thevenot (the last coordinator of Mersenne's group) was forced to close for lack of funds for experimentation. Being attached to mortal men rather than to the eternal Crown, the private societies suffered along their patrons' fortunes. (H. Roger, The Anatomy of a Scientific Institution. 67.) Mersenne had already foreseen these problems and asked for an academy under the Pope, for internationality, and the Kings for security, continuity and financial support. M. Purver ( The Roval Societv: Concept and Creation (Cambridge, Mass.: MIT 1967), 327) asserts that Mersenne's proposal of an academy was to "offset Bacon's proposal of colleges on an international scale for arts and sciences at large. . . . Mersenne's institution was to be for sciences according to the scholastic system, not for sciences in the modern system." But when he was writing the Ouestiones celeberrimae in Genesim. where the proposal of the Academy was first explained, Mersenne could not have read Bacon's Oraanum Novum (published 1623). Mersenne remained faithful to Aristotle's philosophy until the end, but in La Verite des Sciences he admitted that in physics Aristotle had made several mistakes in his books. About Mersenne's concept of science, Lenoble calls him the father of modern science, before Bacon, Descartes and any other of his "century of scientists." (Lenoble, Mersenne. ou la Naissance du Mechanisme. 510). 137

certain one can discover the great secrets of nature by the examination of such a collection.’*®

The second problem with the Academia Parisiensis was its limited radius of action, even though visitors were welcome to participate in the meetings. Mersenne, in

several letters as well as in his publications expressed his desire that the Academy could be universal in content and geography. His dream was that local academies should be part of a national, if not continental, federation, where communication could be exchanged by mail. For this dream to become a reality, the nationalism of each individual country in Europe would have to make room for continental understanding. Mersenne, therefore, wrote to his friend

Peiresc:

I wish we could enjoy such peace that one could institute an Academy, at least in all France, not just in one city only, as we are doing here and elsewhere, if not on a European level. Such academies could carry on their communication by mail, which at times is even more profitable than by direct encounters, where, too often, participants spend more time in useless and heated confrontation of ideas, which in return discourages many from attending such gatherings.’**

Even though Mersenne died before having the satisfaction of seeing his academy receive official status

’"Mersenne, M., Questions Theoloaicmes. 5-6.

’** Mersenne, M., ouaestiones Celeberrimae in Genesim. col. 1687. 138

at a national level, the cell continued to operate’*'® until

1662 when the steps to form the Académie Royale des Sciences

were formally taken. In 1666 a decree by King Louis XIV

approved the official charter. Mersenne's cell could then

present five members to the new Academy, thus continuing the

dream of the founder.

The initial documents of the French Académie

strongly suggest that the British Royal Society may have

been inspired and highly influenced by the academy

established by Mersenne.’*” Even though scientific

gatherings of scholars in Britain had a long established

precedent, it is known that Mersenne's correspondent

Theodore Haak, whom we have already cited in this paper, had

’^According to H. Rager fThe Anatomv of a Scientific Institution. 23), supported by Sprat ( The History of the Roval Society. 213): "Meetings at Mersenne's academy, after the founder's death, were taken as occasions to set competing philosophies in direct opposition. They often degenerated into sophistic wrangling."

’^Fontanelle, Histoire de 1 'Académie Royale des Sciences. 325: "It may be that these assemblies of Paris may have influenced the birth of many academies in Europe. It is, however, certain that the noblemen who started the Royal Society in London had visited France, and had met at the Mon(t)mor's and Thevenot's. When they returned to England they gathered at Oxford, and continued the practices they had already grown accostumed to in Paris." G. D. Cassini, De 1*Origine et du Progress de L'Astronomie. 121: "Among the many foreigners, Mr. Oldenburg could be singled out. When he went back to England, he inspired the British his project to hold similar assemblies, and so he gave them an opportunity to establish the Royal Society of England." 139 his own group of doctors who met regularly."^ Eventually this group merged with the Gresham College group and the

London groups to form the Royal Society of London. The first historian of the Society wrote;

Their [the Royal Society] meetings were frequented, as their affairs permitted; their proceedings rather by action, than discourse; chiefly attending some particular Trials, in Chymistry or Mechanicks: they had no Rules or Method fix'd: their intention was more, to communicate to each other, their discoveries, which they could make in so narrow a compass, than a united, constant, or regular inquisition. And me thinks, their constitution did bear some resemblance, to the Academy lately begun at Paris: where they have at last turned their thoughts, from Words, to experimental Philosophy, and perhaps in imitation of the Royal Society. Their manner is likewise to assemble in a private house, to reason freely upon the works of Nature; to pass Conjectures, and propose Problems, on any Mathematical or Philosophical Matter, which comes in their way. And this is an Omen, on which I will build some Hope, that as they agree with us in what was done at Oxford, so they will go on further and come by the same degrees, to erect another Royal Society in France."*

Sprat wrote the above before the Académie Rovale des Sciences in Paris had been instituted, but the cell of

"^Unfortunately most of the correspondence between Mersenne and the scholars of England has been lost. However, the existing letters do show the influence of Mersenne. The communications of the results obtained in studying the cycloid or the barometric experiment, as fruit of the concerted efforts of the academicians, must have been inspirational to the British. Writing to Mersenne in August 6, 1647, T. de Haak said: "What could be the purpose of Mr. La Maire to be so reserved about his inventions for the common good? To whom or what does a hidden chandelier benefit? It is far better to know and have much less than to miss the real joy which is in communicating, and making others part of our joy?" I Correspondance. vol. 15, 354-55). Mersenne had been writing exactly along the same lines before he had his Academy.

148Sprat,, T. History of the Roval Society. 134. 14 0

Mersenne (even after his death) and the other cultural

centers we mentioned before were still active. Therefore

Sprat's mention of "Academy at Paris" cannot refer to the

official academy that eventually absorbed the former. He

cannot be referring to the Académie de France which was

chartered in 1633, because the goal of this latter academy

was purely in the humanities, and besides, by this time it

had already stopped functioning. It cannot be established

that he was referring to Mersenne's academy either, but, on

the other hand, the acquaintances of Mersenne in London were

too well known to exclude the possibility of this reference.

The fact that the meetings in London "bear resemblance to

the Academy lately begun at Paris" on the one hand, and

"they [the people in Paris] have turned to experimental

Philosophy, in imitation of the Royal Society . . . they

will come to erect another Royal Society in France" on the

other hand, may cause some confusion. One reasonable

explanation could be that at the beginning the British

academy may have looked at the Parisian cells as model, but when in 1662 the Royal Society was formally chartered by

Charles VII, the roles of the two groups may have switched.

One more reason to see Mersenne's influence is the familiarity that the members at the Royal Society had with

Mersenne's publications. In fact, years after Mersenne's death, his Les Mechanicmes de Galilée and the Nouvelles

Pensees de Galilée were read at the meetings of the Royal 141

Society. Furthermore, Boyle’^ was assigned to read and

study the two books and to the present their contents to the

Society.

One could also try to establish how much the

Academy of Sweden was influenced by the fervor of

cooperative research in Paris. In fact, the two main

"°The editors of Mersenne's Correspondance hint at the possibility that De Haak might have introduced R. Boyle and his activity in chemistry to Mersenne and sent him his portrait, but he did not mention Mersenne by name. I Correspondance. vol. 15, 248)

’“it is certain, however, that the two cultural centers were corresponding with each other and therefore influenced each other, even though the two countries were at war. Not only that, but some scholars, like Huygens, Sorbiere, and Newton, belonged to both groups, thus reinforcing the cooperation. A letter from the first secretary of the Royal Society to R. Boyle shows how academic internationalism was pervading these circles: "Mr. Auzout was elected into the Society, nemine contradicente. and a diploma is to be dispatched to him, as was done to Mr. Hevelius. The same, I find from my last dispatch from Paris, is nominated for one of those choice personnes that are to constitute their academy; some of the rest that are pitched upon, being Mr. Roberval, Mr. Carcavy, Mr. Frenicle, Mr. Picard, Mr. Huygens, all very able men, appointed to meet and to consider the best way of framing a philosophical Society, and the best method of carrying on its design. ..." And he continues: "Something similar could also be started in Denmark, because there are there some very capable and intelligent elements. ... as Erasmus Bertholin the mathematician, Thomas Bartholin the physician and Steno the anatomist. . . . I hope our Society will in time ferment all Europe at least. I wish only we had some more zeal, and a great more deal of assistance, to do our work thoroughly, as I am apt to believe The French will study to do theirs (they being likely to be endowed) were it but out of emulation. So good be done to our generation, and a ground laid to do the like to posterity, no great matter what passions do concur for the performance." (T. Birch, The Works of Robert Bovle. vol. 6, 227) The ground laid by R. Bacon in England and Mersenne in France was at last starting to bear fruit. 142

figures of this Academy, Descartes and Chanut, were on very

good relations with Mersenne (Descartes) and with Pascal

(Chanut). However, due to the lack of an individual willing

to play the role of Mersenne in the Academy of Sweden, its

short lifespan makes the efforts of no relevance.’®’

The above study has described the effectiveness of

Mersenne's role as an inspiration to the scientists of his

time and the success of cooperative efforts.

’®’Actually according to Fontanelle (Histoire de 1 'Académie Rovale des Sciences. 231), even the Academia dei Lincei was inspired by the Académie Rovale des Sciences: "Finally the renewal of the true philosophy has made the academies of mathematics and physics so necessary, that they have been installed even in Italy (even though these types of Sciences do not prosper in those lands both because of the sensitivity of the Italians, who cannot suffer this kind of pains, and because of the ecclesiastical authority who has reduced the need of these studies to useless, and possibly dangerous, exercises. The most famous Academy of this type in Italy is the one of Florence, founded by the Grand Duke. It has produced Galileo, Torricelli, Borelli, Redi, Bellini, illustrious names who gave witness to the talents of the nation." The historical value of Fontenelle's testimony has already been contested before. The above statement could, however, be interpreted to mean that the French cultural revival through Mersenne's Academy has stirred in the Italian scientists a spirit of emulation. In fact, as already mentioned, the Academia del Cimento was started to revitalize the extinct Florentine academy. CHAPTER FOUR

MERSENNE, THE EDUCATOR

AND THE MENTOR

The end of the sixteenth and the beginning of the

seventeenth centuries are characterized by a resurgence of

interest not only in natural sciences but also in the pedagogical sciences. During this period the most notable men are Juan Luis Vives in Spain, Sir Francis Bacon in

England (with his Novus Oraanum. which Mersenne read and critiqued in his La Vérité des Sciences), the Italian Tomaso

Campanella, and, more than anyone, the Moravian Bishop

Komenski. Many other figures of lesser prominence also promoted changes in the educational system of the epoch.’

All of these were concentrating their attention not only on

’Deserving mention is the Polish Rabbi Loew of Prague (born between 1512 and 1520, and died 1609). He did not leave any treatise on education, but expressed his pedagogical theory in the expositions and commentaries of the Bible. H. Stransky writes about him that "despite his associative style he was a systematic thinker. Underlying his expositions and commentaries we find a profound and systematic pedagogic theory, from which he draws logical inferences regarding aims, potentialities and limitations of education, the curriculum and teaching methods." Stransky, "Rabbi Judah Loew of Prague and Jan Amos Comenius — two reformers in Educacation," Comenius. Edited by V. Busik (New York: CSASA, 1972), 104-16.

143 144

the subject matter to be delivered in the classrooms but

also on the forms of communication between student and

teacher.

The editors of Mersenne's correspondence admitted

that the educational efforts of Mersenne are still an open

field for researchers.® Mersenne was in the classroom as a

teacher for only two years; his works do not include any

treatise on education nor did he ever write books for

classroom use. In fact, it should be pointed out

immediately that Mersenne is not an educator with interest

in general in youth on the primary or secondary level of

education. One would wish that the requests for an equal,

universal, and free education which can be found in his

contemporary education-specialists like Comenius,* would

find an echo in Mersenne's books or correspondence;

unfortunately, nothing like that appears. Mersenne's

contacts ranged from people at the royal court, at the

^Correspondance. vol. 15, 351.

*Comenius and Mersenne shared mutual esteem. Even though their correspondence is minimal, nevertheless Mersenne appreciated Comenius' innovative ideas. He even offered his services to help him introduce his methods in France. Mersenne thought that in England the authorities, as well as the public, would be more generous than the French. So he advised him to concentrate his efforts in England first. On his part, Comenius once sent a reply to Mersenne's letter and at other times asked his disciple, Samuel Hartlib, to write on his behalf to Mersenne and express his appreciation. He also asked Hartlib to purchase all of Mrsenne's books for him. 145

Archepiscopal Curia of Cardinal Richelieu, and other high

ranking clergy, down to local government officials in the

towns outside Paris. He must have been aware, therefore, of

the differing goals of education set by the political

interests of the three forces in society: the royal court in

Paris, the municipal authorities — struggling to keep their

colleges open at any cost for the prestige they represented

for the town^ — and the private educational administered by

the Jesuits, the Oratorians and other groups.®

In fact, in 1626, a tract summing up the fears of

the royal authority insisted that there were far too many

colleges in France:

even in the smallest towns of the kingdom, to the great detriment of the state, since, by such means, merchants and even peasants find ways of getting their children to abandon trade and farming in favor of a new profession.®

^"France is the only place in Europe equipped with a network of colleges that is entirely owned and administered by the municipalities. Among the chief characteristics of this network are its phenomenal size and density and its independence from the state and clerical control." (George Huppert, Public Schools in Renaissance France. University of Illinois Press, Urbana and Chicago, 1984), 58.

®It is probably because of the problems the Jesuits were facing with the municipal authorities about their way of running the public schools that Mersenne in La Vérité des Sciences (752) pleads for their case.

®F. Dainville, L*education des Jesuites. Paris, 1978, 127 146

One year later another tract addressed to the

assembled French clergy pointed directly to the worst danger:

It is the ease of access to this bewildering number of colleges that has enabled the meanest artisans to send their children to these schools, where they are taught free of charges - and that is what has ruined everything!

These children, "once they put their nose in a book,

are incapable of productive and useful work from that moment

onward.

Louis XIV's government continued to make efforts to

close down colleges and extend its vigilance to elementary

education, embracing a policy, which, if actually enforced, would have limited the access even to simple literacy: "One

should teach reading, writing and counting only. Writing

should not be taught to those whom providence caused to be born peasants: such children should learn only to read."®

Most of Mersenne's books are addressed to philosophers and scholars as well as to skilled workmen.

But it would have brought great honor to Mersenne if in La

Vérité des Science or in L'lmoiete des Deistes or other tracts, besides defending truth he would have also found new ways of spreading knowledge to the lower strata of the populace, giving them the first means (through education) to

'Ibid.

®Ibid. 134. 147 reach it.® Besides defending the cause of the Jesuits, bons

Catholiques et aussi bon Mathématiciennes.’® he should have reacted to the official position of the royal court and to that of the higher ranks of the clergy. It is disturbing that he, the son of a modest workman, who was educated by means of a scholarship, did not side in his publications explicitly with the bourgeoisie in fighting against the machinations of the royalists and, presumably, the high ranking clergy by decrying the goals of restricting the curriculum in the primary schools to reading, writing, and counting. It was not enough to address books to the scholars, clergy and skilled workmen, and to encourage cooperation among these groups, without explaining how the ordinary workmen were to become educated if the cooperation was to be fruitful.

The above remarks are much more appropriate when one realizes that in the very community to which Mersenne

®One of the very original features of Comenius is his demand for a single school for all. This democratic wish ran against all institutions of the time, particularly those of the Jesuits, who aimed only at society's elite. In his main work on education. The Great Didactic he wrote: "All children, boys and girls, those of the rich and those of the poor, those of the nobles and those of the ploughman, from the great cities and the small towns, from the villages and the hamlets should enter the schools on a firm basis of equality." (chapter IX, 23). This demand arises from his connections with the school as "a workshop where the product is men" and where, to everyone, is taught everything that is worthwhile knowing (chapter XI, 30).

’®M. Mersenne, La Vérité des Sciences. 752. 148 belonged, the Place Royal, a young friar, namely Nicholas

Barre (1621-1670), who was born of a wealthy bourgeois

family, in 1644 was allowed to start two associations of religious women to teach children of peasants and workers in the elementary grades at no cost. These associations, started at Rouen, soon spread to Reims, Dijon, Charlon-sur-

Saon, Marseille and even Paris," much earlier than the famous Congregation of the Brothers of the Christian

Doctrine. We have no records that Mersenne encouraged Barre in his project.

One has to conclude that Mersenne was interested not so much in mass education as he was in the advancement of the sciences; his main concern was not so much the primary education of children, as it was the progress of the talented young men who were already at an advanced stage of their education, or had just completed their formal education. Because of his need for financial support to publish his books, it is very likely that Mersenne favored his contacts with the nobility and the rich. Following the custom of the times, his publications were dedicated to people in high position in the Order, in the Church, and in society; and the dedicatory letters are very flattering to the patrons' interests in higher learning and research. He

"p. S. F. Whitmore, The Order of the Minims. 325; see Chanoine Farcy, Le R. P. Barre. Religieuse Minime (Paris: Privately printed, 1942), 15. 149

never mentioned any efforts on the part of any patron to

make education available to everyone. However, it must be

said that Mersenne was far more moderate in his flattery

than many of his contemporaries.

In Mersenne's defense, one has to admit that none of

the above-mentioned educators come close to Mersenne's

stature in the natural and mathematical sciences; the

talented students and young scholars of the time did not

attract the attention of the classical educators like

Comenius. Besides, the educational value of his works and

correspondence cannot be minimized. Finally, he was

influential in inspiring the career choices of many young

students as well as in motivating the progress of their

studies, as it will be shown later in the case of Christiaan

Huygens. Even though he did not go out of his way in search

of such talents, once he estabilished contact with them, he

continued to write to them, encourage them, and recommend

them to the closest scholar he knew in their vicinity. To

be of greater assistance to them, he sought to know and

appreciate many educators, and to follow closely their

innovative ideas on education. In particular, he

corresponded with Komenski and Ruarus, the most advanced promoters of education of youth. To all this, one can add that he was soon interested in the use of language as a 150

means to promote education, and in the advancement of women’®

and the handicapped (in particular the deaf’*) through

education. However, Mersenne's greatest contribution to

education consists of fostering the establishment of modern

scientific standards of methodology; this gives him a unique position among the outstanding educators of the Renaissance.

Mersenne's Scientific Methodology

Though Mersenne did not write a treatise on scientific methods as did his friends Descartes and

Comenius, it is not hard to sketch his approach to scientific research from his publications and correspondence.

’®Here again Mersenne's attention is limited to a few talented women who were writing books in mathematics and poetry. (See Correspondance. vol. 7, 213, 218; vol 14, 609, 638 footnote)

’*In 1630 Mersenne read Jacques Roland's book Aalossostomographia. ou Description d'un Bouche sans Langue. (Claude Girard et Daniel I'Erpiniere, 1630) who described a young man without a tongue, but with the vocal organs so formed that he could speak almost regularly. Mersenne wrote to Gassendi, Peiresc and Beeckman to learn if they knew any similar cases. He himself knew of a certain Mr. Bene, born deaf, who could speak, and read, and write like anyone else. After seven years of research, Mersenne concluded that a deaf person could be taught to speak if he were shown the object, and he could first touch it, smell it and taste it, if possible, and then was trained to move his tongue in the right way. See Correspondance. vol. 2, 278; vol. 3, 367, 379; vol. 4, 590; vol. 16, 375; "Traite de la Voix" vol. 2, prop. 51, 77; "Dé L'Utilite de L'Harmonie", 44, both of them in Harmonie Universelle. 151

First of all, Mersenne felt the urgency of rooting

the sciences on a firm foundation. For this reason, he saw

mathematics and philosophy as sciences whose first

principles or axioms are laid down arbitrarily by the

scientist himself. The structure built on these principles,

therefore, will be solid only if the theorems and

corollaries follow logically from the initially estabilished

axioms."

However, the laws of the natural sciences are

independent of human choices or decisions. The following two passages from the dedicatory letter of Les Mechanicmes to the brother of the King, explain Mersenne's approach to such sciences:

I think that the order and the wonderful laws that nature follows in the motive forces will provide you with great pleasure, because you will be struck by perpetual equilibrium and justice that one can notice among force, reactions, time, velocity and displacement. Such an equilibrium is constantly protecting itself.’®

"For Mersenne arithmetic, even before geometry, is a divine attribute. Before Leibniz crystallized the idea with the famous statement: Deus dum calculât fit mundus. Mersenne had already stated: "God had mathematics as a first model and the prototype of his reasoning when he was creating the world." (La Vérité des Sciences. 283). The next chapter will deal more in detail with Mersenne's philosophy of science, with special emphasis on mathematics.

’®M. Mersenne, Les Mechanicmes de Galilée, dedicatory letter, non numbered pages. This concept of equilibrium in nature is a leit motif in Mersenne's books and is the guiding principle in his research. In the "Book of the Instruments of Percussion" he laid down the principle: "Nature, which does not lose on one side if it is not gaining on another, is like a balance which lowers one arm by the same amount that it raises the other, so that everything keeps the equilibrium established 152

He concluded his letter saying; "I believe that if

justice could speak up it would manifestly confess that

there is no other natural science that is closer to it than

Mechanics.

From this follows that in physics as well as in the

other applied sciences, scientists are not free to choose

the first principles and build their speculations on them.

Mersenne therefore renounced a purely speculative approach

to abstraction." His advice to physicists is to observe

nature, try to describe it in a quantitative way, and to

deduce some useful applications from it. Especially later

in life, he tended to withdraw from the circles of

philosophers with their long and useless discussions on

abstract speculations and on metaphysical interests. He

preferred, like Diogenes with the atomist Zeno, to spend his

time examining the anatomy of a gnat:

. . . which, by itself contains and encloses more wonders than the skills of all men together can reproduce; so that if scientists could understand all the resources enclosed in this small animal, or learn

by God at the beginning." (Harmonie Universelle, book 7, 20). I

’®M. Mersenne, Les Mechanicmes de Galilée, introductory pages. In 1625 he had already stated about physics: "Statics, hydraulics and pneumatics produce such wonderful effects that one is moved to say that humans can imitate the most amazing works of God ..." (La Vérité des Sciences. I.e.. The pages are also not numbered in the reprinted edition of Stuttgart-Bad Cannstatt.

"See L. Auger, "Le R. P. Mersenne et la Physique" Revue d*Histoire des Sciences. 2, (1948), 32-52; R. Lenoble, Mersenne. ou la Naissance du Mechanisms. 388-390. 153

how to produce automata and machines with such ability to move, everything that humanity has so far produced in fruit, gold or silver would not be enough to pay for the simple view of such machinery.’®

It is not that philosophy is thoroughly useless. In

fact he admits that it can contribute to the advancement of

science, but only if philosophers would record faithfully

their experiences and experiments. Unfortunately, he

concluded, "this will never happen until I'honestes hommes

(i. e. the educated men) will accept this role. . . ."® For

example, Mersenne respected and admired Aristotle above any

other philosopher; he called him "an eagle on philosophy,

while the others are only like chickens trying to fly before

growing their wings. He in fact was able to transcend

everything that was sensible and imaginable"®®. He wrote

more than three chapters analyzing his philosophy, in

response to the objections of the alchemist (the third

person, besides the philosopher and the skeptic present in

the dialogue described in La Vérité) to Aristotle's personal

life and his work.®’ On the other hand though, he did not

’®R. Lenoble, Mersenne. ou la naissance du mechanisme. 335.

’®M. Mersenne, Questions inouves. 125-126.

®°M. Mersenne, La Vérité des Sciences. 109-110.

®’The first point that is evident in his studies is that Mersenne's scientific position does not represent an abrupt break with the past. According to him, in science we do not have unjustified rejections of the established principles. The latest discoveries and hypotheses are generally brought about from existing principles. For a new law to be acceptable it must be able to explain the phenomena that the old law explained, at least 154 hesitate to acknowledge how, in the applied sciences,

Aristotle had made several mistakes. In particular, he accused him of applying to the natural sciences the principles of the speculative science of mathematics.®® To

with a comparable convincing argument. This is true in philosophy, in music, in geometry as well as in any other science. Therefore, before starting any new project, Mersenne suggested one should inquire how much has been accomplished in the past. This was, in fact, the main objection that Mersenne had to Bacon's Novum Oraanum: "I say that he should have consulted the scholars of various countries and kingdoms, before stating or fixing so many laws." His caution did not preclude the possibility of further steps, both in theory as well as in practice. Hence, contemporary scientists could outperform any of the past. Scientists and nature have many more secrets to reveal to humanity. It was reasonable, therefore, to admit the limitations of Aristotle himself, as well as those of any other scientist. This was particularly apparent in his relation with Galileo, Descartes, or even Aristotle. Mersenne did not feel overwhelmed by the personality of anyone: a scientist gained authority only by the validity of his argumentations.

®®For example, Mersenne insisted that the concept of infinity did not allow one to extend it to an infinite universe. In La Vérité des Sciences and the L'Impieté des Deistes. Mersenne did not accept the assertion by Giordano Bruno: "An infinite effect of a infinite Cause" because, Mersenne said: the Cause acts freely and not by necessity." But later he started to have doubts. For example in 1634 in the Questions Inouves he stated: "I leave aside many other points of which I doubt I will ever be able to find a certain solution. For example, whether the earth or the sun is at the center of the Universe, if this has an infinite extension to justify their position that a passive infinite being may reflect an active infinite being, that the real space, which mahy think of as infinite, is full of matter . . . and thousands of other questions that others take for granted, even though they could be doubtful, and even false." (See Correspondance. vol.3, 273-289). But then he had a complete change of mind. In 1636, writing to Doctor Rey, he wrote: "Why cannot you admit that an infinite Cause may not have an infinite effect. In the past I held a different position, but the solution is not difficult to prove." (Ibid.. vol. 4, 58). 155

excuse his mentor, Mersenne made a distinction between the

philosophe sensuel (represented by Aristotle in natural

science) and the philosophe intellectuelle et reasonable®*.

The reason Mersenne was opposed to any philosophical

principle in physics is because he had a basic mistrust of

our senses' ability to reach the truth; he felt they could

be deceived very easily. In the Questions Theoloaiques he

wrote:

We cannot ever reach the real reason, or the knowledge that attains certitude. In fact there will always be some circumstances or instances which make us doubt whether the causes that we hypothesize are really true even if there are true answers; or if other answers are equally true. Therefore I think that no one could request from the most educated people anything other than their observations and the remarks that they have made on the various effects or phenomena of nature.*

This attitude of dissatisfaction with and mistrust in his own ability to judge and interpret the results of his experiments made him a victim of the very skepticism that he had fought against for many years. The incapacity of the experimenter to discern the causes from the effects he

®*Mersenne himself, though, is culpable of the same mistakes he condemned in Aristotle. For example, he applied the model of social behaviour to the physical inanimate universe. In the monastery the Superior is expected to visit often the houses of the Qrder. Mersenne asked: "The sun was created by God to preside over the movements of the planets. It is therefore its duty to travel in the sky and visit its subalterns. Why should not the sun move? Maybe because of its nobler state? But this is exactly why it should move: the nobler it is the more it should move." (Questions theoloaiques. 243).

®^. Mersenne, Questions Theoloaiques. 18. 156 produced in the laboratory left him a deeply committed pessimist. In a letter to John Pell, writing about the properties of light, he said:

But why it refracts following this or that ratio, or through this or that angle rather than any other angle, or why a mirror reflects the rays of light; and why the radius AB is not absorbed and consumed at the point B, repelling the mirror BD, but rather it is reflected at C? Videbimus effectum. ignoramus causam.

But then why does a stone fall, and what is a heavy body, and why is it attracted by the earth? Here also we hesitate to give an answer: maybe in order for the part to join the whole? Or maybe there is another reason... lanari vivimus. ianari moriemur until the day of eternity will dawn." (emphasis added).®®

®®M. Mersenne, Correspondance. vol. 9, 54. Actually Mersenne did try to explain the cause of the laws of reflection. In the Coaitata (213) he wrote: "It seems that there is a very great difficulty to understand why the radius AB is reflected fron B to C, and why it is not absorbed there (like a plane well which rejects any more water)." In his posthumous L'Optioue et la Catoptricme he vacillates between an atomistic concept and a "less dense mass." In the above mentioned letter to Pell (55) he continued: "If light is an Aristotelian quality of the mass, what makes it reflect? But if we take it to be the movement of very tiny particles with a spherical shape and with a very hard surface, then it is easier to understand the phenomenon of reflection. Because we know by experience that the balls of the tripod and of ivory, silver, gold or marble bounce backward more strongly and the farther if they collide against a wall or any other impediment on their way." Note that Mersenne here takes the atomistic explanation of reflection from his friend Gassendi just as a simple mental concept, without any positive reason in its favor. Mersenne cannot endorse it because, he wrote, if it were true, we should have a grey region where two opposite light rays intercept. A true remark, which his young 157

This mistrust in his own ability to find the true

values in the experiments very often made him change the

numerical values he had found. For example in Ballistics he

gave the velocity of sound to be 230 royal feet per second

(about 300 meters per second), which he confirmed in the

L'Harmonie Universelle; but in a letter to Rivet he gave the

value of 300 toises per second (about 450 meters per

second), which he reported in his Novarum Observationum

Tomus III.”

Partially to overcome this lack of certitude,

Mersenne said that the scholar has at his disposal three

means: the use of experiments, the light of reason and the

use of physico-mathematical processes together with the

analogies in one field of physics with another. Physics, or

any other natural science, is just the science of the

phenomena, where experiments substitute for syllogistic

argumentation and mathematical formulation takes the place

of the philosophical principles.

We may not know what light is, why it reflects,

refracts or is absorbed, but we can make use of this

correspondent Huygens used later to explain the interference effect by his wave-like theory of light. Using the same approach he explained also the cause of the non-reflecting surfaces: the ray of incident light or the surface on which it collides get so deformed that the ray is not reflected but is absorbed.

“see Mersenne, Ballistics 138-140; L'Harmonie Unverselle. 427; Correspondance vol. 15, 669; Novarum Observationum 127-128. 158

phenomenon;®' we know that colors are the effect of the

reflection and refraction of light, but we cannot know the

details of the particular reflection that causes the

different colors.®* The nature of heat is a mystery, and the

secret force that moves the magnetic needle is unknown [we

call it an occult force], but we can describe the phenomena

and repeat them as often as we choose by exact experiments.®*

For Mersenne, therefore, experiments are the means

to conquer the secrets of nature and to break the barrier of

communication between nature and man.** A qualitative

description of a phenomenon, or its casual observation does

not build up science: the scientist must be able to produce

the phenomena in the laboratory, quantify the causes and

effects, and report the findings with honesty. To use vague

and unquantifiable terms to explain the effects is

equivalent to accepting defeat. He often said that

explanation by sympathy or antipathy is a sign of human

limitation. In the Harmonie Universelle, he wrote:

People have introduced the concept of sympathy and antipathy, or the occult qualities in the arts and

®'M. Mersenne, Questions Theoloaiques. 63.

“Ibid.

®°R. Lenoble, Mersenne. ou la Naissance du Mécanisme. 354.

*°A scientist who would presume to investigate science independently from experiments would be dubbed by Mersenne as "chartaceos philosophes who pretends to learn not from the inspection of nature but from books only." (Correspondance. vol.8, 722, footnote 6.) 159

sciences to cover up their own faults and as an excuse for their ignorance, or as a way to admit that they do not know anything: because it is exactly the same thing to say that the chords that play at unison are oscillating according to a ratio of the sympathy that they have in common, or to admit that one does not know the cause. ...svmoathv is confused with ignorance as I can show you in the case of the cords playing in unison. (Emphasis added)

But experiments, to be useful, must attain certain

standards that determine their convincing force. Here are

few of the standards set by Mersenne:

I. To be reliable an experiment must be all-encompassing,

i.e., it must study and test all the variables of the

problem. We can deduce a rule only when our

observations have covered "all that appears to have some

sort of regulated motion." Lenoble sees in this

statement a perfect description of the modern concept of

physical phenomenon. In order to study the effect of

all the variables involved in a phenomenon, Mersenne

suggested that in each trial only one variable be

changed in the first moment, keeping the others

constant. In a second moment, two or more variables can

be changed and the test repeated to verify that the

phenomenon can be duplicated. Finally, a relation

between the effect and all the variables can be

*’M. Mersenne, "Traite des consonances" in Harmonie Universelle. 63. 160

researched. Mersenne here gaves a rule that anticipated

the empiricists by at least a dozen years.“

“Lenoble, Mersenne. ou la Naissance du Mécanisme^ 361. This is how Mersenne found the relation between the frequency of a musical note, the length of a string and the density of the material of which the string is made, the tension to which it is subjected, and its cross sectional thickness. The Greeks already had conjectured that the pitch of the sound emitted by a vibrating string or any other instrument depended on the frequency of the impulse transmitted through the air. In the sixteenth century many musicians and music theoreticians (like Giambattista Benedetti and Vincenzo Galilei, Galileo's father, the Capuchin Zorzi of Venice), had established that pitch was proportional to the frequency of the vibrating string and hence the traditional music intervals (octave 2:1, fifth 3:2, fourth 4:3 and so on) were ratios of frequencies of vibrations, whatever instrument produced them. Mersenne wanted to verify these rules experimentally for the string instruments and compare the results with the wind and the percussion instruments. For this purpose he used two strings stretched on a common frame. One was the control and the other was subjected to all the alterations, one at a time, until it sounded in unison with the control. In this way he could establish quantitatively that frequency was inversely proportional to the length and the density, and directly proportional to the square root of the tension from which the well known formula can be written:

f = (1/L) J T/d where f is the frequency, L the length of the string, d is its density, and T its tension. Thus, he was able to correct a centuries-old formula proposed by Pythagoras and Ptolemy. In the process Mersenne experimented with strings of gold, steel, cat's intestines, and other materials. He drew a table of densities of about fifty different metals and other materials. The vibrations of a small pendulum, that he kept always handy, or his pulse beat served as a measurement of the time of the vibrations. In this way he established experimentally for the first time that musical pitch was determined by the frequency of the vibrating string. 161

II. The experiment must be repeatable and verifiable in the

laboratory as many times as necessary to reach a

consensus among researchers:

One cannot think that an experiment is acceptable if one cannot detect the same results many times, because one effect could be obtained through many accidental ways . . . I wanted to remark this point because the experimenters should be very careful before they try to deduce some generalizations from their experiments.“

III. The results of an experiment must be verified or

witnessed by other scientists reliable for their

scholarship and honesty. For this reason, Mersenne

dared to challenge Galileo about his deductions of

motion on inclined planes and asked that other

scientists perform the experiment themselves and prove

him correct. He wrote: "I would like that many others

would perform this test on planes of various

inclinations with every possible care, to see whether

their results correspond to ours.

IV. In a dispute among different scientific positions, the

outcome of the experiments should settle the arguments.

When arguing about certain statements made by Galileo on

the motion along an inclined plane and free-falling

33Ibid.. 365.

*M. Mersenne, Questions Inouyes. 153-154; see L. Auger's "Le R. P. Mersenne et la Physique," Revue d'Histoire de la Phvsioue 2 (1948): 33-52. In his reports Mersenne was careful to name the witnesses present. 162

bodies, Mersenne challenged him to support his claims

with the outcomes of experiments:

I doubt that Mr. Galileo performed the experiments of free-falling bodies along the plane, since he does not mention them at all, and the proportions that he gives are against common experience.*®

V. The laws derived experimentally cannot be generalized

indiscriminately to include cases not contemplated in

the experiments. They can help, however, to establish

some reasonable working hypotheses. This holds true in

particular in the case of generalizing the laws of

infinity proved in a finite situation.*® Lenoble in

reporting the above last rule comments: "The rule is

excellent and even modern."*' The comment may be

extended to the other rules as well.

An accurate experiment is one step only, in fact,

not even the first one, because before performing an

experiment one has first to feel the need for the experiment

itself. The history of science shows that science resides

primarily in the mind and in the interpretation of

experimental facts, and only in the second instance in the

*®M. Mersenne, Les Mechanicmes de Galilée. 126.

*®In Novarum Observationum Tomus III, vol 3, 84, Mersenne stated: "I do not think that the large or small ratios that prove to be true in experimental physics hold also in infinity." And again: "Nature does not observe the same proportions in infinity." Ibid.. 85.

*'R. Lenoble, Mersenne. ou la Naissance du Mécanisme. 357. 163

perception of the human senses alone or with the help of

external instruments. Human senses can be deceived by the

most accurate experiment. Therefore, there is a need to

interpret the outcomes of experiments and to justify the

deductions, because nature is rational. Mersenne questioned the claim of historians that the people of Syracuse defended themselves by burning the attacking fleet of the Romans using some parabolic mirrors. He concluded: "These historians offer too doubtful reports and cannot be trusted without detriment of reason." (stress added). The fact is that the witnesses - if the historians did in fact interview them - did not know whether it could be possible, at least at that time, to burn a whole fleet using parabolic mirrors, and, Mersenne added, thev could not know it. Therefore, it must be the task of the historian to check the validity of the claims.*

The same applies to the frequently adopted practice of proving a person guilty of an alleged homicide by having the defendant pass the body of the victim. If the body started bleeding, that was a sign of culpability of the accused and on this basis he was sentenced to death or torture. The justification was that there was a natural antipathy between the victim and the defendant even after

“Reported by R. Lenoble "A propos du tricentenaire de la Mort de Mersenne," Archives Internationelles d'Hisotire de Science 28 (1948): 583. 164

death. Mersenne, conceding that the bleeding may have been

observed by testifying witnesses, questioned the

interpretation of the phenomenon; whereupon he not only rejected the method, but asked the judges to look seriously

into the alleged bleeding and make sure that someone was not trying to deceive the whole audience.

Mersenne had one more point to clarify: the interrelation among the various branches of science. He began his discussion on music with the declaration: "Nobody can reach perfection in music, nor understand it nr discuss it, unless he is willing to combine the principles of physics and medicine with mathematical reasoning."*

Actually, Mersenne was not the first to use and require from scientists, consideration of the intimate relation among the various fields of science. The use he made of this principle made him the first to experiment with and to quantify mathematical formulations with various frequently- accepted statements.

The correct application of the above described methodology helps the experimenter choose which experiment to perform and which to discard beforehand, because he can foretell its usefulness. In the above-mentioned dedicatory letter of les Mechanicrues. Mersenne explains the limits of nature:

*M. Mersenne, Ouaestiones celeberrimae in Genesim. 1696b. 165

Nature cannot be deceived nor give up its rights; no resistance can be overcome in any way but by a stronger force. . . . Consequently machines cannot be used to lift a greater weight than an equal force could lift without the machine. This is why one has to explain the real usefulness of the machines, so that people would not tire themselves uselessly, and so that their reseach could come to a happy conclusion.*

This basic principle helped Mersenne to prove that perpetual motion, for example, is impossible. And this is not, as others tried to explain, because of a "defect of nature" but because "nothing can be lifted beyond the source of its motion" in a frictionless path of motion; friction in fact is, by its very nature, a dissipative force. However, friction can never be ruled out; therefore perpetual motion in actual space is impossible. One needs an internal renewable force (like the heart to keep blood circulation).

A pendulum in motion, for example, does not keep its initial amplitude unless it oscillates in a vacuum; the air resistance, which cannot be avoided, will dampen the oscillations until the pendulum eventually will come to a complete rest.*'

Mersenne could lay down these fundamental laws in science because he himself was a refined experimenter

. . .who set out to lay these laws and improve the whole conception and practice of what he called "well arranged and well made experiments." By deliberately combining the methods of philosophy with those of technology in a

*M. Mersenne, Les Mechanicrues de Galilée, page without numeration.

*'M. Mersenne, L'Harmonie Universelle, book 2, 135; R. Lenoble, Mersenne. ou la Naissance de Mechanisme. 358-59. 166

systematic quantitative exploration of a whole field, looking at once for a common form of explanation and for accurately reproducible results .... His insistence on the careful specification of the experimental procedures, use of control, repetition of the experiments, publication of the numerical results of actual measurements as well as those from theory, recognition of approximations, and estimation of experimental errors marked a notable step in the experimental method.*

His experiments at the top of the dome of St.

Peter's in Rome and with Doraison at the Chataeu d'Orange, on free-falling bodies, for example, led him to the concept of the limiting speed.*

Not only did he conduct his own experiments, but he often checked claims by other experimenters, or was requested to perform experiments for others. Descartes

*A. C. Crombie, "Marin Mersenne and the Seventeenth Century Problem of Scientific Acceptability," in Phvsis 17 (1975), 186-204. See for example the accuracy with which he determined the error in the frequency of oscillations of two strings with different amplitudes in L'Usage du Ouadran. 16. It is instructive, for example, to see the method Mersenne used to measure the speed of sound: he first had a canon fired at a known distance. He carefully measured with his pendulum the interval between the instant he saw the spark of light and the instant the sound of the blast reached his ears. He improved the experiment by first training himself to shout the syllables in the verse Benedicam Domino in exactly one second. He then found a cave where the echo of the first syllable would reach his ears at exactly the instant he had finished the verse. The distance from his position to the reflecting walls had the same value as the speed of sound. The results obtained ranged from 320 m/sec to 450 m/sec, and he discovered that the speed was affected by atmospheric conditions. He therefore suggested that the experiments be performed under identical conditions. See also letter of Aime de Gaigneres. Correspondence vol. 6, 196.

*See R. Lenoble, Mersenne. ou la Naissance du Mechanisme. 389-390. 167

often made use of his friend's services to check his own

theories.

Until the end of his life he never lost the

enthusiasm for the most daring experiments. It has been

already pointed out that if the barometric discovery is not

linked to Mersenne's name, this was only because his friend

Le Tenneur refused to carry out the experiment at the top of

the Puy-Dome. Two weeks after Mersenne's death, P. Petit

performed the experiment for Blaise Pascal, who now is given

credit for having provided the final proof that height of

the column of mercury in the barometer is a function of the

atmospheric pressure. But one year before the experiment

was performed, Mersenne had written to Constantin Huygens

about his willingness to perform the experiment himself;

If we had here a high mountain like the Pic Tenarif [in South America] I would climb it with some mercury and some glass tubes to see for myself whether the vacuum theory could be decisively proved in the way I have stated it in my books of observation.**

With the same enthusiasm he also described to

Christiaan Huygens and to many other young scientists the news that reached him from the youn Polish Desnouyers about a "flying dragon." He already was planning, rather naively, to be among the first to use the machine, and, to travel to

China, America, and elsewhere to test it. One can easily

**M. Mersenne, correspondance. vol. 16, 12; see Cogitata Phisico-Mathematica, 109. 168

imagine the effects of this enthusiasm on his young

correspondents throughout Europe.

Christiaan Huygens

The relation between Mersenne and the young Christiaan

Huygens represents a case of the warmest and most fruitful

friendship that Mersenne ever had. Mersenne was in

correspondence with Christiaan's father Constantin for a

long time. Perhaps he was introduced to Constantin Huygens by Descartes, during the letter's self-imposed exile in

Holland; or by Beeckman, during Mersenne's visit to Holland.

Costantin shared with Mersenne a keen interest in music and

in experimental physics. He had five children; four sons

(Constantin, Christiaan, Ludwijck and Philip) and a daughter

(Susan). After the death of his wife, Constantin was more involved in his children's education, which was the best that his resources as engineer in the Dutch artillery could afford. Descartes was a friend of the family and visited them quite often. In fact, he perceived the singular acumen of Christiaan for science and encouraged him to pursue his studies, but not quite as enthusiastically as the boy's father had expected. After his local schooling, Christiaan, with his older brother, was sent to the University of Leyden to pursue his education. In mathematics the brothers had the disciple of Descartes, Franz van Schooten, for their 169

teacher. Afterwards they moved to Breda where Christiaan

could attend some of the public lectures given by the

British mathematician John Pell.

In 1646, while he was attending lectures at Leyden,

Christiaan wrote a paper about mathematics to his brother

Constantin. Mocking Constantin for his weakness in the

Galilean theory of motion, Christiaan stated some theorems that he had found by himself. When he showed them to his

father, the father was so pleased by his son's achievement that he showed the paper to Mersenne, saying;

I have two great sons, the eldest and the one next to him, who have a great desire to see your quadrature of the hyperbola and your center of percussion. And to show you that they are capable of understanding it, I have made a copy of the letter that the ybunger (aged 17) wrote to the above mentioned elder (who is here with me in charge) about his studies on a mathematical theme which he is studying with great success - as he does in all other things, which would be beyond what you could suspect, if I were to enumerate them. I do not know if during the coming winter I will decide to send the two young men to France, but if I do, may I count on your giving them your attention. And, please, do not be surprised by their conversations.*

Christiaan's paper listed four theorems that he claimed to have proven independently. They were;

I. The ratio between the volume generated by the

revolution of a half parabola and the inscribed triangle

is 3/2;

II. The ratio of the volumes generated by the revolution

of a parabola and the circumscribed triangle is 1/2;

45Correspondance., vol. 15, 451. 170

III. The rate of change of position at equal intervals of

times, by a freely falling body in a vacuum, follows the

same proportion of the sequence 1, 3, 5, 7, . . . But if

air resistance is considered, and if the height is

considerable, there is an instant after which the object

will reach a limiting speed, independently of its mass;

IV. If a body at a certain height is given an initial

horizontal velocity (literally a horizontal impulse),

the path it will trace is a parabola.

In reply, Mersenne, in a letter written to

Constantin, the father, simply added an easy mathematical theorem for Christiaan, as a kindness towards the two

"prodigies": Given two [integer] numbers which are the squares, respectively, of the sum and difference of two other numbers, their difference is a perfect square. He gave an outline of the proof, considering first certain special cases fe.g. if a® - b® = 1 then a® - b® = a + b) .

After two weeks, Mersenne wrote a second letter requesting the proof of the third theorem contained in Constantin's paper. He stated two special cases of the problem of the center of percussion of a circular section and an isoceles triangle hanging from the vertex A. To stimulate the interest of Christiaan and his father, he added: "Since you 171

did not answer my previous two letters, I assume that you

have not had much success with the proofs."*

Finally, on October 28, Mersenne addressed his

letter directly to Christiaan, challenging him to provide

the proofs of the third and fourth theorems in the paper

intended for his brother. He presented several objections

which Mersenne thought Huygens might not have considered:

Can a body receive an initial velocity equal to the

instanteneous velocity of a body falling from one or two

miles? Mersenne suggested a negative answer, citing the

case of straw or wool shot from an arquebuse: its smaller

initial velocity cannot be attributed to air resistance.

Besides, if the rate of change in position of a falling body

were to follow the proportion of the odd numbers, one would

need to say that an object falling from rest had to go

through all the "degrees of velocity" (in modern language we

would say that the velocity would increase uniformly), which

cannot be, no matter what Galileo said.* The reason was

*Ibid.

*Mersenne is trying to make Huygens aware of the errors that he had already rejected in his Harmonie Universelle (book 2, proposition VI, 103-107; Proposition VII, 108-112); Nouvelles Observations p. 2-8 and in Liber Praelusoriis proposition 2. 172

that a falling object had already an intrinsic initial

velocity.*

The assertion, that in a vacuum a falling object with an initial horizontal velocity describes a semi­ parabola, is not correct. This would imply that the

impressed horizontal velocity would keep constant, which is

false because of the axiom: violentum non durabile.

*Mersenne is careful to make the young scholar aware of his corrections of Galileo's theory of motion. Mersenne was so strongly convinced of his position that in his "translation" of Galileo's Dialoao he replaced the whole article 1 of book 5 with his own ideas that he entitled On Forced (violent) Movements. According to P. Constabel and P. Lerner in their critical edition of the Nouvelles Pensees de Galilée. (Edizione Nazionale. vol. 7, 191) Galileo stated that the real trajectory of a heavy body released from a tower described a circle whose diameter was equal to the radius of the earth, and he assumed that the readers could transpose the principle to the case of an object on which an initial horizontal velocity has been impressed. In Harmonie Universelle book 2, "Mouvements des corps," proposition 3, 93-96 he rejected the statement and replaced the curve with à second one closer to a parabola, using the Galilean law for the velocity rate. To do this he considered a terrestrial arc subtended by a central angle of 6 and a half degrees. He divided the arc in five equal parts which he joined to the center. A mass free to fall from point A, on which an initial horizontal velocity has been impressed, is subjected to two motions: the constant horizontal motion and the gravitational force towards the center of the earth. If the odd number proportion for the rate of displacement is true, then after the first interval of time the falling object will be at B (see diagram in Appendix Two), after the second at C, and so on until it reaches the center of the earth. The correct trajectory is described by the curve joining the points A, B, C, D, E, and O. In the case that the radii can be considered to be parallel to each other (fall through a short distance) the trajectory will be a parabola. (See "Utilité de l'Harmonie" in Harmonie Universelle, ch. 6, 38). 173

Mersenne stressed his warning: generally Galileo's

principles on motion hold true only for short distances;

over long distances, they tend to fail almost every time.

The letter was a revelation to Christiaan: the proof

by the principle of authority holds only if the statement is

valid on its own.* Christiaan slowly developed a capacity

to critique the statements of other mathematicians and to

stand alone, if necessary, to claim the veracity of his

statements.

The exchange of correspondence between Mersenne and

Christiaan continued for three years, until Mersenne died.

During this period Mersenne opened to him new fields im

mathematics and physics which were developing with great

strides in Paris. He introduced him to Briggs' logarithms.

*°Mersenne was a strenuous defender of "academic independence" in research. Lenoble (Mersenne. ou La Naissnce du Mécanisme. 224, note 3; 360, note 1) reports the following comment by Mersenne about Descartes: "Denying the principles generally accepted and which are taken for granted, is like cutting the Gordian knot without even trying to unfasten it. It is much easier to reject his principles, because we have our own: cmi tenet possessio valet. I am willing to renounce to my principles only when I shall be shown through valid reasoning that they are wrong and untenable. But asking me to dispose of my positions and to endorse his in obseouium fidei Cartesianae. without acceptable reasons, and against my reasoning and my sense . . Mr. Descartes will be understanding enough to excuse us . . . . He has not yet worked any miracles to force us to accept his doctrine, lacking as it is, of sound reasoning, and which is not supernatural." 174 the problem of the vacuum created by a mercury column,* the inverse relation between frequency and chord length in the lute, and to number theory." In theory of music, Mersenne is definitely the main inspirateur of Christiaan through his letters, and in particular through his Harmonie Universelle, without, of course, minimizing the merits of Christiaan's father in teaching his son to play various instruments.*

“Christiaan Huygens made his contribution in proving the existence of a vacuum when he correctly interpreted the phenomenon of the expanding bladder in the vacuum tube as the effect of pressure (see Correspondance. vol.10, 351 note).

"See Oeuvres de Christiaan Huvaens vol.l, 19-20, 46-49, 53-54; vol. 2, 57-558. One of the problems proposed by Mersenne that bewildered Huygens for a long time was the question of how to find the surface area and the volume of an infinite "cylinder" generated by the revolution of a hyperbola equivalent to a finite cylinder. This problem required concepts that the infinitesimal calculus of Cavallieri, Roberval, and Fermat was unable to address yet.

“Everyone in the Huygens family could play some instrument, and at times they gave concerts in their home, attended by Descartes. Christiaan knew how to play the viola da gamba, the harpsichord and the lute. Constantin Huygens, the father, was also a good theoretician in music. He, in fact, contributed to the Traite de l'Harmonie Unverselle with a proposition on p. 354. Mersenne used to call him "the father and patron of music" and refer to him for consultation. In particular on Jan. 12, 1647, Huygens wrote to Mersenne (Correspondance. vol. 15, 44) asking: "What is it, how much and why is it that when a voice, taken at the lowest tone, or many voices together make a sound, one can hear beside that tone the twelfth and the double fifth? This happens also with the chords of the viola . . . But if the voice is raised by one or two tones, one hears only the natural tone, exactly as it happens with the viola." Mersenne gave the solution in the Harmoniae Libri XII (book IV "De Instrumentis Musicis", proposition V). In recognition of his merits, Mersenne dedicated to him the "Traitez des la voix" in the Harmonie Universelle. 175

Christiaan's answers to Mersenne's questions in mathematics

and physics were often surprising and beyond the ordinary

level for his age, to the astonishment and satisfaction of

his mentor, Mersenne. Hence at a certain point Mersenne did

not hesitate to introduce him to the most advanced research conducted in Paris: the search for the centers of percussion,* gravity,®* and oscillation;®® the study of the cycloid and its properties, as well as the investigation of other new curves.®® These were topics discussed by the

®®Mersenne defined the center of percussion of a rigid body rotating about an axis as the point of the body where it would exert the largest force on striking an obstacle placed in its path; or, equivalently, that point in the object about which the force elements in the rotating body are in equilibriun. See Alan Gabey "Huygens and Mechanics" on Studies on Christiaan Huvaens. 198.

“Note that Lucas Valerius had already written a book De Centro Gravitatis which Mersenne had incorporated in his Svnopsis Mathematics in 1626, and reprinted again in his later edition of 1644.

®®The center of oscillation of a rotating object is that point in the center line whose distance from the axis of oscillation is equal to the length of a simple pendulum which is isochronous with that figure. (Christiaan Huygens The Pendulum Clock. 107)

®®About ten years after Mersenne's death, Christiaan, in writing to his teacher F. van Schooten, said: "The discoveries of Mr. Heuraet, that you said are something you never heard about before, seem to be very nice. But I would like to know more details about which curves they refer to, because even though it may be that some lines are "quadrable," certainly this is not true about all curves; but I did not feel like giving an opinion about curves that do not have a quadrature. There are in fact a large number of parabolas and paraboloids, found in the introduction of Mersenne's book on mechanics (the Tractatus Mechanicus Theoricus et Practicus) which are "quadrable" by the rule given there. Right now I find the demonstrations of these rules quite easy, but I did not before. Besides these, I 176 professors of mathematics at the colleges in Paris, at the

Academia of Paris, and by the most learned researchers throughout Europe. Actually, Christiaan, at that stage, was not ready to give an answer to many of the questions proposed by Mersenne. However, he carefully kept the letters for they represented a continuous challenge to him.

He worked for years until he found the solutions. In fact, when he found them, he wrote them on the margins of some of the letters, and often quoted them in his own letters or books, showing how much he treasured them.

Christiaan owed to Mersenne, directly or indirectly, some of his most brilliant discoveries, a fact that he dutifully aknowledged. In particular, a quick look at topics dealt with in Huygens's Horologium Oscillatorium* shows that the basic principles and ideas were sown in his mind by Mersenne at an early age.* Most importantly than did not find any other curve whose dimensions (i. e., the length, area, surface and volume of revolution about an axis and the center of gravity), have been found." (Huygens, C., Oeuvres de Christiaan Huygens, vol. 2) And he expressed the same idea to R. F. Sluse (ibid.) adding that he had developed on his own the demonstration of the rules given in Mersenne's book. Christiaan mentioned and developed Mersenne's section on the curves in his Horologium Oscillatorium. 90 in the original edition.

The book was published in 1673 in Paris, but a shorter edition had already appeared in 1658 under the title Horologium, again in Paris.

* For example, the use of the funeoendule to measure time, the concepts of the isochronous oscillations, of the center of percussion of a plane or solid figure to find the length of an ideal simple pendulum of equal frequency; and the need of constraining the pendulum to oscillate between 177

anything else, the basic principle attributed to Torricelli

or to Mersenne underlies the whole book and is the first

hypothesis listed in the Horologium Oscillatorium: Two or

more bodies joined together cannot start moving unless their

common center of gravity descends.*

The influence of Mersenne's letters on the young

Huygens affected decisively the choices he made later in his

life. His academic education, following his father's plan

for both sons, prepared him for a career in law, but during

his school years his preferences were different.* Later,

his interest in mathematics and in Mersenne's challenges

caused him to ignore law; and he devoted himself totally to

scientific research, while the elder brother pursued his

father's plans.®’ The effectiveness of Mersenne's influence

two fixed jaws to obtain isochronous oscillations.

*See Huygens, The Pendulum Clock. 108; P. Duhem, Les Origins de la Statioue. vol. 2, 1-6. This principle is the most important single axiom in Huygens' entire mechanics.

®®In 1649 Christiaan wrote to his brother Constantin: I believe that My Father wishes us to visit the Rolle (the law chamber in the Court of Justice) but I hope that this will not happen for a long time." Oeuvres de Christiaan Huvgens vol. 22, 408.

"The two Russian Academicians, U. Frankfurt and A. Frenk, in their Christiaan Huvgens (48) wrote: "Archimedes, Mersenne, Descartes are the three names that have to be mentioned at this early stage of Christiaan's debut in the scientific career. No matter what the difference of the amount of influence of each one of them in his written work, or by how much it changed during the years, this is certain that they determined in great measure his future orientation." See also A. Beaulieu "Huygens et Mersenne, l'animateur" in Table Ronde sur Huygens et la France. 30. 178

can be attributed to the relationship that developed between

Mersenne and Huygens. When Christiaan started receiving

mail from Mersenne, he was corresponding with few but his

brother and, very rarely, with his teacher van Schooten. As he himself later wrote;

I was not able to write in French, yet, at the time when I started writing a few letters to Mersenne. I was studying in Breda at the time this letter was dated, i.e.. April 1648. I was 19 years old.*

Mersenne's letters were full of admiration, encouragement and challenge. After Christiaan sent a second paper proving that a chord hanging from both its ends was not a parabola, contradicting what Galileo had thought,

Mersenne was convinced that he had discovered a mathematical genius. In the letter he wrote to Christiaan's father he said:

. . . Your son has surpassed himself. I cannot answer his letter right away because his paper deserves a longer time of consideration than the first one, which has still some disputable points. I am convinced that if he continues in the same way, he will one day surpass Archimedes . . .*

From then on, Constantin started referring to his second child as Archimedes," not only to Mersenne but to

62'Huygens, Ouevres de Christiaan Huvaens. vol. 14, 217,

63See, Correspondance. vol. 14, 214.

“To stimulate Christiaan to be more daring in his research, Mersenne sent him Blaise Pascal's Experiences Nouvelles first, and then Traite du vide on the theory of vacuum. At the same time, he called Pascal "another Archimedes (like you)." (Correspondance vol. 16). Other times, Huygens wrote, Mersenne "promised a quite large and enviable reward for my work if I could satisfactorily 179

others as well*. Of Christiaan, Mersenne asked permission to

publish the paper he had written on the shape of a chain

suspended from its both ends.* The admiration was sincere

and honest.

Each of the letters contained pieces of advice

revealing the affection that moved Mersenne to write." They

provide what he sought." (Christiaan Huygens, Pendulum Clock. 105). This practice soon became common practice among scientists (see Pascal, Histoire de la Roulette). Even though Mersenne wrote to Huygens about Pascal, they did not correspond until 1658 when Pascal put out a continent- wide competition concerning the cycloid. Of six problems proposed, Huygens correctly answered four. After that, Pascal sent to Huygens his Lettres Provinciales. In return, Huygens sent his first edition of the Horloge. In his letter, Pascal wrote: "I never thought that anyone in your parts of Europe would ever know about me, but I always wished that you would have a place for me among your friends ..." and Huygens' answer was: "I cannot feel any happier now that I have received the gift of your friendship." They cooperated, through the mail, on the problem of chance and Pascal's calculating machine (of which Huygens was to receive one of the first models), with very detailed description. (U.Frankfurt, F. Frenk Christiaan Huvaens. 382). Mersenne's spirit of cooperation did affect his disciples.

“The comparison is very apt. Already at his young age, Christiaan had shown the combination of deep mathematical insight and feeling for technical construction which Archimedes must have had. At the age of thirteen he built himself a lathe and in 1655, together with his brother, he started grinding lenses. Studies in Huvaens. 11.

66See Correspondance. vol. 16, 314.

"The style of the letters shows how Mersenne enjoyed writing to Christiaan. The desire to keep him informed on the developments in Paris, the excuse that there was still some free space on the bottom of the page, just the desire to give him a small interesting problem to solve, were all reasons to write long letters. In some of them one could find three or four conclusions, which in fact did not close the letter. It is possible that this interest manifestly 180

were also expressing his educational methods. We have

already pointed out the questioning of the principle of

authority, as well as the detailed way with which Mersenne

analysed Christiaan's first mathematical paper; we can add

to these a few more points. Mersenne often instructed

Huygens to refer to the primary, rather than to the

secondary sources. Writing about some mathematical

principles formulated by Descartes, he said:

Since I know that you can refer to this source at will, and since he is a very close friend of yours, it would be doing you and him a disservice . . . to give you water from the river to drink, when you can get from the spring.*

In the same spirit, he would advise him to show his

results to F. van Schooten or John Pell, so that he could

become more confident and widen his horizon. Mersenne also

balanced his enticements to do research on some questions

with the promise of generous rewards, while at the same time

advising Christiaan not to overdo it.* For example, after

expressed on the part of Mersenne must be what impressed the young student of the Hagues, and attracted him to further work in mathematics.

“Correspondance. vol. 15, 564.

* Giving the same advice to Howelcke of Danzig after the death of Torricelli (24 October, 1647), Renieri (5 November, 1647), Cavalieri (27 November, 1647) and Giambattista Doni (1 December, 1647) he wrote: "We are afflicted by the deaths of young and old learned mathematicians in Italy. I therefore exhort you to take care of your health so that the fatigues of your work do not harm you. Experiments are better performed during Spring and Summer, while the rigour of the winter, especially by night, is very dangerous, and I do not think your winters are any milder than ours here" (Correspondance vol. 15, 181

receiving the paper on the catenary and expressing his

appreciation, Mersenne went on to suggest that he not

overwork himself: if he would produce only one paper a year

as good as the one on the catenary, he would surely outshine

the others. Mersenne once asked Christiaan's father not to put too much pressure on his son merely to obtain quick

results/* Another time,Mersenne suggested that Christiaan work on the problems he had proposed only after taking his

imminent examinations. And Mersenne's remarks would apply even to the writing style of a scientific paper.

Mersenne's care would go so far as to talk to

Chriatiaan as he would to a mature scientist and to send him papers to evaluate even before Descartes and other well known mathematicians would see them. In fact, more than once, Mersenne sent Christiaan books and papers which he asked him to read and appraise. Afterwards, he was to pass them on to Descartes, or to Franz van Schooten, Chriatiaan's teacher. The famous paper on the catenary gave Mersenne the occasion to introduce Christiaan to the community of scholars throughout Europe." Mersenne's letters were

578).

^Fortunately Christiaan's father could write back to him: "Do not be afraid that I may pressure him too much. I never do it to any of my children, just as my parents never put pressure on me." (Correspondance. vol. 14, 635-38).

"p. De Carcavi in 1656 (Oeuvres vol.l, 418) wrote to Christiaan: "The first time I heard about you was long ago through the late Father Mersenne, who also showed me some of your papers." See also Correspondance. vol. 15, 408 about 182

anxiously expected and read with great respect and pleasure.* Years later he wrote to Carcavi with pride;

I had the honor of being in correspondence with Father Mersenne. He used to encourage me in the study of mathematics towards which he saw that I had a natural tendency. Often he would send me papers of your illustrious mathematicians, and in particular those of Mr. Fermat, which I was starting to understand as I was making progress in these sciences.*

Constantin had planned to send Christiaan to Paris in 1646 in order to have his son close to Mersenne. In fact, he wrote to Mersenne asking him to keep an eye on

Christiaan during his stay in that city. Unfortunately, the plan never materialized during Mersenne's lifetime, so the two never met. But once Christiaan was in Paris, he did not feel a complete stranger, especially in the circle of the learned. He was soon asked to join the newly established

Académie des Sciences and was often admitted to present his

Mersenne writing to Howelke of Huygens.

^Constantin to Mersenne: "My scholar (Christiaan) was here when I received your letters, with great joy and expectation. But since he had to leave immediately for the school (in Leyden) we did not have the time to talk about the points you were raising. You are quite right to call him clairvoyant, because he penetrates easily what others judge to be difficult and obscure. He enjoys making fun of those who go through much trouble to make things look difficult by appearances and embellishment." (Correspondance. vol.14, 635-638). The same feelings of joy can be detected in the first draft of a letter by Christiaan to Mersenne (Ibid.).

*C. Huygens, Oeuvresde Christiaan Huvaens vol.l, 418. Actually in the first draft of this letter, conserved in Leiden, Collectio Huvaeniana. Huygens had initially stated: "which I understood only many years afterwards." 183

discoveries personally before the King and his ministers.

In general he was so welcome everywhere that he spent nearly

thirty years in Paris. His memory of and affection for

Mersenne never faded. After Christiaan's death, they found

in his personal collection of books the Harmonie

Universelle, the Novarum Observâtionum Physico-

Mathematicarum Tomus Tertius. the Nouvelles Pensees de

Galilée, and La Vie du Pere Mersenne. In his works, he quoted from almost all of his mentor's books."

" Christiaan's feelings towards Mersenne become more significant when comparing them with his attitude towards Descartes, who, as mentioned earlier, was very close to his father. When the Vie de Mons. Descartes by Baillet was published, Christiaan commented: "He [Descartes] wanted to propose his own system for physics, in an essay intended to be very similar to this subject, and admitting only the principles of mechanics. Others were asked to join him in this endeavor. That was very praiseworthy. But trying here, as in anything he did, to show that he had discovered the truth, and was basing his approach in the form and presentation of his exposition, (and he was boasting about this) he ended up ruining the project of his philosophy. As a matter of fact those who trusted him and became his followers, thought they had in their possession the knowledge of the causes, as if it were something they could achieve. They, in fact, wasted much of their time in defending the doctrine of their teacher, rather than trying to examine the true reasons of such a great number of natural phenomena of which Descartes did not have but a chimeric knowledge .... But despite the little truth that I find in the Book of Principles of Mr. Descartes, I have nothing but praise for him presenting him always busy in generating this new system, and giving him an appearance of which many people are satisfied and pleased with.(Reported by Lenoble, Mersenne. ou la Naissance du Mécanisme. 447) This description certainly did not enhance the fame of a family friend. However, Christiaan was pleased with Queen Christine's and Chanut's project to bury Descartes in a mauseleum of marble. During his trip to Sweden he tried, unsuccessfully, to visit the tomb. 184

Mersenne and his Young Correspondents;

an Analysis

Analysing the incomplete list of Mersenne's

correspondents left by Hilarion De Coste, one could see that

about half of them were young scholars, some of them not yet

out of school, and others just in their first years on the

profession. They were from all over Europe.

Among those writing to him in France, from outside

Paris, one could count Blaise Pascal,* his brother-in-law

Perrier Petit, Adrien Auzout,* Francois De Verdus, Tomas

Martel,* Raoul Halle de Monflaines,* Theodore Deschamps.*

*There is no letter preserved of the correspondence between Mersenne and Blaise. In part this might be because they did not need to write each other very often. In fact, Blaise moved to Paris with his father and his two sisters, Gilberts and Jacqueline, at the age of eight and remained there for eight more years. Blaise often accompanied his father, Etienne, to the various cultural events in Paris, and there he got to know Mersenne very early. Even when he left Paris for Rouen, where his father had been sent as the Royal Tax Commnissioner, Blaise used to return to Paris with his brother-in-law Petit. However, from the correspondence of his sister Jacqueline, it is certain that Blaise and Mersenne did correspond by letter.

"Because of his visit to Italy during the first experiment with the vacuum, Auzout cooperated with Mersenne in this field with great enthusiasm. In a document dated in 1671, he is credited with the idea of experimenting with the creation of a vacuum in a vacuum, as well as the suggestion of experimenting the vacuum tube at the top of the Mount Puy-de-Dome (see Correspondance. vol. 16, 482 482).

*Tomas Martel was introduced to Mersenne probably by Hobbes, Tomas' teacher. Afterwards Mersenne was instrumental in convincing Hobbes to accept Martel again in the circle of his friends. (See Correspondance. vol. 16, 435; also vol. 11, 278 and vol.12, 360 and 366). 185

Samuel Hartlib, John Pell, Theodore Haak and the Cavendish

brothers were writing from Great Britain. This group is

particularly important because they belonged to the Comenian

mouvement which was trying to bring to the British

educational system the innovations proposed by the Polish

Moravian Bishop, Comenius. Mersenne was also familiar with

the works of White, Oughtred Williams, John Twysden, and

Walter Warner.

Mersenne's correspondence with the young John Pell

(1611-1685) included a vast range of topics: music, the

variation of the declination of the compass needle in

London, the force of percussion, perpetual motion, magic

squares and the problem of possible combinations that could

arise, problems of probabilistic combinations, the cycloid

and the solid generated by its revolutions, and the

quadrature of the conics. In particular, Mersenne

*A close friend of Pascal, Auzout and Martel. He helped Mersenne buy the necessary instruments from the glass factory in Rouen. He also tried to perform the barometric experiment in the dark, after sealing the room for a few days, as Descartes had suggested to Mersenne. (Correspondance. vol. 16, 427 and 469-470).

*Mersenne wrote to Dechamps on July 8, 1647: "It would be very good if, in each province, there would be someone charged with keeping daily records of the winds and other changes in the weather, of the abundance or shortage of fruits, about the health or sickness of the people and animals. These statistics would help to recognize any recurrence of the disorder, from which the future atmospheric conditions could be predicted." (Correspondence vol. 11, 205). In another letter Mersenne described to him the principle of conservation of energy. (Ibid., 319). 186

encouraged Pell to pursue his design to prove the

impossibility of squaring the circle against the repeated

claims of the Danish mathematician Christiaan Sorensen

(generally known by his Latin name Longomontanus}.. To

provide him with the support of the greatest mathematicians,

Mersenne granted Pell's request that Mersenne write a letter

of recommendation to his friends throughout Europe. In 1647

Pell published a book on the topic, in which he announced

the trigonometrical identity:

tan 2a = 2 tan a /(I - tan^a)

Mersenne asked Pell to provide him with real data on

the submarine that De Repeler was said to have demonstrated

in front of the King of Holland in 1620.“ He sent to Pell

and De Haak special and rare books, and even took care that

“Pell devised the symbol " : " for division, while their mutual friend Oughtred devised the symbols " x " for multiplication and the symbolism for inequalities as well as the symbol - for subtraction. Actually, Pell is better known for the formula named after him:

X® = 1 + ky® where x and y are real variables and k is a real constant. Mersenne met Pell in Paris during the letter's visit to Paris from Breda, where he was serving as an Ambassador of the United Kingdom to the Continent. 187

one of Fermat's manuscripts, which was circulating in Paris,

was copied and sent to Pell.®’

The same interest is shown in the correspondence

with all of his other young friends. Some of the reasons

for the Minim's success with these young scholars were:

I. He was able to inspire in them a strong confidence in

their academic gifts. All his letters started always

with an appreciation of his correspondents for their

abilities and interest in science. The conclusion

invariably expressed Mersenne's desire to hear from them

again. In this way, he won their confidence and

friendship. The correspondence he exchanged with them

proved that they felt they could write to him about

anything.*

"Mersenne, Correspondance. vol. 11, 306.

®^When Raoul Halle de Monflains heard that the Jesuit P. Noel was sending the press the letter he had written to Blaise Pascal refuting the theory of the vacuum, he wrote Mersenne the following letter (suggesting, in a derisive manner, that it could be used as the introduction to the proposed book): "I cannot understand under what new flavor he can present it now! From a personal letter he made a volume, and just as if this "metamorphosis" was not enough, from French he translated it into Latin. Such a waste! And he may expect that the public should be grateful to him for his care to instruct them! As for me, I have to say that if the Latin version is not clearer than the French, I will count it as if I had already read it, because I do not think that it is worth the pain of deciphering his ideas. God has given such a singular talent in being particularly obscure. I cannot understand for what purpose Mother Nature thought to use him when she chose such an advocate . . . He can use this statement in the first page of his publication as a statement of approval ..." (Correspondance. vol. 16, 221-222). 188

II. He was able to keep their interest in science growing

by giving them an important part in the discussions of

ideas and in the performance of experiments, even if

they resided outside Paris. Auzout and Pascal, both

still young students, were allowed to attend the

meetings of Mersenne's academy” side by side with the

most renowned scholars of Paris such as Roberval,

Desargues and Hobbes.

III. He was particularly appreciative of the views and

proposals coming from these novice scientists. Quite

often Mersenne's enthusiasm for the youth was not shared

by older scientists. For example, when Pascal wrote his

first paper on the locus of Pappus, Descartes wrote to

Mersenne:

I do not find it exciting that someone has proved some theorems about the conics more easily than Apollonius, because they are only long and boring proofs. . . . But there are other points about the conics which a 16 year old boy will find hard to work on .

“of particular importance was the meeting of September 23-24, 1647, in Which Descartes took part. It was probably at this meeting that it was agreed upon that the height of the mercury should be measured at various sea levels.

“Mersenne, Correspondance. vol. 8. According to Mersenne, however, the situation was different. Writing to Constantin Huygens Mersenne said: "If your young Archimedes will come to us, we will show him one of the best treatises on geometry that he could have ever seen written by the young Pascal. This is the solution of the problem involving the locus of Pappus in three or four lines, which so far was never proven by Mr. Descartes with all his great skill on the subject . . . ." Commenting on Descartes' letter, Taton writes: "One can feel in the comment Descartes' fear that abstract geometry could challenge his method of analytical 189

IV. He often offered his name and reputation to admit his

young scholars to the tightly guarded circles of

scientists. He felt honored, for example, to announce

Huygens' paper on the catenary to Torricelli and his

academy, thus introducing Huygens to the circle of the

European intelligentia.

V. He allowed his students to claim credit for certain

discoveries and research he had already done himself.

VI. He took time to update his young protégés on the most

recent new discoveries, publications, and results. He

even went so far as to visit some of them in the most

remote parts of France. His cell in the monastery was

always open to the many young men, from every part of

geometry. This negative judgment of Descartes, not convinced of the originality of the Essay of Pascal, hurt the reputation of this new enfant-orodiae." (Taton, Desargues, 46). The same distrust of Descartes towards Pascal is found also in Constantin Huygens. In fact when he sent Descartes Pascal's first booklet on the vacuum, he advised him in the accompanying letter to wait before he stated his view until: "this young author will have given all his remarks on this topic. Because after you. Sir, no one else should dare to add anything to your words. It seems to me that this young man that has written this booklet has a too large empty space in his head and that he is rushing too much. I wish that the final work that he has announced would be already out to see all his points of view. If I am not mistaken, they are much less solid than what he has proposed to prove." (Correspondance vol. 15, 564). But in the letter he wrote to Mersenne about the booklet, he asked him to encourage Pascal to carry out the project he had proposed (Correspondance. vol. 16, 219). This was an attitude of unfairness to Blaise, much more painful since it came from someone who was having his son treated with the same care and attention by Mersenne. 190

Europe, who would come to him with recommendations from

his friends. He would entertain them with the same

attention that he gave other well-established

academician and tried to offer them all the assistance

he was able to provide.“

VII. He would adapt himself to the interests of his

correspondents without forcing them to bend to his own

preferences. Most of all, he avoided using his academic

authority and priestly ministry to influence the

preferences and the personal beliefs of those youths who

did not ask for it.

Without doubt, Mersenne's greatest satisfaction came from seeing his attitude of cooperation passed on to the youths with whom he worked. When in 1662 Isaac Vossius, already in correspondence with Mersenne,®® printed his De

®®To Rivet, who was considering sending his son to Paris, Mersenne promised: "If your second child needs anything that I myself or my friends can provide while he will be here, rest assured that I will be entirely at his service." rCorrespondance. vol. 7, 213). He had students visiting him from Poland (e.g. Georges Fhelav, who later became a Protestant minister and a well known jurist (see his letter to Hevelius in Correspondance. vol. 16, 131, 167), from Norway fe.a. Petrus Scavenius from Copenhagen recommended to Mersenne by the famous Frommius), and too many to mention from Switzerland, Italy, England, Germany and Belgium.

®®The letter that Mersenne wrote to Vossius is an example of the warmth with which he wrote to his young friends. After expressing his admiration for the book he had written, he points out some mistakes in the translation from Greek, but then excuses him by attributing the mistakes to the printer. He continues: "Of course you might have felt compelled to prefer a literal translation and to keep some barbarisms in your expressions for the sake of fidelity 191

Lucis Natura et Proprietate Liber. Petit suggested they work together so that they could "give the public the curious observations which I have made on the refraction of light in diaphanous and opaque bodies as well as in liquids. ZThis is a work that the late Pere Mersenne and many others have very much expected.

Once again Mersenne is ahead of his time. Three hundred years after his death, it must be said that his method of cultivating these young scholars is in line with our contemporary educational approach. Modern psychology puts great emphasis on the role of the mentor.“

A recent article in the New York Times reported the poor results of the national exams of the city's high school students. Bernard Mecklowitz, New York City's School

Chancellor, attributed the failure mainly to the difficulty in attracting and maintaining qualified mathematics teachers and to the frecpient lack of enthusiasm among many of those who continue to teach. Susan Zalaluk, director of the Board

to the text. I know though that I could expect far better publications from you . . . . I am anxious to know what other work I can expect from you and from your young friends in the near future. Besides this, what other interests do you cultivate, so that I know what to write to you about in the future?" (Correspondance. vol.15, 500-505).

“See Correspondance. vol. 7, 171.

“see Rosabeth Moss Kanter, Men and Women of the Corporation. 1977; John Feldhussen and Donald Treffinger. Creative Thinking and the Problem Solving in Gifted Education (Dubuque, Iowa: Kendall/Hunt Publ. Co., 1980), 312. 192

of Education Mathematics Unit, admitted that "there is no

substitute to the training of teachers and to fostering

their enthusiasm." She said that the districts that performed better have mathematics supervisors who coordinate

and monitor teaching, while the school districts in the city with the worst scores this year had no mathematics supervisors last year.®® The role of the supervisor cannot be limited to controlling the performance of the novice teachers in the classroom. He is expected to bring out the teachers' potential, to encourage them to perform to the best of their abilities providing them with appreciation, encouragement and motivation. Suggestions for new approaches, recognition for achievements, and promotion for the personal enrichment of the teachers are primary components of the duties of the supervisor. Experience shows that quite often supervisors play the role of a mentor for many teachers in his district.

Even outside the classroom, mentoring is considered a significant predictor of career-success.“ It is the quality or intensiveness of the relationship, not the quantity or extensiveness, that principally affects the mentor's protégé.®’ The truth of this assertion is born out

°®New York Times. July 25, 1989, 68.

“Jerry Willbur, "Does Mentoring Breed Success?" Training and Development Journal 41 (Nov. 1987): 38-41.

“ibid. 193

in the case of the young scholars who were assisted by

Mersenne. In addition to the cases célébrés of Huygens,

Pascal, and Pell, most of Mersenne's young correspondents

succeeded quite well, not only in their studies, but also in their professional careers.

A mentor is expected to be available to talk at all times, providing career counselling on an informal basis, supporting the proteges' efforts to advance in their studies and careers, and influencing them positively in their choices. These study shows that when an organization defines its standards of success very clearly, the implementation does not necessarily appear to be equally successful. Perhaps quite often a cheerleader, a coach, and a counsellor, is needed more than an exact model or example.“ Therefore Mersenne's limited mathematical background as compared to Descartes', Desargues', etc. did not hinder his effectiveness in playing a role model for other more successful protégés. Perhaps it was his lack of a personal mentor at the beginning of his own scientific career that influenced him in his attitude toward providing career counselling to others.”

92 Ibid.

“j. F. Feldhusen, and D. J. Treffinger, Creative Thinking. 89. CHAPTER FIVE

SOME MATHEMATICAL AND GEOMETRICAL

RESULTS BY MERSENNE

As Mahoney states in his study of Fermat, in spite of all the variety and problems among its practitioners, the status of mathematics in the seventeenth century was not a chaotic Tower of Babel. Following the division that

Mersenne made in La Vérité des Sciences, mathematics could be split into six different fields: the classical geometrical tradition, with Federigo Commandino (1509-1575) as its main representative; the predominantly Italian and

German tradition of cossist mathematics — the school of

Cardano (1545), Rudolff (1525), and Clavius (1608); the applied mathematicians; the mystics; the artists and artisans; and the analysts. Quite often the works of one author would cross the borders of a field and include topics of several other fields. However, each category distinguished itself from the others by characteristic attitudes towards the nature and purpose of mathematics, its

194 195

problems, and its methods of solution.’ Many of the

conflicts between the great mathematicians of these

centuries could be reduced to merely different approaches to

a problem.

Mersenne's Philosophical Approach to Mathemathics

It is difficult to identify Mersenne with any of

these schools, mainly because he was, by training and

profession, a philosopher-theologian. He turned to

mathematics and science, at least at the beginning of his

career, chiefly for apologetical reasons. One needs to keep

in mind constantly this framework in which Mersenne acted

and wrote to understand his monumental editorial output.

His education at the college of the Jesuits at La

Flèche and at the Sorbonne imbued him with a scholastic

philosophy of life, to which he remained faithful to the end

’m . Mahoney, The Mathematical Career of Pierre Fermat. 2; see Mersenne, La Vérité des Sciences. 228-30. 196

of his life.® The innovative philosophy of Descartes never

appealed to him, even though Mersenne appreciated more than

anyone else Descartes' mathematical genius. Likewise,

whoever tried to disengage himself from Aristotle aroused

his suspicions. To his Protestant friend, the theologian

Rivet of Leyden, he wrote with sadness about the Jesuit P.

Noel, whose book had just been published:

These scholars, following the example of the others, are starting to desert Aristotle; maybe not you nor I, but the following generation will see, whether they will come out with anything better than the old Peripatetic in philosophy. I just pray God that he may give us a peaceful heart in the place where truth only will reign.®

In his early apologetical books against the

Pyrrhonism, especially in La Vérité des Sciences and in the

L'usage de la Raison as well as later in the Question

Theologigues. he set himself to the task of laying the

philosophical bases for the sciences, and mathematics in

particular.

®This did not keep him from dissenting from the contemporary interpretation of the Aristotelian, as in the case of the maxim, Natura abhorrit vacuum about the space above the mercury column in the barometer. As Rochot points out in his comment on the letter of Mersenne's correspondent Le Tenneur: "I do not pretend any more to convince the Aristotelian that the space at the top of the barometer is a true vacuum. It would be enough to prove to them that the space is not occupied by any of the ordinary corpuscles that we know;"neither"air nor any other thing." Mersenne was following the same line, against most of the Scholastic Aristotelian as well as his close friend Roberval of the College Royale du France and Descartes himself (Correspondance. vol 14, 61).

®Ibid.. vol 14, 45 197

The very first problem with which he was faced was

of an epistemolbgical nature: Can we know reality, and, how

much do we know? His was a practical approach and the goal was to fight agnosticism in all its forms.

First of all: what was knowledge for Mersenne? The

scholastic monk explained it by giving the relation between the way our mind perceives reality and the real objects producing ideas:

Understanding forms a living image which expressly resembles the object, in such a way that if one saw that object represented in the intellect, one would say that there was in some way a greater union between the object and the mental image than there is between "matter and form", and that the axiom intellectus et res intellecta sunt unum et idem, would be true for the representative being of the object (the idea of the object)

This reflects the scholastic definition of true knowledge: Adequatio mentis ad rem coanitam. a relation of conformity between the concept that we have formed about the object of our knowledge and the knowledge itself. The object of our knowledge is usually external to ourselves and is often corporeal. How then can our mind, which is incorporeal, grasp it and assimilate itself to it? For

Mersenne:

The notions of logic, of physics, of metaphysics, of the circle and of the other figures, and even those that you have of stars, of stones, of houses and of any other thing that you know, are intellectual and insensible:

* The intellect and the thing known (or perceived) are one and the same. Mersenne, M., L'usage de la raison. 137 reported by Robert Dear, Mersenne and the Learning of the School. 49. 198

otherwise it would be necessary to say that the understanding was sensible, material and corporeal inasmuch as all that is perceived by a subject is perceived in a way proportionate to the perceiver, according to the axiom cmidouid recioitur ad modum recipientis recioitur. (whatever is received, is received in the mode of the receiver).®

To the next question of the skeptic, "How can I say whether a statement is true or false ?" the philosopher in

La Vérité answered:

We could say that [the external] truth itself is the judge of our understanding. If we find that all that we understand and believe to be true is wrong, our thought process will abandon the opinion it previously had and accept the true concept of the idea. This is nothing but the comprehension of our mind to as what is presented.®

The fact that one could compare the concept in his mind with the external object and find an adecmatio or a non-adecfuatio justified the claim that we can attain the knowledge of something. In fact, regardless of the variations in experience and opinions in every field of knowledge, some things are known to be true: for example, that the whole is greater than its parts; that the light at noon is brighter than that of the stars; and that it is not possible for the same thing to both have and not have the same property. Mersenne goes on citing additional examples from other fields of knowledge like physics and ethics.

(There might be different standards of good and evil but

® Mersenne, La Vérité des Sciences. 274-75, see R. Dear, Mersenne and the Learning of the Schools. 51.

°Ibid.. 195. 199

every-body agrees that "evil should be avoided and good

sought.")

Knowledge of an object, he claimed, does not mean that we know everything about the object, or even that we know its essence. It can be said that Mersenne himself was more radical than the skeptics about things that he assumes to be beyond human comprehension, as religious mysteries and some abstract metaphysical concepts of no practical use in science. Popkins, therefore, calls him a "moderate skeptic." His assertion was that the amount of knowledge we have, based on the few basic first principles accepted by everybody, can be expanded immensely. He showed this by devoting 800 pages of La Vérité des Sciences to list and derive the principles, theorems, and applications of mathematics. The skeptic was compelled to say that, with this practical approach, his attitude was completely changed.^ He had discovered in mathematics the most excellent means of overturning all his initial positions and had returned to common sense.®

However, the essence of the phenomena will always elude us:

About some phenomena that could be proposed in philosophy, one has to say that we cannot think that we are able to penetrate the nature of the individuals, or (know) what happens in their interior. In fact, our

'ibid.. 730

®Ibid.. 751. 200

senses — without which our comprehension could not grasp anything — can perceive only what is external.®

In Questions Inouves (1634, Paris) in response to the question •— Can we know anything certain in physics or mathematics? — Mersenne was even more decisive. He stated:

"There is nothing certain in physics, or rather, there are so few things that are certain that it is most difficult to find them."’®

This fact, though, does not endanger the objectivity of mathematical truths. In both La Vérité des Sciences and

Traite de l'Harmonie Universelle. Mersenne explained our ability to make abstractions from concrete visualizations: in imagination we can abstract from accidental qualities and still our concept can reflect the object that has inspired our abstraction. Mathematical abstractions are not accidental: one can imagine a black swan, but one cannot ever have a swan that is at the same time two swans. And that is exactly why mathematics is a science: "The world may be contingent. But the concept that two times two men are four men is always the same, exactly as the concept we had

® Ibid.. 212. The same principle is repeated also in Questions Theoloaioues. Phvsiques. Morales et Mathématiques. (224-225) where he draws the conclusion: "Therefore one cannot request from the scientists anything else but their observations and the considerations that they have made from them."

’°Mersenne, Questions inouves ou Recreations des Scavans. (1634, Paris), 72-74. 201

conceived"” thanks to mathematics. The independence of

mathematics from the contingency of physical creation

elevates it to a state of pure science.

It is sufficient that the mathematical sciences and all the notions "have a being in reason" (i.e., avent un estre de raison, i.e. exist in the intellect) to which they are true (or in harmony with), provided that they do not contradict the external objects and their properties.

It does not matter whether or not numbers are

objectively independent from concretely existing numbered

objects. Numbers are produced by the position alone, and by

the actual or possible existence of things that are distinct

and separated from each other; for it is not necessary that

there exist different things to establish numbers; it

suffices to establish that they can exist, just as it

suffices that there can be several men in order to establish

the truth of human nature, even though they may actually not

exist.’® This is exactly how the Pythagoreans justified the

existence of irrational numbers, as we do today.

Once the objective concept of numbers is

established, we are required to define the value of each

’’Mersenne, La Vérité des Sciences. 277.

’® Ibid.. 275, translation adapted from Dear's Mersenne and the Learning of the Schools. Again in Questions Inouves (question 18, 53-54) Mersenne explains how mathematics: "is a science of imagination or of pure intelligence, like metaphysics. If one would abstract quantity from mathematics, it would be like taking away its foundations, on which the proofs are based."

13Ibid.. 272. 202

number. Every integer is defined as the n-tuple multiple of

ONE and every rational number as a part of unity. In light

of the objections of the contemporary philosophers it was,

therefore, necessary to establish whether or not ONE is a

number. In La Vérité des Sciences, therefore, the skeptic

was prompted to ask:

What is ONE? Some say it is a pure privation or negation of multiplicity, just like a point on a line. Therefore ONE is nothing and arithmetic will be reduced to nothing according to this opinion, quite common among the scientists. On the other hand, if ONE is something it must be a number. However, that cannot be, unless one wants to contradict the maitre des elements (Euclid), who in the second definition of the seventh book said: "Number is a multitude composed of many units.""

After proving by means of at least five quotations

that Euclid was not opposed to the concept that ONE is a

number, Mersenne distinguished between arithmetic unity and

the geometric point and finally showed how unity is its own

square, radical, nth power, and has the properties of all

the regular perfect numbers; therefore, it is the most

perfect number and even the largest, because it contains all

its powers. Moreover, the number one is one half of two,

one third of three, one nth part of any integer n. Hence,

even though the philosophical concept of unity is

indivisible, mathematical unity is divided into parts by a

fraction whose denominator is always the whole quantity, while the numerator is a definite portion of the unity taken

"ibid.. 254. 203

into consideration. The number ONE is, therefore, a part

and factor of any other quantity. From this follows that

unity is the meeting point of the increasing quantities made up of ONES, which can be made arbitrarily large, and the

infinitely many portions that can be arbitrarily small, but still different from zero. In today's terminology it could be said that one is an "accumulation point," which tends to infinity and shrinks to zero. No other number is justified, if ONE is not a number.

Not satisfied with this strict reasoning the skeptic further inquired whether unity is a concept present in nature and whether what we call ONE is what it is because it partakes of this natural unity or if it exists by itself.

If it exists by itself, continued the skeptic, does everything that is ONE contain the entire unity or only a part of it; e.g.. if a single individual has all the unity of human nature then no other individual can be one any more. In other words, if there are many naked persons no one mantle can cover them all. But if it is by participation, then unity has parts, and hence it is not unity any more because the elements of unity cannot be units. Therefore there is no unity.

In this way, not only can we know something with certainty in mathematics, but even more: mathematics has a philosophical superiority to physics because, according to

Aristotle, a science should have its own unioue principles. 204

from which it is developed bv rigorous demonstrations.

Arithmetic and geometry, more than any other subjects,

satisfy these two principles. Mersenne used this principle

of independence to justify the scientific basis of music:

"It has true demonstrations which are rooted in its own

principles."’® But, of course, Mersenne was aware that the

principles of music are not independent of mathematics and

acoustics.’®

Furthermore, mathematics, above all other fields of

knowledge, is based on the fact that its demonstrations are certain and are causal. The syllogistic method of its demonstrations, rather than the inferential demonstrations used in physics or in the other applied sciences, made their proofs irrefutable, according to Aristotle's approach. A logical connecting thread runs through the various theorems.

’® Mersenne, Traite' de l'Harmonie Universelle. 6.

’®Mersenne following the Scholastics, justify their claim of scientific character of music because one defines music as the set of proportionalities of numbers. The same can be said about optics, which, until Huygens explained it in terms of waves, was considered only a phenomenon of reflecting and refracting straight lines. Mersenne clearly understood that mixing arithmetical or geometrical principles to explain, e.g., the phenomenon of the octave was against the law of homogeneity invoked by Aristotle (See Questions theologigues. 182). He further explained, " if they have a common formal object, they will be subordinates and subordinating, although their material object be different. This happens with mathematics (the subordinating) when optics or music (the subordinates)make use of demonstrations from geometry and arithmetic, for when demonstrations are joined to sensible matter, they do not lose their formal matter." fTraite de l'Harmonie Universelle. 5). 205

and their proofs justify each step in the development of the

subject in a rigorous way. Physics being essentially an

experimental subject fails, by its very nature, to achieve

the same degree of certainty that mathematics achieves. In

fact, Mersenne said:

It could be said that we see only the bark and the surface of nature, because we cannot penetrate in its interior secrets. Similarly, we cannot have any other science but that of the external effects, without ever being able to penetrate its reasons. Nor can we know the way events happen until the day, in which it will please God to free us from such a misery and to open our eyes by the light that he reserve to his true worshippers.”

Therefore, one cannot even be sure of the results of his experiments, because he is not aware of all the facts that may have caused a result.

After going along with the then standard position that physics was not a "science"’® because its subject matter was motion, which involves change, Mersenne contrasted physics to mathematics as the subject whose: __

quantity is invariable, for it cannot come about that a triangle is not composed of three lines, and of three angles conjoined by three indivisible points. It does not matter if there may not be a perfect triangle in the

’' Mersenne, Questions Teolooioues. 11.

’® Clearly what is meant here is the science according to the Aristotelian definition of self-contained principles, developed with syllogistic proof from the theorems. Otherwise Mersenne does, in fact, refer to physics, chemistry, simply as sciences, i.e., applied sciences. 206

world: it suffices the fact that it can exist to establish the truth of this science.’®

Finally, mathematics is superior to the other

subjects because neither philosophy nor jurisprudence nor

any other subject can attain its perfection without them.”

Unfortunately, Mersenne skillfully avoided replying

directly to the last objection of the skeptic:

If all the truth were as clear as the arithmetical and geometrical truth, then I would bid goodbye to Pyrrhonism; but how can the principles of astronomy, astrology, music and perspectives, which are generated from geometry and arithmetics, be wrong? Do we have to admit that a true premise implies a false consequence?®’

Number Theory

Like most of his contemporary mathematicians,

Mersenne too was attracted by the properties of the natural

numbers. He, together with Fermat and Frenicle, can rightly

be considered the fathers of modern number theory.®®

’® Mersenne, La Vérité des Science. 226; see R. Dear, Mersenne and the Learning of the Schools. 54.

®® See La Vérité. 225. The title of the first chapter of the second book of this work deals with de Mathématiques en general et de leur nécessite.

®’ Ibid. . 730.

“Mersenne was characteristically generous in acknowledging the value of his correspondents in their specific field of knowledge. In the preface to the Harmonie Universelle. 9 he wrote: "Now, if I wished to speak of the men of high birth or quality who have thrived so well in this area of mathematics that no one can teach them anything, I would repeat the name of him to whom the book on the organ is dedicated [Etienne Pascal], and I would add Mr. Fermat, Councillor of the Parliament of Toulouse, ... He knows the infallible rules and the analysis for finding an infinity of others [numbers] of this sort." 207

The first five chapters of the La Vérité des

Sciences are, in fact, a general introduction to the theory

of numbers. His analytical mind clearly accepted and

apparently delighted in the numerous subdivisions of the

natural numbers developed since the time of the Greek mathematicians. If nothing strikingly new appears in this

first book of science produced by him, the clarity of his exposition made the book valuable to the readers of his time. To the modern historian of science it is an excellent reference with which to follow the course of mathematical education of an average French learned person in the seventeenth century.”

His subsequent works, such as the Coaitata (1644) and the last great work, the Tomus III (1647), contain

About Frenicle he wrote in the Prospective Curieuse. (1669, 182): "Someone should persuade Mr. Frenicle, who, I believe, is the most advanced in this subject [number theory], to publish his numerous and excellent books." While both Fermat and Frenicle did not make a mystery about their awareness of their abilities, the monk's professed choice of a life of humility prevented him from boasting of his own contributions. In a letter of July 2, 1646, Bonnell wrote to Mersenne in these terms: "I cannot help expressing my utmost admiration for the sharpness of the properties of numbers that you and some other excellent minds have been discovering lately." (Mersenne, Correspondence. vol.14, 328).

”0f interest is the inclusion of many improvements and corrections introduced by the recent mathematicians, both in France and in Europe. However,a constant refrain in all his letters, books and unpublished papers is: "There are thousands of difficulties when dealing with numbers ..." and he would start enumerating one problem after another (e.g.. see Hvdraulica. 54). 208

important results, some of which were fruits of his own

research or of communications that he received from his

correspondents. In particular, Fermat, Frenicle, and

Descartes were the major scholars who exchanged information with him about properties of numbers.

Finally, his correspondence, during the more than thirty five years of active exchange of information, provides an invaluable source about the growing interest in this field among European mathematicians. Fermat, Frenicle, and Descartes excelled through the results of their work; but others referred to Mersenne to be introduced to this mysterious science. On July 2, 1646, Bonnell, full of admiration, wrote to Mersenne: "I cannot admire enough the great subtlety on which you and some other genius have being discovering lately... .

To the modern mathematician, Mersenne is linked mainly with the set of numbers that bear his name, the

Mersenne numbers and the Mersenne primes. They are the result of his interest in finding a general rule to define the prime numbers. The twin concepts of divisibility and

“Mersenne, Correspondance. vol. 14, 328. This letter is very interesting because it gives credit to Mersenne for finding the sum of a finite series of cubes. It also informs us about the role of Mersenne's correspondence in informing and updating amateur mathematicians in small towns of France. Besides, it may be that, with this letter, Bonnell was the first to attract Mersenne's attention to Bongus' error on perfect numbers, which Mersenne corrected in the Svnopsis Mathematics of 1644. 209

primality peculiar to the domain of integers form perhaps

the most important and central part of Mersenne's research.

They, in turn, derive from the first set of problems Fermat

investigated: namely, problems involving the sum of the

aliquot parts (i.e. proper divisors) of the number.

Descartes in a letter dated November 15, 1638,

replying to a request of Mersenne about the prime numbers,

wrote:

I do not know a general rule for recognizing whether a number is prime or not, except that I look at the last digit. This must be l or 3 or 9 (for the number to be prime). If it is 3, I check whether the number can be written as the results of two others, one of which ends with a 1 and the other with a 3, or the first with a 7 and the second with a 9. I do this check starting from the last digit at the extreme right. This shows that the process is really long, but I do not know any shorter way.”

Mersenne, therefore, kept searching for an easier

formula. Fermat claimed that numbers resulting from the

formula

N = 2" + 1

were prime. They, however, could not satisfy Mersenne's aim

because in order for N to be a prime number the exponent of

“Mersenne, Correspondance. vol. VIII, 194. One has to admit that Descartes, when writing the above letter, must have overlooked the prime numbers such as 7, 17, and others ending with 7. In 1644 Mersenne had still to admit that one of the greatest problems of the mathematicians was how to recognize whether a number of fifteen or twenty digits was a prime or not. In fact in the Coaitata he wrote: "one whole century would not be enough to find it out following the methods known until now" fPraefatio qeneralis. non numbered pages, folio XI). 210

2 cannot have any odd factor. If, for example, n = ab,

where b is odd, one would obtain an algebraic factorization

given by

2" + 1 = (2")® + 1

= ( 2“ + 1) ( 2“'®'” - 2“’®^®’+ 2“'®^®’ - . . . + 1)

Which shows that the number is not prime.However, a number without odd factors must be a factor of 2 so that n = 2%

and the numbers take the form

F, = 2® + 1.

Some of the results obtained through this formula are the following; Fq = 3, F, = 5, Fg = 17, Fg = 257, F^

= 65,537, which, indeed, are all primes, but the next values are so large that they were difficult to factor. In a letter to Frenicle (August 1640)®®, Mersenne stated that he did not have the exact proof for his statement, but that he had excluded so many possible divisors, and could see it so clearly, that he found it very difficult to believe that it could be otherwise.

Euclid, when studying perfect numbers, (which is practically what motivated Mersenne to search for a formula

®®"I am almost convinced that all the numbers in the (geometric) progression, whose exponents are the powers of two, to which one is added, are prime numbers." (Mersenne, Correspondance. vol 10, 94; see also ibid.. 348). Fermat worked to prove this conjecture for over twenty years, asking the cooperation of Frenicles, Mersenne, Blaise Pascal, Digby, and others. It was Euler, however, that proved it wrong for z = 5 and z = 7. 211

to determine the prime numbers)” had stated that if the

number 2" -l is prime then (2® - 1)2^’ is perfect. Mersenne,

probably following a lead given by Descartes described

below, or lured by a seemingly successful result for the

first prime values, concluded that;

I. If (2® - 1) is prime then p is prime;

II. If p is a prime such that it differs at most by three

from a number which is an even power of 2, then (2® -

1) is prime also.

Unfortunately, his generalization in the second case

was wrong, as will soon become clear. However, the numbers

Mp = 2® - 1 are still known as Mersenne numbers or Mersenne

primes, if they are indeed prime.

Mersenne left no proof of his conjectures, but his

ability to work with extremely large numbers makes us

suspect that he himself, or his friends Frenicle and/or

Fermat, had a device for finding easily the sum of aliquot

parts (all the positive factors of a number excluding the

number itself).” In the absence of direct evidence from

”The search for perfect numbers was one of the goals of the ordinary mathematicians in Mersenne's times. In fact, he wrote; "One of the greatest difficulties for mathematicians is to tell how many perfect numbers there are." Ibid.

” In his letter to Bruslart, Mersenne asked him to find a solution to the following problem; given a number, find a method to learn how many aliquot parts it has, without listing them, and find their sum. Furthermore, find the number which has 49 aliquot parts, such that it is the smallest among all those with the same number of aliquot parts, and another one which has 360 aliquot parts. (Mersenne, Correspondence. vol.9, 410). The answer is found 212

Mersenne, it may help to follow Descartes' derivation in order to shed some light on the method followed at that time.” In solving problems concerning the alicmot parts of numbers, we imagine the numbers to be composed of either mutually prime factors, or of factors that result from repeated multiplication of some prime numbers by themselves, or by a combination of the two.

in a hand-written note by Mersenne in his personal copy of the L'Harmonie Universelle, on the last blank pages of the book: "Given a number, say 108, find how many aliquot parts it has." Mersenne solves the problem for the particular example provided: "Find two numbers whose product is 108, i.e. 4 and 27, which are analogous to 2® and 3®. Add 1 to both exponents and take their product; subtract one. You obtain 11, which is the answer requested." In the "Praefatio ad Lectorem" to the Hvdraulica Phaenomena in the Coaitata Mersenne also proposed and solved the following problem: "Example of aliauot parts; if someone wants to know what is the smallest number with 59 aliauot parts". The problem is solved by adding 1 to 59 to obtain 60, whose prime factors are 2.2.3.5. Subtract 1 from each prime factor; 1, 1, 2, 4. These are the exponents to the factors 5, 7, 3, 2 in the given order (Mersenne does not explain why and how) and the result is 5040." He lists the 59 factors.

®®The editors of Mersenne's correspondence attribute the method to Fermat, or more probably, to Frenicle, and, together with A. T. Tannery, who edited the Oeuvres Complets de Descartes, strongly oppose the suggestion made by Oystein Ore in Number Theory and Its History that the method is due to Descartes. Because of the modern notation adopted by Ore, though, his procedure has been adopted here. About Frenicle's method see Mersenne, Correspondence. vol.9, 610; see "Le Pere Christopher Clavius," Revue de questions scientifiques. 154 (1983), 344; Harmonie Universelle. Observation XIII, 26, Descartes tried to show his method with several examples, without explicitly describing his method, see Mersenne, Correspondance. vol. 6, 385-386; vol. 7, 245, 270; vol. 8, 314. 213

In this way Descartes started considering any number

as the product of (unique) prime factors.

I. A prime number itself has only one aliouot part,

namely, 1.

II. A number formed by a single prime factor raised to an

integral power, has n aliouot parts, namely 1, p, p®,

p®, . . . p""’. The sum of these parts forms a geometrical

progression and amounts to

(P" -1)/(P-1) III. In the third case, to determine the value of the sum of

the aliouot parts. Descartes operated recursively,

beginning with the case of a prime number multiplied by

another number the sum of whose aliouot parts is known.

Let n be a number, p denote a prime number, and s the

sum of the aliquot parts of n. The aliquot parts of np

are, then, those of n taken alone, those of n multiplied

by p, and n and p themselves since they are evidently

factors of np. The sum of these terms gives (p+l)s + n,

or ps + s + n. Introducing modern notation, let s(n)

be the sum of the aliquot parts of n. Then*

*0n his personal copy of the L'Harmonie Universelle, on the last blank pages, Mersenne gave several examples of the problem: " 'Given a number find the sum of its aliquot parts.' Let the number be 12. I take its prime factors or analogues, i.e., 3 and 4. 3 and its aliquot part make 4 ; 4 and its aliouot jnart make 7. The product of 4 and 7 is 28. Subtract 12, yow will get 16 which is the sum of the aliouot parts." In modern notation, we could formulate it as: If

n = ab. 214

s(np) = p s(n) + s(n) + n.

IV. Generalizing the result, let N be a number with two

different prime numbers raised to any integral powers,

i.e., N = p'^g™. Then the sum of the factors leads to

the recursive formula

s(p"q™) = s(p™) s(q”) + p^s(q™) + q^s(p^)*

Substituting for s(pH) and s(q™) the sum of the

geometric progressions given above, Descartes derived the

explicit formula:

s(p^q®) = (pn+lgM+1 + pHgrn+l _ qm+1 _ pU+l _ pn^m +x)/

(pq - p - q + 1)

In the case where a prime power p^ is multiplied by a

number y, relatively prime with respect to p but whose sum

of aliquot parts is known, the above computation leads to

the recursive formula:

s(p*y) = s(pH) s(y) + p^s(y) + ysfp*).

Descartes concludes this with the rule:

If we have two mutually prime numbers and their aliquot parts, we also have the aliquot parts of t their product: if one is x and the aliquot parts ': are s, the other is y and its aliquot parts are z, the aliquot parts of xy will be sx + yz + sz.

V. In the general case of a positive integer formed by the

product of any arbitrary number of prime factors raised

-where n is the given number, and a and b its prime factors, then _ ” _ - -

. - s(n) = [s(a) + a][s(b) + b] - ab " ‘ which is exactly the formula given above. 215

to some power, one first computes the sum of the aliquot

parts of each of the prime factors separately and then

applies the recursive formula to compute that of the

first two, followed by that of the product of the first

two by the third, of the first three by the fourth, and

so on.®’

Applying this method to Euclid's formula, let N be a perfect number, i.e., one where the sum of the aliquot parts is equal to the number itself, or s(N) = N. Let N = 2"’p, where p is a prime number. Then

s(N) =s(2"-’p) = s(2"’)s(p) + 2"’s(p) + ps(2"’)

®’See Mahoney, The Mathematical Career of Pierre Fermat. 287 ff.; a more direct method derives the same by the formula: Given an integer of the form

N = Pi"P2bP3e • • • Pnm

Where Pi are primes and a, b, c, ..., m are positive i n t e g e r s ,

s(N) = [ (pr'- 1)/(P1-1) ] [Pz'"'-!)/(P2-I) ].. .

[(Pn"'-l)/(Pn - 1)] - N see Mersenne, Correspondence. vol. 9, Appendix 3, 580, where Mersenne gave the following example: Take the number 588 which corresponds to 17685. It is composed of the factors 8, 3, and 37. The exponents of their corresponding prime factors are 3, 1, and 1. Adding a unity to each of them gives 4, 2 and 2. ... The sum of the aliquot parts is: [(2"-l)/(2-l)][(3® - l)/(3-l)][(37:-l)/(37-l)] - 588 = 1392 216

or S(N) = 2^ - 1 + 2"-’ + p(2"’ - 1)

= 2(2"‘’) - 1 + p(2"’ - 1)

2" - 1 + p(2"'’ -1)

which is to say, since s(2"*’p) = 2"*’p

2"*’p = 2(2"'’) - 1 + p(2""’ - 1)

or, solving for p

p = 2" - 1

Hence, N = 2"’p is a perfect number if p is a prime

of the form 2" - 1, and the problem of determining perfect

numbers reduces to that of determining the primalty of 2" -

1 for the various values of n.

For Mersenne's second conjecture, let us analyze his

rule in order to find the method he followed to arrive at

it. In the Coaitata he first noted how difficult it was to

detect a prime number by saying:

The prime numbers are difficult to know, even though there are some rules that can alleviate the work. For example, any prime number, except 2 and 3, differs from unity by 6 or a multiple of 6; likewise [or else], they differ by 4 and its multiples, except 2.

He then passed to give a recula admodum admirabilis

to determine the primalty of certain numbers:

*I.e. let p be any prime number different from 2 and 3, and n be an integer. Then

p - 1 = 6n or, for p different from 2,

p - 1 = 4n 217 The power of 2 from 1 has been subtracted; such that the exponent is a prime number which differs at most by 3 from a power of 2, and is even, is a prime number. For example, 7 differs by 3 from 4, a power of 2 whose exponent is even. Therefore 127 is prime.33

If the rule is tabulated for the first twenty one prime numbers it becomes easier to find how Mersenne arrived at the formula and why it did not work for certain cases. p -3 -2 -1 +1 +2 +3

2 - - 1 3 4 =2^ 5

3 - 1 2 4=2% 5 6

5 2 3 4=2% 5 7 8

7 4=2% 5 6 8=2^ 9 10

11 8=2^ 9 10 12 13 14

13 10 11 12 14 15 16=2*

17 14 15 16=2^ 18 19 20

19 16=2* 17 18 20 21 22

23 20 ' 21 22 24 25 26

29 26 25 24 30 31 32

31 28 29 30 32=2® 33 34

37 34 35 36 38 39 40

39 36 37 38 40 41 42

41 38 39 40 42 43 - 44

33Mersenne, Tomus III, 181-82. The Latin is: "Numerus binarii analogicus unitate decurtatus, cuius exponens primus, ternario vel minore mumbero ab aliquo inarii analogo, cuius exponens sit par,- e s t numerus _ primus; verbi gratia 7 differt ternario a 4 binario analogo, cuius exponens-est par, ideoque 127 is . primus. " _ _ - -- In other words, 2^ - 1 is prime if n = 2? ± 1, or 2™ ± 2, or 2™ ± 3. ~ 218

43 40 41 42 44 45 46

47 44 45 46 48 49 50

51 48 49 50 52 53 54

53 50 51 52 54 55 56

57 54 55 56 58 59 60

61 58 59 60 62 63 64=2®

67 64=2® 65 66 68 69 70

From this table it appears that the first prime for which Mersenne's rule failed to detect a prime is for p = 31

= 2® - 1, where the exponent of 2 is not even as the rule required. The same holds true for 127 = 2^-1. However,

Mersenne listed both values as factors of perfect numbers.

Instead, the rule works perfectly for p= 61, but this escaped Mersenne's attention, probably because he mistakenly thought he could factor the resulting Mersenne number.

Similarly, the rule applies also to 67 and 257 as Mersenne conjectured," without actually finding the perfect numbers they generate. But neither of the two primes generates perfect numbers. Pervouchine detected these errors in 1883 and he also found the perfect number generated by the prime

61.

Mersenne in Coaitata wrote:

"The case of 67 is not a misprint for 61, because Mersenne calculates the case explicitly for 67 and finds it prime. C. Archibald already mentioned this case in his article on Mersenne's numbers, Scrinta Mathematica 3 (1935): 113. So did S. Drake in Phvsis 13 (1971): 241. 219

There is no other perfect number beside those eight unless you want to go beyond the exponent 62 in the progression of the number 2 starting from 1. The ninth perfect number is, in fact, generated by 2*"; the tenth is the perfect number generated by 2 ’“'^; the eleventh is 2^ % i.e., the exponent 257 from which 1 is subtracted and multiplied by the exponent 256."

It was, probably, using this formula that Mersenne was able to correct the error of Peter Bungus (+1601), who claimed to have found 20 perfect numbers with 24 or fewer digits.” Tartaglia had already committed the same mistake.

How can we explain Mersenne's errors of commission and omission, at least up to the value 127, which he "guessed" to be a perfect number generator? The above table shows that p = 31 and 127 are both a unit less than an odd power of 2.

Except for 2, not one of our numbers is under the column +2 or -2; the other values are in the columns +3 and -3. We can, therefore, conjecture how Mersenne was thinking and generalize: The prime exponents of the binary must differ by 1 from an odd exponent, and not more than 3 from an even exponent. With this generalization, the inclusion of 31 and

"Mersenne, Coaitata Phvsico-Mathematica. Praefatio Generalis, non numbered page, folio XI. In modern notation: 2"®(2"^ - 1). Oystein Ore fails to give credit to Mersenne for the case of p = 127 and p = 257, fNumber Theorv and Its Historv. 71). Mersenne warned anyone who wanted to undertake the task of searching for other perfect numbers that there would not be any between the numbers 2 raised to 17000 and 2 raised to 32 000, again a wrong conjecture.

"See however the letter of Bonnell to Mersenne, Correspondance. vol. 14, 331-332. 220

127 and the omission of 89 and 107 from Mersenne's list is

explained/*

D. Slovinski, who found the last three known

Mersenne primes, wrote that "there will always be more

guesses about how Mersenne devised his prime numbers than

the numbers themselves,"” because of the many unexplained

questions. One has to admit, however, that the attention

given to such numbers has been instrumental in finding the

largest prime numbers so far known. They have been declared

"the most wanted factorizations" and the Department of

Energy of the United States under Contract no. DE-A C04-

76DP00789 is financing a project at the Sandria National

*S. Drake, , claims that the error might be due to the printers who could have skipped a line in the manuscript. Since Mersenne stated the problem in two different places three years apart, he probably would have pointed out the mistake to the printers as well as to the readers (or at least to his correspondents), as he did on other occasions. It seems that it is more likely that Mersenne did not work out all the cases and hence did not realize the contradictions. See also "New Mersenne Conjecture," American Mathematical Monthlv 34 (February 1989); 134.

”d . Slovinski, "Searching for the 27th Mersenne Prime," Journal of Recreational Mathematics 11, (1978-79): 258-261. The three largest Mersenne numbers are for p = 23209, 44497, 86243, the last one has 24,065 digits. However, during his lifetime people were questioning Mersenne insistently about his method of finding perfect numbers. See G. Thibaut's letters on January and February 1648, as well as the above- mentioned letter of Bonnel, even though both correspondents did not tell him explicitly that his formulas were not general. 221

Laboratory, California, to search for other possible

Mersenne primes and their use in cryptology.”

Multiply Perfect Numbers

Studying the properties of perfect numbers,

Mersenne, contradicting Descartes, claimed that perfect

numbers are always even, ending in 6 or 8. However, he did

not leave a proof for his statement. He also realized that

many more of them could still be found, using his formula.^®

Knowledge of the aliouot parts and the primes led

Mersenne and his team of number theorists to look in greater

”From a lecture given by Simmons to the Eurocryptology Society in 1983. The lecture was published by Brette, Conjectures et résultats mathématiques in Education et Informatique 11 (1984): 24-27. The discovery of the last Mersenne numbers captured the front pages of the American mass media, (see Time. February 13, 1935). The fact that there are so few perfect numbers made Mersenne conclude the section on perfect numbers in the Coaitata Phvsico- Mathematica in the same way he had concluded La Vérité des Science with a remark that was quite common among the mystic mathematicians: From this it is evident how few are the perfect numbers, and how riahtlv thev are compared with the perfect people (Praefatio generalis," non numbered page.)

^See (Correspondance vol. 8, 194, 282) Descartes' letter to Mersenne, where he asserts that he could provide proof that there exists no other even perfect numbers, besides the ones of Euclid, but that odd perfect numbers do exist. In fact, he said, if a prime p is multiplied by the square of a composite number c the resulting value is a square. As an example he provided the number 22021, which he thought to be prime by mistake, multiplied by 9018009, a square of the product of 3, 7, 11 and 13, obtaining the value of 198585576189, which he claimed to be a square. In a later letter to Frenicle he again repeated his conviction that odd perfect numbers could be found (Adam and Tannery Oeuvres des Descartes, vol 2, 476). However, this is still one of the unsolved problem in modern number theory. 222

depth into the properties of the natural numbers. Following

the definition of Euclid and the ancient mathematicians, he

called deficient (or defective) the numbers smaller than the

sum of their aliquot parts, and abundant (or superfluous) those larger than the sum of their factors. Using the notation introduced above, we can say that n is a perfect number if s(n) = n, deficient if s(n) > n and abundant if s(n) < n.

Then he proposed again the question: for what numbers, if any, is s(n) = mn for a positive integer m?

More generally, how can one find numbers the sum of whose aliquot parts is in a given proportion to the number itself?"

If the number itself is considered as its own aliquot part, it was already known that s(120) + 120 = 360 =

3x120. Let

P„(n) = s(n) + n = mn, e.g. Pg(120) = 3 X 120. Such numbers are called multiply perfect of order m.^

^’Th i s problem is proposed in a paper kept in the Bibliothèque Nationale of Paris (f. fr., nouv. acquis. 5160, f.o 65 verso). reprinted in Correspondance. vol. 9, Appendix 2, 570-73). Mersenne's handwriting was on the back of a letter of Descartes addressed to Meyssonnier.

^Mersenne was the first to define such multiply perfect numbers. He wrote about it in the La Vérité des Sciences (1625), remarking that s(120) = 240 and consequently Pg(120) = s(120) + 120 = 3X120. 223

Mersenne proposed to Descartes in 1631 the problem

of finding other numbers with the same property. Descartes

neglected the proposal for seven years but undertook it then

to compete with the rising reputation of Fermat." Fermat, who delighted in numerical problems, answered Mersenne*s problem as soon as he was proposed the question and suggested 672 as a Mersenne entered this value in his

Nouvelles Pensée de Galilée, giving Fermat due credit, of course. In fact, in the Nouvelles Observations he described the method that Fermat had refused to make public to find all such Pg.

Write in a line the elements of the sequence 2, 4, 8, ...,2", subtract 1 from each term and write it above the corresponding value, e.g. 1, 3, 7, ..., 2" - 1 and below each term of the progression write the corresponding term increased by 1, e.g. 3, 5, 9, ..., 2" + 1. If the ratio of the (n+3)th term on the top of the three lines and the nth term on the third line is a prime number, then its triple of the ratio multiplied by the (n+2)th term in the second line is a Pg*

This was how Fermat had found the value for 672, and he verified that it also worked for 120. He then

"About Descartes' neglect of number theory, he wrote to Mersenne in 1631: "You ask me which numbers have the property you are interested in. My answer is that I do not know it and I do not care to know it. The reason being that it is more a matter of patience than of originality; besides they do not bear any new contribution to mathematics." (Correspondance vol. 8, 279).

" Mersenne, "Nouvelles Observations" in Harmonie Universelle vol. 3, 1123. In modern notation,

2"+*x3xp is a Pg if p = [2" - 1] / [2" + 1] is prime. 224

generalized that the method worked for infinitely many other

numbers. In the Coaitata. Mersenne subsequently gave the

following numbers: 120, 672, 533 776, 1 476 304 896,"

459 818 240. In Reflectiones (ch. 21, 180), he added the number 51001180160.

Descartes, after reading Fermat's results, claimed that 120 and 672 were only two special cases, and that

Fermat, in fact, did not have any general rule but had obtained his result only by trial and error." A very heated exchange of letters among Fermat, Frenicle and Descartes followed. Sometimes the Prior of the Monastery of Saint

Croix, A. Andre Jumeau, also took part in the discussions."

All this kept Mersenne extremely busy and excited about the positive results.

Descartes gave to Frenicle, via Mersenne, his general rule:"

"These four numbers are included in the Nouvelles Pensees de Galilée. "Preface au Lecteur", non-numbered pages.

"Later Fermat acknowledged the weakness of his own method (Correspondance. vol. 6, 158).

"saint Croix, the name by which Jumeau was commonly referred, is well known for the many problems in number theory that he addressed to Descartes, Fermat and others. He found the third value Pg. See Correspondance. vol. 6, 385; vol. 7, 179; O. Ore, Number Theorv and its Historv. 97.

"Mersenne, Correspondance. vol. 8, 192. 225

I. Any number whose aliquot parts are Pg, if it divisible

by 3 but not by 7 or 9 or 13, when multiplied by 273

gives a Pg,*

II. Any number whose aliquot parts give a Pg, if it

divisible by 3 but not by 5, or 9, multiplied by 45

gives a Pg;

III. Any number n divisible by 31 and 512 but not by the

square of 31 nor by 1024 or 43 or 127, when divided by

31 and multiplied by 87376, results in another number N

with the same proportion with its aliquot parts as the

original number with its aliquot parts. In modern

notation; let N = 87376n/31, then

s(n)/n = s(N)/N

He claimed that by using this rule he had been able to derive six Pg, four Pg, after Mersenne had assured him that no one else in Paris had been able to find any. But

Frenicle laughed at Descartes, calling his rules sterile."

On his part, Mersenne listed a series of multiply perfect numbers of his own deductions:

I. If a number is not divisible by 3, then 3Pg = P^;

II. If a number is not divisible by 5, then 5P^ = Pg.

"See Mersenne, Correspondence. vol. 7, 345; vol. 8, 192-93, 281-282. 226

Mersenne himself said that there had been found

thirty four P^, eighteen Pg, ten Pg, seven P^ but not any Pg.”

The Cycloid

The cycloid attracted the attention of the

mathematicians of the seventeenth century more than any

other curve®’. Galileo with Cavallieri and Torricelli in

Italy; Roberval, Descartes, Fermat and Mersenne in France;

Wallis and Wren in England studied the curve. They were

interested in its shape, tangent, area, and in its volume of

revolution around the base, its axis of symmetry, or around

some of its tangents. They also tried to find the center of

gravity of the curve and the center of gravity of the plane

area and of the solid of revolution of the curve. The

priorities of the discoveries caused some of the most bitter

exchange of words, first within the community of French

mathematicians and later among the Italians, French, and

English. Tracing the whole history has been a task that

began in Mersenne's time and continued throughout the

century. Mersenne himself had planned to publish an account

of the whole process. Unfortunately, his premature death

®®See the handwritten notes at the end of Harmonie Universelle transcribed in Correspondance vol. 9, 573.

®’"N o other problem has ever made so much fuss in the circle of scientists as that of the Roulette..." (B. Pascal, L'Histoire de la Roulotte. 1). 227 prevented him from doing so. The aim of this section is to do him justice in the key role he played in the study of the curve from early in his career until the end of his life.”

Both the controversy and the study of the cycloid did not end with Torricelli's or Mersenne's death, though, by then, it had lost most of its initial interest. For the sake of completeness this study will refer briefly to the efforts exerted prior to Mersenne; it will then describe in great

" T h e very first printed presentation of the mathematical aspect of the cycloid is due to Evangelista Torricelli in De Dimensione Parabolae which at the end has an Appendix de dimensions cvcloidis. followed by Scholion de cvcloidibus aliarum specierum (in Qpere di Evangelista Torricelli t. 1, 1.2, 163-172). The correspondence between the various parties involved and in particular that exchanged among Torricelli, Roberval and Mersenne provides a primary source in reconstructing Mersenne's role. Later on, in 1658, Blaise Pascal published a Histoire de la Roulette in French and a Latin translation of it. Because of his closeness to Roberval and a considerable degree of nationalism, Pascal's version is practically a back-up of Roberval's position. In reaction to him John Wallis, who had failed a contest on the cycloid indicted by Pascal, wrote in 1658 the De Cvcloide Tractatus. (London, 1659), which interpreted the series of events in favor of the Italian mathematicians. Again a disciple of Galileo came out with a Lettere ai Filaleti di Tommaso Antiate. della vera storia della Cicloide. supporting, naturally, the Italian version. But the problem with these opposing positions is that not one of them could have the complete picture of the progress of events at the time. Not until 1921 did Cornelius de Waard produce key evidence that may have finally settled the dispute. Other sources for this research include Baillet's vie de Mons. Descartes (1691), the notes taken by Francois de Verdus during Roberval's classes which were seen and approved by the master: Traite des Indivisibles (the translation by Evelyn Walker in A Study of the TRAITE DES INDIVISIBLES of Gilles Personnes de Roberval will be used here) and Christiaan Huygens' Horologium Ondulatorium (Paris, 1663). 228 detail the stages study of the cycloid went through, thanks to Mersenne's efforts.

Pre-Mersennian Study of the Cycloid

Pascal in the above mentioned Histoire de la

Roulette wrote:

The roulette is such a common curve and is so often traced before the eyes of the whole world, that we have reason to wonder why it was not studied by the ancient scholars. In fact, we do not find any study of it in their books.”

Actually this is only partially true. Paul Tannery in his history of science",^, that the Greek "mechanician" of Antioch, Karpos, may have found the curve already in the first century A.D. To support his assertion he quotes lamblicus that Karpos had "squared the circle," and described the curve in the plane by means of a line of double motion, i.e., a point that moves in a plane as a result of two forces acting on it. In fact this is how

Roberval later described his "trochoid." Actually, any curve in the plane that can be described by parametric equations satisfies lamblicus' description. So Tannery's supposition is very weak. If Karpos did indeed find the curve, he did not leave any trace of his results.

53Pascal, L'Histoire de la Cvcloide. 194.

“p. Tannery, Les Sciences Exactes dans I'Antiouite. vol. 2, 7-9 (reported by E. Walker, A Studv of the TRAITE DES INDIVISIBLES Of Giles Personne de Roberval. 55). 229

H. Bosman also claimed that the school of Aristotle must have studied the cycloid.” However, he says, it was the philosophers and not the mathematicians who were interested in the curve. But it was proved before that until the seventeenth century the mathematician and the philosopher in a given school were the same person.

Besides, in this case, the mathematicians were more interested in the base line as the successive points of the moving wheel came in contact with it, rather than in the path in the plane of a single point on the circumference.”

The Dutchman Nicholas, Cardinal of Cusa, also is credited with having attempted the study of the curve. John

Wallis wrote that while the Cardinal was in Paris he noticed a nail on the rim of a moving wheel. Back at home he wanted to describe the curve in mathematical terms, and arrived at the following results: Let AF be the chord on the generating circle subtending an angle of 120 degrees; fix a point Q on the vertical diameter of the circle so that AQ is equal to

AF. Draw a circle centered at Q and with radius AQ, cutting the horizontal line at H and K. According to him the section of the circle HAK describes the motion of the fixed point on the wheel, while this point moves along the

”H. Bosmans, "Andre Tacquet et son Traite d'Arithmétique théorique et pratique," ISIS 4 (February 1927): 53.

“E. Walker, A studv of the TRAITE DES INDIVISIBLES of Giles Personne de Roberval. 56. 230 horizontal line (see graph), and HK is half of the circumference of the generating circle BADF.

HK5

However, this is, of course, not a cycloid.

According to Walker, it is possible that the works of Cusa might have inspired, among others, the French mathematician Charles de Bouvelles. His description of the curve traced in the air by a point on the rim of a circle rotating on a horizontal line is: Divide the vertical diameter AC of the circle BAC in eight equal parts; continue the line AC until I, such that Cl can be split into six more parts, each equal in length to the above parts. AI is the radius of the circle centered at I which describes the desired curve. Furthermore, he concludes, FA is equal to half the circumference of the rolling circle AC. A better approximation, but not yet exact. 231

In the midst of the controversy between Roberval and

Torricelli, Galileo informed Cavallieri, in a letter dated

September 22, 1643, that, about fifty years earlier, he himself had thought about the curve for a long time, but without success.^ He had conjectured that the area enclosed by the curve and the horizontal basis was larger than three times that of the generating circle, even though his experiments would repeatedly yield a slightly lower value.

Wallis took the statement as a conclusive proof in claiming that Galileo deserved the honor of being the first to propose the curve. Later in this paper Galileo's result will be analyzed in detail; for the moment it suffices to report Cavallieri's own version of how he came to know about the curve:

It was actually Fr. Nicerone who proposed this query to me, but I did not spend much time on it, scared by a letter from Galileo. He in fact told me that he had uselessly spent much, very much, time, because, I believe, I had written to him previously. But whether Galileo was the first to formulate the problem, or others had proposed it to him, this I really do not know.” (emphasis added).

In fact, it is quite possible that Galileo tried to investigate the curve at such an early date, since he was

^Mersenne, Correspondance. vol. 16, 540.

58 See Qpere di Evangelista Torricelli. Favaro ed., vol. 3, 479. 232

quite familiar with the mathematical works of De Cusa;” but he probably was not able to obtain the correct curve,

otherwise his empirical method should have given him the correct result of the area.

Mersenne and the Cvcloid

In the above mentioned Histoire de la Roulette. Pascal wrote;

The late Fr. Mersenne, Minim, was the first to think about this problem around the year 1615, while considering the rolling circles. This is why he called it la Roulette. Later, he wanted to study the nature and the properties of the curve, but to no avail.”

He repeats the same statement also in the Latin version, where, however, he mentions that Fr. Mersenne led to it when rotarum motus attentius consideraret.

Several authors, Baillet®' among them, accepted

Pascal's testimony of Mersenne's priority at this early stage. However, Mersenne never claimed such priority and did not leave any record of his interest in the curve at that time.” Besides, as we have already seen, in 1615

Mersenne had just started teaching philosophy and theology

”E. Walker, A studv of the TRAITE DES INDIVISIBLES of Giles Personne de Roberval. 58.

” Pascal, L'Histoire de la Roulette. 194.

®’A. Baillet, La Vie de Mons. Descartes, vol. 2, 158.

”lf Mersenne thought he deserved credit for this, he had a chance to claim his rights during the dispute between Roberval and Torricelli when he was called to be an arbiter. 233

in his monastery at Nevers, so that even if he did give a thought to the curve, he probably did not have the time or the knowledge to arrive at any results.”

Roberval's letter to Torricelli by the end of 1645 read: "However, already twelve years ago now, after being

insistently prodded by our Reverend Mersenne, I ran into its demonstration, . .

Again in a second letter to Torricelli in 1647 he wrote: "Since I was having great success with my method for the indivisibles, in 1634 the very famous Father Mersenne reminded me of the trochoid."“

From this it can be deduced that Mersenne had already discussed the curve with Roberval before 1634.

Pascal suggests that Mersenne might have approached Roberval to propose to him the study of the problem as early as 1628.

Be that as it may, Mersenne "started proposing the search of the nature of this line to all those in Europe

“Pascal's assertion may have been based on the fact that in Ouaestiones in Genesim (1623, col 67-70) Mersenne dedicated four columns to the study of the roue d'Aristote. which is so close to the cycloid that one would think it would lead to it. He mentioned the curve again in his Svnopsis mathematica (1625, 183), so it may well be that is how he became interested in the curve.

“Mersenne, Correspondance. vol. 15, 362.

“ Divers Ouvrages de Mathématique.(Paris. 1693), 286, reported in Mersenne's Correspondance. vol. 6, 212. 234

whom he thought could achieve it, among them Galileo.” But

no one could succeed, and everybody was losing hope".” The

Latin translation says that Galileo was among the first to be asked to study the curve. This may explain to Pascal

Galileo's claim in 1641 that he had studied the curve as far back as fifty years before (conceding a confusion of dates

in Galileo). This, in turn, might explain Cavallieri's admission of ignorance as to whether Galileo thought about the cycloid by himself or at someone else's suggestion.”

Roberval had arrived in Paris in 1628 as a teacher of mathematics and philosophy at the College of Maitre

Gervais. In 1634 he won the chair of Ramus at the College

”ln fact, in the same letter where Galileo talked to Fr. Cavallieri about the cycloid, he mentioned also the letter that Mersenne had sent to him (Correspondance. vol. 9, 115-16).

”Pascal, L'Histoire de la Roulette. 194.

“in a letter to Mersenne on December 23, 1630, he talked about certain lines which he promised to mention in his future book on geometry, but which he thought are of such small relevance that he wondered how Mersenne even thought them to be of some interest. Since his attitude towards the cycloid always had been belittling, the conclusion of the editors of Mersenne's correspondence, that Descartes is referring to the cycloid appeard valid. Descartes insistently asked Mersenne to keep, with care, all the letters he sent to him. In fact, Mersenne, who had a great sense of the value of primary documents, very carefully kept all his correspondents' papers. Sometimes, in fact, he even copied Descartes letters addressed to other scientists interested in some research. Unfortunately, he was not equally thoughtful to keep copies of his own letters, nor were his correspondents equally careful to keep his letters. In fact, we do not have any of his correspondence to Descartes, which would have clarified so many points about his position on the dispute. 235

Royal. As mentioned in a previous section, this chair was

awarded every three years to the mathematician who had won

the contest proposed by the incumbent holder of the chair.

Roberval was able to keep the chair until the time of his

death. The problem proposed by Mersenne of describing one

cycle of the curve traced on a plane by a fixed point of a

circle together with its properties” was very interesting

and new, and no one so far had claimed to have the solution.

Roberval worked on it zealously for a long time. In fact,

after his first attempt, he gave it up as too difficult to

solve; in the meantime he discovered the method of

indivisibles and was successful in solving a lot of other

problems. After Mersenne's repeated requests that he try to

solve the problem of the rotating circle, he applied his new

method to the curve and was able to describe the curve

exactly. Later, he found the tangent at any point of the

curve and succeeded in proving that the area enclosed by the

first cycle of the curve and the basis was exactly three

times that of the generating circle. He, therefore, decided to keep his solutions secret and propose the problem for the contest of the Ramus chair for the following three years.

“According to Mersenne's view in 1624 a curve was thoroughly studied if one could describe the trace, the area it enclosed and the solids it generated farammicon. epioedon. sereon are the Greeks words he used). La Vérité des Sciences, book 2, 427. 236

Not even Mersenne learned of Roberval's solution.

In his Questions Inouves (1634) Mersenne stated the problem

and advanced the hypothesis that the curve might be the

upper half of an ellipse.” Seeing that Roberval was not

getting a definite result, he insistently asked his friends

Cornu and Villers to try to study his problem." Both of

them realized that the problem was way out of their reach

and both knew their efforts would be fruitless. The circle,

the ellipse and, at the beginning, even an isosceles

triangle were their choices for an answer, so that, by June

they ignored Mersenne's promptings. In 1636, Mersenne

published his Harmonie Universelle and in the "Livre des

Movements de Tous les Sortes de Corps" he again suggested

that the curve is an ellipse.” Finally Roberval decided to

disclose to him, in private, the real shape of the curve.”

In the personal copy on the margin of page 120 and 121

Mersenne wrote by hand: "It is not a semi-ellipse, but it is

”Mersenne, Questions Inouves. question 6, 22.

"Between March 12 and June 10, 1635, we have six letters from Mersenne to Cornu (13 May) and to Villers (March 12, March 25, May 1, May 15, June 10) asking them to work together, if necessary, but to find a solution to the problem of the circle (See Correspondance vol. 5, 439-440).

”Mersenne, "Livre des Mouvements de Tous Sortes de Corps" in Harmonie Universelle. Book 2, proposition 9, 120.

”Mersenne, Correspondance. vol.6, 167-76. 237

very similar to it. It is the trochoid, or the cycloid,

which encloses a space three times the generating circle.""

In the "Nouvelles Observations Physique et

Mathématique," however, he corrected himself:

In the ninth proposition of the second book on motions, I said that the circle or sphere rolling on a horizontal plane in a complete turn describes an ellipse, because effectively the described line, as I said, is very close to half of it. But if one examines geometrically such a line, it is not an ellipse, nor a helix, nor a quadratrix, but something in between and can be described in many ways, the first of which is the same as I described it in the above-mentioned ninth proposition. Another way of describing it is by means of the sinusverse or the sine of the right, just as the locus of points which show the way of describing them.”

He went on to describe a third method devised by an excellent Geometre, by which term he was referring to

Roberval.” Furthermore, he gave an accurate description of the oblate and prolate form of the cycloid. But this was exactly what was contained in a letter from Roberval to

74 Ibid.. 167, endnote.

”Mersenne, "Nouvelles observations physiques et mathématiques" 11th observation: De la liane descrite par la revelation d'un circle sur un plan droit in Harmonie Universelle. This is the method of double motion mentioned above. It is found also in the Traite des Indivisibles by Roberval, in the chapter on the cycloid. It easily leads to the current parametric equation of the cycloid:

X = r(0 - sin 0)

y= r sineversus 0 = r(l-cosO)

” For Mersenne's reference to Roberval as "Geometre" see Le Mechanicmes de Galilée, preface, non numbered pages. 238

Fermat^ dated February 6 or 7, 1637. Around Easter of that

year, since his holding of the chair of Ramus was about to

expire, he withheld the proofs of his results.

That year, Roberval was awarded the chair without

any competition, so he allowed Hersenne to disclose the

results to his correspondents, and in particular to his

arch-rival competitor Descartes. From the answers of both

Fermat and Descartes, it seems that Mersenne not only

withheld some results from them but might even had hinted

that the "excellent geometre" was not able to find the

tangent to the curve.^ Attracted by the prospect of

describing a completely new curve, both Fermat and Descartes

rushed to analyze it. On August 5, 1638, Fermat, who had

already distributed copies of his De maximis et de minimis,

wrote Mersenne a letter that was certainly intended for

Roberval. Here he showed his method "mechanique" of finding

the tangent to any point of the cycloid. Let the following

data be given: the "roulette" AB6F, whose axis of vertical

^See C. De Waard, Boulettin des Sciences Mathématiques. 1921, 42, and Mersenne, Correspondance. vol 6, 167.

™This was a strategy used quite often by Mersenne to solicit new solutions from his scholars. A clear example is given by the case of the booklet of Roberval, the Arsitarcus. that Mersenne published anonymously in his book Synopsis Mathematica. In 1645, writing to Torricelli from Rome, Mersenne asked him to study the booklet and tell him explicitly his objections against the book, if he had any, since "Roberval did not find anything to object to the book." ( Correspondance. vol. 14, 412). 239

symmetry is the diameter of the rotating circle, G is the

point where the tangent is expected to pass through.

The line GH, parallel to the base AP will intersect

the circle at C. The segment GH, parallel to the chord CB

which joins the maximum point of the cycloid to the point c,

is the tangent to the cycloid at G.

And he generalized: If the base AF is twice the

circumference of the circle BCE, then one needs to establish

the following proportions:

2CD/BD = GD/DH

If, instead, the base is three times the circumference, then

the proportion becomes

3CD/BD = GD/DH

Descartes, who until that time had ignored the problem, was pleased that "the one you consider the greatest of your geometers admits that he does not know it. But I could, if only I would give it the necessary time, prove that my analysis is as good as his."” On August 23, 1639,

” Mersenne, Correspondance. vol 8, 152. 240

he sent Mersenne his solution. His approach made use of the

instantaneous center of rotation, which actually was the

same as Roberval's method. It proceeds as follows:

Let G be the point on the cycloid where the desired tangent

is to pass; let AF be the base of the cycloid generated by

the circle BCE. If one draws a line starting from G and

parallel to AE, let C be the point where the line intersects

the circle. Then CE joins the point C of the circle to its

lowest point E. From G draw the parallel to CE, thus

determining the point N on the basis AE. (This is the

instantaneous center of the osculating circle passing

through G.) Since the tangent to a circle at any point is

always perpendicular to the radius, the perpendicular to GN,

say GH, is the tangent to the cycloid at the desired point

G.(See graph)

Descartes added that the proof was "fort court et

fort simple."” He claimed that he also had a second proof

"plus belle a mon are et plus aeometricme" but did not give

it to save himself the pain of writing it down. But after his death there was found among the papers that he had

80 Ibid.. 341. 241

copied in his own handwriting Fermat's proof.®’ With this he

claimed to be dropping the study of geometry, having

exhausted all there was to discover in it.®®

In the meantime, Fermat kept refining his method

until the cycloid became to him another example of his

general method. He could then claim:

In the same way that Mr. Descartes can claim having mastered all the curves and tangents, so can I; in particular, the comparison of the portions of the

®’c. Adam and P. Tannery, Ouevres de Descartes, t. 10, 305-307, quoted in Mersenne's Correspondance. vol 8, 37, despite the fact that in his letter to Mersenne he wrote: "I have examined carefully the pretentious demonstration of the roulette mailed by Mr. Fermat .... But this is the most ridiculous aalimakia that I have ever seen. Effectively all he proved was that, having found nothing good about this roulette, but at the same time unwilling to send nothing, he put together a completely inconclusive argument. He hoped that the scholars would not be able to understand it, and the others would think that he had proved something interesting. If Mr. Roberval is satisfied with that, one could say in perfect Latin: mulus mulum fricat [A mule scratches another mule.]" (Mersenne, Correspondance. loc. cit.. 531).

®® Actually, one of the reasons that made Descartes disenchanted with geometry was the refusal on the part of Roberval, who was holding two chairs of mathematics in Paris, to adopt his Geometrv as a textbook in his classes. Roberval himself could not stand Descartes, and he would always take the opposite position in any controversy. The rivalry between the two was well known in the academic environment both in France and abroad. As a matter of fact, a note of the Roman mathematician Ricci to Torricelli explains that Descartes, who always thought very little of the cycloid, devoted attention to its study using his geometrical methods solely to embarrass Roberval (Mersenne, Correspondance. vol 15, 93). In his letter to Mersenne on June 29, 1638 Descartes expressed his bitterness about Fermat's reaction and the attitude of Mersenne's academy towards his geometry (Mersenne, Correspondance vol. 7, 312-313). On the other hand, in Utrecht, Reineri used to read publicly excerpts of Descartes' Geometrv at the local academy. 242

diameter to the normal faoolioue) is mixed with the above lines, I can unravel just as easily as with simple triangle problems.“

By 1639, the French mathematicians had mastered the

problem of the tangent and the area of the cycloid.

Mersenne then began to disseminate these results among other mathematicians of lower caliber, such as Beaugrand and his

former student Pere Niceron; he tried to see whether he

could interest the English mathematicians, such as Pell and his friend De Haak from London, in this problem.* Roberval, continued the study of the various other volumes of rotation and their proportion to the circumscribing cylinders, but kept the results secret to avoid further clashes with

Descartes and his followers. The others considered the chapter closed, even though the bickering among Roberval,

Descartes, and Fermat continued.

The secondary benefits that followed from this painstaking analysis of the cycloid were that

I. The newly developed method of indivisibles by Roberval

and the analysis of Fermat proved to be valid techniques

for studying anv curve;

“Mersenne, Correspondance. vol. 8, 249.

*It is quite possible that in the two letters that Mersenne wrote to Galileo around this time he may have mentioned to him the problem. However Galileo "preferred" to ignore them with the excuse that no one in his Academy in Florence was able to read Mersenne's handwriting (See Mersenne, Correspondance. vol 6, 16). 243

II. Other new curves appeared, like the Folium of

Descartes. or garland, which was proposed by Fermat, but

whose mathematical equation was first expressed by

Descartes as:

kxy = X® + y®

where k is a constant of proportionality;

III. The generalization of the parabolic equation to any

integral order:

y = x"'

IV. Most important of all is the appearance of the inverse

problem: to find the equation whose curve is given.“

Some of these curves were completely new and neither

Descartes nor Roberval had the mathematical tools to

find their equations. However, it stimulated discussion

among the next generation of mathematicians such as

L'Hôpital, J. Bernoulli. Such problems, eventually, led

Leibneiz and Newton to develop the differential and

integral calculus;

V. The finding of the largest or smallest solid that can

be inscribed or circumscribed around a given solid;

VI. The exchange of mathematical papers became so frequent,

and the need of factual proofs to ascertain the priority

of a discovery so acute, that one mathematician would

“C. Adam and P. Tannery, Oeuvres de Mr. Descartes, vol. 2, 514. 244

communicate his results to several other mathematicians.

Hence, the current mathematical notation needed to be

simplified much more than Viete had already done some

fifty years before. Mersenne was the one who was doing

most of the secretarial work for this invisible and

international community. Each single letter that

reached him was leaving his desk in ten copies to all

the corners of France and many places in Europe. He

worked hard to devise a simpler, more manageable and

expressive mathematical symbolism. Even here, the

credit would escape him and would go to the more

prominent name of the century, Descartes.®® He devised a

personal shorthand so that he could put in two or three

®® John Wallis in his mathematical work testifies that Descartes, and others with him, afraid of becoming boring with the continuous repetition of characters, denote roots with a certain symbol. Similarly, they use some hanging numbers to denote the exponents, like, a, a®, a®, a , etc. I therefore thought of giving here a sample in the following table:

Name German Vieta Ought Harriot Descartes Power or symbol red degree

Radix R A a a Quadr. Q Aq aa a, 2 Cubus C Ac aaa 3 Q.Quadrat. QQ Aqq aaaa 4 Surdesolid, S Aqc etc. 5

D. J. F.Scott, The Mathematical Work of John Wallis (London: Taylor and Francis, 1938), 67-68. 245

lines what would ordinarily take a whole page of writing

for others."

Internalization of the Discussions of the Cvcloid.

When the bitterness of the priority struggle over

the cycloid seemed to have subsided in France, in May 1643, the year after the death of Galileo, a letter from

Torricelli to the disciple of Mersenne, Roberval, re-kindled the controversy. It contained "short notes of some observations in geometry, . . . without any proof."®®

As was customary at that time, the letter was read in one of the meetings of the Mersenne's Academy, where

Roberval was one of the prominent members. The second part of the eleventh proposition and the one following it dealt with the cycloid. They read:

Definition: If a fixed point on a circle rotates in a plane starting from the point of contact with a straight line until the point intersects again with the straight line, the curvilinear line that it traced in the plane is called a cycloid.

12th Proposition:

Every cycloidal space, that is, the area contained under the curve, and the straight line which forms its basis, can be shown to be three times the area of the

“Mersenne, Correspondance. vol. 8, 191.

“Reported in Correspondance. 14, 216. 246

generating circle, or 3/2 the area of the inscribed triangle, and this can be shown in five different ways.®®

Roberval, who considered himself the only undisputed

authority on the subject, wrote the official reaction of the

academy to Torricelli in a letter addressed to Mersenne.

About the linea cicloide he wrote:

In Torricelli's cycloid I can recognize our trochoid and I cannot figure out how it reached the Italians, without our knowing it, unless maybe it was sent there by Jean De Beaugrand. He, in fact, used to take the discoveries of others, change their names and spread them among other mathematicians after suppressing the authors' names; but if such a discovery [of mine] pleased such a great man (I mean Torricelli) I am very satisfied.®®

When Torricelli read the letter that Mersenne

forwarded to Florence, he commented about Roberval's

astonishment:

But I am not surprised that our discovery reached France from Italy because I know that the theorem was discovered and spread among his friends by the well- known Galileo, even though he did not prove it. There

“Torricelli later published the five demonstrations in his Opera aeometrica. (1644). (See Qpere di Evangelista Torricelli. FAvaro ed., vol 1, 162-169).

®°Mersenne, Correspondance vol. 12, 253. In a letter he wrote to Torricelli on Nov. 1, 1645 he stresses even more strongly his contempt of the alleged behavior of Beaugrand: "Mr. Beaugrand took the demonstration sent by Mr. Descartes. After having copied it in his own handwriting - Mersenne, myself and many others have read the copy of it - sent it to Galileo. . . but it happened that excellent man, Mr. Desargues, could not bear such plagiarisms, wrote about it in clear terms in one of his works published already six years ago, in clear terms, so that everybody would be informed about what was due to whom .... (Mersenne, Correspondance. vol 14, 6-7). 247

are witnesses still alive and there are quite a few papers of his to prove it.®’

Actually the reaction of both mathematicians is quite difficult to explain. We have already mentioned that

Mersenne had published the results concerning the cycloid in the Nouvelles Observations. (1638, 24) and in the Nouvelles

Pensees de Galileo (1639) ®® He had written about it to many mathematicians throughout Europe; in fact, he had used the word cycloid, to designate the curve for the first time in

1639 in his letters to Pell and Der Haak in London.

Beaugrand was in Italy in 1635, before the solutions of Roberval were made public, while Fr. Niceron, Mersenne's disciple and a professor of mathematics at the Convento della Trinita in Rome, used to visit Fr. Cavallieri in

91 Ibid.

“Galileo was among the recipients of complimentary copies of Mersenne's Harmonie Universelle. Roberto Galilei (no relation to Galileo), who was stationed in Lyon, was given the copy for Galileo - he in turn gave it to a traveller to Italy, a Mr. Gondi, asking him to turn over the book to Galileo's hands. Actually one year later the book had not yet reached Galileo. In fact, Mr. Gondi lost it somewhere after he arrived in Pisa. But Roberto Galilei assured Galileo that Diodati and Mersenne had promised to send him another copy for Galilei. One may presume that both Mersenne and Galileo must have made sure that the book reached its destination the second time. See Correspondance vol. 6, 351; vol. 7, 44, 48, 118, 204, 287, 339. The Italian mathematician in Rome, Michelangelo Ricci, writing to Torricelli his reaction to the above-mentioned letter of Roberval, acknowledges that: "Mr. Raffaello (Maggiotti) tells me that Fr. Mersenne deals with the cycloidal curve in his book on music." (Qpere di Evangelista Torricelli. Favaro ed., vol. 3, 32). 248

Bologna frequently. As a matter of fact, it seems that it was through Niceron that Cavallieri learned about the problem. On February 14, 1640, Cavallieri wrote to Galileo from Bologna that Niceron had asked him on behalf of the

French mathematicians to look into some problems they were working on, one of them being the cycloid. Worried about the reflection on his integrity, he appealed to Galileo, to whom he remembered that he had mentioned the problem a few years back.“ Ten days later Galileo answered Cavallieri:

I do not know whether any of the queries that they sent to you from France were ever solved. I agree with you that they are very difficult to solve. It has been for more than fifty years that I thought of describing that curved line, and I appreciated it as a very pretty line and useful for the arcs of a bridge. I made various experiments on it and on the space enclosed by it and its chord to demonstrate its properties. Initially, in fact, I thought that such a space could have been three times the area of the generating circle, but I was wrong, even though the difference is not really big.*

This proves that Roberval's surprise and his singling out of Beaugrand as the only one who could have possibly reported to the Italians the problem of the cycloid was not justified. After all, this was not the first time

“Cavallieri's letter to Galileo is dated February 14, 1640. See Mersenne, Correspondance. vol 9, 115.

*Ibid.. 125. 249 that Torricelli had mentioned the problem of the rolling circle and used the word cvcloid.”

On the other hand, Torricelli preferred to ignore the two letters sent to him by Cavallieri, in which the latter told him that he had come to know of the cycloid from

Niceron. Also from the already-mentioned letter by

Maggiotti, Torricelli had learned that Mersenne had dealt with the cycloid in his Harmonie Universelle. In his direct reply to Roberval, Torricelli he went se far as to claim that Galileo had thought about the curve some "forty five" years before and had even coined the name, cycloid, for it.

The real problem was that Roberval was not known to the Italians. Even though Giovanni Rocca, a diplomat from

Rome stationed in France, had sent to Bologna a long report about the mathematicians in Paris and their major works,”

Roberval's name was not listed. We have already mentioned that Roberval had studiously avoided any public appearance of his mathematical work to secure his position at the

College de France. On the other hand, Torricelli's name also was new to many in Paris, but not to Mersenne.

Therefore, they did not have much confidence in each other

” In a letter to Maggiotti on Jan. 5, 1641, Torricelli had already hinted that he had found unexpectedly the solution of the cycloid. (See Mersenne, Correspondence. vol 10, 393).

”Cavallieri to Torricelli, see Mersenne, Correspondence. vol. 9, 435. 250

and both parties thought they were deprived of their right to priority.

However, Torricelli sent a very conciliatory letter to Mersenne first, and then directly to Roberval, full of admiration for his mathematical ingenuity; but, at the same time, Torricelli reaffirmed his claims for priority. In this letter;

I. he explained how Galileo was the first to coin the name

cycloid for the curve as long as 45 years ago;

II. he stated that he ran into the solution by mere chance:

pene non cmaerenti. and he had five different ways to

prove his theorem;

III. he admitted that he did not have any result on the

solids generated by the curve; and that the young

disciple and secretary of Galileo, Viviani, had shown

him how to find the tangent to the cycloid;

IV. he acknowledged that Roberval might have discovered the

curve by himself, without the help of any other

mathematician. However, he claimed that there still

were people who could bear witness that Galileo

entertained them with this curve about forty years ago.

Unfortunately, no evidence remains to prove the assertions of Galileo's successor to the chair of mathematics in Florence. As a matter of fact, it raises many more questions than it succeeds in answering. If

Galileo did coin the word cycloid, why did he not use it in 251

his correspondence with Cavallieri? Cavallieri was very

familiar with Galileo's work, but both men referred to the

curve as one suitable for bridges, and Cavallieri, who had

rallied behind his countryman, would have been one of

Torricelli's witnesses. Besides, if Torricelli did in fact

solve the problem in 1641, why did he not share it with

Cavallieri when this was challenged by the French

mathematicians through Niceron? Was it just coincidental

that Torricelli, who had been helped by Viviani to find the

tangent of the curve, succeeded in solving the problem only

after it had already been solved in France? Finally, were

the documents that he proclaimed to be ready to produce the

same that Cavallieri thought he had but could not find?

Pascal had a peculiar explanation of the whole

matter: Galileo had died in January, 1642, and Torricelli

was given custody of all his papers. Among them he found

Descartes or Roberval's solution of the curve which had been

copied by Beaugrand. Roberval had baptized the curve

"trochoid," while Descartes always referred to it as the

"curve of Roberval." Beaugrand, Pascal claimed, did not

mention the author of the proof, but made one believe that

he himself might have been the one, although sham modesty

prevented him from stating so explicitly. Beaugrand had

died even before Galileo. Torricelli appropriated the papers and authorship, hoping no one could accuse him of plagiarism. In fact, soon after his controversy with 252

Roberval started, he printed the papers,“ attributing to

Galileo what belonged to Mersenne” and to himself what

” Torricelli, Opera aeometrica (Florence, 1644, see Qpere di Evangelista Torricell. Favaro, vol. 3.) Even though in his letter to Roberval he acknowledges that the method of finding the tangents was shown to him by Viviani, in the book he does not give him credit for it.

“According to Pascal, Mersenne is credited with coining the term cycloid. We have seen that he was the first to use it in his correspondence and in printed works. However Mersenne never claimed any priority in all this controversy. Was it because of his monastic modesty? But in the field of music he often stated that he would never attribute any of his works to anyone, not even to the great Galileo. More probably he did not want to brag about something that was not worth bragging about, and certainly his role as moderator did not induce him to add more fuel to an already inflammatory situation. Against Galileo's claim of priority it is sufficient to mention that Galileo would not have humiliated himself in coining names of curves that he could not define and whose properties he could not foresee. The editors of Mersenne's correspondence use the same argument to prove that Pascal's attribution to Mersenne is unfounded. In fact, they conjecture that the term "trochoid" is Roberval's. Mersenne and Roberval together might have coined the word cycloid, while Fermat and Descartes referred to it always as the roulette or la liane de Roberval. However, Mersenne often liked to show off his mastery of the Greek language. In his books he makes abundant use of it, always trying to explain the etymology of a word of Greek origin; therefore, Pascal's attribution might be quite correct. Even if he was not able to produce the mathematical properties of the curve, he certainly understood quite soon that "it was not a semi-ellipse, nor a helix or any other familiar curve," as he explained several times to his correspondents. His interest in the curve, shown by the insistence with which he solicited mathematicians everywhere to study it, could very well have induced him to coin a name for it. Besides, why would Roberval want a second name to indicate the same curve? In fact he used it only reluctantly in his correspondence with Torricelli, just as a sign of reconciliation. Had he really been involved in the coinage of the word, it would not have been according to his nature to give up something that he knew belonged to him, as the second letter to Torricelli shows. 253

Roberval deserved.... This, Pascal concluded, "was not only

inexcusable, but even the cause of his misfortune...."99

It is easy to detect how strongly Pascal was manipulated by Roberval, who had even convinced him to make changes in the first edition of his history of the c y c l o i d . 190 in fact, from the argument of Beaugrand's silence, Roberval could not prove that Beaugrand attempted to attribute to himself the credit of discovering the trochoid. It probably was Roberval*s temperament which caused him to attack so vindictively Beaugrand who had sided with his arch-rival Descartes. On the other hand, of the five proofs of Torricelli, the first one is exactly like

Fermat's method, 191 the second similar to Descartes' method, 192 and the third identical to Roberval's method.193

Roberval answered Torricelli's argument two years later, after repeated solicitations from Mersenne, with a very carefully worded letter that took him two months to complete. In it

99pascal. Histoire de la Roulette. 195.

199ibis.. 196.

191ppere di Evangelista Torricelli, vol. 1, 315.

: 102ibld. . 106-107.'

lQ3walker.-A Studv of the TRAITE DES INDIVISIBLES of Gilles -Personne de Roberval. 139. 254

I. He complained about Torricelli's failure to admit

clearly Roberval's priority of finding the tangent and

the value of the area of the trochoid;

II. He traced back the events that led him to be interested

in the curve, recalling how Mersenne approached him more

than 12 years before, (i.e., before 1633), how it took

him one year to solve it, but he did not share the

results with anyone, reserving it for the contest of the

Ramus professorship; but then after Fermat and Descartes

also found the solution, the results were made public;

III. He gave his conjecture of how the solution had reached

Italy through Beaugrand, and how Desargues had stood up

to defend the priorities of Mersenne and himself;

IV. Referring to a letter of Torricelli to Mersenne about

his inability to find a value for the volume generated

by the curve, Roberval had given Mersenne the solution,

including a method to find the tangent, from which

Torricelli now could advance his claims;

V. He explained how it had been a while now that he had

found a general law by which given the ratio of two

planes, the center of gravity of one of them being

known, the other center of gravity could be deduced;

VI. He strongly reacted to Torricelli's letter to Mersenne

where Torricelli stated that he had just found the value

of various solids generated by the cycloid, as well as

the value of the ratio of this solid with the 255

circumscribing cylinders. Both solids implied the

knowledge of the center of gravity of the cycloid

plane;’*

VII. He stated the correct value for the second ratio: If

from a square whose sides are equal to half the base of

the cycloid, one subtracts one third of the height, the

following proportion will follow: the ratio between the

square — whose side is one third of the first — is

equal to the solid cycloid and the circumscribing

cylinder revolving around the axis of symmetry;

VIII. He admitted that he did not have any result about the

center of gravity, but he knew that the axes of the

solids divide the centers proportionately, so that if

one had the value of the ratio then he would be able to

find the quadrature of the circle; the solid generated

by rotation around the tangents depended completely on

the base, the axis and the plane of the trochoid.

’*The ratio found by both Roberval and Torricelli is 5/8 for the cycloid and the circumscribing rectangle rotating about the base and 11/18 for the solid obtained from the revolution aren't the axis of symmetry. The first value is correct. Torricelli revealed his results to Mersenne on May 1, 1644, but the latter had already written to him about Roberval's results on January 13 of the same year (see Mersenne, Correspondance. vol. 13, 127 and 116 respectively). This value appears also in Mersenne's Synopsis. The second value is not correct and Roberval realized it very soon that it was too small. CONCLUSION

In the introduction to the English translation of

the Harmonicorum Libri; The Book on Instruments [(Amsterdam:

N. Nijkhoff, 1957], xiii), the editors complained that

. . . in 1948 was celebrated the third centenary of the death of Mersenne. Observance of the anniversary took the form of the dedication of an issue of a French periodical devoted to the history of Science, the Revue d 'Histoire et de leur Applications (July-December 1948). That no similar recognition appeared in any of the musicological journals of that year is evidence of the neglect of Mersenne in musical studies. It is the opinion of this writer that such lack of attention to Mersenne's writing is unwarranted.

Even though the above remark refers to the neglect

of Mersenne in music, one can safely generalize that thirty years later Mersenne's cause among the English speaking

scholars of science history has not advanced. The above mentioned translation remains the only work of Mersenne available in English; as far as it could be learned by this author, there is one book only about Mersenne published in

English, the Mersenne and the Learning of the School by

Peter Dear. In contrast, in France most of Mersenne's books were reprinted during the past twenty years (some of them by more than one publisher). Similarly, the main study on

Mersenne by Lenoble: Mersenne ou la Naissance du Mechanisms was re-edited, even though the author died in 1963. What is

256 257

more, the publication of Mersenne's correspondence by the

Centre Nationale de la Recherche is a sign of the interest

in the person of Mersenne. On the third centenary of the

death of Mersenne, the French Government produced a

commemorative coin with Mersenne's face on one side and the

names of Descartes, Galileo and Huygens on the other. It

seems that in the English speaking countries the image of

Mersenne is still a stereotype: he basks in the light

reflected by his major contemporaries like Galileo, Pascal,

and especially Descartes. It would seem that the derogatory names given by the cabalist Fludd fie tres Minime Mersenne) or by Voltaire fMinimorum Minimiseimus), together with the

image of him portrayed by Baillet fL'Homme de M. Descartes), are still present in the descriptions of Mersenne by contemporary scientific encyclopedias and scientific dictionaries. The only recognition given him is associated with the prime numbers named for him.

One of the reasons for the lack of recognition of

Mersenne may have been derived from the fact that, unlike his contemporaries Descartes, Galileo and others, Mersenne never looked upon himself and his work as innovative, not because of low self esteem, but because he never really intended to break away from the Aristotelian and Scholastic approach to science, as Galileo and Descartes did. In a way, this gave him the liberty to be himself without any pre­ assumed positions that he felt bound to defend. For this 258

reason, his writings could well be less rhetorically layered

works of analysis. He never felt compelled to hide his

intellectual indebtedness; indeed, he often displayed it as

an endorsement of his own arguments. His style of

presentation relied more on the persuasiveness of familiar

doctrines than on those of his better known contemporaries.

Certainly Mersenne is very different from Galileo

and from Descartes; yet the difference between him and them

is not much greater than that between the two of them. What distinguishes Mersenne from his friend and correspondent,

Descartes, is the originality and the creativity of the letter's approach to science through logic and deduction.

Instead, Mersenne's leading principle in the discovery of the orderliness of the natural phenomena was the correct interpretation of the observations of long and repeated experiments he himself performed. Both men, however, were committed to combat the credulity of the astrologers, the magic occultism of a Paracelsus and Fludd, on the one side, and the formalistic rationalism of the old Scholastic philosophical approach on the other.

As this study has tried to show, Mersenne is an historical figure whose incalculable service to various sciences cannot be overlooked without dimming the picture of the historical development of science during and after the seventeenth century. His academic contributions certainly go beyond the restricted borders of France. His formal 259 education prepared him to be an abstract theoretician, but his natural inclination to science soon prevailed and manifested itself in all his activities. Because of this, he tried to give to everything he wrote an orderliness and reasonableness that is strictly scientific.

Music is the field in which he excelled: his

Harmonie Universelle and the Harmonicorum Libri XII are not only some of the most valuable sources for the musical contributions of the seventeenth century, but monumental symbols of his demanding level of scientific rigor. He taught himself how to become proficient in performing experiments in physics and derived valid laws that stood the test of time and are still accepted in our modern books. The law of the period of an oscillating string, the laws in ballistics about the range of a bullet shot, the compositions of forces in a plane, the conditions for equilibrium, the effects of friction in motion, the difference between the speeds of falling bodies in a vertical direction and on an inclined plane, and the composition of white light are only a few of the many results that he obtained. With time he became so proficient, that he could deduce his own "scientific method" for valid experiments and could even challenge Galileo's claims when he thought that his own experiments proved Galileo wrong.

Descartes, Constantin Huygens, Baliani, and others acknowledged his superiority in experimental physics and 260

often asked him to perform some experiments for them or to

check their results. To this, one has to add that his

editorial output in the other fields, despite all the

shortcomings inevitable for a self-taught scientist, place him among the best known scientists of the century. In twenty five years he published about twenty books, most of which are of his own creation. A few are translations of the best known scientists like Euclid, Theodose, and Galileo, while others contain material edited for his great friends, such as Roberval, Desargues and La Mothe.

Mersenne is as typical a representative of the scientists of his century as any of his contemporaries because his scientific activities bear much the same characteristics typical of the century. Mersenne's promotion of the mathematical sciences as the most effective means of quantifying natural phenomena demonstrates his alignment with the major development in seventeenth century natural philosophy. His chief significance, in fact, lies in his intellectual sympathy with the mathematical and even the mechanical approaches to natural philosophy and in his attempts to make them the basis of a new philosophical consensus. This quantitative and positivistic outlook on natural phenomena moved Lenoble to declare him the "father of the modern mechanistic view of nature." His activity as an intelligencer, which formed a crucial aspect of his agenda, helped consolidate an emerging community which acted 261

as the center of these new approaches.

The picture of Mersenne that has emerged from this

study is that of a scholar who did not allow traditional philosophical ideas to bind him to the past, but who took the liberty to develop a new agenda responding to the changing role of knowledge of the natural world in his cultural environment. His value as a focus of historical examination lies in his clarity of purpose and his unbridled scientific creativity. He described in detail the possibility of building a reflecting telescope, a submarine for defensive purposes, as well as a means of communication between a station on the ocean floor for scientific research with headquarters on the seashore; he enthusiastically received from Poland the project of designing a "flying dragon"; he proposed the development of portable musical organs, machines that could reproduce and magnify the human voice and project moving pictures. At the same time he rejected the abstract and sterile philosophy of some

"learned circles" among the nobility of Paris. However he did not reject a priori the extraordinary observations reported to him by his correspondents. Yet, he was not quick to accept their results if he could not repeat their experiment or obtain confirmation from a third party.

The second chapter of this study has pictured the grim situation in which mathematics found itself in France at the outset of the seventeenth century. It was 262

characterized by the lack of creativity and research in the

field. By the time of Mersenne's death, the scientific

community in France had taken a lead in the scientific

revolution that had come to represent a historical watershed

dividing the ancients from the moderns. Newton and Leibniz

did not create the calculus in a stroke of unprecedented

genius. No matter how novel their achievement, in a profound

sense it was merely the realization of the inherent

potential in a new approach to mathematics, of which they

were not the originators but rather the inheritors and

extenders. Their achievements can be fully grasped only if

they are placed in the context of the approach given to

mathematics by the previous generation of mathematicians,

including Descartes, Fermat, Roberval, Pascal, and Mersenne,

among many others. This approach transcends calculus in its

importance for the development of modern mathematics. It

created a movement that fundamentally altered the practice

of mathematics by freeing it from the strict geometric model

of the classical Greek approach and reformulating it in terms of a new algebraic model. Like any art, it could not rest content with admiring past masterpieces, but rather had to analyze them to discover how they had been achieved and to use them as a starting point for more work. Since past masters had obscured their techniques, contemporary artists had to devise their own methods. Only in that way could science progress. 263

As a result of such a renewed interest and committed dedication to his scientific research, Mersenne, together with the community of scholars whom he was able to rally around him in the Academia Parisiensis. multiplied greatly the number of curves known by and accessible to the mathematicians of his times. He analyzed their properties and found their tangents, the area they enclosed, and the volume of the solids they generated. The case presented in the study of the cycloid is a characteristic example; the conchoid, the loxodrome, and the spiral are other cases. But the failure to perceive a connection between the methods to find the area and the tangents of such curves prevented them from discovering the calculus that immortalized Newton and

Leibniz.

Mersenne's search for the properties of prime numbers and for their use in the discovery of additional perfect numbers elevated number theory to a respectable science. Mersenne's primes are now "the most wanted numbers." A few computers and a large number of scientists are searching for them. Fermat, with his analysis of

Diophantus and his famous conjecture, Frenicle and the Abbe

Saint Croix with their conjectures on the multiple perfect numbers contributed to the progress of this field of mathematics.

Where Mersenne's contribution outdistanced his contemporary scientists most clearly is in his commitment to 264 the service of science and other scientists. Without his dedication to act as clearinghouse for the scholars in and outside France, it is difficult to see how science would have spread so quickly and so widely. Scientific groups, if and when they could be formed, would have remained in isolated oases, with great loss of potential contributions, and important results often would have eluded them if the problems were not tackled with the most varied approaches by an international community. The barometric experiment provides a typical example. This service opened for Mersenne an exchange of correspondence with a great number of experimental scientists, amateur and would-be scientists from all over Europe. Mersenne's correspondence,retrieved from the library of the Ecole des Artes et Metiers in Paris and from book collections throughout Europe, fills sixteen thick volumes, even though many of Mersenne's own letters were unfortunately lost.

Mersenne is probably one of the first European scientists who felt compelled to give to scientific research a continental dimension and a unified vision and goal. If nowadays this goal has been partially reached through hundreds of common projects and research centers, Mersenne deserves part of the credit and has to be counted among the

Fathers of the Scientific Europe.

The main goal of this paper was to show Mersenne's role in fostering the work of the most creative scientific 265

minds of the first half of the seventeenth century. Through

him, Descartes could communicate with his rivals Roberval

and Fermat, and the French mathematicians were informed of

new developments in their field abroad. Thanks to Mersenne's

entrepreneurship Galileo's interdicted books could be read

in France, England, and throughout the rest of Europe even

when they were not available in Italy itself. The only works

of Roberval in mechanics and Desargues in music were

published by Mersenne. Information on the contributions of

many lower caliber scientists would have been lost forever

were it not for the records kept by Mersenne's

correspondence. Had Beeckman found an equally dedicated and

tireless editor as Galileo found in Mersenne, perhaps

Galileo's discoveries would not have been taken to be so

innovative, and the course of science would have been

completely different.

The everlasting legacy of Mersenne, the self-taught

scientist, to our computer-age civilization is this; science

cannot progress without the services of individuals thoroughly dedicated to its development and expansion.

Certainly, educators are the ones most directly involved in this task.

The time for teachers in the classroom and for mentors outside of school is not over yet. Neither television, nor teach-yourself books, nor computer software can ever be substitutes for the services a teacher or a 266

mentor has to offer. On the contrary, in order to utilize

their talents in the service of science fully, the gifted,

creative and talented students and new graduates need the

human touch. This can only be provided through the guidance,

encouragement and challenge of a tactful mentor. It is

obvious that average and marginal students demand even more

dedicated educators, if they are to succeed in their

education. More than anyone else, young educators themselves

need mentors. The article from the New York Times quoted in

chapter four of this study shows how timely the role model

of Mersenne is for our contemporary educational system.

Of course, guidance by mentors is only one of the

factors determining the career success of promising

students. The sociological and educational research

mentioned in chapter four indicates that mentoring for those who lack motivation for high achievement is doomed to

failure. Conversely, it appears that those who posses a high-achievement motivation but receive no mentoring at the beginning of their careers, often appear as overly aggressive and therefore, at times, as alienating. Those who receive both mentoring and exhibit a high level of achievement behavior are the ones most likely to succeed.

Mersenne's program of counseling the young Huygens and other young scientists can serve as a model. The first task of educators should be to perceive the strengths and weaknesses of the protege by exposing him to a large store 267

O f information about diverse topics and by offering him a

variety of potential fields of study and occupations. A

second step could be to motivate students by offering both

short and long term goals that are attainable. This would

help them to develop an awareness of their own potentials

and interests and would also help to develop a program of

studies which would meet their needs.

Mersenne's approach to his proteges was effective because it succeeded in stirring in them a positive attitude towards creative thinking. The young Pascal's first publications on the vacuum, and Christiaan Huygens' on the catenary are the results of Mersenne's encouragement. The objective of the long term goals projected by the educator is to develop in the student an aspiration to aim at higher results. This is done by stimulating in him independence and self-direction in learning, and by cultivating in him a need to experience helpful relations with other professionals and colleagues. All this was achieved by the un-assuming

Mersenne, who did not even pretend to pass into history as a model mentor.

As part of the solution of the many problems that the education of science is facing in the United States of

America, as well as in other parts of the Western World, educators need models that could inspire them to be effective mentors. The objective of such model setting could be: 268

I. To encourage mentoring for students and young

professionals in setting academic and career goals;

II. To develop courses in effective and useful mentoring.

This might include training in ways to encourage and

praise protégés, in ways to promote their strengths, and

in ways to promote career counselling, on the one hand,

and to train protégés on how to attract and benefit most

effectively from working with mentors, on the other;

III. To inspire cooperation between both experienced

professionals (educators in particular) and new recruits

in the techniques of the profession. Any profession

needs to be rejuvenated by the fresh ideas of youths, as

much as the latter need the expertise of seasoned

professionals. Even though nowadays scientific journals

can, in many ways, keep professionals abreast of

innovations in their field, as well as stimulate

research, nothing can compare with the human contact

between scholars of similar interests at different

stages of development in their profession.

Mersenne is such a model for educators.

In the realm of the physical sciences the name of

Mersenne does not evoke the same interest, enthusiasm and admiration as do the names of Galileo, Descartes and Pascal, among others. Although the latter achieved their recognition and fame as the result of scientific discoveries, a considerable amount of their popularity and notoriety 2 6 9

resulted from their clashes with Ecclesiastical authority

and Scholastic philosophy. Mersenne did not create such

notoriety; hence history has forgotten him, even though he

was equally well known to his peers in the scientific world

of his day. Today - on the fourth centenary of Mersenne's birth - is the logical time at last for the humble Minim

of the Place Royale to gain the recognition and reputation which he enjoyed among his fellow-scientists. His mere association and correspondence with men like Galileo,

Descartes, Roberval, Pascal, and Huygens should have insured him lasting and deserving fame. After all, as Pascal pointed out, Mersenne had an outstanding talent for perceiving the crucial point of many problems and of formulating them in such clear terms that it led to some discoveries that otherwise might not have been made. (Histoire de la

Roulette. 2)

He therefore has a right to be in the Hall of Fame, where I. Newton placed him in the company of Kepler and

Galileo. As A. Kircher said in 1650 (two years after

Mersenne's death), Mersenne was truly a Vir inter paucos summus [the top among the few] (Maanes. sive de Arte

Maanetica [Rome: Society of Jesus, 1641], 15). SELECTED BIBLIOGRAPHY

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