Pierre Maurice Marie Duhem Alan Aversa Galileo’s Precursors Translation of Studies on Leonardo da Vinci (vol. 3) ∗ June 29, 2018 Springer ∗ Duhem (1906) To my wife, who made this translation possible, to the Blessed Virgin Mother, and to the Holy Trinity, Who makes all things possible Foreword Note on the Translation Everything translated from the original French of Duhem (1906). Additions in [brackets] belong to the translator. Alan Aversa June 29, 2018 ix Acknowledgements The translator would like to thank Gery and Rebecca Aversa for helping to make this work open access. xi Preface To the third series of our Studies on Leonardo da Vinci , we give a subtitle: Galileo’s Parisian Precursors . This subtitle announces the idea of which our previous stud- ies had already discovered a few aspects and which our new researches place in full light. The Science of mechanics inaugurated by Galileo—by his followers, by his disciples Baliani, Torricelli, Descartes, Beeckman, Gassendi—is not a creation; modern intelligence did not produce it from the very start and from all the pieces which reading the art of Archimedes of applying geometry to natural effects had re- vealed to it. Galileo and his contemporaries used the mathematical skill acquired in the trade of the geometers of Antiquity to clarify and develop a mechanical Science of which the Christian Middle Ages had posed the principles and the most essential propositions. The physicists who taught in the 14 th century at the University of Paris had designed this Mechanics by taking observation as their guide; they had sub- stituted it for the Dynamics of Aristotle, convinced of its ineffectualness to “save the phenomena.” At the time of the Renaissance, the superstitious archaism, where the wit of the Humanists and the Averroist routine of a retrograde Scholasticism are complacent, rejected this doctrine of the “Moderns.” The reaction was strong, particularly in Italy, against the Dynamics of the “Parisians,” in favour of the in- admissible dynamics of the Stagirite. But, despite stubborn resistance, the Parisian tradition found, outside schools as well as in Universities, masters and scholars to maintain and develop it. It is of this Parisian tradition that Galileo and his followers were heirs. When we see the science of Galileo triumph over the obstinate Peripateti- cism of Gremonini, we believe, uninformed in the history of human thought, that we are witnessing the victory of the young modern Science over the medieval Philoso- phy, obstinate in its psittacism; in truth, we behold the long-prepared triumph of the science that was born in Paris in the 14 th century on the doctrines of Aristotle and Averroes, revived by the Italian Renaissance. No movement can perdure if it is not maintained by the continuous action of a motive power, directly and immediately applied to the mobile. This is the axiom on which the whole Dynamics of Aristotle rests. According to this principle, the Stagirite wants to apply to the arrow that continues to fly after leaving the bow a motive power which carries it; he believes this power xiii xiv Preface is found in the disturbed air; it is the air, hit by the hand or by the ballistic machine, which supports and guides the projectile. This assumption, which seems to push its improbability even to ridicule, seems to have been admitted almost unanimously by the physicists of Antiquity; one of them spoke out clearly against it, and he, in the last years of Greek philosophy, is, by his Christian faith, almost separated from this Philosophy; we have named John of Alexandria, nicknamed Philoponus. After showing what was unacceptable in the Peripatetician theory of the motion of projectiles, John Philoponus declares that the arrow continues to move without any mover applied to it, because the rope of the bow has created an energy that plays the role of the motive force. The last thinkers of Greece and even Arab philosophers failed to mention the doc- trine of this John the Christian, for whom Simplicius or Averroes had only sarcasm. The Christian Middle Ages, taken by the naïve admiration which inspired the peri- patetic Science when it was revealed, shared first, with respect to the assumption of Philoponus, the disdain of the Greek and Arab commentators; St. Thomas Aquinas only mentions it to warn those it could seduce. But following the condemnations in 1277 by the Bishop of Paris, Étienne Tem- pier, against a group of theses that supported “Aristotle and those of his suite,” a great movement emerges, which will release the Christian thought from the yoke of Peripateticism and Neoplatonism and produce what the archaism of the Renaissance called the Science of the “Moderns.” William of Ockham attacks, with his customary vivacity, the theory of the mo- tion of projectiles proposed by Aristotle; he is content, besides, to simply destroy without building anything; but his critics, with some followers of Duns Scotus, re- store the honor of the doctrine of John Philoponus; energy , the motive force of which he had spoken, reappears under the name of impetus . This hypothesis of the impe- tus , impressed in the projectile by the hand or by the machine which lauched it, is seized upon by a secular master of the Faculty of Arts in Paris, a physicist of genius; Jean Buridan, toward the middle of the 14 th century, takes it as the foundation of a Dynamics with which “all the phenemena agree.” The role that the impetus plays in the Dynamics of Buridan is very exactly what Galileo will assign to the impeto or momento ; Descartes to the quantité de mouve- ment ; and finally Leibniz to the living force . So exact is this correspondance that, for explaining in his Academic Lessons the Dynamics of Galileo, Torricelli will often take up the reasoning and almost the very words of Buridan. This impetus , which would remain unchanged within the projectile if it were not incessantly destroyed by the resistance of the medium and by the action of grav- ity contrary to the movement, Buridan takes, at equal speed, as proportional to the amount of primary matter that the body contains; he conceived this quantity and de- scribed in terms almost identical to those which Newton will use to define mass. At equal mass the impetus is as much greater as the speed is greater; prudently, Buridan refrains from further clarifying the relationship between the size of the impetus and its speed; more daringly, Galileo and Descartes would agree that this relationship is reduced to a proportionality; they will also obtain an erroneous assessment of the impeto and momentum which Leibniz will have to rectify. Preface xv Like the resistance of the medium, gravity reduces constantly and eventually de- stroys the impetus of a mobile that is launched upward, because such a movement is contrary to the natural tendancy of this gravity; but in a mobile that falls, the move- ment is in line with the trendancy of gravity; the impetus also must be constantly increasing, and the speed, in the course of the movement, must grow constantly. This is, according to Buridan, the explanation of the acceleration observed in the fall of a body, an acceleration that the science of Aristotle already knew, but for which the Hellenic commentators of the Stagirite, Arabs or Christians, had given unacceptable reasons. This Dynamics expressed by Jean Buridan presents in a purely qualitative but always exact way the truths that the notions of live force and work allow us to for- mulate in quantitative language. The philosopher of Béthune is not alone in professing this Dynamics; his most brilliant disciples, Albert of Saxony and Nicole Oresme, adopt it and teach it; the French writings of Oresme make it known even to those who are not clerics. When no resistant medium and when no natural tendancy analogous to gravity is opposed to movement, the impetus maintains an invariable intensity; the mobile to which a movement of translation or rotation is applied continues to move with constant speed indefinitely. It is in this form that the law of inertia presents itself to the mind of Buridan; it is in this same form that Galileo will receive it From this law of inertia, Buridan draws a corollary, the novelty of which we must now admire. If the celestial orbs move eternally with a constant speed, it is, according to the axiom of the dynamics of Aristotle, because each of them is subject to an eternal mover of immutable power; the philosophy of the Stagirite requires that such a mover is an intelligence separate from matter. The study of the motive intelligences of the celestial orbs is not only the culmination of Peripatetic Metaphysics; it is the central doctrine around which all the Neoplatonic Metaphysics of the Greeks and of the Arabs revolve, and the Scholastics of the 13 th century do not hesitate to receive, into their Christian systems, this legacy of pagan theologies. Now, Buridan has the audacity to write these lines: From the creation world, God has moved the heavens with movements identical to those which currently move them; he impressed on them then an impetus by which they continue to be moved uniformly; these impetus , indeed, meeting no contrary resistance, are never destroyed nor weakened... According to this conception, it is not necessary to pose the existence of intelligences that move celestial bodies in an appropriate manner. Buridan stated this thought in various circumstances; Albert of Saxony explains it in turn; and Nicole Oresme, to formulate it, finds this comparison: “Except for violence, it is in no way similar to when a man has made a clock i and lets it go to be moved by itself.” If one wanted, by a precise line, to separate the reigns of the ancient Science of the reign of modern Science, it would have to be drawn, we believe, at the moment when Jean Buridan developed this theory, at the moment when one stopped looking at the stars as moved by divine beings and when it has been admitted that the celestial and sublunary movements depend on the same mechanics. xvi Preface This Mechanics—both heavenly and earthly, to which Newton had to give the shape that we admire today—is, besides, that which, from the 14 th century, is trying to be built. During this century, the testimonies of Francis of Meyronnes and of Al- bert of Saxony teach us, one finds physicists upholding that by supposing the earth as moving and the fixed stars as immobile, an astronomical system more satisfac- tory than that where the earth is deprived of movement would be constructed. Of these physicists, Nicole Oresme developed the reasons with a fullness, clarity, and precision that Copernicus will be far from reaching; to the earth he attributes a nat- ural impetus similar to what Buridan attributed to the celestial orbs; to account for the vertical fall of bodies, he admits that one must compose this impetus by which the mobile revolves around the Earth with the impetus generated by gravity. The principle that he lucidly formulates, Copernicus will simply indicate in a dark way, and Giordano Bruno will repeat it; Galileo will use Geometry to draw the conse- quences, but without correcting the wrong form of the law of inertia with which he is implicated. While Dynamics was being established, little by little the laws that govern the fall of bodies are discovered. In 1368, Albert of Saxony offers these two assumptions: 1. the speed of the fall is proportional to the time elapsed since is departure; 2. the speed of the fall is proportional to the path traversed. Between these two laws, there is no choice. The theologian Pierre Tataret, who taught in Paris towards the end of the 15 th century, reproduced verbatim what Albert of Sax- ony said. The great reader of Albert of Saxony, Leonardo da Vinci, after admitting the second of these two hypotheses, endorses the first; but he fails to discover the law of the spaces traversed by a falling body; from a reasoning that Baliani will resume, he concludes that the spaces traversed in equal and successive periods of time are as the series of integers, whereas they are, in truth, as the series of odd numbers. However, the rule which allows the evaluation of traversed space, in a certain time, by a mobile moved by an evenly varying movement, was known for a long time; that this rule was discovered at Paris, in the time of Jean Buridan, or at Oxford, in the time of Swineshead, is clearly formulated in the book where Nicole Oresme poses the essential principles of analytic geometry; in addition, the demonstration he employed to justify it is identical to what Galileo will give. From the time of Nicole Oresme to that of Leonardo da Vinci, this rule was not forgotten; formulated in the majority of treatises produced by the subtle Dialectic of Oxford, it is discussed in the many commentaries to which these treatises had been subjected, during the 15 th century in Italy, then in various books of Physics composed at the beginning of the 16 th century by the Parisian Scholastics. None of the treaties of which we have just spoken contains, however, the idea of applying this rule to falling bodies. We meet this idea for the first time in the Ques- tions on the Physics of Aristotle , published in 1545 by Domingo de Soto. A student of Parisian Scholasticism, of which he was the patron and from which he adopts most of his physical theories, the Spanish Dominican Soto admits that the fall of a body is uniformly accelerated, that the vertical ascent of a projectile is uniformly Preface xvii retarded, and to calculate the path taken in each of these two movements, he cor- rectly uses the rule formulated by Oresme. This is to say that he knows the laws of falling bodies whose discovery is attributed to Galileo. Moreover, he not only claims their invention; rather, he seems to give them as commonly received truths; without doubt, they were commonly admitted by the masters whose lessons Soto followed in Paris. Thus, from William of Ockham to Domingo de Soto, we see the physicists of the Parisian school posing all the foundations of the Mechanics that Galileo, his contemporaries, and his followers will develop. Among those who, before Galileo, received the tradition of Parisian Scholasti- cism, there is none who deserves more attention than Leonardo da Vinci. At the time when he lived, Italy was opposed with a firm resistance to the penetration of the mechanics of the “ Moderni ,” of the “Juniors;” there, among the masters of the Uni- versities, those who looked to the terminist doctrines of Paris have been limited to reproduce, in an abbreviated and sometimes hesitant form, the essential claims of this Mechanics; they were quite far from producing any fruit of which it was the flower. Leonardo da Vinci, on the contrary, is not content to admit the general principles of the Dynamics of the impetus ; he ruminated on these principles constantly, turning in all directions, urging them, somehow, to give what they contained. The essential assumption of this Dynamics was like an early form of the law of the live force; Leonardo sees the idea of the conservation of energy, and he finds this idea, to ex- press it, in terms of a prophetic clarity. Between two laws of falling bodies, the one exact and the other inadmissible, Albert of Saxony had left his reader in suspense; after some trial and error that Galileo, too, will know, Leonardo knew how to settle on the exact law; he happily extends it to the fall of a body along an inclined plane. Through the study of the compound impeto , he is the first to attempt to explain the curvilinear trajectory of projectiles, an explanation which will receive its completion from Galileo and Torricelli. He sees the correction that should be made to the law of inertia stated by Buridan and prepares the work that Benedetti and Descartes will carry out. Doubtless, Leonardo did not always acknowledge all the riches of the treasure accumulated by Parisian Scholasticism; he left out some of them whose borrowing would had given his mechanical doctrine the most happy complement; he ignores the role of the impetus in the explanation of the accelerated fall of bodies; he ignores the rule that allows the calculation of the path traveled by a body with uniformly accelerated motion. It is no less true that his whole Physics is numbered among those that the Italians of his time called Parisian. Such a title, moreover, would be justly given to him; indeed, he takes the prin- ciples of his physics from his assiduous reading of Albert of Saxony and probably also from his meditation on the writings of Nicolas of Cusa; but Nicolas of Cusa was, too, a follower of the mechanics of Paris. Leonardo is thus in his place among the Parisian precursors of Galileo. xviii Preface Until recent years, the Science of the Middle Ages was considered non-existent. A philosopher, who admirably knows the history of Science in Antiquity and in modern times, once wrote 1 : Suppose that printing press had been discovered two centuries earlier; it would have helped strengthen the orthodoxy and served to propagate, outside of the Summa of St. Thomas and a few books of this kind, the bulls of excommunication and the decrees of the Holy Office. Today, we believe, we are allowed to say: If the printing press had been discovered two centuries earlier, it would have published, gradually and when they were composed, the works which, on the ruins of the Physics of Aristotle, have laid the foundations of a Mechanics of which modern times are rightly proud. This substitution of modern Physics for the Physics of Aristotle was the result of a long-term effort of extraordinary power. This effort took support on the oldest and the most resplendent of medieval Uni- versities, on the University of Paris. How could a Parisian not be proud of it? Its most prominent promoters were the Picard, Jean Buridan, and the Norman, Nicole Oresme. How could a Frenchman not feel a legitimate pride? It resulted from the stubborn fight that the University of Paris, true custodians, at that time, of the Catholic orthodoxy, led against the Peripatetic and Neoplatonist paganism. How could a Christian not give thanks to God for it? The studies that follow have appeared either in the Bulletin Italien or in the Bulle- tin Hispanique ; to Mr. G. Radet, Dean of the Faculty of Letters of Bordeaux, to our colleagues, Mr. E. Bouvy and Mr. G. Cirot, we are indebted for the generous hospi- tality given to our research; may they deign to receive the homage of our gratitude. Bordeaux, Pierre 24 May 1913 DUHEM. 1 G. Milhaud, Science grecque et Science moderne ( Comptes rendus de l’Académie des Sciences morales et politiques , 1904). — G. Milhaud, Études sur la pensée scientifique chez les Grecs et les Modernes , Paris, 1906, pp. 268-269. Contents Dedication Preface Part I Jean I Buridan (of Béthune) and Leonardo da Vinci 1 A date concerning Master Albert of Saxony . . . . . . . . . . . . . . . . . . . . . . . 3 2 Jean I Buridan (of Béthune) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 That the theory of the center of gravity, taught by Albert of Saxony, is not borrowed from Jean Buridan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4 The Dynamics of Jean Buridan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5 That the Dynamics of Leonardo da Vinci proceeds, via Albert of Saxony, from that of Jean Buridan. — To what extent it deviates and why. — The various explanations of the accelerated fall of weights that have been proposed before Leonardo. . . . . . . . . . . . . . . . . . 39 Part II The Tradition of Buridan and Italian Science in the 16 th Century 6 The Dynamics of the Italians at the time of Leonardo da Vinci, the Averroists, Alexandrists, and Humanists . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7 The spirit of Parisian Scholasticism in the time of Leonardo da Vinci 89 8 The Parisian Dynamics in the time of Leonardo da Vinci . . . . . . . . . . . 95 9 The decadence of Parisian Scholasticism after the death of Leonardo da Vinci. The attacks of Humanism. Didier Erasmus and Luiz Vives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 xix xx Contents 10 How, in the 16 th century, the Dynamics of Jean Buridan spread in Italy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 11 On the early progress accomplished in Parisian Dynamics by the Italians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Giovanni Battista Benedetti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Giordano Bruno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Part III Domingo Soto and Parisian Scholasticism 12 Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 13 The life of Domingo Soto, Friar Preacher . . . . . . . . . . . . . . . . . . . . . . . . . 185 14 Domingo Soto and Parisian Nominalism . . . . . . . . . . . . . . . . . . . . . . . . . . 189 15 Potential infinity and actual infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 16 The equilibrium of the Earth and seas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 17 The Dynamics of Jean Buridan and the Dynamics of Soto . . . . . . . . . . . 199 18 Soto tries to make the views of Aristotle and St. Thomas agree with the hypothesis of impetusla]impetus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 19 The origins of Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 The treatise De proportionalitate motuum et magnitudinum . . . . . . . . . . . . 209 Thomas Bradwardine. John of Murs. Jean Buridan. . . . . . . . . . . . . . . . . . . . 212 Albert of Saxony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 20 Albert of Saxony and the law according to which the fall of a weight accelerates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 De intensione et remissione formarum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 21 Nicole Oresme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 22 The Dynamics of Oresme and the Dynamics of Buridan . . . . . . . . . . . . 251 23 The center of gravity of the Earth and the center of the World . . . . . . 259 24 The plurality of worlds and the natural place according to Nicole Oresme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 25 Nicole Oresme, inventor of analytic Geometry . . . . . . . . . . . . . . . . . . . . . 271 Contents xxi 26 How Nicole Oresme established the law of uniformly varying motion 281 The influence of Nicole Oresme at the University of Paris. — The treatise De latitudinibus formarum . Albert of Saxony. Marsilius of Inghen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 The Oxford School in the middle of the 14 th century. — William Heytesbury. — John Dumbleton. — Swineshead. — The Calculator. — The treatise De sex inconvenientibus . — William of Colligham. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 27 The spirit of the Oxford School in the middle of the 14 th century . . . . 307 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 28 The law of uniformly varied movement at the School of Oxford . . . . . 327 The De primo motore of Swineshead and the Dubia parisiensia . . . . . . . . . 327 The Summa of John Dumbleton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 The Regulæ solvendi sophismata and the Probationes of William Heytesbury . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 The Tractatus de sex inconvenientibus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 The opuscule entitled: A est unum calidum . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 The Liber calculationum of Riccardus of Ghlymi Eshedi . . . . . . . . . . . . . . 343 29 How the doctrines of Nicole Oresme spread in Italy . . . . . . . . . . . . . . . . 347 30 How the doctrines of the Oxford school spread into Italy . . . . . . . . . . . 357 31 Leonardo da Vinci and the laws of falling bodies . . . . . . . . . . . . . . . . . . . 369 32 The study of latitude forms at the University of Paris at the beginning of the 16 th century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 John Majoris, John Dullaert of Ghent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Alvaro Thomas of Lisbon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 The Spanish masters. Juan de Celaya. Luis Coronel. . . . . . . . . . . . . . . . . . . 393 Domingo Soto and the laws of falling bodies . . . . . . . . . . . . . . . . . . . . . . . . 401 33 Conclusion. The Parisian tradition and Galileo. . . . . . . . . . . . . . . . . . . . 407 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Part I Jean I Buridan (of Béthune) and Leonardo da Vinci