Pietro Daniel Omodeo and Jürgen Renn

Science in Court Society

Series Editors
Ian T. Baldwin, Gerd Graßhoff, Jürgen Renn, Dagmar Schäfer, Robert Schlögl, Bernard F. Schutz

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Lindy Divarci, Samuel Gfrörer, Klaus Thoden, Malte Vogl

Editor-in-chief
Matteo Valleriani
Max Planck Institute for History of Science, Berlin

Other Editors
Stephen P. Weldon
Department of History of Science, University of Oklahoma
Esther Chen
Library of the Max Planck Institute for the History of Science, Berlin
Kerry V. Magruder
History of Science Collections, University of Oklahoma Libraries
Anne-Laurence Caudano
History Faculty, The University of Winnipeg
Massimiliano Badino
Facultat de Ciéncies, Universitat Autònoma de Barcelona
Robert G. Morrison
Department of Religion, Bowdoin College

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First published 2019
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Edition Open Sources
Sources 11

Benedetti: Life, Work, Contexts

Introduction

Giovanni Battista Benedetti is today a lesser known figure in the history of early modern science. This relative oblivion is in striking contrast to the fame he enjoyed during his lifetime as a prominent mathematician and mathematical philosopher of Venetian origin and reputable courtier to the Savoy dukes in Turin. Among his admirers, the astronomer Johannes Kepler regarded him as one of the few Italians to significantly contribute to the advancement of mathematics in his time: “The Italians are asleep with the sole exceptions of Commandino and Giovanni Battista Benedetti. And in fact, Clavius is German.”[1] For his part, the mathematician of the Collegio Romano, Christopher Clavius, extolled Benedetti’s scientific merits in the 1589 edition of his reputed commentary of Euclid, Euclidis Elementorum Lib[ri] XV. In the dedicatory letter to Carlo Emanuele I of Savoy (Serenissimo Principi ac Domino D. Carolo Emmanueli Sabaudiae Duci), he praised “his court mathematician” Benedetti as “very expert in mathematics” (scientissimus rerum Mathematicarum).[2] In contrast to the eulogies of the Imperial mathematician Kepler and the most prominent astronomers of the Jesuit order, the Urbino school gathered around the mathematical purist Federico Commandino was rather reluctant to acknowledge his achievements, probably due to rivalry, reciprocal misunderstandings, and different philosophical and cultural choices. Among Commandino’s pupils, Guidobaldo Del Monte severely criticized Benedetti’s approach to mechanics and his claim to originality, as documented by his manuscript annotations on mathematical issues.[3] Another member of the Urbino school, Bernardino Baldi, gave an extremely negative and reductive portrait of Benedetti in his collection of short biographies of mathematicians from all epochs, Cronica de matematici.

The Venetian Giovanni Battista Benedetti occupied himself with mathematics, a field in which he served the dukes of Savoy. He wrote a book on gnomonics, which dealt with many proofs belonging to this discipline. However, he has been criticized by the most exquisite scholars for not having respected the method and the purity of explanation which mathematics requires and which was respected by the Greek masters and by their followers. He also wrote some light things of no great import.[4]

Such a harsh judgment can only be explained on the basis of a profound enmity held by Commandino’s followers against Benedetti. This deserves attention since it also influenced the reception of his work. Therefore, we will offer a reconstruction of Benedetti’s cultural environment and the scientific culture of Renaissance Italy in order to understand his work, its roots, and legacy.

Although Benedetti was recognized by his contemporaries, in many respects the theoretical and historical relevance of his work remains obscure. The obscuration of Benedetti is also the result of the prominence of the Urbino school and their bias towards mathematical purism. Moreover, the hagiographic historiography of science sought out the precursors to Galileo and therefore almost naturally adopted the criticism of Benedetti by Galileo’s patron Guidobaldo del Monte. Intrinsic factors also played a role in the eclipse of Benedetti’s fame, principal among which is the relative rarity of his major work, Diversarum speculationum mathematicarum et physicarum liber (Book Including Various Mathematical and Physical Speculations, 1585), not to mention that of most of his earlier publications. Benedetti’s fragmentary style is a special difficulty faced by the reader, a defect that was overemphasized by Baldi and denounced by early modern scholars such as Claude-François Milliet Dechales. Dechales dealt with Benedetti’s geometry in his Cursus seu mundus mathematicus (1690), observing that “the end of his work [the Diversae speculationes] comprises many miscellaneous geometrical remarks, some of which are good, in particular with regard to their special concern, but [they are] disordered.”[5]

Following historiographic commonplaces and their nineteenth-century crystallization, recent historians of science have tended to neglect Benedetti’s work due to the lack of extensive translations of his writings into modern languages, with the exception of the excerpts included in Drake and Drabkin, Mechanics in Sixteenth-Century Italy (1969). This anthology isolated passages that were exclusively devoted to what the editors saw, in hindsight, as the most relevant contributions for the progress of mechanics. Due to its selective nature, this publication did little justice to the complexity and richness of Benedetti’s stature in the history of science. In fact, isolating certain results elides recognition of the multilayered architecture of a book such as the Diversae speculationes, which is characteristic of Renaissance science. Benedetti’s showcase of mathematical erudition and scholarship is thus omitted and obscured. Drake and Drabkin’s selection could only yield a reductive and rather misleading image of Benedetti as a scientist and thinker. As we will argue, entire paragraphs or chapters expunged from Drake and Drabkin’s translation, for instance those concerning philosophy and cosmology, are relevant for an understanding of the author’s general conception of mechanics and physics. From a historical viewpoint, it is hardly possible to trace disciplinary boundaries in the Renaissance that fit those established today. Mechanics was an emerging discipline at the crossroads of mathematics, engineering, and natural philosophy. Hence, a reassessment of Benedetti’s work in its entirety is necessary not only to understand his personality but also to grasp the scientific culture of his age as the result of interdisciplinary controversies.

This open access edition makes the Diversarum speculationum mathematicarum et physicarum liber accessible to a large scholarly readership. Benedetti’s volume is a major contribution to Renaissance science, especially due to its insights into mechanics, the mathematization of (or geometrical approach to) natural investigation, and the connection of celestial and terrestrial dynamics in a post-Copernican perspective. The first edition of this work was an elegant folio, which included heterogeneous writings on technical and philosophical issues as well as on mathematics and physics. Benedetti presented them as short treatises (tractatus) or letters (epistolae) addressed to gentlemen, courtiers, scholars, engineers, and practitioners of different arts. The volume was printed by Niccolò Bevilacqua’s heir (apud haeredem Nicolai Bevilacquae), who was the owner of the main printing house in Turin, which was directly supported by the dukes of Savoy. The Diversae speculationes appeared in a series of prestigious volumes aimed at celebrating the magnificence of the court and the capital, including the works of the court historian Emanuele Filiberto Pingone: Augusta Taurinorum (1577), on Turin, Inclytorum Saxoniae Sabaudiaeque principum arbor gentilitia (1581), on the genealogy of the ruling family, and Sindon evangelica (1581), on Christ’s shroud, which had been recently transferred from Chambéry to Turin. The Diversae speculationes exhibits the same celebratory intention. The volume aimed to make the quality of the court mathematician’s research and skills publicly appreciable. It also bore witness to the intensity of the cultural debates going on in Turin, and connected this city with other centers of learning, especially Venice.[6] Its miscellaneous and epistolary form was suitable for displaying the variety of the author’s interests and for praising his patrons, friends, and colleagues by naming them as dedicatees or addressees of the different sections and letters.

The structure of the Diversae speculationes—its occasional and fragmentary character, its celebratory purpose, and the epistolary display of a network of personal connections—mirrors the socio-cultural embedment of Benedetti’s work. We regard it as exemplary of “science in court society.” As Norbert Elias has pointed out, the höfische Gesellschaft, or court society, is a particular social configuration (gesellschaftliche Figuration) characteristic of the transitional phase to an industrial and capitalist Europe, which we conventionally refer to as the early modern period.[7] During the Renaissance and the Ancien Régime, the court was (or became) a hegemonic center from which powerful elites mediated between the ruler and the socio-political environment. Benedetti’s activities as a court mathematician exemplify such a function. In his role he was expected to interact with the upper classes and respect aristocratic etiquette, and to act as the Savoy “expert” on a wide range of pedagogical and technical issues linked to his profile as a mathematician and mathematical philosopher. He was required to teach geometry to the offspring of the ducal family, to supervise engineering and architecture projects, to produce instruments or machines for practical purposes, warfare, and recreation (such as fountains, sundials, or nautical instruments). He had to adhere to shared court values, norms, and behaviors, primarily those linked to honor and prestige. These courtly principles are reflected in the epistemic values permeating his scientific production, for instance in the value of scientific disinterestedness that marks his theoretical approach to practical as well as to speculative problems. In a hierarchical and aristocratic society, his theoretical attitude marked at once the continuity and the distance between his role as a court mathematician and those involved in practical activities. Moreover, the primacy of courtly interests over those of science as a purely scholarly endeavor (as it was pursued at universities and academies) is evident from Benedetti’s networking strategies, which were aimed at not so much exchange with other scholars as at giving advice to a wide range of people, beginning with the ruling elites of the country. In other words, he was not primarily concerned with establishing a réseau, as was typical for the Republic of Letters. As we will show, he did not regard himself and his activity as part of a learned network but rather as the center of courtly interaction. This center-periphery structuring of his network mirrors—in two senses—the “knowledge economy” his work is embedded in. Sociologically, the centralizing character of court society is reproduced in scientific policies through Benedetti’s function as a technocrat; epistemologically, the fluid style and fragmentary form of the Diversae speculationes is an expression of the expert-advice character of his scientific work. Thus, in order to properly understand his work, we deem it necessary not only to investigate the technical and theoretical dimensions of Benedetti’s science, but also to analyze evidence concerning the ties between these dimensions and the social and cultural environment.

Among the studies on Benedetti, Giovanni Bordiga’s monograph Giovanni Battista Benedetti: filosofo e matematico veneziano del secolo XVI (1926) still stands out as one of the most important references as far as prosopographical information is concerned.[8] The excellent research accomplished by Carlo Maccagni and the proceedings of the conference on Benedetti held in Venice in 1985 at the Istituto Veneto di Scienze, Lettere ed Arti investigated many aspects of Benedetti’s contribution.[9] In spite of the accuracy of these relatively recent Italian studies, Alexandre Koyré’s evaluation of Benedetti’s role in the first stages of modern science, in the Études galiléennes (1939), has had a more direct influence on his international reception. Koyré emphasized the link between the incipient mathematical science of motion and heliocentrism in Benedetti’s speculations. On account of this, Paul Lawrence Rose regarded him as a herald of the “Italian Renaissance of mathematics”.[10] Koyré’s grand narrative of the Scientific Revolution, which he conceived as a development with a “prologue” (Copernicus) and “epilogue” (Newton) in the heavens, included Benedetti as a precursor to Galileo because of the interconnection of mathematical and physical themes in the former’s work. Koyré’s main thesis was that classical physics (the mathematical science of nature of Galileo, Descartes, and Newton) emerged as a direct consequence of Copernicus’s geokinetic system, which undermined the traditional (Aristotelian and Ptolemaic) worldview. Although Benedetti’s relevance as a source for Galileo cannot be denied, scholars now view Koyré’s narrative as dubious due to its abstract treatment of the history of science, conceived of as an internal development of ideas. In the years of the Cold War (or shortly before it), this viewpoint embodied the ideological reaction to Marxist or materialist-oriented accounts, which stressed the technological, empirical, and social roots of modern mechanics, as was the case with Leonardo Olschki and Edgar Zilsel.[11] Bendetti’s approach to mechanics and post-Copernican astronomy therefore appears to be an appropriate case study for reconsidering this general historical problematique, beginning with a reassessment of the relation between mechanics and astronomy in early modernity. This implies a reconsideration of the basic questions of the historiography of science and of historical epistemology, such as the role of material and intellectual factors in the so-called Scientific Revolution.

In this edition of the Diversae speculationes, we aim to present Benedetti’s achievement in its rich complexity. Benedetti is emblematic both of his time and of the non-linearity of the historical process of Renaissance science with its multicentric institutions and scientific networks. We will show that the apparently fragmentary nature of his work is expressive of the peculiar character of science in court society and, in spite of this form, it conceals a fundamental unity of his conception of nature and method, both of which rest on geometry. To be sure, Benedetti regarded mechanics as a model, but he enlarged his perspective to include the most varied fields of investigation in order to concretely demonstrate the fruitfulness of his approach to universal knowledge about astronomy, physics, meteorology, and even literature and ethics.

Overview and Acknowledgments

Our first chapter is devoted to Benedetti’s biography and comprises an overview of his publications. We consider the various dimensions of his identity, such as his nobility as both a Venetian patrician and a Savoy aristocrat in the retinue of Emanuele Filiberto and Carlo Emanuele at the Turin court. In particular, we discuss his role as a courtier, the tasks he was entrusted with, and the relations he established in this context. We also deal with his extra-academic education and his attitude towards mathematics, which he initially saw as an intellectual instrument to be used against the “bookish” culture of the universities and the “idle” rhetoric of the humanists. Later he softened the polemical tone that characterizes his early writings. Moreover, since he saw himself as a philosopher, more precisely as a court philosopher to the Dukes of Savoy, a discussion of Benedetti’s mathematics cannot be separated from his philosophical project. Benedetti claimed for himself the right to discuss in mathematical terms issues of natural philosophy that traditionally belonged to the rather qualitative and conceptual approach of the peripatetics. The Diversae speculationes is an altogether magisterial example of this merging of philosophical and mathematical perspectives.

Chapter 2 is a reconstruction of the cultural life of Renaissance Turin, the town in which Benedetti spent his mature years and where he composed his major work. His achievement was embedded in the cultural ferment of the new capital of Savoy, a place of ambitious town planning and civil reforms. It was a time in which the arts, literature, and philosophy received a new impetus. Editorial projects were launched; the university was reopened and illustrious scholars were attracted there. The dukes’ religious politics was informed by a sense of pragmatism, which is mirrored in the fluctuating relations between the ruling family, the Jesuits, and Rome. Benedetti’s secular attitude towards science and philosophy mirrors the cultural politics of his patrons. In addition we discuss his involvement in various scientific debates divided into courtly conversations, academic controversies, and controversies going beyond the settings of the court and the university. Among such extra-academic public controversies, the most important was Benedetti’s public defense of the reliability of astronomical calculation against a polemist, Benedetto Altavilla, who indirectly attacked his and others’ astrological practice. Newly discovered documents show that Benedetti’s successor as court mathematician, Bartolomeo Cristini, continued that polemic after Benedetti’s death. Cristini discredited Benedetti’s use of astronomical tables to cast horoscopes, in order to ingratiate himself with the dukes and successfully start a career at court. We trust that this chapter offers new insights into the scientific culture of the Renaissance by bringing Turin into focus, a cultural centre that has so far escaped in-depth consideration by historians of early modern science.

In chapter 3 we offer an overview of the structure of the Diversae speculationes. We introduce Benedetti’s mathematical sections in general terms, focusing on his geometrical demonstrations for the solutions of problems of arithmetic—which were the result of his private teaching of mathematics to the Savoy prince—his sketchy annotations on the theory of proportions based Book 5 of Euclid’s Elements, and his considerations on linear perspective aimed at supporting the work of painters and architects. The sections on physics, mechanics, and natural philosophy are not discussed in this chapter as they receive special treatment in other chapters. The last part of the Diversae speculationes was a miscellanea of scientific letters. We discuss them vis-à-vis their significance as a mirror of Benedetti’s social capital. His epistles were mostly directed to aristocrats, beginning with his patrons, other courtiers, and diplomats, especially those from Venice. He also corresponded with professors, artists, engineers, and practitioners, some of whom lived north of the Alps. His network was markedly a center-periphery one, in which the court expert shared his views on the most varied topics with others seeking his advice or opinion. Thus, it was not a scholarly network implemented for the sake of exchange and the advancement of knowledge. Rather than a networking activity establishing a Republic of Letters, Benedetti’s correspondence reflected court-society centralism.

Chapter 4 addresses Benedetti’s epistemology on the basis of passages regarding the certitude of mathematics and his effective use of mathematics in physics. His role as an early champion of what would later become known as “physico-mathematics” is understood here against the background of the philosophy of mathematics in the Renaissance. Moreover, we deem the modal epistemology underlying his science of particular interest: his treatment of nature in mathematical terms did not imply the necessary or deterministic nature of physical processes. Rather, he embraced an ontology and an epistemology of contingency that constituted a bridge between medieval scholastic views on nature and the mathematical physics of the time of Galileo and Descartes. We dedicate an excursus to the vision of nature as the realm of contingency in the period extending from the medieval science of weights to seventeenth century mechanics, and ascribe to Benedetti a central position in this intellectual process.

Chapter 5 deals with the field in which Benedetti has received the most credit from historians: mechanics. Actually, Benedetti himself emphasized the importance of his contribution to mechanics as what would secure his fame in posterity. We summarize his theories on equilibrium and his critical reworking of earlier theories such as those developed by Jordanus Nemorarius and Niccolò Tartaglia. We consider Guidobaldo Del Monte’s negative reaction to Benedetti’s mechanics in detail, as well as the weaknesses and strengths of both authors. We regard this pluralism of clashing and integrating views as revealing the complex paths of discovery undertaken by students of mechanics in a period of the utmost relevance to its modern systematization. Moreover, the subterranean conflict of views and approaches between Benedetti and Del Monte affected Galileo’s work. His mechanics drew from both authors, although he did not acknowledge Benedetti explicitly due to circumstances and opportunity.

Chapter 6 summarizes Benedetti’s astronomical work. Although he did not see himself as an astronomer, his contribution is quite interesting. He should be acknowledged for his effort to develop a new mathematical physics in accordance with post-Copernican astronomy. His discussion of astronomical theory against the background of a general philosophical reform was strikingly innovative. His specific polemics on the reliability of astronomical calculation also receive close treatment here. Furthermore, in an appendix Günther Oestmann offers an assessment of Benedetti’s astrological calculations on the basis of so-far neglected manuscript sources containing two of his horoscopes.

In chapter 7 we deal with Benedetti’s natural philosophy as he presented it in Book 4 of the Diversae speculationes. Although he entitled it “Disputations on Some Opinions Held by Aristotle” (Disputationes de quibusdam placitis Aristotelis), it was a polemic directed “against” fundamental Aristotelian theses on motion, time, space, matter, and cosmology. This is the section in which Benedetti’s commitment to “the system of Aristarchus and Copernicus” most clearly emerges. It is also a fundamental section on the existence of the physical void as the necessary presupposition of any local displacement and on free fall through different media. We see this book of the Diversae speculationes as a major contribution to the Renaissance debate on the foundations of physics, going far beyond the treatment of mechanics and cosmology strictu sensu. Hence, we take into consideration Benedetti’s definition of space as an “inter-bodily gap” (intervallum corporeum), his defense of the possibility of actual infinity in nature against Aristotle’s veto, his understanding of time as an absolute frame complementary to space and its place in the philosophical debates of the Renaissance, the revision of the concepts of natural and violent motion, and finally, the surprising conclusion of the “Disputations on Some Opinions Held by Aristotle” with a Copernican note.

This volume is a continuation of an Edition Open Access project aimed at the publication and scholarly reassessment of the fundamental sources of Renaissance mechanics. This project began with Jürgen Renn and Peter Damerow’s Guidobaldo del Monte’s Mechanicorum Liber in 2010. Elio Nenci’s open-access publication of Bernardino Baldi’s In mechanica Aristotelis problemata exercitationes appeared in 2011 and, in 2013, Matteo Valleriani’s Metallurgy, Ballistics and Epistemic Instruments, including a transcription and an English translation of Nicolò Tartaglia’s Nova scientia. Ideas that were crucial for the writing of this introduction to Benedetti’s Diversae speculationes are derived from another volume by Renn and Damerow, The Equilibrium Controversy: Guidobaldo del Monte’s Critical Notes on the Mechanics of Jordanus and Benedetti and their Historical and Conceptual Backgrounds (2012).

We are very thankful to our colleagues at the Max Planck Institute for the History of Science in Berlin for their insightful remarks and the precious exchanges about early modern science that took place in the many workshops and regular meetings over the past years: Angela Axworthy, Sonja Brentjes, Jochen Büttner, Giulia Giannini, Peter McLaughlin, Matthias Schemmel, Joyce Van Leeuwen, Matteo Valleriani, and many others. We are especially thankful to Antonio Becchi and Elio Nenci for their accurate reading of the manuscript of this book and their expert advice. Rodolfo Garau kindly took pictures of manuscripts and rare books that are preserved in the Biblioteca Nazionale Universitaria di Torino and are published here. Stefan Paul Trzeciok helped us with the translation of several Latin passages. We would like to acknowledge the Collaborative Research Centre “Episteme in Motion” (Freie Universität Berlin), which has constituted an important intellectual framework for this collaboration, and the Deutsche Forschungsgemeinschaft for financing Pietro Daniel Omodeo’s research in Berlin for several years as well as a research trip to Turin in 2012. We are grateful to the librarians of the Max Planck Institute for the History of Science for their valuable support with bibliographical research and the digitization of sources, and to Louis Berger, Lindsay Parkhowell, and Charlie Zaharoff for their copyediting work and linguistic support. The last phase of the authors’ collaboration on the manuscript of this volume is part of a project based at the Ca’ Foscari University of Venice that has received funding from the European’s Horizon 2020 Research and Innovation Programme (GA n. 725883 EarlyModernCosmology).

Footnotes

[1] Kepler 1937–2001, 390: “Itali somniant (preter unum Commandinum et Joh[annem] Baptistam Benedictum, Clavius enim Germanus est).”

[2] Clavius 1589, ff. *4r–*5r. The list of Benedetti’s admirers also includes the Pisa philosopher Jacopo Mazzoni, the Venetian intellectual leader Paolo Sarpi, and the French scholar Marin Mersenne. See Cappelletti 1966, 262.

[3] See Renn and Damerow 2012 and Renn and Omodeo 2013.

[4] Baldi 1707, 140: “GIO[VANNI] BATTISTA Benedetti veneziano attese alle matematiche, nelle quali servì i Duchi di Savoia. Scrisse un libro di gnomonica, il quale toccò molte cose appartenenti alle dimostrazioni della detta disciplina, se non che viene ripreso da più esquisiti di non haver’osservato quel metodo, e quella purità dell’insegnare, che ricercano le matematiche, et è stato osservato dagl’ottimi Greci, e dagl’imitatori loro. Scrisse anco alcune altre cose leggiere, e di nessun momento.” Here and in the subsequent pages, Italian and Latin grammar (e.g., capitalization and punctuation) has been modernized.

[5] Ventrice 1985, 188: “in fine sui operis multa habet miscellanea geometrica, quorum nonnulla ad sectiones praesertim pertinentia bona sunt, sed inordinata.”

[6] Cecchini and Roero 2004.

[7] As Norbert Elias put it (Elias [1969] 2002, 73): “Durch das Bemühen um die Struktur der höfischen Gesellschaft und damit um das Verständnis einer der letzten großen nicht-bürgerlichen Figurationen des Abendlandes eröffnet man sich also mittelbar zugleich einen Zugang zum erweiterten Verständnis der eigenen berufsbürgerlich-städtischen-industriellen Gesellschaft.”

[8] Bordiga 1985.

[9] Bordiga 1926, repr. Bordiga 1985, Maccagni 1967b, Maccagni 1967a, Maccagni 1983, and Istituto Veneto di Scienze 1987.

[10] Rose 1975, 154–156.

[11] The cultural-political intentions of Koyré’s approach emerge most vividly from his 1943 article on Galileo’s Platonism, see Koyré 1943. Lefèvre stresses it in Lefèvre 2001.

1. Prosopography

1.1 The Life and Career of a Renaissance Man

Giovanni Battista de Benedetti came from a patrician family of Venice. Although the title of nobility may appear superfluous to the historian of science, it was not so for him and his contemporaries. Benedetti often noted this in his publications, adding to his name the honorific “Patritius Venetus.” Evidence for Benedetti’s noble origins can be found in a document dated January 14, 1570. This is a patent through which Duke Emanuele Filiberto of Savoy conferred upon Giovanni Battista the privileges of Imperial nobility in addition to his previous titles:

We make, create, and constitute the aforementioned Giovanni Battista Benedetti as a true noble of the Holy Roman Empire and of our Empire forever, alongside all his legitimate and natural sons and daughters (those who are already born and those that will be born). We will call and fully declare them such [nobles of the Holy Roman Empire]—although he and his predecessors are noble and were born from an ancient and noble progeny, as we are very well informed.[1]

In those years, the establishment of the Savoy court in Turin brought about a general transformation of the urban patriziato into an aristocratic class gravitating around the dukes.[2] This trend was parallel to the more general political-social shift from the civil humanism of the medieval municipalities toward the courtly culture of centralized territorial States.

On the occasion of the conferral of the patent on Benedetti, the cross of Savoy was added to his heraldic design along with the motto “sic vita veritas.”[3] This motto, which indicated a conduct of a life dedicated to the search for truth, was the acknowledgment of his mathematical and philosophical excellence. In the preamble to the duke’s patent of nobility, it was precisely Benedetti’s devotion to the mathematical disciplines, the humanae litterae, and the philosophy that was extolled as an example to be imitated and a reason for the conferral of aristocratic privileges on him and his heirs.[4] In this case, scientific distinction led to higher social recognition and even served as a legitimation for it.

During the Renaissance, nobility was more important than professional appurtenances or academic titles. For instance, the celebrated Danish astronomer Tycho Brahe, himself an appreciative reader of Benedetti, held aristocratic lineage in higher esteem than any status linked to university professorship, including the position of imperial mathematician—an appointment which, by contrast, raised the status of his fellow countryman and opponent Nicolaus Reimarus Ursus, who was of low extraction.[5] Accordingly, Brahe always emphasized Benedetti’s lineage when citing his work, for instance his letter on the superlunary location of the supernova of 1577. The capitalization as well as the reverence in this passage from the Astronomiae instauratae progymnasmata (posthumous, 1602) is telling:

The small star of Cassiopeia would not shine as brightly as this nova over the whole surface of the Earth because of the dry fumes placed in-between, if they had been only under that one, and did not affect in the same manner the other stars next to it and augmented that unusual light. But the most excellent philosopher GIOVANNI BATTISTA BENEDETTI, THE VENETIAN PATRICIAN, eminently and skillfully demonstrated this with geometric arguments, in [his] outstanding work concerning mathematical and physical speculations (around the end of his letters). Writing to Annibale Raimondo […] he clearly showed the absurdity which necessarily follows from his false assumption [i.e., the sublunary position of the nova].[6]

Fig. 1.1: An example of the titles Benedetti added to his name in his publications. In the title page of De gnomonum umbrarumque solarium usu (1574), he called himself “Venetian Patrician, Philosopher.” (Max Planck Institute for the History of Science, Library)

The prominence accorded to lineage is evident from Brahe’s self-representation in the portrait at the beginning of his Epistolarum astronomicarum libri (1596), a collection of epistles that arguably took Benedetti’s collection in the Diversae speculationes as a model. Brahe’s image is encircled by the heraldic designs of the family and makes the signs of his nobility very visible. In the same epistolary, Brahe’s letters directed to aristocrats appear more prominently than those addressing “simple” professors or practitioners. He attached greater importance to his correspondence with the patron of sciences, Landgrave William IV of Hesse-Kassel, than to exchanges with the latter’s court mathematician Christoph Rothmann.[7] Similarly, in the Diversae speculationes, Benedetti published with pride his letters to dukes or to illustrious aristocrats.

Apart from his nobility, we do not know much about Benedetti’s origins. According to a horoscope that he cast for himself (Figure 1.2), and was printed by the Neapolitan astrologer Luca Gaurico in Tractatus astrologicus (Astrological Treatise, 1552),[8] Benedetti’s father was a learned Hispanus, or Spaniard. Based on this thin evidence, his biographer, Giovanni Bordiga, speculated that his family could have been merchants trading with Spain.[9] Other archival documents caused him to speculate about Benedetti’s marriage, around 1585, and about the existence of a daughter called Lodovica from an earlier relationship or marriage. She married a certain Domenico Pipino of Racconigi. Benedetti built a sundial for this son-in-law (magnificus Dominus Dominicus Pipinus generus meus), as indicated in De gnomonum… usu (1574). Lodovica died young, long before her father, in 1580.[10]

Fig. 1.2: Benedetti’s own horoscope, in Luca Gaurico, Tractatus astrologicus (1552), f. 76r. (Bayerische Staatsbibliothek)

For the greater part of his life Benedetti was a courtier. For several years he served duke Ottavio Farnese of Parma, whom he joined in 1558 as “lettore di filosofia e mathematica.”[11] Later, from 1567 up to his death on January 20, 1590, Benedetti served the Dukes of Savoy Emanuele Filiberto and Carlo Emanuele I. His duties were typical for a Renaissance court mathematician and are akin to those of Leonardo da Vinci in Milan, Guidobaldo del Monte in Urbino, Galileo in Florence, and Kepler in Prague, to mention only a few well-known names.[12] Benedetti was required to advise his patrons on issues of mathematical expertise. His fields of competence included engineering and architecture.[13] In Parma and Turin he built sundials (such as the modern one in Figure 1.3). He was also responsible for the construction of a fountain in the ducal park (Parco di Viboccone, later Parco Regio), which was destroyed by the French army during the siege of 1706.[14] Moreover, he was consulted on astronomy and music, both traditionally considered mathematical disciplines. In Parma he carried out astronomical observations, which he also reported on in the Diversae speculationes. In two letters to the Parma choirmaster de Rore, Benedetti explained musical consonance and dissonance of two tones by the ratio of oscillations of waves of air generated by the strings of musical instruments.[15] He claimed that the frequency of two strings of equal tension must have an inverse ratio to the lengths of the strings, and thus proposed to describe the degree of consonance or dissonance of two tones mathematically. In Turin he wrote a proposal for the calendar reform in 1578, De temporum emendatione, later reprinted in the Diversae speculationes as the first of his epistles (to Duke Emanuele Filiberto).[16] This proposal was also sent to Rome and was meant as an aid to Clavius’s efforts to correct the calendar.[17] At the same time, he taught mathematics to Emanuele Filiberto and his son Carlo Emanuele I.

Fig. 1.3: A modern sundial on the Church of San Lorenzo in Turin reminiscent of those designed by Benedetti. (Own photography)

Courtly life included participation in literary culture. Baldassar Castiglione, in his idealization of the court of Urbino in Il libro del Cortegiano [The Book of the Courtier] (1528), launched the model of a courtier with a refined literary education.[18] Following such cultural dispositions, a courtier versed in mathematics could advocate the usefulness of his expertise for the interpretation and assessment of “scientific” questions raised by classical sources, even poems. This attitude explains the inclusion of a letter on Ovid in the collection of epistles in the Diversae speculationes.[19] It was addressed to a certain Pancrazio Mellano, perhaps a courtier, asking Benedetti’s opinion about the astronomical references in Book 2 of the Metamorphoses, in which Ovid tells the myth of Phaeton. According to the myth, Phaeton rode his father Apollo’s chariot one day but he was unable to control the horses and keep the sun on its regular path. Finally, he was thrown out of the chariot, took a bad fall, and died. In the poem Ovid described the solar path in some detail but, according to Benedetti, he mixed up daily rotation and annual motion along the ecliptic: “Ovid unduly passes from the daily motion to the annual” (Quod Ovidius transcurrit a motu diurno, ad motum annuum praeter rem). To make his point clear, Benedetti listed the passages dealing with one or other of the two motions ascribed to the sun in ancient astronomy.[20]

As an exponent of the Turin elite, he was himself devoted to poems. For instance, the Milanese painter and writer Giovanni Paolo Lomazzo, who was linked to Savoy’s court, celebrated Benedetti in verse as a philosopher, mathematician, and astrologer. In the first lines of a poem dedicated to him, Lomazzo declared himself delighted that Benedetti appreciated his paintings and cast his birth horoscope. Lomazzo’s poem paints a vivid picture:

Prudence and knowledge descend
From Philosophy into [human] intellects;
Which are perfect as far as their disposition is concerned,
As each one receives its part of justice and reason.
To Benedetti, he so wise
And precious in the world,
Belongs so much of this [philosophy]
That it would be vain to try to equal him:
So sublime does his value shine.
All the more am I delighted that he appreciated
My painting so much so that he considered
The time and the point in which I was born in the world.
Oh splendor of our time, the sound [of your voice] silenced
Every scholar of your art, who had to direct his judgment elsewhere,
As it was overshadowed by yours, which is so deep.[21]

Benedetti received no formal or academic education. Like other Renaissance self-taught men (e.g., Niccolò Tartaglia and Tommaso Campanella), Benedetti was even proud of being removed from the academic habitus and training centers. This is particularly evident from the anti-academic tone of some of his polemics. In the preface to his first scientific treatise, Resolutio omnium Euclids problematum (1553) (On the solution to geometrical problems using a compass with a fixed opening), the twenty-three year old Benedetti emphasized the fact that he had not had a “common” (quod vulgus solet) education at some gymnasium or school. He boldly wrote to his patron, the Dominican abbot and diplomat Gabriel Guzman, that:

Until now I have advanced without any mentor or teacher (under the guidance of God). I have never frequented any gymnasium or school. I have not learned what the vulgar (I mean this word without arrogance) use to estimate erudition, [such as limiting it] to the time spent at school, thus setting an end to learning when the seven years [of regular studies] are ended. As long as I live, I will continue [learning].[22]

It is possible that Benedetti was educated privately by his father, depicted in Gaurico’s Tractatus astrologicus as “Hyspanus, Philosophus, et Physicus” (see Figure 1.2). For his part, Benedetti acknowledged only one teacher, namely the reputed mathematician and scientist-engineer Niccolò Tartaglia (ca. 1500–1557), for introducing him to the first four books of Euclid’s Elements, probably between 1546 and 1548. In the Diversae speculationes mathematicae et physicae, Benedetti mentioned Tartaglia again as one of the very few authors of mathematical works whom he deemed worth reading.[23] However, in the Resolutio omnium Euclidis problematum, he was quick to add that he had learned the rest of the Elements by himself:

As it is honest and right to attribute to everybody his own merit, [I should acknowledge that] Niccolò Tartaglia taught only the first four of Euclid’s books to me. I studied the rest alone with effort and diligence. In fact, for the one who wants to know, nothing is [too] difficult.[24]

Bordiga described such self-celebration as a sign of Benedetti’s “pride in the assumed independence of his own thinking” (orgoglio di creduta indipendenza del proprio pensiero).[25] This is the same pride that would later lead to animosity with other prominent mathematicians such as Del Monte.

Moreover, in the preface to the Resolutio, Benedetti contrasted the simplicity of mathematics with the vanity of rhetoric. He went so far as to accuse learned and eloquent doctors of corrupting the sciences.

Furthermore, mathematics does not require much [stylistic] splendor. If some language expert tried to improve its elegance, this would have no value, because a change of the mathematical language and of the scientific terminology could easily confuse the sense [of the reasoning] and render everything obscure. Therefore, I will follow the scholarly tradition and use plain words in my demonstrations, as I disapprove of deceptive elegance. In this respect, I follow the steps of the ancients who taught the sciences and the subjects themselves using plain words. Petty teachers (indeed, charlatans and babblers) corrupted this manner of teaching. Although they do not understand the subject, their babbling obtains the highest praise by the vulgar who regard them as learned scholars. This should not be surprising, considering that the most perfect and distinguished expertise in the sciences is attained by very few—despite the fact that many people write a great deal in all kind of sciences and arts, babbling a lot and capturing the attention of the uneducated with illusions and bombastic words.[26]

The same tone characterized Benedetti’s next publication. Its title was intentionally polemical: Demonstratio proportionum motuum localium contra Aristotilem et omnes philosophos (1554). In fact, this booklet put forward a novel theory of motion. He argued that bodies of the same material fall through a given medium with the same speed, and not with speeds proportional to their weights, as Aristotle held. This is the reason for Benedetti’s declaration of war “against Aristotle and all philosophers” in the title. Benedetti employed the Archimedean concept of buoyancy to account for the dependence of the motion of fall on their specific rather than absolute weight. As we shall see, these ideas played an important role in the Diversae speculationes. The use of Archimedean notions to improve on Aristotle’s physics was probably stimulated by Tartaglia’s Italian translation (1543) of Book 1 of Archimedes’s treatise on bodies in water.[27] Benedetti’s challenge to Aristotle must have raised considerable discussion, as is shown by the fact that, in his Demonstratio, he discussed Aristotle’s views and responded to his critics at length. In the second edition of the Demonstratio (13 February 1554 more veneto, in fact, 1555), he showed that the resistance encountered by a falling body in a medium depends not on its volume, but on its surface area. Benedetti moreover explained the acceleration of the motion of fall in terms of an increasing impetus of the falling body. He had already outlined his theory of fall in the dedicatory letter of the Resolutio, explaining this anticipation as a means of avoiding plagiarism.[28] Still, in spite of his efforts to secure priority for his ideas by repeated publication, they were plagiarized by the Flemish polymath Jean Taisner in 1562 and spread through Europe with no clear acknowledgement of their origin.[29] This prompted Benedetti to express his indignation and rage at Taisner in the dedicatory letter of his De gnomonum… usu (1574).[30]

As was to be expected by his irreverent tone, some of the first reactions to Benedetti’s early writings were rather critical. As he reports in the preface to the second edition of the Demonstratio (1555), some Roman scholars objected that his treatment of motion was in disagreement with Aristotle (illam [meam propositionem] neutiquam esse iuxta mentem Aristotelis). Benedetti was informed about their disapproval by a Dominican friend, Petrus Arches, an expert of Hebrew and Greek letters cultivated in philosophy and mathematics.[31] Benedetti replied that those scholars worshipped Aristotle like a pagan god (veluti coeleste quoddam numen) and did not admit that their auctor could make mistakes. He claimed that he had not misunderstood Aristotle; rather, that he simply disagreed with him.

I remember that he [the very educated Doctor Peter Arches]—after many different conversations on various subjects—told me that many in Rome considered that proposition of mine (which I sent to you, Reverend Mr. Guzman, among other ones) and they mostly reacted with surprise for I did not specify that it was by no means in accordance with Aristotle’s mind. Such was the reaction of those who considered my demonstration very attentively.

They could not concede that Aristotle was mistaken in any way, because they do not regard him as a human being. Rather, they confer upon him the celestial condition of a pagan divinity. And they see even slight disagreement as a sin. Therefore [they believe that] I committed (and still commit) heresy if, according to their judgment, I do not follow the pure and authentic mind of Aristotle’s doctrine in any manner.

Thus, in order to escape the allegation of such an error or [the rumor] that I am dissimulating and hiding something, especially as far as this issue is concerned, I decided to publish this new booklet in which I present my opinion more clearly. In this manner, everybody should become aware that I correctly understood Aristotle and that I disagree with him on a particular issue with considered reason. This is an unpleasant task for me. In fact, it is only unwillingly that I dissent with such a great man. I know nobody who could rival his excellence in all kind of doctrines. Nevertheless, his teaching is to take as true that which is supported by stronger reasons. He himself followed this precept, as he stated in the Ethics: “Plato is my friend, Socrates is my friend, but truth is even more friend to me.”[32]

It is evident from these passages that Benedetti regarded mathematics as a support for conclusive rational argumentation in the treatment of natural issues. Therefore, as a mathematicus he claimed for himself the right to be called a philosophus. Already in the short biographical indication accompanying his birth horoscope, he was said to be a “Phylosophus, Musicus, atque Mathematicus” (see Figure 1.2). In his publications, Benedetti often stressed his quality as “philosophus” or “filosofo.” Galileo would later add the title of “philosopher” to that of “court mathematician” in Medici’s Florence.[33] However, in Benedetti’s case, it is evident that adding the title of “philosophus” was not part of a strategy aimed at social advance but rather mirrored his cultural and philosophical commitment to a mathematical philosophy of nature with all its consequences, among them that Aristotelian physics was open to critique by means of mathematical reasoning.

Thus, Benedetti not only dealt with fields of mathematical inquiry that traditionally belonged to the domain of mathematics (such as mechanics, optics, mathematical astronomy, and musical theory), but also addressed issues considered beyond the limitations of mathematics, especially terrestrial and celestial physics. The title of the Diversae speculationes mathematicae et physicae is itself provocative, as it brings together mathematics and natural philosophy (or physica), considered to be separate fields, one dealing with the quia (the “phainomena”) and the other with the propter quid (the “causes”). In this respect, Benedetti’s methodology is very close to that of Nicolaus Copernicus, whose heliocentric system he admired. In Book 1 of De revolutionibus orbium coelestium (1543) and in the Narratio prima (1540), Copernicus and his pupil Georg Joachim Rheticus (1514–1574) reversed the Peripatetic hierarchization of physics over mathematics, urging a reform of natural philosophy and celestial and terrestrial physics in order to bring them into accord with the geokinetic and heliostatic innovations in mathematical astronomy. Beyond astronomy, the issue of the status of mathematics and its role in natural investigations was heatedly debated by philosophers and mathematicians during the Renaissance.[34] One ancient predecessor to praise mathematical physics was the Hellenistic “prince of astronomy and geography,” Claudius Ptolemy. In the beginning of the Almagest, he pointed out the superiority of mathematics over theology and physics, and even argued for a possible extension of the method of mathematical astronomy to include the treatment of local motion in general, as well as theology and ethics.

Only mathematics can provide sure and unshakeable knowledge to its devotees, provided one approaches it rigorously. For its kind of proof proceeds by indisputable methods, namely arithmetic and geometry. Hence we were drawn to the investigation of that part of theoretical philosophy, as far as we are able to the whole of it, but especially to the theory concerning divine and heavenly things. For this alone is devoted to the investigation of the eternally unchanging. For that reason it too can be eternal and unchanging (which is a proper attribute of knowledge) in its own domain, which is neither unclear nor disorderly. Furthermore it can work in the domains of the other [two divisions of theoretical philosophy, physics and theology] no less than they do. For this is the best science to help theology along its way, since it is the only one which can make a good guess at [the nature of] that activity which is unmoved and separated; [it can do this because] it is familiar with the attributes of those beings which are on the one hand perceptible, moving and being moved, but on the other hand eternal and unchanging, [I mean the attributes] having to do with motions and the arrangements of motions. For almost every peculiar attribute of material nature becomes apparent from the peculiarities of its motion from place to place. [Thus one can distinguish] the corruptible from the incorruptible by [whether it undergoes] motion in a straight line or in a circle, and heavy from light, and passive from active, by [whether it moves] towards the centre or away from the centre.[35]

Even after Copernicus, Ptolemy’s methodological insights maintained their full importance and could guide scholars who intended to expand the realm of the application of mathematics far beyond the limits established by traditional philosophy. In the Diversae speculationes, Benedetti deepened the discussion of issues of natural philosophy such as the concepts of space, time, and motion, claiming for a mathematician a better and clearer insight into foundational problems of physics.

Astrology was another area of expertise for Benedetti. During the Renaissance, astronomy and astrology were never separated. Benedetti was expected to cast horoscopes and give astrological advice to his patrons, just as Brahe astrologically advised the King of Denmark, Kepler the Emperor, and Galileo the grand dukes of Tuscany.[36]

In Venice Benedetti frequented celebrated exponents of the astrological culture of the time, among them Annibale Raimondo of Verona and Francesco Giuntini. Raimondo reported about a meeting they had in the residence of the senator and poet Domenico Venier. On that occasion he and Benedetti tested Giuntini’s astrological preparation:

We gathered at Mr. Domenico Venier’s place; his magnificence [came] first, followed by the most excellent Mr. Giovanni Battista Benedetti, many other gentlemen, myself (Annibale Raimondo), and finally the ex-reverend father Pacifico of Florence (now, as an ex-friar, known as ‘excellent Mr. Francesco Giuntini’). As soon as the latter arrived, he was given the simple astrological chart of the revolution of the magnificent Venier, without any written indication around or below. The good father took countless and endless texts and aphorisms out of his scapular. He related them to the revolution as good as a physician might give prescriptions to sick people by saying ‘God might help you.’ Since the most excellent Mr. Benedetti and myself laughed uncontainably—thereby making the father believe that he could not have better done—the good father, who was already trotting, was spurred by our laugher to gallop so quickly that it became extremely difficult to bring him back to silence and prevent him from telling more stupidities.[37]

An astrological report by Benedetti, cast for Carlo Emanuele I (Turin, 19 October 1589), is still extant and preserved in the Civic Library of Turin (Coss. 4, ff. 1r-2v). It contains a day-by-day personalized astrological forecast for the month of November 1589. The days are qualified with adjectives such as “buono” (good), “mediocre,” or “cattivo” (bad), but some are treated more specifically (the 9th of November is indicated as apt to “negotii ingeniosi,” ingenious endeavors, whereas the 10th of November as “buono in cose femminili ma nel resto cattivo,” that is, bad except for women’s affairs). Benedetti signed this astrological letter as “Matematico e Astrologiaro.”[38] This signature shows that his “professional” profile could vary depending on circumstances, since it depended in part on the kind of advice requested from him.

In the concluding letter of the Diversae speculationes, Benedetti envisaged a reform of astrology. He directed this letter to a German correspondent whose name he awkwardly Latinized as Volfardus Aisestain.

As for the question whether or not I regard as true all that is written in the books of judicial astrology, I respond that I do not. I even believe that much is wrong […]. But you will be informed about all this in a special tract of mine, about which I told you on another occasion. In it, you will find many things I have proven through the evidence of many observations. I intend to publish that tract along with some other speculations of mine, if only I will have enough time to do that, before I meet the body of the adverse Mars as indicated by my horoscope. This is going to happen in 1592.[39]

This passage concludes his major work. In it, Benedetti predicted, using astrological means, his own death for the year 1592, but he actually died in January 1590.[40] This fact aroused some doubts about his proficiency as an astrologer, especially from his successor as court mathematician, Bartolomeo Cristini.[41]

To sum up, Benedetti’s persona and work had various facets, his interests ranging from mathematics to cosmology and from natural philosophy to literature. In a certain sense, he can be seen as a Renaissance polymath. However, his profile can be better encompassed by the title of “mathematicus,” as long as we do not take it too restrictively. A Renaissance mathematician like Benedetti was an engineer and a technical inventor, as well as a theoretician and a natural philosopher; someone with teaching and civil duties who served as a counsellor, also for astrological matters. Being a court mathematician implied benefiting from high recognition and visibility in society. Thus, this professional and intellectual appurtenance had nothing to do with the rather low acknowledgment that mathematicians received at universities, where physicians, lawyers, and theologians were higher placed and received better salaries.[42] The cultural environment of Turin, with which Benedetti interacted in the most important years of his career, shall be addressed in the next section.

1.2 Benedetti’s Works and Publications

Benedetti published his first work at the age of 23, the Resolutio omnium Euclidis problematum (Resolution to All of Euclid’s Problems, Venice 1553), which offered the solution to “all” geometrical problems using a compass with a fixed opening. The work reacted to a challenge that emerged from a controversy between Niccolò Tartaglia and Lodovico Ferrari in the years 1546–1548 and inserted Benedetti into the scientific debates of his time. One year earlier the astrologer Luca Gaurico had already paid homage to him, including in his Tractatus astrologicus a horoscope of the promising mathematician cast by Gaurico himself.

In 1554 Benedetti published a Demonstratio proportionum motuum localium contra Aristotilem et omnes philosophos (Demonstration Concerning the Proportions of Local Motions against Aristotle and All Philosopers), which is not as famous for its polemical verve as for the presentation of an innovative theory of fall. As we have discussed in the preceding section, in this treatise Benedetti developed a theory of the motion of fall, first proposed in the dedicatory letter of the Resolutio of 1553. Benedetti maintained that bodies of the same material fall through a given medium with the same speed and not with speeds in proportion to their weights, as Aristotle and his followers claimed. Benedetti tried to overcome the fallacies of the Aristotelian theory of fall by employing the Archimedean concept of buoyancy, assuming that the motion of fall depends on their specific rather than absolute weight. As we have also discussed above, in the second edition of the Demonstratio, published in Venice in 1555,[43] Benedetti argued that the resistance incurred by a falling body in a medium depends not on its volume, but on its surface area. This is also the view that he presented in the Diversae speculationum mathematicarum et physicarum liber, published in Turin in 1585. He explained the acceleration of the motion of fall in terms of an increasing impetus of the falling body. Such examples show how he dealt with new challenging problems, which were difficult and sometimes impossible to solve using the mainstream theories of his time, by bringing forth and promoting new ideas.

After the Resolutio omnium Euclidis problematum and the Demonstratio proportionum motuum localium, composed when Benedetti was still in Venice, the next extant works stem from the time when he had already settled in Turin. First, he composed two works on gnomonics, one in Italian and one in Latin. The former is a manuscript preserved in the Civic Library of Carignano (Turin, Italy), entitled La generale et necessaria instruttione per l’intelligentia et compositione d’ogni sorte [di] Horologij Solari, which was presumably written between 1567 and 1573. The latter was printed under the title De gnomonum umbrarumque solarium usu liber (1574). Here Benedetti dealt at length with the construction of sundials with faces of varying inclinations and also with cylindrical and conical surfaces. At ff. 107r-v one finds a discussion of a sundial that perhaps can still be seen today on a wall of the Royal Palace in Turin.[44]

In 1574 Benedetti also wrote about a trigonometrical measuring instrument of his own invention, Descrittione, uso, et ragioni del Trigonolometro. It was never printed and is preserved in manuscript form in the Civic Library of Carignano along with the Italian work on sundials, Intelligentia et compositione d’ogni sorte [di] Horologij Solari.[45] His next scientific treatise, De temporum emendatione opinio (1578), proposed correcting and reforming the calendar. In 1578 the duke initiated a public disputation at the University of Turin where Benedetti argued with Antonio Berga about whether there was more water or more land on the earth, following an argument by Alessandro Piccolomini. The views which Benedetti brought forth against his opponent were published in Turin in 1579 under the title Consideratione… d’intorno al discorso della grandezza terra et dell’acqua del eccellent[e] sig[nor] Antonio Berga. This polemic was renowned, as can be seen in the Italian translation and commentary of Sacrobosco’s Sphere by the theologian, astronomer, and astrologer Francesco Giuntini in Lyon: “The excellent philosopher, Mr. Giovanni Battista Benedetti, mathematician to the serene duke of Savoy, resolved this question very aptly, arguing against the philosopher Berga, a famous reader at the University of Turin. The latter argues against Mr. Piccolomini that there is more water than earth. Benedetti defends the opposite view, which corresponds to truth: that there is less water than earth.”[46]

Next came Benedetti’s defense of the reliability of the mathematical computations underlying astrological predictions in the context of a heated polemic on this issue that burst out in Turin 1580–1581. Benedetti first communicated his views in Italian, in epistolary form: Lettera per modo di discorso… all’illustre sig. Bernardo Trotto. Intorno ad alcune nuove riprensioni, et emendationi contra alli calculatori delle effemeridi (Letter in the Form of a Discourse… Addressed to the Illustrious Mr Bernardo Trotto Concerning Some New Criticism and Corrections against the Ephemerides Calculators) (1581). Benedetti later included a Latin translation of this letter in the Diversae speculationes (1585).[47] His commitment to astrological practice is testified to by an astrological report he wrote for Carlo Emanuele I, a handwritten letter (Turin, 19 October 1589) preserved in the Civic Library of Turin (Coss. 4, ff. 1r-2v).[48]

Finally, Benedetti had his major work, Diversarum speculationum mathematicarum, et physicarum liber, printed in 1585. It was issued again under slightly different titles in Venice in 1586 (Speculationum mathematicarum et physicarum tractatus) and, posthumously, in 1599 (Speculationum liber).

Two of Benedetti’s manuscripts, preserved in the Biblioteca Nazionale Universitaria of Turin until 1904, are irreparably lost due to a fire that burst out in that year, destroying many valuable manuscripts. The first one was a collection of his letters, Lettere di Giovanni Battista Benedetti, Veneziano, matematico del Duca Emanuele Filiberto e Carlo Emanuele I, in risposta ai quesiti fattigli dal Duca e da altri personaggi intorno alla matematica, fisica, musica e filosofia.[49] The second one held similar content and was entitled Lettere di Giovanni Battista Benedetti in risposta a quesiti di fisica e matematica (Letters by Giovanni Battista Benedetti answering questions on physics and mathematics).[50]

Reprints of Benedetti’s works are rather scarce. Excerpts on mechanics from Benedetti’s work were included by Stillman Drake and Israel Edward Drabkin in their Mechanics in Sixteenth-Century Italy: Selections from Tartaglia, Benedetti, Guido Ubaldo and Galileo (Madison, Wisc.-Milwaukee-London, 1969). Carlo Maccagni’s Le speculazioni giovanili “de motu” di Giovanni Battista Benedetti (Pisa, 1967) includes excerpts from the dedicatory letter of the Resolutio omnium Euclidis problematum and the text of the two editions of the Demonstratio proportionum motum localium contra Aristotilem et omnes philosophos.

Footnotes

[1] Bordiga 1985, 752: “Habbiamo creato, fatto et costituito, facciamo creamo et costituiamo il detto Giovan Battista de Benedetti con tutti i suoi figliuoli maschi e femine legittimi, et naturali, nati et che nasceranno, et saranno procreati di legittimo matrimonio, con tutti loro posteri et heredi et successori in perpetuo veri nobili del Sacro Romano Imperio et nostri, et per tali li chiamiamo et dicchiariamo per dabondante (ancora ch’egli insieme coi suoi predecessori siano nobili e nati di antica prole nobili come siamo benissimo informati).”

[2] Stumpo 1998, 138.

[3] Bordiga 1985, 601.

[4] Bordiga 1985, 752: “Emanuele Filiberto per gratia di Dio Duca di Savoia Principe di Piemonte etc. Essendoche le attioni che tendono alla Virtù, come che da quella prendano accrescimento et perfettione, sono ammirate et havute in pregio: così gl’huomini che in quelle di continuo si essercitano vengono da ogniuno istimati et tenuti in particolare consideratione, la onde havendomi sempre fatto conto delle persone che dirizzassero ogni loro pensiero al bene operare, et quanto più si potrà, cercassero col mezo delle scienze, et arti liberali sicure et vere guide alla virtù di venire alla cognizione di esso doppo l’haver noi ricercato che in questo ne sotisfacesse, massime nelle discipline matematiche. Al fine ci è pervenuto nelle mani il nobile messer Giovanni Battista de Benedetti venetiano, nostro mattematico il quale havendo consumato la maggior parte dell’età sua nelle bone lettere et studij di filosofia, et fatto professione delle dette mattematiche, et così divinamente et per eccellenza riuscito che si può dire in quelle (tra gl’altri) essere singolare cosa che si porge tal contento, et la sua servitù a noi molto grata tale soddisfattione che lo giudichiamo degno che partecipi de gl’honori dovuti alle sue virtù acciò che gl’accresca l’animo di perseverare et altri siano invitati a seguitare li suoi vestigij.”

[5] This is why Brahe was not and could not desire to be imperial mathematician to Rudolph II, as has often been wrongly thought. See Voelkel 1999.

[6] Brahe 1916, 250: “Accedit et hoc, quod Stellula illa Cassiopeae in toto Orbe Terrarum ob siccas illas fumositates interpositas non tam splendide apparuisset atque haec Nova, si sub hac sola constitissent, et non reliquas illi vicinas pari modo attingissent, lumineque insueto auxissent. Hoc vero ultimum egregie et solerter ex excellentissimo Philosopho IOHANNE BAPTISTA BENEDICTO PATRICIO VENETO in praeclaro illo Opere quod de speculationibus Mathematicis et Physicis inscripsit, circa finem inter Epistolas eius evidenter et dilucide, Geometricis rationibus demonstratur. Ubi ad hunc ipsum Annibalem Raimundum scribens, absurdum, quod ex eius falsa assumptione necessario sequitur, dilucide ostendit.”

[7] See Mosley 2007.

[8] Gaurico 1552, f. 76r.

[9] Bordiga 1985, 588.

[10] Bordiga 1985, 604–605.

[11] Bordiga 1985, 593–595.

[12] For the broad European context of patronage and the arts in the Early Modern Period, see Bedini 1999, Moran 1981, and Moran 1991.

[13] See Roero 1997 and Mamino 1989.

[14] Maccagni 1967a, 353–354.

[15] Benedetti 1585, 277–278.

[16] Benedetti 1585, 205–210.

[17] Benedetti’s advice on the calendar reform is preserved in the Biblioteca Apostolica Vaticana under the signature cod. Vat. lat. 5645, 148r–150r. See Ziggelaar 1983, 211–214.

[18] Baldassar Castiglione, Il libro del Cortegiano, ed. Walter Barberis (Torino: Einaudi, 2017).

[19] Benedetti 1585, 417–418.

[20] Omodeo 2012b.

[21] Lomazzo 2006, 177–178, III, 19, “Del Sig. Gio. Battista Benedetti Matematico”:

“De la Filosofia nasce e discende
La prudenza e ’l saper de gli intelletti;
Co’ quali essendo nel dispor perfetti,
A ognuno suo diritto e sua ragion si rende
Di questa sì gran parte se ne prende
Il saggio e raro al mondo Benedetti,
Che d’agguagliarlo in vano è chi s’affetti:
Tanto sublime suo valor s’estende.
Però tanto godo io che sì gli piacque
La mia pittura, e perciò egli volse
L’ora et il punto nel qual nacqui al mondo.
Splendor di questa etade al tuo suon tacque
Ogn’un de l’arte tua, e altrove volse
Il suo dir vinto dal tuo sì profondo.”

[22] Benedetti 1553, f. 5r: “[…] huc usque progressus sum (Deo duce) sine monitore praeceptoreque ullo, nullum gymnasium unquam, nullamque scholam frequentavi, neque hoc studui, quod vulgus solet (sed absit verbo arrogantia) pro tempore in scholis transacto, eruditionem estimare, ac septennario finito finem studiis imponere, sed dum vivo, illa prosequi.”

[23] One reads in the preface ad lectorem of the Diversae speculationes the following declaration: “In his autem meditandis, ex arithmeticis authoribus quos inspexi praecipuus fuit Nicolaus Tartalea, quippe quem fere omnia ab aliis scripta collegisse constat, nec alios ex praecipuis quos legere potui omittendos duxi, inter quos sunt Hieronymus Cardanus, Michael Stifelius, Gemma Frisus, Ioanna Novimagus, Cuthbertus Tonstallus, caeterique huiusmodi.”

[24] Benedetti 1553, f. 5v: “Caeterum quia cuiusque quod suum est reddi debet, nam et pium et iustum est, Nicolaus Tartalea, mihi quatuor primos libros solos Euclidis legit, reliqua omnia, privato et labore et studio investigavi, volenti namque scire, nihil est difficile.”

[25] Bordiga 1985, 588 (4).

[26] Benedetti 1553, f. 5v: “Adde quod Mathematicae disciplinae, neque tantum requirunt splendorem, neque si quis peritus linguarum contendat ad elegantiam rem reducere, egregium quid effecerit, quia mutato usu Mathematicae loquendi, ipsiusque scientiae terminis, sensum facile perturbaverit, et ex nihilo nihil apprehensum obtinuerit. Quare morem scholarum sequutus, obstentatione elegantiae explosa, verbis nudis in demonstrationibus usus sum, hac in parte veterum vestigia sequutus, qui nudis verbis scientias resque ipsas docebant, quem modum docendi, nobis devastarunt scioli vel potius circulatores, garruli, rebus ipsoque iudicio destituti, garrulitate siquidem apud vulgus, laudem summam consequuntur, et pro doctis circunferuntur, nec mirum, cum scientiarum perfecta exquisitaque perita, paucissimis detur, non obstante quod multi permulta de omnis generis et scientiis et artibus scribant, permultaque garriant, fucis suis, et ampullis imperitorum oculos perstringentes […].”

[27] Archimedes 1543.

[28] Benedetti 1553, f. 10v. See Maccagni 1967a, 338–340 and Maccagni 1967b, 14–15.

[29] Taisner 1562, see the discussion in Maccagni 1967a, 344–455, n. 13.

[30] Benedetti 1574, f. 4v.

[31] Maccagni 1967b, 20–21, and 20, n. 14.

[32] Maccagni 1967b, 20–21: “Memini eum [eruditissimum Doctorem Petrum Arches], post varia et diversa colloquia utro citroque inter nos habita, mihi retulisse quamplurimos Romae, conspecta mea illa propositione quae ultra reliquas tuae R[everende] D[omine] [Guzman]a me mittebatur, valde mirari solitos me addidisse illam neutiquam esse iuxta mentem Aristotelis, idque ab eis dictum ubi meam demonstrationem attentius considerarunt.

Ne vero Aristotelem ullo modo errasse concederent, cum illum non infra humanae conditionis terminum habeant, sed potius veluti coeleste quoddam numen sibi proponant, censeantque nefas esse si vel latum quidem unguem ab eo quis dissentiat, in hac potius haeresi fuisse, ac etiamnum esse, ut me germanum et genuinum sensum Aristotelicae opinionis nequaquam ex authoris mente assecutum existiment.

Ego vero ne mihi diutius talis impingatur error, neve quid maxime super hac re sentiam, aut dissimulem, aut reticeam, statui, hoc novo libello edito, meam sententiam clarius aperire, ut omnes intelligant me et Aristotelem ipsum antea recte intellexisse, et non temere hoc in loco ab eo discrepare, quod sane quanquam invitus facio (nec tamen libenter a tanto viro diversum sentio, quippe qui norim quam ille praeclarus extiterit in omni doctrinarum genere), docet tamen maiorem ratione veritatis habere, quo ipsemet facendum censuit, quam inquit in Ethicis: ‘Amicus Plato, amicus Socrates, at magis amica veritas.’”

[33] Biagioli 1989, 49–50.

[34] De Pace 1993.

[35] Ptolemy 1984, 35–37.

[36] A very informed case study on astrology at Italian Renaissance courts is Azzolini 2013.

[37] Raimondo 1574: “Ritrovandosi nella camera del Clariss. M. Dominico Veniero prima la sua Mag.[,] lo eccellentissimo M. Gio. Battista Benedetti, molt’altri gentilhuomini, et Annibale Raimondo, che son quel io, vi sopraggiunse al’hora il Reverendo Padre Frate Pacifico Fiorentino de gli bene inculati, adesso per essersi sfratato lo Eccellente M. Francesco Giuntini, alquale, subito giunto, fu dato in mano la figura simplice del cielo della Revolutione del detto Mag. Veniero, senz’altra scrittura intorno, né appresso, il buono padre alhora mise mano al suo scapolario et cavò fuori testi, et afforismi senza fine, et senza fondo, allegandoli tanto a proposito della Revolutione, quanto facea quel buon medico le ricette che ’l dava ai suoi infermi, quando le dicea Dio te la mandi buona, et perché lo Eccell. M. Gio. Battista Benedetti et io se smassellavamo dalla risa, ben però in modo di maravigliarsi, come non fusse possibile a dir meglio di quello che dicea sua paternità, il buon padre per il nostro ridere sì come prima andava trottando, si misse a correr’ de modo che fu gran fatica a poterlo tenere et farlo tacere che’l non dicesse più minchionerie.” Cf. Corradeschi 2009, 111, n. 46. On Raimondo and Giuntini, see Ventrice 1989, 140–145.

[38] Roero 1997, 57–58.

[39] Benedetti 1585, 425–426: “Circa vero id de quo me interrogas, scilicet, utrum putem omnia vera esse, ea quae scripta reperiuntur in libris Astrologiae iudiciariae, respondeo quod non, imo puto plurima falsa esse […]. Sed diffusius haec omnia videbis in meo illo particulari tractatu, de quo tibi alias dixi, in quo multa videbis, quae omnia ab experientia, ex multis a me observatis, comprobata sunt, quem quidem tractatum cum quibusdam aliis meis speculationes in lucem producere cupio, si fieri poterit, antequam ad directionem mei Horoscopi cum corpore Martis Anaeretae perveniam, quae quidem directo circa annum millesimum quingentesimum nonagesimum secundum eveniet.”

[40] Benedetti was not the first mathematician who tried to forecast his own death. Among his predecessors are famous the cases of Johannes Stöffler and Girolamo Cardano. Cf. Omodeo 2014b, 3–4.

[41] Vernazza 1783, 16–18.

[42] On the lower status of mathematicians, see Henry 2011.

[43] Benedetti [1554] 1555, see Benedetti 1985.

[44] Roero 1997, 47.

[45] Clara Silvia Roero published Benedetti’s letter to Carlo Emanuele I (Turin, 19 October 1589), the index of the manuscript on gnomonics, as well as an excerpt from the manuscript on the mathematical instrument trigoniometro as appendices II and III of Roero 1997.

[46] Giuntini 1582, 95–96: “La qual questione ha resoluta molto dottamente lo eccellente filosofo, il signor Giovambattista Benedetti mathematico del serenissimo signor Duca di Savoia, contra il filosofo Berga, famoso lettore nella università di Turino: il quale contra l’opinione del signor Piccolomini defende che l’acqua è maggiore della terra: e il Benedetti defende il contrario in favore della verità: cioè che l’acqua è minore della terra.”

[47] Benedetti 1585, 228–248, “Defensio ephemeridum.”

[48] See Roero 1997, Appendix I.

[49] Peyron 1904, 73–74, Codex 83, N. II. 50.

[50] Peyron 1904, 95, Codex 94, N. III. 27.

2. The Subalpine Environment

Benedetti’s life, work, and reception are indissolubly linked to Turin and the Duchy of Savoy. As one reads in the Diversae speculationes, he chose to live in this town until the end of his life. There he benefited from the patronage of Duke Emanuele Filiberto (Figure 2.1) and, as a consequence, from a prominent social position and recognition. In the dedicatory epistle of his major work, addressed to Filiberto’s successor, Carlo Emanuele I, Benedetti extolled the merits of the deceased duke, who had invited him to Turin almost two decades earlier:

Nineteen years have passed since I was sent for by a letter of the most serene [Emanuele Filiberto] father of Your Highness [Carlo Emanuele I] and I moved from the town Parma to this municipality. Upon my arrival, he received me so humanely, and later I met with so much generosity as a reward for my services, that I began to desire vehemently that I could spend the rest of my life under his authority.[1]

As one reads, Benedetti and Emanuele Filiberto were so close that the patron even wanted his court mathematician to accompany him during his periods of residence in the countryside. On such occasions they often discussed scientific matters:

His benevolence toward me, as well as my respect toward him, consolidated through the time we spent together, and our familiarity [grew] to the point that the duke wanted me to accompany him when he resided in the countryside. [He] often [even invited me] to stay with him overnight. In that time he discussed mathematics with me. He used my work in order to learn those sciences, asking questions on arithmetic, geometry, optics, music and astronomy [astrologia].[2]

Emanuele Filiberto’s passion for mathematics was well known in his day. The Venetian ambassador to Turin, Giovanni Correr, reported on this singular aspect of his personality in 1566:

That Duke is no man of letters but he loves the virtuosi. Hence, he has many of them by him; he likes to listen to their reasoning and he asks them questions. However, there is no subject that delights him more than mathematics, as this science is not only apt but also necessary to the profession of military commander.[3]

Fig. 2.1: Portrait of Emanuele Filiberto from Tonso, De vita Emmanuelis Philiberti (1596). (Biblioteca Nazionale Universitaria di Torino)

The duke’s passion for science and his special relation with his court mathematician is further confirmed by the Venetian ambassador Giovanni Francesco Morosini, who mentioned Benedetti in a speech delivered to the Senators of Venice in 1570:

The duke of Savoy has a wonderful mind apt to every kind of science. However, he did not learn the sciences [le lettere] with the diligence that is necessary to become an expert, as his passion has always been the profession of war […]. But since mathematics is very useful and [even] necessary to professional warfare, His Excellency [Emanuele Filiberto] learns [mathematics] with much pleasure and knows more of it than the average man. He is aware that to receive substantial knowledge in any science one has to be in contact with it and learn it continuously; therefore a certain Mr. Giovanni Battista Benedetti of Venice imparts to him a lesson either on Euclid or on another writer of those sciences every day. In my opinion, as well as according to many other gentlemen, he is the most excellent scholar in this discipline in our times. The duke likes him very much. In fact, not only has [Benedetti] mastered this science, but he is also able to transmit it very well to others in his lessons.[4]

However, Benedetti’s activities in Turin cannot be fully grasped if we limit our consideration to his relationship with the dukes. Rather, we should consider the wider political and cultural environment in which this relationship was established.

2.1 Turin’s Economy and Politics between Italy and Europe

From the point of view of economic exchanges as well as of the European balance of power, Turin was located in a delicate and strategic position. It was in fact an obligatory station on the commercial road connecting Italy and France through the Val di Susa. For many centuries it had served as a transit point for merchants from Liguria, Lombardy, and Piedmont on their way to Lyon and the French and Flemish markets, and vice versa. Merchants were not the only visitors, as scholars from France, Flanders, and the British Islands began their iter Italicum from Turin. Turin was also the first station in Italy of Erasmus of Rotterdam, a key figure of the European Renaissance. On that occasion, on September 4, 1506, he received an “Italian” degree in theology from the University of Turin.

Its intermediate position between Italy and France made the town relevant not only from the point of view of economics and culture but also for military reasons. When Francis I of France and Charles V of Spain fought over Italian and European supremacy, Turin acquired fundamental strategic importance. The French army conquered it in 1536, together with most of Savoy and Piedmont, at the expense of Charles II of Savoy, brother in law to Charles V of Spain. The King of France made Turin the most important center in the region and a bulwark that was fundamental for consolidating his position on the Italian peninsula. Some of the political and administrative reforms promulgated by the new ruler were long-lasting. The most important of them were the creation of a parliament and of a Camera dei Conti for the supervision of public finances.[5]

Emanuele Filiberto, known as “testa di ferro” for his energy and capacity in military affairs, retook Turin on the battlefields. He conducted the campaign against the French as a captain in the service of the Habsburgs. In 1553 he was the supreme commander of Charles V’s imperial army in Flanders and was nominated governor of the Netherlands by Philip II in 1556. His victory in the battle of Saint-Quentin led to the Peace of Cateau-Cambrésis (1559), according to which the Savoy and Piedmontese territories had to be restored. The French agreed to give them back to the dukes of Savoy with the significant exception of five fortified towns, occupied by their troops.

Turin was one of them. Therefore, it took some years before it was eventually returned to Emanuele Filiberto in 1562. In 1563 the duke entered the town and choose it as the new capital of his duchy instead of Chambéry. In this manner, he conferred an Italian identity to his duchy. This transfer set in motion political, social, and economic transformations, which were still in progress when Benedetti arrived in Turin in 1567. Moreover, the Piedmontese territories were politically fragmented. Apart from the centers under French control (Chieri, Pinerolo, Chivasso, and Villanova d’Asti), the region included the Marchesati of Monferrato and of Saluzzo. Moreover, the county of Tenda, connecting Piedmont with the Savoy possession of Nice, was an imperial fief. As for Geneva, a former possession of Savoy, it had become the “Jerusalem” of the Calvinists and would never be regained.

Within this difficult territorial and political constellation it was imperative that Emanuele Filiberto reestablish his authority after years of wars and foreign domination. In the European context, this meant striking a balance between the interests of Spain and France, who both wanted to annex the territories of the duchy either as a part of France or as a continuation of the Milanese territories. Piedmont was already split into a faction favorable to the French and one favorable to the Spaniards during the years of the war, and this division would also continue during the reigns of Emanuele Filiberto and Carlo Emanuele I.[6]

International diplomacy was comprised of marriage politics. Emanuele Filiberto received a French spouse, Margret of Valoys, daughter of Francis I of France and sister of King Henry II. This meant a strong political and cultural link to Paris. Margret was well known for her patronage of literati and artists, among them the poets of the Pléiade, Pierre de Ronsard, and Joachim Du Bellay. However, her son Carlo Emanuele I married a Habsburg, the daughter of Philip II of Spain, infanta Catherine Michelle, who arrived in Turin in 1585. This liaison was strongly encouraged by the pro-Spain party. Its leader was Andrea Provana of Leyní (1511–1592), with whom Benedetti was well acquainted. Four of the letters included in the epistolary of the Diversae speculationes are addressed to him. Benedetti judged the importance of his correspondence with this exponent of the Savoy aristocracy to be second only to those with Emanuele Filiberto (first epistle of his collection) and Carlo Emanuele I (second epistle). We can assume, taking his origins as a guide, that Benedetti supported Provana’s pro-Spain party.

Fig. 2.2: Portrait of Carlo Emanuele I by Francesco Maria Ferrero di Labriano, Augustae Regiaeque Sabaudae Domus Arbor Gentilitia (Turin, 1702), p. 174. (Biblioteca Nazionale Universitaria di Torino)

In his relations to other Italian States the duke also followed a politics of balance. He was particularly keen on having good relations with Venice, which he visited in 1566 and 1574. On the latter occasion he was even endowed with the title of patrizio of the town. In turn, a Venetian embassy was established in Turin. The Savoy relation with Rome was also cordial. The papacy regarded Turin as a bulwark to stop the dissemination of reformed ideas in Italy, especially from the Swiss cantons. For his part, Emanuele Filiberto saw “heresy” as a danger to the unity of his state and his authority. Therefore, on matters of faith, the Roman interests and his own converged. Against the background of the confessional tensions of those years, his support for the Jesuits is comprehensible. Yet he was no fanatic of orthodoxy. He was influenced by the Imperial policy of mediation, as is shown by his ratification of a compromise with the Valdesans in 1561, in which he accorded to them religious freedom in their valleys.

The ties with Rome and Venice were reinforced through Savoy support for expeditions against the Turks. In 1565 Andrea Provana was sent with three galleys to Malta, as the court historian Pingone recounted in his history of Turin, Augusta Taurinorum:

When Malta was besieged by the Turks, in June 1565, duke Emanuele [Filiberto] sent Andrea Provana of Leinì with four well-equipped triremes to bring supplies to the isle together with triremes from the Pope, Spain, and other [states]. First, Provana [Leniacus] arrived and assessed the difficulties. Then, he conveyed others [to the battle] and broke the siege with divine favor. The holy and vigorous order of the knights of Jerusalem was liberated under the superior command of the French Jean of Valetta. Public demonstrations of immense joy and pious celebrations of thanks to God for the victory were displayed in Turin.[7]

In 1571 Provana was enlisted to defend Cyprus and contributed to the “holy” victory in the battle of Lepanto.

In 1571, when duke Emanuele [Filiberto] ruled over Turin and a confederation was established between Pope Pius V, the king of Spain and the Venetian Republic, he was asked to command the fleet with everybody’s agreement. But he had to renounce the offer owing to the present danger to his country engendered by local conflicts. [In his place] John of Austria, offspring of emperor Charles V, of great spirit and promising youth, was made commander. Chief Andrea Provana of Leyní joined this expedition with three triremes. It was fought near Nauplia with the support of the Greeks. The Christians had hardly two hundred triremes and the Ottomans more than three hundred. The battle [Mars] was undecided for a long time but finally victory was given to the Christians, with the favor of God or even as a miracle. Provana, who fought bravely in the commanding trireme, was hit by a gun bullet and could hardly escape under the protection of a galley. One of the [Savoy] triremes, named Margara, was scattered and sunk into the depth; [another one], Pedemontana, was saved many times from the enemy. That victory was celebrated in Turin with thanks given to God and holy days set aside for the people.[8]

On these occasions Benedetti served as an advisor to Provana. Three of the four epistles of the Diversae speculationes addressed to him deal with mathematical issues related to navigation. As one reads, Benedetti undertook to give Provana suggestions concerning navigation and the employment of navigational instruments.[9] The first epistle is entitled Per eundem parallelum absque correctione semper navigari non posse ubi notantur Petri Nonii lapsus in correctione erroris navis et alii Petri Medinae errores (That one cannot always navigate along the same parallel without correction, where an error by Petrus Nonius concerning the correction of the deviation of a ship and other [errors] by Petrus Medina are considered). The second and the third letter deal with a navigation instrument invented by Benedetti based upon the design of Gerardus Mercator (Figure 2.3). They are a description of the instrument accompanied by technical drawings and an explanation of its use. They are entitled De armilla nautica (On the armillary nautical sphere) and De usu armillae nauticae (The utilization of the armillary nautical sphere), respectively.[10] As one reads, the letters follow private discussions with Provana on the difficulties linked to navigation using nautical maps.

Fig. 2.3: An armillary nautical sphere invented by Benedetti for Andrea Provana for navigation purposes, presumably in the Savoy military expeditions against the Turks. (Max Planck Institute for the History of Science, Library)

2.2 Civil Reforms and Military Policy

Emanuele Filiberto and his son were very different rulers. While the court of the former has been depicted as “funzionale, ristretta e popolata di homines novi” (functional, small and composed of homines novi) the latter’s court was “fastosa, aristocratica, centro propulsore di una politica culturale oramai intensamente barocca” (pompous, aristocratic, irradiating center of a deeply baroque cultural politics).[11] Their common efforts were directed towards the consolidation of their state. Emanuele Filiberto implemented profound administrative, financial, and military reforms. He issued tax reforms and imposed the use of the vulgar tongue in official documents. As to his military policy, Emanuele Filiberto introduced the obligatory conscription of all men aged between 18 and 50 years. Thanks to this reform, which followed the Swiss example and Machiavelli’s theory, Emanuele Filiberto trained his subjects to defend their territories in case of invasion, disposed of many thousands of soldiers, and limited the use of mercenary troops.[12] Part of his defensive strategy was the erection of new fortifications in Nice, Bourg-en-Bresse, Saint-Julien, and Montmélian (see Figure 2.4). The construction of the cittadella of Turin was particularly important and rapid. It was achieved in only two years, between 1564 and 1566, and was celebrated by the official town historian Emanuele Filiberto Pingone in the following terms:

In that year [1564], the duke began building a fortification, which is commonly called the citadel, in the most sacred part of the town on the ruins of the temple of the divine Savior.[13]

On March 15, 1566, the citadel of Turin was finished after a few months of work. It was admirable with its five bulwarks, serving all military purposes and built according to the art of architecture. He [the duke] let it be blessed with religious and pious blessings (Archbishop Della Rovere was in charge of the rite). Soon he organized the defenses, entrusting them to Giuseppe Caresana of Vercelli, a subject of his [benemeritus] and a man very expert in the military art.[14]

Francesco Horologi was responsible for the construction of the citadel, designed by engineer Francesco Paciotto, whom Emanuele Filiberto recruited in Flanders. Its pentagonal structure, responding to recent developments in warfare, was the model for later fortifications, such as the citadels created by the same Paciotto in Antwerp (1567) and Parma (1591).

As often occurred during the Renaissance, the military-political function of the citadel had two sides. On the one hand, it served to defend the town from possible assaults from outside. On the other, it affirmed the supremacy of the dukes over the new capital and had the function of dissuading the subjects from claiming too much autonomy.[15] As Martha Pollak remarked, “Paciotto proposed a five-sided fortress, with three bastions oriented towards the countryside, defending the approach to the city from the west, and two bastions facing the city, ready to bring it under control in case of riotous uprisings against the duke.”[16] The new urban arrangement transformed Turin along with its political balance of power. During these changes the relations between the dukes and the local patriziato were often strained. In fact, all decisions had to be negotiated between civic administrators, state functionaries, and the court. Whereas the town council was eager to keep its medieval privileges, the dukes made the opposite effort of centralizing power in order to grasp control firmly in their hands.

Fig. 2.4: Fortification projects in a drawing by Benedetti’s follower as court mathematician, Bartolomeo Cristini. (Biblioteca Nazionale Universitaria di Torino)

A thorough knowledge of the surrounding territory through cartography, alongside fortification and military reforms, was also seen as an important element of defense. The intensity of mapping efforts in the Savoy in the second half of the sixteenth century is a noteworthy example. A large number of maps of great quality were made, for instance Forlani’s Savoia (1552), Boileau de Boullion’s work on the road from Lyon to Turin (1556), Nicolais’s maps Boulonnais (1558), Berry (1566) and Bourbonnais (1569)[17] and, above all, Giacomo Gastaldi’s Pedemontanae vicinorumque regionum… descriptio (1574). Many of the maps made in those years are still preserved in Turin, in the Biblioteca Reale, the Biblioteca Nazionale Universitaria, the Archivio di Stato, and in the wide collection of the Archivio Storico della Città.[18] Benedetti shared this interest in geography and topography. This especially emerges from some of his epistles, for instance those to the architect Gabriele Busca on topography and measuring instruments, to the imperial land surveyor (agrimensor) Anselm Rosenburg (presumably of the Bohemian aristocrat family Rožmberk) on measuring techniques, and to the Turin physician and natural philosopher Giovanni Battista Femello concerning cartographic errors about the position of islands, in particular Iceland.[19]

2.3 Engineering and Architecture

Countless engineers worked in Turin under Emanuele Filiberto and Carlo Emanuele I to implement the defenses and the hydraulic system, among them Francesco Paciotto, Ferrante Vitelli, Ascanio Vitozzi, and Vitozzo Vitozzi.

The leading Italian architect of that time, Andrea Palladio, visited Turin between May and June 1568. He might have been the architect behind the park of Viboccone (later known as Regio Parco) between the rivers Dora, Po, and Stura. Benedetti is said to have constructed his fountain there. It is also likely that Palladio gave Emanuele Filiberto advice on the organization of his popular militia according to the classical Roman model.[20] Later, he dedicated to the duke of Savoy the third of his four books on architecture, Quattro libri dell’architettura (Venice, 1570). This section deals with public constructions, streets, bridges, squares, basilicas, and gymnasia. In the letter to the reader, the author stressed the prestige deriving from public buildings, and the fruitful collaboration between Renaissance princes and architects: “[Public buildings] are bigger and more embellished with rare ornaments than the private ones; and they aim to be used by everybody. Therefore, through them, princes can display the greatness of their generosity to the world and architects have the occasion to show how capable they are through beautiful and wonderful inventions.”[21] The dedication to Emanuele Filiberto by Palladio was motivated by his “heroic spirit”[22] as well as by his interest in and deep understanding of architecture:

As your Highness is familiar with the most noble arts and sciences related to these issues [concerning architecture], you will have much pleasure and relief by considering the subtle and beautiful inventions of humankind as well as the true science of this art, which you understand very well and which has been brought to the most rare and almost absolute perfection. This is witnessed by the illustrious and royal buildings that have been constructed in many parts of your large and most happy state.[23]

Urban and military developments were accompanied by a flourishing literature on war and defense theory. Emanuele Filiberto was a great supporter and collector of such writings:[24] among other examples, Benedetti’s correspondent Busca authored the treatise Della espugnazione et difesa delle fortezze (On the conquest and defense of fortresses, Turin, 1585), which followed the Istruttione de’ bombardieri (Education of the bombardiers, Carmagnola, 1584). He would later publish the tract Architettura militare (1601) in Milan. Another acquaintance of Benedetti’s, Giacomo Soldati composed Discorso intorno al fortificare la città di Torino (Discourse on fortifying the town of Turin).[25]

In this context of military reforms and architectural changes aimed at transforming Turin into the capital of an absolutist state, the skilled engineers implementing the dukes’ vision gained high social recognition. An example is the career of the mathematician Cristini. In 1569 he was courtly librarian and later “calculatore” or “controlore delle fabriche,” that is, supervisor of architectural projects. In this capacity, he become closely connected to the celebrated military and civil architect Ascanio Vitozzi. On December 13, 1582, he became “re d’arme dell’ordine dell’Annunziata,” a honorific and administrative title that implied responsibility for the organization of courtly tournaments, feasts, and balls.[26] He entered the court as Benedetti’s successor in the position of ducal mathematician. Cristini’s career shows the enhanced status of Renaissance scientist-engineers bringing together mathematical and natural expertise, technical skills, administrative duties, and courtly honors. Benedetti addressed one of the letters of the Diversae speculationes to him (the one dealing with geometrical problems encountered in Ptolemy’s Geography) calling him “Bartholomeo Christino Serenissimi Sabaudiae Ducis apparitor.” The term apparitor can mean either “servant” or “functionary.” An unknown hand corrected this title in a copy preserved in the Royal Library of Turin, substituting it for the more emphatic “P[rim]o Feciali,” that is to say, “First Herald.”[27] This was in fact the most appropriate title for the “Roy d’armes.”[28] It is possible that this correction was inserted by Benedetti himself.

Fig. 2.5: Map of Turin in Benedetti’s times, from Pingone’s Augusta Taurinorum (1577). (Staatsbibliothek zu Berlin)

Benedetti interacted with architects and engineers, as can be seen in his correspondence. Four of the scientific letters included in the Diversae speculationes are addressed to the architect Busca. Their topics, however, are not strictly related to building or engineering. In fact, their topics range from meteorology to instruments, topography, and natural philosophy.[29] Benedetti was closely related to the ducal “architect and cosmographer” (Architetto e Cosmografo) Soldati, who had worked as a hydraulic engineer and architect in Milan and in Lombardy, and joined the Turin court in 1576. Benedetti held him in great esteem, judging by the dedication to him in one of the most technical parts of the Diversae speculationes, that is, the entire second book, which is entitled “Explanation on Operations of Perspective” (De rationibus operationum perspectivae).[30] Besides, Benedetti’s interest in engineering and measuring instruments emerges from many writings, especially from his work on gnomonics (1574) and from a manuscript analysing a measuring instrument that is preserved in the Biblioteca Civica di Carignano, entitled Descrittione, uso, et ragioni del Trigonolometro (1578).[31]

2.4 Intellectual Ferment: Arts, Literature, and Philosophy

Renaissance Turin was as appealing to mathematicians as to all other types of intellectuals, including artists, literati, and scholars in general. One could mention the names of two reputed theorists of art who were closely linked with the court: Federico Zuccari and Lomazzo. The former wrote L’idea de’ pittori, scultori, et architetti (The idea of the painters, sculptors and architects, 1607) and the latter Trattato dell’arte della pittura (Treatise on the art of painting, 1584), dedicated to Carlo Emanuele I. Lomazzo also dedicated to the duke of Savoy his collection of poems, Rime (1587), including one for Benedetti. Among the artists appointed by the duke, the Flemish Jan Kraeck or “Carracha,” who resided in Chambéry, not only painted; he also worked as a cartographer and made a large map of Turin in 1572 (385x397 cm) as part of a wider cartographic program.

Illustrious scholars came to Emanuele Filiberto’s court or to the reopened university, first in Mondovì and later in Turin. Among them, the man of letters from Ferrara, Giovanni Battista Giraldi Cinzio, was appointed to teach humanities. His collection of novelle, the Ecatommiti (Mondovì, 1565), ended with a long celebratory poem mentioning the most visible intellectuals of the Peninsula. Many of them were linked to the duke of Savoy, either as professors or as courtiers.[32] For instance, Francesco Ottonaio of Florence, who taught mathematics at the university in Benedetti’s years and exchanged views with him, is extolled for his expertise in astronomy, meteorology, and astrology:

My Ottonaio moves his feet towards you along with the others.
He received the gift of scrutinizing the heavens,
of knowing the reasons for warmth and coldness,
why the days are short or long,
and what layer veils the Sun making it dark,
the manner in which the year becomes adorned of beautiful flowers again,
what nativity is a sign of honor and merit
or of shame and disgrace,
and what is the star presiding over
a man’s state from his birth
until his vital light is extinguished
one circle after the other.[33]

Giraldi Cinzio did not mention Benedetti as he had not yet come to Savoy, but he included the Turin physician and professor of medicine Antonio Berga, with whom Benedetti would later enter a controversy over the proportion of water and earth in the terraqueous globe:

With his gentle and beautiful works
he tries to subtract his name from the oblivion,
defeating the stealing forces of greedy time.
I refer to my gentle Antonio Berga,
who shows the way to those who wish to learn
by writing his papers for the common good.[34]

Two famous authors who visited Turin between 1576 and 1579 are the poet Torquato Tasso and the philosopher Giordano Bruno. Both arrived in the town as fugitives and both enjoyed their stay. Tasso interacted with the cultivated elites. The Turin philosopher and physician Agostino Bucci appears as persona dialogans in three dialogues of his (Il Forno ovvero della nobiltà, Della dignità, and Della precedenza). His connection with the court is further confirmed by the dedication of the 1581 edition of the Gerusalemme liberata to the prince of Savoy.[35]

As for Bruno, he did not establish lasting contacts in town. He visited Turin in 1576 (or at the beginning of 1577) for the first time after abandoning the Dominican cloister of Naples, where he was accused of heresy. However, as he stated later in his Inquisition trial, “non trovando trattenimento a mia satisfattione, venni a Venezia per il Po [as I did not find sufficient means, I came to Venice along the Po].”[36] He visited Turin again in 1578 and went to Chambéry, where he spent the winter of that year as a guest of the Dominicans. On that occasion he possibly carried a booklet, now lost, entitled De’ segni de’ tempi, that he had printed in Venice and that probably dealt with the comet of that year.[37] We will refer to his possible involvement in some polemics on the comet of 1577–1578 later. It is hard to say whether Bruno and Benedetti ever met or were informed of each other’s views. In spite of the fact that they belonged to very distant milieus, there is some affinity between their outlooks. Both shared an aversion to Aristotle, the project of reforming natural philosophy, the support for the Copernican system, and other cosmological views.

The case of the philosophical poet Pandolfo Sfondrati also deserves our attention. He was active in Turin as a poet at the same time as Benedetti and authored poems that were often inserted in the first pages of books printed by the Bevilacqua printing house. Pandolfo made himself visible in Turin with celebratory poems that were included in important local publications, in particular in the works of the court historian Emanuele Filiberto Pingone: Augusta Taurinorum (1577), Inclytae Saxoniae Sabaudiaeque principum arbor gentilitia (1581) and Sindon evangelica (1581).[38] Hence, Sfondrati frequented the same courtly and cultural environment as Benedetti. It is likely that they discussed natural issues together, especially when considering that Sfondrati composed philosophically minded poems such as the Copernican Inferiora regi dum syderis omnia motu, which opens the Animadversiones in Ephemeridas by Benedetto Altavilla (Turin, 1580). There is also evidence that Benedetti was familiar with the Sfondratis, in particular with Paolo Sfondrati, who was a senator of Milan and an ambassador of Filippo II in Turin.[39] Moreover, Pandolfo Sfondrati authored the atomistic poem Democriti prohibent nosci corpuscula formas and a treatise on the tides, which he explained in mechanical terms as the result of the interaction of water particles warmed up by solar rays.[40]

2.5 Religious Policy

2.5.1 Pragmatic Counter-Reformation

The relics were moved from the old to the new capital: Christ’s shroud traversed the Alps together with the court. Religion was an essential stabilizing factor. According to the report of the political thinker Giovanni Botero, Emanuele Filiberto declared that piety was essential to guarantee his authority in the state: “Those people who are zealous in their devotion [to religion] are more moderate: in consequence, they obey their Prince better than licentious people.”[41] His religious politics were characterized by pragmatism. He undertook measures against the dissemination of the Reformation in his country and repressed the reformed communities only insofar as they jeopardized the integrity of the state or its relations with Rome or with other Catholic countries. The persecution of the Valdesans in the Alpine valleys, between 1559 and 1560, was part of a wider endeavor aimed at establishing a Catholic league that could favor his conquest of Geneva. However, when the prospect of an anti-Protestant confederation vanished, Emanuele Filiberto interrupted the aggression. The resistance of the Valdesans had been strong and persistent. It was a destabilizing factor for the duchy. A compromise was reached on June 5, 1561, when Emanuele Filiberto promulgated an edict, known as the Edict of Cavour, conceding to the Valdesans freedom of worship in their mountains. In exchange, he imposed the construction of new fortifications and strict military control of the Alpine passes.[42]

The other pole of Savoy religious politics was external. It concerned the regulation of jurisdictional issues with the Roman Church. A reason for friction was the taxation of the ecclesiastics, which Rome was reluctant to grant. The economic stake was high, since the estates belonging to the Church amounted to a third of the land; in some areas, it even reached values comprised between 40 and 70 percent. The ancient privileges of the dukes to select the candidates for the principal ecclesiastical positions had to be negotiated anew. It was only in 1573, under Gregory XIII, that these were confirmed. Finally, the introduction and the reinforcement of the Inquisition in the duchy fostered confessional uniformity but interfered with secular jurisdiction.[43]

The new Inquisition, established in the wake of the Council of Trent in order to counter the Reformation and reaffirm Catholic hegemony in Italy, had a centralized structure with ramifications for the entire Peninsula. The Holy Office lay at its heart. The various Italian states accepted it as a preventative measure against public disorder, especially against confessional conflicts such as the civil wars affecting France and other European areas. The Inquisition was a repressive control system binding Rome and the local powers. It was a compromise in which, in Adriano Prosperi’s words, “l’aiuto era offerto e richiesto in nome della conservazione del potere, quello politico dei principi e quello della corporazione ecclesiastica” (The aid was offered and requested in order to preserve the power—the political one of the princes as well as that of the ecclesiastical corporation.)[44] Nonetheless, the jurisdiction problem remained acute: what was the legitimacy of a foreign tribunal—the Roman one—trying and condemning the subjects of other countries? In Piedmont, the interests of Turin and Rome were guaranteed through the mediation of the Papal nuncio, who was obliged to inform both the duke and the Holy Office in Rome about Inquisition trials.[45] For their part, these authorities could intervene in the trials and had the right to give their consent concerning the opportunity to carry them out. However, the opposition to the establishment of the Inquisition was strong, particularly in the French-speaking areas. Relying on its Gallican tradition, the local church in Savoy did not accept a form of direct control from Rome.[46]

The relations between Emanuele Filiberto and the Papacy were not always calm, and became strained after the Cavour edict of tolerance in 1561. Its promulgation provoked the protests of Rome and the commission of the intransigent Cardinal Inquisitor, Michele Ghisleri, to the diocese of Alexandria as Bishop of Mondovì, with the aim of reinforcing religious control. The relations between this champion of orthodoxy and the pragmatic politics of the Savoy dukes were tense, including after Ghisleri was elected pontiff as Pius V in 1566.

2.5.2 Jesuit Colleges in Piedmont

In 1561 Emanuele Filiberto supported the foundation of the first Jesuit college in his territories, in Mondovì, where the university was located at that time. A second college was established in Chambéry (1564), which was the former capital of the duchy. Among the Jesuit teachers, Antonio Possevino is one of the most renowned. The future compiler of the Bibliotheca selecta (1593) sojourned in Piedmont between 1560 and 1562.[47] From 1564 to 1567 the future cardinal and inquisitor Robert Bellarmine was also there but could not be appointed professor of philosophy in the Jesuit college of Turin, opened in 1567, due to the intransigent opposition of the university against the attempts to transfer this chair from the university to the Jesuit institution.[48] Both in Mondovì and in Turin the Jesuits took over the so-called “public schools,” which essentially taught the rudiments of grammar.[49]

In those years, the chair of letters belonged to the Ferrara humanist Giovanni Battista Giraldi Cinzio, whose religious tendencies could be called Erasmian.[50] At the moment of the establishment of the Jesuit college in Turin, he sided with the humanistic legacy against their pedagogy. He paid the price of their expansion, as his chair was suppressed and transferred to the Jesuit college.[51] From 1567 to 1574, the Jesuits received 200 scudi per year to teach Greek and Latin grammar, humanities, and rhetoric to the youth (half of Giraldi’s salary). Thus, the humanist had to abandon Turin for Pavia in 1569. In a letter to the Florentine philologist Pietro Vettori (Pavia, March 20, 1569), he lamented his expulsion, although he expressed his gratitude to the dukes for the donation of 500 scudi for his leave.[52] He particularly protested that his teaching as a learned humanist was being substituted by the teaching of elementary grammar.

The opening of the Turin college set off enduring hostilities between the Jesuits, on the one side, and the university and the municipality on the other. According to Grendler’s reconstruction, between 1570 and 1572 the duke and the rector, Achille Gagliardi, made an agreement according to which nine chairs would be given to Jesuit professors. However, the project did not succeed owing to obstruction on the part of the town and the university.[53] In these struggles, the Archbishop Gerolamo della Rovere was among the most strenuous opponents of the Jesuits. The position of Emanuele Filiberto fluctuated. Initially, he supported the “reverend fathers” but later distanced himself from their educational projects. In 1575 he even reintroduced the teaching of humanities in the university. There is indirect evidence that Benedetti sided with the humanists in this anti-Jesuit controversy. In 1583 Francesco Patrizi, who belonged to the same Ferrara cultural environment as Giraldi Cinzio, asked him to support the candidacy of his friend Giovanni Giacomo Orgiazzo for the position of professor of humanities in 1583.[54]

Apart from the political interests at stake (the privileges of the town and of the university), the professors’ resistance concerned the contents of the teaching, as one reads in a document from 1593, “Raggioni perché non sia bene che gli Rev[erendi] Padri Gesuiti leggano la filosofia tutta, et la logica nel loro Comento, et si lasci a leggerli nello Studio et pubbliche scuole, come sempre insino a qui si è fatto” (Reasons why it is not good that the Jesuit Reverend Fathers teach all philosophy and logic in their commentary and are allowed to teach at the university and in public schools, as has been the case until now).[55] According to the academics, philosophy should be imparted to students as the fundamental tenet of the study of medicine. Therefore, the focus should be set on Aristotle’s natural philosophy and not on metaphysics and logic, as was the case with the Jesuits. Metaphysics, as one reads in the document, is the “last” and not the “first” part of philosophy. By contrast, the Jesuits began their teaching with the most abstract issues, e.g., the statute of ideas and universals, and divine ineffability (“utrum Deus sit in praedicamento” or “utrum Deus sit infinitus”).

2.5.3 Benedetti and the Counter-Reformation

What can be said about Benedetti’s attitude toward the culture of the Counter-Reformation emerging after the Council of Trent? We can assume that his scorn for Aristotelian philosophy was not only a dispute with the university professors of his day, but also with the theology-oriented Aristotelianism propagated through the cultural apparatus of the post-Tridentine Church. We have no evidence that Benedetti frequented exponents of the Jesuit order, in spite of their presence in Turin. Rather, we have evidence of his connections with scholars who were not in the mainstream of the official Catholic culture of those years.

Among others, he corresponded with Francesco Patrizi of Cherso, whose Platonism was regarded with suspicion and even censured in Rome. The closeness between them is witnessed by the fact that Benedetti acted as a mediator between the philosopher and the duke of Savoy on at least one occasion. When Patrizi issued his Della nuova geometria libri XV (Fifteen books on the new geometry, Ferrara, 1587) with a dedication to Carlo Emanuele I, Benedetti passed on a copy to his patron. Patrizi expressed his gratitude with the following words (Ferrara, April 6, 1587):

Very Magnificent and Excellent Signore,

I rejoice with your Lordship that you recovered from sickness quicker than believed. And I am very thankful to you for presenting my book to the very serene Prince and promising to inform me about his remarks after he has read it. If by chance the book will be forgotten, due to his many duties [negozii], I hope at least that you will remember me. If his High Serenity will give some sign that he appreciated it [my book], I will be very glad and I will be particularly grateful to your Lordship for your benevolence.[56]

In exchange, Benedetti sent him a copy of his discussion on the relative sizes of the elements of earth and water, as witnessed by a letter from Patrizi (Ferrara, 18 January 1588).[57]

The two scholars shared views on cosmology that were to be censured by the Inquisition in the 1590s. It is thus expedient to briefly recall Patrizi’s natural and cosmological views, as they are close to those Benedetti expressed in his writings, especially in the Diversae speculationes. Already in his De rerum natura libri I priores. Alter de spacio physico, alter de spacio mathematico (First Two Books on Nature, One on Physical Space and One on Mathematical Space, 1587), Patrizi embraced the neo-Stoic doctrine of the fluidity of the heavens, the infinity of space beyond the sphere of the stars, and the free motion of planets through cosmic space following an inner drive. He later expanded on that in Pancosmia, which is the cosmological section of his philosophical masterwork, Nova de universis philosophia (New Universal Philosophy, 1591). In it, he did not limit himself to asserting the infinitude of luminous space beyond the visible stars, to repeating the thesis of planetary self-motion, and to rejecting the existence of celestial spheres responsible for the transportation of the heavenly bodies. He also ascribed to Earth the daily motion around its axis without renouncing its central position in the cosmos and in the planetary system.[58] All of these theses—which, as we shall see, are also present in Benedetti’s Diversae speculationes—were censured by the Inquisition in the 1590s, in the course of an attack directed against Patrizi’s Nova de universis philosophia.

The censure of Patrizi’s philosophy occurred after Benedetti’s death—he was lucky enough to avoid witnessing the prohibition of theses to which he subscribed. On October 8, 1592, the Master of the Sacred Palace, Bartolomé de Miranda, and his fellow, Pedro Juan Saragoza, wrote a document of censure, attacking many passages and doctrines of the Nova philosophia. The same Saragoza would later be one of the two censors of Bruno’s work during his Inquisition trial in Rome.[59] The two censors of Patrizi did not limit their criticism to theology (especially Patrizi’s theses on the Trinity) but also scrutinized natural philosophy and cosmology. They rejected the idea that there is only one unique heaven and judged this to be an opinion at odds with accepted philosophical doctrines and against theology (In lib. 13 Pancosmias tenet unum tantum esse caelum). Furthermore, Patrizi was accused of following Copernicus, whose doctrine of terrestrial motion was seen as incompatible with the Bible:

In Pancosmia, Book 17, f. 103, p. 1, column 2a, he [Patrizi] states ‘that the motion of the Earth is by far in better agreement with reason than the motion of the heavens or the uppermost celestial bodies.’ And he refers to Nicolaus Copernicus’s sentence according to which the sidereal heaven is immobile, along with the stars, while the Earth moves.[60]

Further theses to be censured were his vitalistic concept of celestial bodies and celestial infinity. The criticism of the latter point goes as follows:

This [to sustain this view] is to dream in very deep obscurity and fall down a precipice after abandoning the common way. In fact, the best and greatest God created everything according to weight, number, and measure. Therefore, everybody agrees that no infinite body is possible in act and no existing multiplicity can be infinite in act. On the empyrean heaven see the Fathers and Thomas Aquinas.[61]

Patrizi replied with an Emendatio in libros suae novae philosophiae (Correction of the Books of His New Philosophy), written before Christmas 1592. As far as Earth’s motion is concerned, he clarified that he advocated its motion but not its eccentricity (Non tamen dixi, eam de medio suo et naturali loco exire). Furthermore, he stressed that terrestrial motion was supported by many arguments and several philosophers, and claimed that this view does not contrast with theology. However, he declared himself ready to erase passages referring to terrestrial motion, if necessary (Delebo tamen si iubetis). He also defended his views about the infinity of space but was ready to renounce this thesis as well, if he was ordered to do so: “Delebo hanc etiam si iubetis.”[62]

The Jesuit Benedetto Giustiniani proved a more open-minded censor of Patrizi’s natural views in 1593.[63] The same person, however, would be one of the eleven theologians who decided that the Copernican theory was not reconcilable with the Catholic religion, in 1616. As to Patrizi’s work, it was downright (omnino) prohibited in 1594, and placed in the Clementine Index of 1596 as well as in later Indexes.[64]

Not only did Benedetti correspond with intellectuals engaged against the mainstream in Rome, but he himself had differences with Roman Aristotelians, as he mentioned in the preface to the second edition of the Demonstratio motuum localium contra Aristotelem (1555). On some occasions, Benedetti even allowed himself to be ironical about theological dogmas. For instance, in one of the letters of the Diversae speculationes, he accompanied his adherence to methodological Pythagoreanism (a mathematical approach to the investigation of nature) with a joke about reincarnation and his earlier life:

If the souls’ transmigration imagined by the father of Italian wisdom, Pythagoras, were true, I believe that your soul and mine were once the souls of hunting dogs.[65]

Another indicator of Benedetti’s attitude towards the Counter-Reformation and the confessional quarrels of his time emerges from his approach to the calendar reform. This was a very divisive issue. The pope imposed upon all Christianity an emendation of the calendar in an age when it was affected by profound divisions. In this climate, the pope’s political and religious legitimacy and his authority in such matters was cast into doubt by many, especially in the reformed countries. Reputed Lutheran astronomers such as Michael Maestlin opposed the calendar reform implemented by Christopher Clavius and promulgated by Gregory XIII in 1582. The elimination of ten days to make the spring equinox date correspond to its date at the time of the Council of Nicaea was particularly controversial, since it symbolically legitimized the universality of the Roman Church through Constantine I. Benedetti wrote a reform proposal on April 1, 1578, and Emanuele Filiberto sent it to Rome on May 31. The text was printed two times, first in 1578, and then reprinted in 1585 as the first of the letters in the epistolary section of the Diversae speculationes. The most striking feature of Benedetti’s proposal is its technical radicalism and extreme rationalism. August Ziggelaar’s perspicuous description of Benedetti’s proposal here follows: “Benedetti prefers the Eastern date to be fixed according to the true motions of Sun and Moon rather than by cycles. He finds that the Prutenic tables are sufficiently exact for this purpose. Furthermore not ten days, not even 14, but 21 days should be left out in order to make the first day of January the winter solstice. The lengths of the months are to be adjusted so that they coincide with the presence of the Sun in each of the twelve zodiacal signs. Surely, these time intervals change their lengths in the course of time because of the motion of the perihelion of Earth, but Benedetti assures us that only after 24,000 years will an adjustment be necessary. The proposal is not only interesting and original but also very rational because, by eliminating all reference to the Moon, it makes the year entirely solar.”[66] The extreme technicality of such a proposal, one can argue, was also a means to de-ideologize the issue. Therefore, it was not only rejected for its “scientific radicalism” but also for its rationalistic distance from confessional struggles. This positioning is in line with late-humanistic universalism and signals Benedetti’s distance from the Counter-Reformation and the militant cultural production of those years.[67]

2.6 Cultural Institutions: University, Academies, Collections, and the Press

The reformation of the Studio was a cornerstone in Emanuele Filiberto’s and his successor’s cultural policy. It was reopened in Mondovì in 1560 after the French closed it as a potential center of dissent and revolt. It was transferred to the new capital on October 12, 1566, after long discussions and a dispute between Turin and Mondovì. According to the new statutes, issued in 1571, the direction of the university was conferred to nine reformatores, among whom were the Archbishop, the ducal chancellor, the first and second presidents of the Senate of Turin, and the court physician. These men were responsible for the scientific and administrative direction, and for academic discipline.[68]

Most of the professors (about thirty people) were jurists. Among them, the most reputed was the professor of civil law Guido Panciròli. The physicians Francesco Valleriola and Giovanni Argenterio were also illustrious professors, known and appreciated by Conrad Gesner and Michel de Montaigne. The reopening of the university offered Argenterio the opportunity to come back to Piedmont after many years of practice as a physician in Lyon, and of teaching in Antwerp, Bologna, Pisa, Rome, and Naples.[69] Teodoro Rendio of Chio was appointed to teach Greek grammar and, as mentioned before, the poet Giraldi Cinzio became professor of humanities. At the University of Turin, the teaching of philosophy on the basis of the corpus Aristotelicum received high recognition, as the professors in this discipline had a better salary than their colleagues of medicine. Giacomo Castagneri taught on Aristotle’s Physica, De generatione et corruptione, and De Anima. Other celebrated scholars in philosophy were Antonio Berga and Agostino Bucci, both Piedmontese educated at Padua.[70]

One of the most reputed professors appointed in Mondovì was Francesco Vimercato of Milan, with whom Benedetti later corresponded. Vimercato was a thoughtful Renaissance commentator on Aristotle, whose work he read in the original language. He published commentaries on De anima (1543), on Metaphysics (1551), in particular on book lambda and on Meteorologica (1556). However, his commentary on Physics has to be seen as his magnum opus. After his studies in renowned Italian universities, he was appointed in Paris as the first royal lector in philosophy. There, he was first involved as a judge in the Ramist disputes and later was a colleague of Petrus Ramus. In 1561 he came to Piedmont to serve the Savoy family as a university professor, a councillor, a tutor to Carlo Emanuele I, and, from 1567 to 1570, as diplomat in Milan.[71]

Bordiga argued that Benedetti might have taught at the reopened university, first in Mondovì and then in Turin, by relying on some indirect sources. In fact, the information about Benedetti’s teaching activity at Mondovì was derived from the sixteenth-century biographer of Emanuele Filiberto, Giovanni Tonso, who included his name on the list of those who made that university eminent (Qui viri insignes publice docuerint).[72] Another contemporary of Benedetti, the mathematician Cristini, mentioned him as one of those whom Emanuele Filiberto called to Turin in order to increase the prestige of his university.[73] Still, the role that he could have played in the academic life of Turin remains obscure.[74]

We could find no direct evidence that Benedetti served as a professor in the documents preserved at the Archivio di Stato di Torino. We considered the acts gathered under the signature “Istruzione Pubblica/ Regia Università di Torino/ Mazzo 1 (1267–1701),” which include the statutes of 1571 and other precious sources concerning the first years of the univeristy. A dossier entitled “1571, Costituzione de’ Riformatori dell’Università dello Studio di Torino, coll’Istruzione da osservarsi da medesimi, colle distribuzioni delle ore per la Lettura, e Rolli de’ Stipendi de’ Lettori” (fascicolo 7 primo) includes decrees concerning the reform of the university, the names of those responsible for accomplishing it, and a list of the chairs with the corresponding salaries and the names of the professors. These documents indicate that the professor of mathematics was Francesco Ottonaio of Florence, who had occupied the same chair at Mondovì. Other documents testify that the same person held the chair of mathematics in 1573 and in 1585/6. It is therefore hard to believe that Benedetti was professor in this subject before 1586, as several secondary sources contend.[75] As to his alleged position at Mondovì, that university was opened by Emanuele Filiberto in 1560 and then transferred to Turin in 1566, that is, before Benedetti’s arrival at the Savoy court. Thus, it must be excluded as a possibility that he ever taught there, contrary to the claim made by Tonso. It is possible though, even likely, that he acted as an external advisor on matters concerning the university.[76]

Turin had fewer academies than other centers such as Rome, Naples, and Florence, although it was a typical Renaissance phenomenon to establish academies, circles of learned men who met to discuss issues pertaining to literature, the arts, or the sciences, and who published works representative of their common intellectual efforts. Apart from two minor academies (“de’ Solinghi” and “degl’Impietriti”), Carlo Emanuele I conceived the project of forming an academy guided by the Jesuits around 1585. Bonifacio Vannozzi, a man of letters from Pistoia, described it as follows:

His Highness, the very serene [duke] of Savoy, had the wish to found an academy in this august town of Turin. He charged three Jesuit Fathers of the renowned College with the task. Although they are generally sober of mind, in this case they were so intemperate as to entrust myself [with this endeavor] although the overwhelming responsibility [machina da incurvar le spalle] would be excessive for even the most competent person. His Highness has made himself Prince, Protector, and Head [of the academy], in order to attract a good deal of his courtiers [into the academy] who are so cultivated and refined that, if one adds to it the splendor of the arts [lettere], there will be no court in Europe more illustrious than this one. Our name is ‘Incogniti.’[77]

In spite of the initial impetus, this academy was not particularly successful and did not leave significant traces of its activities. Perhaps it was negatively affected by the fluctuating relations between the Crown and the Jesuit order.

Emanuele Filiberto also created a Theatrum omnium disciplinarum, which was realized for him by Ludovic Demoulin de Rochefort. It is not completely clear what this theater contained. It was probably a Wunderkammer, as Mamino argued.[78]

In March 1572, duke Emanuele [Filiberto] established in Turin a museum [theatrum] of all disciplines [organized] in marvelous order and at a very high cost. Archbishop Gerolamo della Rovere and the philosopher Ludovic Demoulin de Rochefort, the most educated men in all fields, cared for it.[79]

Moreover, the dukes supported editorial activities. Emanuele Filiberto first called the Flemish printer Laurens Leenaertsz van der Beke, better known as Torrentinus, to Piedmont. Torrentinus had already initiated a printing house in Florence under Cosimo I., but he died shortly after his arrival in Mondovì (1561). Hence, in 1570, Emanuele Filiberto employed another printer, Niccolò Bevilacqua. This pupil of Manuzio founded the Compagnia della Stampa (Printing Company), which benefited from ducal privileges (a sort of editorial monopoly). The activity of Bevilacqua and his heirs culminated in 71 editions produced between 1578 and 1580, most of them on juridical subjects. Scientific issues were not neglected in Turin. Among the publications issued during the years of Emanuele Filiberto, between 1563 and 1580, books on scientific and natural subjects constitute about one fifth of the entire production, that is, almost as much as literary publications.[80] They also printed several books by Benedetti, including the Diversae speculationes.

2.7 Scientific Debates

2.7.1 Courtly Conversations

Renaissance Turin was a center of lively cultural and scientific debates taking place in different institutional settings, at court, at the university, and in town. The printing press was a powerful means for public discussion, dissemination of ideas, and criticism. Printed sources are evidently our main source of information about the intellectual debates or polemics that animated Turin in Benedetti’s time.

First of all, we should consider courtly debates. A circle of intellectuals gathered around the Savoy family. The professor of jurisprudence, Bernardo Trotto, depicted the relationship between the rulers and their learned courtiers as follows:

These learned men, played by the Prince like well-tuned musical instruments, immediately give out their specific sounds with words. And they give it their best to be clearly understood in conversations, to please the others with good arguments and to convince them of their opinions. It is like the consonance of truth. In fact, everyone says what one knows or, at least, considers to be true. Hence they discuss natural issues and at times moral ones and mathematical ones. In conclusion, one can regard him [the prince] as Apollo surrounded by the Muses near the water spring that was born from the hoof of Pegasus.[81]

A reflection of the intellectual climate and the topics addressed in such informal meetings is a poem by the court physician Arma, Proposte tenute co’l Serenissimo Prencipe (Issues Discussed with the Very Serene Prince), printed in Turin in 1580. In this tiny book, addressed to Emanuele Filiberto, Arma reported a discussion on meteorological problems that took place between Carlo Emanuele I, Benedetti, Ottonaio, Berga, and himself during the prince’s meal for three successive days. On the first day, Carlo Emanuele I asked the reason why the sun heats. Arma used Plato to argue that its body is not warm but that it heats through the percussion of its rays on terrestrial elements.[82] The next day, Arma had a quarrel with Benedetti, Berga, and Ottonaio concerning “solar attraction.”

The day after, Mister Benedetti
And Mister Berga, along with Ottonaio
Expressed opinions that are far from mine:
That the Sun attracts everything to itself with its great brightness
As if it had hands.[83]

Arma objected that, if the sun attracts other bodies, this would be very inconvenient for its inhabitants, the solarians (solari). The argument is not clear and the reader is only informed about the fact that Benedetti agreed with him.

During the conversation, Ottonaio mentioned the vapors brought upward by the sun’s action. Accordingly, the next topic of discussion concerned natural places and elemental displacements with reference to the behavior of vapors.

On the third day, the prince asked about the origin of lightning, and why we perceive their light before we hear the thunder. Arma answered that our sight is much quicker than our hearing, but this remark was received with skepticism by his colleagues. No details are reported about the objections that should have concerned the theory of perception, light, and optics.

The next issue was colors and the rainbow; Benedetti asked about the center of the rainbow’s arc and Arma offered the following answer:

Benedetti, as an expert master of his art,
Asked me about the center of the arc [of the rainbow].
I answered that it was on the vertical line
Descending downward from the center of the heavenly body,
As was the opinion of Zoroaster.
And with this answer I got rid of him.[84]

At the end of this three-day conference, all opinions were written down for the prince and signed by the ducal advisors:

All of this was presented in written form
To His Highness, reporting all speeches.
Dr Berga confirmed it [the rightness of the report].
Benedetti did the same.
After that we discussed other issues,
Occult things and their effects.[85]

Other publications also mention such table talks at court. For instance, the physicist and philosopher Bucci wrote in the preface to his book on physiology, Disputatio de principatu partium corporis (Disputation on the Superiority of Parts of the Body, 1583), that his discussion about whether the heart or the brain held superiority in the body, and about the localization of the soul, originated from a table talk. On a certain occasion, in fact, Carlo Emanuele I had gathered his learned courtiers and addressed these issues. Among the participants, including several physicians, Bucci also mentions the “mathematicians,” Ottonaio and Benedetti.[86]

2.7.2 Academic and Scholarly Controversies

Scholarly controversies and polemics on various issues and with very different tones were printed in Benedetti’s years. While courtly debates had a polite and entertaining character, academic disputes could be more vehement. However, the two contexts were not always neatly divided. In 1572 two professors of philosophy, Berga and Bucci, held divergent opinions concerning the interpretation of Aristotle’s De anima.[87] As Simone Mammola has argued, their disputes on that issue should be understood against the background of the Averroistic-Simplician debates on Aristotle’s psychology started at Padua and continued at Turin by scholars such as Filateo and Vimercato. The Turin controversy over Aristotle’s soul doctrine, which developed through academic disputations and publications, ranged from cognitive problems linked to the functions of the soul (such as the problem of the relationship between imagination and cogitation) to metaphysical and theological issues, e.g., the legitimacy of a reading of Parmenides’s theory of being as a form of prisca theologia. The real issue at stake was the correct approach to the Aristotelian corpus based on different commentators.[88]

The court physician Arma was at the center of several public controversies as well. In 1575, he defended the scientific status of medicine, traditionally considered “only” an art, in the programmatic book Quod madicina sit scientia et non ars (That Medicine is a Science and Not an Art). This writing was considered worthy of a second edition in 1585. In 1579, Arma entered a dispute against the professor of medicine Giovanni Costeo over the healthiness of bread made out of rice (“pane fatto col decotto di riso”). The two parties then issued a series of publications on this controversial topic.[89]

Another polemic opposed Berga and Benedetti regarding the proportion between the earthly and the watery element of our globe. The debate was initiated by some questions Carlo Emanuele I asked his courtiers. While Berga stuck to the Scholastic view that the elements have an increasing quantity proportional to their distance to the center, Benedetti favored the merging of empirical and mathematical arguments as proposed by Alessandro Piccolomini. Piccolomini had come to the conclusion that the quantity of the earthly element is superior to that of the water element, as one can read in his Della grandezza della terra e dell’acqua (Venice, 1558).[90] Although Benedetti regarded Piccolomini’s arguments as conclusive, Berga undertook to compose a Scholastic refutation of them, Discorso… della grandezza dell’acqua e della terra contra l’opinione dil S. Alessandro Piccolomini (Turin, 1579). Part of his strategy was to eliminate arguments derived from the “misure dei cieli e della terra, dalla Scuola dei matematici immaginate,” that is, from mathematical and empirical methods applied to this issue (“measurements of the heavens and the earth imagined by the school of the mathematicians”). Benedetti reacted with his Considerazione… d’intorno al discorso della grandezza della Terra, et dell’Acqua. Del Eccellent. Sig. Antonio Berga Filosofo nella Università di Torino (Consideration… on the discourse concerning the dimensions of earth and water by the excellent Mr. Antonio Berga, philosopher of the University of Turin, Turin, 1579). In it, he applied mathematics to show that, even if the entire earth were covered by water, the volume of the terrestrial element would not be inferior to that of water since the depth of seas and oceans is small in comparison to the terrestrial radius. The oceanic navigations, the geographical explorations, and the European colonization of the globe during the fourteenth and fifteenth centuries had indisputably demonstrated that water and earth constitute one single globe, a unique globus terraquaeus.[91] This cosmographical advance had been the basis for Piccolomini’s considerations, which met with considerable success and agreement among scholars. On such issues, as Benedetti stressed, experience and mathematical reasoning should be joined:

Very serene Prince, the discovery, after two thousand years, that the [element] earth is much more than the [element] water (for which we are greatly indebted to the very learned Mr. Alessandro Piccolomini) very much pleased the spirits of the most renowned philosophers of our time. In the past, they did not dare to depart from the false doctrine they had imbibed for many centuries, although it was sustained by implausible reasons. Today they are glad to embrace the opposite opinion [concerning water and earth], because both the senses and reason are in accordance with the [new] demonstration of the truth. The ancient mistake has been unveiled by the mathematical school with very certain proofs that offer a firm foundation of the measurement of the heavens and the Earth.[92]

The dispute continued with the Latin translation of Berga’s writing by Francesco Maria Vialardi (1580) and a skeptical intervention by Arma. The latter was a poem dedicated to Carlo Emanuele I, entitled Stanze del dottore Arma al serenissimo Carolo Emanuele di Savoia et Piemonte Prencipe, etc. suo signore sempre osservandissimo. Che l’acqua e la terra non si possono a modo alcuno misurar (Doctor Arma’s Stanzas to the Most Serene Prince Carlo Emanuele of Savoy and Piedmont, His Perpetually Honored Lord, 1580). The composition is poor both from a stylistic viewpoint and a scientific one. The courtly physician could only point out the uncertainty of human knowledge and the wisdom in measure gifted by God to creation, although he also stated we cannot grasp the latter: “The Creator gave it a measure,/ Which cannot be grasped by any creature.”[93] Benedetti addressed the issue again in one of the letters of the Diversae speculationes.[94]

2.7.3 Astronomical-Astrological Polemics

In Renaissance Turin, astronomical and astrological issues were at the center of intense debates and even polemics. In 1578 the protophysician Arma was the target of a denigrating pamphlet in twenty points. An anonymous author attacked a booklet of his on the comet that had just appeared. As we know from indirect evidence, Arma had written one or two treatises on this issue. One was entitled De significatione stellae crinitae.[95] Perhaps it has to be identified with a composition in verses, referred to as πωγωνία (bearded comet) in the anonymous pamphlet. The only remaining evidence of the cometary polemics is a defence of Arma, La Stravagantographia del Sig. Filosofo stravagante, in difesa de la πωγωνία d’il Dottore Arma (The Stravagantography of Mr. Stravagant Philosopher in Defence of Dr Arma’s Bearded Comet).[96] The apologist was one Monsignor Della Torre linked to the court. His apology offers an insight into the controversy: It concerned astronomical and meteorological issues relating to the nature of the comet, its location below or above the sublunary sphere, the criticism of astrological interpretations of the celestial phenomenon, as well as personal insults. The provocateur who had published against Arma is depicted as follows:

I cannot stop wondering who this person is. I cannot understand why he sometimes presents himself as a scholar, sometimes as a cook, as a Roman courtier, or as a practicing friar [frate osservantino] (as he speaks about the osservantini). I cannot believe that he is a practicing [man of religion], as the ecclesiastics speak in a correct manner and not heedlessly like him (who behaved heedlessly). Moderation has always been praised. Therefore, moderate people will always damn this person. I will never believe that he is a scholar. In fact, today’s scholars are well-educated and would never indulge in such excesses, especially against such a man [Dr. Arma] from whom they did not receive anything but pleasure, honor, and courtesy. Although he seems to come from the area of Rome, in the end he shows himself to be a dishwasher because even a cook would behave better than him. Whoever the hell he is, if he will not control himself better in the future, I will repay him as he deserves.[97]

Possibly the identity of this mysterious denigrator was the philosopher Giordano Bruno, who was in Turin in 1578 on his way to Chambéry, in Savoy. At that time he wore the Dominican habit and had just published, as mentioned above, a booklet on meteorology in Venice entitled De’ segni de’ tempi (On the Signs of the Times) that presumably dealt with the comet.[98] The reference to the provocateur as a Roman courtier could correspond to an episode of Bruno’s life. In Paris, in 1585, he told the librarian Guillaume Cotin that he had been once received at the Roman court by Pius V and the Cardinal Rebiba to whom he demonstrated his technique of the art of memory.[99] Moreover, the Turin episode resembles a querelle that burst out in Geneva in 1579, after the publication of a pamphlet in twenty points ridiculing a professor of philosophy. The detractor, Giordano Bruno, was discovered and condemned to exclusion from communion. As one reads in a document (6 August 1579), “Philippe Jordan, dit Brunus, Italien [était] détenu pour avoir faict imprimer certaines responses et invectives contre Mr. de la Faye, cottans 20 erreurs d’iceluy en une de ses leçons.”[100]

Only one year later, between 1580 and 1581, Benedetti was involved in an astronomical-astrological quarrel with a certain Benedetto Altavilla of Vicenza concerning the reliability of ephemerides and astrological prognostication. The controversy was sparked by the publication of Altavilla’s Animadversiones in ephemeridas (Remarks against Ephemerides, 1580) and was continued with two further publications by the same author.[101] They cast the reliability of ephemerides’ calculations and astrological forecasting into doubt, based on the alleged theoretical flaws of mathematical astronomy. Since they appear to have been directed against scientists linked to the court, Benedetti eventually intervened with a printed letter, Lettera per modo di discorso… intorno ad alcune nuove riprensioni, et emendationi, contra alli calculatori delle effemeridi (Turin, 1581). He later translated it into Latin as Defensio ephemerides and included it in the miscellanea of epistles of the Diversae speculationes. We will expand on this polemic later, in the section on Benedetti’s astronomy.

2.7.4 Posthumous Criticism: Cristini on Benedetti

Benedetti died on January 20, 1590, two years before his own astrological prediction. This untimely death did not leave him the time to complete the astrological work that he announced at the end of the Diversae speculationes. What is worse, the fact that his own prediction was wrong awakened doubts and rumours about his scientific talent. The mathematician Cristini was quick to compose a critical essay, entitled “Examination of the mistake and emendation of the nativity of the very excellent mathematician, Mr. Giovanni Battista Benedetti, now deceased, to account for his [wrong] prognostication of his own death two years later than it in fact occurred” (Essaminatione dell’errore, della rettificatione de tempo della natività del fu S[ignor] Gio[vanni] Battista Benedetti mathematico eccellentissimo, per cagion del quale esso s’era pronosticato morte due anni appresso in circa al tempo che gl’è avvenuta). Although the manuscript was lost in the fire of the Biblioteca Nazionale of Turin in 1904,[102] a transcription of significant parts by the eighteenth-century biographer of Cristini, Giuseppe Vernazza, is still extant in the Biblioteca Reale of Turin among the documents that Vernazza gathered for the composition of his Notizie di Bartolommeo Cristini (Notes on Bartolomeo Cristini, 1783).[103]

Cristini started his examination of Benedetti’s errors with specific reference to his mistaken self-prognostication in the Diversae speculationes. He also reported that Benedetti himself acknowledged an error in his nativity, amounting to four minutes. After that, Cristini took upon himself the task of recalculating that horoscope on the basis of the figure published by Gaurico and came to the conclusion that the mistake was even bigger. It amounted to eight minutes.

Benedetti published his prognostication of the moment of his death in the work entitled Diversarum speculationum mathematicarum et physicarum liber (published in 1585), in a letter to the most illustrious Wolfhard Eisenstein [Volfardus Aisenstain], which is to be found at the end of this work. After a brief assessment of those things of the judicial art that he regarded as vain or false, and after announcing to Wolfhard that he would expand [on astrology] in that tract with his astrological observations, which he wished he could publish before his death, he added the indication of the time in which, according to him, [his death] was to happen (that is, [the date] before which he wished he could publish the aforementioned tract). These are his words: “antequam ad directionem mei horoscopi cum corpore Martis anaeretae perveniam, quae quidem directio circa annum millesimum quingentesimum nonagesimum secundum evenienti” [as indicated by my horoscope, before I meet the body of the adverse Mars. This is going to happen in 1592].

As we can see, he was certain that he would die when the direction of his ascendant and Mars would meet. He calls [Mars] “anaereta,” that is, giver or announcer of his death. He confirmed this when […], just before his death, he felt that the disease was attacking him and declared that he made a mistake of four minutes in the rectification of the time of his birth horoscope [natività]. This is as if he would say that, by augmenting by four minutes the time of his birth horoscope, he would have predicted the direction [of his ascendant sign meeting Mars] at about the time when he became sick. Hence, he believed he was dying, and this [his death] in fact occurred at the end of the ongoing year 1590, at 17:00 of 20th January according to [the calendar of] Gregory, which corresponds to the 10th of the old [calendar]. I had to know the time in which he believed he was born in order to assess by how much time he was mistaken in the rectification of his birth horoscope, so that the direction of his horoscope relative to Mars corresponded to the days when he left this world. Therefore, at Benedetti’s death, I immediately began to compute the error of the aforementioned time, though only approximately, because I did not know Mars’s latitude. And I found that it [the mistake] amounted to eight minutes […]. Later, when the same person who told me that Benedetti had acknowledged a mistake of only four minutes according to his calculations, openly accused me of not being able to do this calculation, as my mistake was two times [that of Benedetti], I began the calculation in the following manner. First, I determined the time attributed to his birth […] Etc.[104]

In his transcription of Cristini’s Essaminatione dell’errore, however, Vernazza omitted numbers and calculations. These can be found in another astronomical-astrological assessment and criticism of Benedetti entitled “Thirty-first revolution of the very serene Sir, Duke Carlo Emanuele of Savoy, for the year 1592, very diligently and reliably calculated and explained by Bartolomeo Cristini, scholar of mathematical disciplines in the service of His Highness, according to the best opinion of the main judiciary astrologers” (Revolutione trentesima prima del Serenissimo Signore il Signor Carlo Emanuel Duca di Savoia corrente dell’anno 1592 con ogni diligenze et fedeltà calculata et decchiarata secondo le migliori intelligenze de più principali autori dell’astrologia giundiciaria per Bartolomeo Crestino studioso delle mathematiche discipline in servitio di Sua Altezza). Although the manuscript was damaged by the fire of the Turin library in 1904, it is still readable. The dedicatory letter shows that, at the time of its composition (June 8, 1592), Cristini was striving to obtain a stable appointment at court:

But your very generous Highness awoke in my spirit the desire of mathematical virtues and of undertaking the present endeavor. Your request woke up and unveiled in me the desire (which is always alive) to serve [Your Highness]. However, my desire has been impeded by the difficulties of my continuous poverty and adverse times owing to the fact that no treasurer (or any monetary and financial administrator) regards me as an ordinary servant of Your Highness. [I have been acknowledged as a servant] only in exceptional cases, when my capacity, readiness and knowledge in making calculations has proved useful—as has happened several times, when I was required to serve Your Highness. […]

Therefore, I place growing hope only in Your Highness the more [you] require my services, the more efforts I make for You and the fewer are the number of [benefactors] by whom I can hopefully be supported[105]

In this case, the allegation against Benedetti is for using the Alfonsine tables to determine the nativity of Carlo Emanuele I, instead of more reliable Copernican tables. Such inaccuracy invalidated his astrological judgments.

I took into account the places where they [the planets] are to be found in the horoscope made according to the true time calculated on the basis of Copernicus, following the teaching of the major authors on astrology. In general, since scholars are in disagreement concerning the employment of different tables to compute their horoscopes [revolutioni] and although I have demonstrated (in the calculations at the beginning of my tract) that only one set [of tables] is true, I calculated the astrological figures of the heavens according to both tables—in fact, false ones were also in use by many and in particular by Benedetti—and I offered double astrological judgments depending on the places assigned according to the different figures. In this manner, your Highness will possibly compare them and see which ones are in better agreement with the truth.[106]

The terms of Cristini’s polemics echoed those of the controversy of the years 1580–1581 between Altavilla and Benedetti. In fact, Cristini examined two astrological figures calculated by Benedetti: a nativity based on the “Copernican” tables of the German astronomer Erasmus Reinhold, Prutenicae tabulae (1551), and a prognostication based on the Alfonsine tables.[107] He pitted these figures against a “figure that is computed from Giovanni Antonio Magini’s tables of the second celestial mobiles” (figura della natività di novo da me calculata con le tavole de secondi mobili celesti di Antonio Magini) and a “figure of the revolution that I calculated according to the time of the real motion indicated in Magini’s’s ephemerides” (figura della revolutione da me calculata sotto il tempo che si trova per il moto vero insegnato nell’effemeridi del Magini).[108] Finally, he discussed the differences between his and Benedetti’s calculations and concluded with an accusation directed against Benedetti for being careless and opportunist:

But I believe that he [Benedetti] followed the calculation of Alfonso X rather than the true one only owing to its simplicity. In fact, before [the publication of] the ephemerides of Magini it was very difficult to establish the true time of the revolution. Before him, nobody calculated the Sun up to the seconds in any ephemerides, which is the presupposition for more exact and true computations […]. It is only in consideration of Benedetti’s authority that I did not omit to compare his horoscope with the other one.[109]

In this second criticism, Cristini continued to discredit Benedetti. This time he cast his capacity as both an astrologer and a mathematical astronomer into doubt. Cristini suggested, in fact, that Benedetti misused his prestige to disguise the lack of accuracy in his astrological computations. Mistakes affected not only the horoscope he carried out for himself but also those cast for his patrons.

Once he had established himself as an expert in the field, Cristini continued to prepare prognostications for the ruling family from 1592 to 1595, as testified to by the titles of several manuscripts, such as “diari” and “revolutioni,” which are for the most part lost.[110] He obtained the position at court that he desired in 1594. Carlo Emanuele I designated him “as our and our princely children’s mathematician, follower of Giovanni Battista Benedetti, who has recently passed away” (per mathematico nostro et dei principi nostri figliuoli in luogo del fu Gio[vanni] Battista Benedetti ultimamente defonto). He moreover accorded to the new court mathematician a “reasonable stipend” (un ragionevole stipendio) of three hundred scudi per year, “so that, according to our wish, he will cover the efforts of his studies, and will be in condition to serve us with more ease and comfort in all the duties we will entrust him” (acciò che possi comportar alle fatiche delli studi, et trattenersi al servitio nostro più agevolmente et commodamente come desideriamo in tutti li carighi che ha da noi).[111]

2.8 Strengths and Limitations of the Institutional Framework of Benedetti’s Science

Benedetti’s life, career, and work, as well as his legacy, fortunes, and misfortunes should be understood against the background of the Renaissance world he was part of, in particular the Italian and Turin environments. His case is paradigmatic of both the strength and the limitations of Renaissance science. On the one hand, the cultural and economic flourishing of centers such as Turin, new and challenging engineering and architectural projects, and the establishment of a court and of a modern state apparatus with its need for technical advice and cultural grandeur created an exceptional environment, favorable also to the pursuit of science and philosophical speculations. The constraints of Counter-Reformation culture did not affect the speculative freedom of Benedetti. This is especially due to the pragmatic cultural and religious politics of the Savoy dukes, who were trying to establish a balance between their state and international diplomacy and confessional tensions. On the other hand, however, the fragility of Renaissance knowledge institutions also comes into view. Universities were teaching institutions instead of research centers. Professors were concerned with the transmission of knowledge rather than with the implementation of new knowledge and theories. The intended mission of early-modern universities was preservation, namely the transmission of traditional knowledge to future generations, not producing change. Thus, epistemic processes in institutional settings were often imperceptible and transformations of knowledge often occurred against the explicit intentions of the historical actors. Lectures and commentaries on authoritative sources—the teaching of which was sanctioned by academic statutes and curricula—were not expected to alter the knowledge preserved in the classics and in the textbooks. The Savoy dukes tried to attract prestigious professors to Turin and also supported, at least initially, the teaching of humanities according to the new standards set by humanistic philology. However, the place for free inquiry and innovation was outside universities. Benedetti’s works, marked by original and unorthodox conceptions in physics, mathematics, and other disciplines, emerged from a courtly environment. Yet, this institutional frame proved ephemeral as it was dependent on patronage. It also had a strongly personal character, as it depended on informal exchanges within a system of unsystematic patronage. In Turin, modern scientific academies, with a stable body of investigators and statutes, had not yet made their appearance. Not even literary and artistic academies met with a particularly favorable environment.

Given this context, Benedetti’s scientific activity, accomplished outside university and institutionalized settings, cannot but appear as occasional. In fact, it was linked to the contingency of courtly life, for instance to the requests for advice by the Savoy rulers or other patrons. This is the case with all of Benedetti’s letters and with other publications, such as his writing on the calendar reform. He appears to have given expert advice on issues of cultural policy, such as university appointments, as well as on technical issues, and not least on matters of astrology. His construction of a fountain and of sundials, as well as his writings on technologies and gnomonics are directly connected to his role as a mathematical expert at the court. In the same function, he also entered debates and polemics animating Turin. Some of his interventions were friendly, for instance his exchanges on meteorology with other courtiers in the presence of members of the ruling family. His controversy with Professor Berga over the quantity of water and earth in our globe was more vehement but never harsh. Astrological polemics were the most virulent ones, as evidenced by Benedetti’s publications against the critic of astrology, Altavilla. A constant feature of Benedetti’s scientific work remains its occasional character. This is also reflected in the lack of systematic order in his magnum opus, the Diversae speculationes. Benedetti probably saw himself primarily as a courtier, participating in the cultural life of Turin as an exponent of the Savoy elite, and not as a scientist pursuing the immaterial glories of scholarly achievements. As a matter of fact, he did not primarily take upon himself the burden of a scientific effort going beyond the deliverance of brilliant booklets, short judgments, and advice on specialistic issues.

One astonishing aspect of Benedetti’s intellectual activities is the lack of an enduring and explicit legacy. On the one hand, his conceptions clearly influenced contemporaries and followers in Italy and abroad. Among others, his impact is reflected in the positive opinions of Brahe and Kepler, in Galileo’s reception of several insights of his mechanics and physics, and in the European circulation of his ideas on physics through Taisner’s plagiarism. On the other hand, the lack of an enduring acknowledgment of his work is equally evident and seems to be linked to the fact that he was not able, and perhaps not even willing, to establish a school like the one set up by Commandino in Urbino, or by those later set up by Galileo in Padua and Tuscany. Not even in Turin did he benefit from lasting recognition. As we have seen above, his immediate successor as court mathematician, Cristini, even saw the denigration of his astrological and astronomical skills as an opportunity to obtain a visible position in town and start a courtly career.

In many ways, Benedetti is the mirror of his world, in particular of the courtly society he belonged to. His work can be seen as the embodiment of this context. His case is different from that of many other Renaissance scholars, who strongly identified themselves with their scientific work and output. For scholars like Galileo, for instance, the publication of their works had a functional aim in accessing the courtly milieus. In the case of Benedetti, he was already part of the patrician and aristocratic milieu for many reasons. His work is the product of courtly life rather than his entry ticket to it. The author disappears (or almost vanishes) and leaves in his place a sort of collective author, which is not the scientific Republic of Letters, but rather the court itself, its institutions, its elites, its participants, and its networks. In this respect, Benedetti differs from the great protagonists of Italian Renaissance science. He is very far from the self-celebration of intellectuals like Cardano and Galileo. His work is no monument to himself but rather to his environment, ranging beyond the local boundaries of Piedmont and the Savoy. The Urbino school was also populated by scholars less concerned with their own ego than with science. However, in contrast with this school Benedetti conceived of himself as an innovator, rather than as a restorer of antiquity and classicism.

Footnotes

[1] Benedetti 1585, f. A2r: “Agitur nonus decimus annus ex quo litteris Serenissimi patris tuae Celsitudinis, accersitus ex urbe Parmensi in hanc me civitatem contuli. Is advenientem tam humane excepit, tanta deinde liberalitate fuit complexus ego vicissim ei deserviendi, tam vehementi cupiditate fui accensus, ut sub eius ditione quod superesset vitae agere constituerem.”

[2] Benedetti 1585, f. A2r: “Cuius in me banignitas, mea in illum observantia mirum in modum mutuo usu, et consuetudine est adaucta, ut idem Dux me secum dum rusticaretur esse vellet, saepe etiam secum pernoctare; quo quidem tempore de Mathematicis scientiis mecum agebat, in quibus perdiscendis mea opera utebatur, quaestiones, Arithmeticam, Geometriam, Opticen, Musicam, aut Astrologiam spectantes proponens.”

[3] Firpo 1983, 123: “Non è quel duca litterato, ma ama li virtuosi, et però ne tiene alquanti appresso di sé, sente piacere a udirli ragionare, egli stesso li fa de quesiti, ma nessun ragionamento più li diletta, che quello delle matematiche, come scientia, che non solo è conveniente ma ancora è necessaria alla professione del capitano.”

[4] Firpo 1983, 211: “Ha il signor duca di Savoja un bellissimo ingegno capace d’ogni scienza: ma non ha atteso alle lettere con quella diligenza, che si converria a chi ne volesse sapere, essendo la sua principal professione il mestiero della guerra […]. E perché la scienza delle matematiche è molto utile e necessaria a chi vuole fare questa professione de l’arme, però se ne diletta assai Sua Eccellenza [Emanuele Filiberto] e di quella sa assai più che mediocremente. Con tutto questo sapendo che l’uomo tanto sa di ogni scienza quanto continua in vederla e studiarla, però usa di udire ogni giorno una lezione o d’Euclide o d’altro scrittore di quelle scienze da un messer Giovan Battista Benedetti veneto; uomo, per opinione non solamente mia, ma di molti valentuomini ancora, il maggiore che oggi faccia professione, e di grandissimo gusto del Signor Duca; perché oltre a possedere lui quella scienza eccellentissima sa anco così bene insegnarla ad altri che con molta facilità ne fa restar capacissimo chi lo ascolta.”

[5] Merlin 1998, 16.

[6] Merlin 1998, 33 and Merlin and Stango 1998, 266–267.

[7] Pingone 1577, 85: “Anno Christi 1565 mense Iunio, Dux Emanuel, obsessa a Turcis Melita, Andream Provanam Leniacum cum triremibus quatuor instructissimis mittit, qui una cum Pontificiis, Hispanis et aliis triremibus suppetias insulae afferret. Prior Leniacus applicuit, difficultates exploravit, alios postea advexit, et soluta tandem faventibus superis obsidione, Hierosolymitanorum militum sacer, et strenuus ordo liberatus, Ioanne Valleta Gallo summum magisterium gubernante. Quam ob victoriam Taurini immensae laetitiae publica significatio reddita, et devotae superis gratiarum actiones.”

[8] Pingone 1577, 88: “Anno Christi 1571 Emanuel Dux Taurini agens, confoederatione inita in Turcam Cypri vastatorem, inter Pium quintum Pontificem, Hispaniarum Regem, et Venetam Rempublicam, qui classi praeesset ab omnibus exposcitur: sed ob imminentiam a vicinis discordiis patriae discriminis, excusatus habetur. Ioannes vero Austriacus Caroli quinti Caesaris soboles, magni animi, et expectationis iuvenis praeficitur. At Dux Andream Provanam Leniacum tribus cum triremibus in eam expeditionem adiungit. Apud Naupactum Achaicum concursum, et decertatum. Christianorum vix ducentum triremes: Turcarum vero plusquam trecentum: Mars diu anceps, tandem Deo maximo favente, et quodam potius miraculo ad Christianos inclinavit victoria. Leniacus ex triremi Praetoriam fortiter dimicans sclopeto ictus in capite vix galeae praesidio evasit: triremium una Margaris nomine dissipata, mersaque penitus, Pedemontana semel atque iterum ab hostibus recepta. Ob eam victoriam, Taurini supplicationes superis, feriae mortalibus indictae..” See also Tonso 1596, 142, 161 and 177–179.

[9] Benedetti 1585, 214–216.

[10] Benedetti 1585, 217–219 and 219–220.

[11] Ricuperati 1998, XXII.

[12] See Stumpo 1993, 561.

[13] Pingone 1577, 85: “Eo anno [1564] Dux in aeditiore parte civitatis, in ipsis templi Divi Solutoris ruinis Acropolis aedificare coepit, Cittadellam vulgo dicunt.”

[14] Pingone 1577, 86: “Anno Christi 1566 idibus Martiis, absoluta paucis mensibus Taurinensi acropoli, quinis propugnaculis admiranda, servata omni rei militaris, et architectonicae artis ratione eam religiosa ac pia benedictione communiri curat, Archiepiscopo Rovereo sacris praeeunte: mox praesidiis firmat, eique praeficit Iosephum Caresanam Vercellensem de se benemeritum, ac rei militaris peritissimum.”

[15] Merlin and Stango 1998, 118–119.

[16] Pollak 1991b, 16.

[17] See Broc and Greppi 1989, 113.

[18] The Archivio di Stato preserves Carracha’s maps of Turin: Augusta Taurinorum (1577) and Turino (ca. 1580)—see Archivio Storico della Città di Torino 1982.

[19] Benedetti 1585, 271–274, 405–408 and 267.

[20] Tessari 1993.

[21] Palladio 1570, III, 5: “Ne’ quali [edificii publichi], perché di maggior grandezza si fanno, e con più rari ornamenti, che i privati, e servono a uso, e commodo di ciascun; hanno i Principi molto ampio campo di far conoscere al mondo la grandezza dell’animo loro; e gli Architetti bellissima occasione di dimostrar quanto essi vagliano nelle belle, et meravigliose invenzioni.”

[22] Palladio 1570, III, 3: “Principe, il qual solo a tempi nostri con la Prudenza, e co’l valore s’assimiglia a quelli antichi Romani Heroi, le virtuosissime operationi de’ quali si leggono con maraviglia nell’historie, et parte si veggono nell’antiche ruine.”

[23] Palladio 1570, III, 3: “Delle qual cose [concernenti l’architettura] essendo l’A[ltezza] V[ostra] dotata delle più nobili arti, e scientie; piglierà non poca contentezza, e consolazione considerando le sottili, e belle invenzioni degli huomini, e la vera scienza di quest’arte, da lei molto bene intesa, e ridotta a rara, e perfetta perfezione; come dimostrano gli illustri, e reali edifici fatti fare, e che tuttavia si fanno in diversi luoghi dell’amplissimo, e felicissimo suo stato.”

[24] Pollak 1991a, 18–26.

[25] See Viglino Davico 2005, Pollak 1991a and Signorelli 1969–1970.

[26] Vernazza 1783, 8 and 11.

[27] The letter is to be found in Benedetti 1585, 330–331. The collocation of the volume in the Biblioteca Reale di Torino is G 43 8.

[28] Vernazza 1783, 37, n. 31.

[29] Benedetti 1585, 271–277.

[30] Benedetti 1585, 119–140. See Mamino 1989.

[31] Mamino 1989, 432–433 and Roero 1997.

[32] Villari 1988, 93–95 and 107–110. See Doglio 1998, 599ff..

[33] Villari 1988, 93–94:

“Move insieme con lor verso te il piede
il mio Ottonaio, a cui scorrere il cielo, per grazia, diede.
Del caldo la cagion saper, del gelo,
e perché breve sia, sia lungo il giorno,
e quale offoschi il sole oscuro il velo;
come ritorni di bei fiori adorno
l’anno e chi debba aver dal nascimento
onore e pregio, e qual ingiuria e scorno;
e da che stella prender de’ argomento
de lo stato suo l’uom, poi ch’egli è nato,
insin che il suo vital lume sia spento
di cerchi in cerchio.”

[34] Villari 1988, 95:

“E quel che, con gentil opre, e leggiadre,
tenta che il nome suo da l’oblio s’erga,
vinte del tempo avar le forze ladre,
i’ dico il mio gentile Antonio Berga,
che addita, a chi imparar cerca, la strada,
mentre, ad util comun, le carte verga.”

[35] Doglio 1998, 621 and 625.

[36] Firpo 1993, 159. See Ricci 2000.

[37] Ernst 1992.

[38] See Griseri 1998.

[39] In a letter to Benedetti, Francesco Patrizi asked him to give his regards to Baron Sfondrati. See Patrizi 1975, 42–43.

[40] See Omodeo 2008b and Omodeo 2012a.

[41] “La gente infervorata di devotione è molto più regolata: e per consequenza più ubidiente al Suo Prencipe, che la dissoluta.” Botero 1608, 241.

[42] De Simone 1958.

[43] See Merlin 1995, 238–267, chap. IX, “Tra Controriforma e Ragion di Stato.”

[44] Prosperi 1996, 57–58.

[45] See Prosperi 1996, III, “Inquisizione romana e stati italiani” and Black 2013, 30.

[46] Prosperi 1996, 103–105.

[47] Longo 1998, 475 and Scaduto 1959, 52.

[48] Grendler 2002, 42.

[49] Vallauri 1846, 19.

[50] For instance, Giraldi Cinzio defended the famous commentator on Aristotle’s Poetics, Ludovico Castelvetro, who was excommunicated in 1560 as “eretico fuggitivo e impenitente” for his alleged bias towards Melanchthon. On this occasion Giraldi Cinzio argued that violence and coercion could only produce the opposite effects than those wished for by the defenders of orthodoxy. See Cinzio 1996, Letter n. 101, 371, n. 3.

[51] Vallauri 1846, 19 and Grendler 2002, 42–43.

[52] Cinzio 1996, Letter n.127, 425: “Sed Taurino iam menses quatuor absum, Ticinique publice profiteor. Nam, praeter iacturam valetudinis, quam ibi quotidie faciebam, me ad abeundum urgentem, natio illa haec nostra studia nihil quidem facit. Hinc Princeps ille, qui oratoriam ac poeticam facultatem profiteretur, in Academia sua habere constituit neminem, quod satis esse censuerit Iesuitas nescio quos, suo in collegio, hoc muneris cum puerilis ac infantibus obire; qui, cum Deuspaterio quodam, barbaro plane auctore, mollia ingenia, obscurissima, ne dicam foedissima, imbuunt barbarie. Me tamen abeuntem, praeter annuam quadrigentorum aureorum nummum stipe, quam liberaliter exsolvit, centum etiam scutatis aureis donavit.”

[53] Grendler 2002, 42–44.

[54] Patrizi to Benedetti (Ferrara, 21 March 1583), Patrizi 1975, 39.

[55] Archivio di Stato di Torino, Istruzione Pubblica/ Regia Università di Torino/ Mazzo 1 (1267–1701), Fascicolo 7/2. The document is included as an appendix to Omodeo 2014d.

[56] Patrizi 1975, XXVII, 53: “Molto Magnifico et Eccellentissimo Signore, mi rallegro con Vostra Signoria, che più tosto che non credea si è rilevata dal male, e li rendo moltissime gratie dell’haver presentato il mio libro a quel Serenissimo Prencipe, e ricevuto il favore, che Ella mi avvisi ciò che haverà detto, dopo che l’havrà letto. Et se per sorte per li molti negozii il libro andasse in oblio, spero da Lei il rimedio di un poco di ricordanza, la quale, se partorirà alcun segno che Sua Altezza Serenissima l’habbia havuto caro, mi sarà carissimo e tutto l’obbligo l’haverò a Vostra Signoria e all’amor suo verso me.”

[57] Patrizi 1975, 57–58. In the letter Ottonaio is also mentioned as a common acquaintance and an intellectual partner.

[58] Seidengart 2006, 116–124 and Omodeo 2014a, 174–175.

[59] See Bruno 2000b, doc. 45, 225.

[60] Baldini and Spruit 2009, Vol. I, 3, 51, doc. 1, 2216: “Lib. 17 Pancosmias fol. 103, pag. 1, col. 2.a ait quod Terram revolvi longe videtur esse rationi consonantius, quam Coelum, vel suprema astra moveri. Et refert sententiam Nicolai Copernici dicentis Coelum sydereum stare simul cum stellis, Terram vero moveri.”

[61] Baldini and Spruit 2009, Vol. I, 3, 51, doc. 1, 2219: “Hoc est somniare per altissimas tenebras, et a via communi declinando in praecipitia ruere, nam cum Deus opt. Max. omnia in pondere, numero, et mensura produxerit, nullum infinitum corpus actu dari nullamque rerum subsistentium multitudinem actu infinitam omnes viri fatentur. De Coelo empyreo consultat Patres, et sanctum Thomam.”

[62] Baldini and Spruit 2009, Vol. I, 3, 51, doc. 7, 2231.

[63] Baldini and Spruit 2009, Vol. I, 3, 51, doc. 10.

[64] For a reconstruction of the anti-Platonic reaction also affecting the reception of Patrizi, see Rotondò 1982. On the censure of 1616, see Bucciantini 1995, Bucciantini, Camerota, and Giudice 2011 and Omodeo 2014a, chap. 7.

[65] Benedetti 1585, 285: “Si vera esset animorum illa transmigratio quam sibi Italicae sapientiae Pater Pythagoras effinxerat; tuam, meamque existimarem animam canis, quandoque venatici fuisse.”

[66] Ziggelaar 1983, 211 and 214.

[67] Steinmetz 2011.

[68] See Bonino 1824–1825, Naso 1993, and Catarinella and Salsotto 1998.

[69] On Argenterio, see Temkin 1974, 141–144 and 149–152 and Mammola 2012, 185–193.

[70] On the philosophical culture of Turin of those years, in particular on Bucci, see Mammola 2013.

[71] See Gilbert 1965.

[72] Tonso 1596, 141: “Neque vero liberalium disciplinarum omniumque artium colendarum quam susceperat cogitationem unquam deposuit: nam et publicum earum Gymnasium pro tempore in oppido Monteregali instituit: et qui viri in quacunque scientia excellerent undique conquisuit. […] Mathematicos illustres Franciscum Othonarium, et Io. Baptistam Benedictum Venetum.”

[73] Bordiga derived this information from a manuscript of Cristini’s preserved in the Biblioteca Marciana in Venice. See Bordiga 1926, 596–597.

[74] The historian of Piedmontese Universities Silvio Pivano complained already in the 1920s about the lack of relevant documents. Pivano 1928, 19–22.

[75] See, e.g., Bauer 1991, 156–157.

[76] Roero 1997, 65, n. 5. Evidence for Benedetti’s role as an advisor in university matters can be found in Patrizi’s correspondence, as already mentioned.

[77] Tiraboschi 1824, 289–290: “L’Altezza di questo Serenissimo di Savoia ha desiderato, che si dia principio a fondar un’Accademia in questa sua Augusta cittá di Turino, et n’ha data la cura a tre Padri del Gesù di questo insigne Collegio, i quali, non so da che allucinati, soliti però a non s’abbagliare, hanno fatto gran fondamento nella persona mia, caricandomi d’una macchina da incurvar le spalle, quantunque gigantesche. S.A. se n’è fatto Principe, e Protettore, e Capo, per tirarvi buon numero de’ suoi Cortigiani, tanto culti e fioriti nel resto, che, se vi si aggiugne l’ornamento delle belle e delle pulite lettere, non sarà Corte in Europa più rilucente di questa. Il nostro nome è degli Incogniti […].”

[78] Mamino 1992 and Mamino 1995. By contrast, Cibrario thought that it was an encyclopedic project. See (Cibrario 1839).

[79] Pingone 1577, 88: “Anno Christi 1572 mense Martio, Emanuel Dux Taurini theatrum omnium disciplinarum miro ordine, nec minimis sumptibus instituit, curantibus Hieronymo Ruvereo Archibiscopo, et Ludovico Molineo Rochefortio Philosopho, viris in omni doctrinae genere absolutissimis.”

[80] On Renaissance publications in Piedmont, see Bersano Begey 1961, especially vol. 1. See also Merlotti 1998.

[81] Trotto 1625, 2–3: “[…] questi huomini saputi, tocchi dal Prencipe, come instrumenti musici bene accordati, subito rendono ciascuno il suo suono con le parole et quanto meglio possono procurare d’essere intesi discorrendo, e di dar diletto con le buone ragioni, et anco di tirare gli altri al suo parere, come ad una consonanza della verità: perché ognuno dice quello ch’egli sa o crede almeno sia vero. E quindi si veggono trattare hor cose naturali, hor morali, hor mathematiche. Sì che egli quasi come uno Apolline si può dire, che sta fra le Muse, intorno al fonte, che uscì dal colpo del piede del cavallo alato.” On Trotto’s teaching, see Vallauri 1846, 28 and 48–49.

[82] Arma 1580a, f. A2r: “Scalda co raggi […]/ Sbattendo la Terra di caldo priva. Sì com’il martel che bate l’incudine,/ Riscalda l’un e l’altr’in certitudine.”

[83] Arma 1580a, f. A2v:

“Il Signor Benedetti l’indomani
Col signor Berga, insiem’ a l’Ottonaglio
Forn’in pensier’ a me d’assai lontani,
Che’l Sol tirass’a sé com grand’abbagio
Ogni cosa si com’havesse mani.”

[84] Arma 1580a, f. A4r:

“Il Benedetti, come degno maestro,
Mi dimandò d’il centro di tal arco.
Dissi, che gliera col centro de l’Astro,
Ne la medema linea giù scarco.
Si com’anchora volse Zoroastro.
E con tal dire di lui mi discarco.”

[85] Arma 1580a, f. A4r:

“E tutto quest’in scritti fu donato
A Sua Altezza, con tutti soi detti.
E fu dal Dottor Berga conformato.
Il che fece’l signore Benedetti.
Fu poi d’altre proposte ragionato
E de gl’occolte cose, e soi effetti.”

[86] Bucci 1583, 7–8. See Mammola 2013, 6–8.

[87] Bucci 1572 and Berga 1573.

[88] See Mammola 2013.

[89] Merlotti 1998, 585: “Come s’è visto per la polemica fra Costeo e Arma […] non si trattava di isolati testi a stampa che generavano dibattiti destinati a rimanere manoscritti e chiusi nell’ambito degli eruditi, ma semmai del contrario: di discussioni, cioè, sorte in circoli ristretti di medici e scienziati, prima affidate a manoscritti e poi trasportate a stampa a vantaggio d’un più vasto pubblico.”

[90] For an accurate reconstruction of the polemic and its cultural and scientific context, see Ventrice 1989, 103–145 and Mammola 2014.

[91] Vogel 1993.

[92] Benedetti 1579, 3: “[…] l’essersi doppò due mila e più anni scoperto con trionfo della verità, che la terra è molto maggiore dell’acqua, (del che si ha da haver grande obligo tra gl’altri al dottissimo Signor Alessandro Piccolomini) ha non poco rasserenato, Serenissimo Principe, l’animo de’ più famosi Filosofi di nostra età; i quali, sì come prima non intendeano dipartirsi dalla già imbevuta falsità, e per molti secoli adietro, benché con inefficaci ragioni difesa, così hora si lasciano volentieri persuadere il contrario; poiché il senso, e la ragione s’accorda alla dimostratione del vero. E nella scuola de Mathematici per certissime prove si scuopre l’antico errore, puotendosi far fondamento stabile delle misure de cieli, e della terra.”

[93] Arma 1580b: “Il Creator gli diede tal misura./ Che saper non si può da creatura.”

[94] Benedetti 1579, 397–405, “Defensio nostra contra Antonium Bergam, et Alexandrum Piccolomineum.” See Ventrice 1989, 131–134.

[95] This information stems from Bonino 1824–1825.

[96] It is preserved in the Biblioteca Reale of Turin, coll. G 25–67.

[97] Della Torre 1578: “[…] non mi posso quietare pensando chi possi esser costui. Non posso capire, perché quando fa d’il scuolaro, quando del cuogo, quando del corteggiano di Roma, quando del frate osservantino, poi che di osservantini parla. Di esser osservante, nol posso pensare, perché li religiosi parlano correttamente, e non si sgovernano nel parlare, come ha fatto costui, il quale mattamente si è sgovernato. Fu sempre lodata la modestia. Sarà donque dalli modesti dannato costui. Che sij scuolaro, non lo crederò mai, perché hoggi dì li scuolari sono ben creati e non farebbono tale scappate specialmente contra di un’huomo tale da cui mai hebbero altro che apiacere, honor e cortesia. Par bene che habbi del Romanesco nel principio, ma il fine dimostra più presto haver del sguattero, perché il cuogo si sarebbe meglio deportato che non fa costui. Sij chi diavol esser si voglia. Se esso per avanti meglio non si governarà, tale e tanto mi ritrovarà, quale e quanto mi ricercarà.”

[98] See Omodeo 2008a. On Bruno’s lost meteorological-cometary work, see Ernst 1992.

[99] Spampanato 1921, 654–655 and Ricci 2000.

[100] Spampanato 1921, 132.

[101] On astronomical-astrological quarrels in Renaissance Italy and Turin, see Omodeo 2008a and Tessicini 2013.

[102] Bordiga 1985, 609, n. 6. See Omodeo 2014c.

[103] Vernazza 1783. Two manuscript copies of Vernazza’s biography of Cristini are still extant. One is preserved in the Turin State Archive (Archivio di Stato di Torino, coll. Miscellanea J.b.VIII. 9), the other is kept in the Biblioteca Reale of Turin (Vernazza manuscript, misc. 67.5). The latter is a good copy, ready for the printer. It contains an appendix of “documents” for the personal use of the author. These are transcriptions or translations of significant passages of documents by Cristini that were lost or seriously damaged after the fire at the Turin National Library in 1904. They comprise the dedication and table of contents of the Revolutione trentesimaterza del Ser[enissi]mo Sig[nor] il Signor Carlo Emanuele duca di Savoia (1596), notes from various astrological diaries, an Italian version of the beginning of La rithmomachia o sia gioco di Pithagora and, most importantly, a long extract from the Essaminatione dell’errore… della natività del fu S[ignor] Gio[vanni] Battista Benedetti mathematico.

[104] From Vernazza’s papers accompanying his manuscript of his Notizie di Bartolommeo Cristini. Biblioteca reale di Torino, Misc. 67.5, Vita di Bartolomeo Cristini con documenti, “M.S. L.1.10, 11.493, di pag. 42.” See Omodeo 2014b: “Ha pubblicato il Benedetti, il pronostico fattosi del tempo di sua morte nell’opera sua titulata Diversarum speculationum mathematicarum et physicarum liber stampata dell’anno ’85 in una lettera scritta all’ill.mo Volfardo Aisestain, posta nel fine d’ess’opera, percioché appresso haver brevemente dichiarato quali cose egli stimava vere nella giudiciaria e quali vane o false, et detto com’esso Volfardo potrà veder poi meglio in quel trattato dell’osservationi sue astrologiche, quale sperava dar in luce avanti la sua morte, soggiunge il tempo il quale giudicava essa doverli avvenire, o sia avanti al quale desiava pubblicar detto trattato, con queste istesse parole: “antequam ad directionem mei horoscopi cum corpore Martis anaeretae perveniam, quae quidem directio circa annum millesimum quingentesimum nonagesimum secundum evenienti.” Donde appare ch’esso teniva per certo d’haver a morire, quando giongerebbe alla direttione del suo ascendente al corpo di Marte, quale chiama anaereta cioè datore, o promissore de la morte sua. Il che pare habbi volsuto confirmare quando che, come dice, poco avanti la sua morte ei si sentì carrigar dal male, disse d’essersi fallato di quattro minute nel rettificare il tempo di sua natività, perché questo è come s’havesse detto che quando egli havesse accresciuto tempo di sua natività per quattro minute havrebbe conosciuto la direttione predetta essere minore di quello [che] l’haveva fatta, et periciò il tempo della sua morte caggionata da essa direttione dover essere circa questo tempo, ch’egli s’era infermato, et credeva di morire come è pur avvenuto, essendosi occorso ciò fare dell’anno presente 1590 circa le 17 hore del 20 giorno di genaro secondo Gregorio, che viene ad essere il dieci dell’anno antico. Perciò volendo io essaminare di quanto tempo egli habbi fallato nella rettificatione di essa sua natività, accioché giustamente la direttione predetta dell’horoscopo suo al corpo di Marte venisse a cadere nel giorno istessi ch’egli partì da questo secolo, m’è stato necessario sapere il tempo ch’egli havea presupposto fosse quando nacque […]. Perciò mi posi subito seguita la morte del Benedetti a far conto dell’errore del tempo predetto, così alquanto alla grossa, per non haver nota la sopradetta latitudine di Marte, et ritrovai detto errore essere di minute otto in circa di hora […] Ma perché ho dipoi inteso che chi mi ha riferto il Benedetti haver confessato il detto fallo di min. 4 et haver solamente ritrovato tanto per calculo ha espressamente detto che io errava del doppio et non sapea far questo conto […] mi posi a calculare di questa maniera. Prima ho ritrovato il tempo presupposto della natività […] Etc.”

[105] Cristini, Revolutione, Biblioteca Nazionale Universitaria di Torino, N. VII. 10, f. 4r–v: “Ma V[ostra] Alt[ezza] benignissima sì come è stata cagione d’eccitar nell’animo mio il desio delle vertù matematiche, et di farmi fare la presente fatica; così ancora co’l chiamarmela adesso ha risvegliato, o riscoperto le sempre vive brame mie di servirla, le quali erano tenute sepolte dai disaggi che queste carestie et mali tempi mi causano maggiori giornalmente, percioché non sono conosciuto per servitore ordinario di V[ostra] Alt[ezza] da Tesoriere alcuno, né da ministro di suoi dinari o finanze; se non ne’ casi che la vertù et prontezza, o cognizione mia ne’ conti, può reccarli qualche giovamento come ha fatto più volte quando per servitio di V[ostra] Alt[ezza] sono stato da loro richiesto […].

Et per questo sempre cresce maggiore la speranza mia, in solo vostra Altezza quanto ch’essa più m’incita a servirla, et che maggior è fatica che faccio per lei, et minor il numero di quelli in quali posso haver spernaza di soccorso.”

[106] Cristini, Revolutione, Biblioteca Nazionale Universitaria di Torino, N. VII. 10, f. 8r: “[Ho] havuto riguardo ancora ai luoghi ne’ quali cadono essi [pianeti] nella figura della revolutione fatta secondo il vero tempo dato dal Copernico, come è insegnato da principalissimi scrittori dell’astrologia. Et nell’universal giudicio perché ho conosciuto tra scrittori essere certa diversità seguendo alcuni un tempo et altri un altro nel fare delle revolutioni delli quali ancor ch’io provi (come per i calculi di ciascuno posti al principio di questa opera) l’uno solo essere il vero, ho fatto le figure del cielo che si mostrano sotto ambi essi tempi (atteso che ancor la falsa era seguita da diversi et particolarmente dal Benedetti), ho radopiato essi giudici per i luoghi che diversi significati fanno havere esse figure. Accioché V[ostra] Alt[ezza] provandole ambidue conosca ancor lei quale meglio secondi la verità.”

[107] Cristini, Revolutione, Biblioteca Nazionale Universitaria di Torino, N. VII. 10, ff. 11v–12r.

[108] Cristini, Revolutione, Biblioteca Nazionale Universitaria di Torino, N. VII. 10, ff. 12v–13r.

[109] Cristini, Revolutione, Biblioteca Nazionale Universitaria di Torino, N. VII. 10, f. 16v–17r: “Ma io tengo ch’egli seguisse più tosto il calculo d’Alfonso che il vero; solo per causa della facilità d’esso percioché avanti l’effemeridi del Magini molto difficil cosa era trovar il tempo vero della revolutione percioché nissuno avanti lui havea nell’effemeridi calculato il Sole sino alle seconde onde ne seguono i calculi più sottili e veri […]. Con tutto ciò solo per l’autorità d’esso Benedetti non ho volsuto lasciar del tutto la consideration delle figura sua con l’altra come vedevasi.”

[110] See Peyron 1904, 617–618.

[111] Vernazza 1783, 20–21.

3. Structure of the Book and Main Issues

3.1 Benedetti Introduces His Physico-Mathematical Speculations

The Diversarum speculationum… liber is composed of six books, indicated as follows in the table of contents:

“Arithmetic Theorems” (Theoremata arithmetica);

“Explanation of the Operations of Perspective” (De rationibus operationum perspectivae);

“Mechanics” (De mechanicis);

“Disputations on Some Opinions Held by Aristotle” (Disputationes de quibusdam placitis Arist[otelis];

“On Euclid’s Fifth Book [of the Elements]” (In quintum Euclidis librum);

“Epistolary Answers on Physics and Mathematics” (Physica et mathematica responsa per epistolas).

It should be noted that this partition of themes does not mirror the relative relevance of the issues according to the author. In fact, the length and the importance of the books does not exactly correspond to the subdivision indicated in the table of contents. Two sections are quite short: Book 2 on perspective and Book 5 on Euclid. By contrast, the first one, on arithmetic, and the last one are much longer. Chapter 6 is an extremely diverse collection of letters on the most different subjects.

The dedicatory epistle of the Diversae speculationes begins with an acknowledgment of the generosity of Emanule Filiberto of Savoy, with whom Benedetti had particularly good relations. As one reads, they often talked about mathematical issues pertaining to arithmetic, geometry, optics, music, and astrology.[1] Since the Diversae speculationes appeared after Emanuele Filiberto’s death, it was dedicated to his successor, Carlo Emanuele I. Benedetti reports that both dukes encouraged his inquiries and their questions motivated his investigation of specific questions. This is the reason why Benedetti’s Speculationes have an occasional character and are not ordered in a systematic manner. The Theoremata arithmetica (book one) clearly originated from conversations with Carlo Emanuele I as the theorems are presented as answers to the patron’s questions. For instance, the first one is introduced as follows: “The very serene duke of Savoy asked me to prove by means of science and speculation (as one says) that the product of two fractions is inferior to either factors.”[2] The curiosity the dukes held for mathematical matters was not idle but rather rooted in a deep comprehension of the importance of practical mathematics in military and civil affairs. However, courtly etiquette required that serious matters be concealed under the mask of aristocratic detachment and disinterestedness. The court was a refined cultural center that also functioned as a political headquarters, where the most important decisions had to be taken. “Reality and imagination—as has been remarked—prescribed that at court the weight of the duty of government be mirrored by the lightness of amusement.”[3]

Following his example [that of the Duke] […] many asked my advice either in person or by mail on those mathematical problems. As I never avoid work in support of friends, it happened that, after so many years, looking at my paper boxes [scrinia] I found so many solved problems that they could be gathered in a fairly big volume.[4]

Benedetti expressed his admiration for the stimulating intellectual environment made possible by the magnificence of Carlo Emanuele I and the legacy of his father. The wide range of scientific interests shared by both dukes is emphasized in the concluding remark of the dedicatory letter:

Therefore, the glory [of your Highness] will equal that of the ancient Persian kings, and we can expect great happiness in this century if Plato’s prophecy is correct: the future State in which princes philosophize will be blissfully happy.[5]

Benedetti’s preface to the reader,[6] following the dedicatory epistle, provides some more information related to his mathematical-physical work. The author repeats that the Diversae speculationes are a miscellanea of thoughts on various subjects brought about by his own curiosity or by that of patrons and friends. He declares himself confident that, in spite of their disordered format, his speculations will be considered useful, and makes a further claim for the absolute originality of his ideas: “non dubitans quin illis [meis scriptis] in illis scientiis aliquid commodi atque utilitatis allatura sint, praesertim cum in eiusmodi quaestionibus investigandis atque perpendendis, nemo (quod sciam) hactenus elaboraverit” (as I have no doubt that these [writings of mine] will bring something pleasant and useful, especially because (to my knowledge) nobody has so far sufficiently investigated and pondered such issues).[7] The enthusiasm for novelty that was to motivate the supporters of modernity in the querelle des anciens et des moderns can be sensed in these words. Moreover, Benedetti does not exclude the possibility that different people, at different times and in different places, could have made the same discoveries by treating similar problems:

In fact, I included [traditum est] nothing in these books that, as far as I remember, I read or heard from others. If I picked up on the suggestions of others, either I offered a somehow different demonstration or I wrote on the same subject more clearly. In the case that somebody else wrote the same ideas [eadem tradidit], either I was not informed about this person’s speculations or the memory of these readings has vanished. Aristotle himself remarked that it can easily happen that many come to the same ideas. It can even happen that, writing so much, I repeat a certain issue having forgotten that I already wrote on it. Indeed, this has happened to me a few times.[8]

Benedetti adds that very few people ever wrote books which are entirely and solely the fruit of their own mind, except perhaps for Archimedes. Following in the footsteps of this illustrious predecessor, he presents the results of his personal investigations as independent from any authority. He only acknowledges some influence from Tartaglia and a few other authors on whose works he relies: “Hieronymus Cardanus, Michael Stifelius, Gemma Frisus, Ioannes Novimagius, Cuthebertus Tonstallus, caeterique huiusmodi.” The omission of Del Monte as a source on mechanics is striking and telling about the enmity between the two men, which is well documented and will be discussed later.

Benedetti envisages possible criticism of his work and seeks to anticipate objections with a remark inspired by a humanistic sense of relativism: “Quot capita, tot sententiae (As many heads, so many opinions).”[9] Since a book will never receive universal approbation, he writes, it must suffice to provide fruitful insights which will encourage others to undertake further investigations. He mentions only three ancient models: Ptolemy, Euclid, and Pythagoras. Whereas the reference to the first two men can be seen as an appreciation of the most valuable ancient sources on geometry, mathematical astronomy (and astrology), and geography, the mention of Pythagoras explicitly refers to astronomy and should be understood as a reference to Copernicus’s “restoration” of heliocentrism.[10]

The letter to the reader ends up with a reflection on and a rebuttal of the principle of authority. Benedetti claims that mathematical and natural investigation should only be inspired by the love for truth. All considerations which do not derive from an open-minded philosophical attitude ought to be dismissed as unfounded:

In order to establish the truth, I occasionally had to oppose the opinions of others in many places, but I do not want you to ascribe it to some vice of mine nor to call me a malevolent and a sycophant as I display the errors of others. They should rather be thankful to me, since I aim to erase wrong opinions while I am dealing with the same issues—according to Antisthenes, it is indeed necessary “to begin by unlearning errors.” I show the truth, which all philosophers, beginning with Aristotle, should hold in higher esteem than any human authority or favor. As you will encounter something of this sort in my volume, I beg you to abandon all passions in your judgement, keeping in mind Sallust’s admonishment: “Those who assess controversial issues should avoid all hate, friendship, rage, and compassion.” Thus, always favor truth, which is worthy of the greatest efforts, instead of some person, as too many do. Hopefully you will benefit from my work and, in case you will pick some fruit, in the first place you should be thankful to Him from Whom all sciences descend.[11]

3.2 Mathematical Sections

Benedetti proved his capacity as a mathematician in his early writings, especially in the Resolutio omnium Euclidis problematum (1553). The relevance of this publication for the ongoing debates of the time, involving Ferrari, Cardano, and Tartaglia has often been stressed. For instance, Moritz Cantor, in his classic history of mathematics, Geschichte der Mathematik, regarded Benedetti as “ein wirklicher Geometer” on account of his treatment of geometrical problems using a compass with fixed opening.[12] Special mathematical problems scattered in the Diversae speculationes have been summarized by Bordiga. We will deal with some of these problems in other sections, in connection with other aspects of Benedetti’s work such as mechanics, natural philosophy, and astronomy. For the time being, we will limit our treatment to the two books of the Diversae speculationes specifically dealing with mathematics.

3.2.1 Geometrical Demonstrations for the Solutions of Arithmetic Problems

Two books of the Diversae speculationes are devoted to mathematics in the strict sense, Book 1 to arithmetics and Book 5 to proportions. Additionally, several letters in Book 6 are dedicated to mathematical problems and some metaphysical and physical issues are also treated as geometrical problems. In Book 4 Benedetti discusses examples such as the perfection of the circle, i. e., whether the circle, owing to its properties, has to be seen as the “first” or the “last” of the geometrical figures, and the possibility of a never-ending motion on a finite line or the possibility of an infinite motion on a finite line.

Book 1 of the Diversae speculationes has the form of a collection of mathematical exercises. Although Benedetti did not give them a systematic order, he claimed that his readers might find many useful explanations and remarks.[13] He chose to call his propositions “theorems” instead of “problems” to stress their originality, as Benedetti believed they deserved higher recognition than mere problem-solving. He also apologized for the brevity of his treatment by saying that he had confidence in the intelligence and expertise of his readers (the same remark can be generalized for many other sections of the work). In Book 1 he approached arithmetic problems geometrically following Euclid’s example in Book 2 of the Elements. This geometrical approach finds a legitimization in epistemological considerations about the role of geometrical visualization for the comprehension of truth: “images are necessary to the intelligent one in order to speculate” (quoniam oportet intelligentem phantasmata speculari). In Benedetti’s eyes, the process of learning and thinking requires images (phantasmata).

Moreover, Benedetti informs his reader about the process through which knowledge is acquired. In other words, his treatment is not restricted to theory but intentionally expands on heurstics. He complains that ancient mathematical knowledge was often transmitted in a very concise manner. It often lacked demonstration and definitions, or a clarification of the fundamental concepts. For this reason modern readers are often forced to investigate the hidden reasons of “the numbers” and “their effects,” with huge effort:

As we know, ancient mathematician-philosophers discovered many properties of the numbers but transmitted them to posterity either without reasons or with too few ones. Hence, several mathematical problems emerged, which have been addressed by the duke of Savoy. I consider some of the ensuing reflections on the ancients’ propositions to be worth transmitting to posterity. In this manner, my speculations will not fall in oblivion. Rather, I will offer to many an occasion to investigate abstruse themes which are entailed in problems and theorems and could hardly find an explanation so far.[14]

In order to examine Benedetti’s geometrical approach to arithmetic problems, let us consider theorem 120. Here is the problem:

The ancients already addressed this problem: three associates have an amount of money. The sum of the [money of] the first and the second is known, as well as the sum of the first and the third and the sum of the second and the third. From such three aggregates [the ancients] derived the particular [amount of money] of each one of them.[15]

Following a method of resolution, which Benedetti ascribes to Gemma Frisus, he offers the solution to a case chosen arbitrarily:

Gemma Frisus solves this problem applying the regula falsi [rule of the false]. I will follow the same path. Suppose that the addition of the first with the second is 50, that of the second with the third 70 and that of the first with the third 60. From those sums take any two, for instance 50 and 70, whose addition is 120. Subtract from this sum the other one, that is, 60. The result is 60. Its half is 30. This is the amount of money of the second associate. It you subtract this number from 70 (which is the addition of the second with the third) you will get 40. This is the amount of the third associate. Finally, from this number taken away from 60 you will be able to derive the amount of the first associate.[16]

If we express the problem in modern form, this is the set of equations Benedetti is dealing with:

The algebraic solution devised by Benedetti is the following:

Thus,

and

After this solution, Benedetti offers a geometrical demonstration of the validity of this procedure, referred to as “Gemma Frisus’s regula falsi.” For this purpose he draws a triangle with an inscribed circle (Figure 3.1) and supposes that the three sides correspond to the sums that we have expressed as a set of equations.

Fig. 3.1: Geometrical demonstration for the solution of a particular set of equations. Benedetti, Diversae speculationes, Book 1, Theorem 120, p. 82. (Max Planck Institute for the History of Science, Library)

Benedetti’s demonstration here follows:

In order to grasp this problem, let us consider the triangle here indicated as ABC each side of which corresponds to the addition of [the money of] two associates—for instance, assume that the side AB corresponds to the addition of the first with the second whereas the side BC to the addition of the second with the third and the side AC to the addition of the first with the third. Thereby AE, or AO, should be the number [i.e., the amount of money] of the first associate; EB, or BU, should be the second associate; and CU, or CO, should be the third. Furthermore, since AE is equal to AO, BE to BU and CU to CO, according to premises, if we derive the sum or side AC resulting from the addition of the side AB with BC corresponding to the other sums, we will know the addition of BE with BU. Hence, we will also know the half, which is BE or BU. Once we subtract it from the sum BA the resultant will give us the number [corresponding to] AE. By subtracting the number of AE, that is AO, from the sum (or side) AC, and [by subtracting] BU from BC we will know as a result OV and CU.[17]

We can (anachronistically) formalize Benedetti’s reasoning in the following manner.

The geometrical visualization of Frisus’s method, the so-called regula falsi, corresponds to the addition of two sides of the triangle and the subtraction of the third one:

This corresponds to the following addition:

3.2.2 Theory of Proportions

Book 5 of the Diversae speculationes deals with the fifth book of Euclid’s Elements, on proportions. This book of the Elements was a notorious case in medieval translations, as the definitions were often misunderstood, inconsistent interpolations had been added, and flaws introduced in the demonstrations. In his Italian version, Tartaglia tried to emend these flaws. Commandino’s and Clavius’s attempts followed suit. Moreover, a certain number of Euclidian definitions (especially the fifth and the seventh) could not be simply taken for granted, but required analysis and justification.[18]

Hence, Benedetti proposed to introduce new postulates which he considered to be clearer (intellectui commodiora) in order to render the theorems that rely upon them more easily understandable (quandoquidem iis nostri postulatis admissis, sequentia Theoremata perfacillima reddentur). Actually, what he undertook was a revision of Euclid’s theory of proportions, based on a reorganization of postulates and theorems. As Enrico Giusti stressed, despite its brevity, this booklet by Benedetti stands out as “the first modern attempt to accomplish an organic reform of the fifth book of Euclid’s elements.”[19] The difficulty that he and his contemporary readers of Euclid had to face was not technical but rather foundational, as it resided in the clarity of the definitions and the internal coherence of the theory. For the sake of intelligibility, Benedetti thus inserted two preliminary axioms and reversed the relation between the most simple of the Euclidean theorems and the less intuitive of the axioms in order to establish a gradual progression from the most simple to the more complex propositions. The first two postulates are derived from Euclid’s common notions II and III concerning the addition and subtraction of equal parts to equal quantities.[20]

Benedetti rephrases Euclid’s common notions translating the addition (or subtraction) of parts as the composition of ratios:

[First:] All additions [composita] of an equal number of equal parts are equal. Similarly, all proportions are equal that result from the addition of an equal number of other proportions, which are equal among themselves. Euclid tries to demonstrate this in [Elements] V 22 and 23.

Second: If one subtracts equal parts from equal wholes, the remaining parts are equal among them. Conversely, if one adds equal parts to equal wholes, the additions will be equal among them. One should consider these considerations as referred to proportions.[21]

The postulates that follow (3–9) are Euclid’s propositions 7–13 with minimal changes in the order (the inversion of Euclid’s propositions 8 and 9). Benedetti adds three additional postulates (10–12) before he tackles an explanation of Euclidian theorems on proportions. What is lacking is an explicit treatment of the definitions underlying Benedetti’s “reform” of the theory of proportions.[22]

Bordiga underscored the didactic character of Benedetti’s treatment of Book 5 of the Elements by remarking that his concern about clarity and simplicity might have emerged from the teaching of mathematics to his princely pupils in Turin.[23] This might be true; however, the theoretical relevance of this section and of Benedetti’s effort to reform the theory of proportions should not be neglected. During the sixteenth century, geometry, in particular the theory of proportions, was gradually becoming the fundamental tool for the treatment of physics in the process toward a mathematical inquiry and explanation of natural phenomena. In this context, Book 5 of the Elements acquired a higher theoretical status by offering a sort of “meta-geometry” or mathesis universalis.[24]

The universal meaning of geometry as the foundation of rationality itself had been emphasized in the generation before Benedetti in an emphatic Encomium geometriae (Eulogy of Geometry) delivered by Girolamo Cardano at the Academia Platina of Milan in 1535. In this talk he presented geometry as the highest science, or as a sort of prisca scientia, by contending that geometrical rationality, based on quantity and proportion, is the source for all arts and disciplines. Indeed, the modus geometricus is the essence of rationality and even the a priori of God’s Creation. In Cardano’s eyes, geometry was also a practical discipline insofar as it included statics, mechanics, and architecture as subordinate disciplines. Actually, Cardano’s list of arts and sciences that depend on geometrical rationality (committed to the study of quantities and proportions) is long. Geometry, as one reads in his Encomium, is fundamental for arithmetic, music, astronomy, and optics. It is the conditio sine qua non of architecture (non aedificare sine illa licet) and of the plastic arts, painting and sculpture. Geometry is necessary for the construction and understanding of clocks and machines (horologiorum, machinarumque structura). It is further presupposed by natural magic, by the science of weights, by aesthetics (pulchritudo… tota geometrica ratione constet), and by countless other fields of human activity and knowledge.[25]

Benedetti agreed on the fundamental relevance of geometry as the cornerstone of natural inquiry. The fact that his treatment of proportions, although it was very short and condensed, was printed as a book in its own right in the Diversae speculationes bears witness to the relevance he attached to this part of mathematics. In fact, it was crucial for his treatment of weights and thus a close examination of Book 5 of the Elements was an indispensable premise of his mechanics. As Giusti emphasized, Benedetti’s treatment of composite propositions (relating to Euclid’s proposition 17) was the most significant aspect of his theory insofar as it offered an elegant computational tool.[26] As a matter of fact, Galileo and his school would follow the same track by applying composite propositions to the analysis of functional relations in physics, in particular to motion. Hence, from the viewpoint of natural inquiry, Benedetti’s Book 5 is not exclusively motivated by an abstract interest in pure mathematics but by the challenging problems of contemporary physics.[27]

3.3 The Geometrical Theory Underlying Linear Perspective

Book 2 of the Diversae speculationes, entitled De rationibus operationum perspectivae (Reasons of Perspective Operations), is a short treatise on linear perspective, that is, the optical discipline dealing with the construction of perspective to give the illusion of depth. Its main task was to recreate the “cone of vision.” Historically, it originated from the practical problems of three-dimensional representation in the fine arts but also had relevance for architecture (e.g., in surveying or in theoretical treatises), the military art (e.g., the derivation of the structure of the enemy’s fortification from a scout’s sketches), and found special application in theatrical stage scenery.

Benedetti’s book on perspective begins with a claim of originality: “To my knowledge nobody has so far taught the true and inner causes of the operations in perspective in an accomplished manner. Thus, I deemed it to be worth undertaking some reflection [disputationem] in this field.”[28] The fact that “nobody” (nullus) has adequately treated perspective before is attested to by the great number of misunderstandings and widespread errors in this discipline: “In fact, many of those who prescribe the rules of such operations ignore the implications of the true causes, therefore they make various mistakes, as for instance in the following plane figure A [etc.].”[29] Benedetti’s treatment was indeed different from the tradition of practical treatises (from Piero della Francesa’s De prospectiva pingendi onwards) explaining how to construct a perspective picture owing to its higher mathematical sophistication and theoretical depth. Moreover, it took a different angle than Federico Commandino’s discussion of perspective in connection with mathematical astronomy and geography in his edition of Ptolemy’s Planispherium (1558), which treats it as mathematically equivalent to stereographic projection. Benedetti’s focus was the explanation of the reasons underlying perspective, as well as errors that might occur in theory and practice. In a manner that is similar to his treatment of problems of arithmetic, geometry offers the conceptual tools to formalize knowledge embedded in practice.

Like most parts of the Diversae speculationes, this book on perspective originated from extemporary sources of inspiration. In fact, it begins ex abrupto with the examination of an error in linear perspective which leads Benedetti to more general considerations. Moreover, the dedication of its seventh chapter to the architect Giacomo Soldati strengthens the impression of the occasional character of the writing.[30]

The error discussed in the incipit of the De rationibus operationum perspectivae is represented in a two-dimensional figure (Figure 3.2).

Fig. 3.2: Figura superficialis, or plane diagram, pointing out an error of linear perspective. From Benedetti, Diversae speculationes II, p. 119. (Max Planck Institute for the History of Science, Library)

Benedetti explains how to correctly construct the linear perspective “inscribed” in the given triangle IDQ:

[…] in the plane figure A (here included), in order to ’degradate’ (as one says) [i.e., to put it in perspective] the rectangle QA [QXAU] in the triangle IDQ, they [the practitioners] draw a line parallel to QD from point B (at the intersection of line OL with the side ID of the triangle) or, indifferently, they draw it from point Z (the intersection of the same OL with the perpendicular XI). They are not aware that the latter method is the only correct one, not the former. Conversely, if the former were correct, the latter could not be true. They might excuse themselves by arguing that one draws the aforesaid parallel line from B supposing that the plane IDQ is inclined horizontally relative to the rectangle QA according to the angle IDQ. But this excuse cannot be accepted because, according to their supposition, if one assumes that IDQ is inclined, the inferior angles of the rectangle put in perspective should not be as acute as IDQ and IQD. This can be easily understood considering their construction [ratio], which I show in the plane figure A here included. However, if one wants to see the rectangle in perspective, one should locate this plane relative to the eye in the same manner as the line ID relative to O. But this is too difficult [to represent].

To summarize, the correct manner [to put a rectangle in perspective] is to draw a line ER parallel to QD from point Z, which is common to OI and XI (perpendicular to LP).[31]

In order to visualize the construction, Benedetti produces an additional diagram (Figure 3.3) offering the tridimensional correspondent of the plane diagram he has just examined. Note that point O is the point of departure of the cone of sight. The observer is thus lying with his feet in point P.

Fig. 3.3: Figura corporea, or tridimensional diagram, visualizing the same problem of linear perspective as in Figure 3.2. From Benedetti, Diversae speculationes II, p. 120. (Max Planck Institute for the History of Science, Library)

Departing from these considerations, Benedetti treats various cases (including the one in which the projection plane is not perpendicular to the observer’s line of sight) using the same approach. That is to say, he first draws bi-dimensional diagrams showing the construction and then offers tridimensional geometrical explanations providing an insight into the physical reality underlying the construction. He even offers a sort of virtual instrument to guide constructions in linear perspective.[32] As Judith V. Field has argued, Benedetti “shows the applied mathematician’s concern with particular problems, rather than the pure mathematician’s interest in theoretical developments. The significance of his work lies not only in its mathematical insight but also in the fact that it shows us innovative work in the purest of sciences, namely pure mathematics, being carried out within an applied tradition.”[33] As has been argued, it is plausible that Benedetti’s constructions were included in later works on perspective, in particular by Simon Stevin and Guidobaldo del Monte, although none of them explicitly referred to Benedetti. Apart from Del Monte’s enmity towards him, it was typical of Renaissance mathematicians who wrote on optics to leave their sources unmentioned.[34]

3.4 Sections on Physics: Mechanics and Natural Philosophy

Book 3 of the Diversae speculationes deals with mechanics. Benedetti remarks that many learned people have already written extensively on this issue, but that their achievements are not sufficient because nature and practice (usus) awaken many doubts concerning the validity and the completeness of previously acquired knowledge. New problems have emerged. His intention is to present many novelties and to propose theses that either have never been treated or have never been adequately demonstrated (nusquam ante hac tentata, aut satis exacte explicata). In the final words of his introduction to Book 3, we can see that he attaches great importance to his speculations on mechanics; he even states that he will be especially remembered after his death for his contribution to the advancement of this discipline:

Many man have written a great deal about mechanics, and they have written most ably. But nature and experience are always wont to bring to light something new or previously unknown. And it is therefore incumbent upon a high-minded and grateful individual, if he happens to discover something previously shrouded in darkness, not to begrudge it to posterity. For he himself gained a great deal from the work of others. Now it is my desire to publish a few items that will, I believe, prove not unwelcome to those who concern themselves with mechanics, items which have never before been dealt with or have not been sufficiently well explained. I may thus either show my desire to be helpful or at least give some evidence of possessing a bit of talent and industry. And perhaps in this way alone may I leave behind me proof that I ever lived at all.[35]

Book 4 essentially deals with Aristotle’s opinions on local motion in Physica and De caelo, although critical references to Metaphysica and Meteorologica are also present. In this section, Benedetti seeks to provide new foundations to the theory of motion from a philosophical perspective based on a mathematical approach to nature (inconcussa mathematicae philosophiae basis). He revises basic concepts of physics, such as place and time, as well as natural issues, for example the existence of a physical void. Eventually, he deals with cosmological aspects linked with the theory of motion, including a defense of the Copernican system.

We will treat books 3 and 4 of the Diversae speculationes separately and with particular attention (in chapters 5 and 6), owing to their historical and theoretical relevance.

3.5 The Epistles: Miscellanea Mirroring a Scientific Network

The last book of the Diversae speculationes is a large collection of letters, “Physica et mathematica responsa” (Epistolary Answers on Physics and Mathematics). The autonomy of this section is underscored by the insertion, at its beginning, of a second preface.

To the reader.

Nothing is more suited to virtue than being active and present through constant motion. Just like a shining star shimmering for the eyes of the spectators. It occurred to me that this or that person invited me with words or stimulated me with letters to dispute on those issues in which I am versed. In fact, I was devoted to mathematics and to highly philosophical speculations while I sojourned in the most splendid princely courts and illustrious cities, where many intelligent people displayed admirable curiosity, desire for knowledge and conversation. I collected part of those disputes and answers, because I judged them to be worth communicating. I planned to reread and revise them, if I had some spare time, [having in mind two goals]: on the one hand, that science itself circulates as much as possible and grows; on the other hand, that the names of those excellent men, who stimulated me with their questions, are made immortal, as far as it is in my power to do so, and that I can lead others to follow their example, abandon the idle sloth (which is able to corrupt even the most talented) and start investigating, exchanging, and discussing serious subjects that could prove useful at some point and worth disseminating. For the time being, please, approach our efforts with a benevolent and judicious attitude. Regards.[36]

Significant elements emerge from this passage: the courtly environment influencing Benedetti’s activity, the interest in applied knowledge, the recognition of the importance of dissemination, as well as the celebratory intention of the work (to render the dedicatees immortal). For us, this epistolary is the most tangible evidence of a network of social relations and its scope.

The epistles are not organized chronologically (actually the dates are almost always omitted) but rather according to the importance of the addressees, some of whom were already dead at the time of publication. The first letter was directed to the duke Emanuele Filiberto, the second to his son Carlo Emanuele I, and the following four letters to the powerful nobleman Andrea Provana de Leyní. The topics are linked to Benedetti’s role as court mathematician and mathematical advisor.[37] The first letter is an expert report on the reform of the calendar, the second deals with a geometrical problem (the determination of the circle circumscribing a given square) that probably emerged from the lessons imparted to the prince, and the letters from three to six address practical problems of navigation (such as the use of astronomical instruments at sea).

Although it is not possible to identify all of Benedetti’s correspondents, one can say with certainty that most of the epistles are tied to the northern Italian environment, specifically to Turin and to Venice. Many correspondents were in fact Subalpine or Venetian patricians or courtiers and artists linked to one of these two centers. Sometimes they were linked to both, as was the case with ambassadors such as Domenico Morosini (Dominicus Moresinus), Pietro Pizzamano (Petrus Pizzamanus), and Francisco Mendoza. There is great variety in the status and professions of the correspondents. Michela Cecchini and Clara Silvia Roero, in their accurate reconstruction, came to the following assessment: “The variety of themes that were discussed and of the professions of the participants in the debates shows that Giovanni Battista Benedetti was a man of culture and practice. He was ready to engage in a fruitful debate with exponents of the scientific world in the broadest sense (such as mathematicians, physicians, jurists, and philosophers) and with politicians, diplomats, and ambassadors, as well as with experts of military art and religion. Moreover, he did not dislike architects, artisans, constructors of instruments and fortifications, surveyors, and astrologers.”[38]

Among his correspondents, the list of Savoy patricians and courtiers is long. Those who emerge most prominently are the orator Francesco Barbaro, who sojourned in Turin between 1578 and 1581, the Turin professor of jurisprudence Bernardo Trotto, the court historian Emanuele Filiberto Pingone, and the functionary and artillery general Giuseppe Cambiani. Benedetti advised Cambiani on ballistics, commenting upon Tartaglia to determine the most effective inclination for a cannon.[39] Another member of the Savoy milieu was Giovanni Battista Femello, philosopher, physician, and ordinary professor of practical medicine at Turin. Benedetti wrote to him on a very special geographical issue: the determination of the position of Iceland. By contrast, the mathematical advice directed to his friend, the jurist Francesco Ferrari, concerned ethics and justice. Benedetti explained to him how to make use of proportions in order to explain “commutative justice.”[40] An epistle on the quantification of the time necessary to spill the water out of a vase through three tubes, alongside another two epistles on geometry and the application of geometry to the study of solar radiation, are directed to the Savoy secretary Ludovico Niccolò Calusio. The letters on mechanics addressed to Giovanni Paolo Capra of Novara are better known to historians of science. They intermingle considerations on turning wheels with others on astronomy and the boundaries of the cosmos. Other noblemen belonging to the Turin milieu who corresponded with Benedetti were Franchino Trivulzio and Demoulin de Rochefort, responsible for the ducal Wunderkammer.

Benedetti had intense exchanges not only with patricians but also with artists, engineers, and practitioners. Four letters are addressed to the architect Gabriele Busca and their issues range from natural philosophy and meteorology to the use of instruments and topography. The Savoy clock-maker Jacopo Mayeto (perhaps Maletto) shared Benedetti’s interest in gnomonics, sundials, and mathematical instruments, as witnessed by one of the letters in the collection.[41] Furthermore, a letter to the Savoy surveyor Angelo Ferrario is at once technical and playful: it is a discussion of the curve described by a hanging rope that Ferrario and Benedetti observed together in the park of the ducal castle of Lucento by Turin.[42] Additionally, Benedetti wrote on music to the Flemish madrigalist Cipriano de Rore, who had been appointed in Venice, Ferrara, and Parma and whom Benedetti had probably already met in his hometown Venice. Among the artists, the Venetian poet Girolamo Fenarolo is the addressee of two letters on geometry.

Some epistles tackle the philosophical and epistemological issues underlying mathematics and their cognition. For instance, those to his Venetian friend Domenico Pisani deal with the issues “de philosophia mathematica” (on mathematical philosophy) and “de imaginatione specierum” (on the imagination of the species).[43] Another philosopher corresponding with him was the Turin professor Francesco Vimercato to whom several letters are addressed. While these mainly deal with optics, theses on natural philosophy and cosmology are discussed in a brief letter to the theologian Gerolamo Cordero (Hyeronymus Condrumerius). It is entitled “Quod recte Arist[otelis] senserit coelum casu non esse productum” (Aristotle correctly saw that the heavens are not produced in a casual manner).[44] Benedetti moreover corresponded with the Paduan professor Pietro Catena, known as a keen supporter of the certitudo mathematicarum (see the next chapter) and the mathematical method applied to natural inquiry.

The identity, biographies, and work of many of Benedetti’s correspondents are still obscure. This is especially true for foreign scholars, presumably Germans, whose names were latinized as follows: Theodosius Raisestaim, Paulus Aemilius Raisestaim, Volfardus Aisestain, and Conradus Neubart. Benedetti does not give these names any titles. An exception to this is the correspondent Anselm Rosenburg, who is designated as Imperial surveyor (agrimensor Cesareus).

Footnotes

[1] Benedetti 1585, f. A2r. Note that the pagination A1–A4 is doubled: the first installment A1–A4 contains frontispiece, table of contents, dedicatory epistle, and preface to the reader, whilst the second installment includes the first eight pages of the Theoremata arithmetica. Since the latter has also a pagination number (1–8) we will quote from the first fascicle indicating the folio and from the second giving the page.

[2] Benedetti 1585, 1: “Interrogavit me Serenissimus Dux Sabaudiae, qua ratione cognosci posset scientifice et speculative (ut dicitur) productum ex duobus fractis numeris, quolibet producentium minus esse.”

[3] Barberis 2017, xvii.

[4] Benedetti 1585, f. A2r: “Illiusque imitatione […] non pauci aut praesentes, aut per litteras me de his, atque illis mathematicis quaestionibus consuluerunt. Cumque ego nunquam laborem amicorum causa defugerim, evenit ut post tot annorum curricula, mea scrinia scrutatus, invenerim tot absolutas quaestiones, ut ex eis corpus mediocre effici posse videretur.”

[5] Benedetti 1585, f. A2v: “Quare, et veterum Persarum Regum gloriam [tua celsitudo] aequavit, et nos veluti in spem certam faelicitatis huius saeculi induxit, si verum est Platonis vaticinium, beatam eam futuram Rempublicam in qua Principes philosophentur.”

The comparison between the Duke of Savoy and the ancient Persian kings is in line with an established Renaissance topos derived from classical sources, in particular Xenophon’s Life of Cyrus, as discussed by Vester 2007, 228-229.

[6] Benedetti 1585, “Ad lectorem,” ff. A3r–A4r.

[7] Benedetti 1585, f. A3r.

[8] Benedetti 1585, f. A3r: “Nihil enim his libris a me traditum est, quod aut legisse, aut ab aliis audivisse meminerim, nam si aliena attigi, ea, aut cum aliqua differentia demonstrationis, aut dilucidius scripsi, quod si forte alius eadem tradidit, aut eius lucubrationes ad me non pervenerunt, aut earum perlectionis memoria excidit. Ut etiam Aristoteles ipse sensit facile fieri potest, ut pluribus eaedem opiniones in mentem veniant. Immo multa scribendi evenire potest, ut cum iamdiu aliquid scripserit, iam oblitus, idem repetat, quod mihi etiam nonnunquam accidit.”

[9] Benedetti 1585, f. A3v.

[10] Omodeo 2014a, “The Invention of the Pythagorean Cosmology,” 167ff..

[11] Benedetti 1585, ff. A3v–A4r: “Quoniam vero multis in locis accidit, ut veritatis iudicandae causa necesse mihi fuerit quorundam sententiis adversari nolim te hoc mihi vitio tribuere, meque hoc nomine carptorem maledicumque habere quod alienos errores aperiam, cum potius habenda sit mihi gratia, quod in iis interdum laborans (quae Antisthenes in disciplinis magis necessaria esse dixit, ut mala scilicet prius dediscantur) falsas opiniones evellere studeam, veritatemque ostendere, quam omnis philosophus, Aristotelis exemplo, pluris quam cuiusvis hominis authoritatem, aut gratiam facere debet. Cumque in hoc volumine aliquid eiusmodi legeris te oratum volo, ut in iudicando, affectum omnes exuas, Sallustianum illud prae oculis habens. Omnes qui de rebus dubiis consultant, ab odio, amicitia, ira, atque misericordia vacuos esse decet. Hinc fiet, ut non personae (ut multi solent) sed veritati, quae summo studio dignissima est, semper potius faveas. Vale nostrisque laboribus utere, si quem inde fructum, sicuti spero tuleris, illi precipue habes gratiam a quo omnes fluunt scientiae.”

[12] Cantor 1892, 521–525.

[13] Benedetti 1585, f. A3v.

[14] Benedetti 1585, 1: “Praeclare multa veteres mathematici philosophi de numeris eorum effectibus excogitata posteris tradiderunt, quorum cum vix ullam rationem reddiderint, aut certe per exiguam, occasione diversorum problematum mihi Serenissimo Sabaudiae Duce propositorum praebita, de iis quae ab antiquis proposita fuerunt contemplanda nonnulla occurrerunt, quae posteritate commendare non inutile arbitratus sum, ne heae meae cogitationes interciderent, et occasionem praeberem quamplurimis abstrusa haec indagandi, quae problematibus et theorematibus involuta, vix aliquem qui evolveret nacta sunt.”

[15] Benedetti 1585, 81: “Supponunt etiam antiqui tres socios nummos habere, quorum summa primi et secundi cognita sit, item summa primi et tertii cognita et summa secundi et tertii item cognita, atque ex huiusmodi tribus aggregatis veniunt in cognitionem particularem uniuscuiusque illorum.”

[16] Benedetti 1585, 81: “Gema Frisus solvit hoc problema ex regula falsi. At ego tali ordine progredior. Sit verbi gratia, summa primi cum secundo 50 et secundi cum tertio 70 et primi cum tertio 60; harum trium summarum accipiantur duae quaevis, ut puta 50 et 70, quae coniunctae simul dabunt 120 a qua summa detrahatur reliqua, idest 60. Et restabit nobis 60, cuius medietas ergo 30, hoc est numerus nummorum secundi socii; quo numero detracto a 70 (hoc est a summa secundi cum tertio) remanebit 40, hoc est numerus tertii socii; et adhic numerus desumptus a 60 residuus ergo numerus primi socii.”

[17] Benedetti 1585, 82: “Pro cuius ratione consideremus triangulum adhic subnotatum abc cuius unumquodque latus significet summam duorum sociorum, ut puta latus ab significet summam primi cum secundo, latus vero bc summam secundi cum tertio, latus autem ac summam primi cum tertio, et ae seu ao sit numerus primi socii, et eb vel bu sit secundi socii, et cu seu co sit tertii, cum autem ae aequalis sit ao et be aequalis bu et cu aequalis co ex supposito si de<su>mpta fuerit summa seu latus ac datum ex aggregato laterum ab cum bc reliquarum summarum, relinquet nobis cognitum aggregatum ex be cum bu. Quare et eius medietas be sive bu nobis cognita erit, qua detracta ex summa ba relinquetur nobis cognitus numerus ae, detracto vero numero ae hoc est ao ex ac summa, seu latus, aut bu ex bc remanebit oc seu cu cognitus.”

[18] In Commandino’s Italian translation (Euclid 1575, ff. 63r–v), these definitions go as follows:

“V. Le grandezze si dicono essere nella medesima proporzione, la prima alla seconda, et la terza alla quarta, quando le ugualmente molteplici della prima, et della terza, o vero insieme avanzano le ugualmente molteplici della seconda, et della quarta secondo qual si voglia moltiplicatione, o vero insieme le pareggiano e vero insieme sono avanzate da loro.”

“VII. Quando delle ugualmente moltiplici, la moltiplice della prima vanterà la moltiplice della seconda, et la moltiplice della terza non avanzerà la moltiplice della quarta, allhora la prima alla seconda si dirà haver maggior proportione che la terza alla quarta.”

[19] Giusti 1993, 22.

[20] In Commandino’s Italian translation, these definitions go as follows. See Euclid 1575, f. 7r:

“II. Se alle cose uguali, si aggiungono cose uguali, tutte sono uguali fra loro.

III. Se dalle cose uguali si traggono cose uguali, etiamdio le rimanenti sono uguali tra loro.”

[21] Benedetti 1585, 198: “[Primum.] Quod tota composita ex aequali numero partium aequalium, sunt invicem aequalia. Ut si quis diceret omnes proportiones quae compositae sunt ex aequali numero aliarum proportionum invicem aequalium, sunt etiam invicem aequales, quod Euclides conatur demonstrare in 22. et 23. quinti libri.

Secundum. Quod si a totis aequalibus detractae fuerint aequales partes, quae remanent eruntque partes invicem aequales. Et e converso si aequalibus aequalia addas composita erunt invicem aequalia. Quod in ipsis proportionibus hoc loco semper intelligendum est.”

[22] Cf. Giusti 1993, 27 quoting from Antonio Nardi: “Il Benedetti, Geometra insigne non si accorse, che volendo riformare il 5° libro di Euclide, trascurò la definitione della uguale, e disuguale ragion, quale principio e fondamento dell’opera. Stupiscomi certo di tale inavertenza.”

[23] Bordiga 1985, 629.

[24] Giusti 1993, 22.

[25] Cardano 1966, vol. 4, 440–445.

[26] Giusti 1993, 33: “D’altronde, l’interesse della teoria di Benedetti non sta nel risultato globale, ma soprattutto nel ruolo chiave della proporzione composta, e in particolare nella creazione di un algoritmo di calcolo agile ed elegante.” See Benedetti 1585, 202.

[27] See Giusti 1993, chap. 2, section 2.

[28] Benedetti 1585, 119: “Cum nullus adhuc (quod sciam) veras internasque causas operationis perspectivae perfecte docuerit, operaeprecium existimavi aliqua de iis disputationem suscipere.”

[29] Benedetti 1585, 119: “Multi enim eorum, qui huiusmodi operationis regulas praescribunt, cum eius effectuum veras causas ignorent, varios diversosque errores committunt, ut exempli gratia in subscripta figura superficiali A [etc.].” See Figure 3.2.

[30] Benedetti 1585, 133: “Superioribus diebus non diu postquam de perspectivis inter nos sermonem habuimus, dum animus totus adhuc in his esset. Illud in mentem venit quod eximius ille vir, et profundissimae doctrinae, nec unquam satis laudatis Daniel Barbarus se accepisse profitetur a Ioanne Zamberto patritio Veneto, qui ad verbum omnia desumpserat a Ioanne Cusino Parisiense. Nec parum mirabar peritissimus illum Cusinum, quod in capite quarto secundae partis perspectivae, ut quod piam planum quadrilatam composuisse. […].” See Mamino 1989.

[31] Benedetti 1585, 119: “[…] in subscripta figura superficiali A volentes degradare (ut dicunt) rectangulum qa in triangulo idq ducunt parallelam ipsi qd a puncto B intersecationis lineae ol cum latere id trianguli, et (idem) indifferenter, eandem quoque a punctoque Z intersecationis ipsius ol cum perpendiculari xi ducunt, nescientes hunc solum esse verum modum, non item alium, quia si alius, talis esset, hic, verus non existeret, nam si vellent sese excusare, quod ducendo dictam parallelam a punctoque B hoc fiat praesupponendo planum ipsius idq versus rectangulum qa orizontale inclinatum, secundum angulus idq, haec excusatio accipienda non esset, quia horum consensu, praesupponendo planum idq inclinatum, anguli inferiores rectanguli degradati, non tam acuti, quae sunt duo idq et iqd esse deberent, quem facile eorum ratione innotescet, quae de figura corporea A hic subscripta mox proponam, praeter id, quod volentes deinde aspicere quadratum degradatum, oporteret huiusmodi planum respectu oculi ita collocare, quemadmodum se habet linea id respectu o quem factu nimis arduum esset.

Vera igitur ratio ergo ducere parallelam er ad qd a puncto Z communi ipsis ol et xi perpendiculari ipsi lp.”

[32] Benedetti 1585, 131.

[33] Field 1987, 247.

[34] See Andersen 2007, 152

[35] Translation from Drake and Drabkin 1969, 166. Cf. Benedetti 1585, 141: “Scripserunt multi multa, et quidem scitissime, de mechanicis, at cum natura ususque, aliquid semper vel novum, vel latens in apertum emittere soleant, nec ingenui aut grati sit animi, posteri invidere, si quid ei contigerit comperuisse priis tenebris involutum: cum tam multa ipse ex aliorum diligentia sit consequutus. Paucula quaedam futura, ut reor, non ingrata his qui in hisce mechanicis versantur, nusquam ante hac tentata, aut satis exacte explicata in medium proferre voluit: quo vel iuvandi desiderium, vel saltem non ociosi ingenioli argumentum aliquod exhiberem: atque vel hoc uno modo me inter humanos vixisse testatum reinquerem.”

[36] Benedetti 1585, 204: “Ad lectorem. Ut nil magis virtutis est proprium, quam agitari, et incessabili motu prodesse. Ac veluti fulgidum sydus ante oculus spectantium commicare. Ita mihi mathematicis iisque maxime philosophicis speculationibus dedito, saepissime, ut in principium summorum aulis, et amplissimis civitatibus degenti, ubi multa semper Nobilium mira curiositate, sciendi desiderio, et conferendi cupiditate referta, versantur, ingenia, contigit, modo ab his, modo ab illis, aut verbis tentari, aut literis provocari ad disserendum, de his, in quorum studiis versamur. Quarum concertationum et responsionum, quoniam non omnino indigna existimavi, quae memoria comendarentur, partem aliquam apud me conseruivi. Ubi vero per ocium licuit, relegi, ac tandem de manu mittere decrevi. Tum ut scientia ipsa quo magis diffundetur, crescat; et quicquid valeo, sine invidia in communem utilitatem conferam. Tum ut virorum praestantissimorum, qui me suis interrogationibus excitaverunt, quantum in me erit, gratitudine ergo, nomina reddam immortalia, et eorum exemplo alios, ocio sordidiore abiecto, quod solet iurialium praecipue excelsa ingenia corrumpere, ad sciscitandum conferendum, et disserendum, de rebus seriis, et quae usui aliquando esse possint, et quandoque evulgari mereantur, alliciam. Tu interim nostris laboribus fruere, et nostram diligentiam boni, et aequi consule, et Vale.”

[37] “Mathematics” has to be here understood in the wide and interdisciplinary Renaissance meaning. It comprised arithmetic and geometry, as well as practical mechanics, architecture and engineering, astronomy and meteorology, optics, physics, and even metaphysical and epistemological issues.

[38] Cecchini and Roero 2004, 32: “Dal quadro variegato dei temi discussi e dalle professioni dei protagonisti dei dibattiti emerge la figura di Giambattista Benedetti come uomo di cultura e di pratica, disposto ad instaurare un dialogo proficuo sia con esponenti del mondo scientifico in senso lato, come matematici, medici, giuristi e filosofi, sia con politici, diplomatici e ambasciatori, come pure con uomini d’arme e di religione, non disdegnando neppure architetti e artigiani, costruttori di strumenti e di fortificazioni, agrimensori e astrologi.” In the following we will often rely on Cecchini and Roero for an overview of Benedetti’s correspondents.

[39] Benedetti 1585, 258–259.

[40] Benedetti 1585, 284.

[41] Benedetti 1585, 423–425.

[42] Benedetti 1585, 361–363.

[43] Benedetti 1585, 298 and 298–299.

[44] Benedetti 1585, 298 and 298–299. For the identification of this person, see Cecchini and Roero 2004, 58–59.

4. Epistemology

One of the most challenging aspects of Benedetti’s endeavor was his attempt to merge mathematical and physical speculations, as is clearly stated in the title of the Diversae speculationes mathematicae et physicae. In order to understand his way to “physico-mathematics,” we will discuss his mathematical epistemology starting from some statements scattered in his major work and then look at the premises implicit in his treatment of nature. We will briefly review the Renaissance reflections on mathematics linked to practical developments in technological fields as well as to eclectic reassessments of Pythagorean and Aristotelian debates on the certainty of mathematics and their applicability to natural philosophy. Focusing on the epistemological premises underlying Benedetti’s mechanics, we will discuss medieval and early modern approaches to natural knowledge, which, in spite of their mathematical rigor, rested on a physics and metaphysics of contingency. For many centuries, it was assumed that the mathematical regularity of the phenomena does not imply their causal necessity.

4.1 The Certainty of Mathematics

In the letter to the Venetian patrician Domenico Pisani included in the collection of the Diversae speculationes and entitled De philosophia mathematica (On Mathematical Philosophy), Benedetti emphasized the philosophical dignity of his discipline, placing it at the same rank as physics, metaphysics, and ethics—if not higher than them, considering the certainty of its demonstrations (certitudo suarum conclusionum):

I am surprised that, although you are well-versed in Aristotelian philosophy, nonetheless you make a distinction between the philosopher and the mathematician in your writings, as if the mathematician were not as much a philosopher as the naturalist and the metaphysician. In fact, as far as the certainty of his conclusions is concerned, he deserves the title of philosopher much more than them.[1]

This reference to mathematical conclusiones reveals Benedetti’s methodological focus on the dignity and validity of his discipline. In his connection of mathematical and physical speculations, he seems to put the emphasis on the method rather than on ontology and to seek for the certainty of mathematics and its applications by way of its specific logic. This was the position of his correspondent, the Paduan professor Pietro Catena.[2] Along with him, Benedetti maintained that the certainty of mathematics has an extra-sensible and intelligible character.[3] As Benedetti added in his letter to Pisani:

Actually, you are not the only one who makes this mistake, but this is more grave in consideration of the fact that, although you [Aristotelians] even label ethics as a philosophical discipline, you do not acknowledge that the divine mathematical sciences also should be adorned with the name of philosophy. In fact, if we consider this name more attentively we will clearly see that it is in itself more suited to the mathematician than to anyone else, since none of the others is more certain in his affirmations than the mathematician. And no one is more driven by the love of science in his cognition. This is evident. In fact, [the mathematician] does not rely on the senses nor accepts any presupposition that is not so true and evident to the intellect that no power whatsoever could show that it is false.[4]

Benedetti was acquainted with scholars quarreling over the status of mathematics, its demonstrative methods, and its legitimacy in the treatment of natural issues.

In his time such debates on the foundations and status of mathematics were intense. As an instance of epistemological reflections on the philosophy of mathematics, historians often mention the controversial theses by the Paduan professor of philosophy Alessandro Piccolomini, with whose work Benedetti was familiar. Piccolomini authored, among other writings, a treatise De certitudine mathematicarum (On the Certainty of Mathematics, 1547) affixed to his paraphrases of pseudo-Aristotelian mechanics, In mechanicas questions Aristotelis paraphrasis. As one reads in this sort of appendix, one ought not to cast into doubt the certainty of mathematics. However, this does not depend on demonstrative methods but rather on the subject of inquiry: “Mathematical disciplines are certain not due to the force of their demonstrations but rather to their subject matter itself.”[5] Their special subject is quantity, connected to matter. Hence, the certainty of mathematics, for an Aristotelian such as Piccolomini, rests on the fact that it deals with universal properties of nature that can be extracted from concrete reality by means of abstraction (res mathematicae sunt ex abstractione).

The cause of the certainty of mathematics is evident from Aristotle’s statements. Simplicius is of the same opinion when he states (in De anima I 11) that the cause of the certainty of mathematics is due to the fact that they refer to quantity. In fact, as he argues, quantities are sensible things, they have sensible causes and they are known to us as such.[6]

This consideration led Piccolomini to argue that motion can become a mathematical object, if one abstracts from materiality:

One could argue that, just like magnitude, motion is a common sensible, too. Moreover, it has its effects and causes (see Physics V and VI). Thus, there can be a science of motion (a natural one), which is certain, similar to the science of quantity, that is, mathematics.

We can answer to this [apparent objection], that if we consider motion in general, as separated from matter and insofar as it is a continuum […], our consideration will be mathematical. This is not in contrast with our principles.[7]

The “ontological” and not only “epistemological” dimension of mathematical physics would concern later scholars such as Kepler and Galileo, going beyond the shared Aristotelian discourse in their investigations of the mathematical properties of material processes.[8] Benedetti was rather concerned with mathematics as an intellectual tool, a sort of “logic of scientific inquiry.” In the above-mentioned letter to Pisani on his mathematical philosophy, he stressed the certainty of mathematical reasoning rather than that of its “objects.” Nonetheless, he was interested in the question raised by Piccolomini as to the usefulness of mathematics in the study of motion. As we will discuss, Benedetti’s insight concerning the generalization of the methods already in use in mechanics, in the science of weights, established the premises for the conceptualization of problems in dynamics.

Benedetti’s interest in mathematics as a conceptual instrument accords with the interest in the demonstrative power of mathematics shown by many scholars entering the debates about mathematical certainty. The publication of Piccolomini’s De certitudine mathematicarum led to a series of negative or sympathetic reactions, among them the criticism made by the translator of Proclus’s Commentary on Euclid, Francesco Barozzi, as well as those by the Paduan professors Pietro Catena and Giuseppe Moletti. Barozzi, in his 1560 Quaestio de certitude mathematicarum, and Catena, in his 1563 Oratio pro idea methodi, argued in favor of the demonstrative certainty of mathematics, contra Piccolomini’s exclusive focus on mathematical objects. The theoretical discussion regarding the status of mathematics, the certainty of its demonstrations, their applicability to the investigation of nature, and the hierarchy between natural philosophy and mathematics continued for a while. It also produced frictions among Jesuit scholars such as the philosopher Benito Pereira and the mathematician Clavius, who were inclined to assign different levels of importance to the study and teaching of mathematics in the colleges of their order.[9]

As far as the institutional side of the defence of mathematics is concerned, it opposed scholars and intellectuals benefiting from varying social status, such as mathematicians, philosophers, and theologians. Benedetti’s self-perception and, later, Galileo’s self-presentation as “philosophers” involved polemical stances. They claimed for their mathematical and physical investigations a wide cultural meaning against critics who downplayed such investigations as merely technical and specialistic.

Early polemics over the viability of the mos geometricus were not purely intellectual and academic but were also rooted in the rising recognition of the practical import of mathematics in engineering, architecture, mechanics, and warfare. A new class of intellectuals was emerging composed of “scientist-engineers,” so to speak, both expert in practical disciplines and trained in letters.[10] Edgar Zilsel already remarked that the Renaissance exaltation of mathematics went far beyond purely Platonic and Pythagorean influences. At that time new mathematical writings were composed and published dealing with the practical problems of commerce, topography, architecture, and the arts.[11] Moreover, the emergence of mathematical and natural conceptions dependent on the advance of technology was reinforced by the growing self-consciousness of new social groups.[12] As an example of the awareness of the status of the practical arts one could mention Filippo Pigafetta’s introduction to the Italian edition of Del Monte’s work on mechanics. Here he reversed the assessment of craftsmen and practical knowledge, which had been marked by the contempt of aristocrats and traditional intellectuals, as follows:

‘Mechanic’ is a very honored title. According to Plutarch it refers to a profession linked with warfare. It is suited to a man of high rank who is also capable of using his hands and his intelligence to realize wonderful works of rare usefulness and pleasure for human life.[13]

This judgment well expresses the shifting opinion on practical knowledge which also marked Benedetti’s environment. We have already stressed the centrality of practical mathematics for the Savoy dukes, in particular Emanuele Filiberto, in their construction of the new capital, Turin.

4.2 Physico-Mathematics

As a direct consequence of this mathematizing epistemology Benedetti dismissed the well-established separation between physics and mathematics in cosmology, that is, he refused to separate the investigation of “causes” and calculation.[14] This anti-fictionalist perspective implied a realist commitment related to the Copernican system and its embedding within a renewed cosmology. As we will discuss in the section on Benedetti’s views on the universe, he praised the system “of Aristarchus and Copernicus” as it avoided the absurdities of an anthropocentric conception according to which the immensity of the firmament was created only for us. Rather, all planets are like Earth or, better, like moons reflecting the solar light. Among the direct consequences of the Copernican view was accepting that the fixed stars do not rotate around the center of the world within one day; rather, they are immobile.[15]

From this viewpoint, Benedetti’s understanding of mathematics is not too removed from that of a mathematician such as Copernicus, who, in Book 1 of De revolutionibus, indicated that the mathematical superiority (simplicity and intelligibility) of his own planetary system was such that natural philosophy had to be subordinated to mathematical astronomy and not vice versa. The theologian who wrote the anonymous introduction to Copernicus’s work, Andreas Osiander, tried to reaffirm the hypothetical character of mathematical astronomy, and its subordinate position as a discipline relative to physics and theology. By contrast, Renaissance scholars who appreciated the physical meaning of the Copernican system called it “Pythagorean” to underscore at once its natural philosophical and mathematical character.[16] As an extreme case one could mention Bruno’s declarations during his Inquisition trial. In order to defend his cosmological views, and in particular the motion of Earth, he did not mention Copernicus but the ancient philosophical school of Pythagoras: “I affirmed [the existence of] infinite individual worlds [i.e., planetary systems] similar to that of the Earth. Following Pythagoras, I regard the latter as a celestial body. The Moon is similar to it, as well as other planets and stars, which are infinite [in number].”[17] Pythagorean cosmology was regarded with suspicion by the Inquisitors and the doctrine of the plurality of worlds became one of the allegations against Bruno, who would be eventually executed as a heretic in Rome. In the same years in which Bruno was a prisoner of the Holy Office in Rome and his works were examined for censure, the censors also attacked Patrizi for his natural views, including the doctrine of terrestrial motion. Although Benedetti shared similar views about the plurality of worlds and the possibility of terrestrial motion, he did not incur any censure. We dare say that he was one of the last Renaissance authors who could freely speculate on nature in Italy before natural philosophy became a highly ideological issue in the religious repression escalating in the 1590s.

Benefiting from his subalpine freedom, Benedetti reflected on Pythagorean cosmology in a section entitled Pythagoreorum opinionem de sonitu corporum coelestium non fuisse ab Aristotele sublatam, where he excluded the possibility that the “sound of celestial bodies” is the production of any physical sounds. Rather, he identified the Pythagorean doctrine of the world harmony with divine providence:

As to motions, dimensions, distances, and influences there is nothing that corresponds to such proportions, but, since all of them depend upon the infinite Divine Providence of God, these velocities, those dimensions, distances, and influences must have the most perfect order and relations among them and relative to the universe.[18]

According to Benedetti’s outlook, the harmony of the heavens does not correspond one to one to musical harmony in the strict sense. From this viewpoint, Kepler’s later effort to translate heavenly geometries into musical melodies in the Harmonices mundi libri V (1619) can be seen as a radicalization of similar “Pythagorean premises.”

Most significantly, Benedetti and Kepler shared a commitment in favor of the fusion of mathematical and physical accounts of nature in the frame of an early modern transformation of natural science in which the methods of the physico-mathematical disciplines gained a paradigmatic status. The epistemological shift also involved well-established disciplines such as astronomy. Kepler’s astrophysics, first illustrated in the Astronomia nova (1609), was a significant step toward the derivation of celestial geometries from physical forces. Kepler translated a geometrical discipline (Ptolemaic and Copernican mathematical astronomy) into a physico-mathematical one. In fact, he explained the elliptical path of planetary orbits as the effect of interactions of forces. He emphasized the double bound of his astronomy, inseparably intertwining physics and mathematics, in the title of the work: Astronomia nova αἰτιολογιτός seu physica coelestis de motibus stellae Martis (New Astronomy Investigating the Causes, or Celestial Physics Concerning the Motions of Mars). As Kepler announced in the introduction: “In this work I mixed celestial physics with astronomy.”[19] He meant to launch a new discipline, “celestial physics,” that merged mathematical modeling with causal physics.[20] Kepler remarked that the ignorance of physical causes compels scholars to settle for conjectures since no choice can be made between mathematically equivalent hypotheses. By contrast, physical arguments are decisive in deciding between mathematically equivalent models. Therefore, celestial physics and astronomy should be unified. The result was a mixed science (scientia mixta) whose data came from the senses and whose demonstrations are expressed in mathematical terms. This physicalization is well shown in Kepler’s physico-mathematical concept of “orbit” (orbitae) substituting that of orbs (orbes) (that is, the material spheres transporting celestial bodies). According to him orbit is “the path together with its physical causes—expressed as physical laws.”[21] Shape and velocity of astronomical orbits depend on the force (vis) emanating from the sun, that is, on a physical cause of geometrical effects.[22]

Descartes’s Traité du monde et de la lumière (completed in 1632–1633, but printed posthumously, in 1664) and the Principia philosophiae (1644) marked a culminating point in the move toward the reduction of natural disciplines (such as optics and astronomy but also physiology) to material interactions of corpuscles in motion. Descartes’s philosophy was particularly influential as it legitimized a mathematical treatment of nature with the advances of physics in his time. At the same time, he connected his explanations to views on matter and causality irreconcilable with the qualitative, essentialist, and teleological accounts of the Scholastic tradition. In particular, his mechanization elevated the results of Renaissance mechanics to a higher and more generalized level.

Benedetti’s place is rather at the beginning than at the end of this process. As the title of his major work hints, he was committed to a mathematical-physical investigation of nature. He did not limit his application of a mathematical method to those fields where this approach was already established, but extended it to the treatment of all realms of natural inquiry.

4.3 The Contingency of Nature and Mechanics

Benedetti’s mathematical approach to nature did not lead him to the belief that physical phenomena are ruled by necessity. Rather, he shared a medieval and early-modern ontology and epistemology of contingency enabling a particular cohabitation of mathematized physics and indeterminism (in other words, formal determination without causal necessity). In order to better understand it one has to look at Scholastic motives informing his physics, in particular his mechanics, and the scientific and philosophical work of his successors. This will require a short excursus.[23]

4.3.1 Scholastic Treatments of Nature as the Realm of Contingency

It would be misguided to think that a mathematical approach to nature in Renaissance science implies the assumption that natural causation is ruled by necessity. This was indeed not the case for well-established medieval and Renaissance views. Only in the course of the seventeenth century would contingency be banned from the realm of natural causation in the developments of post-Cartesian mechanism. For philosophers such as Baruch Spinoza and Gottfried Wilhelm Leibniz contingency marked the limitations of our knowledge and not an ontological limitation of nature. As one reads for instance in Spinoza’s Ethica ordine geometrico demonstrata (Ethics, demonstrated in geometrical order) I 29: “There is no contingency in nature. All natural beings are determined by divine necessity to exist and operate in a special manner.” (In rerum natura nullum datur contingens, sed omnia ex necessitate divinae naturae determinata sunt ad certo modo existendum et operandum). By contrast, in the Renaissance a mathematical treatment of natural phenomena underlaid no principle of sufficient reason, hence it did not imply the necessity of natural causation. In particular, mixed mathematical disciplines that had received a Scholastic embedment or systematization rested on a well-established Aristotelian conception, according to which sublunary phenomena are determined without necessity.

Historically, contingentia is the Latin variant translation of the Aristotelian concept of “possibility,” both as modal logical endechomenon as well as physical-metaphysical dynamis within a hylemorphic framework. In the context of the Christian reception, this terminus received an onto-theological connotation in a frame of creationist theology. In late Scholasticism, contingentia came to signify the worldly reality, or nature as Creation. Nature was deemed to be contingent. It exists de facto but could also not exist because it depends on God’s will. As John Duns Scotus put it,

So then, the first issue has become clear: how there is contingency in things—because it comes from God—and what is in God which is the cause of this contingency—because it is his will.[24]

In Aristotle, there was a tension between two meanings of “possibility.” According to Analytica Priora (13: 32 a 18–20) the possible is that which is “neither necessary nor impossible,” whereas according to De interpretatione (13: 22 a 14–13 a 26) possibility is exclusively that which is opposed to “impossibility” and therefore includes also that which is necessary. As a reminiscence of this original tension, one can find in Scholastic philosophy two different definitions of contingency either as “quod est nec impossibile nec necessarium” (that which is neither impossible nor necessary) or “quod non est impossibile” (that which is not impossible).[25] Both meanings were kept in the Latin rendering of the Aristotelian possibility as contingentia by Gaius Marius Victorinus (III–IV cent. CE) and Boethius (IV–V cent. CE), but the Latin expression also suggested affinity between that which is contingent (contingit) and that which occurs (evenit or accidit).[26] This third connotation would eventually prevail through the late-Scholastic differentiation between contingentia and possibilitas and its reception in the philosophical systems of the seventeenth century (and most notably by Leibniz).[27] Unlike abstract (purely logical) possibility, contingency referred only to that which is real but not so by necessity: “id, quod [est sed] potest non esse” (that which [is but] could not be). In the Christian perspective of the Almighty’s Creation, contingency happened to include all that is not God himself, that is to say, nature, or the universe.

This background is fundamental to understand not only theological disputes but also natural philosophical and scientific developments during the Middle Ages and the Early Modern Period. The connotation of nature as contingent—as that “which could not be”—is theological and metaphysical in its essence, since it points to the dependency of the world on God. However, from the point of view of natural conceptualizations, not only the “vertical” dimension of metaphysics is relevant but also the “horizontal” dimension of causality within nature. On the horizontal plane of the interrelation among finite beings, contingency refers to a degree of indetermination, and a certain unpredictability in the connection between causes and effects. Moreover, whereas a theological perspective focuses on the radical contingency of that which exists as created being, natural philosophy addresses the relationship between contingency and necessity within nature, that is, between divine order and phenomenal imperfection. This relationship between that which is not necessary and that which is necessary had to be conceptualized and indeed was conceptualized as the relationship between the absolutum and the conditionale or secundum quid.

In Book 1 of the Summa contra gentiles, Thomas Aquinas defined contingency through its distinction from necessity. In the case of the contingent beings, as one reads in Summa contra gentiles I 67, a cause can produce its effect or not, whereas in the case of necessary beings, their cause cannot not produce them:

The contingent differs from the necessary according to the way each of them is found in its cause. The contingent is in its cause in such a way that it can both not-be and be from it; but the necessary can only be from its cause. […] Just as from a necessary cause an effect follows with certitude, so it follows from a complete contingent cause if it be not impeded.[28]

A contingent cause, as one reads, will fulfill its tendency to produce a certain effect “si non impediatur,” that is, if no impediment hinders its realization.

In Book 2 of the Summa contra gentiles, Thomas dealt extensively with the contingent being (“omne quod est possibile esse et non esse” and “[id quod] ad utrumlibet se habet”).[29] According to him, the world is contingent insofar as it is created. In this general sense, “God is to all things the cause of being” (Summa contra gentiles II 15).[30] In particular, God’s free will is the origin of this world. Nonetheless, Thomas does not exclude that natural reality is populated by both necessary and contingent beings. Absolute necessity (necessitas absoluta), he writes in Summa contra gentiles II 29, does not pertain to God, since His decision and action is independent from any constriction (debitum). Rather, absolute necessity pertains to the immaterial, or “separated” beings as well as to those bodies in which the form fulfills all potentialities of their matter, as is the case with the heavenly bodies transported in circles. As for terrestrial (sublunary) bodies, their forms are imperfectly realized. Matter, as the potentiality to take different forms, is at the origin of their contingency, that is, it is the source of the possibility to realize or not to realize a certain effect: “But in things whose form does not fulfill the total potentiality of the matter, there still remains in the matter potentiality to another form.”[31] For the low realm of birth, corruption, and change, Thomas speaks of conditional necessity (necessitas conditionalis). In the sublunary sphere, contingency cohabits with absolute necessity (e.g., the inevitability of death for all animals and the hylemorphic composition of all bodies). Whereas necessity pertains to the formal determinations of natural phenomena, contingency is the partial fulfillment of necessary tendencies.[32]

According to Scholastic terminology, there is always a “quid” producing the deviation of material phenomena from their formal rule. We will call this outlook an “ontology and epistemology of contingency.”[33] The Pythagoreanism of many Renaissance scholars such as Benedetti did not depart from a view stressing the contingent character of natural phenomena in general. As we will argue, one encounters in Benedetti’s physics and mechanics a systematic use of theoretical tools implying natural contingency in the form of a distinction and interrelation between formal mathematical necessity and its material realization. In order to understand Benedetti’s mathematical treatment of contingency it is useful to consider the medieval approaches to contingency, especially the science of weights (scientia de ponderibus) he relied upon.

The idea of contingency informing physics and mechanics was related to its use in other disciplines, even ethics. Whereas there can be no obstacle impeding the realization of God’s will, which is therefore “absolute” (voluntas absoluta), human will, or voluntas secundum quid, is conditioned by circumstances. In other words, the realization of the highest aims of humankind is intrinsically contingent, as Dante expressed in the Divine Comedy:

But utterance and feeling among mortals,
For reasons which are evident to you,
Have different feathers making up their wings.
I, too, as man feel this disparity […].[34]

Apart from ethical contingency, Scholastic authors also used secundum quid in logic. For instance, Petrus Hispanus explained the meaning of the so-called secundum quid fallacy in his Tractatus sive summule logicales, commenting on Aristotle’s On Sophistical Refutations V (166b36–167a14).[35]

In logic, secundum quid meant either a “diminution” of a concept through restriction of its definition (secundum quid et simpliciter), or the designation of a subject through one of its parts or characteristics (denominatio totius per partem). A secundum quid fallacy occurs if an identity is established between something considered in a particular respect and the same thing considered absolutely (or simpliciter). For instance, the existence of a depicted animal does not imply the existence of the animal simpliciter. Thus, the argument “est animal pictum, ergo est animal” is not correct. In this case, there is a quid signalizing the gap between universal necessity and particular or concrete contingency.

4.3.2 Contingent Causation in the scientia de ponderibus

The scientia de ponderibus heavily drew on the idea of the conditional limitation of natural necessity depending on circumstances (secundum situationem, also literally meant as “depending on the position”). In particular, the concept of gravitas secundum quid, or positional heaviness, had a powerful explanatory function, most notably in the Aristotelian treatment of weights by Jordanus Nemorarius, and continued to be essential during the Renaissance in the reflections on mechanics by scholars such as Tartaglia, Cardano, and Benedetti himself.[36]

In mechanics the “limitation” or “determination” secundum quid implied that the dynamic tendency of a body was reduced or enhanced depending on intervening constraints or circumstances, in particular mechanical ones. The rotations of a lever around a pivot or of a balance around its fulcrum were conceptualized as constrained motions. In such displacements, the inherent (“necessary”) vertical tendency of a weight resulted in a circular motion due to external constraints. Similarly, the heaviness (gravitas) of the bodies suspended at the extremities of a simple machine varied in relation to their changing positions within the system. In such cases, a “necessary” straightforward motion in accordance with natural order resulted contingently in a circular one. The implicit mental model for this kind of displacement was that “circular motion is constrained rectilinear motion.” This means that, in the sublunary sphere of contingency, straightforwardness and rectilinear tendency had a higher onto-epistemological status than circularity since straightforwardness was necessarily rooted in natural order. By contrast, circularity, as the deviation from such order, had to be explained. As a consequence, circularity (in the elementary sphere) was allotted a derived and subordinated onto-epistemological status. In other words, circularity was an instance of nature departing from necessity owing to some rather elusive factor or secundum quid. From this viewpoint, it was seen as a deviant realization of given potentialities similar to moral deviation from the necessary laws of uprightness. In order to stress that the mechanical treatment of the scientia de ponderibus was embedded in the framework of contingency, we could also formulate the principle in this way: “circular motion is rectilinear motion modified by a contingency.”

Almost at the beginning of his small treatise “on the weights,” Nemorarius stressed his Aristotelian commitment. In fact, his approach was based on the opposition between the natural vertical motion of the elements and the violent hindrances producing circular deviation. At the same time, he introduced the key concept of gravitas secundum quid (in some cases, also levitas secundum quid), which we will refer to as “positional heaviness.”

[…] if equal arcs are taken on a greater circle, and on a smaller one, the chord of the arc of the greater circle is longer. From this I can then show that a weight on the arm of a balance becomes lighter, to the extent that it descends along the semicircle. For let it descend from the upper end of the semicircle, descending continuously. I then say that since the longer arc of the circle is more contrary to a straight line than is the shorter arc, the fall of the heavy body along the greater arc is more contrary to the fall which the heavy body would have along the straight line than is a fall through a shorter arc. It is therefore clear that there is more violence in the movement over the longer arc than over the shorter one; otherwise the motion would become heavier. Since something moves with more violence in the ascent [along the arc], it is apparent that there is more positional heaviness [gravitas secundum situm] and, as it is like that depending on position [secundum situationem], one can aptly call it ‘positional heaviness’ [gravitas secundum situm].[37]

In its circular descent along a circular path, a weight deviates from its natural tendency, or intentio, the more the arm of the balance departs from the horizontal position. Therefore, the “violence” is greater when the arc of displacement is longer, while the weight progressively loses its weight insofar as the vertical component in its motion is reduced.

According to Nemorarius, a weight that reaches the bottom of the circular arc described by the arm in its displacement is not “at rest” but only “lighter.” In fact, a natural being is at rest only if it is fully accomplished, that is, once it has realized the aim, or act, toward which its power is directed teleologically. By contrast, a body is always in motion, or striving to move, until it has reached its end: “All motion strives toward its aim—indeed the whole nature strives towards actuality and is realized [in it]—hence the opposition occurs against [a displacement] contrary [to the natural tendency].”[38]

A body on one arm of the balance becomes lighter during its downward motion than an equal one located on the other extremity. Thus, as Nemorarius assumes, or tries to demonstrate, a balance removed from its state of equilibrium will tend to restore the original state. As one reads in the propositio secunda (with reference to the diagram in Figure 4.1), which is the second of a series of propositions developing the details of Nemorarius’s doctrine of weights,

Suppose now that the descent occurs on the side B and the ascent on the side C. I say that both will go back to the [horizontal] position of equality. In fact, B will not further descend, because its descent towards D is more oblique than the ascent of C towards the [horizontal position of] equality; in fact, B and C are equidistant from the place of equality.[39]

Fig. 4.1: Diagram accompanying preposition two in Apianus’s 1533 edition of Nemorarius’s Liber de ponderibus (1533, f. B2r). (Bayerische Staatsbibliothek)

Nemorarius’s reasoning becomes clearer in light of propositions four and five:

Fourth [proposition]: It is positionally heavier, insofar as its descent, in the same position, is less oblique.

Fifth [proposition]: But a more oblique descent partakes less of the straight [descent], for the same quantity [of the path].[40]

In proposition five, it is suggested that the vertical components of the potential descents of the two beams could be identified and compared. This was the source of the idea that the variation of heaviness could also be determined by comparing the straightness of the descents. A similar procedure was later taken up and explained in detail in Niccolò Tartaglia’s considerations in the Questiti et inventioni diverse (1546) about the manner of ascertaining the positional heaviness of two weights on the basis of the so-called angles of contact. These are the “curvilinear” or “mixed” angles between the circular path of the arms of a balance and the vertical lines connecting the weights to the cosmological center of gravity (see Figure 4.2). Tartaglia compared the angles of contact of two equal weights located on the extremes of a balance, and argued that the lifted one is always smaller than the lowered one. Thus, the lifted weight would face a descent that is more oblique. It would acquire a greater positional heaviness than its lowered counterweight and, as a further consequence, the inclined system would reestablish its horizontal balance, if not hindered to do so.

Fig. 4.2: In the Quesiti et invenzioni diverse, Tartaglia argued that the relative positional heaviness of the weights A and B on a balance could be determined on the basis of the “mixed” angles of contact HAF and DBF. Since it is argued that DBF < HAF, the weight B will be heavier than A. Thus, the inclined system will strive toward the restoration of a horizontal equilibrium. (Max Planck Institute for the History of Science, Library)

In spite of his attempt to quantify the quid accounting for the alleged restorative motion of the inclined balance, Tartaglia’s geometrical quantification maintained a margin of indeterminancy. As he stated, the ratio between the two mixed angles is less than that between any determined quantities. Therefore, it is impossible to stabilize the system in its inclined position by adding a small (no matter how small) weight on the lowered side of the balance. According to Tartaglia, it is impossible to counterbalance the positional heaviness of the lifted weight. Quite on the contrary, any additional weight added to the lowered side would make the balance rotate and reach the vertical position.[41]

4.4 The Epistemological Import of Benedetti’s Generalization from Weights to Forces

As we have argued so far, in the medieval scientia de ponderibus circular motion is conceived of as constrained linear motion. Yet, within an Aristotelian cosmology, this mental model is restricted to the sublunary sphere, where motions cannot fulfill their nature. This is indeed the sphere of contingency, where a gap is to be witnessed between the necessary order of things (or “nature” as actuality) and the effective phenomena (subjected to “violence” or to external constraints). The four elements naturally tend toward their places through a straightforward descent or ascent. Heavy bodies, for instance, strive toward the center of gravity, which is, at the same time, the center of the cosmos. If their motion is hindered, as is the case with mechanical constraints, a certain factor or quid has to be taken into account, which explains the deviation from the rule. In this theoretical context, contingency is the concept expressing the relationship between the natural law and phenomenal reality, which follows a norm while deviating from it. The secundum quid is that which explains this deviation. Possibly, it has to be expressed through geometrical means, although it might prove unintelligible or infinitesimal, as was the case with Tartaglia’s ratio between mixed angles accounting for the gravitas secundum quid of the weights of a balance. In the treatment of weights, in particular of those on a balance, Nemorarius and his followers made a limited use of the mental model of curvilinear motion as constrained linear motion. In fact, they employed it to account for phenomena linked to gravity (i.e., the vertical fall of bodies explained in Aristotelian terms). It was Benedetti who made the decisive step toward the generalization of this model in the direction of inertial dynamics. Let us consider his application of it first to balances and then to centrifugal forces.

In the section on mechanics of the Diversae speculationes, Benedetti picked up and revised the Scholastic concept of gravitas secundum quid. Guidobaldo del Monte had already criticized Nemorarius’s and his followers’ conclusion that an inclined balance hinged on its fulcrum as its center of gravity would return to the horizontal position, but his criticism went so far as to renounce the concept of positional heaviness altogether.[42]

Relying on the Archimedean concept of the center of gravity of a body, Del Monte concluded that an equal-arms balance hinged on its fulcrum would remain stable in any position (a correct conclusion only if it is assumed, in modern terms, that the gravitational field is homogeneous): “Propositio IV: Libra horizonti aequidistans aequalia in extremitatibus, aequaliterque a centro in ipsa libra collocato, distantia habens pondera; sive inde moveatur, sive minus, ubicunque relicta manebit.” (Fourth Proposition: Take a balance that is equidistant from the horizon and that has weights in its extremities which have the same weight and equally distant from the center (the latter being located in the balance itself). Whether it is displaced or not, it will remain in the same position in any position.)[43]

Benedetti shared the criticism of Nemorarius and Tartaglia with regard to their specific argumentation about the tendency of such an inclined balance to reach the horizontal position but based his judgement on a novel treatment of positional heaviness. The first chapter of Benedetti’s De mechanicis begins with the statement: “Every weight placed at the end of an arm of a balance has a greater or a lesser heaviness depending on differences in the position of the arm itself.”[44]

Hence, he clearly committed himself to a mechanical theory of equilibrium based on positional heaviness. Benedetti’s technical terms are not always employed in a rigorous and consistent manner. He treats the pondus at times as the varying quantity to be taken into consideration, as is shown by expressions like “proportio ponderis in C ad idem pondus in F” and “unde fit… pondus magis aut minus grave,” in De mechanicis II (Benedetti 1585, 142). Given these semantic fluctuations, we will translate pondus as “body” or as “weight” and gravitas as “heaviness” or as “weight,” depending on the context. At the beginning of chapter 1 of his book on mechanics, Benedetti talks of a varying quantity of heaviness, or gravity (gravitas), belonging to a weight (pondus) or a body placed on a balance beam. Hence, he makes a terminological distinction between pondus, as a kind of absolute weight or heavy thing, and gravitas, as a downward tendency that can act with more or less force on the body (depending on the inclination of the beam). In this case (as in most cases in the text), pondus has the essentialist meaning of a substance (a substratum or ὑποκείμενον). It is the body or weight on the balance, whose special property of being heavy, namely the gravitas, varies depending on a quid. This quid is the position, or situm.

Benedetti seeks to quantify it by means of a method he invented. He considers the line, which he calls linea inclinationis or linea itineris, connecting a weight on an inclined balance beam to the cosmological center of gravity. Note that Benedetti calls the elementary downward tendency an iter from a merely kinematic viewpoint, but also an inclinatio from a physical and more proper one. According to him, the major or minor heaviness of the weight can be assessed through the projection of the linea inclinationis on the horizontal line passing through the fulcrum (Figure 5.1). The more distant it is from the fulcrum, the heavier the positional heaviness becomes. Thus, the weight reaches a maximum of heaviness when the balance is horizontal, and its minimum when it is vertically resting (nititur) on the fulcrum or hanging (pendet) from it. Notably, this approach anticipates the one based on the determination of the torque in classical physics, and comes to the same conclusions.[45]

Additionally, Benedetti equates the heaviness to a virtus, vis, or vigor, i.e., a force, which might also act in different directions (in De mechanicis, Ch. 3) and is applied to the extremity of a constrained mechanical system, like a lever or a balance. This is a significant generalization from weights to forces, but for our present discussion the most important generalization relates to rectilinear tangential tendencies in systems set in circular motion.[46]

The relevant treatment is the epistle to Capra and is included in the Diversae speculationes. It deals with the rotation of a millstone and the question of whether its motion could be perpetual. Benedetti denies this by arguing that the rotation is impeded first by the friction of the air and, second and more importantly, by the resistance of the millstone’s parts. The latter have a straightforward tendency, an inclinatio recte eundi, along the tangential lines of their rotation (Figure 4.3). As one reads, this rectilinear inclination or impulse (impetus) can be bent only by violence. Moreover, the centrifugal tendency grows in proportion to the augmentation of the velocity, as witnessed by other cases, among them the rotation of a catapult or a sling (machina missilis). A centrifugal tendency is seen as a rectilinear natural inclination (naturalis inclinatio recte eundi).

You ask me this question in your letter. Suppose a millstone rested on a virtually mathematical point and was set in circular motion, could that circular motion continue without end, assuming that the millstone is perfectly round and smooth?

I answer that this kind of motion will certainly not be perpetual and will not even last long. For apart from the fact that the wheel is constrained by the air which surrounds it and offers resistance to it, there is also resistance from the parts of the moving body itself. When these parts are in motion, they have by nature a tendency [impetus] to move along a straight path. Hence, since all the parts are joined, and any one of them is continuous with another, they suffer constraint in moving circularly and they remain joined together in such motion only under compulsion. For the more they move, the more there grows in them the natural tendency to move in a straight line, and therefore the more contrary to their nature is their circular motion. And so they come to rest naturally: for, since it is natural to them, when they are in motion, to move in straight line, it follows that, the more they rotate under compulsion, the more does one part resist the next one and, so to speak, hold back the one in front of it.[47]

Fig. 4.3: A diagram showing Benedetti’s considerations on the rotating millstone stressing the centrifugal tendencies of its parts. (Drawing by Irina Tupikova)

The mental model of circular motion as constrained straight motion receives in Benedetti’s treatment a higher degree of generalization. In this case, he argues that, since it contrasts with a natural inclination, it cannot be eternal. Note that this assumption (violent motion cannot be eternal) is Aristotelian but emerges in a context in which this legacy is meant to be rejected.[48]

Another Aristotelian echo looms over Benedetti’s statement that the linear tendency makes a body “lighter,” since if it were freed from the constraint hindering its projection, it would not fall vertically but rather travel through a more or less rectilinear trajectory tangent to the circular motion of the constrained rotation. In the conclusion of his reflection on the natural rectilinear striving of the parts of a body set in circular motion, Benedetti stressed the originality of his treatment “without precedents” and its opposition to Aristotelian dynamics (according to which the projection of a body through a medium presupposes the support of the medium itself).

But if you wish to see this truth more clearly, imagine that while the body, i.e., the top, is spinning around very rapidly, it is cut up or divided into many parts. You will observe not that those parts immediately fall toward the center of the universe, but that they move in a straight line, and, so to speak, horizontally. No one, so far as I know, has previously made this observation on the subject of the top.

From such motion of the top or of a body of this kind it may be clearly seen how mistaken are the Peripatetics on the subject of the forced motion of a body. They hold that the body is driven forward by the air which enters [behind it] to occupy the space left by the body. But actually the opposite effect [that is to say, resistance] is produced by the air.[49]

We have so far observed two instances in Benedetti’s work on mechanics in which a tension between mathematical laws of nature and their empirical realization emerges: his treatment of the rotation of a beam about its pole and that of a turning wheel. In both cases, natural straightforward tendencies are constrained and deviated into violent circular ones. The epistemological meaning of these concepts lies in the possibility of a geometrical treatment of natural contingency seen as the connection between the necessity of the rules and of the principles and their necessitation, that is, their deviation, as witnessed by the empirical reality of curvilinear motions.

4.5 From inclinatio to inertia and Beyond: Mechanistic Perspectives

René Descartes generalized the insights implicit in the idea that curvilinear motion is contingent rectilinearity at an epistemic level (through the expansion of their realm of application) as well as at an epistemological and ontological level (giving them a foundational meaning). In Le Monde, circular motion is treated as a deviation from rectilinear motion. Descartes develops a general theory of the world in which circularity is the main characteristic of the motions of both the particles of matter as well as of planets revolving about the centers of their orbits.[50]

[…] when a body is moving, even if its motion most often takes place along a curved line and, as we said above, it can never make any movement that is not in some way circular, nevertheless each of its parts individually tends always to continue moving along a straight line. And so the action of these parts, that is, the inclination they have to move, is different from their motion.[51]

This is the third of Descartes’s three laws of nature (loix or règles de la Nature) as exposed in chapter 7 (“Des loix de la nature de ce nouveau Monde”). It follows the inertial law of conservation of the state of the bodies and that of the conservation of the quantity of motion. The third law is particularly relevant from the viewpoint of our epistemological inquiry into mathematics without necessity, since it clearly expresses the gap between law and effective reality, between the straightforward tendency of all bodies and their real circular motions, in a manner that is akin to medieval and Renaissance predecessors such as Benedetti. Note that Descartes calls the rectilinear tendency “inclination” just as Benedetti called it “inclinatio recte eundi.” This terminological choice is apt to express its character as a natural inner tendency. The examples that Descartes choses to illustrate his claim are familiar to readers of Renaissance sources on mechanics: the wheel (une roue) and the sling (fronde) (Figure 4.4).

Fig. 4.4: Descartes’s visualization of the centrifugal tendency of bodies thrown by a sling, in Le Monde, Ch.7. (Bayerische Staatsbibliothek)

In the Études galiléennes, Koyré affirmed the complete independence of the law of inertia, which is only in nuce in Galileo’s physics, from experience, since rectilinear motion is never observed in nature. “Contrairement à ce qu’on affirme bien souvent, la loi d’inertie n’a pas son origine dans l’expérience du sens commun et n’est ni une généralisation de cette expérience, ni même son idéalisation. Ce que l’on trouve dans l’expérience, c’est le mouvement circulaire ou, plus généralement, le mouvement curviligne. On n’est jamais—sauf le cas exceptionnel de la chute, qui n’est justement pas un mouvement inertial—en présence d’un mouvement rectiligne.”[52]

In light of our reconstruction, this statement proves quite inaccurate. As we have seen, the vertical fall of a heavy body is not the only observable straight motion: the beginning of the trajectory of a projectile thrown with great speed also looks rectilinear. Slings and catapults are in fact the instruments with which turning wheels and rotating millstones were compared, and it was from these instruments that Benedetti, Descartes, and also Galileo in the Second Day of the Dialogo sopra i massimi system del mondo, derived the centrifugal tendencies of the parts of rotating objects. Is this not a generalization from experience? Such generalization went so far as to include the explanation of the behavior of bodies on a rotating Earth, in the case of Galileo, and the conceptualization of corpuscular and planetary motions, as was the case for Descartes. Moreover, before the classical law of inertia was defined, what took center stage was the observation of rectilinear motions—either the vertical fall or centrifugal tendencies—and of their circular deviations. A major physical problem faced by Scholastic and post-Scholastic mechanics was precisely that of conceptualizing the relationship between curves and straight lines. In particular, against the backdrop of Aristotelian philosophy, curvilinear motion appeared as constrained. It was a derived displacement resulting from a violent external intervention bending the straightforward natural tendency of a moving body. In such an Aristotelian and post-Aristotelian context, circular motion was seen as contingent. That is to say, it was the deviation from natural order depending on an obstacle which was called the “secundum quid.” As we have argued, the concept of “secundum quid” is embedded in the Scholastic reflections upon natural necessity, order, and contingency. It was referred to as a model of causality in which the observed phenomena represent a partial fulfillment of an underlying order, or of natural laws. Accordingly, elementary bodies express their necessary laws in a limited manner, that is, they have to be explained through the so-called necessitas conditionata or necessitas secundum quid. Contingency is the relation between necessary order and phenomenal reality. The gap has to be explained, and was explained with a quid, a factor, or a determination. Accordingly, a quid was introduced into mechanics to account for circular motions in terms of mechanical constraints.

In the medieval scientia de ponderibus, two determinations were considered for the equilibrium of a balance: first, the circle resulting from the inclusion of the vertical motions of the weights in a mechanical system, and second, the situm (location) of the weights in a mechanical system determining a variation in heaviness. The reflection on gravitas secundum situm (positional heaviness) from Nemorarius to Benedetti presupposes this twofold quidditas and focuses on the latter aspect (the variation of the heaviness).

The conviction that circular motion, as a violent motion, requires an explanation is based on the mental model that “circular motion is constrained (or contingented) straight motion.” Although they were embedded in the medieval discourse on contingency, the several attempts to quantify the quid accounting for the deviation testify to the common effort to overcome the qualitative and indeterminable characterization of contingency as a form of causality. What was maintained, for instance in Descartes, was the idea of a gap between law and phenomenon. Yet, if the deviation from the law can be perfectly quantified, then the separation between the order of nature and its realization is virtually eliminated, that is, the fracture between absolute necessity and conditional necessity is recomposed. To be sure, this step toward the necessitation of nature, resulting from the abandonment of contingency in both senses (causal and epistemological), was accomplished only later, in the course of the seventeenth century.

The work of Benedetti and his onto-epistemology of contingency are representative of an age of transition from Scholastic and Renaissance natural philosophies to the various instantiations of the classical science of the next century. Benedetti’s Pythagorean commitment to mathematics, seen as the most powerful logical means applied to all fields of knowledge and to nature in particular, is an illustrative case of the complex and non-linear history of scientific thought. His efforts to overcome Aristotelian conceptions could not really renounce the crucial assumption of the Aristotelian outlook under attack. This particularly concerns the ontology and epistemology underlying his scientific theories and practices. Mathematical determination, both in science and nature, did not imply necessity, neither at the level of material causation nor of explanation. The gap between the laws of nature and the effective processes reflected a Scholastic distinction between formal necessity and material imperfection. Such philosophical assumptions underpinned medieval treatments of phenomena, including statics, and Renaissance developments, especially in the line connecting Tartaglia and Cardano to Benedetti and Descartes. The distinction between formal necessity and phenomenal contingency offered them a horizon within which they could conceptualize general laws as well as their empirical instantiation. In particular, Benedetti could extend the area of application for the mental model that circular motion is a constrained (violent) deviation from the law of rectilinear motion. He did this by applying a model originating from statics to the area of dynamics, thus paving the way for the classical concept of inertia. However, we should not neglect the practical roots of his work in a Scholastic-embedded science of weights, which generalized observations of mechanical systems in order to make universal statements about nature.

Footnotes

[1] Benedetti 1585, 298: “Miror quod cum in Aristotele sis versatus, in tuis tamen scriptis philosophum a Mathematico separes, quasi mathematicus non sit adeo philosophus, ut est naturalis, et metaphysicus, cum multo magis quam ii philosophus sit appellandus, si ad veritatem suarum conclusionum respiciamus.”

[2] Benedetti includes a letter to Catena in Benedetti 1585, 371.

[3] See on this De Pace 1993, 228–229.

[4] Benedetti 1585, 298: “Verum quidem est, te in huiusmodi errore solum non versari; sed gravius est, quod cum vos videatis etiam res morales sub philosophiae appellationem cadere, non animadvertatis divinas scientias mathematicas etiam philosophiae nomine ornandas esse. Quod si eiusdem nomen penitius considerare velimus, inveniemus aperte, mathematico magis illud ipsum quam cuilibet alio convenire, cum nullus ex aliis tam certo sciat id quem affirmat quam mathematicus, neque aliquis sit, qui in cognitionis, et scientiae cupiditatem magis ducantur, ut aperte patet, cum nec etiam ipsi sensui det locum, neque aliquid praesupponat, quem non sit ita verum et intellectui notum, ut nulla quaevis potentia, illud esse falsum ostendere queat.”

[5] Piccolomini 1565, f. 107v: “Mathematics disciplines esse certas non vi demonstrationis, sed ex subjecti ipsius ratione.”

[6] Piccolomini 1565, 106v: “Patet igitur ex dictis Aristotelis causa certitudinis mathematicae. Hoc idem sensit Simplicius, qui primo de Anima 11. dicit causam certitudinis mathematicarum esse, quia versantur circa quantum. Quantitates enim ut dicit ipse, sunt res sensatae, et causas sensatas habent, et ideo nobis notas.”

[7] Piccolomini 1565, 107r: “Si vero adhuc replicaretur, quod motus etiam est sensibile quoddam commune, sicut magnitudo; habet autem motus suas passionet, et suas causas, ut patet 5. et 6. Phys. ergo ita erit certa de motu scientia, naturalis scilicet, sicut scientia de quantitate, quae Mathematica est. Ad hoc respondere possumus, quod si motum consyderabimus, in communi, abstractu a materia quatenus continuum quoddam est, […] tunc consyderatio erit mathematica, et nihil contra nos.”

[8] As Ofer Gal and Raz Chen-Morris recently stressed: “It is not epistemology that worries the two court mathematicians here, but ontology. Neither of them questions the power of mathematics to provide the knowledge they seek; it is the objects that mathematics can be true about that they both feel forced to establish.” See Gal and Chen-Morris 2013, 118–119.

[9] The literature on the Renaissance debates on the philosophical status of mathematics is wide. Among other sources, see Giacobbe 1972, Giacobbe 1973, Carugo 1983, Jardine 1990, 693–697, De Pace 1993, Cozzoli 2007, and Axworthy 2016, chap. 2. For the Jesuit debates on mathematics, see Romano 1999. For the seventeenth century, cf. Mancosu 1996, 8–33.

[10] See Valleriani 2010 and Valleriani 2013.

[11] Zilsel 1942.

[12] See Lefèvre 1978.

[13] Pigafetta in Del Monte 1581: “Mechanico è vocabolo honoratissimo, dimostrante, secondo Plutarco, mestiero alla Militia pertinente, et convenevole ad huomo di alto affare, et che sappia con le sue mani et co’l senno mandare ad esecutione opre maravigliose a singulare utilità et diletto del vivere humano.”

[14] Hypotheses on conventionalism already emerged from the debate on the conflict between Ptolemy’s geometrical models and Aristotle’s homocentric cosmology. See Di Bono 1990 and Granada and Tessicini 2005.

[15] Most of these cosmological views are discussed in Benedetti 1585, Book 4. We deal with the details in chapter 6 as well as, partly, in chapter 7.

[16] Omodeo 2014a, 167–170.

[17] Bruno 2000b, doc. 13, 67: “Ho dechiarato infiniti mondi particulari simili a questo della Terra; la quale con Pittagora intendo uno astro, simile alla quale è la Luna, altri pianeti et altre stelle, le qual sono infinite.”

[18] Benedetti 1585, 191: “Quod autem attinet ad motus, ad magnitudines, ad distantias et ad influxus, nihil est, quod hisce proportionibus conveniat, sed quia haec omnia dependent ab infinita et divina providentia Dei, necessario sit ut istae velocitates, eae magnitudines, distantiae et influxus, talem ordinem et respectum inter se ipsa et universo habeant, qualis perfectissimus sit.”

[19] Kepler 1937–2001, vol. 3, 19.

[20] Gingerich 1975, 261–278.

[21] Goldstein and Hon 2005, 76.

[22] On Kepler’s discovery, see Donahue 1988, Donahue 1993 and Wilson 1968.

[23] We have first discussed contingency and mechanics in the Renaissance in Omodeo and Renn 2015. A volume entirely devoted to ontological and epistemological contingency in the natural debates of early modernity is Omodeo and Garau 2019.

[24] Duns Scotus 1994, 140: “Sic igitur apparet primum, quomodo est contingentia in rebus, quia a Deo, – et quid est in Deo quod est causa huius contingentiae, quia voluntas eius.”

[25] Cf. Vogt 2011, 52. The entire first chapter is relevant for a historical overview of the reception and transformation of the Aristotelian concept of “possibility” as “contingency” in the Latin tradition.

[26] Vogt 2011, 50.

[27] Schepers 1965.

[28] Aquinas 1975, 221f: “Contingens a necessario differt secundum quod unumquodque in sua causa est: contingens enim sic in sua causa est ut non esse ex ea possit et esse; necessarium vero non potest ex sua causa nisi esse. […] Ex causa necessaria certitudinaliter sequitur effectus, ita ex causa contingenti completa si non impediatur.”

[29] Thomas, Summa contra gentiles II,15. Cf. Aquinas 1975, 48: “everything that can be and not-be” and “it is indifferent to either.”

[30] Aquinas 1975, 46: “Deus est omnibus causa essendi.”

[31] Summa contra gentiles II 30: “In quibus [rebus] vero forma non complet totam potentiam materiae, remanet adhuc in materia potentia ad aliam formam.” Cf. Aquinas 1975, 87.

[32] Summa contra gentiles II 23: “Omnis enim agentis per necessitatem naturae virtus determinatur ad unum effectum. Et inde est quod omnia naturalia semper eveniunt in eodem modo, nisi per impedimentum: non autem voluntaria. Divina autem virtus non ordinatur ad unum effectum tantum […]. Deus non agit per necessitate naturae, sed per voluntatem.” Cf. Aquinas 1975, 68: “For the power of every agent which acts by natural necessity is determined to one effect; that is why all natural things invariably happen in the same way, unless there be an obstacle; while voluntary things do not. God’s power, however, is not ordered to one effect only […]. Therefore, God acts, not out of natural necessity, but by His will.”

[33] Omodeo and Renn 2015.

[34] Alighieri 1984, 94; Dante Alighieri, Paradiso XV 79–83:

“Ma voglia e argomento ne’ mortali,
per la cagion ch’a voi è manifesta,
diversamente son pennuti in ali;
ond’ io, che son mortal, mi sento in questa
disagguaglianza […].”

Also, see Paradiso IV, 87, IV, 109, IV, 113, and Purgatorio VII, 57.

[35] Hispanus 1972, 157–158.

[36] See Renn and Damerow 2012, especially the sections from 3.6 to 3.8.

[37] Nemore 1533, f. A3v (emphasis added): “[…] si sumantur de circulo maiori et minori arcus aequales, corda arcum maioris circuli longior est. Propeterea posset ex hoc ostendi, quod pondus in libra tanto sit levius, quanto plus descendit in semicirculo. Incipiat igitur mobile descendere a summo semicirculi, et descendat continue. Dico tunc quod maior arcus circuli plus contrariatur rectae lineae quam minor, et casus gravis per arcum maiorem, plus contrariatur casui gravis, qui per rectam fieri debet, quam casus per arcum minorem. Patet ergo maior est violentiam in motus secundum arcum maiorem, quam secundum minorem. Aliter enim fieret motus magis gravis. Cum ergo plus in ascensu aliquod movetur violentiae, patet, quam maiore est gravitas secundum situm, et quia secundum situationem talium sic sit, dicatur gravitas secundum situm.”

[38] Nemore 1533, ff. A3v–A3r: “In termino enim cuiscunque motus intenditur, intenditur et viget tota natura in actu, qui in motu sit quasi in potentia, secundum quem fiebat contrarietatis suae oppositio.”

[39] Nemore 1533, ff. B2r–v: “Ponatur nunc, quod fiat descensus a parte B, et ascensus a parte C, dico quod redibunt ad situm aequalitatis. Non enim ulterius descendet B, eo quod descensus eius versus D magis obliquus est, quam ascensus C ad aequalitatem; B enim et C iam aequaliter distant a situ aequalitatis.”

[40] Nemore 1533, f. A3r: “Quarta [propositio]: Secundum situm gravius esse, quanto in eodem situ minus obliquus est descensus. Quinta [propositio]: Obliquiorem autem descensum minus capere de directo, in eadem quantitate.” Translation from Renn and Damerow 2012, 63. For proposition four, see Nemore 1533, f. B3v–B4r and, for proposition five, Nemore 1533, f. B4r–C2v.

[41] Tartaglia’s approach was controversial, already in his time. See Renn and Omodeo 2013, sec. 3.6.

[42] Renn and Damerow 2012, 86–92. We will discuss the divergent interpretations of Benedetti and Del Monte later, in chapter 5.

[43] Damerow and Renn 2010, 65.

[44] Drake and Drabkin 1969, 166. Benedetti 1585, 141: “Omne pondus positum in extremitate alicuius brachii librae maiorem, aut minorem gravitatem habet.”

[45] Renn and Damerow 2012, 138. We will deal with the details of Benedetti’s mechanics in the next section.

[46] Cf. Büttner 2008.

[47] Drake and Drabkin 1969, 229. Benedetti 1585, 285 (emphasis added): “Quaeris a me literis tuis, an motus circularis alicuius molae molendinariae, si super aliquod punctum, quasi mathematicum, quiesceret, posset esse perpetuus, cum aliquando esset mota, supponendo etiam eandem esse perfecte rotundam, et laevigatam. Respondeo huiusmodi motum nullo modo futurum perpetuum, nec etiam multum duraturum, quia praeterquam quem ab aere qui ei circumcirca aliquam resistentiam facit stringitur, est etiam resistentia partium illius corporis moti, quae cum motae sunt, natura, impetum habent efficiendi iter directum, unde cum simul iunctae sint, et earum una continuata cum alia. Dum circulariter moventur patiuntur violentiam, et in huiusmodi motu per vim unitae manent, quia quanto magis moventur, tanto magis in iis crescit naturalis inclinatio recta eundi, unde tanto magis contra suammet naturam volvuntur, ita ut secundum naturam quiescant, quia cum eis proprium sit, quando sunt motae, eundi recta, quanto violentius volvuntur, tanto magis una resistit alteri, et quasi retro revocat eam, quae antea reperitur habere.”

[48] On Benedetti’s anti-Aristotelianism, see Maccagni 1983.

[49] Drake and Drabkin 1969, 229–230. Benedetti 1585, 285: “Sed si clarius, hanc veritatem videre cupis, cogita illud corpus, trochum scilicet, dum velocissime circunducitur secari, seu dividi in multas partes, unde videbis illas omnesque, non illico versus mundi centrum descendere, sed recta orizontaliter, ut ita dicam, moveri. Id quem a nemine adhuc (quem sciam) in trocho est obseruatum. Ab huiusmodi motu trochi, aut huius generis corporis, clare perspicitur, quam errent peripatetici circa motum violentum alicuius corporis, qui existimant aerem qui subintrat ab occupandum locum a corpore relictum, ipsum corpus impellere, cum ab hoc, magis effectus contrarius nascatur.”

[50] On the Cartesian cosmos, see Aiton 1972, 30–64 and Gaukroger 2006, 304–317.

[51] Descartes 1998, 29. Descartes 1986, 43–44: “Lors qu’un corps se meut, encore que son mouvement se fasse plus souvent en ligne courbe, et qu’il ne s’en puisse jamais faire aucun, qui ne soit en quelque façon circulaire […], toutesfois chacune de ses parties en particulier tend toujours à continuer la sien en ligne droite. Et ainsi leur action, c’est à dire l’inclination qu’elles ont à se mouvoir, est differente de leur mouvement.”

[52] Koyré 1986, 206.

5. Mechanics

The book on mechanics, De mechanicis, the third of the Diversae speculationes, is divided into twenty-five chapters. Mechanical issues and references to mechanics can also be found in the epistles. As to the discussion of the motion of fall through media and of hydraulic problems, these are not part of this book. De mechanicis begins with a brief preamble in which Benedetti claims that he treats topics that have never been dealt with before or have not been sufficiently explained. In this section we will discuss the positioning and controversies implicit in this strong statement in an age when mechanical studies were very lively in the Italian peninsula and abroad. We will first offer an overview of Book 3 of the Diversae speculationes. Second, we are going to look more closely at the first foundational chapters of the treatise. Third, we will consider the rivalry with Del Monte, emerging from the latter’s harsh criticism of Benedetti and, in part, his misunderstanding of some crucial elements of Benedetti’s theory. The context of these lively disputes is the reaction to the publication of Tartaglia’s eclectic work on this subject, the Quesiti, et inventioni diverse (1546), and his re-issue of the medieval classic on the science of weights. Benedetti, as a critical pupil of Tartaglia, could not sympathize with the absolute rejection of Tartaglia and the medieval tradition his approach rested upon. At the same time, he felt the need to distance himself from several aspects of Tartaglia’s treatment, as we will reconstruct in detail in this section. The debates between Benedetti and Del Monte arguably culminated with Galileo’s work, which stands out as a sort of synthesis of earlier positions. Understanding these historical developments, as well as the intellectual triangle Benedetti-Del Monte-Galileo, is fundamental in order to trace Benedetti’s influence on his contemporaries and on the young Galileo.[1]

5.1 An Overview of De mechanicis

5.1.1 The Foundations of the Theory of the Balance

Chapters 1 to 6 of De mechanicis contain a systematic account of the foundation on which Benedetti built his mechanics. Chapter 1 clarifies qualitatively how the variable weight changes depending on the obliqueness of the balance beam. While a body attached to the end of the beam has a maximum weight if the beam is in a horizontal position, it vanishes when the beam is in a vertical position. Benedetti explained this behavior as a consequence of the different extent to which the attached weight rests on the center of the balance. If the position of the beam is close to the vertical, the weight of a body attached to the end of the beam is close to zero since it rests nearly completely on the center of the balance.

Chapter 2 clarifies the positional changing of the weight quantitatively. Benedetti related the balance with an oblique position of the beam to a bent lever with one horizontal and one oblique arm, thus providing the precondition for a generalization of his result. A generalization of this kind is indeed required if the lines of inclination of the bodies at the end of a balance are conceived as being directed to the center of the earth and hence no longer as being parallel to each other. Benedetti mentioned this possibility at the end of this chapter, but considered the angle between the two directions as being too small to be measured and thus not necessary to be taken into account.

In chapter 3 Benedetti generalized from the downward inclination of a body attached to the balance beam to forces acting upon the body not vertically but making an acute or obtuse angle with the horizontal beam. Accordingly, he replaced the bodies at the end of the balance beam with two weights or two moving forces (duo pondera, aut duae virtutes moventes), as he formulated somewhat ambiguously. His derivation of their quantities was based on a reinterpretation of the horizontal distances between the center of the balance and the vertical projections of the bodies at the end of a beam in an oblique position (Figure 5.1). He interpreted these distances as perpendicular distances from the center of the balance to the lines of inclination, and was thus also able to apply the result he achieved for vertically descending weights to lines of inclination caused by forces that are not vertical.

Fig. 5.1: Benedetti’s diagram showing a balance CBD or FBD. The lines CO and FUEM are the so-called lines of inclination connecting the weights C and F with the center of the elements. The length of the projection on the horizontal is proportional to the positional heaviness. (Max Planck Institute for the History of Science, Library)

Benedetti maintained that his arguments in chapters 1 to 3 clarify all the causes operating on balances and levers. To demonstrate this, he discussed in chapters 4 and 5 the validity of his results if applied to material balances and levers, taking into account that they have a beam with finite extension. This, however, does not imply that he calculated the influence of the weight of the beam itself. His discussion was rather restricted to a justification of his claim that the geometry of a rectangular beam does not require a modification of his propositions. In chapter 5 he treated the case of a lever whose fulcrum is at one of its ends.

Finally, in chapter 6, Benedetti added the description of an instrument used in bakeries for treating the dough. He explained the function of the instrument by applying his proposition from chapter 3.

The systematic approach used by Benedetti in this first part of his treatise is complemented by chapter 9, in which he explained the division of the scale of a steelyard into equal intervals.

5.1.2 Criticism of Tartaglia and Nemorarius

In chapters 7 and 8 Benedetti criticized the theorems of his former teacher Tartaglia, in particular those that Tartaglia adapted from Jordanus Nemorarius. Both chapters deal exclusively with some propositions of Book 8 of Tartaglia’s Quesiti, et inventioni diverse,[2] which is concerned with the science of weights and is entitled, accordingly, Sopra la scientia di pesi. In those cases in which Tartaglia’s propositions are adapted from Nemorarius, Benedetti mentioned explicitly the corresponding proposition in the edition of Nemorarius’ De ratione ponderis, corrected and illustrated by Tartaglia, and published under the title Iordani opusculum de ponderositate Nicolai Tartaleae studio correctum novisque figuris auctum.[3]

Chapter 7 starts with some brief critical remarks on Tartaglia’s propositions 2 to 5. Tartaglia’s proposition 2 essentially paraphrases and modifies the Aristotelian claim that the speed of moving bodies is proportional to the driving force. Following Nemorarius, Tartaglia maintained that the velocities of descending heavy bodies of the same kind are proportional to their power (potentia), while in the case of ascending bodies their velocities are inversely proportional to their power. For bodies of the same kind their power is conceived here as proportional to their sizes, that is, to their weights. Descending bodies are thus simply falling bodies with velocities proportional to their weights, while in the case of ascending bodies their weight acts as a resistance. Tartaglia’s proposition 3 generalizes proposition 2 for bodies with equal weights but unequal positional heaviness. His proposition 4 maintains that in the latter case the power of bodies attached to a balance is proportional to the distances from the center.

Benedetti’s critical remarks are somewhat eclectic. He argues that Tartaglia, in his second proposition, does not take into account the quantity of external resistance (quanti momenti sint extrinsecae resistentiae). With regard to Tartaglia’s third proposition, Benedetti points to its assumptions, namely that the bodies have to be homogenous and must have the same shape. He criticizes Tartaglia’s proof as it does not actually require these assumptions, but would also be true for heterogeneous bodies or for bodies with differing shapes. Concerning the fourth proposition, he criticizes Tartaglia for not proving what he claimed to prove. Instead, he should have followed Archimedes’s proof of the law of the lever.

Benedetti’s chapter 7 continues with a detailed discussion of the second part of Tartaglia’s proposition 5 and the following two corollaries and is thus directly concerned with the equilibrium controversy, that is, the controversy about whether or not a balance in equilibrium removed from its horizontal position will automatically return to this position. Tartaglia maintained in this proposition that a balance that is in equilibrium in a horizontal position will necessarily return to this horizontal position when moved into an oblique position. In a first corollary, he claimed that the more the balance beam is brought into an oblique position, the more the bodies attached to it become positionally lighter. In a second corollary, he claimed that while both bodies in this case become positionally lighter, the lifted body loses less of its positional heaviness than the body moving down. He concluded that the beam will return to a horizontal position. Benedetti questioned Tartaglia’s approach by referring to the first three chapters of his own treatise, arguing in particular that Tartaglia’s second corollary must be wrong. He discussed once more the balance beam in an oblique position, but now without the assumption that the lines of inclination of bodies attached to the balance beam are parallel. Rather, he considered the case that these lines are directed to the center of the world, showing, as we have discussed, that it is not the lifted body, but rather the body that is moved down, which loses less of its positional heaviness.

Benedetti continued in chapter 8 with critical comments on Tartaglia’s propositions 6, 7, 8, and 14. Tartaglia’s proposition 6 contains the proof of his fallacious claim that the lifted body of an oblique balance beam loses less of its positional heaviness than the body moving down, now modified by the further claim that the difference is smaller than any finite quantity. Tartaglia claimed:

[…] that the differences between the heaviness of these two bodies is impossible to give or find between two unequal quantities.[4]

Like Del Monte had done before him, but with different results, Benedetti criticized Tartaglia for not taking into account that the lines of inclination are not parallel.

Tartaglia’s proposition 7 contains the simple statement that if the arms of a balance are unequal and bodies with equal weights are attached to the ends of the beam the balance will tilt on the side with the longer arm. Benedetti criticized Tartaglia again for not taking into account that the lines of inclination are not parallel, and claimed that in any case Tartaglia did not give the correct cause of the effect.[5]

Tartaglia’s proposition 8 formulates, following Nemorarius, the law of the lever in terms of positional heaviness, stating that if the lengths of the parts of the balance beam with unequal arms are inversely proportional to the weights of the bodies attached to them, their positional heaviness will be equal. Benedetti criticized that this proposition is much better demonstrated by Archimedes.

Finally, Tartaglia’s propositions 14 and 15 concern Nemorarius’s proof of the law of the inclined plane, which from a modern perspective is essentially correct. Benedetti criticized Tartaglia’s argument by attributing to it an interpretation of the inclined plane as a balance, with the top of the plane being its center. His criticism, based on the propositions of his chapters 1 to 3, thus completely missed the point of Tartaglia’s argument.

5.1.3 Criticism of Aristotle’s Mechanics

Benedetti’s treatise on mechanics continues mainly with critical notes on the Aristotelian Mechanical Problems.[6] His notes are as diverse as the Aristotelian Mechanical Problems themselves.

Before he embarked on this criticism, Benedetti dealt with the problem of why a steelyard carries a linear gradation in chapter 9.[7] He took into account the weight of the beam and that of the scale by postulating the equilibrium of the balance when no extra weight is added. Then he added weights of one pound on both sides, arguing that, by common science (scientia communis),[8] the balance stays in equilibrium if they are placed at equal distances from the fulcrum. He had thus found the mark on the beam that indicates a magnitude of one pound. He then successively placed further weights onto the scale, now arguing from the law of the lever that they must be compensated by distances proportional to their number. He thus avoided the problem of applying the law of the lever directly to a material steelyard, just as one does in practice when gauging such a balance.[9]

In chapters 10 and 11 Benedetti started with critical remarks on Aristotle’s first problem. Aristotle asked why larger balances are more accurate than smaller ones.[10] Actually, this concrete physical question is not the focus of the extensive answer the author gave to this problem. Instead, he provided a long proof of the basic explanatory principle which plays a major role in the whole treatise. At the end of the proof Aristotle argued that the same load will move faster on a larger balance, thus making such balances more accurate.[11]

The criticism Benedetti applied to Aristotle’s argument has two parts. In chapter 10 Benedetti began by rejecting Aristotle’s claim that the circumference of a circle combines concavity with convexity. He then argued against a specific part of Aristotle’s proof of his principle which involves the superposition of motions. In this part Aristotle showed that:

[…] whenever a body is moved in two directions in a fixed ratio it necessarily travels in a straight line.[12]

He concluded:

[…] if a body travels with two movements with no fixed ratio and in no fixed time, it would be impossible for it to travel in a straight line.[13]

For the Aristotelian author this proposition served as a means to describe circular motion as a result of two movements with no fixed ratio. Benedetti, however, did not relate his criticism to this context. He argued only that Aristotle’s inference concerning movements in two directions is not sufficient since a straight movement can result from two quite different motions. This criticism does not really relate to the Aristotelian argument, other than by showing that his entire attempt to derive the behavior of a balance from a principle of circular motion is misguided.

In the same vein, Benedetti’s criticism in chapter 11 then deals directly with Aristotle’s answer to the question of why larger balances are more accurate than smaller ones. He argued that Aristotle’s argument is not well founded since the greater accuracy has nothing to do with the motion of the balance beam but only with the geometrical constellation.[14] To conclude he added a consideration of material balances, arguing according to his own principles that a weight on the larger balance will be positionally more effective.

Benedetti’s chapter 12 concerns problems 2 and 3 of the Aristotelian Mechanical Problems.[15] Problem 2 raises the question that forms the starting point of the equilibrium controversy:

If the cord supporting a balance is fixed from above, when after the beam has inclined the weight is removed, the balance returns to its original position. If, however, it is supported from below, then it does not return to its original position. Why is this?[16]

Aristotle implicitly assumed that the balance beam has a certain thickness and weight. It follows as a result of the geometry of the balance in an oblique position that if the beam is fixed from above, a greater part of the beam is on the lifted side of the perpendicular line across the suspension point. Consequently the beam will move back by itself into the horizontal position. The opposite is true for a beam fixed from below. In this case, the greater part of the beam is on the lower side so that it cannot move back into a horizontal position by itself.

Benedetti criticized the first case by arguing that it is not only the weight of the beam that causes it to return to the horizontal position, but also the different distances of the weights in an oblique position from the vertical through the point where the beam is fixed. According to his theory of the dependency of the weight on the obliqueness of the beam, the weights must be different on both sides. Benedetti thus generalized Aristotle’s argument to the case of a balance without a material beam carrying weight itself.

In the second case of a beam supported from below, he argued that Aristotle is completely mistaken. Benedetti maintained that the beam will not remain in its oblique position, but that the lower part will move down until the beam is in the vertical position.

Problem 3 of the Aristotelian Mechanical Problems[17] concerning an explanation of the effect of a lever is, for Benedetti, not worth the effort of a detailed criticism. He only briefly notes that Aristotle did not give the true cause, which one will find in his own theory presented in chapters 4 and 5.[18]

In the very short chapter 13, Benedetti criticized problem 6 of the Aristotelian Mechanical Problems:

Why is it that the higher the yard-arm, the faster the ship travels with the same sail and the same wind?[19]

The Aristotelian answer provided in the Mechanical Problems is based on an interpretation of the yard-arm as a lever that has its base at the point where the yard-arm is fixed as the fulcrum. Benedetti maintained that this interpretation of the yard-arm as a lever:

[…] does not give the true explanation. For on this kind of explanation the ship would have to move more slowly rather than more swiftly. For the higher the sail that is struck by the force of the wind, the more the ship’s prow will be submerged in the water.[20]

Benedetti added one sentence with his own explanation, according to which the ship with a higher sail moves more swiftly because the wind blows more strongly in the higher region.

Chapter 14 provides a long discussion of problem 8 of the Aristotelian Mechanical Problems. The question posed in this problem is why round and circular bodies are easiest to move. Three examples are mentioned and later discussed: the wheels of a carriage, the wheels of a pulley, and the potter’s wheel. Benedetti claimed that Aristotle’s answer to the question he posed is not sufficient. Nevertheless, Benedetti himself argued essentially in a similar manner, only somewhat more extensively. Both of them argued that the circle, contrary to differently shaped bodies, touches a plane only at one point which can be considered as the fulcrum of a lever. But Benedetti added a further argument which is not given by Aristotle. He argued that a circle can be pulled along a plane without difficulty and resistance:

[…] because in such a case the center will never change its position by moving upward from below, i.e., will never change its position with respect to the distance or interval which lies beween it and line AD.[21]

At the end of the chapter, Benedetti discussed the question of why a potter’s wheel set into motion by an external force will continue to rotate for a time but not forever. In his response he took into account the friction with the support of the wheel and with the surrounding air. But he also discussed reasons that are more deeply concerned with the nature of such motion, as we have discussed above. He claimed, in particular, that the rotational motion is not a natural motion of the wheel, evidently making reference to the Aristotelian distinction between natural and violent motions. He also claimed that a body moving by itself because an impetus has been impressed upon it by an external force has a natural tendency to move along a rectilinear path. This statement seems to comes close to the principle of inertia of classical physics, but it actually deals with rectilinear motion as a forced motion and does not involve any assertion about its uniformity. Benedetti seems to suggest, in any case, that this natural tendency is in conflict with the forced rotational motion of the wheel, which in turn slows it down. The smaller the wheel and the more its parts are constrained to deviate from the rectilinear path, the greater the decrease in speed will be.[22]

In chapters 15 and 16 Benedetti dealt with issues of scale as they are brought up by the Aristotelian Mechanical Problems. In chapter 15, consisting merely of one short sentence, Benedetti referred to his own earlier treatment of Aristotle’s question of why larger balances are more exact (erroneously citing chapter 10 instead of chapter 11 of his treatise) in order to deal with the ninth problem of the Aristotelian Mechanical Problems, which reads:

Why is it that we can move things raised and drawn more easily and more quickly by means of greater circles?[23]

In chapter 16 he discussed the tenth problem of the Aristotelian Mechanical Problems, which reads:

Why is a balance moved more easily when it is without a weight than when it has one?[24]

In his detailed response to this problem—indeed much more detailed than the one found in the Aristotelian text—Benedetti compared balances that are alike with different sets of weights on their scales, one with two weights of one ounce, the other with two weights of one pound. He then added a half-ounce weight on one side of each balance and observed that the balance with the smaller weights moves more rapidly. He explained this effect by referring to the dynamical assumption that one always has to consider the ratio of the moving force to the body moved.

In chapter 17 Benedetti addressed the twelfth problem of the Aristotelian Mechanical Problems, which reads:

Why does a missile travel further from the sling than from the hand?[25]

Benedetti’s response is based on the concept of impetus, conceived as an intrinsic cause of motion originally acquired by the action of an external force that then gradually decreases after separation from the original mover. He argued that a greater impetus can be impressed by the sling due to the repeated revolutions which evidently lead to an accumulation of this intrinsic force. He observed that the impetus would lead, if not impeded by the sling or the hand, to a straight motion of the projectile along the tangent to the circle of its forced motion. He also noted—distancing himself from a claim made by Tartaglia—that the motion due to the impressed force can mingle with the projectile’s natural motion downward, thus leading to a curved trajectory. It may well be the case that it was this claim that later convinced Galileo and Del Monte to perform their experiment on projectile motion from which they drew the conclusion that such a mixture of motions indeed takes place.[26]

In chapter 18 Benedetti considered problem 13 of the Aristotelian Mechanical Problems dealing with the question of why larger handles can be moved more easily around a spindle than smaller ones.[27] In his short response Benedetti simply referred to the fourth and fifth chapters of his own treatise, stressing that everything depends on the lever. He was evidently convinced that the Aristotelian reduction of such problems to properties of the circle is superfluous, if not misguided.

In chapter 19 he handled problem 14 of the Aristotelian Mechanical Problems in the same way. It reads:

Why is a piece of wood of equal size more easily broken over the knee, if one holds it at equal distance far away from the knee to break it, than if one holds it by the knee and quite close to it?[28]

Again, Benedetti just referred to the earlier chapters of his treatise.

In chapter 20 Benedetti reconsidered problem 17 of the Aristotelian Mechanical Problems, which reads:

Why are great weights and bodies of considerable size split by a small wedge, and why does it exert great pressure?[29]

In the Aristotelian text, the answer is based on interpreting the wedge as two levers opposite to each other, their fulcra being placed at the entry points of the wedge into the wood. Benedetti, however, disagreed with the identification of the two levers allowing the action of the wedge to be interpreted in terms of force, fulcrum, and resistance. He claimed that the fulcrum is actually placed just underneath the deepest point of the opening produced by the wedge entering a block of wood.

In chapter 21 Benedetti claimed to provide the true explanation of compound pulleys. He reduced a compound pulley to a chain of balances by appropriately identifying forces and fulcra, each wheel of the pulley corresponding to one balance.

In chapter 22 Benedetti discussed Aristotle’s wheel, i.e., problem 24 of the Aristotelian Mechanical Problems, which reads:

A difficulty arises as to how it is that a greater circle, when it revolves, traces out a path of the same length as a smaller circle, if the two are concentric.[30]

While the author of the Mechanical Problems referred to dynamical reasons in explaining this apparent paradox, Benedetti resorted to a kinematic argument, a pointwise reconstruction of the trajectory of the motion of a point on the circumference, arguing that it results from a superposition of two motions. In the case in which the motion is controlled by the larger circle, a point on the circumference of the smaller circle traverses a path resulting from an addition of two motions. In the case in which the motion is controlled by the smaller circle, a point on the circumference of the larger circle traverses a path resulting from a subtraction of two motions.

Chapter 23 of Benedetti’s treatise does not exist.[31] In chapter 24 Benedetti discussed problem 30 of the Aristotelian Mechanical Problems, which reads:

Why is it that when men stand up, they rise by making an acute angle between the lower leg and the thigh, and between the trunk and the thigh?[32]

In his response Benedetti suggested that the reason for this behavior is to create an equilibrium of the body with regard to the line that serves as support underfoot.

In chapter 25 Benedetti addressed the last problem, problem 35 of the Aristotelian Mechanical Problems, which reads:

Why do objects which are travelling in eddying water all finish their movement in the middle?[33]

Benedetti’s answer simply referred to the fact that whirlpools are depressed in their middle without giving an explanation of this phenomenon. He could thus restrict himself to arguing that the motion of an object to the center of such a whirlpool is simply its natural downward motion. The final comment by Benedetti is a remarkable conclusion to his criticism of Aristotle as well as his treatise on mechanics:

But in the case of all those other problems that I have omitted, Aristotle’s explanations are correct.[34]

5.2 The Beginning of Benedetti’s Mechanics

After our overview of Benedetti’s book on mechanics, we concentrate on the theses he expounded in the first chapters because they have a foundational character and proved particularly controversial, at least in light of Del Monte’s criticism, which we are aware of from the comments he made in one of his notebooks and from marginal notes in his own copy of Benedetti’s book.

5.2.1 De mechanicis I: “On the different positions of balance beams”

In chapter 1, Benedetti notes that “a body (pondus) […] acquires a larger or smaller weight (gravitas) depending on the different ratio of the beam’s position” (pondus… maiorem, aut minorem gravitatem habet, pro diversa ratione situs ipsius brachii). According to him, a body has the greatest heaviness when the beam at whose extremity it is loaded is in the horizontal position. His idea is based on a simple common-sense intuition: if one considers an equal-arms balance suspended at its center, the weight of a loaded body is:

borne entirely by the fulcrum when resting vertically upon it,

entirely hanging on the fulcrum when suspended vertically below it,

not supported in any way by the fulcrum when the beam is in the horizontal position.

In the first case, the body completely rests or leans on the center (nititur), and the center in turn hinders (impellet) the downward tendency of the weight. In the second case, the body is suspended vertically (pendet) and the center “attracts” it (attrahet), in the sense that it hinders its natural tendency to fall down (inclinatio). Hence, the body attains its maximum weight in the third case. If the balance beam moves upward, departing from the horizontal position, the weight slowly decreases and reaches its minimum at the top when the beam is in the vertical position. If the rotatory motion around the fulcrum continues, now downward, the weight increases again until it reaches its maximum in the horizontal position. It then diminishes until it is suspended entirely below the fulcrum. Benedetti visualizes these variations of weight depending on the position (situs) in a diagram comparing the lines connecting the weight to the center of the world in different cases, more specifically if the beam is:

horizontal,

raised upward, or

moved downward with the same angle as in the second case (which is equivalent to 2).

The parallel lines, called lineae inclinationis or lineae itineris, indicate the direction in which a body would fall if it were free. The closer these lines are to the center of the beam, Benedetti says, the “less heavy” the body becomes.

In his own copy of Benedetti’s book, Del Monte wrote a brief annotation in the margin of chapter 1: “this first chapter is derived entirely from our treatise on the balance in the Mechanicorum liber.”[35] Clearly, he sought to assert the relevance of his treatise for Benedetti’s speculations, in spite of the latter’s claims of originality. It should be remarked, however, that Del Monte’s treatment of the balance, based on the concept of center of gravity, was significantly different from Benedetti’s, which was based on an original reworking of positional heaviness. Del Monte merely reassessed a concept received from authors such as Jordanus Nemorarius, Tartaglia, and Cardano, all of whom he personally opposed. In his book on mechanics, Del Monte had in fact criticized the concept of positional heaviness. Downplaying Benedetti’s theory as a repetition of his predecessor’s theories, he could therefore claim that his own treatment already included a summary (as well as a criticism) of Benedetti’s approach.

5.2.2 De mechanicis II: On the proportion of weights at the extremities of a balance beam in a position other than the horizontal

In chapter 2, Benedetti deals with the proportions of a weight placed at the extremity of a balance beam if its position is not horizontal (De proportione ponderis extremitatis brachii librae in diverso situ ab orizontalis). The thesis to be demonstrated is the following: “The proportion between [the weight of] a body (pondus) at C and [the weight of] the same body (pondus) at F corresponds to that between the whole beam BC and its part BU, which is [set on the beam BC and is] delimitated by the fulcrum and the [intersection between the beam and the] inclination line FUM that connects the weight at F to the center of the world” (Benedetti 1585, 142). For the sake of simplicity, we will represent these relations symbolically in modern terms:

where C is the weight in the horizontal position and F in the inclined position; BC is the beam and BU the part of the beam BC between the center B and the perpendicular line drawn from F.

Benedetti’s demonstration is as follows. He imagines placing a weight D on the other extremity of the balance that has the same proportion to C as F, that is, the following proportion expressed in modern terms:

In accordance with Archimedes’s De ponderibus I.6, the balance will be stable if the weight C is loaded at U, since weights and distances from the fulcrum are proportional by supposition.

The next step is to show that F : C = BF : BU (where BF is the beam, hence BF = BC). In order to demonstrate this, Benedetti resorts to the mental model (imaginemur) of a string hanging vertically from F, to which a weight equal to C is suspended. He claims that it is visually evident that the weight has the same effect at F as at U. The same is valid for the case in which the weight is suspended from U and intersects the circumference described by the rotation of the beam at a point E. In both cases, the balance would remain horizontal since the weight C at F, U, or E would balance the weight at D. Benedetti further argues that the balance under consideration can be treated like a bent lever with a horizontal and an inclined arm (FBD or EBD): “si brachium BE consolidatum fuisset […]” (If the beam BE was made solid […]).

The author concluded that his reasoning has satisfactorily demonstrated his thesis: “A body (pondus) is more or less heavy (grave) the more or less it hangs from (pendet) or rests on (nititur) the fulcrum” (Benedetti 1585, 142). And he deems this resting on or hanging from the fulcrum to be the most direct cause (haec est causa proxima, et per se) of the positional changing of a weight.

As an additional commentary, Benedetti remarks that in his diagram he supposes the inclination line CO to be perpendicular to CB and parallel to BQ, whereas CO and BQ in fact converge at the center of the sphere of the elements (centrum regionis elementaris), that is, the earth. But for the sake of his present argumentation, this angle is negligible and one may simply assume perpendicularity and parallelism. Benedetti thus developed a method to quantify positional heaviness that corresponds to the modern concept of “torque.”

5.3 Del Monte’s Criticism Concerning the Non-Negligibility of the World’s Center

As will be shown in the following section, it was only in his initial treatment of the inclined balance, in chapter 1 of De mechanicis, that Benedetti neglected to consider the convergence of the inclination lines to the center of the elements. This omission gave rise to criticism. Del Monte severely criticized both this assumption and Benedetti’s reasoning in general in De mechanicis, in his handwritten notes on scientific and technical matters known as Meditatiunculae de rebus mathematicis. In his notes he assessed Benedetti’s arguments from his perspective, relying on the concept of the center of gravity as it was developed in his own book on mechanics.

In a marginal note to the Diversae speculationes (Figure 5.2), Del Monte expressed his disagreement with Benedetti’s conclusion: “Thus, in this manner, a weight (pondus) more or less hangs from or rests on the center; this is the next cause and the [cause] in itself [of the variation in heaviness].”[36] His disagreement reads as follows:

because that [that is, the greater or smaller extent to which a weight rests at the center] is neither the next [cause] nor the [cause] in itself. For the weight at F of the arm BF is not equally heavy as the weight U of the arm BU; nor is the weight at E of the arm BE equally heavy as the weight at U of the arm BU. Thus, this entire demonstration is false.[37]

Fig. 5.2: Del Monte’s marginal note to De mechanicis, II. (Max Planck Institute for the History of Science, Library)

This means that Del Monte did not accept the claim that a weight is equally heavy in different positions on the balance beam, provided the projections of the beam along the horizontal are the same length or rather, as Benedetti writes, the distances between the projections of the beam on the horizontal and the center have the same lengths.

To find Del Monte’s counter-arguments, one must look to the Meditatiunculae, f. 145, Contra Cap. 2 Jo. de Benedicti de Mechanicis. As mentioned, he basically rejected Benedetti’s perspective by objecting that he did not take into due account the finite distance of the weights from the center of the world and hence the fact that the plumb lines are not parallel to each other, as Benedetti assumed in this part of his treatise.

In his diagram (Figure 5.3) Del Monte compared the line LUS (parallel to the line AQ, connecting the fulcrum B of the balance with the center of the world M) with the line FM (connecting the upper weight F and the lower weight E with the center of the world M). S is the point at which the line LUS meets the circle that the beam makes around the fulcrum, which is above the position of the lower weight E.

Fig. 5.3: Del Monte’s critical reworking of Benedetti’s diagram in Meditatiunculae, f. 145. (Bibliothèque Nationale de France)

Next, he considered a bent lever made of the oblique arm BS, rigidly connected to the straight arm BD, assuming that BU is half BD. If a weight is now placed at S that is double the weight at D, the bent lever will be in equilibrium, as Del Monte showed with reference to his book, because the center of gravity of the weights at S and at D will be at the point R, which will be in its lowest place on the vertical line BQ. He therefore concluded that it is the weight at S, but not the lower weight E, that will be equally heavy as the weight at U.

He proceeded to demonstrate this in greater detail by considering the proportions into which the line connecting the two weights is cut by the perpendicular BQ for the two cases, that is, the weight placed at S and the weight placed at E. Del Monte concluded that the same weight is heavier at S than at E. He then turned to a closer consideration of the upper weight F. Again he constructed a bent lever LBD in equilibrium in order to compare it with the bent lever formed with the upper weight F. Again he showed that the weight is heavier at L than at F.

Del Monte concluded by summarizing that the entire fallacy is due to Benedetti assumption that the weight at F would gravitate in the same way as at U, which would only be the case, according to Del Monte, if it were to hang freely.

5.4 Benedetti on Weights and Forces Acting on a Balance

Chapter 3 of Benedetti’s De Mechanicis contains a generalization of the results of chapter 2 or, rather, presents a general rule concerning the action of forces (virtutes) on balance beams, including in the case that they do not act vertically downward but also with an acute or obtuse angle. Benedetti moves forward from the result of the previous chapter as follows: the length of the line perpendicularly connecting the center to the line of inclination (the line BU in the diagram) allows the quantity of the positional force (quantitas virtutis… in… situ) of a weight (F in the diagram) to be established. Thus, Benedetti calls the positional weight a force, and this is the presupposition that allows him to generalize from gravitas the action which he calls virtutes moventes, or “moving forces.” The thesis of this chapter is summarized in its title: “That the quantity of any given weight (pondus) or moving force in relation to another quantity can be determined thanks to the perpendicular projections connecting the center of the balance to the line of inclination.”

Benedetti draws two diagrams showing a balance at whose extremities two weights or forces act in different directions (Figure 5.4). At the left extremity B, a weight E has a downward tendency, while at the right extremity, a weight C acts making an acute or an obtuse angle. According to Benedetti, the length of the perpendicular projection drawn from the center to the inclination line, OT, permits the determination of the distance OI on the beam at which the same force acting vertically downward produces the same effect. Given this equation, Benedetti can determine how much the force acting in a non-perpendicular direction has to be augmented in order to balance an equal weight acting perpendicularly on the opposite beam. This measure is given according to the following proportion (expressed in modern terms):

where E is the weight acting vertically on the extremity B; C is the virtus movens acting on the opposite extremity A at an angle; BO is the left beam and OI the part of the right beam OA determined as explained above.

In his argumentation, Benedetti thus equates a balance (BOI) with a bent lever (BOT). Accepting this equation, he concluded that, according to commonly shared knowledge (communi quadam scientia), the weights or forces that are required to obtain a perfect balance can easily be calculated.

The chapter ends with a cosmological corollary: “The closer the center O of the balance is to the center of the elementary sphere, the less heavy (minus grave) it becomes.” In fact, the angles between the beam and the inclination lines become progressively smaller.

Fig. 5.4: Benedetti’s representation of forces acting on a balance in arbitrary directions. (Max Planck Institute for the History of Science, Library)

5.5 Del Monte’s Misunderstanding

In his notes on folio 146 of the Meditatiunculae, Del Monte grappled with Benedetti’s instructions on how to determine positional heaviness in the case of forces acting in an arbitrary direction. These he refuted at length under the erroneous assumption that Benedetti had claimed forces can be indiscriminately replaced by weights. Like Benedetti, Del Monte considered a bent lever BOAC with fulcrum O, weights E and C, a straight arm BO, and a bent arm OAC to discuss the two cases of an acute and an obtuse angle BAC (Figure 5.5).

Fig. 5.5: Del Monte’s critical reworking of Benedetti’s representation of forces acting on a balance in arbitrary directions. (Bibliothèque Nationale de France)

He first recapitulated Benedetti’s procedure, assuming that a vertical line OT drawn from the fulcrum to the line AC represented the oblique arm of the bent lever. He stated that when the weight C is placed at the end of the horizontal line OI, whose length is the same as that of the perpendicular OT, according to Benedetti it will be in equilibrium with the weight E if the weight C is to the weight E as is BO to OT or OI. Del Monte then summarized Benedetti’s claim that when a force represented by the weight C acts along the line TC, the bent lever formed by the straight arm BO and the oblique arm OTC will also be in equilibrium, which he doubted.

Del Monte reformulated this claim by stating that the same weight C will be in equilibrium with the weight E whether it is placed on the straight balance BOI or on the broken bent lever BOTC. He thus replaced Benedetti’s conception of a force acting along an oblique line with that of a weight always tending downward and as a result arrived at absurd conclusions.

Del Monte then showed that the same weight will be heavier on the horizontal at point I than along the bent lever at T, demonstrating that the bent lever TOB will not be in equilibrium if the straight lever BOI is in equilibrium. To show this, Del Monte again proceeded by finding the center of gravity of the weights E and C placed at T. More precisely, Del Monte determined a position for the weight C where the bent lever is in equilibrium, a position, however, that is distinct from T. Thus it follows that T cannot be the position of equilibrium. For this purpose, he extended the line BT to D, just beneath I, so that it is immediately evident that if the weight C is placed at D, the center of gravity of the two weights will be just beneath the fulcrum.

Using the same pattern, he continued by showing that the bent lever BOC cannot be in equilibrium because its center of gravity S can never fall on the perpendicular line OU through the fulcrum. Finally, he applied this argument to the broken bent lever BOTC. Del Monte next addressed the case in which the bent lever is characterized by an obtuse angle BAC, showing that the weight at T is lighter than the weight at I. In his concluding remarks, however, he began to waver. Once again, he stated that Benedetti is completely mistaken when applying his procedure to weights. But he did admit that this may be true when dealing with a force.

As an afterthought, Del Monte once again criticized Benedetti’s appeal to common sense: he did not feel this to be worthy of an expert mathematician. And as a second afterthought, he constructed an extreme case in which it is immediately clear that the broken bent lever cannot be in equilibrium if weights are attached to it rather than forces.

The following considerations enable Del Monte’s marginal annotations to Benedetti’s De Mechanicis III to be understood. These are not perfectly legible, but nonetheless their meaning becomes clear in light of the Meditatiunculae:

If we understand that a weight is at C, as we can assume from his own words, then CT must also be understood as being solid [and connected with] the solid lines TO […] If we hence understand that C is a weight and not moving, [the proposition] is false. If it is understood that C moves as […] of a man, it can be true, since what moves is not a weight. [But] if he himself assumes in the following that [this] can be demonstrated [also for a weight], nothing […] therefore as is evident in chapter 7. All demonstrations of the author are founded on these two chapters inasmuch as they are the first fundaments of mechanics; once their falsity is recognized, everything is rejected.[38]

5.6 Diverging Approaches to Tartaglia

Del Monte’s and Benedetti’s criticisms of Tartaglia’s conception of positional heaviness help us to understand where these two scholars converge and diverge on the issue of the equilibrium (or lack of equilibrium) of a balance deflected from its horizontal position, and also the reasons for the presumed equilibrium or tendency to restore it. Moreover, their arguments reveal a different attitude toward the medieval tradition of the scientia de ponderibus and the gravitas secundum situm.

5.6.1 The Tradition of Nemorarius, Tartaglia, and Cardano

The concept of gravitas secundum situm, or positional heaviness, was extensively employed in Jordanus Nemorarius’s Liber de ponderibus. Del Monte owned and annotated a sixteenth-century Nuremberg edition of the book, commented, and illustrated by Petrus Apianus. Del Monte’s handwritten annotations document his general disagreement with the approach of this medieval scholar, who did not know the Archimedean concept of the center of gravity and therefore tried to develop a deductive science of weights relying solely on the Aristotelian theory of motion and its development in the Arabic tradition of the science of weights. We have already hinted at the Aristotelian framework underlying the concept of gravitas secundum situm. In his book, Jordanus stated that a deflected balance would return to the horizontal position (his second proposition) (Nemore 1565, B2 r). According to Jordanus, the upper weight acquires more positional heaviness than the lower one due to the fact that its descent is less oblique. In fact, he postulated that positional heaviness depends on the obliqueness of descent of a weight (his fourth postulate) and that “a more oblique descent partakes less of the straight [descent] for the same quantity [of path]” (fifth postulate) (Nemore 1533, A4 r). The determination and possibly the quantification of obliqueness was therefore essential to establish the behavior of a deflected balance.

In the sixteenth century, Tartaglia in Quesiti, et inventioni diverse (1546), and Cardano in Book 1 of De subtilitate (first edition, 1550) and in Opus novum de proportionibus (1570), expounded their own versions for determining descent and reinforced Jordanus’s second proposition (that the deflected balance returns to the horizontal position). A brief account of three ways to determine positional heaviness is given in the following pages. The first two are derived from Tartaglia and the last from Cardano.

Descent: A first method of dealing with positional heaviness consisted in comparing the lengths of the projections of the equal arcs described by the motion of opposite balance beams—one ascending and one descending—on the vertical line of descent to the center of the world.

As Tartaglia’s diagram in Figure 5.6 shows, the vertical component of descent of the upper weight is always larger than that of the lower. Thus, the former acquires more heaviness (secundum situm) than the latter and the balance returns to the horizontal position.

Fig. 5.6: According to Tartaglia, the body at I is positionally heavier than the body at V, since the projection of the arc IL on the vertical XY is greater than the projection of VF, WF. (Max Planck Institute for the History of Science, Library)

Angle of contact: Tartaglia’s second method of determining positional heaviness consists in comparing the angles between the circular path of the beams and the perpendicular lines connecting the weights to the center of the elements (as already mentioned in chapter 4). These angles “of contact” are also called “curvilinear angles” or “mixed angles” since they result from the intersection of a straight line downward and a curved line, that of the circle circumscribing the balance (Figure 5.7).

Fig. 5.7: Tartaglia’s representation of the angle of contact for the determination of positional heaviness. (Max Planck Institute for the History of Science, Library)

By comparing the angles of contact of the two weights, Tartaglia could establish that the higher angle is always smaller than the lower; therefore, the higher weight has a straighter descent and is positionally heavier. The inclined balance would therefore return to the horizontal position. It should be noted that Tartaglia perceived the comparison of curvilinear angles as problematic. He considered the ratio of two such angles to be less than any ratio between determined quantities. As a consequence, no weight placed on the positionally lighter side of the deflected balance could compensate for the other weight and keep the balance inclined. On the contrary, any additional weight—no matter how small—would have produced an opposite displacement of the balance beam toward the vertical.

The angle between the support and the beams: We have so far considered two ways of determining positional heaviness on the basis of Tartaglia’s Quesiti. Assuming that positional heaviness depends on the obliquity and straightness of descent, positional heaviness can be determined either from the projections of the descents on the vertical, or the curvilinear angles that are produced by the intersection of the descent arcs and the lines connecting the weights to the center of gravity. Cardano considered three criteria for establishing positional heaviness which he mistakenly regarded as equivalent: first, the distance of the beam from the vertical; second, its distance from the horizontal; and third, an angle that he called meta. This was the angle between the support of the balance and the beam. Commenting on the diagram that is reproduced here as Figure 5.8, he explained:

Aristotle says that this happens when the support is above the balance, because the angle QBF of the meta is larger than the angle QBR. And similarly, when the support is QB, the meta will be AB, and thus the RBA will be larger than the angle FBA, but the larger angle will render the weight heavier. […] The general reason is hence this: the more the weights are removed from the meta or from the line of descent along a straight or an oblique line, that is, [as measured] by an angle, the heavier they are.[39]

Fig. 5.8: According to Cardano, there are three ways to determine positional heaviness. The positional heaviness at point F, for instance, may be determined by the horizontal FP, by the vertical FL, or by the angle QBF. (Max Planck Institute for the History of Science, Library)

Given these premises, Cardano contended that a weight will reach its maximum positional heaviness in the horizontal position. He therefore shared Nemorarius’s and Tartaglia’s opinion about the return of an inclined balance to the horizontal position.

5.6.2 Del Monte’s Critical Remarks on Positional Heaviness

Del Monte’s criticism of Benedetti, in the Meditatiunculae as well as in the marginal remarks of his copy of Diversae speculationes, are closely related to his criticism of Nemorarius, Cardano, and Tartaglia in his Mechanicorum liber (1577). Here he dealt extensively with the balance and provided a detailed discussion of the theories of these scholars which he judged to be irremediable. These theories supported the idea that an inclined balance returns to the horizontal and were thus at odds with his own treatment of the matter, which he based on the Archimedean concept of center of gravity. Del Monte believed that an ideal balance would remain in any position as long as it had equal arms, was hinged on its fulcrum and was loaded with equal weights. The only difficulty in testing this theory, he asserted, was the technical difficulty in constructing a perfect balance. It should be noted, moreover, that he assumed that a center of gravity meeting the requirement of his (and Pappus’s) definition of the center of gravity always exists:

The center of gravity is a certain point within it, from which, if it is imagined to be suspended and carried, it remains stable and maintains the position which it had at the beginning, and is not set to rotation by that motion.[40]

Apart from the conceptual irreconcilability between his own approach and that of the Nemorarius school, Del Monte tried to demonstrate the inconsistencies of positional heaviness also within the conceptual framework of his adversaries. One of his main objections was based on a consideration of the cosmological context, which he considered relevant to correctly treat the inclined balance, at least with regard to positional heaviness. Of course, this aspect indeed matters when considering Tartaglia’s remark that the difference in positional heaviness is infinitesimally small and cannot be compensated by any finite weight resulting from the infinitesimal difference between curvilinear angles.

Contrary to the assumptions of Nemorarius and his successors, Del Monte noted that the downward tendencies of the weights are not parallel but converge at the center of the world. Since the directions toward the center of the world from different points on the circular path of the end of the beam cannot be parallel, they are inappropriate for representing positional heaviness. From the fact that those lines converge, he argued further that the lower weight should actually become positionally heavier than the higher one. His idea is clearly illustrated by the diagram in Figure 5.9.

Fig. 5.9: According to Del Monte, if S represents the center of the world, then the mixed angle SEG between the circular path of the weight at E and the direction to the center of the world is less than the mixed angle SDG. Thus, contrary to what his adversaries claim, by their own suppositions the weight placed at E must be heavier than that at D. (Max Planck Institute for the History of Science, Library)

Del Monte objected that, from the point of view of positional heaviness, it is not in the horizontal position that a body weighs the most but at that point where a straight line drawn from the center of the world touches the circle described by the balance arm. Certainly, if the center of the world were infinitely distant and all lines of direction converging at it were perpendicular and parallel to each other, then the extreme point would mark the horizontal position of the balance arm. Still, for a finite distance from the center of the world, the point where the weight is heaviest lies instead slightly below the horizontal through the fulcrum. Del Monte even demonstrated that the closer the balance is to the center of the world, the further this “extreme point” (where the weight is heaviest) will lie from the horizontal position of the balance arm (as seen from the fulcrum).

Del Monte’s crucial objection to the Nemorarius school was that one should not consider both weights separately, but rather in terms of their connection by the balance beam. He drew attention to the fact that one must not compare two descents, but rather a descent on one side with a rise on the other. With regard to their positional heaviness the two weights are then equal. Thus Del Monte could claim, using the premises of his adversaries, that the deflected balance does not return to the horizontal.

5.6.3 Benedetti on Tartaglia’s and Nemorarius’s Shortcomings

Benedetti addressed the ideas of Tartaglia and Nemorarius on positional heaviness in section seven of his De mechanicis. There, Benedetti stressed that his approach to positional heaviness, focusing on the distance from the fulcrum to the line of inclination, was distinct from and superior to Tartaglia’s approach in the Jordanus tradition of straightness of descent.

More specifically, Benedetti refuted several of Tartaglia’s claims. In particular, he disputed the central thesis that when a balance is moved from its horizontal position, it will return to this position because the body that has moved upward will attain greater positional heaviness than the body which has moved downward. As we have seen above, Jordanus’s and Tartaglia’s arguments were based on a comparison of the descents of the two weights. In other words, the balance would have to break in the middle to visualize these descents. Benedetti now pointed to the simple fact, already emphasized by Del Monte, that when one weight descends, the other must ascend, and that the corresponding arcs will always be similar to each other and positioned in the same way. He concluded that no positional difference in heaviness can be produced in the way that Tartaglia argued.

Nevertheless, Benedetti did not believe in an indifferent equilibrium of such a balance when considered in a cosmological context. In the continuation of his argument, he came to the conclusion that when such a balance in equilibrium is displaced from its original horizontal position, the weight that has been lowered will actually assume a greater positional heaviness than the one that has been lifted up:

Therefore the weight of A in this [lower] position will be heavier than the weight of B.[41]

He reached this conclusion by taking into account that the lines of inclination of the two weights are not parallel to each other but must converge at the center of the elements. The effective lever arms of the two weights must hence be determined by perpendicular lines drawn from the center of the balance to these lines of inclination. It now turned out that the perpendicular line corresponding to the weight that had been lowered is longer than the line corresponding to the weight that had been lifted. Consequently, the lower weight had become heavier positionally, so that one would expect the balance to tilt into a vertical position.

Benedetti added some more critical remarks on Tartaglia’s consideration of positional heaviness. As we have seen, Tartaglia had argued in Quesiti that the upper weight attains a greater positional heaviness than the lower one, but that this difference is arbitrarily small and can therefore not be compensated by any finite weight. This conclusion was reached by comparing curvilinear angles of contact on each side of the balance. In his analysis of this argument, Benedetti again emphasized that the lines of inclination are not parallel to each other but must converge toward the center of the elements, just as Del Monte had done before him. Clearly, since Tartaglia’s argument hinges on angles of contact, which are infinitesimally small compared to ordinary angles, even such a small deviation from the parallel must be relevant. Taking this into account, Benedetti was able to construct a contradiction, thus refuting Tartaglia’s argument. He concluded:

Now the whole error into which Tartaglia and Jordanus fell arose from the fact that they took the lines of inclination as being parallel to each other.[42]

In summary, Benedetti introduced a way of determining the positional effect of a weight or a force that, in the cases he considered, essentially produces the same results as the application of the modern concept of torque. In particular, Benedetti had managed to go beyond the consideration of weights tending downward to include forces acting in an arbitrary direction. In this way, he was also able to take into account the fact that, on a spherical earth, the lines of inclination of weights on a balance are not parallel. He did not manage, however, to successfully apply his measure of positional heaviness to challenging objects such as the inclined plane.

5.7 The Triangulation Benedetti-Del Monte-Galileo

In this chapter, we have dealt with Del Monte’s and Benedetti’s different approaches to mechanics emerging from their reflection on the balance and their treatment of earlier authors. Relative to the issue of positional heaviness, Del Monte’s self-positioning was essentially external whereas Benedetti positioned himself (albeit critically) within the tradition of the Nemorarius school. He explicitly mentioned Tartaglia and Cardano as relevant sources for his treatment, whereas he omitted any mention of Del Monte.[43] In spite of their opposite intentions and mutual suspicion, Benedetti and Del Monte shared several opinions and sometimes reached the same conclusions, albeit following different paths: both considered the cosmological center of gravity relevant for an evaluation (and criticism) of Tartaglia’s concept of positional heaviness, and both remarked that one cannot treat the two balance beams separately, but rather emphasized that they must be considered simultaneously. Moreover, both stressed the ambiguity of the concept of a mixed angle and the difficulty of its determination. Nevertheless, their approaches were quite different. As mentioned, Benedetti still worked within the framework of the gravitas secundum situm, while Del Monte renounced it in favor of the concept of centrum gravitatis. For Del Monte, the displacement of the balance toward the vertical position was an absurdity that revealed the untenability of Tartaglia’s premises. Benedetti deemed this vertical tilt to be the consequence of a correct analysis of the balance based on a concept close to the modern idea of torque, in consideration of the cosmological context. Furthermore, one should stress the importance of Benedetti’s attempt to determine the quantity of positional heaviness, a fact that distinguishes him from his predecessors. Additionally, unlike Del Monte, he treated the balance by also taking into consideration the general case of forces acting arbitrarily on the beams.

In conclusion, it may be useful to recall the problems linked to the triangulation Benedetti-Del Monte-Galileo, on which the equilibrium controversy sheds new light. The remarkable proximity of these authors on several issues is well known in the history of mechanics. Nevertheless, recent accounts tend to neglect or even deny a possible influence of Benedetti on Galileo.[44] By contrast, the influence of Benedetti on Galileo was assumed and underscored by earlier scholars like Raffaello Caverni, Pierre Duhem, Emil Wohlwill, and Ernst Mach.[45] It is helpful to mention the most important issues common to these authors: the attempt at a theory of motion based on Archimedean hydrostatics, the treatment of the acceleration of fall and its causes, the formulation of what in hindsight appear as proto-inertial principles, a similar treatment of the bent lever, the analysis of the relation between vibrating strings and musical tones, their views on the irradiation of surfaces and on thermal and hydrostatic phenomena, and, last but not least, their support of the Copernican world system.[46] Although many of these themes and ideas belonged to the shared knowledge of preclassical mechanics, in some respects the agreement of their approaches is so striking that one may suspect that this is not mere coincidence.[47] Another potential intermediary was Galileo’s friend Paolo Sarpi who discussed Benedetti’s theory of fall in Pensieri naturali e metafisici. In any case, the strongest evidence of Galileo’s acquaintance with Benedetti’s insights is provided by Del Monte’s Meditatiunculae.

An important clue is page 145bis of the Meditatiunculae (Figure 5.10), which is the page opposite the one containing the detailed criticism of Benedetti dealt with in this chapter. This page shows Galileo’s construction of the inclined plane, reducing it to a bent lever.

Fig. 5.10: Del Monte, Meditatiunculae, p. 145bis showing Galileo’s construction relating the bent lever to the inclined plane. (Bibliothèque Nationale de France)

Fig. 5.11: Del Monte’s construction related to the inclined plane on p. 64 of his notebook. The construction was adapted from Pappus’s erroneous solution. (Bibliothèque Nationale de France)

This fact is all the more noteworthy since Del Monte’s notebook, on an earlier page, also contains his own problematic adoption of Pappus’s analysis of the inclined plane (Figure 5.11).[48] In his writings, Galileo had criticized this analysis, substituting it with his own solution of the problem, which makes use of the bent lever conceptualized in the same way as Benedetti had done.[49] Del Monte therefore must have learned about this proof from Galileo, and he must also have seen the connection to Benedetti’s methods. In any case, it is likely that the two scientists discussed this connection and quite plausible that Galileo became familiar with Benedetti’s work through Del Monte. Galileo began to correspond with Del Monte in 1588, three years after the publication of Benedetti’s Diversae speculationes and shortly before he embarked on the writings that later became known as De Motu.[50] Galileo first wrote a dialogue version of De Motu and then an essay in twenty-three chapters. Only the second essay version of these writings contains his proof of the law of the inclined plane, the argument about continuity of motion along a straight line, and a mention of Copernicus. This version was most likely written after Galileo became familiar with Benedetti’s work. His treatise on mechanics, which for the first time discussed explicitly the problem of the effective lever arm, was written much later, certainly after he had visited Del Monte in 1592 during his journey to Padua. Hence, it seems most likely that Galileo was already familiar with Benedetti’s key ideas at the time of writing these works.

Recent research into Del Monte’s biography has shown that Del Monte and Galileo must have met as early as 1589 in Tuscany.[51] They might even have met jointly with Galileo’s teacher, Mazzoni, who, as mentioned earlier, cited Benedetti in his work. Thus, Del Monte, Mazzoni, and Galileo may have discussed Benedetti’s Diversae speculationes, leading Galileo to reconsider his work in progress on motion and, in particular, his treatment of motion along inclined planes, making use of Benedetti’s theory of the bent lever that was mentioned in Del Monte’s notebook. But Benedetti’s impact on Galileo probably went even further than that. Galileo may have started taking the Copernican hypothesis much more seriously after his encounter with Benedetti’s work, discussing this as well as other subjects with Mazzoni. In the above-mentioned letter of 1597, Galileo praised Mazzoni for his Praeludia and reminded him of the controversial issues on which they had meanwhile reached an agreement, and also tried to press him on the Copernican hypothesis.

In particular, Galileo’s concept of momento[52] and his analysis of the bent lever—crucial to both his mechanics and his theory of motion—evidently emerged from the midst of the controversy about positional heaviness. In that debate, Galileo took a position much closer to Benedetti than to Del Monte. Rather than gravitas secundum situm, Galileo used the concept of momento or momentum that Del Monte had introduced in his book by quoting Commandino’s definition of the center of gravity. But while Del Monte made no further use of this in his mechanics, Galileo took this concept from the respected Urbino school, gave it a new meaning that was taken from Benedetti, and made it a pillar of his own conception, which included Commandino’s definition of the center of gravity:

Center of gravity is defined as that point in every heavy body around which parts of equal moments are arranged.[53]

The evidence for this claim concerning Benedetti’s legacy in Galileo’s work derives from the marginal notes Del Monte made in his copy of Benedetti’s book, as well as from his entries in the Meditatiunculae which contain traces of Galileo’s intervention in this controversy.[54]

According to Benedetti and Galileo (and contrary to Tartaglia and Del Monte), the effective length of the lever arm, obtained by drawing a perpendicular from the fulcrum of the balance to the line of inclination, determines the effectiveness of a weight or a mechanical constellation. In his Mechanics, Galileo later stressed how important it is to carefully define the effective distances of weights from their support:

There is one thing that must be considered before proceeding further, and this concerns the distances at which heavy bodies come to be weighed; for it is very important to know the sense in which equal and unequal distances are to be understood, and in what manner they must be measured.[55]

In his analysis of the inclined plane using the bent lever, Galileo also made clear that this procedure is critical for determining the momento of a given weight.[56] As discussed earlier, in his Diversarum speculationum […] liber, Benedetti convincingly demonstrated the efficacy of this method for determining the magnitude of a force or weight according to its position.

In conclusion, the very existence of Del Monte’s annotations on his copy of Benedetti’s Diversae speculationes provides a definitive answer to the question of whether Del Monte had read this book or not.[57] It is also difficult to imagine that he did not discuss his views on Benedetti’s mechanics with Galileo, views that he considered both misguided and profoundly challenging, as is made evident in his handwritten notes. It was most probably Del Monte, Benedetti’s fervent opponent in matters of mechanics, who served as a conduit to Galileo. At the same time, he also made it virtually impossible for Galileo to openly admit to Benedetti’s influence if he did not also want to jeopardize the protection of the most important patron of his early career.

Footnotes

[1] Section 5.1 is derived from Renn and Damerow 2012, chap. 6.1–6.3 and section 5.2 from Renn and Omodeo 2013.

[2] Tartaglia 1546.

[3] Nemore 1565.

[4] Tartaglia 1546, 91r: “[…] che la differenzia ch’è fra le gravità de questi dui corpi egli è impossibile a poterla dar, over trovar’ fra due quantità inequali.” Translation in Drake and Drabkin 1969, 130.

[5] We will discuss Benedetti’s criticism in more detail later.

[6] Aristotle 1980. See Rose and Drake 1971 and also the introduction to Nenci 2011.

[7] Benedetti 1585, 152. See Drake and Drabkin 1969, 178.

[8] In the sixteenth century the term scientia communis was used to designate knowledge common to all mathematical sciences, its core being the Euclidean theory of proportions. See Sepper 1996, 153–154.

[9] See the discussion in Damerow et al. 2002.

[10] Aristotle 1980, 1, 848 b 1–850 a 2 (337–347).

[11] Aristotle 1980, 1 (347).

[12] Aristotle 1585, 507: “Quandoquidem igitur in proportione fertur aliqua id, quod fertur, super rectam ferri necesse.” Translation in Aristotle 1980, 1, 848 b 11–848 b 13 (337).

[13] Aristotle 1585, 508: “Si autem in nulla fertur proportione secundum duas lationes nullo in tempore, rectam esse lationem est impossibile.” Translation in Aristotle 1980, 339.

[14] Benedetti 1585, 153; Drake and Drabkin 1969, 180–182.

[15] Aristotle 1980, 347–355; Drake and Drabkin 1969, 182–183.

[16] Aristotle 1585, 511: “Cur siquidem sursum fuerit spartum, quando deorsum lato pondere, quispiam id admovet, rursum ascendit libra: si autem deorsum constitutum fuerit, non ascendit, sed manet?” Translation in Aristotle 1980, 347–349.

[17] Aristotle 1980, 353–355.

[18] Benedetti 1585, 154; Drake and Drabkin 1969, 183.

[19] Aristotle 1585, 515: “Cur quando antenna sublimior fuerit, iisdem velis, et vento eodem celerius feruntur navigia?” Translation in Aristotle 1980, 361.

[20] Benedetti 1585, 155: “[…] verum non est. Huiusmodi enim ratione navis tardius potius, quam velocius ferri deberet, quia quanto altius est velum, vi venti impulsum, tanto magis proram ipsius navis in aquam demergit.” Translation in Drake and Drabkin 1969, 183.

[21] Benedetti 1585, 155: “[…] quia huiusmodi centrum ab inferiori parte ad superiorem, nunquam mutabit situm respectu distantiae seu intervalli, quae inter ipsum lineamque AD intercedit.” Translation in Drake and Drabkin 1969, 184.

[22] For the historical context, see Büttner 2008.

[23] Aristotle 1585, 517: “Cur ea, quae per maiores circulos tolluntur et trahuntur, facilius et citius moveri contingit […]?” Translation in Aristotle 1980, 365.

[24] Aristotle 1585, 517: “Cur facilius quando sine pondere est, movetur libra, quam cum pondus habet?” Translation in Aristotle 1980, 365.

[25] Aristotle 1585, 518: “Cur longius feruntur missilia funda, quam manu missa […]?” Translation in Aristotle 1980, 367.

[26] See the discussion in Renn, Damerow, and Rieger 2001.

[27] Aristotle 1980, 367.

[28] Aristotle 1585, 518: “Cur eiusdem magnitudinis lignum facilius genus frangitur, si quispiam aequi diductis manibus extrema comprehendens fregerit, quam si iuxta genu?” Translation in Aristotle 1980, 369.

[29] Aristotle 1585, 520: “Cur a parvo existente cuneo magna scinduntur pondera, et corporum moles, validaque sit impressio?” Translation in Aristotle 1980, 371.

[30] Aristotle 1585, 525: “Dubitatur quam ob causam maior circulus aequalem minori circulo convolvitur lineam, quando circa idem centrum fuerint positi.” Translation in Aristotle 1980, 387.

[31] In Drake and Drabkin 1969, 193; chapter 22 is erroneously numbered as chapter 23.

[32] Aristotle 1585, 532: “Cur surgentes omnes, femori crus ad acutum constituentes angulum, et thoraci similiter femur, surgunt?” Translation in Aristotle 1980, 403–405.

[33] Aristotle 1585, 533: “Cur ea quae in vorticosis feruntur aquis, ad medium tandem aguntur omnia?” Translation in Aristotle 1980, 409.

[34] Benedetti 1585, 167: “[…] a quo aliarum omnium quaestionum, quas ego omisi rationes sunt bene propositae.” Translation in Drake and Drabkin 1969, 196.

[35] “Hoc primum caput to[tum] desumptum est a n[ostro] Mechanicorum libri tractatu de lib[ra].”

[36] “[…] unde fit ut hoc modo pondus magis aut minus a centro pendet aut eidem nititur: atque haec est cause proxima, et per se […].”

[37] See Renn and Damerow 2012, 207: “non est neque proxima neque per se; nam [pond]us in F brachii [BF] non est equegrave ut pondus in U brachii BU; [nec] pondus in E brachii BE est equegrave ut pondus [in] U brachii BU. Unde tota haec demonstratio falsa est.”

[38] See Renn and Damerow 2012, 213: “si intelligamus p[ondus] in C, ut supponi p[otest] ex verbis ipsius, intelligendum est C[T] quoque consolidatam consolidatis TO […]. Unde si intelligamus C pondus et non movens, falsa est i[ta]que si intelligatur C movens ut homi[…] vera esse pote[st] quod [deleted: non] moveat non esse pondus s[i…] ipse [vero] in sequenti accipiat [hoc atque ponderi?] posse demonstratum quare nihil […] ut patet in 7 cap. In his duobus cap. fundantur omnes authoris demonstrationes ita ut sunt praecipua mechanicorum fundamenta quorum cognita falsitate omnia rem[oventur].”

[39] Cardano 1550, 17–18: “Aristoteles dicit hoc contingere, quum trutina est supra libram, quia angulus QBF metae, maior est angulo QBR. Et similiter quum trutina fuerit QB, erit meta AB, et tunc angulus RBA, maior erit angulo FBA, sed maior angulus reddit gravius pondus. […] Generalis igitur ratio haec sit: pondera quo plus distant a meta seu linea descensus per rectam aut obliquum, id est, per angulum, eo sunt graviora.”

[40] Del Monte 1577, 1r: “Centrum gravitatis uniuscuiusque corporis est punctum quoddam intra positum, a quo si grave appensum mente concipiatur, dum fertur, quiescit; et servat eam, quam in principio habebat positionem: neque in ipsa latione circumvertitur.” Translation in Drake and Drabkin 1969, 259, revised in Damerow and Renn 2010, 57.

[41] Benedetti 1585, 148: “Pondus igitur ipsius A in huiusmodi situ, pondere ipsius B gravius erit.” Translation in Drake and Drabkin 1969, 176.

[42] Benedetti 1585, 150: “Omnis autem error in quem Tartalea, Iordanusque lapsi fuerunt ab eo, quod lineas inclinationum pro parallelis vicissim sumpserunt, emanuit.” Translation in Drake and Drabkin 1969, 177.

[43] Benedetti 1585, f. A3r.

[44] See the discussion by Ventrice in Bordiga 1985, 732–736. He mentions Drake, Drabkin, Fredette, and Galluzzi among those who are skeptical about a concrete influence of Benedetti on Galileo. Notable exceptions are the commentaries by Carugo and Geymonat in their edition of Galileo’s Discorsi, see Carugo and Geymonat 1958. Bertoloni Meli even considers the possibility of Del Monte and Galileo discussing Benedetti, but nevertheless rejects any substantial influence by the latter on Galileo’s thinking because that influence supposedly would have arrived too late, see Bertoloni Meli 2006, 61–65.

[45] Cozzi and Sosio 1996. For an overview of such potential connections, see the discussion in Bordiga 1985, 732–736 who also mentions Mersenne, Clavius, and Cardinal Michelangelo Ricci as possible intermediaries.

[46] For an overview, see Bordiga 1985.

[47] See, for instance, Drake and Drabkin 1969, 36. Yet, the question of Benedetti’s direct impact on Galileo remains unclear, in particular as Benedetti’s work was never mentioned by Galileo.

There are several possible connections between Benedetti and Galileo that have been considered in the past. For instance, Benedetti is referred to by Galileo’s Pisan colleague Jacopo Mazzoni in In universam Platonis et Aristotelis philosophiam praeludia from 1597. See Mazzoni 1597. He is often mentioned in the Galileo Studies as the addressee of a famous letter by Galileo arguing for the Copernican system (May 30, 1597) . See Galilei 1968, vol. 2, 194–202 . In his book Mazzoni referred to Benedetti’s discussion of the possibility that motion along a straight line can be continuous. See Benedetti 1585, 183–184. For a historical discussion of the context of this argument in contemporary technology, see Freudenthal 2005, a theme that was later taken up by Galileo in chapter 20 of De Motu, which also refers explicitly to Copernicus. See Mazzoni 1597, 193 and Galilei 1960, 326. It is conceivable that such issues had been discussed, inspired by Benedetti’s work, between Galileo, Mazzoni, and Del Monte during Del Monte’s stay in Tuscany in 1589. We would like to thank Pier Daniele Napolitani for drawing our attention to this possibility and to the above-mentioned passages.

[48] Del Monte 1587, 64.

[49] Galilei 1960, 172.

[50] Galilei 1960. For a thorough discussion of the chronology of these writings, see Giusti 1998.

[51] Menchetti 2012.

[52] See the extensive discussion in Galluzzi 1979.

[53] Galilei 1968, vol. 2, 159: “Centro della gravità si diffinisce essere in ogni corpo grave quel punto, intorno al quale consistono parti di eguali momenti.” Translation in Galilei 1960, 151. See also Galilei 2002.

[54] Del Monte 1587.

[55] Galilei 1968, vol. 2, 164: “Un’altra cosa, prima che più oltre si proceda, bisogna che sia considerata; e questa è intorno alle distanze, nelle quali i gravi vengono appesi: per ciò che molto importa il sapere come s’intendano distanze eguali e diseguali, ed in somma in qual maniera devono misurarsi.” Translation in Galilei 1960, 156–157.

[56] See Galilei 1968, vol. 2, 181. Translation in Galilei 1960, 173.

[57] The knowledge that he had read it, however, is not entirely new. See Renn, Damerow, and Rieger 2001, 74.

6. Astronomy

6.1 Benedetti as an Astronomer

Benedetti’s astronomical considerations are not systematic. They are scattered throughout the volume in different sections. In spite of the difficulty of ordering them and obtaining an overview, they were very much appreciated among his contemporaries. Apart from Kepler’s eulogy of Benedetti’s ingenuity, the broad European success of the astronomical parts of this work is documented in other references. A few years after the publication of the Diversae spaeculationes, Brahe must have had a copy of it in Denmark, as he quoted it extensively and accurately on two occasions. In his correspondence with Landgrave William IV and the Hesse-Kassel court mathematician Christopher Rothmann, he referred to Benedetti’s observation of the light of Venus reflected on the part of the lunar disc not presently enlightened by the sun:

In fact, I sometimes saw that Venus illuminated in a rather sensible manner that part of the Moon that was most distant and opposed to the Sun, although the Moon is by far more distant from Venus’s circuit than the comet. I remarked that the Venice patrician Giovanni Battista Benedetti, the most excellent philosopher and mathematician, noted something similar in that erudite work which he wrote on mathematical and physical speculations. At the end of an epistle to a certain Savoy baron, Filiberto, he says: “[…] that the part of the Moon which is deprived of the Sun’s light is sometimes partially illuminated by Venus’s light. I observed this often and showed it to many people.”[1]

Brahe quotes this passage correctly from Benedetti’s letter to Baron Emanuele Filiberto Pingone “De Luce, Lumine, et Colore, De obiectu oculi, De lumine Lunae, et Rubedine nubium” (On light, lumen, and color; on the eye’s object, on the lunar lumen, and the redness of the clouds).[2]

A second long direct quotation of Benedetti can be found in Brahe’s book on the nova of 1572, which was part of the Astronomiae Instauratae Progymnasmata, posthumously published in Prague in 1602.[3] The Danish astronomer here praised Benedetti as a “philosophus et mathematicus inprimis excellentem,” and his work as “praeclarum Opus.” He entirely reproduced Benedetti’s letter and diagrams on the star in Cassiopeia.[4] This letter was directed against Annibale Raimondo—an author whom Brahe also criticized—and demonstrated that the nova appeared above the sublunary sphere. Brahe commented:

Here follows the epistle which I referred to. It is taken from the aforementioned book by [Giovanni] Battista Benedetti alongside the demonstrative diagrams offered by the same author. Afterwards I will consider others, who discussed that star [i.e., the nova of 1572] in an extraordinarily incompetent manner. This [quotation from Benedetti] (as mentioned) will cast light on these issues through a synthetic and wise geometrical truth, so that no significant doubt will survive.[5]

Another reader of the Diversae speculationes was the English scholar of magnetism William Gilbert. In De mundo nostro sublunari philosophia nova (New Philosophy on Our Sublunary World, written about 1600 but published long after the author’s death, in Amsterdam in 1651), he in fact discussed Benedetti’s views on the spots on the surface of the moon, in a chapter trying to determine which parts of it were seas and continents.[6] It is evident that the Diversae speculationes had a wide European circulation, and that the astronomical part attracted the attention of many scholars dealing with mathematical and physical issues.

Benedetti’s treatment of astronomical matters ranges from the calendar reform to the nova of 1572, sundials, and astrology. We would like to focus on a special issue: Benedetti’s defense of ephemerides, Defensio ephemerides, and the quarrel that motivated its writing. This defense of ephemerides figures as one of the epistles of the Diversae speculationes. It is the Latin translation of an Italian letter, Intorno ad alcune nuove riprensioni… contra alli calculatori delle effemeridi (Letter in the Form of a Discourse… Addressed to the Illustrious Mr Bernardo Trotto Concerning Some New Criticism and Corrections against the Ephemerides Calculators, Turin, 1581), addressed to Trotto, which Benedetti had already published when a heated quarrel on the reliability of ephemerides burst out in Turin between 1580 and 1581. In the following pages we will give an account of these facts.[7]

6.2 The Controversy over the Reliability of Ephemerides

The ephemerides controversy began with the publication of Altavilla’s Animadversiones in ephemeridas (Remarks against Ephemerides, Turin, 1580). This lesser-known author from Vicenza intended to denounce the inexactitude of all existing astronomical computations.[8] For this purpose he compared predictions and horoscopes cast using different sets of tables and ephemerides. In particular, he pointed out that ephemerides diverged from each other even more than the astronomical tables, Alfonsine or Copernican, from which they were derived. In his opinion, this fact undermined the reputation of astronomy in general, regardless of whether its cause was the inaccuracy of the compilers (calculatores) or the inexactitude of the tables themselves: “We consider nothing to be more odious than an unreliable person who is regarded by many as trustworthy.”[9] Altavilla declared himself unwilling to decide between Alfonsine or Copernican computations. However, he himself was probably interested in the cosmological issue, judging by the fact that the Animadversiones were introduced with a poem by Pandolfo Sfondrati in favor of a new world system with the earth in motion.[10]

Altavilla had established by observation that both Alfonsine ephemerides and Johannes Stadius’s Copernican computations were in disagreement with the heavens. Still, Stadius’s computations proved to be in better agreement with the heavens. The reference to Stadius is not casual, since the Flemish astronomer had been a protégé of duke Emanuele Filiberto of Savoy, as one can read in the Ephemerides novae of 1556, where the author gave himself the title “mathematician to the King [of Spain] and the Duke of Savoy” (Regius et Ducis Sabaudiae mathematicus). Altavilla listed predictive errors of Ptolemaic astronomers (Regiomontanus, Stöffler, Leowitz) as well as those of post-Copernican ephemerists (Stadius and Giuntini). This led him to skepticism toward predictions in general: “You see, dear reader, how reliable ephemerides are.”[11] Altavilla invited scholars (magistri) to trust only their eyes and to correct astronomy through observational campaigns with no regard for any authority: “Posterity should learn how dangerous it is to blindly adhere to the opinions of the ancients without [perfecting the art through] daily observations of the heavens, and to prefer their opinions to truth.”[12]

The Animadversiones were soon followed by a second publication in Italian: Breve discorso intorno gli errori dei calculi astronomici (Brief discourse on the mistakes of astronomical calculations, 1580). A poem by a certain Francesco Onto of Pinerolo, inserted as a preface to the Breve discorso, made its polemical target explicit: “Altavilla has unveiled the astrologers’ fallacy, as they think to cast certain [astrological] judgments about our lives relying on flawed ephemerides.”[13] Altavilla’s criticism was directed mainly against astrology, whose validity he considered to be doubtful due to the inaccuracy of predictions. His argumentative strategy was no different than that of Pico della Mirandola in books 8 and 9 of the Disputationes in astrologiam divinatricem (Disputations against divinatory astrology, 1496): an attack on mathematical astronomy aimed to discredit astrological forecasting. Altavilla even claimed that astrologers and ephemerists should renounce their activity, as they were not capable of superseding the flaws of their discipline: “Since it is impossible for the scholars in those sciences (especially those who are not capable of using the tables) to renounce ephemerides, and they know that they will encounter irremediable errors, they should be forced to abandon their studies.”[14]

In his second publication, the Discorso, Altavilla complained that many scholars (who were not named) pretended to ignore his criticism. He explained that the decision to write another booklet, this time in Italian instead of Latin, originated from the desire to reach readers outside academic and scholarly circles, probably also at the Savoy court: “In these few pages, I aimed at demonstrating not only to the learned man, but also to everybody else, that the errors [of the ephemerides] are worthy of consideration.”[15] He first reassessed the inadequacy of Alfonsine tables and Alfonsine ephemerides (those of Peuerbach, Prugnerus, Bianchini, Regiomontanus, Stöffler, Schöner, Gaurico, Pitati, Simi, Carelli, Moletti, Leowitz, and others). He moreover stressed the superiority of the Copernican tables in order to show the inconsistency of some unnamed Turin ephemerists who used Alfonsine ephemerides for their predictions although they claimed to prefer Copernicus. To illustrate this inconsistency, he analyzed some astrological figures on the basis of Stadius’s and Giuntini’s tables. In the last section Altavilla turned on the Copernican ephemerists, denouncing the excessive difference between computations based on Stadius and Giuntini: “And the difference between one computation and the other is really great and monstrous.”[16]

This attack on the reliability of astronomical computations and astrology provoked negative reactions both at the university and at the court. Altavilla thus felt compelled to challenge his critics to an academic debate on August 14 and 15, 1581, announcing it through a broadside that is still preserved in the libraries of Turin, along with copies of his Animadversiones.[17] The public dispute concerned the theory of Mars for which, as one reads, some scholars blamed him. He maintained, in fact, that Mars cannot stay in a zodiacal sign for more than two months, considering that its entire revolution lasts twenty-four months. He argued that ephemerides are wrong if they forecast that it would spend six or even seven months in the same zodiacal constellation. This incorrect opinion presented the court mathematician and philosopher Benedetti with an occasion to intervene and criticize Altavilla on this and other issues related to astronomical theory, computation, and astrological prediction.

Soon after Altavilla’s public dispute, Benedetti published an epistle “on some recent remarks and emendations directed against ephemerists” (Turin, 1581). At the beginning, Benedetti indicated Altavilla’s intentions: “I assume […] that his intention was only to demonstrate that [different] ephemerides assigned a different place to the planet at the same point of time […] and that, as a consequence, they offer no certain ground on the basis of which the future can be judged or predicted.”[18] In his account, Benedetti rejects Altavilla’s complaint that Copernican and Alfonsine ephemerides diverge from each other more than the tables from which they are derived. He assures the reader that “the people who calculated have been very accurate and trustworthy” (i calcolatori sono stati diligentissimi e fedeli) and they are exact in their calculations, although some minor and accidental mistakes can occur.[19]

Moreover, he accuses Altavilla of misunderstanding Ptolemy’s astrology, interpreting it in light of Abu Ma’shar and Al-Qabisi (Alcabitius). In particular, Altavilla draws from these sources the rule of the “triplicity” of the conjunctions of Jupiter and Saturn, according to which these planets meet four times in the same three astrological signs, or trine, before they can meet in the next trine. However, although the mean motions of two planets should meet in the triplicity sign, nonetheless their “real” motions (those observed and calculated by the ephemerides upon which astrological predictions rely) may meet elsewhere. This is an obvious consequence of planetary theory. In fact, it distinguishes between “mean” motions, which correspond to the revolutions of the deferents, and “real” motions, which correspond to observable phenomena and are the product of moving epicycles. Benedetti calculates the period of triplicity to be 794 years and 138 days, whereas the Arabs on whom Altavilla relies overestimated it at 960 years.[20] He furthermore remarks that Altavilla neglected planetary theory by criticizing those who let Mars run too fast or too slowly along the signs of the zodiac. Simple observations would show the correctness of the theory according to which the planet can remain in the same sign for six or even seven months. Benedetti explains that the amplitude of Mars’s epicycle accounts for its complex phenomenology, in particular the long period of retrograde motion. On this account, he reports an observational campaign accomplished between 1565 and 1566 in order to check Stadius’s ephemerides:

Yet, he [Altavilla] dared too much, seeking to reprimand so many talented ancient and modern men who, as is required by diligent observers of the heavens, checked with their own eyes these appearances of Mars as well as of the other [planets]. From those [observations], they were forced to “imagine” such a large [Martian] epicycle. By contrast, he has never observed the motions of either this or any other planet, but rather limited himself to look at what is written in the ephemerides. In fact, if he had at least said that he observed Mars’s journey for a certain period, and that he found that the others’ opinion was false, he would have at least given some “color” to his opinion. In my assessment, however, if he had made an observation of the path of Mars, he would not have held the contrary view. In fact, the truth is the following: in every revolution of its epicycle, Mars in the lower part of its epicycle always stays many months (six or seven, or more) in a twelfth [duodecatemerio] of the zodiac. I observed this many times, for instance, in the years 1565 and 1566. First, consulting Stadius’s ephemerides, I found that Mars would finish its retrograde motion on about 12 January 1566, in 16° of Gemini, and that, equally, Mars would be in the same place on the last day of August 1565, before it began its retrograde motion. Second, I found that, after that retrograde motion, on 11 April 1566, Mars would be in 16° of Cancer, so that it would take [Mars] seven months and eleven days [to move] those thirty degrees, from 16° of Gemini to 16° of Cancer. After these computations, I took the instruments and got ready to make a test. And I found that the last night of August of the year 1565 Mars was in the aforesaid 16° of Gemini, as Stadius had noted. I then made observations every week, in order to see the retrograde motion, and I saw that, at about the end of October, the [planet] began its retrograde motion and that retrograde motion lasted until January (or about January) 1566. I later observed the position of that planet on 11 April, and I found it in 16° of Cancer, that is, the place where Stadius had located it. Thus, my experience confirmed Stadius’s computations and I found that he was not mistaken. In the same manner, everybody can ascertain the truth every two years by carrying out observations.[21]

Benedetti thus demonstrated not only the theoretical incompetence of his opponent, but also his lack of empirical verification. Altavilla’s appeal to base astronomy using observation backfired. Benedetti challenged his opponent to observe Mars’s backward motion in Cancer which, according to Stadius’s tables, would begin on November 20, 1582 and last until the end of February 1583. He furthermore observed that everyone familiar with planetary theory would understand the reasons for the orbit of Mars and other planets. For the theory, he added, it did not matter whether one relied on Ptolemy’s Almagest or on the “Rivolutioni de gl’orbi celesti dell’eccellentissimo Copernico.”[22] Of course they were only equivalent as far as the understanding of a system of deferents and epicycles was concerned, but not in their general hypotheses, since Benedetti himself tended toward heliocentrism.

As to the difference between Leowitz’s and Stadius’s computations, Benedetti traced this back to the contrast between the theories underlying the Alfonsine and the Copernican tables. Nonetheless, he ensured that ephemerides never diverged by more than three degrees. Thus, if Altavilla detected greater discrepancies, this was due only to false computations. Benedetti added that Stadius’s superiority over Leowitz was a consequence of him employing better parameters. He advised Altavilla to always rely on the most recent observations and tables.[23] In fact, he judged the progress of astronomy to be such that more recent tables would inevitably be superseded by new ones, augmented and perfected through new observations, just as Copernicus had superseded Alfonso’s astronomers. Divergence between ephemerides was not a shortcoming, but a necessary and desirable sign of the advancement of knowledge and predictive accuracy.

As a courtier expert of mathematics, Benedetti defended the validity of some astrological figures that Altavilla criticized in his second published work, Breve discorso. These horoscopes had probably been cast by somebody that he knew well. Altavilla complained that some astrological figures had not been calculated on the basis of Copernican tables. Benedetti replied that it was not always necessary to use the best tables for predictions, especially if a generic horoscope was expected and if the astrologer had no Copernican tables to consult. He showed, moreover, that Altavilla himself was not able to employ Giuntini’s tables properly and made mistakes of computation. He concluded: “And such monsters [those denounced by Altavilla] are not generated by different tables or ephemerides but, instead, they are the offspring of this author.”[24] He added as a remark: “As to the difference of the Sun according to Copernicus and Alfonso, no learned man, [expert] in these sciences, ignores it, and, as a consequence [everybody knows] the different place [assigned to it] in the heavens during the annual revolutions.”[25] In 1581, the general views of De revolutionibus were so well known in Benedetti’s environment that he deemed it unnecessary to expand on them in the context of a polemic on the accuracy of heavenly computations. The cosmological implications of these different hypotheses were not addressed explicitly in this dispute. However, the defense of mathematical astronomy could not avoid a reference to Copernicus as a source for tables (Reinhold, Stadius, Giuntini) and theory. In this context, “Copernican” and “not Copernican” are expressions that merely mean “based on Copernican tables” or not. Altavilla’s criticism would have been more effective if it had been directed against astrological beliefs as such, rather than attempting to show the inconsistency of the mathematical basis of astrology without sufficient preparation. On the other hand, Benedetti, in his Lettera, focused on the mathematical aspects and cautiously avoided expanding on ethical issues related to astrology.

Altavilla never responded to the court mathematician who had rebutted his arguments so forcefully. The epilogue to their quarrel was the inclusion of a Latin translation of the Lettera, as Defensio ephemeridum (A defense of Ephemerides), in Benedetti’s Diversae speculationes.[26]

6.3 The System of the World

Benedetti did not limit himself to considering astronomy from a computational point of view, but also expanded on cosmological aspects. The epistle “De fine corporum coelestium, et eorum motu” (On the Aim of Celestial Bodies, and their Motions),[27] addressed to Pingone, bears witness to his interest in cosmology and his realist interpretation of Copernicus’s hypotheses. Benedetti remarks that it is not reasonable (si […] humanam rationem sequi volueris) to believe that the heavens were created only for the sake of terrestrial life, “as these [celestial] bodies are divine, uncountable, and endowed with the greatest dimensions” (cum ea corpora sunt divina, in numero incompraehensibilia, maximis magnitudinibus, et motibus velocissimis praedita).[28] This absurdity can be avoided, as Benedetti claims, if one accepts the planetary doctrine of Aristarchus and Copernicus:

[…] this will hardly be believed by those who embrace the doctrine of Aristarchus of Samos and Nicolaus Copernicus. Following their approach it is impossible to make them believe that the rest of the universe has no other aim than to rule over this center of the lunar epicycle [the earth] (to use their way of speaking).[29]

Although he speaks in the third person, as if he were reporting the views of someone else, these are his own views. He is inclined to accept the Copernican system or some variation of it, as the following pages of the letter and the force of the arguments show. Firstly, he assumes a principle of cosmological homogeneity according to which there is no reason why other planets should not be subjected to alterations (ab ortu, et interitu), as the Aristotelians suppose. The peripatetic argument that no change in the heavens was ever observed is not valid, because the distance does not permit verification of whether there is any life or alterations on distant bodies (unde etiam fieri potest, ut in coelo sint particulares alterationes, quae a nobis tamen, qui ab illis longe distamus, non compraehendantur).[30] Benedetti even surmises that other planets are moons reflecting the solar light to dark planets invisible to us.[31] He ascribes this opinion to the followers of Copernicus. This is a free interpretation on his part. Perhaps he aimed to explain the epicyclic motions of other planets through an analogy with the lunar epicycle around the earth. Benedetti also rejects Ptolemaic and Aristotelian arguments against terrestrial motion. Following Copernicus (De revolutionibus I 8), he stresses that the axial rotation avoids the otherwise enormous motion of the fixed stars: “which is eliminated by the rotation of the Earth about its axis (as they say) as it is sufficient to receive the light and the influences of the [celestial] bodies.”[32] Moreover, the annual revolution respects the dignity of the “divine body of the Sun” (divinum corpus solare), which stands still at the center of the planetary circles.[33] Note Benedetti’s astrological concern. In the final passage of his letter, he reassesses Copernicus’s objection to Ptolemy’s view of how bodies suspended in the air are affected by terrestrial motion:

Ptolemy’s objections are not valid for them [astronomers who assume that the earth moves]. As they say, every part maintains the nature of the whole, apart from the fact that the air and water circumscribing the earth receive the same natural impulse of motion [impetum motus]. This is slower the further the air is distant from the earth. According to the same doctrine, there is no necessity that the place of the fixed stars has (either convex or concave) superficial boundaries.[34]

According to this passage, the air close to the earth is transported by the motion of the planet and slows down the more it is distant from it. The fixed stars are placed in a motionless air whose place (locus) has no boundaries, either convex or concave.

In a letter to the courtier Capra, Benedetti confronts the issue of the form of the heavens.[35] This is said to be a sphere encompassed by infinite space. Accordingly, Benedetti distinguishes between spacium (space) and coelum (heavens), a distinction that can be traced back to Stoic cosmology or to the more recent views of Marcellus Palingenus Stellatus. The idea of the infinity of space beyond the starry vault can be found also in Patrizi’s Nova de Universis Philosophia (1591).[36]

Furthermore, Benedetti rejects the existence of material spheres with the role of transporting the planets:2

That you do not accept that distinction of spheres, which was well-established in the past, but rather that you believe that the whole is a continuum accommodating the stellar bodies, this is not new. In fact, some philosophers of solid doctrine were of the same opinion.[37]

The motion of celestial bodies is accompanied by that of transparent bodies similar to vapors (fumi). Their motion is the cause of the apparent sparkling of the most distant stars.[38] The sparkling of the new star in Cassiopeia in 1572 bears witness to its great distance above the moon, which Benedetti also demonstrates through geometry.[39]

One of the books of the Diversae speculationes entails a discussion and a refutation of Aristotelian physical and celestial theses de motu. It has the rather neutral title Disputationes de quibusdam placitis Arist[otelis] (Disputations on Some Opinions Held by Aristotle) but it is indeed an attempt to revise basic concepts of natural philosophy such as locus (place) and tempus (time). We shall deal with this issue in detail in the next section. For now, it is important to anticipate that this anti-Aristotelian section entails Benedetti’s most explicit defense of Copernican planetary hypotheses. Another remarkable thesis of these Disputationes on Aristotle is the statement of a principle of relativity according to which planets appear to us as we appear to them:

Aristotle did not consider that one could affirm the same about the Earth as seen from great distance. There is no doubt that, even if the Earth had the light of the Sun and somebody tried to observe it from the eighth sphere, he would not be able to perceive it. In fact, those celestial bodies that are said to be of the first magnitude and that are believed to be more than a hundred times bigger than the Earth look just like points.[40]

Benedetti supports the plurality of worlds as well (Minus sufficienter explosam fuisse ab Aristotele opinionem credentium plures mundos existere). Every planet should be regarded as another Earth with its elements and natural places: “If those worlds existed, each of them would have its own center and its own circumference and the earths and fires would have an inclination towards the centers and the circumferences of their worlds, respectively.”[41]

6.4 Appendix: An Assessment of Benedetti’s Horoscopes (by Günther Oestmann)

For the recalculation of a historical horoscope, the same methods and means the author had at his disposal must be employed, that is, the use of modern parameters or tables is not allowed.[42] In the following disposition, planetary positions are rendered in ecliptic longitude (degrees ; minutes) for each zodiacal sign (0–30°), geographical coordinates likewise in degrees ; minutes, and time in hours ; minutes. Latitude is denoted as φ.

6.4.1 Nativity Cast by Benedetti for Duke Carlo Emanuele I of Savoy

January 11, 1562 (Julian date), 16;23 p.m., φ = 45°; Planetary positions according to the Prutenicae Tabulae by Erasmus Reinhold (1551).

Fig. 6.1: The horoscope cast by Benedetti for Duke Carlo Emanuele I of Savoy. This was calculated with the Prutenic tables, as transcribed by Bartolomeo Cristini in Revolutione trentesima prima del Serenissimo Signore il Signor Carlo Emanuel Duca di Savoia corrente dell’anno 1592 con ogni diligenze et fedeltà calculata et decchiarata secondo le migliori intelligenze de più principali autori dell’astrologia giundiciaria, Turin. (Biblioteca Nazionale Universitaria: Coll. N VII 10, f. 11v)

Carlo Emanuele I of Savoy was born in the Castle of Rivoli (c. 15 km west of Turin) on January 12, but here the “noon epoch” is indicated in the manner commonly used by astronomers/astrologers: the date changes at 12:00 local time, and the hours are counted from there to 24—contrary to civil use, where sunrise or sunset often marked the change of day. With the proliferation of mechanical clocks in the late Middle Ages, the date change at midnight gradually became predominant and hours were counted from 1–12.[43]

The geographical coordinates of the nearest town to Turin listed in the Catalogus locorum in Reinhold’s Prutenicae Tabulae is Venice, which is 0 h 50 m (12;30) west of Königsberg, the reference meridian of the Prutenic tables. In Petrus Apianus’s Cosmographicus liber (1533), the following specifications are given: Turin 30;30 and Königsberg 42;16 east of the island Porto Santo near Madeira (f. XLIIr, XXXIXr) → the difference of longitude is 11;46 (modern value: 12;46). In the following recalculation, a longitude of 12;00 west of Königsberg has been assumed:

Planets | Original Source | Recalculation
Sun | 1;27 Aq | 1;27 Aq
Moon | 29;09 Ar | 29;16 Ar
Saturn | 28;54 Ge retrograde | 28;55 Ge
Jupiter | [missing] | 21;02 Ta
Mars | 18;42 Ar | 18;41 Ar
Venus | 0;58 Cp | 0;58 Cp
Mercury | 15;48 Cp retrograde | 15;48 Cp
Lunar node (asc.) | 15;19 Aq | 15;16 Aq

6.1 Planets

Houses (Regiomontanus) | Original Source | Recalculation
X | 10;23 Li | 10;16 Li
XI | 5;05 Sc | 4;59 Sc
XII | 24;35 Sc | 24;32 Sc
I | 15;57 Sa | 15;54 Sa
II | 17;11 Cp | 17;07 Cp
III | 2;05 Pi | 1;58 Pi
Lot of Fortune (Night) | 18;15 Vi | 18;05 Vi
Lot of Fortune (Day) | 13;39 Pi | 13;43 Pi

6.2 Houses

The Lot of Fortune (Pars Fortunae; named for the Roman goddess of luck and wellbeing) is calculated in diurnal charts by subtracting the ecliptic longitude of the sun from the longitude of the moon. Then the difference is added to the longitude of the Ascendant: Lot of Fortune = Ascendant + Moon – Sun. For nocturnal charts, the calculation is Ascendant + Sun – Moon.

Although this is a night-time birth chart, Benedetti has marked the Lot of Fortune for night and day.

The sign and degree occupied by the moon when crossing the ecliptic from southern to northern latitude is the ascending node (Caput Draconis). When the moon is moving in the opposite direction (crossing the ecliptic from north to south), the point of intersection is called the South Node (Cauda Draconis). The nodes are not fixed, but have a retrograde movement (a complete revolution of the nodes in the ecliptic takes 6798 days/18.61 years). To both points (which are important in the interpretation of a chart), the strength of a planet has been assigned. The Dragon’s Head is considered beneficial, the Dragon’s Tail malefic. (In Hindu astrology, the ascending node is called Rāhu and the descending node Ketu; both are considered malefic planets.[44].)

Benedetti forgot to inscribe Jupiter. Apart from this flaw everything has been calculated accurately.

6.4.2 Revolution or Solar-Return Horoscope

January 21, 1592 (Gregorian Date), 23 h 15 m 30 s p.m., φ = 45°.

Geographical coordinates of Turin according to Petrus Apianus in Cosmographicus liber (1533): f. XXXVr – Toledo 9;04 East of Porto Santo; f. XLIIr –Turin 30;30 → 21;26 East of Toledo (the reference meridian of the Alfonsine tables).

Planets | Original Source | Recalculation (Alfonsine tables) | Recalculation (Prutenic tables)
Sun | 1;27 Aq | 2;08 Aq | 1;12 Aq
Moon | 9;27 Ta | 11;20 Ta | 8;12 Ta
Saturn | 7;29 Ca retrograde | 10;30 Ca | 7;30 Ca
Jupiter | 11;44 Sa | 10;31 Sa | 11;42 Sa
Mars | 3;05 Ar | 4;23 Ar | 3;00 Sa
Venus | 25;09 Sa | 23;00 Sa | 25;08 Sa
Mercury | 9;36 Aq | 5;43 Aq | 9;25 Aq
Lunar node (asc.) | 4;54 Ca | 5;05 Ca | 4;53 Ca

6.3 Planets

Houses (Regiomontanus) | Original Source | Recalculation I (φ = 45;00; 23;15 p.m.) | Recalculation II (φ = 45;00, 23;17 p.m.)
X | 21;00 Ca | 20;30 Cp | 20;59 Cp
XI | 1;00 Aq | 10;36 Aq | 11;11 Aq
XII | 17;00 Pi | 16;26 Pi | 17;18 Pi
I | 10;49 Ta | 9;56 Ta | 10;43 Ta
II | 15;00 Ge | 14;36 Ge | 15;08 Ge
III | 5;00 Ca | 4;18 Ca | 4;45 Ca
Lot of Fortune (Day) | 18;49 Le | 16;55 Le | 17;44 Le

6.4 Houses

Fig. 6.2: Benedetti’s horoscope for Carlo Emanuele I, calculated with the Alfonsine tables, as transcribed by Cristini in Revolutione trentesima prima (1592), f. 12r. (Biblioteca Nazionale Universitaria di Torino, coll. N VII 10)

The second of Benedetti’s horoscopes to be considered is also taken from Bartolomeo Cristini, Revolutione trentesima prima del Serenissimo Signore il Signor Carlo Emanuel Duca di Savoia corrente dell’anno 1592 con ogni diligenze et fedeltà calculata et decchiarata secondo le migliori intelligenze de più principali autori dell’astrologia giundiciaria, Turin, Biblioteca Nazionale Universitaria: Coll. N VII 10, f. 12r (Figure 6.2).

This is a chart constructed for the moment in which the sun returns to the degree and minute of its longitude at nativity (i.e., transiting the position of the “natal” sun) for the respective location. A revolution horoscope indicates the course of events during the ensuing year.

Contrary to Benedetti’s caption (Figura della revolutione […] cavato dalle tavole d’Alfonso per trovar il tempo), he has obviously used the Prutenic tables for calculating the planetary positions of this chart. But the moon’s position is off by c. 1°, and the cusps of the houses deviate somewhat. Calculating with a time of 23;17 p.m. gives a reasonably good compliance, however. The cusp of house XI (1;00 Aq instead of 11 Aq) is most likely a scribal error.

It is noteworthy that minutes for an arc are only provided for the first house (i.e., the ascendant). For the other cusps, only whole degrees are noted. Benedetti simply cut off the minutes, which was a common rounding procedure at his time.

6.4.3 Natal Horoscope of Giovanni Battista Benedetti

August 14, 1530 (Julian Date), 13 h 13 m p.m., Venice; planetary positions according to the Alfonsine tables. Geographical coordinates of Venice according to Petrus Apianus (1533): Toledo 9;04 East of Porto Santo (f. XXXVr); f. XLIIr: Venice 32;30, Latitude φ = 44;50 → 23;26 East of Toledo (the reference meridian of the Alfonsine tables).

Fig. 6.3: Benedetti’s own horoscope, detailed in Luca Gaurico’s Tractatus astrologicus (1552, f. 76r). (Bayerische Staatsbibliothek)

Planets | Original Source | Recalculation
Sun | 0;43 Vi | 0;42 Vi
Moon | 17;19 Ta | 15;02 Ta
Saturn | 14;23 Ge | 14;23 Ge
Jupiter | 27;47 Vi | 27;46 Vi
Mars | 1;43 Vi | 1;45 Vi
Venus | 23;03 Vi | 23;02 Vi
Mercury | 6;25 Vi | 6;25 Vi
Lunar node (desc.) | 22;51 Ar | 22;52 Ar

6.5 Planets

Houses (Regiomontanus) | Original Source | Recalculation
X | 20;16 Pi | 20;14 Pi
XI | 28;47 Ar | 28;36 Ar
XII | 13;28 Ge | 13;17 Ge
I | 14;34 Ca | 14;23 Ca
II | 5;53 Le | 5;46 Le
III | 25;26 Le | 25;23 Le

6.6 Houses

Apart from the moon’s position (which is about 2° off) the horoscope is correct. In all three horoscopes the houses have been constructed according to the so-called “rational method,” commonly—but erroneously—attributed to Regiomontanus:[45] Circles of position joining in the north and south point of the observer’s horizon are laid at distances of 30° through the celestial equator, thus giving unequal sections of the ecliptic. This method of house division was widely used by astrologers during the sixteenth and seventeenth centuries.

Footnotes

[1] Brahe 1919, 172: “Veneris enim Stella, visa est mihi aliquando eam partem Lunae, quae a Sole aversa erat, et ipsi obiecta, satis sensibiliter illuminare, utut Luna longe remotius a Veneris circuitus distiterit, quam Cometa. Simile quid Ioannem Baptistam Benedictum, Patricium Venetum Philosophum et Mathematicum inprimis excellentem, animadvertisse reperio, in erudito illo Opere, quod de Mathematicis et Physicis speculationibus inscripsit. Sic enim in fine Epistolae, ad Baronem quendam Sabaudarum Philibertum scribens, ait: ‘[…] quod pars Lunae lumine Solis destituta, a lumine Veneris aliquantulum illustratur, quod ego saepe vidi, et multis ostendi.’”

[2] Benedetti 1585, 256–257.

[3] Brahe 1916, 251–253.

[4] Benedetti 1585, 371–374.

[5] Brahe 1916, 251: “Nunc igitur epistolam, quam pollicitus sum, subiungam, verbotenens e praedicto Baptistae Benedicti libro desumptam, una cum demonstrationum delineationibus, quas ipse author assignavit. Deinde ad caeteros qui de hac stella nimis incompetenter, sententiam tulerunt, calamum dirigam. Ex quo (uti dixi) haec adeo succinte et scite geometricam veritatem redoleant, ut nullum, quod alicuius sit momenti, super esse queat, dubium.”

[6] Gilbert 1651, 173: “Luna maculas quasi ostendit substantiae et peripheriae differentia: ita Tellus erga Lunam maculas repraesentat, terrarum continentium minus relucentium; aquarum vero et Oceani, propter laeviorem et luminis apprehensivam naturam magis splendentem. […] Non enim maculae Lunae existunt a partibus Lunae magis perspicuis, ut Iohannem Benedictus contendit, in quibus lumen non reflexum sed penetrans nobis occultatur.” See Pumfrey 2011, 193–203.

[7] Section 6.2 is a revision of Omodeo 2014a, chap. 3.8–9 and chap. 6.3 of Omodeo 2014a, chap. 4.7.

[8] This Benedetto Altavilla could be the same person involved many years later, in 1606, in a gunpowder plot in Venice; he pretended to have discovered it by astrological means and was tortured by the Venice authorities in order to obtain information about the perpetrators. Cf. Smith 1907, vol. 1, 364–365.

[9] Altavilla 1580a, f. A2r.

[10] See Omodeo 2008b and Omodeo 2012a.

[11] Altavilla 1580a.

[12] Altavilla 1580a.

[13] Altavilla 1580b, 2.

[14] Altavilla 1580b, 4–5.

[15] Altavilla 1580b, 3.

[16] Altavilla 1580b, 6.

[17] In Turin: Biblioteca Nazionale di Torino, coll. Q.V.191, and Biblioteca Reale di Torino, coll. G.25.12.

[18] Benedetti 1581, 5.

[19] Benedetti 1581, 6.

[20] See Bonoli 2012, 49–55.

[21] Benedetti 1581, 17–19.

[22] Benedetti 1581, 20.

[23] Benedetti 1581, 32–33.

[24] Benedetti 1581, 37.

[25] Benedetti 1581, 37–38.

[26] Benedetti 1585, 228–248, “Defensio ephemeridum.”

[27] Benedetti 1585, 255–256.

[28] Benedetti 1585, 255.

[29] Benedetti 1585, 255: “[…] id etiam minus putabunt hii, qui opinionem Aristarchi Samii, et Nicolai Copernici sequuntur, quorum ratione fieri non potest, ut credant eius, quod ex universo reliquum est, alium finem non habere, quam regimen huius centri [Tellus] epicycli Lunaris, ut illorum more loquar.”

[30] Benedetti 1585

[31] The same thesis is presented in Benedetti 1585, 195–196.

[32] Benedetti 1585, 255–256: “quae quidem omnia [phaenomena], cum simplici gyro terrae circa suum axem (ut dicunt) tolluntur, quod sufficit ad recipiendum lumen, et influentias illorum corporum.”

[33] Benedetti 1585, 256.

[34] Benedetti 1585: “Rationes autem a Ptolomeo in contrarium adductae apud ipsos, nullae sunt, quia quaelibet pars (ut inquiunt) retinet naturam totius, praeterquam quod aer, et aqua, quae ipsam terram circundant, plane eundem naturalem impetum motus obtineant, qui tanto lentior est, quanto longius distat aer, ab ipsa terra, secundum etiam talem opinionem, nulla necessitas, ut locus fixarum terminaretur aliquibus superficiebus, convexa scilicet, et devexa.”

[35] Benedetti 1585, 285–286, “De motu molae, et trochi, de ampullis, de claritate aeris, et Lunae noctu fulgentis, de aeternitate temporis, et infinito spacio extra Coelum, Coelique figura.”

[36] For Benedetti’s correspondence with Patrizi, see Claretta 1862.

[37] Benedetti 1585, 411: “Quod eam distinctionem orbium, quae iam invaluit, non teneas, sed putes totum esse quoddam continuum excipiens corpora stellarum, novum non est, nam nonnulli solidae doctrinae philosophi idem confuerunt.”

[38] Benedetti 1585, in the section entitled “Disputationes de quibusdam placitis Arist[otelis],” n. 38: “Occultam fuisse gravissimo Stagiritae causam scintillationis stellarum,” 186: “Scintillatio ergo stellarum, neque aspectus nostri ratione, neque alicuius mutationis earundem stellarum, sed ab inaequalitate motus corporum diaphanorum mediorum nascitur, quemadmodum clare cernitur, quod si inter aliquod obiectum, et nos, aliquis fumus, qui ascendat, intercesserit, videbimus obiectum illud quasi tremere. Hoc autem tanto magis fiet, quanto magis distabit obiectum ab ipso fumo; unde admirationi locus non erit, si stellas fixas magis scintillare, quam errantes cernamus. Lumen stellae ad oculum nostrum accedens, perpetuo per diversas diaphaneitates penetrat, medio continuorum motuum corporum mediorum, unde continuo eorum lumen variatur, et hoc in longitudinis magis, quam in propinquis stellis apparet.”

[39] Benedetti 1585, 371–374.

[40] Benedetti 1585, 197, “Disputatio XXXIX, Examinatur quam valida sit ratio Aristotelis de inalterabilitate Coeli: Aristo[teles] non consideravit, quod similiter de terra dici posset, quando ipsa ita eminus prospiceretur, imo absque dubio putandum est, quod si terra luce Solis praedita esset, et aliquis ipsam ab octavo orbe vellet videre, nullo pacto cerneret, cum sidera illa quae primae magnitudinis vocantur, et quae plusquam centies maiora ipsa terra putantur non nisi ut puncta videantur.”

[41] Benedetti 1585, 195: “Si essent dicti mundi, eorum quilibet suum proprium centrum, suamque propriam circunferentiam haberet, terraeque et ignes haberent inclinationem ad centra circunferentiasque suorum mundorum.”

[42] Here a convenient, unfortunately little-known computer program created by Peter Schiller especially for the needs of historians has been used. See Schiller 2001. There is not sufficient room here for a detailed analysis of the choice of appropriate historical parameters; for a concise description, see Oestmann 2002 and Eade 1984, 1–103.

[43] For details, see Bilfinger 1888, 262–286 and Ginzel 1914, 94–96.

[44] See Hartner 1938, 131–134

[45] It was already known in the Maghreb in the eleventh century, see Kennedy 1996, 543. For a profound treatment of the history of house division, see North 1986, although the way this text coins new designations is awkward and may lead to confusion.

7. Foundations of Physics

In this chapter we analyze Book 4 of the Diversae speculationes, entitled Disputationes de quibusdam placitis Aristotelis (Disputations on Some Opinions Held by Aristotle). We will refer to this section as Physical Disputations. Benedetti here developed his theory of motion and clarified his physical conceptions by means of a discussion and criticism of Aristotle’s physics. He dealt with fundamental concepts such as place and time. Moreover, it is here that the interdependency of physics and cosmology in his conception most clearly emerges.

7.1 Sections of the Physical Disputations

The Physical Disputations are a discussion of and an objection to Aristotle’s theses on local motion and cosmology as presented in Physics and De caelo, and partly also in Meteorologica and Metaphysica. Benedetti does not limit himself to criticism but rather seeks to provide a new approach to and foundation of physics and cosmology, beginning with the theory of motion. He calls his approach mathematical (inconcussa mathematicae philosophiae basis). In actual fact, his conceptual tools for the treatment of motion are principally derived from Archimedes’s Floating Bodies and Euclid’s Book 5 on proportions. As we have seen, the reader of Benedetti’s Diversae speculationes is provided with a brief book dedicated to Book 5 of Euclid’s Elements, directly following the Physical Disputations.[1] Benedetti moreover reworks and transforms basic concepts of physics, such as place and time, and propounds anti-Aristotelian natural views such as spatial infinity and the existence of the void. His treatment culminates with cosmological speculations, including a defense of the Copernican system.

One can conveniently subdivide the Physical Disputations into five sections (see table 7.1 below). The first twelve chapters are a lengthy discussion of Aristotle’s Physics IV 8. This section deals with the ratio of velocities of bodies moving through different media or the void. Secondly, from chapter 13 to chapter 18, Benedetti challenges Physics VII 5 on further problems linked with the theory of motion. The third subdivision (chapters 19 to 22) deals with basic philosophical matters (the void, infinity, place, and time). This section is extremely important for an overarching understanding of Benedetti as it connects the investigation of motion with a general reform of natural philosophy. Whereas Drake and Drabkin, in Mechanics in Sixteenth-Century Italy, translated the first two subdivisions (chapters 1–18), they neglected the foundational one, except for chapter 19 on the void. Their choice to expunge this part of the Physical Disputations deprived the English-speaking readership of some of the most daring pages in Benedetti’s work.

Another subdivision (chapters 23–26, entirely translated in Drake and Drabkin 1969) deals with local motion and the shortcomings of the Aristotelian theory of natural places. The fifth and last subdivision, only partly included in the English edition, presents cosmological ideas. It deals with the “sphere” as a geometrical-cosmological figure, as well as with the (apparent) motion of the sun, with stars, meteorological aspects linked to the sun, the propagation of light in the cosmos, and other issues connected with astronomy in a broad sense. The Copernican system is discussed in the second part of this last section (chapters 35 to 39), along with other innovative theses such as the plurality of inhabited worlds akin to the earth and the reciprocity of the observational points in the universe.

We will concentrate on sections 1 (on motion), 3 (on the foundations of physics), and 5, part 2 (on cosmology). As one sees from our overview (table 7.1), the Physical Disputations are no less complex and heterogeneous than the volume of which they are part, the Diverse speculationes. Therefore, we will review Benedetti’s arguments in enough depth to enlighten the thematic interdependency and his approach in general, deliberately leaving aside an excessive analysis of details that would not help to understand his intellectual endeavor as a whole.

Sections and their subjects | Chapters | Details | Presence in Drake and Drabkin 1969
1. Phys. IV 8 | §1–12 | Discussion on the void and down- and upward motion in different media | ✓
2. Phys. VII 5 | §13–18 | On the proportions of velocities, volumes, and surfaces | ✓
3. Foundations of physics revised | §19–22 | On the void (XIX ≈ Phys. IV 8), place (XX ≈ Phys. IV 4), the infinite (XXI ≈ Phys. III 5, De Caelo I,9 etc.,), and time (XXII) | Only §19
4. Violent and natural motion | §23–26 | Rejection of the Aristotelian theory of natural places | ✓
5. The sphere (geometrical and cosmological) | §27–34 | On the sphere (§29), on starry light (§30), on infinite motion (§31), on the Sun, its warmth and seasonal changes (§30–31 and §34), celestial music, and harmony (§32–33) | Only §28 and §29
| §35–39 | On the composition of circular and rectilinear motion and Copernicus’s hypotheses (§35), plurality of worlds (§36), cosmic propagation of light (§37), geometrical aspects relative to elements (§38), and relativity of the point of observation (§39) | ✓ (only §38 missing)

7.1 An overview of the Physical Disputations and of their English translation in Drake and Drabkin 1969.

7.2 An Archimedean Theory of Motion

The Renaissance rediscovery of Archimedes’s work can be seen as a crucial contribution to the theoretical advance of modern physics.[2] The wide dissemination of this ancient work produced a renewed interest in mathematical methods for the investigation of nature. In the Questiti et inventioni diverse of 1554, Tartaglia’s Archimedean perspective implied a critical approach to Aristotle’s mechanics. By contrast, Del Monte had a rather syncretistic approach merging Archimedean and Aristotelian elements. He even argued for the possibility of deriving terrestrial motion from Aristotelian premises (probably referring to geological motions rather than to an astronomical phenomenon).[3] Benedetti’s feelings toward Aristotelian philosophy are rather hostile; nonetheless, it is clear that his anti-Aristotelian theory of motion is embedded in the Renaissance discourse on natural philosophy, mechanics, and motion among mathematically-trained scholars.[4]

7.2.1 Aristotle’s Physics IV 8

Benedetti’s considerations on motion are presented as a criticism of Aristotle’s Physics IV 8. In that section Aristotle argued against the existence of the void and infinity of space and presented some reflections on the motion of bodies through different media. According to Aristotle, the void and infinity would undermine any reasonable theory of natural motion (indeed, his own theory of the natural and violent motions). His polemic is directed against “those,” probably Democritus and his followers, who held the void to be the condition of motion. Aristotle claims that in an infinite and void space there would be no absolute directions; an up- or downward motion would be conceptually impossible. By contrast, he keeps to the “observation” that the elements display natural tendencies upward or downward (water and earth downward, air and fire upward), which, according to him, falsifies the thesis of an infinite and void space. Additionally, this assumption would lead to “absurd” consequences:

But in vacancy […] nothing could go on moving unless it were carried. Nor (if it did move) could a reason be assigned why the projectile should ever stop—for why here more than there? It must therefore either not move at all, or continue its movement without limit, unless some stronger force impedes it.[5]

As far as motion is concerned, Aristotle remarks that the difference of speeds between two bodies depends either on their different weight or lightness or on the different density of the media through which they travel. He adds that the ratio of the speeds is reciprocally proportional to that of the densities, whereas it is directly proportional to that of the times. For the sake of simplicity, we could represent these relations symbolically, in a modern way. Heath, in his study on Mathematics in Aristotle, put it as follows:[6]

Furthermore:

On this basis, Aristotle concludes that motion in the void would be impossible, since “a body would move through the void with a speed beyond any ratio.”

According to Aristotle, differences depending on the weights and on the dimensions of the bodies would disappear in vacuo with “very unpleasant” consequences:

What reason can be assigned for this greater velocity [of a heavier falling body]? If the passage is through a medium, there must be such a difference [in the velocity of fall between heavier and lighter bodies]; for when there is anything there to cleave, the body superior in force of its thrust will necessarily cleave the medium faster, since either its more suitable shape or the natural thrust it exercises, whether following its natural movement or being thrown, makes it cleave the better. Where there is nothing to cleave, therefore, all bodies will move at the same velocity; which is impossible.[7]

Aristotle assumes that the speed of falling bodies is proportional to their weight (or dimension). For instance, one reads in De caelo:

His must surely be a careless mind who does not wonder how it is that a small particle of the earth, if raised to a height and then set free, should refuse to remain where it was but begin to travel, and travel the quicker the bigger it is, whereas if one held the whole Earth in the air and let it go, it would not move. But in fact, for all its weight, it is at rest.[8]

This argument was repeated by Ptolemy in Almagest I 7: “If the Earth had a single motion in common with other heavy objects, it is obvious that it would be carried down faster than all of them because of its much greater size.”[9]

In summary, Physics IV 8 provides Benedetti with a series of entangled problems involving the void, infinity, and motion. He begins his reform of physics by dealing with the last issue and then moving to a treatment of the other ones. In relation to motion through a medium, he enlists some commonly accepted assumptions (primo verissima et obiecta intellectus per se cognita) in chapter 2:

Therefore, whenever two bodies are subjected to or receive one and the same resistance to [the motion of] their surfaces, [the speed of] their motions will turn out to be to each other in precisely the same proportion as their motive forces. And, conversely, whenever two bodies have one and the same heaviness or lightness, but are subject to different resistances, [the speed of] their motions will have the same ratio to each other as the inverse ratio of the resistances.[10]

The cases that have to be considered are basically two: first, different bodies in the same medium, and second, equal bodies in different media. A decisive difference between two bodies is their gravitas (gravity, weight) or levitas (lightness). For Benedetti, gravitas and levitas act as moving forces. He calls them virtutes moventes or facultates moventes. For the sake of brevity, we will refer to them as virtus/virtutes.

If we use V like velocitas for speed, P like pondus for virtus, and R like resistentia for resistance, we can formalize the previously mentioned general assumptions in the following way:[11]

I. Case in which R is constant:

II. Case in which P is constant

7.2.2 Speed in Different Media

Benedetti regards the Aristotelian theory of motion as inadequate and prefers to rely on Archimedes. He conceives of motion as analogous to the behavior of bodies in water. Following the Floating bodies, he holds that weight and lightness are not absolute but relative properties because they depend on the medium: “quia in medio se densiore si poneretur, non grave esset, sed leve, quemadmodum Archimedes ostendit” (in fact, if it is put in a denser medium it is not heavy but light, as Archimedes demonstrates).[12] Thus, the direction of a vertical motion and, as we will see, the speed depend on the matter of a body as well as on the fluid (air, water, or whatever) in which it is merged.

In the second disputatio, Benedetti declares that the virtus (weight or lightness)[13] of a body varies depending on the densitas (density) of the medium. Here he has in mind the three different cases considered by Archimedes in Book 1 of the Floating bodies: a body the weight of which is equal to that of the fluid, lighter, or heavier. Archimedes’s seventh proposition, concerning the last case, might illustrate Benedetti’s mental model for motion in a medium:

Solids heavier than the fluid, when thrown into the fluid, will be driven down as far as they can sink, and they will be lighter in the fluid by the weight of a portion of the fluid having the same volume as the solid.[14]

The actual virtus (either weight or lightness) of a body results by subtracting from the total virtue (virtus totalis) a quantity which Benedetti calls resistentia extrinseca (resistance, for instance in disp. 9) and depends on the densitas (density, for instance in disp. 3), which we could regard as an expression indicating the specific weight:[15]

virtus in medio = virtus totalis – resistentia

Accordingly, he holds the position against Aristotle that the ratio of the speeds in different media is not inversely proportional to the densities of the media but directly proportional to the virtutes in different media.

Given this equation, Benedetti shows (chapters 3–4) that the thesis of Physics IV 8, namely that the ratio of the speeds is reciprocally proportional to that of the densities of the media, is only true in a special case, which can be constructed ad hoc: if the ratio of an object’s weight (or the corresponding speed) in one medium (for instance air) to its weight in another medium (for instance water) is equal to the ratio of the first medium (air) to the second (water). However, it is incorrect to claim that the ratio of the speeds of all bodies moving through different media remains the same. In chapter 6, Benedetti demonstrates moreover that the ratios of the weights (or speeds) of a body through different media are not maintained as constant in different media (Quod proportione ponderum eiusdem corporis in diversis mediis proportiones eorum mediorum densitatum non servant. Unde necessario inaequales proportiones velocitatum producuntur).[16]

We should now add some considerations on Benedetti’s argumentative strategy. He begins chapter 2 with a reference to principles generally taken for granted: primo verissima et obiecta intellectus per se cognita. The adverb “primo” can have two meanings in this context: it could indicate either that the author is proposing the “first” principles of the motion theory or, quite on the contrary, that he is presenting theses that are universally true “only at a first glance.” These “very true assumptions,” these primo verissima, are in fact the Aristotelian statements concerning the proportion of the ratios of velocities and weights as well as the proportion of the ratios of velocities and resistances. In the second part of chapter 2, Benedetti brings forward his own conception of the proportion between the ratio of velocities and that of weights in a medium. He begins the paragraph on his theory with these words: “Aliud quoque supponendum est.” This incipit can be interpreted in two different and rather opposite ways: as “Another proposition must also be presupposed” (which is Drake and Drabkin’s translation) or as “We must presuppose a different proposition.” As a matter of fact, the thesis that follows these words, that on the subtraction of the resistances from the total virtutes, is meant as a general truth. Benedetti himself intends to show that the Aristotelian laws are not true universally but only for special cases. Benedetti presents his theory of motion in opposition to that of Aristotle, so we tend to interpret chapter 2 as an opposition between two contrasting perspectives. Yet a certain ambiguity in his words cannot be denied. Perhaps it is a rhetorical strategy consciously employed by Benedetti: he first presents to the reader some propositions with which natural philosophers should be familiar, and then leads his reader to reject these common statements as not universally true rather than absolutely false.

7.2.3 Motion in pleno and in vacuo

After these considerations on motion in different media, Benedetti deals with the motion of bodies with different dimensions (but of the same material) through a medium as well as through the void. Chapter 7 considers the case in which two bodies A and O, made of the same matter and having the same figure, traverse the same medium. According to Aristotle the ratio of their speeds should be directly proportional to that of their weights. Benedetti demonstrates that this is false, since one could imagine a third body U, made of a different material, with the same virtus (here: gravitas) as O and the same dimensions (magnitudo et figura) as A. If V is the speed, M the magnitudo corresponding to a certain weight (or more in general, a virtus), then:

(according to Aristotle)[17]

(case of two bodies with equal dimensions and different weights).

Since by assumption MO = Mu, it follows that Vu = VO, but this is not consistent with the assertion that two bodies of equal weights and different dimensions will move with different speeds in a medium because a bigger body needs more “force” to penetrate the medium than a smaller one:

The body which, when compared to the other, is of equal weight or lightness, but is subject to smaller resistance, moves [in natural motion] more swiftly than the other in the same proportion as its surface is subject to a smaller resistance than that of the other body, by reason of its being able more readily to divide the continuity of the air or water.[18]

Therefore, the ratio of the velocities of two bodies having equal material composition and figure but different dimensions is untenable, pace Aristotle.

Chapter 9 introduces the issue of motion through the void, asserting, against Aristotle, that the ratio of two bodies in pleno and in vacuo cannot be the same:

In a plenum the ratio of the external resistances in the case of these bodies is subtracted from the ratio of the weights, so that the ratio of the speeds remains. And this last ratio would be annuled if the ratio of these resistances were equal to the ratio of the weights.[19]

The argumentation is not immediately intelligible to the modern reader. Contrary to appearance, in fact, this passage does not describe the following subtraction: \frac{P_1}{P_2} - \frac{R_1}{R_2} (V is the speed, P the weight, and R the resistance).

According to the theory of proportions, the addition and subtraction of a ratio to or from another ratio can be conveniently represented in a modern fashion as follows:[20]

I. Meaning of ratios addition:

II. Ratios subtraction:

Given this, the relation indicated by Benedetti in the above-mentioned passage can be rendered through the following symbolic expression:

This relation expresses in a concise form the Aristotelian perspective concerning the relation of velocities, weights, and resistances, as already taken up by Benedetti at the beginning of the disputatio number two. Benedetti notices that the ratio of the velocities is annuled (nulla esset) if the ratio of the virtutes is equal to that of the resistances ( \frac{P1}{P2} \sim \frac{R1}{R2}). In fact, in this case (and only in this case) the speeds of bodies with different weights would be the same. As a consequence, there would be “no proportion” between different quantities but rather an equality of speeds. Benedetti indicates that this special case of motion in pleno, in which bodies with different weights have equal speeds, is the rule for motion in vacuo.

Chapter 10 expands on motion in vacuo. It presents the famous thesis that bodies of the same material move with the same speed in vacuo, although they might have different dimensions. Benedetti remarks that a body O can be divided into halves A and E, each of the same weight (or virtus). If an ideally weightless bar connects them, the weight at the center of the connection should be the sum of the partial weights and thus equal O. Therefore, bodies composed of the same material will fall with the same speed, independently of their weight. In chapter 11, one reads that motion in pleno is different as a consequence of the friction of the medium, except for the special case in which the parts travel through media whose resistances have “the inverse proportion of the weighs” (si duo corpora… suas resistentias ad invicem proportionatas haberent, ut sunt eorum pondera).[21]

7.2.4 A Note on the Historical Relevance of Benedetti’s Reflections on Motion

Our analysis of the first part of the Physical Disputations should be sufficient to understand Benedetti’s approach to motion and the scope of his investigation. Subdivision two tackles Aristotle’s Physics VII 5 and deals with the proportions of velocities, volumes, and surfaces.[22] We are not going to analyze in further detail Benedetti’s views on motion. Instead, we deem it important to consider the “philosophical” part of the Physical Disputations (which we have indicated as section 3), as it introduces novel prospects on the foundations of physics. It extends from chapter 19 to chapter 22 and is an attempt to revise basic concepts of physics from an anti-Aristotelian perspective.

Before we tackle these foundational aspects, we would like to recount the influence that Benedetti’s theory of motion exerted on the young Galileo. We have already hinted at Benedetti’s triangulation with Galileo via Del Monte. At a theoretical level, one of the strongest pieces of evidence of this influence is the affinity between Galileo’s early speculations on motion and the theses that Benedetti propounded in the Diversae speculationes. This link is evident and well known, beginning with the hydrostatic analogy to explain the motion through a medium, the relativity of heavy and light, and the subtraction of resistance from weight, which allows motion through a vacuum to be accepted and makes it physically plausible.[23] For instance, several chapters of Galileo’s first manuscript among those gathered by Favaro under the title of De motu are very close to Benedetti’s treatment; above all Galileo’s chapter 8, “in which it is shown that different bodies moving in the same medium maintain a ratio [of their speeds] different from that attributed to them by Aristotle”; chapter 10: “in which, in opposition to Aristotle, it is proved that, if there were a void, motion in it would not take place instantaneously, but in time”; chapter 12: “in which, in opposition to Aristotle, it is concluded that the absolutely light and the absolutely heavy should not be posited; and that even if they existed, they would not be earth and fire, as he believed”; and chapter 15: “in which, in opposition to Aristotle, the conclusion is reached that rectilinear and circular motions have a ratio to each other.” This connection between the work of the young Galileo and Benedetti’s insights on motion is significant, the more so since Galileo’s patron Del Monte was skeptical about the possibility of quantifying motion.[24] Actually, in his manuscripts, Del Monte took some annotations on falling bodies in different media. This might have been an issue of discussion between him and Galileo.[25]

Moreoever, it should be remarked that the Copernican cosmological element is absent from Galileo’s early manuscript De motu, although this would become a crucial aspect of his later investigations. Also, the alliance of mechanics and Copernican astronomy, which emerged only later in Galileo, bears witness to Benedetti’s influence on his work.[26]

7.3 On the Void: Atomistic Prospects

Benedetti’s considerations on motion are followed by a treatment of the void which, from the perspective of Aristotle’s physics, is directly connected with the former issue. His theory of motion through media and through the void is the basis upon which he casts Aristotle’s rejection of the void into doubt. Chapter 19 of the Physical Disputations, Quam sit inanis ab Aristotle suscepta demonstratio quod vacuum non detur, is a transition from the Archimedean theory of motion (chapters 1–18) to the reconceptualization of natural philosophy in general. The discussion on the void is directed against Aristotle’s Physics IV 8, in which the theory of motion serves to reject the physical void and infinity. Benedetti claims that the void is conceptually possible as a consequence of his previous demonstration that the Aristotelian assumptions on the relation between speed and density of the medium are wrong or, at least, not universally valid. “Ex iis, quae superius demonstravimus facile cognosci potest irritam esse eam rationem, quam Aristoteles 8 cap. lib. 4 Physicorum ad destruendum vacuum, confinxit” (From the demonstrations above it is easy to see that the argument that Aristotle devised in Physics VIII 4 is vain).[27] According to Benedetti’s Archimedean dynamics, motion through a void is not absurd at all. He explains that such a motion is simply quicker than one taking place through any medium, because no resistance has to be subtracted from the body’s virtus.

Hence, Benedetti reintroduces the concept of the void into physics, which was excluded by Aristotle as a criticism of Democritean atomism.[28] Benedetti’s move is in line with the rehabilitation of atomistic philosophy during the Renaissance and the early modern period. The void and atoms are the two ingredients of the same matter theory inspired by the rediscovery of Lucretius and a reassessment of Epicureanism.

Among Benedetti’s contemporaries the most committed natural philosopher supporting atomism was Bruno. Although they probably developed their theories independently of each other, they have many points of contact. In his Italian philosophical dialogues (published in London in the years 1584–1585), Bruno widely drew on ancient atomistic doctrines. He called Democritus and Epicurus those “who contemplated nature with open eyes” (con occhi più aperti han contemplata la natura).[29] He derived from them (often relying on Lucretius) a vision of the universe as infinite, the idea of a countless number of worlds (similar to the solar system), as well as the principle of cosmological homogeneity and the atomic theory of matter and of minima moving through a void. In De l’infinito universo e mondi, he celebrated Epicurus’s theory of matter, summarizing it as follows:

Epicure similarly nameth the whole and the universe a mixture of bodies and of the void; and in this universe and in the capacity thereof to contain the void and the empty, and furthermore in the multitude of the bodies contained therein he maintaineth that the nature of the world, which is infinite, doth exist.[30]

However, the concept of the void assumes in Bruno a very special meaning since, in his philosophical terminology, the terms ‘space,’ ‘void,’ and ‘aether’ are used as synonyms. The ethereal void is the medium that makes the motion of bodies possible.[31] In the second dialogue of De l’infinito, he recapitulates the following three meanings of vacuo:

First, the void refers to all which is not bodily and thus does not have the property of resisting penetration. In this sense, there is a “boundless ethereal region” (eterea regione inmensa), in which the worlds are plunged. The countless worlds populating the universe are themselves composed of matter and the void.[32]

Second, the void has the meaning of infinite space (spacio infinito). In it everything is included. It is the container that cannot be included in anything else.

Third, space can be assumed to be nothingness, in accordance with Aristotle. Bruno calls this a metaphysical meaning, or rather lack of meaning. According to him, this extra-physical meaning was a sophism that served Aristotle to refute the possibility of the void in nature. For Bruno, only the first and the second meaning of the void make sense. They could and should be embraced in natural philosophy. By contrast, the metaphysical void has to be rejected as meaningless.

Bruno’s reflections on the physical void are part of his program to revive an atomistic conception of nature. He combined it with cosmological infinity and the post-Copernican idea that a plurality of worlds exists, each one of them a solar system centered on a star similar to ours. This vision, which shows similarities to Benedetti’s, paved the way for seventeenth-century cosmologies, in particular for the Cartesian multi-centric universe.

Other Renaissance scholars supported combinations of atomism and heliocentric astronomy as well. A case in point is the circle of scientists gathered around Henry Percy of Northumberland in London. Just like Benedetti and Bruno, they brought together heliocentrism, atomism, an empirical and mathematical approach to nature, Renaissance naturalism, and anti-Aristotelianism.[33] Thomas Harriot, for example, was an atomist and a supporter of the infinity of the universe.[34] Nicolas Hill, another member of the “Northumberland circle,” authored an apology of Epicureanism entitled Philosophia Epicurea (1601), which was directly inspired by Bruno. It included Copernican arguments in favor of terrestrial motion and many others in favor of the earth’s magnetism, in accordance with Gilbert.[35] Following ancient and modern atomists, Hill affirmed the boundlessness of the universe and the plurality of worlds.[36]

Atomism was to be embraced by celebrated exponents of sixteenth-century science and philosophy such as Galileo and Pierre Gassendi. At the same time, corpuscular alternatives were spread by Descartes and his followers.[37] Atomism and, more generally, corpuscular theories of matter encountered much censorship, especially owing to theological difficulties, since they appeared to be irreconcilable with Scholastic accounts of the Eucharist.[38] Nonetheless, many modern scholars were attracted, just as Benedetti was, to the prospect of connecting the advances of mechanics with a physics and a theory of motion rooted in corpuscularism alongside a post-Copernican cosmological outlook. As Galluzzi has recently pointed out, this was a guiding idea for Galileo already in his Iuvenilia.[39] Theories of motion, atomism, and Copernican astronomy are three interconnected issues in his as well as Benedetti’s work. In De motu, Galileo was committed to the homogeneity of matter and reflected on it in connection with motion, in particular with the Benedettian thesis that “si vacuum esset, motuum in instant non contingere, sed in tempore” (If the void exists, motion does not occur instantaneously but rather in time).[40] In the tenth section of this manuscript, Galileo followed in Benedetti’s footsteps and came to the same conclusion: “Hence it follows, not that motion in a void is instantaneous, but that it takes place in less time than the time of the motion in any plenum.”[41]

Benedetti shared many ideas with contemporary and later scholars in his criticism of Aristotle’s natural philosophy as well as in his effort to construct a new physics. Whereas he developed his conceptions on matter and the cosmos independently of Bruno, and probably did not directly influence the English circle of Harriot and Hill, nonetheless he must have influenced the views of Italian scholars such as Galileo who were familiar with the Diversae speculationes. Benedetti showed them that a revision of dynamics could not be fulfilled independently of a broader program of philosophical reform.[42]

7.4 On Place: Space as intervallum corporeum

Chapter 20 of Benedetti’s Physical Disputations deals with the concept of place. Aristotle, in Physics IV 4, defines it as the surface in which a body is included and denies that it could be conceived as an intervallum. By contrast, Benedetti defines it precisely as intervallum corporeum, an expression which could be roughly translated as an inter-bodily gap. Aristotle says that locum (the place) and locatum (the placed) must be coextensive but, as Benedetti notices, a spherical surface contains more “extension” (intervallum) than any other figures with an equal surface. As a consequence, one can imagine two places (in the sense given to the term by Aristotle) occupied by bodies of different dimensions or, the other way round, two bodies of different dimensions which suit the same surface. By contrast, an intervallum of space contains only bodies of equal volume, thus respecting the bijective relation between the volume of the place and that of the placed body: “But equal bodily extensions [intervalla] delimitated by any figure will always contain equal bodies.”[43] The definition of place as intervallum has the advantage that it allows us to assign a place to every body, “in the world or outside it, pace Aristotle.”[44] This remark anticipates the cosmological treatment of infinite space, or of the possibility of space beyond our worldly system (mundus). In chapter 20, on place, Benedetti remarks that Aristotle is wrong when he rejects the concept of place as intervallum because it could lead to infinity. Benedetti objects that this is not inconvenient at all, since one could conceive of infinita loca. To grasp this infinity is no more difficult than to understand the never-ending process which Aristotle calls “potential infinity” (infinitum in potentia) and concerns the division of a body ad infinitum:

That infinite places must exist, if place is [conceived of as] intervallum, is not inconvenient. In fact, just as any body can be [divided into] infinite bodies (which he [Aristotle] would call “in potentia”), so can any intervallum be [divided into] infinite intervalla as well.[45]

Benedetti is always brief in his treatment of philosophical issues and leaves much implicit. Therefore, the reader is challenged to reconstruct the complete vision implied in his short statements and critical points. Reference to Renaissance philosophical sources discussing the same theses from an Aristotelian viewpoint can help us to better understand Benedetti’s conception. In this case, Alessandro Piccolomini’s refutation of the conception of space as a receptacle of bodies can shed light on the opposite viewpoint defended by Benedetti. The following quotation is taken from the first part of Piccolomini’s Della filosofia naturale (On natural philosophy), a very successful introduction to natural philosophy in Italian, which Benedetti might have known:

Other philosophers (and not unimportant but very reputed ones) asserted that there is void space among natural beings and that it is not only distinguished in essence [per sua natura] from the bodies it accommodates, but also it is so separated from them that it can remain completely naked and really deprived of them. Furthermore, many supposed that such a space (which is completely void, without any bodies) is mixed and situated between the beings of this world and their parts. Additionally, they believed that it infinitely extends outside the heavens. Thus, these philosophers considered the void to be that being which not only can be deprived of all bodies and substances […] but also is apt and capable of receiving the bodies, giving them a place (something which does not occur to lines, points, and other accidents). One of the main reasons for holding such doctrines was, as they said, that, if there is no space in nature (or a void place without bodies), the motion from place to place would be impossible, although the motion of alteration would be possible. In fact, all beings, in order to move to some other place, need some in-between space through which they can move. If this space were not void, it would be necessary that, as it is filled with other bodies, different bodies would penetrate each other while crossing that full space. And this is impossible.[46]

Thus Piccolomini presented the views of the supporters of a natural void in order to reject them. Yet his discussion is helpful as it indirectly presents us with the views of such thinkers as Benedetti who defended void and infinity. Benedetti was in fact favorable to the notions of the void and infinity. For him, space is prior to and independent of bodies. It should be remarked that, in Benedetti’s perspective, infinite space does not imply cosmological infinity or the boundlessness of the material universe, precisely because space is independent of matter. In this respect, his conception is different from the one that Bruno defended in those years. Rather, it is closer to that of his correspondent Patrizi. As De Risi has pointed out, Patrizi’s conception of space is marked by its ontological autonomy from bodies. It is in fact an “incorporeal and immaterial extension, tridimensional and infinite, which receives and precedes all created bodies.”[47] Patrizi saw space as a sort of Plotinian hypostasis (extensio hypostatica), that is, a pre-corporeal quantity quantifying reality. He saw this idea as the lever to be employed in order to radically reform Scholastic metaphysics, a project that he developed in a series of publications, De spacio physico et mathematico (On physical and mathematical space, 1587), Della nuova geometria (On the new geometry, 1587), and eventually in the Nova de universis philosophia (1591).[48] It also served him to set the foundations of an epistemology of mathematics according to which the ancient science dealing with figures had to be substituted for a new science of space itself. This is the concept of his New Geometry. Patrizi sent a copy of this book to Benedetti, probably before the publication, to gain the favor of Carlo Emanuele I, to whom the book was dedicated.

To sum up, Benedetti defines the locus, against Aristotle, as intervallum corporeum, or bodily extension. He regards it as an invariable empty extension capable of being occupied by material bodies, an idea that becomes clearer from the related discussion on time. He basically rejects the Aristotelian definition of place as the superficies corporis ambientis (surface of the containing body), remarking that the place is a measure, actually an intervallum, and not a surface. Moreover, he explains that only his understanding of locus as a kind of objective space guarantees that two bodies of equal material constitution necessarily occupy the same quantity of “place” as intervallum or empty extension, independently of their form. By contrast, the definition given by Aristotle is inconsistent with the assumption of a bijective relation between locus and locatus. In fact, as geometry shows, if one accepts Aristotle’s definition “a great absurdity [maximum inconveniens] would follow, namely that equal places can contain unequal bodies or equal bodies can occupy unequal places.”[49] Additionally, Benedetti remarks that his concept of place admits an infinite universe, since it is capable of containing infinita corpora (infinite bodies).

7.5 In Defense of Infinity

Chapter 21 is devoted to infinity: Utrum bene Aristoteles senserit de infinito (Whether Aristotle judged correctly about infinity). Needless to say, Benedetti holds that Aristotle’s arguments on this issue are idle. The Greek philosopher rejects the possibility of an infinite body (infinitum corpus), that is to say, the infinity of the universe because—as the argument goes—there is no locus infinitus which could include it. Benedetti protests that this is a petitio principi since this rejection is based on a questionable definition of locus: “cum Aristoteles debuerit beneficio loci destruere infinitum, ordine perverso de infinito prius” (Since Aristotle had to destroy infinity with the benefit of [a correct understanding of] place, he wrongly started with [a discussion of] infinity).[50] By contrast, Benedetti’s redefinition of place as intervallum entails no conceptual hindrances to the acceptance of the infinity of the universe. As one reads: “hoc modo nullum inconveniens sequeretur, quod extra caelum reperiri possit corpus aliquod infinitum” (In this manner it is not inconvenient to assume that one can find an infinite body outside the heavens).[51]

Additionally, Aristotle writes (Physics VIII 8 and De caelo I 9) that a continuum can include infinite parts only in potentia (potentially) and not in acto (in act). Benedetti does not agree with him. He argues that, if the continuum is in acto, its infinite parts should be in acto as well, because it is foolish (stultum) to believe that something actually existing could be composed of parts which exist only potentially (quae potentia existunt).

The weakest argument brought forward by Aristotle is, according to his critic, that the infinite cannot be considered to be a quantity (Physics III 5), because only a finite quantity is thinkable, for instance the dimensions of a square or a cube. Benedetti objects that the definition of number (which he does not provide, however) does not include finitude (necessitas terminorum). Numbers are not necessarily terminated (determinati) and one can conceive an infinite multitude exactly like a finite one:

Aristotle’s arguments in the same part of Physics III 5 are even worse. He denies that infinity can be regarded as a quantity by saying that only one defined quantity is intelligible, such as a cubit, a three-cubit, etc. Thereby, he does not consider that in the same manner the quantity of infinite cubits is intelligible as well. Moreover, the definition of quantity does not imply limitation by necessity. For instance, the definition of number does not imply the necessity of any determined number, since an infinite multitude is not less intelligible than a finite.[52]

The last false Aristotelian assumption on infinity is the equation of infinitum and vacuum in Physics IV 8. Benedetti’s commentary is harsh: “Later, in Physics IV 8, he says that there is no difference between infinity and the void. Indeed, he could not assert and imagine anything more absurd than this.”[53]

Like Bruno, the infinitist par excellence, in De l’infinito universo e mondi (1584), Benedetti remarks that Aristotle’s arguments are not compelling. Aristotle denied the possibility of an unbounded space on the basis of a finite cosmology (i.e., the theory of the spherical and geocentric heavens along with the theory of the natural places), which excludes cosmological infinity by definition. Yet his assumptions are not self-evident.

In summary, in chapter 21 of the Physical Disputations, Benedetti defends the mathematical and physical possibility of the infinite. The question of the title “Whether Aristotle Judged Correctly about Infinity” is rhetorical. It calls for a negative answer.

7.6 On Time: Toward an Absolute Frame for Physics

Benedetti’s definition of tempus (time) is closely connected with that of locus (place). He deals with it in Physical Disputations, chapter 22, Exagitatur ab Aristotele adducta temporis definitio (Rejection of Aristotle’s Definition of Time). Benedetti questions the definition of time as motus mensura numerusque (the measure and number of motion) and offers an alternative conceptualization. But before we discuss his opinion, we will recount standard conceptions of time from antiquity.

It should be noted as a preliminary remark that the understanding and definition of time was regarded as a natural as well as a psychological issue. This should not be surprising, since the doctrine of the soul, or psychologia, was an integral part of natural philosophy, or physica, in standard university curricula. A standard reference on time was the Timaeus, where Plato called time, rather poetically, the “moving image of eternity” (εἰκὼ κινητός αἰῶνος).[54] According to his myth, the divine Demiurge created time together with the heavens (οὐρανὸς), making them inseparable. Plato’s pupil Aristotle, in Book 4 of the Physics, then proposed the definition of time as the “dimension of movement in its before-and-afterness.”[55] He also noticed that χρόνος (time) had generally been connected with the motion of the celestial sphere and was referred first and foremost to the heavens because all measurements of motion and time depend on celestial regularities:

Neither qualitative modification nor growth nor genesis has the kind of uniformity that rotation has; and so time is regarded as the rotation of the sphere, inasmuch as all other orders of motion are measured by it, and time itself is standardized by reference to it.[56]

For both philosophers, Plato and Aristotle, the tie between time and cosmic order was unquestionable. This interconnection was later refuted by a third authoritative source. At the downfall of the ancient world, Augustine, in Book 11 of the Confessiones, denied the commensurability of time and local motion: “The motion of a body is one thing, and that by which we measure how long it is, another thing.”[57] He pointed out the transient character of the temporal flux and noticed that the three dimensions of time are a product of our mind (animus). In the mind, time is constituted as the memory (memoria) of the past, the intuition (attentio) of the present, and the expectation (expectatio) of the future.[58] In other words, Augustine underscored the subjectivity of time, conceived of as a distensio animi, a “stretching” of the mind independent of heavenly cycles.

According to Benedetti’s criticism of Aristotle, the definition of time as “the measure and number of motion” is intrinsically wrong because measuring presupposes commensurability. But because time and motion are heterogeneous, they cannot be compared. A line is measured by a line, a surface by a surface, and a corpus—that is, a three-dimensional body—by a corpus. Similarly, motion is measured by motion and not by time: “Time cannot be the measure of motion, but only motion can measure motion, precisely a faster one [measures] a less fast one, and a shorter one [measures] a longer one, whereas a number is measured by a number, and time by time inasmuch as it is long or short, and not inasmuch as it is fast or slow.”[59] Hence, time can only be measured extrinsically (per accidens) through motion, as is the case with common expressions like “two hours, or two days, or two years,” referring to astronomical displacements. These are only metaphors that refer to heavenly motions as “placed” in the interval of time that corresponds to their motion. In the following passage, presenting these reflections, time is called the “place of motion” (locus motus):

It could seem to somebody that, to indicate [significare] a certain quantity of motion, one has to assume as much time as if one says, for instance, that a certain operation has been carried out in the space of two hours, or two days, or two years. Still, it should be remarked that this is not literally [simpliciter] true, since the imagination conceives the interval of two hours, days, or years as the motion of celestial bodies without which neither years, not days, not hours would exist, even though time is placed, so to speak, in time, just as a body in a place. Thus, motion is measured by motion and time by time, and not the one by the other.[60]

Benedetti claims that time, unlike motion, is eternal: “Necessarily, from a philosophical point of view, time is eternal and motion not.”[61] Whereas a motion can be extinguished and a body can be at rest after a displacement, time goes on inexorably. It is always present to our senses and always escapes them because it is the never-ending passing of a single instant. Benedetti makes clear that he intends the instans to mean “one in species” (unum in specie), i.e., the essence underlying the vanishing flux of time that cannot be experienced in itself as a given and measurable succession (non in numero).[62] This Augustinian paradox helps Benedetti to stress the heterogeneity of time and motion. Still, he does not renounce an objective meaning, which is essential to his dynamics. Motion and time, he writes, are continua successiva, continuous and successive quantities. Their relation can be conveniently described as that between a place and the bodies that it contains. His explanation sheds light on the concept of locus as well as on that of tempus:

Just as a dense body occupies a lesser interval of place than a less dense [rarus] body, similarly a quick motion is accomplished [peragetur] in a shorter space of time than a slower motion.[63]

From this passage it is clear that Benedetti’s idea of place as intervallum corporeum goes in the direction of an empty homogeneous space which can be occupied by bodies of greater or lesser density. Time has the same absolute character as place. It can contain more or less rapid motions without being affected by them. Space and time or, more precisely, intervallum corporeum and tempus are objective and independent coordinates of natural phenomena.

In order to understand Benedetti’s considerations on time, it is useful to delve into the views of his immediate forerunners, in particular Cardano and Scaliger.

7.6.1 Cardano’s Subtleties on Time

In the letter to the reader of the Diversae speculationes, Benedetti indicated Cardano as one of his principal sources.[64] However, concerning the concept of time their opinions are not in agreement. Cardano tackles the issue tempus quid sit (What is time?) in Book 18 of De subtilitate. Unlike Benedetti, he does not explicitly question Aristotle, but rather quotes his definition as a common truth: “in fact, motion is the measure of time” (motus enim tempus est mensura).[65] Still, he reworks the Aristotelian concept within a rather Augustinian perspective, which leads to original results.

He introduces the problem of time in connection with his treatment of the dream (somnium), beginning with the question: “Why does time appear much longer while dreaming than in reality?”[66] He reports that once he dreamed that he visited an unknown city far away from his home in Milan. He traveled through mountains, valleys, and fields. In order to cover that distance six days of travel would be barely sufficient. Thus, when he woke up, he thought that he had slept for a long time but in actual fact his nap had taken less than one hour. The reason for this expansion of time, Cardano explains, is due to the fact that dreamed activities (operationes) are accomplished independently of any bodily effort (absque corporis labore) and therefore very rapidly. A correct judgment about time depends on bodily movement. This is why the perception of time is not altered in our mind when we imagine long-lasting processes while awake. “During sleep, time is contracted in the opposite manner than if we are awake: in fact, motion is the measure of time.”[67] Surprisingly, Cardano’s Aristotelian conclusion, “motion is the measure of time,” does not refer to heavenly motions, but to corporeal activity. In other words, he extrapolates and isolates the peripatetic definition from its original context. In fact, from an objective, physical, and cosmological meaning, time acquires a rather subjective meaning, connected with physiology and perception. Time, as Augustine would say, is an “expansion of the mind.”

It should be added that there is a passage of the Physics where Aristotle also hinted at the subjective dimension of time, even at how the soul grasps it. This passage might have been a source of inspiration for Cardano:

Time cannot be disconnected from change; for when we experience no changes of consciousness, or, if we do, and are not aware of them, no time seems to have passed, any more than it did to the men in the fable who ‘slept with the heroes’ in Sardinia, when they awoke; for under such circumstances we fit the former ‘now’ to the later, making them one and the same and eliminating the interval between them, because we did not perceive it. So, just as there would be no time if there were no distinction between this ‘now’ and that ‘now,’ there appears to be no time between two ‘nows’ when we fail to distinguish between them. Since, then, we are not aware of time when we do not distinguish any change (the mind appearing to abide in a single indivisible and undifferentiated state), whereas if we perceive and distinguish changes, then we say that time has elapsed, it is clear that time cannot be disconnected from motion and change.[68]

These considerations are not aimed at isolating a subjective meaning of time by eliminating any objective references. Still, it is plausible that Cardano took into account such passages in order to freely speculate on time, in Book 18 of De subtilitate, from a perspective that owes more to Augustine than to Aristotle.

This Augustinian influence is particularly evident from the following passage, in which Cardano tries to better define time:

But what is time? Although nothing of it is ever [given], nonetheless everything is in it and it accompanies [assistit] everything always. It itself generates and destroys everything; it is the source [auctor] of life and death. Its expectation is always very long, while its memory is always very short. Although it is always with us, we never grasp it. Even though there is such an abundance of it, nonetheless no restoration [reparatio] of it is ever conceded, thence the waste of no other thing is greater or worse.[69]

In this passage, Cardano brings together ideas derived from erudite lectures, as well as from commonsense, experience, and even trivial commonplaces. Additionally, he recounts the Augustinian paradoxes: time is everywhere and nowhere, it is made out of expectation and memory, and it is for us the most familiar and unknown mental presence. To quote from the Confessiones: “What is time then? If nobody asks me, I know: but if I were desirous to explain it to one that should ask me, plainly I know not.”[70] Cardano adds to this paradox a popular sense of the caducity of life, according to which a discourse on time is a kind of memento mori. Time itself is said to be the cause of life and death.

Hence, for Cardano, the assumption that “time is the measure of motion” does not mean maintaining the cosmological dependency of time on astronomical cycles. Quite to the contrary, time transcends motions and changes. In fact, we do not perceive it in itself but rather that which happens in it. What we know about time is a product of the mind, precisely of its imaginative faculty:

Thus, we do not comprehend time but rather that which occurs, or occurred, and endures in it. But time itself per se is out of reach [ignotum] to the senses. In fact all that we know [about it] is constituted through imagination.[71]

Like Benedetti, Cardano denied the interdependency of cosmological space and time, although from a different standpoint. He was not concerned with reformulating the physical space and time framework of motion in mathematical terms. Instead, he concentrated on the psychological and physiological dimension of time as a construction of the imaginatio. Therefore, this treatment was connected with that of the mind and was introduced by considerations on sleeping and dreaming. This even led to considerations on altered states of mind such as ecstasies and hallucinations, as well as the divine and demoniac visions of the hermits—Cardano went so far as to report some hallucinations that he had when he was a child. His mental treatment of time, as well as this excursus on altered states of mind, were both harshly criticized by the Flemish humanist Julius Caesar Scaliger, as we shall now discuss.

7.6.2 Scaliger’s Aristotelian Restoration

Scaliger conceived of his Exotericae exercitationes as a critical confrontation with Cardano’s De subtilitate, although he formally presented them as a reverent discussion of some points that arose from the lecture on the “subtleties” of that doctissimus vir, as one reads in the dedicatory epistle, “who will never be praised enough” (nunquam satis laudatus). Among the most notable theses indicated in the Index acutiorum sententiarum at the end of the book, one is directly relevant to our discussion: the exercitatio number 352.2, which explains why “time is not the measure of motion.” That exercitatio deals with the passage of De subtilitate on time that we have discussed, but reverses Cardano’s viewpoint.

Scaliger tries to answer the question of “whether time is the measure of time,” remarking that most people just repeat this definition without properly understanding its meaning.[72] The allusion is clear: Cardano repeats a commonplace without any thorough reflection on its meaning. In fact, even though time might be regarded as the measure of “our motion,” it is definitely not the measure of worldly motion, in particular not of the first motion of the heavens, i.e., the daily one.[73] Scaliger therefore denounces Cardano’s subjective reading of the Aristotelian definition. According to him, the objective dimension of time cannot be dismissed. From a cosmological-ontological perspective it is in fact a dependent (affectus) of heavenly motion;[74] to be precise, it descends from the “first motion,” or the daily rotation of the starry sphere. The first body (primum corpus), that is, the heaven of the fixed stars, is the measure of all bodies. Similarly, its motion, the primus motus, is the yardstick of all motions. In accordance Book 12 of the Metaphysica, where Aristotle advocates a spherical and geocentric cosmology of concentric spheres, Scaliger states that “time is either the same thing as motion or its affection.”[75] In other words, he intends to restore an Aristotelian objective conception of time in accordance with a metaphysical perspective that attaches an ontological priority to motion over time.

Scaliger acknowledges that Aristotle ambiguously defined the interrelation between time and motion by accepting both possibilities: “time is the measure of motion and, in turn, motion is the measure of time.”[76] In his attempt to dispel this paradoxical interdependency, Scaliger distinguishes between two fundamental dimensions of time: the objective and the subjective dimension or, as one reads, the understanding “according to nature” (a Natura) and that “according to us” (mensura nostra). In nature, motion is the measure of time as well as its source. By contrast, time is the measure of motion only for us, which is an aspect that Cardano allegedly overemphasizes. This is clearly documented by expressions like “the equinoxial circle accomplishes its motion in this much time.”[77] In actual fact, things are the opposite of what our way of speaking suggests: time is derived from the translation from “here” (ubi) to “there” (ubi) in space (in loco). The ubi (where), to which people improperly refer in order to express a quantity of time, is a naive way of thinking that reduces time to certain spatial determinations. As to the definitions: time is a transient “now” (nunc), whereas place (locus) is a continuous “where” (ubi):

Since the quantity of time corresponds to that of a motion between a ‘here’ [ubi] and a ‘there’ [ubi], the ‘where’ [ubi] that we use for time is deduced [transumptum est], without inconvenience, from that ‘where’ [ubi] which, in fact, pertains to [a determination of] place [locus]. Actually, time is a transient now, whereas place is an enduring where [ubi].[78]

Scaliger’s criticism of Cardano is not limited to a vindication of the objective meaning of time, its interconnection with space (or more accurately, place), and the priority of cosmological motions over time, i.e., time as something derived from celestial motions. He additionally criticises Cardano’s hint that time is the cause of generation and corruption, dealing with the question of “Whether time generates and corrupts [things]” (An tempus generet et corrumpat). In Scaliger’s assessment—and according to the Aristotelian authority on which he relies—this is impossible. Time cannot generate or corrupt anything, since it is not a substance but a quantity, that is, a property of a substance. Therefore, it cannot produce any effects.[79] Scaliger opposes his “very subtle” (subtilissime) opinion to Cardano’s superficiality: “Our life is the act of the soul. In it, time has neither jurisdiction nor power.”[80] Scaliger adds that Cardano’s words “are suited to the vulgar” (vulgo proprior), not to philosophers. What’s more, his references to visions and alternative states of mind should be regarded as only insignificant phenomena which concern children and melancholici like Cardano himself.[81]

7.6.3 Benedetti and the Renaissance Concepts of Time

Our excursus on Renaissance concepts of time is far from exhaustive, but it helps us to grasp the scientific context out of which Benedetti’s own position emerged. It also permits us to point out some major problems in the conceptualization of time, in particular its subjective and objective dimensions. The interrelation between tempus and locus was at the center of the reflections, the debates, and even the polemics of scholars investigating nature from various angles. The debate on time and on its relation to motion has meanings that are, at the same time, physical (Benedetti), psychological-physiological (Cardano), and cosmological-metaphysical (Scaliger). Finally, for a more traditional cosmological perspective on time as the measure of celestial motions and of the motion of the first mobile as the standard measure for all other measurements of time, one can refer to Alessandro Piccolomini’s Della filosofia naturale, where one finds the following definition:

Hence, time, which is the measure of all movements, mainly has to measure a motion that is the most regular and enables the measurement of all other [motions] that do not have the same regularity in their components. As this motion is that of the first heaven, one has to conclude that time (which is like [a property] of a substance [in proprio soggetto],) is first of all measured by it through that motion of the first mobile. Through it all other motions are ruled. Hence, although time can be called the rule and measure of any motion, it will be reasonable not to pluralize it alongside the plurality of motions. Rather, it will remain one and the same for the whole world, just as the first motion, which is its proper and true subject [soggetto], is singular.[82]

In the generation before Benedetti, Cardano affirmed that time is independent of cosmological space on the basis of his assumption that time is a “stretching of the mind” connected with the perception of corporeal activity. Cardano formally accepted the traditional Aristotelian definition of time as the measure of motion, but only as the consciousness of physiological motion. Scaliger criticized this perspective and accused Cardano of misunderstanding Aristotle. In the Exercitationes he tried to restore a peripatetic metaphysical conception of time as a product of celestial motions. In a sense, his conception of time has a “conservative” character for his commitment to the Aristotelian tradition. However, the distinction of the subjective and objective dimensions of the issue permits us to highlight a problematic aspect of the Renaissance definition of time. According to Scaliger, time can be regarded as the measure of motion only from a subjective viewpoint, not from a natural one. According to nature, the relation between time and motion is the reverse: motion is the measure and time the measured thing.

For Benedetti, as a mathematician advancing a new Archimedean theory of motion and a post-Aristotelian physics, time is the locus motus, the place of motion, that is to say, an objective—we dare say, absolute—measure independent of its content and of spatial determinations. He shared with Cardano the independence of time from matter but not his subjectivism, while he maintained with Scaliger the objectivity of time without assuming the Aristotelian dependency of time on motion.

7.7 Natural and Violent Motions Revisited

After his revision of the physical concepts of the void, infinity, space, and time, Benedetti moves on to discard the Aristotelian theory of natural and violent motions in the section from chapter 23 to chapter 26. Chapter 23, “Motuum rectum esse continuum, vel dissentiente Aristotele” (Rectilinear motion is continuous although Aristotle is in disagreement), shows that continuity is not exclusive to circular motion, therefore Aristotle’s distinction between circular and rectilinear motion must be corrected. In Physics VIII 8 one reads that “the motion of a body moving on a finite straight line cannot be continuous.”[83] By contrast, Benedetti demonstrates that the projection of a circular motion has the same continuity as the circular motion itself, although it is a forward and backward motion on a line. This is the case with planetary appearances produced by the displacement onto an epicycle as seen from the earth.[84] In a way, this section is a rehabilitation of the epicyclic model against Averroist-Aristotelian criticisms concerning the physical tenability of a non-concentric model of heavenly circles. This contrast between mathematical and physical astronomy received much attention from sixteenth-century Aristotelian scholars who received their education at Padua. In the 1530s Giovan Battista Amico and Girolamo Fracastoro promoted a fleeting rebirth of homocentric astronomy through the publication of De Motibus corporum coelestium iuxta principia peripatetica, sine eccentricis et epicyclis (Venice, 1537 and Paris, 1540) and Homocentrica (Venice, 1538), respectively.[85] More directly, Benedetti’s assertion that a continuous rectilinear motion is possible seems to be directed at the opposite statement in Piccolomini’s Aristotelian Della filosofia naturale (Book 2, Part 2, chapter 3, “Come esser non possa infinito corpo alcuno di quelli che per natura loro si muovano per retta linea” (There can be no infinite body among those that move straightforeward following their own nature)). Just as in the Diversae speculationes, this section is complemented with diagrams.[86]

Chapter 24 is a refutation of a series of Aristotelian assumptions on natural and violent motion. In the title of this chapter, Aristotle is called vir gravissimus, but this attribution sounds quite ironic, since the theses of this “very grave man” are here refuted. The first criticism concerns the idea that a projectile is transported by air once separated from its thrower. According to Benedetti, it is rather the contrary: air is a hindrance to motion because it resists the penetrations. Secondly, Aristotle writes in De caelo I 8 that a body accelerates the closer it is to its aim. Instead, one should say that a body moves (e.g., falls) quicker the further it is from its aim (e.g., the ground). In fact, the longer the distance it covers, the more it is pushed (maior fit semper impraessio) by its inner impetus, which is due to the spontaneous inclination toward its proper place (inclinatio ad locum suum eundi). Chapter 26 is a rejection of Aristotle’s statement that a body is not “heavy” in its proper place. In fact, air in air, or water in water, has no weight at all, except for when one artificially compresses an element (for instance inflating air in a bladder). The difference in density of the compressed element produces a difference in weight. Chapter 25 denies that vertical motion could legitimately be called natural. In fact, only perpetual circular motion is natural. An entire (i.e., spherical) body and its parts spontaneously move in circles. By contrast, rectilinear motion is that of a part separated from its whole. The cosmological significance of these remarks should be stressed. It is implicit but can be demonstrated by comparison with Copernicus’s De revolutionibus I 8, where the same distinction between the motion of the whole and that of its parts accounts for the difference between the natural circular motion of a planet, basically the earth, and the vertical fall of heavy bodies.[87]

Copernicus presented his considerations on natural and violent motions in De revolutionibus I 8, which is the chapter aimed at rejecting Ptolemaic and Aristotelian arguments against terrestrial motion. There Copernicus attacked the Aristotelian theory of natural and violent motion and sought to abandon the doctrine of natural places. Benedetti’s undertaking is very close, even though the cosmological theme has not emerged yet.

7.8 The Cosmological Perspective of the Physical Disputations

The cosmological dimension of Benedetti’s anti-Aristotelian discussion is documented in the last part of the Physical Disputations.

7.8.1 Physico-Mathematical Astronomical Issues

The astronomical-cosmological section begins (chap. 28) with a reflection on the sphere that goes against the opinion of Aristotle, ironically called Princeps Peripateticorum. Whereas the ancient philosopher regarded the circle as the “first plane figure” (prima figurarum superficialium) and the sphere, the form of the heaven, as the “first body” (prima corporearum figurarum, that is, the first three-dimensional figure), Benedetti claims that they are the “last” figures. In fact, they can be regarded as polygons with infinite sides: “the triangle is the first plane figure and the circle the last one.”[88] He adds that the principium, the beginning and the origin of everything, is its center and those figures which equally encircle it in all directions can be said to be perfect. The author concludes as follows: “That which is perfect, although it is [qualitatively] first as to its essence [natura], is last as to its generation.”[89] The circle, according to him, is perfect because it is, in a sense, an “infinite figure.” If one considers it as a polygon of infinite sides, one can say that the sum of its angles is equal to an infinite number of right angles. One can interpret this statement as follows: every polygon inscribed in a circumference can be divided into several triangles whose vertices touch the center of the circumference and whose bases coincide with the sides of the polygon. In the case of the circle, taken as the “last” polygon, the triangles decomposing it are infinite in number. Since the angles at the vertices are zero and the sum of all internal angles must be 180°, it follows that the angles at the base must be two square angles. Thus, Benedetti feels vindicated: “The circle and the sphere are not constituted of one single angle, as Aristotle believes […]. Rather, these are figures of infinite right angles. For that reason I call them last and perfect, because one cannot add anything to infinity.”[90] To sum up this reasoning, Benedetti shares Aristotle’s opinion that the sphere is the perfect figure, but adduces geometrical-metaphysical reasons. For both authors the sphere is the form of the world (Benedetti would say, “of the caelum” surrounded by an infinite empty space) for aesthetic and metaphysical reasons.

In the following chapters, Benedetti reviews a series of astronomical and meteorological issues on which he accused Aristotle of being wrong. Chapter 28 deals with the sparkling of the stars, which is, according to Benedetti, the effect of the motion of heavenly transparent media (ab inaequalitate motus corporum diaphanorum mediorum nascitur).[91] Among other things, Benedetti denies (chap. 30) that the warmth of the sun can be produced by its motion rather than by its light and subsequently (chap. 31) explains the seasonal variations. Chapter 33 reassesses, against Aristotle, the plausibility of the Pythagorean doctrine of celestial harmony. This has nothing to do with the production of sounds, nor with any harmonic proportions between the “aspects” of the heavenly bodies. Rather, it is the secret order imparted by to the world by divine providence.[92] Chapter 33 comprises a lengthy discussion on meteorology, in which atmospheric phenomena are essentially explained through the variations of air density.

7.8.2 The Copernican Conclusion of the Physical Disputations

Chapter 35, Motum rectum curvo posse comparari (Straight and curvilinear motions are comparable), is a crucial chapter for our analysis, since it is here that Benedetti, almost at the end of his Physical Disputations, introduces the Copernican theory. Although we have already dealt with Benedetti’s astronomical views in the previous chapter, it is useful to recount here the most important features of his cosmology in the context of the philosophical section of the Diversae speculationes (IV) and add some more details.

The subject matter is the comparability of rectilinear and circular motion: “[Aristotle] is wrong when he says that straight motion cannot be compared to the curvilinear (Physics VII 4), where he mistakenly also says that one cannot find any lines equal to the circumference of a circle.”[93] It is directed against Aristotle’s denial that a straight and a circular motion could be compared, thus hinting at the qualitative difference between celestial circular motions and the vertical tendency of the elements in the sublunary sphere. From a Copernican perspective, Aristotle’s words could be considered to be an implicit rejection of terrestrial motion. In fact, if the earth rotates, one should assume that the trajectory of a falling body is rectilinear for an observer on the earth but has a circular component as well, if considered in relation to the outside world.

Benedetti first appeals to Archimedes’s De quadratura circuli (On the quadrature of the circle) to argue that the circle and the straight line are comparable: “If, then, this quadrature can exist, there can also exist, for the reason already given, a straight line equal to the circumference of that circle.”[94] Thus, a geometrical problem, the squaring of the circle, attains a direct cosmological meaning. If the issue at stake is the distinction between celestial and elementary motions, they are of course different, but this difference does not lie in the circularity of the former and the straightness of the latter, but rather in the uniformity of speed opposed to acceleration.

These considerations offer Benedetti the occasion to expand on the velocity of celestial motions. According to the commonly held opinion (secundum opinionem cummunem), the heavens would have to cover an immense distance within the 24 hours of the daily rotation. Close to the equator, the sun would cover 1,000 Italian miles per minute and Saturn 260,000 miles per minute, not to speak of the rapidity of the fixed stars. The assumption of this inconvenient velocity would of course be avoided if one assumed “the most beautiful theory” (pulcherrima opinio) of Aristarchus, “divinely” restored by Nicolaus Copernicus:

And as for the speed of the fixed stars situated near the equator, one may make one’s estimate, and, in fact, this will seem very difficult to some. But this difficulty does not occur in the most beautiful system of Aristarchus of Samos that has been so divinely expounded by Nicolaus Copernicus.[95]

From a Copernican perspective, the sun would cover “only” 48 miles per minute and Saturn 24, whereas the heavens would be stationary.

In the subsequent chapter (chap. 36), Benedetti reworks the doctrine of the doctissimus Aristarchus. It is entitled “Minus esse explosam ab Aristotele opinionem credentium plures mundos existere” (The view of those who hold that many worlds exist was not adequately refuted by Aristotle) and deals with the plurality of worlds. According to Aristotle, a universe with a plurality of worlds similar to the earth would be unstable and eventually collapse, since the earthly parts of the other worlds would fall toward the cosmological center and the fiery parts would eventually become part of the fiery sphere of our sublunary world. This Aristotelian objection is based on an a priori assumption of the theory of the natural places. It is therefore easy for Benedetti to contradict him by arguing that all worlds (that is, planets) would have their elements and their places.[96]

Apart from that, as we have already seen, Benedetti proposes a bizarre transformation of the Copernican system based on an analogy between the moon and the other planets. Like our satellite, all these light-mirroring and wandering bodies are supposed to turn around dark earths which, in turn, spin about their axis:

If the system of the learned Aristarchus is correct, it will be perfectly logical for that which takes place in the case of the Moon to take place also in the case of any of the five other planets. Thus, just as the Moon with the help of its epicycles revolves around the Earth as if on the circumference of a certain other epicycle of which the Earth is like a natural center (i.e., in the middle), carried around the Sun by the sphere of annual motion, so too may Saturn, Jupiter, Mars, Venus and Mercury revolve about some body situated in the center of their major epicycle. And this body, also having some motion about its axis, may be opaque, possessing conditions like those of the Earth, with conditions on the epicycle in question similar to those on the lunar epicycles described.[97]

This conception could provide an explanation for the existence of planetary epicycles, whose physical tenability has been already stressed in the disputatio 23. We could also regard these views of Benedetti as a cosmological reading of Copernicus focused on possible cosmological and physical consequences of the planetary theory. The plurality of worlds and the analogy between the moon and the planets are not the only innovative elements in comparison with the theses of De revolutionibus. After a section on the motion of light through the cosmic void (chap. 37) and one on the geometry of the elements (chap. 38), the conclusive section of the Physical Disputations (chap. 39) attacks a Peripatetic dogma: the unalterability of the heavens. In De caelo I 22 Aristotle remarked that no change in the heavens was ever observed. This is, according to Benedetti, not a valid argument. One should rather assume a principle of relativity of the point of observation. In fact, the earth would be invisible from the eighth heaven (that of the fixed stars), even though, by supposition, it was endowed with a light equal to that of the sun. The distance hinders us from perceiving changes that occur on other worlds.[98]

With this rejection of the distinction betwee a sublunary and a heavenly realm, Benedetti’s criticism of Aristotelian physics is complete. It should be noticed that this final objection hits at the core of the Peripatetic natural philosophy, since it is a denial of the fundamental distinction between a terrestrial and a celestial physics, on which the entire physics and cosmology of the Aristotelians relies.

7.9 An Evaluation: Benedetti’s Path to Natural Philosophy

The Disputationes de quibusdam placitis Aristotelis is a complex book within the larger book. It concerns at least three main fields of investigation: motion, the foundations of physics, and astronomy, in particular cosmology. It begins with a rejection of the theory of the natural places (violent and natural motion) based on an Archimedean relativization of heaviness and lightness as well as on a mathematical approach derived from the Euclidean theory of proportions. It deals subsequently with basic concepts of physics. It defines space and time anew as an absolute framework for the investigation of natural phenomena, in particular motion. This part of the Physical Disputations also aims at demonstrating actual infinity and the void, which are Democritean theses rejected by Aristotle in Physics and De caelo. The astronomical part then follows, which confronts many special issues and illustrates what we shall call a “post-Copernican cosmology.” Benedetti advocates the heliocentric system (albeit modified relative to the model of Copernicus’s De revolutionibus), the plurality of worlds, the inter-changeability of the observational viewpoint in the universe, and, last but not least, the homogeneity and continuity of the sublunary and the heavenly realm, contrary to one of the most fundamental assumptions of Aristotelian philosophy.

Let us summarize the Copernican considerations that could have influenced Benedetti and consider the extent to which he went beyond them. First of all, Copernicus abandons the theory of natural and violent motions because, “if anyone believes that the Earth rotates, surely he will hold that its motion is natural not violent.”[99] Additionally, the daily rotation of the heavens is more absurd than that of the earth because it would be excessive compared to that required of the relatively small earth. A third Copernican remark concerns the infinity of space. It is directed against the Aristotelian assumption that there is nothing, “no space, no body, no void,” outside the heavens (dicunt quod extra caelum non esse corpus, non locum, non vacuum). Copernicus remarks that the axial rotation of the earth undermines the strongest argument in favor of cosmological finiteness: “For the chief contention by which it is sought to prove that the universe is finite is its motion.”[100] As to the objections against the earth’s motion based on considerations of the effects to be expected for flying and thrown objects, Copernicus assumes, against Aristotle’s claim for the simplicity of motion, that things on Earth participate in the planetary motion and, therefore, the vertical displacement of light and heavy bodies (cadentium vero et ascendentium) is a composite motion (duplicem) relative to the whole (mundi comparatione), with a rectilinear and a circular component. Copernicus holds that only circular motion is natural and it does not only pertain to celestial bodies but also to the elements in their natural place. As we have seem, he defines rectilinear motion as the tendency of bodies to reach their whole if they have been separated from it. This vertical appetency is not uniform but accelerated. Copernicus also criticizes Aristotle’s opinion that bodies are heavy (or light) in their proper place, since weight depends exclusively upon the tendency of the part towards their whole.

Many of these Copernican ideas and suggestions appear also in Benedetti’s Physical Disputations: the rejection of the theory of natural places and of violent and natural motions, the excessive rapidity of the rotation of the heavens, the void, infinity, the naturalness of circular motion against the unnaturalness of the vertical motion of the parts separated from their whole, and the criticism of Aristotle’s assertion about the weight of the bodies in their natural place. However, it should be remarked that Copernicus does not expand on these ideas for the most part and cursorily presents them only for the sake of his apology for terrestrial motion. Benedetti’s treatment is much more explicit and, what is more, his motivations and presuppositions appear to be quite different. His Archimedean and Euclidean treatment of motion is the basis of his rejection of the distinction between natural and violent motions. No consideration of this kind is present in Copernicus’s work. Moreover, the reference to spatial infinity in De revolutionibus is limited to a remark. Copernicus himself does not explicitly support this thesis and leaves the discussion to the natural philosophers or, as he calls them, the physiologi. Actual infinity receives a substantially different treatment in Benedetti since it is closely related to the attempt to define space anew as intervallum corporeum. It is precisely this broad, natural philosophical dimension which is absent in Copernicus’s work and which, in our opinion, Benedetti did not derive from his reading of De revolutionibus or from general astronomical concerns. It seems, by contrast, that he was primarily interested in the physical issue of a mathematical treatment of motion and that the criticism of the Aristotelian philosophy led him in a quite natural way to also confront cosmology. Nor could issues like the void and atomism be reasonably derived from Copernicus. Even the planetary theory of Benedetti departs from De revolutionibus as it includes theses like the plurality of worlds and the corruptibility of the heavens. However, it is clear that the heliocentric and geokinetic theories fit perfectly into Benedetti’s worldview. In light of his general theory, as he writes, Aristotelian and Ptolemaic arguments against Copernicus’s theory appear extremely weak: “contra quam [doctrina] nil plane valent rationes ab Aristotele, neque etiam a Ptolomeo propositae.”[101] Koyré wrote that Bruno’s La cena de le Ceneri (London, 1584) was the best defense of the Copernican system from a natural and physical point of view before Galileo’s Dialogo sopra i due massimi sistemi del mondo (Florence, 1632). However, this statement seems to underestimate the force of Benedetti’s Physical Disputations, which are perhaps less speculative than Bruno’s dialogues but should nonetheless be regarded as an extremely strong apology for the physical tenability of the Copernican system. A reciprocal influence between Bruno and Benedetti cannot be excluded, since the wandering philosopher from Nola stayed for a period in Turin and the Savoy around 1578 and probably participated in a debate concerning the comet of 1577.[102] At any rate, the Diversae speculationes encountered much more acknowledgment among astronomers of the time than Bruno’s work. As we have seen, Brahe extensively quoted Benedetti both in his Epistolae astronomicae of 1596 and in his book on the nova of 1572. Kepler’s admiration for Benedetti was no less and was only equaled by his admiration for Commandino and Clavius.[103] The proximity of many themes of the Diversae speculationes and those of the young Galileo are a well-known issue in the history of mechanics; in light of our discussion, it is plausible to assume that Benedetti’s influence on Galileo also concerned the insight into the close relation between the heliocentric theory and a new mechanics.[104]

Our analysis has shown that the heliocentric system is not the main issue at stake in the Physical Disputations, although that theory becomes part of a general program of reform for natural philosophy. Far from being a mere “Copernican enterprise,” Benedetti’s visionary project is much more complex. It is an ambitious attempt to build a new physics, in the wide Renaissance meaning of the term, out of a criticism of Aristotelian physics. Concerning Aristotle, it is clear that the princeps peripateticorum provides him with a model, albeit a negative one, in which the theory of motion, cosmology, astronomy, meteorology, natural philosophy, and metaphysics are closely interrelated. Benedetti’s undertaking is precisely a revision and a restructuring of these interdisciplinary ties on the basis of new insights and a mathematical approach. Although his investigation intentionally and explicitly departs from Peripatetic physics, it is historically possible only in the form of a thorough confrontation with Aristotelian themes. Indeed, the Physical Disputations have the form of a dispute on Aristotelian places. Benedetti’s familiarity with Aristotle’s Physics, De caelo, and Meteorologica should also be underscored. This apparently contradictory aspect of early modern physics in its ambiguous relation to Aristotelianism has already been stressed by Anneliese Maier in her studies on the medieval contributions to classical science.[105] In a sense, the development of a new physics required a thorough confrontation with Aristotle and his concepts, as also the examples of Bruno and Galileo bear witness to in different ways.

Footnotes

[1] Benedetti 1585, 198ff..

[2] Paolo D’Alessantro and Pier Daniele Napolitani have emphasized the impact of its sixteenth-century reedition in their studies on the Latin reception of Archimedes: “Nel 1544 usciva a Basilea l’editio princeps greco-latina di gran parte dei testi di Archimede oggi noti. Senza tema di esagerare, si può sostenere che questo avvenimento – al pari della pubblicazione del De revolutionibus orbium coelestium di Niccolò Copernico, apparso l’anno precedente a Norimberga – abbia fornito impulso alla nascita della scienza moderna.” D’Alessandro and Napolitani 2012, 9.

[3] See Omodeo 2015.

[4] As has been argued in Exploring the Limits of Preclassical Mechanics, the wider conceptual framework of Archimedean theories of motion was in fact deeply rooted in Aristotelianism. See Damerow et al. 2004.

[5] Aristotle 1995, IV 8, 215a 17–22 (351).

[6] Heath 1949, 116.

[7] Aristotle 1995, IV 8, 216a 17–21 (357).

[8] Aristotle 1986, II 13, 294 a (223–225).

[9] Ptolemy 1984, 44.

[10] Drake and Drabkin 1969, 198. Cf. Benedetti 1585, 169: “Quotiescunque igitur duo corpora unam eandemque resistentiam ipsorum superficiebus, aut habebunt aut recipient, eorum motus inter seipsos eodem plane modo proportionati consurgent, quo erunt ipsorum virtutes moventes: et e converso, quotiescunque duo corpora unam eandemque gravitatem, aut levitatem, et diversas resistentias habebunt, eorum motus inter seipsos eandem proportionem sortientur, quam habebunt eorum resistentiae converso modo […].”

[11] It should be remarked that this algebraic rendering of Benedetti’s thoughts does little justice to contemporary efforts to represent functional dependencies among different magnitudes with the help of the available mathematical techniques, such as compound proportions, as will become clear from the following discussion; see also Damerow et al. 2004.

[12] Benedetti 1585, 170.

[13] In this passage, Benedetti employs the word pondus, but in the discussion he also talks of “lightness”; therefore, we prefer to use the term virtus movens or, more briefly, virtus, which is the term used by Benedetti.

[14] Dijksterhuis 1956, 376.

[15] See Benedetti 1585, 174: “dictis corporibus subtrahitur proportio resistentiarum extrinsecarum.” For a discussion of Benedetti’s employment of the termini ‘pondus,’ ‘densitas,’ and ‘resistentia,’ see Helbing 1987, 155–168.

[16] The details, the text, and the notes in the English edition by Drake and Drabkin are clear enough, so we will not expand on these aspects here.

[17] It corresponds to the above-mentioned equation: \frac{V1}{P1} = \frac{V2}{P2}.

[18] Drake and Drabkin 1969, 198. See Benedetti 1585, 169: “Corpus illud quod alteri comparatum, aequalis erit ponderis, aut levitatis, sed minoris resistentiae, existet velocius altero, in eadem proportione, cuius superficies resistentiam suscipit minorem ea quae alterius est corporis, ratione facilioris divisionis continuitatis aeris, aut aquae […].”

[19] Drake and Drabkin 1969, 205. See Benedetti 1585, 174 (emphasis is ours): “In pleno dictis corporibus subtrahitur proportio resistentiarum extrinsecarum a proportione ponderum, ut velocitatum proportio remaneat, quae nulla esset, si dictarum resistentiarum proportio, ponderum proportioni aequalis esset, et hanc ob causam diversarum velocitatum proportionem in vacuo haberent ab ea, quae est in pleno.”

[20] The same concept of addition and subtraction of ratios can be found in the manuscripts of Thomas Harriot, who called them “compositio” and “subductio” (or “compositio contraria”). See for instance Schemmel 2008, 635, reference letters g and o.

[21] See Drake and Drabkin 1969, 206, n. 119: “Benedetti asserts that the speeds are equal only in the void, on the ground that in a plenum there would be an additional frictional resistance that would disturb this equality (unless this resistance itself were proportional to the weights of the bodies: Ch. 11).”

[22] See Helbing 1987, 162.

[23] Apart from Koyré 1986, see also Drabkin, “Introduction” to Galilei 1960, 9–10.

[24] The ongoing debate between Galileo and Del Monte on the possibility of developing a mathematical dynamics is seen in Galileo’s letter of November 29, 1602: “V.S. Ill.ma scusi la mia importunità, se persisto in voler persuaderle vera la proposizione dei moti fatti in tempi uguali nella medesima quarta del cerchio; perché essendomi parsa sempre mirabile, hora viepiù mi pare, che da V.S. Ill.ma vien reputata come impossibile: onde io stimerei grand’errore e mancamento il mio, s’io permettessi che essa venisse repudiate dalla di lei speculazione, come quella che fusse falsa.” See Galilei 1968, 97.

[25] See Tassora 2001, 281–283.

[26] Drabkin 1964, Drake 1976, and Renn and Damerow 2012, 142–155.

[27] See Benedetti 1585, 179.

[28] On the medieval debates on the void and on the historical place of Galileo, and Benedetti before him, see Grant 1981, 60–66.

[29] Bruno 2000a, 374.

[30] Bruno 1968, 272–273.

[31] See Del Prete 1999, 61 and Michel 1962.

[32] Bruno 2000a, 348: “[…] perché questo spirito, questo aria, questo etere non solamente è circa questi corpi, ma ancora penetra dentro tutti, e viene insito in ogni cosa.”

[33] Kargon 1966.

[34] On his intellectual stature and achievement, see Schemmel 2008, who stressed that Harriot’s unpublished manuscripts reveal that his research activity was similar to that of Galileo in the same years. Moreover, Harriot’s reflections on infinity and the minimum bear witness to his familiarity with the work and ideas of Bruno. See Fox 2000 and Henry 1982.

[35] Hill 2007, 155–157. See Plastina 1998, Plastina 2004 and Omodeo 2014a, 372–377.

[36] For an overview of English science in that period, see Hill 1997, 15–76 and 118–200.

[37] On medieval and early-modern corpuscular matter theories, see Lüthy, Murdoch, and Newman 2001. On Galileo, see Galluzzi 2011.

[38] See Redondi 1983, chap. 7, 203–226. Also see Ariew 2013.

[39] Galluzzi 2011, 9.

[40] Galilei 1968, vol. I, 276.

[41] Galilei 1960, 47. See Galilei 1968, vol. I, 282: “[…] et ita non est necessarium, motum in vacuo fieri in instanti, sed in tempore minori quam sit motus in quovis pleno.”

[42] Thus, Galluzzi’s remarks on the scientific production of the young Galileo could be conveniently extended to Benedetti (Galluzzi 2011, 19): “Il laboratorio del De motu antiquiora servì a Galieo per prendere definitivamente coscienza che la cosmologia e la fisica aristoteliche non potevano essere riformate con interventi limitati ad ambiti di indagine circoscritti. In particolare, lo scritto pisano evidenziava la raggiunta consapevolezza dell’importanza strategica di una radicale riforma della concezione aristotelica del movimento. Era questo, tra l’altro, l’ostacolo più serio da superare perché potesse essere accolta la visione copernicana, intesa non come semplice ipotesi matematica, ma come dottrina fisica. […] Né si può escludere che abbia tratto anche da esse [da tali problematiche] l’impulso per dedicarsi, con l’impegno proprietario esibito dai documenti dell’attività giovanile, all’impresa ambiziosa di una radicale riforma della concezione tradizionale del movimento e della struttura della materia.”

[43] Benedetti 1585, 180: “Sed intervalla corporea aequalis a quavis figura terminata, continebunt semper corpora aequalia.”

[44] Benedetti 1585, 181: “Et hoc modo nullum est corpus, quod in mundo aut extra mundum (dicat autem Aristoteles quicquid voluerit) locum suum non habeat.”

[45] Drake and Drabkin 1969, 198. Cf. Benedetti 1585, 180: “Quod si locus intervallum esset, infinita loca existerent […] inconveniens [non] existit, quia eodem plane modo quo aliquod corpus potest esse infinita corpora (quod ipse diceret in potentia), sic etiam intervallum aliquod posset esse infinita intervalla.”

[46] Piccolomini 1585, I 3, chap. 5, f. 44r: “Altri filosofi sono stati poi, et non già minimi, ma di gran fama, che han voluto che tra le cose della natura, si trovi spatio voto, non solamente distinto per sua natura dai corpi ch’egli riceve; ma intanto separato da quelli, che ignudo in tutto, et attualmente spogliato ne può restare. Né son mancati molti, che cotale spatio, voto in tutto d’ogni corpo, han posto, non solo meschiato, et interposto tra le cose che sono al mondo, e tra le parti loro; ma ancora fuor dei Cieli, infinitamente han creduto che si distenda. Questi filosofi adunque per il voto intendevano quella cosa, che non solo fusse privata d’ogni corpo et d’ogni sostanza […] ma fosse ancora atta, et capace, a ricevere i corpi, et dar lor luogo, il che delle linee, et de punti, et altri accidenti, non avviene. Tra le ragioni principali, che gli spingeva a credere una tal cosa, l’una era questa nella qual dicevano, che se non si potesse trovare tra le cose della natura spatio alcuno, o luogo voto d’ogni corpo; allora, se bene il movimento dell’alteratione potrebbe forse restare al mondo, tuttavia il movimento di luogo a luogo, non potrebbe senza ’l voto restar giammai. Conciosia c’havendo bisogno quella cosa, che ha da muoversi ad alcuno altro luogo, di alcuno spazio di mezo, per il quale si muova; se cotale spatio non fosse voto, saria forza che essendo pieno di altro corpo, nel passar per quello spatio pieno, venissero a penetrarsi più corpi insieme; adunque è cosa al tutto impossibile.”

[47] De Risi 2014, 282.

[48] De Risi 2014, 276–277.

[49] Benedetti 1585, 180: “maximum inconveniens sequeretur, scilicet aequales locos capere inaequalia corpora, aut corpora aequalia, locos inaequales occupare.”

[50] Benedetti 1585, 181.

[51] Benedetti 1585, 181.

[52] Benedetti 1585, 181: “Sed peius etiam sensit Aristotles eodem loco capitis quinti lib. 3 Physicorum, negando infinitum posse connumerari inter quantitates, dicens unam aliquam quantitatem intelligi ut cubitum, tricubitum, et cetera; ubi non considerat eadem etiam ratione intelligi posse aliquam quantitatem infinitorum cubitorum, et in quantitatis definitione nullam esse necessitatem terminorum, ut exempli gratia in definitione numeri, non est necessitas alicuius determinati numeri, quia multitudo, non minus infinita, quam finita, intelligi possit.”

[53] Benedetti 1585, 181: “Ubi postea cap. 8 lib. 4 Physicorum ait nullam esse differentiam inter infinitum, et vacuum, revera nihil absurdius hoc dicere fingereve poterat.”

[54] Plato Timaeus 37C–39E

[55] Aristotle 1995, IV 11, 220 a 25–26 (395).

[56] Aristotle 1995, IV 14, 223 b 21–22 (425).

[57] Augustine 1988, XI 14, 265: “aliud sit motus corporis, aliud quo metimur quamdiu sit.”

[58] Augustine 1988, XI 28.

[59] Benedetti 1585, 183: “Tempus non erit mensura motus, sed motus quidem potest mensurare motum, videlicet velocior minus velocem, et brevior longiorem; et numerus mensuratur numero, et tempus tempore in quantum longum est, aut breve, non in quantum velox, aut tardum.”

[60] Benedetti 1585, 182: “Si alicui videtur, quod ad significandam aliquam quantitatem motus, dicere huiusmodi operationem duarum horarum, aut duorum dierum, aut duorum annorum spatio completam esse, sit ponere tantum tempus, animadvertere debet hoc simpliciter non esse verum, quia horarum, dierum, et annorum intervalla, imaginatione concipiuntur ut motus corporum caelestium, sine quibus neque anni, neque dies, neque horae existerent, etiam si omnis motus sit (ut ita dicam) locatus in tempore, ut corpore in loco, unde motus motu, et tempus tempore, non autem aliud ab alio mensuratur.”

[61] Benedetti 1585, 182: “Tempus ex necessitate, philosophice tamen loquendo, res est aeterna, motus non item.”

[62] Benedetti 1585, 182: “Tempus igitur potius locus motus erit dicendum, quam numerus aut mensura eius, et tale est, ut consumatum videatur a continuo quodam fluxu unius instantis […] et cum dico ab uno instanti, unum in specie, et non in numero intelligo, quod a sensibus nostris percipi non potest, neque etiam notari, quia novum semper instans nobis occurrit.”

[63] Benedetti 1585, 183: “Quemadmodum corpus densum occupat minus intervallum loci, contra quam fiat in corpore raro; sic etiam motus velox breviori temporis spatio peragetur, quam tardus.”

[64] Benedetti 1585.

[65] Cardano 1966, vol. 3, 651.

[66] Cardano 1966, vol. 3, 651: “Cur somnium tempus longius multo ostendat quam sit.”

[67] Cardano 1966, vol. 3, 651: “Contraria ratione tempus in somnio contrahitur, cum vel non somniamus […]: motus enim tempus est mensura.”

[68] Aristotle 1995, IV 11, 218 b 20–219 a 1 (383).

[69] Cardano 1966, vol. 3, 651: “Sed quid tempus est? Cuius cum nihil unquam sit, omnia tamen in illo sint et semper omnibus assistit. Illud idem omnia generat et occidit, auctor vitae ac mortis. Utque illius expectatio longissima, ita semper memoria brevissima. Cumque nos semper comitetur, nunquam ipsum tamen agnoscimus. Nec cum eius tanta sit copia, reparatio tamen ulla conceditur: unde fit, ut nullius alterius rei iactura sit maior et vilior.”

[70] Augustine 1988, XI 14, 238: “Quid est ergo tempus? Si nemo ex me querat, scio; si querenti explicare velim, nescio.”

[71] Cardano 1966, vol. 3, 651: “Ergo nos non tempus, sed quod in eo fit, factumque est, atque manet, comprehendimus. Tempus vero ipsum per se est sensui ignotum: nam imaginatione constat quod a nobis cognoscitur.”

[72] Scaliger 1557, f. 458v: “Hoc quidem ab omnibus iactatur: pauci introspectant.”

[73] Scaliger 1557, f. 458v: “Nostri sane motus mensura sit: primi motus mensura non erit.”

[74] Scaliger 1557, f. 458v: “At tempus est posterius motu primo. Est nimirum affectus eius, ab illo pendens.”

[75] Scaliger 1557, f. 458v: “Tempus, inquit, aut est idem, quod motus, aut affectus eius.”

[76] Scaliger 1557, f. 458v: “Tempus esse mensuram motus, et vicissim motum mensuram temporis.”

[77] Scaliger 1557, f. 458v: “[…] dicimus tot partes aequinoctialis tanto moveri tempore.”

[78] Scaliger 1557, f. 458v (punctuation and emphases have been standardized and modernized): “Quia tempus tantum est, quantus est motus inter ‘ubi’ et ‘ubi’ in loco. Iccirco ‘ubi’ quod pro tempore usurpamus, transumptum est, haud iniuria, ab eo ‘ubi’ quod est vere loci. Erit ergo tempus nunc fluens; et locus erit ubi continuatum.”

[79] Scaliger 1557, f. 458v: “Cum enim tempus sit quantitas, nihil agit.”

[80] Scaliger 1557, f. 458v: “[…] vita nostra est actus animae: in quem nihil habet tempus aut iuris, aut imperii.”

[81] Scaliger 1557, f. 459r, Exercitatio 353, “De tua visione.”

[82] Piccolomini 1585, I 3, chap. 7, ff. 52v–53r: “Di qui è, che’l tempo, c’ha da esser la misura di tutti i movimenti, bisogna che principalmente si applichi a misurare alcun movimento regolato sopra tutti gli altri, mediante il quale possa poi far da misura de gli altri tutti, che tal regola […] non hanno nelle parti loro. E tal movimento […] essendo […] quello del primo Cielo: si deve concludere, che’l tempo sia come in proprio soggetto da lui primamente misurato, nel detto movimento del mobil primo: mediante il quale, regolandosi tutti gli altri movimenti: ragionevol cosa sarà, che’l tempo, se ben si può chiamar regola, et misura d’ogni movimento, come di sopra habbiam lungamente detto, tuttavia egli non si multiplichi con la multiplicatione delli movimenti, ma resi uno stesso per tutto il mondo, sì come uno solo è quel movimento primo, che è il proprio, et vero soggetto suo.”

[83] Aristotle 1995, VIII 8, 261b 32–33 (369)..

[84] Benedetti 1585, 183: “[…] ut accidit lineae directionis, & retrogradationis planetarum, unde circulus uan erit ut epiciclus et b ut terrae centrum.”

[85] Pierre Duhem pointed out the tension between physical and mathematical astronomy, albeit in a very inadequate way. See Duhem 1908. For a better understanding of the matter, see Di Bono 1990.

[86] Piccolomini argues as follows: “Non è a punto manco sconvenevole il concedere cotale infinità in qual si voglia corpo semplice, che o salendo, o scendendo si muova per retta linea; che si sia veduto disopra esser di quello che in circolo si rivolge. Percioche essendo il partirsi salendo dal mezo dell’universo, e’ l venir discendendo a quello, che movimenti di luogo a luogo tra di lor contrarii: e ricercando li contrarii movimenti, luoghi contrarii parimenti; confessaremo esser opposti tra di loro il luogo disopra e questo qua giù da basso. E perché sempre tra due contrarii, se l’uno e per natura sua finito, e determinato, non sopporta il giustissimo governo della natura, che l’altro sia infinito e senza termine, secondo che discorrendo per tutte le contrarietà si può vedere.” See Piccolomini 1585, II 2, chap. 3, f. 24v.

[87] One can compare the text of Benedetti, Diversae speculationes, disp. XXV, with that of Copernicus, De revolutionibus, I 8. The latter wrote: “Igitur quod aiunt, simplicis corporis esse motum simplicem (de circulari in primis verificatur) quamdiu corpus simplex in loco suo naturali, ac unitate sua permanserit. In loco, siquidem non alius, quam circularis est motus, quo manet in se totus quiescenti similis. Rectus autem supervenit iis, quae a loco suo naturali peregrinantur, vel extruduntur, vel quomodolibet extra ipsum sunt. Nihil autem ordinationi totius et formae mundi, tantum repugnat, quantum extra locum suum esse. Rectus ergo motus non accidit, nisi rebus non recte se habentibus, neque perfectis secundum naturam, dum separantur a suo toto, et eius deserunt unitatem.”

[88] Benedetti 1585, 186.

[89] Benedetti 1585, 185: “Quod autem perfectum est, licet natura sit primum, est tamen ultimum generatione.”

[90] Benedetti 1585, 185–186: “Circulus sphaeraque non ex uno solo angulo recto constat, ut idem Aristoteles putat […] sed sunt figurae infinitorum angulorum rectorum, et hanc ob causam a me dicuntur ultimae et perfectae, quia infinito nihil addi potest.” The authors would like to thank here Irina Tupikova for suggesting this interpretation of Benedetti’s argument.

[91] Benedetti 1585, 189.

[92] The Pythagorean harmony was ridiculed by Alessandro Piccolomini among others. See Piccolomini 1585, II 4, chap. 12, “Del suono, et armonia, che i pitagorici pensavano, che nascesse per li movimenti de’ corpi celesti,” f. 105v: “[…] Quando i corpi celesti movendosi facesser suono avvenir dovrebbe a noi poscia che da si alto, et soverchio strepito, et sproportionato all’odito nostro quasi assordati, né quello né altro suono che qua giù si faccia, odire in modo alcuno dovremmo giamai. Per la qual cosa essendo manifesto che tante diversità di strepiti, che tra questi corpi inferiori si fanno, ancora che piccolissimi sieno, son da noi odite distintamente, è forza dire, per la ragione ultimamente fatta che né suono, né armonia, non può causarsi per li movimenti de gli Orbi, o delle stelle che volgendosi faccin la su in cielo.”

[93] Benedetti 1585, 194: “[Aristoteles] recte dicere non potest motum rectum ad curvum comparabilem non esse 4. cap. lib. 7 Physicorum ubi errat quoque dicens reperiri non posse lineam aliquam rectam alicuius circuli circunferentiae aequalem.”

[94] Drake and Drabkin 1969, 220. Cf. Benedetti 1585, 194: “Si igitur dicta quadratura dari potest, potest etiam dari una recta linea aequalis circunferentiae eiusdem circuli.”

[95] Drake and Drabkin 1969, 221. Cf. Benedetti 1585, 195: “Et amplius de stellis autem fixis circa aequatorem posits quivis cogitet; quod revera difficillimum quibusdam videbitur, quod quidem non occurrit secundum pulcherrimam Aristarchi Samii opinionem, divinitus a Nicolao Copernico expressam, contra quam nil plane valent rationes ab Aristotele, neque etiam a Ptolomeo propositae.”

[96] Benedetti 1585, 195.

[97] Drake and Drabkin 1969, 222. Cf. Benedetti 1585, 195–196: “Si doctissimi Aristarchi opinio est vera, rationi quoque consentaneum erit maxime, ut quod Lunae contingit, cuilibet etiam ex aliis quinque planetis eveniat, idest, ut quemadmodum Luna suorum epicyclorum ope circum terram volvitur, quasi per circunferentiam alterius cuiusdam epicycli, in quo terra sit instar centri naturalis (idest sit in medio) delati ab orbe annuo circa Solem; sic etiam Saturnus, Iupiter, Mars, Venus, atque Mercurius, circum aliquod corpus in medio sui epicicli maioris, situm habens, volvantur; quod quidem corpus, et aliquem quoque habeat motum circa suum axem, sit opacum, iis conditionibus, quae terrae sunt similes, praeditum existat, et in dicto epyciclo sint res similes istis lunaribus.”

[98] Benedetti 1585, 197.

[99] Copernicus 1978, 15

[100] Copernicus 1978, 15–16.

[101] Benedetti 1585, 195.

[102] Omodeo 2008a.

[103] Kepler 1937–2001, 390.

[104] Damerow and Renn 2010.

[105] Maier 1951, 304–305.

Concluding Remarks

Giovanni Battista Benedetti, the Renaissance scientist, has received ambivalent historical judgements by scholars in the past. The historian of medieval science and philosophy Anneliese Maier, for one, viewed him with mixed feelings. To her, Benedetti appeared to be a sort of intellectual companion of Galileo Galileo, at the same time his “forerunner” in mathematical physics and an epigone who was disrespectful to his own medieval Vorläufer or predecessors. Maier wrote that Parisian scholastics such as Nicole Oresme and Jean Buridan had provided Benedetti and Galileo with the concepts they needed to inquire into physics—she particularly had the concept of impetus in mind—which they did not acknowledge in their fierce attacks on Aristotelian philosophy.[1] Maier shared Koyré’s view that modern mechanics was constructed around a few central concepts and authors relevant for Newton’s Principia mathematica. They were perplexed by the concomitant reception and rejection of medieval physics by Renaissance scientists. In our view, however, it is too narrow a point of view to just consider individuals and sets of ideas and their genealogies. Instead, one should consider the wider intellectual currents and the shared knowledge they generated. The incipient querelle des anciens et des moderns[2] is an example of a debate transcending specific questions and problems, even approaches and methodologies, towards a larger reflection on the relation between past and present. The problems inherent in this gap between the individual perception of change and the intellectual transitions of the time are exhibited by the astronomy of Nicolaus Copernicus, a sort of “unaware revolutionary,”[3] who saw himself (or at least presented himself) as a Renaissance restorer of planetary theories defended in antiquity by the legendary Pythagoreans. By contrast, his scholastic counterpart, the Padua-trained physician and natural philosopher, Girolamo Fracastoro, presented his homocentric reform of mathematical astronomy as a radical innovation, comparable with Amerigo Vespucci’s discovery of the New World.[4] Fracastoro’s work was based on the modeling of all celestial motions through concentric spheres (in line with a well-established Aristotelian tradition). In Benedetti’s case, the rejection of the philosophia naturalis taught in the universities was achieved with intellectual means descending from that very philosophical tradition. Rather than viewing this fact as a paradox, it should be regarded as a sign of a profound tension in Renaissance science between past and present and a hallmark of what we have called preclassical mechanics.[5] The intention to outdo traditional authorities in order to move beyond their legacy had to rely on the shared knowledge of the time, which was marked by Aristotelian thought. In our introduction we delved into Benedetti’s conceptions and reconstructed their socio-cultural coordinates, characterized by the Renaissance tension between conceptual heritage and novelty. Maier’s perplexity thus rests upon a lack of reflection on the embedment of intellectuals and their theories in socio-cultural processes. Benedetti in particular ought not to be seen as a link in a chain, but rather as one representative of a complex and comprehensive knowledge economy.[6]

In order to correctly locate Benedetti in the knowledge economy of the Renaissance, it is expedient to consider him against the background of the material and intellectual conditions of early-modern science, and as a figure between the intentions and identities of a new genre of intellectuals who formed the archetype for modern scientists. Benedetti’s case helps us to reflect upon the social position and intellectual identity of these new types of scholars as well as on the way socio-cultural coordinates penetrated science, as far as its demarcation, content, form, and justification are concerned. With social coordinates we refer to the institutional setting involving Benedetti’s role as a courtier and thus to his function as a court mathematician, which, in turn, was linked to the wider socio-economic interests of a Renaissance territorial state.[7]

In his seminal work on the sociological roots of modern science, Zilsel discussed the scientific relevance of the social transformations taking place in the late Middle Ages and the Renaissance. In particular, he argued that the emergence of modern science depended on the rise of capitalism. We could aptly refer to this phase as a pre-capitalistic or early-capitalistic “knowledge society.” Technical knowledge proved to be a key element in the organization of life and production while the status of the artisans, those whom Zilsel called the “artist-engineers,” increased and received high recognition among civil and political authorities. The town of Florence is prototypical for these changes, as Leonardo Olschki has forcefully demonstrated in his studies on science and vulgar literature.[8] A wide range of artist-engineers transformed Florentine society and its mentalities. Filippo Brunelleschi, most representatively and symbolically, forever changed the skyline of the same town in which, at the end of the Italian Renaissance, Galileo composed works that irreversibly modified the landscapes of science and scientific culture.[9] In Florence and Europe more generally, in the passage from the Middle Ages to early modernity the “artisan-practitioners” were confirmed as a new class. The codification of their experience and knowledge profoundly changed epistemology and science, most evidently in mechanics. This practical art was first codified as a physico-mathematical discipline, and then as a science in its own right, and was later adopted as a methodological and ontological point of reference in the shift toward the mechanistic world views of the seventeenth century.[10]

According to a corollary of the Zilsel thesis about the social origins of modern science, scientific culture was reshaped by the merging of three intellectual strands: the artisanal/technical, the scholastic/logical, and the humanistic/rhetorical. This fusion was accomplished by mediators, who were social actors with an in-between status bridging different intellectual and social realms. “Hybrid experts” became increasingly necessary because of their capacity to bring together the technical and the theoretical dimensions of knowledge. Their socio-cultural relevance would never diminish from the late Middle Ages to the Industrial Revolution and beyond.[11]

During the Renaissance, this mediation was secured by a new group of “scientist-engineers,” a series of court mathematicians of which Galileo is the best-known figure and which also included his protector, Guidobaldo Del Monte. Actually, the description of the Renaissance figure of the “scientist-engineer” suits the intellectual and social profile of Benedetti very well.[12] Galileo and his like were well versed both in the technical as well as in the intellectual dimensions of knowledge production. Renaissance “scientist-engineers” underwent a period in apprenticeship of practical mathematics, in some field of application like architecture or the art of war, but later distanced themselves from artist-practitioners as they aspired to gain higher social recognition, especially as courtiers. They had a high degree of education, as they mastered theoretical mathematics, the language of the learned, Latin, as well as the courtly language, for instance by acquainting themselves with the elegant Italian of the literature of the time. Scientist-engineers thus acted as mediators connecting the centers of power and decision on the one hand and the workshops and building sites on the other. As was the case with Benedetti, these experts could supervise artisanal work or give advice on technical issues.[13] As courtiers they were additionally required to participate in the refined dialogical and literary culture of the elite, to serve as educators as well as to use their astronomical expertise to cast horoscopes for the rulers.

The most specific socio-political aspect of Benedetti’s time is the affirmation of court society as a particular social formation whose features show continuities and fractures both with the earlier aristocratic setting of the feudal society and the later capitalist one. A distinguishing feature is the centralization of power and administration around the court. As Norbert Elias argued, this formation culminated in the absolutism of the Ancien Régime but was preceded, on a smaller scale, by early attempts at territorial centralization.[14] Although such social formations apparently gravitated around an individual sovereign who made all decisions (as much of the literature of the time on the Principe and its privileges boasted), it was in fact a hierarchical system in which the group of experts surrounding the princely ruler constituted an oligarchy who operated the complex organization of modern states. The Duchy of Savoy is one such case. The dukes strove to create a “modern” capital city partly by following the model of Florence, insofar as culture and prestige are concerned, but also the Spanish and French models, insofar as the suzerainty of the ruling family is concerned. Other models played a role, too, for instance the Netherlands for military technology and Switzerland for military conscription and discipline.

Benedetti shared the enthusiasm of his patrons (especially Emanuele Filiberto) for mathematics and its perceived powerfulness as an instrument for successful navigation in war and peace. He also shared the aristocratic values of the court such as disinterest and prestige. Adherence to these values largely explains his bias toward theory despite the practical origins of his knowledge and the fields of application of his mathematics (ranging from mechanics to navigation, architecture, and perspective). He also ventured into the most general fields, such as cosmology and philosophy (as seen through his criticism of Aristotelian natural conceptions, his favorable opinion on Copernican astronomy and post-Copernican cosmology, and his remarks on “Pythagorean” philosophy of mathematics). Actually, he did not hesitate to call his wide and unsystematic work “speculations,” an expression that stresses the theoretical character of the endeavor.

From the perspective of a court scientist such as Benedetti, mathematics was the key to practice and theory. It was his specific field of expertise among the Turin courtly elites; through it he acquired a central epistemological status in line with the exaltation of the certitudo mathematicarum by many of his contemporaries, among them his correspondent Pietro Catena. At the same time, the practical context surrounding the mathematical approach in many fields such as mechanics led him to emphasize the contingent element of natural phenomena. Thus, the centrality of mathematics in Benedetti’s work has a multilayered meaning, including the theoretical, practical, epistemological, and social. The limits of validity and applicability of Benedetti’s mathematical science mirror the boundaries of his field of competence in the division of intellectual labor within his courtly environment. Although he used geometry as a sort of universal key, he could not impose his views on other courtiers who were experts in fields such as philosophy and medicine. In this context of enforced openness, Benedetti’s criticisms of Aristotelianism appear as a sort of defense of his professional position in the framework of a courtly dialogical pluralism. Such an environment explains the occasional (and fragmentary) character of the Diversae speculationes, which brings together occasional materials such as texts for private teaching, letters, short treatises, expert advice, and polemical essays (among others), in which Benedetti made his mathematical expertise manifest and showed its usefulness.

The intellectual distribution of labor in the Renaissance ensured that Benedetti was at the heart of the courtly milieu by virtue of his family’s social status and not through his ambition alone. His work exhibits many similarities with the work of other Italian court mathematicians, most eminently that of the aforementioned Del Monte and Galileo, as far as the range of their interests and the overall approach are concerned. Benedetti’s most daring passages, which open up unconventional solutions to technical and theoretical problems, and his remarkable disregard for authority qualify him as one of the Italian novatores, although he did not make explicit his natural conception as an all-encompassing alternative to the well-established Aristotelian philosophy. His fierce attacks on crucial aspects of the Aristotelian conception—relating to motion, the void, infinite space, time, infinity, and planetary theory—did not result in a systematic new natural philosophy. Rather, he limited himself to collecting results in different areas and to working on the most varied aspects without finding their common denominator. He also made elliptical references to Pythagoreanism and implicitly rehabilitated some aspects of atomist and stoic conceptions, for instance the plurality of worlds and the fluidity of the heavens. Cardano, whom he appreciated, went much further in the inquiry of the common foundations of the sciences (specifically mathematics, practical arts, and medicine) while Benedetti’s correspondent Patrizi advanced a systematic natural philosophy inspired by neo-Platonism. In the same years in which Benedetti finished and published his physico-mathematical speculations in Turin, Bruno published philosophical dialogues in London expounding a natural philosophy and an anthropology that led to far more radical consequences for the premises of cosmology, similar to those reached by Benedetti. Another contemporary of Benedetti, Telesio, had offered the first modern attempt to build up a conception of nature on new principles. His Natura iuxta propria principia paved the way for the next generations of scholars searching for new foundations in natural science. Among them was his direct follower Campanella, who brought his philosophy to France in the seminal years of the mechanical philosophies of Pierre Gassendi and René Descartes. Benedetti participated in this wide cultural transformation; he contributed to advancing the mathematical and physical disciplines and discarding consolidated theories—but without offering a systematic alternative.

To summarize the most evident features of Benedetti’s endeavor: it was courtly, secular, anti-Academic, unsystematic, occasional, elitist, learned, abstract, pleasant, and useful. It was secular, that is, non-theological, as it was linked to the interests of the ruling class and the state. It was a useful and pleasant science: on the one hand, it was practice-oriented but not purely empirical; on the other hand, it proved witty and fit for courtly sociability. It was abstract and disinterested: superior to the vulgar and tuned to aristocratic values. Learned: fit to be exhibited at court alongside the other arts. Elitist: Benedetti elevated mathematics from a practical discipline of scientist-engineers to a refined cultural activity. Occasional: linked to the variegated political and cultural interests of the court. Unsystematic: fragmented, lacking the inner coherence of scholasticism. Anti-Academic: free from concerns about respect for university scholarly traditions. All of these characteristics of Benedetti’s science were the hallmark of court science: it was technical and abstract without losing contact with practice and experience—a mathematical-empirical science in nuce; it was (relatively) free from bookish tradition and theology but not from the contingencies of courtly life.

What is the common denominator of the great variety of subjects dealt with by Benedetti? What is the center around which they all gravitate? Is there one unifying principle behind the apparent disorder and heterogeneity? It should be emphasized that Benedetti first established his fame as a mathematician. His early treatment of motion by mathematical means was explicitly directed “against Aristotle and all philosophers” (contra Aristotilem et omnes philosophos). In his time “mathematics” had a wide scope. It comprised arithmetic and geometry, astronomy and astrology, as well as music, but also reached far beyond the boundaries of the quadrivium by encompassing optics, practical mechanics, architecture, and engineering. The expansion of mathematics into the fields of physics, natural philosophy, meteorology, and even metaphysics and epistemology was a crossing of the disciplinary boundaries. Benedetti’s time bears witness to several attempts to expand the boundaries of mathematics. Cardano, for one, claimed that geometry had the function of a universal logic fundamental to rational thought, and that the practical disciplines including statics, mechanics, and architecture were its subordinate fields of inquiry.[15]

Benedetti’s intellectual identity, however, proves much more complex than his corporate identity as a mathematician.[16] His pronounced titles vary. In a short biographical note accompanying the birth horoscope published by Gaurico, he was referred to as “Phylosophus, Musicus, atque Mathematicus”; on October 19, 1589, he signed an astrological report cast for Carlo Emanuele I as “Matematico e Astrologiaro”;[17] contemporary admirers of his such as the Milan painter and poet Lomazzo and the Danish astronomer Brahe called him “matematico” and “philosophus et mathematicus inprimis excellentem,” respectively.[18] Probably, Brahe’s designation of Benedetti as both philosopher and mathematician best captures the poles of his intellectual activity. Intriguingly enough, Benedetti generally dropped the title of “mathematician,” keeping only that of “philosopher” in his publications. On the title page of his magnum opus of 1585, the Diversae speculationes, he appears as “patritius Venetus philosophus,” exactly the same epithet that appears in De gnomonum umbrarumque solarium usu liber (1574). In the publications in the vernacular, he correspondingly appears as “filosofo del sereniss. duca di Savoia,” e.g., in the Consideratione… d’intorno al discorso della grandezza terra et dell’acqua (1579). In the last publication, his self-presentation as court philosopher is interestingly opposed to the designation of his intellectual opponent, Antonio Berga, as “filosofo nella Università di Torino,” that is, “university philosopher”—which is equivalent to scholastic philosopher. These references are telling for Benedetti’s self-perception or, to use an in-vogue expression, his self-fashioning.[19] In both cases, the image of court philosopher was his intended identity, whether reflected or purposely constructed (or a mixture of both). As was the case with Galileo, the Florentine courtier, the philosopher’s social status and reputation was higher than that of the mathematician. This is why, among the conditions for Galileo’s appointment as a courtier to the Medicis, he regarded the designation “philosopher” as relevant.[20]

As for the epistemological debates mirroring the disciplinary and social divides and hierarchies of the time, heated controversies began over the “certainty of mathematics.” The determination of the degree of certainty of mathematics also concerned the legitimacy of using mathematics in physics. In the case of Benedetti, the tension between his function as court mathematician and his identity as philosopher—and patrizio—lies beneath his science. While philosophical legitimacy was essential for the acknowledgment of the intellectual dignity of his endeavor, the practical dimension of mathematics remained fundamental for the social justification of his function as a court expert.

One could single out the social and the political-cultural coordinates of Benedetti’s science as two complementary drives. On the one hand, his position as a court mathematician directly determined much of the content of his writings, occasioned by the requests addressed to him as a court expert in technical issues pertaining to mathematics. His position also determined formal aspects of his work, in particular its occasional character and fragmentation. On the other hand, Benedetti’s identity as a philosopher was directly related to his cultural ambitions and his engagement aimed to affirm mathematical philosophy in the intellectual arena against scholastic thinkers and humanistic literati. His political identity as a lay aristocrat made him an organic part of the centralizing project of the court and marked his distance from Counter-Reformist drives which sought to impose Roman universal interests over territorial states’ autonomy. His support for a sort of party of the politiques resulted in treatises advising on politically relevant technical and cultural issues (e.g., navigation on the occasion of the battle of Lepanto or the calendar reform). His activity as a lay educator, e.g., his arithmetic teaching to the prince, Carlo Emanuele I, is found in his pedagogical writings, some of which were published in his scientific miscellanea. In summary, both content and form, as well as the demarcation of the fields of his scientific competence as a mathematician and philosopher, depended on social settings and cultural engagement.

The fact that Benedetti never established a scientific school around himself can be seen as an indication of the precarity of patronized science, linked to the person of a particular ruler and not institutionalized at the level of an academic body. In the course of the seventeenth century, these limitations of early court society would be solved by securing scientific continuity for patronized science through the foundation of scientific societies. These societies constituted an improvement over the volatility of Renaissance patronage, which depended on the humors and interests of a prince, by replacing him with a corporative persona ficta deputed to protect, credit, and promote science. This did not imply a diminution of the political relevance of science. As has been argued, the institution of the Académie Royale des Sciences as a means to patronize all of the sciences also meant the conquest of a new kingdom, la république des lettres tout entière, for Louis XIV.[21]

Montesquieu was a perspicacious observer of the courtly society in which Benedetti lived and worked. In his opinion, the “courtly air,” or the ethos of the ruling elites of a monarchic state, “consists in putting away one’s own greatness for a borrowed greatness. This greatness is more flattering to a courtier than is his own.”[22] Such grandeur empruntée, or borrowed greatness, was a function of a person’s distance from the ruler. Benedetti’s greatness could have solely consisted in his mathematical acumen, in his mechanical insights and demonstrations, or in his philosophical discernment; these are the virtues that the historian of science is inclined to observe as principal. However, Benedetti saw himself as a court gentlemen, and only valued his capacity as a mathematician as subordinate. He presented himself as a court intellectual, more precisely, as a “philosopher to the Dukes of Savoy.” He “borrowed his greatness” (in Montesquieu’s words) from his proximity to the rulers. In the courtly milieu, it was honor and rank, together with their corollary, ambition, rather than skill, diligence, and measure that marked the character of a nobleman who belonged to the hegemonic class of the new state. Greatness is a major motivation for Benedetti’s science, which cannot be confined to technical demonstrations or the solution of specific problems. Rather, his treatment of details never departed from concerns about the big picture; in his work, special issues were constantly elevated and received their meaning on the level of a grand overview, natural and epistemological.

Greatness is not the only courtly quality to enter Benedetti’s science. As Montesquieu further observed: “At court one finds a delicacy of taste in all things, which comes from continual use of the excesses [superfluités] of a great fortune, from the variety, and especially the weariness, of pleasures, from the multiplicity, even the confusion, of fancies, which, when they are pleasing, are always accepted.”[23] To be sure, one cannot say that Benedetti’s knowledge was superfluous in the sense that it had no concrete application. In the Renaissance, it was evident to anybody how closely mathematics was connected to practical realms ranging from war technology to fortification, navigation, and administration. Benedetti’s work and activities related to these realms; even his astrological consultancies can be appreciated for their practical orientation—as astrology notably coincided with the so-called astronomia practica, as opposed to mathematical astronomy, or astronomia theorica. Still, Benedetti insisted on his lineage as a “philosopher” (connected with his claims about the Pythagorean universality of his method and the fragmentation of its applications) despite the attention given to practice and concreteness in Renaissance mathematics. Such a contention was aimed at confirming his superiority over the immediate application of knowledge or the material origin of arts such as mechanics.

His stress on theory—on “speculation”—is well attuned to the spirit of court society, which was centered on nobility, that is, on disinterest and rank, rather than efficacy. The “superfluity of Benedetti’s science” corresponds to the leisure character of knowledge in general, due to fact that its bond with materiality and practice was sublimated. Whereas corporative and merchant societies like those of the Italian Quattrocento (or, more generally, bourgeois and democratic ones like those emerging in the seventeenth century) would emphasize the practical origin and meaning of science, a court society stresses its symbolic value rather than direct usefulness and economical importance.

Besides the superfluité (which applies to Benedetti only if it is not taken too literally), all of the other qualifications Montesquieu attached to the court atmosphere suit his endeavor: good taste (we can add, “wit”), variety, pleasure, multiplicity, even confusion. The main virtue of a court society rested on the sense of honor and ambition: “Honor, meddling in everything, enters into all the modes of thought and all the ways of feeling and even directs the principles.”[24] Norbert Elias, who agreed with this assessment, also pointed out the fatal consequences for budgetary issues of a mentality that is so distanced from a modern bourgeois economy. From an economic viewpoint, court society was intrinsically flawed. It was destined to bankruptcy because form, ritual, and etiquette counted more than parsimony. Similarly, courtly science displayed detachment from monetary return. Elias has also emphasized the centrality of etiquette for this detachment. At court, formal etiquette was decisive, as it served to maintain and reinforce distances and hierarchies.[25]

The sense of honor and superiority typical of such social formations appears in Benedetti’s intended distance (social, intellectual, moral, and epistemological) from artisanal practice and the erudition of university professors. He appropriated the results and methods of both fields, in particular those of the practical arts, but at a higher level of generalization. He particularly envisaged a reformed natural philosophy as the most cherished fruit of his “mathematical-physical speculation.” Such theoretical distance from immediacy is the epistemological parallel of the sense of honor and social distance and, as such, it became an essential ingredient of Benedetti’s science and added symbolic value. As a court intellectual, he did not identify himself with traditional forms of higher culture such as Scholastic Aristotelianism or humanistic rhetoric. He proudly affirmed himself as a courtier, free to think and philosophize in the protected space of the court, independent of the most immediate material needs, of academic constraints dictated by tradition, and concerns about systematicity and completeness. Ambition, the companion of aristocratic honor, “meddled in everything” and directed Benedetti’s search for the most general principles of a new vision of nature, both mathematical and physical. The court protected and promoted a science and philosophy in which disinterestedness was foremost. In its favorable womb, a daring mind could venture out to explore new realms beyond established disciplinary boundaries. The speculative freedom of the court also determined the specific form of Benedetti’s work, its occasional character, and the amazing variety exhibited by his diversae speculationes mathematicae et physicae.

The economy of honor in the court society left an enduring epistemological imprint on the social fabric of science. Symbolic capital governed modern science long after it became coupled with economic capital and, in many ways, it still significantly influences science and research. The legacy of courtly ingenuity and leisure has to be acknowledged as a lasting influence upon scientific practice as well. Moreover, the topos of a protected space, so attractive to the emergent category of philosopher-scientists in the sixteenth and seventeenth centuries, contributed to creating the myth of the independence of pure science. Constant claims and controversies about scientists’ autonomy have accompanied the modern path to science in its migration from the court to the scientific academy and from the scientific academy to the laboratory. The connections of modern science to the economy and society at large, politics, and cultural structures can be appreciated by considering the complex historical ties that link knowledge with its material and cultural conditions reaching far beyond the perception of the individual historical actors. The spirit of Benedetti’s science can be seen as typical of an age of profound social transformation and political reconstitution, which is reflected in the exceptional re-structuring of knowledge and the transition to novel forms of scientific acquisition, legitimation, and transmission.

Footnotes

[1] Anneliese Maier established a connection between Benedetti’s treatment of motion and that of Galileo in Maier 1951, 304–305.

[2] Lehner and Wendt 2017.

[3] Copernicus’s revolutionary role malgré soi already puzzled Thomas S. Kuhn, who called him at once “radical” and “conservative” and regarded De revolutionibus orbium coelestium, the book propounding the first modern heliocentric theory in mathematical astronomy, “revolution-making” rather than “revolutionary.” Cf. Kuhn 1959, 135 and 148.

[4] Goddu 2010 and Granada and Tessicini 2005. Also see Omodeo 2017.

[5] See Renn et al. 2018.

[6] For a recent study accomplished in this vein, see Trzeciok 2016.

[7] For further considerations on Benedetti in light of a discussion on methodological and historiographical approaches, see Renn et al. 2018.

[8] Olschki [1919–1927] 1965.

[9] On the Florentine prototype, see Renn 2014. Cf. Zilsel 2000, 941.

[10] On artisanal knowledge and its codification, see Smith 2004 and Long 2001. On practical knowledge, see Valleriani 2017. On the elevation of mechanics to a worldview, see Renn and Damerow 2010.

[11] Ursula Klein has made this point most forcefully in Klein 2015.

[12] The figure of the “scientist-engineer” has been introduced into the history of science by Renn 2001, particularly in the contributions by Lefèvre (Lefèvre 2001) and Renn, Damerow, and Rieger (Renn, Damerow, and Rieger 2001). Valleriani discusses it in detail in Valleriani 2010, chap. 6.

[13] Valleriani 2010, 208: “Except for the period of the apprenticeship, an engineer-scientist was almost never personally employed in workshops or building sites, but he was aware of the work procedures followed in these locations and was able therefore either to commission craftsmen or other persons involved with practical activities, to supervise or teach them, or simply be consulted to evaluate their works.”

[14] As already discussed in the introduction. The reference work is Elias [1969] 2002.

[15] Girolamo Cardano, Encomium geometriae recitatum anno 1535 in Academia Platina Mediolana in Cardano 1966, vol. 4, 440–445.

[16] By “corporate” we refer here to the esprit de corps of a group that considers itself a bounded entity whose interests are marked as separate from other groups. The guild culture of the Middle Ages originated this particular meaning of corporation, which precedes the modern sense of a professional group or legal body.

[17] Roero 1997, 57–58.

[18] Lomazzo 2006, 177: “Del Sig. Gio. Battista Benedetti Matematico” Brahe 1916, 251–253.

[19] Greenblatt 1980.

[20] Biagioli 1993. Also see Biagioli 1989.

[21] Biagioli 1995, 1418 and 1438.

[22] Montesquieu 1989, 33.

[23] Montesquieu 1989, 33.

[24] Montesquieu 1989, 33.

[25] Elias [1969] 2002, 173.

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A Facsimile of Giovanni Battista de Benedetti’sDiversarum speculationum mathematicarum, et physicarum liber (Turin 1585)

Facsimile page Facsimile of folio A1rFacsimile page Facsimile of folio A1vFacsimile page Facsimile of folio A2rSerenissimo Carolo Emanueli Sabaudiae Ducis / Dedicatory letter to Carlo Emanuele of Savoy Facsimile page Facsimile of folio A2vFacsimile page Facsimile of folio A3rAd lectorem / To the reader Facsimile page Facsimile of folio A3vFacsimile page Facsimile of folio A4rFacsimile page Facsimile of folio A4vFacsimile page Facsimile of page 1Theoremata arithmetica Facsimile page Facsimile of page 2Facsimile page Facsimile of page 3Facsimile page Facsimile of page 4Facsimile page Facsimile of page 5Facsimile page Facsimile of page 6Facsimile page Facsimile of page 7Facsimile page Facsimile of page 8Facsimile page Facsimile of page 9Facsimile page Facsimile of page 10Facsimile page Facsimile of page 11Facsimile page Facsimile of page 12Facsimile page Facsimile of page 13Facsimile page Facsimile of page 14Facsimile page Facsimile of page 15Facsimile page Facsimile of page 16Facsimile 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