Homage to Galileo Galilei 1564-2014 Reading Iuvenilia Galilean Works Within History and Historical Epistemology of Science

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21-1 | 2017 Homage to Galileo Galilei 1564-2014 Reading Iuvenilia Galilean Works within History and Historical Epistemology of Science

Electronic version URL: http://journals.openedition.org/philosophiascientiae/1229 DOI: 10.4000/philosophiascientiae.1229 ISSN: 1775-4283

Publisher Éditions Kimé

Printed version Date of publication: 15 February 2017 ISBN: 978-2-84174-801-3 ISSN: 1281-2463

Electronic reference Philosophia Scientiæ, 21-1 | 2017, “Homage to Galileo Galilei 1564-2014” [Online], Online since 15 February 2019, connection on 30 March 2021. URL: http://journals.openedition.org/ philosophiascientiae/1229; DOI: https://doi.org/10.4000/philosophiascientiae.1229

Gerhard Heinzmann Laboratoire d’Histoire des Sciences et de Philosophie, Archives H.-Poincaré, Université de Lorraine, CNRS, Nancy (France)

Le premier cahier de Philosophia Scientiæ est paru il y a plus de vingt ans, en juin 1996. Il a été édité par une jeune équipe des Archives Henri-Poincaré à Nancy et dirigé par le même rédacteur en chef dans un souci constant : promouvoir la recherche en philosophie des sciences, tout particulièrement en ce qui concerne la logique, l’informatique, les mathématiques et la physique, en s’inscrivant dans la tradition analytique, en prenant en compte l’histoire et la pratique des sciences, en préservant le français, l’anglais et l’allemand comme langues « scientifiques » à titre égal. En passant la main à la nouvelle équipe, Manuel Rebuschi comme rédacteur en chef et Baptiste Mélès comme rédacteur en chef adjoint, j’exprime ma gratitude à Manuel, rédacteur en chef adjoint depuis 2008, et à Sandrine Avril, secrétaire de rédaction, pour leur travail efficace et leur dévouement. Je remercie Madame Charrié des Éditions Kimé, le comité de rédaction et le comité scientifique pour leur coopération facile et collégiale et je suis reconnaissant à l’Université de Lorraine et au CNRS pour leur soutien sans faille. L’émergence du monde numérique et l’accès en ligne ont changé la dimension de la distribution du savoir et exigent des expertises de plus en plus poussées. Je souhaite à la nouvelle équipe de savoir utiliser ses larges compé- tences pour saisir l’occasion des nouvelles techniques – tout en conservant les anciennes – afin de consolider l’ancrage national et international de la revue.

* **

The ﬁrst issue of Philosophia Scientiæ was published more than twenty years ago in June 1996. It was edited by a young team at the Henri-Poincaré Archives in Nancy and directed by the same editor-in-chief with the same

Philosophia Scientiæ, 21(1), 2017, 3–4. 4 Gerhard Heinzmann constant objective. This was to promote research in philosophy of science, particularly with regard to logic, informatics, mathematics and physics, in line with the analytical tradition and taking into account the history and practice of science. Another aim was to preserve French, English and German as “scientific” languages on an equal basis. Now I am handing over to the new team—Manuel Rebuschi as editor-in- chief and Baptiste Mélès as managing editor. I express my gratitude to Manuel, the managing editor since 2008, and to Sandrine Avril, editorial secretary, for their efficient work done with loyalty. I would also like to thank Madame Charrié of Kimé Editions, the editorial board and the scientific committee for making their cooperation easy and convivial and I am grateful to the University of Lorraine and the CNRS for their unfailing support. The emergence of the digital world and online access have changed the dimension of the distribution of knowledge and mean more and more expertise is required. My wish for the new team is that they know how to use their broad palette of skills to seize the opportunity offered by new techniques—while preserving the old ones—in order to consolidate the national and international position of the journal.

Gerhard Heinzmann Homage to Galileo Galilei 1564-2014.

Reading Iuvenilia Galilean Works within History and Historical Epistemology of Science

Introduction. 1564-2014. Homage to Galileo Galilei

Raﬀaele Pisano Lille 3 University (France)

Paolo Bussotti Udine University (Italy)

1 The Iuvenilia–Early Galilean works

When Galileo Galilei (1564-1642) published the Sidereus Nuncius in 1610 [Galilei 1890-1909, III, pt. I, 51–96], he was a famous enough scientist, who was not young: for, he was 46. Nevertheless, this little book represented the fundamental turning point in Galileo’s life and scientific production. The Sidereus Nuncius was very successful and gave rise to numerous discussions. Some scholars defended Galileo—the most important was Kepler—, many others, with a series of different arguments, criticized the content of the Sidereus. Galileo became the most famous and discussed European scientist. All his most important contributions, among them we remind the reader Il Saggiatore [Galilei 1623], the Dialogo [Galilei 1632] and the Discorsi e dimostrazioni [Galilei 1638], appeared after the Sidereus Nuncius. All these works are contributions which aim at reaching theoretical conclusions, although the experimental method plays a fundamental role to reach such conclusions. While, if we go back to Galileo’s production before 1610, three facts are surprising, enough: a) the relatively scarce amount of written and published works; b) most part of these works have a practical approach, that is, they are dedicated to the military operations or military fortifications or to instruments usable in a military context; c) some contributions are commentaries to important works of ancient authors, above all Archimedes. There are also some more theoretical contributions, like those relating to the new star appeared in

Philosophia Scientiæ, 21(1), 2017, 7–15. 8 Raﬀaele Pisano & Paolo Bussotti

1604 [Galilei 1890-1909, II, 267-306]. However, they are a minority. On the other hand, it is known that Galileo claimed more than once that the years he spent in Padua, working for Venetia Republic, were his most happy and fruitful period, from a scientiﬁc as well as a personal standpoint. It seems therefore only natural that he developed many of the ideas explained from 1610 forward in the period spent in Padua. Hence, the reason of interest connected to Galileo’s production preceding 1610 is twofold:

1. The speciﬁc content of Galileo’s writings.

2. The attempt to understand which ideas he had developed in that period, but clearly expounded after 1610.

Usually the word Iuvenilia is reserved for Galileo’s works reported in the initial pages of the first volume of Favaro’s National Edition [Galilei 1890-1909, I, 7– 178, hereafter EN]. According to the outlined panorama, we propose to extend the word Iuvenilia to the whole production of Galileo published before the Sidereus Nuncius. Thus, we refer to the content of the first two volumes of the Galilean National Edition. We consider all these contributions as Iuvenilia, identifying the separation line with the fact that, before the publication of Sidereus Nuncius, Galileo published, in substance, no theoretical work, though his theoretical activity had already been rather profound, as is well known. Here we report the content of the first two volumes of the National Edition:

Iuvenilia. – Theoremata circa centrum gravitatis solidorum. – La bilancetta. – Tavola delle proporzioni della gravità in specie de i metalli e delle gioie pesate in aria e in acqua. – Postille ai libri de sphaera et cylindro di Archimede. – De motu. [Galilei 1890-1909, I]

Breve instruzione all’architettura militare. – Trattato di fortiﬁ- cazione. – Le mecaniche. – Lettera a Iacopo Mazzoni (30 maggio 1597). – Trattato della sfera ovvero cosmograﬁa. – De motu accelerato. – La nuova stella dell’ottobre 1604. – Frammenti di lezioni e di studi sulla nuova stella dell’ottobre 1604. – Considerazione astronomica circa la stella nova dell’anno 1604 di Baldesar Capra, con postille di Galileo. – Dialogo de Cecco di Ronchitti da Bruzene in perpuosito de la stella nuova. – Il compasso geometrico e militare. – Del compasso geometrico e militare, saggio delle scritture antecedenti alla stampa. – Le operazioni del compasso geometrico e militare. – Usus et fabrica circini cuiusdam proportionis, opera et studio Balthesaris Caprae, con postille di Galileo. – Difesa contro alle calunnie et imposture di Baldessar Capra. – Le matematiche nell’arte militare. [Galilei 1890-1909, II] Introduction. 1564-2014. Homage to Galileo Galilei 9

The Galilean literature is wide, so that it is impossible to list all references.1 In the secondary literature one can read reflections, which are sometimes interesting, but not always historically proved such as clear hypotheses, reasoning and assumptions. In general, Galileo’s adolescent life and scientific activity are not explored and known, enough [Heilbron 2010]; i.e., it seems that his father also proposed, or simply had in mind, a sort of experimental method for his musical instruments [Cohen 2010, 85 sq.]. Other probably alleged— or not—stories concern his father Vincenzo who would give Galileo the idea of joining the Camaldolese order, though this story does not look accurate because the religious order, at that time, did not accept young boys. Some authors pointed out Vincenzo’s opposition to his son’s mathematical studies. There is also a debate on Galileo’s home-birthplace in Pisa. Furthermore, a new edition of Galileo’s Opere (such as also Torricelli’s) works seems to be necessary. Therefore, there are several unsolved aspects and historical problems as to Galileo’s young life. We hope that this special issue offers to the historians, philosophers and scientists a right provocation, an intellectual stimulus and a sufficient grit to face again Galilean studies in the 21st century.

2 On the special issue

The early Galilean works [Galilei 1890-1909, I–II] are historically crucial to better understand both philological aspects and early foundational Galilean convictions-doubts-methods before his definitive study on parabolic trajectories (1608-1609) and his mechanics. A cautious attitude sometimes shines, in some circumstances Galileo appears not completely confident with physical subjects, as it is the case, e.g., in some parts of the Trattato di Fortificazione [Ivi, II; see also Pisano & Bussotti 2015, and below Pisano’s works on the subject], where there are pictures which show rectilinear trajectories for the projectiles. The influence of the studies developed in this period clearly appears in conspicuous parts of later Galileo’s production: for example, when Galileo addressed technical subjects as the strength of materials in the Discorsi e dimostrazioni matematiche, he also faced architecture and fortresses. He conceived these parts not only as researches for military expert architects, but also as lectures for students: i.e., two speeches on fortifications stem from teaching speeches collected by his pupils on indications of Galileo himself. In this sense, the experience he had made in the period spent in Venice Republic, as a professor at Padova university and as an expert, who worked for the Republic, was fundamental. The relatively limited amount of secondary literature (on mechanics correlated to fortifications) is a matter of fact, testifying the modest interest in these topics with respect to (the more relevant) Galileo’s

1. In the References, we mention a selected list of the primary and secondary sources connected to the content of our Introduction. 10 Raﬀaele Pisano & Paolo Bussotti researches on mechanics, instead largely commented by many scholars. This is why we propose a special issue that, starting from the period in which Galileo produced his works on fortiﬁcations and military architecture, is the occasion to rethink, more profoundly, of the whole production by Galileo preceding his discoveries of the law of uniformly accelerated motion (1608- 1609) and the publication of Sidereus Nuncius in 1610.

3 The papers

All of the papers in this special issue have been independently blinded refereed. We have respected different individual ideas, historical, philosophical and epistemological accounts from each of the authors. The authors’ contributions appear in alphabetical order. Each of the authors is responsible for his or her own opinions, which should be regarded as a personal point of view based on his scientific background. Crapanzano (Italy) analyses Galileo’s dialogue De motu and related contributions under a particular perspective: the reading offered by Raffaello Caverni (1837-1900) of these Galilean works. The author reminds us that Caverni wrote, at the end of the 19th century, the monumental work Storia del metodo sperimentale in Italia, a text, which, although imprecise for many aspects, continues to represent an important source. After that, Crapanzano summarizes the history of De motu and its relations with De motu antiquiora. The author points out that Caverni played an important role in establishing the text of De motu for Galileo’s National Edition. He actively collaborated with Antonio Favaro (1847-1922), though Caverni is not mentioned in the Edition. The section 2 is an interesting and successful attempt to frame Caverni’s Storia del metodo sperimentale in Italia within the context in which this work was conceived. Finally, the author faces the difficult problem to catch the role of De motu within history of mechanics. In his paper, Drago (Italy) deals with a broad problem: what was the role of Galileo in the birth of modern science. To face the question, Drago first presents the most accredited positions in the literature as to this subject. Following his line of reasoning, the author identifies three dialectic dichotomies, which allow us to enter the problem: 1) experiment/ mathematical hypotheses; 2) potential infinity/ actual infinity; 3) axiomatic organization/ problem-based organization. As to this third aspect, the dialectic classical logic/ non-classical logic, identified by Drago, also plays a fundamental role. With regard to the birth of science, the author first analyses the interplay of these three aspects in their general form. Afterwards (8th and final section) the role of Galileo is analysed in the light of the explained dichotomies. Fredette (Canada) analyses De motu antiquiora published in the first volume of Galileo’s EN. The author traces a continuous line between this early work and the most mature and important contributions given by Galileo Introduction. 1564-2014. Homage to Galileo Galilei 11 to science: the Dialogo sopra i due massimi sistemi del mondo and Discorsi e dimostrazioni matematiche intorno a due nuove scienze. Fredette precisely identifies eight issues developed in details by Galileo in the Dialogo and in the Discorsi, which were already present in De motu antiquiora. The author, given any single issue he has posed, refers to the solution offered by Galileo in De motu and to the corresponding idea fully developed in the Dialogo and in the Discorsi. In this manner, a useful lens, by which to look at the whole of Galileo’s production is offered starting from one of his initial contributions to physics. In his contribution, Gatto (Italy) interprets Le mecaniche by Galileo as a conceptual bridge between the old science of weights and the modern statics. The author traces an ideal itinerary to explain the novelties of Galileo’s text relying upon what he calls “the comprehension principle”, namely the notion “according to which force, resistance, time, space, speed, continuously compensate each other: if the power increases, the speed decreases because, in moving the resistance through a given space, in the same time the force crosses a space longer than it”. In modern terms: work cannot be created ex nihilo, but only transformed; the nature cannot be deceived. Gatto faces such an important problem in this early work by Galileo offering a profound picture both from a philological-historical and conceptual standpoint. Gorelik (USA) deals with an extended form of the so-called Needham Question, that is: what prevented Greco-Roman and medieval civilizations from developing science following the line indicated by Archimedes and what prevented the Eastern civilizations from giving significant contributions to physics for centuries after Galileo (Needham’s questions were restricted to the Eastern civilizations)? Along his itinerary to find an answer, the author points out the mathematical and experimental method of modern physics and tries to identify a line of separation between Galileo’s and Archimedes’ approach. He then identifies an optimistic feeling in the regularity of nature and in our capability to penetrate such regularity. After that, the possible role of Christian religion is also taken into consideration. Lévy-Leblond (France) addresses a particular subject within Galileo’s production: the Due lezioni all’Accademia Fiorentina circa la figura, sito e grandezza dell’Inferno di Dante (1588). The author explains that Galileo entered into two different interpretations of Dante’s Inferno, the one by Manetti and the other by Vellutello. Galileo’s lectures, in which he also ex- ploited his mathematical knowledge to interpret some passages of the Inferno, fully succeeded and contributed to make Galileo famous in the academic environment. Probably these lessons played a role when Galileo became professor in Pisa university (1589). Beyond specifying all these aspects in details, the author also frames the Due lezioni in the context of Galileo’s early scientific production. However, a great part of this paper is dedicated to the specification of Galileo’s arguments developed in the Due lezioni. This allows the reader to fully appreciate Galileo’s pedagogical and argumentative capabilities, which characterize the whole of his production. 12 Raffaele Pisano & Paolo Bussotti

Martins & Cardoso (Brazil) present Galileo’s Trattato della Sfera ovvero Cosmografia, a treatise of geocentric astronomy, probably written in 1600 or few years earlier, as a textbook. The author compares Galileo’s Trattato with the text which was a reference point for this kind of books: the Sphaera by Sacrobosco. The similarities and differences are pointed out both as far as the conceptual and the stylistic-methodological aspects are concerned. The authors also look for the sources of Galileo, thus mentioning Peuerbach’s Theorica Planetarum and above all a series of vernacular treatises on the sphaera written in the second half of the 16th century. In this context, Galileo’s annotations of Piccolomini’s Sfera del mondo appear particularly significant. Other sources are mentioned, too, although Piccolomini’s work seems the most important one. The two authors also offer profound insights on the literature concerning Galileo’s Trattato della Sfera. Massai’s (Italy) contribution represents an ideal chronological conclusion of this special issue because the author analyses the Sidereus Nuncius, work which we assumed as the conclusion of the Iuvenilia and the beginning of Galileo’s mature phase. Massai intends to point out the importance of the instruments, the observations and the experiments in Galileo’s scientific praxis. The author underlines that Galileo was an expert instrument-maker. This was fundamental for him to realize the telescope could have an astronomical utilization. Furthermore, Massai enters the logic of many argumentations developed by Galileo in the Sidereus Nuncius, showing that Galileo reaches the right conclusion as he excludes a series of hypotheses because they are not consistent with the observations. Therefore, Massai’s is both a descriptive and a methodological paper. Mottana (Italy) examines Galileo’s treatise La bilancetta, taking into account the first draft of this work and the successive additions. The thesis of the author is clear: experiments guided the whole of Galileo’s production and scientific procedures. Thus, Mottana presents early Galileo’s studies from the very beginning of his scientific activity. After having described the environment in which La bilancetta was conceived, the author enters into an analytical and exhaustive examination of the subjects dealt with in the first draft, describing in detail all the operations carried out by Galileo with the instrument. The same approach is used to describe the additions. Here the figure of Guidobaldo dal Monte plays an important role. The whole paper is enriched by numerous and perspicuous references to primary and secondary sources.

Acknowledgments

The genesis of this issue relies upon our studies in history of science and, particularly, history of physics. We enthusiastically express our appreciation and gratitude to contributing authors for their eﬀorts to produce papers of interest and of high quality. We also address our acknowledgments to Gerhard Introduction. 1564-2014. Homage to Galileo Galilei 13

Heinzmann (Editor in Chief), Manuel Rebuschi, Baptiste Mélès (Managing editors), and Sandrine Avril (Editorial Assistant) for their good job and positive reception of our project to publish an issue on the Galilean Iuvenilia Works in the prestigious Philosophia Scientiæ.

Primary sources

Galilei, Galileo [1610], Sidereus Nuncius, [Galilei 1890-1909, III, pt. I, 51–96].

—— [1623], Il Saggiatore, [Galilei 1890-1909, VI, 197–372].

—— [1632], Dialogo sopra i due massimi sistemi del mondo, [Galilei 1890-1909, VII, 21–520].

—— [1634], Le mecaniche, Firenze, [Galilei 1890-1909, II, 155–191], 1649.

—— [1638], Discorsi e dimostrazioni matematiche intorno a due nuove scienze attenenti alla mecanica e i movimenti locali, [Galilei 1890-1909, VIII, 4–458].

—— [1890-1909], Le opere di Galileo Galilei. Edizione Nazionale sotto gli auspice di Sua Maestà il Re d’Italia, Florence: Tipograﬁa di G. Barbèra, edited by A. Favaro.

—— [ﬂ. 16th-a], Breve instruzione all’architettura militare, [Galilei 1890-1909, II, 15–75].

—— [ﬂ. 16th-b], Trattato di fortiﬁcazione, [Galilei 1890-1909, II, 77–146].

—— [fl. 16th-c], Ms. A – Trattato di fortificazioni e modo d’espugnare la città con disegni e piante di fortificazioni, Biblioteca Ambrosiana. D: 328 Inf.– 38640.

—— [ﬂ. 16th-d], Ms. B – Trattato delle fortiﬁcazioni, con disegni diversi, Biblioteca Ambrosiana. D: 296 Inf.–38499.

—— [fl. 16th-e], Ms. m – Trattato di fortificazioni, con disegni e figure (acephalous), Biblioteca Ambrosiana. D: 281 Sup.–2–80277.

Selected secondary sources for this introduction

Biagioli, Mario [1990], Galileo’s system of Patronage, History of Science, 28, 1–61. 14 Raﬀaele Pisano & Paolo Bussotti

Bussotti, Paolo (ed.) [2001], Galileo Galilei: Sidereus Nuncius, Pisa: Pacini.

Cohen, Floris [1984], Quantifying Music: The Science of Music at the First Stage of Scientiﬁc Revolution 1580-1650, Dordrecht: Springer.

—— [2010], How Modern Science Came into the World: Four Civilizations, One 17th-Century Breakthrough, Amsterdam: Amsterdam University Press.

Favaro, Antonio [1883], Galileo Galilei e lo Studio di Padova, Firenze: Le Monnier.

Gatto, Romano [2002], Galileo Galilei. Le mecaniche, Firenze: Olschki.

Heilbron, L. John [2010], Galileo, New York: Oxford University Press.

Pisano, Raﬀaele [2009a], Il ruolo della scienza archimedea nei lavori di meccanica di Galilei e di Torricelli, in: Da Archimede a Majorana: La ﬁsica nel suo divenire, edited by E. Giannetto, G. Giannini, D. Capecchi, & R. Pisano, Rimini: Guaraldi Editore, 65–74.

—— [2009b], Galileo Galileo. Riﬂessioni epistemologiche sulla resistenza dei corpi, in: Relatività, Quanti Chaos e altre Rivoluzioni della Fisica, edited by E. Giannetto, G. Giannini, & M. Toscano, Rimini: Guaraldi Editore, 61–72.

—— [2009c], On method in Galileo Galilei’s mechanics, in: Proceedings of ESHS 3rd Conference, edited by H. Hunger, Vienna: Austrian Academy of Science, 147–186.

—— [2009d], Continuity and discontinuity. On method in Leonardo da Vinci’ mechanics, Organon, 41, 165–182.

—— [2011], Physics–mathematics relationship. historical and epistemological notes, in: European Summer University History and Epistemology In Mathematics, edited by E. Barbin, M. Kronfellner, & C. Tzanakis, Vienna: Verlag Holzhausen GmbH, 457–472.

—— [2015], A Bridge between Conceptual Frameworks, Science, Society and Technology Studies, edited by R. Pisano, Dordrecht: Springer

—— [2016], Details on the mathematical interplay between Leonardo da Vinci and Luca Pacioli, BSHM Bulletin: Journal of the British Society for the History of Mathematics, 31(2), 104–111, doi:10.1080/17498430.2015. 1091969.

Pisano, Raﬀaele & Bussotti, Paolo [2015], Galileo in Padua: architecture, fortiﬁcations, mathematics and “practical” science, Lettera Matematica (Springer International Edition), 2(4), 209–222, doi:10.1007/s40329-014- 0068-7. 15

—— [2016]. A Newtonian tale details on notes and proofs in Geneva Edition of Newton’s Principia. BSHM Bulletin: Journal of the British Society for the History of Mathematics, 31(3), 160-178, doi:10.1080/17498430.2016.1183182.

Pisano, Raffaele & Capecchi, Danilo [2010], Galileo Galilei: Notes on Trattato di Fortificazione, in: Galileo and the Renaissance Scientific Discourse, edited by A. Altamore & G. Antonini, Roma: Edizioni nuova cultura, 28–41.

—— [2015], Tartaglia’s Science of Weights and Mechanics in the Sixteenth Century: Selections from Quesiti et inventioni diverse: Books VII-VIII, Dordrecht: Springer, chapter 1.

Wallace, William A. [1992], Galileo’s Logic of Discovery and Proof: The background, content, and use of his appropriated treatises on Aristotle’s posterior analytics, Dordrecht; Boston: Kluwer Academic.

Per quelle confuse carte... The Galilean De motu in Raﬀaello Caverni’s Reading

Francesco Crapanzano University of Messina (Italy)

Résumé : On peut, semble-t-il compter parmi les contributions liées au De motu et à ses enjeux, le texte obscur, verbeux et parfois inexact de Raﬀaello Caverni (1837-1900), l’Histoire de la méthode expérimentale en Italie. L’œuvre monumentale du prêtre toscan fut publiée en six grands volumes (le dernier à titre posthume) et suscita aussitôt autant d’éloges que de critiques. Au- delà des jugements extravagants et des hypothèses audacieuses, les idées de Caverni sur le De motu peuvent tout à fait s’avérer utiles, non tant comme « revue critique », que comme reconstruction du cadre théorique dans lequel s’inscrivait le travail de Galilée. On en tirera des hypothèses sur la façon dont Galilée a entrepris la construction de la nouvelle physique et d’une « histoire des eﬀets » désormais plus précise.

Abstract: In my view, among the contributions related to De motu and its related issues, we may include Raﬀaello Caverni’s (1837-1900) confused, “lengthy” and not always accurate writing on the subject in his Histoire de la méthode expérimentale en Italie. The monumental work of the Tuscan priest was published in six formidable volumes (the last posthumously) and immediately met both praise and criticism. Leaving aside certain odd judgments and risky assumptions, the Caverni’s ideas on De motu can legitimately be considered as useful material. This is not so much the case for his “review” but rather for his reconstruction of the theoretical framework within which Galileo worked in order to hypothesize how he undertook the construction of the new physics and for a more accurate “history of eﬀects”.

Philosophia Scientiæ, 21(1), 2017, 17–33. 18 Francesco Crapanzano 1 Introduction: De motu and its fortune

The so-called De motu appeared in 1642, after Galileo’s death when Vincenzo Viviani, the last follower of the great Pisan scientist, decided to collect all his writings in order to: [...] reprint all Galileo’s works in the form of sheets with greater fullness and magnificence in two columns for the two languages, the former Tuscan, in which the author wrote, the latter Latin to be translated by some of our compatriots and lastly with the addition of a great number of writings by the same author which I gathered from different parts with great difficulty. [Viviani 1674, 104–105], [Favaro 1885, 19, 158–159, my translation] This was an onerous but rather heartfelt task1 and led him to happen on some papers by Galileo collected in quinternions entitled De motu antiquiora while he was working on the bibliography of a major study on the fifth book of Euclid’s Elements. Viviani immediately thought they were early studies of the Master about motion, already showing an independent judgment with respect to the predominant Aristotelianism and, above all, they would be included appropriately in later works [Viviani 1674, 104–105].2 The manuscript in question, along with all the Galilean papers, ended up in the hands of the abbot Jacopo Panzanini, nephew of Vincenzo Viviani (his sister’s son), who ordered its bequest to the heirs Carlo and Angelo Panzanini. The latter were not particularly involved in the conservation of the manuscripts and probably happy to sell them to Giovanni Battista Clemente de’ Nelli (1725- 1793), a Florentine noble senator, bibliographer and scholar.3 Besides having

1. At the time when Viviani expressed this need a collection of Galileo’s works had already been published in two volumes edited by Carlo Manolessi and published in Bologna in 1655-1656. [Favaro 1883b, 311-342], [Favaro 1888, 1908-1909], [Galluzzi & Torrini 1984, 160–161, 190–192, 253–254, 301–308]. 2. On this and other events related to De motu, see the accurate reconstruction by [Camerota 1992, 17–62]. 3. Nelli explained that he bought the manuscripts despite being the beneficiary of them, through inheritance, thanks to a primogeniture resulting from the testamentary disposition of Viviani. Due to the unavailability of the testamentary trusts, the legacy would go to the de’ Nelli family [Nelli 1793, II, 761, 874–875], [Favaro 1887, 258– 260]. As recalled by Camerota, “l’acquisto del Nelli si è rivelato decisivo per le sorti della Collezione Galileiana, tanto che al marchese può senz’altro riconoscersi l’enorme merito di aver salvato da una probabilissima, completa dispersione la raccolta dei manoscritti di Galileo. Sembra infatti che i documenti fossero stati conservati— non si sa bene se dallo stesso Viviani o dai fratelli Panzanini—in una buca da grano della cosiddetta Casa dei Cartelloni, un tempo dimora del Viviani e, in seguito, abitazione degli eredi Panzanini. Questa impropria sistemazione, insieme alla completa mancanza di vigilanza o di scrupolo mostrata dai due Panzanini, sarebbe all’origine della scomparsa di un buon numero di carte galileiane, che, sottratte dal loro “nascondiglio”, cominciarono a circolare liberamente, alimentando un vero e proprio mercato” [Camerota 1992, 21–22]. The Galilean De motu in Raffaello Caverni’s Reading 19 patiently bought and collected the Galilean manuscripts, Nelli also had the admittedly lesser merit of having published a rich and documented biography [Nelli 1793].4 The texts of De motu antiquiora were also reported by Nelli thus: I keep some studies carried out by the Florentine Philosopher in his youth and transcribed by him in different quinternions above one of which there is the writing De motu antiquorum [...]; in others there are some errors included in Aristotle’s works. [Nelli 1793, II, 759] And then referring to the 1589-1592 period in which Galileo was a professor of mathematics in Pisa, he adds: [...] some scientific dialogues, in which [the great scientist] introduces Alessandro and Domenico as interlocutors who examine various propositions of Mechanics and especially about Motion, scattered in the Aristotle’s works [...] showing them to be for the most part erroneous and false. [Nelli 1793, I, 42]5 The biography written by Nelli quite closely follows that of Vincenzo Viviani, especially with regard to the classification of the writings in question which are correlated, by both of them, with the dominant Aristotelian conceptions about motion. In his early work, Galileo criticized erroneous theories with these criticisms containing early arguments and opinions which he was to recall in the most famous mature works [Nelli 1793, II, 759].6 Once the Florentine nobleman died, the papers passed to his sons, who decided to sell them because they were not very well-off. Thus the danger of dispersion reoccurred but this was avoided thanks to the purchase of the entire collection by the Grand Duke of Tuscany in 1818. It then formed the core of the Galilean Collection, now kept in the National Library of Florence. Other scholars, who had access to the manuscripts in the course of the following years, agreed with Viviani and Nelli that De motu antiquiora was a set of various Latin texts about motion dating from the early period of Galileo [Venturi 1818-1821, II, 330], [Antinori 1839], [Libri 1841, 12– 13];7 and in 1854 Eugenio Albèri included some manuscripts of De motu

4. On which, in a positive light [Micheli 1988], [Hall 1979], [Favaro 1883a] while conversely underlining a lot of mistakes and errors. 5. It is easy to recognize by the Nelli’s indication the dialogue that will merge in the Galilean early writings included in Edizione Nazionale by Favaro [Galilei 1890- 1909, 367–408]. 6. It is not a matter of chance that there has been so much discussion of both Viviani and Nelli’s biographical reconstructions reporting a “continuist” character. See [Micheli 1988], [Camerota 1992, 22–25, and passim]. 7. In particular, Guglielmo Libri, Florentine mathematician and historian of science, was the first scholar to have noticed the similarity between the Galilean 20 Francesco Crapanzano in the eleventh volume of Galileo’s works he edited, presenting them as Sermones de motu gravium [Galilei 1854, 9–80].8 A few decades later Antonio Favaro began the work on the Edizione Nazionale examining the texts of De motu which were not included in the eleventh volume of Albèri; and he did so with great care, despite the difficulties and the “annoying characteristics” of the manuscripts.9 However Favaro also initially made an error with respect to a part of the text included in the Edizione Granducale which he could not find in Galileo’s manuscripts. This and many other problems concerning the right location and timing of the texts in question were at the basis of the collaboration between the student from Padua and the priest Raffaello Caverni (1837-1900). In view of the future Edizione Nazionale, Favaro took advantage of his help, recognizing him as an expert on Galileo’s writings.10 Caverni collaborated very actively and can therefore be legitimately considered as an editor of the De motu writings to the same extent as Favaro. However his name is not present in the first volume of the Edizione Nazionale, due to a misunderstanding between the two authors which we shall refer back to later. It should be noted at this point that the Latin texts collected by Favaro with the help of Caverni would make up what was to subsequently be called De motu. As is clear in the first volume of the Galileo’s works, the scholar from Padua considered the arguments present in De motu very close to those exhibited extensively in the major works to the extent that:

[...] we can say that in them there are in seed and sometimes speciﬁcally explained, the wonderful discoveries which already put the author much above other contemporary philosophers and show the blossoming fruits later set out in the Discourses and Mathematical Demonstrations Relating to Two New Sciences. [Galilei 1890-1909, 246]

De motu and that of Giambattista Benedetti. This similarity was later taken up by Raffaello Caverni according to whom Galileo did nothing more than reiterate Benedetti’s thesis on motion. As we will show later, this sometimes frantic search for the genesis of Galileo’s theories is one of the main limits of the research conducted by the Florentine Abbot. 8. This edition of Galileo’s writings is particularly important for us as it will be the one used by Caverni for his analysis. 9. When Favaro presented the writings in question in 1883—well before they were published in the first volume of the Edizione Nazionale, he noted: “Frequentissime e spesso anche capricciose le abbreviazioni che si incontrano ad ogni pie’ sospinto; poco chiara la scritturazione, con frequenti cancellature e interpolazioni; omissioni di parole che rendono talvolta difficile l’afferrare il senso; la punteggiatura quasi del tutto mancante o irregolare; e finalmente non pochi errori di lingua, i quali contribuiscono in non lieve misura ad aumentare le incertezze nella interpretazione dei luoghi di dubbio significato” [Favaro 1883b, 3]. 10. This is well illustrated by the correspondence between Favaro and Caverni whose regesta is in the appendix to the excellent article by Cesare Maffioli on Storia del metodo sperimentale in Italia [Maffioli 1985, 23–85]. The Galilean De motu in Raffaello Caverni’s Reading 21

In the light of this consideration, it would seem natural to place Favaro along the interpretative axis that had indicated the seeds of the fundamental principles of dynamics in these early manuscripts which were to be subsequently reported. In fact, this view was contested because there would be no clear evidence of this insofar as Favaro gave no precise indications on the elements that were common to De motu and Discorsi. Similarly, he would have been very wary of declaring the connection between the two works.11 There is indeed no specification of those themes of early doctrines which Galileo would entirely recall in 1638 but the same cannot be said about Favaro’s alleged caution and, ultimately, it is impossible to definitively establish whether he considered De motu as the first immature text written on dynamics or as containing in nuce the foundations of the future science of motion. In any case, my own view is that we can exclude an understanding that tends to isolate the early writings on motion with respect to the Discorsi and this is confirmed in the foregoing publication where Favaro considered that: Those unpublished writings scraped together in the Galilean collection and primarily those related to the motion, have seemed very important to me, since the beginning. Although in some of them you find only youth studies, however like Michelangelo’s first scribbled drawings, in others and overall, I perceived much more relevant characters which need to be taken into account to portray the blossoming fruits in the Discourses. We can feel the Galilean verbosity which while perhaps not having the attraction of the familiar eloquence in the Two New Sciences, sickens a little but you can see, even in the form, the young man who, disdaining the succinct and concise Aristotelian doublet, went on to prefer the broad Platonic covering. [Favaro 1887, 229] Even in this case, however, the historian from Padua does not specify the connections between the old works and the mature reports on the new science of motion thus supporting a suggestion by Caverni who in 1889 (approximately a year before the release of De motu in the first volume of the Edizione Nazionale) wrote to Favaro that if he wanted to proceed by prefixing a short foreword, then it would be better to publish the writings in fragments: [...] but if you really persist in thinking of publishing them in that wonderful order divided by us and in which we read, as in a mirror and since the first steps, the processes of the speculations by Galileo in the science of motion and then, in order to prevent criticism [...] you cannot do so without an erudite Preface, as long as it is needed, without the ministry intervening to act as precisely as Procuste. [Lettera di Caverni a Favaro, dated 20 March 1889] [Maffioli 1985, 79]

11. Raymond Fredette, in particular, questioned Favaro’s alignment with the views of the previous criticism expressed in various diﬀerent ways and tones by Viviani, Nelli, Libri, Antinori and Albèri [Fredette 1969]. 22 Francesco Crapanzano

Despite Caverni’s good advice and the choice to present the writings in the order agreed, Favaro published only abbreviated information. Rather than indicating a deterioration of the relationship between the two or a simple diﬀerence of opinion, this probably reveals that there was not enough critical work on Galileo’s Pisan period which prevented the construction of an exact logical chronological order of De motu, as called for by Caverni when he suggested the “learned preface”.

2 A “maverick”: Raﬀaello Caverni and his Storia del metodo sperimentale in Italia

Why did Antonio Favaro decide to collaborate with a little-known priest? To answer this question, it is useful to point out two or three stages of Caverni’s biography. In 1874, Caverni’s first book was published, Problemi naturali di Galileo Galilei e di altri autori della sua scuola. The text was mainly for students of physics and, while not widely used in schools, was loved by scholars. The subject is doubly interesting because it testifies to the priest’s interest in the history of science12 and also expressed what would become a constant feature of his research—a kind of nationalism applied to the genesis of scientific discoveries or inventions: My intention was to open as a window to help young people find a broader field and explain how some inventions and doctrines, which we think came from France and Germany, were taught us much earlier by our compatriots and to show a small part of Italy’s hidden treasures to those who accuse the country of poverty. [Caverni 1874, 1] In the same year De’ nuovi studi della filosofia. Discorsi a un giovane studente [Caverni 1877, 40, 41, 159, 160], [Pagnini 2001, 40–50] was published, maybe “the greatest of his works because of intensity, originality and density of thought” [Favaro 1899-1900, 379]. In it, the priest from Quarate discusses the compatibility of the new theory about Darwinian evolution with the Holy Scriptures, by touching on a sore point of the relationship between science and faith. He does so with the aim of finding a settlement between the two parties and this was possible provided that something was sacrificed of the Catholic hermeneutic “prerogatives”, namely to recognize and accept the distinction: [...] in the books between the divine and the human: the former’s object is the truth of faith and it is infallible while the latter’s object is the science concepts acquired for the natural study of

12. In confirmation of this, ten years earlier, Caverni mentions in his diary: “L’amore che mi sento avere grande per gli studi della fisica e della matematica, scegliendo come guida negli studi presenti il divino Galileo” [Pagnini 2001, 21]. The Galilean De motu in Raffaello Caverni’s Reading 23

the hagiographers and which may be true or false like all things learned by the natural use of reason. [Caverni 1877, 24] In these words, everyone will have recognized the adoption of Galilean hermeneutic criterion in which the great Pisan scientist, who was fully convinced of Copernican heliocentrism, tried to argue that the Bible was not in contradiction with scientific evidence. Indeed he claimed that such evidence was actually compatible with Holy Scriptures, provided that these are not read literally and instead are interpreted in the light of such evidence,13 because: Writing often is not able simply to give an explanation but actually it requires an explanation which is different from the apparent meaning of words. [Galilei 1890-1909, 282] Our priest would not have expected the volume to be placed on the Index of prohibited books by July 1, 1878 and, though he tried to impose his point of view, finally he had to give up. Indeed it is probable that the Sacred Congregation of the Index did not prohibit the text because it presented the evolution theory. Instead this occurred because of the Church’s growing hostility to science (at least from the process to Galileo onwards), an antipathy towards the Jesuits (they always had opted for Aristotle when the new science was born under the sign of Plato) and, not least, because they had found in Galileo’s work the hermeneutic canons required to interpret the Holy Scriptures [Caverni 1877, 40, 41, 159, 160], [Pagnini 2001, 40–50]. Despite being put on the Index, among the few readers of De’ nuovi studi della filosofia were Antonio Favaro, who wrote a letter to Raffaello Caverni from Padua dated March 25, 1877. They were yet to become friends but it seems that his own reading of the text had led the scholar from Padua to deepen his friendship with the pastor from Quarate: You seem to me a good teacher, a very good teacher, but a teacher deeply convinced of the saying: Who loves well, castigates well. Was I mistaken? Meanwhile this book has increased my desire to meet you personally and perhaps before long you will find me knocking at your door. If my health allows me to rest and carry out a long work which I thought about for several years before deciding, I will also want to ask for your advice. [Lettera di Favaro a Caverni of March 25, 1877] in [Pagnini 2001, 53–54] This is the beginning of the friendship between Favaro and Caverni which is testified by a prolific correspondence of more than a hundred letters over more

13. This is what Galileo says, for example, in the Letter of 21 December 1613 to Benedetto Castelli when he writes that the truths of faith and truths of reason cannot be in conflict with each other, at most it is “ofizio de” saggi espositori affaticarsi per trovare i veri sensi de’ luoghi sacri, concordanti con quelle conclusioni naturali delle quali prima il senso manifesto o le dimostrazioni necessarie ci avesser resi certi e sicuri” [Galilei 1890-1909, 283]. See also the Lettera alla Gran Duchessa Madre Maria Cristina di Lorena [Galilei 1890-1909, 309–348]. 24 Francesco Crapanzano than a decade [Maffioli 1985, 54–71].14 As Maffioli noted “Favaro’s friendship helped to break Caverni’s isolation from the scientific and academic world” [Maffioli 1985, 28]; and although the priest was shy and tormented by many doubts and problems, the scholar from Padua’s bibliographic information often proved very valuable. On the other hand: [...] the friendship of Favaro was pretentious and required continuous exchanges involving information and searches for manuscripts and documents, reviews, contacts with publishers and editors of periodicals and so on. From 1883 onwards, Favaro completely drew Caverni into the great project of a new edition [...] of the Galileo’s works. [Maffioli 1985, 29] The pastor was enthusiastic and did not hesitate to advise and collaborate on these topics (Galileo) which he had shared an interest in since he was young. He had a different historiographical approach from Favaro. Both preferred the documentary aspect, but for the former this served as a platform to show the foundations and evolution of thought and tradition which had allowed Galilean science to blossom fully. The latter’s true goals were represented by the chronological reconstruction and the indisputable attribution of the writings. However, in 1887, Caverni was unexpectedly excluded by the editors of the Galilean works. He considered Favaro responsible for this which meant their personal relationship cooled somewhat.15 Although the scholar from Padua managed to convince the priest to continue their collaboration, shortly afterwards their uncertain friendship was to lead Caverni to abandon the challenge because he became increasingly convinced that he was being tricked in some way. From then on, he was to work hard on his most important and debated work—the majestic Storia del metodo sperimentale in Italia, to which he had already devoted part of his time.16 Until then he had worked following the indications of Favaro on the order of Galileo’s writings De motu, printed in

14. For a comprehensive evaluation of the intensity of the relationship between these two, all private notes and quotes in the papers and letters addressed to others should be added to these papers. Between them, besides the cultural exchanges, there were “le manifestazioni di affetto, i segni di una lunga consuetudine: il classico invio della fotografia (benché entro un opuscolo scientifico!) prima di poter fare una diretta conoscenza, i pranzi in comune, le discussioni nei caffè sul Lungarno, i fiori di lavanda per la moglie del Favaro preparati dal parroco di Quarate” [Maffioli 1985, 28]. 15. The disappointment following the failed official recognition as curator along with Antonio Favaro and Isidoro Del Lungo of the Edizione Nazionale is a more complex affair than it may first appear. It is unclear, for example, the role of Favaro in this exclusion since it appears that the publisher Le Monnier who first worked on this job had said he was very content with Caverni. However the work of guardianship had to be paid for by the Ministry which then would have to decide on appointments made. We do not know why (simple oversight or failure to report?) two were nominated, instead of three [Maffioli 1985, 32–33]. 16. As resulting from the part of correspondence between Favaro and Caverni, promptly reported in [Maffioli 1985, 37]. The Galilean De motu in Raffaello Caverni’s Reading 25 part in the edition of Galileo’s works written by Albèri and then, for another part, by Favaro himself, in a journal.17 In a letter dated 22 August 1884, the scholar from Padua clearly asked Caverni to take care of the above-mentioned writing in order to understand:

I. If the writings to which belong the extracts contained in the ﬁrst part of that work [the unpublished works published on the magazine Editor’s Note] are to be published II. With respect to the Sermones De motu gravium, how are the fragments contained in the second part of the same work to be distributed? [Lettera di Favaro a Caverni of 22 August 1884] in [Maﬃoli 1985, 62–63]

As we have noted, the priest paid attention both to the right order to be given to the manuscripts and also to their placing to correctly portray the inner evolution of Galileo’s thought.18 Although Favaro preferred a philological approach in the wake of positivism, he still consulted Don Raffaello who was to provide an order for De motu in five books which would have required an important critical apparatus. Towards the middle of April 1887, although Caverni had already learned of his exclusion from the editors of the Edizione Nazionale, he still believed this to be perhaps due to a simple misunderstanding and declared himself willing to invite Favaro to his rectory to discuss “the best appearance to give to early Galileo studies on motion” [Lettera di Favaro a Caverni of 12 April 1887] [Maffioli 1985, 67].19 Just two weeks later, he read an article in La Nazione (April 11–12, 1887) which stated he had been officially excluded and wrote to Favaro that he had learned “some things which caused him to greatly change his mind and [require] important clarifications” [Lettera di Caverni a Favaro of 30 April 1887] in [Maffioli 1985, 67]. For almost two months there was no exchange of letters—a sign that no clarification was forthcoming or that it was insufficient. In the following months, Favaro again asked the order of De motu but Caverni replied coldly, until in July of the same year he made some unacceptable requests.20 Therefore, after this change of course,

17. This is reported by [Maffioli 1985, 72]; where it is the precise bibliographical indication of [Favaro 1883a, 43–97, 135–157]. 18. With regard to the curators of the Edizione Nazionale, he had no hesitation in reproaching their choice not to closely analyze the thought of Galileo for the purpose of restoring a more correct ordering of the unpublished works and to limit themselves “a mettere i punti e le virgole al loro posto, a restituire le dieresi e altri segni esquisiti, come farebbe un accademico della Crusca, a cui fosse dato a curare qualche prezioso testo di lingua” [Caverni 1895, 341]. 19. Caverni, in fact, answers, that “non ha alcuna difficoltà ad ospitare Favaro nella sua canonica. Ha eseguito lo studio dei primi scritti di Galileo sul moto colla maggior diligenza possibile, per servirsene nella Storia della Meccanica”[Lettera di Caverni a Favaro of 17 April 1887, in [Maffioli 1985, 67]. 20. Caverni quite curtly declared that he had thought about the work Favaro intended to give him and, given the difficulty, was hesitating before starting work, preferring to set certain conditions beforehand: “1o Che gli sia assegnato il tempo 26 Francesco Crapanzano the priest inevitably became quite critical of the Edizione Nazionale and, conversely, Favaro did not hesitate to criticize him more or less explicitly which fuelled—or even actually created—the widespread view that Caverni was Anti- Galilean. This does not mean that Caverni would not refrain from expressing debatable opinions on Galileo in his Storia del metodo sperimentale in Italia. However, according to Favaro, this work remains “the greatest collection of material for history of the Galilean School” [Favaro 1899-1900, 379].

3 A diﬃcult task: the role of De motu in the history of mechanics

It was extremely important for Caverni to put the tangled web of studies, events and discoveries in the history of the experimental method into an order. The difficulties he encountered in putting the Galilean Ms. 71 and Ms. 46 into order are more understandable. Caverni sensed that those zibaldate carte (“mixed-up papers”) played a role in the birth and evolution of Galilean science. He was therefore committed to determining when they were written and to situating them in an ideal historian’s path of scientific ideas. As such a path they were positively considered as the roots upon which Galileo founded the new physical science while also showing how the great Pisan scientist skilfully used existing research lines and insights. The priest from Quarate mainly exposes his research on De motu in a chapter of the fourth volume of the Storia del metodo sperimentale in Italia which is primarily dedicated to the Scienza del moto dei gravi.21 At the beginning of it, as if to stress the importance of the topic, there are some general methodological considerations. Firstly, it explains how: [...] the attribute that is given to the discoveries of Physicists or the speculations of Philosophers and is deduced from the name of a man is in all fairness an impropriety which can only be kept in the speech convention [...]. This is because the intellectual order is a consortium which is as closely linked and necessary as in the civil orders and therefore a part of science may be named after an author in the same way as a religious society or family takes its name from the hierarch or father. So saying the laws of falling bodies belong to Galileo is not to be understood conveniente e compatibile con gli altri lavori che sta curando. 2o Che si accettino prefazioni e note da lui sottoscritte e nella misura che reputerà necessaria. 3o Che gli si assegni una congrua ricompensa. 4o Che detta ricompensa sia versata dagli editori galileiani all’atto della consegna del manoscritto” [Lettera di Caverni a Favaro of 17 July 1887] in [Maffioli 1985, 68]. 21. Though Caverni mentions the propositions of De motu in other parts of the volume, he methodically reports his considerations about this in chapter VI entitled Delle discese dei gravi lungo i piani inclinati [Caverni 1895, 328–381]. The Galilean De motu in Raffaello Caverni’s Reading 27

in the way that many do, as if they were spontaneous ideas from Galileo’s mind alone, [but] instead that there was a well- founded basis upon which he contributed by building the ediﬁce. [Caverni 1895, 328–329]22

The goal is just to prove that such a building “on solid foundations” [Caverni 1895, 328–329] and, in particular, the theory of motion expounded in Discourses and Mathematical Demonstrations Relating to Two New Sciences, should not be considered “a fascinating appearance” but rather that it was necessary to recognize “the bud and the little plant which engendered it” [Caverni 1895, 328–329]. It is an indisputable fact that Caverni derives several Galilean demonstrations from other mathematicians and natural philosophers such as Leonardo, Tartaglia, Descartes, etc., through not always in clear and direct ways. He recognizes, however, how Galileo’s eﬀort consists in making its dynamics independent from the principles of an essentially previous statics. Between theorems and corollaries, the pastor from Quarate encounters the demonstration on equality of spaces and times of a body in motion on an inclined plane. Torricelli had noticed how proposition VI could have been obtained in a more simple and elegant way than Galileo’s [Torricelli 1644, 107], [Galilei 1890-1909, 215–226], but Caverni tells us that:

[...] carrying out the second volume of Part ﬁve of the Galilean manuscripts, [he noticed a proposition] that showed itself the properly divided order by Torricelli: it was proved therein that the times are proportional to the lengths of the surfaces and are equally high after the mechanical theorem concluded the isochronism of circles. [Caverni 1895, 328–329]

The pastor tried to ﬁnd out if the script was signed or added later by a follower. Given the skilful writing of a man endowed with good eyesight and a hand which was still steady, he undoubtedly believed the ﬁrst hypothesis. Studying these papers further, with increasing surprise he observed another statement about motion along inclined planes23 and it seemed to him “to distinguish the practice of young wings, before spreading freely their own wings and the fragment published in volume XI page 56–62 by Albèri” [Caverni

22. A similarity can be noted between Caverni’s work and Koyré’s idea about Galileo: “Modern science did not spring perfect and complete, as Athena from the head of Zeus, from the minds of Galileo and Descartes. On the contrary, the Galilean and Cartesian revolution [...] had been prepared by a strenuous eﬀort of thought [...]. Yet, in order to understand the origin, the bearing and the meaning of Galilean- Cartesian revolution, we cannot dispense with at least a glance backwards at some of the contemporaries and predecessors of Galileo” [Koyré 1943, 333]. It is possible that Koyré was impressed by this concept thanks to Caverni’s work of which he was one of the few readers. 23. More precisely the one showing how “le tardità di due gravi scendenti per due varie obliquità di piani ugualmente elevate erano proporzionali alle lunghezze discese” [Caverni 1895, 338]. 28 Francesco Crapanzano

1895, 338–339]; therefore, ﬁnding in this the conﬁrmation that the material he had encountered belonged to the early period of Galilean production.24 Caverni then just needed to understand why Galileo, by publishing the treaty Dei movimenti locali, had repudiated the other previously processed demonstrations to adopt a new one based on an assumption; so, working hard on comparing “those mixed-up papers”:

the principles of statics were as mutually agreed with the principles of dynamics to give the right measure [when] suddenly we abandon the statics and see that the author avoids it in the manner of a person who is hiding a weapon under his clothing. [Caverni 1895, 340]

The reason for Galileo’s choice seemed inexplicable at ﬁrst glance but thus it becomes clear. Sensing that the statics would not allow him to make the desired steps forward in physics, he turned his attention elsewhere and demonstrated the propositions by making exclusive use of dynamics—solution that seemed less elegant and consistent but which was to turn out to be incredibly proliﬁc in future results. Having thus restored a sense of continuity in the production of the great Pisan physicist, Caverni seized the opportunity to criticize the Edizione Nazionale and its editors (without naming them) in which he said there was not enough space for all the papers in a chronological order. He also criticized the editors for being more interested in “putting full stops and commas in place” [Caverni 1895, 341] than in considering Galileo’s inner intense activity. He then makes explicit one of his key investigative tools—the comparison of calligraphic forms with the various colours of ink because:

[...] as is common knowledge, a writing hand feels the eﬀect of the years in the same way as the movements of all the other limbs, and similarly each one can gain experience in himself by comparing the papers written at the age of thirty with those written at ﬁfty. [Caverni 1895, 341]

It follows a long analysis of the ten De motu propositions in the order presented by the manuscripts of the “First Book”25 which ends with the conﬁrmation of what had been repeatedly announced, i.e., the ways:

[...][for which] Galileo arrived in the history of science which had yet to be recorded without violating the mechanical terms, in order to demonstrate his undisputable conclusions. They were those

24. The text was actually included in the edition made by Albèri and entitled De proportionibus motu ejusdem mobile super diversa plana inclinata [Galilei 1854, 56–62]. 25. Analysis that we cannot discuss here in a detailed way but which is very precise and rich in references including those necessary to Albèri’s edition of the Galilean works [Caverni 1895, 342–349 passim]. The Galilean De motu in Raﬀaello Caverni’s Reading 29

mechanical terms reduced to the statics and the author does not use the ten propositions making up his first treaty [...] and could not use another topic. However as the new dynamics were established in 1604, other broad paths were opened for science making it possible to reach the same desired goal through more direct and plainer paths namely to demonstrate the brachistochronism of descending bodies for many inflected strings and underlying a fourth circle. [Caverni 1895, 349] The pastor of Quarate analyzed the second book with his usual attention and did the same with the third Galilean book De motu. We find the propositions gradually amended by the father of modern science and the same basic idea, namely that the manuscripts in question represent the central seed from which Galileo manages to make a new physics flourish, a physics whose most complete and mature expression is represented by the Discorsi of 1638 [Caverni 1895, 350–366]. According to Caverni, in 1630, the great Pisan scientist opted for the third version of the manuscript which he considered to be the best. However almost all the propositions were recast and the result was not the best it might have been with some resulting theorems declassed to the status of corollary while some corollary points were elevated to theorems. Also the order of the demonstrations had not always been respected while appearing rather obscure and long winded to some. In particular, as has been reported, Descartes was to notice that in order to execute these demonstrations, it was not necessary to be a great surveyor. However while this is true, Caverni points out that we must also note that Descartes read the Discorsi in 1638 without considering that the proved propositions dated from more than forty years earlier when the geometry used was still the work of the likes of Commandino, Valerio and Guidobaldo. It is these works that the Pisan scientist’s work should have been compared with and certainly not with the newly-invented analytic geometry.26 Furthermore, [...] it is important to note [...] that, in the Galilean theorems, it is not simple pure geometry but geometry applied to motion science which involved a greater difficulty because the predecessors had only provided a few examples for certain points. For this reason, to make more certain of the truth of these new findings, Galileo often reduced the abstract generalities to concrete cases and called for Arithmetic to be matched with Geometry. [Caverni 1895, 372] So while Galileo might not have been a great mathematician in elaborating the theorems on motion, Descartes himself could not deny that he was “the first to apply geometry and arithmetic [...] to prove the new properties and the several and complicated effects of motion” [Caverni 1895, 372]. Still, if Descartes and others had seen the demonstrations as they were in their first

26. On the contrary, with regard to these terms of comparison, Galileo—says Caverni—surpasses them “per una certa elegante facilità” [Caverni 1895, 371]. 30 Francesco Crapanzano version, they would certainly appreciate them. For that reason Caverni can say proudly that: We believe we have done something for the above-mentioned mathematicians and think we have also partly contributed to the fame of Galileo in publishing the three De motu books in their original forms. [Caverni 1895, 373] At the end of the chapter, it is clear that Galileo’s readers reproached the erudite priest from Quarate for the excessive prolixity and verbosity of his writing. Carlo Del Lungo [Del Lungo 1920, 274–277] expanded particularly this point. This problem is actually verifiable throughout the Storia del metodo sperimentale in Italia but it would not be the most relevant point. The main error (and the most shameful for a historian) is that the writer expressed his ideas in different tones depending on the source and displayed the “insatiable urge of persecuting Galileo in each of his activities” [Del Lungo 1920, 281], or allowing himself to be misled by his personal events during the historical reconstruction. In this paper we cannot examine whether and how the judgment is generally deserved but chapter VI on the science of motion definitely shows something of interest. The great Pisan scientist developed his science based on the “mechanical theorem by Tartaglia” [Caverni 1895, 373] but while this origin was the most famous, it was not the most important. Claudio Beriguardi and Giovan Battista Baliani were: [...] considered by many people to be not only imitators or emula- tors, but actual plagiarists of Galileo. They now appear in history in their true appearance representing the various provisions of the minds open to the seed of science that the superior provident hand spread widely [...] rather than in some closed and privileged small garden. [Caverni 1895, 374] Baliani should indeed be considered closer to Torricelli for the contribution that he gives the science of motion which consisted, among other things, in retracing more strictly and extending (despite not having read or seen it) what Galileo had produced in De motu.27 On the other hand it should be noted that Caverni consistently maintains a certain “nationalistic” line which leads him to question the work of Huygens and the transalpine physicist Edme Mariotte. He considered that they had the merit of making some Galilean principles and theorems prolifically known but also “to a small degree, take away Galileo’s merit of being [...] the first to put those theorems into work” [Caverni 1895, 381].

27. Baliani, in particular, shows in an elegant and clear manner the mechanics theorem of which Galileo “in principio si fa Autore [...] e poi lo repudia come sospetto” [Caverni 1895, 376]. The Genoan mathematician “non dà alla sua scienza tutta l’estensione della scienza galileiana, a confronto della quale, se rimane inferiore rispetto alla materia, vince però l’esaltato emulo suo rispetto alla elegante semplicità della forma. Dicemmo, e lo ripetiamo, che in ciò il Baliani, meglio che a Galileo, si rassomiglia al Torricelli” [Caverni 1895, 377]. The Galilean De motu in Raﬀaello Caverni’s Reading 31 4 Conclusion

Overall, regarding the pages thus far considered, I think we cannot charge Caverni with acrimony against Galileo. However, in the last chapter entitled Delle libere cadute dei gravi, there is a certain “oddity”. Among other things, in fact, we read that anyone:

[...] who even suspected that those first Galileo writings, De motu, are really exercises about Benedetti’s books, as we qualified them, could also easily be persuaded when re-reading the chapter “In quo causa accelerationis motus naturalis in fine, in medio affertur” which is a long and bright comment on the words [written on the subject in] the book Delle disputazioni. [Caverni 1895, 293]28

From that point, when returning to the text, there is a surprising reconstruction of Galileo. After reading the books recommended by the teacher and his erstwhile colleague Jacopo Mazzoni during the years spent in Pisa, Galileo learnt to think freely with a mind of his own rather than following the Aristotelian authority:

At the time, among the young auditors in Pisa, there was also Galileo, in whom Mazzoni recognized an unusual intellectual ability to penetrate the science of motion. He recommended Benedetti’s book to Galileo, explaining the speculation thereon to him in private. The young pupil, in those lively words and in the reading suggested by the teacher, felt the first ineffable taste of freedom of thought. Also, because the intense advice and effective examples convinced him to no longer believe in the authority of Aristotle, he therefore concluded he should not follow the authority of any other philosopher either, even Benedetti himself. [Caverni 1895, 275]

So, what might seem to be a complaint about the Galilean plagiarism of Benedetti through a hurried reading proves to be a real source of inspiration which Galileo opposes in those matters that do not convince him. Perhaps Aldo Mieli is right in saying that the Storia del metodo sperimentale needed “to be redone” not so much because of its content or hasty judgments but rather because of its complex and obscure style. A “reading which is not critical and careful enough may often lead the reader into error” [Mieli 1920, 262]. It is worth emphasising the judgements about De motu that Raﬀaello Caverni shows in his pages, paying the necessary attention to these youthful writings of Galileo.

28. The pages of Galileo that Caverni referred to are part of De motu published in [Galilei 1890-1909, 315–323]; the writing of Giovanni Battista Benedetti, Disputationes de quibusdam placitis Aristotelis, is included in [Benedetti 1585, 168– 203]. 32 Francesco Crapanzano Bibliography

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What Was the Role of Galileo in the Century-Long Birth of Modern Science?

Antonino Drago Federico II University, Napoli (Italy)

Résumé : Quel est le rôle de Galilée dans la naissance séculaire de la science moderne ? Je réponds à la question ci-dessus à la lumière de deux nouveaux éléments. Dès le xvie siècle, Nicolas de Cues, quoiqu’il ne pratiquât pas la science expérimentale, a anticipé une partie substantielle de la révolution copernicienne et de la naissance de la méthodologie galiléenne. Le deuxième élément est l’introduction d’une nouvelle conception des fondements de la science ; ils sont définis comme constitués de trois dialectiques. Après cette définition, la naissance de la science moderne correspond à un très long proces- sus historique qui se conclut à notre époque. Sur cette longue période, Galilée ne fut pas seulement le premier à recourir à une méthodologie scientifique, mais aussi presque le seul à jamais avoir été conscient de l’ampleur des enjeux intellectuels de l’entreprise scientifique.

Abstract: I answer the above question in the light of two new elements. Firstly, in the 15th century while Cusanus did not practice experimental science, he substantially anticipated both the Copernican revolution and the birth of Galilei’s methodology. Secondly, I shall introduce a new conception of the foundations of modern science which are constituted by three dialectics. In this light, the complete birth of modern science, whose scope was so broad, required a very long historical process, which was completed in recent years. Within this long time span, Galileo was not only the first to practice a scientific methodology, but also almost the only scientist ever to be aware of the intellectual breadth of scientific enterprise.

Philosophia Scientiæ, 21(1), 2017, 35–54. 36 Antonino Drago 1 Introduction: What are the foundations of science? The three dialectics

The role played by Galileo in the birth of modern science depends on two questions: What is the definition of modern science, in particular its foundations? How long was the process of birth of modern science? Without doubt, the foundations of science are constituted by the well-known dialectic1 between experimental data and mathematical hypothesis. But after that, what in precise terms? I suggest the following definition of the foundations of science that I obtained on the basis of four decades of historical works on the main theories of logic, mathematics and physics; they are constituted by two more dialectics of a theoretical nature; one between the two kinds of infinity (either potential, or actual) and a dialectic between the two types of organization of a theory (either deductive form principles-axioms, as Aristotle theorized in ancient times, AO, or aimed at the solution of a crucial problem, PO). The infinity dialectic was formalized, by, on one hand, classical mathematics relying on the actual infinity (AI) (e.g., Zermelo’s axiom) and on the other hand by constructive mathematics relying on (almost) only potential infinity (PI) [Markov 1962], [Bishop 1967]. The organization dialectic was formalized by means of the kind of logic managing it; on the one hand classical logic, managing AO (e.g., in Euclid’s Elements) and, on the other hand, non-classical logic, in particular intuitionist logic [Dummett 1977] managing PO [Drago 1990, 2012b]. When a scientist builds a theory, each theoretical dialectic appears under its formal aspect; a formal alternative is opposite to and incompatible with the other alternative; the scientist has to choose one. In sum, both dialectics constitute two scientific dichotomies. Once the choices are made, each dialectic is dissolved and the scientist proceeds within the scientific realm to establish the notions and principles of his theory. However, when we consider all the theories of a scientific discipline—e.g., physics—we again recognize these theoretical dialectics in the different couples of choices on which the various theories rely. In the light of the above illustrated foundations of science, the question of the birth of modern science receives a first, partial answer: modern science was born when not only its method for producing accurate results was recognized, but also its foundations, i.e., the two theoretical dialectics. Since the alternative choices of these dialectics were not formalized and recognized as relevant for science before the second half of the last century, we obtain a surprising result: the birth of modern science is a historical process spanning

1. I use the word “dialectic” in the intuitive sense, but in a Platonic sense as both authors [Hopkins 1988], [Counet 2005] do. It is not the Hegelian dialectic (that transcends reality through a dynamic of the Absolute Spirit); roughly speaking, it is given by two polarities that are not necessarily on the same plane. It may be resolved by either a reconciliation (as in the first dialectic) or by a choice between the two alternatives of the dialectic (as in the cases we will see in the following). What Was the Role of Galileo in the Birth of Modern Science? 37 several centuries, at least half a millennium. In the following sections, we will detail this long process in relation to Galileo. Notice that the three dialectics of the foundations of science are mutually independent. Thus, according to the above definition, the question about the birth of modern science has to be disentangled into the three questions about when each of these dialectics emerged and was then established, at first in intuitive and then in formal terms. In what follows, I will deal with them by considering each dialectic at a time.

2 The dialectic experiment/ mathematical hypothesis. Its historical birth and its problematic deﬁnition

Certainly, the historical start of modern science was marked above all by the birth of the experimental method through its innumerable applications giving new scientific results. Even a first, rough definition of this method was enough to mark an unprecedented novelty with respect to the philosophical world of the ancient Greeks. Rightly, historians of science attributed a crucial importance to this method. However, regarding this dialectic it is no longer possible to assume a positivist viewpoint, which attributes an eminent role to Galileo because he, more than any other, set hard experimental facts against the idealistic philosophical tenets of Aristotelian philosophers. If the experimental method was what positivist historians claim, that is, dealing above all with hard, experimental facts, it would be easy to determine a date for the births of both scientific activity and the method governing it. Instead, Grosseteste, Francis Bacon, Cusanus, Lavoisier, etc., suggested different aspects of this dialectic. Moreover, Galileo is a complex figure; in contradiction to the above appraisal, he also declared that he could obtain scientific results without experiments [Galilei 1638, II Day]. In addition, the question of whether Galileo’s activity conformed to a well-defined experimental method is largely undecided. For instance, about Galileo’s capital achievement—the discovery of the law of the accelerated motion—Galluzzi wrote: [...] it is not still made irrefutably clear which element of the Galilean investigation was decisive in such an enterprise: either the natural deduction or the observation, either the geometrico- mathematical analogy or the experiment. [Galluzzi 1979, 158] Notwithstanding having pondered for three centuries on this dialectic, philosophers of science did not suggest any common agreement on its main features. We know that not only positivist, but also neo-positivist philosophy, although supported by excellent scholars, failed in this task. No surprise if this unsat- isfactory situation is presented by Feyerabend as no method at all in science 38 Antonino Drago

[Feyerabend 1975]. Recently, two scholars wrote an even more discouraging appraisal: [...] More recent debate has questioned whether there is anything like a ﬁxed toolkit of methods which is common across science and only science. [Andersen & Hepburn 2015] The plain conclusion to be derived from these unsuccesses is that this dialectic is philosophy-laden. Indeed, both Koyré and Lindberg stressed that the establishment of the experimental method was the result and at the same time the cause of a profound change not only in the methodology aimed at obtaining answers from nature, but also in the metaphysical conception of reality [Koyré 1957], [Lindberg 1992].

3 The dialectic experiment/ theoretical hypothesis. Its historical birth and Cusanus’ contributions

In such a context of uncertainty about this dialectic, new results concerning how Cusanus suggested combining experimental data with mathematical hypotheses are important. This relationship of combination was essentially new with respect to Greek philosophy, which considered a conjunction between mathematics and reality to be impossible. It was new also according to the idea that was staunchly emphasized by Koyré, i.e., modern science overcame the finiteness of the Greek world (Aristotle) by introducing a mathematics explicitly involving infinity, i.e., either potential infinity (i.e., unlimited), or actual infinity. According to some scholars this dialectic started, although in a reduced form of a mere programme, in the 13th century with Robert Grosseteste’s (1175-1253) book De Luce [Crombie 1994, II, 319], [Lewis 2007]. However, subsequently, in the 15th century Cusanus (1401-1484) suggested important aspects of this method, although he intermixed them with theological, philosophical and cosmological subjects, all treated in a somewhat obscure way [Cusanus 1972]. First, he provided an accurate philosophical basis. According to him the two words mens and mensura share the same root; the mens connects itself to reality through a mensura; the mens mutually compares through proportions the numbers obtained (remember that a proportion was the only mathematical technique used in physics before the time of Galileo and Descartes) and moreover it mutually combines the corresponding concepts in order to obtain theoretical constructs.2 2. The “Introduction” by Graziella Federici Vescovini to [Cusanus 1450] is very informative about current scientific knowledge in Cusanus’ time. Apart from her—in my opinion too severe—criticisms, she cleverly summarizes Cusanus’ conception of human knowledge: “The mind is thus specular simplicity that all obtains from its capability of measuring, numbering and representing”, [Federici Vescovini 2003, xxx]. What Was the Role of Galileo in the Birth of Modern Science? 39

In addition, in the book De staticis experimentis3 Cusanus indicated the main features of an experimental science through the measurements of a particular physical magnitude, weight: And thus, by means of experiments done with weight-scales he [the physician] would draw nearer, through a more precise surmise, unto all that is known. [Cusanus 1450, 608, 164] [...] Experimental knowledge requires extensive written records. For the more written records there are, the more infallibly we can arrive, based on experiments, at the art elicited from the experiments. [Cusanus 1450, 615, 178] Notice that in ancient times weight, being considered a quality of a body, was extraneous to geometry. Its quantification as a quantity decisively introduced scholars to the quantifying of a multitude of other physical properties [Crombie 1994, 423].4 Furthermore, in the history of physics the transition from the theoretical concept of absolute weight to the concept of relative weight played an exceptional role. According to Koyré, Galileo considered only absolute weight, because he thought there was one single centre of the universe, i.e., the Sun, from which he could not free himself [Koyré 1978, 25 ff.]. Before him, Cusanus had already abolished any centre of the universe whatsoever as well as any fixed locations for celestial bodies.5 In fact, in the above mentioned book Cusanus used not so much weight but weight difference. Remarkably, by means of a Roman scale together with a hourglass Cusanus wanted to determine other physical quantities: e.g., the volume of a body, its specific weight, the weather, the temperature, the sound, the magnetic force, the weight of the air by means of the so-called leaning tower of Pisa experiment, etc.; so that he verbally introduced the indirect measures and, ultimately, the doubly artificial apparatus of measurement. One historian evaluates Cusanus’ work in the following terms:

3. In the following, I will refer to Jasper Hopkins’ English translations of almost all Cusanus’ online books http://www.jasper-hopkins.info/. 4. About this point, Koyré, strangely enough ignored Cusanus’ suggestions which constituted those advances that Koyré himself considered a crucial step in the history of the scientiﬁc method: “It is ironic that two thousands years beforehand, Pythagoras had proclaimed that number is the same essence of things, and the Bible had taught that God had founded the World on “number, weight, measure” [Wisdom, 11, 21]. All people reiterated that, nobody believed that. Surely, nobody before Galileo took that seriously. No one attempted to determine these numbers, these weights and these measures. No one attempted to count, to weigh, to measure. Or, more exactly, no one had the idea of counting, of weighing and of measuring. Or, more exactly, no one ever sought to get beyond the practical uses of number, weight, measure in the imprecision of everyday life. [...]” [Koyré 1961, 360]. It is apparent that Cusanus did exactly what Koyré stressed as lacking and he did it inspired by the same Biblical passage. 5. See Santinello for innovations and advances of Cusanus in science [Santinello 1987, 105–109]. In particular, notice that Copernicus knew Cusanus’ modern view of Heaven without any center [Klibansky 1953]. 40 Antonino Drago

In the mid-fifteenth century appeared what should have been the crowning work of this genre, the Idiota de staticis experimentis, from the pen of Cardinal Nicholas of Cusa. This incidental piece from one of the best-known philosophers and churchmen of the time does not appear to have attracted much attention. It advocates the use of balance and the comparison of weights for the solution of a wide range of phenomena, ranging from mechanics to medicine, and including those of chemical. He advocates differences in weights as a guide to the evaluation of natural waters, the condition of blood and urine in sickness and health, the evaluation of the efficacy of drugs and the identification of metals and alloys. It would be difficult to find a more specific prescrip- tion for what scientists were actually doing two centuries later. [Multhauf 1978, 386]

A century and a half after Cusanus, the first of Galileo’s three periods of activity, was devoted especially to the study of physical phenomena relating to weight (e.g., a rolling ball on an incline, a pendulum) [Wisan 1974, 136–161]. This is the physics of the Earth, which, unlike the celestial one, which is based only on observation, is manipulative, as is modern science. Later, Newton’s use of the new, marvellous mathematics of infinitesimal analysis assigned instead the highest role to celestial physics, despite the fact that the foundations of this mathematics were unknown and manifestly linked to the metaphysics of actual infinity [Newton 1687]. Yet, three centuries after Cusanus and one century after Newton, chemistry was born. This event was a “postponed revolution” (a chapter title of [Butterfield 1949]), since, being influenced by his idealistic mathematics, Newton has led chemists in a misleading direction, relying on several metaphysical notions: absolute space, absolute time, fixed and perfectly hard atoms, gravitational force as constituting the intermolecular links. Lavoisier freed chemistry from all of a priori notions by appropriately re-defining the experimental method for his field of research, i.e., all that concerned the complete methodology of chemical reactions. Lavoisier founded chemistry by means of only those physical measurements which had been suggested by Cusanus, i.e., weight differences that he considered between the reagent substances and the compounded substances [Drago 2009b]6. He linked these weight differences to weight-mass conservation. This was a trivial law according to Cusanus, since God pervades matter, which therefore cannot change in quantity [Crombie 1959, 296], [Drago 2009b].— One may object that Cusanus’ book constitutes a mere program of research, since he never practised experimental science to the point of obtaining scientific results (he however invented the hygrometer). Yet, recall that at least among the musicians a very ancient and widespread practice obtained new results from artificial

6. Kuhn devotes a page to stressing the importance of physical magnitude weight for the birth of chemistry [Kuhn 1957, 262]. Yet, strangely enough he attributes this inﬂuence to Newton. What Was the Role of Galileo in the Birth of Modern Science? 41 instruments. It is not by chance that Galileo’s father, his brother and he also were musicians. Moreover, through the method which Cusanus had described, several physical laws were established, ﬁrst those of acoustical instruments. In particular, Galileo’s father, Vincenzo, inductively inferred from experiments on the string tensions obtained by charging weights more accurate and general laws than Pythagoras’ law [Cohen 1984, 84].7 Hence, Cusanus did not need to put his program into practice, since artisans were already implicitly applying it to several kinds of phenomena.

4 The dialectic experiment/ mathematical hypothesis. Its historical birth and Galileo’s determinant achievements

A century and a half after Cusanus, Galileo extensively applied the experiment/hypothesis dialectic to several fields of physical phenomena. However, Galileo had a particular conception of this dialectic, as his celebrated words emphasize: Philosophy [read: science] is written in that great book which ever lies before our eyes—I mean the universe—but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth. [Galilei 1890, vi, 232] This mathematical language is the same as God’s; and whereas we have to go step by step, He sees all in the blink of an eye. Hence, geometry was already written in the world since eternity. In this conception of Galileo we recognize—as Koyré correctly emphasized, scandalizing the positivist historians of science—an ontology: not only is mathematics outside our minds (which however can understand it), it is essentially inherent in both reality and God; that is, Galileo considers geometry such a perfect theory as to be a supernatural theory that structures reality.8 This Platonist conception of mathematics

7. He improved Pythagoras’s law, successfully suggesting the quadratic and even more complex relationships between two variables; this was a mathematical breakthrough, which later his son Galileo performed for the law regarding the case of the falling bodies; an initial law stating a linear relationship between space and time, was discarded by Galileo in favour of a law stating a square relationship. 8. For the same reason, Koyré assumed a prejudice of a mathematical-physicist, i.e., an absolute accuracy in the results of measurements [Koyré 1961]. Therefore, he opposed the historians (e.g., Crombie) whose conceptions of the history of science do not attribute a basic role to actual infinity. 42 Antonino Drago influenced the entire history of theoretical physics. The subsequent physical theory, geometrical optics, made use of points at infinity as if they were ordinary points. Moreover, Newton’s mathematical technique was infinitesimal analysis, which is informed by a Platonist philosophy of infinitesimals. In the 19th century, this conception reached a climax through the dominant role played by mathematical physics, a role which in 20th century was countered by the discovery of quanta and then symmetries. Mathematics indeed may be joined with experiment also according to an instrumental conception of it, as is the case in both chemistry and thermodynamics [Drago 2009b, 2012a]. In fact, a century and a half before Galileo, Cusanus had suggested an alternative philosophical conception of mathematics, which was summarized by McTighe [McTighe 1970]. According to him, ideal mathematical entities are built from reality (and not, as Galileo claimed, recognized as the ideal elements of reality), i.e., they are constructs of our minds. Whereas ordinary knowledge of reality is always approximate and hence it is not subject to the principle of non-contradiction (through a sharp and exhaustive division into true and false), instead, mathematics is perfect knowledge because it enjoys a specific feature; it strictly obeys this principle. Owing to this logical property, mathematics is considered by Cusanus the most certain knowledge we have; for this reason, we apply it to understanding reality. Moreover Cusanus conceived numbers as the explicit forms of unity, i.e., a number enjoys the property of being both as great and as small as it can be; thus it is through numbers that all things are best understood, i.e., referred to unity. For these reasons, the creation of our minds, mathematics, can direct our interpretation of reality. Indeed, by combining experimental data with mathematics the intellectus conjectures the mathematical formula of a physical law. This Cusanus’ way of describing the relationship between mathematics and experiment more adequately than Galileo’s previous quotation describes the activity of an experimental physicist (e.g., in such a way—by means of a conjecture—the law of falling bodies actually originated in Galileo [Wisan 1974, 207–222]). Furthermore, Galileo knew only the mathematics of geometry and proportions and did not share in the effort—elicited, e.g., by his disciples Cavalieri and Torricelli—to extend the realm of mathematics [Drago 2003]. Also for this reason, the theory of mechanics was born later, through Newton, since it required the invention of a very different mathematics, calculus (just as at present time the advances in theoretical physics require the new mathematical technique of symmetries). Before Galileo, Cusanus had, on the other hand, been able to expand the mathematics of his time (geometry and proportions) to new achievements (we will see them in the next section). I conclude that Cusanus offered to theoretical physics a more adequate metaphysics of the scientific methodology than Galileo’s, owing to latter’s Platonist view of mathematics. Yet, most historians disregarded the entire metaphysical and mathematical change brought about by the introduction of the scientific method. For an instance, Cohen gave a totally objectivist What Was the Role of Galileo in the Birth of Modern Science? 43 definition [Cohen 1984, 85–86]. The experimental method developed by Galileo enjoys the following two features:

• it is mathematical in that the relationships between the parameters are quantitative, in that the proofs are geometrical, and above all the properties of falling and projecting bodies are logically derived from a set of a priori postulates;

• it is experimental in that not [only] daily experience but nature subjected to artiﬁcial manipulation provides both the starting point and the ﬁnal empirical check of the axiomatic system.

Cohen’s words “a priori postulates” and “axiomatic” attribute to Galileo’s thinking the choice AO; in the following section 6, I will disprove it. I remark that all Cohen’s features of the scientific method are present in Cusanus, including “the final empirical check”, in the cases of a physician wanting to cure a patient and a musician wanting to perfect his musical instruments [Cusanus 1450, 622–623]. Yet, his illustration does not follow a precise order, somewhat inattentive and also fanciful. It is apparent that this deficiency of Cusanus is due to his lack of experimental practice, which he left to others (e.g., artisans, musicians).9 The reason is that Cusanus is interested in exploring only the mind’s faculties, not natural phenomena. Hence, he sees the elements of knowledge but he does not apply them to producing knowledge from finite reality; he is too interested in infinite reality. At most, he is interested in physical principles; e.g., the principle of relativity (through the celebrated observation that over a ship sailing in a calm sea the phenomena are the same as those on the land), the impossibility of a perpetual motion, the inertia principle [Cusanus 1462-1463]. Galileo played a key historical role in the introduction of the dialectic experiment/mathematical hypothesis, since i) he qualified this dialectic, ii) he introduced the mathematical description of the evolution of phenomena (kinematics in space and time); iii) through it, he established breakthrough theoretical laws, in particular the decisive experimental law of falling bodies, which described reality in contrast with everyday experience. This result established a new kind of truth, since it was objective in two respects, i.e., the experimental and the mathematical joined in an objective unity. Moreover, iv) he decisively propagandized his historically important advances notwithstanding the harsh opposition of Aristotelian philosophers of his time; v) he practised and made this method productive to the extent of achieving an essential part of a mechanical theory; vi) his great and long activity produced so many scientific results that it eventually established a stable tradition of experimental research that, subsequently followed by several other scholars,

9. He contented himself with illustrating his experimental method by describing the simplest artiﬁcial instrument, a wood spoon [Cusanus 1450, IV, chap. II, IV]. 44 Antonino Drago systematically accumulated new commonly recognized results.10 From all the above I conclude that Cusanus anticipated all the elements of Galileo’s experimental method; however, this consideration in no way detracts from Galileo’s glory in having organized all these elements into a consistent set of rules and systematically applied them to a multitude of natural phenomena, so that he established a new tradition in science. Rather, the above anticipations by Cusanus present the historical birth of this dialectic as a less clear-cut change than that commonly presented by historians in the following words: “Galileo had no forebears and stands apart from history” [Wallace 1998, 27].

5 The second dialectic: potential inﬁnity/ actual inﬁnity. Its historical birth

Ancient Greek mathematicians deliberately avoided the use of the infinity. Late, first Cusanus introduced a conception of the infinity into mathematics11 and developed it (e.g., he presented the intuitive notion of a mathematical limit as a series of multilateral polygons approaching a circle). Owing to this innovation, Cassirer considered Cusanus to be the first modern philosopher of knowledge [Cassirer 1950].12 In addition, I have discovered [Drago 2009a, 2012a] that he defined (without mathematical formula, yet through exact words, the infinitesimal: “of which there cannot be a lesser [positive] number” [Cusanus 1440, I, 4, 11], i.e., the basic notion, the hyper-real number, of non- standard analysis [Robinson 1960, chap. X]. On the other hand, two centuries after Cusanus, Galileo’s conception of mathematics was mostly that of the ancients, also regarding infinity. In the book concluding his scientific career [Galilei 1638, First Day] he discusses the two ideas of infinity (AI and PI) in order to understand how to apply one of them to physics. Having dissected the problem, Galileo concludes that in mathematics one cannot define the usual arithmetical operations on infinite objects. Moreover, by recognizing that he is unable to decide which kind of infinity he has to choose and even when one

10. This appraisal on Galileo’s contributions to this dialectic is comparable with similar appraisals suggested by eminent scholars on Galileo’s method, i.e., Wisan, Machamer and McMullin [Wallace 1992, 7–12]. 11. It was recognised also by G. Cantor, the inventor of the Theory of (infinite) Sets [Cantor 1883, fn. 2]. 12. “This one’s stand facing the problem of knowledge of Cusanus does him the first modern thinker” [Cassirer 1978, I, 39]. He presents Cusanus as: “[...] the founder and champion of modern philosophy” [Cassirer 1978, I, 39]. Note also [Vanstenbeerghe 1920, 279]: “The key to the philosophical system of Nicholas of Cusa [...]—and this is very modern—is its theory of knowledge”. Unfortunately, Koyré overlooked Cusanus’ role [Federici Vescovini 1994], although Cusanus more than any other scholar broke— precisely the fortunate title of the major Koyré’s book—the closed Cosmos of the Ancient Greeks and opened minds to the infinite Universe. What Was the Role of Galileo in the Birth of Modern Science? 45 of them is useful in formulating his laws, he asks to be allowed in any case to make use of the notion of infinity in theoretical physics.

6 The third dialectic: axiomatic organization/ problem-based organization. Its historical birth

During the birth of modern science, Cusanus organized his theological theories in a different form from the Aristotelian organization, i.e., on problems (PO) [Drago 2009c, 2012b]. This point was very clear to him as a theologian: in theology to choose AO means to derive all the truths from a priori dogmas; however Cusanus wanted to solve problems, e.g., the best name of God, the double nature of Christ, the Trinity, the constitution of both the Universe and matter, the new logic, peace in the world, etc. Also his scientific programme of weight measuring is aimed at solving problems, ultimately the problem of how to acquire the knowledge of the natural world. In sum, also in scientific subjects Cusanus’ choice is for PO. A century and a half after Cusanus, Galileo’s theoretical formulations of his scientific results did not conform to the traditional way of organizing a theory, AO, either, which he knew well;13 in particular, he never appeals to some general principle from which to deduce physical laws [Clavelin 1996, 66]. Yet, he does not assume a definite position on this dialectic. In each of his last two books he illustrates their contents by alternating two kinds of organization—deductive and dialogical-inductive, the latter recalling the dialogues of Aristotle’s adversary, Plato. In sum, he did not choose a specific model (also because he did not complete any theory). As a matter of fact, Galileo’s experimental method of producing science is in contrast with the Aristotelian model of organizing a scientific theory (see its elementary presentation in [Beth 1959, § 1.2]), but subsequent scientists, owing to his inconclusive position on this subject, eventually lessened the import of his innovations; they changed only one postulate of the Aristotelian model, the evidence postulate, which was attributed no longer to axioms, but to the data of the theory.14 In sum, the result was a mere reform of this organization, not an alternative model to AO.

13. It is a merit of W.A. Wallace to have emphasized Galileo’s Juvenilia manuscripts concerning the illustration of Aristotle’s apodictic organization of a theory. Yet, Wallace wanted to exploit this fact in order to link Galileo to Aristotelian philosophy, although Galileo’s last two books manifestly show that he did not share Aristotle’s model of organization [Wallace 1992], [Coppola & Drago 1984]. 14. As a consequence the dialogical parts of Galileo’s last two books have been misinterpreted as no more than odd presentations of the subject to the reader. 46 Antonino Drago 7 The third dialectic: classical logic/ non-classical logic. Its historical birth

The dialectic regarding the kind of organization is formalized by means of two different kinds of logic, respectively classical logic—governing the deductions of a AO theory—and non-classical logic—governing the inductions of a PO theory. Some scholars have remarked that the logical ways of arguing of both Cusanus and Galileo are different from those of classical logic. These differences were attributed to their attendance at the Padua school of logic, where Zabarella made the greatest effort to theorize a different way of reasoning from that of Aristotle. Actually, after the Padua period, Cusanus wanted to found a new kind of logic and he was successful in this aim. He believed that Aristotle’s logic represented a specific activity of the ratio, a particular way of arguing of the mens; yet, according to him there exists another activity of the mind, i.e., the intellectus, which generates coniecturae according to a new kind of logic. He wanted to characterize the specific laws of this new logic. He suggested a celebrated “coincidence of opposites”, which he later abandoned as ineffective. Rather he implicitly changed logic. An accurate inspection of his texts shows that he made use of the characteristic propositions of non-classical logic, those for which the double negation law fails; i.e., the doubly negated propositions whose corresponding affirmative propositions lack evidence (DNPs). Through them he solved his main problem—the best to name God—, by suggesting names pertaining to non-classical logic (Not-Other, Posse=est). Moreover, he was capable of developing the logical arguments, which are specific to the ideal model of a PO theory, i.e., ad absurdum arguments [Drago 2012a]. Some authors have remarked, on the other hand, that Galileo paid little attention to logic. In particular, he never attacked Aristotle’s legacy in logic. Moreover, his last two books present a strange logical approach; when expounding theorems, he makes use of propositions in Latin in accordance with classical logic;15 when illustrating his investigations he makes use of a Platonist-like dialogue among three people all speaking the vulgar Italian language; an inspection of the logic of these dialogues recognizes several DNPs and hence an implicit use of non-classical logic [De Luise & Drago 1995, 2009]. As an instance, let us consider the first 50 pages of his De motu locali. An examination of them shows that: 1. The parts of his text written in Latin and concerning formal theorems do not include DNPs; 2. The other parts—written in vulgar Italian—include more than 100 DNPs;

15. On this basis, Wallace states that “if one takes reliance on Aristotle’s logical canons to be the sign of a Peripatetic, one [Galileo] can rightfully be called a Peripatetic himself”, [Wallace 1998, 51]. Yet he ignores the inductive parts of Galileo’s theoretical work. What Was the Role of Galileo in the Birth of Modern Science? 47

3. However, some of them are uncertain because some are dubious;

4. Moreover, there is no DNP in the part illustrating rectilinear uniform motion;

5. Instead there are around 90 DNPs in the 25 pages dedicated to solving the problem of naturally accelerated motion;

6. The latter DPNs compose 8 cycles of reasoning, which are illustrated by table 1;

7. Three of these cycles are ad absurdum proofs;

8. His reasoning eventually obtains a DNP which is a universal predicate [Galilei 1638, 190–191]; it represents an hypothesis solving the problem;

9. Which then the author translates—by merely dropping the two nega- tions of this DPN—into the corresponding aﬃrmative proposition in order to deductively derive from it, now considered as a postulate subject to classical logic, all possible consequences to be tested by experiment. Galileo is the only scientist apart from Einstein to have illustrated this logical step through admirably precise words [Einstein 1905a, 891]:

Let us then, for the present, take this as a postulate, the absolute truth of which will be established when we ﬁnd that the inferences from this hypothesis correspond to agree perfectly with experiment [Galilei 1638], [Galilei 1638, 183, period before Theorem 1, Proposition I of the Naturally accelerated motion; emphasis added]16

However, Galileo ignored the alternative logical nature of both the DNPs and the ad absurdum arguments, maybe because his way of reasoning by means of DNPs was sometimes invalid (actually, it was invalid, according to recent studies, also in the deductive reasoning) [De Luise & Drago 2009, fn. 15]. I conclude that, his genius was capable of achieving exceptional, but irregular results in logic.

16. This author’s translation is justified by appealing to the principle of sufficient reason, which Galileo applies, but not in the right place, in the conclusion of the next cycle of reasoning: “where the mobile moves indifferently to either the motion or the rest, and by itself has no inclination to the motion to no location, nor any resistance to be removed; because in the same way it is impossible that a heavy body or a compound of such bodies naturally increases its height, by going away from the common center towards which all heavy bodies converge, in the same way it is impossible that it spontaneously moves, if such a motion does not approach the above-mentioned common center” [Galilei 1638, 203]. 48 Antonino Drago

1o 2o 3o 4o 5◦ 6◦ 7◦? 8◦ (3 →⊥) (the case (v = v = 0) ks →⊥) Hp of a Hp of Objection: Refutation: Refutation Falsity Laws on law v = inﬁnite Is time As many of the of the falling kt degrees inﬁnite? degrees of objection: other bodies of velocity Even in law on velocity as the the case v = ks inclines degrees v = 0? of time Legenda: Hp = hypothesis; ⊥ = absurd; k = constant; s = distance; t = time; v = velocity

Table 1.The eight cycles of reasoning in De motu locali

8 Conclusion: What is Galileo’s role in this centuries long historical process?

First, one has to take into account that Galileo’s methodological change and his several scientific results were not enough to give rise to modern science because he did not achieve any complete physical theory. Rather both Cusanus and Galileo anticipated scientific theories. On this point a comparison is not easy because their advances anticipated two very different scientific theories, respectively chemistry [Drago 2009b], which was born three centuries after Cusanus; and mechanics, which was born a few decades after Galileo’s death. Strangely enough, neither anticipated geometrical optics; it was instead anticipated by Grosseteste four centuries before its birth, which occurred in the last period of Galileo’s life. Hence, an appraisal on Galileo’s role cannot refer to a complete scientific object, but only to the elements of physical theories. We will consider their most important elements, i.e., the foundational choices. Let us compare Cusanus and Galileo with regard to the two theoretical dialectics. Cusanus was the first to conceive mathematics non-Platonically and he improved the mathematics of the time by first introducing infinity; and regarding infinity, he formulated, albeit in verbal terms, the notion of limit in PI mathematics and a basic notion of AI mathematics. First Cusanus made them manifest through the invention of new mathematical notions and new (theological) theories which relied on the alternative choices—respectively, AI and PO—to the dominant ones, PI and AO. He introduced a new logic, which he qualified as non-Aristotelian, at present recognized as intuitionist. Moreover, he intensively discussed the two dialectics regarding infinity and logic. Already Cassirer stressed that no one more than Cusanus, after the first of his main books, De docta ignorantia of 1444, accomplished such a metaphysical change [Cassirer 1950, 277]. One century and a half after Cusanus came Galileo, who discussed the relevance of the two kinds of infinity in theoretical physics, any way he did What Was the Role of Galileo in the Birth of Modern Science? 49 not decide whether and how this dialectic had to be solved by a scientist; but he did not want to appeal to AI. In addition, he knew well both the organization of a theory and the logic suggested by Aristotle. He looked for an alternative logic, yet his Paduan period did not lead him to suggest a specific novelty; although as a matter of fact he adapted some of his theoretical works to the characteristic features of the model of a PO theory, e.g., the DNPs, he considered an alternative organization of a theory to be no more than a Platonic dialogue.

I conclude that regarding the two theoretical dialectics Cusanus was more advanced than Galileo.

In conclusion, it is to Galileo glory to have given birth to science in its first dialectic through systematic experimental practice; he was aware of the other two theoretical dialectics, but was inconclusive about them. For these reasons he has rightly been characterized as “The first modern scientist and the last of the ancient Greeks”. Cusanus played a somewhat complementary role; regarding the first dialectic, he has suggested only a programme for an experimental science of nature; however, regarding the latter two theoretical dialectics he preceded and was more advanced than Galileo, so that he anticipated the discovery of them. Which allows us to characterize Cusanus by means of a similar definition to the one above: “The last of the medieval scientists and the first philosopher of the foundations of modern science”.17 Although Cusanus had a considerable influence on the Italian intellectual milieu, e.g., on Leon Battista Alberti, Leonardo da Vinci and Giordano Bruno, no direct connection with Galileo is known. Had Galileo assumed Cusanus’ priorities, he would represent the culmination of a long intellectual effort started more than one century before. If, alternatively, he did not know Cusanus’ works, Galileo’s genius proves to be even greater, but also little explained.

After Galileo, without any discussion about the said dialectics, Newton decided the theoretical dichotomies with a pair of choices, which became a paradigm throughout the two subsequent centuries of development of theoretical physics. Due to this fact, a winter set in thinking about the two theoretical dialectics; in this winter only two scientists, i.e., L. Carnot [Carnot 1803, xiii– xvii, 3] and independently Einstein [Einstein 1905], made manifest the four choices, without receiving attention on this subject [Drago 2013]. Hence, one more merit has to be attributed to Galileo, i.e., at the very beginning of modern science to have presented through his discussions almost all the foundational issues of the scientiﬁc enterprise which have been re-discovered only after a very long period of time.

17. Koyré acknowledge him as “the last great philosopher of the dying Middle Age” [Koyré 1957, 6]. Cassirer as “the ﬁrst modern philosopher” [Cassirer 1950, 31–56]. 50 Antonino Drago Acknowledgments

I am grateful to Prof. David Braithwaite for having revised my poor English and to an anonymous referee for some important suggestions.

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Wisan, Winifred L. [1974], The new science of motion: A study of Galileo’s De motu locali, Archive for History of Exact Sciences, 13(2), 103–306, doi: 10.1007/BF00327483. Les De motu antiquiora de Galileo Galilei : le lancement de la carrière du ﬁlosofo-geometra

Raymond Fredette Université du Québec à Montréal (Canada)

Résumé : Dans cet article, j’analyse avec passion le De motu antiquiora (également De motu) dans les Edizione Nazionale [Galilei 1890-1909, ci-après EN] par Galileo Galilei [ci-après Galilée] (1564-1642). Cette œuvre de jeunesse de Galilée a été écrite entre 1589 et 1592. Elle traite du mouvement de la chute des corps et rejette clairement la physique du mouvement d’Aristote et ses conceptualisations cosmologiques. Elle n’a jamais été composée comme une version ﬁnale et a été publiée seulement après la mort de Galilée. Le De motu antiquiora présente quelques incertitudes mathématiques typiques de cette période dont je vais discuter dans cette article.

Abstract: In this paper I passionately analyse the De motu antiquiora (hereafter De motu) of the Edizione Nazionale [Galilei 1890-1909, hereafter EN] by Galileo Galilei (1564-1642). This work by the young Galilean was written between 1589 and 1592. It deals with the motion of falling bodies and clearly rejects Aristotle’s physics of motion and his cosmological conceptualizations. It was never composed as a ﬁnal draft and was published only after his death. The De motu antiquiora presents a few mathematical uncertainties typical of that time which I discuss in this paper.

1 Introduction

En soumettant publiquement le présent texte, j’aimerais reproduire à mon compte ce qu’Henri-Irénée Marrou, le grand historien de la ﬁn de la culture antique et spécialiste de saint Augustin, produisait chez moi durant ses cours. La fascination qu’on éprouve à coller au texte, à l’archive, au vestige fragile,

Philosophia Scientiæ, 21(1), 2017, 55–70. 56 Raymond Fredette fugace, fragmentaire du passé, dans le but d’appuyer le récit, le nourrir comme de sa nécessité. Je me propose de rendre le récit impératif par sa référence toujours explicite à la trace factuelle, à la prétendue donnée. La donnée pure n’existe pas. Ce sont les aléas du hasard qui nous « donnent » un reste qui, lui, doit toujours faire l’objet d’un choix pour sa pertinence et qui par là est nécessairement l’objet d’une manipulation jamais neutre. Je fais le pari que les gens que passionne la lecture de récits historiques veuillent peut-être me suivre pour voir comment on traite le passé-objet comme une cible et la connaissance historique comme une lampe de poche ou un rayon laser qui vise la cible sans jamais pouvoir savoir si la cible ne cache pas encore quelque chose à révéler et qu’on avait oublié. La vérité historique est de même nature que toute vérité : ce mot en grec se dit « ἀλήθεια » ; « aléthéia », le a est privatif. Léthé est la déesse de l’oubli. Le vrai est ce qui échappe à l’oubli. D’où l’idée que dire vrai c’est dévoiler, enlever le voile qui empêche de voir, révéler. J’aimerais cependant éviter le récit mythique, c’est-à-dire la narration des événements comme si un journaliste-témoin-présent revenait nous parler après coup, suite à un voyage astral dans le passé. Les journalistes sont des professionnels habituellement très compétents, mais ce ne sont pas des historiens. Pour essayer d’éviter la mystification, je me propose de toujours tenir à bout de bras deux discours : d’une part, celui qui présente la prétendue donnée, qui est toujours fragmentaire, mais qui sert de fondement et de source et de base à l’interprétation des événements, et d’autre part, celui qui narre cette interprétation elle-même. Je suis animé ici par l’idée que je me fais de ce que doit être l’activité scientifique. Cela me sert de modèle et c’est d’histoire des idées scientifiques dont il va être question. En écrivant l’histoire de cette façon, mon but est d’aller toujours au-devant de la critique, d’inviter l’introduction de nouvelles données fragmentaires ou de nouvelles perspectives sur elles. Pourquoi cela ? Tout simplement pour que se poursuive en pleine conscience inlassablement le travail inéluctablement provisoire et artisanal des beaux métiers de scientifique et d’historien, d’historienne. Sans autre détour, voici l’amorce de ce récit à double entrée ! Autour de 1590, Galilée écrit :

Nombreux ceux qui seront, après lecture de mes travaux, non pas à examiner avec attention si mes dires sont vrais, et changer d’avis, mais seulement à chercher soigneusement comment, à tort ou à raison, ils pourraient saper mes arguments. [EN, I, 412, 19–22]

Lorsqu’il consigne cette réflexion parmi d’autres dans un petit cahier qu’on retrouvera par hasard longtemps après sa mort, Galilée n’a pas encore trente ans. Et, s’il donne déjà à l’époque des signes de sa nature ambitieuse et fière, il ne peut quand même pas connaître à l’avance le destin qui l’attend. Nous, après coup, nous le savons. Que Galilée ait eu une telle réflexion est un fait. Les De motu antiquiora de Galileo Galilei 57

Ce fait appartient au passé, à l’histoire ; en allemand, on dirait Historie. Mais ce fait aurait cependant très bien pu échapper à l’histoire, c’est-à-dire à la connaissance historique, Geschichte. Nous avons ici un même mot qui désigne deux réalités à ne jamais confondre : une réalité disparue, morte, un passé à jamais révolu, par définition hors d’accès en lui-même si ce n’est à travers des traces ; et une réalité actuelle, vivante, fruit de l’exercice du métier toujours à recommencer qui consiste à faire parler ces traces. Ce que vous allez lire est l’histoire parfois loufoque et invraisemblable, mais pourtant véridique et passionnante, d’une de ces traces : un manuscrit de jeunesse dans lequel Galilée a écrit qu’il s’attendait à rencontrer chez de nombreux lecteurs une résistance féroce à se ranger à son avis. Si cette réflexion témoigne de la conscience claire qu’il a déjà que ses idées vont soulever la controverse, on aurait tort de penser qu’il s’agit là d’une prémonition applicable à l’ensemble de sa carrière mouvementée. Cette réflexion est une note qu’on ne doit appliquer qu’aux travaux auxquels elle se réfère. Et ces travaux sont ceux qui contiennent ses premières tentatives de philosophe, de mathématicien, de physicien portant sur la question du mouvement. Comment, en vertu de quoi, des corps – disons un morceau de bois, un caillou, qui diffèrent par la pesanteur – se déplacent-ils dans des milieux différents comme l’air et l’eau ? Or, les De motu antiquiora (les DMA sous-entendu scripta mea, comme il les appellera), ses Travaux plus anciens sur le mouvement, de son vivant, Galilée ne les soumettra jamais à personne pour lecture.

2 Sur le manuscrit

En effet, ses premiers travaux sur le mouvement, Galilée ne les a jamais publiés. Et pour autant qu’on sache, il ne les a jamais laissés circuler. De plus, ils ne sont jamais mentionnés explicitement, ni dans les œuvres qu’il a publiées ou laissées circuler ni dans sa volumineuse correspondance. La note que j’ai citée ici est la seule et unique remarque explicite qu’on connaisse de Galilée à propos de ces premiers De motu. Et on la trouve parmi d’autres notes qu’il avait accumulées dans un cahier à part en vue d’apporter des révisions et des corrections importantes aux résultats de ses recherches alors en cours de rédaction. Et pourtant, aujourd’hui, on peut lire ces travaux de jeunesse en dynamique. En effet, avec grands soins, selon une disposition très particulière, il les a conservés dans ses papiers. Et jusqu’à ce que le manuscrit contenant l’autographe de ces travaux soit trouvé, par hasard longtemps après sa mort, lui qui avait eu un moment l’intention d’en publier les résultats, mais qui s’est ensuite abstenu de le faire, a été, vraisemblablement, toute sa vie, en silence, le seul à en connaître l’existence. Par contre, ne les ayant pas détruits, c’est pour ainsi dire lui-même qui les a livrés à l’histoire. Et quelle histoire ! Cette histoire est celle d’une vieille idée d’Aristote revisitée par Galilée, la notion de pesanteur. De fait, c’est le réexamen par le jeune Galilée de cette 58 Raymond Fredette vieille idée aristotélicienne qui va mettre en chantier et nourrir toute sa carrière de chercheur. À partir de 1585, date à laquelle il abandonne l’université de Pise où depuis cinq ans son père espérait faire de lui un médecin, jusqu’à sa mort en 1642, Galilée va passer plus de 50 ans, comme en tête-à-tête, avec l’idée de pesanteur, la gravitas. Et en 1632, dans son fameux Dialogue sur les deux grands systèmes du monde, le Dialogo [EN VII 260.7–261.10], discutant avec l’aristotélicien Simplicio, voici ce qu’il va mettre dans la bouche de Salviati, son porte-parole. Nos deux compères tiennent une discussion de principe autour de « deux auteurs qui ex professo écrivent contre Copernic » : SIMPLICIO : Mais l’auteur va faire une objection, comme vous le voyez ; il demande de quel principe dépend ce mouvement circulaire des lourds et des légers : est-ce d’un principe interne ou d’un principe externe ? SALVIATI : Si l’on s’en tient à son problème, je dis que le principe qui fait tourner le boulet sur la concavité de la Lune est le même que celui qui maintient sa circulation pendant la descente ; je laisse ensuite à l’auteur le soin d’en faire un principe interne ou externe, à sa guise. SIMPLICIO : L’auteur va prouver qu’il ne peut être ni interne ni externe. SALVIATI : Je répondrai, moi, que le boulet sur la concavité n’avait pas de mouvement ; je serai donc dispensé d’avoir à expliquer comment il se fait qu’au cours de la descente, il reste toujours à la verticale du même point, puisqu’il n’y restera pas. SIMPLICIO : Bien, mais, comme les lourds et les légers ne peuvent avoir aucun principe, ni interne ni externe, de mouvement circulaire, le globe terrestre n’aura pas non plus de mouvement circulaire : nous avons ainsi atteint le but. SALVIATI : Je n’ai pas dit que la Terre n’a aucun principe, ni externe ni interne, de mouvement circulaire, mais je dis que je ne sais pas lequel des deux elle possède ; et ce n’est pas parce que je ne le sais pas que cela peut supprimer le principe. Mais cet auteur sait en vertu de quel principe d’autres corps du monde tournent – comme ils le font assurément –, alors je dis que ce qui fait mouvoir la Terre, c’est quelque chose de semblable à ce qui fait se mouvoir Mars, Jupiter et aussi, à ce qu’il croit, la sphère étoilée ; qu’il m’indique le moteur d’un de ces mobiles, et je m’engage à lui dire ce qui meut la Terre. Bien plus, je prends le même engagement s’il réussit à m’apprendre ce qui meut les parties de la Terre vers le bas. SIMPLICIO : La cause de cet effet est bien connue, chacun sait que c’est la pesanteur [gravità]. SALVIATI : Vous vous trompez, signor Simplicio. Ce que vous devez dire, c’est que chacun sait qu’on l’appelle pesanteur. Mais Les De motu antiquiora de Galileo Galilei 59

ce que je vous demande, ce n’est pas le nom, c’est l’essence de la chose : de cette essence, vous n’en savez pas plus que de l’essence de ce qui fait tourner les étoiles, si ce n’est le nom qu’on y a attaché et qui est devenu familier, banal, parce qu’on en a fait l’expérience fréquemment, mille fois par jour ; mais ce n’est pas cela qui nous fait mieux comprendre quel est le principe, ou la vertu, qui meut la pierre vers le bas, pas plus que nous ne savons ce qui la meut vers le haut quand elle s’est séparée du lanceur, ou ce qui meut la Lune en rond, sauf (comme je l’ai dit) le nom : nous lui avons donné comme nom propre et singulier le nom de gravità, alors que nous utilisions dans le second cas le terme plus générique de virtù impressa et dans le dernier cas le nom d’intelligenza ou d’assistente ou d’informante ; à l’inﬁni des autres mouvements, nous donnons comme cause la natura. [EN VII, 260.7–261.10] Fréreux & De Gand, dont je transcris la traduction, notent, avec à-propos, et précisent ici qu’à l’époque de cette discussion la tradition voulait que « les anges qu’on assignait à la conduite des planètes dans leur course étaient dits assistants ; informant est le principe moteur interne, comme l’âme dans les êtres animés » [Galilei 2000].

3 La méthode de travail de Galilée

Le succès prodigieux de la carrière de Galilée repose sur des méthodes de travail qu’il a pratiquées dès le départ. C’est à Aristote que l’Occident doit la première codification des règles logiques de la pensée. Le Stagirite a donné à ce code le nom grec d’Organon qui veut dire L’Instrument. Et pour ce faire, Aristote a abondamment puisé dans ce qui, peut-être, est le plus beau cadeau que d’autres Grecs nous ont légué : les mathématiques. Il y avait là en effet pour lui comme une façon empirique de notre pensée à se pratiquer à être rationnelle, logique. Aujourd’hui la logique fait partie du domaine de la mathématique. À l’époque de Galilée, logique et géométrie, pour nommer la branche alors la plus importante de ce domaine, constituaient deux territoires que la philosophie tenait bien distincts. En 1638, date de publication de ses Discours concernant deux sciences nouvelles, les Discorsi, Galilée a derrière lui 50 ans de pratique du métier de philosophe-géomètre. C’est l’association et l’intégration de la logique à la géométrie au sein du travail quotidien du philosophe de la nature qui permet à celui-ci « d’inventer », i.e., de détecter les défectuosités, tant logiques qu’empiriques, des arguments en cours sur une question, et ainsi de se mettre en piste pour en trouver de meilleurs. Sur la différence entre la logique et la géométrie en rapport à la philosophie naturelle, voici un passage clef des Discorsi. Aux personnages de Simplicio l’aristotélicien et de Salviati, porte- parole de Galilée, l’auteur, en fin pédagogue, ajoute le personnage de Sagredo, 60 Raymond Fredette qui, lui, représente l’honnête homme de l’époque à l’esprit curieux, libre et indépendant.

SALV. [...] Ainsi comprend-on que quand le support F s’appro- chera de l’extrémité D, il faudra nécessairement augmenter à l’inﬁni la somme des moments des forces appliquées en E et en D si l’on désire équilibrer ou surmonter la résistance placée en F . SAGR. Que dirons-nous, seigneur Simplicio ? Ne convient-il pas d’avouer que la géométrie est le plus puissant de tous les instruments pour aiguiser l’esprit et pour le mettre en mesure de raisonner et d’analyser de la meilleure façon ? Platon n’avait-il pas grandement raison quand il désirait que ses élèves eussent d’abord de solides bases mathématiques ? J’avais très bien compris d’où vient le pouvoir du levier, et comment selon qu’augmente ou diminue sa longueur, le moment de la force et de la résistance augmente ou diminue de son côté ; et pourtant, dans la solution du présent problème je me trompais, non pas un peu, mais du tout au tout. SIMP. Vraiment je commence à comprendre que la logique, bien qu’elle soit d’excellent usage pour régler le raisonnement, ne possède nullement, pour ce qui est d’éveiller l’esprit à l’invention, l’acuité de la géométrie. SAGR. Il me semble à moi que la logique nous apprend à re- connaître si les raisonnements et les démonstrations déjà éla- borées et inventées procèdent de façon concluante ; mais qu’elle nous apprenne à inventer les raisonnements et les démonstrations concluantes, cela vraiment je ne le crois pas. Mieux vaudrait cependant que le seigneur Salviati nous fasse voir selon quelle proportion augmentent les moments des forces nécessaires pour surmonter la résistance du même cylindre de bois quand varient les points choisis pour la rupture. [EN VIII 175, 12–34]

Comme nous pouvons le constater, la science nouvelle est ici concernée par des problèmes d’ingénierie, ceux de résistance des matériaux ; l’autre science nouvelle est la science du mouvement de translation des corps dans l’espace. Voici un commentaire de Maurice Clavelin :

Un des passages des Discorsi qui montre le mieux ce que la géométrie, pour Galilée, apportait à la philosophie naturelle : un langage permettant de caractériser les faits physiques dans leur spéciﬁcité, et en même temps une science capable, par la richesse de ses propres propositions, d’« éveiller l’invention » du physicien en l’aidant à pressentir et à expliciter les relations susceptibles de conduire à l’intelligibilité des phénomènes naturels. [Galilei 1970, 111, note 61] Les De motu antiquiora de Galileo Galilei 61

Or, il se trouve non moins de 8 passages du Tractatus des DMA en 23 chapitres qui illustrent sans équivoque la clarté de cette pratique dès le départ de la carrière, autour de la fin des années 1580 : = EN I 263, 25–31 : l’expérience ne fournit pas les causes des phénomènes ; = EN I 267, 2–16 : se tromper avec rigueur permet de se corriger ; = EN I 277–278. 32–5 : il y a des synergies du logique/mathématique/expé- rimental ; = EN I 289-290, 25–4 : réfuter Aristote permet de produire du nouveau ; = EN I 294, 26–33 : le vide révélerait la pesanteur vraie des choses ; = EN I 300–301, 14–10 : Archimède confirme notre apprenti dans son métier de physicien ; = EN I 314, 4–21 : on doit tester empiriquement la validité d’une assertion ; = EN I 325, 15–19 : un argument réfuté mérite l’examen de son contraire. Voyons donc ces passages des DMA dans leur détail. 1/8 = EN I 263.25–31 : l’expérience ne fournit pas les causes des phénomènes. Mais, afin que toujours nous utilisions des raisons davantage que des exemples (en effet nous cherchons à savoir les causes des effets, lesquelles ne sont pas rapportées par l’expérience), nous apporterons sous les yeux de tout le monde notre façon de penser, dont la confirmation fera s’écrouler l’opinion d’Aristote. Nous disons donc que, les mobiles de même espèce (que soient appelées de même espèce ces choses qui sont produites de la même matière, comme du plomb ou du bois, etc.), bien qu’ils diffèrent en grandeur, sont cependant mus avec la même rapidité, et une pierre plus grande ne descend pas plus rapidement qu’une plus petite. Nous sommes ici au chapitre 8, celui qui contient l’exposé des résultats théoriques les plus importants de toute l’entreprise des DMA : on y trouve une première approximation de la loi de la chute. Le passage suivant également appartient au même chapitre. 2/8 = EN I 267.2–16 : se tromper avec rigueur permet de se corriger. Il est ainsi donc évident comment, donné le rapport des mouvements des mobiles qui diffèrent seulement en pesanteur et non en grandeur, sont donnés aussi les rapports de ceux qui diffèrent selon n’importe quelle autre manière. Ainsi {1} donc, afin que nous trouvions ce rapport et, contre la façon de penser d’Aristote, que nous montrions que d’aucune manière les mobiles, même 62 Raymond Fredette

s’ils sont d’une espèce différente, ne respectent le rapport de leurs pesanteurs, nous démontrerons des choses desquelles non seulement la réponse à cette question dépend, mais également à la question du rapport des mouvements des mêmes mobiles dans différents milieux ; et nous examinerons l’une et l’autre question simultanément. Tout l’intérêt de ce passage se trouve dans la note marginale {1} qui lui est liée dans l’édition critique et que Favaro signale. Voici une transcription partielle de cette note avec mes commentaires. {1} Le passage cité jusqu’à « et nous examinerons l’une et l’autre question simultanément », se trouve dans la marge du folio 72r du manuscrit Gal. 71 de la BNCF, la Biblioteca Nazionale Centrale di Firenze, en remplacement cependant de près de trente-trois lignes de texte, dont les 16 premières se trouvent au folio 71v, et que Galilée avait d’abord rédigées. Les voici : Afin donc que nous trouvions ce rapport, on doit en venir à la cause de la rapidité et de la lenteur du mouvement [...]. Nous disons que les mobiles entre eux respectent dans leurs mouvements le rapport qu’ont leurs pesanteurs : pourvu qu’ils soient pesés dans ce milieu dans lequel le mouvement doit avoir lieu. [...] en effet, deux corps pesants ne respectent pas le même rapport dans des milieux différents. Mais ici surgit une difficulté très grande : en effet nous voyons par l’expérience que de deux boules égales en grandeur, dont l’une est plus pesante du double de l’autre, envoyées depuis une tour, celle qui est plus pesante ne percute pas terre deux fois plus rapidement ; qui plus est, même qu’au début du mouvement la plus légère devance la plus pesante, et qu’à travers un certain espace elle est portée plus vite. Or ceci est une matière pressante ; et de la plus haute importance, laquelle, parce qu’elle dépend de certaines choses qui n’ont pas encore été expliquées, sera mise en réserve, jusque là où sera donnée la cause de l’augmentation de la rapidité du mouvement naturel : où il sera démontré, que c’est par accident que le mouvement naturel est plus lent au début ; et à partir de cela que c’est aussi par accident qu’il sera reconnu qu’un mobile deux fois plus pesant ne descend pas d’une tour deux fois plus rapidement ; et il sera ensuite expliqué également pourquoi un mobile plus léger est porté plus rapidement, au début du mouvement, qu’un plus pesant. Relevons cette double affirmation étonnante selon laquelle, contrairement à ses prévisions théoriques, Galilée affirme avoir observé « non seulement qu’un mobile deux fois plus pesant ne descend pas d’une tour deux fois plus rapidement, mais qu’un mobile plus léger est porté plus rapidement, au début du mouvement, qu’un plus pesant1 ».

1. Il faut lire l’article de Thomas B. Settle, « Galileo and early experimentation » [Settle 1983] pour comprendre quel sens donner au fait que Galilée va consacrer un Les De motu antiquiora de Galileo Galilei 63

Le fait d’avoir accès à l’autographe des Travaux plus anciens sur le mouvement permet de voir notre jeune Galilée dans l’intimité même de son travail. Sans nous perdre trop dans les détails, examinons comment il réagit face à une situation fort embarrassante. Galilée ici a corrigé l’erreur d’Aristote en montrant que les corps chutent non pas en proportion de leurs pesanteurs absolues, mais de leurs pesanteurs spécifiques relatives. Il s’attend donc à ce que des corps de même matière tombent à des vitesses relativement égales et de surcroît uniformes. Or, ce n’est pas ce dont on fait l’expérience. Ces corps homogènes, c’est un fait, accélèrent en tombant. S’il fait tout à fait confiance à sa correction d’Aristote, il ne peut quand même pas penser à cette époque que la pesanteur ne soit pas pour quelque chose dans la chute. Vers quelles autres propriétés de la matière voulez-vous qu’il se tourne ? Il ne peut pas ne pas s’accrocher à cette propriété. Au-delà des innombrables propriétés dont les choses matérielles sont faites, s’il y en une qui leur est commune à chacune, n’est-ce pas le fait que toutes sont plus ou moins pesantes ? Et ce n’est qu’à regret, longtemps plus tard, nous l’avons signalé ci-dessus, qu’il avouera qu’il ne sait pas ce que c’est vraiment que la pesanteur. Et il se contentera alors de faire confiance aux derniers résultats, tant ceux des expérimentations avec le plan incliné que ses démonstrations de la Troisième Journée des Discorsi. Ce n’est pas tout, cependant. Non seulement les corps homogènes accé- lèrent en tombant, mais « [...] au début du mouvement la plus légère devance la plus pesante, à travers un certain espace elle est portée plus vite ». Anna De Pace, à propos de l’affirmation du jeune Galilée, écrit [ma traduction] : Ce qui rend perplexe sur la véracité de cette qualification, comme cela a été signalé à plusieurs reprises, c’est l’impossibilité d’observer ce que Galilée déclare pourtant avoir observé, à savoir que de deux corps en chute libre le plus léger précède durant un certain bout, le plus pesant. [De Pace 1990, 54] Est-ce à dire que Galilée est un halluciné ou encore un menteur ? Faire de la science, c’est, entre autres choses, faire des expérimentations. Celles-ci, par définition, pour être valables, doivent pouvoir être répliquées par les collègues. La science, disait Gaston Bachelard, c’est l’union des travailleurs de la preuve. Thomas B. Settle a publié les résultats d’une réplication vraisemblable de ce que Galilée rapporte avoir observé par expérimentation [Settle 1983, 12–15]. On a photographié les débuts de chutes de lâchers simultanés de poids. Les 51 sujets recrutés tenant dans une main une boule de fer, dans l’autre une en bois. chapitre tout entier à ces problèmes ; il s’agit de l’avant-dernier chapitre du livre II de son Tractatus, le chapitre 22 dans la traduction partielle anglaise de Drabkin [EN I, 333–338]. Il faut lire aussi Galileo lettore di Girolamo Borri nel De Motu [De Pace 1990, 3–69] pour un compte rendu d’une « littéraire » par opposition à celui d’une « scientifique » de certains aspects des problèmes en cause ; voir également la note 14 de ma communication de Liège en 1997 [Fredette 2001, 125–130]. 64 Raymond Fredette

[...] the signiﬁcant ﬁnding was that in only 12 % of the drops did the heavy ball either precede or fall alongside, the light one. In 88 % of the trials [il y a eu 4 fois 51 lâchers] the light ball clearly preceded the heavy one [...]. [Settle 1983, 12–15]

Vouloir produire de la connaissance nouvelle, c’est très compliqué. Et il n’y a pas de « logique de la découverte scientiﬁque », comme l’a montré Popper, il n’y a pas une recette à suivre. On le voit bien, Galilée ne trouve pas du premier coup la loi de la chute des corps. Mais il est cohérent. Sa critique d’Aristote le guide, mais il faut plus. Et c’est une grande sensibilité à ces écarts entre ses prédictions théoriques et les phénomènes qui vont guider sa recherche. Peu à peu, l’accélération dans la chute des corps va s’imposer à lui comme étant « naturelle ». Il n’y a rien là d’évident. Il y a même là quelque chose de profondément incompréhensible pour Galilée. Aristote avait imaginé comme étant intrinsèque aux corps deux propriétés contraires, la pesanteur et la légèreté, expliquant qu’en l’absence d’obstacles à leur mouvement, lesdits corps tendent naturellement et au plus vite à joindre, soit vers le haut soit vers le bas, leur lieu naturel. Galilée renoue avec les Anciens et élimine la légèreté qui n’est plus que de la pesanteur moindre. Mais alors comment comprendre qu’un corps dont la pesanteur constante fait qu’il tombe ne tombe pas selon un mouvement uniforme, mais va en s’accélérant ? Galilée ne le comprendra jamais. La loi d’attraction universelle des corps qui dit que les corps sont des masses qui tombent les unes sur les autres en raison directe de leur quantité de matière et en raison inverse du carré des distances qui les séparent, comme Isaac Newton va l’établir en 1687 dans ses Principes mathématiques de la philosophie naturelle, une telle loi aurait répugné profondément à Galilée, qui avait horreur des aﬃrmations aux odeurs de forces occultes.

3/8. = EN I 277.32–278. 5 : les synergies du logique/mathé- matique/expérimental. [...] Ceci est la démonstration d’Aristote : celle-ci certes conclurait fort à propos et par nécessité, si les choses qu’il a tenues pour acquises, Aristote les avait démontrées, ou encore, si elles n’ont pas été démontrées, si elles eussent du moins été vraies ; mais il a été trompé en ceci, [entre les lignes, au-dessus du mot « trompé » Galilée ajoute le mot, allucinatus « induit en erreur »] que ces choses, qu’il a tenues pour acquises comme étant des axiomes bien connus, non seulement ne sont pas ≤278 ≥ manifestes à la perception des sens, mais n’ont jamais été démontrées, et ne sont même pas démontrables, parce qu’elles sont entièrement fausses. En eﬀet, il a supposé que les mouvements d’un même mobile dans diﬀérents milieux respectent entre eux, en rapidité, le rapport qu’ont les subtilités des milieux : ce qui certes est faux, ayant Les De motu antiquiora de Galileo Galilei 65

été abondamment démontré ci-dessus. [Voir les démonstrations du chapitre 8, 267–270, 17–14]

Nous sommes ici au cœur de l’essentiel de la portée épistémologique de l’entreprise de Galilée. Notre jeune physicien est convaincu qu’Aristote a fait erreur sur le plan des axiomata. C’est cette conviction qui va le guider et le motiver tout au long d’une longue carrière de cinquante ans de recherche durant laquelle, ne l’oublions pas, il ne réussira jamais – et ce n’est que normal – à se sortir complètement de la formidable emprise exercée par le génial eﬀort d’intelligibilité du monde que fut l’œuvre du Stagirite. Et la pratique de la rigueur des rapports entre les axiomes et les démonstrations qu’on peut en déduire, c’est de la géométrie euclidienne que Galilée la tient en premier lieu, bien davantage que de l’Organon aristotélicien. Et c’est, pourrions-nous dire, en laboratoire que Galilée met ses théorèmes à l’épreuve2.

4/8. = EN I 289.25-290.4 : réfuter Aristote permet de produire du nouveau. Ceux qui sont venus avant Aristote n’ont pas considéré le pesant et le léger si ce n’est par comparaison à propos de corps moins pesants ou moins légers ; et à mon avis, cela est tout à fait à bon droit : or Aristote, au livre IV du De Caelo, s’efforce de réfuter l’opinion des anciens et de confirmer la sienne qui lui est contraire. Or comme c’est cette opinion des anciens que nous allons suivre, nous examinerons tant les réfutations d’Aristote que ses confirmations, en en confirmant les réfutations, et en en réfutant les confirmations ; et nous accomplirons cela lorsque nous aurons exposé l’opinion d’Aristote.

Voici une stratégie épistémologique simple qui témoigne de cet esprit nouveau de découverte qui marque l’avènement de la pensée scientiﬁque moderne naissante à la Renaissance en Italie. Et constatons que cette épistémologie est bien en place aux DMA. Et que c’est Aristote mieux que tout autre qui renseigne Galilée sur les anciens. Constatons également que c’est avec grand soin, clarté et en détail qu’Aristote critique ses prédécesseurs. L’activité scientiﬁque est communautaire, mais nécessite la liberté d’esprit. Galilée lit Aristote avec un tel esprit et développe un grand respect envers le Stagirite qui lui fait penser souvent que si ce dernier était là, il se rangerait à son avis contre les aristotéliciens de son époque. Et ici, dans la substance du contexte du passage cité, c’est du concept essentiel d’espace vide dont il s’agit.

5/8. = EN I 294.26-33 : le vide révèlerait la pesanteur vraie des choses. Thémistius, suivant l’opinion d’Aristote, en parlant du vide, au

2. À témoin, la numérisation du monumental manuscrit Gal. 72, BNCF : http: //www.imss.fi.it/ms72/INDEX.HTM. 66 Raymond Fredette

sujet du texte #74 du livre IV des Physiques 216a9–21, a écrit ceci : « C’est ainsi que puisque le vide cède uniformément, mais à la vérité il ne cède aucunement (puisqu’en effet il n’est rien, bien subtil homme celui qui se figure le vide céder), d’où il résulte que les différences des choses pesantes et des choses légères, c’est-à- dire les variations (momenta) des choses, sont supprimées, et, ce qui s’ensuit, c’est que la vitesse de toutes les choses mues arrive à être égale et sans discrimination. » Or combien ces paroles sont fausses sera bientôt connu, quand nous aurons mis en lumière de quelle manière c’est uniquement dans le vide que peuvent être données les vraies différences des pesanteurs et des mouvements, et qu’il n’est aucunement possible de les trouver dans le plein. Il y a ici trois éléments fort intéressants. Il y a tout d’abord la présence chez Thémistius du mot momenta. L’usage du mot momentum ici est évidemment, sous la plume du philosophe grec, di- recteur de l’université de Constantinople, ami de Julien l’Apostat au ive siècle, tout à fait traditionnel. Et Galilée n’a pas encore pris conscience du besoin du concept capital que sera celui de « moment » [Galluzzi 1979, 172, note 73], pour un long commentaire très pertinent qui demanderait des développements ici. J’y reviendrai. Il y a ensuite une affirmation d’Aristote en Phys. 216a 9– 21, reprise par le commentaire de Thémistius, qui est déduite explicitement de l’argumentation, à savoir que dans l’hypothèse de leurs mouvements dans le vide tous les corps tomberaient à la même vitesse. Il y a enfin le désaccord de Galilée ici aux DMA avec une affirmation qu’il finira par faire sienne beaucoup plus tard. Il faut prendre ici la mesure du chemin à parcourir pour clarifier ce qui est en jeu. Je me contente de faire remarquer que l’acquisition aux DMA du concept de vide pour penser le mouvement est un pas épistémologique gigantesque. Et que ce pas, Galilée le franchit en faisant simultanément deux choses : exploiter les vertus inventives de la pensée géométrique et mettre les résultats obtenus au service de la confrontation avec les phénomènes observables sous contrôle. 6/8. = EN I 300.14–301.10 : Archimède confirme notre apprenti dans son métier de physicien. D’autre part, je n’ignore pas ici que quelqu’un puisse m’objecter que j’ai présupposé comme vrai pour ces démonstrations ce qui est faux : à savoir que les poids suspendus à partir de la balance maintiennent avec la balance des angles droits ; alors que pourtant les poids, tendant vers le centre, seraient convergents. À ceux-là je répliquerais que je me recouvre des ailes protectrices du surhumain Archimède (que je ne mentionne jamais sans admiration). En effet, lui-même a présupposé la même chose dans sa Quadrature de la Parabole ; et cela, peut-être, afin de montrer qu’il était de loin tellement en avant des autres, qu’il pouvait tirer le vrai du Les De motu antiquiora de Galileo Galilei 67

faux : et cependant, il ne faut pas se mettre à douter comme s’il avait conclu lui-même au faux, puisqu’il avait prouvé cette même conclusion dans une autre démonstration géométrique antérieure. Voilà pourquoi ou bien on doit dire que les poids suspendus maintiennent vraiment avec la balance des angles droits, ou bien il n’est nullement important s’ils maintiennent ou non des angles droits, qu’il suffise qu’ils soient égaux ; ce qui d’aventure sera plus probable : à moins que nous voulions dire, ceci est plutôt une licence géométrique ; comme lorsqu’Archimède présuppose que les surfaces ont de la pesanteur, et que l’une est plus pesante que l’autre, alors que pourtant elles sont effectivement toutes exemptes de pesanteur. Et les choses que nous avons démontrées, comme nous l’avons dit ci-dessus, doivent être comprises à propos de mobiles à l’abri de toute ≤301≥ résistance extrinsèque : mais puisqu’il est d’aventure impossible de les trouver dans la matière, que quelqu’un ne soit pas étonné, en faisant à propos de ces choses un test [de his periculum faciens], si l’expérimentation déçoit, et qu’une grande sphère, même si elle est sur un plan horizontal, ne puisse pas être mue par une force minime. En effet, aux causes déjà dites, il s’ajoute encore ceci : à savoir qu’un plan ne peut pas vraiment être parallèle à l’horizon. En effet, la surface de la Terre est sphérique, ce à quoi un plan ne peut pas être parallèle : voilà pourquoi, un plan ne touchant la sphère qu’en un point seulement, si nous nous éloignons d’un tel point, il est nécessaire de monter : voilà pourquoi avec raison la sphère ne pourra pas être écartée d’un tel point par la moindre force qui soit.

Tout d’abord, à propos du renvoi par Galilée à la Quadrature de la parabole : ce que notre jeune philosophe partisan d’une physique mathématique laisse entendre n’est pas quelque chose qu’Archimède a dit explicitement en autant de mots dans son livre. Il est cependant vrai qu’Archimède suppose là que les poids suspendus depuis les extrémités des leviers horizontaux en équi- libre pendent perpendiculairement au levier. Drabkin & Drake signalent que Benedetti, dans ses Diversarum speculationem [...] liber, Turin, 1585, 148–151, aurait critiqué une telle présupposition chez Jordanus et Tartaglia. Guidobaldo dal Monte aussi a discuté ces choses dans ses Mechanicorum (1577) et tout cela a été largement débattu au siècle suivant [Drabkin & Drake 1960, 67, note 10]. Et dans le Dialogus, sa version dialoguée des DMA, s’inspirant pour la première fois du traité Des corps ﬂottants, Galilée avait d’abord donné à ses démonstrations équivalentes à celles du chapitre 8 des ﬁgures qui tiennent compte de la rotondité du plan de l’horizon [EN I, 381–384]. Or, à ce sujet, on peut rappeler également ce que dit Charles Mugler dans la Notice qui précède sa traduction :

La proposition 2 du premier livre, où Archimède déduit de l’hy- pothèse de la convergence des verticales la forme sphérique de la 68 Raymond Fredette

surface du liquide en état d’équilibre, est la première, et pendant des siècles la seule, tentative de rendre compte par le raisonnement mathématique du fait d’observation, familier aux riverains de la Méditerranée, de la courbure de la mer. [...] Aristote, [Du Ciel, 287b 4–14], manque de rigueur [...]. [Mugler 1970, II, 3–4] Et ceci, en passant, alimente la thèse de Lucio Russo, qui croit qu’Archimède appartient à une véritable culture technoscientifique analogue à celle qui sera mise au point à l’époque de Galilée et qui aurait existé autour d’Alexandrie, de la mort d’Alexandre jusqu’à l’extinction de cette culture sous l’hégémonie de Rome. À propos, ensuite, de l’idée de test : l’expression que Galilée utilise, periculum facere, est épistémologiquement chargée à l’époque [Schmitt 1969, 80–138]. Faire une expérience, c’est observer, c’est ressentir, mais faire un test, c’est mettre à l’épreuve sa perception, son observation, son sentiment et surtout c’est courir un danger, un péril, celui de s’être trompé, illusionné. Aristote est et a toujours été un champion de l’expérience, de l’observation contre, par exemple, les élucubrations imaginatives des idées platoniciennes. Mais la science moderne émergente met au point une attitude critique nouvelle. Provando e riprovando sera la devise des disciples de Galilée à l’Accademia del Cimento. Enfin, à propos du plan parallèle à l’horizon : on doit se mettre d’accord avec Drabkin qui suggère que Galilée veut dire ici qu’aucune surface plane sauf celle d’une sphère ne peut avoir tous ces points à une égale distance du centre du Monde [Drabkin & Drake 1969, 68, note 13]. Et voilà comment Galilée fait d’Archimède son maître de physique mathématique. 7/8. = EN I 314.4–21 : tester empiriquement la validité d’une assertion. Or, il arrive quelquefois que certaines opinions, bien qu’elles soient fausses, subsistent très longtemps chez les gens ; parce qu’à première vue elles présentent en leur faveur une apparence de vérité, et pour cela personne ne se met en peine de chercher à savoir si c’est digne d’être cru. Une illustration d’une telle chose est ce qui est cru concernant les choses qui se trouvent sous l’eau, et dont l’opinion commune affirme qu’elles apparaissent plus grandes qu’elles sont vraiment. Or lorsque je n’ai pas pu trouver une cause à un tel effet, à la fin, ayant recours à l’expérimentation, j’ai trouvé qu’une pièce de monnaie reposant au fond de l’eau n’apparaît d’aucune manière plus grande, mais plutôt plus petite : voilà pourquoi moi j’estime que celui qui le premier a énoncé cette façon de penser doit avoir été amené à cette opinion durant l’été, lorsque les prunes ou d’autres fruits sont parfois mis dans une coupe en verre pleine d’eau, dont la forme rappelle une surface conoïde ; ceux-ci assurément apparaissent de loin plus grands qu’ils ne sont, à ceux qui ainsi les regardent alors que les rayons de lumière sont laissés passer à travers le verre. Mais c’est la forme de la Les De motu antiquiora de Galileo Galilei 69

coupe, et non pas l’eau, qui est la cause d’un tel eﬀet, comme nous l’avons plus amplement mis en lumière dans les commentaires à l’Almageste de Ptolémée, lesquels (à la faveur de Dieu) seront publiés bientôt. Or un indice de cela est que, l’œil posté au-dessus de l’eau, en sorte qu’on puisse observer sans qu’intervienne le milieu du verre, la prune n’apparaît pas plus grande.

4 Conclusion

Simpliste vous allez me dire, ces observations optiques ? Après coup, en science, une fois établie la vérité d’une chose, tout paraît simple. N’empêche que ce petit exemple de prunes vues à travers de la vitre révèle le genre de culture d’observation contrôlée pratiquée par Galilée bien longtemps avant d’arpenter les quais de l’Arsenal de Venise, bien avant de fabriquer des lentilles pour observer « ce que personne n’a encore vu ». Et les commentaires du copernicien Galilée à l’Almageste de Ptolémée – possiblement des notes de cours liées à son enseignement pisan – n’ont jamais été trouvés. On peut penser qu’ils ont été détruits en cours de carrière. Nous n’en savons cependant rien.

8/8. = EN I, 325.15-19 : un argument réfuté mérite l’examen de son contraire. La manière dont la solution de l’argument d’Aristote peut être convenablement tirée des propositions de l’argument lui-même est évidente : aussi, vu qu’il ne nous accable pas plus longtemps, voyons si nous pouvons construire des arguments davantage acca- blants et plus pénétrants en faveur du contraire.

Voilà de la pensée logico-mathématico-physique vivante et en pleine action. La carrière du ﬁlosofo-geometra, ce sont bien ses Travaux plus anciens sur le mouvement qui la lance. Épistémologiquement parlant, le texte de 1638 des Discorsi dont nous sommes partis prend solidement racine à Pise 50 ans plus tôt.

Bibliographie

De Pace, Anna [1990], Galileo lettore di Girolamo Borri nel De Motu, dans De Motu. Studi di storia del pensiero su Galileo, Hegel, Huygens e Gilbert- Quaderni di Acme, édité par Università di Milano. Dipartimento di ﬁlosoﬁa, Milan : Cisalpino, Istituto Editoriale Universitario, t. 12, 3–69.

Drabkin, Israel Edward & Drake, Stilman [1960], Galileo Galilei. On Motion and On Mechanics, Madison : The University of Wisconsin Press. 70 Raymond Fredette

Drabkin, Israel Edward & Drake, Stilman (éds.) [1969], Mechanics in Sixteenth-Century Italy : selections from Tartaglia, Benedetti, Guido Ubaldo, & Galileo, t. 13, Madison : The University of Wisconsin Press.

Fredette, Raymond [2001], Notes pour une traduction intégrale du Traité contenu dans le de motu antiquiora de Galilée, dans Proceedings of the XXth International Congress of History of Science : medieval and classical traditions and the Renaissance of physico-mathematical sciences in the 16th century, Turnhout : Brepols, 125–130.

Galilei, Galileo [1890-1909], Le opere di Galileo Galilei. Edizione Nazionale sotto gli auspice di Sua Maestà il Re d’Italia. Esposizione e disegno di Antonio Favaro, Firenze : Tipograﬁa di G. Barbèra.

—— [1970], Discours et démonstrations mathématiques concernant deux sciences nouvelles, Paris : A. Colin, Introduction, traduction et notes par M. Clavelin.

—— [2000], Dialogue sur les deux grands systèmes du monde, Galilée, publié en 1632, Points Sciences, Paris : Seuil, traduction par R. Fréreux et Fr. de Gandt.

Galluzzi, Paolo [1979], Momento, Studi galileiani, Roma : Edizioni dell’Ate- neo.

Mugler, Charles [1970], Archimède, t. II, Paris : Les Belles Lettres.

Schmitt, Charles B. [1969], Experience and experiments : a comparison of Zabarella’s view with Galileo’s in De Motu, Studies in the Renaissance, 16, 80–138, doi :10.2307/2857174.

Settle, Thomas B. [1983], Galileo and early experimentation, dans Springs of Scientiﬁc Creativity : Essays on Founders of Modern Science, édité par R. Aris, T. H. Davis & R. H. Stuewer, Minneapolis : The University of Minnesota Press, 3–20. “It Is Impossible to Deceive Nature”. Galileo’s Le mecaniche, a Bridge between the Science of Weights and the Modern Statics

Romano Gatto Basilicata University (Italy)

Résumé : Il est impossible de tromper la nature. C’est l’avertissement que Galilée adresse aux « ingénieurs ignorants » qui étaient convaincus que les machines pouvaient vaincre la nature en réalisant l’impossible. Dans Le mecaniche il montre que les mouvements mécaniques ne peuvent pas se produire contre la nature. Les machines doivent obéir à certaines lois de la nature auxquelles on ne peut pas déroger, comme le principe de compensation qui établit une liaison entre la force motrice et la résistance déplacée ; formulé en termes de mouvements virtuels, il devient le principe fondamental d’une nouvelle approche de la dynamique, au moyen de laquelle Galilée construit une nouvelle science de l’équilibre, première forme de la statique moderne.

Abstract: It is impossible to deceive nature. This is Galileo’s warning against the “ignorant engineers” who were convinced that machines could overcome nature by undertaking impossible projects. In Le mecaniche the Pisan scientist shows that mechanical movements cannot happen against nature. In fact, machines must obey some inviolable laws of nature like the compensation principle, which establishes a connection between the moving force and the moved resistance. This principle, formulated in terms of virtual movements, becomes the fundamental principle of a new approach, the dynamic one, by means of which Galileo constructs a new science of equilibrium, which shapes the ﬁrst form of modern statics.

Philosophia Scientiæ, 21(1), 2017, 71–91. 72 Romano Gatto 1 Introduction: the two versions of Le mecaniche

On September 26, 1592, the “Serenissima Repubblica” of Venice assigned Galileo the chair of Mathematics at the University of Padua.1 On accepting the position, Galileo left the University of Pisa, where he had taught for three years, and moved to Padua, where he remained for 18 years. Galileo himself defined these years as “the most beautiful” ones of his life. During this time, his prestige, as both a scientist and a teacher, grew beyond all bounds. Besides public teaching, he also provided private teaching in his house to numerous scholars coming from every part of Europe. At the University, he held two courses every year, as we can see in the Rotuli artistarum dello Studio di Padova Pars Prior 1520-1739 [Viviani 1717], a very precious source in the University of Padova Archive, despite suffering from lacunae and incompleteness. By means of this and other sources, we know that during his debut year (1592-1593), Galileo taught “science of the fortifications” and “mechanics”. Vincenzo Viviani (1622-1703), who assisted him in the final years of his life, writes in his Racconto istorico della vita del signor Galileo Galilei2 that during his stay in Padova, Galileo [...] wrote for his students some treatises, among which one on Fortifications, according the use of that times, one on Gnomonic and practice Perspective, an epitome of the Sphere and a treatise on Mechanics which circulates in manuscript form and that P. Marin Mersenne translated into French and, in 1634 published in Paris, and lastly, in 1649, Cavalier Luca Danesi published in Ravenna. [Viviani 2001, 39] Viviani’s indications are so clear that they leave no room for interpretation: the text on mechanics he is speaking about is the work Le mecaniche that Antonio Favaro (1847-1922) published in 1891 in the second volume of the National Edition of Galileo’s Opera [Galilei 1890-1909]. In the National Library in Florence, there are two autographed manuscripts of Viviani’s Racconto and in one of them, close to this quotation, the author, in his own hand, added the following gloss: In 1593 he wrote Mechanics and other things. [Galilei 1890-1909, XI, cc. 22–68]; [Viviani 2001, c. 35v] This evidence would lead us to conclude that in 1593 Galileo wrote “the treatise on mechanics, which circulates in manuscript form”, Le mecaniche. Yet as I demonstrated in my essay, Tra la scienza dei pesi e la statica. Le

1. A chair left vacant since March 1588 when his regular professor, Giuseppe Moletti (1531-1588), passed away. 2. This work was published for the first time in [Viviani 1717], but here we will quote from the recent edition [Viviani 2001]. “It Is Impossible to Deceive Nature”. Galileo’s Le mecaniche 73 mecaniche di Galileo Galilei, published together with the critical edition of this work in 2002 [Galilei 2002, IX–CXLIV], the date of 1593 refers not to Le mecaniche, but to another text, which, in an abridged way, deals with the same matter. Evidently, Viviani knew only the text published for the first time in French by Marin Mersenne (1588–1648), so he claims that it was the one Galileo wrote in 1593. Cornelio Will, who was the archivist and advisor of the Prince of Thurn- Taxis, brought the existence of this shorter version to Favaro’s attention. In April 1898, Will wrote to the Italian professor, indicating that in the Prince’s library, he had discovered a code containing some Italian manuscripts, one of which was attributed to Galileo. Favaro verified that, alongside the text of Le mecaniche, there was a second text, entitled Delle meccaniche lette in Padova dal Signor Galileo Galilei l’anno 1594 [Favaro 1889]. This text concerned nearly the same matters, but was written in a more concise form. In my aforementioned critical edition, I considered them to be two versions of the same work, and without much imagination, referring to their length, I called them respectively versione breve, that is, short version, and versione lunga, long version. I also demonstrated how the text of the short version relates to the course of mechanics Galileo held during the academic year 1592-1593, and how the long version concerns the lessons from the academic year 1598-1599. From a formal point of view, the two versions are characterized by the different development of their texts: the short version is concise to the point of almost being reduced to the essential concepts with only some demonstration. The long version, however, ranges widely with great linguistic and stylistic accuracy, and includes exhaustive and beautiful demonstrations. This helps us understand that, while the first was merely a lesson guide Galileo used in the first year of his teaching, the second was the result of a well-thought-out and more accurate revision whose aim was the completion of a real treatise on mechanics. We can recognize this fact by comparing the introductions of these two works: in the short version it runs to a few lines concerning the definition of mechanics and the list of five simple machines; in the long version there is a very interesting dissertation on mechanics including unexplored aspects on the theory of machines. One of the most relevant aspects of the long version is the fact that it had been conceived as a geometrical treatise. The introduction is followed by a chapter of definitions and axioms, which is lacking in the short version. The presence of this chapter is clear proof of Galileo’s intent to compile a rigorous treatise in which any concept had to be introduced if it had not been defined and explained beforehand, similar to Euclid’s Elements. This desire to be clear and precise stands out also in the insertion (after the treatment of the lever) of the chapter “Alcuni avertimenti circa le cose dette” [Some notices about the aforesaid things], also missing from the short version. In this chapter, Galileo sheds light on a few important questions concerning the angular lever and the second-class lever, which are not in any other treatise of that time. However, in comparing the two texts, an analysis of the technical- scientific contents shows that we are examining two successive phases in the 74 Romano Gatto development of the same ideas. The contents of the long version are the same as those of the short version—apart from the sections concerning the so-called Baroulkos, namely, a train of shafts into a wheel, and concerning the perpetual screw, which are present only in the short version. In the long version, Galileo deals with these subjects in a wider and more detailed fashion, adding a few pertinent and remarkable demonstrations, including many good examples, and he introduces some innovative ideas on the theory of machines. But the most peculiar aspect that distinguishes this version and makes it a new original treatise in mechanics is its systematic use of the concept of the moment.

2 The deﬁnition of mechanics in the short version

One of the most important questions we want to emphasize is the fact that Galileo provides a new definition of mechanics within the few lines of the short version: The science of mechanics is the faculty that teaches us the reasons and makes us understand the causes of the miraculous effects that we see happen with some tools, in moving and lifting heavy weights with a small force. [Galilei 2002, 5, ll, 1–4] This definition marks a caesura with the Aristotelian tradition still alive at that time. At the beginning of his Μηχανικά Προβλήματα, the pseudo-Aristotle writes: Among the events that occur in accordance to nature, those of which the cause is unknown arouse wonder, while, among the events that occur against nature, arouse astonishment all those are realized with art for the benefit of mankind. [Aristotele 2000, 847a, 11–13] According to the theory of natural places, the pseudo-Aristotle distinguishes mechanical phenomena according to nature (κατὰ φύσιν), from those against nature (παρὰ φύσιν). He maintains that, while our advantage is susceptible to change in many ways, nature always acts consistently and plainly in the same direction. So, in many cases, nature can produce disadvantageous effects for our utility. Therefore, if we wish to do something against nature we must resort to τέχνη, to art, to the help of some tool. For this reason, the pseudo-Aristotle defines mechanics as follows: The part of the art which comes to our aid in such difficulties. [Aristotele 2000, 847a, 17–19] For the pseudo-Aristotle, and the long tradition that followed him and still dominated at the time of Galileo, mechanics does not explain the nature of “It Is Impossible to Deceive Nature”. Galileo’s Le mecaniche 75 the bodies or their natural motions; it is art that, by means of machines (artificials), allows us to make things against nature (παρὰ φύσιν) and that for this reason arouses wonder. The numerous Renaissance commentators on the Μηχανικά Προβλήματα have interpreted the above-mentioned opening passage of this work in various ways, according to the different ways to translate παρὰ φύσιν. These interpretations are very important in establishing their epistemological position towards the Aristotelian theory of natural places. In my previously mentioned critical edition, I pursued a wider and deeper analysis of these positions, concluding that only Galileo fully develops the theory of natural places. Galileo first subjected this theory to severe criticism in De motu [Galilei 1890-1909, I, 243–419], but only in the two versions of Le mecaniche does he complete his project to revise and re-establish the bases of the mechanics.

3 The compensation principle in the long version

As we have seen, in the short version, Galileo deﬁnes mechanics as the science that teaches the reasons and explains the causes of the eﬀects that may appear to be miracles only to those who ignore the true laws of nature. It explains arcane ideas concerning “miraculous” phenomena connected to the use of tools, and causes them to lose any attribute of irrationality. Galileo thus divests mechanics of any metaphysical element, and shows that the machines do not possess any supernatural virtue but operate according to precise laws of nature. In the long version, he makes it even more explicit that nothing can happen which nature does not allow. Because of the ignorance of the laws of nature, he writes:

I have seen the generality of mechanicians deceive themselves in going to apply machines to many operations of their own nature impossible [...]. Of which mistakes I think to have understood the cause to be essentially the belief that these artiﬁcers had and still have to be able to lift great weights with a small force, in a certain manner deceiving nature, whose instinct, yet unchanging constitution it is, that no resistance can be overcome but by a force that isn’t more powerful than it. [Galilei 2002, 45, ll, 8–17]

This passage is remarkable. Here Galileo plainly reaﬃrms that nature cannot in any way be deceived, that it is impossible to operate against the natural order. The machines have to observe the unavoidable law that “no resistance can be overcome but by a force that isn’t more powerful than it”. So where does the beneﬁt of the machines lie? In order to answer this question Galileo explains that in the working of the machines we must consider four things: 76 Romano Gatto

The ﬁrst one is the weight to move from place to place; the second is the force or power that has to move it; the third is the distance between both terms of the motion; the fourth is the time in which this change must be made; and it is the same thing if instead of this time we consider the quickness and the speed of motion, as we deﬁne that motion faster than another which covers that distance in a shorter time. [Galilei 2002, 45–46, ll, 28–34]

Previously in the short version Galileo had pointed out that, in the working of a machine, not only were the moving forces in play, but the speed of the moved resistance also had to be considered:

But it is necessary to notice that the more the eﬀort is in using the lever, the more, on the opposite, we spend time; and the less the force is with respect to the weight, the greater is the space through the force will move with respect to the space through which the weight moves. [Galilei 2002, 7, ll, 75–79]

The working of a lever, and of all the other simple machines that rely on it, follows a principle according to which force, resistance, time, space, and speed continuously compensate each other: if the power increases, the speed decreases because, in moving the resistance through a given space, the force crosses a space longer than it in the same time. This is the reason why we call this statement the compensation principle. In the introduction to the long version, Galileo explains that, with no machine, it is still possible to move a great weight with a small force from one place to another by dividing it into a certain number of parts, with no part being superior to the moving force, and transferring them one by one until the whole weight is carried to the assigned place. He ﬁrmly remarks, at the end of this work, that nobody can aﬃrm that a great weight has been transported by a force less than it, but only that it has been moved by a force that is reiterated many times along that space which the whole weight, all together, would have traversed only once. In so doing, it is clear that the velocity of the moving force is many times superior to that of the moved weight, as this weight is superior to the force indicated. While this force has crossed the given space many times, the whole resistance has crossed it only once. Thus, Galileo correctly concludes:

Therefore, we ought not to aﬃrm that a great resistance has been overcome by a small force, contrary to the constitution of nature. Then only we will able to say that the natural constitution has been overcome, when the lesser force would transfer the greater resistance with the same speed of motion with which the force moves; that we absolutely aﬃrm to be impossible to be done with any imaginable machine. [Galilei 2002, 7, ll, 75–79]

The compensation principle appears again and again in Le mecaniche, from beginning to end. Upon reading the introduction, we soon recognize that one “It Is Impossible to Deceive Nature”. Galileo’s Le mecaniche 77

Figure 1. of Galileo’s previous aims was to show that every machine is subject to this same rule. In fact, this principle appears to be a thread running through the entire treatise, which Galileo expressly reasserts when he concludes the discussion of each machine considered. At the end of the exposition on the balance and the lever in the long version, he remarks that

The beneﬁt drawn from this instrument is not that of which the common mechanics are persuaded, that nature could be overcome and, in a certain manner deceived, a small force overpowering a very great resistance by means of a lever, because we will demonstrate that, without the help of the length of the lever, the same lever, the same force, in the same time obtains the same eﬀect. [Galilei 2002, 55, ll, 392–398]

Here, he shows the real function of the tools, thereby demonstrating that nature cannot be deceived. Given the lever BCD (Fig. 1) with the fulcrum at C, the force applied at D and the resistance at B, let us suppose the distance CD, for example, to be five times the distance CB. It is clear that the force is able to lift a weight five times greater than itself. Now let this lever move until it assumes the position ICG. The force has now passed through the space DI while the weight has crossed the space BG. Since DI = 5CB and BCGˆ = DCIˆ , then necessarily ID = 5BG. If we then consider the distance CD at point L, so that CL = BC, the same force applied at L will be able to lift from point B only at an equivalent of one fifth of the weight that was placed there at first. Now given that CL = BC, the distances LM and BG, crossed respectively by the moving force and the resistance, will be equal. Reiterating this action five times, we may observe the same effect: the whole weight will be lifted from B to G. At this point Galileo concludes:

But the repeating of the space LM is certainly more than the single measurement of the space DI, that is ﬁve times LM. Therefore, the transferring of the weight from B to G requires neither less force, nor less time, nor a shorter way if the weight were placed at D, than it would need if the same force had been applied at L. [Galilei 2002, 55, ll, 392–398] 78 Romano Gatto

Therefore, the beneﬁt derived from the use of this tool is only the possibility to move a given body all at once. But the time the tool itself takes in moving the weight will be just the same it would take if, having divided the space into a certain number of parts, each one equal to the given force, this force had transported them the same distance one at a time. After having explained the law of the shaft into the wheel, Galileo considers the case of a winch in which the diameter of the wheel is ten times that of its shaft; so this tool is able to lift a weight ten times that of the force applied.

Figure 2.

He explains that, when the force moves once along the circumference FCG (Fig. 2), the shaft EAD, to which the weight is tied, winds around the rope and lifts the weight in completing only one turn. Therefore, the weight H will cover only the tenth part of the way covered in the same time by the moving force. At this point, reproducing the same reasoning of the introduction, Galileo points out:

If therefore, by means of this machine, the force for moving a resistance greater than itself, along a given space, must of necessity move ten times as far, there is no doubt that, dividing the weight into ten equal parts, each of them will be equal to the force, and consequently, it might have been transported one at time, as great a space as that which itself will move. So that, making the journeys, each equal to the circumference EAD, it will not have gone any further than if it had moved only once around the circumference F GC and had carried the same weight H to the same distance. [Galilei 2002, 50, ll, 545–550]

In concluding, he reasserts that the beneﬁt derived by the use of this tool is only that it is possible to carry the given weight together across the given distance, but not with less labor, or with greater speed, or a greater distance that the same force might have used if carrying parcels. The validity of this “It Is Impossible to Deceive Nature”. Galileo’s Le mecaniche 79 principle was also the focus for the second-class lever and the block and tackle. In the ﬁrst case, referring to Fig. 3, Galileo writes: And by moving the weight, with the lever used in this manner, it is gathered in this also, as well as in the other tools, that what is gained in force is lost in speed. In fact, when the force C raises the lever and transfers it to AI, the weight crosses the space BH, which is as much lesser than the space CI passed by the force, as the distance AB is lesser than the distance AC; that is, as the force is less than the weight. [Galilei 2002, 61, ll, 600–606]

Figure 3.

In the second case, in determining the behavior of a block and tackle with four pulleys (Fig. 4), he says: And we will likewise note, that to make the weight ascend, the four ropes BL, EH, DI, and AG ought to pass, whereupon the mover will be to begin, as much as those four ropes are long; and yet, nevertheless, the weight shall move but only as much as the length of one of them. So that we say this by way of advertisement, and for conﬁrmation of what has been many times spoken, namely, that look with what proportion the labor of the mover is diminished, the length of the way, on the contrary is increased with the same proportion [Galilei 2002, 65, ll, 756–765].

4 The Deﬁnition of moment

It should be noted that in the case of the inclined plane, and consequently of the screw, Galileo also shows that the compensation principle is valid. But in order to properly understand the role this principle plays in Galileo’s theory of the inclined plane, it is necessary to first introduce the concept of moment. In the chapter entitled Diffinizioni [Definitions], Galileo defines gravity, moment and center of gravity. Of the first, he writes: Therefore, we call gravity the propensity of moving naturally downwards, which in solid bodies is caused by the greater or less 80 Romano Gatto

Figure 4.

quantity of matter, whereof they are constituted. [Galilei 2002, 48, ll, 139–143]

As concerns the second, he writes:

Moment is the propensity of descending, caused not so much by the gravity of the moveable, as by the disposure which diverse grave bodies have in relation to one another; by means of which moment, we oft see a body less grave counterpoise another of greater gravity: as in the steelyard, a great weight is raised by a very small counterpoise, not through excess of gravity, but through the remoteness from the point whereby the beam is upheld, which conjoined to the gravity of the lesser weight adds thereunto moment, and impetus of descending, wherewith the moment of the other greater gravity may be exceeded. Moment then is that impetus of descending compounded of gravity, position, and the like, whereby that propensity may be occasioned. [Galilei 2002, 48–49, ll, 143–145]

Therefore, Galileo deﬁnes both gravity and moment similarly, that is, as a “propensity” to descend. The ﬁrst is a natural intrinsic propensity of the bodies due to the “quantity of matter whereof they are constituted.” The second is an extrinsic propensity due to the gravity of the moving body and to “It Is Impossible to Deceive Nature”. Galileo’s Le mecaniche 81

“the disposure which diverse grave bodies have in relation to one another”. It is by virtue of this “disposure” that a body with less gravity is able to balance one of greater gravity. The example of the steelyard, in which a small weight is able to counterbalance a greater one, is very appropriate. It clearly shows that this effect occurs “not because of excess of gravity, but of the distance from the point whereby the steelyard is supported”. Therefore, the moment is a quantity composed of gravity and distance joined together which particularly comes into play in the use of machines. In fact, “the disposure which diverse grave bodies have in relation to one another” is simply the distance between the acting power and the center of rotation of a balance, such as that of a shaft into a wheel or the fulcrum of a lever, etc. This is the definition of the static moment that we analytically represent as force × distance.3 But, in concluding the definition of moment, Galileo affirms that this quantity can be composed, not only by gravity and position, but also by other elements which may cause the propensity to descend. This statement is particularly remarkable because it provides insight into the possibility of some different understandings of the definition of moment. Everything becomes clear in the chapter “Alcuni avertimenti circa le cose dette”, in which he shows

how the velocity of the motion is able to increase moment in the moveable, according to that same proportion by which the said velocity of the motion is augmented. [Galilei 2002, 53, ll, 332–335]

He considers the lever AB divided in unequal parts by the fulcrum C and two weights at the points A and B such that the lever is in balance (Fig. 5).

Figure 5.

He has already explained that equilibrium occurs when the ratio between the weights equals the inverse ratio of the respective distances. At this point, he ﬁrst points out that a very small moment of gravity added to one of the two extremities is enough to break the equilibrium, and then he considers how the lever moves to arrive at the new position DCE. We will ﬁrst examine this

3. Really, Galileo never expressed moment with this formulation, as he did not define in mathematical terms any derived physical quantity. 82 Romano Gatto last question, then return to consider the first crucial question a little later. In considering the motion that the weight B makes when descending into E, and that the weight A contemporary makes in ascending into D, Galileo remarks: we shall without doubt find the space BE to be so much greater than the space AD, as the distance BC is greater than CA. [Galilei 2002, 53, ll, 313–314] In fact, given DCAˆ = ECBˆ , the ratio of the arcs BE and AD will be equal to that of the semi-diameters BC and CA of the circumferences to which, respectively, they belong. So that the velocity of motion of the descending grave B comes to be so much superior to the velocity of the other ascending mobile A, as the gravity of this exceeds the gravity of that; and, as it is not possible that the weight A should be raised to D, although slowly, unless the other weight B moves to E swiftly, it will not be a surprise, or averse to the order of nature, that the velocity of the motion of the grave B should compensate for the greater resistance of the weight A, so long as it slowly moves to D, and the other one swiftly descends to E. And so on the contrary, the weight A being placed at the point D, and the other B at the point E, it will not be unreasonable that the first one slowly falling to A, should be able quickly to raise the other to B, recovering by its gravity what it had lost by the slowness of motion. And by this discourse we may come to know how the velocity of motion is able to increase moment in the mobile, according to that same proportion by which the said velocity of motion is augmented. [Galilei 2002, 53, ll, 317–335] Here Galileo gives a new definition of moment in which the quantities at play are gravity (by which he means force) and, instead of distance, the virtual velocity of the motion that gravity generates. It is a dynamic definition, which he explicitly came back to some years later, in the second edition of the Discorso intorno alle cose che stanno in su l’acqua o che in quella si muovono, published in 1612, in which he specifies: In the opinion of the mechanicians, moment is the virtue, the force, the effect, through which the moving moves and the mobile withstands; this virtue depends, not only on the simple gravity, but on the velocity of the motion, on the different obliquities of the spaces on which the motion takes place, because a grave descending in a more oblique space makes impetus greater than in a less inclined. And, in short, whatever the cause of this virtue is, nevertheless it holds name of moment. [Galilei 1612, 6] Together with this new definition of moment, he enunciates the following two principles: “It Is Impossible to Deceive Nature”. Galileo’s Le mecaniche 83

1. Weights absolutely equal, moved with equal velocity, operate with equal forces and moments. 2. The velocity of the motion increases the moment and the force of the gravity so that weights absolutely equal, but joined with unequal speeds, have unequal force, moment and virtue, and the fastest one is more powerful according to the proportion of its velocity to that of the other one [Galilei 1612, 6–7]. Here he has replaced distance with velocity considering that, in relation to the deﬁnition of moment, these two quantities are equivalent. What legitimates this substitution? In order to show the perfect equivalence of this dynamic deﬁnition with the static one in Le mecaniche, Galileo gives the following example of a balance with equal arms:

Two weights of absolute equal gravity, placed in a balance with equal arms, are balanced, and no one of them inclines lifting the other one, because the equality of the distances of both from the center, on which the balance is sustained and around which it moves, makes sure that, moving the balance, these weights would cross at the same time equal spaces, that is they would move with the same speed. Therefore there is no reason because one of the two weights has to incline more than the other one, so that they equilibrate and their moments are of the similar and equal virtue. [Galilei 1612, 6–7]

This is clearly inspired by the principle of the circle by the pseudo-Aristotle. If the diameter of the circle rotates around its center at a certain angle, at the same interval of time, its extremes cross equal arcs, that is, they move with the same speed. This shows that the equality of the moments of equal weights at the extremes of the balance’s arms is assured both by the fact that these arms have the same length, and by the fact that they move with the same speed. The greatest velocity of a given weight increases its moment as the greatest distance does. In a balance ACB with unequal arms, AC < CB, if B rotates following the arc BE, A will cross the arc AD at the same time. Galileo very easily demonstrates that the ratio of the previously mentioned arcs is equal to that of the distances CB and AC. Since these arcs are crossed at the same time, it is clear that the speed of B is much greater than that of A, as the gravity of B exceeds the gravity of A. Now, given BE/AD = BC/AC, in considering the moment, it is possible to substitute the ratio of velocities to that of the distances. At the same time, BE/AD represents the ratio of the speeds, justifying Galileo’s conclusion: By this discourse we may come to know how the speed of the motion is able to increase moment in the moveable, according to that same proportion by which the said speed of the motion is increased. [Galilei 2002, 53, ll, 332–335] 84 Romano Gatto 5 An “audacious assertion”. The angular lever

At this point, we are able to come back to the crucial undecided matter concerning the breaking of a lever’s equilibrium. The question arises in the chapter “Alcuni avertimenti circa le cose dette”, where, with regard to the balance ABC in equilibrium (Fig. 5), Galileo writes:

It is already manifest, that the one [weight] will counterbalance the other, and consequently, that if a very small moment of gravity were added to one of the weights, it would move downwards, raising the other. So that adding an insensible weight to the weight B, the balance would move and descend from the point B towards E, and the other extremity A would ascend into D. And as in order to make the weight B come down every small gravity added to it is enough, therefore if we don’t keep any account of this insensible moment, we will make no diﬀerence between one weight sustaining, and one weight moving another. [Galilei 2002, 53, ll, 299–310]

Galileo expresses the idea that, once the amount of force necessary for the equilibrium of a lever is established,4 in order to move or to lift the resistance it is enough to increase the power by any amount, no matter how small, which, in current mathematical terms, we would say, “converges to zero”. It is by the virtue of this idea that we can forego considering the addition of this “insensible moment” and affirm that there is no difference between the power that counterbalances the resistance and the one that moves it. Considering Galileo’s conclusion, which states that “if we do not keep any account of this insensible moment, we will make no difference between one weight sustaining and one weight moving another”, Marshall Clagett speaks of an “audacious assertion” [Clagett 1972, 179]. He is of the opinion that this idea was also implicit in the Μηχανικά Προβλήματα by the pseudo-Aristotle and in the Mechanics by Heron, in which “the same proportionality of the weights and of the distances (or of the times)” is applied either in the case of a balance in equilibrium, or of a machine in motion. However, Clagett maintains that the benefit of making it explicit for the first time decidedly has to be credited to Galileo. My opinion is that Heron, in his Mechanics, expressed the exact same concept just as clearly, namely, that in order to move a given resistance by means of machine, it is first necessary to determine the force counterbalancing it, and then to increase this force by a small amount. After having shown that, in order to keep a fixed pulley in equilibrium, a power equal to the resistance is needed, Heron adds:

4. Really it is implicit that this fact is valid for every tool. “It Is Impossible to Deceive Nature”. Galileo’s Le mecaniche 85

If however a small amount is added to the weight, then the other weight is pulled upwards. If therefore the force moving the weight is greater than it, then this force will be strong enough for it and will move it except if any friction occurs in the turning of the pulley or a stiﬀness of the rope, so that they would cause a hindrance to the motion. [Ferriello, Gatto et al. 2016] It is the same “insensible moment” of which Galileo speaks. The real diﬀerence between Heron and Galileo is the idea of limit the latter expresses. Another remarkable question Galileo examines in the aforementioned chapter “Alcuni avertimenti circa le cose dette”, which concerns the concept of distance in considering the moment of the force acting on the angular lever. He explains that we must assume as distance the segment perpendicular to the direction of the force. He points out that in the balance ACB (Fig. 6), the weights G and

Figure 6.

H “make their impulse and would descend, in case they were freely moved, describing the lines AG and BH”. Thus, if the arm CB rotates to assume the position CD, then the weight hanging on point D “will make its moment and impetus according to the line DF ”, given that F is the point at which CB meets the perpendicular line to it from D. Therefore, in considering the moment of the gravity on D, we must assume as distance the segment CF that is the orthogonal projection of CD on CB. Now given CF < CB, the moment of the gravity at D is smaller than at B.

6 The compensation principle in the inclined plane

At this point we are able to consider the question of the compensation principle in the inclined plane. But before getting to the heart of the matter, we can aﬃrm that the treatment of the inclined plane is one of the most relevant subjects in Le mecaniche. In this treatise, for the ﬁrst time in the history of mechanics, Galileo enunciated the exact formulation of the law which connects power to resistance in relation to the dimensions (height and length) of the inclined plane, namely, that 86 Romano Gatto

the ratio between the weight and the force is the same of that between the length of the elevated plan to the perpendicular height. [Galilei 2002, 11, ll, 191–192] In the long version, this law is demonstrated in a very rigorous manner. In the short version, it is introduced and enunciated more simply as “from the light of nature and for experience”, which teaches that the lower the elevation of the inclined plane, the smaller the force necessary to raise a given weight. Starting from these considerations, in this version Galileo already comes to the remarkable conclusion that if we had a plane without any inclination, the heavy bodies put on it would not move in themselves, but it is true that every very small force would be enough to move them from the place. [Galilei 2002, 10, ll, 163–167]5 After having introduced and demonstrated the law of the inclined plane, Galileo then focuses on the screw, that is, an inclined plane wound around its height. He questions whether or not, for this particular tool, the compensation principle occurs. In fact, he says: It seems, that in this case the force is multiplied without the movers moving a longer way than the moveable. [Galilei 2002, 73, ll,1079–1081] Galileo is referring to the inclined plane ABC on which the movable weight E, joined by means of a rope EDF to another smaller weight F , is placed (Fig. 7). The equilibrium law of this device says that

E : F = BC : CA

Figure 7.

The weight F , falling along the vertical CB, causes the weight E to ascend along the inclined plane AC. Since the two weights are connected to one

5. Really, already in De motu, correcting Pappus, who had aﬃrmed that in order to move a body on the horizontal plane a force equal to its weight is needed, Galileo had shown in an admirable way that “Quocumque mobile super planum horizonti aequidistans a minima vi movetur, imo et a vi minori quam quaevis alia vis [...]” [Galilei 1890-1909, I, 209]. “It Is Impossible to Deceive Nature”. Galileo’s Le mecaniche 87 another, the distance covered by E necessarily has to equal that covered by F . But Galileo observes that, for the equilibrium, we must consider not the lengths of the distances covered, but the lengths of their projections on the vertical axle, namely, as we would say today, the vertical component of the aforementioned movements. Now, while the weight F moves vertically in the direction of CB, the weight E moves obliquely along the plane AC. So, when E has covered the whole distance AC, it has crossed a vertical space equal to CB, and a horizontal space equal to AB, which is as if, in order to move from A to C, the weight would have crossed the horizontal space AB and then the vertical space CB. But since a force however small is enough to move a body on the horizontal plane, with regard to the oblique motion of the weight, we must consider only the vertical movement. Consequently, in order to move the weight on the inclined plane AC only the fraction BC/AC of the weight itself is needed. Therefore, the greatest distance AC requires a force smaller than that the shorter distance BC. Galileo’s conclusion is:

It is therefore very interesting to consider by what lines the motions are made, especially in exanimate grave bodies, the moments of which have their total vigor and entire resistance in the line perpendicular to the horizon; and in the other transversally elevated and inclined they feel the more or less vigor, impetus, or resistance, the more or less those inclinations approach unto the perpendicular inclination. [Galilei 2002, 74, ll, 1101–1107]

The reference to Jordanus’ De ratione ponderis [Jordanus Nemorarius 1952] is clear. Just in analyzing this aspect of Galileo’s theory of the inclined plane, Maurice Clavelin states that:

While in De ratione ponderis the virtual moves occur only to provide the indirect demonstrations of certain peculiar problems, with Galileo they become an occasion to formulate a general principle to apply to all the simple machines. But it is also indisputable that the general principle so enunciated for the ﬁrst time enveloped the possibility of a new exposition of the statics. [Clavelin 1968, 168–169]

There is no doubt that the compensation principle, formulated in terms of virtual movements, is a fundamental principle of the Galilean theory of machines that indicates the possibility for constructing a new science of the equilibrium, namely the statics. Galileo appears to be aware of the innovative signiﬁcance of this principle. He turned to it in the already mentioned Discorsi to explain the phenomenon of the hydrostatic equilibrium in communicating vases. If, due to pressure, the water’s level CD goes down to QO (Fig. 8), the water’s column CL must necessarily rise up to the level AB, and the ascent LB will be greater than the descent GQ, insofar as the width of the vase GD is greater than the width of the pipe LC. So Galileo wonders: 88 Romano Gatto

Figure 8. Source: [Galilei 1980]

Being the moment of the velocity of a mobile compensate for that of the gravity of another one, which wonder will it be if the fast ascent of the little water CL withstands the slow descent of the great water GD? [Galilei 1612, 16] He explains that the same effect occurs for a balance in which one arm is 100 times longer than the other one. He reasserts once more that, by means of this tool, a weight P of 2 pounds is able to counterbalance the weight P ’ of 200 pounds only because, in the same time, P is obliged to cross a space 100 times greater than P ’. It is interesting to remark that Galileo says that the example shown above will be useful to get out of error some practical mechanicians, who embark on some impossible enterprises basing on a false foundation. [Galilei 1612, 16] This does not differ at all from what he says in the introduction of the long version, where, after having affirmed to have seen “the generality of Mechanicians deceive themselves in going about to apply Machines to many operations of their own nature impossible”, and after having explained the real nature of the machine and the laws to which they are subject, he concludes: These then are the benefits that may be derived from mechanical instruments, and not those which ignorant engineers dream of, with the deception of so many principles, and with their own shame, while they undertake impossible enterprises. [Galilei 2002, 48, ll, 123–126]

7 Conclusion

Galileo was truly the first author in the modern age who introduced the compensation principle [Gatto 2015] in a clear and unequivocal way, and who “It Is Impossible to Deceive Nature”. Galileo’s Le mecaniche 89 used virtual movements to establish a new scientific base for the science of weights. Yet he was not the very first in the history of mechanics to do so. In ancient times, another scientist introduced this principle into the theory of machines. We are speaking of Heron of Alexandria who, in his Mechanics, at the end of the treatment of the Baroulkos, writes: This tool and those of great power similar to it are slow, because the smaller the moving force is relating to the weight to be moved, the longer is the time the work needs. The force to the force and the time to the time are in the same inverse ratio. [Ferriello, Gatto et al. 2016, 113–114] We immediately realize that this statement is similar to Galileo’s. We likewise recognize that, as we have seen in Le mecaniche, Heron also reasserts this principle at the end of the discussion on the train of blocks and tackles and that of the levers, which are the other compound tools he considers.6 The strong similarity in considering and developing this aspect of the problem, the idea that the addition of an amount of force, however small, to the power which makes the balance is enough to break the equilibrium of a lever, and some other questions we cannot discuss here, would lead us to think that Galileo could have known the Mechanics by Heron7. Clagett excludes this possibility, sharing the common opinion that this work by Heron was unknown at the end of the sixteenth century. But Clagett himself, describing the dynamic approach in Le mecaniche, in which Galileo first uses the virtual movements to confirm the law of the lever and subsequently to derive the law of the inclined affirms that the similarity with Heron’s Mechanics is manifest.8

6. About the train of blocks and tackles, he writes: “It is clear that a delay occurs with this tool because the process takes the same ratio [of the Baroulkos] [...] The ratio of the times equals the [inverse] ratio of the moving forces”. Similarly, in concluding the exposition of the train of levers, he writes: “Here too the delay occurs in the same ratio, because there is no difference between these levers and the shaft into the wheels”. 7. This treatise has reached us, for the first time, in 1893 through an Arabic manuscript of the 9th century, whose text, with its French translation, was published by Camille Carra de Vaux [Carra de Vaux 1983]. 8. Also Roberto Marcolongo, although surprised because of the strong similarity of some theories by Leonardo with the corresponding by Heron, was convinced that it was impossible that Leonardo could have known Heron’s Mechanics. In fact, he writes: “Heron’s considerations on the centres of gravity of the plane figures, those on the theory of the simple machines, and particularly that on the blocks and tackles, are very important in the history of mechanics. But it is obvious that this book persisted to be unknown to the Western scholars, and, therefore, Leonardo couldn’t know it. But I think that we can also assert that, by some ways that we continue to ignore yet to day, some idea by Heron has reached the Western scientists and it is preserved there” [Marcolongo 1937, 138]. It is also interesting to note what Giovanni Vailati wrote in a letter to Ernest Mach (Crema, 3 August 1897): “The way in which this last author [Heron] enunciates and applies the aforesaid principle [the principle of the virtual movements] is such to suggest a period of the development of statics corresponding to that expressed in Galileo’s early works” [Vailati 1971, 115]. 90 Romano Gatto

We have already explored the possibility that Galileo and other Italian authors of the Renaissance were familiar with Heron’s Mechanics [Gatto 2015]. The influence of Heron’s work on Galileo and its power to inspire him cannot be underestimated. The principle of virtual movements is the key difference between Heron and Galileo’s approaches. Heron applied it to the lever, the block and tackle, and later to the shaft into the wheel, which are the machines he limited to the Archimedean principle of the balance. In so doing, he provided the evidence that confirmed the relationship between the efficiency of a machine and its related delay. He never applied this principle to the inclined plane, which he dealt with another way without making any reference to the above-mentioned simple machines. Galileo, beginning with the Archimedean law of equilibrium, followed by the concept of the moment, was able to conceive of a form of the principle of virtual movements as a general principle for the theory of machines, thereby granting mechanical science a new perspective. This accomplishment is so significant that Le mecaniche should be considered as the first modern treatise on statics.

Bibliography

Aristotele [2000], Problemi meccanici, a cura di Maria Elisabetta Bottecchia Dehò, Catanzaro: Rubettino.

Carra de Vaux, Camille Marie Bernard [1983], Les Mécaniques ou l’élévateur de Héron d’Alexandrie, publiées pour la première fois sur la version arabe de Qustâ’ Ibn Lûqâ, et traduit en français par M. Le Baron Carra de Vaux, Journal Asiatique, n.s., 386–472 (1re édition); 152–269 (2e); 420–514 (3e).

Clagett, Marshall [1972], La scienza della meccanica nel Medio Evo, Milano: Feltrinelli.

Clavelin, Maurice [1968], La Philosophie naturelle de Galilée, Paris: Armand Colin.

Favaro, Antonio [1889], Delle Meccaniche lette in Padova dal Signor Galileo Galilei l’anno 1594, Memorie del Reale Istituto Veneto di Scienze, Lettere ed Arti, 26(5), 1–26.

Ferriello, Giuseppina, Gatto, Maurizio, & Gatto, Romano [2016], The “Baroulkos” and the Mechanics by Heron, Firenze: Olschki.

Galilei, Galileo [1612], Discorso intorno alle cose che stanno in su l’acqua o che in quella si muovono, Firenze: Apresso Cosimo Giunti.

—— [1890-1909], Le opere di Galileo Galilei. Edizione Nazionale sotto gli auspice di Sua Maestà il Re d’Italia. Esposizione e disegno di Antonio Favaro, Firenze: Tipograﬁa di G. Barbèra. “It Is Impossible to Deceive Nature”. Galileo’s Le mecaniche 91

—— [1980], Discorso intorno alle cose che stanno in su l’acqua o che in quella si muovono, in: Opere di Galileo Galilei, edited by F. Brunetti, Torino: UTET, vol. I, 419–517.

—— [2002], Le mecaniche, Firenze: Olschki, Edizione critica e saggio introduttivo di Romano Gatto.

Gatto, Romano [2015], La meccanica di Erone nel Rinascimento, in: Scienza e rappresentazione, edited by P. Caye, R. Nanni, & Napolitani P.D., Firenze: Olschki, 151–172.

Jordanus Nemorarius [1952], Liber Jordani de Nemore de ratione ponderis, in: The Medieval Science of Weights, edited by E. A. Moody, M. Clagett, Madison: The University of Wisconsin Press, 167–227.

Marcolongo, Roberto [1937], Memorie sulla geometria e la meccanica di Leonardo da Vinci, Napoli: S.I.E.M.

Vailati, Giovanni [1971], Epistolario 1891-1909, Torino: Einaudi.

Viviani, Vincenzo [1717], Racconto istorico della vita del signor Galileo Galilei, in: Fasti consolari dell’Accademia Fiorentina di Salvino Salvini consolo della medesima e Rettore generale dello Studio di Firenze, Firenze: Stamperia di S.A.R. per Gio. Gaetano Tartini e Santi Franchi, 393–431.

—— [2001], Vita di Galileo, Roma: Salerno Editrice.

A Galilean Answer to the Needham Question

Gennady Gorelik Boston University (USA)

Résumé : Pour pouvoir la résoudre, nous généralisons la question posée par Needham : Qu’est-ce qui a empêché la science gréco-romaine et médiévale de franchir la prochaine grande étape après Archimède, et qu’est-ce qui a empêché les savants orientaux de contribuer à la physique moderne des siècles encore après Galilée ? Pour répondre à cette question, on propose comme distinction- clef entre physique moderne et science néo-galiléenne le droit d’inventer des concepts fondamentaux « illogiques », vériﬁables par expérimentation. Ce droit se fonde sur la croyance en un univers régi par des lois fondamentales cachées, mais accessibles à la connaissance humaine. Cette croyance prend sa source dans la vision biblique du monde, qui devint le socle des cultures européennes à l’époque de la révolution scientiﬁque à l’occasion de l’invention de l’imprimerie et de la Réforme.

Abstract: To make the Needham question answerable it is extended thus— What hindered Greco-Roman and Medieval science from making the next major step after Archimedes, and what hindered Eastern scientists from contributing to modern physics for centuries after Galileo? To answer this question the key distinction between modern physics and pre-Galilean science is suggested: the right to invent “illogical” fundamental concepts which can be veriﬁed by experiments. This right is based on the belief that the Universe is governed by hidden fundamental laws which Man is capable of knowing about. The source of this belief was the biblical worldview which became the basis for European cultures by the time of the Scientiﬁc Revolution thanks to book printing and the Reformation.

Philosophia Scientiæ, 21(1), 2017, 93–110. 94 Gennady Gorelik 1 Introduction

The strongest question on Galileo’s role in history was put by the British biochemist and sinologist Joseph Needham:

Why did modern science, the mathematization of hypotheses about Nature, with all its implications for advanced technology, take its meteoric rise only in the West at the time of Galileo? Why modern science had not developed in Chinese civilization [which in the previous centuries] was much more eﬃcient than occidental in applying human natural knowledge to practical human needs? [Needham 1969, 16, 190]

Evidently he had in mind physics, since in his view

[...] the birth of the experimental-mathematical method, which appeared in almost perfect form in Galileo, [...] led to all the developments of modern science and technology. [Needham 1959, 156]

So, Needham’s “Grand Question” is to be coupled with the question: What was the actual innovation of Galileo, that had changed science so much, accelerated its progress a hundredfold, though only in the West? Hence, hereafter the term “modern science” means “modern physics”.

2 An extended Needham question

By the time when Needham came to his heuristic question, the birth of modern science had already been named “the Scientific Revolution” and explained in a few ways: by needs of capitalist economy, by Protestant ideology, by “mathematization of nature”, by contacts between scholars and craftsmen facilitated by the capitalist economy, etc. [Cohen 1994], but none of those explanations satisfied Needham [Needham 2000]. Indeed, all the achievements of the new physics had no economic value in the 17th century. All the greatest “revolutionaries”—Copernicus, Galileo, Kepler, and Newton—used both empirical and mathematical tools. Only two of them were Protestants. And in China contacts between scholars and craftsmen did not result in modern physics. While the discussion about the Scientific Revolution continues with no consensus in sight [Cohen 2010], [Huff 2011], the Needham question was sometimes dismissed as a counterfactual question about a unique event [Sivin 1982]. However, Needham didn’t ask why modern physics emerged in Italy rather than in England, and he would hardly have been so puzzled if Eastern scientists had contributed into modern physics in the 19th century. Anyway, A Galilean Answer to the Needham Question 95 the Needham question is debated in China and in the West [Dun 2000], [Ducheyne 2008]. To make this question historically answerable, I will extend it in cultural time and space. Indeed, refuting Aristotelian physics, Galileo relied on “superhuman”, “the most divine” Archimedes [Galilei 1590]. The new Galilean science was eagerly accepted in France, Holland, England, and even in backward Russia, but failed to reach beyond Europe for centuries, although medieval Europeans used to assimilate important Eastern innovations like Hindu-Arabic numerals. Arabic science adopted Greek science much earlier than the Europeans, advanced optics and astronomy, but did not contribute into post-Galilean science. The real question is not why modern science emerged in the West at the time of Galileo but why it took so long since the time of Archimedes and why after the birth of modern science it was not adopted in Eastern civilizations for so long. So, an extended Needham question is:

What hindered Greco-Roman and Medieval scientists from making the next major step after Archimedes, and why didn’t Easterners contribute into modern science for centuries after Galileo?

To answer this question we are to find commonality among the cultures where the new (modern) science took roots and fructified, and to explain the timing of the Scientific Revolution and the social forces, which brought it about. We are to look not for a single cause, but for the decisive one. Some contributory causes—a system of higher education and Greco-Roman intellectual tradition—were present in Islamic civilization but it did not adopt the science of Galileo. And European universities had been around for four centuries before modern science was born.

3 Modern physics as a fundamental science

First of all, what is the key distinction between modern physics and pre- Galilean science? The scientiﬁc tools of experiment and mathematics are vital but not uniquely modern, since back in the 13th century Roger Bacon stated that “without experiment it is impossible to know anything thoroughly” and “no science can be known without mathematics” [Bacon 1268]. In fact, both of the tools were used by Archimedes, who was not only the ﬁrst real physicist but also a great engineer and mathematician. Galileo’s experiments and mathematics didn’t go beyond what was feasible for Archimedes. Of course measuring experiment and mathematics are indispensible tools to verify or disprove a theory expressed in quantitative language. But in modern physics no less important is the third tool, described by Einstein as “the boldest speculation [to] bridge the gaps between the empirical data” 96 Gennady Gorelik

[Einstein 1953]. Such a speculation results from inventive imagination rather than from mathematical or empirical inferences [Cohen 1995]. The real novelty of modern physics can be seen in the scheme depicted by Einstein in his letter of 1952 [Einstein 1993, 137] (cf. Fig. 1).

Figure 1. Einstein’s explanation how modern physics works

In Fig. 1, axioms A—the fundamental concepts of theory—are invented by intuition taking off from the ground of experience E: the fundamentals are “free inventions of the human spirit (not logically derivable from what is empirically given)” [Einstein 1949]. Then some statements Sn derived from A are to be verified by landing in the E. And if the landing is soft, the theory is endorsed. There is the key difference between Galileo’s science and Archimedes’ one and the principal similarity between Galileo’s and Einstein’s. In Archimedean physics all the notions are visible and tangible (weight, density, geometrical form), whereas in modern physics fundamentals do not have to be evident, and their validation is a result of the whole scientific enterprise joining theory and experiment. Einstein emphasized, that “concepts can never be derived logically from experience [...]. Unless one sins against logic, one generally gets nowhere” [Einstein 1993, 147], apparently meaning “against the logic of previous theory or common sense”, since there is no other logic when a theorist’s inventive intuition is just taking off. The first “illogical” fundamental notion invented by Galileo was “vacuum”, or rather “motion in vacuum”. He defied the authority of Aristotle, who, as philosophers believed, had “logically proved” the nonexistence of void, or the vacuum [Galilei 1590, 34]. Galileo introduced “vacuum” as a physical notion, rather than a logical one. In logic, a notion is validated by pure reason, whereas in physics it is validated by reason coupled with experience. To invent a new fundamental notion, a scientist has to believe that: The Universe is governed by profound exact laws which are hidden like the foundation of a building [in Latin, fundamentum], but humans are able to probe into and comprehend these fundamental laws by inventing new concepts to be validated empirically. A Galilean Answer to the Needham Question 97

Such a belief is the prerequisite, or the postulate, of modern— fundamental—science. Human ability to comprehend the working of the Universe was a “miracle” for Einstein although he himself took part in such miracles. All the boldest inventions of modern science were encouraged by fundamental worldview coupled with cognitive optimism. It was the key novelty which let Copernicus initiate the Astronomical Revolution and Galileo invent modern physics [Gorelik 2012]. The boldest idea of Copernicus, which, in his words, “seemed absurd” [Copernicus 1543, 5], was to take a careful look at the planetary motions from the Solar point of view. Kepler’s boldest idea was that all the planetary motions are governed by a fundamental law. For both of them, fundamental cognitive optimism supported laborious mathematical processing of their astronomical data, and they could be named fundamental astro-mathematicians. Galileo became the first fundamental physicist by establishing the method of modern physics: believing in fundamental unity of terrestrial and celestial phenomena, he launched his boldest speculations by taking off from terrestrial physical experiments, invented the concept of “motion in vacuum”, employed mathematical language and landed his speculations in the empirical reality of both terrestrial and celestial phenomena. He never experienced vacuum by his senses, but having compared motions in air and water, he felt free to invent the notion of vacuum as a “medium totally devoid of resistance” and came to the idea that in such a medium “all bodies would fall with the same speed” [Galilei 1914, 72]. It was the notion of invisible vacuum that helped Galileo to discover the law of free fall, the law of inertia, and the principle of relativity. According to Needham, Galileo’s “experimental-mathematical method” included, as a key element, “formulation of a hypothesis involving a mathematical relationship”, and—just in a footnote—Needham mentioned “concepts of the unobserved and the unobservable” [Needham 1959, 156]. However, at the turning points in history of modern science, to formulate a new hypothesis a theorist had to invent a “concept of the unobservable” [Needham 1959, 156]. To formulate a rational hypothesis and to invent a somewhat “irrational” fundamental concept are quite different acts. The next fundamental concepts invented in Galilean way were universal gravity, electromagnetic field, quanta of energy, photons, curved space-time, etc. Introducing a new fundamental notion a theorist usually has to dismiss some of the old ones, and it could be no easier than to accept the new notions. It was the Galilean way of making physics that became the main engine to propel the whole of science. 98 Gennady Gorelik 4 The source of fundamental cognitive optimism

Reflecting on making science, Einstein remarked: “one cannot build a house or construct a bridge without using a scaffolding which is really not one of its basic parts” [Einstein 1993, 147]. What kind of scaffolding did the first constructors of modern science use? Copernicus began his heliocentric thinking being

[...] annoyed that the movements of the world machine, created for our sake by the best and most systematic Artisan of all, were not understood with greater certainty by the philosophers”. [Copernicus 1543, 4]

A half-century later, when Kepler was thinking about the same machine,

[...] the very existence of general lawfulness of natural processes was not assured at all. How great must his faith in such lawfulness have been to give him the strength to devote decades of patient hard work to the empirical investigation of planetary motion and to formulate its mathematical laws! [Einstein 1930b]

To Kepler’s mind

[...] astronomers are priests of the highest God with respect to the Book of Nature, [they] do not promote the praise of the intellect but above all behold the glory of the Creator. He who is convinced of this does not easily bring to light anything other than what he himself believes, nor does he abruptly alter anything in [astronomical] hypotheses unless he hopes that from them the phenomena can be demonstrated with greater certainty. [Boner 2013, 40]

All the originators of modern science shared the faith in fundamental lawfulness and intelligibility of Nature, and it was this faith, as cognitive optimism, that encouraged their research and resulted in brand new scientiﬁc knowledge. What was that encouragement? An inkling of the answer was found by Edgar Zilsel, who traced the usage of the phrase “physical law” and discovered that it emerged in the 17th century within biblical worldview as a transformation of the idea of the Nature governed by God’s laws. Earlier, the notion “law” had only juridical and theological meanings [Zilsel 1942]. Galileo did not use the phrase “physical law” in his books. Instead, he used (Italian) words ragione [reason, ratio, proportion] or principio [principle]. The transformation began in Galileo’s theological letters of 1613-1615, and here is a summary of his views: A Galilean Answer to the Needham Question 99

The Scripture and Nature both derive from God, the Scripture as His dictation, the Nature as the obedient executrix of His orders. The purpose of the Scripture is to persuade humans of those propositions which are necessary for service of God and salvation. To adapt to the understanding of unlearned people, the Scripture speaks many things which diﬀer from the bare meaning of words, and it would be blasphemy to accept them literally by attributing to God human feelings like anger, regret, or forgetfulness. Nature, on the other hand, never transgresses the laws imposed upon her, or cares a whit whether her recondite reasons and ways of operating are understandable to men. God has endowed us with senses, language, and intellect not to bypass their use and give us by other means the knowledge we can obtain with them. Therefore, whatever sensory experience and necessary demonstrations prove to us concerning natural phenomena, it should not be questioned on account of Scripture’s words which appear to have a diﬀerent meaning. This is especially so for those sciences about which we can read only very few words in the Scripture which does not contain even the names of all the planets, and so it was not written to teach us astronomy. [Galilei 1613-1615]

In short, there are unbreakable laws of the abstruse reasons in Nature, and humans are able to comprehend them. This is actually the postulate of the fundamental science (formulated in chap. 2). By the end of the 17th century Galileo’s “laws imposed upon Nature by God” transformed into seemingly secular “laws of Nature”, due to inﬂuential writings of Descartes and Newton. Since then the term was used by both believers and nonbelievers, and by the 20th century its biblical origin had been forgotten. For a Marxist atheist, Zilsel, “the law-metaphor originates in the Bible”, but for the theistic originators of modern science most ways to talk about God were metaphorical. The origin of the expression “physical law” reveals the role of biblical worldview in their mentality and hints at the connections between the postulate of fundamental science and the basic biblical ideas/images of God who created the lawful Universe for the humankind, and of humans made as His likeness with the purpose to rule over all the Earth. Such supernatural wording sounded quite natural for the originators of modern science, all of whom were true believers: Copernicus was a cleric, Galileo and Kepler in their adolescence intended to become clerics, and Newton wrote about the Bible more than about physics. All of them were profound biblical theists, thought in religion as freely and boldly as in science, and felt free to interpret the Bible by themselves. Basing their religious thinking on their understanding of the Bible, they came to be at odds with Church canons: 100 Gennady Gorelik

Galileo could not accept the Pope’s opinions as ﬁnal truths; Newton could not accept the doctrine of the Trinity. The faith in the Creator-Lawgiver and in humanity as His purpose, together with experience of faith as “the conviction of things not seen”, encouraged the originators’ “boldest speculations” to invent new invisible fundamentals to gain the knowledge about “all the Earth” to be able to rule over it. The same faith was expressed by the third great inventor of invisible fundamental (electromagnetic ﬁeld), Maxwell, who wrote to his friend:

[...] Christianity—that is, the religion of the Bible—is the only scheme or form of belief which [makes an explorer free indeed]. You may search the Scriptures and not ﬁnd a text to stop you in your explorations.1 [Campbell & Garnett 1884, 96]

The fourth fundamental inventor was an openly religious person [Planck 1950], and according to the ﬁfth one:

Our moral leanings and tastes, our sense of beauty and religious instincts, are all tributary forces in helping the reasoning faculty toward its highest achievements. [Einstein 1930a, 375]

Of course, there were atheists back in the time of Archimedes, as well as in the 17th century (an atheist, astronomer E. Halley, was a colleague and friend of Newton), but there were no atheists among the originators of modern science.

5 Modern science in the biblical civilization

Returning to the extended Needham question, we can see the commonality between the countries where Galilean science did take roots, or, rather, the commonality between these countries’ sociocultural minorities from which the future scientists emerged. All the would-be scientists had to be readers. By the 17th century, the most widely read book in Europe was the Bible, as a result of Gutenberg’s invention, Reformation, and the print explosion. Since then the Bible became the most inﬂuential text for European cultures as diﬀerent as Italian and Scandinavian, British and Russian. It was the main common factor uniting all these cultures into modern European civilization, which therefore could be called the biblical one.

1. A prayer found in Maxwell’s papers reads: “Almighty God, who hast created man in Thine own image, and made him a living soul that he might seek after Thee and have dominion over Thy creatures, teach us to study the works of Thy hands that we may subdue the Earth to our use, and strengthen our reason for Thy service [...]” [Campbell & Garnett 1884, 160]. A Galilean Answer to the Needham Question 101

There were a few correlations between the history of modern science and the sociocultural role of the Bible. The first correlation between religious and scientific postulates was revealed back in the late 16th century by missionaries who brought to China both the Bible and European science. The Chinese emperor welcomed missionaries, but they failed to implant the European science into Chinese soil. One of the later missionaries explained: The Chinese atheists are not more tractable with relation to Providence, than with regard to the Creation. When we teach them that God, who created the universe out of nothing, governs it by general Laws, worthy of his infinite Wisdom, and to which all creatures conform with a wonderful regularity, they say, that these are high-sounding words to which they can affix no idea, and which do not at all enlighten their understanding. As for what we call laws, answer they, we comprehend an Order established by a Legislator, who has the power to enjoin them, to creatures capable of executing these laws, and consequently capable of knowing and understanding them. If you say that God has established Laws, to be executed by Beings capable of knowing them, it follows that animals, plants, and in general all bodies which act conformable to these Universal Laws, have a knowledge of them, and consequently that they are endowed with understanding, which is absurd. [Needham 1969, 308] It was absurd for those who did not believe in the biblical Creator-Lawgiver. The most alien for non-biblical cultures is the notion of Man made as God’s likeness to rule over all the Earth. Islam, being the closest to biblical tradition historically and geographically, rejects the biblical status of Man, since the Quran states: “Nothing is as God’s likeness” [Michot 2005]. On the other hand, in mainstream Islam, after Al-Ghazali’s “renewal of the faith” in the 12th century, the very idea of unbreakable fundamental laws of Nature was considered incompatible with the omnipotence of God, and the decline of the Islamic Golden Age of science followed [Hoodbhoy 1991, 105]. In the time of the quote about the “Chinese atheists” (1737), a young Russian, Mikhail Lomonosov, a fisherman’s son, after having graduated from the Slavic-Greek-Latin Academy in Moscow, was getting education in Germany. Then he returned to Russia and became the first prominent native Russian scientist. On his path to science he overcame many barriers, but among them there were no Chinese or Islamic ones. Like Galileo and Newton, Lomonosov was a biblical theist and happened to be at odds with clerical officialdom. He greatly contributed into higher education in Russia and European ways to do science. A result was the first Russian world-class achievements like Lobachevskian geometry and Mendeleev’s periodic table. Thus, modern science took roots in Russia with no native scientific tradition, but failed in China, India and the Islamic world, whose innovations in science and technology Europe assimilated up until the 16th century. 102 Gennady Gorelik

To explain the European birthplace of modern science, the factor of Christian culture was employed more than once. However, by the time of the Scientific Revolution Christianity had been around for sixteen centuries. The medieval Church taught mainly about the corrupted state of humanity after the Fall of Man, rather than about human dignity and freedom endowed by God. The key new factor of modernity emerged in the 16th century due to book printing and Reformation, when the Bible became socially much more accessible and a very powerful guide. This guide made clear that human fallibility was an element of divinely endowed freedom of Man. So, this social factor correlated with post-Gutenberg time and European space of the Scientific Revolution. Another correlation manifested in the shifting of leading role in modern science from scientists of Catholic background to those of Protestant one. This shift, discovered by A. de Candolle in 1870s and emphasized by R. Merton in the 1930s [Cohen 1990], could be explained by quite different roles of the Bible in cultures of the two denominations, rather than by theological differences. The principle “Sola Scriptura” made the reading of the Bible the central factor in Protestant culture, whereas Catholic Church discouraged the laity to read the Bible on their own. In the 20th century, de Candolle’s disparity is supported by the statistics of Protestant vs. Catholic backgrounds of Nobelists, about 8:1 per capita. Some pre-Guttenberg clerics who did read the Bible, such as Robert Grosseteste and Roger Bacon, manifested that biblical theology was quite compatible with a genuine interest in natural sciences, though it was too much for a cleric to concentrate on physics. Any belief is a preconception, but it can be more or less influential, helpful or harmful. The biblical preconceptions, inculcated by accessible vernacular translations of the Bible, proved to be very helpful for exploring Nature (as well as for advancing technology and economy). The key factor was the basic belief in divine human purpose in the divinely lawful Universe, rather than the array of theological subtleties, different in Christian denominations. Of course, just reading the Bible didn’t make a scientist out of any person, but for religious adolescents amply endowed with pro-scientific abilities—intellectual curiosity, independent insight and persis- tence (like the pioneers of modern science)—the key biblical preconception informed their cognitive optimism. At the same time, laconic biblical stories provoked questions in truth seeking, even if led to question the Church’s canons. That was why medieval churches stood against making vernacular translations accessible to laity. Thus, the biblical answer to the Needham question explains also other factual correlations: the time and space of the Scientific Revolution, theism of its originators, and the disproportion of scientists of different religious backgrounds. A Galilean Answer to the Needham Question 103

There is, however, another important fact—prevailing atheism of scientists in the 20th century [Larson & Witham 1998].

6 Scientiﬁc thinking and religious feeling

In Einstein’s words, “In the temple of science are many mansions, and various indeed are they that dwell therein and the motives that have led them thither” [Einstein 1918]. Different also are the types of problems that attract theorists of different mental styles, such as “birds” and “frogs” [Dyson 2009]. The difference between “intuitive” and “analytical” styles helps to understand the role of theists in originating modern physics and to see a room for atheists’ contributions. The quoted Einstein’s scheme of making modern science includes three phases of scientific enterprise, or three kinds of problems to solve: 1. to invent new fundamentals-axioms (E ⇒ A), 2. to derive from them specific testable statements (A ⇒ S), and 3. to test these statements empirically (S ⇒ E). Only the first phase requires an intuitive “leap of faith”. The other two phases require creative using of the new fundamentals and devising new experiments. So, atheists, like L. Boltzmann, P. Dirac, S. Weinberg, have enough room for creativity. In fact, this room is much bigger than one for the first phase: it takes just one or very few persons to blaze a new trail into unexplored territory, but it takes many people to develop a trail into a highway (of applied science and technology). All the three phases are necessary to accomplish a cycle of establishing a new fundamental theory. To start a new cycle in the expanding spiral of quest in modern science, a new leap of inventive intuition is necessary. However, in the very first cycle, in the 17th century, when inventing new fundamentals had no precedent, the Einsteinian “free inventive spirit” needed unprecedented support, which was provided by scientists’ spiritual/religious faith. According to Harvard psychologists, personal theistic belief correlates with intuitive (vs. analytic) cognitive style rather than with such factors as education level, IQ, and familial religiosity [Shenhav, Rand et al. 2012]. So, the power of a physicist’s intuition could be responsible both for his theism and for his type of creativity. An additional source of creative successes in modern physics was the paradoxical combination of cognitive audacity and personal humility. Exploring Nature’s “recondite reasons”, Galileo found himself in a sea “with vacua and infinities” and questioned his ability to reach dry land. He believed, nevertheless , that it was “possible to arrive at the true and primary causes” of natural phenomena and perceived his work as “merely the beginning, 104 Gennady Gorelik ways and means by which other minds more acute than [his] will explore remote corners” of the “vast and most excellent science” he had just opened up [Galilei 1953, 485], [Galilei 1914, 44, 153–154]. The same humble audacity could be seen in Newton who likened himself to a boy on the seashore, who had found a few smooth pebbles, with the sea of undiscovered truths before him. Such a combination of bold creativity and personal humility in Galileo and Newton stemmed apparently from their belief in biblical connection between Almighty Creator of the whole World and mortal humans made as His likeness to rule over all the Earth. To be able to rule over the world, humankind has to explore the world to understand how it works. It was the occupation of Galileo’s successors, who invented new fundamentals of science. Even in the 20th century the effectiveness of physics seemed miraculous to Einstein and unreasonable to E. Wigner [Einstein 1993, 131], [Wigner 1960]. It was much more unreasonable four centuries ago, when no fundamental law of physics had been discovered. Well known are both Einstein’s credo: “Subtle is the Lord, but malicious He is not” and Bohr’s apparently secular saying that a new fundamental theory which is not crazy enough has no chance of being correct. Both ways to encourage a theorist are based on the belief in the right to invent “crazy” fundamentals to comprehend the lawful Universe. Cognitive optimism complemented by personal humility corresponds to the golden mean between a belief in Full and Final theory of Everything and the disbelief in the fundamental lawfulness of the world. As F. Dyson put it: If it should turn out that the whole of physical reality can be described by a finite set of equations, I would be disappointed. I would feel that the Creator had been uncharacteristically lacking in imagination. [Dyson 1988] Einstein had [...] found no better expression than “religious” for confidence in the rational nature of reality insofar as it is accessible to human reason. Wherever this feeling is absent, science degenerates into uninspired empiricism. [Einstein 1993, 119] If, in the 20th century, eloquent and religiously unaffiliated physicists choose theistic wording to express their cognitive belief, apparently it is the most adequate wording, and the root of this belief was indeed theistic. It is supported by theism of all the founders of modern physics. Their preconception of personal freedom and cognitive optimism of Godlike creative creatures—an idea of biblical descent—transformed into a “self-evident” secular worldview. The secular nature of scientific knowledge was most clearly expressed by a Catholic priest and astrophysicist Georges Lemaître, who discovered the expansion of the Universe and suggested that it began with the explosive birth. Thirty years later (and two years before becoming the President of the A Galilean Answer to the Needham Question 105

Pontifical Academy of Sciences) this astrophysicist in soutane, at a conference on astrophysics, stated that the theory of Big Bang [...] remains entirely outside any metaphysical or religious question. It leaves the materialist free to deny any transcendental Being. [...] For the believer, it removes any attempt at familiarity with God [...]. It is consonant with the wording of Isaiah speaking of the “Hidden God”, hidden even in the beginning of creation [...]. There is no natural limitation to the power of mind. The Universe does not make an exception, it is not outside of its grip. [Lemaître 1958, 7] Whereas the results of scientific quest are indeed metaphysically neutral, the motive vigor of research results from specific metaphysics, or, rather, from pre-physical belief that “there is no natural limitation to the power of mind”, that the Universe is fundamentally lawful, and that free humans are capable of discovering those laws. Lemaître’s religious feeling was as compatible with his scientific thinking as it was for Einstein and Galileo.

7 Secular fruits of religious culture

Compatibility, however, is too weak a word to describe the connection between modern science and biblical metaphysics/prephysics. Dignity of free Man as the purpose of God’s creation was the basis for the manifesto of early modern humanism “On the dignity of Man” (1486) by Pico della Mirandola. He put the following words into the mouth of “God the Father, Supreme Architect of the Universe” after his “last creative act”: We have placed you [Man] at the world’s center so that you may survey everything else in the world. [Pico della Mirandola 1486, 261] This manifesto as well as Pico’s critique of astrology had been published before Copernicus started to think about astronomy. Just like the European arts and humanities owe so much to biblical images, stories and ideas, the most rational and empirical domain of human knowledge—modern physical science—also appears to have biblical roots. Due to the Printing Revolution and the Reformation, the Bible informed the cultural background of the modern Western civilization, including atheists, who relied on their self-evident personal freedom secured by law. In fact, they assimilated their cultural postulates in their cultural upbringing, even if in their adulthood, they believed they needed no metaphysical—or, rather, pre- physical and pre-ethical—support. Those biblical atheists could hardly see the biblical stories as the reason to accept any belief but they could appreciate the Bible as a historical source of humanitarian postulates. These atheists may 106 Gennady Gorelik be the most selﬂess children of the biblical civilization, since they adhere to the central biblical tenet without rational natural substantiation and without irrational hope for supernatural approval. Religion used to be instrumental in changing culture to be later shared by both believers and unbelievers. The biblical view on human freedom developed into the idea of freedom of conscience via separation of Church and state and into the notion of unalienable human rights, which was constitutionally self-evident to the founders of the United States. One of them, Thomas Paine, rejected all the churches as tools to enslave mankind, but, in his book The Rights of Man, to answer the question “What are those rights?” he referred to “the Mosaic account of the creation, whether taken as divine authority or merely historical” [Paine 1791, 49]. This argument, however, could be strong only for those whose upbringing was as biblical as it was for founders of the United States. The history of modern Western view on human rights is similar to the transformation of Galileo’s “laws imposed by God upon Nature” into the secular “laws of nature”. Biblical preconception of unalienable rights of Man resulted in the Universal Declaration of Human Rights (1948) which became a social framework for all cultural traditions compatible with human dignity. Western civilization was open to Eastern cultural innovations since pre- modern time. As to openness of other civilizations, exemplary was the medieval Islamic Golden Age when actual separation of science/philosophy and mosque let adherents of various faiths to freely cooperate. In the 20th century, a few Far-Eastern countries assimilated Western technological and social innovations.

8 Conclusion

In the 21st century two Chinese historians have made an assessment: Compared with the huge system of universities and research institutes and the large number of researchers in contemporary China, the quantity of original scientiﬁc work accomplished is embarrassingly small and asked: “What is responsible for this situation?” [Hao & Cao 2009]. It looks like a sign of intellectual freedom, which is indispensable for advancing science. If China is to catch up with the West in fundamental science without assimilating the idea of unalienable human rights, it will invalidate the suggested answer to the Needham question. And there is a real problem waiting for invention of new fundamental concepts,—the problem of quantum gravity. It remains unsolved a century after Einstein had discovered it, 80 years after Matvei Bronstein (1906-1938) predicted that its solution might require “the rejection of our ordinary concepts of space and time, replacing them by some much deeper and nonevident concepts” [Gorelik 2005], and after thousands of articles on the subject in the last ﬁfty years. A Galilean Answer to the Needham Question 107 Acknowledgments

I am grateful to Robert S. Cohen for many years of enlightening, Lanfranco Belloni for help with Galileo’s Italian, Chia-Hsiung Tze for introduction to the Needham question, Freeman Dyson, Toby E. Huﬀ, Silvan S. Schweber, and Sergey Zelensky for discussions.

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Jean-Marc Lévy-Leblond Université de Nice (France)

Résumé : En 1587, le jeune Galilée est invité à donner Due lezioni all’Ac- cademia Fiorentina circa la figura, sito e grandezza dell’Inferno di Dante (ci- après Leçons sur l’Enfer) [Galilei 1587] afin d’éclairer une vive controverse sur l’interprétation de la géographie de l’Enfer dantesque. Ce travail d’exégèse littéraire permet à Galilée de faire reconnaître ses talents mathématiques comme ses qualités pédagogiques. Mais la portée de ces leçons va bien au-delà, car on peut y voir apparaître plusieurs thèmes majeurs de l’œuvre ultérieure de Galilée : au plan mathématique, l’importance de la géométrie d’inspiration archimédienne, au plan physique, l’étude des questions de similitude que pose la résistance des matériaux – sans oublier l’intérêt constant du scientifique pour la langue et pour la culture littéraire. Abstract: In 1587 the young Galileo was invited to give Due lezioni all’Accademia Fiorentina circa la figura, sito e grandezza dell’Inferno di Dante (hereafter Leçons sur l’Enfer) [Galilei 1587] aimed at settling an intense controversy regarding the geography of the Dantean Hell. This study in exegetics enabled Galileo to bring his mathematical talents and didactic qualities to the knowledge of the Tuscan scholars. But these lessons have a much greater importance, in that they reveal several of the major themes of Galileo’s further work from both mathematical and physical standpoints, such as the question of scale linked to the strength of materials, as well as the scientist’s unremitting interest in language and commitment to literary culture.

1 Introduction

Le texte des Leçons sur l’Enfer ﬁgure parmi les tout premiers travaux de Galilée [Galilei 1587]. Ce travail est fort méconnu sauf des experts, et souvent

Philosophia Scientiæ, 21(1), 2017, 111–130. 112 Jean-Marc Lévy-Leblond considéré au mieux comme un exercice de virtuosité en même temps qu’un travail de circonstance destiné à faire connaître son auteur. La relégation au second plan de ces Leçons sur l’Enfer fut d’ailleurs très précoce. Galilée semble n’avoir guère voulu en faire état par la suite et se montrait réticent à en communiquer le texte1. Viviani, son élève et premier biographe, n’en fait même pas mention. Pourtant, loin de constituer un élément négligeable et latéral de l’œuvre galiléenne, les Leçons sur l’Enfer recèlent des éléments préfigurant plusieurs thèmes essentiels des contributions majeures de Galilée à la mécanique [Settle 2002]. Ce texte annonce, d’une façon parfois paradoxale, au moins l’une des « sciences nouvelles » [Galilei 1638], que Galilée rendra publiques à la fin de sa vie, soit plus de quarante ans après les Leçons sur l’Enfer, et qui constituent son apport essentiel à la physique moderne. Le jeune Galilée, après avoir commencé ses études de médecine à Pise en 1580, découvre les mathématiques à partir de 1583 (il n’a pas vingt ans), abandonne l’université où cette discipline n’était pas tenue en haute estime, et se consacre à l’étude personnelle des maîtres anciens, Euclide et Archimède au premier chef [Geymonat 1963]. Il acquiert vite une considérable maîtrise en la matière, ce qui lui permet de l’enseigner à l’occasion pour gagner sa vie. Dès 1586, Galilée fait circuler un petit opuscule, La bilancetta, d’inspiration typiquement archimédienne, sur un instrument de pesée capable de mesurer les densités des objets et donc la qualité des alliages – dans la tradition de la fameuse anecdote d’Archimède détectant la fraude du joaillier qui avait fondu la couronne de Hiéron. À la même époque, il rédige quelques théorèmes, toujours dans un style très archimédien, sur les centres de gravité des solides de révolution. Remarqué par des savants de premier plan, tel Guidobaldo del Monte, il est invité par l’Académie florentine à donner ces Leçons, afin d’éclairer la controverse qui opposait depuis des décennies deux interpré- tations de l’Enfer de Dante dues respectivement à Manetti et Vellutello. Ce travail d’exégèse littéraire permet au jeune et ambitieux Galilée de faire reconnaître ses talents mathématiques comme ses qualités pédagogiques, et constitue une fructueuse opération de promotion. Sa réputation croissant, il finit par obtenir en 1589 une chaire de mathématiques à l’université de Pise, poste qui inaugure la carrière institutionnelle de Galilée. Mais quelle est la place de ces Leçons dans l’œuvre de Galilée, ou plutôt, dans le projet général qu’il élabore dès ces années de jeunesse ?

2 La pure langue toscane

Certes, il ne s’agissait nullement, ni au début du xvie siècle pour Manetti ou Vellutello, ni à la ﬁn de ce même siècle pour Galilée, de prendre au

1. Dans une lettre de 1694 (dix ans après la parution des Leçons), un certain Luigi Alamanni se plaint de n’avoir pu en obtenir communication par l’auteur [Galilei 1890- 1909, IX, 7]. Galilée, de l’Enfer de Dante au purgatoire de la science 113 sérieux la description de Dante du point de vue théologique. Tout simplement, l’importance de la Divine Comédie dans la culture toscane rendait évidente la nécessité de la comprendre sous tous ses aspects – y compris topogra- phiques – de façon à en rendre la difficile lecture plus aisée [Agnelli 1891], [Orlando 1993]. Par-delà la démonstration de ses compétences mathématiques personnelles, Galilée a dans ses Leçons une ambition culturelle plus haute. Sa formation au sein de l’élite pisane lui avait donné une excellente éducation littéraire, artistique et musicale (son père, Vincenzo Galilei, était l’un des premiers musiciens de son temps, ami de Monteverdi, et son frère sera aussi instrumentiste et compositeur). Aussi Galilée s’engagera profondément dans les débats littéraires et artistiques de son temps : dès les années suivant ses Leçons sur l’Enfer, il prend part aux controverses acharnées qui opposaient les partisans de l’Arioste et ceux du Tasse, comme aux discussions fort à la mode sur les mérites comparés de la peinture et de la sculpture [Galilei 1587], [Panofsky 2016], [Raffin 1992]. Mais les Toscans cultivés, tels ceux qui formaient l’Académie florentine, étaient évidemment loin de tous posséder les connaissances scientifiques, mathématiques en particulier, du jeune Galilée. Si la nouvelle science naissait, comme Galilée en offre l’exemple emblématique, du plus profond de la culture de son temps, il s’en fallait que les porteurs de cette culture puissent spontanément reconnaître et assimiler ce surgeon. Et de fait, la suite des siècles connaîtra le divorce progressif entre science et culture dont notre époque est victime [Lévy-Leblond 2004]. Peut-être conscient de ce risque, Galilée veut en tout cas montrer dans ses Leçons sur l’Enfer que la physique mathématique n’est pas simplement pourvoyeuse de calculs tech- niquement efficaces, mais peut apporter sa contribution aux débats culturels les plus nobles, et acquérir ainsi un statut intellectuel comparable à celui des humanités classiques2. C’est dans ce contexte que l’on doit comprendre le recours de Galilée à la langue italienne, et non au latin, pour la plupart de ses œuvres majeures – c’est le cas du Dialogo sopra i due massimi sistemi del mondo (ci-après Dialogue) [Galilei 1632] et des Discorsi e dimostrazioni matematiche intorno a due nuove scienze attenenti alla mecanica e i movimenti locali (ci-après Discours) [Galilei 1638] sans parler du Saggiatore (ci-après L’Essayeur) [Galilei 1623]. On a souvent et justement souligné l’importance de cette décision. Mais il ne s’agit pas, comme on le dit habituellement, d’un choix essentiellement politique par lequel Galilée viserait un lectorat plus vaste que celui des seuls érudits afin d’obtenir un plus large appui dans ses batailles intellectuelles. En vérité, Galilée ne considère nullement l’italien (qui, à l’époque, est en fait le toscan) comme la langue vulgaire, qu’il serait nécessaire, bon gré, mal gré, d’utiliser pour être

2. C’est pourquoi on ne saurait souscrire au sévère jugement du poète et écrivain allemand Durs Grünbein [Grünbein 1999] quand il affirme qu’en ne s’intéressant qu’aux dimensions de l’Enfer – ce qui, d’ailleurs, lui était demandé dans ses Leçons –, Galilée « congédie » le contenu sensible, poétique et métaphysique du poème de Dante, ouvrant ainsi la voie à l’actuel (et certes bien réel) « divorce entre les sciences naturelles et les arts ». 114 Jean-Marc Lévy-Leblond entendu de tous. C’est, pour lui, la langue même de la haute culture de son temps, celle qui à la fois exige et permet la plus grande clarté et la plus grande subtilité dans l’expression. Et c’est le latin, au contraire, qu’il considère sans complaisance comme un jargon technique, sans doute souvent utile entre gens du métier, mais impropre à une élucidation du sens. Les Leçons sur l’Enfer sont à cet égard parfaitement révélatrices et inaugurales. Contraint d’employer, dans ses explications scientifiques, certains termes savants (géométriques en particulier), Galilée s’en justifie et même s’en excuse auprès des membres de l’Académie florentine : [...] espérons que vos oreilles, accoutumées à entendre ce lieu résonner toujours des paroles choisies et distinguées que la pure langue toscane nous offre, puissent nous pardonner lorsque parfois elles se sentiront offusquées par quelque mot ou terme propre au domaine dont nous traitons, et tiré de la langue grecque ou latine, puisque le sujet que nous abordons nous oblige à faire ainsi. [Galilei 1587, 32] On comprend mieux alors que, même dans ses grandes œuvres scientifiques, la décision de Galilée d’écrire en italien est bien plus qu’une manœuvre tactique de « communication », comme on dirait aujourd’hui, mais exprime la ferme volonté d’inscrire son travail dans la culture de sa société et de son temps. Galilée n’est pas ici un cas isolé. Contrairement à une conception répandue mais simpliste, la révolution scientifique du dix-septième siècle n’est nullement liée à l’existence en Europe d’une langue de communication scientifique unique qui aurait été le latin. Bien au contraire, elle coïncide avec le développement, au sein des activités intellectuelles les plus exigeantes, des langues nationales, désormais considérées comme vecteurs de la culture moderne. En Italie déjà, Guidobaldo del Monte, cité plus haut, avait en 1585, deux ans à peine avant les Leçons sur l’Enfer, publié un ouvrage de mécanique à la fois en latin et en toscan. Pendant la première moitié du dix-septième siècle qui voit l’accomplissement de la révolution scientifique, Descartes pour le français, Harvey pour l’anglais, et jusqu’à Leeuwenhoek pour le néerlandais, offrent des exemples probants de cette légitimation des langues nationales [Lévy-Leblond 1996].

3 Les intervalles entre les ciels

Il n’est pas question de faire de ces modestes Leçons sur l’Enfer les prolégo- mènes de toute l’œuvre ultérieure de Galilée. Ce n’est pas le futur astronome et cosmologue, l’auteur du Dialogue [Galilei 1632], que l’on peut entrevoir dans ces Leçons. Au contraire même, puisqu’elles s’ouvrent sur une apologie de la cosmologie archaïque, faisant l’éloge des résultats obtenus dans la mesure des « intervalles entre les ciels » et de leurs mouvements, ce qui renvoie clairement à la représentation antique et médiévale d’un univers composé de plusieurs Galilée, de l’Enfer de Dante au purgatoire de la science 115 sphères célestes géocentriques. Galilée est loin d’être à l’époque le copernicien militant qu’il deviendra. Depuis sa chaire de Pise, il enseigne sans réserves le système ptoléméen des sphères. Ce n’est qu’en 1597, donc dix ans après les Leçons sur l’Enfer, qu’il fera état, en privé (dans une lettre à Kepler), de son ralliement à la théorie copernicienne, et bien plus tard encore, après 1610, qu’il commencera à la défendre publiquement [Galilei 2004]. Il faudra sans doute qu’il prenne ses distances avec une épistémologie quelque peu naïve qui lui fait écrire, au début des Leçons, que cet arpentage céleste qu’il célèbre, aussi « difficile et admirable » soit-il, concerne des « choses qui, totalement ou en grande partie, tombent sous le sens ». C’est en aban- donnant l’illusion commune d’un monde immédiatement donné à l’observation que Galilée pourra fonder en raison une nouvelle vision du cosmos : que ce soit par l’instrumentation (la lunette) ou par la théorisation (les mathématiques), la science ne comprend le monde que de façon médiate. Notons cependant, dans ce début des Leçons, un exemple typique de l’une des ressources rhétoriques constantes de Galilée, dont il fera grand usage dans ses œuvres majeures – l’ironie. C’est dans cette veine qu’il se permet d’expliquer la difficulté d’évaluer les dimensions de l’Enfer, puisque « ce lieu où il est si facile de descendre et dont il est pourtant si difficile de sortir » est « enseveli dans les entrailles de la Terre, caché à tous nos sens » – au contraire des « ciels » qui « tombent sous le sens » –, et que « la difficulté d’une telle description est considérablement accrue par l’absence de toute étude venant d’autres personnes ».

4 Par quelques raisons qui nous sont propres

Les Leçons sur l’Enfer de Galilée sont avant tout un exercice de géomé- trie. C’est à décrire et évaluer les structures spatiales sous-jacentes chez Dante qu’il s’attelle d’abord. La mathématisation de la physique dont Galilée est à juste titre considéré comme l’un des initiateurs se fera toujours chez lui sous l’égide de la géométrie, dans une filiation directement archimédienne. Rien encore, chez Galilée, de l’algébrisation que Descartes, si peu de temps après lui, commencera à mettre en œuvre. Et dans la très fameuse citation de Galilée, selon laquelle le grand livre de la Nature « est écrit en langage mathématique » [Galilei 1623], il faut prendre garde de ne pas oublier la suite, à savoir que « les caractères [dans lesquels ce livre est écrit] sont des triangles, des cercles, et d’autres figures géométriques », car il ne s’agit nullement des formules littérales de l’algèbre et de l’analyse modernes. En tout cas, Galilée, dès sa jeunesse, est un géomètre accompli comme le démontrent d’emblée les Leçons. Choisissant avec soin et commentant les vers adéquats de La Divine Comédie, Galilée commence par confirmer la description de Manetti : l’Enfer 116 Jean-Marc Lévy-Leblond est une cavité approximativement conique dont le sommet est au centre de la Terre, et dont l’axe perce la surface de la Terre à Jérusalem (évidemment...). La base du cône infernal, à la surface de la Terre, est un cercle de un diamètre égal au rayon du globe ; en coupe, la corde correspondant à un diamètre de ce cercle est donc la base d’un triangle équilatéral ayant le centre de la Terre pour sommet, ce qui revient à dire que l’angle au sommet du cône est de 60°. Et c’est là que Galilée met en œuvre son savoir mathématique, d’abord pour invalider une opinion erronée : Si l’on veut connaître [la] grandeur [de l’Enfer] par rapport à tout l’agrégat d’eau et de terre, nous ne devons pas suivre l’opinion de ceux qui ont écrit sur l’Enfer, estimant qu’il occupait la sixième partie de l’agrégat. [Galilei 1587, 34] Sur une coupe centrale de la Terre passant par l’axe du cône, le secteur infernal occupe en effet un sixième de l’aire du disque – en négligeant provisoirement la voûte de l’Enfer. D’aucuns, profanes en géométrie à trois dimensions, pourraient donc penser que la même proportion vaut pour les volumes. Mais, poursuit Galilée, si nous faisons nos calculs selon ce que démontre Archimède dans ses livres De la sphère et du cylindre, nous trouverons que l’espace de l’Enfer occupe un peu moins de la quatorzième partie du [volume] de l’agrégat. Je dis cela même si cet espace arrivait jusqu’à la surface de la Terre, ce qu’il ne fait pas ; car son embouchure reste couverte par une très grande voûte de Terre, au sommet de laquelle se trouve Jérusalem, et qui a pour épaisseur la huitième partie du demi-diamètre. [Galilei 1587, 34] Les traités d’Archimède faisaient alors partie des mathématiques les plus érudites, que les commentateurs précédents de Dante, littéraires purs, ne maîtrisaient certainement pas [Archimède 2003, I]. L’apport de Galilée fait appel à une expertise toute particulière, dont il peut légitimement se targuer. C’est sans doute ainsi qu’il faut comprendre sa prétention à éclairer la controverse « par quelques raisons qui nous sont propres ». Il n’est pas sans intérêt de vérifier les estimations de Galilée à l’aide tant des formulations archimédiennes que des expressions algébriques aujourd’hui usuelles (mais qui restent du niveau universitaire – voir Annexe 1). Notons que si l’on rétablit la voûte de l’Enfer, avec son épaisseur d’un huitième du rayon terrestre, le volume total de l’Enfer en est considérablement diminué, puisqu’il n’est plus que légèrement inférieur à 1/22 du volume terrestre (au lieu de 1/14). Galilée, de l’Enfer de Dante au purgatoire de la science 117 5 Une ligne qui conduit naturellement vers le centre ?

Galilée, dans son commentaire, ne se veut pas seulement mathématicien, et convoque également son expertise de physicien débutant. C’est à ce titre qu’il va émettre une critique sévère contre les commentaires de Vellutello. Ce dernier, en effet, concevait les gradins successifs de l’Enfer comme des portions de cylindre aux parois parallèles à leur axe commun, à l’instar des gradins d’un amphithéâtre antique. Galilée conteste cette interprétation, en arguant que de telles parois ne seraient nullement verticales, puisqu’elles devraient avoir pour génératrices des rayons tirés depuis le centre de la Terre, car en deux points distants, les directions des verticales ne sont pas parallèles, mais convergentes. Ainsi, selon Galilée, les falaises qui borneraient de tels gradins cylindriques seraient-elles en fait obliques par rapport à la (ou plutôt aux) verticale(s) locale(s) ; les arêtes externes de ces gradins seraient en surplomb prononcé, donc absolument instables : Si [Vellutello] suppose que le ravin se dresse entre des berges équidistantes entre elles, on aura des parties supérieures sans supports pour les maintenir, et de ce fait, immanquablement, elles s’effondreront. On sait en effet que les corps pesants suivent en tombant une ligne qui les conduit directement vers le centre, et si sur cette ligne ils ne trouvent rien qui les arrête et les soutienne, ils continuent à descendre et à tomber. [Galilei 1587, 52] Dans l’architecture de Manetti, en revanche, les parois des gradins sont tron- coniques, segments de cônes emboîtés ayant le centre de la Terre pour sommet, de sorte que ces parois, obliques par rapport à l’axe de l’Enfer, sont dirigées vers le centre géométrique du globe terrestre, considéré par Galilée comme déterminant la direction de l’attraction terrestre. De très peu postérieure aux Leçons de Galilée, l’édition de la Crusca offre une représentation sans doute fondée sur ces Leçons, ou en tout cas confortée par elles, parfaitement claire à cet égard. La caution apportée par Galilée à ce point de vue semble de prime abord fondée sur un convaincant raisonnement physicien qui conforte la pertinence scientifique de son discours. À mieux y réfléchir cependant, le physicien d’aujourd’hui est obligé de prendre quelques distances par rapport à l’argumentation de Galilée. Tout d’abord, l’Enfer selon Vellutello est très petit : sa profondeur comme son diamètre maximal ne dépassent pas le dixième des valeurs que prennent ces dimensions dans la version de Manetti (son fond se situe aux environs du dixième du rayon terrestre à partir de la surface, bien loin du centre de la Terre), et son volume est donc mille fois plus petit. Dans ces conditions, la variation de la direction de la pesanteur d’un point à l’autre de l’Enfer est faible, de quelques degrés au maximum, et peut être pratiquement assimilée à la direction de l’axe de l’Enfer. C’est alors une approximation tout à fait 118 Jean-Marc Lévy-Leblond raisonnable que de considérer les gradins comme des portions de cylindres, à parois parallèles donc, comme le propose Vellutello. Mais si l’on veut être plus précis, ce qui est nécessaire dès lors que l’Enfer, selon Manetti appuyé par Galilée, a des dimensions de l’ordre du rayon terrestre, le raisonnement de Galilée se heurte à une sérieuse objection. C’est que les verticales (entendez : les directions de la force de pesanteur) en chaque point ne sont dirigées vers le centre du globe terrestre que si celui-ci est une sphère complète, uniformément pleine. Or, dès lors que l’on évide le globe en lui retirant le vaste espace conique qu’exige l’Enfer, le champ de gravité intérieur de cette sphère incomplète est affecté et les directions des verticales locales perturbées. Galilée n’avait certes pas les moyens d’évaluer ces modifications que seule la théorie de la gravitation newtonienne permettra de calculer. Il est alors quelque peu ironique de constater que, d’après cette théorie, la situation se rapproche plutôt de celle décrite par Vellutello ! En effet, l’attraction due à la partie évidée étant supprimée, la force de pesanteur en un lieu quelconque des bords de l’Enfer serait dirigée non vers le centre géométrique de la sphère, mais vers un point situé plus bas sur l’axe du cône infernal. Ainsi les parois verticales des gradins ne devraient-elles pas être des troncs de cône ayant le centre de la Terre pour sommet commun, mais des troncs de cône nettement plus fermés, plus proches de segments cylindriques. Vues depuis le centre de la Terre, les parois des gradins apparaîtraient donc en surplomb plus ou moins prononcé. De fait, le calcul selon la théorie newtonienne de la gravitation montre que, dans l’Enfer selon Manetti, la direction de la pesanteur serait en fait partout presque parallèle à l’axe de l’Enfer (Annexe 2). Ainsi, et fort curieusement, au regard de la physique moderne, la situation serait finalement intermédiaire entre celle que décrit Manetti, et celle, retoquée par Galilée, que proposait Vellutello, et manifestement plus proche de cette dernière – y compris dans la géographie de Manetti ! La critique de Galilée est donc beaucoup moins dévastatrice qu’il ne le pensait et qu’une première lecture ne le laisse croire. Le point nodal de l’argumentation développée ci-dessus consiste à distin- guer le centre géométrique du globe terrestre du centre gravitationnel attractif. Dans une perspective aristotélicienne, où le géocentrisme est absolu et où le centre de la Terre est le centre intrinsèque de l’Univers, lieu naturel des graves, cette distinction est évidemment impensable. Galilée, dans ses Leçons sur l’Enfer, passe à côté du problème, mais on peut légitimement se demander si une réflexion ultérieure sur la situation n’a pas compté parmi les sources de sa remarquable discussion anti-aristotélicienne de la première Journée du Dialogue. On pense ici au passage où, bien avant le développement de la théorie newtonienne, la forme sphérique de la Terre (et des autres astres) est expliquée, non par le pouvoir attractif du centre de la Terre pensé comme une propriété intrinsèque d’un point privilégié de l’univers, mais comme la résultante des forces d’attraction mutuelles des parties du globe :

[...] les parties de la Terre se meuvent non parce qu’elles tendent vers le centre du monde, mais pour se réunir avec leur tout, et Galilée, de l’Enfer de Dante au purgatoire de la science 119

c’est pour cela qu’elles ont une inclination naturelle vers le centre du globe terrestre, en vertu de laquelle elles conspirent à former et à conserver ce globe. [Galilei 1632, 57–58] Nul doute en tout cas que le Galilée de 1632, s’il eût repris la question de la verticalité des gradins de l’Enfer, aurait compris que sur et dans une Terre privée d’un important volume, les corps pesants ne suivraient pas en tombant « une ligne qui les conduit directement vers le centre » du globe.

6 À la recherche de la grandeur d’un géant

Le géomètre Galilée convoque enﬁn la théorie des proportions pour préciser certains aspects de la description de Dante. Il s’agit d’abord de connaître, au fond de l’Enfer, la profondeur du puits de glace où Lucifer est enfoncé jusqu’à mi-poitrine, son nombril coïncidant pile avec le centre du monde. Galilée commence par évaluer la taille des « géants » décrits par Dante comme ayant une face aussi haute que la fameuse énorme « pigne » en terre cuite qui décore une cour du Vatican (où elle est encore aujourd’hui visible, avec ses trois mètres et quelques de hauteur). Considérant que le rapport de la tête au corps est le même chez les géants que chez les humains (soit de 1 à 8), Galilée, par une simple règle de trois, attribue aux premiers une taille d’environ 25 mètres. Quant à Lucifer, il est si grand, d’après Dante, que le rapport entre la longueur de l’un de ses bras et la taille d’un géant est supérieur au rapport entre la taille d’un géant et celle d’un humain. D’où, par deux nouvelles règles de trois, la longueur du bras de Lucifer, environ 340 mètres au moins, et la taille de Lucifer lui-même, pas loin de 1200 mètres. Mais le problème est que Galilée raisonne ici en pur géomètre, ne s’intéres- sant qu’aux formes des objets et des êtres, et pas du tout à leur constitution physique. Or, la résistance des matériaux suit des lois d’échelle qui ne sont pas celles des simples proportions géométriques. C’est là un phénomène empiriquement bien connu dans la pratique artisanale : si, partant d’un objet de modestes dimensions, barque, charpente, chariot, on en augmente toutes les cotes dans un même rapport pour fabriquer un objet semblable mais plus grand, on se rend compte que sa fragilité augmente rapidement avec le facteur d’agrandissement. Le premier à avoir attiré l’attention des physiciens sur ce point capital, jetant les bases de la théorie moderne de la résistance des matériaux, n’est autre que Galilée lui-même ! C’est le résultat essentiel de l’une des deux « sciences nouvelles » qu’il développe dans les Discours. Le tout début de l’ouvrage annonce avec force cette conception : SALVIATI : [...] Ne croyez donc plus, seigneur Sagredo [...] que des machines et des constructions faites des mêmes matériaux, reproduisant scrupuleusement les mêmes proportions entre leurs 120 Jean-Marc Lévy-Leblond

parties, doivent être également ou, pour mieux dire, proportionnellement aptes à résister ou à céder aux chocs venus de l’extérieur, car on peut démontrer géométriquement que les plus grandes sont toujours moins résistantes que les plus petites ; de sorte qu’en fin de compte toutes les machines et constructions, qu’elles soient artificielles ou naturelles, ont une limite nécessaire et prescrite que ni l’art ni la nature ne peuvent dépasser, – étant bien sûr entendu que les proportions et les matériaux demeurent toujours identiques. [Galilei 1638, 51] Galilée applique ces idées au cas des êtres vivants, en démontrant clairement qu’une homothétie spatiale ne respecte pas les contraintes physiques, et qu’un grand animal a besoin de membres plus épais par rapport à sa taille qu’un petit pour soutenir son poids (que l’on compare, à titre d’exemple, un éléphant, un chien et une souris) : SALVIATI : [...] il serait impossible, aussi bien en ce qui concerne les hommes que les chevaux ou les autres animaux, de fabriquer des squelettes capables de durer et remplir régulièrement leurs fonctions, en même temps que ces animaux croîtraient immensé- ment en hauteur – à moins bien entendu d’utiliser une matière beaucoup plus dure et résistante que la matière habituelle, et de déformer leurs os en les agrandissant démesurément, ce qui aboutirait à les rendre monstrueux par la forme et par l’aspect. [Galilei 1638, 169] Écho peut-être et en tout cas rectification de ses considérations dans les Leçons sur l’Enfer, il envisage dans les Discorsi le cas des géants : SALVIATI : [...] si l’on voulait conserver chez un géant particu- lièrement grand la même proportion qu’ont les membres chez un homme ordinaire, il faudrait ou trouver une matière bien plus dure et plus résistante pour en constituer les os, ou bien admettre que sa robustesse serait proportionnellement beaucoup plus faible que celle des hommes de taille médiocre ; sinon, à augmenter sans mesure sa hauteur, on le verrait plier sous son propre poids et s’écrouler. [Galilei 1638, 169–170] À plus forte raison, Lucifer ne saurait-il avoir les proportions d’un être humain, avec des dimensions simplement multipliées par un facteur d’échelle unique. Ou bien il doit être singulièrement disproportionné, avec des membres d’une épaisseur relative monstrueuse, ou bien il est d’une grande fragilité. Cette dernière conclusion pourrait d’ailleurs se défendre puisqu’aussi bien Lucifer est apparemment immobile et dans une zone de gravité pratiquement nulle, ne risquant donc guère de chute fatale. À quelques décennies près, c’est la thèse que Galilée aurait pu défendre... Dans les Leçons, Galilée reprend encore le raisonnement de proportion- nalité géométrique pour développer un argument bien plus crucial encore que Galilée, de l’Enfer de Dante au purgatoire de la science 121 celui touchant à la taille de Lucifer, puisqu’il s’agit tout bonnement de la résistance de la calotte terrestre servant de voûte à l’Enfer. Il écrit :

[Selon certains], il ne semble pas possible que la voûte qui recouvre l’Enfer, aussi mince qu’elle doit être avec un Enfer aussi haut, puisse tenir sans s’écrouler et tomber au fond du gouffre infernal [...] si elle n’est pas plus épaisse que le huitième du demi-diamètre [...]. On peut facilement répondre à cela que cette taille est tout à fait suffisante : en effet, si l’on considère une petite voûte, fabriquée selon ce raisonnement, qui aurait un arc de 30 brasses, il lui resterait comme épaisseur 4 brasses environ [...] ; [mais si] on 1 avait ne serait-ce qu’une brasse, ou 2 , au lieu de 4, elle pourrait déjà se maintenir. [Galilei 1587, 54–55]

La comparaison utilisée par Galilée entre la calotte de l’Enfer et une voûte ma- çonnée renvoie sans aucun doute aux rapports entre la structure de l’Enfer de Dante et l’architecture de la célèbre coupole du Dôme de Florence conçue par Brunelleschi, et qui joua un rôle emblématique dans la Renaissance italienne3 [Toussaint 1997]. Mais si cette analogie possède un sens culturel profond, sa valeur scientifique est nulle, pour les raisons mêmes que Galilée développe dans les Discorsi : une voûte aussi gigantesque que celle de l’Enfer, si elle avait les mêmes proportions géométriques qu’une petite voûte maçonnée, n’aurait certainement pas la même solidité. Au regard des conceptions modernes sur la pesanteur et la résistance des matériaux, le couvercle de l’Enfer serait inéluctablement appelé à s’effondrer, selon les arguments mêmes que Galilée a initiés. En effet, la résistance d’une voûte, comme celle d’une poutre ou d’un os, croît comme l’aire de sa section alors que son poids varie comme son volume. Si toutes les dimensions sont multipliées par un même facteur d’échelle, 10 par exemple, le poids sera multiplié par 1000 mais la résistance à l’effondrement par 100 seulement ; elle sera proportionnellement 10 fois plus fragile. Il y a donc nécessairement une limite à la solidité d’une structure obtenue par simple changement d’échelle à partir d’une structure solide plus petite. Et dans le cas de la voûte de l’Enfer comparée à la petite voûte maçonnée envisagée par Galilée, où le facteur d’échelle est de plusieurs centaines de milliers, cette limite est plus qu’évidemment dépassée et de beaucoup. Nous touchons ici à un point crucial concernant les Leçons et leur rôle dans le développement de la pensée de Galilée. Il est en effet très vraisemblable qu’il ait rapidement compris son erreur de raisonnement, découlant d’une conception purement géométrique, ne tenant pas compte des lois d’échelle concernant les propriétés physiques de la matière. Et c’est la prise de conscience de cette méprise qui aurait été à l’origine de ses travaux sur la résistance des matériaux, qu’exposent les Discorsi. Cette thèse, avancée par Mark Peterson [Peterson 2002, 575], s’appuie sur de sérieux arguments. Que Galilée ait vite

3. Notons que Manetti, dont Galilée prend la défense dans ses Leçons sur l’Enfer, fut aussi le biographe de Brunelleschi. 122 Jean-Marc Lévy-Leblond réalisé le caractère fallacieux des changements d’échelle mis en œuvre dans ses Leçons sur l’Enfer expliquerait en particulier la discrétion, voire la réticence, dont il a fait preuve presque immédiatement à l’égard de ce travail, ainsi que nous l’avons rappelé. Il a certainement consacré à ces questions d’intenses réflexions dans les années 1590 et 1600. On peut même conjecturer que Galilée, en comprenant son erreur, subit un véritable choc psychologique, dont on trouve écho dans les Discorsi, lorsque immédiatement après l’énoncé fondateur de Salviati rappelé ci-dessus, son interlocuteur réagit avec une émotivité surprenante : SAGREDO : Déjà la tête me tourne, et mon esprit, comme un nuage qu’un éclair déchire brusquement, se remplit pour un instant d’une lumière inhabituelle qui de loin me laisse entrevoir, pour les estomper et les cacher aussitôt, des idées étranges et désordonnées. Car de vos propos il me semble que l’on de- vrait conclure à l’impossibilité d’exécuter à l’aide d’un même matériau deux constructions, à la fois semblables et inégales, et dont les résistances seraient proportionnellement identiques. [Galilei 1638, 51–52] Ces lignes constituent une réfutation explicite de l’argument des Leçons assimilant la coupole de l’Enfer à une petite voûte maçonnée. On a tout lieu de croire que Galilée ait rapidement compris son erreur. De fait, si les Discorsi n’ont été publiés qu’en 1638, le matériau en était prêt dès avant 1610, époque à laquelle Galilée se consacre aux observations astronomiques et publie son premier ouvrage majeur, le Sidereus Nuncius [Galilei 1610]. Ainsi, dans une lettre de 1609 à Antonio de Medici, il énonce ce qui est, avec près de trente ans d’avance, un résumé explicite des Discorsi, tout au moins de leur première Journée : J’ai récemment réussi à obtenir tous les résultats, avec leurs dé- monstrations, concernant les forces et les résistances de morceaux de bois de diverses longueurs, tailles et formes [...], laquelle science est absolument nécessaire pour fabriquer des machines et toutes sortes de constructions, et qui n’a jamais auparavant été traitée par quiconque. [Peterson 2002, note 15] Il est donc loisible de considérer les Leçons sur l’Enfer comme le creuset où s’est initié le travail fondamental de Galilée dans les Discorsi. On peut encore mettre en évidence une autre difficulté physique de la description de Dante, liée comme la précédente à la rupture de symétrie que la cavité infernale imposerait à la distribution des masses terrestres. Galilée, s’il s’était, quelques décennies plus tard, reposé la question de la cohérence physique du modèle dantesque, n’aurait pas manqué de percevoir le problème. C’est qu’une Terre largement évidée par l’Enfer conique, verrait son centre d’inertie déplacé : il ne coïnciderait plus avec le centre géométrique du globe, mais se trouverait décalé sur l’axe du cône, dans la direction opposée à celle Galilée, de l’Enfer de Dante au purgatoire de la science 123 de Jérusalem. Un calcul rapide montre que ce décalage serait de l’ordre de 3 % du rayon terrestre, soit environ 200 km. La Terre tournerait sur elle-même autour de ce centre d’inertie ; la distribution des masses n’étant plus isotrope (à symétrie sphérique), mais à symétrie axiale, elle se comporterait comme une toupie. Puisque son axe de symétrie (passant par Jérusalem) ne coïnciderait pas avec l’axe des pôles, le mouvement de rotation diurne de la Terre sur elle-même serait considérablement perturbé par une précession simultanée de l’axe hyérosolomitain, ce qui est contraire à l’observation la plus banale. Ainsi donc, la mécanique suffirait à elle seule à invalider l’idée d’une cavité infernale au sein de la Terre. Mais, bien entendu, placer l’apex de l’Enfer, demeure de Lucifer, au centre de la Terre, n’a de sens que si ce lieu est aussi le centre du monde, ce qui suppose la validité du géocentrisme (que le jeune Galilée, rappelons-le, ne récusait pas encore). Dans le système du monde copernicien, il serait plus naturel de transporter l’Enfer dans le Soleil – ce qui assurerait au demeurant le fonctionnement de ses fournaises. Une telle théorie a d’ailleurs été fort sérieusement avancée en 1727 par Tobias Swinden [Swinden 1733].

7 En l’invitant à presser le pas

Reste une énigme dans le commentaire galiléen de l’Enfer de Dante. C’est qu’il n’y est guère question de la temporalité du voyage de Dante et de Virgile. Or, étant donné les distances si précisément établies par Galilée, on voit que ce sont des milliers de kilomètres que le poète et son guide sont amenés à parcourir – à pied, et même au prix de marches et d’escalades assez pénibles. Mais le parcours entier des deux hommes dans l’Enfer dure moins de trois jours – de la nuit du jeudi saint 7 avril 1300 au soir du samedi saint ! Notons que dans le petit Enfer de Vellutello, où les distances se chiffrent en centaines de kilomètres plutôt qu’en milliers, la difficulté serait (un peu) moins sérieuse... Le paradoxe est d’autant plus grand qu’on trouve quand même dans le texte de Galilée une allusion au temps du voyage, précisément utilisée comme argument contre la géométrie de Vellutello. Virgile, ayant amené Dante au premier cercle, le presse : « Poursuivons, car un long chemin nous pousse. » Galilée en conclut que la distance qu’il leur reste à parcourir est bien plus longue que celle déjà franchie, et que par conséquent l’Enfer est certainement bien plus profond que le dixième du rayon terrestre proposé par Vellutello. Comment donc comprendre que Galilée, si précis sur les mesures numériques des distances, n’ait pas procédé à une estimation quantitative des temps de parcours ? Cela est d’autant plus étrange que Galilée s’occupait déjà à l’époque du mouvement des corps. Le plus probable est qu’il a bien effectué ces calculs, et s’étant rendu compte de leur incompatibilité avec une interprétation réaliste de la narration dantesque, a décidé de les passer sous silence – sans pouvoir s’empêcher pour autant d’en tirer un argument qualitatif, fort douteux au demeurant, contre Vellutello. Sans doute faut-il nous résigner, avec Galilée, à admettre que la 124 Jean-Marc Lévy-Leblond description de Dante, si elle permet une interprétation géographique cohérente de l’Enfer, ne laisse la chronologie du voyage relever que de la licence poétique.

8 Donner à d’autres l’occasion de s’ingénier tant et plus

Il ne saurait pour autant être question de faire des Leçons sur l’Enfer la clé de toute l’œuvre de Galilée. On a tenté par exemple de voir dans le passage de L’Enfer où le monstre Géryon emporte dans les airs le poète et son guide, sans qu’ils perçoivent le mouvement autrement que par la caresse de l’air au passage, une prémonition du principe de relativité, dont Galilée aurait pu s’inspirer [Ricci 2005, 717]. Une idée semblable a été avancée par Primo Levi dans l’un de ses tout derniers textes, où il interprète ce même passage comme une prémonition de la sensation d’apesanteur éprouvée par les astronautes en orbite inertielle [Levi 1987]. Mais ces suggestions ne résistent pas vraiment à l’examen, ni du texte du poème de Dante, ni de son éventuelle signification scientifique. Dans le registre littéraire, il serait peut-être plus intéressant d’évoquer la visite que le jeune Milton dit avoir rendue en 1638 au vieux Galilée alors reclus dans sa résidence forcée d’Arcetri – rencontre non attestée au demeurant, mais immortalisée par un groupe statuaire représentant le poète penché sur l’épaule du physicien, que l’on peut voir au rez-de-chaussée du département de physique de l’université « La Sapienza » de Rome. En tout cas, dans son grand poème Paradise Lost [Milton 1667], Milton mentionne explicitement Galilée à plusieurs reprises [Henderson 2001]. On ne peut que rêver au dialogue qu’ont peut-être eu les deux hommes sur l’œuvre de Dante, une hypothétique source de la vision miltonienne de l’Enfer [Steggle 2001]. On s’en voudrait de passer sous silence un intéressant travail récent, d’une inspiration fort proche de celle des Leçons sur l’Enfer de Galilée, dans lequel est proposée une subtile interprétation géométrique moderne de la description assez obscure que donne Dante du Paradis, et plus généralement de son univers spatial entier. Le physicien Mark A. Peterson, déjà cité, y montre qu’il est tout à fait cohérent de comprendre l’univers dantesque comme intrinsèque- ment courbe et fermé, présentant la topologie d’une sphère tridimensionnelle (évidemment irreprésentable au sein d’un espace tridimensionnel euclidien infini et plat) [Peterson 1979]. On aimerait connaître le sentiment de Galilée sur cette analyse !

9 Conclusion

Ces commentaires trop rapides sur un épisode peu connu de l’histoire des débuts de la science moderne semblent permettre de renouveler quelque peu Galilée, de l’Enfer de Dante au purgatoire de la science 125 le vieux débat sur les critères de scientificité que nous appliquons à tel ou tel énoncé afin de lui attribuer ou de lui refuser le label « qualité science ». Ni la logique de l’argumentation – contradictoire, selon les bonnes règles méthodo- logiques –, ni la rigueur des calculs ou l’exactitude des faits d’observation ne font défaut aux Leçons sur l’Enfer de Galilée, pas plus qu’aux sérieuses études infernales de la théologie naturelle ou aux canulars (plus ou moins drôles...) de la physique moderne sur le même sujet4. Où l’on voit que ce qui caractérise l’admissibilité d’un énoncé dans le corpus scientifique relève moins d’une appré- ciation de sa validité que d’un jugement sur sa pertinence. La contre-épreuve est d’ailleurs aisée, puisque nombre des assertions de la science telle qu’elle se fait – la majorité, sans doute – se révèlent erronées sans pour autant disqualifier les recherches qui les ont produites, du moment que leur intérêt est reconnu par la collectivité savante. De ce point de vue, l’opposition classique entre une histoire des sciences internaliste, privilégiant les dynamiques intrinsèques aux travaux disciplinaires, et une histoire externaliste, mettant en avant les effets de leur environnement social, perd beaucoup de sa vigueur. Car la question de la pertinence reconnue à tel ou tel programme de recherche permet justement de faire lien entre l’organisation conceptuelle d’un champ de recherches et ses déterminations culturelles, idéologiques, économiques ou politiques. Ainsi, la question classique de la validation ou de la réfutation des idées scientifiques le cède-t-elle en importance aux considérations sur leur qualification ou leur disqualification. De fait, bien des travaux sont abandonnés sans jamais être tombés sous le coup d’une critique explicite et rédhibitoire ; le plus souvent, c’est insidieusement que les voies de la recherche prennent une autre orientation, laissant en jachère au bord de la route abandonnée des terrains à moitié explorés. Si peu de travaux scientifiques accèdent au Paradis de la reconnaissance définitive, peu aussi sont condamnés à l’Enfer de l’oubli ou du rejet absolus. La plupart se retrouvent au Purgatoire – comme les Leçons sur l’Enfer de Galilée, justement.

Remerciements

C’est avec plaisir que je remercie Françoise Balibar, Lucette Degryse et Jean- Paul Marmorat pour leurs apports à ce travail, ainsi que Raﬀaele Pisano pour sa précieuse aide à sa mise en forme.

4. On trouvera une étude de divers traitements scientiﬁques de l’Enfer in [Lévy- Leblond 2006, 70–93]. 126 Jean-Marc Lévy-Leblond Annexe 1. Le volume de l’Enfer

Selon Dante lu par Galilée, l’Enfer est un gouﬀre conique dont le sommet est au centre de la Terre, le demi-angle au sommet valant θ = 30°, borné par une calotte sphérique et recouvert par une voûte dont l’épaisseur d vaut 1/8 du rayon terrestre (voir la Figure 1 pour les notations). Si l’Enfer « arrivait jusqu’à la surface de la Terre », et que l’on ne tienne pas compte du volume de la voûte qui le recouvre, le volume de la cavité (délimitée en coupe par la ligne OAJBO) serait : 2 v = πR3(1 − cos θ) 3 à comparer au volume total de la Terre : 4 V = πR3 T erre 3 soit, en proportion : 1 v/V = (1 − cos θ) T erre 2 numériquement, ici √ (2 − 3) 1 v/V = = T erre 4 14, 92 ... un peu moins que la quatorzième partie, comme l’écrit Galilée. Mais en tenant compte de la voûte, la hauteur de l’Enfer est réduite par un facteur 7/8, et son volume par un facteur (7/8)3, d’où maintenant : √ V 2 − 3 7 3 1 Enfer = = ... VT erre 4 8 22, 3 Galilée ne disposait certes pas des notations modernes, et encore moins des ressources du calcul intégral sur quoi se fondent les expressions ci-dessus. Mais les résultats d’Archimède, dans son Traité de la sphère et du cylindre, lui permettaient d’aboutir aux mêmes conclusions sans diﬃcultés.

Annexe 2. La pesanteur en Enfer

Lorsque l’on évide le globe terrestre pour faire sa place à l’Enfer sur le mode proposé par Dante et commenté par Galilée, la distribution des masses perd sa symétrie sphérique. Or c’est cette symétrie, comme l’a montré plus tard Newton, qui entraîne que les forces attractives exercées sur un objet quelconque par les diﬀérentes parties de la Terre se combinent en une force totale dirigée vers le centre de la Terre et d’une intensité égale à celle qu’exercerait une masse située en ce point égale à la somme des masses des parties de la Terre plus proche du centre que l’objet considéré. Ce résultat dépend crucialement Galilée, de l’Enfer de Dante au purgatoire de la science 127

Figure 1. La Terre et l’Enfer en coupe Le disque de centre O (et de rayon OA0 = R) et le triangle courbe OAB (de demi- angle au sommet θ = 30°) représentent respectivement la Terre (sphérique) et l’Enfer (conique) vus en coupe. Les arcs AHB et A0JB0 délimitent la calotte qui couvre l’Enfer, d’épaisseur HJ = d = R/8. KM et LN dessinent la coupe du cylindre circonscrit à la sphère terrestre. De même que l’aire totale de la sphère est égale à l’aire latérale du cylindre de hauteur KM, l’aire de la calotte de section A0JB0 est égale à l’aire latérale de la portion du cylindre circonscrit ayant pour hauteur HJ (Archimède). du fait que la force de gravitation entre deux masses varie comme l’inverse du carré de leur distance, ce qui, évidemment, le mettait hors de portée de Galilée. Dans le cas d’une sphère évidée, le calcul de la force gravitationnelle en un point quelconque est loin d’être immédiat, même pour un évidement de forme géométrique élémentaire comme le cône infernal, et il faut recourir à des calculs numériques assez lourds. Un modèle simple va pourtant nous fournir un résultat exact intéressant qui confirme nos conclusions sur la perturbation des verticales. Supposons un Enfer non plus conique, mais sphérique. Dans tout l’espace intérieur de la sphère infernale ainsi que sur ses bords, la force de pesanteur serait alors uniforme, ayant en tout point la direction de la droite joignant les centres de l’Enfer et de la Terre et la même valeur (c’est là un petit théorème de la théorie newtonienne de la gravitation qui se démontre fort simplement). Ainsi donc, dans ce modèle sphérique de l’Enfer, quelle que soit sa taille, c’est paradoxalement la conception de Vellutello (gradins cylindriques à bords parallèles) qui prévaudrait sur celle de Manetti (gradins en troncs de cône) ! Dans le cas dantesque d’un Enfer conique, la situation est certes plus compliquée. La résolution numérique des équations qui déterminent le potentiel et le champ de gravitation dans ce cas confirme pourtant que la direction de la pesanteur au sein de l’Enfer et jusque sur ses bords varie fort peu. La figure 2 montre le résultat d’un tel calcul. 128 Jean-Marc Lévy-Leblond

a b

Figure 2. La pesanteur en Enfer Les ﬂèches indiquent la direction et la valeur (relative) du champ de pesanteur a) dans une Terre pleine ; b) dans une Terre creusée d’un Enfer sphérique ; c) dans une Terre creusée d’un Enfer conique. On constate que le champ de pesanteur montre, pour les deux formes de l’Enfer une direction constante – exactement en b), approximativement en c). [Calculs et graphisme par Jean-Paul Marmorat, communication privée] Galilée, de l’Enfer de Dante au purgatoire de la science 129 Bibliographie

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Roberto de Andrade Martins Universidade de São Paulo (Brazil)

Walmir Thomazi Cardoso Pontifícia Universidade Católica de São Paulo (Brazil)

Résumé : Dans cet article nous étudions le Trattato della sfera de Galilée, écrit avant 1600. C’est un traité d’astronomie géocentrique qui suit la structure du Tractatus de sphæra de Johannes de Sacrobosco. Nous analysons quelques particularités du traité, en le comparant à d’autres travaux astronomiques du xvie siècle, et nous discutons ses sources probables. Nous soutenons que l’influence du commentaire de Christoph Clavius sur la Sphæra de Sacrobosco ne peut pas être considérée comme son influence unique ou principale. Le traité de Galilée était probablement inspiré par la Sfera del mondo de Piccolomini, un travail qui anticipait plusieurs particularités du Trattato della sfera. Cette influence est établie par de nombreuses annotations de Galilée trouvées dans un exemplaire du livre de Piccolomini.

Abstract: This paper studies Galileo Galilei’s Trattato della sfera ovvero cosmografia, which was written before 1600. It is a geocentric astronomical treatise that follows the main structure of Johannes de Sacrobosco’s Tractatus de sphæra. This paper analyzes some peculiarities of Galileo’s treatise, comparing it to several other vernacular astronomical works of the sixteenth century and discussing its likely sources. Contrary to previous claims, we argue that Christoph Clavius’ commentary on Sacrobosco’s Sphæra cannot be regarded as its only or main influence. A likely inspiration for Galileo’s treatise is Alessandro Piccolomini’s Sfera del mondo, a work that anticipated several specific features of the Trattato della sfera. This influence is corroborated by Galileo’s copious marginal notes found in a copy of Piccolomini’s book.

Philosophia Scientiæ, 21(1), 2017, 131–147. 132 Roberto de Andrade Martins & Walmir Thomazi Cardoso 1 Introduction

One of the earliest works attributed to Galileo is called Trattato della sfera ovvero cosmografia [Treatise on the Sphere, or Cosmography]. Its date of composition is unknown (probably before 1600) and it was published posthumously by the priest Urbano d’Aviso [Galilei 1656]. It is a short and elementary geocentric astronomical treatise. Its content and structure generally follow Johannes de Sacrobosco’s medieval Tractatus de sphæra. A first look at the contents and style of the Trattato della sfera ovvero cosmografia (hereinafter called Trattato in brief) provides no internal evidence that it was written by Galileo. It includes no reference to Copernicus or his ideas ; it accepts and defends the main geocentric astronomical ideas : that the Earth does not move in any way, that it is at the center of the universe, and that the Sun, the Moon, the planets and the stars move around it. Also, it contains no recent information that became available during the sixteenth century, such as new evaluations of the size of the Earth, the knowledge of stars visible from the Southern hemisphere, or the existence of people living in the tropical zone. Due to its contents, it is understandable that after its publication the Trattato did not attract much attention, and up to the nineteenth century, there were strong doubts concerning its authenticity. Of course, it provides no information concerning Galileo’s later ideas ; however, it is an important source for studying his early acquaintance with the astronomy of his time.

2 Comparison between the Trattato della sfera and Sacrobosco’s Tractatus

Most of the treatises belonging to the tradition of Sacrobosco’s Tractatus de sphæra (or Sphæra, in short) contained the whole text of the medieval book, adding explanations, images, and new information. The original text was usually printed in a different (larger) typeface so that the readers would be able to identify it. The Trattato cannot be regarded as a summary of Sacrobosco’s work, as claimed by Wallace [Wallace 1977, 255]. The medieval Sphæra was a short work (about 9,000 words). The Trattato della sfera has nearly 16,000 words. For its size, it could comprise the whole content of Sacrobosco’s work, as well as substantial explanations and additions. However, its author chose a different approach. It does not contain a translation of the Sphæra, and not even cites Sacrobosco’s name. Sacrobosco’s work contains a preamble and four chapters. The Trattato begins with a long methodological introduction (with no correspondence in the Sphæra) followed by 24 small chapters. Generally, the chapters of the Trattato follow the order of the content of Sacrobosco’s work. There are, Galileo’s Trattato della sfera ovvero cosmografia and Its Sources 133 however, important omissions and additions. At the end of the first chapter, the Sphæra describes the ancient measurements of the size of the Earth ; this subject is lacking in the Trattato. Most of the content of the second chapter of Sacrobosco’s book does appear in the Trattato, but in a different sequence ; and it omits the final part of chapter two, about the five terrestrial zones. The third chapter of the Sphæra explains the several types of astronomical risings and settings ; the Trattato omits this subject altogether. The rest of the third chapter is largely followed by the Trattato, but it adds a more detailed elucidation of latitude and longitude and describes 22 geographical climes, instead of the seven classical ones. The beginning of the fourth chapter of Sacrobosco’s work explains the motions of the Sun and planets ; the Trattato omits several parts of this content. The Trattato contains a description of the phases of the Moon and its visibility—subjects that do not appear in the Sphæra, although they are discussed in Sacrobosco’s De anni ratione [Sacrobosco 1573, 211–214]. The final part of Sacrobosco’s work expounds the miraculous eclipse of the Sun during Christ’s passion ; the Trattato does not refer to this miracle but includes a discussion of the motion of the 8th sphere and its trepidation that is lacking in the Sphæra. Of course, there are several other differences between the contents of these two works. The editions and commentaries of the Sphæra frequently contained several illustrations. The Trattato contained only one single figure [Galilei 1656, 24]. Another curious difference is the complete lack of classical literary citations in the Trattato, whereas Sacrobosco’s book contained several citations of Virgil, Lucan, and Ovid [Martins 2003]. The overall tone used in the Trattato is very similar to Sacrobosco’s approach. It simply exposes accepted knowledge ; there is no polemical character in any part of the book, in strong contrast with Galileo’s later famous works. Notice that some former authors, such as Francesco Barozzi and Francesco Maurolico, had maintained a polemical attitude in their astronomical writings [Barozzi 1585], [Maurolico 1543].

3 Language and aim of the Trattato della sfera

Another relevant difference between the two works was their languages. Since the Tractatus de sphæra was used as a textbook in European universities, most of its versions were written in Latin. All of the 30 editions published during the fifteenth century, for instance, were in that language.1 Among the 92 renderings of the Sphæra published between 1501 and 1550, we find 12 that

1. A survey of editions and commentaries of the Sphæra, produced by one of the authors of this paper (Martins) is available at . 134 Roberto de Andrade Martins & Walmir Thomazi Cardoso were not in Latin. The Portuguese and Spanish versions were produced for the use of pilots and other people involved in navigation ; others were created for the general public, who was not familiar with Latin. This raises a question concerning the aim and the target readers of the Trattato ovvero cosmografia. Galileo’s teaching subjects at the universities of Pisa and Padua included Sacrobosco’s Sphæra and the use of Latin in the lectures was obligatory [Favaro 1888, vol. 1, 177]. One of the reasons was that many of the students attending those universities came from other countries and the only lingua franca available at that time was Latin. Galileo must have used some Latin version of the Sphæra in his public teaching activities, and it would have been useless to prepare an Italian textbook for those students. Stillman Drake and William Shea conjectured that the Trattato was composed for private teaching in 1586-1587 after Galileo left his medicine studies and before he became a professor at Pisa [Drake 1978, 12], [Shea 1990, 51]. However, there is no evidence that he knew or taught astronomy during this period [Favaro 1888, vol. 1, 15]. Also, Drake’s claim that “it is safe to assume that Galileo’s first-year lectures on astronomy were based on his Treatise on the sphere” [Drake 1978, 19] is groundless. Besides his official duties at the universities, Galileo delivered private classes on several subjects, as a way to improve his financial income. It is certain that he taught astronomy to some students in the early years of the seventeenth century—at the time when the extant manuscript copies of the Trattato were produced. It is unlikely, however, that he wrote a textbook for those private lessons. Firstly, many of his students were foreigners. According to extant records, in the period 1601-1607, about 10 private students studied the sphere under Galileo. One of them was British, two were Hungarian and seven were Polish [Favaro 1888, vol. 1, 186–191]. It is doubtful whether they would have appreciated a textbook in Italian. Secondly, Galileo could deliver private classes on astronomy without the inconveniency of writing a textbook for his students : he could use any of numerous available treatises—both in Latin and in Italian. Composing a new textbook would be understandable if Galileo intended to impart new knowledge, not available in the usual treatises. The Trattato, however, could not fulfill this purpose, as all its content could be found in works published one century earlier. It did not include updated information about the size of the Earth, or the stars visible from the Southern hemisphere, or the existence of people living in the Torrid Zone, for instance. It did not address the practical navigational use of astronomy that was very important during the sixteenth century, or the use of new astronomical instruments. Neither did it discuss recent astronomical theories (such as those of Copernicus and Tycho Brahe) or fresh observational findings (such as comets and Tycho’s nova). Perhaps Galileo did not compile the Trattato for his students. It might be just a set of notes he wrote for himself, when he was learning the traditional astronomy he was going to teach in Pisa. William Wallace conjectured that it Galileo’s Trattato della sfera ovvero cosmografia and Its Sources 135 was composed towards the end of 1590, when he wrote to his father requesting his copy of the Sfera [Wallace 1998, 35].2 Maybe he used these annotations as guidelines for his classes, later ; and he could have lent the manuscript to some students who wanted to copy it.

4 Suggested sources of the Trattato della sfera

Some authors have claimed that the main source of the Trattato was the famous commentary on Sacrobosco’s Sphæra written by Christoph Clavius [Wallace 1977, 255, 257], [Wallace 1984, 257], [Lattis 1994, 5]. In the late sixteenth and early seventeenth centuries, Clavius’ bulky book (about 500 pages and 170,000 words) was one of the most respected and widely used astronomical textbooks. After its first edition in 1570, it underwent successive improvements and was printed in 1575, 1581, 1585, 1591, 1593, 1594, 1596, 1601, 1602 (twice), 1603, 1606, 1607 (three times), 1608, 1611 and 1618. Of course, most of the contents of the Trattato can be found in Clavius’ book, or in some other large commentary on the sphere. Any generic textual parallel between the two works, such as the one published by Wallace is pointless [Wallace 1977, 258]. For deciding whether a specific work was the main source of the Trattato it is necessary to analyze some of its unusual features. William Wallace and Alistair Crombie have noticed that part of Galileo’s early manuscript on Aristotle’s De Cælo (MS 46), might have been copied from Clavius’ commentary on the Sphæra [Wallace 1977, 258], [Wallace 1990, 28], [Crombie 1996, 181], and this part is also similar to a section of the Trattato. They claimed that the methodological introduction of the Trattato was based on Clavius’ book [Crombie 1996, 178]. However, in his later publications Wallace suggested that Galileo could have only indirect acquaintance with Clavius’ work, through Mutius de Angelis [Wallace 1990, 36]. Stillman Drake, on the other hand, concluded that this introduction was added to the work at a later time (around 1602) and that Galileo’s source for that part was Ptolemy’s Almagest [Drake 1978, 52–53]. It might also have been derived from Caspar Peucer, a book owned by Galileo [Peucer 1573, 1–8]. One peculiarity of the Trattato is the inclusion of 22 geographical climes, supplying a table with their data. The descriptions contained in the table are in Latin, not in Italian, suggesting that it was copied from a Latin book. The declination of the ecliptic used for the computation was 23° 29’, a figure that was not very common during the sixteenth century. An almost identical

2. In november 1590 Galileo asked his father Vincenzio Galilei to send to him a few books, including a Sfera (notice that he did not call it “Sphæra”) [Galilei 1890-1909, vol. 10, 44–45]. 136 Roberto de Andrade Martins & Walmir Thomazi Cardoso table can be found in Clavius’ treatise [Clavius 1585, 429–430]. Hence, William Wallace concluded that it was copied from that book [Wallace 1977, 258], thus establishing a strong link between the Trattato and the famous commentary. However, this evidence is not conclusive, as we have already shown [Martins 2010].

In all editions of Clavius’ commentary on Sacrobosco, beginning with the first one (1570), we do find a table closely similar to the one reproduced in the Trattato. All descriptions are in Latin ; the declination of the ecliptic was also 23° 29’ ; and the numbers are almost identical to those reproduced by Galileo. However, there are a few numerical differences. Notice that Clavius’ table is reproduced without any numerical change in the several editions of his book—we have compared the editions of Rome (1570, 1581, 1585, 1606) ; Venice (1591) ; Lyon (1593, 1594, 1602, 1607), and they are exactly equal. Three of the numbers that appeared in the Trattato are different from those of Clavius. These might be simply copying mistakes, but they could be due to his using a different source. Two of the three differences between Galileo’s and Clavius’ numbers also appear in the table published by Francesco Pifferi—and that cannot be ascribed to a mere coincidence [Pifferi 1604, 348–349]. It is significant that Pifferi was a mathematics professor at the University of Padua before Galileo and that the manuscript of his book could have been available several years before its publication, in 1604.

Besides that, the table published by Clavius was not original. An identical one was printed by Caspar Peucer [Peucer 1569, 268–269], one year before the ﬁrst edition of Clavius’ book. Other published books might also have tables very similar to those of Peucer and Clavius. Therefore, the description of the 22 geographical climes and the corresponding table appearing in the Trattato could have been copied from another work and they do not establish a strong connection with Clavius’ book.

Another particularity of the Trattato is its description of the phases of the Moon and the condition of its visibility. Although these are elementary astronomical topics that could have been introduced by Sacrobosco in his Sphæra, he did not include them in that work, and Clavius’ commentary also does not contain this subject. Therefore, his book could not be the source from which the Trattato drew its description of the phases and visibility of the Moon.

Analyzing further details of the Trattato we find several other relevant differences between its content and Clavius’ work—for instance, some of the specific arguments concerning the immobility of the Earth [Martins & Cardoso 2008]. We may certainly reject the claim that the Trattato was a summary of Clavius’ book. The general style of the Trattato and the very language that was chosen for its composition also suggest that its main source should be sought elsewhere. Galileo’s Trattato della sfera ovvero cosmografia and Its Sources 137 5 Sixteenth-century Italian astronomical works

Taking into account its language, the Trattato della sfera ovvero cosmografia might have been inspired by some previous vernacular publication. The Italian versions of Sacrobosco’s Tractatus de sphæra that were published before 1600 were those of Mauro Mattei, Antonio Brucioli, Piervincenzo Danti de’ Rinaldi, and Francesco Giuntini. We have examined all of them, and their description will be presented in a forthcoming paper. None has many similarities to the Trattato della sfera. During the sixteenth century, there appeared several astronomical books inspired by Sacrobosco’s work but that did not follow it in a strict way. Most of them were in Latin, but there were also vernacular ones. The most famous one was Alessandro Piccolomini’s De la sfera del mondo, first issued in 1540. This work was published again in 1548, 1550, 1552, 1553, 1554, 1558, 1559, 1561, 1564, 1566, 1573, 1579, 1584, 1595. It was translated into French and printed in 1550, 1608 and 1619 ; and a Latin version appeared in 1568. The analysis of Piccolomini’s work is particularly relevant for us because, according to Domenico Berti, Galileo wrote numerous annotations in a copy of Piccolomini’s book on the sphere [Berti 1876, 100]. Antonio Favaro reported that Galileo had a copy of the 1572 edition of De la sfera del mondo, remarking that this was probably the copy that belonged to the library of Meucci [Favaro 1886, 251].3 Alessandro Piccolomini (1508-1578) was a famous humanist of the sixteenth century, born in Siena [Fabiani 1759]. He wrote poetry, plays, and books on philosophy, astronomy, and other subjects. He became a priest and later was nominated bishop of Patras (Greece), although he never visited that country. He belonged to a group of writers who defended the replacement of Latin by Italian (or, more exactly, the Tuscan dialect) as a scholarly language and he was one of the main writers who contributed to the early development of a vernacular scientific prose [Suter 1969, 210]. He lived for four years in Padua, where he attended the university. During this period he developed a strong interest in astronomy and published his famous treatise De la sfera del mondo, that was accompanied by his treatise Delle stelle fisse, the first celestial atlas presenting all the Ptolemaic constellations in their real configurations, with the indication of the magnitude of each star. The first edition was dedicated to a lady, Laudomia Forteguerri de Colombini. One of Piccolomini’s purposes, in his vernacular works, was to make philosophy, poetry, and astronomy available to everyone (including women) who did not know Latin and who had no access to university courses. Piccolomini did not regard the classics as sacred writings. He thought that the ideas contained in the old books were more important than their

3. There is no edition of 1572. Favaro was probably referring to the 1573 issue. 138 Roberto de Andrade Martins & Walmir Thomazi Cardoso original language and for that reason, instead of producing rigorous and literal translations, he preferred to create vernacular paraphrases of Aristotle’s Rhetoric and Physics, for instance. For the same motive, he produced his own vernacular books inspired by Sacrobosco’s Tractatus de sphæra and by Peurbach’s Theorica planetarum, rather than translating them to Italian. The first edition of Piccolomini’s Sfera del mondo had four parts or books. He kept improving it in the following editions, adding new topics and expanding his explanations. In 1564, he published his final revised and enlarged version in six books. It contained 252 pages and over 100,000 words—that is, it was more than ten times larger than Sacrobosco’s Sphæra. Alistair Crombie has already remarked that Galileo’s style is somehow similar to Piccolomini’s, from whom he borrowed some relevant phrases such as “sensate esperienze e certe dimostrazioni” [Crombie 1996, 236–237]. Several characteristics of Piccolomini’s work have counterparts in the Trattato. First of all, the use of Italian as a language for exposing astronomical knowledge. Secondly, the absence of a rigid correspondence between Piccolomini’s composition and Sacrobosco’s Sphæra. Thirdly, a preoccupation with epistemological and methodological issues. Fourth, the inclusion of the phases of the Moon among its topics, adjacent to the explanation of the eclipses. Both treatises lack the discussion of the miraculous eclipse at the time of Christ’s passion. Another curious similarity is the deficiency of citations of the classic poets in the Sfera del mondo. Perhaps Piccolomini dropped out most of the quotes that appeared in Sacrobosco’s work because of his campaign for Italian (not Latin) as the preferred language.

6 Galileo’s annotations in Piccolomini’s Sfera

In 1886, when Antonio Favaro wrote his work on Galileo’s library, he was not sure about the location of the copy of Piccolomini’s book containing Galileo’s annotations, although he thought that it might belong to the Meucci library. He wrote to Ferdinando Meucci, on January 6th 1890, asking him if he had that work or if he knew who owned it. Meucci answered to Favaro on the following day, telling him that he did have the book.4 We have obtained no information about further contacts between Favaro and Meucci concerning this copy of Piccolomini’s Sfera del mondo.

4. Museo Galileo. Carteggio Meucci I : Carteggio cronologico. 1865-1893. Lettere (1890) I, 1890, #3. We are grateful to Mrs. Alessandra Lenzi, librarian of the Museo Galileo, for providing copies of those two letters (private communication, November 14, 2015). Galileo’s Trattato della sfera ovvero cosmograﬁa and Its Sources 139

It is known that Favaro asked Ferdinando Meucci to bequeath some of his books to the National Library of Florence.5 It is also known that Meucci had a copy of Nicolò Tartaglia’s La Nova Scientia with marginal notes by Galileo [Favaro 1886, 268], and that this copy now belongs to the library of the Museo Galileo (shelfmark MED 0976/01).6 A heavily annotated copy of the 1573 edition of Piccolomini’s De la sfera del mondo now belongs to the library of the Museo Galileo (shelfmark MED 0719). This specific item was displayed at the 1929 Prima esposizione nazionale di storia della scienza, in Florence (registration number 6952). Although the staff of the Museo Galileo was not aware that this item was the one containing Galileo’s notes described by Favaro,7 it is fairly probable that it was. One important evidence is the comparison between the leather book covers of MED 0976/ 01 (Tartaglia’s book) and MED 0719 (Piccolomini’s book). The design of both book covers, in blind stamping, is identical, although they were printed by different publishers. This shows that both belonged to the same library. They also have identical stamps at the title page : one from the Laboratorio di Fisica in Arcetri ; the other one from the Museo degli Strumenti Antichi di Astronomia e di Fisica, R. Istituto di Studi Superiori, Firenze, with the handwritten number 31 in the case of Piccolomini’s book, and number 32 for Tartaglia’s book. Since Tartaglia’s book came to the Museo Galileo through Meucci’s library, it is highly probable that Piccolomini’s book had the same origin. Favaro suggested that the marginal notes of this copy of Piccolomini’s work could have been made by Galileo, but at that time he had not been able to examine its handwriting. Following our contact with the library of the Museo Galileo, Dr Patrizia Ruffo inspected that book and concluded that the annotations might indeed have been made by Galileo.8 A detailed analysis by one of us (Walmir Cardoso) has shown that the calligraphy of the marginal notes found in the 1573 copy of De la sfera del mondo belonging to the library of the Museo Galileo displays striking similarities and no conspicuous differences when compared to several samples

5. See, for instance, Favaro’s letter to Meucci, March 29, 1890. Museo Galileo. Carteggio Meucci I : Carteggio cronologico. 1865-1893. Lettere (1890) I, 1890, #14. 6. We are grateful to Mrs. Alessandra Lenzi for calling our attention to this book (private communication, November 10, 2015). 7. Mrs. Alessandra Lenzi, librarian of the Museo Galileo ; private communication, November 10, 2015. 8. “La dott.ssa Ruffo ha fatto una prima analisi sulla calligrafia delle postille del nostro MED 0719 e secondo lei potrebbero in effetti essere autografe di Galileo” (Mrs. Alessandra Lenzi, private communication, November 14, 2015). We have been informed that Ilaria Poggi and Patrizia Ruffo plan to transcribe those annotations and to publish them in a future update of the National Edition of the works of Galileo ; and that Patrizia Ruffo and Michele Camerota began a comparison between those annotations and the Trattato della sfera (Mrs. Alessandra Lenzi, private communication, November 28, 2015). We have no further information about their work, which was prompted by our interaction with the librarian of the Museo Galileo. 140 Roberto de Andrade Martins & Walmir Thomazi Cardoso of Galileo’s handwriting (letters, manuscripts, and marginal annotations) around 1590. A small part of the available evidence is presented here.9 Galileo owned and made annotations in a 1570 copy of Piccolomini’s De le stelle fisse [Favaro 1886, 251–252]. This item belonged to the National Library of Florence and can now be found in the Museo Galileo (shelfmark Rari 111/02). Let us compare some of those marginal notes to the ones found in the 1573 copy of De la sfera del mondo.

Figure 1. Detail of Galileo’s annotation at fol. 2r, De le stelle ﬁsse [Piccolomini 1570]

Figure 2. Detail of annotation at p. 51, Sfera del mondo [Piccolomini 1573]

It is possible to notice the strong similarity between Galileo’s authentic calligraphy10 in Fig. 1 (“opinione de i pittagorici”) and the notes on the Sfera del mondo, Fig. 2 (“opinione de pittago/ rici”) and Fig. 3 (“ragione de pitagorici”). Galileo’s notes to the preface of De le stelle ﬁsse present three times the word “opinione” (Figs. 1, 4, 5). In all of them, the word is broken twice, opi- ni-one, and the letter “n” after the break has a characteristic drawing. We

9. All images reproduced here are details of page scans available at the website of the Museo Galileo, De le stelle fisse, 1570 : http ://bib- dig.museogalileo.it/Teca/Viewer ?an=323989 ; La sfera del mondo, 1573 : http ://bib- dig.museogalileo.it/Teca/Viewer ?an=300237. 10. We have used the marginal annotations to the preface of the 1570 copy of Piccolomini’s De le stelle fisse (fols. 2r, 2v), because Antonio Favaro explicitly recognized them as authentic [Favaro 1886, 251–252]. Galileo’s Trattato della sfera ovvero cosmografia and Its Sources 141

Figure 3. Detail of annotation at p. 52, Sfera del mondo [Piccolomini 1573]

Figure 4. Another detail of Galileo’s annotation at fol. 2r, De le stelle ﬁsse [Piccolomini 1570]

Figure 5. Detail of Galileo’s annotation at fol. 2v, De le stelle ﬁsse [Piccolomini 1570]

find the same characteristics in the marginal notes on pages 51 and 74 of the Sfera del mondo (Figs. 2 and 6). Other significant features are Galileo’s way of writing “che”, “del”, and letters “q”, “d” and “z”. These and other traits establish such a strong similarity between Galileo’s authentic calligraphy at De le stelle fisse and the handwriting of the marginal notes at Sfera del mondo that they leave no doubt that the later ones were also written by Galileo, probably around the same time. The comparison with other autograph writings by Galileo, of the period from 1588 to 1597, strongly suggests that he wrote the marginal notes to Piccolomini’s Sfera del mondo around 1590—that is, about the same time when he requested his father to send him his copy of the Sfera. Additional evidence will be published by us in a forthcoming paper. 142 Roberto de Andrade Martins & Walmir Thomazi Cardoso

Figure 6. Detail of annotation at p. 74, Sfera del mondo [Piccolomini 1573]

7 The inﬂuence of Piccolomini’s Sfera on Galileo

Galileo’s marginal annotations to Piccolomini’s Sfera del mondo do not contain any discussion, criticism, or comparison with other works, nor any original matter. For the most part, Galileo underlined or otherwise marked parts of the book that seemed relevant to him, and wrote at the margins some words or sentences that replicate information contained in the printed text itself. This characteristic way of making notes suggests that, at that time, Galileo was just beginning to learn the foundations of astronomy and that Piccolomini’s work was one of the first (or the very first) astronomical treatises he ever studied. Therefore, the marginal remarks can show which specific topics and ideas called the attention of the young Galileo. The only marginal note found in the first book of Piccolomini’s treatise appears at page 23 (chapter 9), where the author discussed the circles on a sphere and states that two maximum circles cannot be parallel. There are no other annotations in the first book, most of which contained geometrical prerequisites to the study of astronomy. Galileo had studied Euclid under Ricci, before 1590. It seems likely that he thought that the first book of the Sfera del mondo did not contain anything new for him. However, chapter 9 did contain an important description of the sphere, its diameter, hemisphere, as well as the axis and poles of a rotating sphere, corresponding to the very beginning of Sacrobosco’s Sphæra. This elucidation is noticeably absent in the beginning of the Trattato, although it is seldom missing in any other elementary astronomical treatise of that time. Galileo’s first annotation in the second book is found at page 44 (chapter 7), which discusses the position of the Earth in the universe. Piccolomini’s original presentation of the proofs for the central position of the Earth is different from Sacrobosco’s. They were already contained in the first edition of his work (1540) and later appeared in Clavius and Giuntini’s treatises. The Trattato Galileo’s Trattato della sfera ovvero cosmografia and Its Sources 143 also presented a condensed version of those arguments [Galilei 1891, 220]. At page 47 we find an annotation of Galileo stressing another argument for the central position of the Earth : if it were not in the middle of the universe, the eclipses of the Moon would not occur when the Sun and the Moon are in diametrically opposite directions. This was not a new contention since Piccolomini himself ascribed it to Ptolemy and Averroes ; but it was a rather unusual one and it was reproduced in the Trattato [Galilei 1891, 221]. Chapters 8, 9 and 10 of the second book are particularly noteworthy [Piccolomini 1573, 48–55]. They present proofs for the immobility of the Earth—a subject that would later become the center of Galileo’s interests. There are altogether fourteen marginal notes on these pages, disclosing the strong appeal of this subject for young Galileo. Among the several arguments, Piccolomini discussed the vertical motion of a stone, comparing it to what supposedly occurs when the person throwing the rock is moving in a ship [Piccolomini 1573, 52]. The ship argument was rather uncommon and it was not reproduced by Clavius. It was annotated by Galileo (“very beautiful reason to prove that the Earth does not move circularly”) and appears in the Trattato [Galilei 1891, 224]. There are many other peculiar topics in Piccolomini’s work that called the attention of Galileo, as shown by his marginal notes. Most of them have a correspondence in the text of the Trattato, as we will disclose in a forthcoming paper. This fact suggests that Galileo was strongly influenced by Piccolomini’s Sfera del mondo and that it was a major source of the content of the Trattato. Of course, many of those points also appear in other astronomical treatises of that time—such as Clavius’ book. However, there is no known copy of Clavius’ work with annotations by Galileo. Hence, there is no direct evidence that Clavius’ commentary on Sacrobosco was read by Galileo and influenced him before the time when he wrote the Trattato della sfera.

8 Other sources of the Trattato della sfera

Although the influence of Piccolomini’s Sfera del mondo upon the composition of the Trattato might have been very strong, it certainly could not be the only source used by Galileo. Indeed, there are some features of the Trattato that did not appear in Piccolomini’s work. The very title Trattato della sfera ovvero cosmografia was extraordinary and implied an equivalence between astronomy and cosmography that was not acceptable to most authors of that time—including Piccolomini. Galileo might have been influenced by Oronce Finé’s De mundi sphaera, sive cosmographia or by Francesco Barozzi’s Cosmographia, a commented version of Sacrobosco’s Tractatus de sphæra. Barozzi’s Cosmographia is especially relevant since Galileo had a copy of this book [Favaro 1886, 260], and some parts of the Trattato may have been inspired by this work [Martins & Cardoso 2008]. 144 Roberto de Andrade Martins & Walmir Thomazi Cardoso

The methodological discussion at the beginning of Galileo’s work is different from Sacrobosco’s first chapter, but similar accounts were published by Francesco Barozzi, Oronce Finé, Francesco Capuano and Christoph Clavius [Martins & Cardoso 2008]. A similar treatment also occurs at the beginning of Peucer’s work owned by Galileo [Peucer 1573, 1–8]. Perhaps he was familiar with one of those sources when he wrote the Trattato. Galileo did certainly study Georg Peurbach’s Theorica planetarum, because he taught it at the universities of Pisa and Padua. In Peuerbach’s work, we find some subjects that were not discussed in Sacrobosco’s Sphæra but are contained in the Trattato, such as the phases and visibility of the Moon [Peurbach 1569, 95]. Galileo could also have used instead Barozzi’s Cosmographia for this part of his work [Barozzi 1585, 288, 295, 300]. The declination of the ecliptic presented in the Sfera del mondo was 24° [Piccolomini 1573, 94, 101]—a rather unusual value ; the Trattato used 23.5° [Galilei 1891, 230, 233] and therefore this figure was copied from another source. Another relevant difference is the treatment of longitude and its deter- mination by observations of the eclipse of the Moon, described in the Trattato [Galilei 1891, 241–242] in a way clearly independent from Piccolomini’s exposition [Piccolomini 1573, 110–115]. Although we have no information about the time when Galileo studied Clavius’ commentary on Sacrobosco’s Sphæra, its influence upon the Trattato cannot be excluded, since several of its passages exhibit strong similarities to Clavius’ work [Martins & Cardoso 2008].

9 Final remarks

This paper analyzed some peculiarities of Galileo’s treatise, comparing it to several other vernacular astronomical works of the sixteenth century and discussing its likely sources. Contrary to previous claims, we argued that Christoph Clavius’ commentary on Sacrobosco’s Sphæra cannot be regarded as its only or main inﬂuence. A likely inspiration for Galileo’s treatise is Alessandro Piccolomini’s Sfera del mondo, a work that was carefully studied by Galileo and that anticipated several peculiarities of the Trattato della sfera. Our claim for this inﬂuence is strengthened by our analysis of Galileo’s copious marginal notes found in a copy of Piccolomini’s book. We admit, however, that there are features of the Trattato that have no counterpart in the Sfera del mondo. Therefore, the issue of Galileo’s sources is very complex and has not been completely solved.

Acknowledgments

One of the authors (Roberto Martins) is grateful for the support received from the Brazilian National Council for Scientiﬁc and Technological Research Galileo’s Trattato della sfera ovvero cosmograﬁa and Its Sources 145

(CNPq) and from the São Paulo Research Foundation (FAPESP) during the development of this work. The authors are grateful to Biblioteca Nazionale Centrale di Firenze and Museo Galileo for authorizing the reproduction of the images of this paper.

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Thoughts on Galileo’s Work

Marco M. Massai Pisa University and Istituto Nazionale di Fisica Nucleare, sezione di Pisa (Italy)

Résumé : J’analyserai des déclarations et des idées proposées par Galilée dans son Sidereus Nuncius et j’en proposerai des interprétations non convention- nelles. Je vais essayer de montrer qu’avec un esprit logique et de nouvelles hypothèses, il est possible de parvenir à de nouvelles interprétations possibles de la pensée de Galilée.

Abstract: I wish to analyse certain statements and ideas proposed by Galileo Galilei in his Sidereus Nuncius and will provide non-conventional interpretations of them. I will try to show that with logical connection and new hypothesis, it is possible to make some new possible interpretations of Galileo’s thoughts.

1 Introduction

Che fai, tu, Luna, in ciel? dimmi, che fai, sileziosa luna? [What do you do Moon in the sky? Tell me, what do you do, silent Moon?]; [Giacomo Leopardi, Canto di un pastore errante dell’Asia, 1830].

Searching for new and original interpretations of Galileo Galilei’s statements, ideas and proposals is an enterprise which is surely more doomed to failure than others. This is because of the quantity and especially the quality of the work developed by many historians of science in an attempt to profoundly grasp Galileo’s thought. New light has been thrown on the interpretation of Galileo’s writings only after many decades of seemingly consolidated positions on Galileo’s contribution to the development of science and the basic work of Stillman Drake in the second half of the twentieth century. According to

Philosophia Scientiæ, 21(1), 2017, 149–164. 150 Marco M. Massai some well-known historiographic interpretations such as the one offered by Alexandre Koyré, Galileo was not a completely experimental scientist.1 Since there have been new interpretations of Galileo’s work, it makes sense to look for new nuances within the scientific contribution. A significant part of Galileo’s interpreters are not experimental physicists. It is thus reasonable to think that the perspective of an experimental physicist could, to an extent, shed new light on Galileo’s research. I have spent decades of my working life constructing and using measuring instruments to determine operating characteristics, collect large amounts of data, measure physical quantities and finally to confirm or disprove the laws of physics. Because of this, I might perhaps be able to provide a new standpoint from which to interpret aspects of Galileo’s work. I shall focus on the Sidereus Nuncius [Starry Messenger], published in Venice on March 12th 1610. Although this work does not present a philosophical elaboration of the expounded observations, it offers a rich series of discoveries and insights for astronomy and science. My general idea is that Galileo is basically an experimental scientist. This idea is connected with the question of whether Galileo was the inventor of his telescope.2 I believe that he was and this position will be clarified in my paper. In fact, the few traces of information that can be found about the “toy” which was patented in September 1608 by three lenses manufacturers in the town of Middelburg in Zeeland would not suggest that it could have been transformed into a powerful instrument to investigate the skies. Indeed, the rudimentary tools found in European fairs for almost a year had aroused no curiosity or interest in many European scientists with the exception of Thomas Harriot. However, even though Harriot could claim the primacy of having first pointed a telescope at the Moon, he only managed to produce and publish a very rough and elementary picture from which no new and useful information can be drawn. Galileo’s intuition in constructing and using (August 1609) the telescope cannot be a pure coincidence.

1. See the classical work [Koyré 1939]. Without claiming to oﬀer an exhaustive series of references on the complex and debated relations between theory and experiments in Galileo, I would also mention: [Agazzi 1994], [Akhutin 1982], [Bucci 2015], [Crombie 1981], [De Caro 1993], [Drake 1970, 1985, 1986, 1990], [Dupont 1965], [Erlichson 1993, 1994], [Fehér 1982], [Feyerabend 1975], [Finocchiaro 1974, 1977, 1980], [Gingerich 1993], [Giusti 1994], [Grigorian 1983], [Gruender 1981], [Maar 1993], [MacLachlan 1976], [Naylor 1980, 1990], [Piccolino & Wade 2014], [Pizzorno 1976], [Prudovsky 1989], [Segre 1980], [Shea 1977], [Takahashi 1993a,b], [Wisan 1981]. 2. The literature on Sidereus Nuncius, the role of the telescope within Galileo’s work, the lunar observations and the Jupiter satellites are simply huge. Interpretations concerning the Sidereus are almost as abundant as the literature. I mention the useful text by [Reeves 2014], in which the whole literature concerning the Sidereus Nuncius from 2000 to 2011 is referred to. I shall only quote here the literature from which I have drawn some inspiration—without fully agreeing—for this paper: [Bucciantini, Camarota et al. 2012], [Fabbri 2012, 2015], [Bussotti 2001, 1–88], [Galluzzi 2011], [Giudice 2014], [Molaro 2013], [Ronchi 1964], [Wootton 2009]. Thoughts on Galileo’s Work 151

Figure 1. The cover of the Sidereus Nuncius (1610)

Galileo’s well-known experience in designing technical instruments is a necessary element to understand his success with the telescope. His great technical experience alone enabled him to connect the construction and functioning of a machine with the theoretical principles underpinning that very functioning. Therefore, Galileo was able to link the experimental dimension with the theoretical—logical and mathematical—aspects. On the other hand, Galileo himself stated in the initial pages that he was not the original inventor of the telescope, referring instead to a “Flemish”.3 However, soon after, he indicates a speciﬁc way to calibrate the instrument he built and invites the reader to try this. However he also suggests they take care and use great skill otherwise failure is assured as he was to repeat later before the description of the discovery of the Medicean Planets:4

2 The aim of this work

This paper’s aim is to ﬁnd a few sentences and hints in this work which was written almost spontaneously by Galileo and thus make a diﬀerent interpretation to the widely shared and consolidated analyses. I will quote and describe

3. “[...] rumor ad aures nostras increpuit, fuisse a quodam Belga Perspicillum elaboratum [...]” [Galilei 1610, III, 60]. 4. “Illos tamen iterum monitos facimus, Perspicillum exactissimo opus esse [...]” [Galilei 1610, III, 80]. 152 Marco M. Massai statements on several issues and I take a free approach to the text without following the order emerging from the pages of the Sidereus Nuncius. In this manner, it will introduce some interpretative hypotheses to fully grasp the meaning of some of Galileo’s statements which, at ﬁrst glance, might appear rather odd. I am aware what I am claiming verges on the provocative but I will provide evidence to back up my view.

3 A methodological question

A first, purely methodological question, emerges through analysis of Galileo’s progress, starting with the first observations of “the body of the Moon”, where he identifies bright spots in the dark and dark shadows on the sunlit parts. He carried out nightly observations and obtained both still images and something which could be interpreted almost as a living picture that he seeks to analyze.5 This is a crucial first step: Galileo builds up a three-dimensional representation in movement of what he can see and observe, each time more carefully. He builds a model that he refines night by night as a consequence of his new observations. These models are confirmed by a long series of observations and also by the logic with which Galileo strongly connects those observations. He is aware that his description of the Moon is completely different from that accepted by the astronomers and supported by the Church. In spite of this, he remains firm in his opinion that the Moon is similar to the Earth in all respects and is not a body made out of ethereal matter—the first of the sovralunare world where perfection reigns. And this is in fact a truly great revolution in thought. However Galileo does not just end with a description, albeit far more refined than previous efforts. He goes beyond this and determines the height of the Moon’s mountains which was a mere hypothesis at the time.

4 Mountains on the Moon

Measuring gives a sense of reality which is diﬀerent from that obtained by descriptions. Measuring involves transforming an “object’s” characteristic into numbers. Initially an object may only exist in thought and with measurement it acquires a connotation of reality. This is the essence of any physical quantity, where the term “physical” expresses the link between our mental model and a series of facts related to external reality. Galileo performed the measurement of the height of a mountain on the Moon and found it was 4 Italian miles in height which was a very surprising ﬁnding.

5. “[...] sed, quod maiorem infert admirationem, permultae lucidae cuspides intra tenebrosam Lunae partem [...] quae, paulatim, aliqua interiecta mora, magnitudine et lumine augentur [...]” citep[III, 64]Galileo1610mass. Thoughts on Galileo’s Work 153

Figure 2. Procedure followed by Galileo to measure the height of the mountains on the Moon

However Galileo needed to go further because although he was proposing a simple but eﬀective three-dimensional, geometric model, this was not easy to accept. In fact, the result of his calculation is a value that surprises and maybe even disconcerts because, as he explicitly points out, it goes far beyond the value estimated at that time for the height of the highest European mountains. However, we must remember that the height of a mountain was measured from the valley bottom and this means the otherwise inexplicable discrepancy with the actually measured value is not actually surprising. These surprisingly high mountains could tarnish the validity of Galileo’s hypothesis which had upset popular opinion. After centuries of stable belief, it was well established that the lunar surface was smooth, although the question of the nature of the Moon’s many, vast and irregular spots remained to be solved and were indeed the subject of many inferences. Nonetheless, the clarity and the simplicity of Galileo’s reasoning supported a measurement which made the most amazing hypothesis reality.

5 The lunar atmosphere

Galileo wrote many pages on the description of the Moon. Indeed, he was aware that he needed to provide a series of evidence to support his hypothesis. However there is one aspect which should perhaps be more thoroughly inves- tigated and may hide a more subtle interpretation. Galileo gives one more recommendation on the quality of the instrument to anyone who wanted to conﬁrm his observations. Immediately afterwards, he deals with a question which could also aﬀect his explanation of the nature of the Moon namely why do the edges of the Moon circle appear so perfectly smooth and not rough “like 154 Marco M. Massai

Figure 3. The Moon in a schematic drawing by Galileo a gear”?6 This objection most certainly was aimed at preventing any critical remarks about his work in the future. However, such an objection has also the aim of reinforcing his own work by showing the reader that he is ready to take possible criticism of his statements into account. Nevertheless there is something more. In fact, Galileo gives two explanations why it is not possible to distinguish the sawtooth mountains’ profile on the Moon’s edge. The first example consists in the vision of several ridges of mountains following each other at increasing distances. This view is also typical on the Earth, which is a confirmation that the nature of the Moon is the same as Earth’s . This similarity is now taken for granted. The yokes appear almost continuous because the alternation of peaks and valleys of a chain are positioned at random with respect to the subsequent chain. The mountains on the Earth thus explain mountains on the Moon. The second explanation is more picturesque but perhaps less conclusive because the scale factor is completely different and because Galileo now compares the mountains’ dimensions with those of sea waves. Galileo uses a beautiful image of a stormy sea with rows of waves advancing parallel to each other but which cannot be perceived individually. Therefore, we see a substantially continuous horizon. These explanations appear logical and strengthen the interpretation of the lunar world as a double of the terrestrial world. Also, they are widely sufficient to justify the observational data, or rather in this case the lack thereof. However, Galileo is not satisfied with this explanation and looks for another. This requires a further hypothesis—the presence on the Moon of a shell-type “atmosphere” similar to the Earth’s and

6. “Praetereundum tandem non est, quanam ratione contigat, ut Medicea Sidera, dum angustissimas circa Iovem rotationes absolvunt, semetipsis interdum plusquam duplo maiora videantur” [Galilei 1610, III, 95]. Thoughts on Galileo’s Work 155 thus once again providing a strong analogy between the Moon and the Earth.7 The lunar atmosphere allows Galileo to justify the lack of definition observed on the edge of the Moon which prevents us from perceiving and distinguishing the individual peaks. Galileo supports this third explanation with one of the few drawings in Sidereus Nuncius. This is not an accurate representation of what he sees but a model. In this picture, a circular ring around the rim of the Moon appears clearly. In other words, Galileo aims to provide an unambiguous description to avoid any misunderstanding. At this point in his work, Galileo is in a hurry to describe his most revolutionary discovery—the Medicean Planets, the last and most important subject of the whole book. Therefore, while this additional explanation may seem superfluous and redundant based on the introduction of a new idea, it could be interpreted as a further similarity between the Moon and the Earth. Perhaps another reason led Galileo to go much further than a mere description of what he saw and also beyond the reference that the reader can share. This is the subject of a hypothesis that I shall propose but first it is necessary to point out another important step in advance.

6 The atmosphere on Jupiter

The last part of Sidereus Nuncius is dedicated to the discovery of the four Medicean Planets. The description is enriched by dozens of observations collected night after night with meticulous precision and reported in the text with exact drawings, one for each day. Much earlier in the book when Galileo reaches the end of his observations, he introduces the hypothesis that an atmosphere surrounds another celestial body—Jupiter in this case (see above note 6). Galileo carefully observes the relative positions of the four satellites whose motion is easily perceivable from one night to the next and measures this with accuracy. At the same time, Galileo also tries to measure the intensity of the satellites’ light and notes this with care. However, it must be said that the relative position and the angular distance of a star or planet is an easily measurable quantity and therefore not ambiguous. Instead, it is diﬃcult to measure the light intensity of an object in the black night sky, particularly if this object is observed very close to another brighter body. This is exactly the case when the satellites of Jupiter are observed by Galileo shortly before or after their occultation by the far larger body of the planet. Galileo has already encountered a similar problem, i.e., the problem of evaluating the magnitudes of a celestial object through his telescope. He deals with it some pages earlier when describing the diﬃculty of measuring the

7. “Huic rationi altera subnecti potest, quod nempe circa lunare corpus est, veluti circa Terram, orbis quidam densioris sunstantiae reliquo aethere [...]” [Galilei 1610, III, 70]. 156 Marco M. Massai apparent magnitude of the stars he had just discovered. Indeed, he describes how the instrument behaves very strangely and shows different magnifications depending on whether the observed object is a planet or a star. In fact, Galileo could not know the physical explanation of the optical aberration phenomenon which is usually associated with the use of an optical instrument. Therefore, he is hesitant to give the description of a mechanism according to which the instrument is apparently able to enlarge the observed objects in different ways, i.e., with different magnifications. For the same reason, he could not give any justification for the apparent malfunctions found in the instrument itself, like, for example, the “brightening rays” he observes in the stars. In addition, Galileo misses another crucial issue in the understanding of what “you really see” in the ocular lens which is related to a complex mechanism that would be called “physiology of vision” in the future. At this point, Galileo did not have the time to pursue this matter because he urgently needed to make public his numerous observations of all visible objects in the sky of Padua as soon as possible. In the winter of 1610, despite not being an astronomer he was about to totally and definitively revolutionize the image of the sky and also suggest the need to overhaul the model of the Cosmos through his rich innovations. Indeed, the Aristotelian-Ptolemaic model endorsed by the Church encountered a great deal of difficulty in facing up to the whole complex set of new ideas discovered by Galileo. This depended on the geometric complexity required to take the new, observational data into account and also on the weakness of some of the principles on which the Aristotelian-Ptolemaic system was based. These principles were now being demolished. First of all, the skies’ immutability (a secular assumption) was a condition which appeared a necessity because the sky was considered a divine work. The Copernican model was published almost 70 years earlier and was still seen only as an original, mathematical exercise to provide new ephemerides. Nevertheless, from a physical and mathematical point of view the discoveries of Jupiter’s satellites was problematic for both the Ptolemaic and Copernican systems (at least in the version expounded by Copernicus where the system was based on circular motion). Though the Astronomia Nova were published in 1609, when Galileo published the Sidereus, Kepler’s work had little influence on astronomers. However, while there were several difficulties with the Copernican system, the situation for its Ptolemaic counterpart was undoubtedly even worse. Thus Copernicus’s theory represented a very new horizon for knowledge of the sky. A further remarkable element is explained in the last pages of the Sidereus Nuncius where Galileo writes about the variable intensity of the light of Jupiter’s satellites when they are close to the planet. This observation may indeed appear superfluous when compared with the astonishing announcement of the existence of four new planets. Moreover, Galileo does not provide many details regarding this problem although he was precise and meticulous in describing these observations. Notwithstanding, these references appear significant. Galileo advances a hypothesis that can explain this apparent Thoughts on Galileo’s Work 157 variability of the light reflected by satellites after a very long thought process. Initially, he provides the hypothesis that their orbits can be highly elliptical, being the major axis along the line of sight of the observer. This meant he could explain the weakening of the light by the increased distance of the satellite. Galileo certainly knew from his previous experience that light intensity decreases with increasing distance from the source. Whatever the function which connects light intensity and the distance of the light source, it seemed implausible that the decrease in light intensity was as strong as Galileo’s observation suggested. Galileo saw that the brightness of the satellites was at its minimum when they came close to the planet. The only theoretical possibility is thus that the satellite moves away far from Jupiter, its center of motion and therefore that the orbit must be greatly elliptical in relation to the weakening of the light output observed and recorded by Galileo. The logic that supports this argument may seem compelling. However, Galileo does not bother to justify the images described although the satellite’s brightness does not change when they depart even slightly from Jupiter. Anyway, this behaviour of the phenomenon is not compatible with the previous hypothesis which seems artificial. However, this hypothesis is a scientific-literal artifice used by Galileo to lead up to a second explanation. In this explanation, he returns strongly to the hypothesis that an atmosphere could surround a celestial body—Jupiter in this case. The presence of a dense shell would weaken the light absorbing a large proportion of the emission from satellites.8 With this explanation, Galileo seeks to make the results of his observations compatible with a model—some demonstrations, which he uses to validate the sensitive experience.

7 A hypothesis

Given the deep differences between the Moon and Jupiter as observed by Galileo, the existence of an atmosphere on both could appear a strange coincidence. This incited me to search for a different, more general motivation underlying Galileo’s words. It is fascinating to imagine that Galileo had a common purpose for giving an account of two different events which acquires the value of a universal statement—planets have a shell of air around themselves. Nevertheless, this statement is not particularly significant if it is limited to an explanation of observational data but if used as a starting hypothesis which is well supported by observations, it could have deep, cosmological implications. It would indeed be linked to the disputes concerning the De revolutionibus orbium coelestium of Nicolaus Copernicus. In fact, if the Moon and the planets

8. “Non modo Tellurem, sed etiam Lunam, suum habere vaporosum orbem circumfusum, tum ex his quae supra diximus [...]: at idem quoque de reliquis Planetis ferre iudicium congrue possumus; adeo ut etiam Iovem densiorem reliquo aethere ponere orbem [...]” [Galilei 1610, III, 95]. 158 Marco M. Massai

Figure 4. Saturn and the phases of Venus as represented by Galileo in Il Saggiatore

have an atmosphere in their circular movement around their center-of-motion, then the motion itself is not incompatible with the existence of an atmosphere. Also, if this logical consequence is valid for the Moon and planets as shown by Galileo and if they are similar to the Earth, this obstacle is overcome by imagining an Earth moving around the Sun. Therefore, the movement of the Earth around its axis and around the Sun does not imply the atmosphere is stripped away. Of course, such a deduction is not evident or explicit in Galileo’s words. Therefore any such proposal should be maintained in the ﬁeld of the inferences which can be made by speculating on Galileo’s words to reconstruct his possible thought. However, if this hypothesis is correct, why was Galileo not more explicit? This is impossible to answer. Again, it should be recalled that the heliocentric system was not accepted by most theologians, astrologers, astronomers and scholars, even if De revolutionibus had not yet been rejected. As is well known, Copernicus’s theory was condemned in 1616 while Galileo was to add many other discoveries to those announced in the Sidereus Nuncius in the 1610-1616 period. All these observations were to be as fundamental as the previous ones. Galileo discovers Saturn to have a strange shape like a small olive. He ﬁrst recognizes the sunspots to be on the Sun’s surface and not far from the Sun.

Finally he made his most important observation. Venus was watched nightly and every week which showed phases similar to those of the Moon. It therefore became geometrically evident that the Mother of Loves was moving around the Sun as Copernicus had suggested, unheeded, for decades.

This, in fact, was the most conclusive observation in support of the Polish astronomer’s revolutionary idea. Thoughts on Galileo’s Work 159 8 On the nature of the nebulous stars

Let us make one last incursion into the sensitive and dangerous ﬁeld of “possible interpretations” of what Galileo wrote in the Sidereus Nuncius. I attempted to read between the lines of this small book, a true cornerstone of the history of astronomy and also a turning point for the whole of science. The next step examined is that in which Galileo describes what he sees pointing his telescope towards the nebulae, or nebulous stars. Immediately he proposes a new radical change in the interpretation of those indistinct spots that had long been considered one of the most mysterious objects in the sky when viewed with the naked eye. Philosophers, priests, astronomers had provided several proposals to explain their nature which were mostly mythologically based. Also the analogy with the Milky Way had been used to ﬁnd some explanations but this was done without “sensible experiments and necessary demonstrations”.9 Galileo does not hide his surprise and also his satisfaction in revealing what he sees in the ocular lens of his telescope. This is, once again, a real revolution. In fact, the interpretation of these objects changes dramatically because Galileo reveals that they are composed of myriad stars which are so weak and apparently mutually close that they cannot be distinguished by the human eye. Thus one has the perception that they are made of a continuous material. This was the “essence” of the nebulous stars before Galileo discovered and announced their discrete nature.10 Certainly, this discovery by Galileo—this invention, because he based this idea on observations—is not among the most remarkable if compared with the others expounded in the Sidereus Nuncius. In fact, the other discoveries represent the foundations of a completely new scheme in the description of the sky. This scheme supports the heliocentric model. But another revolution which would have to wait several more centuries to be accepted was smouldering under the ashes. It was proposed more than two thousand years ago and it concerns the real composition of nature of matter around us. Does matter have a continuous structure or is it composed by the atoms that Democritus and Leucippus imagined in a remote corner of Greece? In those years the implications of the atomistic hypothesis were rather hazardous. Asking yourself this question at that period had a dangerous implication because the atomic theory was strongly opposed by the Church. Consequently, philosophers who supported it were considered blasphemous and

9. “Amplius (quod magis miraberis), Stellae ab Astronomis singulis in hanc usque diem NEBULOSAE appellatae, Stellarum mirum in modum consitarum greges sunt [...]” [Galilei 1610, III, 78]. 10. “Quod tertio loco a nobis fuit observatu, est ipsiusmet LACTEI Circuli essentia, seu materies, quam Perspicilli beneﬁcio adeo ad sensum licet intueri. [...] Est enim GALAXIA nihil est aliud, quam innumerarum Stellarum coacervatim consitarum congeries” [Galilei 1610, III, 78]. 160 Marco M. Massai

Figure 5. Nebulous stars in the Galileo’s drawings heretical and therefore under the menace of the Inquisition. Why not therefore imagine that Galileo, who was always ready to make daring and surprising logical connections and analogies, had found this idea attractive. After the Milky Way and the nebulous stars, i.e., celestial matter and terrestrial matter, too, could he reveal his true, discrete nature?

9 Conclusions

My references to Sidereus Nuncius and to the logical deductions developed in this work by Galileo have been based merely on what Galileo himself explicitly wrote. Certainly, they are read today more than 400 years later at the dawn of the twenty-first century. Obviously, the meanings that we now assign to the words of Galileo, have in some cases been completely revolutionized by the historical and scientific background. Profound differences are also evident from an epistemological point of view. However, many aspects of Galileo’s thought are still alive and his teaching is still of great significance and relevance particularly his reference to sensible experiences and necessary demonstrations. Galileo left his revolutionary message relying upon experiments and observations, rather than upon the Thoughts on Galileo’s Work 161 theoretical, dialectic sophistry which characterizes the speculation of several other philosophers. As I have tried to show, it is not impossible to search for and maybe even find different interpretations from the classical one which is accepted by history of science. It is very difficult to determine whether they are correct and really match with the thought of Galileo and I think no one can state this with reasonable certainty. However nor perhaps can this be completely denied.

È la prima volta che la luna diventa per gli uomini un oggetto reale [...]. [Calvino 1968]

Acknowledgments

I wish to express my gratitude to Paolo Bussotti for a great deal of precious advice.

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Annibale Mottana Roma Tre University (Italy)

Résumé : Cette étude montrera que la confirmation expérimentale des conjectures constituait chez Galilée l’objet d’un souci constant, et ce dès le début de son activité scientifique. À l’appui de cette thèse, je citerai ses premiers travaux expérimentaux et montrerai avec quelle profondeur il était capable d’analyser par lui-même ses résultats. Je démontrerai ensuite qu’il a discuté de ses résultats avec d’autres scientifiques afin de confirmer ses intuitions initiales. C’est en les consultant et en bénéficiant de leurs conseils qu’il put mener ses intuitions à leur état final.

Abstract: This study aims to show that the experimental confirmation of his conjectural ideas was a constant concern for Galileo from the very beginning of his scientific activity. In order to support my view, I shall quote his early experimental work and first show the extent to which he was able to analyse his results in depth by himself. Then I shall give evidence that he discussed his results with other scientists to help confirm his initial intuitive ideas. He consulted with them and their advice helped him to come to his final conclusion regarding his insight.

1 Introduction

Albert Einstein called Galileo Galilei “the father of modern physics and in fact of the whole of modern natural science” [Einstein 1933, 164]. He justiﬁed his judgement with a rather shocking statement: “pure logical thinking can give us no knowledge whatsoever of the world of experience; all knowledge about reality begins with experience and terminates in it”. Indeed, this is what Galileo is credited to have done. This study aims to show that the

Philosophia Scientiæ, 21(1), 2017, 165–179. 166 Annibale Mottana experimental confirmation of his conjectural ideas was a constant concern for Galileo from the very beginning of his scientific activity. In order to support my view, I shall quote his early experimental work and first show the extent to which he was able to analyse his results in depth by himself. Then I shall give evidence that he discussed his results with other scientists to help confirm his initial intuitive ideas. He consulted with them and their advice helped him to come to his final conclusion regarding his insight.

2 Preliminaries

Galileo’s approach to philosophia naturalis1 was the following: he strongly believed that Nature is the work of God and only needed a correct interpretative approach to be fully understood. Moreover, he thought that the scholars he called “filosofi in libris” [Galilei 1890-1909, XI, 11–12, hereafter OG], did not have the correct approach in basing their understanding on reading Aristotle’s scientific books.2 Rather, he believed that the correct interpretation of a natural fact should stem from direct, close observation and from well-planned and carefully performed experiments to verify it. Finally, he believed that a true scientist should be able to work out the collected evidence “in lingua matematica, e i caratteri son triangoli, cerchi ed altre figure geometriche, senza i quali mezzi è impossibile a intenderne umanamente parola; senza questi è un aggirarsi vanamente per un oscuro laberinto”3 [OG, VI, 219]. In fact, Galileo drew the best from the books of Euclid, Archimedes and other ancient mathematicians,4 whose greatness he could fully appreciate. However, he constantly favoured exploring “questo grandissimo libro, che essa natura continuamente tiene aperto innanzi a quelli che hanno occhi nella fronte e nel cervello”5 [OG, XI, 109]. In other words, he pushed himself to engineer new experiments that mimicked and explained what he could directly observe in

1. At that time, one would call philosophia naturalis (= natural philosophy) the entire scientific knowledge of the Earth because nobody had yet had the idea that making artificial materials was possible. God had created everything from chaos. 2. These include both Aristotle’s original or attributed Greek works and also all interpretations by his followers, spanning from the peripatetic school to the scholastic one which takes into due consideration the Islamic interpretations available in Latin translation. Galileo was careful not to mention the Bible in this context but could not avoid involving it implicitly. 3. “In the language of mathematics, the characters of which are triangles, circles and other geometrical figures; without these it is impossible for a man to understand a word; without them it is like roaming in vain through a dark maze.” All translations from Italian are by me. 4. He knew only those translated into Latin and Italian, as well as their medieval and Renaissance comments because his knowledge of Greek was only “non mediocre” (= just ordinary) [Viviani 1674, 2], [OG, XIX, 601]. 5. “This very large book which Nature itself constantly keeps open in front of those who have eyes on their forehead and in their brain.” Galileo’s La bilancetta: The First Draft and Later Additions 167 the surrounding world before applying mathematics to their interpretation. Moreover, his statement above also clarifies the fact that he felt a definite separation existed between the simple observation of natural events through sight and the understanding thereof conceived in and by the brain. Valuable results could only be achieved when scientific observation and technological experience were directly combined. He began practicing science in this way as early as in his youth [Fredette 1972], [Drake 1978, 1986], around the time when he started his training at Florence’s newly founded Accademia delle Arti del Disegno, an institution whose purpose was to link science concepts to engineering practices. There, he met some of the best mathematicians, natural scientists and technologists of late Italian Renaissance and became confident in considering nature as just one of the two means chosen by God to display his immense power. The result of this form of training was the development of the approach that has only recently been named “technoscience” [Gorokhov 2015]. In Florence at the age of 22, Galileo gave the first demonstration of his new ideas on how to practice science, i.e., his first demonstration of what we would now call “technoscience”. He wrote a short essay on a new method of using scales that is now entitled La bilancetta (= The little balance) [OG, I, 215–220],6 and used this instrument to make measurements of selected metals and gemstones. This whole way of thinking was already rooted in Galileo’s mind long before he made it clear in his famous “theological letters”, i.e., the two letters he wrote to his pupil Benedetto Castelli on December 21, 1613 [OG, V, 282– 288] and to grand-duchess Cristina di Lorena on June 1615 [OG, V, 309– 348]. In both those letters, Galileo claims that God makes his work known to man using two equal sources of evidence—the Bible and nature. The Bible had to be interpreted by a human language and obviously, as a pious Catholic, he meant Latin. This often makes God’s words difficult to grasp and easily misunderstood. The evidence inherent in nature must derive from data obtained through laborious observations and planned experiments which need to be supported by a skilled use of mathematics. Therefore, the path towards understanding which proceeds through science can never fail because

6. According to Vincenzo Viviani, Galileo’s last pupil and the collector of most if not all his papers, he wrote this essay in 1586 [Viviani 1674, 9], [OG, XIX, 605]. Stillman Drake challenges this dating which is given in a marginal note added to a letter by Viviani to Cardinal Leopoldo de’ Medici on April 29, 1654. He supports his challenge [Drake 1986, 437–439] by citing the watermark on a paper sheet near the end of the unfinished dialogue De motu, which is bound in the same volume just before La bilancetta. This watermark could show that Galileo wrote the dialogue on sheets acquired while teaching at Siena during the 1586-1587 academic year. In such a dialogue, Galileo refers to have recorded previously two sets of measurements of gold-silver alloys by the hydrostatical method. Drake infers from this that Galileo wrote La bilancetta later than those measurements namely when he had left Siena in late spring 1587 and moved back to Florence. Notably, Drake contradicts himself in the table that follows at p. 440: there, he lists La bilancetta as written at Florence in 1586 and the dialogue De motu at Siena in 1586-1587. 168 Annibale Mottana it consists of logical figures. Galileo’s juvenile writings have fortunately been preserved in the Florence National Library and they provide excellent proof on how early and firmly he adopted this belief [Fredette 1972], [Drake 1978, 6–7], [Camerota 2004, 45–48], [Gorelik 2012], etc. In particular, La bilancetta is the first of Galileo’s writings in Italian, the language in which could best express his ideas, and thus deserves detailed study.

3 The edited text of La bilancetta

La bilancetta was Galileo’s ﬁrst original full scientiﬁc essay but it is also a very innovative text because it alone enabled him to open up a new approach to upgrade hydrostatics and make it a method for studying solid matter. His innovation occurred almost two millennia after Archimedes of Syracuse had laid the fundamentals of that branch of Physics.7 Therefore, it was a real misfortune for the development of philosophy of science that most historians underrated La bilancetta for a variety of reasons:

1. It is a minor contribution to science, when compared to the later novelties on astronomy and general physics that made Galileo foremost among contemporary scientists. 2. The fact that it is in Italian could perhaps suggest that the author himself did not consider it worthy of particular attention. Indeed, at that time, high-quality scientiﬁc papers were written in Latin to reach the largest possible audience within the scholarly community. Galileo was an exception but he too often agreed to do so. 3. It is the work of a youth who was not yet completely trained. Galileo had written it after leaving Pisa University without a degree and before meeting Guidobaldo del Monte who he could have consulted on the mathematical aspects of the subject. 4. It contains the description of a long-known instrument, which he used without introducing any special modiﬁcations. Even the weighing system is the usual one. He just operated using a sequence of steps which made the new results possible.

7. Marcelin Berthelot had a completely diﬀerent idea [Berthelot 1891, 477–481]. To his mind, Galileo had invented nothing new but only taken advantage of a long tradition. He supported his evaluation by referring to what is reported in the pseudo- Priscian (4th-5th century), in Mappae clavicula (an anonymous treatise he believed to be of 12th century) and in Heraclius (13th century). Conclusively, he argued that “l’emploi de la balance hydrostatique pour analyser les alliages d’or et d’argent repose sur une tradition certaine [...] transmise au Moyen Âge depuis le temps des Grecs et des Romains” [Berthelot 1891, 481] (= the use of hydrostatical balance for the analysis of gold and silver alloys is based on a solid tradition [...] that Greeks and Romans had transmitted to the Middle Ages). Galileo’s La bilancetta: The First Draft and Later Additions 169

5. Finally and most importantly for science historians who consider experimental results a priority, the essay apparently does not contain data which supports its conclusions.

The last prejudiced view is definitively wrong. The table of weights in air and water which Galileo used did exist but it was not known to most early science historians [Berthelot 1891]8 because the autographs were recognized and published only after two and a half centuries by a local academic journal [Favaro 1879]. By contrast, the text of the essay had its editio princeps just after Galileo’s death. It appeared first as an extensive quotation hidden in a book published by another author [Hodierna 1644, cc. 2A1–8A]. However it was well known to Galileo’s followers and was sent to press under his name some ten years after the editio princeps in the first print of Galileo’s selected works which appeared at Bologna in 1656. Indeed, the title La bilancetta (although circulating before as the nickname for the apparatus) was chosen for the essay by Carlo Manolessi, the curator of the Bologna edition. The fact that Galileo9 had neglected to give a title to the essay would possibly indicate the minor importance he himself accorded it with respect to others later but he never refrained from distributing copies of it to his visitors while confined at Arcetri. After having considered all those objections, I reached the conclusion that the autograph and all the contemporaneous copies of La bilancetta were worth direct and careful analysis. I did this to both edit the essay in a better form which corresponds to the original text Galileo had in mind10 and also to better understand the conceptual framework developed by the young Galileo. Indeed, all handwritten versions are worth studying closely in all their details right up to their last words since the essay chronologically precedes the full development of the epistemological thought of a “father of modern Physics” or in fact of “the whole of [...] modern science” [Einstein 1933, 164].

4 The ﬁrst draft

Volume 45 of the collection “Mss. Galileiani” kept in the Biblioteca Nazionale Centrale of Florence (BNCF) is titled “GALILEO | GALLEGGIANTI | E | BILANCETTA | P. II. T. XVI” and contains most of the handwritten texts

8. Positively, Galileo’s table on weights in air and water was unknown to Richard Davies, who assembled all data available in literature from Francis Bacon’s time to his own [Davies 1748]. It was also unknown to Mathurin-Jacques Brisson, who measured the speciﬁc gravity of over 5000 materials [Brisson 1787]. 9. As Hodierna also did in the editio princeps. However, in the long title given by Hodierna, he makes it clear that Galileo himself had called “bilancetta” the new instrument he had invented [Hodierna 1644, 1]. 10. Which I did, by producing an updated diplomatic edition of all the texts mentioned here [Mottana 2016]. 170 Annibale Mottana

I intend to analyse in detail. The best-known one is “manuscript G”, which consists of a single, large sheet of paper (c. 55), 281 × 205 mm in size, densely written on both sides. It shows a small, rather crude drawing on top of the verso depicting the essential features of a balance or scales. Shortly after the pages in the volume were numbered in pencil by Atto Vannucci, the BNCF librarian from 1862 onwards [Procissi 1959, 119], a clever binder made this lone sheet conspicuous because he enclosed it between two cardboard sheets for protection. Then, some years later, Antonio Favaro ascertained that the handwriting of G was Galileo’s own [OG, I, 234] and the national edition which he later managed took this for granted. All subsequent scholars of the work also subscribed to this palaeographic identification [Procissi 1959, 119], [Crombie 1975, 157–175]. In continuation, volume 45 contains a clean copy of the same text, named “copia A” by Favaro [OG, I, 210–220] and written on cc. 56r-58r. These sheets are definitely not in Galileo’s handwriting [OG, I, 212]. Isidoro del Lungo, Gilberto Govi and Umberto Marchesini, the three philologists who helped Favaro edit the first volume of Edizione Nazionale—the time-table of all surviving writings by Galileo and his relations [Castagnetti & Camerota 2001, 359–361]—believed A to be coeval to G. Moreover, they also believed that A was the archetype for the copies Galileo distributed to his followers and this despite another copy (C) existing which might appear more authoritative because it is written by Vincenzo Viviani, Galileo’s last pupil and the first careful collector of most of Galileo’s writings and books. Copy A is followed by a numbered but blank sheet (c. 59) and then another script closely related to the same subject, titled Tavola delle proporzioni delle gravità in specie de’ i metalli e delle gioie pesate in aria, et in aqua. (cc. 60r-62r) This kind of table is inconspicuous within volume 45 because it is just at the end of it and looks like a bunch of sheets containing random numbers which could have been used by the binder to buffer the binding. Favaro deserves further credit for recognizing the relationships between their content and the text of La bilancetta [Favaro 1879]. Later, he published the Tavola... following the philologically reconstructed version of the main essay [OG, I, 224–227]. However, nobody took much notice of this list of 39 metal and mineral names added with numbers written in the fractional form used at Galileo’s time. Only a German doctorate student, Heinrich Bauerreiß, mentioned the table in his dissertation [Bauerreiß 1914, 62–64] and realized that he could calculate the specific weights from Galileo’s given weights in air and water. Nevertheless, he did not try to interpret their meanings. The interpretation of Galileo’s experimental weights in air and water was not carried out for another century. I made this interpretation both for gemstones (“gioie”) and for metals (“metallic”), with reference to the data printed in the Edizione Nazionale [Mottana 2014, 24–31], [Mottana 2014- 2015, 187–241]. However, I later realized [Mottana 2016] that the original, Galileo’s La bilancetta: The First Draft and Later Additions 171 handwritten Tavola, like the current text of La bilancetta itself, consists of two parts. The first part is a partial clean copy written in smooth-running, accurate but unknown handwriting. It begins with the afore-mentioned title and lists eight measurements covering recto and verso of one sheet (c. 60). The following two sheets are an untitled sequence of words and numbers scribbled in Galileo’s own handwriting (cc. 61r-62r) and list the weights of several metals and gemstones measured in air and water. On these sheets, somebody (probably the unknown copyist) ticked off the eight items listed in the clean copy, i.e., on c. 60. Thus, there is no doubt that this second section is Galileo’s original working draft and the first section of the Tavola, including the long title, is a partial clean copy of it made at an undefined moment but certainly later than the experimental session.11 To conclude, then, given the unequivocal evidence of what is signed and what is not, we can distinguish two time-sections in the composition of La bilancetta. There was an early, rough section (G: comprising cc. 55, 61 and 62) which was handwritten by the young Galileo. This was followed by a clean copy in unknown handwriting but which contains interesting additions that we assume to have been devised by Galileo after consulting his mentors (A: cc. 56, 57 and 60). Reading c. 55, i.e., the hastily written first draft of the essay is not a big problem. Actually, G shows numerous word abbreviations of all types, including tachygraphic symbols and special graphs inherited from the Middle Ages (e.g., ¯o,¯a,¯ufor on, an, un; ß for per; ßeè for perché; d¯uqfor dunque; arch.de for Archimede, etc.) and has no capital letters anywhere. However, the ductus is always easy and there is no attempt at cryptography. Galileo was writing for himself, putting his ideas down with no restraint whatsoever as he did not intend to divulge what he was devising. Ideas were popping up freely in his mind and he put them down on paper just as fast as they came. Therefore, he used the sheet of paper that he had available in the most condensed way he could—cutting words, mostly using standard abbreviations, even omitting vowels, sometimes. At the end, to show that he had written all that he had in mind at that moment, he shortened a word he had frequently spelled out before either in full (argento = silver) or cut in the usual way (arg.). He used a semicolon (arg:). To my understanding, this very unusual abbreviation—Galileo uses it nowhere else—is a clear indication that he had reached the end of the topic or of his reasoning thereon and of the entire essay rather than just the individual sheet of paper. Yet, copy A continues for

11. Drake points out that sheets 60–62 have a watermark (a circled rooster or basilisk), which is “undoubtedly Florentine”. Based on this, he speciﬁes “a dating of 1585-1586 that implies priority with respect to La bilancetta” [Drake 1986, 437]. I could not verify this information, and, in particular, whether the watermark is located in cc. 61–62, handwritten by Galileo, or in c. 60, which the copyist wrote. Nevertheless, I strongly doubt that Galileo performed the experiments before writing his theoretical essay, which shows all evidence of being an ex abrupto script. 172 Annibale Mottana several lines and consequently Favaro believed that G is a surviving fragment from a longer paper. I should like to suggest another possibility.

5 The additions

Galileo wrote his essay as a youngster, when he was a student in Florence Academia. Cosimo I de’ Medici created this school to supply his newly granted Duchy with technically trained architects, engineers, gunners and so forth. The teachers were some of Florence’s best technicians and scientists. Furthermore, occasionally, certain scholars who were qualified at university level (but who had not continued their career in universities) taught at the school. One of them was Ostilio Ricci, a friend of Galileo’s father Vincenzo, who was the man who had enabled Galileo to enter the academic world. Ricci was not a university man but was nonetheless among the best teachers of mathematics of his time. He himself had been the pupil of the very best teacher—Niccolò Fontana called Tartaglia (= the Stutterer), another exceptionally gifted man who had been rejected by learned society because of his speech defect. Ricci gladly advised Galileo about mathematical problems. Another academic who was even better disposed toward Galileo was Guidobaldo del Monte, a nobleman who had studied mathematics at Urbino in the private advanced school set up there by Federico Commandino, the famous humanist and mathematician who had edited the Latin translation of Archimedes’ works. Guidobaldo had not continued with a university career because it was inadequate to his high social rank and also because he preferred the practical application of mathematics (particularly ballistics) to developing abstract concepts. He was a technologist with strong mathematical basis, a “technoscientist”, if we use a modern word [Gorokhov 2015]. He often lived in Florence because he was the Chief Officer for Fortifications of the entire Duchy. Guidobaldo became acquainted with the young Galileo, found him to be a clever pupil, invited him to his mansion near Urbino and exchanged letters and discussed problems with him for several years. During the 1587-1592 period, Guidobaldo was noting down theorems and ideas at random in a bulky manuscript titled Meditatiunculae de rebus mathematicis (= Small meditations on mathematical matters).12 Like most learned men of his time, Guidobaldo wrote his notes in Latin. Therefore, it comes as a surprise to read a sentence in Italian following a long Latin evaluation of how the lever rule applies to a balance (cc. 232–234) [Tassora 2001, 540–543]. This sentence is written in a smaller character and with different ink but is clearly in Guidobaldo’s handwriting, and tells:

12. This manuscript is in the Bibliothèque nationale de France in Paris (Ms. latin 10246 ). Roberta Tassora transcribed and commented on it in her unpublished dissertation which is in open access online [Tassora 2001]. Galileo’s La bilancetta: The First Draft and Later Additions 173

Potrà forse accadere, che np, pl uenghino linee tanto piccole, che non si possi comodamente trouar la loro proportione. Et però si potrà pigliar il punto q, doue si voglia, e tirar le linee qnr, qps, qlt, e tirar la linea rst parallela a npl, che hauerà rs à st la medesima proportione, che ha np à pl. Poi si potrà pigliar un bastone diritto, et avvolgergli atorno una corda di citara ben sottile, e che le spire si tocchino l’una l’altra, che per esser pari, si potrà ueder quante spire siano np pl, ouero rs st. E così, quanto comporta l’atto pratico, si hauera in numerj la proportion dell’oro e dell’argento della magnitudine di ac.13 [Tassora 2001, 543] The first lines of this passage are a rigorous mathematical demonstration explaining a figure not reported here which Guidobaldo had carefully drawn using square and rule on the top part of the sheet. The final lines are of no mathematical interest and are not particularly rigorous either. Guidobaldo describes a practical method to make rapid but rough measurements of the ratio between gold and silver in a mix. This script is valuable however because it refers to the same method as that set out in G, the first draft handwritten by Galileo. Thus, it was Galileo who had the idea of making a wire coil around one arm of the balance and Guidobaldo simply corroborated the idea in his statement using more or less the same words as Galileo. However as a pure mathematician, he does not bother with technical peculiarities such as using two different kinds of metal wires to speed counting up as Galileo suggested in the last lines of his manuscript G. The sentence mentioned above by Guidobaldo is of dual historical importance. 1. It corroborates Viviani’s [Viviani 1674] idea of dating La bilancetta to 1586, a year that a number of critics [Crombie 1975, 167], [Settle 1983] have suggested to be too early in the development of Galileo’s scientific thought. In fact, we can be sure that Guidobaldo wrote his Meditatiunculae after 1587, when he released his main work commenting on Archimedes’ hydrostatic essays for printing and received the handwritten notes of his teacher Commandino on Pappus’ work [Tassora 2001, 36–38] and before 1592 at the latest, when he finished an essay

13. “It could possibly happen that np, pl became lines as small as not make it easy to ﬁnd their proportion. One could then project from a point q anywhere and draw the lines qnr, qps, qlt, and draw the line rst parallel to npl, which will have the same proportion rs to st, which np to pl has. Then, one could take a straight stick, and roll it up with a very thin guitar string so that the coils touch one other, and because they are equal one shall see how many coils are np pl and how many rs st. In such a way, in the short time taken by the operation, one will have in numbers, what the proportion of silver and gold will be in the bulk of ac”. My translation should compare to what Cecil Stanley Smith wrote at page 140 of the Appendix concerning the structure of La bilancetta [Fermi & Bernardini 1961, 133–141]. Smith’s translation, which is the only translation available in English, covers the entire passage as given by the oﬃcial edition [OG, I, 217] which in turn mostly derives from copy A. 174 Annibale Mottana

on prospectics [Tassora 2001, 42–44]. By contrast, for La bilancetta, the previous ante quem documented date is 1641, when Galileo wrote a letter presenting a copy of it among his works donated to the Polish Monsignor Stanislao Pudlowski (rectius: Stanisław Pudłowski) [Burattini 1675, 3]14 [Tancon 2006, 112–115]. Alternatively and by inference only, other scholars have suggested the composition of La bilancetta to be around 1611-1612 when Galileo argued bitterly with the Florentine Aristotelians and used similar arguments to contrast them eﬀectively in his Discorso al serenissimo don Cosimo II. Gran duca di Toscana Intorno alle cose, che stanno in su l’acqua, ò che in quella si muouono. The Discorso [...], which he printed twice in 1612 with a few minor changes [De Salvo 2010, 83–153], is the cornerstone of Galileo’s conceptions on hydrostatics and takes the slim book on the little balance into full account although it never mentions the experimental data from the Tavola. Hence, the text of La bilancetta should predate the Discorso [...] by some years. By inference, it also suggests the possible composition of G before 1587-1592. 2. It shows that Guidobaldo, one of the most brilliant mathematical minds of his time, had taken the young Galileo’s problem very seriously. He provided the correct mathematical solution [Tassora 2001, 543] and also actually adopted the instrumental set up that would make the little balance a rapid and eﬀective tool to measure the proportion of gold to silver in a mix even though he does not mention the other practical advice contained in the last lines of G.

That is enough regarding the main text. However, copy A, from which most printed editions of La bilancetta derive, contains the description of another technical operation which is most practical to speed up counting but also so simple as not to appear even worth mentioning by Guidobaldo. Galileo may have considered it inappropriate to submit it to him for his support as there is nothing mathematical in this operation. It consists of rubbing a thin “stiletto” (= stylet) above the coils to count them by the sounds made because of the contrast between metals [OG, I, 220 ll. 4–18]. We should consider it a bold suggestion which probably derived from Galileo’s deepest memories of teaching from his father Vincenzo, a musician, unless Vincenzo himself directly suggested it (he was still alive at that time). Galileo’s good ear for music made him an excellent lute player [Viviani 1674, 2], [OG, XIX, 601] and therefore we cannot wonder if he proposed a sonic method to count the coils in order to determine the gold to silver ratio. Instead,

14. Burattini testifies that the Monsignor allowed him to see the copy in 1642. He made use of this copy in an essay titled La bilancia sincera [...] con la quale per teorica e pratica con l’aiuto dell’acqua, non solo si conosce le frodi dell’oro e degl’altri metalli, ma ancora la bontà di tutte le gioie e di tutti i liquori. This essay, written in 1645, was never printed but survives as a manuscript in the Bibliothèque nationale de France (Mss. Ital. n. 448, Suppl. fr. 496 ). Galileo’s La bilancetta: The First Draft and Later Additions 175 it is legitimate to wonder when he added this operation to G, as it first occurs in copy A and in all other surviving handwritten copies. If Vincenzo suggested it to him, this would have occurred before July 2, 1591 when Galileo’s father died in Florence. Otherwise, we could consider any time before 1612 when Galileo had to reconsider all his ideas on hydrostatics because of the quarrel he had with the Florentine Aristotelian scientists [De Salvo 2010]. There were also at least three instances when he had the chance to rethink his early work on hydrostatics. In 1597, while a professor in Padua, Galileo devised a proportional compass, called “compasso geometrico et militare” (= geometric and military compass), which made it possible to rapidly perform all calculations needed by practical stoneworkers, land-surveyors, moneylenders, gunners, etc. [Drake 1977, 35–54], [Camerota 2004, 124–129], [Bertoloni Meli 2004, 166–169]. He had a number of such compasses built by an artisan, Marcantonio Mazzoleni, and made good profit by selling them. When the popularity of the instrument grew to the point that he could no longer personally teach all buyers how to use it, in August 1606 he wrote a handbook titled Le operazioni del compasso geometrico e militare [OG, II, 363–424], had it printed privately and started distributing it together with each new compass he sold. Among the seven proportional scales drawn onto the compass made up of two movable brass rulers (four on the recto and three on the verso), the outermost one concerns the proportionality existing between pairs of seven materials (gold, lead, silver, copper, iron, tin, marble and stone), each of them marked as a point on both rulers. Chapters XXI–XXIV of the handbook explain how to operate the compass. With the two rulers set at any opening, the intervals between any correspondingly marked pair of points give the size of spheres (or of other solid bodies such as cylinders and cubes) which are similar to each another and equal in weight. In other words, the compass solves the calculation that arises when comparing materials of different specific gravity, irrespective of whether this is determined hydrostatically or by dividing the weight of the corresponding sphere, cylinder, or cube for its volume.15 Galileo had other chances to rethink hydrostatics, because his ideas were subject to the envy that always surrounds a successful scientist in the university world.16 Less than six months after his handbook on his compass

15. The developer, if not the inventor, of this purely geometric method was Marino Ghetaldi, from Ragusa in Dalmatia. He published it in 1603 in a book titled Promotus Archimedes seu de variis corporum generibus gravitate et magnitudine comparatis [Napolitani 1988], [Bertoloni Meli 2004, 168–170]. Galileo and Ghetaldi met, possibly in Rome in 1600 and certainly in Padua in 1607. They exchanged letters for several years. 16. He speaks of “false imposture [...] fraudolenti inganni [...] temerari usurpa- menti” [OG, II, 518] (= false imposture, fraudulent deceits and reckless usurpations). I am currently studying the happenings related to the compass affair within the framework of Galileo’s attempt at creating a factory of scientific instruments in Padua. 176 Annibale Mottana appeared, Baldassarre Capra, a former pupil of him, translated it into Latin [OG, II, 425–511] and attempted to claim the invention as his own. This compelled Galileo to take legal action against him (which he won in April 1607) and to act firmly to protect his own reputation. He also did this by writing a pamphlet [OG, II, 515–599] which however adds nothing to the scientific question. Thus, we find no further clues about when he added the sound method to measure the gold to silver ratio in copy A of La bilancetta. Possibly, the emotional stress of his father’s death may have led Galileo to propose such a peculiar idea.

6 Final remarks

La bilancetta never mentions the measures reported in the Tavola, not even as a simple reference. Similarly, there is nothing about gemstones and metals in Guidobaldo’s Meditatiunculae. Possibly Galileo did not show Guidobaldo his measurements carried out in air and water or perhaps Guidobaldo simply did not consider them worth of his attention and thought them technically too low-level. In any case, La bilancetta referred to the reference works on hydrostatics without the support of all the experimental data that Galileo had made. That is possibly the reason why two and a half centuries passed before the signiﬁcance of speciﬁc weight as a reliable indication to identify gemstones was recognized [Haüy 1817] and it took two more centuries before Galileo’s pioneering contribution to gemology was rightly acknowledged [Mottana 2014].

Acknowledgments

For their critical reading and sharp, stimulating suggestions, I would like to thank Michele Camerota, Paolo Galluzzi, Marco Guardo, Massimo Peri and a very careful unknown reviewer.

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Adresses des auteurs

Paolo Bussotti Gennady Gorelik Udine University – Italy Center for Philosophy and [email protected] History of Science Boston University Walmir Thomazi Cardoso 745 Commonwealth Avenue Pontifícia Universidade Católica Room 506 de São Paulo Boston, MA 02215 – USA Faculdade de Ciências Exatas e [email protected] Tecnologia – PUC–SP Rua Marquês de Paranaguá, 111 Jean-Marc Lévy-Leblond São Paulo – SP Université de Nice-Sophia Antipolis 01303-050 – Brasil 06000 Nice – France [email protected] [email protected]

Francesco Crapanzano Roberto de Andrade Martins University of Messina Extrema, MG – Brazil Dip. COSPECS [email protected] Via Concezione, 6 98121 Messina – Italy Marco M. Massai [email protected] Dipartimento di Fisica “E. Fermi” Largo Bruno Pontecorvo, 3 Antonino Drago 56127 Pisa – Italy via Benvenuti 5 [email protected] Calci Pisa 56011 – Italy [email protected] Annibale Mottana Roma Tre University Raymond Fredette Faculty of Science CIRST Largo S. Leonardo Murialdo 1 Université du Québec à Montréal 00146 Roma – Italy C.P. 8888, succ. Centre-ville [email protected] Montréal (Québec) H3C 3P8 – Canada Raﬀaele Pisano [email protected] Université Charles-de-Gaulle-Lille 3 CIREL Romano Gatto Domaine universitaire du “Pont de University of Basilicata Bois” Campus di Macchia Romana BP 60149 via dell’Ateneo Lucano 59653 Villeneuve d’Ascq Cedex – 85100 Potenza – Italy France [email protected] [email protected]