RIVKA FELDHAY

3         The use and abuse of mathematical entities: Galileo and the Jesuits revisited

INTRODUCTION

On the second day of the Two New Sciences1 the three interlocutors Sagredo, Simplicio, and Salviati suspend their learned conversation on forces of fracture and resistance to indulge in yet another digression among many that have become well known as characteristic marks of Galileo's texts. Sagredo, the aristocratic amateur of natural philosophy and mathematics addresses Simplicio, the Aristotelian philosopher, with the following remark:

What shall we say, Simplicio? Must we not confess that the power of geometry is the most potent instrument of all to sharpen the mind and dispose it to reason perfectly, and to speculate? Didn't Plato have good reason to want his pupils to be first well grounded in mathematics? (133)

Simplicio, portrayed in this not very polemical text as an open-minded scholar, graciously responds:

Truly I begin to understand that although logic is a very excellent instrument to govern our reasoning, it does not compare with the sharpness of geometry in awakening the mind to discovery.

This unexpected agreement encourages Sagredo to further elaborate his position by saying:

It seems to me that logic teaches how to know whether or not reasonings and demonstrations already discovered are conclusive, but I do not believe that it teaches how to find conclusive reasonings and demonstrations.

The edge of Galileo's ambitious project is enfolded in this brief exchange. Suggesting that geometry is a tool of discovery, whereas logic serves for assessing and criticizing arguments already known, Sagredo hints at the need to restructure the body of natural knowledge, substituting mathematics for logic as the organon of philosophy.

Ever since the nineteenth century, the historiography of science has fruitfully oscillated between different interpretations of what really constituted the core of Galileo's project. Experimental practices,2 mathematical Platonism,³ Aristotelian method,⁴ or some kind of a combination between experiment and mathematical deductivism⁵ are just a few among many alternative clues suggested by scholars along the years, by means of which the “essence” of Galileo's enterprise was thought to be captured. Whatever may be the angle through which Galileo's theory and practice are to be examined, it is beyond doubt, however, that the transition from traditional natural philosophy to the new science was much effected by the role assigned to mathematics in Galilean discourse, though not necessarily by its actual mathematical techniques.

Many questions have been asked about the new status of mathematics in Galileo's scientific program. Some historians were most interested in the origins of Galileo's mathematical orientation, which could be found in classical mathematical texts, or perhaps among the medieval calculators, or the Parisian School, or among the mathematical practitioners of fifteenth- and sixteenth-century Italian courts.⁶ Other historians were more interested in the contents of the justifications, devised by Galileo for using mathematics in the investigation of nature, and in their philosophical validity.⁷ Yet others preferred to emphasize the compatibility or incompatibility of mathematical arguments with the established method of the official science of sixteenth-century universities.⁸

To this variety of points of view I would like to add yet another aspect. Assuming a breach within Galileo's scientific project (which has already been pointed out by other historians), but also taking into consideration the context of Galileo's project in the field of practicing mathematicians, Galileo's justification of the status of mathematics may be better understood if we realize and analyze the complexity of its various functions: on the one hand to create a bridge between the different and sometimes incompatible directions of his own inquiries, conferring upon them the coherence of a research program; on the other hand to construct for himself a differentiated position among other mathematicians working in the same cultural field.

My essay, then, is a preliminary attempt to provide a framework of some less discussed aspects of Galileo's politics of knowledge. The justification of the status of mathematics is not examined here as the source from which a coherent research project necessarily emerged but as a necessary strategy of creating coherence for a project whose inner connections were not yet clear. Furthermore, the cultural context within which such strategy was mainly practiced consisted in a newly reconstructed community of mathematicians whose field of research was in the process of being defined.

My point of departure is the debate over the mathematical sciences that broke out in Italy in the middle of the sixteenth century and has been known in historical literature as the controversy on the certainty of mathematics (De certitudine mathematicarum disciplinarum).9 The cultural significance of this debate emerged as it began to play a role in the actual practices of mathematicians attempting to gain for their project a central educational role. Jesuit mathematicians made the first institutionally organized effort to place the mathematical disciplines at the heart of a broad cultural program.¹⁰ Therefore, my focus in the first part of this essay is on the appropriation and development of the main themes of the debate as strategies of legitimizing their field of knowledge.

Galileo's science sprang from the same roots as the Jesuits’ program, and it shared much of its spirit with Jesuit mathematicians. The dynamics of Galileo's own development, however, pushed him into formulating a different agenda. At the same time, Galileo never detached himself completely from his roots, which assumed the form of a counter-discourse, insisting in his texts but split from his later agenda. The second part of this paper comprises an analysis of various passages from Galileo's Dialogue that exemplify the structural split within his own scientific program.

The third part follows the traces left in Galileo's Dialogue by the debate on the certainty of mathematics. Galileo's final annihilation of the discourse on mathematical entities – used by the Jesuits as their main legitimation – was a way of covering up the split in his own program, as well as strategy of differentiating his position from that of the Jesuit mathematicians.

1. THE CONSTRUCTION OF A FIELD FOR MATHEMATICIANS

In 1547 Alessandro Piccolomini, a member of the Accademia degli Infiammati, which was active in transmitting humanistic and Renaissance learning to the University of Padua, published a paraphrase of Aristotle's Mechanical Questions, appendiced by a commentary on the certitude of mathematics (Commenatrium de certitudine mathematicarum disciplinarum).11 Piccolomini's treatise challenged the accepted interpretation of Averroes and the Latin commentators, according to which in the hierarchy of the speculative sciences (scientiae) the mathematical disciplines were the first in the order of certainty, because their demonstrations were the model for demonstrations potissimae, perceived as the strongest and most certain of all other forms of demonstration.¹² Following his reading of Proclus's Commentary on the First Book of Euclid's Elements, and of the Greek Commentators, whose recently translated work started to transform the reading of Aristotle in those same years,¹³ Piccolomini changed his adolescent opinion on that matter and claimed that geometrical demonstrations had nothing to do with scientific demonstrations potissimae.¹⁴

Piccolomini's treatise is a natural point of departure for understanding the appropriation and rejection of the ancient discourse on mathematical entities in the cultural context of early modern science. Its arguments, recently represented and analyzed in detail by Anna De Pace amount to a strategy that attempted to establish a clear boundary between mathematics and natural philosophy, while still legitimizing mathematics as an autonomous but inferior science.

The ontology of mathematical entities delineated by Piccolomini consisted in a combination of his Averroistic interpretation of “quantity” as the most general accident of primary matter, his Aristotelian theory of abstraction, and his Aristotelian reading of Proclus's thesis about the middle position (medietas) of mathematics. The quantity that inheres in primary matter before it is embodied in any substantial form was, in Piccolomini's words, “quantum phantasiatum,”¹⁵ the most common and basic of all sensible accidents¹⁶ and undetermined by any specified form. Piccolomini calls it “indeterminate quantity” and describes it thus:

… since matter is by its own nature devoid of any substantial form, and nevertheless has in it the possibility and readiness for all forms: thus the quantity which is proper for it is likewise bare and devoid of any determination or figure; and is nevertheless able and disposed to receive all terms or figure…17

Quantity is the most immediate and manifest property of matter and hence the easiest to abstract. Mathematical entities are easily liberated from matter by simple abstraction. The certainty with which they may be known is connected to their simple being. Devoid of complexity and depth, they are the most accessible for human cognition.¹⁸

Piccolomini, however, denied that the certainty achieved by mathematical demonstrations, whose subject matter is quantity, can be identified with scientific demonstrations. Using Proclus's analysis of Euclid I, 32 [In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles are equal to two right angles] as an example for a noncausal demonstration he claimed it could not be identified with demonstration potissima, and he generalized this critique to all Euclidean proofs. Thus, his conclusion was that mathematical demonstrations were not really scientific in the Aristotelian sense.¹⁹

The separation of mathematical objects from substance, which explains, according to Piccolomini, their capacity to be known with a high degree of certainty, but which differentiates them from the objects of natural philosophy entangled in the reality of matter and form, also accounts for the difference between geometrical and philosophical demonstrations. Together these differences justify the distinction between mathematics as a science of abstract being – “quantum phantasiatum” – and natural philosophy, the science of reality. Even the mixed sciences, which apply mathematics to the investigation of nature, are devoid of real scientificity, and though they are extremely useful for humanity they still represent an inferior form of knowledge compared to natural philosophy. One example used by Piccolomini to substantiate this difference concerned the sphericity of the Earth and heaven. Whereas the natural philosopher attempts to discover the essential causes inherent in natural things, the astronomer considers their mathematical properties without asking about their true nature:²⁰

…the astronomer…even while considering that the heaven is spherical, or that the earth round, does not need, for this [purpose] to know the true nature and their substance, but solely from the positions, figures and aspects seen in the heaven argues that they are of such form…for this [reason] it can be concluded, that though the science of natural things often overlaps with other sciences in treating a certain subject, or in demonstrating a certain conclusion, nevertheless the natural philosopher differs from all the others in that never separating the concepts of the forms from those of their proper matter, he treats both natures as related to each other, namely the matter and the form: which are the two principles, and the intrinsic causes of natural things.

In the flourishing community of Paduan mathematicians and Averroist philosophers which sustained a rich technical and scientific tradition at the time,21 including among its members people such as J. Contarini, D. Barbaro, G. Moleto, and N. Tartaglia, Piccolomini's treatise was received with much surprise and irritation. This was the natural audience for Francesco Barozzi, a young Venetian patrician. He had been lecturer of mathematics since 1559 and immersed in the study of Proclus's Commentary for some years, publishing (in 1560) his Opusculum – consisting in an oration and two questions on the certainty and the middle position of mathematics – in response to Piccolomini's startling innovation.²² The work was dedicated to D. Barbaro, seeking his protection for daring to challenge Piccolomini's recent publication. Barbaro responded with a letter – thus leaving some testimony for the hostility toward Piccolomini in his circle – in which he expressed his long-term expectation for a refutation of Piccolomini's opinion as “new and unfounded” (“nova et non fondata”).²³ In 1559 (the first year of his lectureship) Barozzi read Proclus's Commentary in his course and left his interpretation in manuscript form.²⁴ As a Venetian patron he corresponded extensively with prominent Italian mathematicians, among them Clavius, Guidobaldo del Monte, Giuseppe Moleto, and other eminent personalities.

Barozzi's work is of interest as a main source of interpretation and transmission of Proclus's Commentary, which he edited and published in the same year as he published his Opusculum.²⁵ Accepting Piccolomini's thesis about the medietas of mathematics (between philosophy and the “divine science” metaphysics) as the basis for reconciling Aristotle and Plato, Barozzi neglected Piccolomini's interpretation of this middle position in terms of abstraction and attempted to ground the certainty and scientificity of mathematical demonstrations in Proclus's claim to the innateness and priority of mathematical entities.

Thus Barozzi was the first to legitimize a reading of Aristotle in terms of a Neo-Platonic ontology of the objects of mathematical discourse. Barozzi, following Proclus, claimed that Plato arranged the sciences according to the perfection of their entities. Therefore, according to him: “Divine philosophy holds the first place, mathematics the second, natural philosophy the third.”26 At first glance it seems that Aristotle refused this order, since he gave priority to natural philosophy. This opposition is superficial, however, according to Barozzi. In truth, Aristotle accepted the middle nature of mathematical entities, since they mediate between matter and the purely abstract entities of metaphysics. “And indeed, this middle essence cannot be anything else but mathematical.”²⁷ But the middle position of mathematics means that the certainty of knowledge of its objects is superior to that of the knowledge of the objects of natural philosophy. And because there must always be a correspondence between the objects of a science and its demonstrations, it follows that mathematical demonstrations are more certain than any other kind of demonstration.²⁸

Two historical facts may echo something about the diffusion and transmission of Barozzi's ideas. That Galileo owned the Opusculum is known from the description of his library by Antonio Favaro.²⁹ Also, in the “Prolegomena” to his commentary on Euclid's Elements discussed below, Christopher Clavius admitted to the inspiration of Barozzi and his work on Proclus. Thus, Barozzi's reluctance to take issue with Piccolomini's Aristotelian theory of abstraction – in spite of his rejection of other parts of Piccolomini's reading – and his preference for blurring Proclus's harsh critique of this theory allowed mathematicians of different convictions to draw upon his ideas without giving full account of the profound differences between the Platonic and the Aristotelian position on the mathematical entities.

A more radical opposition to Piccolomini, entangled with a more radical reading of Aristotle in Proclean terms, characterizes the work of Pietro Catena, who held the chair of mathematics in Padua for almost thirty years (1547–1576) and developed his ideas in three works, all touching upon the relation between mathematics and philosophy, their objects, their demonstrations, and their status in the hierarchy of the speculative sciences.³⁰

The main thesis common to Piccolomini and Barozzi, but rejected by Catena, was that of the middle position of mathematical entities, for which Catena substituted a view of mathematical universals as predicates of the rational soul that he derived from his Platonic reading of the Posterior Analytics. Unlike physical phenomena, which are perceived primarily through sense experience, mathematical entities are pure intelligibles, constituted only through a rational process of thought and in no need of the senses to be recognized.31 Catena believed in their innateness and invoked the theory of reminiscence to justify the pure intellectual nature of their recognition.³² But more generally, Catena subscribed to the view that all knowledge was first anchored in universals preexisting in the intellect, rather than in abstraction from particulars. Above all, the objects of geometricians – lines, points, and planes – originate in the soul, not in sense images.³³

The clue to Catena's position may be his presupposition that any particular participates in a universal mathematical nature, although particulars cannot be reduced to such entities, because they also contain other elements in which they are distinguished as particulars. Science, according to Catena consisted essentially in the application of universal intelligibles to particulars, thus transforming recognition into actual knowledge.

To illustrate this process Catena used Aristotle's example of a bronze triangle recognized as participating in a universal (in this case a mathematical triangle) through an examination of the equality of the sum of its angles to two straight ones. The mind presupposes a bronze triangle and gradually excludes some of its properties (its bronzeness, for example) until it realizes that with the elimination of the three sides the property of the sum of the angles disappears.³⁴ Aristotle and Euclid agreed on this idea of science, but they used different logical procedures – syllogism and mathematical demonstration respectively – as their practice. Therefore, Aristotle, in his Posterior Analytics, referred to two kinds of inductions,³⁵ although he did not clarify the differences between them. Catena took upon himself to do just that, and this was probably the most original part of his contribution. Definition played an essential role in a syllogistic procedure, but a mere classificatory role in a geometrical one. To show this Catena picked up again Proclus's and Piccolomini's example of Euclid 1, 32,³⁶ only to claim the opposite of their conclusion, namely that in spite of its difference from a demonstration potissima, it still led to true, certain, and actual knowledge of the particular.37 Thus, Catena agreed with Piccolomini about the difference between mathematical and syllogistic demonstrations, but insinuated – without actually articulating this conclusion clearly – that only mathematical demonstrations could serve in the discovery of new truths, whereas syllogisms were effective in the orderly presentation of old ones.

Catena's interpretation of the concept of universal science, which he deemed common to Aristotle and Euclid, enabled him to include the mixed mathematical disciplines within this framework, without any need to distinguish their status from the rest of the sciences. The basic difference was that in the constitution of corporeal entities (such as rays of light, for example, compared to geometrical lines) as objects of science not only pure rational thought but also experience played a major role. Still, the principles of the science as well as its basic concepts were universal intelligibles and the causes discovered were the product of rational discourse, not of the senses.³⁸

The positions of Piccolomini, Barozzi, and Catena – who were the first to construct some archetypal strategies in early modern politics of knowledge – may be summed up as follows: Piccolomini recognized the superior certainty of mathematics but was most interested in bounding it within a separate, autonomous domain. The high degree of certainty attributed to mathematics was related by Piccolomini to the inferiority of its objects, which were, in his perception, the most simplistic in the ontological sense and therefore the easiest to acquire knowledge about. In this sense they were radically differentiated from the objects of natural philosophy, representing a higher degree of complexity and allowing for intrinsic knowledge of their essence through a much more complex rational process culminating in the demonstration potissima. The boundary between mathematics and philosophy was clearly at the center of Piccolomini's interest. Barozzi, using a Proclean reading of mathematical entities, focused his interest on proving the scientificity – and not just certainty – of mathematics by stressing its middle position (medietas) between philosophy and metaphysics, both in the order of nature (ontology) and in the order of knowing (epistemology). This doctrine he deemed common to Aristotle and Plato and was the source of recognizing mathematical demonstrations as equivalent to demonstrations potissimae. The compatibility between the objects of a science and the kind of demonstrations it used led, according to Barozzi, to the inescapable recognition of the higher status of mathematics relative to philosophy. However, the question of the use of mathematics in natural philosophy was not really predominant in his writing. In fact, De Pace's interpretation emphasizes that it played a minor role in his mind.39 But this was the main focus of Catena's arguments. Attributing a common ideal of science to Aristotle and Euclid – in spite of a deep divergence of methods – Catena thought that mathematical demonstrations were superior to demonstrations potissimae as instruments of acquiring new knowledge. Hence, he claimed that knowledge of the world was only possible through the use of mathematical methods.

All three writers presupposed some kind of agreement between Aristotle, Plato, and sometimes Proclus on the certainty of mathematics, in spite of their different and sometimes oppository readings of their sources: Piccolomini stressed the agreement of Aristotle and Proclus on the middle position of mathematics – wrongly attributing to Aristotle a theory of abstraction – and used Proclus's occasional critique of some Euclidean proofs to claim the incompatibility of mathematical demonstration and demonstration potissima. Barozzi attempted to reconcile the Aristotelian and Platonic position on the medietas of mathematics but ignored the Aristotelian theory of abstraction. Catena attempted a Platonic reading of the Posterior Analytics in order to prove an idea of science common to Aristotle and Plato. It is thus clear that all three writers attributed a major role to the ancient authorities in their attempts to gain legitimation for their respective positions.

The debate over the status of mathematics in the sixteenth century signaled the beginning of a structural shift on the medieval map of knowledge toward a different understanding of the place of mathematics. The change was initiated by a variety of separate developments such as the activities of mathematical practitioners in Italian courts, the renaissance of Greek mathematical texts, the spread of Archimedean discourse, the emergence of Copernican astronomy, and the rise of the new algebra. But it was among the Jesuits that the first efforts were made to assimilate all these changes into an institutionalized research program with a special cultural and educational vocation. It is also in the context of the Jesuit program that the debate over the certainty of mathematics was appropriated and developed as a source for a variety of practices in the politics of knowledge.

The history of the efforts of Jesuit mathematicians to create for themselves a separate identity within a humanistic-scholastic educational project and to secure their status vis-à-vis the theologians and philosophers of the society has not yet been written. Recent historical scholarship, however, points to tensions between philosophers and mathematicians – concerning the scope and place of mathematics in the Jesuit curriculum, the interpretation of cosmological phenomena such as the nova of 1604, the motion of the Earth, the critique of Archimedes, etc.40 – all of which touched upon the problematic boundary between mathematical and philosophical discourse. In some of my previous work I have argued that the Jesuit policy of constructing boundaries between fields of knowledge functioned as a cultural mechanism of control enabling the reproduction of a Thomistic framework, in spite of the transgression of its boundaries which became common practice among Jesuit mathematicians.⁴¹ The traditional mathematical disciplines were the science of numbers, the science of continuous magnitudes, and pure and mixed mathematics. Two examples may illustrate the kind of dynamics created by the Jesuit appropriation of new areas of mathematical research that tended to undermine these traditional boundaries within the mathematical disciplines (or between mathematics and natural philosophy) and the practice of keeping the boundaries, which was also exercised by Jesuit mathematicians.

The Jesuits’ involvement is particularly interesting in two areas. The first concerns their role in the reception, assimilation, and transition of Vietà’s algebra. Whereas Clavius himself praised algebra, publishing his textbook on the subject in 1608, his work did not really assimilate the “new art” and the innovations of recent Italian algebraists.⁴² His student Staserio, however, who dedicated much of his life to building up the mathematical program of studies in the Jesuit college in Naples, succeeded in integrating the new algebra into the Jesuit curriculum.⁴³ Baldini's historical researches have taught us that around 1600 the Collegio Romano became a center of debate over the innovations springing from the use of algebra in solving geometrical problems.⁴⁴ The intense preoccupation with algebra could not take place, however, without challenging the boundary between discrete numbers and continuous magnitudes, as has been shown by Jacob Klein, and more recently by Lachterman.45 Nevertheless, the famous controversy between Paulus Guldin and Cavallieri over the method of indivisibles⁴⁶ echoes the tendency of many Jesuit mathematicians to defend the traditional disciplinary divisions, in spite of their interest and even promotion of research topics that clearly endangered them.

No less significant was the involvement of Jesuit mathematicians in the Archimedean revival that took place exactly in the same years. The origins of this involvement go back both to Torres – the first professor of mathematics at the Collegio Romano – who was a student of Maurolico, and to Clavius who was in close contact with Maurolico and planned to publish his manuscripts.⁴⁷ It is Baldini, again, who has pointed out the impact of these connections with Maurolico and through him also with Commandino's work, which was felt both through the emphasis on geometry and on Archimedean problems of measuring as well as through an interest in the Archimedean statical tradition of mechanics, thoroughly brought into contact with the medieval dynamical tradition in the context of the Jesuit “mixed mathematical science.”⁴⁸ A most compelling piece of evidence for the new horizons opened up by the integration of the Archimedean tradition may be found in the plurality of works on centers of gravity, written by Jesuit mathematicians at the turn of the seventeenth century and later on.

It is well known that Clavius wrote on centers of gravity, but his work was not preserved. Staserio, Villalpando, Luca Valerio, and later on Guldin and Saint Vincent (all of them trained by Clavius in Rome) wrote on centers of gravity, testifying to the continuation of that tradition in Jesuit circles. Work on centers of gravity, however, was situated exactly on the borderline between mathematical and physical discourse. In fact, the concept of “weight” itself was conceptualized in qualitative terms in the context of Aristotelian physics and in terms of “quantity” in the Archimedean mathematical one. This exemplifies a clear point of interference and of potential tensions between mathematicians and philosophers.

The institutionalization and success of a mathematical program of studies and research was the context in which the debate on the certainty of mathematics was replicated, intensified, and developed in Jesuit circles. Benedictus Perera was the first to elaborate and deepen Alessandro Piccolomini's arguments with clear implications for the status of the mathematicians of the society. This strategy was countered by a rather intense campaign of Christopher Clavius, the architect of the program and the founder of a Jesuit mathematical tradition, who replicated some of Barozzi's arguments in his effort to buttress the position of the mathematicians. A certain climax was achieved, though, in the work of Josephus Blancanus who developed an ontology of mathematical entities in an attempt to ground the mathematical disciplines in a firm philosophical basis. In briefly reconstructing these three positions as strategies in the Jesuit politics of knowledge, my aim is to clarify the background against which Galileo's later rejection of the discourse on mathematical entities should be understood.

Perera developed his position in long passages of the widely circulated De Communibus Omnium Rerum Naturalium Principus, first published in 1576, and reprinted nine times until the end of the century.49 Departing from Piccolomini's suggestion that “quantity” inheres in prime matter as indeterminate extension independently of any substantial form, Perera securely anchored this contention in the Greek commentators and in Averroes.⁵⁰ He thus deepened the ontological dimension of Piccolomini's thesis and inferred the fully consistent conclusions from it. Stressing the radical separation of quantity not only from sensible substances – as did Albertus Magnus, Thomas Aquinas, and other Latin commentators⁵¹ – but from any substance, he then attempted to prove their complete disjunction from real physical or metaphysical essences. Quantity thus became fully extrinsic to form. Hence it was the most superficial dimension of things, easy to separate and abstract, although instrumental for understanding certain aspects of them. Perera illustrated his contention through the example of the mathematical property of the sphere touching the plane in one point only. Whereas this is true for the sphere as a mathematical – or abstract – extension, he argued, it is not true for the sphere as physical extension.⁵²

Perera's rejection of the theory of mathematical “medietas” – adopted by Piccolomini from Proclus but interpreted in an Aristotelian sense as abstraction from sensible matter – was effectively carried out through an attack on the Platonic doctrine of reminiscence, essential for the idea that mathematical entities are innate in the human soul. God has given us a human soul that is “tabula nuda,” not inscribed with any contents and capable of learning all sciences, Perera argued.53 For if knowledge is truly acquired through reminiscence, how would one explain the necessity of the senses – which even the Platonists cannot deny? And if the senses are necessary for acquiring knowledge, then it is not possible to maintain the theory of reminiscience.⁵⁴

Perera's radical rejection of the middle position of mathematical entities and his elaboration of the ontology of indeterminate extension as inhering in prime matter – independently of any substance – constitute his main contributions to the development of Piccolomini's position. If quantity was disconected from substance, then it had nothing to do with the explanation of causes, not even formal causes. Furthermore, Perera followed in the footsteps of Piccolomini denying mathematical demonstrations the status of a model for demonstration potissima and criticizing Euclid I, 32 as a non-causal and nonessential proof. Who cannot see, he argued, that the geometer proves the sum of the angles of a triangle equaling two right ones through a construction of the external angle, which cannot be considered a cause since it is completely accidental to the essential nature of the triangle.⁵⁵

Perera's negation of the innate nature of mathematical entities together with his peculiar understanding of geometrical demonstrations led him to a clearer and more radical distinction between the certainty of mathematics, which he explains by its rigorous structure, accepting and even strengthening the arguments to substantiate it, and the scientificity of demonstrations potissimae, which are the only ones capable of treating real, material, physical substances and heading to true conclusions. Thus, in the order of the nobility of the sciences, mathematics was the most inferior, according to him, both because of the simplicity of its subject matter, and because of the kind of demonstrations it used. Moreover, the rigorous structure of mathematics secures its status as a discipline, but its objects and demonstrations excluded it from the realm of the sciences:

For the mathematician neither considers the essence of quantity, nor treats of its affections as they flow from such essence, nor declares them by the proper causes on account of which they are in quantity, nor makes his demonstrations from proper and per se but from common and accidental predicates. It is my opinion, that the mathematical disciplines are not proper sciences.⁵⁶

No wonder that Perera's conception of the mixed sciences was completely instrumental. When the astronomer thinks of the magnitude, shape, form, and motion of the heavens, he is not preoccupied with true causes that explain the nature of things, Perera claimed, but with some reasonings that can save the appearances. This, according to him, was the nature of eccentrics, epicycles, some irregular motions of celestial bodies, trepidation, etc.57

The first chapter of Clavius's “Prolegomena” to his Commentary on Euclid's Elements⁵⁸ reads as a direct and concise answer to Perera's arguments. First, he argued, the meaning of the word Mathesis in Greek was discipline, or doctrine, for only the arts of quantity used causal and potissimae proofs. The Pythagoreans and the Platonists believed that rational souls in some sense contained determined number, and therefore they could acquire these disciplines. Countering Perera's rejection of the theory of reminiscence, Clavius, quoting from the Meno, suggested that the process of remembering, was, in fact, a process of disciplining. This was understood by Plato in terms of a Socratic interrogation – which he exemplified in the story of Meno – and led to the ascent of the soul toward eternal truths. Clavius expressed a certain ambivalence toward Plato's theory, which presupposed, according to him, the migration of souls from one body to another – a possibility condemned as erroneous and false by Christian doctrine. Nevertheless, he massively relied upon quotations from Plato and from Proclus with which he became acquainted through the edition and interpretation of Barozzi. Following Barozzi too, however, he did not exclude Aristotle from his list of authorities, emphasizing the compatibility of mathematical disciplines with the canons of the Posterior Analytics and their rigorous structure – using only preknown principles and proved propositions – which justified their status as doctrine or discipline.⁵⁹

Praising the nobility of the mathematical sciences in the third chapter, Clavius emphasized the certainty of their demonstration, which he contrasted with demonstrations practiced in the other sciences. Whereas those were incapable of actually demonstrating their claims (a fact resulting in endless unresolved disputations and in the plurality of philosophical sects) Euclid's propositions were unambiguous, and the certainty of mathematical demonstrations led to the pure truth.⁶⁰ Clavius supported this contention with a quotation from Plato's Philebus, where the truth of geometry is connected to supreme goodness.61

Moving, in the fourth chapter of the “Prolegomena,” to the utility of mathematics, Clavius departed from their utility in administrating and governing the public sphere to their necessity for the study of all other disciplines. First he quoted Proclus showing how mathematics facilitated the passage from physical, sensible, and thus murky reality to the clear, enlightened reality of metaphysics.⁶² In Platonic terms, the passage from the sensible to the intelligible world was called ascent to the contemplation of divine things, and for this ascent the mathematical disciplines prepared the soul.⁶³ Last, Clavius turned to the educational context, quoting both from Philebus and from the seventh book of the Republic, to stress again the necessity of mathematics as a basis for all other studies, as well as for leadership of political life in a city state.⁶⁴

It is Clavius's strategy, throughout the “Prolegomena,” to indicate the basic agreement between Plato and Aristotle on the nobility, utility, and necessity of the mathematical sciences, even though their respective justifications may sometimes be formulated by different vocabularies or anchored in different philosophical world views. This means that the simple dichotomization between Platonists as lovers of mathematics and Aristotelians as ignorant in this realm did not hold true for Jesuits mathematicians,⁶⁵ who refused to choose between Platonic and Aristotelian legitimation of their sciences, preferring to recruit both in the process of constructing their professional identity.

The controversy between Perera and Clavius represented in the hidden (but obvious) counterarguments of Clavius's “Prolegomena” testifies to the need of both philosophers and mathematicians to recruit ancient authorities for strengthening their positions. Plato and Aristotle were read and interpreted in accordance with contemporary needs, and their works functioned as imaginary constructions. Rather than a source of inspiration for mathematical innovation, they were used as topics for the symbolic capital contained in their figures.

In response to Piccolomini's and Perera's attempts to introduce a breach between mathematical entities and real, substantial forms, Clavius, relying upon Proclus's judgment, contended that the objects of mathematics, although considered in abstraction from matter, treated things immersed in matter. Adopting Barozzi's thesis of the medietas of mathematics he conceptualized mathematical entities as ontologically bridging between the complete abstractness of metaphysical objects and the full sensibility and materiality of physical ones:

Since the mathematical diciplines deal with things which are considered apart from any sensible matter, although they are immersed in material things, it is clear that they hold a place intermediate between metaphysics and natural science, if we consider their subject matter. For as has been rightly shown by Proclus, the subject of metaphysics is seperated from any matter, both from the point of view of the thing itself, and from the point of view of reason. The subject of physics is truly connected to sensible matter, from the point of view of the thing itself as well as from the point of view of reason. And since the mathematical disciplines consider their subject separately from any matter, even though it [matter] is found in the thing itself, it is established that they are intermediate between two.66

The chapter on the division of the mathematical sciences in Clavius's “Prolegomena” aimed at redrawing and broadening the traditional map of knowledge, to fit better the project of Jesuit mathematicians. In restructuring the field Clavius drew upon the argument about mathematical entities, being immersed in material things, although considered in abstraction from it. The Pythagoreans and quite a number of philosophers believed that the mathematical disciplines essentially consisted of four branches, each having a specific subject: arithmetics with discrete numbers, geometry with continous magnitudes, music with numbers in relation to voices, and astronomy with continous magnitudes in relation to the motion of celestial bodies. However, there was another division, anchored in the writings of other ancient authors – especially Geminus and Proclus, according to Barozzi's interpretation. The first considered mathematical entities as purely intellectual and absolutely seperated from matter. But in truth, mathematical entities belonged to things connected with matter.⁶⁷ Without explicitly stating this, Clavius's juxtaposition of “intellectibles”versus mathematical entities immersed in material things seems to provide the justification for augmenting the number of mathematical diciplines concerned with physical phenomena to six, namely astrology, perspective, geodesy, canonics (music), suppotatrics (practical arithmetics), and mechanics, each being further divided into more specific branches.

There is a sense in which Clavius's practice of restructuring the map of knowledge can be derived from his (quasi)theoretical conception of mathematical entities as inherent in things immersed in matter. His theoretical arguement, however, was not anchored in any wide philosophical framework. Rather, it was an isolated insight, a reworking and reinterpretation of one passage from Proclus. His real justification came from the practice of mathematics itself. His elaborate descriptions of the various branches of knowledge pertaining to the physical world that have been successfully treated by mathematicians with mathematical methods was his proof. His insistence on the necessity and utility of mathematics for studying all other diciplines, which he supported with quotations from many ancient writers, Christians (St. Peter and St. Augustine) as well as non-Christians (Plato, Aristotle, Proclus, and others), was rheotrical by nature, based on repetition and accumulation of historical evidence, not on scholastic subtleties.68

More than anything else it is Clavius's style of arguing in many contexts, measured against what is known about his scientific career, that justifies the interpretation of the “Prolegomena” in terms of a cultural practice more than in terms of a philosophical justifications of the status of mathematics. His text comprised an attempt to restructure the map of knowledge so that more space be allowed for the discourse of mathematicians and thus deepening and stabilizing their authority compared to that of the philosophers. Departing from the Aristotelian premise that a science is defined by its specific subject matter, and by the kind of demonstrations it uses, he interpreted the nature of mathematical entities as a bridge between physical and metaphysical ones, being immersed in sensible matter and considered in abstraction from it. But although a boundary was thus created between natural philosophy and the mathematical sciences on the one hand, and between mathematics and metaphysics on the other hand (a boundary necessary for securing the autonomy of mathematics), Clavius's main strategy was to narrate the successes of mathematics in dealing with problems of the concrete physical world throughout history and in the present and to label anew as many mathematical subdisciplines as he could. Furthermore, although Clavius identified arithmetic and geometry as the two main mathematical fields of knowledge, he abstained from drawing too clear a boundary between pure and applied mathematics, suppressing the term “mixed sciences,” which he had used in his preface to Sacrobosco's Sphere.

Compared with Clavius's “Prolegomena,” Josephus Blancanus's “Treatise on the Nature of Mathematics”69 was a much more comprehensive attempt to rebut the attacks of opponents in an articulated, well-informed way, relying upon philosophical and metaphysical thinking of the period. Blancanus's text signals the crystallization of a meta-discourse among Jesuit mathematicians concerning the status of their field of knowledge and its justification.

Blancanus's point of departure, like that of Clavius's, was the subject matter of the mathematical disciplines, which he attempted to distinguish both from that of natural philosophers as well as from that of the metaphysicians. However, the content of his arguments differed substantially from that of his mentor. Recognizing Perera's contention that the subject matter of metaphysical discourse is quantity but rejecting Perera's judgement about the nonessential nature of that quantity and hence his denial of the status of mathematics as science, Blancanus defined a special kind of quantity called “delimited” or “finite” quantity (quantitas terminata), which he distinguished from Perera's “indeterminate quantity” (quantitas indeterminata). The entities considered by the mathematicians, according to him,

are entirely different from those that the natural scientist and the metaphysician consider in quantity absolutely…from this delimitation there result the various figures and numbers which the mathematician defines and of which he demonstrates various theorems.⁷⁰

Drawing upon Clavius's insight that mathematical entities inhere in things immersed in matter, even though they are considered separately from it, but following much more closely Aristotle's own argumentation about the problem, Blancanus used the Aristotelian terminology concerning the abstract matter of mathematical entities which Aristotle had called “intelligible matter”:

But this [delimited quantity] is the quantity that is usually called intelligible matter, in contradistinction to sensible matter, which concerns the natural scientist, for the former is seperated by the intellect from the latter and it is perceived by the intellect alone.⁷¹

However, it was precisely because of the abstract nature of intelligible matter that mathematicians had been attacked for the nonexistence of mathematical entities. Blancanus answered to such a projection in the following terms:

… many [people] object to mathematicians that mathematical entities do not exist, except only by the intellect. However, we should know that even if these mathematical entities do not exist in that perfection, this is merely accidental… Therefore, even though these [perfect mathematical figures] do not exist in the nature of things, since in the mind of the Author of Nature, as well as in the human mind, their ideas do exist as the exact archetypes of all things, indeed, as exact mathematical entities, the mathematician investigates their ideas, which are primarily intended per se, and which are [the] true entities.72

To the contention of some philosophers that mathematicians use suppositions and argue in a mere accidental way, Blancanus responded that mathematical definitions were essential – not just nominal – and that only in mathematics is it possible to give definitions in which

the entire nature of the subject is primarily given to us: So it follows that the mathematical sciences proceed from what is better known to us as well as from what is better known by nature… And this is the reason why geometrical demonstrations are always so efficient and possess the highest degree of certitude.⁷³

Arguing for the reality of mathematical entities and the essentiality of mathematical definitions constitutes the core of Blancanus's “apologia.” The certainty and scientificity of mathematical demonstrations stem naturally from the nature of the objects, which, he emphasized, no writer had ever doubted before Piccolomini, who had very few followers, nobody other, in fact, than Perera, Fonesca, and the Coimbran commentators. The rest of the tradition – Aristotelians and Platonists alike (and here Blancanus was following Clavius's narratological techniques) – all admitted that mathematical proofs were the strongest given in any science.

The implications of Blancanus's insistence on elaborating a sound “metaphysical” foundation for justifying the mathematical disciplines were uncertain from the point of view of the mathematicians’ politics of knowledge. No doubt Blancanus's “apologia” was a much stronger response to the philosophers’ critique than Clavius's pragmatic arguments. In Jesuit culture it could have meant a real resource for legitimation. However, Blancanus also tied up the fortunes of the mathematicians’ project to a philosophical discourse and to an ontology that would soon become obtrusive to major trends developing within the mathematics of his time, especially to the use of indivisbles and infinitesimals in the practice of mathematicians. One immediate effect of his vision was already apparent in his own text: The boundaries imposed in his treatise between mathematics and philosophy and between pure and applied mathematics were much more effectively constructed.

First we are going to discuss pure mathematics, i.e., geometry and arithmetic, which differs in kind from applied mathematics, namely, astronomy, optics, [perspectiva], mechanics and music. Quantity abstracted from sensible matter is usually considered in two ways. For it is considered by the natural scientist and the metaphysicians in itself… but the geometer and the arithmeticians consider [quantity] not absolutely, but insofar as it is delimited…74

This may have expressed the need to conform to the general policy of the Jesuit order, already implemented in the Ratio studiorum, which used the construction of boundaries as a strategy of control. In any case, the policy endorsed in Blancanus's text differed in nuance from Clavius's philosophically less committed solutions, which enabled both conformity with the policy of the order and maneuvering of the boundaries according to the needs of the mathematicians.

II. GALILEO'S MATHEMATICAL STRATEGIES: BETWEEN “MIXED MATHEMATICS” AND MATHEMATICAL PHYSICS

Galileo's early work should be read against the background of the debate on the certitude of mathematics and its appropriation by Jesuit mathematicians in the attempt to legitimize their ever broadening interests. The work on centers of gravity (Theoremata Circa Centrum Gravitatis Solidorum, 1585–7), the Bilancetta (1585–6), and even the project of De Motu (1590), which intended to combine Aristotelian dynamics with Archimedian statics, perfectly suited the spirit of the field of knowledge delineated by Jesuit mathematicians. As mentioned above, writing on centers of gravity was rather popular among mathematicians of the Jesuit Society, and at this stage Galileo was not exceptional in choosing this topic. Neither did Galileo's range of problems and applications exceed the realm of pure geometry. No attempt was made to cope with gravity in a physical sense or even with the effect of weight at different distances from the fulcrum. Rather, Galileo restricted himself to treatment of pure geometrical entities.

Slightly different was the case of Bilancetta, which used the theory of the lever and was concerned with its application to various practical problems. Here the objects of discourse were real and material, having weight and varying in volume and even in the medium in which they were immersed. In their different styles of arguing Galileo's first two texts corresponded to the two main directions in which Archimedes's work was received in sixteenth-century Italy: one axiomatic and purely geomatrical, springing from Archimedes's On the Equilibrium of Planes, and the other more physical, local, ad-hoc, and stemming from the discussion of On Floating Bodies.75

Galileo's project of studying motion as it emerged in the premature text of the De Motu, however, already transgressed, or even broke through, the boundaries between mathematics and natural philosophy which had only started to become a sensitive issue in the Jesuit politics of knowledge during the same years. Natural motions of terrestrial bodies were certainly not typical subjects of mathematical discourse in the last decade of the sixteenth century. At the same time, expanding the field of application of Archimedean models was not as unknown strategy.

Galileo's project consisted in an attempt to offer a unified explanation of all motions – natural up and down motions, as well as violent projectile ones – in mathematical terms, by using Archimedean models to cope with problems in the sphere of Aristotelian dynamics. Eventually this attempt failed to explain one basic feature of the motions, namely acceleration. Galileo tended to use one Archimedean model – the hydrostatic – to explain the difference in the velocities of bodies moving up and down as a difference between their specific weights in relation to the mediums in which they moved. At the same time he used the balance model to visualize the analogy between rest (equilibrium) and up and down motions and to experiment with the same body along differently inclined planes. The two models were not compatible, as the one considered specific weights, whereas the other dealt with absolute weights. Moreover, the full effect of weight on motion was not taken into consideration, as Galileo did not include in his balance model the distance of the weight from the center of the system. The velocity, in any case, came out of the theory as directly proportional to the body's (specific) weight. Hence, it could only be uniform. But that was incompatible with the facts known from experience.

Galileo proposed two ways of coping with this difficulty, which in retrospect read more as excuses for a failure rather than as real solutions to his problems. First, he suggested that acceleration was an accidental feature of motion, caused by the levity impressed in the body externally (either by the hand throwing a projectile or a property thought to be kept in the body from previous elevation) and intensifying its motion in its first stages.76 This explanation pushed him back to treating levity as a substance, not as a state relative to gravity, thus undermining his radical critique against the Aristotelian physics that was one of the main targets of the his text. But the second way of treating the supposed “accident” of acceleration was even worse, because it put in doubt the rationale of his whole enterprise. The direct proportion between the velocity of a body and its weight could not be observed by the one doing the experiment, Galileo claimed.⁷⁷

In admitting his failure to identify the mathematical results expected from his theory in experience, Galileo, in fact, invoked the main objection to the scientificity of mathematics which had first been used by Piccolomini and entered the circles of Jesuit philosophers mainly through Perera. In the context of the complicated field of positions concerning the status of mathematics, and the arguments adopted by the different participants, Galileo's admittance of the difficulty of mathematical reasoning to capture processes pertaining to material objects could be read as a declaration of defeat.

Against this background, the project of his Mecaniche,⁷⁸ seems as a return to the boundaries of mathematics accepted within the original discourse of the “mixed sciences,” neglecting the problem of natural motion and free fall and concentrating on a problem in the traditional realm of mechanics, namely, the force necessary to elevate a weight along planes of different inclinations. This force was now differentiated into two components: the weight of the body and the distance from the center of the system, expressed the body's inclination to fall. The inclination to fall – conceptually differentiated from gravitas by the term moment (momento) – was measured by setting two limits: maximum momento on the perpendicular plane and minimum on the horizontal plane. Thus, the law of the moment stated that moments on the inclined plane and the perpendicular plane related to each other as the perpendicular line is to the inclined line. Moment, then, was constructed as a purely geometrical entity.79

Following Galluzzi, I would like to emphasize that the Mecaniche embodies a different type of project than the De Motu: Unlike De Motu, Galileo's Mecaniche did not present a quest for a unified explanation of all motions. Rather, it was an attempt to ground the study of motion and build it upon mechanical foundations, rooted in the traditions of the “mixed sciences,” which acquired their renewed legitimacy in the environment of Jesuit mathematicians.⁸⁰ This means that the question of velocity remained on the margins of the discussion, coming up either as an addition of momento to weight or as an effect of a force that was constant. Velocity, then, if discussed at all, could only be conceived as uniform velocity. Galileo's use of the term momento, however, points to the possibility of translating it into dynamical terms. Thus translated, the law of the moment would entail that in determinate periods of time the body would pass distances on the inclined plane that relate to the distances on the perpendicular like the inclined line is to the perpendicular. Still, because the velocity was conceived as a product of a constant force, the prominent fact of acceleration could not be integrated into this framework. That, probably, was the origin of the dead end that forced Galileo to go beyond his original association of velocity with constant forces and beyond mechanical motions toward a different type of mathematical analysis of natural motion.⁸¹

Galileo's split from the “mixed sciences” and his conscious attempt to create an alternative science of mechanics – which brought about his growing estrangement from the discourse of Jesuit mathematicians – will be illustrated, in this paper, by a detailed analysis of his treatment of local motion in some passages of the Dialogue (1632).⁸² Traditionally these passages have been read as an expression of the miraculous birth of modern science in Galileo's text, a reading that used to emphasize the break between Galileo's early and mature science, and obviously between the old and the new science. Recent readings, however, criticizing, developing, and documenting suggestions already made in the late nineteenth century have tended to stress Galileo's embeddedment in Aristotelian science.83 Whereas nineteenth-century historians such as P. Duhem discovered the continuity between Galileo's work and that of the fourteenth-century calculators, for example, contemporary historians have stressed his anchorage in the work of Jesuit philosophers and mathematicians.⁸⁴ Continuing this last line of argument, my reading of selected passages of Galileo's Dialogue aims to represent, and interpret in a more contextual way, suggestions first made by Galluzzi in his Momento and then developed by Renn and others.⁸⁵

In this reading, the attempt to broaden the discourse on mechanical motion by applying some of its concepts and techniques to the study of natural motion eventually led Galileo to a theory of acceleration in which weight was neglected as a cause and velocity moved into the center of discussion. But velocity was now thought of as the sum total of degrees of velocity, and it was represented geometrically by the infinity of parallel lines making up the surface of a geometrical figure. Thus, Galileo's project may be seen as an Aristotelian-Archimedean synthesis that violated the basic rules of both discourses. A reading of passages from the Dialogue in terms of this “problematique” is the focus of the second part of the paper.

As is well known, the first day of the Dialogue opens with a discussion of Aristotle's “general discourse upon universal first principles” (18), which leads, rather quickly, to a critical examination of his fundamental distinctions between two kinds of natural motions – along straight and circular lines – and also between two kinds of motions along straight lines: natural up and down motions on the one hand and violent motion on the other. Salviati raises many objections against this discourse, complaining that it seemed as if “he [Aristotle] was pulling cards of his sleeve, and trying to accommodate the architecture to the building instead of modeling the building after the precepts of architecture” (16) and that “whenever defects are seen in the foundations, it is reasonable to doubt everything else that is built upon them” (18). Suggesting that “basic principles and fundamentals must be secure, firm, and well established, so that one may build confidently upon them” (ibid.), he raises the reader's expectations for a foundational discourse built upon alternative “basic principles with sounder architectural precepts” (ibid.). What follows, however, does not really meet such expectations.

Every body constituted in a state of rest but naturally capable of motion will move when set at liberty only if it has a natural tendency towards some particular place; for if it were indifferent to all places it would remain at rest, having no more cause to move one way than another. Having such a tendency, it naturally follows that in its motion it will be continually accelerating. (20, my emphasis).

This passage opens a long digression from the critique of Aristotle that constitutes the major part of the first day to a modified, but still Aristotelian, discussion of accelerated motion, a digression in which Galileo's alternative is condensely presented for the first time. The passage consists of two statements: 1) The cause of motion is a natural inclination toward a place. 2) Natural motion is accelerated. In another famous passage in the Dialogue, Galileo reveals to the attentive reader that invocation of “nature” in scientific practice is always an indication for lack of explanation:

… we do not really understand what principle or what force it is that moves stones downward, any more than we understand what moves them upward after they leave the thrower's hand, or what moves the moon around. We have merely … assigned to the first the more specific and definite name “gravity” … and as the cause of infinite other motions we give “Nature.” (235)

In the light of this confession it looks as if the creation of an alternative mathematical science of motion involves resignation of the effort to suggest causal explanation either to motion or to acceleration. Instead, already at this early stage Salviati offers a conceptual analysis of the continuum that he applies to accelerated motion:

Beginning with the slowest motion, it [a moving body] will never acquire any degree of speed without first having passed through all the gradations of lesser speed – or should I say of greater slowness? For, leaving a state of rest, which is the infinite degree of slowness, there is no way whatever for it to enter a definite degree of speed before having entered into a lesser, and another still less before that. It seems much more reasonable for it to pass first through those degrees nearest to that from which it set out, and from this to those farther on. But the degree from which the movable body began to move was that of most extreme slowness, that it to say from rest. (21)

Salviati suggests that acceleration involves a continuous increase or decrease of degrees of speed (or slowness). Sagredo, however, demands an explanation to the obvious paradox (finally formulated by Salviati) such a description entails: How can a body pass infinite degrees of slowness (or speed) in finite time? Salviati tries to “solve” this difficulty by saying that “the movable body does pass through the said gradations, but without pausing in any of them” (20). This “solution” conceals a lifetime of reflection on problems of infinity, the continuum and indivisibles that Galileo could not settle. Used here as a strategy of excluding further discussion, Salviati, however, takes up the opportunity to make a very condense presentation of the core of Galilean innovations in the field of the new science of motion, stemming from his new conceptualization of impetus and from the choice to focus on acceleration as the central phenomenon of the analysis of motion.

This choice leads to the privileging of a few limited areas of research of local motion, especially the falling of a stone, namely, free fall, and the motion of a cannon ball, namely, projectile motion. Through a short discussion touching upon these subjects Salviati attempts to engage his hearers’ and interlocutors’ interest and consent by claiming the following:

(1) That free fall and projectile motion are accelerated or deccelerated. He acquires quick consent for this claim by translating his concepts of “acceleration” and “degrees of speed” (and slowness) into the well-known but poorly defined traditional terms of impetus and velocity: “Tell me,” he asks Sagredo, “if you have any trouble granting that the ball, in descending is always gaining greater impetus and velocity.” The obvious answer to which is: “I am quite confident of that” (22).

(2) That the impetus acquired in fall is enough to lift the falling body up to the same height from which it started falling. This is a much more problematic assumption, for which Galileo acquired a real proof only after publication of the Two New Sciences, but which he used as a postulate there. Here the claim is justified by pointing out experiments that could confirm it.

From these two statements Salviati concludes that two equal bodies falling from the same height, one in free fall and another on an inclined plane, will arrive with the same “impetus” – which we have seen him treating as synonymous with “velocity”:

Figure 3.1.

So will you not put an end to your difficulty by conceding that two equal movable bodies, descending by different lines and without any impediment, will have acquired equal impetus whenever their approaches to the center are equal? (23, my emphasis)

Sagredo's difficulty in understanding the claim leads to Salviati's explication and to the use of a geometrical figure (see Figure 3.1) to represent free fall by the perpendicular and descent along the inclined plane by the oblique. The geometrical representation, serving here as a tool for clarifying the meaning of concepts,⁸⁶ allows Salviati to try and disperse the ambiguity with which the term “impetus” has traditionally been stricken, and which still pervades Galileo's texts:

I ask you to concede that the impetus of that which descends by the plane CA, upon arriving at point A, would be equal to the impetus acquired by the other at point B after falling along the perpendicular CB. (23)

If before impetus and velocity were interchangeable, here equal impetuses seem to unambiguously mean that the two equal bodies acquire the same degree of speed upon arrival. Sagredo, however, responds by first conceding the conclusion, and then, bringing back the ambiguity: “In fact,” he says, “they have both advanced equally toward the center” (Ibid.). Equally in what sense?

At first it seems obvious that the claim about the equal impetus acquired at the point of arrival by the free-falling body and the body on the inclined plane entails that they both move with the same velocity. Salviati, however, takes this opportunity to point out a fundamental incompatibility between the two concepts of velocity – one deriving from Aristotle's Physics and the other from Archimedes,87 applied, however, to accelerated motions. According to the first definition of equal velocity – reiterated by Simplicio as equal spaces passed in equal times (24) – the body on the perpendicular moves faster than that on the inclined plane. According to the second definition – the equal proportion between spaces traversed and times elapsed – their velocities are equal. Salviati's explanation of this situation tends to calm down Sagredo's initially strong doubts. Still, he demands a real proof of the last conclusion, that the times of fall of both bodies relate to each other as the distances they traverse, which Salviati promises to supply from the mouth of his academic friend. Indeed, this had been Galileo's key theorem in his work on inclined planes. It is to be found in De Motu⁸⁸ and has been labeled by some scholars as the length-time theorem.

Four statements concerning acceleration have been established by Salviati up to this point:

1) That free-fall motion and motion on the inclined plane are accelerated.

2) That the impetus acquired in accelerated motion starting from the same point suffices to lift the bodies to the same height.

3) That the impetus or degree of speed acquired at the point of arrival is equal for both bodies.

4) That the velocity of both bodies is equal according to an Archimedean concept of velocity.

Nevertheless, the claim for the continuity of acceleration made immediately afterwards goes back again to the Aristotelian concept of velocity, relying on the growing slowness of motion as the body moves on lesser and lesser inclined planes, until it comes to rest on the horizon. Yet, this is combined with an Archimedean argument according to which: the degree of velocity acquired at a given point of the inclined plane is equal to the velocity of the body falling along the perpendicular to its point of intersection with a parallel to the horizon through the given point of the inclined plane. (28)

The strong tensions characteristic of Galileo's discourse emerge even in this cryptic presentation of some of his major discoveries. The recognition (pointed out above⁸⁹) of a gap in our knowledge concerning the cause of local motion and its acceleration seems to lead to the attempt to understand acceleration first on a phenomenological level: by a conceptual analysis of the continuum and the geometrical representation of continuous acceleration and by comparing different accelerated motions. This presentation, however, raises two fundamental problems. First, while Saliviati's argument is unfolding, its origins in the old mechanics understood as a “mixed science” crop up with greater clarity. They finally become evident in the following passage:

Let us remember that we agreed that bodies descending along the perpendicular CB and the incline CA were found to have acquired equal degrees of velocity at the point B and A. Now, proceeding from there, I believe you will have no difficulty in granting that upon another plane less steep than AC – for example, AD – the motion of the descending body would be still slower than along the plane AC. Hence one cannot doubt the possibility of planes so little elevated above the horizontal AB that the ball may take any amount of time to reach the point A. If it moved along the plane BA, an infinite time would not suffice, and the motion is retarded according as the slope is diminished. (27)

In Galileo's Mecaniche, the speed of motion depends upon the body's weight and on the distance from the center of the system, called the moment of weight. The same bodies, therefore, moving on lesser and lesser inclined planes, acquire lesser and lesser momenti. The way to measure this motion is by assigning maximum moment to the perpendicular and minimum to the horizontal. Therefore, the speed on the less inclined planes is considered smaller.

Salviati's analysis in the Dialogue suppresses the traditional mechanical considerations in terms of weight and moments of weight. It leaves the notion of velocity connected with this discourse, in spite of its basic incompatibility with the definition of velocity as the proportion of times elapsed and traversed distances, which is used in the attempt to convince us that the velocity of a falling body and that of a body moving on the inclined plane are equal. This example clearly shows that the decomposition of velocity into infinitesimal degrees presented at the beginning of the text and the substitution of the traditional Aristotelian definition of velocity with the Archimedean definition cannot in fact be conceptually truncated from the traditional discourse of mechanics, in which weight played a major role.

The same uncertainty is evident in Salviati's formulation of the comparison between two equal bodies, one free falling and the other moving on an inclined plane. Speaking about two equal bodies means that weight is considered relevant to the discussion. At the same time the main thrust of the argument points to the horizon of the constancy of “impetus” – that is, the increase or decrease of the degree of velocity – and the equal velocity of the two bodies according to the Archimedean definition. If the degrees of velocity acquired by two bodies in free fall and on the incline are always the same, and if their velocities are also the same, what is the relevance of their equal weight?

The second fundamental problem raised by Salviati's presentation concerns the decomposition of velocity into infinite degrees of velocity and its relation to the Archimedean definition of velocity as a proportion between times elapsed and distances traversed. In fact, decomposition of velocity means that the proportion is not between lines, but rather between infinite sets of points. Galileo, however, lacked the philosophical justification to deal with such proportions. Furthermore, the switch between velocity decomposed into infinitesimal degrees for conceptual analysis and the application of the Archimedean proportion has the effect of blurring the distinction between degrees of velocity and velocity altogether. Salviati's conclusion from the following two statements – that the motion downwards is accelerated and that the impetus gained suffices to lift the bodies to the same height – reads as follows:

Two equal movable bodies, descending by different lines and without any impediment, will have acquired equal impetus whenever their approaches to the center are equal.

However, the next two references to the same issue – “In fact, they [the free falling body and the one moving on the inclined plane] have both advanced equally toward the center” (23) and “the speeds of the bodies falling by the perpendicular and by the incline are equal” (24) – remain ambiguous. Such ambiguity bordering on the obscure, culminates in Salviati's summary, which reiterates both that the motion is slower as the inclination above the horizon gets smaller and simultaneously that:

We may likewise suppose that the degree of velocity acquired at a given point of the inclined plane is equal to the velocity of the body falling along the perpendicular to its point of intersection with a parallel to the horizon through the given point of the inclined plane. (28)

Simplicio's failure to understand this opaque formulation brings about Sagredo's last attempt at clarification:

Whence no doubt can remain that the ball [namely, a cannon ball projected upwards which starts to lose its velocity and continues with slower and slower motion until it stops] before reaching the point of rest, passes through all the greater and greater gradations of slowness, and consequently through that one at which it would not traverse the distance of one inch in a thousand years. Such being the case, as it certainly is, it should not seem improbable to you, Simplicio, that the same ball, in returning downward, recovers the velocity of its motion by returning through those same degrees of slowness through which it passed going up. (31)

At first glance it seems that Sagredo's explanation is based upon a complete nonsequitur. How is the continuous nature of acceleration connected to the need of the body to “recover” its velocity? In fact, however, this passage, coming from the mouth of Sagredo, testifies to the model of accelerated motion lurking in Galileo's mind, in spite of its being erased from the text. In this model acceleration is still considered as the effect of an external force that the body loses while going up (in the De Motu hydrostatic model it is called levity, in analogy to the loss of weight in a medium of smaller specific gravity) and that it regains while going down. In contradistinction to Salviati's arguments, in Sagredo's explanation the abstract mathematical considerations are substituted with a picture easy to imagine and clearly present to the senses, which appeals to Simplicio's discursive habits and squeezes his long-awaited consent: “This argument convinces me much more than the previous mathematical subtleties” (ibid.).

The second and last discussion of free fall in the Dialogue taking place on the second day exhibits a very similar structure. This time the digression is made in response to Simplicio's quotation from a recent book written by a Jesuit mathematician,90 who tried to calculate the velocity of a cannon ball falling from the orbit of the Moon to the center of the Earth. Salviati's quest to understand the rules underlying this calculation is met with the answer that the falling ball continues to move at uniform velocity equal to the motion along the Moon's orbit. Salviati's and Sagredo's sarcastic dismissal of this answer is followed by the presentation of alternative principles for analyzing the fall, and by an alternative calculation, including an explanation of the method by which it could be arrived at.

“The movement of descending bodies is not uniform,” claims Salviati, “but…starting from rest they are continually accelerated” (221). There follows a straightforward statement – missing in the previous presentation – of the law of fall, that is the exact mathematical ratio of acceleration: “The acceleration of straight motion in heavy bodies proceeds according to the old numbers beginning from one” (222). This acceleration is then explicitly said to be equal to all falling bodies, without any connection to their weight: “for a ball of one, ten, a hundred, or a thousand pounds will all cover the same hundred yards in the same time” (223).

To understand the principles of the cannon ball's fall from the orbit of the Moon, Salviati quotes one more theorem and some “conjectures.” The theorem he refers to is the “double distance rule,”91 which establishes the relationship between accelerated and uniform motions. In accordance with this rule the falling cannon ball would acquire a degree of speed equal to the velocity of a body uniformly traversing double the space at the same time (255). This means that the cannon ball whose motion was calculated by the Jesuit in fact moves much faster than he had claimed in his book.

The “conjectures” to which Salviati then refers consist of observations of pendulums, conclusions from the work on inclined planes, a geometrical demonstration of the double distance rule based on medieval techniques of proving the mean speed theorem, and the representation of velocities by the infinity of lines making up the surfaces of a triangle and a parallelogram. All these provide the broad framework in which Salviati wishes to anchor his mathematical conclusions concerning the cannon ball and its velocity.

The example of a pendulum leads Sagredo to report of an impression he formulated to himself as a result of observation: “I have sometimes thought that the ascending arc [of a ball of lead suspended by a thread and removed from the perpendicular] would be equal to the descending one” (226). From this observation Salviati concludes that the impetus in both cases – descent and ascent – is the same: “…The impetus acquired in the descending arc, in which the motion is natural, is able by itself to drive the same ball upward by a forced motion through as much space in the ascending arc” (ibid.). But whereas impetus, in this case, is expressed in terms of the equal space traversed by the body going down and up, immediately afterwards impetus is expressed in terms of velocity: “… Just as in the descending arc the velocity goes on increasing to the lowest point of the perpendicular, so in the ascending arc it keeps diminishing all the way to the highest point” (ibid.). Moreover, the increase and decrease of velocity is also said to be in the same ratio, and thus: “…The degrees of speed at points equally distant from the lowest point are equal to each other” (227).

The pendulum serves Salviati as a model for another kind of accelerated motion: that of a cannon ball imagined to be descending down to the center of the Earth and ascending to the other side through a hole perforated at the center. The model of the pendulum applied to the cannon ball yields the further conclusion that because the velocity upon ascent diminishes in the same ratio as it increased along descent, and because the spaces passed by the ball on its motion down and on its motion up are in the same ratio, so is the time of descent.

This also leads to the understanding of accelerated motion in terms of an equivalent uniform motion: “it certainly seems reasonable that if it were always to move with this highest degree of speed, it would pass through both these distances in an equal amount of time.” Salviati thus formulates the “double distance rule,” stating that a falling body passing from accelerated motion to uniform motion would traverse double the space while continuing with the highest degree of speed for an equal time. The purpose of all these steps becomes clear as Salviati at last moves to his final conclusion: “Therefore all the space passed through with all the degrees of speed, increasing and decreasing…must be equal to the space passed in as many of the maximum speeds as number one half the total of the increasing and decreasing ones” (ibid.).

But Salviati does not stop here. Rather, he declares the degrees of speeds to be “indeterminate” infinitesimals: “the increases in the accelerated motion being continuous, one cannot divide the ever-increasing degrees of speed into any determinate number; changing from moment to moment, they are always infinite” (228) and suggests a powerful geometrical analogy through which they can be imagined. Representing the time continuum of the fall by the perpendicular of a rectangle triangle, he imagines the degrees of speed as all the lines parallel to the base (see Figure 3.2):

Figure 3.2.

Therefore, to represent the infinite degrees of speed…there must be understood to be infinite lines, always shorter and shorter…this infinity of lines is ultimately represented here by the surface of the triangle… (229).

However, the representation serves as more than just illustration. By completing the triangle into a parallelogram the surface of which represents uniform degrees of velocities equal to the maximum degree achieved by the falling body, he proceeds to drawing the comparison between accelerated and uniform motion through a kind of geometrical demonstration, although he avoids assigning to it the status of a proof:

…While the whole surface of the triangle was the sum total of all the speeds with which such a distance was traversed in the time AC, so the parallelogram becomes the total and aggregate of just as many degrees of speed but with each one of them equal to the maximum BC. This total of speeds is double that of the total of the increasing speeds in the triangle, just as the parallelogram is double the triangle. And therefore if the falling body makes use of the accelerated degrees of speed conforming to the triangle ABC and has passed over a certain space in a certain time, it is indeed reasonable and probable that by making use of the uniform velocities corresponding to the parallelogram it would pass with uniform motion during the same time through double the space which it passed with the accelerated motion. (229)

A careful reading of the two digressions on local motion in the first and second day of the Dialogue reveals some of the conceptual difficulties symptomatic of Galileo's project, which were, at the same time, also problems in the politics of knowledge. As usual, it is Sagredo who dares – in the first digression – to pose a challenge which signals these difficulties: “A great part of your difficulty consists in accepting this very rapid passage of the movable body through the infinite gradations of slowness antecedent to the velocity acquired during the given time…” (22) Salviati's “solution” to this problem is then given in terms of the “infinite instants” contained in every “single instant of time”: “The movable body does pass through the said gradations, but without pausing in any of them. So that even if the passage requires but a single instant of time, still, since a very small time contains infinite instants, we shall not lack a sufficiency of them to assign to each its own part of the infinite degrees of slowness, though the time be as short as you please.” (Ibid., my emphasis, R.F.) Another aspect of the same difficulty is raised by Salviati himself on the second day, and is immediately silenced by recognizing the impossibility of dividing continuous motion into discrete units: “For the increases in the accelerated motion being continuous, one cannot divide the ever-increasing degrees of speed into any determinate number, changing from moment to moment, they are always infinite.” (228) This, however, does not deter him from imagining velocity – in the very next passage – in terms of the sum total of all the lines making up a geometrical figure: “And just as BC was the maximum of all the infinitude in the triangle, representing the highest degree of speed acquired by the moving body in its accelerated motion, while the whole surface of the triangle was the sum total of all the speeds with which such a distance was traversed in the time AC, so the parallelogram becomes the total and aggregate of just as many degrees of speed but with each one of them equal to the maximum BC.”(229)

Two features characterize, then, Galileo's analysis of the continuous nature of acceleration in both digressions: first, decomposition of continuous magnitudes in terms of discrete units is attempted, in spite of the serious critique of such attempts by a long tradition, streching back to the Greeks. Galileo offered no philosophical justification for this daring analysis. Second, in both digressions Galileo made no distinction between physical and mathematical arguments. In fact, he conflated both spheres of knowledge in a seemingly non-problematic way. Salviati's cryptic “excuses” for the conceptual difficulties: that the body passes all the gradations of velocity and slowness without pausing in any of them, and that the “infinitesimals” of time, velocity, and mathematical magnitudes are not “determinate” numbers could not – in fact – rid the text from the anxiety of paradoxes, and remained enormously problematical for his science. At the same time Galileo exposed himself to the blame of transgressions of two kinds of boundaries: between the sciences of continuous magnitudes and discrete number within mathematical discourse on the one hand, and between physical and mathematical science on the other. These boundaries, however, were invested with disciplinary interests, and became a sensitive area of dispute among philosophers and mathematicians after the debate on mathematical certitude. Thus, the conceptual analysis of the continuum, and its application to a mathematical science of motion were heavily involved in the contemporary politics of knowledge.

But it was precisely the analysis of the continuum which served as a necessary assumption for comparing two accelerated motions of two bodies, one free falling along the perpendicular, the other rolling along an inclined plane, and thus for realizing the constant ratio of acceleration in all naturally accelerated motions. Granted that their grades of velocity increase and decrease continuously and that their distances relate to each other as the times of fall (the “lengh-time rule”), Salviati demands consent for his conclusion that their velocities are equal. Again it is Sagredo who points out the difficulty such an inference purports. And again it is Salviati who offers a solution by pointing out a contradiction. “The speeds of the bodies falling by the perpendicular and by the incline are equal. Yet this proposition is quite true, just as it is also true that the body moves more swiftly along the perpendicular than along the incline.” (24) Salviati explains this gap by the difference between two definitions of velocities: a narrow, Aristotelian definition stipulating that bodies moving with equal velocities traverse equal spaces in equal times; and its broadened and Archimedean version, according to which equal velocities of two moving bodies entails equal proportion of distances to times in accelerated motion. Modern scholarship, however, has shown that these two definitions are in fact incompatible in the case of accelerated motions, and that Galileo was not unaware of the difficulty involved in applying the Archimedean definition to the phenomenon of acceleration. Thus, Galileo's identification in nature of two accelerated motions with different distances and different times – which he interpreted mathematically in terms of the “double distance rule” – actually involved an uneasy co-existence and unbearable tensions between the Archimedean and the Aristotelian approaches that guided his investigations, and had to be transgressed in order to give birth to his mathematical-physical discourse.

All the tensions involved in the analysis of the continuum on the one hand, and in the Aristotelian-Archimedean synthesis when applied to the investigation of naturally accelerated motions reappear in any attempt to understand the status and function of “degree of velocity” in its relation to “impetus” and “speed.” As we have seen, these terms are often used by Galileo interchangeably. Thus, Salviati asks his interlocutors to grant that “the impetus of that which descends by the plane CA upon arriving at the point A would be equal to the impetus acquired by the other at point B after falling along the perpendicular” (23); at the same time both bodies “have as much impetus (that is, the same degree of velocity)” (24); and also, the “impetus of each should be equally sufficient to carry it back to the same height,” (23) and the “speeds of the bodies falling by the perpendicular and by the incline are equal.” (24)

The terminological confusion between “degrees of velocity,” “impetus” and “speed” has been subject to endless debates among historians culminating in “historiographical traditions” around this problem. The Duhem-Clagett tradition stresses Galileo's anchorage in fourteenth-century development of “impetus physics” by Buridan, and the development of tools for the mathematical representation of degrees of intensity of qualities (among them velocity) by Oresme, – Koyré and his followers tend to emphasize the “deductive” character of Galileo's discourse stemming from a new mathematical metaphysics which guided the minds of the great scientists of the seventeenth century, and especially that of Galileo; and the Drake-Wisan-Naylor tradition built a lot upon the results of experimental work used by Galileo to corroborate his mathematical deductions, thus characterizing his method as hypothetico-deductive in different senses. More convincing to me, however, is Galluzzi's account in Momento. Galluzzi anchors Galileo's inconsistencies in the inner development of his science from its earliest beginnings in the theorems on centers of gravity, through On Motion, the Mechanics, the Discourse on Floating Bodies, the unpublished manuscripts of 1600–1609, and up to the Dialogue and the Two New Sciences. In very broad terms (and therefore unfaithful to the subtlety of his discussion), Galluzzi's thesis is that around 1610 a break occurred in the very heart of Galileo's project, which had aimed at a causal and mathematical explanation of all motions in terms of weight, force, and velocity and at a synthesis of statics and dynamics into a new science of mechanics. The concept of moment, according to Galluzzi, in fact allowed for a mediation between a geometrical science of weight (statics) stemming from Archimedean sources, and the more dynamical approach of the Aristotelian Mechanical Questions, allowing for the interchageability and compensation of weight by motion. Galileo's ambition was to apply a combination of these approaches to the study of natural motion by modeling his dynamical concept of “moments of velocity” upon the static concept of “moments of weight,” and by an attempt to understand acceleration in free fall and projectile motion in terms of a series of increasing and decreasing “moments of velocity” reduced to a series of moments of uniform velocity. Within this broad framework “moments of velocity” were never divorced from “moments of weight,” since weight and motion always compensate for each other. On the other hand, the justification for the constancy of acceleration expressed in the proportion between distances and the square of times was only to be found in the Merton rule, and thus in the Buridan-Oresme tradition of impetus theory, which did not aspire for causal explanation of natural motions, and was unrelated to considerations of weight or gravity of bodies. The strongest proof for Galluzzi's thesis can be found in one fragment of the famous Ms. 72, where Galileo identifies between “moment” and “grade of velocity” (speaking about “momentum seu gradus velocitatis”). After 1609, however, Galileo apparently gave up upon his broadest and most ambitious project, and developed his analysis of accelerated motion around the decomposition of “velocity” into “degrees of velocity” independently of weight, and around a concept of velocity as the sum total of “infinitesimal” (instantaneous) velocities. Thus, the concept of “moment” was more or less suppressed in his published texts. Instead, he developed his “length-time” theorem already proved for uniform velocity in the De Motu, and applied it to accelerated motion combined with the decomposition of velocity into its degrees.

In the light of Galluzzi's thesis it is now possible to read Galileo's confusion of “degrees of velocity,” “impetus,” and “speed” as a result of the split characterizing his discourse after he had to give up his ambition to find new foundations for a science of all motions based on the combination between the Archimedean statical approach, the Aristotelian tradition of the Mechanical Questions and the Oxfordian-Parisian development of impetus physics. Thus, “degrees of velocity” were never wholly divorced from “moments of velocity,” closely connected to “moments of weight.” Velocity, or speed remained undifferentiated from its degrees in any explicit way, although such differentiation is implied in many of the texts. Likewise, “impetus” remained immersed in ambiguities, sometimes conflated with momento and thus expressing some kind of “energy” accumulated along the fall and sufficient to elevate the body to the same height from which it started the fall, other times expressing velocity translated from dynamic to kinematic terms.

The Dialogue Concerning the Two Chief World Systems was written when Galileo's investigations of motion began to look like a science without foundations, a project whose coherence was torn by an inner split between proper physical considerations of weight in relation to velocity which survived only in the form of a subtext and were mainly confined to the application of results achieved in the framework of his old Mechanics, (especially the length-time rule), and a conceptual-mathematical analysis of accelerated motion and its geometrical representation on the other hand. Excluding either a philosophical justification of the analysis of the continuum, as well as causal explanations of motion in terms of weight or force, it also conflated different types of discourses, and transgressed the boundaries between mathematical and physical science. Thus, when the Dialogue was finally being written, Galileo badly needed some kind of justification both for the inner coherence of his project as well as for his peculiar position within the cultural field of mathematical and philosophical discourse.

III. MATHEMATICAL ENTITES AND THE POLITICS OF KNOWLEDGE

A clue to Galileo's reflective perspective on his own project may be found in a short exchange among Simplicio, Salviati, and Sagredo, which takes place on the second day of the Dialogue: “I have frequently studied your manner of arguing,” says Simplicio, “which gives me the impression that you lean toward Plato's opinion that nostrum scire sit quoddam reminisci.” Salviati's response to this challenge is complex. Neither explicitly affirming, nor else denying Simplicio's observation, he chooses to remain indirect about his debt to Plato, stressing instead his commitment to deeds, which do not, however, exclude words:

How I feel about Plato's opinion I can indicate to you by means of words and also by deeds. In my previous arguments I have more than once explained myself with deeds. I shall pursue the same method in the matter at hand, which may then serve as an example, making it easier for you to comprehend my ideas about the acquisition of knowledge if there is time for them some other day, and if Sagredo will not be annoyed by our making such a digression.

Sagredo, of course, graciously expresses his intense interest in probing into any discourse that may provide an alternative to the one practiced in the schools:

Rather, I shall be much obliged. For I remember that when I was studying logic, I never was able to convince myself that Aristotle's method of demonstration, so much preached, was very powerful. (190–1)

Traditionally, such passages as the one quoted above have been interpreted in terms of the epistemological revolution that necessarily accompanied the new “scientific” – mainly mathematical – contents suggested by Galileo. Some philosophers and historians of sceince have cherished the idea that it was Platonic (mathematical) epistemology, or even ontology, which actually enabled – not just accompanied – the emergence of mathematical physics. The old-new Platonic epistemology, they have claimed, substituted for the Aristotelian – logical, but nonmathematical – epistemology, accepted, for many centuries as the adequate framework for practicing Aristotelian physics.

A close reading of some more passages in the Dialogue, however, may suggest a different view. In conformity with the spirit of Salviati's words, such a view will accentuate practice, in speechacts rather than epistemology, as the basic clue to understanding the process by which Galileo's dispersed insights crystallized into a project that seemed coherent at the time. Such reading will also point out the futility of any attempt to reduce Galileo's options into the dichotomy of a Platonic or Aristotelian discourse. Again, Salviati's reluctance to commit himself to any given epistemology may provide a hint in this direction. This does not mean that Platonism and Aristotelianism had no ideological role in Galileo's politics of knowledge. It means, however, that the labels must be deconstructed, in order to understand their function as one cultural practice among others used by many sixteenth- and seventeenth-century intellectuals, among them Galileo.

The Dialogue Concerning the Two Chief World Systems offers relatively easy access to the map of options that constituted the cultural field in which Galileo operated and where he attempted to create for himself a recognized, legitimate, and well-specified position as a mathematical philosopher. In the remainder of this paper an attempt is made to interpret Galileo's position and the way he differentiated it from others’ by reconstructing the network of debates among mathematicians and philosophers lurking behind the text or on its surface. Galileo's position, so the argument goes, can only be understood in the context of the positions he is aligning himself with or differentiating himself from. It was determined at the same time by considerations stemming from the inner problematics of his science (analyzed in the previous section) as well as by the dynamics of the field of positions in which he was trying to play.

Galileo needed to justify a discourse that conceptualized velocity as the sum total of an infinity of degrees of speeds and stated the irrelevance of weight for their mathematical determination. Nevertheless, it still left open queries about the role of weight in relation to natural motion and acceleration, which were suppressed but not totally excluded from the text. The preliminary exchange among Simplicio, Salviati, and Sagredo prepares the stage for the differentiation of an intersubjective field out of which Galileo's position emerges, and whose initial conditions are: a vague association with a Socratic mode of inquiry, an understanding of discourse as a set of practices, including speech (“deeds” and “words,” in Galileo's terminology), and a vision of some alternative to the Aristotelian method of logical demonstration in natural philosophy.

Two kinds of arguments immediately followed Salviati's general comments on the “acquisition of knowledge” (191). The first argument was a purely geometrical refutation of objections to the motion of the Earth, making use of the geometrical notion of the “angle of contact” (the angle between the tangent and the curve). Indirectly and nonexplicitly, it implied a position on the two major conceptual problems discussed in the previous section: the translation of a (non-commonsensical) mathematical construct (angle of contact parallel to degrees of velocity) into a claim on the conditions of possibility of physical motion on the one hand and a very problematic assumption about the relation between two incommensurables, finite and “infinite” quantity, on the other. The second reflection on the acquisition of knowledge concerned the “point of contact between two spheres” and suggested a direct, explicit, and radical position on the nature of mathematical entities, their relation to physical objects, and the consequences for the justification of a mathematical knowledge of nature.

The “angle of contact” was used by Galileo in the context of a counterargument to the objection of Ptolematic astronomers to the diurnal motion of the Earth, whose whirling was claimed to cause stones, animals, and other heavy bodies on its surface to be ejected as a result of the impetus created by this kind of movement. Salviati contends that such projection is a physical nonpossibility, and he offers a subtle geometrical reasoning to support this claim. First he proves that the proportion between the tangent and the secant grows infinitely toward the point of contact. Then he argues that if a physical body were to be separated from the surface of the Earth it would be subject to two opposing motions: a projection along the tangent outside the circumference of the Earth's orbit and the tendency of the body to fall toward the center of the Earth along the secant. The velocity along the tangent, he concludes, would necessarily be smaller than the velocity along the secant. This conclusion is based upon the analysis of velocity in terms of moments of velocity, represented by the parallel lines lying between the two sides of an angle: “the degrees of speed infinitely diminished… correspond to the parallels included between the two straight lines meeting in an angle” (200–201). Since the angle of contact is always smaller than the angle between the tangent and the secant (see Figure 3.3), it is clear that the parallel lines between its two sides (corresponding to the degrees of velocity) are smaller than those between the tangent and the secant. Therefore, the motion along the tangent will never prevail over the motion along the secant, for, according to Salviati: “To have projection occur, it is required that the impetus along the tangent prevail over the tendency along the secant” (196).

Figure 3.3.

In fact, Salviati's argument is far more obscure that the summary presented here. I believe, however, that this summary is not distorting and will prove useful for the initiation of my discussion. It is Galileo's use of a geometrical notion – the angle of contact as smaller than any other angle – and its translation into physical reality – the condition of possibility of motion – rather than the details of an argument that captures my attention right now, and it is the emergence of a position relative to other positions on the same question that constitutes the core of this story, parallel, and in connection, with the internal story about the problematic structure of Galileo's mathematical-philosophical discourse.

Between 1579 and 1589, the rich knot of controversies associated with a Euclidean proposition concerning the angle of contact began to take place among mathematicians all over Europe and became a “topos” around which logical, philological, methodological, and mathematical issues of fundamental importance to the organization of the map of knowledge were articulated.92 A long list of participants in these debates, which spread widely up to the end of the seventeenth century and beyond, is quoted by L. Maieru. Among them are some of the greatest early modern mathematicians, such as Galilei, Borelli, Wallis, and Jacob Bernoulli.

The core of the polemics sprang from the work of J. Peletier, one of the earliest critics of Euclid in the modern era. Peletier was the first to point out the conceptual incompatibility between Euclid III. 16, which implied that a geometrical magnitude – the angle of contact – should be considered “minimal quantity” – a discrete – and the two major rules that had been guiding the Euclidean project almost uninterruptedly since antiquity: namely, the principle of the continuous nature of geometrical magnitudes, implied by X, I, and the principle of homogeneity invoked by the definition of ratio and proportion in the fifth book (V.3).⁹³ If indeed, Peletier argued, the angle of contact is smaller than any acute rectilinear angle, it cannot be multiplied and exceed an acute rectilinear one. In other words, there cannot be a ratio between the angle of contact and a rectilinear angle. In response to such difficulty, Peletier suggested – in two early texts from 1557 and 1563 – excluding the angle of contact from the realm of mathematical discourse. He then elaborated his critique in a public response – labeled an Apology (1579) – to Clavius's defenses of Euclid, which he first included in the first edition of his Commentary (1574) and repeated in the next two editions of 1586 and 1589. In the same spirit Peletier also argued that Euclid's proofs by superposition (I.4, for example) should be discarded, since there was something “mechanical” about moving triangles that did not fit the “nobility” of geometry.

Peletier's strategies testify to the crystallization of one position among sixteenth-century mathematicians that tended to privilege the norms behind the traditional boundaries implied by the Euclidean project: between discrete and continuous entities as exclusive objects of arithmetic and geometry, respectively, as well as between the objects of mathematical discourse as separate from matter and motion on the one hand and those of natural philosophy on the other hand. The application of the theory of proportion and the Eudoxian method of exhaustion enabled relations between the different kinds of mathematical entities to be established, and allowed for the application of mathematical methods to physical phenomena under specific conditions, but created many constraints against blurring the boundaries. Peletier's insistence that the angle of contact not be considered a quantity was a strategy of exclusion considered by him as an act of defense of necessary boundaries, even against Euclid himself, in a place where his own writings seemed to violate the norm of his own discourse. It was easier to argue for exclusion, however, than to actually practice it while still doing Euclidean geometry. This becomes very obvious when one looks into Peletier's attempt to provide a universal proof of the problem of constructing a curvilinear angle equal to a rectilinear one and is forced to add the angle of contact to another angle, treating it, then, as a quantity.94

The conflict between declared norms of the Euclidean discourse on the one hand and the need to solve geometrical problems on the other, which arose in the context of Peletier's critique of Euclid, should be remembered when Clavius's position in the polemic is being reconstructed and interpreted. As in most of his other polemics,⁹⁵ here too Clavius took a middle position, trying to defend both Euclid's proposition (III, 16) as well as the accepted boundary between continuous and discrete quantities, while still preserving the status of mathematical entities as separate from matter and motion. Thus, against Peletier he argued that the angle of contact was a quantity. However, he also contended that it was not a “minimal quantity,” since it could be divided endlessly by a curve of the same type, so that there were, in fact, infinite angles of contact greater than a fixed one, and they could be compared to each other by superposition.⁹⁶ Still, even while defending Euclid's technique of superposition, Clavius was careful not to mix motion with geometrical entities. While speaking about superposition, he claimed, Euclid was referring to an operation of the mind, not to any mechanical moving of triangles or angles.⁹⁷

No doubt Clavius was motivated by pragmatic reasons in his defense of Euclid. He rightly pointed out that because in geometry propositions depended on each other, exclusion of any Euclidean proposition meant that many others had to be excluded too. Peletier's own difficulties in discarding the angle of contact was a living demonstration to the legitimacy of Clavius's concerns, remembering that in his prominent position as the leading mathematician of the age he had far greater stake in practicing geometry than did Peletier. Nonetheless, Clavius's enormous caution in defending the boundaries of the mathematical sciences, and between its various disciplines, should not be interpreted solely from the mathematical point of view, for it also reflected the politics of knowledge peculiar to the Society of Jesus.

The complexity of the Jesuit attitude toward the boundaries of the mathematical sciences has already been pointed out in the first section of this paper. In their attempts to gain a higher status than had traditionally been assigned to mathematicians within the context of medieval and renaissance universities, Jesuit mathematicians were concerned with securing the autonomy of their field of knowledge. The quest for autonomy, however, often involved destabilization of the old boundaries.

New developments within the body of knowledge, such as the integration of algebra and of Archimedean materials, seemed to be leading toward the reconceptualization of the boundaries between pure and mixed mathematics and between mathematics and natural philosophy. It also began to destabilize the boundary, within pure mathematics, between the fields treating discrete and continuous quantities. The analysis of the continuum, however, had a long and controversial history connected to deep philosophical and theological issues.98 No wonder that some of the philosophers were suspicious of those innovations and insisted on the subordination of the “mixed sciences” to the higher parts of speculative philosophy and on the inadequacy of mathematics to solve problems in physics. Clavius's position, which had a deep impact on the official policy of the order was marked by the conviction that the traditional boundaries should be reproduced and by the acceptance of a kind of compromise between philosophers and mathematicians about their division of labor, in spite of many transgressions on both sides.⁹⁹

It should be stressed, however, that the implementation of the boundaries of mathematical discourse was just one aspect of the Jesuits’ broader attempt to structure post-Tridentine culture in accordance with their theological and educational orientation. Clavius's tendency to reproduce the traditional boundaries – even though they were not always maintained in practice – conformed with the order's policy, applied in other spheres of knowledge, which combined an openness toward innovation with sophisticated means of control.¹⁰⁰

Galileo's refutation of the argument about projection of bodies from the surface of the Earth and his interpretation of the “angle of contact” as minimal quantity thus pushed him to take a position against Clavius and many of the mathematicians who followed him, not only against philosophers or theologians. Simplicio's comment, however, that the argument may be very subtle, but that “these mathematical subtleties do very well in the abstract, but they do not work out when applied to sensible and physical matters,” develops into a far more explicit discussion of the application of mathematical truths to material reality, provoked by the contention that: “mathematicians may prove well enough in theory that sphaera tangit planum in puncto, a preoposition similar to the one at hand; but when it comes to matter things happen otherwise”(203).

What follows is a surprisingly poor discussion of the relationship between mathematical abstractions and the concrete reality of physical, material bodies, which contrasts enormously with the rich philosophical, methodological, mathematical, and even theological and philological debates around the same topic in the sixteenth and seventeenth century, relying upon a long tradition since Greek antiquity.101 Salviati's arguments proceed in three steps: First of all, he claimed, doubting that a material sphere is a sphere amounts to stating a contradiction, similar in kind to the saying that a sphere is not a sphere. This first step relies on the most basic agreement among human beings about the use of terms, necessary to maintain a community of speakers. Reverting, then, to the definition of the sphere as that form upon whose surface all points are equally distant from the center, Salviati provides the geometrical proof of this proposition – two spheres touch each other in one point – showing that to assume that two spheres touch each other at more than one point means assuming points on the surface that are not equally distant from the center, which is absurd. Simplicio easily accepts the proof, but rightly clings to his original problematics, which concerns the application of abstract concepts, not a proof in the abstract. Salviati, then, on the edge of impatience, argues last by analogy:

It would be novel indeed if computations and ratios made in abstract numbers should not thereafter correspond to concrete gold and silver coins and merchandise. Do you know what does happen, Simplicio? Just as the computer who wants his calculations to deal with sugar, silk, and wool must discount the boxes, bales, and other packings, so the mathematical scientist (filosofo geometra), when he wants to recognize in the concrete the effects which he has proved in the abstract, must deduct the material hindrances, and if he is able to do so, I assure you that things are in no less agreement than arithmetical computations. The errors, then, lie not in the abstractness or concreteness, not in geometry or physics, but in a calculator who does not know how to make a true accounting. (207–8)

The task of the philosopher-geometrician is analogous to that of the merchant in the market, Salviati argues. Both are calculators, the first of physical effects in nature and the last of goods in the market.

At first reading it is indeed hard to accept that Salviati's discussion of the most fundamental feature of Galileo's project is so dull, simplistic, and unconvincing, especially in comparison with the polemical background against which it was originally written. A second reading is therefore required. This second reading will focus on two aspects: Following the allusions dispersed along the text, which were certainly clear to contemporaries, but much less so to historians, I shall point out the political implications of Galileo's choice to structure his self-justification in this particular form. I shall then point out how his self-justification functioned to fill a void (unresolved conceptual problems and inner split) within his own scientific discourse.

A closer look at the text reveals that it begins by delineating two positions regarding the role of mathematics in the investigation of nature. The first is represented by Sagredo, who concludes from Salviati's argument on the angle of contact that: “It must be admitted that trying to deal with physical problems without geometry is attempting the impossible” (203). Simplicio, then, is presented as a philosopher who chooses a middle way: He is not “one of those Peripatetics who discourage their disciples from the study of mathematics,” but somebody who still agrees with Aristotle “that he [Plato] plunged into geometry too deeply and became too fascinated by it” (ibid.).

Contrary to common belief Simplicio is far from being represented as a simpleton. In fact, he expresses a well-differentiated position that casts doubt upon the role of mathematics in the investigation of nature, without, however, being too blunt about it. Simplicio is said to differ from those philosophers who “discourage their disciples from the study of mathematics as a thing that disturbs the reason and renders it less fit for contemplation” (ibid.). The accusation cannot but echo Clavius's words in his De Mathematicis (one of three treatises written in the 1580s as part of the preparatory work for the Ratio Studiorum), where he commented that: “It will also contribute much…if the teachers of philosophy abstained from those questions which do not help in the understanding of natural things and very much detract from the authority of the mathematical disciplines in the eyes of the students.”102 One Jesuit notorious for becoming the target of Clavius's complaints was Benedictus Perera, from whom Simplicio seems to be distinguished at first. However, two of Simplicio's arguments inevitably bring Perera to the mind of the reader. First, he sounds skeptical about Salviati's Platonic orientation invoked by his understanding knowledge in terms of Plato's theory of reminiscence (190–1). Second, he quotes a great Peripatetic philosopher who accused Archimedes of assuming something instead of proving it (204). Both complaints resonate with Perera's interpretation of a passage from Plato's Republic VII, where he had written that “mathematicians dream about quantity, and in treating their demonstrations proceed not scientifically, but from certain suppositions.”¹⁰³

The dense web of allusions lingering over Simplicio's positioning, as it is crafted by Galileo, suggests the need to probe further into the cultural field in which the text was embedded. Simplicio's attitude toward the role of mathematics, which is affirmative in a sense, but insists on the clear boundaries of the mathematical disciplines and their limitations in dealing with sensible matter clearly alludes to Perera. Galileo's choice to focus the discussion on “sphaera tangit planum in puncto” also follows Perera, who had selected the most commonplace topos in a long and continuous tradition of writing-originating from the Platonic texts – on the objects of mathematical discourse.¹⁰⁴ Are mathematical spheres real? What is the difference between a mathematical sphere and a bronze sphere? What is the significance of the ontological status of mathematical objects for the kind of principles, arguments, and proofs produced by mathematicians? These were recurring questions raised and answered in different ways by the ongoing debate within a tradition, which nevertheless shared one basic assumption: Platonists, Aristotelians, and even Archimedeans believed in the uniqueness of mathematical objects and their difference from physical ones. This assumption was also reproduced in the Jesuit Ratio Studiorum of 1599, which explicitly mentioned the difference between a physical and a mathematical point as a subject of studies to be inculcated during the second year of the philosophical cycle in Jesuit universities.105

The rich intertextuality of Galileo's Dialogue enables him to differentiate various positions among Jesuit mathematicians and philosophers on that question and allowed for their (tacit) representation. Thus, Perera was represented by Simplicio. Clavius's middle position on the angle of contact – considering it a quantity, but not infinitesimal – may also be said to exist through Galileo's presentation of a debate in which Clavius was one of the most outstanding participators. Scheiner was represented by the Disquisitiones Mathematicae, quoted by Simplicio in the second day. What is missing from Galileo's text, however, is Blancanus's justification of the role of mathematics which he attempted to ground philosophically in his Treatise on the Nature of Mathematics.¹⁰⁶

To understand this omission one should look at the exact configuration of problems in the midst of which Galileo chooses to locate his argument on mathematics and physics.

Salviati has just presented his refutation of the anti-Copernican claim concerning the projection of objects from the Earth's surface as a result of its speedy whirling. Galileo modeled the situation of a body on the surface of a moving Earth upon his analysis of a stone attached to a stick moving in a circle around a center. According to this model the impetus of the circular motion is impressed on the body which leaves the notch and starts moving along the tangent from the point of separation. The weight of the body is then the cause of a downward motion in the direction of the center. However, the tendency downwards – along the secant – always prevails over the motion along the tangent, because the velocity at the beginning of the latter motion is extremely slow: “For the distance traveled being so extremely small at the beginning of its seperation (because of the infinite acuteness of the angle of contact), any tendency that would draw it back toward the center of the wheel, however small, would suffice to hold it on the circumference.” As mentioned before, the geometrical argument consists of showing that the angle of contact is always smaller than any acute rectilinear angle, and therefore the motion along the tangent will never prevail over the motion downwards.

Many of the characteristics of Galileo's discourse, pointed out above, crop up in this analysis. At first, both motions are described in terms of their causes: The tangential one is perceived as caused by the impetus of the whirling; the downward motion by the weight of the body. Immediately afterwards, however, the discussion shifts into another conceptual framework, and the motion is analyzed in terms of velocity and moments of velocity, which have nothing to do with weight and cannot offer a causal explanation of fall and acceleration. Within such a framework there is no way to contend that the velocity of the motion along the tangent might in some circumstances prevail: “Saying this is false; not from any deficiency in logic or physics or metaphysics, but merely in geometry.” But the framework chosen by Galileo for this discussion does open the door to an objection coming from the mouth of Sagredo, which points out a major difficulty. The objection relates to the weight of the body, which has been presented as the cause of the downward motion at the beginning of the argument. Just as speed diminishes infinitely, so may weight be susceptible to the same analysis, in the case of very light bodies on the surface of the Earth.

Sagredo's objection sets the stage for Salviati's final clarification of the situation of a stone on the surface of the Earth, in terms of the diminishing degrees of velocity toward the point of rest and the diminution of speed as the weight of the body is diminished infinitely. As against this “twofold diminution ad infinitum” (200) he analyzes the diminishing degrees of speed of the body moving along the tangent, which are represented as those parts of the parallels lying between the rectilinear and the curve of the circle (i.e., between the sides of the angle of contact):

They grow always less than these parallels of which they are parts, and diminish in an increasing ratio as they approach the point of contact…Thus the shortness of such lines is reduced until it far surpasses what is needed to make the projectile, however, light, return to (or rather be kept on) the circumference. (201)

Sagredo, however, remains unhappy with this final clarification. Targeting his last objection at the weakest link in Salviati's argument – his complete ignorance of the mathematical relation between weight and velocity – he raises his last question. It is possible to imagine, he contends, that the weight of the body diminishes in a greater proportion that the speeds. Wouldn't the speed along the tangent suffice then to carry the body away? Salviati's response to this last challenge is threefold. First, he denies, but in a rather ambivalent way, that weight is really relevant for his discussion: “I have been taking it as true that the speeds of naturally falling bodies follow the proportions of their weights out of regard to Simplicio and Aristotle, who declares this in many places as an evident proposition.” This ambivalence is rather incompatible with his very clear position regarding irrelevance of weight for analyzing free fall, which we have seen him stating in another passage of the dialogue. Yet, he claims, even if weight is relevant, it certainly is the case that the proportion of the speed is much less than that of the weights, which he can easily show by experiment. The third reaction, however, is the most interesting. For, he contends, even if the speed would decrease in a much greater ratio, even the lightest materials would not be projected:

Now weight never does diminish clear to its last term, for then the moving body would be weightless; but the space of return for the projectile to the circumference does reduce to its ultimate smallness, which happens when the moving body rests upon the circumference at that very point of contact, so that no space whatever is required for its return. Therefore let the tendency to downward motion be as small as you please, yet it will always be more than enough to get the moving body back to the circumference from which it is distant by the minimum distance, which is none at all. (203)

Salviati's arguments, I claim, exhibit in an exemplary form the split between two incompatible discourses that is visible all along the Dialogue: one rooted in mechanics as a “mixed science” but unable to provide the mathematical conceptualization of accelerated motion and the other phenomenological and mathematical but unable to integrate the physical cause of acceleration into its framework. As has been pointed out before, Salviati's preliminary analysis is performed within the conceptual framework of the old mechanics, in which the motion along the tangent is caused by some force impressed upon the body – the impetus. Downward motion is caused by the weight of the body, and the speed of the motion is measured in relation to the distance traversed by the body at equal times.

Posed in these terms, however, the solution is not clear, for Salviati cannot make any mathematical claim about the relationship between the impetus and the weight. Salviati, then, switches to his analysis of accelerated motion in terms of degrees of velocity, and he succeeds in offering a brilliant solution, unable, however, to incorporate weight into this explanation although it had been the point of departure of the whole argument.

It is exactly at this point that the discussion is interrupted by Sagredo's remark, claiming that “it must be admitted that trying to deal with physical problems without geometry is attempting the impossible” (203). Exactly at the moment when the failure to incorporate weight into the physical-mathematical construction of quantity is most transparent, Salviati's radical position about the complete reducibility of physical entities to mathematical ones is inserted, and Blancanus's intricate deliberations about mathematical objects and physical objects are erased from the text.

Now, in a way, Blancanus's arguments about mathematical entities, although cast in a different language and drawing upon the tradition, carried much of the same message as Galileo's. By arguing that “if there were given a material sphere and plane which were perfect and remained so, they would touch one another in a single point” he expressed his belief in the ideal nature of mathematical entities and in the exact correspondence between these ideal forms and material conditions in the physical world. Clavius, when arguing that mathematical entities are separated from matter although they are immersed in it, and Blancanus, in stressing the materiality of mathematical entities and the essentiality of mathematical definition, were likewise expressing the same vision. However, this vision functioned very differently in Galileo's discourse and in the Jesuits’. To cover up the gap between his physical causal discourse and his mathematical analysis, Galileo attempted to deny any boundary between mathematics and physics in the Dialogue and to erase the traditional discourse on mathematical entities. Comparing the mathematical philosopher to the calculator, he posed the ideal of a man of deeds who opted for practical solutions and who knew how to construct ideal realities that would also be true in the world of matter. In contradistinction Clavius and Blancanus attempted to use the discourse of mathematical entities for legitimating their discipline within a cultural project whose boundaries they were forced to accept and reproduce, even while sometimes committing their own transgressions.

NOTES

  1 G. Galilei, Two New Sciences, translated with introduction by S. Drake, Madison, Wisconsin: University of Wisconsin Press, 1974. Page numbers of original texts will be inserted within parentheses.

  2 See, for example: T. Settle, “An Experiment in the History of Science,” Science, 1961, 133:19–23; S. Drake, Galileo at Work: His Scientific Biography, Chicago & London: Dover Publications, 1978; D. K. Hill, “Galileo's Work on 116v: A New Analysis,” Isis, 1986, 77:283–91; idem, “Dissecting Trajectories: Galileo's Early Experiments on Projectile Motion and the Law of Fall,” Isis, 1988, 79:646–68.

  3 A. E. Burtt, Metaphysical Foundations of Modern Physical Science, revised ed. London: Humanities Press, 1949; A. Koyré, Etudes Galiléennes, Paris: Hermann, 1939; idem, “Galileo and Plato,” reprinted in P. P. Wiener and A. Noland, eds., Roots of Scientific Thought, New York: Basic Books, 1957, 147–75; P. Galluzzi, “Il Platonismo del tardo cinquencento e la filosofia di Galileo,” in P. Zambelli, ed., Ricerche sulla Cultura dell'Italia Moderna, Bari: Laterza, 1973, 39–79; M. De Caro, “Galileo's Mathematical Platonism,” in J. Czermak, ed., Philosophie der Mathematik, Wien: Hölder-Pichler-Tempsky, 1993.

  4 The major writer in this tradition is: W. A. Wallace, Galileo and His Sources: The Heritage of the Collegio Romano in Galileo's Science, Princeton, NJ: Princeton University Press, 1984.

  5 See, for example: W. L. Wisan, “The New Science of Motion: A Study of Galileo's De Motu Locali,” Archive for the History of Exact Sciences, 1974, 13:103–306; idem, “Galileo's Scientific Method: A Reexamination,” in R. E. Butts and J. C. Pitt, eds., New Perspectives on Galileo, Boston: Kluwer, 1978; idem, “Galileo and the Process of Scientific Creation,” Isis, 1984, 75:269–86; R. H. Naylor, “Galileo and the Problem of Free Fall,” British Journal for the History of Science, 1974, 7:105–34; idem, “Galileo's Theory of Motion: Processes of Conceptual Change in the Period 1604–1610,” Annals of Science, 1977, 34:365–92; idem, “Galileo's Theory of Projectile Motion,” Isis, 1980, 71:550–70 and others by the same author.

  6 See, for example: Koyré’s Etudes for classical texts; P. Galluzzi, Momento. Studi Galileiani, Rome: Allneo and Bizzarri, 1979; W. A. Wallace, Prelude to Galileo: Essays on Medieval and Sixteenth-Century Sources of Galileo's Thought, Boston: Kluwer, 1981; E. Sylla, “Galileo and the Oxford Calculatores: Analytical Languages and the Mean-Speed Theorem for Accelerated Motion,” in W. A. Wallace, ed., Reinterpreting Galileo, Washington: Catholic University of America Press, 1986; E. W. Strong, Procedures and Metaphysics. A study in the Philosophy and Mathematical-Physical Science in the Sixteenth and Seventeenth Centuries, Hildesheim, Germany: Georgolms, 1966 (1st ed., Berkeley, 1936); S. Drake and I. E. Drabkin, Mechanics in Sixteenth-Century Italy, Madison, Milwaukee, & London: University of Wisconsin Press, 1969; P. L. Rose, The Italian Renaissance of Mathematics. Studies on Humanists and Mathematicians from Petrarch to Galileo, Geneve: Droz, 1975; M. Biagioli, “The Social Status of Italian Mathematicians 1450–1600,” History of Science, 1989, 27:41–95.

  7 J. Pitt, Galileo, Human Knowledge, and the Book of Nature: Method Replaces Metaphysics, Dordrecht: Kluwer, 1992.

  8 J. H. Randall, The School of Padua and the Emergence of Modern Science, Padua: Antenore, 1961; D. W. Edwards, “Randall on the Development of Scientific Method in the School of Padua – a Continuing Reappraisal,” in J. P. Anton, ed., Naturalism and Historical Understanding. Essays on the Philosophy of f. H. Randall, Jr., Albany: State University of New York Press, 1967, 42–5; C. B. Schmitt, “Experience and Experiment: A Comparison of Zabarella's View with Galileo's in De Motu,” Studies in the Renaissance, 1969, XVI:80–138; W. A. Wallace, Galileo and His Sources … (1984a).

  9 The first historian who drew attention to the reflections of sixteenth-century humanists, mathematicians, and philosophers on the status of mathematics was N. Gilbert in his Concepts of Method in the Renaissance, New York: State University of New York Press, 1960, 86–91. Other historians who followed Gilbert's original suggestions were: G. C. Giacobbe, “Il commentarium de certitudine mathematicarum disciplinaram di Alessandro Piccolomini,” Physis, 1972, XIV, 2:162–93; idem, “Francesco Barozzi e la Quaestio de certitudine mathematicarum,” Physis, 1972, XIV, 4:357–74; idem, “Lariflessione metamatematica di Pietro Catena, Physis, 1973, XV, 2:178–96; idem, “Epigoni nel seicento della ‘Quaestio de certitudine mathematicarum’: Giuseppe Biancani,” Physis, 1976, XVIII, 1:5–40; idem, “Un gesuita progressista nella ‘Quaestio de certitudine mathematicarum’ rinascimentale: Benito Pereyra,” Physis, 1977, XIX, 51–86; idem, Alle Radici della Rivoluzione Scientifica Rinascimentale: Le Opere di Pietro Catena Sui Rapporti tra Matematica e Logica, Pisa: Domus Galileana, 1981; A. Carugo, “Giuseppe Moleto: Mathematics and the Aristotelian Theory of Science at Padua in the Second Half of the 16th-Century,” in L. Olivieri, ed., Aristotelismo Veneto e Scienza Modema, Padua, 1983, 509–17; idem, “L'insegnamento della matematica all'università di Padova prima e dopo Galileo,” in Storia della Cultura Veneta: Nen Pozza, 1984, 151–99. The most comprehensive treatment, however, upon which I have heavily drawn in A. De Pace, Le Matematiche e il Mondo: Ricerche su un Dibattito in Italia nella Seconda Metà del Cinquecento, Milano: Francoangeli, 1993.

10 Some sleeted references relevant for studying the institutionalization of the Jesuit mathematical program are: F. de Dainville, L'Education des Jésuites XVIe-XVIIIe Siècles, Paris: Editions de Minuit, 1978; G. Cosentino, “L'insegnamento delle matematiche nei collegi Gesuitici nell'Italia settentrionale: Nota introduttiva,” Physis, 1971, 13:205–17; idem, “Le matematiche nella ‘Ratio studiorum’ della Compagnia di Gesù,” Miscellanea Storica Ligure, 1970, 2/2:171–213; W. A. Wallace, Galileo and His Sources … (1984a); P. Dear, Mersenne and the Learning of the Schools, Ithaca, NY: Cornell University Press, 1988; idem, Discipline & Experience: The Mathematical Way in the Scientific Revolution, Chicago, IL: University of Chicago Press, 1995; U. Baldini, Legem Impone Subactis: Studi su Filosofia e Scienza dei Gesuiti in Italia 1540–1632, Rome: Bulzoni, 1992; R. Gatto, Tra Scienza e Immaginazione: Le Matematiche Presso il Collegio Gesuitico Napoletano (1552–1670 ca.), Firenze: Olschki, 1994; S. J. Harris, “Les chaires de mathématiques,” in L. Giard, ed., Les Jésuites a la Renaissance, Paris: Presses Universitaires de France, 1995, 239–161; R. Feldhay, Galileo and the Church: Political Inquisition or Critical Dialogue), Cambridge, NY: Cambridge University Press, 1995.

11 A. Piccolomini, In Mechanicas Quaestiones Aristotelis … Eiusdem Commentarium de Certitudine Mathematicarum Disciplinarum, Rome, 1547. On Piccolomini, see R. Suter, “The Scientific Work of Allesandro [sic] Piccolomini,” Isis, 1969, 60:210–22.

12 De certitudine …, c. 69r, quoted by De Pace, 22, n. 3:

     Mathematicas demonstrationes, in primo esse ordine certitudinis … testatur Averroes 2. Metaph. com. 16. super illis verbis Aristotelis, videlicet: “Certitudo mathematica non in omnibus expetenda.” Quam quidem Averrois authoritatem, omnes fere latini, quos ego viderim, veluti ex antiquioribus, Divus Albertus, Divus Thomas, Marsilius, et Egidius; ex recentioribus vero, Zimarra, Suessanus, Acciaiolus, et plerique alii; si quando in eam inciderunt, uno ore, quasi alius alium sequens, ita interpretati sunt, ut propterea Averroes illud asserat, quia Mathematicus ex notioribus et nobis et naturae demonstrat, quippe qui vel solus, vel maxime, demonstratione illa, quam potissimam appellant, utatur, qua scilicet simul, et quod effectus sit, et cur sit liquido innotescit.

13 R. Sorabji, Aristotle Transformed. The Ancient Commentators and Their Influence, Ithaca, NY: Cornell University Press, 1990.

14 De certitudine …, cc. 69r–69v, quoted by De Pace, 26, n. 14:

     Ego vero, quamvis in adolescentia mea, tantorum virorum authoritate ductus, in eorum opinionem … descenderim, deinceps tamen, dum mathematicas disciplinas assidue versans, intimius pertractavi, tantum abest, ut in scientia diutius permanserim, ut non solum demonstrationes Geometrarum, reliquorumque mathematicorum, non esse illas potissimas, sed ne vix ad illas accedere, existimaverim. Verum enimvero, hanc meam sententiam, quamvis mihi constantissime probaretur, ac quampluribus rationibus fulciretur,in me ipso tamen, ne quid, quasi a me dictum videretur, eousque comprimendam duxi, donec Proclum ipsum, hoc idem sentire cognoscens, maxima animi laetitia affectus, testem tam locupletem nactus, id ipsum dehinc clara voce frequenter asserui.

15 Ibid, c. 95r, De Pace, 39, n. 42:

     res ipsae mathematicae, de quibus fiunt demonstrationes, nec omnino in subiecto, sensibiles sunt, nec penitus ab ipso liberatae, sed in ipsa phantasia reperiuntur figurae illae mathematicae, habita tamen occasione a quantitatibus in materia sensibili repertis… Materia ergo harum scientiarum, erit quantum ipsum, hoc modo, ut ita dicam, phantasiatum, et id a plerisque, quamvis non satis proprie, materia intelligibilis nuncupatur.

16 Ibid., c. 106v, De Pace, 43: “… quantitas vero est omnium sensatissimorum sensatissimum.”

17 A. Piccolomini, Parte Prima della Filosofia Naturale …, Venice, 1576, cc. 17v–18r, De Pace, 42.

18 De certitudine …, c. 107r, De Pace, 43:

     … res illae … abstrahibiles erunt maxime, et iccirco faciles cognitu, certae, ac manifestae. Quantitas igitur, quia … nulli materiae limatatae adducitur, iccirco nihil habet arcani, seque totam nobis explicat et manifestat.

19 Ibid., 102r, De Pace, 30, nn. 24, 25:

     Quod autem [demonstrationes Mathematicae] non invenianturm etiam in hac causa formali, arguo primum. Omnis demonstrationis potissimae, est medium diffinitio, vel passionis, vel subiecti. Demonstrationum mathematicarum, non est tale medium … Praeterea, omnis demonstratio potissima, medium habet, quod est causa immediata, ipsius effectus, idest passionis. Sed nulla mathematica reperiturtalis … si Theorema … 32. primi Elem. perpendatur, cognoscetur quod angulus extrinsecus, qui ponitur ibi medium, ad declarandam passionem, quae est habere tres, de triangulo, non est diffinitio, neque trianguli (ut patet) nec passionis. Tam enim triangulus, quam habere tres, non indiget in sui diffinitione angulo extrinseco. Quo non existente, etiam est triangulus, et habet tres. Idem patebit in omnibus fere aliis Euclidis Theorematibus et Problematibus.

20 Parte Prima della Filosofia Naturale, cc. 24v–25r, De Pace, 56:

     … l'astrologo…quantonque consideri il Cielo essere sferico, o la terra rotonda; non per questo ha egli dibisogno di conoscere la vera natura, et sostanza loro, anzi solamente da’ siti, figure, et aspetti che si veggano in Cielo, argomenta esser di tal figura…Per la qual cosa si puo concludere, che se ben la scientia delle cose naturali, convien molto volte con altra scientia, in trattar d'alcun soggetto, o in dimostrare alcuna conclusione, nondimeno in questo da tutti gli altri è differente il filosofo naturale, che non separando mai i concetti delle forme da quei delle proprie materie loro, ambedue queste nature abbraccia come rispettiva l'una dell'altra; ciè la materia, et la forme: le quali sono i due principji, et le cause intrinseche delle cose naturali.

21 M.Tafuri, Venezia e il Rinascimento, Torino: Einaudi, 1985.

22 F. Barozzi, Opusculum, in quo una Oratio, et duae Quaestiones: altera de certitudine, et altera de medietate Mathematicarum continentur, Padua, 1560. On Barozzi see: P. L. Rose, “A Venetian Patron and Mathematician of the Sixteenth Century: Francesco Barozzi (1537–1604),” in Studi Veneziani, New Series, 1977, I:119–77.

23 Quoted by De Pace, p. 126, n. 18.

24 Transcribed with an introduction by De Pace, Le Matematiche …, 339–430.

25 Procli Diadochi Lycii Philosophi Platonic iac Mathematici probatissimi in Primum Euclidis Elementorum librum Commentariorum … Libri IIII. A Francisco Barocio Patritio Veneto summa opera, cura, ac diligentia cunctis mendis expurgati, Padua: 1560.

26 Opusculum, c. 38v, De Pace, 129.

27 Ibid., 38r, De Pace, 131.

28 Ibid., c. 20V, De Pace, 139, n. 52:

     in unaquaque scientia tum iuxta subiectam materiam, tum etiam iuxta scientiae illius methodum certitudinem requirendam esse, ut doctrinae methodus subiectae materiae correspondeat. Si igitur subiectam Mathematicarum materiam maximum in se se habere certitudinem fatemur, cur demonstrationes etiam mathematic as certissimas esse non dicemus?

29 A. Favaro, “La libreria di Galileo Galilei descritta ed illustrata,” in Bulletino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche, 1886, XIX:219–93.

30 P. Catena, Universa loca in logicam Aristotelis in mathematicas disciplinas, Venice, 1556; idem, Super loca mathematica contenta in Topicis et Elenchis Aristotelis, Venice, 1561; idem, Oratio pro Idea Methodi, Padua, 1563. On Catena see: P. L. Rose, “Professors of Mathematics at Padua University 1521–1588,” Physis, 1975, XVII:302–33.

31 Universa loca, 70–71, De Pace, 191:

     natura enim et per sensum notum est quoniam calidum est, ideo non est opus praecipere mente et suppositione aliqua intellectuali, et quadam scrupolosa indagine suum quia de calido, quando calidum est subiectum, seu datum vel genus; hoc casu, quando est notum quia est dati, despicitur praecognoscere mentis indagatione de dato, an sit. Quod non contingit similiter de numero, quando numerus est datum: de eo est necesse mente et intellectuali actu praeaccipere quia numeri, videlicet quod numerus actu est mente conceptus, ac si existeret, vel aptitudinem ad existendum habeat, et hoc quidem propter hoc, quod numerus neque natura neque sensu actualiter percipitur quod sit, sed tantum intellectu dignoscitur.

32 Ibid., 25, De Pace, 200.

33 Ibid., 72, De Pace, 197:

     si illa linea, quae altramento pingitur, vel penna aut stilo protrahitur recta non sit, non ob id tamen dicendum est Geometram errare, quia non ad id intentionem dirigit Geometra quod oculis subijcitur, sed ad id potius, quod intus animo concipit, dirigit intentionem.

34 Ibid., 44, De Pace, 210–11, n. 52.

35 Ibid., 26, De Pace, 212, n. 54: “Verbum hoc inducens, duas inductiones significat: Alteram Geometricam, reliquam syllogisticam.”

36 Ibid., 25–6, De Pace, 213. One should note, however, that Catena changes a few of Euclid's terms – speaking about a triangle inscribed in a semicircle – in order to make this proposition comparable with Aristotle's discussion of the sum of angles of the triangle in the Posterior Analytics, 71, 19–27.

37 Ibid., 28:

     et simpliciter scitur per Geometricam inductionem, quae semper ex veris, primis, causis illativis conclusionis, et ex magis notis procedit, non autem ex immediatis semper, neque ex causis quae dant esse, sed ex his tantum, quae dant propter quid illationis, tale instrumentum quod inductionem Geometricam voco,non est una consequentia, sed plures, ut plurimum, neque per immediata semper procedit, sed alternatim per immediata, et per ea qua probata sunt procedit, immediata autem, voco propositiones per se notas, et etiam illas propositiones demonstratas, quae immediate probant sequentes.

     Quoted from Giacobbe, Le opere di Pietro Catena…, 130.

38 Ibid., 97–8, De Pace, 240–1, n. 120.

39 See De Pace, especially 170–85.

40 Baldini, “La nova del 1604 e i matematici e filosofi del Collegio Romano,” Annali dell'Istituto e Museo di Storia della Scienza, 1981, VI: 63–98; idem, Legem Impone Subactis….

41 See Feldhay, Galileo and the Church … Chapter 8.

42 Baldini, Legem Impone Subactis…, 56.

43 Gatto, Tra Scienza e Immaginazione…, 153–8.

44 Baldini, Legem Impone Subactis…, 54–5.

45 J. Klein, Greek Mathematical Thought and the Origin of Algebra, transl. by E. Brann, New York: Dover, 1992 (second ed.); D. R. Lachterman, The Ethics of Geometry: A Genealogy of Modernity, New York: Routledge, 1989, Part 2.

46 See E. Giusti, Bonaventura Cavallieri and the Theory of Indivisibles, Rome, 1982; P. Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century, New York: Oxford University Press, 1996.

47 Baldini, Legem Impone Subactis…, 54; Gatto, Tra Scienza e Immaginazione..., 83.

48 Ibid., 44, 54. It should be noted, however, that Baldini has been very cautious in drawing conclusions from these facts. His tendency is to see them as promising beginnings that never developed into real fruitful research. My aim, however, is to show that exactly in the years of Galileo's formation an institutional structured space existed where the type of problems he was interested in were discussed. Further research on Jesuit mathematics currently under quite vigorous pursuit will show whether Jesuit science was seriously constrained and for which reasons. Alternatively an excavation of the sources will point out the results of all these beginnings, which are still obscure in modern historical research.

49 The edition quoted by De Pace is that of Rome, 1585.

50 Ibid., 209 col. b, quoted by De pace, “Interpretazione di Aristotele e comprensione matematica della natura,” in G. Canziani and Y. C. Zarka, L'Interpretazione nei Secoli XVI e XVII, Milano: Franco angeli, 1993, p. 284–5, n. 45: “Vocatur autem quantitas haeproxime inhaerens materiae primae ab Averre indeterminata.”

51 Ibid., 373 col. b-374 col. a, in De Pace, Le Matematiche…, 88, n. 148: “Albertus, D. Thomas, Aegidius et alii, existimant quantitatem Mathematicam non posse abstrahi ad omni substantia, sed tantum a sensibili.”

52 Ibid., 375 col. b, De Pace, 88.

53 Ibid., 86 col. a, De Pace, 92–3, n. 164:

     A principio… donatur a Deo animus, immortalis ille quidem et capax omnium disciplinarum, sed rudis expersque scientiarum, et ut pulchre inquit Aristoteles ceu tabula quaedam nuda, in qua nihil omnino pictum est.

54 Ibid., 86 col. b, De Pace, 94, n. 166:

     Si animus noster olim habuit scientiam omnium rerum, quam postea dum est in corpore per disciplinam institutionem exercitationemque reminiscitur, cur ad reminiscendum adeo egemus sensibus, ut his subaltis nulla quoat scientia comparari? Nos enim experimur, cum quidpiam reminisci volumus quod antea cognovimus, non valde indigere aut uti opera sensuum exteriorum; et sine his non solum posse sed interdum etiam fieri solere multarum rerum reminiscentiam sola vi et ope imaginatricis vel cogitatricis facultatis.

55 Ibid., 26 cols, a-b, De Pace, 90, n. 154.

56 This passage has been frequently quoted by historians of science interested in Jesuit mathematics. See A. Crombie, “Mathematics and Platonism in the Sixteenth-Century Italian Universities and in Jesuit Educational Policy,” in Y. Maeyama and W. G. Saltzer, eds., Prismata, Naturwissenschaftsgeschichtliche Studien, Wiesbaden, 1977, 67. See also Mancosu, Philosophy of Mathematics…, 13, from which I am quoting.

57 Ibid., 51 col. a, in De Pace, “Interpretazioni di Aristotele…,” 287.

58 C. Clavius, Commentaria in Euclidis Elementorum Libri XV…, 3d ed., Rome, 1591.

59 Ibid., 4: “Cur sic ditae sint.”

60 Ibid., 5: ‘Nobilitas atque praestantia scientiarum mathematicarum’:

     Si vero nobilitas, atque praestantia scientiae ex certitudine demonstrationum, quibus utitur, sit udicanda: haud dubie Mathematicae disciplinae inter caeteras omnes principem habebunt locum. Demonstrant enim omnia, de quibus suscipiunt disputationem, firmissmis rationibius, confirmantque, ita ut vere scientiam in auditoris animo gignant, omnemque prorsus dubitationem tollant… Theoremata enim Euclidis, caeterorumque Mathematicorum, eandem hodie, quam ante tot annos, in Scholis retinent veritatis puritatem, rerum certitudinem, demonstrationum robur, ac firmitatem.

61 Ibid.: “Hue accedit id, quod Plato ait in Philebo, seu Dialogo, qui de summo bono inscribitur. Earn scientiam esse digniorem, praestantioremque, quae magis synceritatis, veritatisque est amans.”

62 Ibid., p. 5:

     Nam si a rebus sensibilibus, quas Physicus considerat, ad res ab omni materia sensibili secretas, seiunctasque, quas contemplatur. Metaphysicus, vires, aciemque nostri intellectus attollere absque ullo medio tentemus: nos metipsos excaecabimus; non secus ac ei contingit, qui e carcere aliquo tenebricoso, in quo dix latuit in lucem Soliis clarissimam emittitur. Quam ob rem, antequam a rebus physicis, quae materiae sensibus obnoxiae sunt coninunctae, ad res metaphysicas, quae sunt ab eadem maxime anulsae, intellectus ascendat; necesse est, ne harum claritate offundatur, prius eum assuefieri rebus minus abstractis, quales a Mathematicis considerantur, ut facilius illas possit comprehendere.

63 Ibid.: “Quocirca recte Divinus Plato Mathematicas disciplinas erigere animum, & ad divinarum rerum contemplationem exdocere mentis aciem affirmat.”

64 Ibid.:

     Immo vero idem Plato in Philebo, omnes disciplinas sine Mathematicis viles esse non dubitavit asserere, Qua de causa in 7. de Republ. praecipit: Mathematicas disciplinas primo omnium esse addiscendas, propter varias, ac multiplices earum utilitates (ut copiose scribit) non solum ad reliquas artes rectius percipiendas, verum etaim ad Rem publicam bene administrandam.

65 This, of course, has already been noticed by most readers of Jesuit mathematical texts. See Crombie, Galluzzi, Wallace, or Dear.

66 Clavius, “Prolegomena,” 5:

     Quoniam disciplinae Mathematicae de rebus agunt, quae absque ulla materia sensibili considerantur, quamvis re ipsa materiae sint immersae; perspicuum est eas medium inter Metaphysicam, et naturalem scientiam obtinere locum, si subjectum earum consideremenus, ut recte a Proclo probatur, Metaphysices etenim subiectum ab omni est materia seiunctum, & re, & ratione; Physices vero subeictum & re & ratione materiae sensibili est coniunctum: Unde cum subiectum Mathematicarum disciplinarum extra omnem materiam consideretur, quamvis re ipsa in ea reperiatur, liquido constat, hoc medium esse inter alia duo.

67 Ibid., 4: “Volunt itaque praedicti auctores, scientiarum Mathematicarum quasdam in intellectibilibus duntaxat ab omni materia separatis, quasdam vero in sensibilibus, ita ut attingant materiam sensibus obnoxiam, versai.”

68 Ibid., 6:

     Non parum etiam conducunt hae artes ad Philosophiam naturalem, moralem, Dialecticam, & ad reliquaas id genus doctrinas, artesque perfecte acquirendas, ut perspicue docet Proclus. His adde, quod omnia volumina antiquorum Philosophorum, maxime Aristotelis, & Platonis quosmerite duces nobis sequendos, ad bene recteque philosophandum proponimus, eorumque fere omnium interpretum cum Graecorum, tum Latinorum, exemplis Mathematicis sunt referta, ea potissimum de causa, ut ea quae alioquin multis obstructa difficultatibus videbantur esse, per exempla huiusmodi clariora, magisque perspicua fierent: … Quantum vero emolumenti hae disciplinae ad sacras literas recte percipiendas, interpretandasque conferant, multis pulcherrime nobis exponit B. Augustinus lib. 2 cap. 16 de Doctrina Christiana demonstrans… Quo item loco, Geometriam magnam asserre Theologis utilitatem perhibet.

69 J. Blancanus, “De Mathematicarum Natura Dissertatio,” Bologona, 1615. The text was published as an appendix to Blancanus's Aristotelis Loca Mathematica. Five years later Blancanus published another treatise on the mathematical sciences, his “Preparation for Learning and Advancing the Mathematical Disciplines” (“Apparatus ad mathematicas addiscendas et promovendas”). The “Treatise on the Nature of Mathematics” was lately translated into English; it is published as an appendix to Mancosu, Philosophy of Mathematics.… I've used this translation in all my quotations from the “Treatise…”.

70 Blancanus, “Treatise … “ in Mancosu, 179.

71 Ibid., 179–80.

72 Ibid., 180.

73 Ibid., 184.

74 Ibid., 179.

75 On the development of these two directions see U. Baldini, “Archimede nel Seicento Italiano,” in C. Dollo, ed., Archimede: Mito Tradizione Scienza, Florence: Olschki, 1992, 248 ff.

76 A. Favaro, ed., Le Opere di Galileo Galilei, 20 Vols. in 21, Florence, 1968 (1st edition 1890), I, 318:

     Quia igitur grave mobile…descendens, tardius movetur in principio, ergo necessarium est, illud minus esse grave in principio sui motus quam in medio vel in fine; cum certo sciamus, ex demonstratis in primo libro, velocitatem et tarditatem, gravitatem et levitatem sequi…Verum naturalis et intrinseca mobilis gravitas certe non est diminuta, quia nec diminuta est moles nec densitas illius: restat ergo, imminutionem illam gravitatis esse praeternaturalem et accidentariam…Videamus ergo et diligenter perscrutemur, an forte virtus ista sit causa diminuendae gravitatis mobilis in principio sui motus.

77 Ibid, 273: “Sed animadvertentum est, quod magna hic oritur difficultas: quod proportiones istae, ab eo qui periculum fecerit, non observari comperientur.”

78 Traditionally dated to 1593, but see also A. Carugo and A. C. Crombie, “The Jesuits and Galileo's Ideas of Science and of Nature,” Annali del Museo di Storia della Scienza di Firenze, 1983, VIII:1–68.

79 See Galluzzi's discussion in Momento, 199–227.

80 See also Harris, “Les chaires des mathématiques,” See note 10.

81 It should be noted, however, that Galluzzi thinks Galileo's project in the Mecaniche is much more ambiguous than I stated here. For him, Galileo's choice of the term “momento” to indicate a concept he could not differentiate before demonstrates the dynamical considerations underlying the Mecaniche in its very foundations.

82 All citations from Galileo's Dialogue are taken from: Dialogue Concerning the Two Chief World System – Ptolemaic and Copernican, transl. with revised notes by S. Drake, Berkeley: University of California Press, 1992.

83 See, for example, P. Damerow, G. Freudentahl, P. MacLaughlin, and J. Renn, Exploring the Limits of Preclassical Mechanics, New York: Springer, 1962.

84 See Wallace, Galileo and His Sources.…

85 Galluzi, Momento…, Renn, Exploring.…

86 See Exploring…; 18–19.

87 See the discussion of the definition of velocity and of “Mirandum Paradox” in Exploring …, pp. 13–5, 194–9.

88 Opere, “De Motu,” ch. 14, 296–302.

89 See above, p. 105.

90 The book referred to is probably a thesis defended by a student of Christopher Scheiner, Disquisitiones mathematicae, de controversiis et novitatibus astronomicis…sub praesidio Christophori Scheiner… Nobilis et Doctissimis iuvenis, Ioannes Georgius Locher, Boius Monacensis, Artium et Philosophiae Baccalaureus, Magisteri) Candidatus, Ingolstadt, 1614.

91 See the fascinating discussion of the rule in Explorations … (note 83), pp. 171–4; 178–85.

92 L. Maieru, “… in Christophorum Clavium de Contactu Linearum Apologia”: Considerazioni attorno all a Polemica tra Peletier e Clavio circa l'angolo di contatto (1579–1589),” Archive for the History of Exact Sciences, 1990,41/1:115–37.

93 Heath, Euclid's Elements, III, 16: “The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed; further the angle of the semicircle is greater, and the remaining angle less than any acute rectilinear angle.”

     X, 1: “Two unequal magnitudes set out, if from the greater there be substracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out.”

     V, 3: “a ratio is a sort of relation in respect of size between two magnitudes of the same kind” (My emphasis, R.F.).

94 Maierù, “… in Christophorum Clavium,” 129.

95 A good example is his middle position in the debate between homocentrists and Copernicans on epicycles and eccentrics; see J. M. Lattis, Between Copernicus and Galileo, Chicago, IL: University of Chicago Press, 1995, Chapter 5.

96 Maierú, In Christophorum clavium … op. cit.

97 Ibid. p. 133.

98 For this history, see N. Kretzmann, ed., Infinity and Continuity in Ancient and Medieval Thought, Ithaca: NY Cornell University Press, 1982.

99 About such compromise common among sixteenth-century astronomers, see N. Jardine, The Birth of History and Philosophy of Science: Kepler's A Defence of Tycho against Ursus with Essays of its Provenance and Significance, Cambridge: Cambridge University Press, 1984, II, 7.

100 See Feldhay, Galileo and the Church, Chapters 7, 8, 11.

101 For the Greek background see R. Feldhay and S. Unguru, “Greek Mathematical Discourse: Some Examples of Tensions and Gaps,” in T. Berggnen, ed. Proceedings of the Third International Conference on Ancient Mathematics, Delphi. Vancouver, British Columbia: Simon Fraser University Press, 1997, 45–7.

102 Quoted from Crombie, “Mathematics and Platonism…,” 66.

103 Perera, De communibus…, 1576, 24, in Mancosu, Philosophy of Mathematics…, 214, n. 12, translation by Crombie, ibid., 67:

     Confirmatio Minoris ducitur ex his, quae scribit Plato in 7, lib. de Republ. dicens Mathematicos somniare circa quantitatem, & in tractandis suis demonstrationibus non scientificè sed ex quibusdam suppositionibus procedere, quamobrem non vult doctrinam eorum appellare intellegetiam aut scientiam, sed tantum cogitationem.

104 See discussion of Perera's position in the De Certitudine…above, 92

105 See the discussion of Ratio in Feldhay, Galileo and the Church, 223–32.

106 See discussion of Blanianus’ treatise, in this essay, pp. 98–100.