Homage to Galileo Galilei 1564-2014 - OpenEdition Journals

Philosophia ScientiæTravaux d'histoire et de philosophie des sciences

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

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

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

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

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Editorial

Gerhard HeinzmannLaboratoire 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-Poincaréà Nancy et dirigé par le même rédacteur en chef dans un souci constant :promouvoir la recherche en philosophie des sciences, tout particulièrement ence qui concerne la logique, l’informatique, les mathématiques et la physique,en s’inscrivant dans la tradition analytique, en prenant en compte l’histoireet la pratique des sciences, en préservant le français, l’anglais et l’allemandcomme langues « scientifiques » à titre égal.

En passant la main à la nouvelle équipe, Manuel Rebuschi comme rédacteuren chef et Baptiste Mélès comme rédacteur en chef adjoint, j’exprime magratitude à Manuel, rédacteur en chef adjoint depuis 2008, et à Sandrine Avril,secrétaire de rédaction, pour leur travail efficace et leur dévouement. Je remer-cie Madame Charrié des Éditions Kimé, le comité de rédaction et le comitéscientifique pour leur coopération facile et collégiale et je suis reconnaissant àl’Université de Lorraine et au CNRS pour leur soutien sans faille.

L’émergence du monde numérique et l’accès en ligne ont changé la di-mension de la distribution du savoir et exigent des expertises de plus en pluspoussées. Je souhaite à la nouvelle équipe de savoir utiliser ses larges compé-tences pour saisir l’occasion des nouvelles techniques – tout en conservant lesanciennes – afin de consolider l’ancrage national et international de la revue.

** *

The first issue of Philosophia Scientiæ was published more than twentyyears ago in June 1996. It was edited by a young team at the Henri-PoincaréArchives 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, inline with the analytical tradition and taking into account the history andpractice of science. Another aim was to preserve French, English and Germanas “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, fortheir efficient work done with loyalty. I would also like to thank MadameCharrié of Kimé Editions, the editorial board and the scientific committee formaking their cooperation easy and convivial and I am grateful to the Universityof Lorraine and the CNRS for their unfailing support.

The emergence of the digital world and online access have changed thedimension of the distribution of knowledge and mean more and more expertiseis required. My wish for the new team is that they know how to use their broadpalette of skills to seize the opportunity offered by new techniques—whilepreserving the old ones—in order to consolidate the national and internationalposition of the journal.

Gerhard Heinzmann

Homage to Galileo Galilei1564-2014.

Reading Iuvenilia Galilean Works within Historyand Historical Epistemology of Science

Introduction.1564-2014. Homage to Galileo Galilei

Raffaele PisanoLille 3 University (France)

Paolo BussottiUdine 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, whowas not young: for, he was 46. Nevertheless, this little book representedthe fundamental turning point in Galileo’s life and scientific production. TheSidereus Nuncius was very successful and gave rise to numerous discussions.Some scholars defended Galileo—the most important was Kepler—, many oth-ers, with a series of different arguments, criticized the content of the Sidereus.Galileo became the most famous and discussed European scientist. All hismost 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 arecontributions which aim at reaching theoretical conclusions, although theexperimental method plays a fundamental role to reach such conclusions.While, if we go back to Galileo’s production before 1610, three facts aresurprising, enough: a) the relatively scarce amount of written and publishedworks; b) most part of these works have a practical approach, that is, they arededicated to the military operations or military fortifications or to instrumentsusable in a military context; c) some contributions are commentaries toimportant works of ancient authors, above all Archimedes. There are also somemore theoretical contributions, like those relating to the new star appeared in


8 Raffaele Pisano & Paolo Bussotti

1604 [Galilei 1890-1909, II, 267-306]. However, they are a minority. On theother hand, it is known that Galileo claimed more than once that the yearshe spent in Padua, working for Venetia Republic, were his most happy andfruitful period, from a scientific as well as a personal standpoint. It seemstherefore only natural that he developed many of the ideas explained from1610 forward in the period spent in Padua. Hence, the reason of interestconnected 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 initialpages 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 extendthe word Iuvenilia to the whole production of Galileo published before theSidereus Nuncius. Thus, we refer to the content of the first two volumes of theGalilean National Edition. We consider all these contributions as Iuvenilia,identifying the separation line with the fact that, before the publication ofSidereus Nuncius, Galileo published, in substance, no theoretical work, thoughhis 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 speciede i metalli e delle gioie pesate in aria e in acqua. – Postilleai 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 maggio1597). – Trattato della sfera ovvero cosmografia. – De motuaccelerato. – La nuova stella dell’ottobre 1604. – Frammentidi lezioni e di studi sulla nuova stella dell’ottobre 1604. –Considerazione astronomica circa la stella nova dell’anno 1604di Baldesar Capra, con postille di Galileo. – Dialogo de Ceccodi Ronchitti da Bruzene in perpuosito de la stella nuova. –Il compasso geometrico e militare. – Del compasso geometricoe militare, saggio delle scritture antecedenti alla stampa. – Leoperazioni del compasso geometrico e militare. – Usus et fabricacircini cuiusdam proportionis, opera et studio Balthesaris Caprae,con postille di Galileo. – Difesa contro alle calunnie et imposturedi Baldessar Capra. – Le matematiche nell’arte militare. [Galilei1890-1909, II]

Introduction. 1564-2014. Homage to Galileo Galilei 9

The Galilean literature is wide, so that it is impossible to list all references.1In the secondary literature one can read reflections, which are sometimesinteresting, but not always historically proved such as clear hypotheses,reasoning and assumptions. In general, Galileo’s adolescent life and scientificactivity are not explored and known, enough [Heilbron 2010]; i.e., it seems thathis father also proposed, or simply had in mind, a sort of experimental methodfor his musical instruments [Cohen 2010, 85 sq.]. Other probably alleged—or not—stories concern his father Vincenzo who would give Galileo the ideaof joining the Camaldolese order, though this story does not look accuratebecause the religious order, at that time, did not accept young boys. Someauthors pointed out Vincenzo’s opposition to his son’s mathematical studies.There is also a debate on Galileo’s home-birthplace in Pisa. Furthermore, anew edition of Galileo’s Opere (such as also Torricelli’s) works seems to benecessary.

Therefore, there are several unsolved aspects and historical problems as toGalileo’s young life. We hope that this special issue offers to the historians,philosophers and scientists a right provocation, an intellectual stimulus and asufficient 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 tobetter understand both philological aspects and early foundational Galileanconvictions-doubts-methods before his definitive study on parabolic trajecto-ries (1608-1609) and his mechanics. A cautious attitude sometimes shines, insome circumstances Galileo appears not completely confident with physicalsubjects, 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 thesubject], where there are pictures which show rectilinear trajectories for theprojectiles. The influence of the studies developed in this period clearlyappears in conspicuous parts of later Galileo’s production: for example, whenGalileo addressed technical subjects as the strength of materials in the Discorsie dimostrazioni matematiche, he also faced architecture and fortresses. Heconceived these parts not only as researches for military expert architects, butalso as lectures for students: i.e., two speeches on fortifications stem fromteaching speeches collected by his pupils on indications of Galileo himself. Inthis 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 theRepublic, was fundamental. The relatively limited amount of secondary liter-ature (on mechanics correlated to fortifications) is a matter of fact, testifyingthe 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 secondarysources connected to the content of our Introduction.


researches on mechanics, instead largely commented by many scholars.This is why we propose a special issue that, starting from the period inwhich Galileo produced his works on fortifications and military architecture, isthe occasion to rethink, more profoundly, of the whole production by Galileopreceding 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 andepistemological accounts from each of the authors. The authors’ contributionsappear in alphabetical order. Each of the authors is responsible for his or herown opinions, which should be regarded as a personal point of view based onhis scientific background.

Crapanzano (Italy) analyses Galileo’s dialogue De motu and related con-tributions under a particular perspective: the reading offered by RaffaelloCaverni (1837-1900) of these Galilean works. The author reminds us thatCaverni wrote, at the end of the 19th century, the monumental work Storiadel metodo sperimentale in Italia, a text, which, although imprecise for manyaspects, continues to represent an important source. After that, Crapanzanosummarizes the history of De motu and its relations with De motu antiquiora.The author points out that Caverni played an important role in establishing thetext of De motu for Galileo’s National Edition. He actively collaborated withAntonio Favaro (1847-1922), though Caverni is not mentioned in the Edition.The section 2 is an interesting and successful attempt to frame Caverni’s Storiadel metodo sperimentale in Italia within the context in which this work wasconceived. Finally, the author faces the difficult problem to catch the role ofDe motu within history of mechanics.

In his paper, Drago (Italy) deals with a broad problem: what was therole of Galileo in the birth of modern science. To face the question, Dragofirst 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-basedorganization. As to this third aspect, the dialectic classical logic/ non-classicallogic, identified by Drago, also plays a fundamental role. With regard to thebirth of science, the author first analyses the interplay of these three aspectsin their general form. Afterwards (8th and final section) the role of Galileo isanalysed in the light of the explained dichotomies.

Fredette (Canada) analyses De motu antiquiora published in the firstvolume of Galileo’s EN. The author traces a continuous line between thisearly work and the most mature and important contributions given by Galileo


to science: the Dialogo sopra i due massimi sistemi del mondo and Discorsie dimostrazioni matematiche intorno a due nuove scienze. Fredette preciselyidentifies eight issues developed in details by Galileo in the Dialogo and inthe 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 inDe motu and to the corresponding idea fully developed in the Dialogo and inthe Discorsi. In this manner, a useful lens, by which to look at the whole ofGalileo’s production is offered starting from one of his initial contributions tophysics.

In his contribution, Gatto (Italy) interprets Le mecaniche by Galileo as aconceptual bridge between the old science of weights and the modern statics.The author traces an ideal itinerary to explain the novelties of Galileo’stext relying upon what he calls “the comprehension principle”, namely thenotion “according to which force, resistance, time, space, speed, continuouslycompensate each other: if the power increases, the speed decreases because,in moving the resistance through a given space, in the same time the forcecrosses a space longer than it”. In modern terms: work cannot be created exnihilo, but only transformed; the nature cannot be deceived. Gatto faces suchan important problem in this early work by Galileo offering a profound pictureboth from a philological-historical and conceptual standpoint.

Gorelik (USA) deals with an extended form of the so-called NeedhamQuestion, that is: what prevented Greco-Roman and medieval civilizationsfrom developing science following the line indicated by Archimedes and whatprevented the Eastern civilizations from giving significant contributions tophysics for centuries after Galileo (Needham’s questions were restricted to theEastern civilizations)? Along his itinerary to find an answer, the author pointsout the mathematical and experimental method of modern physics and triesto identify a line of separation between Galileo’s and Archimedes’ approach.He then identifies an optimistic feeling in the regularity of nature and inour capability to penetrate such regularity. After that, the possible role ofChristian religion is also taken into consideration.

Lévy-Leblond (France) addresses a particular subject within Galileo’sproduction: the Due lezioni all’Accademia Fiorentina circa la figura, sito egrandezza dell’Inferno di Dante (1588). The author explains that Galileoentered into two different interpretations of Dante’s Inferno, the one byManetti 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 academicenvironment. Probably these lessons played a role when Galileo becameprofessor in Pisa university (1589). Beyond specifying all these aspects indetails, the author also frames the Due lezioni in the context of Galileo’searly scientific production. However, a great part of this paper is dedicatedto the specification of Galileo’s arguments developed in the Due lezioni. Thisallows the reader to fully appreciate Galileo’s pedagogical and argumentativecapabilities, which characterize the whole of his production.


Martins & Cardoso (Brazil) present Galileo’s Trattato della Sfera ovveroCosmografia, a treatise of geocentric astronomy, probably written in 1600 orfew years earlier, as a textbook. The author compares Galileo’s Trattato withthe text which was a reference point for this kind of books: the Sphaera bySacrobosco. The similarities and differences are pointed out both as far asthe conceptual and the stylistic-methodological aspects are concerned. Theauthors also look for the sources of Galileo, thus mentioning Peuerbach’sTheorica Planetarum and above all a series of vernacular treatises on thesphaera written in the second half of the 16th century. In this context, Galileo’sannotations of Piccolomini’s Sfera del mondo appear particularly significant.Other sources are mentioned, too, although Piccolomini’s work seems the mostimportant one. The two authors also offer profound insights on the literatureconcerning Galileo’s Trattato della Sfera.

Massai’s (Italy) contribution represents an ideal chronological conclusionof this special issue because the author analyses the Sidereus Nuncius, workwhich we assumed as the conclusion of the Iuvenilia and the beginning ofGalileo’s mature phase. Massai intends to point out the importance of theinstruments, the observations and the experiments in Galileo’s scientific praxis.The author underlines that Galileo was an expert instrument-maker. Thiswas fundamental for him to realize the telescope could have an astronomicalutilization. Furthermore, Massai enters the logic of many argumentationsdeveloped by Galileo in the Sidereus Nuncius, showing that Galileo reachesthe right conclusion as he excludes a series of hypotheses because they are notconsistent with the observations. Therefore, Massai’s is both a descriptive anda methodological paper.

Mottana (Italy) examines Galileo’s treatise La bilancetta, taking intoaccount the first draft of this work and the successive additions. The thesis ofthe author is clear: experiments guided the whole of Galileo’s productionand scientific procedures. Thus, Mottana presents early Galileo’s studiesfrom the very beginning of his scientific activity. After having described theenvironment in which La bilancetta was conceived, the author enters into ananalytical and exhaustive examination of the subjects dealt with in the firstdraft, describing in detail all the operations carried out by Galileo with theinstrument. The same approach is used to describe the additions. Here thefigure of Guidobaldo dal Monte plays an important role. The whole paper isenriched by numerous and perspicuous references to primary and secondarysources.

Acknowledgments

The genesis of this issue relies upon our studies in history of science and,particularly, history of physics. We enthusiastically express our appreciationand gratitude to contributing authors for their efforts to produce papers ofinterest and of high quality. We also address our acknowledgments to Gerhard


Heinzmann (Editor in Chief), Manuel Rebuschi, Baptiste Mélès (Managingeditors), and Sandrine Avril (Editorial Assistant) for their good job andpositive reception of our project to publish an issue on the Galilean IuveniliaWorks 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 scienzeattenenti alla mecanica e i movimenti locali, [Galilei 1890-1909, VIII, 4–458].

—— [1890-1909], Le opere di Galileo Galilei. Edizione Nazionale sotto gliauspice 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 cittàcon 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 thisintroduction

Biagioli, Mario [1990], Galileo’s system of Patronage, History of Science, 28,1–61.


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

Cohen, Floris [1984], Quantifying Music: The Science of Music at the FirstStage 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: LeMonnier.

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 dimeccanica di Galilei e di Torricelli, in: Da Archimede a Majorana: Lafisica 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 deicorpi, in: Relatività, Quanti Chaos e altre Rivoluzioni della Fisica, editedby E. Giannetto, G. Giannini, & M. Toscano, Rimini: Guaraldi Editore,61–72.

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

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

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

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

—— [2016], Details on the mathematical interplay between Leonardo daVinci and Luca Pacioli, BSHM Bulletin: Journal of the British Societyfor 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.

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

Pisano, Raffaele & Capecchi, Danilo [2010], Galileo Galilei: Notes onTrattato di Fortificazione, in: Galileo and the Renaissance ScientificDiscourse, edited by A. Altamore & G. Antonini, Roma: Edizioni nuovacultura, 28–41.

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

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

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

Francesco CrapanzanoUniversity of Messina (Italy)

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

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


18 Francesco Crapanzano

1 Introduction: De motu and its fortuneThe so-called De motu appeared in 1642, after Galileo’s death when VincenzoViviani, the last follower of the great Pisan scientist, decided to collect all hiswritings in order to:

[...] reprint all Galileo’s works in the form of sheets with greaterfullness and magnificence in two columns for the two languages,the former Tuscan, in which the author wrote, the latter Latinto be translated by some of our compatriots and lastly with theaddition of a great number of writings by the same author which Igathered 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 somepapers by Galileo collected in quinternions entitled De motu antiquiora whilehe was working on the bibliography of a major study on the fifth book ofEuclid’s Elements. Viviani immediately thought they were early studies of theMaster about motion, already showing an independent judgment with respectto the predominant Aristotelianism and, above all, they would be includedappropriately in later works [Viviani 1674, 104–105].2

The manuscript in question, along with all the Galilean papers, ended upin the hands of the abbot Jacopo Panzanini, nephew of Vincenzo Viviani (hissister’s son), who ordered its bequest to the heirs Carlo and Angelo Panzanini.The latter were not particularly involved in the conservation of the manuscriptsand 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 workshad already been published in two volumes edited by Carlo Manolessi and publishedin 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 reconstructionby [Camerota 1992, 17–62].

3. Nelli explained that he bought the manuscripts despite being the beneficiary ofthem, through inheritance, thanks to a primogeniture resulting from the testamentarydisposition of Viviani. Due to the unavailability of the testamentary trusts, the legacywould 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 sortidella Collezione Galileiana, tanto che al marchese può senz’altro riconoscersi l’enormemerito di aver salvato da una probabilissima, completa dispersione la raccolta deimanoscritti 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 dagrano della cosiddetta Casa dei Cartelloni, un tempo dimora del Viviani e, inseguito, abitazione degli eredi Panzanini. Questa impropria sistemazione, insieme allacompleta mancanza di vigilanza o di scrupolo mostrata dai due Panzanini, sarebbeall’origine della scomparsa di un buon numero di carte galileiane, che, sottratte dalloro “nascondiglio”, cominciarono a circolare liberamente, alimentando un vero eproprio 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 theadmittedly lesser merit of having published a rich and documented biography[Nelli 1793].4 The texts of De motu antiquiora were also reported by Nellithus:

I keep some studies carried out by the Florentine Philosopher inhis youth and transcribed by him in different quinternions aboveone of which there is the writing De motu antiquorum [...]; inothers there are some errors included in Aristotle’s works. [Nelli1793, II, 759]

And then referring to the 1589-1592 period in which Galileo was a professorof mathematics in Pisa, he adds:

[...] some scientific dialogues, in which [the great scientist] in-troduces Alessandro and Domenico as interlocutors who examinevarious propositions of Mechanics and especially about Motion,scattered in the Aristotle’s works [...] showing them to be for themost 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 whichare correlated, by both of them, with the dominant Aristotelian conceptionsabout motion. In his early work, Galileo criticized erroneous theories withthese criticisms containing early arguments and opinions which he was to recallin the most famous mature works [Nelli 1793, II, 759].6

Once the Florentine nobleman died, the papers passed to his sons, whodecided to sell them because they were not very well-off. Thus the dangerof dispersion reoccurred but this was avoided thanks to the purchase of theentire collection by the Grand Duke of Tuscany in 1818. It then formedthe core of the Galilean Collection, now kept in the National Library ofFlorence. Other scholars, who had access to the manuscripts in the courseof the following years, agreed with Viviani and Nelli that De motu antiquiorawas a set of various Latin texts about motion dating from the early periodof 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] whileconversely underlining a lot of mistakes and errors.

5. It is easy to recognize by the Nelli’s indication the dialogue that will merge inthe 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 bothViviani 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 ofscience, was the first scholar to have noticed the similarity between the Galilean


in the eleventh volume of Galileo’s works he edited, presenting them asSermones de motu gravium [Galilei 1854, 9–80].8

A few decades later Antonio Favaro began the work on the EdizioneNazionale examining the texts of De motu which were not included in theeleventh volume of Albèri; and he did so with great care, despite the difficultiesand the “annoying characteristics” of the manuscripts.9 However Favaro alsoinitially made an error with respect to a part of the text included in theEdizione Granducale which he could not find in Galileo’s manuscripts. Thisand many other problems concerning the right location and timing of the textsin question were at the basis of the collaboration between the student fromPadua and the priest Raffaello Caverni (1837-1900). In view of the futureEdizione Nazionale, Favaro took advantage of his help, recognizing him asan expert on Galileo’s writings.10 Caverni collaborated very actively and cantherefore be legitimately considered as an editor of the De motu writings tothe same extent as Favaro. However his name is not present in the first volumeof the Edizione Nazionale, due to a misunderstanding between the two authorswhich we shall refer back to later. It should be noted at this point that theLatin texts collected by Favaro with the help of Caverni would make up whatwas to subsequently be called De motu. As is clear in the first volume of theGalileo’s works, the scholar from Padua considered the arguments present inDe motu very close to those exhibited extensively in the major works to theextent that:

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

De motu and that of Giambattista Benedetti. This similarity was later taken upby Raffaello Caverni according to whom Galileo did nothing more than reiterateBenedetti’s thesis on motion. As we will show later, this sometimes frantic search forthe genesis of Galileo’s theories is one of the main limits of the research conductedby the Florentine Abbot.

8. This edition of Galileo’s writings is particularly important for us as it will bethe one used by Caverni for his analysis.

9. When Favaro presented the writings in question in 1883—well before they werepublished in the first volume of the Edizione Nazionale, he noted: “Frequentissimee spesso anche capricciose le abbreviazioni che si incontrano ad ogni pie’ sospinto;poco chiara la scritturazione, con frequenti cancellature e interpolazioni; omissioni diparole che rendono talvolta difficile l’afferrare il senso; la punteggiatura quasi del tuttomancante o irregolare; e finalmente non pochi errori di lingua, i quali contribuisconoin non lieve misura ad aumentare le incertezze nella interpretazione dei luoghi didubbio 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 Storiadel metodo sperimentale in Italia [Maffioli 1985, 23–85].


In the light of this consideration, it would seem natural to place Favaro alongthe interpretative axis that had indicated the seeds of the fundamental prin-ciples of dynamics in these early manuscripts which were to be subsequentlyreported. In fact, this view was contested because there would be no clearevidence of this insofar as Favaro gave no precise indications on the elementsthat were common to De motu and Discorsi. Similarly, he would have beenvery wary of declaring the connection between the two works.11 There is indeedno specification of those themes of early doctrines which Galileo would entirelyrecall in 1638 but the same cannot be said about Favaro’s alleged caution and,ultimately, it is impossible to definitively establish whether he considered Demotu as the first immature text written on dynamics or as containing in nucethe foundations of the future science of motion. In any case, my own view isthat we can exclude an understanding that tends to isolate the early writingson motion with respect to the Discorsi and this is confirmed in the foregoingpublication where Favaro considered that:

Those unpublished writings scraped together in the Galileancollection and primarily those related to the motion, have seemedvery important to me, since the beginning. Although in someof them you find only youth studies, however like Michelangelo’sfirst scribbled drawings, in others and overall, I perceived muchmore relevant characters which need to be taken into account toportray the blossoming fruits in the Discourses. We can feel theGalilean verbosity which while perhaps not having the attractionof the familiar eloquence in the Two New Sciences, sickens a littlebut you can see, even in the form, the young man who, disdainingthe succinct and concise Aristotelian doublet, went on to preferthe broad Platonic covering. [Favaro 1887, 229]

Even in this case, however, the historian from Padua does not specify theconnections between the old works and the mature reports on the new scienceof motion thus supporting a suggestion by Caverni who in 1889 (approximatelya year before the release of De motu in the first volume of the EdizioneNazionale) wrote to Favaro that if he wanted to proceed by prefixing a shortforeword, then it would be better to publish the writings in fragments:

[...] but if you really persist in thinking of publishing them inthat wonderful order divided by us and in which we read, as in amirror and since the first steps, the processes of the speculationsby Galileo in the science of motion and then, in order to preventcriticism [...] you cannot do so without an erudite Preface, as longas it is needed, without the ministry intervening to act as preciselyas Procuste. [Lettera di Caverni a Favaro, dated 20 March 1889][Maffioli 1985, 79]

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


Despite Caverni’s good advice and the choice to present the writings in theorder agreed, Favaro published only abbreviated information. Rather thanindicating a deterioration of the relationship between the two or a simpledifference of opinion, this probably reveals that there was not enough criticalwork on Galileo’s Pisan period which prevented the construction of an exactlogical chronological order of De motu, as called for by Caverni when hesuggested the “learned preface”.

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

Why did Antonio Favaro decide to collaborate with a little-known priest? Toanswer this question, it is useful to point out two or three stages of Caverni’sbiography. In 1874, Caverni’s first book was published, Problemi naturalidi Galileo Galilei e di altri autori della sua scuola. The text was mainlyfor students of physics and, while not widely used in schools, was loved byscholars. The subject is doubly interesting because it testifies to the priest’sinterest in the history of science12 and also expressed what would become aconstant feature of his research—a kind of nationalism applied to the genesisof scientific discoveries or inventions:

My intention was to open as a window to help young people finda broader field and explain how some inventions and doctrines,which we think came from France and Germany, were taughtus much earlier by our compatriots and to show a small partof Italy’s hidden treasures to those who accuse the country ofpoverty. [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 ofthought” [Favaro 1899-1900, 379]. In it, the priest from Quarate discussesthe compatibility of the new theory about Darwinian evolution with the HolyScriptures, by touching on a sore point of the relationship between science andfaith. He does so with the aim of finding a settlement between the two partiesand this was possible provided that something was sacrificed of the Catholichermeneutic “prerogatives”, namely to recognize and accept the distinction:

[...] in the books between the divine and the human: the former’sobject is the truth of faith and it is infallible while the latter’sobject 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 hagiographers and which may be true or false like all thingslearned by the natural use of reason. [Caverni 1877, 24]

In these words, everyone will have recognized the adoption of Galileanhermeneutic criterion in which the great Pisan scientist, who was fully con-vinced of Copernican heliocentrism, tried to argue that the Bible was not incontradiction with scientific evidence. Indeed he claimed that such evidencewas actually compatible with Holy Scriptures, provided that these are not readliterally and instead are interpreted in the light of such evidence,13 because:

Writing often is not able simply to give an explanation butactually it requires an explanation which is different from theapparent meaning of words. [Galilei 1890-1909, 282]

Our priest would not have expected the volume to be placed on the Index ofprohibited books by July 1, 1878 and, though he tried to impose his pointof view, finally he had to give up. Indeed it is probable that the SacredCongregation of the Index did not prohibit the text because it presentedthe evolution theory. Instead this occurred because of the Church’s growinghostility to science (at least from the process to Galileo onwards), an antipathytowards the Jesuits (they always had opted for Aristotle when the new sciencewas born under the sign of Plato) and, not least, because they had foundin Galileo’s work the hermeneutic canons required to interpret the HolyScriptures [Caverni 1877, 40, 41, 159, 160], [Pagnini 2001, 40–50]. Despitebeing put on the Index, among the few readers of De’ nuovi studi della filosofiawere Antonio Favaro, who wrote a letter to Raffaello Caverni from Padua datedMarch 25, 1877. They were yet to become friends but it seems that his ownreading of the text had led the scholar from Padua to deepen his friendshipwith the pastor from Quarate:

You seem to me a good teacher, a very good teacher, but a teacherdeeply convinced of the saying: Who loves well, castigates well.Was I mistaken? Meanwhile this book has increased my desireto meet you personally and perhaps before long you will find meknocking at your door. If my health allows me to rest and carryout a long work which I thought about for several years beforedeciding, I will also want to ask for your advice. [Lettera di Favaroa Caverni of March 25, 1877] in [Pagnini 2001, 53–54]

This is the beginning of the friendship between Favaro and Caverni which istestified 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 toBenedetto Castelli when he writes that the truths of faith and truths of reason cannotbe in conflict with each other, at most it is “ofizio de” saggi espositori affaticarsi pertrovare i veri sensi de’ luoghi sacri, concordanti con quelle conclusioni naturali dellequali prima il senso manifesto o le dimostrazioni necessarie ci avesser resi certi esicuri” [Galilei 1890-1909, 283]. See also the Lettera alla Gran Duchessa MadreMaria Cristina di Lorena [Galilei 1890-1909, 309–348].


than a decade [Maffioli 1985, 54–71].14 As Maffioli noted “Favaro’s friendshiphelped to break Caverni’s isolation from the scientific and academic world”[Maffioli 1985, 28]; and although the priest was shy and tormented by manydoubts and problems, the scholar from Padua’s bibliographic information oftenproved very valuable. On the other hand:

[...] the friendship of Favaro was pretentious and required continu-ous exchanges involving information and searches for manuscriptsand documents, reviews, contacts with publishers and editors ofperiodicals and so on. From 1883 onwards, Favaro completelydrew Caverni into the great project of a new edition [...] of theGalileo’s works. [Maffioli 1985, 29]

The pastor was enthusiastic and did not hesitate to advise and collaborate onthese topics (Galileo) which he had shared an interest in since he was young.He had a different historiographical approach from Favaro. Both preferredthe documentary aspect, but for the former this served as a platform to showthe foundations and evolution of thought and tradition which had allowedGalilean science to blossom fully. The latter’s true goals were representedby the chronological reconstruction and the indisputable attribution of thewritings. However, in 1887, Caverni was unexpectedly excluded by the editorsof the Galilean works. He considered Favaro responsible for this which meanttheir personal relationship cooled somewhat.15 Although the scholar fromPadua managed to convince the priest to continue their collaboration, shortlyafterwards their uncertain friendship was to lead Caverni to abandon thechallenge because he became increasingly convinced that he was being trickedin some way. From then on, he was to work hard on his most important anddebated work—the majestic Storia del metodo sperimentale in Italia, to whichhe had already devoted part of his time.16 Until then he had worked followingthe 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 betweenthese two, all private notes and quotes in the papers and letters addressed to othersshould be added to these papers. Between them, besides the cultural exchanges,there were “le manifestazioni di affetto, i segni di una lunga consuetudine: il classicoinvio della fotografia (benché entro un opuscolo scientifico!) prima di poter fare unadiretta conoscenza, i pranzi in comune, le discussioni nei caffè sul Lungarno, i fioridi 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 morecomplex affair than it may first appear. It is unclear, for example, the role of Favaro inthis exclusion since it appears that the publisher Le Monnier who first worked on thisjob had said he was very content with Caverni. However the work of guardianship hadto 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].


part in the edition of Galileo’s works written by Albèri and then, for anotherpart, by Favaro himself, in a journal.17 In a letter dated 22 August 1884, thescholar from Padua clearly asked Caverni to take care of the above-mentionedwriting in order to understand:

I. If the writings to which belong the extracts contained in thefirst part of that work [the unpublished works published on themagazine Editor’s Note] are to be publishedII. With respect to the Sermones De motu gravium, how are thefragments contained in the second part of the same work to bedistributed? [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 givento the manuscripts and also to their placing to correctly portray the innerevolution of Galileo’s thought.18 Although Favaro preferred a philologicalapproach in the wake of positivism, he still consulted Don Raffaello who wasto provide an order for De motu in five books which would have required animportant critical apparatus. Towards the middle of April 1887, althoughCaverni had already learned of his exclusion from the editors of the EdizioneNazionale, he still believed this to be perhaps due to a simple misunderstandingand declared himself willing to invite Favaro to his rectory to discuss “the bestappearance to give to early Galileo studies on motion” [Lettera di Favaro aCaverni of 12 April 1887] [Maffioli 1985, 67].19 Just two weeks later, he readan article in La Nazione (April 11–12, 1887) which stated he had been officiallyexcluded and wrote to Favaro that he had learned “some things which causedhim to greatly change his mind and [require] important clarifications” [Letteradi Caverni a Favaro of 30 April 1887] in [Maffioli 1985, 67]. For almost twomonths there was no exchange of letters—a sign that no clarification was forth-coming or that it was insufficient. In the following months, Favaro again askedthe order of De motu but Caverni replied coldly, until in July of the same yearhe made some unacceptable requests.20 Therefore, after this change of course,

17. This is reported by [Maffioli 1985, 72]; where it is the precise bibliographicalindication 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 purposeof 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 preziosotesto 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 maggiordiligenza possibile, per servirsene nella Storia della Meccanica” [Lettera di Cavernia 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


the priest inevitably became quite critical of the Edizione Nazionale and,conversely, Favaro did not hesitate to criticize him more or less explicitly whichfuelled—or even actually created—the widespread view that Caverni was Anti-Galilean. This does not mean that Caverni would not refrain from expressingdebatable opinions on Galileo in his Storia del metodo sperimentale in Italia.However, according to Favaro, this work remains “the greatest collection ofmaterial for history of the Galilean School” [Favaro 1899-1900, 379].

3 A difficult task: the role of De motu inthe 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 anorder. The difficulties he encountered in putting the Galilean Ms. 71 andMs. 46 into order are more understandable. Caverni sensed that those zibaldatecarte (“mixed-up papers”) played a role in the birth and evolution of Galileanscience. He was therefore committed to determining when they were writtenand to situating them in an ideal historian’s path of scientific ideas. As such apath they were positively considered as the roots upon which Galileo foundedthe new physical science while also showing how the great Pisan scientistskilfully used existing research lines and insights. The priest from Quaratemainly exposes his research on De motu in a chapter of the fourth volume ofthe Storia del metodo sperimentale in Italia which is primarily dedicated tothe Scienza del moto dei gravi.21 At the beginning of it, as if to stress theimportance of the topic, there are some general methodological considerations.Firstly, it explains how:

[...] the attribute that is given to the discoveries of Physicists orthe speculations of Philosophers and is deduced from the nameof a man is in all fairness an impropriety which can only be keptin the speech convention [...]. This is because the intellectualorder is a consortium which is as closely linked and necessaryas in the civil orders and therefore a part of science may benamed after an author in the same way as a religious society orfamily takes its name from the hierarch or father. So saying thelaws of falling bodies belong to Galileo is not to be understood

conveniente e compatibile con gli altri lavori che sta curando. 2o Che si accettinoprefazioni e note da lui sottoscritte e nella misura che reputerà necessaria. 3o Chegli si assegni una congrua ricompensa. 4o Che detta ricompensa sia versata daglieditori galileiani all’atto della consegna del manoscritto” [Lettera di Caverni a Favaroof 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 entitledDelle discese dei gravi lungo i piani inclinati [Caverni 1895, 328–381].


in the way that many do, as if they were spontaneous ideasfrom 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 inDiscourses and Mathematical Demonstrations Relating to Two New Sciences,should not be considered “a fascinating appearance” but rather that it wasnecessary to recognize “the bud and the little plant which engendered it”[Caverni 1895, 328–329]. It is an indisputable fact that Caverni derives severalGalilean demonstrations from other mathematicians and natural philosopherssuch as Leonardo, Tartaglia, Descartes, etc., through not always in clear anddirect ways. He recognizes, however, how Galileo’s effort consists in makingits dynamics independent from the principles of an essentially previous statics.Between theorems and corollaries, the pastor from Quarate encounters thedemonstration on equality of spaces and times of a body in motion on aninclined plane. Torricelli had noticed how proposition VI could have beenobtained 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 Galileanmanuscripts, [he noticed a proposition] that showed itself theproperly divided order by Torricelli: it was proved therein thatthe times are proportional to the lengths of the surfaces andare equally high after the mechanical theorem concluded theisochronism of circles. [Caverni 1895, 328–329]

The pastor tried to find out if the script was signed or added later by afollower. Given the skilful writing of a man endowed with good eyesight anda hand which was still steady, he undoubtedly believed the first hypothesis.Studying these papers further, with increasing surprise he observed anotherstatement about motion along inclined planes23 and it seemed to him “todistinguish the practice of young wings, before spreading freely their ownwings 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 aboutGalileo: “Modern science did not spring perfect and complete, as Athena from thehead of Zeus, from the minds of Galileo and Descartes. On the contrary, the Galileanand 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 someof the contemporaries and predecessors of Galileo” [Koyré 1943, 333]. It is possiblethat Koyré was impressed by this concept thanks to Caverni’s work of which he wasone 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].


1895, 338–339]; therefore, finding in this the confirmation that the materialhe had encountered belonged to the early period of Galilean production.24

Caverni then just needed to understand why Galileo, by publishing thetreaty Dei movimenti locali, had repudiated the other previously processeddemonstrations to adopt a new one based on an assumption; so, working hardon comparing “those mixed-up papers”:

the principles of statics were as mutually agreed with the prin-ciples of dynamics to give the right measure [when] suddenlywe abandon the statics and see that the author avoids it in themanner 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 thusit becomes clear. Sensing that the statics would not allow him to makethe desired steps forward in physics, he turned his attention elsewhere anddemonstrated the propositions by making exclusive use of dynamics—solutionthat seemed less elegant and consistent but which was to turn out to beincredibly prolific in future results. Having thus restored a sense of continuityin the production of the great Pisan physicist, Caverni seized the opportunityto criticize the Edizione Nazionale and its editors (without naming them) inwhich he said there was not enough space for all the papers in a chronologicalorder. He also criticized the editors for being more interested in “putting fullstops and commas in place” [Caverni 1895, 341] than in considering Galileo’sinner intense activity. He then makes explicit one of his key investigativetools—the comparison of calligraphic forms with the various colours of inkbecause:

[...] as is common knowledge, a writing hand feels the effect of theyears in the same way as the movements of all the other limbs, andsimilarly each one can gain experience in himself by comparingthe 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 presentedby the manuscripts of the “First Book”25 which ends with the confirmation ofwhat had been repeatedly announced, i.e., the ways:

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

24. The text was actually included in the edition made by Albèri and entitledDe 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 Galileanworks [Caverni 1895, 342–349 passim].


mechanical terms reduced to the statics and the author does notuse the ten propositions making up his first treaty [...] and couldnot use another topic. However as the new dynamics were estab-lished in 1604, other broad paths were opened for science makingit possible to reach the same desired goal through more directand plainer paths namely to demonstrate the brachistochronismof descending bodies for many inflected strings and underlying afourth circle. [Caverni 1895, 349]

The pastor of Quarate analyzed the second book with his usual attention anddid the same with the third Galilean book De motu. We find the propositionsgradually amended by the father of modern science and the same basic idea,namely that the manuscripts in question represent the central seed from whichGalileo manages to make a new physics flourish, a physics whose most completeand mature expression is represented by the Discorsi of 1638 [Caverni 1895,350–366]. According to Caverni, in 1630, the great Pisan scientist opted for thethird version of the manuscript which he considered to be the best. Howeveralmost all the propositions were recast and the result was not the best it mighthave been with some resulting theorems declassed to the status of corollarywhile some corollary points were elevated to theorems. Also the order of thedemonstrations had not always been respected while appearing rather obscureand long winded to some. In particular, as has been reported, Descartes wasto notice that in order to execute these demonstrations, it was not necessaryto be a great surveyor. However while this is true, Caverni points out that wemust also note that Descartes read the Discorsi in 1638 without consideringthat the proved propositions dated from more than forty years earlier whenthe geometry used was still the work of the likes of Commandino, Valerio andGuidobaldo. It is these works that the Pisan scientist’s work should have beencompared with and certainly not with the newly-invented analytic geometry.26

Furthermore,

[...] it is important to note [...] that, in the Galilean theorems, it isnot simple pure geometry but geometry applied to motion sciencewhich involved a greater difficulty because the predecessors hadonly provided a few examples for certain points. For this reason,to make more certain of the truth of these new findings, Galileooften reduced the abstract generalities to concrete cases and calledfor Arithmetic to be matched with Geometry. [Caverni 1895, 372]

So while Galileo might not have been a great mathematician in elaboratingthe theorems on motion, Descartes himself could not deny that he was “thefirst to apply geometry and arithmetic [...] to prove the new properties andthe several and complicated effects of motion” [Caverni 1895, 372]. Still, ifDescartes and others had seen the demonstrations as they were in their first

26. On the contrary, with regard to these terms of comparison, Galileo—saysCaverni—surpasses them “per una certa elegante facilità” [Caverni 1895, 371].


version, they would certainly appreciate them. For that reason Caverni cansay proudly that:

We believe we have done something for the above-mentionedmathematicians and think we have also partly contributed to thefame of Galileo in publishing the three De motu books in theiroriginal forms. [Caverni 1895, 373]

At the end of the chapter, it is clear that Galileo’s readers reproached theerudite priest from Quarate for the excessive prolixity and verbosity of hiswriting. Carlo Del Lungo [Del Lungo 1920, 274–277] expanded particularlythis point. This problem is actually verifiable throughout the Storia del metodosperimentale in Italia but it would not be the most relevant point. The mainerror (and the most shameful for a historian) is that the writer expressed hisideas in different tones depending on the source and displayed the “insatiableurge of persecuting Galileo in each of his activities” [Del Lungo 1920, 281], orallowing himself to be misled by his personal events during the historical re-construction. In this paper we cannot examine whether and how the judgmentis generally deserved but chapter VI on the science of motion definitely showssomething of interest. The great Pisan scientist developed his science based onthe “mechanical theorem by Tartaglia” [Caverni 1895, 373] but while this originwas the most famous, it was not the most important. Claudio Beriguardi andGiovan Battista Baliani were:

[...] considered by many people to be not only imitators or emula-tors, but actual plagiarists of Galileo. They now appear in historyin their true appearance representing the various provisions of theminds open to the seed of science that the superior provident handspread widely [...] rather than in some closed and privileged smallgarden. [Caverni 1895, 374]

Baliani should indeed be considered closer to Torricelli for the contributionthat he gives the science of motion which consisted, among other things, inretracing more strictly and extending (despite not having read or seen it) whatGalileo had produced in De motu.27 On the other hand it should be notedthat Caverni consistently maintains a certain “nationalistic” line which leadshim to question the work of Huygens and the transalpine physicist EdmeMariotte. He considered that they had the merit of making some Galileanprinciples and theorems prolifically known but also “to a small degree, takeaway 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 mechanicstheorem of which Galileo “in principio si fa Autore [...] e poi lo repudia comesospetto” [Caverni 1895, 376]. The Genoan mathematician “non dà alla sua scienzatutta l’estensione della scienza galileiana, a confronto della quale, se rimane inferiorerispetto alla materia, vince però l’esaltato emulo suo rispetto alla elegante semplicitàdella forma. Dicemmo, e lo ripetiamo, che in ciò il Baliani, meglio che a Galileo, sirassomiglia al Torricelli” [Caverni 1895, 377].


4 Conclusion

Overall, regarding the pages thus far considered, I think we cannot chargeCaverni with acrimony against Galileo. However, in the last chapter entitledDelle 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 “Inquo causa accelerationis motus naturalis in fine, in medio affertur”which is a long and bright comment on the words [written on thesubject in] the book Delle disputazioni. [Caverni 1895, 293]28

From that point, when returning to the text, there is a surprising recon-struction of Galileo. After reading the books recommended by the teacherand 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 theAristotelian authority:

At the time, among the young auditors in Pisa, there was alsoGalileo, in whom Mazzoni recognized an unusual intellectualability to penetrate the science of motion. He recommendedBenedetti’s book to Galileo, explaining the speculation thereonto him in private. The young pupil, in those lively words andin the reading suggested by the teacher, felt the first ineffabletaste of freedom of thought. Also, because the intense adviceand effective examples convinced him to no longer believe in theauthority of Aristotle, he therefore concluded he should not followthe authority of any other philosopher either, even Benedettihimself. [Caverni 1895, 275]

So, what might seem to be a complaint about the Galilean plagiarism ofBenedetti through a hurried reading proves to be a real source of inspirationwhich Galileo opposes in those matters that do not convince him. PerhapsAldo 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 butrather because of its complex and obscure style. A “reading which is notcritical and careful enough may often lead the reader into error” [Mieli 1920,262]. It is worth emphasising the judgements about De motu that RaffaelloCaverni shows in his pages, paying the necessary attention to these youthfulwritings of Galileo.

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


Bibliography

Antinori, Vincenzo [1839], Della vita e delle opere di Galileo Galilei. LibriQuattro, Firenze: Biblioteca Naz., Ms. II, V, 111.

Benedetti, Giambattista [1585], Diversarum Speculationum Mathematicarumet Physicarum. Haeredem, Torino: Nicolai Bevilaquae.

Camerota, Michele [1992], Gli scritti De motu antiquiora di Galileo Galilei:il Ms. Gal. 71 , Cagliari: CUEC.

Caverni, Raffaello [1874], Problemi naturali di Galileo Galilei e di altri autoridella sua scuola, Firenze: Sansoni.

—— [1877], De’ nuovi studi della filosofia. Discorsi di Raffaello Caverni a ungiovane studente, Firenze: Carnesecchi.

—— [1895], Storia del metodo sperimentale in Italia, vol. 4, Firenze: G. Civelli.

Del Lungo, Carlo [1920], La “Storia del metodo sperimentale in Italia” diRaffaello Caverni, Archivio di storia della scienza, I, 272–282.

Favaro, Antonio [1883a], Alcuni scritti inediti di Galileo Galilei tratti daiManoscritti della Biblioteca Nazionale di Firenze, Bullettino di bibliografiae di storia delle scienze matematiche e fisiche, XVI, 135–157.

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

—— [1885], Documenti inediti per la storia dei Manoscritti Galileiani dellaBiblioteca Nazionale di Firenze, Bullettino di bibliografia e di storia dellescienze matematiche e fisiche, XVIII, 1–112, 151–230.

—— [1887], A proposito di “Alcuni scritti inediti di Galileo Galilei”,Miscellanea galileiana inedita, 229–239.

—— [1888], Per la Edizione Nazionale delle Opere di Galileo Galilei sotto gliauspici di S.M. il Re d’Italia. Esposizione e disegno, Firenze: Barbèra.

—— [1899-1900], Raffaello Caverni: nota commemorativa, Atti del RegioIstituto Veneto di scienze, lettere ed arti, LIX(2), 377–379.

—— [1908-1909], Galileo e le edizioni delle sue opere, in: Atti della R.Accademia della Crusca, 27–72.

Fredette, Raymond [1969], Les De motu “plus anciens” de Galileo Galilei:prolégomènes, Ph.D. thesis, Université de Montréal.

Galilei, Galileo [1854], Le Opere di Galileo Galilei, vol. XI, Firenze: Societàeditrice Fiorentina, edited by Albèri, E.


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

Galluzzi, Paolo & Torrini, Maurizio (eds.) [1984], Le opere dei discepoli diGalileo Galilei: Carteggio 1649-1656, vol. II, Firenze: Giunti Barbéra.

Hall, Rupert A. [1979], Galileo nel xviii secolo, Rivista di filosofia, 15, 369–390.

Koyré, Alexandre [1943], Galileo and the scientific revolution of the seven-teenth century, The Philosophical Review, LII(4), 333–348.

Libri, Guglielmo [1841], Galileo, sua vita e sue opere, Milano: Soc. Editoridegli Ann. Universali delle Scienze e dell’Industria.

Maffioli, Cesare S. [1985], Sulla genesi e sugli inediti della Storia del metodosperimentale in Italia di Raffaello Caverni, Annali dell’Istituto e Museo diStoria della Scienza di Firenze, 10(1), 23–85.

Micheli, Gianni [1988], L’idea di Galileo nella cultura italiana dal xvi alxix secolo, in: Galileo, La sensata esperienza, edited by P. et al. Galluzzi,Milano: Amilcare Pizzi, 163–186.

Mieli, Aldo [1920], L’opera di Raffaello Caverni come storico, cenni prelimi-nari, Archivio di storia della scienza, I, 262–265.

Nelli, Giovan Battista C. [1793], Vita e commercio letterario di GalileoGalilei, Losanna [but Florence]: S.N. [but Moucke].

Pagnini, Sara [2001], Profilo di Raffaello Caverni (1837-1900) con appendicedocumentaria, Firenze: Pagnini e Martinelli editori.

Torricelli, Evangelista [1644], Opera geometrica, Firenze: Typis A. Masseet L. De Landis.

Venturi, Giambattista [1818-1821],Memorie e lettere inedite finora o dispersedi Galileo Galilei, Modena: G. Vincenzi e Comp.

Viviani, Vincenzo [1674], Quinto libro degli Elementi d’Euclide, ovveroscienza universale delle Proporzioni, spiegata colla dottrina del Galileo, connuov’ordine distesa, e per la prima volta pubblicata da Vincenzo Vivianiultimo suo Discepolo. Aggiuntevi cose varie, e del Galileo, e del Torricelli; iragguagli dell’ultime Opere loro, che dall’Indice si manifesta, Firenze: AllaCondotta.

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

Antonino DragoFederico II University, Napoli (Italy)

Résumé : Quel est le rôle de Galilée dans la naissance séculaire de la sciencemoderne ? Je réponds à la question ci-dessus à la lumière de deux nouveauxéléments. Dès le xvie siècle, Nicolas de Cues, quoiqu’il ne pratiquât pas lascience expérimentale, a anticipé une partie substantielle de la révolutioncopernicienne et de la naissance de la méthodologie galiléenne. Le deuxièmeélément est l’introduction d’une nouvelle conception des fondements de lascience ; ils sont définis comme constitués de trois dialectiques. Après cettedéfinition, la naissance de la science moderne correspond à un très long proces-sus historique qui se conclut à notre époque. Sur cette longue période, Galiléene 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 enjeuxintellectuels 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 experimentalscience, he substantially anticipated both the Copernican revolution and thebirth of Galilei’s methodology. Secondly, I shall introduce a new conception ofthe 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 ascientific methodology, but also almost the only scientist ever to be awareof the intellectual breadth of scientific enterprise.


36 Antonino Drago

1 Introduction: What are the foundationsof science? The three dialectics

The role played by Galileo in the birth of modern science depends on twoquestions: What is the definition of modern science, in particular its foun-dations? How long was the process of birth of modern science? Withoutdoubt, the foundations of science are constituted by the well-known dialectic1

between experimental data and mathematical hypothesis. But after that, whatin precise terms?

I suggest the following definition of the foundations of science that Iobtained on the basis of four decades of historical works on the main theories oflogic, mathematics and physics; they are constituted by two more dialectics ofa theoretical nature; one between the two kinds of infinity (either potential, oractual) and a dialectic between the two types of organization of a theory (eitherdeductive form principles-axioms, as Aristotle theorized in ancient times, AO,or aimed at the solution of a crucial problem, PO). The infinity dialecticwas formalized, by, on one hand, classical mathematics relying on the actualinfinity (AI) (e.g., Zermelo’s axiom) and on the other hand by constructivemathematics relying on (almost) only potential infinity (PI) [Markov 1962],[Bishop 1967]. The organization dialectic was formalized by means of the kindof logic managing it; on the one hand classical logic, managing AO (e.g., inEuclid’s Elements) and, on the other hand, non-classical logic, in particularintuitionist logic [Dummett 1977] managing PO [Drago 1990, 2012b]. Whena scientist builds a theory, each theoretical dialectic appears under its formalaspect; a formal alternative is opposite to and incompatible with the otheralternative; the scientist has to choose one. In sum, both dialectics constitutetwo scientific dichotomies. Once the choices are made, each dialectic isdissolved and the scientist proceeds within the scientific realm to establishthe notions and principles of his theory. However, when we consider allthe theories of a scientific discipline—e.g., physics—we again recognize thesetheoretical dialectics in the different couples of choices on which the varioustheories rely. In the light of the above illustrated foundations of science, thequestion of the birth of modern science receives a first, partial answer: modernscience was born when not only its method for producing accurate results wasrecognized, but also its foundations, i.e., the two theoretical dialectics. Sincethe alternative choices of these dialectics were not formalized and recognizedas relevant for science before the second half of the last century, we obtain asurprising 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 asboth authors [Hopkins 1988], [Counet 2005] do. It is not the Hegelian dialectic (thattranscends reality through a dynamic of the Absolute Spirit); roughly speaking, it isgiven by two polarities that are not necessarily on the same plane. It may be resolvedby either a reconciliation (as in the first dialectic) or by a choice between the twoalternatives 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 willdetail this long process in relation to Galileo. Notice that the three dialecticsof the foundations of science are mutually independent. Thus, according tothe above definition, the question about the birth of modern science has tobe disentangled into the three questions about when each of these dialecticsemerged 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 atime.

2 The dialectic experiment/ mathematicalhypothesis. Its historical birth andits problematic definition

Certainly, the historical start of modern science was marked above all by thebirth of the experimental method through its innumerable applications givingnew scientific results. Even a first, rough definition of this method was enoughto mark an unprecedented novelty with respect to the philosophical world ofthe ancient Greeks. Rightly, historians of science attributed a crucial impor-tance to this method. However, regarding this dialectic it is no longer possibleto assume a positivist viewpoint, which attributes an eminent role to Galileobecause he, more than any other, set hard experimental facts against theidealistic philosophical tenets of Aristotelian philosophers. If the experimentalmethod was what positivist historians claim, that is, dealing above all withhard, experimental facts, it would be easy to determine a date for the birthsof both scientific activity and the method governing it. Instead, Grosseteste,Francis Bacon, Cusanus, Lavoisier, etc., suggested different aspects of thisdialectic. Moreover, Galileo is a complex figure; in contradiction to the aboveappraisal, he also declared that he could obtain scientific results withoutexperiments [Galilei 1638, II Day]. In addition, the question of whetherGalileo’s activity conformed to a well-defined experimental method is largelyundecided. For instance, about Galileo’s capital achievement—the discoveryof the law of the accelerated motion—Galluzzi wrote:

[...] it is not still made irrefutably clear which element of theGalilean investigation was decisive in such an enterprise: eitherthe 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, philoso-phers of science did not suggest any common agreement on its main features.We know that not only positivist, but also neo-positivist philosophy, althoughsupported 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 discouragingappraisal:

[...] More recent debate has questioned whether there is anythinglike a fixed toolkit of methods which is common across scienceand only science. [Andersen & Hepburn 2015]

The plain conclusion to be derived from these unsuccesses is that this dialecticis philosophy-laden. Indeed, both Koyré and Lindberg stressed that theestablishment of the experimental method was the result and at the sametime the cause of a profound change not only in the methodology aimed atobtaining answers from nature, but also in the metaphysical conception ofreality [Koyré 1957], [Lindberg 1992].

3 The dialectic experiment/ theoreticalhypothesis. Its historical birth andCusanus’ contributions

In such a context of uncertainty about this dialectic, new results concerninghow Cusanus suggested combining experimental data with mathematical hy-potheses are important. This relationship of combination was essentially newwith respect to Greek philosophy, which considered a conjunction betweenmathematics and reality to be impossible. It was new also according to theidea that was staunchly emphasized by Koyré, i.e., modern science overcamethe finiteness of the Greek world (Aristotle) by introducing a mathematicsexplicitly involving infinity, i.e., either potential infinity (i.e., unlimited), oractual infinity. According to some scholars this dialectic started, althoughin a reduced form of a mere programme, in the 13th century with RobertGrosseteste’s (1175-1253) book De Luce [Crombie 1994, II, 319], [Lewis 2007].However, subsequently, in the 15th century Cusanus (1401-1484) suggested im-portant 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. Accordingto him the two words mens and mensura share the same root; the mensconnects itself to reality through a mensura; the mens mutually comparesthrough proportions the numbers obtained (remember that a proportion wasthe only mathematical technique used in physics before the time of Galileo andDescartes) and moreover it mutually combines the corresponding concepts inorder to obtain theoretical constructs.2

2. The “Introduction” by Graziella Federici Vescovini to [Cusanus 1450] is veryinformative about current scientific knowledge in Cusanus’ time. Apart from her—inmy opinion too severe—criticisms, she cleverly summarizes Cusanus’ conception ofhuman knowledge: “The mind is thus specular simplicity that all obtains from itscapability of measuring, numbering and representing”, [Federici Vescovini 2003, xxx].


In addition, in the book De staticis experimentis3 Cusanus indicated themain features of an experimental science through the measurements of aparticular 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 wecan arrive, based on experiments, at the art elicited from theexperiments. [Cusanus 1450, 615, 178]

Notice that in ancient times weight, being considered a quality of a body, wasextraneous to geometry. Its quantification as a quantity decisively introducedscholars to the quantifying of a multitude of other physical properties [Crombie1994, 423].4 Furthermore, in the history of physics the transition from thetheoretical concept of absolute weight to the concept of relative weight playedan exceptional role. According to Koyré, Galileo considered only absoluteweight, 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 asany fixed locations for celestial bodies.5 In fact, in the above mentioned bookCusanus used not so much weight but weight difference. Remarkably, by meansof a Roman scale together with a hourglass Cusanus wanted to determineother physical quantities: e.g., the volume of a body, its specific weight, theweather, the temperature, the sound, the magnetic force, the weight of the airby means of the so-called leaning tower of Pisa experiment, etc.; so that heverbally introduced the indirect measures and, ultimately, the doubly artificialapparatus of measurement. One historian evaluates Cusanus’ work in thefollowing terms:

3. In the following, I will refer to Jasper Hopkins’ English translations of almostall Cusanus’ online books http://www.jasper-hopkins.info/.

4. About this point, Koyré, strangely enough ignored Cusanus’ suggestions whichconstituted those advances that Koyré himself considered a crucial step in the historyof the scientific method: “It is ironic that two thousands years beforehand, Pythagorashad proclaimed that number is the same essence of things, and the Bible had taughtthat God had founded the World on “number, weight, measure” [Wisdom, 11, 21].All people reiterated that, nobody believed that. Surely, nobody before Galileo tookthat seriously. No one attempted to determine these numbers, these weights andthese 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, noone ever sought to get beyond the practical uses of number, weight, measure in theimprecision of everyday life. [...]” [Koyré 1961, 360]. It is apparent that Cusanus didexactly what Koyré stressed as lacking and he did it inspired by the same Biblicalpassage.

5. See Santinello for innovations and advances of Cusanus in science [Santinello1987, 105–109]. In particular, notice that Copernicus knew Cusanus’ modern view ofHeaven without any center [Klibansky 1953].

40 Antonino Drago

In the mid-fifteenth century appeared what should have been thecrowning work of this genre, the Idiota de staticis experimentis,from the pen of Cardinal Nicholas of Cusa. This incidentalpiece from one of the best-known philosophers and churchmenof the time does not appear to have attracted much attention. Itadvocates the use of balance and the comparison of weights for thesolution of a wide range of phenomena, ranging from mechanicsto medicine, and including those of chemical. He advocatesdifferences in weights as a guide to the evaluation of naturalwaters, the condition of blood and urine in sickness and health, theevaluation of the efficacy of drugs and the identification of metalsand 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 ofactivity, was devoted especially to the study of physical phenomena relatingto 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 basedonly on observation, is manipulative, as is modern science. Later, Newton’s useof the new, marvellous mathematics of infinitesimal analysis assigned insteadthe highest role to celestial physics, despite the fact that the foundations ofthis mathematics were unknown and manifestly linked to the metaphysics ofactual infinity [Newton 1687]. Yet, three centuries after Cusanus and onecentury after Newton, chemistry was born. This event was a “postponedrevolution” (a chapter title of [Butterfield 1949]), since, being influenced byhis idealistic mathematics, Newton has led chemists in a misleading direction,relying on several metaphysical notions: absolute space, absolute time, fixedand perfectly hard atoms, gravitational force as constituting the intermolecularlinks. Lavoisier freed chemistry from all of a priori notions by appropriatelyre-defining the experimental method for his field of research, i.e., all thatconcerned the complete methodology of chemical reactions. Lavoisier foundedchemistry by means of only those physical measurements which had beensuggested by Cusanus, i.e., weight differences that he considered between thereagent substances and the compounded substances [Drago 2009b]6. He linkedthese weight differences to weight-mass conservation. This was a trivial lawaccording to Cusanus, since God pervades matter, which therefore cannotchange in quantity [Crombie 1959, 296], [Drago 2009b].— One may object thatCusanus’ book constitutes a mere program of research, since he never practisedexperimental science to the point of obtaining scientific results (he howeverinvented the hygrometer). Yet, recall that at least among the musiciansa very ancient and widespread practice obtained new results from artificial

6. Kuhn devotes a page to stressing the importance of physical magnitude weightfor the birth of chemistry [Kuhn 1957, 262]. Yet, strangely enough he attributes thisinfluence to Newton.


instruments. It is not by chance that Galileo’s father, his brother and he alsowere musicians. Moreover, through the method which Cusanus had described,several physical laws were established, first those of acoustical instruments. Inparticular, Galileo’s father, Vincenzo, inductively inferred from experimentson the string tensions obtained by charging weights more accurate and generallaws than Pythagoras’ law [Cohen 1984, 84].7 Hence, Cusanus did not need toput his program into practice, since artisans were already implicitly applyingit to several kinds of phenomena.

4 The dialectic experiment/ mathematicalhypothesis. Its historical birth andGalileo’s determinant achievements

A century and a half after Cusanus, Galileo extensively applied the experi-ment/hypothesis dialectic to several fields of physical phenomena. However,Galileo had a particular conception of this dialectic, as his celebrated wordsemphasize:

Philosophy [read: science] is written in that great book whichever lies before our eyes—I mean the universe—but we cannotunderstand it if we do not first learn the language and graspthe symbols, in which it is written. This book is written in themathematical language, and the symbols are triangles, circles andother geometrical figures, without whose help it is impossible tocomprehend a single word of it; without which one wanders invain through a dark labyrinth. [Galilei 1890, vi, 232]

This mathematical language is the same as God’s; and whereas we have togo step by step, He sees all in the blink of an eye. Hence, geometry wasalready written in the world since eternity. In this conception of Galileo werecognize—as Koyré correctly emphasized, scandalizing the positivist histori-ans of science—an ontology: not only is mathematics outside our minds (whichhowever 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 supernaturaltheory that structures reality.8 This Platonist conception of mathematics

7. He improved Pythagoras’s law, successfully suggesting the quadratic and evenmore complex relationships between two variables; this was a mathematical break-through, which later his son Galileo performed for the law regarding the case of thefalling bodies; an initial law stating a linear relationship between space and time, wasdiscarded 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, heopposed the historians (e.g., Crombie) whose conceptions of the history of science donot attribute a basic role to actual infinity.

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influenced the entire history of theoretical physics. The subsequent physicaltheory, geometrical optics, made use of points at infinity as if they wereordinary points. Moreover, Newton’s mathematical technique was infinitesimalanalysis, which is informed by a Platonist philosophy of infinitesimals. In the19th century, this conception reached a climax through the dominant roleplayed by mathematical physics, a role which in 20th century was counteredby the discovery of quanta and then symmetries.

Mathematics indeed may be joined with experiment also according toan instrumental conception of it, as is the case in both chemistry andthermodynamics [Drago 2009b, 2012a]. In fact, a century and a half beforeGalileo, Cusanus had suggested an alternative philosophical conception ofmathematics, which was summarized by McTighe [McTighe 1970]. Accordingto him, ideal mathematical entities are built from reality (and not, as Galileoclaimed, recognized as the ideal elements of reality), i.e., they are constructsof our minds. Whereas ordinary knowledge of reality is always approximateand hence it is not subject to the principle of non-contradiction (througha sharp and exhaustive division into true and false), instead, mathematicsis perfect knowledge because it enjoys a specific feature; it strictly obeysthis principle. Owing to this logical property, mathematics is consideredby Cusanus the most certain knowledge we have; for this reason, we applyit to understanding reality. Moreover Cusanus conceived numbers as theexplicit forms of unity, i.e., a number enjoys the property of being both asgreat and as small as it can be; thus it is through numbers that all thingsare best understood, i.e., referred to unity. For these reasons, the creationof our minds, mathematics, can direct our interpretation of reality. Indeed,by combining experimental data with mathematics the intellectus conjecturesthe mathematical formula of a physical law. This Cusanus’ way of describingthe relationship between mathematics and experiment more adequately thanGalileo’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 bodiesactually originated in Galileo [Wisan 1974, 207–222]). Furthermore, Galileoknew only the mathematics of geometry and proportions and did not sharein the effort—elicited, e.g., by his disciples Cavalieri and Torricelli—to extendthe realm of mathematics [Drago 2003]. Also for this reason, the theory ofmechanics was born later, through Newton, since it required the invention ofa very different mathematics, calculus (just as at present time the advancesin theoretical physics require the new mathematical technique of symmetries).Before Galileo, Cusanus had, on the other hand, been able to expand themathematics of his time (geometry and proportions) to new achievements (wewill see them in the next section).

I conclude that Cusanus offered to theoretical physics a more adequatemetaphysics of the scientific methodology than Galileo’s, owing to latter’sPlatonist view of mathematics. Yet, most historians disregarded the entiremetaphysical and mathematical change brought about by the introductionof the scientific method. For an instance, Cohen gave a totally objectivist


definition [Cohen 1984, 85–86]. The experimental method developed by Galileoenjoys the following two features:

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

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

Cohen’s words “a priori postulates” and “axiomatic” attribute to Galileo’sthinking the choice AO; in the following section 6, I will disprove it. I remarkthat all Cohen’s features of the scientific method are present in Cusanus,including “the final empirical check”, in the cases of a physician wantingto 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 ofCusanus 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 exploringonly the mind’s faculties, not natural phenomena. Hence, he sees the elementsof knowledge but he does not apply them to producing knowledge from finitereality; he is too interested in infinite reality. At most, he is interested inphysical principles; e.g., the principle of relativity (through the celebratedobservation that over a ship sailing in a calm sea the phenomena are the sameas those on the land), the impossibility of a perpetual motion, the inertiaprinciple [Cusanus 1462-1463].

Galileo played a key historical role in the introduction of the dialecticexperiment/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 breakthroughtheoretical laws, in particular the decisive experimental law of falling bodies,which described reality in contrast with everyday experience. This resultestablished 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 advancesnotwithstanding the harsh opposition of Aristotelian philosophers of his time;v) he practised and made this method productive to the extent of achieving anessential part of a mechanical theory; vi) his great and long activity producedso many scientific results that it eventually established a stable tradition ofexperimental research that, subsequently followed by several other scholars,

9. He contented himself with illustrating his experimental method by describingthe simplest artificial instrument, a wood spoon [Cusanus 1450, IV, chap. II, IV].

44 Antonino Drago

systematically accumulated new commonly recognized results.10 From allthe above I conclude that Cusanus anticipated all the elements of Galileo’sexperimental method; however, this consideration in no way detracts fromGalileo’s glory in having organized all these elements into a consistent set ofrules and systematically applied them to a multitude of natural phenomena, sothat he established a new tradition in science. Rather, the above anticipationsby Cusanus present the historical birth of this dialectic as a less clear-cutchange 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 mathematicallimit as a series of multilateral polygons approaching a circle). Owing to thisinnovation, Cassirer considered Cusanus to be the first modern philosopher ofknowledge [Cassirer 1950].12 In addition, I have discovered [Drago 2009a,2012a] that he defined (without mathematical formula, yet through exactwords, 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 centuriesafter Cusanus, Galileo’s conception of mathematics was mostly that of theancients, also regarding infinity. In the book concluding his scientific career[Galilei 1638, First Day] he discusses the two ideas of infinity (AI and PI) inorder to understand how to apply one of them to physics. Having dissected theproblem, Galileo concludes that in mathematics one cannot define the usualarithmetical operations on infinite objects. Moreover, by recognizing that heis 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 withsimilar 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 andchampion of modern philosophy” [Cassirer 1978, I, 39]. Note also [Vanstenbeerghe1920, 279]: “The key to the philosophical system of Nicholas of Cusa [...]—and this isvery 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 theAncient Greeks and opened minds to the infinite Universe.


of them is useful in formulating his laws, he asks to be allowed in any case tomake use of the notion of infinity in theoretical physics.

6 The third dialectic: axiomaticorganization/ problem-basedorganization. Its historical birth

During the birth of modern science, Cusanus organized his theological theoriesin 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: intheology 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, thedouble nature of Christ, the Trinity, the constitution of both the Universe andmatter, the new logic, peace in the world, etc. Also his scientific programmeof weight measuring is aimed at solving problems, ultimately the problem ofhow to acquire the knowledge of the natural world. In sum, also in scientificsubjects Cusanus’ choice is for PO. A century and a half after Cusanus,Galileo’s theoretical formulations of his scientific results did not conform tothe 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 deducephysical laws [Clavelin 1996, 66]. Yet, he does not assume a definite positionon this dialectic. In each of his last two books he illustrates their contents byalternating two kinds of organization—deductive and dialogical-inductive, thelatter recalling the dialogues of Aristotle’s adversary, Plato. In sum, he didnot 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 incontrast with the Aristotelian model of organizing a scientific theory (see itselementary presentation in [Beth 1959, § 1.2]), but subsequent scientists, owingto his inconclusive position on this subject, eventually lessened the import ofhis innovations; they changed only one postulate of the Aristotelian model, theevidence postulate, which was attributed no longer to axioms, but to the dataof the theory.14 In sum, the result was a mere reform of this organization, notan alternative model to AO.

13. It is a merit of W.A. Wallace to have emphasized Galileo’s Juveniliamanuscripts concerning the illustration of Aristotle’s apodictic organization of atheory. Yet, Wallace wanted to exploit this fact in order to link Galileo to Aristotelianphilosophy, although Galileo’s last two books manifestly show that he did not shareAristotle’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 twodifferent kinds of logic, respectively classical logic—governing the deductionsof a AO theory—and non-classical logic—governing the inductions of a POtheory. Some scholars have remarked that the logical ways of arguing of bothCusanus and Galileo are different from those of classical logic. These differ-ences were attributed to their attendance at the Padua school of logic, whereZabarella made the greatest effort to theorize a different way of reasoning fromthat of Aristotle. Actually, after the Padua period, Cusanus wanted to found anew kind of logic and he was successful in this aim. He believed that Aristotle’slogic represented a specific activity of the ratio, a particular way of arguing ofthe 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 acelebrated “coincidence of opposites”, which he later abandoned as ineffective.Rather he implicitly changed logic. An accurate inspection of his texts showsthat he made use of the characteristic propositions of non-classical logic, thosefor which the double negation law fails; i.e., the doubly negated propositionswhose corresponding affirmative propositions lack evidence (DNPs). Throughthem he solved his main problem—the best to name God—, by suggestingnames pertaining to non-classical logic (Not-Other, Posse=est). Moreover, hewas capable of developing the logical arguments, which are specific to the idealmodel of a PO theory, i.e., ad absurdum arguments [Drago 2012a].

Some authors have remarked, on the other hand, that Galileo paid littleattention to logic. In particular, he never attacked Aristotle’s legacy inlogic. Moreover, his last two books present a strange logical approach; whenexpounding theorems, he makes use of propositions in Latin in accordancewith classical logic;15 when illustrating his investigations he makes use ofa Platonist-like dialogue among three people all speaking the vulgar Italianlanguage; an inspection of the logic of these dialogues recognizes several DNPsand 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. Anexamination of them shows that:

1. The parts of his text written in Latin and concerning formal theoremsdo not include DNPs;

2. The other parts—written in vulgar Italian—include more than100 DNPs;

15. On this basis, Wallace states that “if one takes reliance on Aristotle’s logicalcanons to be the sign of a Peripatetic, one [Galileo] can rightfully be called aPeripatetic himself”, [Wallace 1998, 51]. Yet he ignores the inductive parts of Galileo’stheoretical work.


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

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

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

6. The latter DPNs compose 8 cycles of reasoning, which are illustrated bytable 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 inorder to deductively derive from it, now considered as a postulate subjectto classical logic, all possible consequences to be tested by experiment.Galileo is the only scientist apart from Einstein to have illustrated thislogical step through admirably precise words [Einstein 1905a, 891]:

Let us then, for the present, take this as a postulate, theabsolute truth of which will be established when we findthat the inferences from this hypothesis correspond to agreeperfectly with experiment [Galilei 1638], [Galilei 1638, 183,period before Theorem 1, Proposition I of the Naturallyaccelerated motion; emphasis added]16

However, Galileo ignored the alternative logical nature of both the DNPsand the ad absurdum arguments, maybe because his way of reasoning by meansof DNPs was sometimes invalid (actually, it was invalid, according to recentstudies, also in the deductive reasoning) [De Luise & Drago 2009, fn. 15]. Iconclude that, his genius was capable of achieving exceptional, but irregularresults in logic.

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

48 Antonino Drago

1o 2o 3o 4o

(3 →⊥)5◦

(the casev = 0)

6◦

(v =ks →⊥)

7◦? 8◦

Hp of alaw v =kt

Hp ofinfinitedegreesofvelocity

Objection:Is timeinfinite?

Refutation:As manydegrees ofvelocityas thedegreesof time

Refutationof theobjection:Even inthe casev = 0?

Falsityof theotherlawv = ks

Laws onfallingbodiesoninclines

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 thiscenturies long historical process?

First, one has to take into account that Galileo’s methodological changeand his several scientific results were not enough to give rise to modernscience because he did not achieve any complete physical theory. Ratherboth Cusanus and Galileo anticipated scientific theories. On this point acomparison is not easy because their advances anticipated two very differentscientific theories, respectively chemistry [Drago 2009b], which was born threecenturies after Cusanus; and mechanics, which was born a few decades afterGalileo’s death. Strangely enough, neither anticipated geometrical optics; itwas instead anticipated by Grosseteste four centuries before its birth, whichoccurred in the last period of Galileo’s life. Hence, an appraisal on Galileo’srole cannot refer to a complete scientific object, but only to the elements ofphysical theories. We will consider their most important elements, i.e., thefoundational choices.

Let us compare Cusanus and Galileo with regard to the two theoreticaldialectics. Cusanus was the first to conceive mathematics non-Platonicallyand he improved the mathematics of the time by first introducing infinity;and regarding infinity, he formulated, albeit in verbal terms, the notion oflimit in PI mathematics and a basic notion of AI mathematics. First Cusanusmade them manifest through the invention of new mathematical notions andnew (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 andlogic. Already Cassirer stressed that no one more than Cusanus, after thefirst of his main books, De docta ignorantia of 1444, accomplished such ametaphysical change [Cassirer 1950, 277].

One century and a half after Cusanus came Galileo, who discussed therelevance of the two kinds of infinity in theoretical physics, any way he did


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 theorganization of a theory and the logic suggested by Aristotle. He looked foran alternative logic, yet his Paduan period did not lead him to suggest a specificnovelty; although as a matter of fact he adapted some of his theoretical worksto 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 aPlatonic dialogue.

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

In conclusion, it is to Galileo glory to have given birth to science inits first dialectic through systematic experimental practice; he was awareof the other two theoretical dialectics, but was inconclusive about them.For these reasons he has rightly been characterized as “The first modernscientist and the last of the ancient Greeks”. Cusanus played a somewhatcomplementary role; regarding the first dialectic, he has suggested only aprogramme for an experimental science of nature; however, regarding thelatter two theoretical dialectics he preceded and was more advanced thanGalileo, so that he anticipated the discovery of them. Which allows us tocharacterize Cusanus by means of a similar definition to the one above: “Thelast of the medieval scientists and the first philosopher of the foundations ofmodern science”.17 Although Cusanus had a considerable influence on theItalian intellectual milieu, e.g., on Leon Battista Alberti, Leonardo da Vinciand Giordano Bruno, no direct connection with Galileo is known. Had Galileoassumed Cusanus’ priorities, he would represent the culmination of a longintellectual effort started more than one century before. If, alternatively, hedid not know Cusanus’ works, Galileo’s genius proves to be even greater, butalso little explained.

After Galileo, without any discussion about the said dialectics, Newtondecided the theoretical dichotomies with a pair of choices, which became aparadigm throughout the two subsequent centuries of development of theoret-ical physics. Due to this fact, a winter set in thinking about the two theoreticaldialectics; in this winter only two scientists, i.e., L. Carnot [Carnot 1803, xiii–xvii, 3] and independently Einstein [Einstein 1905], made manifest the fourchoices, without receiving attention on this subject [Drago 2013]. Hence, onemore merit has to be attributed to Galileo, i.e., at the very beginning of modernscience to have presented through his discussions almost all the foundationalissues of the scientific enterprise which have been re-discovered only after avery 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

AcknowledgmentsI am grateful to Prof. David Braithwaite for having revised my poor Englishand to an anonymous referee for some important suggestions.

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Les De motu antiquiora de Galileo Galilei :le lancement de la carrière du

filosofo-geometra

Raymond FredetteUniversité 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-aprèsEN] par Galileo Galilei [ci-après Galilée] (1564-1642). Cette œuvre de jeunessede Galilée a été écrite entre 1589 et 1592. Elle traite du mouvement de lachute des corps et rejette clairement la physique du mouvement d’Aristoteet ses conceptualisations cosmologiques. Elle n’a jamais été composée commeune version finale et a été publiée seulement après la mort de Galilée. Le Demotu antiquiora présente quelques incertitudes mathématiques typiques decette période dont je vais discuter dans cette article.

Abstract: In this paper I passionately analyse the De motu antiquiora (here-after De motu) of the Edizione Nazionale [Galilei 1890-1909, hereafter EN]by Galileo Galilei (1564-1642). This work by the young Galilean was writtenbetween 1589 and 1592. It deals with the motion of falling bodies and clearlyrejects 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 ofthat time which I discuss in this paper.

1 Introduction

En soumettant publiquement le présent texte, j’aimerais reproduire à moncompte ce qu’Henri-Irénée Marrou, le grand historien de la fin de la cultureantique 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,


56 Raymond Fredette

fugace, fragmentaire du passé, dans le but d’appuyer le récit, le nourrir commede sa nécessité. Je me propose de rendre le récit impératif par sa référencetoujours explicite à la trace factuelle, à la prétendue donnée. La donnée puren’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 estnécessairement l’objet d’une manipulation jamais neutre.

Je fais le pari que les gens que passionne la lecture de récits historiquesveuillent peut-être me suivre pour voir comment on traite le passé-objet commeune cible et la connaissance historique comme une lampe de poche ou unrayon laser qui vise la cible sans jamais pouvoir savoir si la cible ne cache pasencore quelque chose à révéler et qu’on avait oublié. La vérité historique est demême 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 àl’oubli. D’où l’idée que dire vrai c’est dévoiler, enlever le voile qui empêche devoir, révéler.

J’aimerais cependant éviter le récit mythique, c’est-à-dire la narrationdes événements comme si un journaliste-témoin-présent revenait nous parleraprès coup, suite à un voyage astral dans le passé. Les journalistes sontdes professionnels habituellement très compétents, mais ce ne sont pas deshistoriens. Pour essayer d’éviter la mystification, je me propose de toujourstenir à bout de bras deux discours : d’une part, celui qui présente la prétenduedonnée, qui est toujours fragmentaire, mais qui sert de fondement et de sourceet de base à l’interprétation des événements, et d’autre part, celui qui narrecette interprétation elle-même.

Je suis animé ici par l’idée que je me fais de ce que doit être l’activitéscientifique. Cela me sert de modèle et c’est d’histoire des idées scientifiquesdont il va être question. En écrivant l’histoire de cette façon, mon but estd’aller toujours au-devant de la critique, d’inviter l’introduction de nouvellesdonnées fragmentaires ou de nouvelles perspectives sur elles. Pourquoi cela ?Tout simplement pour que se poursuive en pleine conscience inlassablement letravail inéluctablement provisoire et artisanal des beaux métiers de scientifiqueet d’historien, d’historienne. Sans autre détour, voici l’amorce de ce récit àdouble entrée !

Autour de 1590, Galilée écrit :

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

Lorsqu’il consigne cette réflexion parmi d’autres dans un petit cahier qu’onretrouvera par hasard longtemps après sa mort, Galilée n’a pas encore trenteans. 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. Maisce fait aurait cependant très bien pu échapper à l’histoire, c’est-à-dire à laconnaissance historique, Geschichte. Nous avons ici un même mot qui désignedeux réalités à ne jamais confondre : une réalité disparue, morte, un passé àjamais révolu, par définition hors d’accès en lui-même si ce n’est à travers destraces ; et une réalité actuelle, vivante, fruit de l’exercice du métier toujoursà recommencer qui consiste à faire parler ces traces. Ce que vous allez lireest l’histoire parfois loufoque et invraisemblable, mais pourtant véridique etpassionnante, d’une de ces traces : un manuscrit de jeunesse dans lequel Galiléea écrit qu’il s’attendait à rencontrer chez de nombreux lecteurs une résistanceféroce à se ranger à son avis. Si cette réflexion témoigne de la conscienceclaire qu’il a déjà que ses idées vont soulever la controverse, on aurait tort depenser qu’il s’agit là d’une prémonition applicable à l’ensemble de sa carrièremouvementée. Cette réflexion est une note qu’on ne doit appliquer qu’auxtravaux auxquels elle se réfère. Et ces travaux sont ceux qui contiennent sespremières tentatives de philosophe, de mathématicien, de physicien portantsur la question du mouvement. Comment, en vertu de quoi, des corps – disonsun morceau de bois, un caillou, qui diffèrent par la pesanteur – se déplacent-ilsdans 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 plusanciens sur le mouvement, de son vivant, Galilée ne les soumettra jamais àpersonne 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, ilsne sont jamais mentionnés explicitement, ni dans les œuvres qu’il a publiéesou laissées circuler ni dans sa volumineuse correspondance. La note que j’aicitée ici est la seule et unique remarque explicite qu’on connaisse de Galiléeà propos de ces premiers De motu. Et on la trouve parmi d’autres notes qu’ilavait accumulées dans un cahier à part en vue d’apporter des révisions etdes corrections importantes aux résultats de ses recherches alors en cours derédaction. Et pourtant, aujourd’hui, on peut lire ces travaux de jeunesse endynamique. En effet, avec grands soins, selon une disposition très particulière,il les a conservés dans ses papiers. Et jusqu’à ce que le manuscrit contenantl’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’estensuite 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’estpour 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, lanotion 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 carrièrede chercheur. À partir de 1585, date à laquelle il abandonne l’université de Piseoù depuis cinq ans son père espérait faire de lui un médecin, jusqu’à sa morten 1642, Galilée va passer plus de 50 ans, comme en tête-à-tête, avec l’idéede pesanteur, la gravitas. Et en 1632, dans son fameux Dialogue sur les deuxgrands systèmes du monde, le Dialogo [EN VII 260.7–261.10], discutant avecl’aristotélicien Simplicio, voici ce qu’il va mettre dans la bouche de Salviati, sonporte-parole. Nos deux compères tiennent une discussion de principe autourde « deux auteurs qui ex professo écrivent contre Copernic » :

SIMPLICIO : Mais l’auteur va faire une objection, comme vousle voyez ; il demande de quel principe dépend ce mouvementcirculaire des lourds et des légers : est-ce d’un principe interneou d’un principe externe ?SALVIATI : Si l’on s’en tient à son problème, je dis que le principequi fait tourner le boulet sur la concavité de la Lune est le mêmeque celui qui maintient sa circulation pendant la descente ; je laisseensuite 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 niexterne.SALVIATI : Je répondrai, moi, que le boulet sur la concavitén’avait pas de mouvement ; je serai donc dispensé d’avoir à expli-quer comment il se fait qu’au cours de la descente, il reste toujoursà la verticale du même point, puisqu’il n’y restera pas.SIMPLICIO : Bien, mais, comme les lourds et les légers nepeuvent avoir aucun principe, ni interne ni externe, de mouvementcirculaire, le globe terrestre n’aura pas non plus de mouvementcirculaire : nous avons ainsi atteint le but.SALVIATI : Je n’ai pas dit que la Terre n’a aucun principe, niexterne ni interne, de mouvement circulaire, mais je dis que je nesais pas lequel des deux elle possède ; et ce n’est pas parce que jene le sais pas que cela peut supprimer le principe. Mais cet auteursait en vertu de quel principe d’autres corps du monde tournent –comme ils le font assurément –, alors je dis que ce qui fait mouvoirla Terre, c’est quelque chose de semblable à ce qui fait se mouvoirMars, Jupiter et aussi, à ce qu’il croit, la sphère étoilée ; qu’ilm’indique le moteur d’un de ces mobiles, et je m’engage à lui direce qui meut la Terre. Bien plus, je prends le même engagements’il réussit à m’apprendre ce qui meut les parties de la Terre versle bas.SIMPLICIO : La cause de cet effet est bien connue, chacun saitque c’est la pesanteur [gravità].SALVIATI : Vous vous trompez, signor Simplicio. Ce que vousdevez dire, c’est que chacun sait qu’on l’appelle pesanteur. Mais


ce que je vous demande, ce n’est pas le nom, c’est l’essence de lachose : de cette essence, vous n’en savez pas plus que de l’essencede ce qui fait tourner les étoiles, si ce n’est le nom qu’on y aattaché et qui est devenu familier, banal, parce qu’on en a faitl’expérience fréquemment, mille fois par jour ; mais ce n’est pascela qui nous fait mieux comprendre quel est le principe, ou lavertu, qui meut la pierre vers le bas, pas plus que nous ne savonsce qui la meut vers le haut quand elle s’est séparée du lanceur, ouce qui meut la Lune en rond, sauf (comme je l’ai dit) le nom : nouslui avons donné comme nom propre et singulier le nom de gravità,alors que nous utilisions dans le second cas le terme plus génériquede virtù impressa et dans le dernier cas le nom d’intelligenza oud’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 « lesanges qu’on assignait à la conduite des planètes dans leur course étaient ditsassistants ; informant est le principe moteur interne, comme l’âme dans lesêtres animés » [Galilei 2000].

3 La méthode de travail de Galilée

Le succès prodigieux de la carrière de Galilée repose sur des méthodes detravail qu’il a pratiquées dès le départ.

C’est à Aristote que l’Occident doit la première codification des règleslogiques de la pensée. Le Stagirite a donné à ce code le nom grec d’Organon quiveut dire L’Instrument. Et pour ce faire, Aristote a abondamment puisé dansce 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 denotre pensée à se pratiquer à être rationnelle, logique. Aujourd’hui la logiquefait partie du domaine de la mathématique. À l’époque de Galilée, logique etgéométrie, 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 sciencesnouvelles, les Discorsi, Galilée a derrière lui 50 ans de pratique du métierde philosophe-géomètre. C’est l’association et l’intégration de la logique àla géométrie au sein du travail quotidien du philosophe de la nature quipermet à celui-ci « d’inventer », i.e., de détecter les défectuosités, tant logiquesqu’empiriques, des arguments en cours sur une question, et ainsi de se mettreen piste pour en trouver de meilleurs. Sur la différence entre la logique etla géométrie en rapport à la philosophie naturelle, voici un passage clef desDiscorsi. 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 etindépendant.

SALV. [...] Ainsi comprend-on que quand le support F s’appro-chera de l’extrémité D, il faudra nécessairement augmenter àl’infini la somme des moments des forces appliquées en E et en Dsi l’on désire équilibrer ou surmonter la résistance placée en F .SAGR. Que dirons-nous, seigneur Simplicio ? Ne convient-il pasd’avouer que la géométrie est le plus puissant de tous les ins-truments pour aiguiser l’esprit et pour le mettre en mesure deraisonner et d’analyser de la meilleure façon ? Platon n’avait-ilpas grandement raison quand il désirait que ses élèves eussentd’abord de solides bases mathématiques ? J’avais très bien comprisd’où vient le pouvoir du levier, et comment selon qu’augmente oudiminue sa longueur, le moment de la force et de la résistanceaugmente ou diminue de son côté ; et pourtant, dans la solutiondu présent problème je me trompais, non pas un peu, mais dutout au tout.SIMP. Vraiment je commence à comprendre que la logique, bienqu’elle soit d’excellent usage pour régler le raisonnement, nepossède 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’ellenous apprenne à inventer les raisonnements et les démonstrationsconcluantes, cela vraiment je ne le crois pas. Mieux vaudraitcependant que le seigneur Salviati nous fasse voir selon quelleproportion augmentent les moments des forces nécessaires poursurmonter la résistance du même cylindre de bois quand varientles points choisis pour la rupture. [EN VIII 175, 12–34]

Comme nous pouvons le constater, la science nouvelle est ici concernée pardes problèmes d’ingénierie, ceux de résistance des matériaux ; l’autre sciencenouvelle 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 lagéométrie, pour Galilée, apportait à la philosophie naturelle : unlangage permettant de caractériser les faits physiques dans leurspécificité, et en même temps une science capable, par la richessede ses propres propositions, d’« éveiller l’invention » du physicienen l’aidant à pressentir et à expliciter les relations susceptibles deconduire à l’intelligibilité des phénomènes naturels. [Galilei 1970,111, note 61]


Or, il se trouve non moins de 8 passages du Tractatus des DMA en23 chapitres qui illustrent sans équivoque la clarté de cette pratique dès ledépart 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 desphénomènes.Mais, afin que toujours nous utilisions des raisons davantageque des exemples (en effet nous cherchons à savoir les causesdes effets, lesquelles ne sont pas rapportées par l’expérience),nous apporterons sous les yeux de tout le monde notre façon depenser, dont la confirmation fera s’écrouler l’opinion d’Aristote.Nous disons donc que, les mobiles de même espèce (que soientappelées de même espèce ces choses qui sont produites de la mêmematière, comme du plomb ou du bois, etc.), bien qu’ils diffèrenten grandeur, sont cependant mus avec la même rapidité, et unepierre plus grande ne descend pas plus rapidement qu’une pluspetite.

Nous sommes ici au chapitre 8, celui qui contient l’exposé des résultatsthéoriques les plus importants de toute l’entreprise des DMA : on y trouveune première approximation de la loi de la chute. Le passage suivant égalementappartient 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 mouve-ments des mobiles qui diffèrent seulement en pesanteur et nonen grandeur, sont donnés aussi les rapports de ceux qui diffèrentselon n’importe quelle autre manière. Ainsi {1} donc, afin quenous 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 deleurs pesanteurs, nous démontrerons des choses desquelles nonseulement la réponse à cette question dépend, mais également àla question du rapport des mouvements des mêmes mobiles dansdifférents milieux ; et nous examinerons l’une et l’autre questionsimultanément.

Tout l’intérêt de ce passage se trouve dans la note marginale {1} qui lui est liéedans l’édition critique et que Favaro signale. Voici une transcription partiellede cette note avec mes commentaires.

{1} Le passage cité jusqu’à « et nous examinerons l’une et l’autre questionsimultanément », se trouve dans la marge du folio 72r du manuscrit Gal. 71de la BNCF, la Biblioteca Nazionale Centrale di Firenze, en remplacementcependant de près de trente-trois lignes de texte, dont les 16 premières setrouvent au folio 71v, et que Galilée avait d’abord rédigées. Les voici :

Afin donc que nous trouvions ce rapport, on doit en venir à lacause de la rapidité et de la lenteur du mouvement [...]. Nousdisons que les mobiles entre eux respectent dans leurs mouvementsle rapport qu’ont leurs pesanteurs : pourvu qu’ils soient pesésdans ce milieu dans lequel le mouvement doit avoir lieu. [...] eneffet, deux corps pesants ne respectent pas le même rapport dansdes milieux différents. Mais ici surgit une difficulté très grande :en effet nous voyons par l’expérience que de deux boules égalesen grandeur, dont l’une est plus pesante du double de l’autre,envoyées depuis une tour, celle qui est plus pesante ne percutepas terre deux fois plus rapidement ; qui plus est, même qu’audébut du mouvement la plus légère devance la plus pesante, etqu’à travers un certain espace elle est portée plus vite. Or ceci estune matière pressante ; et de la plus haute importance, laquelle,parce qu’elle dépend de certaines choses qui n’ont pas encore étéexpliquées, sera mise en réserve, jusque là où sera donnée la causede l’augmentation de la rapidité du mouvement naturel : où ilsera démontré, que c’est par accident que le mouvement naturelest plus lent au début ; et à partir de cela que c’est aussi paraccident qu’il sera reconnu qu’un mobile deux fois plus pesantne descend pas d’une tour deux fois plus rapidement ; et il seraensuite expliqué également pourquoi un mobile plus léger est portéplus rapidement, au début du mouvement, qu’un plus pesant.

Relevons cette double affirmation étonnante selon laquelle, contrairementà ses prévisions théoriques, Galilée affirme avoir observé « non seulementqu’un mobile deux fois plus pesant ne descend pas d’une tour deux fois plusrapidement, mais qu’un mobile plus léger est porté plus rapidement, au débutdu 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


Le fait d’avoir accès à l’autographe des Travaux plus anciens sur lemouvement permet de voir notre jeune Galilée dans l’intimité même de sontravail. Sans nous perdre trop dans les détails, examinons comment il réagitface à une situation fort embarrassante. Galilée ici a corrigé l’erreur d’Aristoteen montrant que les corps chutent non pas en proportion de leurs pesanteursabsolues, mais de leurs pesanteurs spécifiques relatives. Il s’attend donc à ceque des corps de même matière tombent à des vitesses relativement égales etde surcroît uniformes. Or, ce n’est pas ce dont on fait l’expérience. Ces corpshomogènes, c’est un fait, accélèrent en tombant. S’il fait tout à fait confianceà sa correction d’Aristote, il ne peut quand même pas penser à cette époqueque la pesanteur ne soit pas pour quelque chose dans la chute. Vers quellesautres propriétés de la matière voulez-vous qu’il se tourne ? Il ne peut pas nepas s’accrocher à cette propriété. Au-delà des innombrables propriétés dont leschoses 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’àregret, longtemps plus tard, nous l’avons signalé ci-dessus, qu’il avouera qu’ilne sait pas ce que c’est vraiment que la pesanteur. Et il se contentera alors defaire confiance aux derniers résultats, tant ceux des expérimentations avec leplan 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 devancela plus pesante, à travers un certain espace elle est portée plus vite ».

Anna De Pace, à propos de l’affirmation du jeune Galilée, écrit [matraduction] :

Ce qui rend perplexe sur la véracité de cette qualification, commecela a été signalé à plusieurs reprises, c’est l’impossibilité d’obser-ver ce que Galilée déclare pourtant avoir observé, à savoir que dedeux corps en chute libre le plus léger précède durant un certainbout, le plus pesant. [De Pace 1990, 54]

Est-ce à dire que Galilée est un halluciné ou encore un menteur ? Faire dela science, c’est, entre autres choses, faire des expérimentations. Celles-ci,par définition, pour être valables, doivent pouvoir être répliquées par lescollègues. La science, disait Gaston Bachelard, c’est l’union des travailleursde la preuve. Thomas B. Settle a publié les résultats d’une réplicationvraisemblable de ce que Galilée rapporte avoir observé par expérimentation[Settle 1983, 12–15]. On a photographié les débuts de chutes de lâcherssimultanés de poids. Les 51 sujets recrutés tenant dans une main une boule defer, dans l’autre une en bois.

chapitre tout entier à ces problèmes ; il s’agit de l’avant-dernier chapitre du livre II deson 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 Pace1990, 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 14de ma communication de Liège en 1997 [Fredette 2001, 125–130].

64 Raymond Fredette

[...] the significant finding was that in only 12% of the drops didthe heavy ball either precede or fall alongside, the light one. In88% of the trials [il y a eu 4 fois 51 lâchers] the light ball clearlypreceded the heavy one [...]. [Settle 1983, 12–15]

Vouloir produire de la connaissance nouvelle, c’est très compliqué. Et il n’y apas de « logique de la découverte scientifique », comme l’a montré Popper, iln’y a pas une recette à suivre. On le voit bien, Galilée ne trouve pas du premiercoup la loi de la chute des corps. Mais il est cohérent. Sa critique d’Aristotele guide, mais il faut plus. Et c’est une grande sensibilité à ces écarts entreses prédictions théoriques et les phénomènes qui vont guider sa recherche. Peuà 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 pourGalilée. Aristote avait imaginé comme étant intrinsèque aux corps deuxpropriétés contraires, la pesanteur et la légèreté, expliquant qu’en l’absenced’obstacles à leur mouvement, lesdits corps tendent naturellement et au plusvite à joindre, soit vers le haut soit vers le bas, leur lieu naturel. Galilée renoueavec les Anciens et élimine la légèreté qui n’est plus que de la pesanteurmoindre. Mais alors comment comprendre qu’un corps dont la pesanteurconstante fait qu’il tombe ne tombe pas selon un mouvement uniforme, mais vaen s’accélérant ? Galilée ne le comprendra jamais. La loi d’attraction universelledes corps qui dit que les corps sont des masses qui tombent les unes sur lesautres en raison directe de leur quantité de matière et en raison inverse du carrédes distances qui les séparent, comme Isaac Newton va l’établir en 1687 dansses Principes mathématiques de la philosophie naturelle, une telle loi auraitrépugné profondément à Galilée, qui avait horreur des affirmations aux odeursde 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 concluraitfort à propos et par nécessité, si les choses qu’il a tenues pouracquises, Aristote les avait démontrées, ou encore, si elles n’ontpas été démontrées, si elles eussent du moins été vraies ; mais il aété trompé en ceci, [entre les lignes, au-dessus du mot « trompé »Galilée ajoute le mot, allucinatus « induit en erreur »] que ceschoses, qu’il a tenues pour acquises comme étant des axiomesbien connus, non seulement ne sont pas ≤278 ≥ manifestes à laperception des sens, mais n’ont jamais été démontrées, et ne sontmême pas démontrables, parce qu’elles sont entièrement fausses.En effet, il a supposé que les mouvements d’un même mobile dansdifférents milieux respectent entre eux, en rapidité, le rapportqu’ont les subtilités des milieux : ce qui certes est faux, ayant


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

Nous sommes ici au cœur de l’essentiel de la portée épistémologique del’entreprise de Galilée. Notre jeune physicien est convaincu qu’Aristote a faiterreur sur le plan des axiomata. C’est cette conviction qui va le guider etle motiver tout au long d’une longue carrière de cinquante ans de recherchedurant 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 effortd’intelligibilité du monde que fut l’œuvre du Stagirite. Et la pratique de larigueur des rapports entre les axiomes et les démonstrations qu’on peut endéduire, 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 dunouveau.Ceux qui sont venus avant Aristote n’ont pas considéré le pesantet le léger si ce n’est par comparaison à propos de corps moinspesants ou moins légers ; et à mon avis, cela est tout à fait àbon droit : or Aristote, au livre IV du De Caelo, s’efforce deréfuter l’opinion des anciens et de confirmer la sienne qui lui estcontraire. Or comme c’est cette opinion des anciens que nousallons suivre, nous examinerons tant les réfutations d’Aristoteque ses confirmations, en en confirmant les réfutations, et en enréfutant les confirmations ; et nous accomplirons cela lorsque nousaurons exposé l’opinion d’Aristote.

Voici une stratégie épistémologique simple qui témoigne de cet esprit nouveaude découverte qui marque l’avènement de la pensée scientifique modernenaissante à la Renaissance en Italie. Et constatons que cette épistémologieest bien en place aux DMA. Et que c’est Aristote mieux que tout autrequi renseigne Galilée sur les anciens. Constatons également que c’est avecgrand soin, clarté et en détail qu’Aristote critique ses prédécesseurs. L’activitéscientifique est communautaire, mais nécessite la liberté d’esprit. Galilée litAristote avec un tel esprit et développe un grand respect envers le Stagiritequi lui fait penser souvent que si ce dernier était là, il se rangerait à son aviscontre les aristotéliciens de son époque. Et ici, dans la substance du contextedu 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 écritceci : « C’est ainsi que puisque le vide cède uniformément, mais àla vérité il ne cède aucunement (puisqu’en effet il n’est rien, biensubtil homme celui qui se figure le vide céder), d’où il résulte queles différences des choses pesantes et des choses légères, c’est-à-dire les variations (momenta) des choses, sont supprimées, et, cequi s’ensuit, c’est que la vitesse de toutes les choses mues arriveà être égale et sans discrimination. » Or combien ces paroles sontfausses sera bientôt connu, quand nous aurons mis en lumière dequelle manière c’est uniquement dans le vide que peuvent êtredonnées 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 duconcept capital que sera celui de « moment » [Galluzzi 1979, 172, note 73],pour un long commentaire très pertinent qui demanderait des développementsici. 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 del’argumentation, à savoir que dans l’hypothèse de leurs mouvements dans levide tous les corps tomberaient à la même vitesse. Il y a enfin le désaccord deGalilée ici aux DMA avec une affirmation qu’il finira par faire sienne beaucoupplus tard. Il faut prendre ici la mesure du chemin à parcourir pour clarifier cequi est en jeu. Je me contente de faire remarquer que l’acquisition aux DMAdu concept de vide pour penser le mouvement est un pas épistémologiquegigantesque. Et que ce pas, Galilée le franchit en faisant simultanément deuxchoses : exploiter les vertus inventives de la pensée géométrique et mettreles résultats obtenus au service de la confrontation avec les phénomènesobservables sous contrôle.

6/8.= EN I 300.14–301.10 : Archimède confirme notre apprenti dansson métier de physicien.D’autre part, je n’ignore pas ici que quelqu’un puisse m’objecterque j’ai présupposé comme vrai pour ces démonstrations ce quiest faux : à savoir que les poids suspendus à partir de la balancemaintiennent avec la balance des angles droits ; alors que pourtantles poids, tendant vers le centre, seraient convergents. À ceux-là jerépliquerais que je me recouvre des ailes protectrices du surhumainArchimède (que je ne mentionne jamais sans admiration). Eneffet, lui-même a présupposé la même chose dans sa Quadraturede la Parabole ; et cela, peut-être, afin de montrer qu’il était deloin tellement en avant des autres, qu’il pouvait tirer le vrai du


faux : et cependant, il ne faut pas se mettre à douter comme s’ilavait conclu lui-même au faux, puisqu’il avait prouvé cette mêmeconclusion dans une autre démonstration géométrique antérieure.Voilà pourquoi ou bien on doit dire que les poids suspendusmaintiennent vraiment avec la balance des angles droits, ou bienil n’est nullement important s’ils maintiennent ou non des anglesdroits, qu’il suffise qu’ils soient égaux ; ce qui d’aventure sera plusprobable : à moins que nous voulions dire, ceci est plutôt unelicence géométrique ; comme lorsqu’Archimède présuppose que lessurfaces ont de la pesanteur, et que l’une est plus pesante quel’autre, alors que pourtant elles sont effectivement toutes exemptesde pesanteur. Et les choses que nous avons démontrées, commenous l’avons dit ci-dessus, doivent être comprises à propos demobiles à l’abri de toute ≤301≥ résistance extrinsèque : maispuisqu’il est d’aventure impossible de les trouver dans la matière,que quelqu’un ne soit pas étonné, en faisant à propos de ces chosesun test [de his periculum faciens], si l’expérimentation déçoit, etqu’une grande sphère, même si elle est sur un plan horizontal, nepuisse pas être mue par une force minime. En effet, aux causesdéjà dites, il s’ajoute encore ceci : à savoir qu’un plan ne peut pasvraiment être parallèle à l’horizon. En effet, la surface de la Terreest sphérique, ce à quoi un plan ne peut pas être parallèle : voilàpourquoi, 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 écartéed’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 laisseentendre n’est pas quelque chose qu’Archimède a dit explicitement en autantde mots dans son livre. Il est cependant vrai qu’Archimède suppose là queles poids suspendus depuis les extrémités des leviers horizontaux en équi-libre pendent perpendiculairement au levier. Drabkin & Drake signalent queBenedetti, dans ses Diversarum speculationem [...] liber, Turin, 1585, 148–151,aurait critiqué une telle présupposition chez Jordanus et Tartaglia. Guidobaldodal Monte aussi a discuté ces choses dans ses Mechanicorum (1577) et toutcela a été largement débattu au siècle suivant [Drabkin & Drake 1960, 67,note 10]. Et dans le Dialogus, sa version dialoguée des DMA, s’inspirant pourla première fois du traité Des corps flottants, Galilée avait d’abord donné àses démonstrations équivalentes à celles du chapitre 8 des figures qui tiennentcompte de la rotondité du plan de l’horizon [EN I, 381–384]. Or, à ce sujet, onpeut rappeler également ce que dit Charles Mugler dans la Notice qui précèdesa 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 pendantdes siècles la seule, tentative de rendre compte par le raisonnementmathématique du fait d’observation, familier aux riverains de laMéditerranée, 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’Archimèdeappartient à une véritable culture technoscientifique analogue à celle qui seramise 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’hégémoniede 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 untest, c’est mettre à l’épreuve sa perception, son observation, son sentimentet 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’observationcontre, 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’Accademiadel Cimento.

Enfin, à propos du plan parallèle à l’horizon : on doit se mettre d’accordavec Drabkin qui suggère que Galilée veut dire ici qu’aucune surface plane saufcelle d’une sphère ne peut avoir tous ces points à une égale distance du centredu Monde [Drabkin & Drake 1969, 68, note 13]. Et voilà comment Galilée faitd’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’ellessoient fausses, subsistent très longtemps chez les gens ; parce qu’àpremière vue elles présentent en leur faveur une apparence devérité, et pour cela personne ne se met en peine de chercher àsavoir si c’est digne d’être cru. Une illustration d’une telle chose estce qui est cru concernant les choses qui se trouvent sous l’eau, etdont l’opinion commune affirme qu’elles apparaissent plus grandesqu’elles sont vraiment. Or lorsque je n’ai pas pu trouver une causeà un tel effet, à la fin, ayant recours à l’expérimentation, j’ai trouvéqu’une pièce de monnaie reposant au fond de l’eau n’apparaîtd’aucune manière plus grande, mais plutôt plus petite : voilàpourquoi moi j’estime que celui qui le premier a énoncé cette façonde penser doit avoir été amené à cette opinion durant l’été, lorsqueles prunes ou d’autres fruits sont parfois mis dans une coupe enverre 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 lumièresont laissés passer à travers le verre. Mais c’est la forme de la


coupe, et non pas l’eau, qui est la cause d’un tel effet, comme nousl’avons plus amplement mis en lumière dans les commentaires àl’Almageste de Ptolémée, lesquels (à la faveur de Dieu) serontpubliés bientôt. Or un indice de cela est que, l’œil posté au-dessusde l’eau, en sorte qu’on puisse observer sans qu’intervienne lemilieu 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 cepetit exemple de prunes vues à travers de la vitre révèle le genre de cultured’observation contrôlée pratiquée par Galilée bien longtemps avant d’arpenterles quais de l’Arsenal de Venise, bien avant de fabriquer des lentilles pourobserver « ce que personne n’a encore vu ». Et les commentaires du copernicienGalilée à l’Almageste de Ptolémée – possiblement des notes de cours liées àson enseignement pisan – n’ont jamais été trouvés. On peut penser qu’ils ontété 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 soncontraire.La manière dont la solution de l’argument d’Aristote peut êtreconvenablement tirée des propositions de l’argument lui-mêmeest 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 ancienssur le mouvement qui la lance. Épistémologiquement parlant, le texte de1638 des Discorsi dont nous sommes partis prend solidement racine à Pise50 ans plus tôt.

Bibliographie

De Pace, Anna [1990], Galileo lettore di Girolamo Borri nel De Motu, dansDe 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 Motionand On Mechanics, Madison : The University of Wisconsin Press.

70 Raymond Fredette

Drabkin, Israel Edward & Drake, Stilman (éds.) [1969], Mechanics inSixteenth-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 Traitécontenu dans le de motu antiquiora de Galilée, dans Proceedings of theXXth International Congress of History of Science : medieval and classicaltraditions and the Renaissance of physico-mathematical sciences in the 16thcentury, Turnhout : Brepols, 125–130.

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

—— [1970], Discours et démonstrations mathématiques concernant deuxsciences nouvelles, Paris : A. Colin, Introduction, traduction et notes parM. Clavelin.

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

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 ofZabarella’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 Springsof Scientific Creativity : Essays on Founders of Modern Science, édité parR. Aris, T. H. Davis & R. H. Stuewer, Minneapolis : The University ofMinnesota Press, 3–20.

“It Is Impossible to Deceive Nature”.Galileo’s Le mecaniche,

a Bridge between the Science of Weightsand the Modern Statics

Romano GattoBasilicata University (Italy)

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

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


72 Romano Gatto

1 Introduction: the two versions ofLe mecaniche

On September 26, 1592, the “Serenissima Repubblica” of Venice assignedGalileo the chair of Mathematics at the University of Padua.1 On acceptingthe position, Galileo left the University of Pisa, where he had taught forthree years, and moved to Padua, where he remained for 18 years. Galileohimself defined these years as “the most beautiful” ones of his life. Duringthis time, his prestige, as both a scientist and a teacher, grew beyond allbounds. Besides public teaching, he also provided private teaching in his houseto 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 delloStudio di Padova Pars Prior 1520-1739 [Viviani 1717], a very precious sourcein the University of Padova Archive, despite suffering from lacunae andincompleteness. By means of this and other sources, we know that duringhis debut year (1592-1593), Galileo taught “science of the fortifications” and“mechanics”. Vincenzo Viviani (1622-1703), who assisted him in the final yearsof 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 onFortifications, according the use of that times, one on Gnomonicand practice Perspective, an epitome of the Sphere and a trea-tise on Mechanics which circulates in manuscript form and thatP. Marin Mersenne translated into French and, in 1634 publishedin Paris, and lastly, in 1649, Cavalier Luca Danesi published inRavenna. [Viviani 2001, 39]

Viviani’s indications are so clear that they leave no room for interpretation: thetext on mechanics he is speaking about is the work Le mecaniche that AntonioFavaro (1847-1922) published in 1891 in the second volume of the NationalEdition of Galileo’s Opera [Galilei 1890-1909]. In the National Library inFlorence, there are two autographed manuscripts of Viviani’s Racconto and inone of them, close to this quotation, the author, in his own hand, added thefollowing 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 “thetreatise 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, GiuseppeMoletti (1531-1588), passed away.

2. This work was published for the first time in [Viviani 1717], but here we willquote 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 ofthis work in 2002 [Galilei 2002, IX–CXLIV], the date of 1593 refers not toLe mecaniche, but to another text, which, in an abridged way, deals with thesame matter. Evidently, Viviani knew only the text published for the firsttime in French by Marin Mersenne (1588–1648), so he claims that it was theone 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. InApril 1898, Will wrote to the Italian professor, indicating that in the Prince’slibrary, he had discovered a code containing some Italian manuscripts, one ofwhich was attributed to Galileo. Favaro verified that, alongside the text of Lemecaniche, there was a second text, entitled Delle meccaniche lette in Padovadal Signor Galileo Galilei l’anno 1594 [Favaro 1889]. This text concernednearly the same matters, but was written in a more concise form. In myaforementioned critical edition, I considered them to be two versions of thesame work, and without much imagination, referring to their length, I calledthem respectively versione breve, that is, short version, and versione lunga,long version. I also demonstrated how the text of the short version relates tothe course of mechanics Galileo held during the academic year 1592-1593, andhow the long version concerns the lessons from the academic year 1598-1599.

From a formal point of view, the two versions are characterized by thedifferent development of their texts: the short version is concise to the point ofalmost being reduced to the essential concepts with only some demonstration.The long version, however, ranges widely with great linguistic and stylisticaccuracy, and includes exhaustive and beautiful demonstrations. This helpsus understand that, while the first was merely a lesson guide Galileo used inthe first year of his teaching, the second was the result of a well-thought-outand more accurate revision whose aim was the completion of a real treatise onmechanics. We can recognize this fact by comparing the introductions of thesetwo works: in the short version it runs to a few lines concerning the definitionof mechanics and the list of five simple machines; in the long version there isa very interesting dissertation on mechanics including unexplored aspects onthe theory of machines. One of the most relevant aspects of the long version isthe fact that it had been conceived as a geometrical treatise. The introductionis followed by a chapter of definitions and axioms, which is lacking in theshort version. The presence of this chapter is clear proof of Galileo’s intentto compile a rigorous treatise in which any concept had to be introduced if ithad 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 thetreatment 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 theangular lever and the second-class lever, which are not in any other treatise ofthat 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 sameas those of the short version—apart from the sections concerning the so-calledBaroulkos, namely, a train of shafts into a wheel, and concerning the perpetualscrew, which are present only in the short version. In the long version, Galileodeals with these subjects in a wider and more detailed fashion, adding a fewpertinent and remarkable demonstrations, including many good examples, andhe introduces some innovative ideas on the theory of machines. But the mostpeculiar aspect that distinguishes this version and makes it a new originaltreatise in mechanics is its systematic use of the concept of the moment.

2 The definition of mechanics in the shortversion

One of the most important questions we want to emphasize is the fact thatGalileo provides a new definition of mechanics within the few lines of the shortversion:

The science of mechanics is the faculty that teaches us the reasonsand makes us understand the causes of the miraculous effectsthat we see happen with some tools, in moving and lifting heavyweights with a small force. [Galilei 2002, 5, ll, 1–4]

This definition marks a caesura with the Aristotelian tradition still alive atthat time. At the beginning of hisΜηχανικά Προβλήματα, the pseudo-Aristotlewrites:

Among the events that occur in accordance to nature, those ofwhich the cause is unknown arouse wonder, while, among theevents that occur against nature, arouse astonishment all thoseare realized with art for the benefit of mankind. [Aristotele 2000,847a, 11–13]

According to the theory of natural places, the pseudo-Aristotle distinguishesmechanical phenomena according to nature (κατὰ φύσιν), from those againstnature (παρὰ φύσιν). He maintains that, while our advantage is susceptible tochange in many ways, nature always acts consistently and plainly in the samedirection. So, in many cases, nature can produce disadvantageous effects forour utility. Therefore, if we wish to do something against nature we must resortto τέχνη, to art, to the help of some tool. For this reason, the pseudo-Aristotledefines 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 stilldominated at the time of Galileo, mechanics does not explain the nature of


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 forthis reason arouses wonder. The numerous Renaissance commentators on theΜηχανικά Προβλήματα have interpreted the above-mentioned opening passageof this work in various ways, according to the different ways to translateπαρὰ φύσιν. These interpretations are very important in establishing theirepistemological position towards the Aristotelian theory of natural places.

In my previously mentioned critical edition, I pursued a wider and deeperanalysis of these positions, concluding that only Galileo fully develops thetheory of natural places. Galileo first subjected this theory to severe criticismin De motu [Galilei 1890-1909, I, 243–419], but only in the two versions of Lemecaniche does he complete his project to revise and re-establish the bases ofthe mechanics.

3 The compensation principle in the longversion

As we have seen, in the short version, Galileo defines mechanics as the sciencethat teaches the reasons and explains the causes of the effects that may appearto be miracles only to those who ignore the true laws of nature. It explainsarcane ideas concerning “miraculous” phenomena connected to the use of tools,and causes them to lose any attribute of irrationality. Galileo thus divestsmechanics of any metaphysical element, and shows that the machines do notpossess any supernatural virtue but operate according to precise laws of nature.

In the long version, he makes it even more explicit that nothing can happenwhich nature does not allow. Because of the ignorance of the laws of nature,he writes:

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

This passage is remarkable. Here Galileo plainly reaffirms that nature cannotin any way be deceived, that it is impossible to operate against the naturalorder. The machines have to observe the unavoidable law that “no resistancecan be overcome but by a force that isn’t more powerful than it”. So wheredoes the benefit of the machines lie? In order to answer this question Galileoexplains 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 secondis the force or power that has to move it; the third is the distancebetween both terms of the motion; the fourth is the time in whichthis change must be made; and it is the same thing if instead ofthis time we consider the quickness and the speed of motion, as wedefine that motion faster than another which covers that distancein a shorter time. [Galilei 2002, 45–46, ll, 28–34]

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

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

The working of a lever, and of all the other simple machines that rely onit, follows a principle according to which force, resistance, time, space, andspeed continuously compensate each other: if the power increases, the speeddecreases because, in moving the resistance through a given space, the forcecrosses a space longer than it in the same time. This is the reason why wecall this statement the compensation principle. In the introduction to the longversion, Galileo explains that, with no machine, it is still possible to move agreat weight with a small force from one place to another by dividing it into acertain number of parts, with no part being superior to the moving force, andtransferring them one by one until the whole weight is carried to the assignedplace. He firmly remarks, at the end of this work, that nobody can affirmthat a great weight has been transported by a force less than it, but only thatit has been moved by a force that is reiterated many times along that spacewhich the whole weight, all together, would have traversed only once. In sodoing, it is clear that the velocity of the moving force is many times superiorto 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 resistancehas crossed it only once. Thus, Galileo correctly concludes:

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

The compensation principle appears again and again in Le mecaniche, frombeginning to end. Upon reading the introduction, we soon recognize that one


Figure 1.

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

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

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

But the repeating of the space LM is certainly more than thesingle measurement of the space DI, that is five times LM .Therefore, the transferring of the weight from B to G requiresneither less force, nor less time, nor a shorter way if the weightwere placed at D, than it would need if the same force had beenapplied at L. [Galilei 2002, 55, ll, 392–398]

78 Romano Gatto

Therefore, the benefit derived from the use of this tool is only the possibilityto move a given body all at once. But the time the tool itself takes in movingthe weight will be just the same it would take if, having divided the space intoa certain number of parts, each one equal to the given force, this force hadtransported them the same distance one at a time. After having explained thelaw of the shaft into the wheel, Galileo considers the case of a winch in whichthe diameter of the wheel is ten times that of its shaft; so this tool is able tolift 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 ropeand lifts the weight in completing only one turn. Therefore, the weight H willcover only the tenth part of the way covered in the same time by the movingforce. At this point, reproducing the same reasoning of the introduction,Galileo points out:

If therefore, by means of this machine, the force for movinga resistance greater than itself, along a given space, must ofnecessity move ten times as far, there is no doubt that, dividingthe weight into ten equal parts, each of them will be equal tothe force, and consequently, it might have been transported oneat time, as great a space as that which itself will move. So that,making the journeys, each equal to the circumference EAD, it willnot have gone any further than if it had moved only once aroundthe circumference FGC and had carried the same weight H tothe same distance. [Galilei 2002, 50, ll, 545–550]

In concluding, he reasserts that the benefit derived by the use of this tool isonly that it is possible to carry the given weight together across the givendistance, but not with less labor, or with greater speed, or a greater distancethat the same force might have used if carrying parcels. The validity of this


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, itis gathered in this also, as well as in the other tools, that what isgained in force is lost in speed. In fact, when the force C raisesthe 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 theforce 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 withfour pulleys (Fig. 4), he says:

And we will likewise note, that to make the weight ascend, the fourropes BL, EH, DI, and AG ought to pass, whereupon the mover will beto begin, as much as those four ropes are long; and yet, nevertheless, theweight shall move but only as much as the length of one of them. So thatwe say this by way of advertisement, and for confirmation of what has beenmany times spoken, namely, that look with what proportion the labor of themover is diminished, the length of the way, on the contrary is increased withthe same proportion [Galilei 2002, 65, ll, 756–765].

4 The Definition of momentIt should be noted that in the case of the inclined plane, and consequently ofthe screw, Galileo also shows that the compensation principle is valid. But inorder to properly understand the role this principle plays in Galileo’s theory ofthe inclined plane, it is necessary to first introduce the concept of moment. Inthe chapter entitled Diffinizioni [Definitions], Galileo defines gravity, momentand center of gravity. Of the first, he writes:

Therefore, we call gravity the propensity of moving naturallydownwards, 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 bythe gravity of the moveable, as by the disposure which diversegrave bodies have in relation to one another; by means of whichmoment, we oft see a body less grave counterpoise another ofgreater gravity: as in the steelyard, a great weight is raised bya very small counterpoise, not through excess of gravity, butthrough the remoteness from the point whereby the beam isupheld, which conjoined to the gravity of the lesser weight addsthereunto moment, and impetus of descending, wherewith the mo-ment of the other greater gravity may be exceeded. Moment thenis that impetus of descending compounded of gravity, position,and the like, whereby that propensity may be occasioned. [Galilei2002, 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 thebodies due to the “quantity of matter whereof they are constituted.” Thesecond is an extrinsic propensity due to the gravity of the moving body and to


“the disposure which diverse grave bodies have in relation to one another”. Itis by virtue of this “disposure” that a body with less gravity is able to balanceone of greater gravity. The example of the steelyard, in which a small weight isable to counterbalance a greater one, is very appropriate. It clearly shows thatthis effect occurs “not because of excess of gravity, but of the distance fromthe point whereby the steelyard is supported”. Therefore, the moment is aquantity composed of gravity and distance joined together which particularlycomes into play in the use of machines. In fact, “the disposure which diversegrave bodies have in relation to one another” is simply the distance betweenthe acting power and the center of rotation of a balance, such as that of ashaft into a wheel or the fulcrum of a lever, etc. This is the definition of thestatic moment that we analytically represent as force × distance.3 But, inconcluding the definition of moment, Galileo affirms that this quantity canbe composed, not only by gravity and position, but also by other elementswhich may cause the propensity to descend. This statement is particularlyremarkable because it provides insight into the possibility of some differentunderstandings of the definition of moment. Everything becomes clear in thechapter “Alcuni avertimenti circa le cose dette”, in which he shows

how the velocity of the motion is able to increase moment in themoveable, according to that same proportion by which the saidvelocity 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 twoweights 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 betweenthe 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 thetwo extremities is enough to break the equilibrium, and then he considers howthe 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 notdefine in mathematical terms any derived physical quantity.

82 Romano Gatto

last question, then return to consider the first crucial question a little later. Inconsidering the motion that the weight B makes when descending into E, andthat the weight A contemporary makes in ascending into D, Galileo remarks:

we shall without doubt find the space BE to be so much greaterthan the space AD, as the distance BC is greater than CA.[Galilei 2002, 53, ll, 313–314]

In fact, given DCA = ECB, the ratio of the arcs BE and AD will be equalto 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 somuch superior to the velocity of the other ascending mobile A,as the gravity of this exceeds the gravity of that; and, as it isnot possible that the weight A should be raised to D, althoughslowly, unless the other weight B moves to E swiftly, it will notbe a surprise, or averse to the order of nature, that the velocityof the motion of the grave B should compensate for the greaterresistance of the weight A, so long as it slowly moves to D, andthe 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 thepoint E, it will not be unreasonable that the first one slowly fallingto A, should be able quickly to raise the other to B, recoveringby its gravity what it had lost by the slowness of motion. And bythis discourse we may come to know how the velocity of motionis able to increase moment in the mobile, according to that sameproportion 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 quantitiesat play are gravity (by which he means force) and, instead of distance, thevirtual 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 theDiscorso 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, theforce, the effect, through which the moving moves and the mobilewithstands; this virtue depends, not only on the simple gravity,but on the velocity of the motion, on the different obliquitiesof the spaces on which the motion takes place, because a gravedescending in a more oblique space makes impetus greater thanin a less inclined. And, in short, whatever the cause of this virtueis, nevertheless it holds name of moment. [Galilei 1612, 6]

Together with this new definition of moment, he enunciates the following twoprinciples:


1. Weights absolutely equal, moved with equal velocity, operate with equalforces and moments.

2. The velocity of the motion increases the moment and the force of thegravity so that weights absolutely equal, but joined with unequal speeds,have unequal force, moment and virtue, and the fastest one is morepowerful according to the proportion of its velocity to that of the otherone [Galilei 1612, 6–7].

Here he has replaced distance with velocity considering that, in relation tothe definition ofmoment, these two quantities are equivalent. What legitimatesthis substitution? In order to show the perfect equivalence of this dynamicdefinition with the static one in Le mecaniche, Galileo gives the followingexample of a balance with equal arms:

Two weights of absolute equal gravity, placed in a balance withequal arms, are balanced, and no one of them inclines lifting theother one, because the equality of the distances of both from thecenter, on which the balance is sustained and around which itmoves, makes sure that, moving the balance, these weights wouldcross at the same time equal spaces, that is they would move withthe same speed. Therefore there is no reason because one of thetwo weights has to incline more than the other one, so that theyequilibrate 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 thesame interval of time, its extremes cross equal arcs, that is, they move with thesame speed. This shows that the equality of the moments of equal weights atthe extremes of the balance’s arms is assured both by the fact that these armshave 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 greatestdistance does. In a balance ACB with unequal arms, AC < CB, if B rotatesfollowing the arc BE, A will cross the arc AD at the same time. Galileo veryeasily demonstrates that the ratio of the previously mentioned arcs is equal tothat of the distances CB and AC. Since these arcs are crossed at the sametime, it is clear that the speed of B is much greater than that of A, as thegravity of B exceeds the gravity of A. Now, given BE/AD = BC/AC, inconsidering the moment, it is possible to substitute the ratio of velocities tothat of the distances. At the same time, BE/AD represents the ratio of thespeeds, justifying Galileo’s conclusion:

By this discourse we may come to know how the speed of themotion is able to increase moment in the moveable, according tothat same proportion by which the said speed of the motion isincreased. [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 matterconcerning the breaking of a lever’s equilibrium. The question arises in thechapter “Alcuni avertimenti circa le cose dette”, where, with regard to thebalance ABC in equilibrium (Fig. 5), Galileo writes:

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

Galileo expresses the idea that, once the amount of force necessary for theequilibrium of a lever is established,4 in order to move or to lift the resistanceit 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 bythe virtue of this idea that we can forego considering the addition of this“insensible moment” and affirm that there is no difference between the powerthat counterbalances the resistance and the one that moves it. ConsideringGalileo’s conclusion, which states that “if we do not keep any account of thisinsensible moment, we will make no difference between one weight sustainingand one weight moving another”, Marshall Clagett speaks of an “audaciousassertion” [Clagett 1972, 179]. He is of the opinion that this idea was alsoimplicit in the Μηχανικά Προβλήματα by the pseudo-Aristotle and in theMechanics by Heron, in which “the same proportionality of the weights andof the distances (or of the times)” is applied either in the case of a balance inequilibrium, or of a machine in motion. However, Clagett maintains that thebenefit of making it explicit for the first time decidedly has to be credited toGalileo. My opinion is that Heron, in his Mechanics, expressed the exact sameconcept just as clearly, namely, that in order to move a given resistance bymeans of machine, it is first necessary to determine the force counterbalancingit, 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 resistanceis needed, Heron adds:

4. Really it is implicit that this fact is valid for every tool.


If however a small amount is added to the weight, then the otherweight is pulled upwards. If therefore the force moving the weightis greater than it, then this force will be strong enough for itand will move it except if any friction occurs in the turning ofthe pulley or a stiffness of the rope, so that they would cause ahindrance to the motion. [Ferriello, Gatto et al. 2016]

It is the same “insensible moment” of which Galileo speaks. The real differencebetween Heron and Galileo is the idea of limit the latter expresses. Anotherremarkable question Galileo examines in the aforementioned chapter “Alcuniavertimenti circa le cose dette”, which concerns the concept of distance inconsidering the moment of the force acting on the angular lever. He explainsthat we must assume as distance the segment perpendicular to the direction ofthe 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 assumethe position CD, then the weight hanging on point D “will make its momentand impetus according to the line DF”, given that F is the point at whichCB meets the perpendicular line to it from D. Therefore, in considering themoment of the gravity on D, we must assume as distance the segment CFthat is the orthogonal projection of CD on CB. Now given CF < CB, themoment of the gravity at D is smaller than at B.

6 The compensation principlein the inclined plane

At this point we are able to consider the question of the compensation principlein the inclined plane. But before getting to the heart of the matter, we canaffirm that the treatment of the inclined plane is one of the most relevantsubjects in Le mecaniche. In this treatise, for the first time in the history ofmechanics, Galileo enunciated the exact formulation of the law which connectspower to resistance in relation to the dimensions (height and length) of theinclined plane, namely, that

86 Romano Gatto

the ratio between the weight and the force is the same of thatbetween the length of the elevated plan to the perpendicularheight. [Galilei 2002, 11, ll, 191–192]

In the long version, this law is demonstrated in a very rigorous manner. Inthe short version, it is introduced and enunciated more simply as “from thelight of nature and for experience”, which teaches that the lower the elevationof the inclined plane, the smaller the force necessary to raise a given weight.Starting from these considerations, in this version Galileo already comes tothe remarkable conclusion that

if we had a plane without any inclination, the heavy bodies puton it would not move in themselves, but it is true that every verysmall force would be enough to move them from the place. [Galilei2002, 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 itsheight. He questions whether or not, for this particular tool, the compensationprinciple occurs. In fact, he says:

It seems, that in this case the force is multiplied without themovers 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 ascendalong 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 orderto move a body on the horizontal plane a force equal to its weight is needed, Galileohad shown in an admirable way that “Quocumque mobile super planum horizontiaequidistans a minima vi movetur, imo et a vi minori quam quaevis alia vis [...]”[Galilei 1890-1909, I, 209].


another, the distance covered by E necessarily has to equal that covered byF . But Galileo observes that, for the equilibrium, we must consider not thelengths of the distances covered, but the lengths of their projections on thevertical axle, namely, as we would say today, the vertical component of theaforementioned movements. Now, while the weight F moves vertically in thedirection of CB, the weight E moves obliquely along the plane AC. So, whenE has covered the whole distance AC, it has crossed a vertical space equal toCB, and a horizontal space equal to AB, which is as if, in order to move fromA to C, the weight would have crossed the horizontal space AB and then thevertical space CB. But since a force however small is enough to move a bodyon the horizontal plane, with regard to the oblique motion of the weight, wemust consider only the vertical movement. Consequently, in order to move theweight on the inclined plane AC only the fraction BC/AC of the weight itselfis needed. Therefore, the greatest distance AC requires a force smaller thanthat the shorter distance BC. Galileo’s conclusion is:

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

The reference to Jordanus’ De ratione ponderis [Jordanus Nemorarius 1952] isclear. 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 toprovide the indirect demonstrations of certain peculiar problems,with Galileo they become an occasion to formulate a generalprinciple to apply to all the simple machines. But it is alsoindisputable that the general principle so enunciated for the firsttime enveloped the possibility of a new exposition of the statics.[Clavelin 1968, 168–169]

There is no doubt that the compensation principle, formulated in termsof virtual movements, is a fundamental principle of the Galilean theory ofmachines that indicates the possibility for constructing a new science of theequilibrium, namely the statics. Galileo appears to be aware of the innovativesignificance of this principle. He turned to it in the already mentioned Discorsito explain the phenomenon of the hydrostatic equilibrium in communicatingvases. If, due to pressure, the water’s level CD goes down to QO (Fig. 8), thewater’s column CLmust necessarily rise up to the level AB, and the ascent LBwill be greater than the descent GQ, insofar as the width of the vase GD isgreater 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 thatof the gravity of another one, which wonder will it be if the fastascent of the little water CL withstands the slow descent of thegreat water GD? [Galilei 1612, 16]

He explains that the same effect occurs for a balance in which one arm is100 times longer than the other one. He reasserts once more that, by meansof 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 space100 times greater than P ’. It is interesting to remark that Galileo says thatthe example shown above

will be useful to get out of error some practical mechanicians,who embark on some impossible enterprises basing on a falsefoundation. [Galilei 1612, 16]

This does not differ at all from what he says in the introduction of thelong version, where, after having affirmed to have seen “the generality ofMechanicians deceive themselves in going about to apply Machines to manyoperations of their own nature impossible”, and after having explained the realnature of the machine and the laws to which they are subject, he concludes:

These then are the benefits that may be derived from mechanicalinstruments, and not those which ignorant engineers dream of,with the deception of so many principles, and with their ownshame, 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 thecompensation principle [Gatto 2015] in a clear and unequivocal way, and who


used virtual movements to establish a new scientific base for the science ofweights. 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 ofmachines. We are speaking of Heron of Alexandria who, in his Mechanics, atthe end of the treatment of the Baroulkos, writes:

This tool and those of great power similar to it are slow, becausethe 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 andthe time to the time are in the same inverse ratio. [Ferriello, Gattoet al. 2016, 113–114]

We immediately realize that this statement is similar to Galileo’s. We likewiserecognize that, as we have seen in Le mecaniche, Heron also reasserts thisprinciple at the end of the discussion on the train of blocks and tackles and thatof the levers, which are the other compound tools he considers.6 The strongsimilarity in considering and developing this aspect of the problem, the ideathat the addition of an amount of force, however small, to the power whichmakes the balance is enough to break the equilibrium of a lever, and someother questions we cannot discuss here, would lead us to think that Galileocould have known the Mechanics by Heron7. Clagett excludes this possibility,sharing the common opinion that this work by Heron was unknown atthe end of the sixteenth century. But Clagett himself, describing the dynamicapproach in Le mecaniche, in which Galileo first uses the virtual movements toconfirm the law of the lever and subsequently to derive the law of the inclinedaffirms 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 delayoccurs 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, inconcluding the exposition of the train of levers, he writes: “Here too the delay occursin the same ratio, because there is no difference between these levers and the shaftinto the wheels”.

7. This treatise has reached us, for the first time, in 1893 through an Arabicmanuscript of the 9th century, whose text, with its French translation, was publishedby Camille Carra de Vaux [Carra de Vaux 1983].

8. Also Roberto Marcolongo, although surprised because of the strong similarityof some theories by Leonardo with the corresponding by Heron, was convinced thatit was impossible that Leonardo could have known Heron’s Mechanics. In fact, hewrites: “Heron’s considerations on the centres of gravity of the plane figures, those onthe 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 bookpersisted to be unknown to the Western scholars, and, therefore, Leonardo couldn’tknow it. But I think that we can also assert that, by some ways that we continue toignore yet to day, some idea by Heron has reached the Western scientists and it ispreserved there” [Marcolongo 1937, 138]. It is also interesting to note what GiovanniVailati wrote in a letter to Ernest Mach (Crema, 3 August 1897): “The way in whichthis last author [Heron] enunciates and applies the aforesaid principle [the principleof the virtual movements] is such to suggest a period of the development of staticscorresponding to that expressed in Galileo’s early works” [Vailati 1971, 115].

90 Romano Gatto

We have already explored the possibility that Galileo and other Italianauthors 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 cannotbe underestimated. The principle of virtual movements is the key differencebetween Heron and Galileo’s approaches. Heron applied it to the lever, theblock and tackle, and later to the shaft into the wheel, which are the machineshe limited to the Archimedean principle of the balance. In so doing, heprovided the evidence that confirmed the relationship between the efficiency ofa machine and its related delay. He never applied this principle to the inclinedplane, which he dealt with another way without making any reference to theabove-mentioned simple machines. Galileo, beginning with the Archimedeanlaw of equilibrium, followed by the concept of the moment, was able to conceiveof a form of the principle of virtual movements as a general principle for thetheory of machines, thereby granting mechanical science a new perspective.This accomplishment is so significant that Le mecaniche should be consideredas the first modern treatise on statics.

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

Gatto, Romano [2015], La meccanica di Erone nel Rinascimento, in: Scienzae 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 diLeonardo da Vinci, Napoli: S.I.E.M.

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

Viviani, Vincenzo [1717], Racconto istorico della vita del signor GalileoGalilei, in: Fasti consolari dell’Accademia Fiorentina di Salvino Salviniconsolo 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 tothe Needham Question

Gennady GorelikBoston University (USA)

Résumé : Pour pouvoir la résoudre, nous généralisons la question posée parNeedham : Qu’est-ce qui a empêché la science gréco-romaine et médiévale defranchir la prochaine grande étape après Archimède, et qu’est-ce qui a empêchéles savants orientaux de contribuer à la physique moderne des siècles encoreaprès Galilée ? Pour répondre à cette question, on propose comme distinction-clef entre physique moderne et science néo-galiléenne le droit d’inventer desconcepts fondamentaux « illogiques », vérifiables par expérimentation. Ce droitse fonde sur la croyance en un univers régi par des lois fondamentales cachées,mais accessibles à la connaissance humaine. Cette croyance prend sa sourcedans la vision biblique du monde, qui devint le socle des cultures européennes àl’époque de la révolution scientifique à l’occasion de l’invention de l’imprimerieet 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 majorstep after Archimedes, and what hindered Eastern scientists from contributingto modern physics for centuries after Galileo? To answer this question thekey distinction between modern physics and pre-Galilean science is suggested:the right to invent “illogical” fundamental concepts which can be verified byexperiments. This right is based on the belief that the Universe is governed byhidden fundamental laws which Man is capable of knowing about. The sourceof this belief was the biblical worldview which became the basis for Europeancultures by the time of the Scientific Revolution thanks to book printing andthe Reformation.


94 Gennady Gorelik

1 Introduction

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

Why did modern science, the mathematization of hypothesesabout Nature, with all its implications for advanced technology,take its meteoric rise only in the West at the time of Galileo? Whymodern science had not developed in Chinese civilization [whichin the previous centuries] was much more efficient than occidentalin 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, whichappeared in almost perfect form in Galileo, [...] led to all thedevelopments of modern science and technology. [Needham 1959,156]

So, Needham’s “Grand Question” is to be coupled with the question: What wasthe actual innovation of Galileo, that had changed science so much, acceleratedits 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 modernscience had already been named “the Scientific Revolution” and explainedin a few ways: by needs of capitalist economy, by Protestant ideology, by“mathematization of nature”, by contacts between scholars and craftsmenfacilitated by the capitalist economy, etc. [Cohen 1994], but none of thoseexplanations satisfied Needham [Needham 2000].

Indeed, all the achievements of the new physics had no economic valuein the 17th century. All the greatest “revolutionaries”—Copernicus, Galileo,Kepler, and Newton—used both empirical and mathematical tools. Onlytwo of them were Protestants. And in China contacts between scholars andcraftsmen did not result in modern physics.

While the discussion about the Scientific Revolution continues with noconsensus in sight [Cohen 2010], [Huff 2011], the Needham question wassometimes dismissed as a counterfactual question about a unique event [Sivin1982]. However, Needham didn’t ask why modern physics emerged in Italyrather than in England, and he would hardly have been so puzzled if Easternscientists 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 cul-tural time and space. Indeed, refuting Aristotelian physics, Galileo reliedon “superhuman”, “the most divine” Archimedes [Galilei 1590]. The newGalilean science was eagerly accepted in France, Holland, England, and evenin backward Russia, but failed to reach beyond Europe for centuries, althoughmedieval Europeans used to assimilate important Eastern innovations likeHindu-Arabic numerals. Arabic science adopted Greek science much earlierthan the Europeans, advanced optics and astronomy, but did not contributeinto post-Galilean science.

The real question is not why modern science emerged in the West at thetime of Galileo but why it took so long since the time of Archimedes and whyafter the birth of modern science it was not adopted in Eastern civilizationsfor so long. So, an extended Needham question is:

What hindered Greco-Roman and Medieval scientists from mak-ing the next major step after Archimedes, and why didn’tEasterners contribute into modern science for centuries afterGalileo?

To answer this question we are to find commonality among the cultureswhere the new (modern) science took roots and fructified, and to explainthe timing of the Scientific Revolution and the social forces, which broughtit 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-Romanintellectual tradition—were present in Islamic civilization but it did not adoptthe science of Galileo. And European universities had been around for fourcenturies 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 arevital but not uniquely modern, since back in the 13th century Roger Baconstated 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 realphysicist but also a great engineer and mathematician. Galileo’s experimentsand mathematics didn’t go beyond what was feasible for Archimedes.

Of course measuring experiment and mathematics are indispensible toolsto verify or disprove a theory expressed in quantitative language. But inmodern physics no less important is the third tool, described by Einsteinas “the boldest speculation [to] bridge the gaps between the empirical data”

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[Einstein 1953]. Such a speculation results from inventive imagination ratherthan from mathematical or empirical inferences [Cohen 1995].

The real novelty of modern physics can be seen in the scheme depicted byEinstein 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 inventedby intuition taking off from the ground of experience E: the fundamentalsare “free inventions of the human spirit (not logically derivable from what isempirically given)” [Einstein 1949]. Then some statements Sn derived from Aare to be verified by landing in the E. And if the landing is soft, the theory isendorsed.

There is the key difference between Galileo’s science and Archimedes’ oneand the principal similarity between Galileo’s and Einstein’s. In Archimedeanphysics all the notions are visible and tangible (weight, density, geometricalform), whereas in modern physics fundamentals do not have to be evident,and their validation is a result of the whole scientific enterprise joining theoryand experiment. Einstein emphasized, that “concepts can never be derivedlogically from experience [...]. Unless one sins against logic, one generallygets nowhere” [Einstein 1993, 147], apparently meaning “against the logicof previous theory or common sense”, since there is no other logic when atheorist’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, asphilosophers believed, had “logically proved” the nonexistence of void, or thevacuum [Galilei 1590, 34]. Galileo introduced “vacuum” as a physical notion,rather than a logical one. In logic, a notion is validated by pure reason, whereasin 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 toprobe into and comprehend these fundamental laws by inventing new conceptsto be validated empirically.


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 in-ventions of modern science were encouraged by fundamental worldview coupledwith cognitive optimism. It was the key novelty which let Copernicus initiatethe 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 motionsfrom the Solar point of view. Kepler’s boldest idea was that all the planetarymotions are governed by a fundamental law. For both of them, fundamentalcognitive optimism supported laborious mathematical processing of their as-tronomical data, and they could be named fundamental astro-mathematicians.

Galileo became the first fundamental physicist by establishing the methodof modern physics: believing in fundamental unity of terrestrial and celestialphenomena, he launched his boldest speculations by taking off from terrestrialphysical experiments, invented the concept of “motion in vacuum”, employedmathematical language and landed his speculations in the empirical reality ofboth terrestrial and celestial phenomena. He never experienced vacuum by hissenses, but having compared motions in air and water, he felt free to inventthe notion of vacuum as a “medium totally devoid of resistance” and came tothe 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 todiscover 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 mathemat-ical relationship”, and—just in a footnote—Needham mentioned “concepts ofthe unobserved and the unobservable” [Needham 1959, 156]. However, at theturning points in history of modern science, to formulate a new hypothesisa 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 universalgravity, electromagnetic field, quanta of energy, photons, curved space-time,etc. Introducing a new fundamental notion a theorist usually has to dismisssome 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 topropel the whole of science.

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4 The source of fundamental cognitiveoptimism

Reflecting on making science, Einstein remarked: “one cannot build a houseor construct a bridge without using a scaffolding which is really not one ofits basic parts” [Einstein 1993, 147]. What kind of scaffolding did the firstconstructors of modern science use?

Copernicus began his heliocentric thinking being

[...] annoyed that the movements of the world machine, createdfor 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 processeswas not assured at all. How great must his faith in such lawfulnesshave been to give him the strength to devote decades of patienthard work to the empirical investigation of planetary motion andto formulate its mathematical laws! [Einstein 1930b]

To Kepler’s mind

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

All the originators of modern science shared the faith in fundamental lawful-ness 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 foundby Edgar Zilsel, who traced the usage of the phrase “physical law” anddiscovered that it emerged in the 17th century within biblical worldview asa 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]. Thetransformation began in Galileo’s theological letters of 1613-1615, and here isa summary of his views:


The Scripture and Nature both derive from God, the Scriptureas His dictation, the Nature as the obedient executrix of Hisorders. The purpose of the Scripture is to persuade humans ofthose propositions which are necessary for service of God andsalvation. To adapt to the understanding of unlearned people,the Scripture speaks many things which differ from the baremeaning of words, and it would be blasphemy to accept themliterally by attributing to God human feelings like anger, regret,or forgetfulness. Nature, on the other hand, never transgressesthe laws imposed upon her, or cares a whit whether her reconditereasons and ways of operating are understandable to men. Godhas endowed us with senses, language, and intellect not to bypasstheir use and give us by other means the knowledge we can obtainwith them. Therefore, whatever sensory experience and necessarydemonstrations prove to us concerning natural phenomena, itshould not be questioned on account of Scripture’s words whichappear to have a different meaning. This is especially so forthose sciences about which we can read only very few wordsin the Scripture which does not contain even the names of allthe 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, andhumans are able to comprehend them. This is actually the postulate of thefundamental science (formulated in chap. 2).

By the end of the 17th century Galileo’s “laws imposed upon Nature byGod” transformed into seemingly secular “laws of Nature”, due to influentialwritings of Descartes and Newton. Since then the term was used by bothbelievers and nonbelievers, and by the 20th century its biblical origin hadbeen 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 Godwere metaphorical. The origin of the expression “physical law” reveals the roleof biblical worldview in their mentality and hints at the connections betweenthe postulate of fundamental science and the basic biblical ideas/images ofGod who created the lawful Universe for the humankind, and of humans madeas His likeness with the purpose to rule over all the Earth.

Such supernatural wording sounded quite natural for the originators ofmodern science, all of whom were true believers: Copernicus was a cleric,Galileo and Kepler in their adolescence intended to become clerics, and Newtonwrote about the Bible more than about physics. All of them were profoundbiblical theists, thought in religion as freely and boldly as in science, and feltfree to interpret the Bible by themselves. Basing their religious thinking ontheir understanding of the Bible, they came to be at odds with Church canons:

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Galileo could not accept the Pope’s opinions as final truths; Newton could notaccept 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 invisiblefundamentals to gain the knowledge about “all the Earth” to be able to ruleover it.

The same faith was expressed by the third great inventor of invisiblefundamental (electromagnetic field), Maxwell, who wrote to his friend:

[...] Christianity—that is, the religion of the Bible—is the onlyscheme or form of belief which [makes an explorer free indeed].You may search the Scriptures and not find a text to stop you inyour 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 religiousinstincts, are all tributary forces in helping the reasoning facultytoward its highest achievements. [Einstein 1930a, 375]

Of course, there were atheists back in the time of Archimedes, as well as in the17th century (an atheist, astronomer E. Halley, was a colleague and friend ofNewton), but there were no atheists among the originators of modern science.

5 Modern science in the biblicalcivilization

Returning to the extended Needham question, we can see the commonalitybetween the countries where Galilean science did take roots, or, rather, thecommonality between these countries’ sociocultural minorities from which thefuture scientists emerged. All the would-be scientists had to be readers. By the17th century, the most widely read book in Europe was the Bible, as a result ofGutenberg’s invention, Reformation, and the print explosion. Since then theBible became the most influential text for European cultures as different asItalian and Scandinavian, British and Russian. It was the main common factoruniting all these cultures into modern European civilization, which thereforecould be called the biblical one.

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


There were a few correlations between the history of modern science andthe sociocultural role of the Bible. The first correlation between religious andscientific postulates was revealed back in the late 16th century by missionarieswho brought to China both the Bible and European science. The Chineseemperor welcomed missionaries, but they failed to implant the Europeanscience into Chinese soil. One of the later missionaries explained:

The Chinese atheists are not more tractable with relation toProvidence, than with regard to the Creation. When we teachthem that God, who created the universe out of nothing, governsit by general Laws, worthy of his infinite Wisdom, and to whichall creatures conform with a wonderful regularity, they say, thatthese are high-sounding words to which they can affix no idea, andwhich do not at all enlighten their understanding. As for what wecall laws, answer they, we comprehend an Order established by aLegislator, who has the power to enjoin them, to creatures capableof executing these laws, and consequently capable of knowing andunderstanding them. If you say that God has established Laws,to be executed by Beings capable of knowing them, it follows thatanimals, plants, and in general all bodies which act conformable tothese Universal Laws, have a knowledge of them, and consequentlythat 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’slikeness to rule over all the Earth. Islam, being the closest to biblical traditionhistorically and geographically, rejects the biblical status of Man, since theQuran states: “Nothing is as God’s likeness” [Michot 2005]. On the otherhand, in mainstream Islam, after Al-Ghazali’s “renewal of the faith” in the12th century, the very idea of unbreakable fundamental laws of Nature wasconsidered incompatible with the omnipotence of God, and the decline of theIslamic Golden Age of science followed [Hoodbhoy 1991, 105].

In the time of the quote about the “Chinese atheists” (1737), a youngRussian, Mikhail Lomonosov, a fisherman’s son, after having graduatedfrom the Slavic-Greek-Latin Academy in Moscow, was getting education inGermany. Then he returned to Russia and became the first prominent nativeRussian scientist. On his path to science he overcame many barriers, butamong them there were no Chinese or Islamic ones. Like Galileo and Newton,Lomonosov was a biblical theist and happened to be at odds with clericalofficialdom. He greatly contributed into higher education in Russia andEuropean ways to do science. A result was the first Russian world-classachievements 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 scienceand technology Europe assimilated up until the 16th century.

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To explain the European birthplace of modern science, the factor ofChristian culture was employed more than once. However, by the time ofthe Scientific Revolution Christianity had been around for sixteen centuries.The medieval Church taught mainly about the corrupted state of humanityafter the Fall of Man, rather than about human dignity and freedom endowedby God. The key new factor of modernity emerged in the 16th century due tobook printing and Reformation, when the Bible became socially much moreaccessible and a very powerful guide. This guide made clear that humanfallibility was an element of divinely endowed freedom of Man. So, thissocial factor correlated with post-Gutenberg time and European space of theScientific Revolution.

Another correlation manifested in the shifting of leading role in modernscience from scientists of Catholic background to those of Protestant one. Thisshift, discovered by A. de Candolle in 1870s and emphasized by R. Mertonin the 1930s [Cohen 1990], could be explained by quite different roles ofthe Bible in cultures of the two denominations, rather than by theologicaldifferences. The principle “Sola Scriptura” made the reading of the Biblethe central factor in Protestant culture, whereas Catholic Church discouragedthe laity to read the Bible on their own. In the 20th century, de Candolle’sdisparity is supported by the statistics of Protestant vs. Catholic backgroundsof Nobelists, about 8:1 per capita.

Some pre-Guttenberg clerics who did read the Bible, such as RobertGrosseteste and Roger Bacon, manifested that biblical theology was quitecompatible with a genuine interest in natural sciences, though it was too muchfor a cleric to concentrate on physics.

Any belief is a preconception, but it can be more or less influential, helpfulor harmful. The biblical preconceptions, inculcated by accessible vernaculartranslations of the Bible, proved to be very helpful for exploring Nature (aswell as for advancing technology and economy).

The key factor was the basic belief in divine human purpose in the divinelylawful Universe, rather than the array of theological subtleties, different inChristian denominations. Of course, just reading the Bible didn’t make ascientist out of any person, but for religious adolescents amply endowed withpro-scientific abilities—intellectual curiosity, independent insight and persis-tence (like the pioneers of modern science)—the key biblical preconceptioninformed their cognitive optimism. At the same time, laconic biblical storiesprovoked questions in truth seeking, even if led to question the Church’scanons. That was why medieval churches stood against making vernaculartranslations accessible to laity.

Thus, the biblical answer to the Needham question explains also otherfactual correlations: the time and space of the Scientific Revolution, theismof its originators, and the disproportion of scientists of different religiousbackgrounds.


There is, however, another important fact—prevailing atheism of scientistsin 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 variousindeed are they that dwell therein and the motives that have led them thither”[Einstein 1918]. Different also are the types of problems that attract theoristsof different mental styles, such as “birds” and “frogs” [Dyson 2009].

The difference between “intuitive” and “analytical” styles helps to under-stand the role of theists in originating modern physics and to see a roomfor atheists’ contributions. The quoted Einstein’s scheme of making modernscience includes three phases of scientific enterprise, or three kinds of problemsto solve:

1. to invent new fundamentals-axioms (E ⇒ A),2. to derive from them specific testable statements (A⇒ S), and3. to test these statements empirically (S ⇒ E).

Only the first phase requires an intuitive “leap of faith”. The other two phasesrequire creative using of the new fundamentals and devising new experiments.So, atheists, like L. Boltzmann, P. Dirac, S. Weinberg, have enough room forcreativity. In fact, this room is much bigger than one for the first phase: ittakes 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 scienceand technology).

All the three phases are necessary to accomplish a cycle of establishing anew fundamental theory. To start a new cycle in the expanding spiral of questin modern science, a new leap of inventive intuition is necessary. However,in the very first cycle, in the 17th century, when inventing new fundamentalshad no precedent, the Einsteinian “free inventive spirit” needed unprecedentedsupport, which was provided by scientists’ spiritual/religious faith.

According to Harvard psychologists, personal theistic belief correlateswith intuitive (vs. analytic) cognitive style rather than with such factors aseducation level, IQ, and familial religiosity [Shenhav, Rand et al. 2012]. So,the power of a physicist’s intuition could be responsible both for his theismand for his type of creativity.

An additional source of creative successes in modern physics was theparadoxical 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. Hebelieved, nevertheless , that it was “possible to arrive at the true and primarycauses” of natural phenomena and perceived his work as “merely the beginning,

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ways and means by which other minds more acute than [his] will explore remotecorners” of the “vast and most excellent science” he had just opened up [Galilei1953, 485], [Galilei 1914, 44, 153–154]. The same humble audacity could beseen in Newton who likened himself to a boy on the seashore, who had founda few smooth pebbles, with the sea of undiscovered truths before him.

Such a combination of bold creativity and personal humility in Galileo andNewton stemmed apparently from their belief in biblical connection betweenAlmighty Creator of the whole World and mortal humans made as His likenessto rule over all the Earth. To be able to rule over the world, humankind hasto explore the world to understand how it works. It was the occupation ofGalileo’s successors, who invented new fundamentals of science. Even in the20th century the effectiveness of physics seemed miraculous to Einstein andunreasonable to E. Wigner [Einstein 1993, 131], [Wigner 1960]. It was muchmore unreasonable four centuries ago, when no fundamental law of physicshad been discovered.

Well known are both Einstein’s credo: “Subtle is the Lord, but maliciousHe is not” and Bohr’s apparently secular saying that a new fundamental theorywhich is not crazy enough has no chance of being correct. Both ways toencourage 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 thegolden mean between a belief in Full and Final theory of Everything and thedisbelief in the fundamental lawfulness of the world. As F. Dyson put it:

If it should turn out that the whole of physical reality can bedescribed by a finite set of equations, I would be disappointed. Iwould feel that the Creator had been uncharacteristically lackingin imagination. [Dyson 1988]

Einstein had

[...] found no better expression than “religious” for confidence inthe rational nature of reality insofar as it is accessible to humanreason. Wherever this feeling is absent, science degenerates intouninspired empiricism. [Einstein 1993, 119]

If, in the 20th century, eloquent and religiously unaffiliated physicists choosetheistic wording to express their cognitive belief, apparently it is the mostadequate wording, and the root of this belief was indeed theistic. It is sup-ported by theism of all the founders of modern physics. Their preconceptionof personal freedom and cognitive optimism of Godlike creative creatures—anidea of biblical descent—transformed into a “self-evident” secular worldview.

The secular nature of scientific knowledge was most clearly expressed bya Catholic priest and astrophysicist Georges Lemaître, who discovered theexpansion of the Universe and suggested that it began with the explosivebirth. Thirty years later (and two years before becoming the President of the


Pontifical Academy of Sciences) this astrophysicist in soutane, at a conferenceon astrophysics, stated that the theory of Big Bang

[...] remains entirely outside any metaphysical or religious ques-tion. It leaves the materialist free to deny any transcendentalBeing. [...] For the believer, it removes any attempt at familiaritywith God [...]. It is consonant with the wording of Isaiah speakingof the “Hidden God”, hidden even in the beginning of creation[...]. There is no natural limitation to the power of mind. TheUniverse 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, themotive vigor of research results from specific metaphysics, or, rather, frompre-physical belief that “there is no natural limitation to the power of mind”,that the Universe is fundamentally lawful, and that free humans are capableof discovering those laws. Lemaître’s religious feeling was as compatible withhis 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 betweenmodern science and biblical metaphysics/prephysics. Dignity of free Man asthe purpose of God’s creation was the basis for the manifesto of early modernhumanism “On the dignity of Man” (1486) by Pico della Mirandola. He putthe following words into the mouth of “God the Father, Supreme Architect ofthe Universe” after his “last creative act”:

We have placed you [Man] at the world’s center so that youmay survey everything else in the world. [Pico della Mirandola1486, 261]

This manifesto as well as Pico’s critique of astrology had been published beforeCopernicus started to think about astronomy.

Just like the European arts and humanities owe so much to biblicalimages, stories and ideas, the most rational and empirical domain of humanknowledge—modern physical science—also appears to have biblical roots. Dueto the Printing Revolution and the Reformation, the Bible informed thecultural background of the modern Western civilization, including atheists,who relied on their self-evident personal freedom secured by law. In fact, theyassimilated their cultural postulates in their cultural upbringing, even if intheir adulthood, they believed they needed no metaphysical—or, rather, pre-physical and pre-ethical—support. Those biblical atheists could hardly see thebiblical stories as the reason to accept any belief but they could appreciate theBible as a historical source of humanitarian postulates. These atheists may

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be the most selfless children of the biblical civilization, since they adhere tothe central biblical tenet without rational natural substantiation and withoutirrational hope for supernatural approval. Religion used to be instrumental inchanging culture to be later shared by both believers and unbelievers.

The biblical view on human freedom developed into the idea of freedom ofconscience via separation of Church and state and into the notion of unalien-able human rights, which was constitutionally self-evident to the founders ofthe United States. One of them, Thomas Paine, rejected all the churches astools to enslave mankind, but, in his book The Rights of Man, to answer thequestion “What are those rights?” he referred to “the Mosaic account of thecreation, whether taken as divine authority or merely historical” [Paine 1791,49]. This argument, however, could be strong only for those whose upbringingwas as biblical as it was for founders of the United States.

The history of modern Western view on human rights is similar to thetransformation of Galileo’s “laws imposed by God upon Nature” into thesecular “laws of nature”. Biblical preconception of unalienable rights of Manresulted in the Universal Declaration of Human Rights (1948) which becamea 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 themedieval Islamic Golden Age when actual separation of science/philosophy andmosque let adherents of various faiths to freely cooperate. In the 20th century,a few Far-Eastern countries assimilated Western technological and socialinnovations.

8 ConclusionIn the 21st century two Chinese historians have made an assessment:

Compared with the huge system of universities and researchinstitutes and the large number of researchers in contemporaryChina, the quantity of original scientific work accomplished isembarrassingly small

and asked: “What is responsible for this situation?” [Hao & Cao 2009]. Itlooks like a sign of intellectual freedom, which is indispensable for advancingscience. If China is to catch up with the West in fundamental sciencewithout assimilating the idea of unalienable human rights, it will invalidatethe suggested answer to the Needham question. And there is a real problemwaiting for invention of new fundamental concepts,—the problem of quantumgravity. It remains unsolved a century after Einstein had discovered it, 80 yearsafter Matvei Bronstein (1906-1938) predicted that its solution might require“the rejection of our ordinary concepts of space and time, replacing themby some much deeper and nonevident concepts” [Gorelik 2005], and afterthousands of articles on the subject in the last fifty years.


AcknowledgmentsI am grateful to Robert S. Cohen for many years of enlightening, LanfrancoBelloni for help with Galileo’s Italian, Chia-Hsiung Tze for introduction to theNeedham question, Freeman Dyson, Toby E. Huff, Silvan S. Schweber, andSergey Zelensky for discussions.

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Galilée, de l’Enfer de Danteau purgatoire de la science

Jean-Marc Lévy-LeblondUniversité 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 surl’interprétation de la géographie de l’Enfer dantesque. Ce travail d’exégèselittéraire permet à Galilée de faire reconnaître ses talents mathématiquescomme 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 ultérieurede Galilée : au plan mathématique, l’importance de la géométrie d’inspirationarchimédienne, au plan physique, l’étude des questions de similitude que posela résistance des matériaux – sans oublier l’intérêt constant du scientifiquepour la langue et pour la culture littéraire.

Abstract: In 1587 the young Galileo was invited to give Due lezioniall’Accademia Fiorentina circa la figura, sito e grandezza dell’Inferno di Dante(hereafter Leçons sur l’Enfer) [Galilei 1587] aimed at settling an intensecontroversy regarding the geography of the Dantean Hell. This study inexegetics enabled Galileo to bring his mathematical talents and didacticqualities to the knowledge of the Tuscan scholars. But these lessons havea much greater importance, in that they reveal several of the major themesof Galileo’s further work from both mathematical and physical standpoints,such as the question of scale linked to the strength of materials, as well asthe scientist’s unremitting interest in language and commitment to literaryculture.

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


112 Jean-Marc Lévy-Leblond

considéré au mieux comme un exercice de virtuosité en même temps qu’untravail de circonstance destiné à faire connaître son auteur. La relégationau second plan de ces Leçons sur l’Enfer fut d’ailleurs très précoce. Galiléesemble n’avoir guère voulu en faire état par la suite et se montrait réticentà en communiquer le texte1. Viviani, son élève et premier biographe, n’enfait même pas mention. Pourtant, loin de constituer un élément négligeableet latéral de l’œuvre galiléenne, les Leçons sur l’Enfer recèlent des élémentspréfigurant plusieurs thèmes essentiels des contributions majeures de Galilée àla mécanique [Settle 2002]. Ce texte annonce, d’une façon parfois paradoxale,au moins l’une des « sciences nouvelles » [Galilei 1638], que Galilée rendrapubliques à la fin de sa vie, soit plus de quarante ans après les Leçons surl’Enfer, et qui constituent son apport essentiel à la physique moderne.

Le jeune Galilée, après avoir commencé ses études de médecine à Piseen 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 Archimèdeau premier chef [Geymonat 1963]. Il acquiert vite une considérable maîtriseen 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’inspirationtypiquement archimédienne, sur un instrument de pesée capable de mesurerles densités des objets et donc la qualité des alliages – dans la tradition dela fameuse anecdote d’Archimède détectant la fraude du joaillier qui avaitfondu la couronne de Hiéron. À la même époque, il rédige quelques théorèmes,toujours dans un style très archimédien, sur les centres de gravité des solidesde révolution. Remarqué par des savants de premier plan, tel Guidobaldodel Monte, il est invité par l’Académie florentine à donner ces Leçons, afind’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 fairereconnaître ses talents mathématiques comme ses qualités pédagogiques, etconstitue une fructueuse opération de promotion. Sa réputation croissant, ilfinit par obtenir en 1589 une chaire de mathématiques à l’université de Pise,poste qui inaugure la carrière institutionnelle de Galilée. Mais quelle est laplace de ces Leçons dans l’œuvre de Galilée, ou plutôt, dans le projet généralqu’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 Manettiou 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 LuigiAlamanni 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 évidentela 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 mathématiquespersonnelles, Galilée a dans ses Leçons une ambition culturelle plus haute. Saformation au sein de l’élite pisane lui avait donné une excellente éducationlittéraire, artistique et musicale (son père, Vincenzo Galilei, était l’un despremiers musiciens de son temps, ami de Monteverdi, et son frère sera aussiinstrumentiste et compositeur). Aussi Galilée s’engagera profondément dansles débats littéraires et artistiques de son temps : dès les années suivant sesLeçons sur l’Enfer, il prend part aux controverses acharnées qui opposaientles partisans de l’Arioste et ceux du Tasse, comme aux discussions fort àla mode sur les mérites comparés de la peinture et de la sculpture [Galilei1587], [Panofsky 2016], [Raffin 1992]. Mais les Toscans cultivés, tels ceux quiformaient l’Académie florentine, étaient évidemment loin de tous posséder lesconnaissances scientifiques, mathématiques en particulier, du jeune Galilée. Sila 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 porteursde cette culture puissent spontanément reconnaître et assimiler ce surgeon.Et de fait, la suite des siècles connaîtra le divorce progressif entre science etculture dont notre époque est victime [Lévy-Leblond 2004]. Peut-être conscientde ce risque, Galilée veut en tout cas montrer dans ses Leçons sur l’Enfer quela physique mathématique n’est pas simplement pourvoyeuse de calculs tech-niquement efficaces, mais peut apporter sa contribution aux débats culturelsles plus nobles, et acquérir ainsi un statut intellectuel comparable à celui deshumanités classiques2.

C’est dans ce contexte que l’on doit comprendre le recours de Galilée à lalangue italienne, et non au latin, pour la plupart de ses œuvres majeures – c’estle cas du Dialogo sopra i due massimi sistemi del mondo (ci-après Dialogue)[Galilei 1632] et des Discorsi e dimostrazioni matematiche intorno a due nuovescienze attenenti alla mecanica e i movimenti locali (ci-après Discours) [Galilei1638] sans parler du Saggiatore (ci-après L’Essayeur) [Galilei 1623]. On asouvent et justement souligné l’importance de cette décision. Mais il ne s’agitpas, comme on le dit habituellement, d’un choix essentiellement politique parlequel Galilée viserait un lectorat plus vaste que celui des seuls érudits afind’obtenir un plus large appui dans ses batailles intellectuelles. En vérité, Galiléene considère nullement l’italien (qui, à l’époque, est en fait le toscan) commela 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 écrivainallemand Durs Grünbein [Grünbein 1999] quand il affirme qu’en ne s’intéressantqu’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 deDante, ouvrant ainsi la voie à l’actuel (et certes bien réel) « divorce entre les sciencesnaturelles et les arts ».


entendu de tous. C’est, pour lui, la langue même de la haute culture de sontemps, celle qui à la fois exige et permet la plus grande clarté et la plus grandesubtilité dans l’expression. Et c’est le latin, au contraire, qu’il considère sanscomplaisance comme un jargon technique, sans doute souvent utile entre gensdu métier, mais impropre à une élucidation du sens. Les Leçons sur l’Enfersont à cet égard parfaitement révélatrices et inaugurales. Contraint d’employer,dans ses explications scientifiques, certains termes savants (géométriques enparticulier), Galilée s’en justifie et même s’en excuse auprès des membres del’Académie florentine :

[...] espérons que vos oreilles, accoutumées à entendre ce lieurésonner toujours des paroles choisies et distinguées que la purelangue toscane nous offre, puissent nous pardonner lorsque parfoiselles se sentiront offusquées par quelque mot ou terme propreau domaine dont nous traitons, et tiré de la langue grecque oulatine, 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 manœuvretactique de « communication », comme on dirait aujourd’hui, mais exprimela ferme volonté d’inscrire son travail dans la culture de sa société et de sontemps. Galilée n’est pas ici un cas isolé. Contrairement à une conceptionrépandue mais simpliste, la révolution scientifique du dix-septième sièclen’est nullement liée à l’existence en Europe d’une langue de communicationscientifique unique qui aurait été le latin. Bien au contraire, elle coïncideavec le développement, au sein des activités intellectuelles les plus exigeantes,des langues nationales, désormais considérées comme vecteurs de la culturemoderne. 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 mécaniqueà la fois en latin et en toscan. Pendant la première moitié du dix-septièmesiècle qui voit l’accomplissement de la révolution scientifique, Descartes pourle 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 cielsIl 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 astronomeet cosmologue, l’auteur du Dialogue [Galilei 1632], que l’on peut entrevoir dansces Leçons. Au contraire même, puisqu’elles s’ouvrent sur une apologie de lacosmologie archaïque, faisant l’éloge des résultats obtenus dans la mesure des« intervalles entre les ciels » et de leurs mouvements, ce qui renvoie clairementà la représentation antique et médiévale d’un univers composé de plusieurs


sphères célestes géocentriques. Galilée est loin d’être à l’époque le copernicienmilitant qu’il deviendra. Depuis sa chaire de Pise, il enseigne sans réserves lesystème ptoléméen des sphères. Ce n’est qu’en 1597, donc dix ans après lesLeçons sur l’Enfer, qu’il fera état, en privé (dans une lettre à Kepler), de sonralliement à la théorie copernicienne, et bien plus tard encore, après 1610, qu’ilcommencera à la défendre publiquement [Galilei 2004].

Il faudra sans doute qu’il prenne ses distances avec une épistémologiequelque peu naïve qui lui fait écrire, au début des Leçons, que cet arpentagecéleste qu’il célèbre, aussi « difficile et admirable » soit-il, concerne des « chosesqui, totalement ou en grande partie, tombent sous le sens ». C’est en aban-donnant l’illusion commune d’un monde immédiatement donné à l’observationque Galilée pourra fonder en raison une nouvelle vision du cosmos : que ce soitpar 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’unedes ressources rhétoriques constantes de Galilée, dont il fera grand usagedans ses œuvres majeures – l’ironie. C’est dans cette veine qu’il se permetd’expliquer la difficulté d’évaluer les dimensions de l’Enfer, puisque « ce lieuoù 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 contrairedes « ciels » qui « tombent sous le sens » –, et que « la difficulté d’une telledescription est considérablement accrue par l’absence de toute étude venantd’autres personnes ».

4 Par quelques raisons qui nous sontpropres

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 chezDante qu’il s’attelle d’abord. La mathématisation de la physique dontGalilée est à juste titre considéré comme l’un des initiateurs se fera tou-jours chez lui sous l’égide de la géométrie, dans une filiation directementarchimédienne. Rien encore, chez Galilée, de l’algébrisation que Descartes,si peu de temps après lui, commencera à mettre en œuvre. Et dans la trèsfameuse citation de Galilée, selon laquelle le grand livre de la Nature « estécrit en langage mathématique » [Galilei 1623], il faut prendre garde de ne pasoublier 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’agitnullement des formules littérales de l’algèbre et de l’analyse modernes. En toutcas, Galilée, dès sa jeunesse, est un géomètre accompli comme le démontrentd’emblée les Leçons.

Choisissant avec soin et commentant les vers adéquats de La DivineComédie, Galilée commence par confirmer la description de Manetti : l’Enfer


est une cavité approximativement conique dont le sommet est au centre de laTerre, 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 diamètreégal au rayon du globe ; en coupe, la corde correspondant à un diamètre dece cercle est donc la base d’un triangle équilatéral ayant le centre de la Terrepour 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 pourinvalider une opinion erronée :

Si l’on veut connaître [la] grandeur [de l’Enfer] par rapport à toutl’agrégat d’eau et de terre, nous ne devons pas suivre l’opinion deceux qui ont écrit sur l’Enfer, estimant qu’il occupait la sixièmepartie de l’agrégat. [Galilei 1587, 34]

Sur une coupe centrale de la Terre passant par l’axe du cône, le secteur infernaloccupe en effet un sixième de l’aire du disque – en négligeant provisoirementla 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 dansses livres De la sphère et du cylindre, nous trouverons que l’espacede l’Enfer occupe un peu moins de la quatorzième partie du [vo-lume] de l’agrégat. Je dis cela même si cet espace arrivait jusqu’àla surface de la Terre, ce qu’il ne fait pas ; car son embouchurereste couverte par une très grande voûte de Terre, au sommet delaquelle se trouve Jérusalem, et qui a pour épaisseur la huitièmepartie du demi-diamètre. [Galilei 1587, 34]

Les traités d’Archimède faisaient alors partie des mathématiques les plusérudites, que les commentateurs précédents de Dante, littéraires purs, nemaîtrisaient certainement pas [Archimède 2003, I]. L’apport de Galilée faitappel à une expertise toute particulière, dont il peut légitimement se targuer.C’est sans doute ainsi qu’il faut comprendre sa prétention à éclairer lacontroverse « par quelques raisons qui nous sont propres ». Il n’est pas sansintérêt de vérifier les estimations de Galilée à l’aide tant des formulationsarchimédiennes que des expressions algébriques aujourd’hui usuelles (mais quirestent du niveau universitaire – voir Annexe 1). Notons que si l’on rétablitla voûte de l’Enfer, avec son épaisseur d’un huitième du rayon terrestre, levolume total de l’Enfer en est considérablement diminué, puisqu’il n’est plusque légèrement inférieur à 1/22 du volume terrestre (au lieu de 1/14).


5 Une ligne qui conduit naturellement versle centre ?

Galilée, dans son commentaire, ne se veut pas seulement mathématicien, etconvoque également son expertise de physicien débutant. C’est à ce titrequ’il va émettre une critique sévère contre les commentaires de Vellutello. Cedernier, en effet, concevait les gradins successifs de l’Enfer comme des portionsde cylindre aux parois parallèles à leur axe commun, à l’instar des gradins d’unamphithéâtre antique. Galilée conteste cette interprétation, en arguant que detelles parois ne seraient nullement verticales, puisqu’elles devraient avoir pourgénératrices des rayons tirés depuis le centre de la Terre, car en deux pointsdistants, les directions des verticales ne sont pas parallèles, mais convergentes.Ainsi, selon Galilée, les falaises qui borneraient de tels gradins cylindriquesseraient-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 bergeséquidistantes entre elles, on aura des parties supérieures sanssupports pour les maintenir, et de ce fait, immanquablement, elless’effondreront. On sait en effet que les corps pesants suivent entombant une ligne qui les conduit directement vers le centre, et sisur 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 dirigéesvers le centre géométrique du globe terrestre, considéré par Galilée commedéterminant la direction de l’attraction terrestre. De très peu postérieure auxLeçons de Galilée, l’édition de la Crusca offre une représentation sans doutefondée sur ces Leçons, ou en tout cas confortée par elles, parfaitement claireà cet égard. La caution apportée par Galilée à ce point de vue semble deprime abord fondée sur un convaincant raisonnement physicien qui conforte lapertinence scientifique de son discours.

À mieux y réfléchir cependant, le physicien d’aujourd’hui est obligé deprendre quelques distances par rapport à l’argumentation de Galilée. Toutd’abord, l’Enfer selon Vellutello est très petit : sa profondeur comme sondiamètre maximal ne dépassent pas le dixième des valeurs que prennent cesdimensions dans la version de Manetti (son fond se situe aux environs dudixième du rayon terrestre à partir de la surface, bien loin du centre de laTerre), et son volume est donc mille fois plus petit. Dans ces conditions, lavariation de la direction de la pesanteur d’un point à l’autre de l’Enfer estfaible, de quelques degrés au maximum, et peut être pratiquement assimiléeà la direction de l’axe de l’Enfer. C’est alors une approximation tout à fait


raisonnable que de considérer les gradins comme des portions de cylindres,à parois parallèles donc, comme le propose Vellutello. Mais si l’on veut êtreplus précis, ce qui est nécessaire dès lors que l’Enfer, selon Manetti appuyépar Galilée, a des dimensions de l’ordre du rayon terrestre, le raisonnement deGalilée 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 lecentre du globe terrestre que si celui-ci est une sphère complète, uniformémentpleine. Or, dès lors que l’on évide le globe en lui retirant le vaste espace coniquequ’exige l’Enfer, le champ de gravité intérieur de cette sphère incomplèteest affecté et les directions des verticales locales perturbées. Galilée n’avaitcertes pas les moyens d’évaluer ces modifications que seule la théorie de lagravitation newtonienne permettra de calculer. Il est alors quelque peu ironiquede constater que, d’après cette théorie, la situation se rapproche plutôt decelle décrite par Vellutello ! En effet, l’attraction due à la partie évidée étantsupprimée, la force de pesanteur en un lieu quelconque des bords de l’Enferserait dirigée non vers le centre géométrique de la sphère, mais vers un pointsitué plus bas sur l’axe du cône infernal. Ainsi les parois verticales des gradinsne devraient-elles pas être des troncs de cône ayant le centre de la Terre poursommet commun, mais des troncs de cône nettement plus fermés, plus prochesde segments cylindriques. Vues depuis le centre de la Terre, les parois desgradins apparaîtraient donc en surplomb plus ou moins prononcé. De fait,le calcul selon la théorie newtonienne de la gravitation montre que, dansl’Enfer selon Manetti, la direction de la pesanteur serait en fait partout presqueparallèle à l’axe de l’Enfer (Annexe 2). Ainsi, et fort curieusement, au regardde la physique moderne, la situation serait finalement intermédiaire entre celleque décrit Manetti, et celle, retoquée par Galilée, que proposait Vellutello, etmanifestement plus proche de cette dernière – y compris dans la géographiede Manetti ! La critique de Galilée est donc beaucoup moins dévastatrice qu’ilne 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 oùle centre de la Terre est le centre intrinsèque de l’Univers, lieu naturel desgraves, cette distinction est évidemment impensable. Galilée, dans ses Leçonssur l’Enfer, passe à côté du problème, mais on peut légitimement se demandersi une réflexion ultérieure sur la situation n’a pas compté parmi les sourcesde sa remarquable discussion anti-aristotélicienne de la première Journée duDialogue. On pense ici au passage où, bien avant le développement de la théorienewtonienne, 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 propriétéintrinsèque d’un point privilégié de l’univers, mais comme la résultante desforces d’attraction mutuelles des parties du globe :

[...] les parties de la Terre se meuvent non parce qu’elles tendentvers le centre du monde, mais pour se réunir avec leur tout, et


c’est pour cela qu’elles ont une inclination naturelle vers le centredu globe terrestre, en vertu de laquelle elles conspirent à formeret 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 laverticalité des gradins de l’Enfer, aurait compris que sur et dans une Terreprivée 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 précisercertains 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 enfoncéjusqu’à mi-poitrine, son nombril coïncidant pile avec le centre du monde.Galilée commence par évaluer la taille des « géants » décrits par Dante commeayant une face aussi haute que la fameuse énorme « pigne » en terre cuite quidécore une cour du Vatican (où elle est encore aujourd’hui visible, avec ses troismètres et quelques de hauteur). Considérant que le rapport de la tête au corpsest le même chez les géants que chez les humains (soit de 1 à 8), Galilée, parune 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 longueurde l’un de ses bras et la taille d’un géant est supérieur au rapport entre la tailled’un géant et celle d’un humain. D’où, par deux nouvelles règles de trois, lalongueur du bras de Lucifer, environ 340 mètres au moins, et la taille de Luciferlui-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 constitutionphysique. Or, la résistance des matériaux suit des lois d’échelle qui ne sontpas celles des simples proportions géométriques. C’est là un phénomèneempiriquement bien connu dans la pratique artisanale : si, partant d’un objetde modestes dimensions, barque, charpente, chariot, on en augmente toutesles cotes dans un même rapport pour fabriquer un objet semblable mais plusgrand, on se rend compte que sa fragilité augmente rapidement avec le facteurd’agrandissement. Le premier à avoir attiré l’attention des physiciens sur cepoint capital, jetant les bases de la théorie moderne de la résistance desmatériaux, n’est autre que Galilée lui-même ! C’est le résultat essentiel del’une des deux « sciences nouvelles » qu’il développe dans les Discours. Le toutdébut de l’ouvrage annonce avec force cette conception :

SALVIATI : [...] Ne croyez donc plus, seigneur Sagredo [...] quedes machines et des constructions faites des mêmes matériaux,reproduisant scrupuleusement les mêmes proportions entre leurs


parties, doivent être également ou, pour mieux dire, propor-tionnellement aptes à résister ou à céder aux chocs venus del’extérieur, car on peut démontrer géométriquement que les plusgrandes sont toujours moins résistantes que les plus petites ; desorte qu’en fin de compte toutes les machines et constructions,qu’elles soient artificielles ou naturelles, ont une limite nécessaireet prescrite que ni l’art ni la nature ne peuvent dépasser, – étantbien sûr entendu que les proportions et les matériaux demeurenttoujours identiques. [Galilei 1638, 51]

Galilée applique ces idées au cas des êtres vivants, en démontrant clairementqu’une homothétie spatiale ne respecte pas les contraintes physiques, et qu’ungrand animal a besoin de membres plus épais par rapport à sa taille qu’un petitpour soutenir son poids (que l’on compare, à titre d’exemple, un éléphant, unchien et une souris) :

SALVIATI : [...] il serait impossible, aussi bien en ce qui concerneles hommes que les chevaux ou les autres animaux, de fabriquerdes squelettes capables de durer et remplir régulièrement leursfonctions, en même temps que ces animaux croîtraient immensé-ment en hauteur – à moins bien entendu d’utiliser une matièrebeaucoup plus dure et résistante que la matière habituelle, etde déformer leurs os en les agrandissant démesurément, ce quiaboutirait 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 Leçonssur 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 unhomme ordinaire, il faudrait ou trouver une matière bien plusdure et plus résistante pour en constituer les os, ou bien admettreque sa robustesse serait proportionnellement beaucoup plus faibleque celle des hommes de taille médiocre ; sinon, à augmenter sansmesure sa hauteur, on le verrait plier sous son propre poids ets’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’uneépaisseur relative monstrueuse, ou bien il est d’une grande fragilité. Cettedernière conclusion pourrait d’ailleurs se défendre puisqu’aussi bien Luciferest apparemment immobile et dans une zone de gravité pratiquement nulle, nerisquant donc guère de chute fatale. À quelques décennies près, c’est la thèseque 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


celui touchant à la taille de Lucifer, puisqu’il s’agit tout bonnement de larésistance de la calotte terrestre servant de voûte à l’Enfer. Il écrit :

[Selon certains], il ne semble pas possible que la voûte qui recouvrel’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 toutà 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] onavait ne serait-ce qu’une brasse, ou 1

2 , au lieu de 4, elle pourraitdéjà 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 deDante et l’architecture de la célèbre coupole du Dôme de Florence conçue parBrunelleschi, et qui joua un rôle emblématique dans la Renaissance italienne3

[Toussaint 1997]. Mais si cette analogie possède un sens culturel profond, savaleur scientifique est nulle, pour les raisons mêmes que Galilée développe dansles Discorsi : une voûte aussi gigantesque que celle de l’Enfer, si elle avaitles mêmes proportions géométriques qu’une petite voûte maçonnée, n’auraitcertainement pas la même solidité. Au regard des conceptions modernes surla pesanteur et la résistance des matériaux, le couvercle de l’Enfer seraitinéluctablement appelé à s’effondrer, selon les arguments mêmes que Galilée ainitiés. 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 parexemple, le poids sera multiplié par 1000 mais la résistance à l’effondrement par100 seulement ; elle sera proportionnellement 10 fois plus fragile. Il y a doncnécessairement une limite à la solidité d’une structure obtenue par simplechangement d’échelle à partir d’une structure solide plus petite. Et dans lecas de la voûte de l’Enfer comparée à la petite voûte maçonnée envisagée parGalilée, où le facteur d’échelle est de plusieurs centaines de milliers, cette limiteest plus qu’évidemment dépassée et de beaucoup.

Nous touchons ici à un point crucial concernant les Leçons et leur rôle dansle développement de la pensée de Galilée. Il est en effet très vraisemblablequ’il ait rapidement compris son erreur de raisonnement, découlant d’uneconception purement géométrique, ne tenant pas compte des lois d’échelleconcernant les propriétés physiques de la matière. Et c’est la prise de consciencede cette méprise qui aurait été à l’origine de ses travaux sur la résistance desmatériaux, 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.


réalisé le caractère fallacieux des changements d’échelle mis en œuvre dans sesLeçons sur l’Enfer expliquerait en particulier la discrétion, voire la réticence,dont il a fait preuve presque immédiatement à l’égard de ce travail, ainsique nous l’avons rappelé. Il a certainement consacré à ces questions d’intensesréflexions dans les années 1590 et 1600. On peut même conjecturer que Galilée,en comprenant son erreur, subit un véritable choc psychologique, dont ontrouve écho dans les Discorsi, lorsque immédiatement après l’énoncé fondateurde Salviati rappelé ci-dessus, son interlocuteur réagit avec une émotivitésurprenante :

SAGREDO : Déjà la tête me tourne, et mon esprit, commeun nuage qu’un éclair déchire brusquement, se remplit pour uninstant d’une lumière inhabituelle qui de loin me laisse entre-voir, pour les estomper et les cacher aussitôt, des idées étrangeset désordonnées. Car de vos propos il me semble que l’on de-vrait conclure à l’impossibilité d’exécuter à l’aide d’un mêmematériau 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 Leçonsassimilant la coupole de l’Enfer à une petite voûte maçonnée. On a toutlieu de croire que Galilée ait rapidement compris son erreur. De fait, si lesDiscorsi n’ont été publiés qu’en 1638, le matériau en était prêt dès avant1610, époque à laquelle Galilée se consacre aux observations astronomiques etpublie 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 detrente ans d’avance, un résumé explicite des Discorsi, tout au moins de leurpremière Journée :

J’ai récemment réussi à obtenir tous les résultats, avec leurs dé-monstrations, concernant les forces et les résistances de morceauxde bois de diverses longueurs, tailles et formes [...], laquelle scienceest absolument nécessaire pour fabriquer des machines et toutessortes de constructions, et qui n’a jamais auparavant été traitéepar quiconque. [Peterson 2002, note 15]

Il est donc loisible de considérer les Leçons sur l’Enfer comme le creuset oùs’est initié le travail fondamental de Galilée dans les Discorsi.

On peut encore mettre en évidence une autre difficulté physique de ladescription de Dante, liée comme la précédente à la rupture de symétrie quela cavité infernale imposerait à la distribution des masses terrestres. Galilée,s’il s’était, quelques décennies plus tard, reposé la question de la cohérencephysique du modèle dantesque, n’aurait pas manqué de percevoir le problème.C’est qu’une Terre largement évidée par l’Enfer conique, verrait son centred’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


de Jérusalem. Un calcul rapide montre que ce décalage serait de l’ordre de3% du rayon terrestre, soit environ 200 km. La Terre tournerait sur elle-mêmeautour de ce centre d’inertie ; la distribution des masses n’étant plus isotrope(à symétrie sphérique), mais à symétrie axiale, elle se comporterait comme unetoupie. Puisque son axe de symétrie (passant par Jérusalem) ne coïncideraitpas avec l’axe des pôles, le mouvement de rotation diurne de la Terre surelle-même serait considérablement perturbé par une précession simultanée del’axe hyérosolomitain, ce qui est contraire à l’observation la plus banale. Ainsidonc, la mécanique suffirait à elle seule à invalider l’idée d’une cavité infernaleau sein de la Terre. Mais, bien entendu, placer l’apex de l’Enfer, demeure deLucifer, au centre de la Terre, n’a de sens que si ce lieu est aussi le centredu 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 audemeurant le fonctionnement de ses fournaises. Une telle théorie a d’ailleursété 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’iln’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 sontdes 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 leparcours entier des deux hommes dans l’Enfer dure moins de trois jours – dela nuit du jeudi saint 7 avril 1300 au soir du samedi saint ! Notons que dans lepetit Enfer de Vellutello, où les distances se chiffrent en centaines de kilomètresplutôt qu’en milliers, la difficulté serait (un peu) moins sérieuse... Le paradoxeest d’autant plus grand qu’on trouve quand même dans le texte de Galiléeune allusion au temps du voyage, précisément utilisée comme argument contrela géométrie de Vellutello. Virgile, ayant amené Dante au premier cercle, lepresse : « Poursuivons, car un long chemin nous pousse. » Galilée en conclutque la distance qu’il leur reste à parcourir est bien plus longue que celle déjàfranchie, et que par conséquent l’Enfer est certainement bien plus profondque le dixième du rayon terrestre proposé par Vellutello. Comment donccomprendre que Galilée, si précis sur les mesures numériques des distances,n’ait pas procédé à une estimation quantitative des temps de parcours ? Celaest d’autant plus étrange que Galilée s’occupait déjà à l’époque du mouvementdes corps. Le plus probable est qu’il a bien effectué ces calculs, et s’étant renducompte de leur incompatibilité avec une interprétation réaliste de la narrationdantesque, a décidé de les passer sous silence – sans pouvoir s’empêcher pourautant d’en tirer un argument qualitatif, fort douteux au demeurant, contreVellutello. Sans doute faut-il nous résigner, avec Galilée, à admettre que la


description de Dante, si elle permet une interprétation géographique cohérentede l’Enfer, ne laisse la chronologie du voyage relever que de la licence poétique.

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

Il ne saurait pour autant être question de faire des Leçons sur l’Enfer laclé de toute l’œuvre de Galilée. On a tenté par exemple de voir dans lepassage de L’Enfer où le monstre Géryon emporte dans les airs le poète etson guide, sans qu’ils perçoivent le mouvement autrement que par la caressede l’air au passage, une prémonition du principe de relativité, dont Galiléeaurait pu s’inspirer [Ricci 2005, 717]. Une idée semblable a été avancée parPrimo Levi dans l’un de ses tout derniers textes, où il interprète ce mêmepassage comme une prémonition de la sensation d’apesanteur éprouvée par lesastronautes en orbite inertielle [Levi 1987]. Mais ces suggestions ne résistentpas vraiment à l’examen, ni du texte du poème de Dante, ni de son éventuellesignification scientifique.

Dans le registre littéraire, il serait peut-être plus intéressant d’évoquerla visite que le jeune Milton dit avoir rendue en 1638 au vieux Galilée alorsreclus dans sa résidence forcée d’Arcetri – rencontre non attestée au demeurant,mais immortalisée par un groupe statuaire représentant le poète penché surl’épaule du physicien, que l’on peut voir au rez-de-chaussée du département dephysique de l’université « La Sapienza » de Rome. En tout cas, dans son grandpoème Paradise Lost [Milton 1667], Milton mentionne explicitement Galilée àplusieurs reprises [Henderson 2001]. On ne peut que rêver au dialogue qu’ontpeut-être eu les deux hommes sur l’œuvre de Dante, une hypothétique sourcede la vision miltonienne de l’Enfer [Steggle 2001].

On s’en voudrait de passer sous silence un intéressant travail récent, d’uneinspiration fort proche de celle des Leçons sur l’Enfer de Galilée, dans lequelest proposée une subtile interprétation géométrique moderne de la descriptionassez obscure que donne Dante du Paradis, et plus généralement de son universspatial entier. Le physicien Mark A. Peterson, déjà cité, y montre qu’il esttout 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 euclidieninfini et plat) [Peterson 1979]. On aimerait connaître le sentiment de Galiléesur cette analyse !

9 ConclusionCes commentaires trop rapides sur un épisode peu connu de l’histoire desdébuts de la science moderne semblent permettre de renouveler quelque peu


le vieux débat sur les critères de scientificité que nous appliquons à tel ou telénoncé afin de lui attribuer ou de lui refuser le label « qualité science ». Ni lalogique de l’argumentation – contradictoire, selon les bonnes règles méthodo-logiques –, ni la rigueur des calculs ou l’exactitude des faits d’observation nefont défaut aux Leçons sur l’Enfer de Galilée, pas plus qu’aux sérieuses étudesinfernales 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 caractérisel’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-épreuveest d’ailleurs aisée, puisque nombre des assertions de la science telle qu’elle sefait – la majorité, sans doute – se révèlent erronées sans pour autant disqualifierles recherches qui les ont produites, du moment que leur intérêt est reconnupar la collectivité savante. De ce point de vue, l’opposition classique entre unehistoire des sciences internaliste, privilégiant les dynamiques intrinsèques auxtravaux disciplinaires, et une histoire externaliste, mettant en avant les effetsde leur environnement social, perd beaucoup de sa vigueur. Car la question dela pertinence reconnue à tel ou tel programme de recherche permet justementde faire lien entre l’organisation conceptuelle d’un champ de recherches et sesdéterminations culturelles, idéologiques, économiques ou politiques.

Ainsi, la question classique de la validation ou de la réfutation des idéesscientifiques le cède-t-elle en importance aux considérations sur leur qualifica-tion ou leur disqualification. De fait, bien des travaux sont abandonnés sansjamais être tombés sous le coup d’une critique explicite et rédhibitoire ; le plussouvent, c’est insidieusement que les voies de la recherche prennent une autreorientation, laissant en jachère au bord de la route abandonnée des terrainsà moitié explorés. Si peu de travaux scientifiques accèdent au Paradis de lareconnaissance définitive, peu aussi sont condamnés à l’Enfer de l’oubli ou durejet absolus. La plupart se retrouvent au Purgatoire – comme les Leçons surl’Enfer de Galilée, justement.

RemerciementsC’est avec plaisir que je remercie Françoise Balibar, Lucette Degryse et Jean-Paul Marmorat pour leurs apports à ce travail, ainsi que Raffaele Pisano poursa 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].


Annexe 1. Le volume de l’Enfer

Selon Dante lu par Galilée, l’Enfer est un gouffre conique dont le sommet estau centre de la Terre, le demi-angle au sommet valant θ = 30°, borné parune calotte sphérique et recouvert par une voûte dont l’épaisseur d vaut 1/8du rayon terrestre (voir la Figure 1 pour les notations). Si l’Enfer « arrivaitjusqu’à la surface de la Terre », et que l’on ne tienne pas compte du volumede la voûte qui le recouvre, le volume de la cavité (délimitée en coupe par laligne OAJBO) serait :

v = 23πR

3(1− cos θ)

à comparer au volume total de la Terre :

VT erre = 43πR

3

soit, en proportion :v/VT erre = 1

2(1− cos θ)

numériquement, ici

v/VT erre = (2−√

3)4 = 1

14, 92 . . .

un peu moins que la quatorzième partie, comme l’écrit Galilée. Mais en tenantcompte de la voûte, la hauteur de l’Enfer est réduite par un facteur 7/8, etson volume par un facteur (7/8)3, d’où maintenant :

VEnfer

VT erre= 2−

√3

4

(78

)3= 1

22, 3 . . .

Galilée ne disposait certes pas des notations modernes, et encore moins desressources du calcul intégral sur quoi se fondent les expressions ci-dessus. Maisles résultats d’Archimède, dans son Traité de la sphère et du cylindre, luipermettaient 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 modeproposé par Dante et commenté par Galilée, la distribution des masses perdsa symétrie sphérique. Or c’est cette symétrie, comme l’a montré plus tardNewton, qui entraîne que les forces attractives exercées sur un objet quelconquepar les différentes parties de la Terre se combinent en une force totale dirigéevers le centre de la Terre et d’une intensité égale à celle qu’exercerait unemasse située en ce point égale à la somme des masses des parties de la Terreplus proche du centre que l’objet considéré. Ce résultat dépend crucialement


Figure 1. La Terre et l’Enfer en coupeLe disque de centre O (et de rayon OA′ = 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 A′JB′ délimitent la calotte qui couvrel’Enfer, d’épaisseur HJ = d = R/8. KM et LN dessinent la coupe du cylindrecirconscrit à la sphère terrestre. De même que l’aire totale de la sphère est égale àl’aire latérale du cylindre de hauteur KM , l’aire de la calotte de section A′JB′ estégale 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’inversedu carré de leur distance, ce qui, évidemment, le mettait hors de portée deGalilée. Dans le cas d’une sphère évidée, le calcul de la force gravitationnelleen un point quelconque est loin d’être immédiat, même pour un évidement deforme géométrique élémentaire comme le cône infernal, et il faut recourir à descalculs numériques assez lourds. Un modèle simple va pourtant nous fournirun résultat exact intéressant qui confirme nos conclusions sur la perturbationdes verticales.

Supposons un Enfer non plus conique, mais sphérique. Dans tout l’espaceintérieur de la sphère infernale ainsi que sur ses bords, la force de pesanteurserait alors uniforme, ayant en tout point la direction de la droite joignant lescentres de l’Enfer et de la Terre et la même valeur (c’est là un petit théorèmede 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 bordsparallèles) qui prévaudrait sur celle de Manetti (gradins en troncs de cône) !

Dans le cas dantesque d’un Enfer conique, la situation est certes pluscompliquée. La résolution numérique des équations qui déterminent le potentielet le champ de gravitation dans ce cas confirme pourtant que la direction de lapesanteur au sein de l’Enfer et jusque sur ses bords varie fort peu. La figure 2montre le résultat d’un tel calcul.


a b

c

Figure 2. La pesanteur en EnferLes flèches indiquent la direction et la valeur (relative) du champ de pesanteura) dans une Terre pleine ; b) dans une Terre creusée d’un Enfer sphérique ; c) dansune Terre creusée d’un Enfer conique. On constate que le champ de pesanteurmontre, pour les deux formes de l’Enfer une direction constante – exactementen b), approximativement en c). [Calculs et graphisme par Jean-Paul Marmorat,communication privée]


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Galileo’s Trattato della sfera ovverocosmografia and Its Sources

Roberto de Andrade MartinsUniversidade de São Paulo (Brazil)

Walmir Thomazi CardosoPontifícia 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 structuredu Tractatus de sphæra de Johannes de Sacrobosco. Nous analysons quelquesparticularités du traité, en le comparant à d’autres travaux astronomiquesdu xvie siècle, et nous discutons ses sources probables. Nous soutenons quel’influence du commentaire de Christoph Clavius sur la Sphæra de Sacroboscone peut pas être considérée comme son influence unique ou principale. Le traitéde Galilée était probablement inspiré par la Sfera del mondo de Piccolomini,un travail qui anticipait plusieurs particularités du Trattato della sfera. Cetteinfluence est établie par de nombreuses annotations de Galilée trouvées dansun exemplaire du livre de Piccolomini.

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


132 Roberto de Andrade Martins & Walmir Thomazi Cardoso

1 Introduction

One of the earliest works attributed to Galileo is called Trattato della sferaovvero cosmografia [Treatise on the Sphere, or Cosmography]. Its date of com-position is unknown (probably before 1600) and it was published posthumouslyby the priest Urbano d’Aviso [Galilei 1656]. It is a short and elementarygeocentric astronomical treatise. Its content and structure generally followJohannes de Sacrobosco’s medieval Tractatus de sphæra. A first look at thecontents and style of the Trattato della sfera ovvero cosmografia (hereinaftercalled Trattato in brief) provides no internal evidence that it was written byGalileo. It includes no reference to Copernicus or his ideas ; it accepts anddefends the main geocentric astronomical ideas : that the Earth does notmove in any way, that it is at the center of the universe, and that the Sun, theMoon, the planets and the stars move around it. Also, it contains no recentinformation that became available during the sixteenth century, such as newevaluations of the size of the Earth, the knowledge of stars visible from theSouthern hemisphere, or the existence of people living in the tropical zone.

Due to its contents, it is understandable that after its publication theTrattato did not attract much attention, and up to the nineteenth century,there were strong doubts concerning its authenticity. Of course, it providesno information concerning Galileo’s later ideas ; however, it is an importantsource for studying his early acquaintance with the astronomy of his time.

2 Comparison between the Trattato dellasfera and Sacrobosco’s Tractatus

Most of the treatises belonging to the tradition of Sacrobosco’s Tractatusde sphæra (or Sphæra, in short) contained the whole text of the medievalbook, adding explanations, images, and new information. The original textwas usually printed in a different (larger) typeface so that the readers wouldbe able to identify it.

The Trattato cannot be regarded as a summary of Sacrobosco’s work, asclaimed by Wallace [Wallace 1977, 255]. The medieval Sphæra was a shortwork (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 wellas substantial explanations and additions. However, its author chose a differentapproach. It does not contain a translation of the Sphæra, and not even citesSacrobosco’s name.

Sacrobosco’s work contains a preamble and four chapters. The Trattatobegins with a long methodological introduction (with no correspondence inthe Sphæra) followed by 24 small chapters. Generally, the chapters of theTrattato 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 ; thissubject is lacking in the Trattato. Most of the content of the second chapter ofSacrobosco’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 astronomicalrisings and settings ; the Trattato omits this subject altogether. The rest of thethird chapter is largely followed by the Trattato, but it adds a more detailedelucidation of latitude and longitude and describes 22 geographical climes,instead of the seven classical ones. The beginning of the fourth chapter ofSacrobosco’s work explains the motions of the Sun and planets ; the Trattatoomits several parts of this content. The Trattato contains a description ofthe phases of the Moon and its visibility—subjects that do not appear inthe Sphæra, although they are discussed in Sacrobosco’s De anni ratione[Sacrobosco 1573, 211–214]. The final part of Sacrobosco’s work expounds themiraculous eclipse of the Sun during Christ’s passion ; the Trattato does notrefer to this miracle but includes a discussion of the motion of the 8th sphereand its trepidation that is lacking in the Sphæra. Of course, there are severalother differences between the contents of these two works.

The editions and commentaries of the Sphæra frequently contained severalillustrations. The Trattato contained only one single figure [Galilei 1656, 24].Another curious difference is the complete lack of classical literary citations inthe 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 ap-proach. It simply exposes accepted knowledge ; there is no polemical characterin any part of the book, in strong contrast with Galileo’s later famous works.Notice that some former authors, such as Francesco Barozzi and FrancescoMaurolico, had maintained a polemical attitude in their astronomical writings[Barozzi 1585], [Maurolico 1543].

3 Language and aim of the Trattato dellasfera

Another relevant difference between the two works was their languages. Sincethe Tractatus de sphæra was used as a textbook in European universities,most of its versions were written in Latin. All of the 30 editions publishedduring the fifteenth century, for instance, were in that language.1 Among the92 renderings of the Sphæra published between 1501 and 1550, we find 12 that

1. A survey of editions and commentaries of the Sphæra, produ-ced by one of the authors of this paper (Martins) is available at<http ://www.ghtc.usp.br/server/Sacrobosco/Sacrobosco-ed.htm>.


were not in Latin. The Portuguese and Spanish versions were produced for theuse of pilots and other people involved in navigation ; others were created forthe general public, who was not familiar with Latin.

This raises a question concerning the aim and the target readers of theTrattato ovvero cosmografia. Galileo’s teaching subjects at the universities ofPisa and Padua included Sacrobosco’s Sphæra and the use of Latin in thelectures was obligatory [Favaro 1888, vol. 1, 177]. One of the reasons was thatmany of the students attending those universities came from other countriesand the only lingua franca available at that time was Latin. Galileo must haveused some Latin version of the Sphæra in his public teaching activities, and itwould have been useless to prepare an Italian textbook for those students.

Stillman Drake and William Shea conjectured that the Trattato wascomposed for private teaching in 1586-1587 after Galileo left his medicinestudies and before he became a professor at Pisa [Drake 1978, 12], [Shea1990, 51]. However, there is no evidence that he knew or taught astronomyduring this period [Favaro 1888, vol. 1, 15]. Also, Drake’s claim that “it is safeto assume that Galileo’s first-year lectures on astronomy were based on hisTreatise on the sphere” [Drake 1978, 19] is groundless.

Besides his official duties at the universities, Galileo delivered privateclasses on several subjects, as a way to improve his financial income. It iscertain that he taught astronomy to some students in the early years of theseventeenth century—at the time when the extant manuscript copies of theTrattato were produced. It is unlikely, however, that he wrote a textbook forthose private lessons. Firstly, many of his students were foreigners. Accordingto extant records, in the period 1601-1607, about 10 private students studiedthe sphere under Galileo. One of them was British, two were Hungarian andseven were Polish [Favaro 1888, vol. 1, 186–191]. It is doubtful whether theywould have appreciated a textbook in Italian. Secondly, Galileo could deliverprivate classes on astronomy without the inconveniency of writing a textbookfor his students : he could use any of numerous available treatises—both inLatin and in Italian.

Composing a new textbook would be understandable if Galileo intendedto 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 workspublished one century earlier. It did not include updated information aboutthe size of the Earth, or the stars visible from the Southern hemisphere, or theexistence of people living in the Torrid Zone, for instance. It did not addressthe practical navigational use of astronomy that was very important duringthe sixteenth century, or the use of new astronomical instruments. Neither didit discuss recent astronomical theories (such as those of Copernicus and TychoBrahe) or fresh observational findings (such as comets and Tycho’s nova).

Perhaps Galileo did not compile the Trattato for his students. It might bejust a set of notes he wrote for himself, when he was learning the traditionalastronomy he was going to teach in Pisa. William Wallace conjectured that it


was composed towards the end of 1590, when he wrote to his father requestinghis copy of the Sfera [Wallace 1998, 35].2 Maybe he used these annotations asguidelines for his classes, later ; and he could have lent the manuscript to somestudents who wanted to copy it.

4 Suggested sources of the Trattato dellasfera

Some authors have claimed that the main source of the Trattato was the famouscommentary on Sacrobosco’s Sphæra written by Christoph Clavius [Wallace1977, 255, 257], [Wallace 1984, 257], [Lattis 1994, 5]. In the late sixteenthand early seventeenth centuries, Clavius’ bulky book (about 500 pages and170,000 words) was one of the most respected and widely used astronomicaltextbooks. After its first edition in 1570, it underwent successive improvementsand 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 textualparallel between the two works, such as the one published by Wallace ispointless [Wallace 1977, 258]. For deciding whether a specific work was themain source of the Trattato it is necessary to analyze some of its unusualfeatures.

William Wallace and Alistair Crombie have noticed that part of Galileo’searly manuscript on Aristotle’s De Cælo (MS 46), might have been copied fromClavius’ 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 basedon Clavius’ book [Crombie 1996, 178]. However, in his later publicationsWallace suggested that Galileo could have only indirect acquaintance withClavius’ work, through Mutius de Angelis [Wallace 1990, 36]. Stillman Drake,on the other hand, concluded that this introduction was added to the work at alater time (around 1602) and that Galileo’s source for that part was Ptolemy’sAlmagest [Drake 1978, 52–53]. It might also have been derived from CasparPeucer, 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 tableare 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 figurethat 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 fewbooks, including a Sfera (notice that he did not call it “Sphæra”) [Galilei 1890-1909,vol. 10, 44–45].


table can be found in Clavius’ treatise [Clavius 1585, 429–430]. Hence, WilliamWallace concluded that it was copied from that book [Wallace 1977, 258],thus establishing a strong link between the Trattato and the famous com-mentary. However, this evidence is not conclusive, as we have already shown[Martins 2010].

In all editions of Clavius’ commentary on Sacrobosco, beginning with thefirst one (1570), we do find a table closely similar to the one reproduced inthe Trattato. All descriptions are in Latin ; the declination of the ecliptic wasalso 23° 29’ ; and the numbers are almost identical to those reproduced byGalileo. However, there are a few numerical differences. Notice that Clavius’table is reproduced without any numerical change in the several editions ofhis 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 thoseof Clavius. These might be simply copying mistakes, but they could be due tohis using a different source. Two of the three differences between Galileo’s andClavius’ numbers also appear in the table published by Francesco Pifferi—andthat cannot be ascribed to a mere coincidence [Pifferi 1604, 348–349]. It issignificant that Pifferi was a mathematics professor at the University of Paduabefore Galileo and that the manuscript of his book could have been availableseveral years before its publication, in 1604.

Besides that, the table published by Clavius was not original. An identicalone was printed by Caspar Peucer [Peucer 1569, 268–269], one year before thefirst edition of Clavius’ book. Other published books might also have tablesvery similar to those of Peucer and Clavius. Therefore, the description of the22 geographical climes and the corresponding table appearing in the Trattatocould have been copied from another work and they do not establish a strongconnection with Clavius’ book.

Another particularity of the Trattato is its description of the phases ofthe Moon and the condition of its visibility. Although these are elementaryastronomical topics that could have been introduced by Sacrobosco in hisSphæra, he did not include them in that work, and Clavius’ commentary alsodoes not contain this subject. Therefore, his book could not be the sourcefrom which the Trattato drew its description of the phases and visibility of theMoon.

Analyzing further details of the Trattato we find several other relevantdifferences between its content and Clavius’ work—for instance, some of thespecific arguments concerning the immobility of the Earth [Martins & Cardoso2008]. We may certainly reject the claim that the Trattato was a summary ofClavius’ book. The general style of the Trattato and the very language thatwas chosen for its composition also suggest that its main source should besought elsewhere.


5 Sixteenth-century Italian astronomicalworks

Taking into account its language, the Trattato della sfera ovvero cosmografiamight have been inspired by some previous vernacular publication. The Italianversions of Sacrobosco’s Tractatus de sphæra that were published before 1600were those of Mauro Mattei, Antonio Brucioli, Piervincenzo Danti de’ Rinaldi,and Francesco Giuntini. We have examined all of them, and their descriptionwill be presented in a forthcoming paper. None has many similarities to theTrattato della sfera.

During the sixteenth century, there appeared several astronomical booksinspired by Sacrobosco’s work but that did not follow it in a strict way. Mostof them were in Latin, but there were also vernacular ones. The most famousone 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 andprinted 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 ofPiccolomini’s book on the sphere [Berti 1876, 100]. Antonio Favaro reportedthat Galileo had a copy of the 1572 edition of De la sfera del mondo, remarkingthat 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 six-teenth century, born in Siena [Fabiani 1759]. He wrote poetry, plays, and bookson philosophy, astronomy, and other subjects. He became a priest and later wasnominated bishop of Patras (Greece), although he never visited that country.He belonged to a group of writers who defended the replacement of Latin byItalian (or, more exactly, the Tuscan dialect) as a scholarly language and hewas one of the main writers who contributed to the early development of avernacular scientific prose [Suter 1969, 210]. He lived for four years in Padua,where he attended the university. During this period he developed a stronginterest in astronomy and published his famous treatise De la sfera del mondo,that was accompanied by his treatise Delle stelle fisse, the first celestial atlaspresenting all the Ptolemaic constellations in their real configurations, with theindication of the magnitude of each star. The first edition was dedicated to alady, Laudomia Forteguerri de Colombini. One of Piccolomini’s purposes, inhis vernacular works, was to make philosophy, poetry, and astronomy availableto everyone (including women) who did not know Latin and who had no accessto university courses.

Piccolomini did not regard the classics as sacred writings. He thoughtthat 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.


original language and for that reason, instead of producing rigorous andliteral translations, he preferred to create vernacular paraphrases of Aristotle’sRhetoric and Physics, for instance. For the same motive, he produced hisown vernacular books inspired by Sacrobosco’s Tractatus de sphæra and byPeurbach’s Theorica planetarum, rather than translating them to Italian.

The first edition of Piccolomini’s Sfera del mondo had four parts orbooks. He kept improving it in the following editions, adding new topics andexpanding his explanations. In 1564, he published his final revised and enlargedversion 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 somehowsimilar to Piccolomini’s, from whom he borrowed some relevant phrases suchas “sensate esperienze e certe dimostrazioni” [Crombie 1996, 236–237].

Several characteristics of Piccolomini’s work have counterparts in theTrattato. First of all, the use of Italian as a language for exposing astrono-mical knowledge. Secondly, the absence of a rigid correspondence betweenPiccolomini’s composition and Sacrobosco’s Sphæra. Thirdly, a preoccupationwith epistemological and methodological issues. Fourth, the inclusion of thephases of the Moon among its topics, adjacent to the explanation of theeclipses. Both treatises lack the discussion of the miraculous eclipse at the timeof Christ’s passion. Another curious similarity is the deficiency of citations ofthe classic poets in the Sfera del mondo. Perhaps Piccolomini dropped out mostof the quotes that appeared in Sacrobosco’s work because of his campaign forItalian (not Latin) as the preferred language.

6 Galileo’s annotations in Piccolomini’sSfera

In 1886, when Antonio Favaro wrote his work on Galileo’s library, he was notsure about the location of the copy of Piccolomini’s book containing Galileo’sannotations, although he thought that it might belong to the Meucci library.He wrote to Ferdinando Meucci, on January 6th 1890, asking him if he hadthat work or if he knew who owned it. Meucci answered to Favaro on thefollowing day, telling him that he did have the book.4 We have obtained noinformation about further contacts between Favaro and Meucci concerning thiscopy 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 theMuseo Galileo, for providing copies of those two letters (private communication,November 14, 2015).


It is known that Favaro asked Ferdinando Meucci to bequeath some ofhis books to the National Library of Florence.5 It is also known that Meuccihad a copy of Nicolò Tartaglia’s La Nova Scientia with marginal notes byGalileo [Favaro 1886, 268], and that this copy now belongs to the library ofthe Museo Galileo (shelfmark MED0976/01).6 A heavily annotated copy ofthe 1573 edition of Piccolomini’s De la sfera del mondo now belongs to thelibrary of the Museo Galileo (shelfmark MED0719). This specific item wasdisplayed at the 1929 Prima esposizione nazionale di storia della scienza, inFlorence (registration number 6952). Although the staff of the Museo Galileowas not aware that this item was the one containing Galileo’s notes describedby Favaro,7 it is fairly probable that it was. One important evidence is thecomparison between the leather book covers of MED0976/ 01 (Tartaglia’sbook) and MED0719 (Piccolomini’s book). The design of both book covers, inblind stamping, is identical, although they were printed by different publishers.This shows that both belonged to the same library. They also have identicalstamps at the title page : one from the Laboratorio di Fisica in Arcetri ; theother one from the Museo degli Strumenti Antichi di Astronomia e di Fisica,R. Istituto di Studi Superiori, Firenze, with the handwritten number 31 inthe case of Piccolomini’s book, and number 32 for Tartaglia’s book. SinceTartaglia’s book came to the Museo Galileo through Meucci’s library, it ishighly probable that Piccolomini’s book had the same origin.

Favaro suggested that the marginal notes of this copy of Piccolomini’swork could have been made by Galileo, but at that time he had not beenable to examine its handwriting. Following our contact with the library of theMuseo Galileo, Dr Patrizia Ruffo inspected that book and concluded that theannotations might indeed have been made by Galileo.8

A detailed analysis by one of us (Walmir Cardoso) has shown that thecalligraphy of the marginal notes found in the 1573 copy of De la sferadel mondo belonging to the library of the Museo Galileo displays strikingsimilarities 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 postilledel nostro MED0719 e secondo lei potrebbero in effetti essere autografe di Galileo”(Mrs. Alessandra Lenzi, private communication, November 14, 2015). We have beeninformed that Ilaria Poggi and Patrizia Ruffo plan to transcribe those annotationsand to publish them in a future update of the National Edition of the works ofGalileo ; and that Patrizia Ruffo and Michele Camerota began a comparison betweenthose annotations and the Trattato della sfera (Mrs. Alessandra Lenzi, privatecommunication, November 28, 2015). We have no further information about theirwork, which was prompted by our interaction with the librarian of the Museo Galileo.


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 Dele stelle fisse [Favaro 1886, 251–252]. This item belonged to the NationalLibrary of Florence and can now be found in the Museo Galileo (shelfmarkRari 111/02). Let us compare some of those marginal notes to the ones foundin 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 authenticcalligraphy10 in Fig. 1 (“opinione de i pittagorici”) and the notes on theSfera del mondo, Fig. 2 (“opinione de pittago/ rici”) and Fig. 3 (“ragionede pitagorici”).

Galileo’s notes to the preface of De le stelle fisse present three times theword “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 atthe 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 explicitlyrecognized them as authentic [Favaro 1886, 251–252].



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 [Piccolomini1570]

find the same characteristics in the marginal notes on pages 51 and 74 of theSfera del mondo (Figs. 2 and 6). Other significant features are Galileo’s wayof writing “che”, “del”, and letters “q”, “d” and “z”.

These and other traits establish such a strong similarity between Galileo’sauthentic calligraphy at De le stelle fisse and the handwriting of the marginalnotes at Sfera del mondo that they leave no doubt that the later ones werealso written by Galileo, probably around the same time.

The comparison with other autograph writings by Galileo, of the periodfrom 1588 to 1597, strongly suggests that he wrote the marginal notes toPiccolomini’s Sfera del mondo around 1590—that is, about the same timewhen he requested his father to send him his copy of the Sfera. Additionalevidence will be published by us in a forthcoming paper.



7 The influence of Piccolomini’s Sfera onGalileo

Galileo’s marginal annotations to Piccolomini’s Sfera del mondo do not containany discussion, criticism, or comparison with other works, nor any originalmatter. For the most part, Galileo underlined or otherwise marked parts ofthe book that seemed relevant to him, and wrote at the margins some words orsentences that replicate information contained in the printed text itself. Thischaracteristic way of making notes suggests that, at that time, Galileo wasjust beginning to learn the foundations of astronomy and that Piccolomini’swork was one of the first (or the very first) astronomical treatises he everstudied. Therefore, the marginal remarks can show which specific topics andideas called the attention of the young Galileo.

The only marginal note found in the first book of Piccolomini’s treatiseappears at page 23 (chapter 9), where the author discussed the circles on asphere and states that two maximum circles cannot be parallel. There areno other annotations in the first book, most of which contained geometricalprerequisites to the study of astronomy. Galileo had studied Euclid underRicci, before 1590. It seems likely that he thought that the first book of theSfera del mondo did not contain anything new for him. However, chapter 9did contain an important description of the sphere, its diameter, hemisphere,as well as the axis and poles of a rotating sphere, corresponding to the verybeginning of Sacrobosco’s Sphæra. This elucidation is noticeably absent in thebeginning of the Trattato, although it is seldom missing in any other elementaryastronomical 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 originalpresentation of the proofs for the central position of the Earth is different fromSacrobosco’s. They were already contained in the first edition of his work(1540) and later appeared in Clavius and Giuntini’s treatises. The Trattato


also presented a condensed version of those arguments [Galilei 1891, 220]. Atpage 47 we find an annotation of Galileo stressing another argument for thecentral 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 arein diametrically opposite directions. This was not a new contention sincePiccolomini himself ascribed it to Ptolemy and Averroes ; but it was a ratherunusual 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 theEarth—a subject that would later become the center of Galileo’s interests.There are altogether fourteen marginal notes on these pages, disclosing thestrong appeal of this subject for young Galileo. Among the several arguments,Piccolomini discussed the vertical motion of a stone, comparing it to whatsupposedly occurs when the person throwing the rock is moving in a ship[Piccolomini 1573, 52]. The ship argument was rather uncommon and it wasnot reproduced by Clavius. It was annotated by Galileo (“very beautiful reasonto 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 theattention of Galileo, as shown by his marginal notes. Most of them have acorrespondence in the text of the Trattato, as we will disclose in a forthcomingpaper. This fact suggests that Galileo was strongly influenced by Piccolomini’sSfera 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 treatisesof that time—such as Clavius’ book. However, there is no known copy ofClavius’ work with annotations by Galileo. Hence, there is no direct evidencethat Clavius’ commentary on Sacrobosco was read by Galileo and influencedhim 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 compositionof the Trattato might have been very strong, it certainly could not be theonly source used by Galileo. Indeed, there are some features of the Trattatothat did not appear in Piccolomini’s work. The very title Trattato della sferaovvero cosmografia was extraordinary and implied an equivalence betweenastronomy and cosmography that was not acceptable to most authors ofthat time—including Piccolomini. Galileo might have been influenced byOronce Finé’s De mundi sphaera, sive cosmographia or by Francesco Barozzi’sCosmographia, a commented version of Sacrobosco’s Tractatus de sphæra.Barozzi’s Cosmographia is especially relevant since Galileo had a copy of thisbook [Favaro 1886, 260], and some parts of the Trattato may have been inspiredby this work [Martins & Cardoso 2008].


The methodological discussion at the beginning of Galileo’s work is dif-ferent from Sacrobosco’s first chapter, but similar accounts were published byFrancesco Barozzi, Oronce Finé, Francesco Capuano and Christoph Clavius[Martins & Cardoso 2008]. A similar treatment also occurs at the beginning ofPeucer’s work owned by Galileo [Peucer 1573, 1–8]. Perhaps he was familiarwith one of those sources when he wrote the Trattato.

Galileo did certainly study Georg Peurbach’s Theorica planetarum, becausehe taught it at the universities of Pisa and Padua. In Peuerbach’s work, we findsome subjects that were not discussed in Sacrobosco’s Sphæra but are contai-ned in the Trattato, such as the phases and visibility of the Moon [Peurbach1569, 95]. Galileo could also have used instead Barozzi’s Cosmographia for thispart of his work [Barozzi 1585, 288, 295, 300].

The declination of the ecliptic presented in the Sfera del mondo was24° [Piccolomini 1573, 94, 101]—a rather unusual value ; the Trattato used23.5° [Galilei 1891, 230, 233] and therefore this figure was copied from anothersource. 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 expo-sition [Piccolomini 1573, 110–115].

Although we have no information about the time when Galileo studiedClavius’ commentary on Sacrobosco’s Sphæra, its influence upon the Trattatocannot be excluded, since several of its passages exhibit strong similarities toClavius’ work [Martins & Cardoso 2008].

9 Final remarksThis paper analyzed some peculiarities of Galileo’s treatise, comparing it toseveral other vernacular astronomical works of the sixteenth century anddiscussing its likely sources. Contrary to previous claims, we argued thatChristoph Clavius’ commentary on Sacrobosco’s Sphæra cannot be regardedas its only or main influence. A likely inspiration for Galileo’s treatise isAlessandro Piccolomini’s Sfera del mondo, a work that was carefully studiedby Galileo and that anticipated several peculiarities of the Trattato della sfera.Our claim for this influence is strengthened by our analysis of Galileo’s copiousmarginal 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 Sferadel mondo. Therefore, the issue of Galileo’s sources is very complex and hasnot been completely solved.

AcknowledgmentsOne of the authors (Roberto Martins) is grateful for the support receivedfrom the Brazilian National Council for Scientific and Technological Research


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

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Thoughts on Galileo’s Work

Marco M. MassaiPisa University and

Istituto Nazionale di Fisica Nucleare, sezione di Pisa (Italy)

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

Abstract: I wish to analyse certain statements and ideas proposed byGalileo Galilei in his Sidereus Nuncius and will provide non-conventionalinterpretations of them. I will try to show that with logical connection andnew hypothesis, it is possible to make some new possible interpretations ofGalileo’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 errantedell’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 failurethan others. This is because of the quantity and especially the quality of thework developed by many historians of science in an attempt to profoundlygrasp Galileo’s thought. New light has been thrown on the interpretation ofGalileo’s writings only after many decades of seemingly consolidated positionson Galileo’s contribution to the development of science and the basic workof Stillman Drake in the second half of the twentieth century. According to


150 Marco M. Massai

some well-known historiographic interpretations such as the one offered byAlexandre Koyré, Galileo was not a completely experimental scientist.1 Sincethere have been new interpretations of Galileo’s work, it makes sense to look fornew nuances within the scientific contribution. A significant part of Galileo’sinterpreters are not experimental physicists. It is thus reasonable to thinkthat the perspective of an experimental physicist could, to an extent, shednew light on Galileo’s research.

I have spent decades of my working life constructing and using measuringinstruments to determine operating characteristics, collect large amounts ofdata, measure physical quantities and finally to confirm or disprove the laws ofphysics. Because of this, I might perhaps be able to provide a new standpointfrom which to interpret aspects of Galileo’s work.

I shall focus on the Sidereus Nuncius [Starry Messenger], published inVenice on March 12th 1610. Although this work does not present a philo-sophical elaboration of the expounded observations, it offers a rich series ofdiscoveries and insights for astronomy and science. My general idea is thatGalileo is basically an experimental scientist. This idea is connected withthe question of whether Galileo was the inventor of his telescope.2 I believethat he was and this position will be clarified in my paper. In fact, the fewtraces of information that can be found about the “toy” which was patentedin September 1608 by three lenses manufacturers in the town of Middelburg inZeeland would not suggest that it could have been transformed into a powerfulinstrument to investigate the skies. Indeed, the rudimentary tools found inEuropean fairs for almost a year had aroused no curiosity or interest in manyEuropean scientists with the exception of Thomas Harriot. However, eventhough Harriot could claim the primacy of having first pointed a telescopeat the Moon, he only managed to produce and publish a very rough andelementary picture from which no new and useful information can be drawn.Galileo’s intuition in constructing and using (August 1609) the telescopecannot be a pure coincidence.

1. See the classical work [Koyré 1939]. Without claiming to offer an exhaustiveseries of references on the complex and debated relations between theory andexperiments in Galileo, I would also mention: [Agazzi 1994], [Akhutin 1982], [Bucci2015], [Crombie 1981], [De Caro 1993], [Drake 1970, 1985, 1986, 1990], [Dupont1965], [Erlichson 1993, 1994], [Fehér 1982], [Feyerabend 1975], [Finocchiaro 1974,1977, 1980], [Gingerich 1993], [Giusti 1994], [Grigorian 1983], [Gruender 1981], [Maar1993], [MacLachlan 1976], [Naylor 1980, 1990], [Piccolino & Wade 2014], [Pizzorno1976], [Prudovsky 1989], [Segre 1980], [Shea 1977], [Takahashi 1993a,b], [Wisan 1981].

2. The literature on Sidereus Nuncius, the role of the telescope withinGalileo’s work, the lunar observations and the Jupiter satellites are simply huge.Interpretations concerning the Sidereus are almost as abundant as the literature. Imention the useful text by [Reeves 2014], in which the whole literature concerningthe Sidereus Nuncius from 2000 to 2011 is referred to. I shall only quote here theliterature from which I have drawn some inspiration—without fully agreeing—for thispaper: [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 anecessary element to understand his success with the telescope. His greattechnical experience alone enabled him to connect the construction andfunctioning of a machine with the theoretical principles underpinning thatvery functioning.

Therefore, Galileo was able to link the experimental dimension with thetheoretical—logical and mathematical—aspects. On the other hand, Galileohimself stated in the initial pages that he was not the original inventor ofthe telescope, referring instead to a “Flemish”.3 However, soon after, heindicates a specific way to calibrate the instrument he built and invites thereader to try this. However he also suggests they take care and use great skillotherwise failure is assured as he was to repeat later before the description ofthe 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 waswritten almost spontaneously by Galileo and thus make a different interpreta-tion to the widely shared and consolidated analyses. I will quote and describe

3. “[...] rumor ad aures nostras increpuit, fuisse a quodam Belga Perspicillumelaboratum [...]” [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 withoutfollowing the order emerging from the pages of the Sidereus Nuncius. In thismanner, it will introduce some interpretative hypotheses to fully grasp themeaning of some of Galileo’s statements which, at first glance, might appearrather odd. I am aware what I am claiming verges on the provocative but Iwill provide evidence to back up my view.

3 A methodological question

A first, purely methodological question, emerges through analysis of Galileo’sprogress, starting with the first observations of “the body of the Moon”, wherehe identifies bright spots in the dark and dark shadows on the sunlit parts. Hecarried out nightly observations and obtained both still images and somethingwhich could be interpreted almost as a living picture that he seeks to analyze.5This is a crucial first step: Galileo builds up a three-dimensional representationin movement of what he can see and observe, each time more carefully. Hebuilds a model that he refines night by night as a consequence of his newobservations. These models are confirmed by a long series of observations andalso by the logic with which Galileo strongly connects those observations. Heis aware that his description of the Moon is completely different from thataccepted by the astronomers and supported by the Church. In spite of this, heremains firm in his opinion that the Moon is similar to the Earth in all respectsand is not a body made out of ethereal matter—the first of the sovralunareworld where perfection reigns. And this is in fact a truly great revolution inthought. However Galileo does not just end with a description, albeit far morerefined than previous efforts. He goes beyond this and determines the heightof 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 bydescriptions. Measuring involves transforming an “object’s” characteristic intonumbers. Initially an object may only exist in thought and with measurementit 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 aseries of facts related to external reality. Galileo performed the measurementof the height of a mountain on the Moon and found it was 4 Italian miles inheight which was a very surprising finding.

5. “[...] sed, quod maiorem infert admirationem, permultae lucidae cuspides intratenebrosam Lunae partem [...] quae, paulatim, aliqua interiecta mora, magnitudineet lumine augentur [...]” citep[III, 64]Galileo1610mass.


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

However Galileo needed to go further because although he was proposing asimple but effective three-dimensional, geometric model, this was not easy toaccept. In fact, the result of his calculation is a value that surprises and maybeeven disconcerts because, as he explicitly points out, it goes far beyond thevalue estimated at that time for the height of the highest European mountains.However, we must remember that the height of a mountain was measured fromthe valley bottom and this means the otherwise inexplicable discrepancy withthe actually measured value is not actually surprising. These surprisinglyhigh mountains could tarnish the validity of Galileo’s hypothesis which hadupset popular opinion. After centuries of stable belief, it was well establishedthat the lunar surface was smooth, although the question of the nature of theMoon’s many, vast and irregular spots remained to be solved and were indeedthe subject of many inferences. Nonetheless, the clarity and the simplicity ofGalileo’s reasoning supported a measurement which made the most amazinghypothesis reality.

5 The lunar atmosphere

Galileo wrote many pages on the description of the Moon. Indeed, he wasaware 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 morerecommendation on the quality of the instrument to anyone who wanted toconfirm his observations. Immediately afterwards, he deals with a questionwhich could also affect his explanation of the nature of the Moon namely whydo 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 criticalremarks about his work in the future. However, such an objection has alsothe aim of reinforcing his own work by showing the reader that he is readyto take possible criticism of his statements into account. Nevertheless there issomething more. In fact, Galileo gives two explanations why it is not possibleto distinguish the sawtooth mountains’ profile on the Moon’s edge.

The first example consists in the vision of several ridges of mountainsfollowing each other at increasing distances. This view is also typical onthe Earth, which is a confirmation that the nature of the Moon is the sameas Earth’s . This similarity is now taken for granted. The yokes appearalmost continuous because the alternation of peaks and valleys of a chain arepositioned at random with respect to the subsequent chain. The mountainson the Earth thus explain mountains on the Moon.

The second explanation is more picturesque but perhaps less conclusivebecause the scale factor is completely different and because Galileo nowcompares the mountains’ dimensions with those of sea waves. Galileo usesa beautiful image of a stormy sea with rows of waves advancing parallel toeach other but which cannot be perceived individually. Therefore, we seea substantially continuous horizon. These explanations appear logical andstrengthen the interpretation of the lunar world as a double of the terrestrialworld. Also, they are widely sufficient to justify the observational data, orrather in this case the lack thereof. However, Galileo is not satisfied with thisexplanation and looks for another. This requires a further hypothesis—thepresence 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 plusquamduplo maiora videantur” [Galilei 1610, III, 95].


thus once again providing a strong analogy between the Moon and the Earth.7The lunar atmosphere allows Galileo to justify the lack of definition observedon the edge of the Moon which prevents us from perceiving and distinguishingthe individual peaks.

Galileo supports this third explanation with one of the few drawings inSidereus Nuncius. This is not an accurate representation of what he sees buta model. In this picture, a circular ring around the rim of the Moon appearsclearly. In other words, Galileo aims to provide an unambiguous description toavoid any misunderstanding. At this point in his work, Galileo is in a hurry todescribe his most revolutionary discovery—the Medicean Planets, the last andmost important subject of the whole book. Therefore, while this additionalexplanation may seem superfluous and redundant based on the introductionof a new idea, it could be interpreted as a further similarity between the Moonand the Earth. Perhaps another reason led Galileo to go much further than amere description of what he saw and also beyond the reference that the readercan share. This is the subject of a hypothesis that I shall propose but first itis 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 fourMedicean Planets. The description is enriched by dozens of observationscollected night after night with meticulous precision and reported in thetext with exact drawings, one for each day. Much earlier in the book whenGalileo reaches the end of his observations, he introduces the hypothesisthat an atmosphere surrounds another celestial body—Jupiter in this case(see above note 6).

Galileo carefully observes the relative positions of the four satellites whosemotion is easily perceivable from one night to the next and measures this withaccuracy. At the same time, Galileo also tries to measure the intensity ofthe satellites’ light and notes this with care. However, it must be said thatthe relative position and the angular distance of a star or planet is an easilymeasurable quantity and therefore not ambiguous. Instead, it is difficult tomeasure the light intensity of an object in the black night sky, particularly ifthis object is observed very close to another brighter body. This is exactly thecase when the satellites of Jupiter are observed by Galileo shortly before orafter their occultation by the far larger body of the planet.

Galileo has already encountered a similar problem, i.e., the problem ofevaluating the magnitudes of a celestial object through his telescope. He dealswith it some pages earlier when describing the difficulty of measuring the

7. “Huic rationi altera subnecti potest, quod nempe circa lunare corpus est, veluticirca 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 describeshow the instrument behaves very strangely and shows different magnificationsdepending on whether the observed object is a planet or a star. In fact, Galileocould not know the physical explanation of the optical aberration phenomenonwhich is usually associated with the use of an optical instrument. Therefore,he is hesitant to give the description of a mechanism according to which theinstrument is apparently able to enlarge the observed objects in different ways,i.e., with different magnifications. For the same reason, he could not give anyjustification 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 understandingof what “you really see” in the ocular lens which is related to a complexmechanism that would be called “physiology of vision” in the future. Atthis point, Galileo did not have the time to pursue this matter because heurgently needed to make public his numerous observations of all visible objectsin the sky of Padua as soon as possible. In the winter of 1610, despite notbeing an astronomer he was about to totally and definitively revolutionizethe image of the sky and also suggest the need to overhaul the model ofthe Cosmos through his rich innovations. Indeed, the Aristotelian-Ptolemaicmodel endorsed by the Church encountered a great deal of difficulty in facingup to the whole complex set of new ideas discovered by Galileo. This dependedon the geometric complexity required to take the new, observational data intoaccount and also on the weakness of some of the principles on which theAristotelian-Ptolemaic system was based. These principles were now beingdemolished. First of all, the skies’ immutability (a secular assumption) wasa condition which appeared a necessity because the sky was considered adivine work. The Copernican model was published almost 70 years earlierand was still seen only as an original, mathematical exercise to provide newephemerides. Nevertheless, from a physical and mathematical point of view thediscoveries of Jupiter’s satellites was problematic for both the Ptolemaic andCopernican systems (at least in the version expounded by Copernicus wherethe system was based on circular motion). Though the Astronomia Nova werepublished in 1609, when Galileo published the Sidereus, Kepler’s work hadlittle influence on astronomers. However, while there were several difficultieswith the Copernican system, the situation for its Ptolemaic counterpart wasundoubtedly even worse. Thus Copernicus’s theory represented a very newhorizon for knowledge of the sky.

A further remarkable element is explained in the last pages of the SidereusNuncius where Galileo writes about the variable intensity of the light ofJupiter’s satellites when they are close to the planet. This observation mayindeed appear superfluous when compared with the astonishing announcementof the existence of four new planets. Moreover, Galileo does not providemany details regarding this problem although he was precise and meticulousin describing these observations. Notwithstanding, these references appearsignificant. Galileo advances a hypothesis that can explain this apparent


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 meanthe could explain the weakening of the light by the increased distance ofthe satellite. Galileo certainly knew from his previous experience that lightintensity decreases with increasing distance from the source.

Whatever the function which connects light intensity and the distance ofthe light source, it seemed implausible that the decrease in light intensity wasas strong as Galileo’s observation suggested. Galileo saw that the brightnessof 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 fromJupiter, its center of motion and therefore that the orbit must be greatlyelliptical in relation to the weakening of the light output observed and recordedby Galileo. The logic that supports this argument may seem compelling.However, Galileo does not bother to justify the images described althoughthe satellite’s brightness does not change when they depart even slightly fromJupiter. Anyway, this behaviour of the phenomenon is not compatible withthe previous hypothesis which seems artificial. However, this hypothesis is ascientific-literal artifice used by Galileo to lead up to a second explanation.In this explanation, he returns strongly to the hypothesis that an atmospherecould surround a celestial body—Jupiter in this case. The presence of a denseshell would weaken the light absorbing a large proportion of the emission fromsatellites.8 With this explanation, Galileo seeks to make the results of hisobservations compatible with a model—some demonstrations, which he usesto validate the sensitive experience.

7 A hypothesis

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

8. “Non modo Tellurem, sed etiam Lunam, suum habere vaporosum orbemcircumfusum, tum ex his quae supra diximus [...]: at idem quoque de reliquis Planetisferre iudicium congrue possumus; adeo ut etiam Iovem densiorem reliquo aethereponere 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 shownby Galileo and if they are similar to the Earth, this obstacle is overcome byimagining an Earth moving around the Sun. Therefore, the movement of theEarth around its axis and around the Sun does not imply the atmosphere isstripped away. Of course, such a deduction is not evident or explicit in Galileo’swords. Therefore any such proposal should be maintained in the field of theinferences which can be made by speculating on Galileo’s words to reconstructhis possible thought. However, if this hypothesis is correct, why was Galileonot more explicit? This is impossible to answer. Again, it should be recalledthat 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 Galileowas to add many other discoveries to those announced in the Sidereus Nunciusin the 1610-1616 period. All these observations were to be as fundamental asthe previous ones. Galileo discovers Saturn to have a strange shape like asmall olive. He first recognizes the sunspots to be on the Sun’s surface andnot far from the Sun.

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

This, in fact, was the most conclusive observation in support of the Polishastronomer’s revolutionary idea.


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 cornerstoneof 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 pointinghis telescope towards the nebulae, or nebulous stars. Immediately he proposesa new radical change in the interpretation of those indistinct spots that hadlong been considered one of the most mysterious objects in the sky whenviewed with the naked eye. Philosophers, priests, astronomers had providedseveral proposals to explain their nature which were mostly mythologicallybased. Also the analogy with the Milky Way had been used to find someexplanations but this was done without “sensible experiments and necessarydemonstrations”.9

Galileo does not hide his surprise and also his satisfaction in revealingwhat he sees in the ocular lens of his telescope. This is, once again, a realrevolution. In fact, the interpretation of these objects changes dramaticallybecause Galileo reveals that they are composed of myriad stars which are soweak and apparently mutually close that they cannot be distinguished by thehuman eye. Thus one has the perception that they are made of a continuousmaterial. This was the “essence” of the nebulous stars before Galileo discoveredand announced their discrete nature.10

Certainly, this discovery by Galileo—this invention, because he based thisidea on observations—is not among the most remarkable if compared with theothers expounded in the Sidereus Nuncius.

In fact, the other discoveries represent the foundations of a completely newscheme in the description of the sky. This scheme supports the heliocentricmodel. But another revolution which would have to wait several more centuriesto be accepted was smouldering under the ashes. It was proposed more thantwo thousand years ago and it concerns the real composition of nature ofmatter around us. Does matter have a continuous structure or is it composedby the atoms that Democritus and Leucippus imagined in a remote cornerof Greece? In those years the implications of the atomistic hypothesis wererather hazardous. Asking yourself this question at that period had a dangerousimplication 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 usquediem 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 enimGALAXIA nihil est aliud, quam innumerarum Stellarum coacervatim consitarumcongeries” [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 thereforeimagine that Galileo, who was always ready to make daring and surprisinglogical connections and analogies, had found this idea attractive. After theMilky 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 inthis work by Galileo have been based merely on what Galileo himself explicitlywrote. Certainly, they are read today more than 400 years later at the dawnof the twenty-first century. Obviously, the meanings that we now assign tothe words of Galileo, have in some cases been completely revolutionized bythe historical and scientific background. Profound differences are also evidentfrom an epistemological point of view.

However, many aspects of Galileo’s thought are still alive and his teachingis still of great significance and relevance particularly his reference to sensibleexperiences and necessary demonstrations. Galileo left his revolutionarymessage relying upon experiments and observations, rather than upon the


theoretical, dialectic sophistry which characterizes the speculation of severalother philosophers. As I have tried to show, it is not impossible to search forand maybe even find different interpretations from the classical one which isaccepted by history of science. It is very difficult to determine whether theyare correct and really match with the thought of Galileo and I think no onecan state this with reasonable certainty. However nor perhaps can this becompletely denied.

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

Acknowledgments

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

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Galileo’s La bilancetta:The First Draft and Later Additions

Annibale MottanaRoma Tre University (Italy)

Résumé : Cette étude montrera que la confirmation expérimentale des conjec-tures constituait chez Galilée l’objet d’un souci constant, et ce dès le débutde son activité scientifique. À l’appui de cette thèse, je citerai ses premierstravaux expérimentaux et montrerai avec quelle profondeur il était capabled’analyser par lui-même ses résultats. Je démontrerai ensuite qu’il a discuté deses 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 sesintuitions à leur état final.

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

1 Introduction

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


166 Annibale Mottana

experimental confirmation of his conjectural ideas was a constant concern forGalileo from the very beginning of his scientific activity. In order to supportmy view, I shall quote his early experimental work and first show the extent towhich he was able to analyse his results in depth by himself. Then I shall giveevidence that he discussed his results with other scientists to help confirm hisinitial intuitive ideas. He consulted with them and their advice helped him tocome to his final conclusion regarding his insight.

2 Preliminaries

Galileo’s approach to philosophia naturalis1 was the following: he strongly be-lieved that Nature is the work of God and only needed a correct interpretativeapproach to be fully understood. Moreover, he thought that the scholars hecalled “filosofi in libris” [Galilei 1890-1909, XI, 11–12, hereafter OG], did nothave the correct approach in basing their understanding on reading Aristotle’sscientific books.2 Rather, he believed that the correct interpretation of anatural fact should stem from direct, close observation and from well-plannedand carefully performed experiments to verify it. Finally, he believed that atrue scientist should be able to work out the collected evidence “in linguamatematica, e i caratteri son triangoli, cerchi ed altre figure geometriche,senza i quali mezzi è impossibile a intenderne umanamente parola; senzaquesti è un aggirarsi vanamente per un oscuro laberinto”3 [OG, VI, 219]. Infact, Galileo drew the best from the books of Euclid, Archimedes and otherancient mathematicians,4 whose greatness he could fully appreciate. However,he constantly favoured exploring “questo grandissimo libro, che essa naturacontinuamente tiene aperto innanzi a quelli che hanno occhi nella fronte e nelcervello”5 [OG, XI, 109]. In other words, he pushed himself to engineer newexperiments that mimicked and explained what he could directly observe in

1. At that time, one would call philosophia naturalis (= natural philosophy) theentire scientific knowledge of the Earth because nobody had yet had the idea thatmaking artificial materials was possible. God had created everything from chaos.

2. These include both Aristotle’s original or attributed Greek works and also allinterpretations by his followers, spanning from the peripatetic school to the scholasticone which takes into due consideration the Islamic interpretations available in Latintranslation. Galileo was careful not to mention the Bible in this context but couldnot avoid involving it implicitly.

3. “In the language of mathematics, the characters of which are triangles, circlesand other geometrical figures; without these it is impossible for a man to understanda word; without them it is like roaming in vain through a dark maze.” All translationsfrom Italian are by me.

4. He knew only those translated into Latin and Italian, as well as their medievaland 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 ofthose 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 definiteseparation existed between the simple observation of natural events throughsight and the understanding thereof conceived in and by the brain. Valuableresults could only be achieved when scientific observation and technologicalexperience were directly combined. He began practicing science in this wayas early as in his youth [Fredette 1972], [Drake 1978, 1986], around the timewhen he started his training at Florence’s newly founded Accademia delleArti del Disegno, an institution whose purpose was to link science concepts toengineering practices. There, he met some of the best mathematicians, naturalscientists and technologists of late Italian Renaissance and became confidentin considering nature as just one of the two means chosen by God to displayhis immense power. The result of this form of training was the developmentof the approach that has only recently been named “technoscience” [Gorokhov2015]. In Florence at the age of 22, Galileo gave the first demonstration of hisnew ideas on how to practice science, i.e., his first demonstration of what wewould now call “technoscience”. He wrote a short essay on a new method ofusing scales that is now entitled La bilancetta (= The little balance) [OG, I,215–220],6 and used this instrument to make measurements of selected metalsand gemstones.

This whole way of thinking was already rooted in Galileo’s mind longbefore he made it clear in his famous “theological letters”, i.e., the two lettershe 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 knownto man using two equal sources of evidence—the Bible and nature. TheBible had to be interpreted by a human language and obviously, as a piousCatholic, he meant Latin. This often makes God’s words difficult to graspand easily misunderstood. The evidence inherent in nature must derive fromdata obtained through laborious observations and planned experiments whichneed to be supported by a skilled use of mathematics. Therefore, the pathtowards understanding which proceeds through science can never fail because

6. According to Vincenzo Viviani, Galileo’s last pupil and the collector of mostif 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 aletter by Viviani to Cardinal Leopoldo de’ Medici on April 29, 1654. He supportshis challenge [Drake 1986, 437–439] by citing the watermark on a paper sheet nearthe end of the unfinished dialogue De motu, which is bound in the same volume justbefore La bilancetta. This watermark could show that Galileo wrote the dialogue onsheets acquired while teaching at Siena during the 1586-1587 academic year. In sucha dialogue, Galileo refers to have recorded previously two sets of measurements ofgold-silver alloys by the hydrostatical method. Drake infers from this that Galileowrote La bilancetta later than those measurements namely when he had left Siena inlate spring 1587 and moved back to Florence. Notably, Drake contradicts himself inthe table that follows at p. 440: there, he lists La bilancetta as written at Florencein 1586 and the dialogue De motu at Siena in 1586-1587.


it consists of logical figures. Galileo’s juvenile writings have fortunately beenpreserved in the Florence National Library and they provide excellent proofon how early and firmly he adopted this belief [Fredette 1972], [Drake 1978,6–7], [Camerota 2004, 45–48], [Gorelik 2012], etc. In particular, La bilancettais the first of Galileo’s writings in Italian, the language in which could bestexpress 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 veryinnovative text because it alone enabled him to open up a new approach toupgrade hydrostatics and make it a method for studying solid matter. Hisinnovation occurred almost two millennia after Archimedes of Syracuse hadlaid the fundamentals of that branch of Physics.7 Therefore, it was a realmisfortune for the development of philosophy of science that most historiansunderrated La bilancetta for a variety of reasons:

1. It is a minor contribution to science, when compared to the laternovelties on astronomy and general physics that made Galileo foremostamong contemporary scientists.

2. The fact that it is in Italian could perhaps suggest that the authorhimself did not consider it worthy of particular attention. Indeed, atthat time, high-quality scientific papers were written in Latin to reachthe largest possible audience within the scholarly community. Galileowas an exception but he too often agreed to do so.

3. It is the work of a youth who was not yet completely trained. Galileohad written it after leaving Pisa University without a degree and beforemeeting Guidobaldo del Monte who he could have consulted on themathematical aspects of the subject.

4. It contains the description of a long-known instrument, which he usedwithout introducing any special modifications. Even the weighingsystem is the usual one. He just operated using a sequence of stepswhich 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 longtradition. He supported his evaluation by referring to what is reported in the pseudo-Priscian (4th-5th century), in Mappae clavicula (an anonymous treatise he believedto 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 reposesur une tradition certaine [...] transmise au Moyen Âge depuis le temps des Grecset des Romains” [Berthelot 1891, 481] (= the use of hydrostatical balance for theanalysis of gold and silver alloys is based on a solid tradition [...] that Greeks andRomans had transmitted to the Middle Ages).


5. Finally and most importantly for science historians who consider exper-imental results a priority, the essay apparently does not contain datawhich supports its conclusions.

The last prejudiced view is definitively wrong. The table of weights in airand water which Galileo used did exist but it was not known to most earlyscience historians [Berthelot 1891]8 because the autographs were recognizedand 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 justafter Galileo’s death. It appeared first as an extensive quotation hidden in abook published by another author [Hodierna 1644, cc. 2A1–8A]. However itwas well known to Galileo’s followers and was sent to press under his namesome ten years after the editio princeps in the first print of Galileo’s selectedworks which appeared at Bologna in 1656. Indeed, the title La bilancetta(although circulating before as the nickname for the apparatus) was chosenfor the essay by Carlo Manolessi, the curator of the Bologna edition. Thefact that Galileo9 had neglected to give a title to the essay would possiblyindicate the minor importance he himself accorded it with respect to otherslater but he never refrained from distributing copies of it to his visitorswhile confined at Arcetri.

After having considered all those objections, I reached the conclusion thatthe autograph and all the contemporaneous copies of La bilancetta were worthdirect and careful analysis. I did this to both edit the essay in a better formwhich corresponds to the original text Galileo had in mind10 and also to betterunderstand the conceptual framework developed by the young Galileo. Indeed,all handwritten versions are worth studying closely in all their details right upto their last words since the essay chronologically precedes the full developmentof 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 NazionaleCentrale 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 RichardDavies, who assembled all data available in literature from Francis Bacon’s time to hisown [Davies 1748]. It was also unknown to Mathurin-Jacques Brisson, who measuredthe specific gravity of over 5000 materials [Brisson 1787].

9. As Hodierna also did in the editio princeps. However, in the long title givenby Hodierna, he makes it clear that Galileo himself had called “bilancetta” the newinstrument he had invented [Hodierna 1644, 1].10. Which I did, by producing an updated diplomatic edition of all the texts

mentioned here [Mottana 2016].


I intend to analyse in detail. The best-known one is “manuscript G”, whichconsists of a single, large sheet of paper (c. 55), 281× 205 mm in size, denselywritten on both sides. It shows a small, rather crude drawing on top of theverso depicting the essential features of a balance or scales. Shortly after thepages in the volume were numbered in pencil by Atto Vannucci, the BNCFlibrarian from 1862 onwards [Procissi 1959, 119], a clever binder made thislone sheet conspicuous because he enclosed it between two cardboard sheetsfor protection. Then, some years later, Antonio Favaro ascertained that thehandwriting of G was Galileo’s own [OG, I, 234] and the national editionwhich he later managed took this for granted. All subsequent scholars of thework 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. Thesesheets are definitely not in Galileo’s handwriting [OG, I, 212]. Isidoro delLungo, Gilberto Govi and Umberto Marchesini, the three philologists whohelped Favaro edit the first volume of Edizione Nazionale—the time-table ofall surviving writings by Galileo and his relations [Castagnetti & Camerota2001, 359–361]—believed A to be coeval to G. Moreover, they also believedthat A was the archetype for the copies Galileo distributed to his followers andthis despite another copy (C) existing which might appear more authoritativebecause it is written by Vincenzo Viviani, Galileo’s last pupil and the firstcareful collector of most of Galileo’s writings and books.

Copy A is followed by a numbered but blank sheet (c. 59) and then anotherscript closely related to the same subject, titled

Tavola delle proporzioni delle gravità in specie de’ i metalli e dellegioie pesate in aria, et in aqua. (cc. 60r-62r)

This kind of table is inconspicuous within volume 45 because it is just at theend of it and looks like a bunch of sheets containing random numbers whichcould have been used by the binder to buffer the binding. Favaro deservesfurther credit for recognizing the relationships between their content and thetext of La bilancetta [Favaro 1879]. Later, he published the Tavola... followingthe philologically reconstructed version of the main essay [OG, I, 224–227].However, nobody took much notice of this list of 39 metal and mineral namesadded with numbers written in the fractional form used at Galileo’s time. Onlya German doctorate student, Heinrich Bauerreiß, mentioned the table in hisdissertation [Bauerreiß 1914, 62–64] and realized that he could calculate thespecific 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 wasnot carried out for another century. I made this interpretation both forgemstones (“gioie”) and for metals (“metallic”), with reference to the dataprinted in the Edizione Nazionale [Mottana 2014, 24–31], [Mottana 2014-2015, 187–241]. However, I later realized [Mottana 2016] that the original,


handwritten Tavola, like the current text of La bilancetta itself, consists oftwo parts. The first part is a partial clean copy written in smooth-running,accurate but unknown handwriting. It begins with the afore-mentioned titleand lists eight measurements covering recto and verso of one sheet (c. 60).The following two sheets are an untitled sequence of words and numbersscribbled in Galileo’s own handwriting (cc. 61r-62r) and list the weights ofseveral metals and gemstones measured in air and water. On these sheets,somebody (probably the unknown copyist) ticked off the eight items listed inthe clean copy, i.e., on c. 60. Thus, there is no doubt that this second sectionis Galileo’s original working draft and the first section of the Tavola, includingthe long title, is a partial clean copy of it made at an undefined moment butcertainly later than the experimental session.11

To conclude, then, given the unequivocal evidence of what is signed andwhat is not, we can distinguish two time-sections in the composition of Labilancetta. There was an early, rough section (G: comprising cc. 55, 61 and62) which was handwritten by the young Galileo. This was followed by a cleancopy in unknown handwriting but which contains interesting additions thatwe 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 abig problem. Actually, G shows numerous word abbreviations of all types,including tachygraphic symbols and special graphs inherited from the MiddleAges (e.g., o, a, u for on, an, un; ß for per; ßeè for perché; duq 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. Galileowas writing for himself, putting his ideas down with no restraint whatsoeveras he did not intend to divulge what he was devising. Ideas were poppingup freely in his mind and he put them down on paper just as fast as theycame. Therefore, he used the sheet of paper that he had available in the mostcondensed way he could—cutting words, mostly using standard abbreviations,even omitting vowels, sometimes. At the end, to show that he had written allthat he had in mind at that moment, he shortened a word he had frequentlyspelled out before either in full (argento = silver) or cut in the usual way(arg.). He used a semicolon (arg:). To my understanding, this very unusualabbreviation—Galileo uses it nowhere else—is a clear indication that he hadreached the end of the topic or of his reasoning thereon and of the entireessay 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 orbasilisk), which is “undoubtedly Florentine”. Based on this, he specifies “a datingof 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 islocated 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 writinghis theoretical essay, which shows all evidence of being an ex abrupto script.


several lines and consequently Favaro believed that G is a surviving fragmentfrom 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 FlorenceAcademia. Cosimo I de’ Medici created this school to supply his newly grantedDuchy with technically trained architects, engineers, gunners and so forth. Theteachers were some of Florence’s best technicians and scientists. Furthermore,occasionally, certain scholars who were qualified at university level (but whohad not continued their career in universities) taught at the school. One ofthem was Ostilio Ricci, a friend of Galileo’s father Vincenzo, who was theman who had enabled Galileo to enter the academic world. Ricci was not auniversity man but was nonetheless among the best teachers of mathematicsof his time. He himself had been the pupil of the very best teacher—NiccolòFontana called Tartaglia (= the Stutterer), another exceptionally gifted manwho had been rejected by learned society because of his speech defect. Riccigladly advised Galileo about mathematical problems.

Another academic who was even better disposed toward Galileo wasGuidobaldo del Monte, a nobleman who had studied mathematics at Urbinoin the private advanced school set up there by Federico Commandino, thefamous humanist and mathematician who had edited the Latin translation ofArchimedes’ works. Guidobaldo had not continued with a university careerbecause it was inadequate to his high social rank and also because he preferredthe practical application of mathematics (particularly ballistics) to developingabstract concepts. He was a technologist with strong mathematical basis, a“technoscientist”, if we use a modern word [Gorokhov 2015]. He often livedin Florence because he was the Chief Officer for Fortifications of the entireDuchy. Guidobaldo became acquainted with the young Galileo, found himto be a clever pupil, invited him to his mansion near Urbino and exchangedletters and discussed problems with him for several years.

During the 1587-1592 period, Guidobaldo was noting down theoremsand ideas at random in a bulky manuscript titled Meditatiunculae de rebusmathematicis (= Small meditations on mathematical matters).12 Like mostlearned 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 Latinevaluation of how the lever rule applies to a balance (cc. 232–234) [Tassora2001, 540–543]. This sentence is written in a smaller character and withdifferent 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 herunpublished dissertation which is in open access online [Tassora 2001].


Potrà forse accadere, che np, pl uenghino linee tanto piccole, chenon si possi comodamente trouar la loro proportione. Et però sipotrà 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 medesimaproportione, che ha np à pl. Poi si potrà pigliar un bastone diritto,et avvolgergli atorno una corda di citara ben sottile, e che le spiresi tocchino l’una l’altra, che per esser pari, si potrà ueder quantespire siano np pl, ouero rs st. E così, quanto comporta l’attopratico, si hauera in numerj la proportion dell’oro e dell’argentodella magnitudine di ac.13 [Tassora 2001, 543]

The first lines of this passage are a rigorous mathematical demonstrationexplaining a figure not reported here which Guidobaldo had carefully drawnusing square and rule on the top part of the sheet. The final lines are of nomathematical interest and are not particularly rigorous either. Guidobaldodescribes a practical method to make rapid but rough measurements of theratio between gold and silver in a mix. This script is valuable however becauseit refers to the same method as that set out in G, the first draft handwrittenby Galileo. Thus, it was Galileo who had the idea of making a wire coil aroundone arm of the balance and Guidobaldo simply corroborated the idea in hisstatement using more or less the same words as Galileo. However as a puremathematician, he does not bother with technical peculiarities such as usingtwo different kinds of metal wires to speed counting up as Galileo suggestedin the last lines of his manuscript G.

The sentence mentioned above by Guidobaldo is of dual historical impor-tance.

1. It corroborates Viviani’s [Viviani 1674] idea of dating La bilancettato 1586, a year that a number of critics [Crombie 1975, 167], [Settle1983] have suggested to be too early in the development of Galileo’sscientific thought. In fact, we can be sure that Guidobaldo wrote hisMeditatiunculae after 1587, when he released his main work commentingon Archimedes’ hydrostatic essays for printing and received the hand-written notes of his teacher Commandino on Pappus’ work [Tassora2001, 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 easyto find their proportion. One could then project from a point q anywhere and drawthe lines qnr, qps, qlt, and draw the line rst parallel to npl, which will have the sameproportion rs to st, which np to pl has. Then, one could take a straight stick, and rollit up with a very thin guitar string so that the coils touch one other, and because theyare equal one shall see how many coils are np pl and how many rs st. In such a way, inthe short time taken by the operation, one will have in numbers, what the proportionof silver and gold will be in the bulk of ac”. My translation should compare to whatCecil Stanley Smith wrote at page 140 of the Appendix concerning the structure ofLa bilancetta [Fermi & Bernardini 1961, 133–141]. Smith’s translation, which is theonly translation available in English, covers the entire passage as given by the officialedition [OG, I, 217] which in turn mostly derives from copy A.


on prospectics [Tassora 2001, 42–44]. By contrast, for La bilancetta, theprevious ante quem documented date is 1641, when Galileo wrote a letterpresenting a copy of it among his works donated to the Polish MonsignorStanislao Pudlowski (rectius: Stanisław Pudłowski) [Burattini 1675, 3]14

[Tancon 2006, 112–115]. Alternatively and by inference only, otherscholars have suggested the composition of La bilancetta to be around1611-1612 when Galileo argued bitterly with the Florentine Aristoteliansand used similar arguments to contrast them effectively in his Discorsoal 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 Salvo2010, 83–153], is the cornerstone of Galileo’s conceptions on hydro-statics and takes the slim book on the little balance into full accountalthough it never mentions the experimental data from the Tavola.Hence, the text of La bilancetta should predate the Discorso [...] bysome years. By inference, it also suggests the possible composition of Gbefore 1587-1592.

2. It shows that Guidobaldo, one of the most brilliant mathematical mindsof his time, had taken the young Galileo’s problem very seriously. Heprovided the correct mathematical solution [Tassora 2001, 543] and alsoactually adopted the instrumental set up that would make the littlebalance a rapid and effective tool to measure the proportion of gold tosilver in a mix even though he does not mention the other practicaladvice contained in the last lines of G.

That is enough regarding the main text. However, copy A, from which mostprinted editions of La bilancetta derive, contains the description of anothertechnical operation which is most practical to speed up counting but alsoso simple as not to appear even worth mentioning by Guidobaldo. Galileomay have considered it inappropriate to submit it to him for his support asthere is nothing mathematical in this operation. It consists of rubbing a thin“stiletto” (= stylet) above the coils to count them by the sounds made becauseof the contrast between metals [OG, I, 220 ll. 4–18]. We should consider ita bold suggestion which probably derived from Galileo’s deepest memories ofteaching from his father Vincenzo, a musician, unless Vincenzo himself directlysuggested it (he was still alive at that time).

Galileo’s good ear for music made him an excellent lute player [Viviani1674, 2], [OG, XIX, 601] and therefore we cannot wonder if he proposed a sonicmethod 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 qualeper teorica e pratica con l’aiuto dell’acqua, non solo si conosce le frodi dell’oro edegl’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 Bibliothèquenationale de France (Mss. Ital. n. 448, Suppl. fr. 496 ).


it is legitimate to wonder when he added this operation to G, as it first occursin copy A and in all other surviving handwritten copies. If Vincenzo suggestedit to him, this would have occurred before July 2, 1591 when Galileo’s fatherdied in Florence. Otherwise, we could consider any time before 1612 whenGalileo had to reconsider all his ideas on hydrostatics because of the quarrelhe had with the Florentine Aristotelian scientists [De Salvo 2010].

There were also at least three instances when he had the chance to rethinkhis early work on hydrostatics.

In 1597, while a professor in Padua, Galileo devised a proportionalcompass, called “compasso geometrico et militare” (= geometric and militarycompass), which made it possible to rapidly perform all calculations neededby practical stoneworkers, land-surveyors, moneylenders, gunners, etc. [Drake1977, 35–54], [Camerota 2004, 124–129], [Bertoloni Meli 2004, 166–169]. Hehad a number of such compasses built by an artisan, Marcantonio Mazzoleni,and made good profit by selling them. When the popularity of the instrumentgrew to the point that he could no longer personally teach all buyers how touse it, in August 1606 he wrote a handbook titled Le operazioni del compassogeometrico e militare [OG, II, 363–424], had it printed privately and starteddistributing it together with each new compass he sold. Among the sevenproportional scales drawn onto the compass made up of two movable brassrulers (four on the recto and three on the verso), the outermost one concernsthe proportionality existing between pairs of seven materials (gold, lead, silver,copper, iron, tin, marble and stone), each of them marked as a point on bothrulers. Chapters XXI–XXIV of the handbook explain how to operate thecompass. With the two rulers set at any opening, the intervals between anycorrespondingly marked pair of points give the size of spheres (or of othersolid bodies such as cylinders and cubes) which are similar to each anotherand equal in weight. In other words, the compass solves the calculation thatarises when comparing materials of different specific gravity, irrespective ofwhether this is determined hydrostatically or by dividing the weight of thecorresponding sphere, cylinder, or cube for its volume.15

Galileo had other chances to rethink hydrostatics, because his ideaswere subject to the envy that always surrounds a successful scientist in theuniversity 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 MarinoGhetaldi, from Ragusa in Dalmatia. He published it in 1603 in a book titled PromotusArchimedes seu de variis corporum generibus gravitate et magnitudine comparatis[Napolitani 1988], [Bertoloni Meli 2004, 168–170]. Galileo and Ghetaldi met, possiblyin Rome in 1600 and certainly in Padua in 1607. They exchanged letters for severalyears.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 theframework of Galileo’s attempt at creating a factory of scientific instruments inPadua.


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. Thiscompelled 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 writinga pamphlet [OG, II, 515–599] which however adds nothing to the scientificquestion. Thus, we find no further clues about when he added the soundmethod 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 sucha peculiar idea.

6 Final remarks

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

AcknowledgmentsFor their critical reading and sharp, stimulating suggestions, I would like tothank Michele Camerota, Paolo Galluzzi, Marco Guardo, Massimo Peri and avery careful unknown reviewer.

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Adresses des auteurs

Paolo BussottiUdine University – [email protected]

Walmir Thomazi CardosoPontifícia Universidade Católicade São PauloFaculdade de Ciências Exatas eTecnologia – PUC–SPRua Marquês de Paranaguá, 111São Paulo – SP01303-050 – [email protected]

Francesco CrapanzanoUniversity of MessinaDip. COSPECSVia Concezione, 698121 Messina – [email protected]

Antonino Dragovia Benvenuti 5Calci Pisa 56011 – [email protected]

Raymond FredetteCIRSTUniversité du Québec à MontréalC.P. 8888, succ. Centre-villeMontréal (Québec) H3C 3P8 –[email protected]

Romano GattoUniversity of BasilicataCampus di Macchia Romanavia dell’Ateneo Lucano85100 Potenza – [email protected]

Gennady GorelikCenter for Philosophy andHistory of ScienceBoston University745 Commonwealth AvenueRoom 506Boston, MA 02215 – [email protected]

Jean-Marc Lévy-LeblondUniversité de Nice-Sophia Antipolis06000 Nice – [email protected]

Roberto de Andrade MartinsExtrema, MG – [email protected]

Marco M. MassaiDipartimento di Fisica “E. Fermi”Largo Bruno Pontecorvo, 356127 Pisa – [email protected]

Annibale MottanaRoma Tre UniversityFaculty of ScienceLargo S. Leonardo Murialdo 100146 Roma – [email protected]

Raffaele PisanoUniversité Charles-de-Gaulle-Lille 3CIRELDomaine universitaire du “Pont deBois”BP 6014959653 Villeneuve d’Ascq Cedex –[email protected]