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Atomism in Late MedievalPhilosophy and Theology



History of Scienceand Medicine LibraryVOLUME 8

Medieval andEarly Modern Science

Editors

J.M.M.H. Thijssen, Radboud University NijmegenC.H. Lüthy, Radboud University Nijmegen

Editorial Consultants

Joël Biard, University of ToursSimo Knuuttila, University of Helsinki John E. Murdoch, Harvard University

Jürgen Renn, Max-Planck-Institute for the History of ScienceTheo Verbeek, University of Utrecht

VOLUME 9



Atomism in Late Medieval

Philosophy and Theology

Edited by

Christophe Grellard and Aurélien Robert

LEIDEN • BOSTON2009



ISSN 1872-0684ISBN 978 90 04 17217 3

Copyright 2009 by Koninklijke Brill NV, Leiden, The Netherlands.Koninklijke Brill NV incorporates the imprints Brill, Hotei Publishing,IDC Publishers, Martinus Nijhoff Publishers and VSP.

All rights reserved. No part of this publication may be reproduced, translated,stored in a retrieval system, or transmitted in any form or by any means, electronic,mechanical, photocopying, recording or otherwise, without prior written permissionfrom the publisher.

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printed in the netherlands

On the cover : Goussin de Metz, l’Image du monde, Paris, 1304; Bibliothèque de RennesMétropole, MS0593, f. 64a. Courtesy of the Bibliothèque de Rennes Métropole.

This book is printed on acid-free paper.

Library of Congress Cataloging-in-Publication DataAtomism in late medieval philosophy and theology / edited by Christophe Grellardand Aurélien Robert. p. cm. — (History of science and medicine library ; v. 8) Includes bibliographical references and index. ISBN 978-90-04-17217-3 (hardback : alk. paper)1. Atomism. 2. Philosophy, Medieval. I. Grellard, Christophe. II. Robert, Aurélien. III.Title. IV. Series.

BD646.A85 2009 189—dc22

2008042420



CONTENTS

Preface ......................................................................................... viiList of Authors ............................................................................ ix

Introduction ................................................................................ 1 Christophe Grellard & Aurélien Robert

Beyond Aristotle: Indivisibles and Infi

nite Divisibility in theLater Middle Ages .................................................................. 15 John E. Murdoch

Indivisibles and Infinities: Rufus on Points ................................ 39

Rega Wood

Richard Kilvington on Continuity ............................................. 65 El żbieta Jung & Robert Podkoński

The Importance of Atomism in the Philosophy of Gerard ofOdo (O.F.M.) .......................................................................... 85

Sander W. de Boer

Nicholas of Autrecourt’s Atomistic Physics ............................... 107 Christophe Grellard

William Crathorn’s Mereotopological Atomism ........................ 127 Aurélien Robert

An Indivisibilist Argumentation at Paris around 1335: Michelof Montecalerio’s Question on Point and the Controversywith John Buridan .................................................................. 163

Jean Celeyrette

John Wyclif ’s Atomism ............................................................... 183 Emily Michael



Blasius of Parma facing Atomist Assumptions .......................... 221 Joël Biard

Bibliography ................................................................................ 235

Index of Ancient, Medieval and Renaissance Authors ............. 247Index of Modern and Contemporary Authors ......................... 249

vi contents



PREFACE

Most of the papers collected in this volume are the result of a confer-ence held at the Maison française d’Oxford in November 2004, which wasorganized by the present editors. This two-days workshop was entitled“Atomism and its Place in Medieval Philosophy” and its first aim wasto assess the different issues in which atomism could have played arole during the Middle Ages. But the contributions generally focused

their target on the physical/mathematical distinction within medievaldebates about the continuum and the indivisible. For this reason, eachchapter has been thoroughly rewritten and we asked other scholars toparticipate to this book. This is the reason why we have decided tochange the title for the publication with an even more general title.

As organizers of the conference, we would like to express ourgratitude to the institutions that sponsored us and helped us with theirfinancial support: first, the Maison française d’Oxford , where the workshoptook place and whose director, Alexis Tadié, generously offered excellent

conditions for the organization; the “Service Science et Technologie” ofthe French Embassy in Great Britain (London); the GDR 2522 “Philoso-phie de la connaissance et de la nature au Moyen Âge et à la Renais-sance” (CNRS, Tours); the ACI “Articulations entre mathématique etphilosophie naturelle (XIV e –XVIe s.)” (CNRS) which all financed themain part of the colloquium; finally, the Center “Tradition de la penséeclassique” (EA 2482) of the University of Paris I Panthéon-Sorbonnewho helped both for the conference and for the publication of thevolume. We therefore thank them all for their participation, without

which this conference and this book wouldn’t have existed.Above all we must thank Alexis Tadié and Stéphane Van Damme

who encouraged us and made the realization of this project possible.Our gratitude also goes to Margaret Cameron and Dallas Denery Junior who helped us to translate into English some of the chapterspresented here which were initially written in French. We must alsothank the people who were present at the conference but who didn’ttake part in this volume: Richard Cross, Luc Foisneau, Gabriele Gal-luzzo, Dan Garber, Isabel Irribaren, Andrew Pyle, Sabine Rommevaux,Cecilia Trifogli.



LIST OF AUTHORS

Joël Biard, Université François Rabelais, Centre d’Études Supérieures de la Renaissance (UMR 6576 du CNRS), Tours, France.

Jean Celeyrette, UMR 8163 Savoirs, Textes, Langage CNRS-Université de Lille III , Lille, France.

Sander W. de Boer, Radboud Universiteit Nijmegen, Nijmegen, TheNetherlands.

Christophe Grellard, Université de Paris I Panthéon-Sorbonne, Paris,

France.

Elbieta Jung, University of Ł ód Ω , ŁódΩ, Poland.

Emily Michael, Brooklyn College and the Graduate Center, City University of

New York , New York, USA.

John E. Murdoch, Harvard University, Cambridge, Mass., USA.

Robert Podkoski, University of Ł ód Ω , ŁódΩ, Poland.

Aurélien Robert, Centre d’Études Supérieures de la Renaissance (UMR 6576

du CNRS), École française de Rome, Tours-Roma, France-Italy.

Rega Wood, Stanford University, USA.



INTRODUCTION

Christophe Grellard & Aurélien Robert

1. Medieval Atomism in Recent Historiography

In the second half of the twentieth century, there has been a greatrenewal in the history of medieval atomism. If the rising of philosophi-

cal debates on the composition and the divisibility of a continuum in theLatin West during the thirteenth and the fourteenth centuries had longbeen considered a mere transition between Ancient and Renaissanceatomism,1 more recent studies have taken the opposite direction andtended to restore the image of a period of intense reflections on indi-visibles. John E. Murdoch is mostly responsible for this turn in recenthistoriography, thanks to his work on a lot of still unedited authors.2 In some respects, this attitude is not that new if we consider that atthe end of the nineteenth century, Kurd Lasswitz’s essay Geschichte der

Atomistik vom Mittelalter bis Newton (1890) and Léopold Mabilleau’s Histoirede la philosophie atomistique (1895) had already attempted to make roomfor medieval atomism, though both of them were unfamiliar with mostof the important authors of the fourteenth century, with the exceptionof Nicholas of Autecourt.3 But it is only recently that some importantmonographs have endeavoured to replace these two pioneering books.Bernhardt Pabst, in 1994,4 and Andrew Pyle in 1995,5 furnished newand detailed studies of medieval atomism, with full chapters dedicatedto fourteenth-century atomism.6 The ambition of this book is not to

replace those essays in the history of medieval atomism, but to raisesome new questions about the commonly accepted view of the nature

1 We should mention earlier monographs on this topic, such as Van Melsen, From Atomos to Atom, where the chapter on medieval atomism is joined to the Renaissanceperiod and is limited to developments on minima naturalia.

2 See the bibliography at the end of the volume.3 In fact, they only knew Autrecourt’s articles condemned in Paris.4 Pabst, Atomtheorien des lateinischen Mittelalters.5

Pyle, Atomism and its Critics from Democritus to Newton.6 For a more detailed historiographical essay on medieval atomism, cf. Lüthy,Murdoch & Newman, Late Medieval and Early Modern Corpuscular Matter Theories, “intro-duction,” pp. 1–17.



2 christophe grellard & aurélien robert

of fourteenth-century atomism in particular. The aim is to go fur-ther into detail about specific authors than do the expansive histories

mentioned above, though we do not intend the present work to be amere collection of single essays on single authors. As this introductionwill try to show, several leitmotivs are present throughout the differentchapters. The principal tasks of this book are, first, to distinguish thesingularity of fourteenth-century atomism, compared to other periods;second, to show that the understanding of the debates over this periodis far more complicated than it is usually asserted in the old as in therecent historiography; and third, to ask whether fourteenth-centuryatomism is rather mathematical, physical or even metaphysical, as some

of the contributors have tried to challenge the prevailing view aboutthe mathematical nature of indivisibilism at that time.

It is a dif ficult task to catch the essence of medieval atomism—ifit exists—for as Gaston Bachelard used to say, the atomist doctrinesbecome more and more confused when one wants to embrace them asa whole.7 It is undeniably true that different forms of atomism existedin the Middle Ages, from the Arabic occasionalist theories of the Mutakallimun to the infancy of modern science in the natural philoso-phy of the Renaissance. One may be tempted to distinguish the theoryof the elements in the medical context, the medieval interpretationsof Plato’s atomism in the Timaeus, the reconstruction of Democritus’sthesis through Aristotle’s critics, etc. For medieval attempts to considerthe existence of atoms or indivisibles are to be found in very differ-ent contexts. In any case, the rather common attitude nowadays is toassert that apart from Arabic theological atomism, the only survival ofatomism in the Middle Ages is the fourteenth-century “indivisibilism.”8 This way of dividing the history of atomism is certainly artificial and

misleading, for as early as the twelfth century some atomistic theoriesof matter were developed by philosophers such as William of Conches.So we are left with Bachelard’s opinion, while historians tend to restrictthemselves to their own specific areas. According to John E. Murdoch,for example, fourteenth-century atomism presents some particularfeatures that allow the historian of philosophy and science to isolatethis period from other traditions. The mathematical—or rather geo-metrical—nature of the fourteenth-century debates on the continuum

7 Bachelard, Les intuitions atomistiques, p. 11.8 Murdoch, “Naissance et développement de l’atomisme au bas Moyen Âge latin.”



introduction 3

and indivisibles differentiates it from William of Conches’s atomism,for example, which is in turn influenced by Plato’s Timaeus and by the

medical school of Salerno and has nothing to do with Aristotle’s geo-metrical attacks against Leucippus and Democritus.In this volume, we will focus on fourteenth-century discussions of

indivisibles and atoms,9 in order to take stock of the situation on recenthistoriography and above all to discuss Murdoch’s hypothesis, which isthe prevailing one today. Indeed, all the chapters presented here try torespond, implicitly or explicitly, to the question: are debates on atomismin the fourteenth century purely mathematical and geometrical?

In order to give the reader a general idea of the history of atomism in

the Middle Ages, let us first describe in a few lines the Ancient sourcesthat were available to the medieval philosophers, since it is partly atthe origin of the main stream in recent historiography.

2. Ancient and Medieval Atomism

It has long been thought that Ancient atomists, such as Epicurus orLucretius, were rediscovered during the Renaissance, notably after the

works of Poggio Bracciolini on the manuscript of the De natura rerum discovered by him in 1417. But the manuscripts on which Poggiobased his edition are from the 9th century, and we know now thatLucretius was still copied during the Middle Ages, as is evident fromthe many manuscripts of the De natura rerum dating after the 9th–10thcenturies.10 Moreover, as J. Philippe has shown in his pioneering study,11 Lucretius’s poem was discussed throughout the Middle Ages with nointerruption from the era of the Church Fathers to the twelfth century.An example of this persistence is William of Conches, who quotes a

passage from the De natura rerum in his Dragmaticon philosophiae, thoughhe didn’t have access to the original text but only knew it from Cicero,Priscian and probably other sources.12 The same is also true for Epicurus,whose works were known through a still longer chain of intermediatesources. One feels this tradition in the medieval encyclopaedias, as inIsidore of Seville, the Venerable Bede and Rhaban Maur, all of whom

9 One exception is Rega Wood’s chapter on Richard Rufus, which serves to clarify

the particular situation of fourteenth-century discussions.10 Philippe, “Lucrèce dans la théologie chrétienne,” (1896), pp. 151–152.11 Ibid.12 William of Conches, Dragmaticon philosophiae [ Ronca], p. 27.




discussed the existence of atoms, a tradition that continues up to Vincentof Beauvais, for example. John of Salisbury also dealt with Epicurism

in his Metalogicon13

and in his Entheticus, where he tried to refute itsprincipal tenets.14 We may multiply the examples, but there is no doubtthat ancient atomist theories were known to the medieval philosophersand associated with names such as Epicurus or Lucretius. The idea ofan eclipse of atomism seems to be confirmed by the violent reactionsof the Church Fathers. Everybody has in mind Lactantius’s attacks, forexample.15 So, even if Augustine said that Epicurism was dead in histimes, it is now well-known that this view has no basis in historical real-ity. It is true, indeed, that Ancient atomists were not discussed for their

theories of matter as such, but rather for the theological consequencesof their views, among them the negation of Divine providence and theimplication of the impassivity of God. Therefore, despite the fact thatmany indirect sources were present during the Middle Ages, there wereno new atomist theories of matter, nor detailed exegesis of ancient ideas,until the 12th century. One of the main reasons for this absence wasthe assimilation of Epicurism with heresy; and even if atomism is notnecessarily connected with the theory of pleasure, its view of matterhas been discredited for several ancillary reasons as well.16 Further, themain theses of Ancient atomism were known through severe critics orthrough partial quotations.17 The first philosophical resurgence of atom-

13 Cf. John of Salisbury, Metalogicon [ Hall ], II, 2, 10–11, p. 58 or IV, 31, 22–27.14 The relevant passages are found in Philippe, “Lucrèce dans la théologie chré-

tienne,” (1896), pp. 158–159.15 For example: Lactantius, De ira dei , 10, 1–33.16 For example, in the 9th and 10th centuries, Epicurism was attached to the Cathar

heresy. Cf. Philippe, “Lucrèce dans la théologie chrétienne,” (1896), pp. 148–160. Inthe 11th century, one can still find violent critics against Epicurus in Marbode ofRennes, Liber decem capitulorum [ Leotta], pp. 54–58. The passage ends with a terrible judgement (p. 58):

Quapropter stultos Epicuri respue sensus, Qui cupis ad vitam quandoque venire beatam; Sperne voluptates inimicas philosophiae, In grege porcorum nisi mavis pinguis haberi Illisa rigidam passurus fronte securim.17 As J. Philippe asserts ( Ibid. p. 161): “Les citations de Lucrèce chez les gram-

mairiens, les extraits de son oeuvre donnés par les Apologistes assurèrent, bien mieuxqu’un enseignement méthodique, la conservation de ses idées qui entrèrent ainsi dans

l’enseignement théologique. Présenté comme système, l’Epicurisme eût été vite proscrit,et, de fait, il l’a été souvent: mais des citations éparses semblaient moins dangereuses,et, comme les idées qu’elles contenaient répondaient souvent à des questions soulevéespar les commentaires bibliques, on les adopta sans défiance.”



introduction 5

ism dates from the 12th century, in the works of William of Conches,but it is a result of different traditions, Platonist and medical, and it is

not a strict reading of Ancient atomism.In the 13th century, there were no developed atomist theories as faras we know—be they mathematical or physical—even if one can findsome corpuscular tendencies in some physicians as Urso of Salernoat the very end of the 12th century,18 or in some philosophers such asRobert Grosseteste and Roger Bacon.19 All of these authors discussed theconcept of minimum, sometimes considering it as a synonym of “atom,”as is explicitly af firmed by Albert the Great, for example.20 Not onlythere were no atomist theories of matter, but providence, indifference

of God, hedonist visions of happiness, etc. were no longer subjects ofdiscussion when authors treated the nature of matter, except for somedetailed critiques of Arabic atomism in Thomas Aquinas’s Summa contra

gentiles for example.21 The target had changed.In addition to the theological motive for the eclipse of Ancient

atomism, the arrival of new Latin translations of Aristotle’s texts onnatural philosophy in the thirteenth century played an important role.Epicurus and Lucretius were no longer the protagonists when peopledealt with the nature and function of atoms, as both were progressivelyreplaced by Democritus, to whom Aristotle devoted many passages inthe Physics, the De generatione et corruptione and De caelo. This is the reasonwhy, according to Murdoch, fourteenth-century atomism is merelya response to Aristotle’s anti-atomism and never a return to Ancienttheories. He wrote:

18 Cf. Jacquart, “Minima in Twelfth-Century Medical Texts from Salerno.”19 Cf. Molland, “Roger Bacon’s Corpuscular Tendencies (and some of Grosseteste’s

too).” Work should be done on David of Dinant, who seems to have been temptedby corpuscular theories too.

20 Albert the Great, De generatione et corruptione [ Hoddfeld ], t. 1, c. 12, p. 120, 44–55:“Democritus autem videbat quod omnia naturalia heterogenia componuntur ex simili-bus sicut manus ex carne et osse et huiusmodi, similia vero componuntur secundumessentiam ex minimis quae actionem formae habere possunt, licet enim non sit accipereminimum in partibus corporis, secundum quod est corpus, quod autem non accipi minusper divisionem, tamen est in corpore physico accipere ita parvam carnem qua si minoraccipiatur, operationem carnis non perficet, et hoc est minimum corpus non in eo quod

corpus, sed in eo quod physicum corpus, et hoc vocavit atomus Democritus.”21 Cf. Aquinas, Summa contra Gentiles, III, c. 65 and 69. For the references to the Arabsin Aquinas, cf. Anawati, “Saint Thomas d’Aquin et les penseurs arabes: Les loquentesin lege Maurorum et leur philosophie de la nature.”




Unfortunately, almost all this indivisibilist literature is devoted to arguingagainst the Aristotelian position and to establishing that continua can becomposed in this or that fashion of indivisibles; very little is said that helpsto explain precisely why this current of indivisibilism arose in the first thirdof the fourteenth century or what function it was held to serve. Thereseems to be no sign of a resurgence of ancient physical atomism amongthese late medieval indivisibilists, nor anything resembling a consciouslyatomistic interpretation of mathematics.22

Murdoch’s statement is partly true, because every philosopher whowrote something in the area of natural philosophy at this time had todiscuss the question of continuity as found in the sixth book of Aristotle’s Physics, and this is not the indivisibilist’s privilege. Moreover, Murdochis perfectly right in saying that one cannot find real hints of ancientphysical atomism in this fourteenth-century indivisibilist literature, forif there were physical theories for the existence of atoms, they hadnothing in common with Democritus’s or Lucretius’s views on the subject.23 However, can we limit our characterization of fourteenth-centuryatomism to its mathematical features?

3. Fourteenth-Century Atomism: Mathematical, Physical

or Metaphysical?

Much recent research is devoted to showing that there was also a morephysical form of atomism in the fourteenth century. In this respect, themost representative philosopher of this physicalist way of thought isundoubtedly Nicholas of Autrecourt, who is admittedly considered as anexception in the philosophical landscape of later medieval philosophy(see Christophe Grellard’s chapter). But there are other, lesser-known

thinkers who developed consistent views about the physical nature ofatoms and their role in the explanation of natural phenomena, amongthem Gerard of Odo, William Crathorn or John Wyclif, to whom someof the chapters in the present volume are dedicated (see the contribu-tions of De Boer, Robert and Michael).

22 J.E. Murdoch, “Infinity and Continuity,” p. 576.23 It is clear that medieval philosophers knew mainly the physical part of Democritus’s

doctrine, because of the mediation of Aristotle’s critics (they also knew some epistemo-

logical elements from Cicero and Aristotle’s Metaphysics, book Γ ). But it is also evidentthat Democritus’s thought cannot be limited to such an aspect. For recent presentationson different facets of his philosophy, cf. Brancacci & Morel, Democritus: Science, the Arts,and the Care of the Soul .



introduction 7

In this context of hesitation concerning the different possible streamsof medieval atomism—physical or mathematical—the main aim of this

book is to assess past and present research, focussing on the differentforms taken by indivisibilist theories in the Latin West, as presented bytheir followers and their critics. It will appear that even at the end ofthe thirteenth century, when no indivisibilist theory was formed, thequestion of the nature of points was neither purely mathematical orgeometrical, nor purely physical. Richard Rufus is a good representa-tive of such a mixed point of view (see Wood’s chapter). And, in thefourteenth century, both attitudes can be found. All the divisibilists usedgeometrical arguments, because they are much stronger than any other.

On the contrary, some indivisibilists tried to show that geometry is notthe right tool to argue against atomism if one considers atoms in a physi-cal or metaphysical way. Others tried to respond to the mathematicalarguments, but always with physical considerations. With the exceptionof Michel of Montecalerio24 and Henry of Harclay, who discussedmore precisely the mathematical arguments, Walter Chatton, WilliamCrathorn, Gerard of Odo, Nicholas of Autecourt, and John Wyclif,demonstrate a strong propensity for the use of physical or metaphysicalconsiderations. According to them, indivisibles must be considered aselemental components of reality, and not as mere unextended points.Nicholas of Autrecourt and William Crathorn even tried to develop areal atomistic physics, and John Wyclif ’s position could be traced backto the Platonist tradition inaugurated by the twelfth-century commenta-tors of Plato. In any case, their positions are never reducible to a merereaction to Aristotle’s arguments, nor to a reconstruction of Democritusthrough Aristotelian doxography.

It is clear that from the divisibilist side, the strongest arguments

against atomism are geometrical. They are presented by Aristotle, butalso by Al-Ghazali’s and Duns Scotus’s works. As other examples ofthis attitude, we can mention Thomas Bradwardine, who dealt with theproblem of the continuum in a mathematical and geometrical fashion,and Richard Kilvington, who tried to mix mathematical elements withphysical thought experiments. Other divisibilists, as Walter Burley andBlasius of Parma, remained suspended between both methods.

Beyond the oppositions between indivisibilists and anti-indivisibilists,the aim of the present book is thus to give an account of the complexity

24 On Montecalerio, see Celeyrette’s chapter in this volume.




of the reflections upon the structure of continuum and matter inmedieval natural philosophy. Indeed, a strict opposition between two

antagonistic sides would be exaggeratedly schematic, even if somepersonal struggles cannot be excluded.25 The chapters presented heremay be considered a starting point for further studies about the dif-ferent atomist traditions in the Middle Ages, about the sources ofmedieval atomism, and about the relevant periods that must be takeninto account by the historian of sciences.26 We hope that the follow-ing contributions will contribute to an understanding of atomism asa continuously—though more or less accurately—present context inmedieval speculations about nature.

4. Overview of the Contributions

In his inaugural chapter, John E. Murdoch gives a general survey of thedivisibilist/indivisibilist debates in the later Middle Ages and a detaileddramatis personae of atomists and their critics. It is argued that the realmotive behind fourteenth-century indivisibilism remains the rejectionof Aristotle’s arguments as formulated in his Physics and his other trea-

tises on natural philosophy. But Murdoch also endeavours to show thatmedieval philosophers went far beyond Aristotle, though he is alwaysthe point of departure for their enquiries. Many of them put forthnew elements and new methods for the analysis of the continuum’sdivisibility that were by no means present in Aristotle’s texts. Murdochinsists that the major issues of this renewal were mathematical and geo-metrical, even if one can also find a new language of analysis derivedfrom logic in such a context.

This first chapter can be considered as a guide through these com-

plicated discussions, written from a standpoint representative of theprevailing historiography that some of the other chapters in this volumewill challenge.

25 On this point see the dramatis personae in Murdoch’s contribution to this volume.26 For example, studies about quantity in the twelfth century would probably

reveal some atomistic preoccupations, as is clear from Peter Abelard’s discussion inthe Dialectica, where he detailed the theory of his master (William of Champeaux?),

who clearly stated that quantities (lines, but even bodies) are made of indivisibles. Cf.Peter Abelard, Dialectica [ De Rijk], pp. 56–60. Therefore, we may find some degreeof similarity between twelfth and fourteenth-century indivisibilism. This would deserveanother volume.



introduction 9

In Rega Wood ’s chapter, one will find a sketch, from a thirteenth-century standpoint, of what would become the main blind alleys

for fourteenth-century philosophers. Are mathematical, physical andmetaphysical points of view on the nature of points alike and are theydirected to the same object? Rega Wood thus presents the interpreta-tion of Aristotle’s arguments against atomism by the thirteenth-centuryphilosopher Richard Rufus of Cornwall, who dealt with this issue inmany of his works. Since it is usually stated that medieval discussionsabout indivisibles are nothing but a mere reaction to the rediscoveryof Aristotle’s texts, the case of Rufus seems important, for he belongsto the first generation of philosophers who commented on the whole

Aristotelian corpus. This chapter then shows how confused Aristotle’spositions were about the definition of point in his different books, andhow this confusion could lead to different kinds of interpretations. Thecentral question in Rufus’s works is to know whether points are meremathematical objects or substances of some sort, since, surprisinglyenough, both assertions can be found in Aristotle. Thus, Rega Woodthoroughly examines the texts in which Rufus distinguishes the respectiveroles and objects of mathematics, physics and metaphysics. FollowingAristotle in the main lines, Rufus strongly denied that a continuumis composed of indivisibles from a mathematical point of view, buthe admits that points are really found in sensible objects, from whichmathematicians abstract their concept of point. Therefore, there isa strict link between mathematical and physical points. Moreover, indistinguishing sensible from intelligible matter, Rufus seems to contendthat intelligible quantity is infinitely divisible, while sensible matter hasa kind of natural minimum. Turning then to the nature of points,Rufus considered them as accidents of matter, because matter can-

not be spatially organized without the disposition of its points, i.e. bytheir respective positions. At the same time, he denies that points areconstitutive parts of a body, even if they can be considered as a quasicause of lines, for example. Therefore, points are primarily defined bytheir position in a line, a surface or a body, but are strictly extensionless.Hence, Rega Wood shows that points should be interpreted in Rufus’sthought as quasi essential parts of a line, but not as constitutive, noras integral or quantitative parts of a substance. The final section ofthe chapter deals with the question of the infinite. If Rufus follows

Aristotle in his criticism of physical atomism, all the same he af firmsthat infinities can be unequal, a position that sounds similar to the laterview of Henry of Harclay.




In their contribution, Robert Podkoński and El żbieta Jung presentKilvington’s curious attitude toward atomism. Indeed, even if he

proposes some geometrical arguments for infinite divisibility, he doesnot seem really interested in the classical debate as it appears in DunsScotus’s reprisal of Avicenna’s rationes mathematicae. Neglecting the mostpopular arguments, he tries to elaborate new geometrical thought-experiments. Podkoński and Jung analyze carefully three of them: thefirst deals with the angle of contingency, the second with the evolutionof a triangle in a cone of shadow, and the last with the possibility ofan infinite line. In the two first problems, Kilvington identifies Euclidas the one who introduced the idea of an infinitely small mathematical

being, in the prop. 16 of the third book of the Elements, and Plato ashis atomist opponent in the Timaeus, but he totally ignores all of hiscontemporaries. In all three cases, however, Kilvington is not reallyconcerned with the confutation of atomism, even if he remains a firmdefender of infinite divisibility as an absolute principle. His first aim israther to examine and solve paradoxical cases linked to the question ofcontinuum and to the Aristotelian thesis. On this point, he underlinestwo dif ficulties against Aristotle: first, it doesn’t seem possible to adoptthe Archimedean and Euclidean principle of continuity; second, in anAristotelian context, we are not able to answer Zeno’s paradox.

Despite these repeated attacks against indivisibilism, some authorstried to escape the threat of geometrical aporia by considering theproblem from a more physical aspect. In his chapter, Sander W. de Boer endeavours to prove that Gerard of Odo (c. 1285–1349) is probablyone of the first consistent atomists in the fourteenth century. Of course,Henry of Harclay and Walter Chatton were indivisibilists before him,but they did not defend a physicalist point of view. From several unedited

texts, Sander W. de Boer shows that atomism occupied a much moreimportant place than past commentators usually assumed, and thatOdo made new ontological claims about the indivisibles, to the effectthat they are kinds of physical parts of the continuum. Odo’s atomismseems to rest on two basic claims: 1) indivisibles are parts prior to thewhole they belong to; and 2) it follows from the mereological assertionin 1) that the number of atoms should be finite. To establish these twoclaims, Odo calls in different physical phenomena, such as the nature ofthe degrees of heat, intension and remission of light, etc. Rejecting the

existence of actual infinities, Odo contends that there must be minimaand maxima in natural phenomena. Sander W. de Boer concludes



introduction 11

that Odo “consistently uses his atomism in explaining reality, and theapplication of this atomism to God’s power and to the inner structure

of continuous physical processes is not provoked by any mathematicalarguments.”More famous than Gerard of Odo is Nicholas of Autrecourt, who

is undoubtedly one of the most studied of the fourteenth-centuryatomists. In his chapter, Christophe Grellard demonstrates that Nicholasdid not limit himself to considerations about the existence and natureof indivisibles, but rather that he explored the possibility of forminga complete alternative physics. The main concern of Nicholas wasto prove the eternity of the world from atomistic explanations of

generation and corruption, i.e. aggregation and segregation, of atomswhich eternally exist. Of course, Autrecourt’s standpoint is physicalin this context when he constructs the conditions of a local motion ina void, or when he explains condensation and rarefaction and othernatural phenomena. But his attitude is also partly metaphysical, whenhe criticizes, for example, the matter/form couple in order to reducematter to a mere atomic flux. As Christophe Grellard shows, there isno need of a substratum in change according to Autrecourt; rather,the atomic flux is enough because atoms are not bare particulars, butqualitative entities. They are the basic substances of the world. Insome respect, these atoms are more similar to Aristotle’s minima naturalia than to Democritean atoms. How then to explain the unity of a thingcomposed of atoms? Nicholas takes for granted that there are essentialand accidental atoms in a natural compound. The essential ones func-tion as kinds of natural magnets and make the others hold together.Indeed, even if he criticizes Aristotle’s distinction between matter andform, such essential atoms are sometimes called ‘formal atoms’, in the

sense that they contain what will be the principle of motion, a sortof virtus. But, since the natural power of these atoms is not alwayssuf ficient, an atomic compound can be helped by a celestial influence.Surprisingly enough, this copulatio between atoms and stars is a sine qua

non condition for a natural change. For example, procreation requiresthree conditions: it occurs through a material condition (the sperm), aformal condition (the man) and by the ef ficiency of a star. Autecourt’scosmology is therefore purely atomistic, from the explanation of thecelestial stars to the natural phenomena in the sensible world. This

genuine theory, as Christophe Grellard shows, takes its origin not onlyin some developments of Aristotle’s corpus itself, especially from the




De generatione et corruptione and the De caelo, but also from the Arabicatomism of the Mutakallimun, known through the Latin translation of

Maimonides’s Guide of the perplexed .At the very same time, around 1330, another philosopher, WilliamCrathorn, developed a similar theory, minus the celestial action onatomic compounds. Aurélien Robert ’s aim is to show that Crathorn putsforth the basic foundations for an atomistic physics, which rests upontwo distinctive features: a mereological interpretation of the continuumdebate, and a systematic use of the notions of space and position. InCrathorn’s theory, indivisibles are conceived as things, actually exist-ing in the continuum as real parts, and occupying a single place in

the universe. Moreover, these entities have a certain nature (there areatoms of gold and atoms of lead, for example) and it must be inferred,as Robert shows, that they also have a certain quantity or magnitude.From this reconstruction of the physical or metaphysical structure ofatoms, it is demonstrated that Crathorn applied this theory to Aristotle’sarguments, giving a new definition of contiguity and continuity fromthe arrangement of parts and from the contiguity of places occupied bythe atoms. Hence, Crathorn reduces all movement to a local motion ofatoms, and explains various physical phenomena with his new analysisof the indivisibles (such as condensation and rarefaction, for example).Finally, even if Crathorn’s attempt to elaborate an atomistic physics israther original, Aurélien Robert brings out some limits to his analysisdue to theological reasons, notably when the Oxford master applieshis analysis to the cases of angels or souls. In conclusion, this tendsto prove that from mereotopological elements, Crathorn endeavouredto think the possibility of an atomistic physics, where atoms resemblemore the minima naturalia than the atoms of Democritus, Epicurus or

Lucretius. Jean Celeyrette takes up a lesser-known Parisian dispute between JohnBuridan and Michel of Montecalerio which took place a short timebefore and after 1335. Buridan’s question De puncto has been editedby Vladimir Zubov in 1961, but Jean Celeyrette makes its contextfar more clear for us. Indeed, he provides a new evidentiary basis forreconstructing the whole debate from unedited manuscript material.It appears that the departure point of the confrontation between bothmasters was a discussion of the Ockhamist view about points. Though

Buridan doesn’t agree with Ockham in his commentaries on Aristotle’s Physics, he followed him in his De puncto. Jean Celeyrette then details thedifferent steps of the dispute, showing that there were probably fourconsecutive disputed questions—and possibly four texts—in which both



introduction 13

masters responded to each other. Montecalerio presents an indivisibilisttheory very different from the ones examined in the other chapters of

this volume, for he doesn’t want to identify points with parts, as Odo,Autrecourt and Crathorn tend to do. Points are considered as accidentsexisting in a substance as in a subject (subiective ), a position similar toRufus’s view. Incidentally, this chapter presents an interesting standpointfor the general historiographical question posed in this book about themathematical or physical nature of atomist debates in the Middle Ages,for Jean Celeyrette concludes: “. . . mathematics are roughly absent . . .One finds no allusion, even to challenge their relevance, to the rationes

mathematicae very fashionable among English scholars since Scot.” This

chapter even demonstrates an appeal to physical practice in Buridan’stext—though he is a divisibilist—when he invokes, for example, experi-mentation and the work of the Alchemists. Therefore, Celeyrette’s studytends to establish the fact that even in a discussion about the possibleexistence of points, some medieval thinkers used to consider it not asa purely mathematical problem, but also as a real need for physics.

The last atomist philosopher presented in this volume is John Wyclif.The main purpose of Emily Michael ’s chapter is to examine how Wycliftried to make an atomistic view of prime matter compatible witha hylomorphist conception of natural beings. As Michael shows, inWyclif ’s cosmology, matter is firstly conceived as a composition of afinite number of atoms defined as real entities occupying each pos-sible place in the world. Though indivisible parts of matter have noquantity, nor quality, their contiguity in space defines the total shapeand quantity of matter in the world. This theory is very similar toCrathorn’s position, except that there is no void in the world accord-ing to Wyclif. If one turns to the reasons for Wyclif ’s adoption of

such a corpuscular theory of matter, Michael contends, one shouldfind that his first motivation was theological, for his view is supportedby a logical interpretation of Scripture. Wyclif asserts that God hascreated a finite world with a finite number of atoms in it. Michaelthus demonstrates that the whole methodology of Wyclif’s cosmologyis directed by the logic of Scripture and by the interpretation of the Book of Genesis. Afterwards, Michael turns to the compatibility of theatomistic view of matter with a pluralistic hylomorphism, inspired bysome of his scholastic predecessors, especially in the Franciscan school.

Finally, Wyclif ’s atomism is evaluated from the question of the mixtio ofthe elements. The traditional view of Aquinas and other scholastics wasthat in the generation of a new compound, its elements do not remainin the mixture. On the contrary, Wyclif thinks that the elements remain




in the compound as small atomic particles. As a result of her analysis,Michael establishes that what Wyclif calls minima naturalia comes from

such a natural mixing process of the elemental atoms. Therefore, thereis a natural hierarchy of beings: indivisibles, i.e. extensionless points;the elemental atoms (air, earth, fire and water), which are determinedby an elemental form; minima naturalia (minimal parts of flesh, bones,etc.), which are the fundamental particles of a body composed ofelemental atoms; bodies, formed by the composition of those minima

naturalia plus substantial forms; and human beings, which are bodiesplus the rational soul. This chapter brings us a new, important viewof a non-mathematical but metaphysical form of indivisibilism which

takes its place in a whole cosmology inspired by Plato’s Timaeus andAugustine’s De Genesi ad litteram.

With Joël Biard ’s contribution on Blasius of Parma’s attitude towardsatomism, we reach the very end of the fourteenth century. It is worthnoting that Blasius reveals the permanency of atomistic problematic,even when most atomist philosophers seem to have disappeared. Indeed,from time to time, the Italian philosopher seems ready to use a kind ofatomistic point of view. Leaving aside some aspects of Blasius’s thought(the question of the void and of the nature of matter) Biard deals pri-marily with the two main questions on the continuum and on the minima

naturalia. About the first point, Blasius first denies the possible existenceof indivisibles, by using classical geometrical arguments. Nevertheless,he seems to admit as rather plausible a kind of indivisibilism, that isinfinite indivisibilism. This partial defense of atomism relies on bothmathematical and physical arguments. In sum, Blasius tries to acceptsimultaneously the infinite divisibility of a line and the infinity of points.Without quoting any atomists (such as Henry of Harclay or Nicholas

of Autrecourt, both of whom accept the existence of an infi

nity ofindivisibles), Blasius finally defends a kind of indivisibilism, but it isgoing too far to consider him as an atomist. Indeed, concerning thequestion of natural minima, he clearly asserts the infinite divisibilityof matter. But, once again, atomism implicitly remains: if there is noabsolute minimum, Blasius concedes, there should be a minimum inthe sense of physical limits of existence; and this limit is determinedby the proportion of matter in a being. Finally, we should say that atthe end of the century, atomist solutions were still more or less pres-

ent as a convenient answer in some context between mathematicsand physics. Blasius of Parma is an interesting witness to this kind of“regional atomism.”



BEYOND ARISTOTLE: INDIVISIBLES AND INFINITEDIVISIBILITY IN THE LATER MIDDLE AGES

John E. Murdoch

The basic text for late medieval Latin atomism and its critics was

Aristotle’s Physics, especially Book VI. Here the atoms or indivisibles

he considered and combatted were extensionless, a conception that

can be found in scholastic debate about atoms all the way to Galileoand his atomi non quanti .1

The medieval atomists were clustered in the fourteenth-century,2 as

were their Aristotelian critics. Figure 1 provides the basic dramatis personae

of the fourteenth-century atomists and their critics. The list of atomists

is nearly complete, save for the followers of Wyclif. The list of their

critics is naturally less complete, being made up of chiefl y those who

name their atomist opponents. Yet even without such identification, we

can often tell other critics, such as John Buridan and his school, because

they oppose specific identifiable atomist arguments.The question of the motives for the late medieval atomism is pretty

murky. The motives for Greek atomism are, at least to some extent,

an answer to Parmenides’s monism and center in attempts to explain

natural phenomena (if not always totally successfully). Equally clear are

the motives for the Arabic atomism of the Mutakallimun: namely, to

put all causal relations into the hands of God through the mechanism

of the doctrine of continuous creation. However, in the case of late

medieval atomism there is not such a wholesale application to nature

or to a God who creates the universe anew at every instant.

1 Galileo Galilei, Discorsi e dimostrazione matematiche intorno a due nuove scienze, vol. 8,p. 72. There is, of course, the quite separate consideration of minima naturalia thatarises out of Aristotle’s criticism of Anaxagoras in Physics, I, ch. 4. For the medievalhistory, as well as the historiography, of this kind of atomism or corpuscularianism, seeMurdoch, “The Medieval and Renaissance Tradition of Minima Naturalia”.

2 For the earlier medieval atomism by the likes of Isidore of Seville, William ofConches, etc., see Pabst, Atomentheorien des lateinischen Mittelalters. For the standard treat-

ments of the medieval atomism of the fourteenth century, see Duhem, Le système dumonde, vol. 7, pp. 3–157; Maier, “Kontinuum, Minima und aktuell Unendliches,” In Die Vorläufer galileis im 14. Jahrhundert 2nd ed., pp. 155–215; and, more briefl y, Murdoch,“Infinity and Continuity.”



16 john e. murdoch

Dramatis personae

INDIVISIBILISTS ARISTOTELIANS

Henry of Harclay

Walter Chatton OFM

Crathorn OP

John Wyclif

ENGLISH

William of Alnwick OFM

Adam Wodeham OFM

Thomas Bradwardine

William of Ockham OFM

Roger Rosetus OFM

Walter Burley

John the CanonGerard of Odo OFM

Nicholas Bonetus OFM

John Gedo

Marcus Trevisano

Nicholas Autrecourt

CONTINENTA L

Single line arrows represent (named) criticismDouble line arrows represent verbatim borrowing

Fig. 1. Fourteenth-Century Indivisibilism and its Critics



indivisibles and infinite divisibility 17

The most frequently occurring “motive” is that of angelic motion,

although it sometimes functions as an excuse to discuss at length the

atomist or indivisibilist composition of continua.3

Alternatively, Henryof Harclay was convinced that the belief in the possible inequality of

infinites was grounds for a related belief in the composition of conti-

nua out of indivisibles.4 Yet one feels that the real motive behind such

fourteenth-century atomists was the scrutiny and consequent rejection

of Aristotle’s arguments against such atomism or indivisibilism.

I now want to turn to my major topic: to measure how both the late

medieval atomists and their critics went beyond (and not just developed)

the Aristotelian base from which they began; beyond in the sense of

providing new conceptions and new arguments for their cause.Before that, however, I want to establish that, in coming up with the

extensionless indivisibles of Book VI of Aristotle’s Physics, the fourteenth-

century medieval atomists skewed the view of ancient atomism. This

meant, of course, the opinion of Democritus, since they either did not

have or failed to appeal to Epicurus or Lucretius. For example, citing

Aristotle on Book I of De generatione, the late medieval indivisibilists

3 Walter Chatton, Reportatio Super Sententias [ Etzkorn e.a.] Liber II, pp. 114–146:“Et quia non potest sciri de motu angeli utrum sit continuus vel discretus in motunisi sciatur utrum motus et alia continua componantur ex indivisibilibus, ideo quaeropropter motum angeli utrum quantum componatur ex indivisibilibus sive permanenssive successivum.” Then Chatton spends the remaining 32 pages investigating thislatter question and never returns to the notion of angelic motion. Similarly Gerardof Odo, Super primum Sententiarum, dist. 37 (MSS Naples, Bib. Naz. VII. B.25, ff.234v–244v; Valencia, Cated. 139, ff. 120v–125v): “Ad quorum evidentiam querendasunt quatuor . . . Tertium utrum motus angeli habeat partem aliquam simpliciter pri-mam.” But then on the very next folio he breaks into what is of real interest to him:

“Utrum continuum componatur ex indivisibilibus;” he continues this inquiry to theend of the question, only devoting a brief paragraph at its end to the problem de motuangeli . Indeed, two other MSS of Gerard’s question, shorn of its concern about angelicmotion, made it appear as if Gerard had written a work on Aristotle’s Physics. BothWalter and Gerard were atomists, but it is worth noting that Duns Scotus espousedan Aristotelian opposition to atomism (and, of course, chronologically preceding thesetwo atomists) and included his discussion of the continuum in the context of angelicmotion ( Comm. Sent , II, dist 2, Q. 9), perhaps encouraging later atomists to do so (oneshould note that both Chatton and Odo were also Franciscans).

4 Henry of Harclay, “Utrum mundus poterit durare in eternum a parte post” (whichamounts to his Quaestio de in fi nito et continuo ), MSS Tortosa, Cated. 88, ff. 87r–v; Florence,Bib. Naz. Fondo princ. II.II.281, fol. 97r: “Preterea, specialiter contra hoc quod dicitur

in auctoritate Lincolniensis: Quod plura sunt puncta in uno magno continuo quamin uno parvo. Contra hoc sunt omnia argumenta que probant continuum non possecomponi ex indivisibilibus; probant enim eciam quod in uno continuo non sint plurapuncta quam in alio.”



18 john e. murdoch

maintained the essential agreement of Plato and Democritus with

respect to the composition of continua out of indivisibles.5

Now the medievalists did not have Theophrastus on Democritus’saccount of the variation of tastes through a corresponding variation

in the shapes of atoms,6 but they did have the basic passage in the

Metaphysics saying that atoms varied in shape, arrangement and posi-

tion (shape being far and away the most important).7 And they also

had the fourth chapter of De sensu where they might have guessed the

Democritean account of tastes.8

Accordingly, the Aristotelian critics of medieval atomism correctly

recognized that the indivisibles of Democritus have magnitude and have

parts.9 Thomas Bradwardine, attempting to refute all sorts of atomismin his Tractatus de continuo, says rightly that Democritus held continua to

be composed of indivisible bodies, though he devotes little space to this

view and claims that what Democritus really had in mind was compo-

sition out of an infinite number of substances. Moreover, he does not

properly refute the Democritean opinion since he sometimes takes it

5

Gerard of Odo, Super primum Sententiarum, I, dist. 37 (MSS citt. Note 3, Naples, fol.235r; Valencia, fol. 120v): “Quantum ad primum sciendum quod, ut recitatur primo De generatione, opinio fuit Platonis et Democriti quod continuum componitur ex indivisibilibus et resolvitur in indivisibilia. Diversimode tamen, quia Democritus asserebatcorpus componi ex atthomis et resolvi in atthomas sive in magnitudines indivisibiles,quod idem est; Plato vero ponebat corpus componi ex superficiebus, superficies ex lineis,lineas ex punctis, et eodem modo resolvi. Convenienebant tamen in hoe quod uterquedicebat primam compositionem continui esse ex indivisibilibus et ultimam divisionemterminari ad indivisibilia.” Much later we find a similar view expressed by the Wyclifite

John Tarteys in his Logica (MS Salamanca 2358, fol. 97v): “In ista materia, sicut in omnialia materia naturali et philosophica, est specialiter credendum illi parti pro qua ratioplus laborat inducendo nuda dicta Aristotelis sonantia in oppositum cum tunicis quas

texerunt sapientes sequaces Platonis et Democriti qui convincerunt ex ratione infallibilicorpora continua ex atthomis, id est, partibus indivisibilibus, integrari.” It is worthnoting that Nicholas Bonetus claims to be following Democritus and opts for atomicmagnitudes in each species of bodies, surfaces, lines, and points, although he does notcite the passage of Aristotle from De generatione I. On the whole medieval history ofthis passage (which some historians have held to be as much Aristotle as Democritus),see Murdoch, “Aristotle on Democritus’s Argument Against Infinite Divisbility in De generatione et corruptione, Book I, Chapter 2.”

6 Theophrastus, De sensu, 49–83 (espec. 65).7 Metaphysics, A, ch. 4, 985b4–22.8 Aristotle, De sensu, ch. 4.9 For example, Albert of Saxony, Quaestiones in octo libros physicorum [Paris, 1518],

VI, q. 1, f. 64v: “Alio modo capitur indivisibile quod, licet habeat partem vel partes,tamen propter eius parvitatem non sunt abinvicem separabiles; talia indivisibilia posuitDemocritus que vocavit corpora atomalia.” Nicole Oresme made a similar distinctionamong indivisibles. (Cf. his Quaestiones Physicorum, MS Sevilla 7–6–30, fol. 66r).




to be included in his extensive refutation of extensionless indivisibles

composing a continuum, be they infinite or finite in number.10

Let us return, however, to the examination of the new notions andarguments beyond the Aristotelian base of Physics VI .

One of the most obvious developments in this regard was the

geometrical arguments added to support Aristotle’s opposition to

indivisibilism. One source for them was clearly the Latin translation

of Al-Ghazali’s Metaphysica.11 Fundamentally, these arguments against

indivisiblism were based on techniques of parallel or radial projection

which entail, if geometrical figures were composed of indivisibles,

the equality, for example, of the sides of a square with its diagonal

or of two concentric circles. Thus, if parallels were drawn betweenevery indivisible or point in the sides of the square they would cut

its diagonal in the same number of indivisibles or points, from which

their equality followed. And the same, mutatis mutandis, for concentric

circles where all radii are drawn. (Figure 2). These kinds of arguments

gained in popularity and prestige when Duns Scotus used them in his

own opposition to indivisibilism, even bringing Euclid into the picture

as Al-Ghazali had not.12

The fourteenth-century atomists’ replies to these geometrical argu-

ments were almost always highly unsatisfactory. Their answers often

employed inappropriately physical ideas into geometry. Thus, Henry

of Harclay, the first clearly established atomist on the English scene,

claimed that, like two sticks, the parallels between the sides of a square

“take more” of the diagonal than they do of the sides.13 Or Walter

10 Thomas Bradwardine, Tractatus de continuo, MS Toruń, Poland, R402, p. 187 (theMS is paginated): “Omnes igitur opiniones erronee specialiter reprobantur, preter

opinionem Democriti ponentem continuum componi ex corporibus indivisibilibus, quetamen per illam conclusionem et eius corollarium suf ficienter reprobatur. Non tamenest verisimile quod tantus philosophus posuit aliquod corpus indivisibile, sicud corpusin principio est dif finitum, sed forte per corpora indivisibilia intellexit partes substantieindivisibiles et voluit dicere substantiam componi ex substantiis indivisibilibus.”

11 Cf. Al-Ghazali, Metaphysica [ Muckle], pp. 10–13. This is a Latin translation ofAl-Ghazali’s Maqā sid al-fal ā sifa which in turn is taken from Avicenna’s Persian work D ā nish-N ā meh. For the most satisfactory translation of this latter, see Avicenne, Le livrede science [tr. Achena & Massé]. The medieval Latin scholars did not know these detailsand, in any case, did not have a Latin translation of this work of Avicenna’s.

12 John Duns Scotus, Comm. Sent . II, dist 2, q. 9. English translations are availableof relevant passages in both Al-Ghazali and Duns Scotus in Grant, A Source Book in

Medieval Science, pp. 314–319.13 Henry of Harclay (MSS citt. note 4: Tortosa, 90v; Florence, 99r): “Voco autem‘recte intercipi’ partem linee intercipi inter duas lineas equidistantes facientem cum lineisequidistantibus angulos rectos. Talis porcio est equalis in omni parte equidistancium



20 john e. murdoch

Chatton maintained that drawing all the radii of two concentric circles

can not be done due to defectus materiae.14 Moreover, to account for the

incommensurability of the diagonal and the side of a square, Harclayamazingly claims (even citing Euclid in the course of his argument)

that the ratio of points in the diagonal and side is as two mutually

prime numbers.15

One may contrast this with Bradwardine, who cites both Harclay

and Chatton in his treatment of the continuum that systematically

denies any indivisibilism anywhere, but highlights its inconsistency with

mathematics, geometry in particular.16 Alternatively, John Buridan, in a

treatment of indivisibilism not nearly so geometrical as Bradwardine’s,

nevertheless claims that mathematical atomism would totally annihilategeometry.17 Bradwardine, however, goes a step further in asking whether

in using Euclid’s geometry to upend indivisibilism he might be guilty

of a petitio. In his questioning, he was in effect inquiring into, we would

now say, the independence of axioms or suppositions.18

linearum. Nam illo modo solo debet intelligi equedistancia linearum. Si porcio lineeoblique caderet inter lineas equidistantes, multo maior intercipitur quam alia recte

cadens. Ita dico quod est de puncto. Nam inter lineas equidistantes et immediate sehabentes non posset intercipi punctus secundum situm rectum, et tamen posset secun-dum obliquum. Et licet istud videtur mirabile de puncto, cum sit indivisibilis, tamenistud est necessario verum.”

14 Walter Chatton, Reportatio super Sententias [ Etzkorn e.a.], II, dist. 2, q. 3, p. 131:“Ad aliud de circulis dico unum generale, quod ubicumque oporteret dividere punc-tum, ponam lineam interrumpi et non procedere. Et quod arguitur contra eum percommunem conceptionem positam, I Euclidis vel petitionem a puncto ad punctumquodcumque lineam ducere, negat illud propter defectum materiae in casu. Unde siprotrahas unam lineam a puncto maioris circuli ad centrum et post velis protraherealiam a puncto proximo mediato prius accepto, illa forte transibit vel veniet ad idempunctum minoris circuli; et ideo si velis protrahere tertiam a puncto intermedio duobus

punctis prius acceptis, illa interrumpetur propter defectum materiae quando veniet adconcursum priorum linearum.”15 Henry of Harclay (MSS citt., note 4; Tortosa, 91r; Florence, 99v): “Ad hoc potest

dici quod Euclides in ista proposicione per quam probat quod diameter et costa suntincommensurabilia, intelligit quod non numerantur communiter per aliquam unamquantitatem vel per aliquod per se divisibile; non quin numerentur per punctum. Ettunc est dicendum quod diameter et costa se habent sicut duo numeri contra se primi,qui non numerantur per aliquem numerum communem, set per solam unitatem.”

16 For a summation of Bradwardine’s procedure, see Murdoch, “Thomas Brad-wardine: mathematics and continuity in the fourteenth century.” This article givesthe Latin text of the enunciations of the de fi nitiones, suppositiones, and conclusiones of thiswork of Bradwardine’s.

17

John Buridan, Quaestiones in octo libros Physicorum [ Paris, 1509], VI, q. 2: “. . . etomnino perirent conclusiones et suppositiones geometrie.”18 Thomas Bradwardine, Tractatus de continuo, (MS Toruń R.40.2, pp. 156, 188).

Bradwardine first mentions the possibility of a petitio in a comment relative to his fourth




Although Aristotle himself on occasion referred to mathematics

and geometry in his analysis of continuous magnitudes, this was not

central to his discussion. Instead, in Physics VI we find him giving twoconceptions or definitions of what it is for something to be continuous.

First, things are continuous the parts of which have their extremities

as one ( ultima sunt unum ).19 Secondly, everything that is continuous is

divisible into divisibles that are always further divisible, that is, it is

infinitely divisible.20

To the latter notion first: Aristotle in Book III of the Physics had given

an extensive investigation of the infinite and had distinguished between

what came to be called an actual infinite and a potential infinite, hold-

ing that the latter was the only kind that was permissible (a distinctionthat he does not bring up in his Book VI inquiry to the continuum).

Yet there is a consideration not brought up at all by Aristotle, but

whose investigation fairly bristles in the Middle Ages—especially in the

supposition (which reads: Omnes scientias veras esse ubi non supponitur continuumex indivisibilibus componi): “Hoc dicit quia aliquando utitur declaratis in aliis scientiisquasi manifestis, quia nimis longum esset hec omnia declarare. Ubi autem tractant de

compositione continui ex indivisibilibus non supponit eas veras esse propter petitionemprincipii evitandam.” He fills this out much later by maintaining: “Posset autem circapredicta fieri una falsigraphia: Avroys in commento suo super Physicorum, ubi dicit,quod naturalis demonstrat continuum esse divisibile in infinitum et geometer hoc nonprobat, sed supponit tanquam demonstratum in scientia naturali, potest igitur inpug-nare demonstrationes geometricas prius factas dicendo: Geometriam ubique supponerecontinuum ex indivisibilibus non componi et illud demonstrari non posse. Sed illud non

valet, quia suppositum falsum. Non enim ponitur inter demonstrationes geometricascontinuum non componi ex indivisibilibus nec dyalecticer indigent ubique, quoniam<non> in 5to Elementorum Eudlidis. Et similiter, nec geometer in aliqua demonstrationesupponit continuum non componi ex infinitis indivisibilibus mediatis, quia, dato eiusopposito, quelibet demonstratio non minus procedit, ut patet inductive scienti conclu-

siones geometricas demonstrare. Verumtamen Euclides in geometria sua supponit, quodcontinuum ex [in] finitis et immediatis athomis non componitur, licet hoc inter suassuppositiones expresse non ponat. Si falsigraphus dicat contrarium et ponat aliquamlineam ex duobus punctis componi, Euclides non potest suam conclusionem primamdemonstrare, quia super huius lineam non posset triangulus equilaterus collocari, quianullum angulum haberet, ut patet per 16am et eius commentum. Similiter, si dicat falsig-raphus, continuum ex athomis immediatis componi, 4am suam conslusionem et 8am nonprobat, ambe enim per su<per>positionem probantur. Similiter in probatione <23>3i. Iste autem conclusiones non demonstrantur per aliquas conclusiones priores, sed eximmediatis principiis ostenduntur. Per has autem conclusiones relique demonstrantur,et ex his 3bus quasi tota geometria Euclidis dependet et in ipsa omnis alia geometriafundatur, quare geometria supponit <continuum> ex [in]finitis et immediatis athomis

non componi.” For more on this problem of a petitio, see the article cited in note 16,pp. 117–119.19 Aristotle, Physics, VI, 1, 231a22–29 (cf. Physics V, 3, 227a10–12).20 Aristotle, Physics, VI, 1, 231b12–15.



22 john e. murdoch

fourteenth century. This is whether or not there can be unequal infi-

nites or, put another way, whether there can be infinites that have a

part/whole relation to one another. To put it yet another way, mustall infinites be equal to one another?21 Initially the possibility (or

impossibility) of unequal infinites appeared relative to the problem of

an eternal world. For instance, given a past eternity, there would be

twelve times as many months as years, therefore, one infinite would

be twelve times another infinite (and, consequently, there could not be

an eternal world).22

Henry of Harclay, who allowed unequal infinites and made it a

central part of his investigation of the infinite and the continuum,

deserves credit for being apparently the first to say that past eternityis the mirror of future eternity (which was theologically all right), and

that any argument that could be made against past eternity would be

effective against future eternity.23 And he also saw (as did many oth-

ers) that unequal infinites were involved in the infinite divisibility of a

continuum (for example, there were more parts in a whole continuum

than in its half ).

21 There were other ancient sources for the notion of unequal infinites, but theydid not receive, one way or another, translation into Latin. See, for example, Plutarch, De comm. not. Adv. Stoicos, 1097a; Philoponus, De aeternitate mundi contra Proclum, I, 3,[Rabe], p. 11, and Apud Simplicium, Phys, VIII, 1, [ Diels], 1179, pp. 15–27; AlexanderAphrodisias, Quaest. natural ., III, p. 12; Proclus, Comm. in Euclidem, Def. 17, Elem. theol .,prop. 1; Lucretius, De natura rerum, I, 615–626. Some medievals imagined Aristotle tohave something like the notion of infinites having part/whole relations Phys. III, 5,204a22–27 (but here clearly has in mind quidditative, not quantitative parts). Cf. forexample, Walter Burley, Super Aristotelis libros de physica auscultatione commentaria [ Venice,

1589], coll. 288–289.22 Archetypical is Bonaventure ( Sent . II, dist. 1, p. 1, art. 1, q. 2): “Prima est haec.Impossibile est infinito addi—haec est manifesta per se, quia omne illud quod recipitadditionem, fit maius; infinito autem nihil maius, sed si mundus est sine principio,duravit in infinitum: ergo durationi eius non potest addi. Sed constat, hoc esse fal-sum, quia revolutio additur revolutioni omni die: ergo etc. Si dicas, quod infinitumest quantum ad praeterita, tamen quantum ad praesens, quod nunc est, est finitumactu, et ideo ex ea parte, qua finitum est actu, est reperire maius; contra, ostenditur,quod in praeterito est reperire maius: haec est veritas infallibilis, quod, si mundus estaeternus, revolutiones solis in orbe suo sunt infinitae; rursus, pro una revolutione solisnecesse est fuisse duodecim ipsius lunae: ergo plus revoluta est luna quam sol; et solinfinities: ergo infinitorum ex ea parte, qua infinita sunt, est reperire excessum. Hoc

autem est impossibile: ergo etc.”23 Henry of Harclay (MSS citt. note 4: Tortosa, 82v; Florence, f. 94v): “Preterea, adprincipale videtur quod eadem argumenta que probant mundum non potuisse fuisse abeterno, eadem possunt fieri ad probandum mundum non posse esse in eternum.”




Relative to the question of unequal infinites, basically three tradi-

tions were operative in the Middle Ages: (1) Unequal infinites are not

allowable and, therefore, any situation which implied them is equallyuntenable (most often such situation as the possibility of an eternal

world).24 (2) Given that no infinite is greater than another and yet that

some infinites are unequal, one concludes that infinites are incompa-

rable to one another.25 (3) Revise the rules about parts and wholes to

include infinite magnitudes and multitudes. This third alternative for

the most part begins with Harclay (who finds a predecessor in Robert

Grosseteste) and is carried further by William of Ockham and reaches its

peak with Gregory of Rimini.26 From the standpoint of understanding

24 For example, Bonaventure ( op. cit ., note 22), Thomas Bradwardine, De causa Deicontra Pelagium, cap. 1 coroll. 40, espec. pp. 124–125. For many others, see Dales, Medieval Discussions of the Eternity of the World and Dales & Argerami, Medieval Latin Textson the Eternity of the World .

25 This appears to be a fourteenth-century Parisian tradition. Nicole Oresme,Quaestiones super libros Physicorum, III, q. 12 (MS Sevilla Colomb. 7–6–30, ff. 37v–39v):“Utrum infinitum sit alio maius aut equale sive minus vel utrum esset, si esset infinitum,

vel utrum infinitum sit infinito comparabile.” Albert of Saxony, Quaestiones in libros decaelo et mund o, I, q. 8 [Paris, 1518]: “Utrum infinitum posit esse maius vel minus alio,

si essent plura infi

nita, seu utrum sit unum comparabile alteri.” This tradition seemsto have a legacy in Galileo, Discorsi . . ., p. 79 and Newton in a 1693 letter to RichardBentley, in Cohen, Newton’s Papers and Letters on Natural Philosophy, pp. 293–299.

26 Henry of Harclay (MSS citt., note 4: Tortosa, 83v; Florence, 95r): “Dicitur quodinfinitum neque est maius neque minus, set maioritas vel minoritas est respectu ali-cuius finite. Contra: Ista proposicio est per se nota: ‘Omne totum est maius sua parte,’et hec adhuc magis nota: ‘Illud quod continet aliud et aliquid ultra illud vel preterillud est totum respectu illius.’ Sic est in proposito. Nam totum tempus futurum abhac die continet totum tempus futurum a crastina die et addit supra illud, igitur esttotum respectu illius.” Reference to Grosseteste is occasioned by texts that say that, forexample, the infinite number of points contained in (but not, like Harclay, composed outof) a whole line is double the infinite number contained in its half (see Comment. In VIII

libros physicorum Aristotelis [ Dales], pp. 91–95 for Ockham, see Murdoch, “William ofOckham and the Logic of Infinity and Continuity.” Gregory of Rimini, on the otherhand, carries the analysis of parts and wholes and greater than and less than muchfurther. Cf. Lectura super primum Sententiarum [ Trapp], dist. 42–44, q. 4, vol. 3, p. 458):In one way, he says, everything functions as a whole “which includes something andsomething else in addition to ( praeter ) that something.” But in a second, more restrictedway that is a whole “which includes something in the first way and also includes agiven amount more times than does that included ( et includit tanta tot quot non includitinclusum ).” An infinite multitude can, Gregory continues, very well function as a wholewith respect to another infinite multitude, if “whole” is taken in sense one; but not insense two. It seems clear, then, that what Gregory intends is, in our terms, a distinctionbetween whole and part in the sense of set and subset (his first meaning) and whole

and part in the sense of unequal cardinality of the sets involved (his second meaning).Should there be any doubt, one need only look at the corresponding distinction hedraws for the term “greater than”. In the strict sense (unequal cardinality) “a multitudeis called greater which contains one more times or contains more units ( pluries continet



24 john e. murdoch

the infinite, this alternative was most fruitful (although no medieval

scholar, as far as I have been able to determine, took the additional

step of de fi ning the infinite by means of the equality of part and whole;that was left to the nineteenth century).

Returning to the composition of continua, which was at least one

of the considerations giving rise to the problem of unequal infinites,

we should note that, whether there were an infinite number of indi-

visibles or merely a finite number of them composing a continuum,

they were, once again, inevitably extensionless. This is clear from the

moves Aristotle made in Physics VI ; but one thing he did not focus on

was the existence of such extensionless indivisibles. Not so the fourteenth

century. Indeed, almost all those embracing a nominalist ontology heldthat, in the strict sense, indivisibles did not exist. Ockham, for example,

claimed that the term ‘point’ signifies the same as ‘a line of such and

such a length’ ( linea tante vel tante longitudinis or linea non ulterius protensa

vel extensa ).27 Many others of nominalist persuasion say similar things.28

On the other hand, it has been objected to this nominalist definition

of a point that, translating it into the geometrically true proposition

that ‘between every two points there is always another point’ we would

get ‘between every two lines of such and such a length there is always

another line of such and such a length’, which would be, of course,

absurd.29 Not so! It has to do with the de fi nition of a point as a “line of

such and such a length” and that is no more absurd than defining a

unum vel plures unitates )”; yet more generally (set/subset only), “every multitude whichincludes all the units of another multitude and certain other units is called greaterthan that (other multitude), even though it does not include more units than it ( includit

unitates omnes alterius multitudinis et quasdam alias unitates ab illis dicitur maior illa, esto quodnon includat plures unitates quam illa ).”27 William of Ockham, Tractatus de quantitate et Tractatus de corpore Christi [Grassi ],

OTh X, p. 22. Cf. William of Ockham, Expositio Physicorum [ Richter e.a.], III, ad tex.71, OPh V, p. 585: “Unde non habent dicere quod punctus sit quoddam indivisibiledistinctum a linea terminans ipsam lineam, sed debent dicere quod si sit res alia a linea,quod terminat lineam ex quo non potest esse sine linea et non est pars lineae.”

28 John Buridan, Questiones super libros Physicorum [ Paris, 1509], VI, 4, f. 97r–v: “Tuncergo est dubitari quare punctum dicitur communiter ab omnibus esse indivisibile;respondetur quod hoc non dicitur quia sit ita vel quia sit verum de virtute sermonis, seduno modo hoc dicitur secundum imaginationem mathematicorum ac si esset punctumindivisibile, non quia debeant credere quod ita sit, sed quia in mensurando revertuntur

eedem conditiones sicut si ita esset.” Also, Thomas Bradwardine, Tractus de continuo (MSTorun, R.40.2, p. 192): “Superficiem, lineam sive punctum omnino non esse. Undemanifeste: Continuum non continuari nec finitari per talia, sed seipso.”

29 Kretzmann, Approaches to Nature in the Middle Ages, p. 215.




point as a pencil of lines (Veronese)30 or as an infinite series of enclosure

volumes (Whitehead and others).31 That is to say, once we (Ockham,

Buridan, Veronese, Whitehead, etc.) have de fi ned a point, then we cango on to speak of points in a normal way in their many occurrences in

mathematics and natural philosophy. The medievals quite well realized

this; such normal definitions of points, lines, surfaces, and instants in no

way meant this had any effect upon points and the like in geometry and

even upon arguments against indivisibilism or atomism of any sort.32

A case in point is at the beginning of Book VI of the Physics: Aristotle

argues here that points or indivisibles have no parts and consequently

they can only touch whole-to-whole; but if they do, there is no increase

in size ( non facit maius ) of the continuous line they supposedly compose.This was one of the crucial arguments the late medieval atomists had

to answer. Points or indivisibles had to have some connecting relation to

one another to allow of faciens maius.

Thus, the indivisibilist Henry of Harclay says that points can very

well cause an increase in size if they touch, or are applied to one

another, secundum diversos situs.33 And Gerard of Odo, who is perhaps

30 Veronese, Fondamenti de geometria.31 Whitehead, An Enquiry Concerning the Principles of Natural Knowledge and The Concept

of Nature.32 For Ockham, Exp. Phys., III, ad text. 71 ( ed. cit . & loc. cit .): “Et si aliquando auc-

tores ponant vocaliter talem propositionem categoricam, per eam intelligent unam ypotheticam. Nunc autem ad veritatem conditionalis non requiritur veritas anteceden-tis, et ideo ad mathematicas non requiritur quod aliquod infinitum sit, sed requiriturquod ex tali propositione in qua ponitur iste terminus ‘infinitum’ sequatur alia velsequatur ex alia, et hoc potest contingere sine hoc quod infinitum possit esse. Et sicutest de infinito, ita est de puncto, linea et superficie. Et recte sensientes in mathematicaet non transgredientes limites mathematice non asserunt quod punctus sit quaedam

res indivisibilis distincta a linea nec linea a superfi

cie nec superfi

cies a corpore, sedponunt conditionales aliquas in quibus subiicitur ‘punctus’ vel ‘linea’ vel ‘superficies’sic accepta.” For Albert of Saxony, see his Sophismata [ Paris, 1495], unfol.; MS, Paris,BNF Lat. 16134, f. 43v): “. . . precisius loqui possumus imaginando instantia indivisibiliain tempore, licet talia in rei veritate non sint; nihilominus expedit ea imaginari . . . itain proposito non plus neque minus dico quod expedit ea imaginari ad explicandumcertas et precisas mensuras motuum et mutationum quas sine imaginatione instantiumindivisibilium ita precise exprimere non possumus; nec ex hoc sequitur aliquod incon-

veniens, quoniam sermones de talibus indivisibilibus per alias longas orationes debiteexponuntur, propter quas etiam orationes prolixas evitandas expedit tales terminosponere quos aliqui (ed. antiquos!) crediderunt supponere pro rebus veris indivisibilibus,licet tales res indivisibiles non sint nisi secundum imaginationem.” For Oresme, see

his Questiones super libros physicorum (MS Sevilla, Colomb. 7–6–30, 67v): “Quod non estnegandum indivisibilia esse, large et equivoce capiendo esse et ymaginando aliter quammathematicus ymaginatur, quia talia sunt significabilia.”

33 Henry of Harclay (MSS citt., note 4: Tortosa, f. 90r; Florence, f. 99r): “Ex istis



26 john e. murdoch

the most famous of the continental atomists, similarly claims that parts

are distinguishable within indivisibles secundum differentias respectivas loci

vel temporis, which are, for points, ante et retro, sursum et deorsum, dextrorsumet sinistrorum, and for instants, initium futuri et fi nis preteriti .34

On the other hand, in his opposition to indivisibilism Bradwardine

grants the devil his due, as it were, and allows the respectable Euclidean

notion of superpositio to function as the connecting relation between

indivisibles.35 But then he establishes that any geometrical relation of

superpositio is absolutely dissociated from continuity (or impositio, as he

calls it).36

As an aside, it must be noted that the brilliance of Bradwardine’s

geometrical and axiomatic treatment and continuity obscures the giveand take of the actual arguments of the indivisibilists and their critics.

Thus, though he opposes both Harclay and Chatton and cites them

by name, it is dif ficult, if not impossible, to tell from his account what

their detailed notions and arguments were.37

omnibus accipio quod necdum punctum, ymo nec linea nec corpus, facit maius extensivenisi applicetur secundum diversos situs. Ita dico quod duo indivisibilia, sicut puncta, si

applicentur ad invicem secundum diversos situs, magis faciunt secundum situm.”34 Gerard of Odo, Sent . I, dist. 37, MSS citt., note 3; Naples, ff. 238r–v; Valencia,ff. 22r–v) sets forth six deffensiva pro compositione continuorum ex indivisibilibus: “Nunc veroponenda sunt quedam deffensiva pro opinione ista, que sunt sex in numero. Primumest indivisibile secundum partes est distinguibile et determinabile secundum differentiasrespectivas loci vel temporis. Istud declare in quinque generibus indivisibilium quan-titative. Primo in superficie que est indivisibilis secundum dimensionem profunditatis:Quoniam ipsa distinguitur et determinatur per intra et extra . . . Idem apparet, si suma-tur punctus in centro spere, et hoc secundum omnem differentiam localem: ante etretro, sursum et deorsum, dextrorssum et sinistrosum. Quod apparet, quia secundumdifferentiam circumvolvatur spera, pars que est sursum moveatur ante, pars deorsummovebitur retro, et sic de aliis oppositionibus.”

35

Superpositio also occurs in Averroes, Henry of Harclay and Gerard of Odo, but witha quite different, non-Euclidean, meaning. On all of this, see Murdoch, “Superposition,Congruence and Continuity in the Middle Ages.”

36 Thomas Bradwardine, Tractatus de continuo (MS Toruń R.40.2, pp. 158–160; Erfurt,Amploniana 40 385, ff. 19r–21r): “Conclusiones 8–13: 8. Inter nullas rectas sibi super-positas puncta alica mediare. 9. Lineam rectam secundum totum vel partem magnamrecte alteri superponi et habere aliquod punctum intrinsecum commune cum ista noncontingit. 10. Linee recte unam partem magnam alie recte imponi et aliam partemmagnam superponi eidem vel ad latus distare ab illa impossibile comprobatur. 11. Uniusrecte duo puncta in alia continuari et per partem eius magnam superponi eidem velad latus distare ab illa non posse. 12. Linee recte unam partem magnam recte alterisuperponi et aliam ad latus distare ab ista est impossibile manifestum. 13. Unius recte

duo puncta alteri superponi vel unum imponi, aliud vero superponi et magnam eiuspartem ad latus distare ab ista non posse contingere.”37 Thomas Bradwardine (MSS in previous note, p. 165, ff. 25v–26r): “Pro intellectu

huius conclusionis est sciendum, quod circa compositionem continui sunt 5 opiniones




To continue to speak of the connecting relation between indivisibles,

there is, however, another side to the late medieval atomists’ and their

critics’ consideration of such a relation. This side involves the order ofthe constituent indivisibles within the continuum they compose. This

question of order was not apparent in Aristotle or Bradwardine or in

any of the relations between indivisibles à la Harclay or Odo. Order

becomes involved when one asks of the relation of some one indivis-

ible, not to a second indivisible, but to all other indivisibles (which are

infinite in number) in the continuum to which they belong. This is

brought out neatly by the reply William of Alnwick gives to one of the

positive arguments that Harclay provides for his belief that there are

(an infinite number of ) indivisibles which are immediate to one another.First the argument of Harclay:

For though my intellect does not understand how a continuum is com-posed of indivisibles that are immediately next to one another, the divineintellect necessarily does. One of my arguments for this view is the follow-ing: it is certain that God knows every point that can be designated in acontinuum. Take, then, the first inchoative point of a line; God perceivesthat point and any point in this line different from it. It follows, then, thateither another line falls between the more immediate point He perceives

or one does not. If not, then God perceives this point to be immediateto another one. If such a line does intercede, then, since points can beassigned in the line (which falls between the first inchoative point and theother point), these mean points have not been perceived by God.38

famose inter veteres philosophos et modernos. Ponunt enim quidam, ut Aristoteles etAverroys et plurimi modernorum, continuum non componi ex athomis, sed ex partibusdivisibilibus sine fine. Alii autem dicunt ipsum componi ex indivisibilibus dupliciter

variantes, quoniam Democritus ponit continuum componi ex corporibus indivisibilibus.

Alii autem ex punctis, et hii dupliciter, quia Pythagoras, pater huius secte, et Platoac Waltherus modernus, ponunt ipsum componi ex finitis indivisibilibus. Alii autemex infinitis, et sunt bipartiti, quia quidam eorum, ut Henricus modernus, dicit ipsumcomponi ex infinitis indivisibilibus immediate coniunctis; alii autem, ut Lyncul <nien-sis>, ex infinitis ad invicem mediatis. Et ideo dicit conclusionem: ‘Si unum continuumcomponatur ex indivisibilibus secundum aliquem modum,’ intendendo per ‘modum’aliquem predictorum modorum; tunc sequitur: ‘quodlibet continuum sic componi exindivisibilibus secundum similem modum componendi’.”

38 Henry of Harclay (MSS citt., note 4: Tortosa, f. 88r; Florence, f. 98r): “Licet enimmeus intellectus non comprehendit quomodo continuum componitur ex indivisibili-bus immediate se habentibus, tamen intellectus divinus hoc necessario comprehendit.Cuius est una racio hec: Certum est quod Deus modo intuetur omne punctum quod

possit signari in continuo. Accipio igitur primum punctum in linea incoativum linee;Deus videt illum punctum et quodlibet aliud punctum ab isto in hac linea. Usque adillum punctum inmediaciorem quem Deus videt intercipit alia linea aut non. Si non,Deus videt hunc punctum esse alteri inmediatum. Si sic, igitur cum in linea possint



28 john e. murdoch

Harclay’s argument stated in slightly different terms amounts to:

(0) God furnishes a manner of actualizing or specifying all the pointsin a given line (He is always conveniently on call to perform such

tasks).

(1) God knows or perceives the initial point of the line and all others

in the line.

(2) Consequently, God knows the relation of the initial point to all others.

(3) If all such relations are of distance, then God does not know all oth-

ers, contra hypothesim.

(4) Therefore, one such relation must be (not that of distance) but of

contact, which is to have indivisibles immediate to one another.

We then turn to William of Alnwick’s reply:

I reply in brief that this is true: (1) ‘between the first point of the line andevery other point of the same line known by God there is a mean line’.For any singular [of this universal ] is true, and, moreover, its contradic-tory is false. And this is so because the term ‘mean line’ in the predicateimmediately following the universal sign [ i.e., ‘every’] has merely confused supposition. On the other hand, this is false: (2) ‘there is [some one]

mean line between the first point and every other point of the same lineperceived by God’, since there is no [one] mean line between the firstpoint and every other point perceived by God. For there cannot be anysuch mean line, for if there were, it would fall between the first point anditself; nor would that line be perceived by God. And therefore, when it isinferred: “if there is [such a mean line], then, as points can be assignedin the line, etc.,” the term ‘line’ there has particular supposition. Andhence an inference is made af firmatively from a superior to an inferiorand thus the fallacy [of af firming] the consequent is committed.39

signari puncta, illa puncta media non erant visa a Deo. Probacio huius consequencie:Nam per positum linea cadit inter hunc punctum primum et quodlibet aliud ab hocpuncto quod Deus videt; et ideo, per te modo inventum punctum medium Deusnon videbat.” The term ‘immediaciorem’ is puzzling, since being immediate is notsusceptible of degrees.

39 William of Alnwick, Determinatio 2 (MS Pal. lat. 1805, f. 14r–v): “Dico autembreviter quod ista est vera: ‘Inter primum punctum linee et omnem alium punctumeiusdem linee cognitum a Deo est linea media.’ Quelibet enim singularis est vera, eteius eciam contradictoria est falsa. Et hoc ideo est, quia ‘linea media’ in predicatosequens mediate signum universale stat confuse tantum. Hec tamen est falsa: ‘Est linea

media inter primum punctum et omnem alium punctum eiusdem linee visum a Deo,’quia nulla est linea media inter primum punctum et omnem alium punctum visum aDeo. Non enim contingit dare aliquam talem lineam mediam; sic enim mediaret interprimum punctum et seipsam, nec illa linea esset visa a Deo. Et ideo cum infertur: ‘Si




The crucial point in Alnwick’s reply is that there is a difference in

supposition that accounts for the fact that proposition (1) is true while

proposition (2) is false. Thus the term ‘mean line’ has merely confused( confuse tantum ) supposition and, if we look in our “logical primer” about

terms having this kind of supposition,40 we learn that no disjunctive

descent can be made from such terms. That is, given proposition (1)

‘between the first point of a line and every other point of the same

line known by God, there is a mean line’, we cannot make a logical

descent to the disjunction ‘either this mean line is between the first point

and every other point or that mean line is between the first point and

every other point or that other mean line is, etc.’ On the other hand,

the false proposition (2), where the term ‘mean line’ has particular ordeterminate supposition, specifically allows such a disjunctive descent,

causing the falsity of proposition (2) on grounds that any of the disjuncts

is false (or to put it in Alnwick’s terms, there is, running disjunctively

throughout the ‘mean lines’ involved, ‘no [one] mean line between the

first point and every other point’).

If, now, we translate what is being said in the medieval language of

supposition into the notions of quantifiers in modern logic, we can

interpret the distinction between these two propositions as follows:

True (1) for all y there is an x such that x falls between the first point

and y.

False (2) there is an x such that for all y, x falls between the first point

and y.41

Here the universal and existential quantifiers are reversed in the two

propositions and we would say that the truth of (1) and the falsity of (2)

derives from the fact that it is a case of multiple quantifi

ers involving a

sic, igitur cum in linea possent puncta signari, et cetera,’ ibi ‘linea’ stat particulariter;et ideo arguitur a superiori ad inferius af firmative et sic facit fallacia consequentis.”

40 For instance, Ockham’s Summa totius logicae, I, ch. 68, conveniently appearing inEnglish translation with facing Latin text in his Philosophical Writings, ed. & tr. PhilotheusBoehner, revised by Stephen Brown, pp. 70–74.

41 That is, in the standard notation:(y)(∃x) 1<x<y(∃x)(y) 1<x<y

Incidentally, Alnwick’s remark that “there cannot be any such mean line, for if therewere, it would fall between the first point and itself ” is quite correct, since it followsfrom proposition (2), without any further distinction in the scope of quantifier, that(∃x) 1<x<x.



30 john e. murdoch

relational predicate (i.e., ‘mean’, ‘between’) where these quantifiers are

shifted in position. The medieval logician would say that the quantifier

shift has to do with a shift in supposition where the same term (‘meanline’) has a different supposition in proposition (1) and (2). Moreover,

in any case, a term having merely confused supposition does not imply

the same term with particular or determinate supposition.

Now the force of Alnwick’s criticism is that Harclay seems to be

unaware of the fact that his argument blurs the distinction between

propositions (1) and (2). Indeed, in Alnwick’s eyes, he would like proposi-

tion (1) to imply proposition (2), which, we have seen, it does not.42

I have dealt more extensively with the question of the order of

atoms or indivisibles within continua, using Harclay and Alnwick asa particularly good instance of what is at stake in this question. My

reason for so doing is that this situation entailing a shift between sup-

positions (or quantifiers) is involved wherever ordered infinite series,

sequences, and processes are in question. And such question of order

with respect to an infinity of elements are met with everywhere in

natural philosophy. Thus we find a multiple quantification or supposi-

tion technique applied to Aristotle’s distinction between the potential

and the actual infinite,43 to the solution of Zeno’s “stadium” paradox,44

42 But since the (false) proposition (2) does imply the (true) proposition (1), then toargue from (1) to (2) means that Harclay has committed the fallacy of af firming theconsequent (as Alnwick says).

43 William of Ockham, Exposit io physicorum, III text. 61, [ Wood e.a.] OPh V,p. 560: “Est autem istis adiciendum quod quamvis haec sit vera: ‘omni magnitudineest minor magnitudo,’ haec tamen est impossibilis: ‘aliqua magnitudo est minor omnimagnitudine.’ Ista enim est vera: ‘omni magnitudine est minor magnitudo,’ quia est unauniversalis cuius quaelibet singularis est vera. Haec tamen est falsa: ‘aliqua magnitudo

est minor omni magnitudine,’ quia est una particularis cuius quaelibet singularis estfalsa. Et est simile sicut de istis duobus: haec est vera: ‘omnis homo est animal,’ ethaec falsa: ‘aliquod animal est omnis homo.’ Et ratio diversitatis est quia in ista: ‘omnimagnitudine est minor magnitudo; ly ‘minor magnitudo’ supponit confuse tantumpropter signum universale praecedens a parte subiecti, et ideo ad veritatem suf ficit quodista magnitudine sit una magnitudo minor et illa magnitudine sit una alia magnitudominor et sic de aliis. Sed in ista: ‘aliqua magnitudo est minor omni magnitudine’ ly‘magnitudo’ supponit determinate, et ideo oportet quod aliqua una magnitudo numeroesset minor omni magnitudine, et per consequens esset minor seipsa.”

44 William of Ockham, Expositio physicorum, III, text. 79, [ Wood e.a.] OPh V, pp.565–66: “Istis visis dicendum est quod intentio Philosophi est solvere rationes Zenonisper istum modum quod quia non sunt ibi partes infinitae quarum quaelibet secundum

se totam sit extra aliam, ita quod sit accipere primam, secundam et tertiam. Sed suntibi partes infinitae quarum nulla est prima. Ideo non est inconveniens per illas partesinfinitas moveri. Unde ad primam rationem Zenonis, quando accipit quod impossibile




to the problem of the motion of a sphere over a plane surface (since

it touches or is in contact with point after point in the plane),45 and

other repetitions and refutations and arguments similar to that givenby Henry of Harclay.46

What is more, in his criticism of Gerard of Odo’s atomism, John

the Canon says that this technique amounts to a traditional rule and

by means of its application recte possunt solvi multe rationes, which rationes

all have to do with problems of infinity or continuity.47 Of course this

est pertransire infinita, verum est si illa infinitae sint secundum se tota distincta, ita

quod sit accipere aliquod de illis quo nullum sit prius. Sed talia non sunt in aliquocontinuo. Si autem illa infinita non sint secundum se tota distincta, ut non sit aliquodprimum distinctum inter ea, possibile est talia infinita pertransiri, immo necesse esttalia infinita pertransiri quandocumque aliquod spatium pertransitur.”

45 Walter Burley, Super Aristotelis libros de physica auscultatione commentaria [ Venice, 1589],col. 729: “Ad hanc formam dico, quod est concedenda, scilicet quod in plano continueest punctus post punctum, et tamen ista est falsa, scilicet quod in plano est punctuscontinue post punctum, quia nullus punctus est continue post alium punctum.”

46 Walter Burley, ibid., col. 730 (to give the essentials of a long text): “He concedesthat sine medio aliud instans post instans A erit is true, but this does not imply the falseproposition that aliud instans post A erit sine medio, since ‘sine medio’ (= ‘immediate’) is asyncategorematic term, and in the inference above would move from suppositio con-

fuse tantum to suppositio determinate.” Gregory of Rimini, answering what is essentiallyHarclay’s argument (above, note 38), Sent . II, dist. 2, Q. 2 [Trapp], p. 292: “Ad tertiumdico, stante praedicta suppositione, quod deus videt quod inter primum punctum etquodlibet aliud eiusdem lineae intercipitur linea, et infinita etiam puncta, non tamenlinea non visa, nec puncta non visa ab eo. Nullam tamen lineam deus videt intercipiinter primum punctum et quodlibet aliud punctum ab eo visum.”

47 John the Canon, Questiones in Physicorum VI, quaestio unica (MSS Florence, Bibl.Naz., conv. Soppr. C. 8 22 fol. 119v; Vat. Lat. 3013, fol. 73r): “Et pro quibusdamaliis rationibus solvendis: pro una, scilicet quod in continuo sunt plures partes quaminfinite, quia quelibet pars est divisibilis in infinitum, applico istam regulam: Quodquandocunque arguitur ab alico termino communi supponente confuse tantum respectualicuius magnitudinis ad eundem terminum supponentem personaliter respectu alicuius

multitudinis, non est bona consequentia, quia sequitur fallacia figure dictionis. Quoddeclaratur in quadam consimili ratione. Solet enim probari a quibusdam quod multitudonon possit crescere in infinitum, quia, si sic, tunc ultra omnem multitudinem finitamdatam esset dare multitudinem finitam maiorem; sed multitudo maior omni multitudinefinita est multitudo infinita; ergo, si ultra omnem multitudinem finitam datam esset daremultitudinem finitam maiorem, sequeretur quod alica multitudo finita esset infinita,quod est impossibile. Quod autem ultra omnem multitudinem finitam datam esset daremultitudinem finitam maiorem, si multitudo posset crescere in infinitum, patet, quiaquelibet singularis huius universalis foret vera, nam ultra hanc multitudinem finitamdatam esset dare multitudinem finitam maiorem, et ultra illam et sic in infinitum. Adistam rationem respondetur quod hic est fallacia figure dictionis, quoniam in maiori isteterminus ‘multitudo’ in predicato prime propositionis supponit confuse tantum et dicit

quale quid; in minori autem supponit tantum determinate respectu eiusdem multitudinisimportate per signum universale; et ideo commutatur quale quid in hoc aliquid. Itarecte possunt solvi multe rationes iuxta istam materiam. Applicate, si vis.”



32 john e. murdoch

technique employing a difference in supposition is older than its newly

found fourteenth-century application to such problems.48 Indeed, an

improper understanding of what is involved in the technique is con-demned by Robert Kilwardby in 1277.49

Of course, questions of order of a different sort were in Aristotle

himself when he asks of the permissible use of the indivisible begin-

nings and endings of successive things (like motion) or permanent

things changing within a continuous time (like something changing

from white to not white).50 These passages in Aristotle led, as many of

you know, to an enormous outburst of literature in the thirteenth- and

particularly fourteenth-century, where some of it was both clever and

fairly thought-provoking.51 This was the literature of “limit decisions”(as I and others have called it). Its concern was determining whether,

to speak in modern terms, things were intrinsically or extrinsically limited

(both successive things and permanent things against a continuous

time).

Now one of the most creative aspects of this “limit decision” litera-

ture was the many sophismata constructed to deal with these decisions.

Often the function of sophisms was something akin to the following:

one might be quite clear about what limit decision might be made in

such and such a situation; but then a sophism might serve to reveal that

things were not so clear after all and that the limit decision in question

deserved further investigation.52

Further, the problem of limit decisions and the late medieval indivisi-

bilists comes clearly into the picture when Bradwardine in his Tractatus

de continuo showed that indivisibilism effectively destroyed the distinc-

tion between intrinsic and extrinsic limits that was at the heart of all

this fourteenth-century literature, something that would be preserved

48 For example, Peter of Spain, Tractatus, called afterwards Summule logicales [ DeRijk], pp. 222–23.

49 Chartularium Universitatis Parisiensis [ Denifle e.a.] vol. I, p. 558: “Item quod non estsuppositione in propositione magis pro supposito quam pro significato, et ideo idem estdicere, cujuslibet hominis asinus currit, et asinus cujuslibet hominis currit.”

50 Physics, VI, ch. 5 and VIII, ch. 8, 263b9–264a6.51 For all of this, see Kretzmann, “Incipit/Desinit”; Murdoch, “Propositional Analysis

in Forteenth-Century Natural Philosophy: A Case Study”; above all, Wilson, William

Heytesbury: Medieval Logic and the Rise of Mathematical Physics.52 See, for example, The Sophismata of Richard Kilvington, both edited and translated byNorman and Barbara Kretzmann; Wilson (above, note 51), chs 2–3; Knuuttila, “Remarkson the Background of Fourtheenth Century Limit Decision Controversies”.




only if continua were infinitely divisible.53 John Buridan also claimed a

quite different inconsistency between maintaining atoms and properly

expounding the beginning and ending limits of a thing. But he resolvedthis apparent inconsistency by making temporal intervals do the work

that atomic instants were supposedly required to do.54

Bradwardine and Buridan were quite right in charging indivisibilists

with, at least, obscuring limit decisions. For example, the indivisibilist

Crathorn supported his contention that of no finite continuum is there

an inifinite number of proportional parts with a slightly humorous

argument (which sounds as if it were drawn from some sophism-like

literature on limits) that, if someone sins and then repents in succeeding

alternate proportional parts of time, there is no way of knowing whetherthat person deserves damnation or salvation. Such a determination can

only be made if there are a finite multitude of proportional parts.55

We do not have time or space to consider the late medieval atomists’

replies to all of Aristotle’s arguments against indivisibilism, let alone

those of their critics in support of Aristotle. Yet I should mention one

of the arguments in Physics, VI, ch. 2 about a mobile moving at various

speeds over a given magnitude, that gave rise to considerable concern

for the fourteenth-century atomist.

The centerpiece here in Aristotle was that, given a mobile moving

over a given space or magnitude (s1 ) in a given time (t

1 ), then (a) a faster

mobile can of course move over the same magnitude in less time and

(b) a slower mobile in the time of the faster mobile will move over less

magnitude, and, therefore, (c) by alternately taking faster and slower

53 Thomas Bradwardine, Tractatus de continuo (MSS citt., note 36, p. 170, fol. 30r):

“Concl. 50, Si sic, (scil. continuum compositum de indivisibilibus immediate conunctis)omni quod incipiet esse aliquale vel desinet esse tale secundum utramque significationemincipiet vel desinet esse tale. Coroll . Cuiuslibet et qualiscumque rei talis esse primumintrinsecum et postremum.”

54 John Buridan, Quaestiones in libros Physicorum [ Paris, 1509], VI, q. 4: “Si autemaliquis vult aliter exponere incipit et desinit, ego dicam quod B incipit esse quia aliquotempore est et immediate ante illud tempus non erat.” This was in line, of course, withBuridan’s nominalist removal of instants.

55 William Crathorn, Sent. I, q. 3, [ Hoffmann] p. 237: “Ponatur quod aliquis sicdisponatur quod peccet in prima parte proportionali temporis et in secunda pae-niteat, iterate in tertia peccet et in quarta paeniteat et sic alternatim in aliis partibusproportionalibus temporis finite, cuius terminus sit a instans, in quo instanti volo quod

moriatur. Isto casu posito aut salvabitur aut damnabitur. Neutrum potest dari, quianon est dare ultimum poenitentiae, quam non sequatur peccatum nec econverso. Igiturtali, qui infinities peccavit et infinities paenituit, non posset deus iusto iudicio aliquampoenam infligere nec aliquod praemium conferre.”



34 john e. murdoch

mobiles, the faster will continuously divide the time, the slower the

magnitude.56 Moreover, this argument assumes that (1) there can be

faster and slower mobiles ad in fi nitum, and (2) that magnitude, time andmotion have a one-to-one correspondence between them (something

that Aristotle had already said in his account of time in Book IV of

the Physics ).57

The atomists’ response to this was the following: Since they did not

wish to give up mobiles moving faster and slower,58 they denied, in effect,

one or the other, or both, of these two assumptions. Thus, some late

medieval atomists maintained that there was a fastest motion (usually

the motion of the primum mobile ), thus denying the first assumption.59

Similarly, the assumption is overturned by pointing out that fast andslow have to be determined by a different measure than moving in a

given time more or less distance.60

56 Aristotle, Physics, VI, 2, 232a23–b20. Bradwardine supports what Aristotle says herein Concl. 24 of his Tractatus de continuo (MSS citt., note 36, p. 164, fol. 25r): “Quocumquemotu locali signato potest motus localis uniformis et continuus omni proportione rectefinite ad rectam finitam velocior et tardior inveniri. Coroll . Quodcumque spatium finitumquocumque tempore finito posse uniformiter et continue pertransiri.”

57

Aristotle, Physics, IV, 11, 219a10–14.58 For they did not maintain the alternative of the equality of speeds for atoms asEpicurus’s had done (Diogene Laertius, X, 61) which, in any case, they did not haveat their disposal. Nor were they cognizant of Epicurus’s answer to Aristotle’s argu-ment in Phys. VI, 2.

59 William Crathorn (ed. cit., note 55), pp. 242–43: “Nona conclusio est quod impos-sibile est aliquem motum esse velociorem motu vere continuo; licet enim unus motus

vere unus et continuus, si aliquis talis sit, sit velocior motibus illis, qui non sunt verecontinui, et motus non continuus sit velocior alio motu non continuo, tamen si aliquid

vere continue moveatur non apparenter tantum, impossibile est aliquem motum talimotu esse velociorem.” He then gives three arguments in support of this conclusion.Nicholas of Autrecourt, Exigit ordo [O’Donnell], p. 215: “Sed illud latet nos ut supradixi,

et tale mobile quod sic se habet quantum ad veritatem est mobile primum quod moveturmotu velocissimo et de tali potest dici quod movetur in ipso nunc.” John Wyclif alsomaintained a fastest motion ( Tractatus de logica, [ Dziewicki ] p. 39) where he said that“nichil potest velocius moveri motu successivo quam movetur equinoccialis”.

60 Gerard of Odo, Sent . I, dist. 37 (MSS citt., note 3) Naples, 239v, 241v; Valencia,123r, 124r: “Probatio consequentie pro cuius evidentia premittuntur tres suppositiones.Prima est quod in omni tempore contingit aliquid moveri velocius et tardius. Secundaquod mobile velocius plus pertransit de spatio in equali tempore mobili tardiori.Tertia quod contingit mobili tardo duplicem sesquialteram seu emioliam longitudinempertransiri a velociori . . . Ad probationem respondeo, primo ad suppositiones: Negoeas omnes tres ad intellectum ad quem inducuntur. Quando enim dicitur in primaquod in omni tempore contingit velocius et tardius moveri, si intelligatur quod in

omni tempore contingit velocius et tardius moveri, hoc est, plus et minus pertransiride spatio, suppositio est falsa simpliciter, quia possible est aliqua duo moveri, semperunum velocius altero, et numquam mobile velocius pertransibit plus de spatio quammobile tardius. Unde si motus primi mobilis duraret per imperpetuum, polus articus




Yet another way was open to the medieval indivisibilists in accounting

for the fast-slow mobile argument. It was to criticize the second assump-

tion of the one-to-one correspondence between motion, magnitude andtime. This assumption could be either according to one measure of

taking time as continuous or by another measure of taking time as discrete.

Thus, under time as continuous, two time intervals measuring motions

contained an equal number of indivisibles when, and only when, the

mobiles in question are moving over the same or equal magnitudes.

However, if we consider time as discrete, then, if one time interval is less

than, equal to, or greater than another time interval, then the infinite

number of indivisibles in the one is, correspondingly, less than, equal

to, or greater than the other. Therefore, faster and slower are measuredby discrete time, in effect answering Aristotle’s argument.61

Finally, there is a totally new element introduced into the debate

about continuity. This consists in the consideration of the geometrical

properties of the horn angle or angle of contingence (like DAC or

EAC in Figure 2) or the angle of a semicircle (like BAD or BAE). The

horn angle is one of the few geometrical objects which are intuitively

in fi nitesimal . The properties of these curvilinear angles were accurately

expressed by Campanus of Novara in comments to his version of

Euclid’s Elements.62 Thus, there are an infinite number of horn angles

(derivable by drawing greater and lesser circles through the same point

of tangency) less than any acute rectilinear angle for example, DAC,

EAC, etc., < FAC, and, mutatis mutandis, the same goes for angles

of a semicircle relative to right angles. These are non-Archimedean

magnitudes since the multiples of any horn angle will not exceed any

rectilinear angle no matter how small. But Campanus noted that such

angles do not obey the continuity principle: “if one moves through

et polus antarticus moverentur continue, non tamen plus pertransirent de spatio quampoli orbis lune, dato quod orbis lune non revolveretur nisi semel in centum annis etquod etiam continue moveretur.”

61 Henry of Harclay (MSS citt., note 4: Tortosa, fol. 93v; Florence, fol. 100v): “Etdico tunc breviter pro argumento quod accipiendo tempus et motum ut consideranturut continua, non sunt plura instantia in duobus diebus quam in uno die, suppositoquod equale spacium mensuretur per duos dies et per unum diem. Sed accipiendotempus ut discretum, sic sunt plura instantia in duobus diebus quam in uno.” Chattonalso criticizes the one-to-one correspondence assumption. For this, and for Harclay as

well, see the discussion and texts in J. Murdoch, “Atomism and motion in the four-teenth century.”62 In his comments to III, 15 (III, 16 of the Greek), where Euclid himself mentions

a horn angle, and X, 1.



36 john e. murdoch

all mean values, from being less than something to being greater than

it, then one moves through an equal.”63 Thus, given rectilinear angle

BAF less than the semicircular angle BAE, by rotating AF about A

toward AC, one will reach the right angle BAC moving through allacute rectilinear angles less than BAC, but never being equal to the

semicircular angle BAE or for that matter BAD. In these observations

Campanus was followed by any number of later medieval works, but

notably by Bradwardine’s Geometria speculativa. Bradwardine also claimed

that indivisibilism would mean that a straight line could divide a horn

angle, contrary to what Euclid, and he himself, had maintained.64

Now these curvilinear angles entered into medieval philosophy and

theology since it was maintained that they afforded a way to measure

the infinite distance or excess between quantities or things within one

continuous latitude or series. Thus, already in 1290 Godfrey of Fontaines

uses the relation between horn angles and rectilinear angles to explain

how a finite caritas viae can be infinitely exceeded by a finite caritas

63 This occurs in the course of Campanus comment to III, 15 [Basel, 1558]: “Hoctransit a minori ad maius, et per omnia media, ergo per aequale.”

64

Thomas Bradwardine, Geometria speculativa [ Molland], pp. 66–75. See alsoBradwardine, Tractatus de continuo, concl. 7–8 (MSS citt., note 36: p. 176, fol. 36r): “Sisic (scil. continuum componatur ex indivisibilibus finitis), angulus contingentie divideturper rectam.”

Fig. 2

B

D

E

F

A C




patriae.65 Similarly, in the 1340’s Gregory of Rimini uses our angles to

explain the infinite distance between the finites albedo and nigredo.66

The champion, however, of applying curvilinear vs. rectilinear angleswas Peter Ceffons, ca. 1348–49. Taking many of the assumptions of this

application from Nicole Oresme,67 he turned to these angles to obtain

a scale of measure that would prove to be effective when set against

the different perfections of radically distinct species. He concludes that

these angles do provide such a measure, filling out what he has in mind

through no less than nineteen corollaries.68

To conclude this account of “angle calculus,” we might note that

John of Ripa, who was no stranger in the application of mathematics

to essentially theological problems,69 evidently thought that Peter Ceffonshad gone too far in applying de proportionibus angulorum.70

This ends my catalogue or, at least, partial catalogue, of the ways in

which in the fourteenth century quite new elements were brought to bear

on Aristotle’s notions and arguments about infinity and continuity. Some

65 Godefrey of Fontaines, Quodlibet VII, q. 12, [ De Wulf e.a.], p. 392: “Et ponitur

exemplum de angulo recto et contingentiae; nam angulus contingentiae, quanto circulusest maior tanto angulus est maior ( lege minor), sed nunquam tamen attingere potest adhoc quod aequetur angulo recto ( lege rectilineo), cum tamen quocumque circulo datomaior posset imaginari circulus et per consequens angulus contingentiae. Et hoc modopotest dici quod comparando caritatem viae ad caritatem patriae secundum hos modosperfectionum secundum suos actus, caritas viae posset augeri in infinitum . . .” As canbe seen, Godfrey’s, or the editor’s, mathematics is questionable in places.

66 Gregory of Rimini, Sent . I, dist. 17, q. 4 [ Trapp], vol. 2, pp. 410–411: “Ad pro-bationem, cum dicitur quod individuum albedinis, quod est A, excedit B individuumnigredinis sine proportione et per consequens in infinitum, . . . simili modo posset probariquod quilibet angulus rectilineus esset infinitus vel infinitae magnitudinis. Sumaturenim unus rectus et sit A, et unus angulus contingentiae et sit B. Tunc probo quod A

excedit B sine proportione.”67 On his relation to Oresme, see now the fundamental article of Mazet, “PierreCeffons et Oresme—Leur relation revisitée”. It should be noted that Ceffons was some-thing of an inveterate “borrower,” often citing, in some cases verbatim, the likes ofThomas Bradwardine’s De proportionibus velocitatum, Roger Swineshead’s De obligationibus,

John Mirecourt’s Comm. Sent ., etc.68 See Murdoch, “ Mathesis in philosophiam scholasticam introducta: The Rise and Develop-

ment of the Application of Mathematics in Fourteenth Century Philosophy and The-ology” and “Sublilitates Anglicanae in Fourteenth-Century Paris: John of Mirecourt andPeter Ceffons.”

69 John of Ripa, Quaestio de gradu supremo [Combes e.a.] and his Conclusiones (i.e. exlibris Sent . [Combes], pp. 70–72.

70

John of Ripa, Commentarius in Sententiarum, I, dist. 2, q. 4 (MS BNF, Lat. 15369, f.147v): “Si arguantur contra premissas conclusiones per exempla mathematica, huius-modi exempla sunt refellenda, et maxime illa que ex proportionibus angulorum et ipso-rum excessibus arguunt consimiles proportiones et excessus inter species entium.”



38 john e. murdoch

of these elements grew from seeds already present in Aristotle’s text,

such as his comments about faster and slower moving mobiles (whose

whole purpose in these comments was to establish the infinite divis-ibility of the continua involved), or such as the tremendous growth of

literature in the fourteenth century of ever more complicated decisions

regarding the limits of continuous processes and events. More strikingly

new were elements entirely foreign to the Aristotelian seedbed including

geometrical arguments against indivisibilism, the very question of the

existence of such indivisibles, the consideration of the possibility of

unequal infinites, the differences of the logical doctrine of supposition

as applied to the infinite and the continuum and, lastly, the intrusion

of curvilinear angles into debates about infinite excesses.



INDIVISIBLES AND INFINITIES:RUFUS ON POINTS

Rega Wood

[ L]et us remember that we are dealing with infinities and indivisibles,the former incomprehensible to our finite understanding by reason oftheir largeness, and the latter by their smallness. Yet we see that humanreason does not want to abstain from giddying itself about them (Galileo,

Discorsi e Dimostrazioni matematiche intorno a due nuoue scienze 1).1

Accounting for indivisibles consistently sometimes seems beyond even

Aristotle’s capacity. His denial that indivisibles are quantitative, con-

stitutive parts of the continua of magnitude, motion, and time is not

completely consistent, despite the fundamental role this tenet plays

in his physics. In De anima he cites uncritically the claim that straight

touches sphere at a point,2 which suggests that points are integral parts

of external bodies. In Physics 4.11, he sometimes describes instants

not as temporal limits, but as units or numbers of time, and hence asconstitutive parts, as Julia Annas has shown.3

As to their ontological status, Aristotle ridicules the suggestion that

points or lines are substances in the Metaphysics (3.5.1002a24–b10).

Yet, in the Posterior Analytics (1.27.87a36) he defines points as substance

with position.

Whatever may be true about Aristotle himself on the relationship

of mathematical objects to objects in the world, his interpreters are far

from agreed. Some claim that the subject of Aristotle’s mathematics

are ideal objects, not inhering in and quite independent of sensible

1 Galileo [ Leiden, 1638], p. 73. Translation: Two New Sciences [ Drake], p. 34.2 Aristotle, De anima 403a12–14.3 Annas, “Aristotle, Number and Time”. In the standard medieval Latin transla-

tion, these passages read as follows: Physica 4.11.4.12.220a3–4: “ipsum autem nuncest sicut id quod fertur, unitas est numeri;” 4.11.220b5–12: “tempus autem numerusest . . . quod numeratur, hic autem accidit prius et posterius semper alterum; ipsa enimnunc altera sunt. Est autem numerus unus quidem et idem qui est centum equorumet qui est centum hominum, quorum autem numerus est, altera sunt, equi ab homi-

nibus”; 4.11.219b24–29 (Annas, pp. 11–12): “eo vero quod fertur cognoscimus priuset posterius in motu, secundum autem quod numerabile est prius et posterius, ipsumnunc . . . secundum enim quod numerabile est prius aut posterius, ipsum nunc est.” See Aristoteles Latinus 7.1.2, pp. 176–179.



40 rega wood

things. Jonathan Lear, by contrast, holds that Aristotle’s mathematics

is about abstract objects only in so far as we consider mathematical

objects in abstraction from their physical instantiation, and not inthe sense that points and lines are not found in external objects.4 By

contrast, thinkers like William Ockham have claimed that for Aristotle

points were merely conceptual; that a point is nothing but a line not

further extended, making it vain to define a sense in which there are

points in a line.5

If, pace Ockham, we suppose that at least for Aristotle, there are

points in lines, indeed, potentially infinitely many of them, it is not clear

whether and how the infinitely many points in one line can be com-

pared to the infinitely many points in lines of different lengths. Aristotleis clear that infinities are not wholes, but in some sense parts ( Physics 3.6.207a26–28; 3.7.208a14). And it is almost but not quite universally

agreed that for Aristotle, all infinities are equal, though he does not

explicitly say so in Physics 3. 6–8,6 which is the passage normally cited

for the view.7 Yet presumably, if pressed, Ockham, who claimed that one

infinity could be greater than another,8 would have credited Aristotle

with this insight too. Other medieval authors who held that one infin-

ity could be greater than another include Robert Grosseteste, Henry

Harclay, Adam Wodeham, and also Richard Rufus of Cornwall, author

of the first known Western commentary on Aristotle’s Metaphysics.

Given the complexity of Aristotle’s views on indivisibles, Rufus faced

a formidable and complex interpretative challenge, which cannot be

considered fully here. Instead I will consider Rufus’s response to only

three questions:

1) Are points parts of external objects?

2) Are points and lines substances and parts of substances?3) Are some infinities of indivisibles greater than others?

4 Lear, “Aristotle’s Philosophy of Mathematics.”5 William of Ockham, Tractatus de quantitate [Grassi], OTh X, 1, p. 22; Expositio

Physicorum [ Wood e.a.], OPh V, 6.2, pp. 452–462.6 Thanks to Henry Mendell for confirming that such an explicit statement is not

found elsewhere.7 See for example John Dorp, in John Buridan, Compendium totius logicae [ Venice,

1499; Frankfurt am Main, 1965], sign. 15 as cited by Ashworth, “An Early Fifteenth

Century Discussion of Infinite Sets,” pp. 232–233.8 William of Ockham, Expositio in libros Physicorum [ Wood e.a.] OPh V, 6.6, p. 565;Quodlibeta septem [ Wey], OTh IX, 2.5, p. 132. For a discussion of Ockham’s views seeMurdoch, “Ockham and the Logic of Infinity and Continuity.”



indivisibles and infinities: rufus on points 41

1. Are there Points in the External World?

On the first question Rufus and his contemporaries agree with JonathanLear. Points, like lines and spheres, are found in sensible objects, but

mathematicians can consider them in abstraction from sensible objects

without distortion.9 Typically this point is stated as a gloss on a text

like Physics 2.2.193b22–25: “The next point to consider is how the

mathematician differs from the student of nature; for natural bodies

contain surfaces and volumes, lines and points, and these are the sub-

ject matter of mathematics.”10 Rufus, like his contemporaries, defines

mathematical abstraction by distinguishing it from physical and meta-

physical abstraction.Many such claims are commonplaces stated by a number of authors

and in several of Rufus’s works; certain precisions occur, however,

only in one work. So we will distinguish carefully positions stated only

in a single work or group of works, since the attribution of Rufus’s

commentaries on Physics (In Phys.) and De Anima (In DAn) has recently

been challenged. Also considered here are two commentaries on

Aristotle’s Metaphysics, Memoriale in Metaphysicam Aristotelis and Dissertatio

in Metaphysicam (henceforth MMet and DMet ) and two commentaries

on Peter Lombard’s Sentences, Sententia Oxoniensis and Sententia Parisiensis

(henceforth SOx and SPar ).11

Firstly, I consider texts on how mathematical abstraction differs from

physical abstraction and secondly texts which compare mathemati-

cal abstraction with metaphysical abstraction. The following general

conclusions will emerge. Mathematicians consider embodied objects,

but do not consider them as such; by contrast metaphysicians con-

sider disembodied or immaterial objects, objects apart from matter.

Mathematical abstraction, unlike physical abstraction, prescinds fromchange; it considers its objects in abstraction from the sensible matter

in which they inhere.

What variation is there among the accounts of the difference between

physicist and mathematician and between physical and mathematical

9 Cf. Aristotle, Physics 2.2.193b34–35.10 Aristotle, Physics [ Barnes], I, p. 331.11 These work will be cited as follows MMet , Erfurt Quarto 290 (henceforth Q290);

DMet , Vat. lat. 4538 (henceforth V4538) and Q290; SOx, Balliol College 62 (henceforthB62), SPar , Vat. lat. 12993; In Phys. [ Wood], Oxford, 2003; In DAn, Madrid, Bibl. nac.,3314 and Erfurt Quarto 312 (henceforth M and Q312). For the controversy see Donati,“The anonymous commentary on the Physics.”



42 rega wood

objects? Unlike the other works, MMet distinguishes between primary

and qualified consideration and between primary and secondary sen-

sible accidents. MMet tells us that the mathematician does not primarilyconsider sensible matter, but she does consider magnitude which con-

cerns sensible matter. She abstracts from primary sensible accidents, the

active and passive qualities that produce substantial change—namely,

heat, cold, wet, and dry—but does not abstract from magnitude as a

secondary quality.12

The Physics commentary tells us that the physicist defines things in

terms of sensible matter; the mathematician, in terms of intelligible

matter.13 Intelligible matter is a puzzling concept about which con-

temporary Aristotelians do not agree.14 According to Lear, Aristotleinvokes intelligible matter “to account for the fact that we are thinking

about a particular object.” Perceptible objects “have intelligible matter

insofar as they can be objects of thought rather than perception; that

is, it is the object one is thinking about that has intelligible matter.”

Albert the Great states a similar view, describing intelligible matter as

conceptual or imaginable quantity.15 What Rufus believes is not entirely

clear, but it seems he wants to define intelligible matter negatively, and

more generally, as unextended matter lacking position, and hence the

appropriate proper subject for unextended accidents.16

12 MMet 6.2: “Dicendum quod mathematicus abstrahit a materia sensibili. Hoc est,non sic considerat materiam sensibilem sicut naturalis, quia non primo considerat materiamsensibilem; considerat tamen magnitudinem quae aliquo modo concernit materiamsensibilem . . . Abstrahit ergo a materia sensibili primo et immediate ut a corpore calido,frigido, humido, et sicco, et ita de aliis qualitatibus sensibilibus; non tamen abstrahita magnitudine quae est sensibilis secundo” (Q290, f. 47vb). Cf. Aristotle, Metaphysics, 11.3. For a similar claim about magnitude as the ultimate subject of geometrical objects

see Jones “Intelligible Matter and Geometry in Aristotle.”13 In Phys. [ Wood ], 2.3.2, p. 121: “Dicendum quod materia dupliciter est: sensibiliset intelligibilis. Mathematicus tangit in definitione suorum accidentium materiam intel-ligibilem; naturalis, sensibilem materiam.” See also In Phys. P8, as quoted below.

14 Cf. Ross, Aristotle’s Metaphysics, II, pp. 199–200, Frede and Patzig, Aristoteles Metaphysik Z , II, pp. 195–196. Bostock in Aristotle, Metaphysics: Books Z and H , pp.156–157, 284–285. Cf. Aristotle, Metaph. 7.10.1036a2–12, 7.11.1036b32–1037a5;8.6.1045a33–36.

15 Lear, “Aristotle’s Philosophy of Mathematics,” p. 182. Albert the Great, Physica [Hossfeld] 1.1.1., Opera omnia 6.1: 2.

16 Rufus, DMet 8 (V4538, f. 65vb): “In aliquibus autem accidentibus, utpote in lineaet in genere generalissimo lineae, ut in quantitate, et in aliis generibus generalissimis,

est compositio ex essentia accidentis quae proprie dicitur esse et ex substantia materiaeprimae. Verbi gratia, linea dicit aggregationem ex essentia lineae et materia intelligibili.Unde linea dicit formam primo, et illa forma est illud quod addit linea super essentiammateriae primae situabilis”. Cf. Rufus, DMet 8 (V4538, f. 63vb): “Ad hoc dicendumquod sensibile unde sensibile situale est. Unde sensibile exigit materiam situalem, et




The claim that mathematicians do and physicists do not abstract from

sensible matter is the single commonest statement on the subject we

will encounter. The De anima commentary remarks that mathematiciansconsider sensible objects but not as such. And though mathematical

objects can be abstracted from natural matter, here a synonym for

sensible matter, they cannot be abstracted from intelligible matter. In

abstracting from sensible matter, the mathematician prescinds from

motion and change.17

DMet repeats the claim found in the Physics and De anima commentar-

ies that though mathematical objects can be abstracted from sensible

matter, they cannot be abstracted from intelligible matter. Unlike In

Phys., DMet suggests that the definitions of the physicist and the mathe-matician can be the same, though the physicist’s definition explicitly

references substance and the mathematician’s only implicitly.18

SOx tells us that natural objects differ from mathematical in that they

include mobility.19 In his Parisian theology lectures, Rufus states many

of the same distinctions listed above, though he expresses considerable

skepticism about their significance. Things which are transmutable are

properly defined in terms of sensible matter and cannot be abstracted

ideo sensibile non est intelligibile ultimum, eo quod intelligibile ultimum situale non est.Hoc praedicatum igitur ‘habere materiam’—dico, situalem—definit sensibile. Prius enimest habere materialem situalem quam esse sensibile, et quod habet materiam situalemest sensibile, et ideo bene dicit cum dicit quod substantia sensibilis habet materiam, etdebetis intelligere situalem; et per istam igitur propositionem innuit materiam esse”.

17 In DAn 1.1.Q1 (Q312, f. 19va): “Naturalia autem quamvis sint sensibilia et itade facili apprehensibilia a nobis, causatur tamen in eis incertitudo. . . . Mathematicaliaautem et sunt sensibilia (et ita facile apprehensibilia) et considerantur non ut cum motu etmateria naturali consistunt , et propterea accipiunt ut bene nata perficere intellectum.”

In DAn 3.3.E3 (M, f. 81vb): “Hoc habito, quia ex iam dictis posset credi mathematicaposse simpliciter abstrahi ab omni materia, incidenter subiungit quod sicut simum nonpotest abstrahi a materia naturali ut a naso, similiter rectum non potest separari a materiaintelligibili, ut a continuo, et similiter de aliis mathematicis passionibus.”

18 DMet 1 (Q290, f. 4rb): “Mathematicus enim qui considerat de curvo si abstrahata materia sensibili, non tamen a materia intelligibili.”

DMet 5 (Q290, f. 10vb): “Unde res mathematica quamvis abstrahatur a materiasensibili, non tamen a materia intelligibili.”

DMet 6 (Q290, f. 13rb): “Ad aliud dicendum quod definitione eadem contingerepotest quod mathematicus et physicus utantur, sed non ut eadem. In definitione cuius-libet accidentis accipitur substantia, vel explicite vel implicite. In definitione autemmathematica, etsi accipiatur substantia, ipse tamen non percipit substantiam esse in

illa definitione. Physicus autem percipit eam.”19 SOx 2.13 (B62, f. 132ra): “Lux non est corpus mathematicum, quia lux natura est et

addit super mathematicum, sicut linea radiosa super lineam mathematicam. Ergo siest corpus, est naturale. Sed omne corpus naturale est naturaliter mobile motu aliquoproprio naturali.”



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from sensible matter; mathematical objects cannot be abstracted from

intelligible matter.

Sent. Par. 2.3: These things are said, but I do not see how they are adequateto the inquiry. I, too, can distinguish [concerning] matter: sensible matterdiffers from intelligible matter . Sensible matter is assumed and appears indefinitions of transmutable things. Intelligible matter is also [assumed]for mathematical objects and they cannot be abstracted from this [intel-ligible] matter, as the Philosopher says.20

In general we are told that the objects studied by the mathematician can

be considered in abstraction from sensible matter, not so those studied

by the physicist. Mathematical objects can be considered in abstraction

from the external objects in which they are embodied, since they are

unconnected with motion and change. Since mathematical objects as

such are not transmutable, they can be defined and abstracted from

sensible matter, but they cannot be abstracted from intelligible matter.

Some texts refer to natural and others to sensible objects, and this

variation in terminology is found in the theological and metaphysical

works as well as in the other Aristotle commentaries. Most different

from the rest is MMet , in which there is no mention of that puzzling

concept, intelligible matter.How does mathematical differ from metaphysical abstraction? MMet

tells us that the metaphysician abstracts from change, and hence since

she considers objects “without sensible matter,” she studies their essences

without change.21 In Phys., too, reserves to the metaphysician the treat-

ment of things removed from motion and matter as such. By contrast

mathematical objects are not actually separate, but are only considered

separately. Though they are actually conjoined with matter, they are

not considered as such.22

20 SPar 2.3 (V12993, f. 143va): “Dicta sunt haec, sed non video qualiter ad remquaesitam satisfaciunt. Possum et ego distinguere materiam, nam alia est materia sensibilis,alia intelligibilis. Materia sensibilis accipitur et apparet in definitionibus rerum transmuta-bilium. Materia intelligibilis est etiam rerum mathematicarum et ab hac materia nonabstrahuntur ut dicit Philosophus.”

21 MMet 6.2 (Q290, f. 48vb): “Sed nota quod metaphysicus considerat <considerasE> primas substantias sine materia sensibili , . . . et considerat eas essentias <pos. motu sedtrp. E> sine motu.”

22 In Phys. [ Wood ], P8, pp. 90–91): “Res enim quaedam sunt secundum actum

exsistendi et modum considerandi separatae a motu et materia; quaedam secundumactum exsistendi et modum considerandi coniunctae cum motu et materia; quaedamsecundum actum exsistendi coniunctae, secundum modum considerandi separatae etabstractae. De primis est metaphysica tanquam de principali . . . De secundis rebus est




Neither In DAn nor DMet compare mathematician and metaphysi-

cian. However they do compare the physicist or naturalist with the

metaphysician. Specifically In De anima compares the treatment of thesoul in psychology with that of the metaphysician. The psychologist is

concerned with the soul as it actualizes a body; while the metaphysi-

cian treats the soul as an absolute or separated substance. DMet makes

the same claim on behalf of the metaphysician: His topic is not what

actualizes a natural body. Rather, he considers the soul in so far as it

is not in the body, the soul as spirit or intelligence.23

As mentioned earlier, Rufus’s take on mathematical abstraction is

quite similar to Jonathan Lear’s, particularly in a couple of precisions

found in In Phys. and In DAn, but at least apparently absent from DMet .Lear makes the point by saying that:

mathematical objects exist, but all this statement amounts to . . . is thatmathematical properties are truly instantiated in physical objects and,by applying a predicate filter, we can consider these objects as solelyinstantiating the appropriate properties.24

The filter in question eliminates “predicates which concern the material

composition of the object,” and allows the mathematician to consider

only “the geometrical properties of objects and to posit objects thatsatisfy these properties alone.” Medievals would have described Lear’s

filter, which he calls a “qua-operator,” in quite similar terms, using such

expressions as ‘qua’ and ‘inquantum’—exempli gratia, ‘a body as a body’,

or ‘a body as such’— reduplicatio, to use the medieval technical term.25

In the proem to In Phys. (paragraph 8), after telling us that

though mathematical objects are conjoined with physical bodies, the

naturalis philosophia. De tertiis rebus est mathematica, sicut de magnitudine et numero.Ista enim licet secundum actum exsistendi sunt materiae coniuncta, sunt tamen secundum modumconsiderandi abstracta; considerantur enim non inquantum huiusmodi.”

23 Rufus, In DAn. 2 (Q312, f. 22va): “Haec enim differentia incorporeum est ipsiusanimae secundum quod anima est in se absoluta substantia, scilicet secundum quodanima est de consideratione metaphysici. Sic autem non intendit hic de anima sedsecundum quod est natura sive actus corporis naturalis.”

Rufus, DMet 1 (V4538, f. 2rb): “Anima duplicem habet considerationem. Consideraturenim ipsa inquantum anima, et hoc est inquantum actus corporis naturalis, et sic etiamunita cum corpore. Consideratur autem alio modo inquantum spiritus vel intelligentia,et secundum istum modum non consideratur ipsa inquantum est in corpore. De ipsa

autem considerata secundum primum modum est scientia de anima. De ipsa autemconsiderata secundum modum ultimum fit perscrutatio in ista scientia.”24 Lear, “Aristotle’s Philosophy of Mathematics,” p. 170.25 Ibid. pp. 168–169, 175.



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mathematician considers them abstractly, Rufus adds a further precision.

He distinguishes between the negation of the mode and the mode of

negation. Rufus says that:

In Phys. P9: In abstracting, he does not consider quality or number insofaras they are not connected with natural passions, but he considers themnot insofar as they are such, as a reference to the negation of the mode andnot the mode of negation.26

To suppose that number, points or other mathematical objects were not

found in conjunction with natural passions would be to use the mode

of negation—falsely, in this case, since in fact we encounter embod-

ied spheres, for example. But to consider numbers as if they were notconjoined with natural passions would be to deny the [physical] mode,

in order properly to understand the properties of spheres apart from

bronze, iron or wood, for example.

Another way to put this is that the mathematician does not consider

mathematical objects in so far as they are connected with physical

objects, but equally does not deny that they are so connected. In the

following snippet from the De anima citation quoted more fully in a

previous note, this idea is expressed by the placement of the bolded

‘not’, which indicates a “negation of the mode.”

In DAn 1.1.Q1: Mathematical objects are both sensible and considerednot insofar as they are comprised with motion and natural matter, andaccordingly they are taken in such a way as is well designed to perfectthe intellect.27

DMet does not make this point in comparing the mathematician and

the physicist. However in comparing the physicist and the metaphy-

sician, it, too, makes the point by the placement of the ‘not’. Since

metaphysician employs the mode of negation, rather than the negationof the [ physical] mode, the negation appears before the verb rather

than before the description of the mode.

26 In Phys. [ Wood], P9, p. 92: “Non enim considerat abstrahens quantitatem velnumerum inquantum est non cum passionibus naturalibus, sed considerat non inquantumillius, ut tangatur negatio modi et non modum negationis.” The significance of the technicalterminology is a bit clearer in Scotus, where he uses it to state his understanding of thedistinction between contraries and contradictories. See John Duns Scotus, Quaestiones

super Praedicamenta Aristotelis [Andrews e.a.], Q. 39, p. 529.27 In DAn 1.1.Q1 (Q312, f. 19va): “Mathematicalia autem et sunt sensibilia . . . etconsiderantur non ut cum motu et materia naturali consistunt, et propterea accipiunturut bene nata perficere intellectum.”




DMet 1: The soul is considered insofar as it is soul . . . united with the body.In another mode, however, it is considered as spirit or intelligence andaccording to this mode it is not considered insofar as it is in the body.28

In sum, by contrast to metaphysical abstraction, mathematical abstrac-

tion is not about objects removed from the sensible world. Its objects

are sensible and perceptible, not separated from physical objects, Since,

however, they can be verbally and conceptually abstracted from physical

objects, they are inseparable only from unsituated intelligible matter.

But whether they are accidents inhering in intelligible or natural, situ-

ated matter, the mathematician considers them without reference to the

natural passions of change and corruption. The mathematician does

not employ the mode of consideration proper to the natural, changingworld, but neither does she posit independently existing, immaterial

mathematical objects.

2. Are Points Substances or Parts of Substances?

This brings us to our second question: Are points substances, parts of

substances, or accidents? The answer should be obvious, since points

do not exist apart from bodies. But alas the answer is far from obvious,since Aristotle defines a point as a substance with position.29 Ordinarily,

Rufus’s scrupulously refrains from citing this definition, though his

contemporaries do not do so.30 Where he finds it dif ficult to avoid,

he modifies it, speaking not of point as substance with position, but

of point being in substance in so far as it has position.31 One work,

however, cites Aristotle’s definition: DMet 5, chapter 15, an exposition

of the term ‘final’ ( fi nis ). Here Rufus’s dubs Aristotle’s definition the

material definition and pairs it with a definition with which he is more

comfortable, called the formal definition. As formally defined, point is

a position of substance—an accident.

28 DMet 1 (V4538, f. 2rb): “Consideratur enim [anima] ipsa inquantum anima, ethoc est inquantum . . . unita cum corpore. Consideratur autem alio modo inquantumspiritus vel intelligentia, et secundum istum modum non consideratur ipsa inquantumest in corpore.” Quoted more fully above.

29 Posterior Analytics, 1.27.87a36. But cf. De anima 1.4.409a6.30 Cf. exempli gratia Anonymous Erfurt I, In De anima (Q312, f. 55rb): “Probatio

minoris: punctus, eo quod est substantia posita, est aliquid in se.” Robert Grosseteste,Commentarius in Posteriorum analyticorum libros [ Rossi ], 1.18, p. 258.31 In DAn 1.4.Q2 (Q312, f. 20ra): “punctus autem est in substantia composita secun-

dum quod habet positionem.”



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DMet 5.15: We should say that a point is substance in position when point isdefined materially, but its formal definition is the position of substance.32

In chapter 1 on the term initial ( initium ), Rufus provides an unqualifieddefinition which is a variation of the formal definition: point is the first

disposition of matter when situated:

DMet 5.1: Hence a point is the fi rst disposition advening on matter when it issubject to site and it is like ( quasi ) the origin of a line.33

But though strictly speaking a point is an accident, an accident of mat-

ter, Rufus also wants to take seriously Aristotle’s claim that points are

substances, substances in position. Similarly, lines are defined materi-

ally as the substance of matter replicated infinitely often in one direc-tion, while formally lines are defined as the replicability of matter in

a single direction. If matter is broadly construed, a point is like ( quasi )

the origin of a line. This explains the sense in which a point is the

cause of a line.

DMet 5.1: Point is the cause of line in the genus of material cause, tak-ing matter broadly.34

In the unqualified definition of point as a quasi origin of the line, the

language of mathematical construction, in which points are said to

cause lines, is construed as a simile; a point is like the material cause.

And the sense in which points and lines are substances is as they pertain

to matter broadly construed. Like intelligible matter, matter broadly

construed is an obscure concept. We know from DMet 7 that matter

broadly construed is not subject to generation and corruption; it is an

element in the composition of incorruptible ideas.35 Conceivably, Rufus

is suggesting that we are in the realm of mental objects. If that sug-

gestion were correct, then the claim might be that in mental geometry,points function as substances and materially cause lines.

32 DMet 5.15 (Q290, f. 10vb): “Ad aliud quod obicitur dicendum quod punctus estsubstantia posita, materialiter definiendo punctum; formalis autem definitio est positiosubstantiae.”

33 DMet 5.1 (Q290, f. 9rb): “Unde punctus est prima dispositio adveniens materiaequando est sub situ et est quasi origo lineae.”

34 DMet 5 (V4538, f. 27va): “Ad aliud dicendum quod punctus est causa lineae in

genere causae materialis, communiter sumendo materiam.”35 DMet 7 (Q290, f. 20va): “Sed modo videtur quod haec consequentia quam sup-ponit Aristoteles—scilicet, quod si ideae sunt particulares, quod sunt corruptibiles—nonteneat, sicut bene verum est quod ideae habent materiam communiter dictam, sed nonmateriam quod est subiectum generationis.”




However, the context of the claim that a point is the cause of a line

shows that it is not a claim about the mental world, but a complicated

metaphysical account of the external world. Points are the dispositionsthat give matter position and fit it for extension and hence ultimately for

corporality. It is as the first disposition of matter in position that point

is like ( quasi ) the origin of a line. Similar accounts were offered by other

authors. Richard Fishacre, for example, one of Rufus’s contemporaries,

agrees that matter is disposed to receive the form of corporality by a

form that gives it the position of a point, punctualis position. Fishacre

would, however, reject Rufus’s definition of point as the first disposition

advening to matter, since he holds that unity is the first disposition.36

Seeing matter broadly construed as matter prior to its disposition bypoints is an account that will serve for DMet 7, as well as DMet 5, of

course. Prior to being disposed to assume position, matter is unextended

and not subject of generation and corruption. Notice that Rufus’s

claim that points are in some sense of the phrase ‘the material causes’

of lines does not commit him to the claim that points are constitutive

parts of lines. Indeed, he makes the statement that if matter is broadly

construed, points cause lines materially, shortly after denying that points

make up a line.

DMet 5.1: [ W ]e should say that a point is the cause of a line in the genusof material cause, taking matter broadly. And he proves this as follows.In its material description a line is nothing but the substance of mattersubject to position infinitely often replicated. For matter subject to posi-tion once generates a point; infinitely replicated, it generates a line. Aline, therefore, is the substance of matter situated infinitely often pointwise.Hence a line is generated from the substance of matter as it exists whensubject to infinitely many points.37

36 Richard Fishacre, Sent. [ Long, p. 9], 2.9: “Haec enim materia nunc sub formacorporali, si spoliaretur usque ad formam corporeitatis, et hac etiam adhuc forte esthabens positionem et situalis et secundum imaginationem punctalis. Sed [si ] adhucspolietur forma qua est situalis, et remanebit nondum omnino nudata. Habet enimadhuc formam primam, quae est tamquam unitas in formis, qua est una et sine posi-tione. Unitas enim est substantia sine positione.”

37 DMet 5.1 (Q290, f. 9rb): “Ad aliud dicendum quod punctus est causa lineae ingenere causae materialis, communiter sumendo materiam. Et hoc declarat sic: linea enimnihil aliud est secundum materialem descriptionem nisi substantia materiae infinitiesreplicata sub situ; ipsa enim semel exsistens sub situ punctum gignit; infinities replicata,

lineam gignit. Linea igitur est substantia materiae infinities situata punctualiter. Undelinea gignitur ex substantia materiae sub infinitis punctis exsistens. Unde punctus estprima dispositio adveniens materiae quando est sub situ et est quasi origo lineae. Exhoc potest dici, communiter loquendo, quod punctus est principium et causa lineaein genere causae materialis communiter acceptae.”



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When Rufus says that a line is the substance of matter situated point-

wise ( punctualiter ), presumably ‘pointwise’ explains how matter is situated

in a line. If matter were not disposed by points it could not composesomething with position. Since matter has no position prior to its

disposition by points, there is a sense in which a point is the origin of

any extended line.

Rufus’s mention of infinite replicability would cause problems if he

had suggested that points replicated infinitely often made up the line,

since that would imply that points were constitutive, integral parts.

Since, however, it is matter as situated pointwise that is replicated, Rufus

is committed neither to the claim that points are constitutive parts of

the line nor to the claim that they can touch. That favors the account,since indivisibles cannot touch or be continuous with each other, both

of which Aristotle defines as a relation of bodies with distinct limits

( Phys. 6.1.231a21–22). Here, Robert Grosseteste appears to be the

source of the discussion:

Physica 4: The replicability, therefore, of matter ad in fi nitum is the principleof sensible things. For things having extension and sensible magnitudeare not made from simple matter except by the infinite replication ofmatter over itself.38

Notice that Grosseteste seems to say, but Rufus does not say, that

simple matter is replicated, where simple matter presumably means

unextended matter.

This is all a bit puzzling, and it would be nice to be clearer. But at

least it should help us to understand how Rufus can in the same passage

both af firm that points are in some sense the material cause of lines and

also deny that points are constitutive parts of lines. The denial appears

just before the passage previously quoted, at the outset of the question

A similar statement is found in the chapter on ‘finis’. See DMet 5.15 (Q290, f. 10vb):“Ad aliud quod obicitur dicendum quod punctus est substantia posita, materialiterdefiniendo punctum; formalis autem definitio est positio substantiae. Et similiter formalisdefinitio unitatis est non-positio substantiae. Et intelligatur ly ‘non’ privative. Lineaautem secundum definitionem materialem est substantia materiae infinities situaliterreplicata secundum unam partem tantum. Definitio autem formalis est replicatio sivereplicabilitas situaliter infinita secundum unam partem tantum.”

38 Physica 3 [ Dales] p. 54: “Replicabilitas igitur materie in infinitum numerus est et

principium rerum sensibilium <add. ipsam replicacionem Dales>. De simplici namque<autem Dales> materia non fierent res habentes extensionem et magnitudinem sen-sibilem, nisi per materie infinitam super se replicacionem.” Corrections of the textcourtesy of Neil Lewis.




Rufus asks about Metaph. 5.1.1012b34–1013a20, which begins: “We call

an origin that part of a thing from which one would start first, e.g. a

line or a road has an origin in either of the contrary directions.” In thepassage that concerns Rufus, Aristotle distinguishes between origins that

are and are not immanent parts. As Rufus understands him, Aristotle

posits two partial descriptions: ‘parts that make things’ and ‘parts that

are in things’ and considers only two combinations: origins that make

something but are outside it and origins that make something and are

inside it. For the case of points in lines Rufus holds that we must allow

a third possibility. Points are intrinsic parts that do not make up the

things in which they are found:

DMet 5.1: We should say that ‘initial’ can be said in a third mode—namely,as that from which a thing is not made and it is in it . . . In another mannerwe should say that a point is the beginning ( initium ) of a line in the thirdmode, and a line is not made from points, but the substance of a linecomes from them—that is, it rests on them ( constat ex ) and properly it isnot made from them.39

Here, there is the puzzling af firmation that lines rest on points, but are

not made from them. I think we can make sense of the odd conjunction

as a consequence of Rufus’s controversial claim that the substance of aline arises from points in that they dispose matter for position, though

a line is not made from points.

Thus despite Rufus’s concession of an extended sense in which

points are material causes of a line, it seems to me that Rufus escapes

the dangers posed by Aristotle’s definition of points from the Posterior

analytics, as well as the trap posed by positing a line generated from

points. But the simile is common and alluring, given the indivisibil-

ist mathematics of the day. In the Physics commentary, too, Rufus is

tempted by it, suggesting that creation is like a line flowing from thepoint which is our extensionless creator. But here too, Rufus escapes

39 DMet 5.1 (Q290, f. 9rb; V4538, f. 27va): “Item, cum ‘initium’ dicitur dupliciter:uno modo ex quo primo fit res et est in ea, et alio modo dicitur initium ex quo fit reset non est in ea, quare non habemus tertium modum ex quo non fi t res et est in ea? . . . Adprimam dicendum quod ‘initium’ potest dici tertio modo, scilicet ex quo non fi t res et est

in ea. Et secundum hoc intelligit per illam litteram “ex quo fit res prius et non est inea.” [5.1.1013a7 Michael Scot tr.]. “Alio modo dicendum quod punctus est initiumlineae secundum tertium modum et non fi t linea ex punctis, sed substantia lineae ex his,id est constat ex his et non fit proprie ex his.”



52 rega wood

the problem, since the line flows rather than the point—in this case,

the extrinsic limit of the flowing line.40

Elsewhere in the Physics Rufus also rejects the claim that points areconstitutive quantitive parts. He is no atomist; his denial is, if anything,

more resolute than Aristotle’s. Commenting on Physics 6, he deals more

consistently than Aristotle with the claim that a sphere touches a line at

a point. Touching, for Rufus, is continuous motion or continuous varia-

tion, and hence what happens at a point can only be part of any touch.

He also tells us that rather than saying that contact is at indivisibile

points, we should say that contact is “according to indivisibles.”

In Phys. 6.1.2: [ P]artial contact will be at a point. Yet not just pointsbut lines touch. In this manner, the total contact that touches a line isnothing but a continuous variation by touching indivisibles. And just aswe speak of the total contact in this manner, so we should say that inthe contact what is touched in the whole contact is some continuum . . .,which is touched according to indivisibles. And just as the continuous variationby touching indivisibles is not touching indivisibles, . . . so it is true that a lineis not its points.41

Note how Rufus carefully reminds us that though there are infinitely

many points in a line, points not only cannot touch, but do not con-

stitute a line.

3. Are some Infinities of Indivisibles Greater than Others?

Specifying the manner in which something can be said precisely is

typical of Rufus. And it has a major (if not infinite) role to play in his

solution to our third question: Are some some sets of infinitely many

indivisibles greater than others?

40 In Phys. [ Wood], 8.1.4, pp. 217–218.41 In Phys. 6.1.2 [ Wood] pp. 189–190: “Videtur quod oporteat ponere lineam esse ex

punctis hac ratione . . . Hoc mobile tetigit hanc lineam totam sed solum tetigit puncta.Ergo tota haec linea est puncta . . . Intelligendum est sic, quod sicut dictum est et pro-batum, quilibet contactus partialis erit in puncto. Nec tamen tanguntur solummodopuncta sed linea. Hoc modo totalis contactus quod tangit lineam nihil est nisi continua

variatio per tangere indivisibilia. Et sicut dicimus hoc modo ex parte totalis contactus,ita debemus dicere ex parte contacti quod illud quod tangitur toto illo contactu estaliquid continuum . . . quod continue tangitur secundum puncta indivisibilia. Et sicut ipsa

continua variatio secundum tangere indivisibilia non est ipsa tangere indivisibilia, sicutnec motus est ipsa mutata sed variatio secundum illa, sic verum est quod linea non estsua puncta”. See also ibid. 6.2.7, p. 199: “Sed si [punctum] moveretur per se, . . . essenttunc puncta partes magnitudinis et continua—quod est impossibile.”




Robert Grosseteste presented, perhaps for the first time in the West,

the claim that infinities come in different sizes, arguing that some infini-

ties contained others. Grosseteste claimed, for example, both that therewere infinitely many points in a one cubit line and that there were twice

that number in a two cubit line. Indeed, he claimed that infinities were

related in every proportion, numerical and non numerical. Such mea-

surement and comparison is impossible to us, to be sure, but possible

to God for whom the infinite is finite as Augustine teaches.42

Rufus first quotes and responds to this position in a discussion of

Metaphysics 10.1.1053a18–24,43 where he is considering Aristotle’s claim

that we come to understand things by knowing their measure. Since

infinity destroys cognition, and continua are infinitely divisible, Rufusasks how we can know them. He quotes Grosseteste claiming that

though we cannot know the infinitely many points in a line, God can.

God, for whom the infinite is finite, can compare infinities and measure

the greater by the lesser infinite.

In response to Grosseteste, Rufus suggests an alternative approach:

the proper measure of a line is potentially quantified prime matter. In

support of this claim Rufus repeats the claim he made in DMet 5 that

a line is prime matter superimposed on by position infinitely often.

In a sense this is an endorsement of Grosseteste’s position, since it

is Grosseteste who holds that replicating indivisibles infinitely often

produces finite quantity. It differs from Grosseteste chiefl y in not claim-

ing that the matter so replicated is simple and in not invoking divine

omnipotence explicitly.44

42 Grossesteste, Commentarius in VIII libros Physicorum [ Dales], 4, pp. 91–93.43 DMet 10 (Q290, f. 32ra): “Ad hoc respondet quis quod ista linea cognoscitur per

numerum infinitorum punctorum in ipsa exsistentium. . . . apud ipsam [causam primam]est cognitio numeri punctorum infinitorum.”

44 God may be invoked implicitly, however. Rufus had established to his satisfactionthat matter, even prime matter, was intelligible. But though Rufus expressed no hesita-tion about our intellect’s capacity to grasp prime matter in Contra Averroem, a doubtis expressed in Speculum animae. On intelligibility, see Contra Averroem (Q312, f. 84vb):“De quaestione quarta et quinta quaerentibus de individuis et de substantia materiaeprimae, an sint simpliciter quantum est de se intelligibilia, an non, patet in tractatuillarum quod sunt intelligibilia, et in tractatu quintae et septimae quomodo sunt intel-ligibilia.” Regarding our intellects see CAv Ad 1 (Q312, f. 84rb): “omne ens et naturaessentialiter est intelligibile ab intellectu primo, similiter <simpliciter E> autem etquantum est de se ab intellectu causato.”

Compare Speculum animae 4 (Q312, f. 109va): “nam omnis <rep. E> creatura [est] vere <animae A > intelligibilis, et ab intellectu Primo et ab intellectu creato, nisi sitdefectus a parte nostri intellectus, propter quem scilicet non possit ipsa principia primaintelligere—quae principia quantum in se ipsis est <!> maxime intelligibilia sunt.”



54 rega wood

DMet 10: Prime matter is the origin of all quantities. Hence it is theorigin of every dimension of continuous quantity whatever. For whenposited under a single position, it makes a point, but when replicatedunder position in a single direction infinitely often, it makes a line. . . .Thus matter as a potentially quantified entity is the proper measure ofany quantity whatever; as a potentially quantified entity in one dimensionit is the measure of a line. Moreover so disposed it pertains to the samecategory as that which it measures.45

Here, it looks rather as if Rufus is trying to explain how something

dimensionless can measure dimensions. Prime matter is one and unex-

tended, lacking any determinate dimensions, but when it is disposed for

position, it is potentially extended and divisible. The suggestion seems

to be that when prime matter’s potential for quantity is actualized in

infinitely many different positions, this results in extension. This seems

a somewhat uncritical response to Grosseteste.

Fortunately, therefore, in his Oxford lectures Rufus has a second look.

This time he is both considerably more critical and at the same time

seemingly more committed to one of Grosseteste’s most controversial

views. Rufus explicitly endorses Grosseteste’s claim that the infinity

of points in a short line is less than the infinity of points in a longer

line.Let us start our examination of the Oxford discussion by looking at

the dilemma as Rufus presents it. It seems that the numbers succeeding

10 are fewer than the numbers succeeding 1. And nothing prevents

indeterminate quantities from being greater than or lesser than each

45 DMet 10 (Q290, f. 32ra): “Sed alio modo potest responderi quod materia prima estorigo omnium quantitatum <qualitatum E>. Unde ipsa est origo cuiuslibet dimensionis

quantitatis continuae. Ipsa enim sub uno situ posita producit punctum; ipsa autem infini-ties replicata sub situ in unam partem est causa lineae. Ipsa autem infinities replicata

secundum duas dimensiones producit superficiem. Ipsa autem secundum triplicemdimensionem producit corpus. Sic igitur materia [ut] ens in potentia quanta est men-sura propria cuiuslibet quantitatis, ut ipsum ens in potentia quanta secundum unamdimensionem est mensura <mensura lineae] mensurabile E> lineae; ipsa [autem] sicdisposita est eiusdem praedicamenti <puncti E> cum eo <eius E> cuius est mensura,et ita <illa E> mensura et mensuratum sunt unigenea.”

For a similar discussion see MMet 10.5 (Q290, f. 52ra): “Ad quod dicendum sineoppositione quod materia, ens in potentia quanta, est unum et minimum et mensurarerum in quantitate exsistentium, sed haec diversificatur per differentias reales. Proutenim est considerata sub situ sic est subiectum puncti; prout vero non sub situ, sic uni-

tatis. Haec eadem iterum materia, <add. non E> prout sub situ considerata, infinitiesreplicata secundum unam extensionem gignit lineam; secundum vero duas, superficiem;secundum vero tres, corpus. Et sic patet qualiter materia est subiectum puncti, unitatis,numeri, lineae, superficiei et corporis.”




other; one not precisely measurable pile can be greater than another.

So it seems that one infinity can be greater than another. Rufus asks:

SOx 1.2H: [ I ]n one line there are infinitely many lines, and in half ofit there are infinitely many lines, and is this infinity more than that?Again, numbers increase infinitely from 10; they also increase from one.Is this infinity greater than that? Is that infinite greater than this by theten added to it? Again, the account ( ratio ) of finite and infinite pertainsto quantity. Great, small, greater, lesser are indeterminate quantities, andnone of them conflicts with the account ( ratio ) of the infinite, thereforenothing prohibits one infinity’s being greater than another.46

Next come quotations of Grosseteste and Augustine supporting this

claim, followed by what looks like a rejection of these authors. Theydo not understand the account of infinity; they seem to have confused

the account of infinity with the account of all, as in God sees all things.

But to see all is to see a whole; so if God sees all, what he sees is finite

not infinite, since an infinite is not a whole but incomplete.

SOx 1.2H: These [thinkers] do not appear correctly to comprehend theaccount ( ratio ) of infinity. For “beyond which there is nothing” is not theaccount of infinity, but rather this is the account of that which is all;however, all and whole and perfect are the same [account,47 and] hence

‘all’ indicates the finite. Therefore, the foregoing seems rather to be theaccount of the finite than the infinite.48

Rufus then explains that ‘what contains everything within itself ’ is

not a correct description of the infinite, rather something is infinite

if whatever quantity we assign to it, there will be a further quantity.

To be beyond an infinite [series] is impossible, since such a series is

unending. And therefore, we expect Rufus to conclude that one infinity

cannot exceed another. Though he does not state it explicitly, Rufus is

46 SOx 1.2H (B62, f. 22rb): “Sed contra hoc videtur: in una linea sunt infinitaelineae, et in eius medietate sunt infinitae lineae. Et nonne haec infinita sunt plura illisinfinitis? Item, ascendat numerus in infinitum a denario, ascendat etiam et ab unitate.Nonne istud infinitum est maius illo? Nonne habet decem unitates additas illi? Item,quantitati congruit ratio finiti et infiniti. Magnum, parvum, maius, minus sunt quanti-tates et indeterminatae, ergo neutri illorum repugnat ratio infiniti, ergo nihil prohibetinfinitum esse maius infinito.”

47 Cf. Aristot., De Caelo 1.6.268a20.48

SOx 1.2H (B62, f. 22rb): “Non videntur isti recte accipere rationem infiniti. Nonest enim haec ratio infiniti ‘extra quod nihil’. Sed est haec potius ratio eius quod estomne; omne autem et totum et perfectum idem, quare ‘omne’ finitum dicit. Ergopraedicta potius est ratio finiti quam infiniti.”



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committed to this conclusion. He holds that exceeding or being outside

( extra ) the infinite is a contradiction in terms.

SOx 1.2H: Again, according to the Philosopher [ Phys. 3.6.207a7–8] whatcontains everything within itself is not the account of the infinite, butrather “something is infinite, if [whatever] we take as its quantity thereis always something further ( extra ).” It is pointless to say that ‘the infiniteis that outside (extra) which there is something’, for ‘outside’ finishes andterminates, hence it conflicts with ( repugnat ) the infinite, and the phrase‘beyond (extra) the in fi nite’ is a contradiction in terms.49

In this respect Rufus’s solution is preferable to Ockham’s, since Ockham,

at least when he is not expressing himself carefully, allows that one

infinity exceeds another.50 Harder to understand is the concession Rufus makes to Grosseteste:

one infinite can be greater than another, though one infinite cannot

exceed another,

SOx 1.2H: Again, the infinite is not a whole, the infinite is not a part,and yet an in fi nite is greater than an in fi nite.51

Rufus explains his concession by reference to a simile found in De anima.

The common sense as it judges sensible species from different senses

is described as a point using the termini of two paths: 3.2.427a10–15.

Rufus compares the common sense to a point at the center of a circle

that is numerically one by substance and subject, yet infinitely many

in being and account ( rationem ).

SOx 1.2H: Again, in a circle there is a point at the center, and it is numeri-cally one in its subject, yet in being and account ( ratio ) it is as multiple andas many as the lines terminated at it are many (Cf. DAn 3.2.427a9–14).Therefore the point itself is (as it were) in fi nitely many points, though only in

49 SOx 1.2H (B62, f. 22rb): “Item, non est haec ratio infiniti quod continet in seomne, sed haec est ratio infiniti, secundum Philosophum: ‘Infinitum est cuius quanti-tatem accipientibus semper est aliquid extra sumere’. Nec est aliquid dictu ‘infinitumest extra quod est aliquid’; nam ‘extra’ finit et terminat. Unde opponitur infinito, etest oppositio in adiecto ‘extra infinitum’.”

50 William of Ockham, Quaestiones variae 1 [ Etzkorn & Kelley], OTh VIII, p. 80:“Tamen revolutiones lunae sunt plures infinitates quam revolutiones solis. Et ideo positahypothesis debet concedi quod in fi nitum est maius in fi nito et exceditur ab in fi nito.” More cau-tiously in Quodlibet 2.5, Ockham claims “tot sunt ista et adhuc sunt multa.” However,

he still does not absolutely deny that one infinity can exceed another, just that suchan excess can be determinate ( certo numero ).51 SOx 1.2H (B62, f. 22rb): “Item, infinitum non est totum, infinitum non est pars

(cf. Physica, 3.6.207a26–28), et tamen infinitum est maius infinito.”




its being and account, and yet by substance and subject it is single (unicus). Andtherefore it is not true that a single point would contain as many pointsas a maximal line or as many as the world machine.52

Rufus tells us that points can be numbered either by substance or

account. There are not as many points at the center of the circle as

there are in a maximal line. But he does not tell us whether this is

because there is only one substantial point at the center of a circle,

or because the being or account ( ratio ) of the point at the center of

the circle can be counted in fewer ways than the points in a maximal

line. Going from a circle to a straight line, Rufus again concludes that

since single substantial points can be many in being, there are more in

longer than in shorter lines.

SOx 1.2H: Points can be numbered in two ways, as is already evident,namely either by substance and subject or by being . At the extremities of asingle line there are two points, two in reality ( rem ) and subject, but in theline itself any one point is one by number and subject , [ yet] twofold by being andaccount (rationem), in that it is the beginning of one line and the end ofanother. In this manner (modum), there are fewer points in the shorter lineand more in the longer, yet there are infinitely many in both.53

Since this conclusion, and in particular the phrase “in this manner,” ishard to understand, we should look at Rufus’s De anima commentary

for further clarification. Of the six early De anima commentaries I know

(and not many more survive) it is the only one which explains Aristotle’s

example in terms of the distinction between the substance of points

and their account ( ratio ) and being, the distinction that is key to Rufus’s

Oxford explanation of how one infinity can be greater than another.54

52

SOx 1.2H (B62, f.22rb): “Item, in circulo est punctus qui est centrum, et est unicusnumero secundum subiectum, tam multiplex tamen sive tam multi secundum esse etrationem quam multae sunt lineae ad ipsum terminatae. Est igitur ipse punctus quasiinfiniti puncti, sed solum secundum esse et rationem, tamen substantia et subiectounicus est. Et ideo non est verum quod unicus punctus contineat tot punctos quot etmaxima linea sive quot et mundi machina.”

53 SOx 1.2H (B62, f.22rb): “Dupliciter enim est numerare punctos, ut iam patet,scilicet secundum substantiam et subiectum, aut secundum esse. In extremitatibusunius lineae duo puncti sunt, duo secundum rem et subiectum, in ipsa vero linea quivisunus punctus unus est numero et subiecto, duplex secundum esse et rationem, eo quodprincipium est unius lineae et finis alterius. Secundum hunc modum sunt in breviorelinea pauciores puncti et in longiore plures, infiniti tamen in utraque.”

54

Others consulted use somewhat different terminology, sometimes not naming thesense in which the point is many. See Anonymous, Sententia super II et III De anima [Bazán],2.26, pp. 343–347: essence and being (hereafter, Anonymous Bodley). Anonymous ErfurtII, In Dan 2, Q312.65va; subject. Anonymous, Lectura in librum de anima a quodam discipulo



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Aristotle, and Rufus following him, employ the analogy to explain how

the common sense can simultaneously be aware of sensibles received

by two distinct senses—yellow and sweet, for example.

In DAn 2.12.E5: But the common sense is not in [ just] any matter what-ever one and diverse by account ( rationem ), but it is one and indivisibleas it is in itself one and indivisible; and because it is indivisible in thismanner, it is one discerning [subject] at the same time. And just as thesame point as a whole [and ] in itself pertains in account ( ratione ) to diversethings in so far as it is the terminus of diverse lines meeting ( concurrentium ),so similarly the common sense, since it is one indivisible by substance, as awhole [and] in itself pertains in account to diverse things in so far as it is theterminus of two diverse paths leading from the particular senses. And in

this manner its being does not pertain to one indivisible by account, sincethe soul uses one indivisible by substance twice at the same time, in sofar as it is the terminus of two paths leading from some two senses. Andso in this way it uses the same terminus according to substance as two, namely[directed] toward two paths of two senses, [and] it judges the diverse sensiblesof two senses. In so far as it is one by substance it judges these at oneand the same time.55

Controversially, Rufus claims that because the common sense is indivis-

ible and unextended, it can simultaneously receive two distinct sensibles,

unlike an extended body which cannot at the same time be white andblack.56 Here we may have switched from the common sense to the

sense of vision, but that will not matter for Rufus’ claim: though a

reportata [Gauthier], 2.25, p. 419: substance and apprehension. Adam Buck field, In DAn2, [ Powell ], p. 190: subject; grateful thanks to Miss Powell for permission to cite Adam.Ps. Buck field, In DAn 2, Merton College 272.17va, subject or substance.

55 In DAn 2.12.E5 (M3314, ff. 78vb–79ra): “Consequenter solvit hanc rationemdicens: Sed sensus communis non est quocumquemodo unum et secundum rationem

diversum, sed est unum et indivisibile sicut punctus <penitus M> in se est unus etindivisibilis; et quia sic est indivisibile, est unum discernens et in eodem tempore. Etsicut idem punctus est secundum se totum in ratione diversorum, prout est terminusdiversarum linearum concurrentium, similiter sensus communis cum sit unum indivisi-bile secundum substantiam, est secundum se totum in ratione diversorum secundumquod est terminus diversarum duarum viarum ductarum a sensibus particularibus. Etsecundum hoc esse eius non est in ratione unius indivisibilis, quia anima utitur unoindivisibili secundum substantiam in eodem tempore bis, secundum quod est terminusduarum viarum a duobus aliquibus sensibus ductarum. Et secundum quod sic utiturtermino eodem secundum substantiam tamquam duobus, scilicet ad duas vias duorumsensuum, sic duorum sensuum sensibilia diversa iudicat. Inquantum autem est unum secundumsubstantiam, sic uno eodemque tempore haec iudicat .”

56

Anonymous Bodley, In DAn 2.26, takes this to be Aristotle expounding an opinionwith which he disagrees. According to Anonymous Bodley the common sense is entirelyindivisible, and it is impossible that it should be divisible in accordance with the diversesensibles it apprehends (ed. Bazán, pp. 343–347).




body cannot be both white and black at the same time (and in the

same respect), a point can. Because unextended entities as a whole

can pertain in account to diverse things, we can simultaneously senseblack and white.

In DAn 2.12.Q2: And by his saying that the common sense is one anddiverse, as is a point (3.2.427a9–12), we should understand that he solvesthe aforesaid doubt suf ficiently. For in this way it is evident that the com-mon sense and a body are not similar. For if it were possible that somebody as whole in itself should have diverse accounts at once, so that itcould by one account receive whiteness and according to another accountblackness, it would be possible for the same body to be white and blackat once. But now that is not possible, but the common sense can be so,

pertaining in account to diverse things at once by itself as a whole. . . .57

When Rufus asserts that because our senses are unextended as points

are, contrary qualities can simultaneously be predicated of them, he

cannot be referring to something conceptual. When he speaks about

numbering points according to their capacity for different accounts,

what is differently numbered will pertain to the external world. In the

case of two spheres colored white and black, for example, the part of

their touch that occurs at a point is at a point that is simultaneously

the limit of something white and the limit of something black, butuncolored because unextended.

Just how this solves the problem of how one infinity can be greater

than, without being able to exceed another, is still not clear, however.

Here is the first of three alternatives: Perhaps Rufus is suggesting that

there will be more points that both begin and end a one inch line in a

four inch than in a three inch line, more connecting points about which

we can claim that they are the beginning and end of one inch lines.

57 In DAn 2.12.Q2 (M3314, f. 79ra): “Et intelligendum quod per hoc quod dixitsensum communem esse unum <idem M> et diversum sicut punctus [3.2.427a9–12],suf ficienter dissolvit dubitationem praedictam. Quia <Et M> per hoc patet quod nonest simile de sensu <sensi M> communi et de corpore; si enim esset possibile aliquodcorpus secundum se totum simul se habere secundum diversas rationes, ita quod possitsecundum unam rationem recipere albedinem et secundum aliam rationem nigredinem,esset possibile idem corpus simul esse album et nigrum. Nunc autem non est illud pos-sibile, sed sensus communis <om. M> potest esse sic in ratione diversorum secundum setotum simul. Intelligatur enim locus quidam in corpore in quo est situm instrumentumipsius sensus communis a quo instrumento exeunt et extenduntur venae sive viae diversae

ad organa singulorum sensuum. Et in isto <add. sed exp. ig M> organo in quo concurrunt huiusmodi viae <corr. (s. lin.) ex: venae M> radicatur sensus communis, qui manensunus et idem secundum substantiam secundum quod ad ipsius organum terminatur

via ducta ab organo visus, secundum se totum immutatur et recipit visibile.”



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That is to say in a two inch line, there is only one point that connects

two one inch lines. In a three inch line, there will be potentially infinitely

many such points, since every point between the one inch and the twoinch mark will do so. In a four inch line, there will be even more such

points, since there will be infinitely many such points between the two

inch mark and the three inch mark, as well as between the one inch

mark and the two inch mark.

If this is correct, Rufus’s notion of numbering by being or account

will result in claims for the differential greatness of various infinities

that are parasitic on the different finite quantities associated with them,

their included parts. Nonetheless, it is the kind of thing that might

make a difference. So far we have looked at intensive infinities thatarise in an unending process of division. Now consider an extensive

infinity, produced by a process of addition or multiplication. The

example proposed by Nicholas Denyer is an infinitely long prison sen-

tence. Given the choice of two such sentences, each of which is for an

unlimited number of years, we prefer a sentence of one day a year in

hell rather than a sentence of 364 days a year,58 since any finite part

of that sentence includes more days. Neither sentence extends beyond

the other, however. Indeed, since the two sentences can be put in one-

to-one correspondence, they have the same cardinality and hence are

equal. But since the one is a proper subset of the other, there is also a

sense in which one is greater than the other.59 The first interpretation

credits Rufus with insight into this second sense of greatness. In the

case that concerns him, neither line can be further divided than the

other, but there will be more of given size line lengths in the longer

than the shorter line, and hence the number of points corresponding to

the connecting points in the shorter line will be a subset of the number

of connecting points in the longer line.A second interpretation is possible, and it seems more likely. Rufus

may not have anything in mind like aliquot parts that divide a line

without remainder, since he does not mention such parts. In fact both

in his Oxford lectures and in his De anima commentary, he speaks of

points that are twofold in account ( rationem ). This suggests that he is

counting and comparing two infinities of points with actual common

end points, and perhaps in comparing one to another, superposition

58 Sorabji, Time Creation and the Continuum, p. 218.59 Maddy, “Proper Classes”, p. 114.




plays an important role. Perhaps Rufus is thinking of comparing two

unequal lines, each of which is composed of infinitely many points: call

them AC (shorter) and AD (longer).60

Intermediate points in both lineswill be both the end of lines beginning at the left and the beginning

of lines ending at the right (or vice versa). If, however, the shorter is

superimposed on the longer, intermediate points in the longer line (AD)

overlapped by the shorter line (AC) will begin both lines that end where

AC ends and lines that end where AD ends. Since those points can be

counted as the termini of more lines, there will be more accounts of

points in AD. Since we can count them as connecting lines to more

actual end points, points in AD will be greater in account than points

in AC. And the same will hold for any superimposed or superimpos-able infinities. I suppose that Rufus considers only lines with actual end

points, because he refers to the intermediate points as the beginning

and end of only two lines.

As in the previous interpretation, the sense in which one infinite is

greater than another depends on their finite relationship. However the

sense in which one is greater than the other will differ; and there is

no sense in which the relationship could be considered proportional,

which is an advantage of this interpretation.

A third interpretation suggests that by saying that the point is one by

substance and subject but multiple by being and account, Rufus may

be making a distinction like that between extension and intension.61

The infinity of points in a longer line does not exceed the infinity of

points in a shorter line extensionally, but it might be considered greater

intensionally, in virtue of the way the infinities are conceived or re-

presented, by using them as endpoints of line segments or measuring

them in some other way.

Against this suggestion there are considerations related to medievaland modern philosophical usage: Rufus has at hand medieval technical

60 I owe much of this interpretation to class discussion and particularly to JoshuaSnyder. He and Shawn Burns wrote a paper based on this interpretation of Rufus,“Rufus on the Comparison of Infinities,” which may subsequently appear in print. Theexample of line segments comes from Josh. A similar suggestion was made by GiorgioPini: given a shorter (AC) and a longer line (AD), point C at the end of the shorterline will be one in account, since it serves only as the end of AC. But the point cor-responding to C in AD will be two in account in AD, since it can be counted both as

the end of AC and the beginning of CD. Thus there will be at least one more pointin account in AD than in AC.61 I owe this suggestion to Gary Ebbs and much of the discussion to Timothy

O’Connor and Krista Lawlor.



62 rega wood

terminology to describe such distinctions and does not use it here.

On a variety of topics, Rufus distinguishes between signification and

supposition or appellation, where signification corresponds roughly tosense, and appellation to extension or reference. Similarly, Rufus speaks

of formal predication when he wants to distinguish different natures in

the same real subject—for example, intellect and will are both identified

as the rational soul, but not with each other, so will and intellect differ

in formal predication or definition. Rufus makes reference to neither

distinction here.

Nor does Rufus’s discussion of ways of counting provide the kind of

explanation normally provided by the intension-extension distinction

today. Moderns often use the distinction between intension and exten-sion when they want to explain changes in reference despite a fixed

intension. Such variation is explained in terms of different concepts.

For example, the description ‘animal with heart’ and ‘animal with

kidney’ are coextensive—that is they pick out the same creatures; the

intensions of the two descriptions differ, however. Or for a medieval

Aristotelian, ‘featherless biped’ and ‘rational animal’ are conceptually

distinct descriptions with the same supposition or appellation. What

Rufus seems to want to explain, however, are not conceptual differ-

ences, but differences in the external world. So for example, I think he

wants to explain the different ways the point at the center of the world

can be numbered if two lines are drawn from it to the circumference

rather than three.

In favor of the suggestion is that Rufus seems to think that what

varies is how we count things. So, for example, he seems to want to

explain how the common sense is one when we only see an object,

but two when we both see and touch it. The common sense counts as

two if it is receiving sensible species from two senses simultaneously.Unlike many modern usages, it is not our knowing more or less that

seems salient here. Rather, the point serves more or fewer functions in

counting or measuring.

Will this explain in what sense the point at the center of a circle

counts as less than all the points in a maximal line, though infinitely

many lines can meet at any point? A point at the center of a circle is

potentially infinitely many and so are the points in a maximal line. There

are in fact more points in a line than at the center of a circle, if only

because there is more than one point in any line, more in subject andsubstance. But, surely, Rufus wants us to understand a sense in which

taken together the points in the line are more in account. So perhaps




he is saying that more things will actually be measured by reference to

the points in a line than to a point at the center of the circle. Again

what will determine how great a point is is not the potentially infinitenumber of lines that could meet at it, but the lines we actually mea-

sure in reference to it. The “greatness” of a point in this sense will

depend on how often it is used in measurement. Whether this should

be described as intensional I must leave to others more versed in the

modern literature. If we describe as intensional what we use in mea-

surement, perhaps it does.

We do not yet, and we may never, have enough evidence to decide be-

tween these three interpretations, though I am inclined to think that

the second is most likely. In any case, however, the emphasis on theclaim that one incomplete infinity cannot exceed another suggests that

the sense of greater-than Rufus wants to admit will involve counting

and comparing the completed parts of infinities, as do each of the

suggested interpretations. It looks as if Rufus wants us to compare

the parts that correspond to each other in different infinities to get an

acceptable sense in which one infinity can be greater than another. In

the first interpretation, it will take more cuts to reach points at any given

distance from each other in a longer than a shorter line. And similarly,

points in AD will count as beginning more lines than points in AC, and

points that appear in three measurements will be greater than those

in two. Rufus’s emphasis on actual cuts or intersections suggests that

he is deliberately looking for a finite number of diverse accounts for a

point and avoiding the incomprehensibility medievals believe would be

involved in considering infinitely many accounts of a point. Hence the

example of the common sense that presumably counts as three when

it judges that honey is yellow, sweet, and sticky as the terminus of the

senses of touch, taste, and sight, but only as two if we are blind-folded.Here having come to a sticky point, we should stop.

4. Conclusion

What have we seen? We have seen that Rufus is sure that points as

mathematical entities are in the world, and equally certain that points

are not quantitative parts. Both are aspects of agreement with Aristotle

but on which Rufus’ exposition shows some progress in the sense of

increased clarity. We have also seen that Rufus does not get beyond

certain problems in Aristotle; his discussion of intelligible matter seems



64 rega wood

equally obscure. On the other hand, it may have been useful to try to

explain how points can be origins or parts of a line. Also an attempt

has been made to provide a consistent account of the troublesomedefinition of point provided by the Posterior Analytics.

The influence of mathematics, with its indivisibilist assumptions, was

as unhelpful to Rufus as to Aristotle.62 But Rufus has been more careful

not to commit himself to problematic assertions based on mathematics.

Spheres touch only partly at a point; not points but lines flow ( In Phys. ).

Lines are composed not of points but of matter arranged pointwise;

points are only quasi origins of the line ( DMet ). These may not be

entirely felicitous solutions, but at least they reflect a consistent aware-

ness of the commitments of the philosophical position.Finally, possibly on account of his Christian commitments, Rufus has

a new problem to deal with: how to find an acceptable sense in which

to af firm that though one infinity does not extend beyond another,

nonetheless one infinity can be greater than another. Whether it had

much influence, we cannot tell at this point. As John Murdoch has

pointed out, however, Olivi may have been aware of a similar posi-

tion.63 But no doubt Grosseteste exercised more influence than Rufus.

Still, Rufus provides an interesting and credible response and a useful

starting point for debate. And this is true for his answers to all three

of our questions.

62

See Michael White and Wilbur Knorr, as cited by White, “Aristotle on the Non-Supervenience of Local Motion,” pp. 154–155.63 Murdoch, “The ‘Equality’ of Infinites in the Middle Ages,” pp. 171–174; Peter

of John Olivi, Quaestiones in II Sent. [ Jansen], pp. 30–40.



RICHARD KILVINGTON ON CONTINUITY*

Elżbieta Jung Robert Podkoński

Although with his solutions to the problem of the possible existence ofindivisibilia Richard Kilvington seems to fit into the main stream of thefourteenth-century considerations, which leaned toward the refutation

of atomism, he attacks and solves the problem in an original manner.In this paper we will focus on two of Kilvington’s questions, respectivelyfrom his De generatione et corruptione and Sentences commentaries, where hepresents geometric proofs for the infinite divisibility of a continuum.Richard Kilvington’s commentary on De generatione et corruptione is a setof ten fully developed questions.1 They were written around 1324–1325

* We want to express our gratitude to Chris Schabel for his help with English.1 All ten questions are contained in Mss: Bruges 503, ff. 20vb–50vb (along withKilvington’s questions on the Ethics and the Sentences ); Erfurt SB Amploniana O–74,ff. 35ra–86va, Sevilla Bibl. Columbina 7.7.13, ff. 9ra–29rb, Paris BNF lat. 6559, ff.61ra–119va. Part of the set is to be found in Cambridge Peterhouse 195, ff. 60r–69rand Krakow Bibl. Jagiellonska, cod. 648, ff. 40ra–53rb. In Ms. Vat. lat. 4353, the manu-script, where Maier (cf. Maier, Ausgehende Mittelalers, pp. 253–54) ‘found’ Kilvington’scommentary on the Physics, there are 40 lines of the fourth question listed below.

This is a list of questions contained in Ms. Paris BNF lat. 6559: 1. ff. 61ra–65rb: Utrum augmentatio sit motus ad quantitatem (q. 3 in Mss. Krakow BJ.

648 and Bruges 503, Erfurt SB Ampl. O–74). 2. ff. 65va–68va: Utrum numerus elementorum sit aequalis numero qualitatum primarum (q.

7 in Mss. Bruges 503, Erfurt SB Ampl. O–74).3. ff. 68va–71rb: Utrum ex omnibus duobus elementis possit tertium generari (q. 9 in Mss.Bruges 503, Erfurt SB Ampl. O–74).

4. ff. 71rb–88rb: Utrum in omni generatione tria principia requirantur (q. 10 in Mss. Bruges503, Erfurt SB Ampl. O–74; part of the question contained in Vat. lat 4353, f. 125r).

5. ff. 89ra–97vb: Utrum continuum sit divisibile in in fi nitum (q. 2 in Mss. Krakow BJ 648,Bruges 503, Erfurt SB Ampl. O–74).

6. ff. 97vb–101va: Utrum omnis actio sit ratione contrarietatis (q. 5 in Mss. Krakow BJ 648,Bruges 503, Erfurt SB Ampl. O–74).

7. ff. 101va–105vb: Utrum omnia elementa sint adinvicem transmutabilia (q. 7 in Mss. Bruges503, Erfurt SB Ampl. O–74).

8. ff. 105vb–112vb: Utrum mixtio sit miscibilium alteratorum unio (q. 6 in Mss. Bruges 503,

Erfurt SB Ampl. O–74). 9. ff. 112vb–119va: Utrum omnia contraria sint activa et passiva adinvicem (q. 4 in Mss.Krakow BJ 648 and Bruges 503, Erfurt SB Ampl. O–74).

10. ff. 131ra–132vb: Utrum generatio sit transmutatio distincta ab alteratione (q. 1 in Mss.Krakow BJ 648 and Bruges 503, Erfurt SB Ampl. O–74).



66 elbieta jung & robert podkoski

and were well known to his contemporaries.2 Kilvington’s last work,Quaestiones super libros Sententiarum, is a set of eight fully developed questions

and four subordinate problems, which are also composed as questions.3

The most probable date for Kilvington’s Sentences lectures is 1332–1334.The problem of the division of a continuum was debated by Kilvingtonat length in the question Utrum continuum sit divisibile in in fi nitum (the fifthquestion in our list) and was later recapitulated in the Sentences, in thequestion Utrum unum in fi nitum sit maius alio.4

In Kilvington’s time, the most popular geometrical proofs againstatomism were those of John Duns Scotus, which were later adoptedby William of Ockham and then developed by Thomas Bradwardine.5

John Duns Scotus presents two anti-atomistic geometrical arguments.He begins the first one with a construction of two concentric circles on

2 The question on reaction (the ninth in our list) inspired Heytesbury and gave him animpulse to debate the problem (on the discussion on the issue see Caroti, “Da WalterBurley al Tractatus sex inconvenientium: la tradizione inglese della discussione medievale Dereactione,” pp. 279–331). Maier suggests that Wodeham’s references are a report of RichardFitzRalph’s arguments against Kilvington’s theory of infinity (on FitzRalph’s polemic withKilvington, cf. Maier, Die Vorläufer Galileis im 14. Jahrhundert , pp. 208–211; Courtenay,Schools and Scholars in Fourteenth-Century England , pp. 76–78; K. Walsh, A Fourteenth-CenturyScholar and Primate: Richard FitzRalph in Oxford, Avignon and Armagh, pp. 19–20).

3 The work, whole or in parts, is contained in the following manuscripts: Bruges503, ff. 79vb–105rb; Bruges 188, ff. 1–56; Bologna, Archiginnasio A–985, ff. 1a–52a;BAV, Vat. lat. 4353, ff. 1–60; Florence, Bibl. Naz., Magliabecchi II. II 281, ff. 43–50,Paris, BNF, lat. 15561, ff. 198–228; Paris, BNF, lat. 17841, f. 1r–v, Erfurt, CA 2o 105,ff. 134–81; Prague, Univ., III. B. 10, ff. 191–227; Tortosa, Cat. 186, ff. 35r–66r. Fordetailed information on secondary literature cf. Kretzmann, The Sophismata of Richard Kilvington, p. XXVI, n. 35.

The titles of the questions are as follows:1. Utrum Deus sit super omnia diligendus;2. Utrum per omnia meritoria augeatur habitus caritatis quo Deus est super omnia diligendus;

a. Utrum aliquis possit augmentare peccatum alteri;3. Utrum omnis creatura sit suae naturae cum certis limitatibus circumscripta; a. Utrum aliquod corpus possit simul et semel esse in diversis locis; b. Utrum unum in fi nitum sit maius alio;4. Utrum quilibet actus voluntatis per se malus sit per se aliquid;5. Utrum peccans solum per instans mereatur puniri per in fi nita instantia interpellata; a. Utrum voluntas eliciens actum voluntatis pro aliquo instanti debeat ipsum actum per aliquod

tempus necessario tenere;6. Utrum aliquis nisi forte in poenam peccati possit esse perplexus in hiis quae pertinent ad

salutem;7. Utrum omne factum secundum conscientiam ab aliquo sitt meritorium;8. Utrum peccatum veniale aggravet mortale mortaliter .4 For detailed information about Kilvington’s works and secondary literature see

Jung-Palczewska, “Works by Richard Kilvington,” pp. 184–225.5 Murdoch, “Thomas Bradwardine: Mathematics and Continuity in the Fourteenth

Century,” pp. 104–110; Podkoński, “Al-Ghazali’s Metaphysics as a Source of Anti-Atomistic Proofs in John Duns Scotus Sentences Commentary,” pp. 614–618.



richard kilvington on continuity 67

the basis of the third postulate of book I of Euclid’s Elements.6 Then,assuming that the circumferences of the circles are composed of points,

Scotus indicates two points of the circumference of the greater circlethat are immediately adjacent to one another. Next, he draws a linefrom each of those points to the centre of the circles, again invokingthe appropriate postulate from the Elements.7 Further, he posits a ques-tion whether these lines intersect the circumference of the smallercircle at one or at two points. If one accepts the latter answer, onemust conclude that there are as many points on the circumference ofthe smaller circle as on the circumference of the greater one, which isobviously absurd. When one agrees, however, that the supposed two

radii intersect the circumference of the smaller circle at the same point,then let us draw—Scotus argues—a tangent to the smaller circle fromthis very point. One of Euclid’s postulates assures us that the tangent isperpendicular to each of the radii.8 Consequently, we obtain two rightangles that are unequal, which is also absurd (fig. 1).

William of Ockham, who generally neglected rationes mathematicae inhis philosophical inquiry, brings up a simplified version of Scotus’s argu-ment in one of his Quaestiones Quodlibetales entitled Utrum linea componatur

ex punctis (Whether a line is composed of points).9 If we draw all of

6 John Duns Scotus, Ordinatio, [ Balic], Liber Secundus, dist. II, pars II, quaestio 5,pp. 278–350: Utrum angelus possit moveri de loco ad locum motu continuo. Actually, Scotusreferred here to the second postulate and it might have been numbered so in the copyof Elements he had at hand, but in modern editions of Euclid the postulate he invoked isthe third one (cf. Ibid ., note T3). Cf. also Euclid, Elements [trans. Heath], vol. 1, p. 199.

7 John Duns Scotus, Utrum angelus. . . ., op. cit., p. 292: “Super centrum quodlibet,quantumlibet occupando spatium contingit circulum designare, secundum illam peti-tionem 2 I Euclidis. Super igitur centrum aliquod datum, quod dicatur a, describanturduo circuli: minor, qui dicatur d,— et maior b. Si circumferentia maioris componiturex punctis, duo puncta sibi immediata signentur, quae sint b c,—et ducatur linea rectaab a ad b et linea recta ab a ad c, secundum illam petitionem I Euclidis ‘a puncto inpunctum lineam rectam ducere’ etc. Istae rectae lineae, sic ductae, transibunt recteper circumferentiam minoris circuli. Quaero ergo aut secabunt eam in eodem puncto,aut in alio?”

8 John Duns Scotus, ibid ., pp. 292–293: “Si autem duae rectae lineae ab et ac secentminorem circumferentiam in eodem puncto (sit ille d), super lineam ab erigatur linearecta secans eam in puncto d, quae sit de,—quae sit etiam contingens respectu minoriscirculi, ex 17 III Euclidis. Ista, ex 13 I Euclidis, cum linea ab constituit duos angulosrectos vel aequales duobus rectis,—ex eadem etiam 13, cum linea ac (quae poniturrecta) constituet de angulos duos rectos vel aequales duobus rectis; igitur angulus ade etetiam angulus bde valent duos rectos,—pari ratione angulus ade et angulus cde, valentduos rectos. Sed quicumque duo anguli recti sunt aequales quibuscumque duobus rectis,ex 3 petitione I Euclidis; igitur dempto communi (scilicet ade), residua erunt aequalia:igitur angulus bde erit aequalis angulo cde, et ita pars erit aequalis toti!”

9 William of Ockham, Quodlibeta Septem [ Wey], Quodlibet I, Quaestio 9, pp. 50–65,Utrum linea componatur ex punctis.




the radii from the centre of any two concentric circles to each of theindivisibles that constitute the circumference of the outer circle—saysOckham—these radii should intersect both circles in the same numberof constituent indivisible points. Therefore any two such circles mustbe equal in circumference, which is obviously false (fig. 2).10

10

William of Ockham, ibid., pp. 54–55; Cf. also: Podkoński, “Al-Ghazali’s Metaphysicsas a Source of Anti-Atomistic Proofs in John Duns Scotus Sentences Commentary,”pp. 614–615.

Fig. 1

an indivisibleline

a tangent to the smaller circle

e

D

B

a

d

right

anglesb c

Fig. 2

Indivisible lines




The second of Scotus’s anti-atomistic geometrical arguments concernsthe incommensurability of the diagonal and the side of a square.Generally, the construction employed in the following part of the proofis similar to the one presented in Roger Bacon’s Opus Maius, written in1266 or 1267. We find here the following reasoning:

[ If ] the world is composed of an infinite number of material particlescalled atoms, as Democritus and Leucippus maintained . . . the diagonalof the square . . . and its side would be commensurable . . . For if the sidehas ten atoms, or twelve or more, then let the same number of lines bedrawn from those atoms to the same number in the opposite side, the sidesof the square being equal; . . . therefore since the diagonal passes throughthose lines, and no more can be drawn in the square, the diagonal mustreceive a single atom from each line, and thus they have an aliquot part asa common measure, and the side has just as many parts as the diagonal,

both of which conclusions are impossible.11

(fi

g. 3).In the beginning of his proof the Subtle Doctor invokes again thepostulates from the Elements that define the notion of geometrical com-mensurability.12 First, he considers the case when parallel lines intersect

11

Roger Bacon, Opus Maius [trans. Burke], p. 173.12 John Duns Scotus, Ordinatio, [ Balic], Liber Secundus, dist. II, pars II, quaestio5, pp. 297–298.

Fig. 3

Indivisible lines




the diagonal of a square at every point of its length. Next, he posits ahypothesis that there are points on the diagonal that do not belong to

any of the parallel lines. If we accept this—Scotus argues—then let usdraw a line from any of these points that is parallel to the nearest ofthe parallel lines assumed before. This new line necessarily crosses theside between the points that constitute it. Consequently, one can indicatea point that lies between two immediately adjoining points. This wayone arrives at a contradiction, therefore one must deny the hypothesisthat any continuum is composed of immediately conjoined indivisibleentities (fig. 4).13 William of Ockham, in the above-mentioned question,limits himself to a short presentation of Bacon’s argument:

[ If a line were composed of points]—Ockham argues—then a side of asquare would be equal to its diagonal and the diagonal would be com-mensurable with the side. The consequence is obvious, because one candraw a line from any point of one side [of a square] up to any point ofthe opposite side . . . and each of these lines contains a certain point ofthe diagonal. Consequently, there exists a line between any point of the

13 John Duns Scotus, ibid., p. 298: “Secunda probatio est ex 5 sive ex 9 X Euclidis.Dicit enim illa 5 quod ‘omnium quantitatum commensurabilium proportio est adinvicem sicut alicuius numeri ad aliquem numerum’, et per consequens—sicut vult9—‘si lineae aliquae sint commensurabiles, quadrata illarum se habebunt ad invicemsicut aliquis numerus quadratus ad aliquem numerum quadratum’; quadratum autemdiametri non se habet ad quadratum costae sicut numerus aliquis quadratus ad aliquemnumerum quadratum; igitur nec linea illa, quae erat diametri quadrati, commensurabiliserit costae illius quadrati. Minor huius patet ex paenultima I, quia quadratum diametriest duplum ad quadratum costae, pro eo quod est aequale quadratis duarum costarum;nullus autem numerus quadratus est duplus ad alium numerum quadratum, sicut patetdiscurrendo per omnes quadratos, ex quibuscumque radicibus in se ductis. Ex hoc patet

ista conclusio, quod diameter est assymeter costae, id est incommensurabilis. Si autemlineae istae componerentur ex punctis, non essent incommensurabiles (se haberent enimpuncta unius ad puncta alterius in aliqua proportione numerali); nec solum sequereturquod essent commensurabiles lineae, sed etiam quod essent aequales,—quod est planecontra sensum. Accipiantur duo puncta immediata in costa, et alia duo opposita inalia costa,—et ab istis et ab illis ducantur duae lineae rectae, aequidistantes ipsi basi.Istae secabunt diametrum. Quaero ergo aut in punctis immediatis, aut mediatis. Siin immediatis, ergo non plura [sunt] puncta in diametro quam in costa; ergo non estdiameter maior costa. Si in punctis mediatis, accipio punctum medium inter illa duopuncta mediata diametri (illud cadit extra utramque lineam, ex datis). Ab illo punctoduco aequidistantem utrique lineae (ex 31 I); ista aequidistans ducatur in continuumet directum (ex secunda parte primae petitionis I): secabit costam, et in neutro puncto

eius dato, sed inter utrumque (alioquin concurreret cum alia, cum qua ponitur aequi-distans,—quod est contra definitionem aequidistantis, quae est ultima definitio positain I). Igitur inter illa duo puncta, quae ponebantur immediata in costa, est punctusmedius.”




diagonal and any point of the side. Therefore, if there were six pointscomposing the diagonal, there would necessarily be six [points] in eachof the sides.14

Kilvington, however, does not mention any of these constructions inhis works. Among twelve principal arguments from his question Utrum

continuum sit divisibile in in fi nitum, which appeal to the problem of infinitedivisibility from mathematical, physical, and metaphysical points ofview, one finds three strictly geometrical proofs. In these argumentsKilvington deals with the following examples: an angle of tangency(angulus contingentiae ), “a cone of shadow” and a spiral line. For thisreason, in order to expose his theory of continuity we will presenthis detailed deliberations on these three study cases. Since Kilvington

goes back to his geometrical proofs only once in his later work (in hisquestions on the Sentences ), we will also attempt to answer the questionabout the consistency of his theory.

14 William of Ockham, Quodlibeta Septem [ Wey], Quodlibet I, Quaestio 9, p. 51:“Si sic, tunc costa esset aequalis dyametro et esset diameter commensurabilis costae.Consequentia patet quia a quolibet puncto dyametri ad costam contingit protraherelineam rectam. Quod patet quia a quolibet puncto costae unius ad quodlibet punc-

tum costae alterius contingit protrahere lineam rectam, immo ita esset de facto positahypothesi; et quaelibet talis linea protrahitur per aliquod punctum dyametri; igitur aquolibet puncto dyametri ad costam est aliqua linea recta. Si igitur sint sex puncta indyametro, erunt necessario sex in utraque costa.”

Fig. 4

Immediate points in a side of a square

a “mediate” point in a side of a square

a “mediate”point in a

diagonal of asquare




1. An Angle of Tangency

Kilvington begins all his analyses of the concept of continuity with theclaim that every continuum, especially a mathematical one, is infinitelydivisible. In the eleventh principal argument he claims that one of thegeometrical “entities” introduced by Euclid himself does not fit thisAristotelian concept of continuity. The case considered is an angle oftangency. In fact, there is only one definition within Euclid’s work thatis devoted to this “species” of angles, i.e. proposition 16, Book III. Itreads:

The straight line drawn at right angles to the diameter of a circle fromits extremity will fall outside the circle, and into the space between thestraight line and the circumference another straight line cannot be inter-posed; further the angle of the semicircle is greater, and the remainingangle less, than any acute rectilinear angle.15

Consequently Euclid, as it seems, introduces an infinitely small math-ematical being that seems to be an indivisible.

While referring to the above definition, Kilvington does not cite itverbatim, but gives his own interpretation, saying that “an angle of

tangency is not divisible by a straight line but by a circular one.”16

First, Kilvington tries to convince the reader that Euclid is wrong, andin order to do so he sets up the following geometrical construction.Let’s take two contiguous circles, one of which is two times greaterthan the other, and a line that is contiguous to both of them in thepoint of contact. When the line is rotated around the point it will firstcross this bigger circle without crossing the smaller one. This will be sobecause—argues Kilvington—all the points of the bigger circumferenceare closer to the line than the points of the smaller one.17 Therefore

15 Euclid, Elements [trans. Heath], vol. 2, p. 37.16 Richard Kilvington, Utrum continuum sit divisibile in in fi nitum, Erfurt SB Ampl. O–74,

f. 41vb: “Si quaestio foret vera, igitur angulus contingentiae foret divisibilis in infinitum.Consequens falsum et contra conclusionem 14 III Euclidis, ubi dicitur quod anguluscontingentiae non est divisibilis secundum lineam rectam sed circularem.”

17 Richard Kilvington, ibid., f. 41vb: “Probo, quod sit divisibilis secundum lineamrectam, quia capio duos circulos quorum unus sit duplus ad alium et contingant se. Etcapiatur linea contingens illos circulos in eodem puncto in quo circuli se contingant, etpono quod illa linea quiescat secundum illum punctum et moveatur secundum aliud

extremum. Quo posito, arguo sic: si illa linea secet utrumque circulorum praedictorum,et propinquior est maiori circulo secundum omnia sua puncta quam minori, igiturcitius secabit maiorem circulum quam minorem. Quo concesso arguo sic: haec lineasecat circulum maiorem et non minorem, igitur illa dividit angulum contingentiaecontentum a circulo minori.”




we will come up with a rectilinear angle that seems to be smaller thanan angle of tangency (fig. 5).

Kilvington’s own response to this argument reveals that he is abso-lutely aware that the conclusion is false. He simply says that the line

will cross both circles, the bigger and the smaller, simultaneously.18 Although Kilvington does not offer any explanation, it seems that heknows that every straight line that crosses the point of tangency andforms an angle with a tangent is a secant (fig. 6.). Consequently, norectilinear acute angle is smaller than an angle of tangency. This is,most likely, why he concludes that an angle of tangency can be dividedonly by circular lines.19

Next, Kilvington observes that both an angle of tangency and arectilinear angle are infinitely divisible. Evidently, Kilvington’s statementstems from the following reasoning: since both angles are infinitely divis-ible, one can recognize them as infinite sets of infinitely small parts thatconstitute the angles; and in accordance with Aristotle’s opinion it wascommonly accepted that all infinities are equal, so in this sense an angle

18 Richard Kilvington, ibid., f. 43rb–va: “Et dico, quod acceptis duabus lineis cir-

cularibus, quarum maior contineat intrinsecus minorem, et capiatur linea in punctocontactus et moveatur secundum aliud extremum, dico quod non prius secabit illalinea circulum maiorem quam minorem.”

19 Richard Kilvington, ibid.: “Ad undecimum principale, quod angulus [contingentiae]est divisibilis secundum lineas circulares et non rectas.”

Fig. 5

α




of tangency and a right angle are equal; therefore, one would say thatthere is a proportion between them, namely an equal proportion.20 But in his answer Kilvington states that there is no proportion between

any rectilinear angle and an angle of tangency, because: “this angleis an infinite part of a right angle, like a point is an infinite part of a[ finite] line”.21 Again, Kilvington leaves the readers with the dif ficultyof reconstructing the possible reasoning that underlies his conclusion.The above statement is in accordance with Campanus of Novara’sclaim that “any rectilinear angle is greater than an infinite number of

angles of contingency.”22 And in this case, as it seems, Kilvington deniesthe existence of any proportion between the multitudes of rectilinearacute angles and of curvilinear angles in order to avoid any methodof constructing the acute angle that would be equal to the angle oftangency. If we accepted that there are as many “parts” of the rightangle as “parts” of an angle of tangency, we might recognize somekind of correspondence between rectilinear acute angles and curvilinearones. And consequently, the “most acute” of these angles would equalthe “smallest” of the curvilinear angles—only when it is agreed that

the “smallest” angles exist. But Kilvington is of the opinion that any

20 Richard Kilvington, ibid., f. 41vb: “Item, si sic, cum angulus rectus sit divisibilis ininfinitum secundum lineas rectas proportionaliter, igitur aliquis angulus rectus haberetproportionem ad angulum contingentiae, et ita angulus rectus et angulus contingentiaeforent aequales.”

21 Richard Kilvington, ibid.: “Item, si angulus contingentiae foret divisibilis secun-dum lineam circularem, igitur angulus contingentiae haberet aliquam proportionemad angulum rectum. Consequens falsum, quia iste angulus est infinita pars anguli recti,

sicut punctus infinita pars lineae.”22 John Campanus of Novara, [in:] Murdoch, “The Medieval Language ofProportions: Elements of the Interaction with Greek Foundations and the Developmentof New Mathematical Techniques,” p. 243.

Fig. 6

α




continuum is infinitely divisible, and consequently there is no last and“smallest” part of it.23

Kilvington accepts Ockham’s definition of a continuum, accordingto which every continuum is regarded as a set containing in actu infinitesubsets of smaller and smaller proportional parts.24 In the case consid-ered a right angle is such a continuum that is divisible into smaller andsmaller acute rectilinear angles. At the same time, each of them is biggerthan an angle of tangency, and an angle of tangency is also infinitelydivisible into proportional parts. The fact that both angles should beconsidered as an infinite set and subset allows us to state that one isbigger or lesser than the other, but it does not allow us to establish any

proportionality between them.25 This claim is also in accordance withOckham’s concept of relations between actual infinities that must beunequal.26 Evidently, Kilvington recognizes an angle of tangency asan actually infinitely small geometrical “entity,” which can be divided,however, in in fi nitum.27

23 Richard Kilvington, Utrum continuum . . ., Ms. Paris, BNF lat. 6559, f. 95ra: “Adrationem, quando quaeritur et cetera, dico, quod continuum est duplex sicut quan-

tum, quia aliquod est continuum per se et aliquod per accidens. Continuum per seest illud, quod per se est quantum et illud est tale, quod habet divisiones quae suntaccidentia sua. Alio modo sumitur continuum per accidens, et sicut dicimus quodalbedo est continuum. Divisio etiam accipitur dupla. Uno modo pro illo, cuius partespossunt actualiter dividi sive separari per divisionem. Alio modo pro eo, quod habetpartes quae possunt separari ab invicem sive per divisionem sive non, et sicut dicimusquod coelum est continuum et non divisibile quia partes eius non possunt ab invicemseparari. Sed primo modo accipiendo continuum et hoc modo continuum qualiter-cumque intelligendo quaestionem universaliter quaestio est falsa, accipiendo secundomodo quaestio est vera.”

24 For a detailed explanation of Ockham’s conception of infinity, see Goddu, The Physics of William of Ockham, pp. 159–176. Kilvington’s considerations on infinity are

presented in Podkoń

ski, “Thomas Bradwardine’s Critique of Falsigraphus’s Concept ofActual Infinity,” pp. 147–153.25 Richard Kilvington, Utrum continuum . . ., Erfurt, SB Ampl. O–74, f. 41vb: “Item,

si sic, cum angulus rectus sit divisibilis in infinitum secundum lineas rectas proportion-aliter, igitur aliquis angulus rectus haberet proportionem ad angulum contingentiae,et ita angulus rectus et angulus contingentiae forent aequales. Consequentia probatur.Sequitur, quod nullus angulus sit maior angulo recto, et per consequens sequitur, quodangulus rectus sit infinitus, cum habeat infinitas partes proportionales quarum nullapars unius est pars alterius, et quaelibet est maior angulo contingentiae.”

26 William of Ockham, Comm. Sent. II, q. 8, as quoted in Murdoch, “William ofOckham and the Logic of Infinity and Continuity,” p. 170: “concedo quod infinitaessent excessa, sicut probat ratio, et quod unum infinitum esset maius alio.”

27

Ricardus Kilvington, Utrum continuum . . ., Erfurt, SB Ampl. O–74, ff. 42rb–43va:“Ad undecimum principale, quod angulus est divisibilis secundum lineas circulares etnon rectas. Et nego consequentiam: igitur angulus contingentiae habet certam propor-tionem cum angulo acuto vel recto. Et dico, quod acceptis duabus lineis circularibus,




2. “A Cone of Shadow”

At the beginning of the ninth principal argument Kilvington says thataccording to Plato there is a kind of figure that is indivisible.28 As amatter of fact, in his Timaeus Plato claims that all beings are made upof solids and solids are made up of indivisible triangles that form theirsides.29 First, Kilvington states that there is a triangle whose apex iscontingent to its base.30 In order to prove his assumption, Kilvingtonpresents the following mental experiment. Let us imagine that a lucidbody A illuminates an opaque, circular, flat body B that is smaller thanA. In effect, we obtain a cone of shadow—C. Then, let’s presume that

while A increases in size, external parts of B are continuously becom-ing transparent. The process of transmutation of B occurs accordingto its proportional parts, i.e in the first period of time one half of allradii of B diminishes, and in the next period of time one half of theremaining parts of radii diminishes, and so on (fig. 7).31

Then Kilvington simplifies the case taking into account only one ofthe vertical sections of the cone C: a triangle formed by one of thediameters of the remaining part of B and two rays of light tangent toit. Thus the height of this triangle is continuously decreasing until B iswholly transparent. Now, it is easier to consider the case as a series ofsmaller and smaller triangles (fig. 8). Eventually, Kilvington observes that,

quarum maior contineat intrinsecus minorem, si accipiatur linea contingens utrumquecirculi et quiescat in puncto contactus et moveatur secundum aliud extremum. Dicoquod non prius secabit illa linea circulum maiorem quam minorem.”

28 Richard Kilvington, ibid., f. 41ra: “Aliqua superficies est indivisibilis sicut Plato ponit.”

29 See Plato, Timaeus, 53c.30

Richard Kilvington, Utrum continuum . . ., Erfurt, SB Ampl. O–74, f. 41ra: “aliquisest triangulus cuius unus punctus est immediatus eius basi, et omnia puncta eius sunt

immediata.”31 Richard Kilvington, ibid., f. 41 vb: “Igitur radii incidentes a partibus circumferen-

tialibus ipsius a per partes circumferentiales ipsius b concurrunt. Consequentia patet,quia illae lineae non sunt aeque distantes, et cum illae non concurrant in b nec citrab, igitur concurrunt ultra b. Capio igitur piramidem ex partibus illarum linearum quaesunt ultra, qua piramidis sit c. Tunc c est obumbrata quia nullae lineae incidunt ab a ad aliquod punctum intrinsecum ipsius c. Tunc sic: si b continue corrumpetur in corpusdyaphanum, et prius secundum partes circumferentiales quam centrales, et a continuequiescat non auctum nec transmutatum, igitur partibiliter illuminatur. Consequentiapatet, quia lineae incidentes ab a per partes b erunt infra lineas incidentes ab a per

puncta extrinseca b, igitur erunt breviores illis. Et secabunt se in aliquo puncto c, etcum in quolibet instanti post initium in quo incipit b corrumpi erunt lineae incidentesab a per partes b; igitur in quolibet instanti postquam b inceperit transmutare eritplus illuminatum de c.”




if the above-described process terminates, one of these triangles mustbe the last and as such it is the smallest and indivisible. It is obviousthat the smallest triangle consists of three points—the apex and twoextremes of the basis—that are immediately adjacent to one another.32

It is clear that this assumption is based on the concept of discontinuitystemming from Aristotle’s Physics, that any two immediately adjacentpoints are contingent and do not constitute a unity.33

In the next paragraph, Kilvington asks whether it is possible for coneof shadow C to disappear totally when there remains some parts of B.

32 Richard Kilvington, ibid., f. 41rb: “Capio tunc aliquod dyametrum partis non

corruptae de b, tunc dyameter cum partibus linearum incidentium per extrema illiusdyametri causant unum triangulum, qui non est divisibilis, quia tunc esset aliqua parsmedii ultra b non illuminata, quod est contra positum.”

33 Cf. Aristotle, Physics, VI, 1, 231a–b.

Fig. 7

Fig. 8

B

r

r/2

r/4

B

C

A




He considers the case in a manner that is typical of his other works,establishing a kind of adequacy between proportional parts of a certain

period of time, namely one hour, and succeeding stages of transmuta-tion of A and of B. Again, he simplifies the case examining separatelythe growth of A, B remaining unchanged, and the diminishing of B,when A does not change its dimensions.34

In his answer Kilvington states that the final result is the same in bothinstances: there must always be a cone of shadow left. Consequently,when A and B are changing simultaneously, as long as B exists there isa cone of shadow, no matter how small it is.35 Eventually, Kilvingtonconcludes that the whole case is founded on a false assumption, because

no surface is indivisible. He does not present, however, any explanationreferring to his—above-described—mental experiment. Most likely hedoes so because it is obvious for him and his audience that, althoughthere will finally be a cone of shadow formed by immediately conjoinedpoints, it does not mean that its section could be recognized as an

34

Richard Kilvington, Utrum continuum . . ., Erfurt, SB Ampl. O–74, f. 41rb: “Et probo,quod totum c illuminabitur antequam b corrumpatur, quia b corrupto et a non auctoin eodem <tempore> corrumpetur b et illuminabitur c. Igitur a aucto et corruptob, prius illuminabitur c quam b corrumpatur. Antecedens de se patet, et probaturconsequentia, quia ex corruptione b tantum illuminabitur de c in uno tempore sicutin alio sibi aequali, et ex augmentatione similiter a, igitur tantum est illuminatumde c per augmentationem a et corruptionem b in uno tempore sicut in tempore sibiaequali. Ponatur igitur, quod b corrumpatur in hora. Tunc sic: in medietatae horaeerit plus quam medietas c illuminata per augmentationem a, igitur corrumpitur b et uniformiter illuminabitur c post medietatem horae sicut in prima parte. Igitur inminori tempore quam sit medietas horae illuminabitur pars residua de c. Consequentiapatet. Et antecedens probo, quia si non, tunc maioretur. Et tunc in medietatae horae

esset medietas c illuminata, et sic plus erit illuminata propter maiorationem a, igiturin eodem tempore erit plus quam medietas illuminata. Antecedens patet, quia si a non augmentetur tunc illuminabitur totum c in hora et c uniformiter illuminabitur;igitur in medietate temporis erit medietas illuminata, igitur erit illuminata proptermaiorationem a. Patet, quia si in principio corrumpatur medietas c illuminata sivetanta pars per quantam corruptionem illuminabitur c et deinde maioretur a, tunc permaiorationem a erit aliquid illuminatum de c et tantum erit illuminatum. Si augereturquando b corrumpetur, igitur et cetera.”

35 Richard Kilvington, ibid., f. 43rb: “Et dico, quod isto casu posito, quod non priusterminabitur c quam corrumpatur b. Et nego istam consequentiam: a non aucto et bcorrupto, c illuminabitur in hora, igitur a aucto et b corrupto illuminabitur c ante finemhorae. Et causa est quia augmentatio ipsius a non facit c citius illuminari secundum

se totum, quam illuminabitur post corruptionem b. Sed quod ante finem horae plusilluminetur per a, si augeretur, et tamen non sequitur, quod c citius illuminabitur peraugmentationem a et corruptionem b, quam per corruptionem b tantum.”




extended surface. In the other part of this question Kilvington statesexplicitly that two immediate lines do not enclose a surface between

them.36

3. A Spiral Line

The third purely geometrical case, Kilvington debates, is based on thefollowing statement: each infinite line, as continuous, is infinitely divis-ible. This claim stands against the commonly accepted opinion that aninfinite line cannot exist, because, if it did, each of its parts would be

infi

nite and thus a part would equal the whole. First, Kilvington pres-ents a procedure for creating an infinite line. Let’s take a column—saysKilvington—and let’s mark out all its proportional parts: a sequence ofhalves of its height. And then we draw a spiral line, winding aroundthis column, starting from a point on a circumference of its base, insuch a way that each succeeding coil embraces one proportional part,i.e. the first coil the first half of a column, the second one fourth, thethird one eighth, and so on in in fi nitum (fig. 9). It is obvious that eachcoil is longer than a circumference of the column, and there are infi-

nitely many coils forming one line. This line is evidently continuousbecause an ending point of one coil is the beginning of the next coil.Consequently, the line is actually infinitely long, for it can be regardedas a sum of infinitely many parts, each of them possessing a certainlongitude.37

36 Richard Kilvington, ibid., f. 38ra–b: “aliqua est linea quae ducitur a punctoassignato ad duo puncta immediata, vel non sed duae. Non secundo modo quia

forent immediatae. Sequitur quod duae lineae immediatae claudunt aliquam superfi

-ciem—quod est impossibile.”37 Richard Kilvington, ibid., f. 39rb–va: “Si quaestio est vera, tunc linea infinita foret

divisibilis in infinitum. Consequentia patet, quia continuum est divisibile in infinitumet linea est continua, igitur et cetera. Minor patet, scilicet quod aliqua linea sit infinita,quia sit aliquod corpus columpnare a, tunc in a est aliqua linea infinita. Quod probosic: quia capio aliquam lineam gyrativam ductam super primam partem proportionalemipsius a, quae sit b; b ergo est quantitas continua cuius convenit addere maiorem partemin infinitum quarum nulla est pars alterius nec econverso, igitur convenit devenire adaliquam lineam infinitam actu. Quod probatur: consequentia prima per Aristotelem IIIPhysicorum et Commentatorem, commento 64, ubi probant additionem in continuoper partes aequales in infinitum, et arguant sic: si talis additio sit possibilis, convenit

devenire ad aliquam aliam magnitudinem infinitam in actu. Sic arguo in propositio.Et primum antecedens probo, quia convenit addere ipsi b lineam girativam secun-dae partis proportionalis, tertiae et quartae et sic in infinitum, quarum quaelibet est




Next Kilvington notes that it is impossible to construct an infiniteline in the described way, because a column circumvolved by a spiralline is of a finite height. Consequently the spiral line must have twoextremes, i.e. its beginning and end points, and as such it must befinite. Thus, Kilvington attacks the problem of labeling an end pointof a spiral line. In the first step he takes, he proves that a spiral linehas to be immediately adjacent to the upper surface of the column,because if it were not, there would be some proportional parts of thecolumn not circumvolved by the spiral line. Consequently, the spiralline would be finite, because it would consist only of a finite number ofcoils. But if a spiral line is immediately adjacent to the upper surface,it should be possible to label its end point. And if there were an endpoint the line would be finite, which is against the main propositionof this argument.38

continua alteri, igitur convenit sibi addere lineas infinitas aequales, quarum quaelibet

est maior c, et quarum nulla pars unius est pars alterius. Consequentia patet, quiacuiuslibet partis proportionalis linea girativa est maior linea circulari secundum quamattenduntur girationes corporis.”

38 Richard Kilvington, ibid., f. 39va: “Item, hoc probo per rationem quod impossibile

Fig. 9

CE

B

A

D

h / 2

h / 4

h / 8




Kilvington proves that although there is “no distance” between the spiralline and the upper surface of the column, there is also no determined

point ending the line. In fact, he observes, one can consider any of thepoints on the circumference of the upper surface of the column as anend point of the spiral line. And if it is so, it has an infinite number ofend points.39 Therefore, there is no end point, and the line is infinite andimmediate to the circumference of the upper surface of the column.

In his answer Kilvington af firms the last conclusion and says thatthe spiral line has two limits. One of them is intrinsic—and this is thestarting point of the line. The other, however, is an extrinsic limit—andthis is the circumference of the upper surface of the column, which does

not belong to this line. The only possible explanation is that Kilvingtonconsiders the spiral line to approach this circular line asymptotically(fig. 10).

Kilvington repeats the above-presented construction of an infinitespiral line in his question Utrum unum inifnitum potest esse maius alio fromhis commentary on Peter Lombard’s Sentences. It serves as one of manyobvious examples of actual infinities that are to be found or are pos-sible in the created world. The difference is that Kilvington shows amethod of constructing the spiral line that is infinite with respect toboth of its extremes. He draws two spiral lines, both starting from thesame point in the middle of the height of a column and going into

est aliquam lineam esse infinitam. Quia pono tunc quod talis foret in corpore colump-nari, et sit illa b. Capio igitur lineam circularem in extremo a corporis circularis, versusquod sit progressio partium proportionalium—qua sit c, tunc si b sit linea infinita estimmediata c extremo. Quia si b et c distarent aliqua esset proportionalis pars inter b

et c —et sic hoc solum componeretur ex partibusfi

nitarum partium proportionaliumet sic b foret finita—quod est contra positum.”39 Richard Kilvington, ibid., f. 39va–b: “Sed probo quod b et c non sunt immediata,

quia b terminabitur ad aliquod punctum ipsius c et non est maior ratio quare magisad unum quam ad aliud; igitur b terminatur ad infinita puncta. Huic dicitur quod b terminatur ad punctum tantum, qui terminat lineam rectam a quam incepit. Verbigratia, posito quod prima giratio incepisset in d puncto de lineae, tunc necessario ter-minabitur ad punctum e terminum eiusdem lineae. Sed contra, probo per rationem,quod terminabitur ad quodlibet punctum, quia giratio primae partis proportionalis secatomnem lineam rectam in superficie illius corporis extrema protensam ab uno extremoin aliud. Capio igitur aliquam linearum—et sit illa linea a, igitur et ponatur quod seceteam in hoc puncto fa lineae. Tunc in hoc puncto incipit una giratio quae terminabitur

ad eandem lineam et punctus girans non recedet a fa linea nisi iterum accedat adipsam. Igitur, si ille punctus omnino girat a corpus per modum puncti, sequitur quodin fine sit d punctus in f puncto fa, igitur b linea terminabitur ad f punctum, et peridem potest probari quod terminaretur ad quolibet punctum.”




opposite directions toward its upper surface and base.40 The laconicmanner of presenting this construction suggests that Kilvington pre-sumes that this argumentation is familiar to his audience. As it wasalready proven in the question Utrum continuum, both halves of the linewould lack an end point, and consequently the whole line would be

infinite utroque extremo.

4. Conclusions

It is clear at the outset, that in his question on the continuum Kilvingtondoes not take part in a discussion on indivisibilism. Although Kilvington,like John Duns Scotus, William of Ockham and Thomas Bradwardine,employs geometry, he is instead only interested in revealing paradoxes

resulting from the Aristotelian defi

nition of continuity. If we take forgranted that a continuum is infinitely divided into proportional parts,we have to accept that there is no last proportional part. Consequently,the process of dividing cannot be completed and lacks its limit.

40 Ricardus Kilvington, Utrum unum in fi nitum potest esse maius alio, Ms. BAV, Vat. lat.

4353, f. 40r–v: “linea sit infinita utroque extremo, ut patet de linea gyrativa in corporecolumpnari quae per utriusque suae medietatae girat singulas partes proportionalesversus extrema illius corporis. Et quod talis sit infinita patet, quia additio fuit sibi perpartes aequales in infinitum, igitur ibi est infinitum tale et in actu.”

Fig. 102π 4π 6π 8π 10π

x

h




All three of the above geometrical examples show that theArchimedean principle of continuity is not valid here. This principle,

repeated later on by Euclid, states that:Two unequal magnitudes being set out, if from the greater there besubtracted a magnitude greater than its half, and from that which is lefta magnitude greater than its half, and if this process be repeated continu-ally, there will be left some magnitude which will be less than the lessermagnitude set out.41

In the first case discussed, Kilvington shows that there is no transi-tion between an angle of tangency and a rectilinear one, althoughboth are certain geometrical magnitudes. The second case—“a coneof shadow”—demonstrates that if a process of the diminution of amagnitude is continuous, it cannot end. It seems that in the third casediscussed one finds the results of the first two, since it employs both aproportional division and a transition between two different kinds ofgeometrical entities—a spiral and circular lines. It also shows that wecannot come to the end, because we cannot find the last point of aspiral line.

These examples are modifications of Zeno’s paradox, namely the

paradox of dichotomy. Even though Aristotle was certain that whileintroducing isomorphism of different kinds of continua he eliminatedthe paradox,42 Kilvington makes it clear that Aristotle failed. As amatter of fact, the Aristotelian definition of continuity seems to be inaccordance with Zeno’s statement that infinite division into propor-tional parts cannot be completed. But Kilvington is not able to solvethis paradox. He only notices contradictions that derive from acceptedprinciples, and he leaves the readers—as it is apparent—with thedif ficulty of interpretating his mathematical arguments. Kilvington,

however, does not accept the alternative solution according to whichthe process of the division of a continuum can be completed becausethere are indivisible entities. But he explicitly argues for the commonlyaccepted Aristotelian concept of continuity.

Only the last of Kilvington’s above-presented geometrical construc-tions exercised the interest of other medieval thinkers. One finds thedebates on properties of a linea girativa in the works of e.g. Roger Roseth,

41 Euclid, Elements [trans. Heath], vol. 3, X, prop. I, p. 258.42 Cf. Aristotle, Physics, VIII, 10, 266b2.




John Buridan, Benedict Hesse and John Major. Most of them, however,took into account only one aspect of Kilvington’s discussion, asking

whether a spiral line is actually or potentially infinite.43

43 Cf. Roger Roseth, Utrum aliqua creatura possit esse in fi nita, in Lectura super Sententias,Quaestiones 3, 4 & 5 [ Hallamaa], pp. 266–272; John Buridan, Utrum linea aliqua gyrativasit in fi nita, et semper accipio in fi nitum categorematice, in: Quaestiones super libros Physicorum, Liber

III, Quaestio 16, edited in Thijssen, John Buridan’s ‘Tractatus de infinito,’ pp. 23–33;Benedict Hesse, Utrum aliqua linea gyrativa sit in fi nita, accipiendo ‘in fi nitum’ categorematice, inQuaestiones super octo libros Physicorum Aristotelis [ Wielgus], pp. 384–387; John Major, Dein fi nito [ Élie], pp. 12–52.



THE IMPORTANCE OF ATOMISM IN THE PHILOSOPHYOF GERARD OF ODO (O.F.M.)*

Sander W. de Boer

Introduction

The Franciscan theologian Gerard of Odo (Giraldus Odonis ) was born

c. 1285 in the village of Camboulit, near Figeac in the South of France,and died in 1349.1 He lectured on the Sentences in the Franciscan studium in Paris in the period 1326–1328.2 Odo wrote an influential commentaryon Aristotle’s Nicomachean Ethics (to which he owes his name of Doctor

moralis ).3 Even though he did not write more commentaries on the worksof Aristotle, Odo did have a great interest in logical and natural-philo-sophical topics. There are several separate questions or tracts in thesefields, most of them anonymous, that have been ascribed to him. Mostof this material can also be found in some form in his commentary

on the Sentences.4

Since Odo has not written any commentary on the

* I would like to thank Prof. Paul J.J.M. Bakker and Prof. Hans M.M.H. Thijssenfor their helpful comments on an earlier version of this article.

1 For biographical details see: Schabel, “The Sentences Commentary of GerardusOdonis, O.F.M.”; De Rijk, Giraldus Odonis O.F.M.: Opera Philosophica I: Logica, pp. 1–5and Weijers, Le travail intellectuel à la Faculté des arts de Paris: textes et maîtres (ca. 1200–1500),III, pp. 79–83.

2 These Parisian lectures were actually the second time Odo lectured on the Sentences.

Thefi

rst time was in Toulouse in the late 1310s. Parts of these Toulouse lectures musthave found their way into the Parisian commentary, but it is unknown how much. Fora detailed description of Odo’s commentary on the Sentences, cf. Schabel, “The SentencesCommentary of Gerardus.”

3 This influence can, for example, clearly be traced in John Buridan’s commen-tary on the Ethica as James Walsh has demonstrated in Walsh, “Some Relationshipsbetween Gerald Odo’s and John Buridan’s Commentaries on Aristotle’s ‘ Ethics’.” Thecommentary survives in about 17 Mss. For a list of the Mss and early prints of thiscommentary see Lohr, ‘Medieval Latin Aristotle Commentaries. Authors G–I’, p. 164.The Venice edition (1500) is accessible on Gallica (http://gallica.bnf.fr/).

4 Cf. De Rijk, “Works by Gerald Ot (Gerardus Odonis ) on Logic, Metaphysics andNatural Philosophy Rediscovered in Madrid, Bibl. Nac. 4229.” For examples of this

connection between the tracts in the Madrid Ms and the Sentences commentary, cf.Bakker, “Guiral Ot et le mouvement. Autour de la question De motu conservée dansle manuscrit Madrid, Biblioteca Nacional, 4229” and De Rijk, Giraldus Odonis O.F.M.:Opera Philosophica II: De intentionibus.



86 sander w. de boer

Physics, these separate tracts combined with his commentary on theSentences are the only sources we have to determine his views on topics

of natural philosophy.In this article I want to examine his position on the question in naturalphilosophy that he is most famous for, both in his own time and ours,namely the question of the structure of the continua: space, time andmotion. Understanding the structure of continua was so important toOdo, that he dedicated two extensive questions to it in his commentaryon the Sentences. In addition, there are two separate tracts on this issuethat are closely related to these questions in the Sentences. All his treat-ments of the problem have a very similar structure and the core of his

solution is always the same: continua are composed of a finite numberof non-extended indivisibles that touch each other whole to whole. Thereason why indivisibles can compose something that is extended in thisway, is that they have, as Odo calls it, certain differences of either placeor time.5 When indivisibles touch each other whole to whole, but notaccording to every difference, they can compose something extended.Besides this notion of ‘difference’, Odo also introduces the principle thatthere can be local motion without the moving thing changing position.This occurs, for example, when a sphere is rotated around its centre,according to Odo. In such a situation only the differences of the pointin the centre would “move”. With this solution, Odo is the first of theParisian atomists in the fourteenth century.6

After the pioneering research on fourteenth-century atomism byPierre Duhem and Anneliese Maier, there have been two scholars inparticular who have significantly contributed to our knowledge of Odo’satomism.7 The first was Vassili Zubov who in 1959 published an article

5 “. . . quod indivisibile secundum partes quantitativas est distinguibile et determinabilesecundum differentias respectivas loci vel temporis.” (Ms Madrid, Bibl. Nac., 4229,f. 183vb)

6 This group includes: Nicholas Bonetus, Marc Trivisano, John Gedeonis, Nicholasof Autrecourt. Nicholas Bonetus includes many parts of Odo’s text almost verbatimin the part on quantity in his Liber predicamentorum (Venice, 1505), even if he eventuallyreaches a somewhat different version of atomism; cf. Zubov, “Walter Catton, Gérardd’Odon et Nicolas Bonet.” Another follower is Marc Trivisano, who includes large partsof Odo’s text in book 2 of his tract ‘ De macrocosmo’; cf. Boas, “A Fourteenth-centuryCosmology.” The otherwise unknown John Gedeonis, to conclude, as John Murdochpointed out, also keeps referring to the arguments of a certain magister (undoubtedly

Odo) when he develops his atomistic position in Ms Vat. Lat. 3092, ff. 113v–124r.7 Duhem and Maier could only consult Odo’s tract on the continuum that is con-tained in Ms Vat. Lat. 3066. The Ms contains only the recto side of the first folium ofa question on the structure of the continuum. In order to determine Odo’s position,




on the atomists in Oxford, he has written about Odo as well. Murdochhas classified fourteenth-century atomism as a doctrine that is not the

result of a physical analysis, but rather of a purely intellectual reactionto Aristotle’s analysis of continuous quantity.12 This would not onlyexplain the fact that the atoms were thought of as similar to math-ematical points, but also the fact that even the relations between theseatoms were conceived in mathematical terms. This view also leads tothe conclusion that fourteenth-century atomism is not a description orexplanation of reality, as was the case in Greek atomism.

Because Murdoch focused on those parts of Odo’s tract that illus-trated the late-medieval application of propositional and mathematical

analysis the other parts were not discussed. In particular, Murdoch stud-ied the responses of Odo and other atomists to the (often mathematical)arguments against their atomistic position. As a result, Odo’s positivearguments in favour of atomism have never been examined in detail.It is precisely on this point that I hope to contribute to our knowledgeof his atomism. Starting my analysis with Odo’s arguments in favourof atomism, I intend to complement the picture of Odo’s atomism thatwe had so far, and also bring out its ontological ramifications.

In this article I will focus on the separate tract on the continuumthat is found (only) in the Ms Madrid, Biblioteca Nacional, 4229, towhich I will refer in the rest of this article as De continuo.13 The tract isclosely related to Odo’s treatment of the continuum in book II of hiscommentary on the Sentences.14

1. My Theses

In this paper, I hope to show two things about the position and impor-

tance of atomism in Odo’s philosophy, namely, first, that his atomismoccupies a much more important and central place in his philosophy

& Synan, “Two Questions on the Continuum: Walter Chatton (?), O.F.M. and AdamWodeham, O.F.M.”

12 Murdoch, “Naissance et développement de l’atomisme,” p. 27.13 I have also looked at Odo’s other texts on the continuum. Although there are

some interesting differences between the different versions, they have no implicationsfor the conclusions of this article.

14

The precise relations between the different versions of Odo’s questions on thestructure of continua are complex. I’m currently preparing an article on this topicwhich will also include a critical edition of the De continuo tract from the Ms Madrid,Bibl. Nac. 4229.



atomism in the philosophy of odo 89

than we assumed, and, secondly, that this atomism is not just a purelyintellectual response to Aristotle’s treatment of continuous quantity,

but also makes an ontological claim. More precisely, in my view Odointended his atomism to be both an accurate description and an expla-nation of real continuous physical processes and structures.

To substantiate these claims, I also want to propose a new interpre-tation of his atomism in which the two following principles play animportant role. (1) In Odo’s atomism there is an ontological priorityof the part over the whole, i.e. the whole is seen as nothing more thanthe sum of its parts, and every property of the whole can be reducedto the properties of the parts. (2) The necessity of a finite number

of indivisibles follows from the ontological primacy of the part andtherefore is not motivated by the necessity of a limit to division, butby the composition of continua out of prior parts combined with the(Aristotelian) idea that the infinite cannot be traversed.

To make clear what I mean by this second principle, let me makea distinction between what I would call a strong and a weak readingof the term `composed’ in the proposition ‘a continuum is composedof indivisibles’. In the weak reading this proposition only means thata continuum can be divided into indivisibles. Therefore, a continuumis in some way composed of them since these indivisibles had to besomehow (at least potentially) contained in the continuum prior tothe division. This weak reading always takes the whole existing con-tinuum as its starting point, and therefore never implies a primacy ofthe indivisible parts over the whole. It is in exactly this same way thatthe non-atomist will claim that a continuum is composed of parts thatare always further divisible, without implying in any way that theseparts exist prior to the whole continuum.15 That is, if we divide an

existing continuum, we can always divide the resulting parts further adin fi nitum. The strong reading, on the other hand, does imply a primacyof the part. In this reading a continuum is composed , that is formed,by a continuous joining of prior existing part to part. It is this strongreading that describes the position of Odo. To avoid any confusion onwhat this primacy entails, we could say the following. The ontologicalprimacy of the part, however crucial in understanding Odo’s atomism,does not imply any form of Democritian atomism where all the atomsfirst exist separately. It does imply that the answer to the question “how

15 Cf. Aristotle, Physics VI, 232b24–26.




do the indivisible parts of the continuum come into being?” is not “by

division of the continuum,” but is “they come into being, one by one,

when the continuum is composed”.It is important to note that even though an atomism that is an expla-

nation of the physical world, such as the atomism of Democritus or

Epicurus, needs some sort of primacy of the indivisible part, an atom-

ism that is only the result of an intellectual analysis of the structure of

continuous quantity does not. Such an “analytical” atomism could be

motivated solely by the need for a limit to divisibility.

2. The ‘De continuo’ Tract

Let me now turn to Odo’s De continuo tract. In this text Odo gives six

arguments in favour of his atomism as well as six arguments against

it. Here I will discuss two of his arguments in favour of atomism, the

first and the sixth.

2.1. The Distinctions between Act-Potency and Quantitative-Proportional

The first argument in De continuo is also the most important one, because

it introduces a number of distinctions that play a major and recur-

rent role in Odo’s refutation of a number of counter-arguments. The

argument also shows the primacy of the part in Odo’s atomism. The

basic argument is very brief and runs as follows: every whole composed

of an infinite number of magnitudes, just as a cubit is composed of

two semi cubits, is an actual infinitely large magnitude. And since an

actual infinite magnitude is impossible, no continuum can be infinitely

divisible.16

The phrasing of the argument is important here. Where thefi

rstpremise speaks of “being composed” ( compositus ), the conclusion speaks

of “being divisible” ( divisibilis ). This implies, if the argument is valid,

a parallelism between composition and divisibility, in the sense that

something is divisible in the same number of parts as it is composed

of. Given this parallelism the continuum indeed cannot be infinitely

16

De continuo, ff. 179rb–va: “Omne totum compositum ex magnitudinibus multitudineinfinitis, sicut componitur cubitus ex duobus semicubitis, est magnitudo actu infinita.Sed non est dare magnitudinem actu infinitam. Ergo nullum continuum est divisibilein infinitum.”




divisible, since an infinite number of composing parts would make thecontinuum infinitely large.17 The addition of the clause “as a cubit is

composed of two semi cubits” is crucial for this line of reasoning, aswe will see.In its basic form the argument is far too crude, as Odo himself also

realizes. He therefore introduces two objections, both of which arguethat some distinction needs to be made. The first objection states thatbeing composed of parts can be understood in two ways. In the firstway we understand the parts to be actually present, and in the secondway we understand the parts to be merely potentially present. Theinfinite number of parts in a continuum must be understood in the

second, potential, way.18 Odo’s analysis of this distinction between act and potency is best

seen in the passage where he distinguishes between two meanings ofa potentiality of parts.

. . . this division by act and potency either distinguishes between themanner in which the infinite number of parts exist in the continuum, inthe sense that the infinite number of parts do not actually exist in thecontinuum but only potentially, or distinguishes between the manner ofmultiplication of the parts of the continuum, in the sense that all the

parts of the continuum taken together are not to be actually multipliedin infinity, but in potency only, because they are not actually multiplegiven the fact that they are not actually divided.19

I will start with the second meaning, which Odo calls “the multiplica-tion of parts”. In this case we take all the parts together, that is westart from the unity of the continuum, and say that there is no actualinfinity of parts. The core of Odo’s defence is that this second read-ing no more denies an actual infinity of parts than it denies an actual

17 There is one hidden premise, which is that there is a smallest part in this infinity.In the way Odo thinks about composition, a composition from a collection of partswhere there is no smallest one is ruled out, as will be seen.

18 De continuo, f. 179va: “Ad hanc rationem dicentur duo secundum duas distinctionescommunes. Primo dicendo quod componi aliquid ex magnitudinibus multitudine infinitispotest intelligi dupliciter. Primo actu, secundo potentia. Nunc autem ita est quod con-tinuum est compositum ex magnitudinibus multitudine infinitis in potentia, non tamenin actu. Et ideo non oportet continuum esse magnitudinem actu infinitam.”

19 De continuo, f. 179va: “. . . haec divisio per actum et potentiam vel distinguit inesse

partium infinitarum, ita quod sit sensus quod partes infinitae non insunt actu continuosed potentia tantum, vel distinguit multiplicationem partium continui, ita quod sit sensusquod omnes partes continui simul sumptae non sunt actu multiplicandae in infinitumsed in potentia, quia non sunt actu multae exquo non sunt actu divisae.”




duality or triplicity.20 In other words, we take the unity of the continuumand claim that there is no actual multitude. Odo simply dismisses this

reading as not responding to his own argument, since this reading doesnot speak of parts but only of unity. It is important to note that thereason why he can so easily dismiss the objection taken in this sense isprecisely his phrasing of his own argument in such a way that it startsfrom the priority of the parts. This priority is secured by the additionof the clause “as a cubit is composed of two semi cubits”.21

The second reading of the difference between actual and potentialin the argument, does start from a primacy of the part. In this read-ing the potentiality concerns the manner in which the infinity of parts

is present in the whole. Odo’s response to this reading is surprisinglyshort. He immediately claims that in this reading the objection is false,because an infinite number of parts would make the whole infinitelylarge. Although he gives no explanation here, we can reconstruct theunderlying idea. For what could the potency of the part mean, if acontinuum is composed out of prior existing parts? It could meannothing more than the parts being continuously joined, and thereforenot actually being distinguishable as parts, since they do not exist asactually separate from the whole continuum. Interpreted in this way,the difference between act and potency of the part cannot solve any-thing, since an infinite number of parts joined together constitutes aninfinite whole, whether the original composing parts are distinguishablein the whole or not.

To summarize, the validity of Odo’s whole argumentation againstthe distinction between act and potency is based on the assumptionthat continua are composed of ontologically prior parts. For if the twosemi cubits were not prior but only arose from the division of the cubit,

Odo could no longer reject the counter-argument that the infi

nity ofparts in the continuum is merely potential. Recall that the only reasonwhy he could so easily reject this reading of the distinction betweenact and potency, was that it started from the unity of the continuumand therefore did not respond to his argument.

20 De continuo, f. 179vb: “Non ergo plus vitatur per illam solutionem infinitas partiumquam dualitas vel quaternitas. Quare illa solutio non valet.”

21

Of course the distinction I made between a weak and a strong reading of “beingcomposed” could also be applied to this clause. It is, however, clear that Odo takesthe clause in the strong reading where we first have two semi cubits and only aftercombining those a cubit. Otherwise his argument would make no sense.




His imaginary opponents, however, have another way out. This is theAristotelian distinction between two types of division, namely between

quantitative and proportional division.22

Where in a quantitative divi-sion a whole is divided into a certain, always finite, number of equallysized parts, in a proportional division, a whole is divided according toa certain factor, usually the factor two, in a potentially infinite numberof parts decreasing ever in magnitude, but without a last and minimalpart. This distinction gives Odo’s opponents a powerful objection, sinceit concerns the manner of division of the continuum and thereforealways starts from the unity of the continuum.

Surprisingly, in his response to the objection, Odo denies that there is

a difference between these two ways of dividing. To understand why hedenies this, we must look at the peculiar way in which he understandsthe proportional division. For Odo, a proportional division means that Itake the whole and divide it into two parts. Then I take both parts anddivide them both into two parts. Then I take the resulting four partsand again divide each into two parts, and so on.23 Now it is evidentthat the result of this procedure (at any stage of the division) is a finitenumber of equal parts. And taken this way, a proportional divisionis little more than an unnecessarily complicated way of describing aquantitative division. Also, since a proportional division now amountsto the same as a quantitative division, an infinite proportional divisionwould result in an infinite number of parts of equal magnitude. Andthis would imply, as Odo says, that the whole continuum would beinfinite in size.24 The emphasis on the resulting equal magnitude of theparts is meant to exclude the possibility of a proportional division in the

22

De continuo, f. 179vb: “Alio modo dicetur ad rationem dicendo quod aliquid com-poni ex magnitudinibus multitudine infinitis contingit intelligi dupliciter. Primo quodinfinitae magnitudines illae sint eiusdem quantitatis. Vel secundo: eiusdem proportionis.Et secundum hoc dicitur quod compositum ex magnitudinibus multitudine infinitis,si sint eiusdem quantitatis, est infinitum actu. Si vero non, sed eiusdem proportionis,non erit magnitudo infinita. Nunc autem ita est de continuo quod componitur exmagnitudinibus infinitis eiusdem proportionis, non eiusdem quantitatis. Ideo non estmagnitudo infinita actu.”

23 De continuo, f. 179vb: “. . . quia infinitae magnitudines eiusdem proportionis neces-sario sunt eiusdem quantitatis. Quod patet, quia accipiatur una quantitas et dividaturin duas quantitates aequales; iterum illae dividantur in alias equales et sic in infini-tum, semper multiplicatio secundum illam proportionem dividetur secundum eandem

quantitatem inter se.”24 De continuo, f. 179vb: “Omne compositum ex magnitudinibus secundum quantita-tem aequalibus et secundum multitudinem infinitis, sicut cubitus componitur ex duobussemicubitis, est magnitudo actu infinita.”




Aristotelian sense. Of course, if my interpretation of Odo’s atomism is

correct, such proportional division has to be excluded on the grounds

that it has no last part, and therefore the parts could never be prior tothe whole but would always be posterior.

In conclusion, we have seen that Odo’s first argument is explicitly

formulated as an argument that starts from the composition of the

continuum out of prior parts of equal magnitude. Without the addition

of the phrase: ‘as a cubit is composed of two semi cubits’ his argument

would fail. In his response to the objections, all distinctions are reduced

to a resulting finite number of parts of an equal magnitude.

2.2. Degrees of Heat

To show that this priority of the parts is fundamental for the whole of

Odo’s atomism and not just implied in his first argument, I want to

discuss a second argument he gives in favour of his atomism. It is the

sixth and last argument in De continuo. It discusses what occurs in the

intension of heat.25 Here Odo argues that given an infinite divisibility

of the continuum, heat would also be infinitely divisible; and therefore

there would be an infinite intensity of heat. For heat is caused by a

qualitative motion, and this motion, being continuous, would includean infinite number of “having changeds” ( mutata esse ). In each of these

mutata esse, the intensity of the heat would increase, no matter how

little. In a similar manner as in the first argument, Odo concludes that

this infinite number of increases of intensity would result in the final

intensity of the heat being infinitely great.26

The phrasing of the argument is again far from innocent. The

premise stating that in each of the mutata esse the intensity of heat will

increase implies first of all that there can be motion in a single moment,

something Aristotle explicitly denies;27 and secondly it also seems to

25 De continuo, ff. 182ra–b: “Sexta ratio principalis est haec: si continuum esset divisibilein infinitum, calor esset divisibilis in infinitum et esset actualiter infinitus. Utrumqueconsequens est impossibile. Ergo et antecedens. Quod primum consequens sit falsum:dicit Philosophus De sensu et sensato quod nulla passio sensibilis est in infinitumdivisibilis. Aliud consequens est manifeste impossibile.”

26 De continuo, f. 182rb: “Arguo sic: cuicumque calori additi sunt infiniti calores faci-entes unum cum ipso, est actualiter calor infinitus. Sed si motus calefactionis <esset>

divisibilis in infinitum habens infinita mutata esse per quae sunt acquisiti infiniti graduscaloris, necesse est quod primo calori sint additi infiniti gradus et per consequens infiniticalores. Ergo quilibet calor intensus per motum esset actualiter infinitus.”

27 Aristotle, Physics [ Barnes], VI, 241a15–16: “Again, since motion is always in timeand never in a now, and all time is divisible (. . .).”




imply an even stronger claim, namely that a change or movement in acertain time-interval is only possible when there is motion or change in

each of the moments in that interval; and therefore that the propertiesof a continuum originate from the properties of its indivisible parts.In other words, if there were no change in each indivisible part of themotion, there could be no change in the whole motion. The parallelwith the first argument for atomism will be clear. Again, Odo triesto show that an infinite number of parts in a continuum will alwaysconstitute an infinite whole.

And just as in the first argument Odo lets his opponents object withthe distinction between actual and potential parts. The primacy of the

parts is perhaps most clearly stated in the following passage:I argue against this: I accept that all the degrees that are acquired bymovement are actually acquired and actually exist in the heat. I acceptfrom the other part <of the syllogism> that these degrees of form areactual. Then <I argue> as follows: an infinite number of actualities thatare actually acquired render that in which they are actually infinite. But asis clear from what was presupposed, these degrees are actual, are actuallyacquired and are infinite in number. Therefore they render somethingactually infinite, because this absence of distinction does not remove the

infi

nity. In fact, if there were an infi

nite number of distinct cubits, <and>if afterwards they were joined and became indistinct, such an absenceof distinction would not at all prevent that this infinite number of cubitswould render the quantity actually infinite.28

In this fragment Odo responds to the objection that we need to distin-guish between act and potency, and that the parts in the intension ofheat are only potential parts. According to Odo, this distinction can-not solve anything, since the infinite number of degrees are at somemoment actually received. And from that moment on they are all

contained in the unity of the resulting heat. That the resulting heat isa unity, in which the composing degrees are no longer distinguishabledoes not detract anything from the fact that, even as undistinguished,they all contribute to the intensity of the resulting unity. And to leave

28 De continuo, ff. 182rb–va: “Contra istud arguo: accipio quod omnes gradus acquisitiper motum sunt actu acquisiti et actu insunt calori. Accipio ex alia parte quod istigradus formae sunt actuales. Tunc sic: infinitae actualitates numero actu acquisitae

reddunt istud cui insunt actu infi

nitum. Sed ut patet per supposita, isti gradus suntactuales et actu acquisiti et secundum multitudinem infiniti. Ergo ipsi reddunt actualiquid infinitum, quia illa indistinctio non tollit infinitatem. Si enim essent infiniticubiti distincti, si postea coniungerentur et essent indistincti, talis indistinctio minimeprohiberet quin illi cubiti infiniti redderent quantitatem infinitam actu.”




no doubt, Odo once again brings his example of a cubit into play. If

I were to combine an infinite number of cubits into a longer length,

then, even supposing that the individual cubits are not distinguishablein the resulting length, this resulting length will still be infinite.

Again we see the ontological priority of the part appear in this

argument. The importance of this primacy will become clear in the

remainder of this article where I intend to show the important role of

atomism in Odo’s philosophy and theology. Still, we might ask at this

stage why this peculiar way of looking at the relations between parts

and wholes in a continuum has not been noted earlier in the scholarly

literature? The answer seems to be that it only becomes apparent in

the arguments Odo gives in favour of atomism. And those were thearguments that had for understandable reasons not yet received any

attention. As for the examined parts of Odo’s atomism, that is his

responses to the various counter-arguments, these parts are, as far as

I can see, indifferent to the question whether the part or the whole of

a continuum is primary.

What is also remarkable about this second argument is that it seems

to describe a concrete physical process, which is something we would

not immediately expect if Odo’s atomism were indeed primarily moti-

vated by a conceptual analysis of continuous quantity. However, it is

yet too soon to draw any conclusions here about possible ontological

commitments of Odo’s atomism, or its explanatory powers, since the

argument occurs in a tract that deals with continua in general and could

therefore just as well be a merely hypothetical example.

2.3. A Missing Argument: God’s Omnipotence

There is one final aspect of De continuo I want to draw attention to in

this context, namely the lack of arguments invoking God’s omnipo-tence, that were both common and important in the fourteenth-century

atomism debate, for example in the writings of the Oxford atomists

Walter Chatton and Henry of Harclay.29 In De continuo there are no such

29 For Chatton, cf. Quaestio de continuo [Murdoch & Synan], § 36, pp. 58–59, 61,68, 72. Also, William of Alnwick attributes two arguments for atomism to Henry of

Harclay, in his Determinatio II, both of which invoke God’s omnipotence. As far asI know the Determinatio II is unpublished, but J. Murdoch gives a partial translation(including these arguments) based on his own private edition in Grant (ed.), A Source Book in Medieval Science, pp. 319–324.




be shown that Odo introduces an atomistic structure of real continuain texts and arguments that do not deal with continua as their main

topic, this would indicate that in the particular case of Gerard of Odowe must qualify Murdoch’s general assessment of fourteenth-centuryatomism.

In the remainder of this article I will examine a total of three con-texts, taken from Odo’s commentary on the Sentences and from anotherseparate tract, in which atomism plays an important and sometimeseven crucial role. The first of these contexts consists of the questionsthat deal with infinity.

3.1. In fi nity

Odo discusses infinity several times in his commentary on the Sentences.His most detailed exposition is found in a question on the possibilityof eternal motion.31 Odo’s answer to this question is af firmative, to theextent that motion could have existed from eternity. In his response tosome of the objections he refines both the concept of ‘equal things’(aequalia ) and the concept of ‘infinite magnitude’ (magnitudo in fi nita ).According to Odo, things can be equal in two ways. First, by being

limited at the same points (conterminari ); second, in a negative formula-tion, by not exceeding each other (non excedi ). Infinities can only be equalin the second way of not exceeding each other. The notion of ‘infinitemagnitude’ can even be understood in three senses. The infinity canresult from the absence of a beginning, from the absence of an end,or from the absence of both. Motion can then be infinite in the senseof the absence of a beginning.32

What this solution shows is that Odo did have some conceptual toolsto deal with the paradoxes that result from comparing infinities. It also

31 Sentences II, dist . 2, q. 2 (Ms Valencia, Cab. 200, ff. 17rb–18ra): “Utrum motuspotuerit esse ab aeterno.”

32 Ibid. (Ms Valencia, Cab. 200, f. 18ra): “Modo dico quod nulla magnitudo infinitaper exclusionem ultimi potest esse actu pertransita, quoniam esse actu pertransitumimponit ei terminum, sed non imponit sibi primum. Magnitudo vero infinita per exclu-sionem primi tantum, non tamen per exclusionem ultimi, potest esse actu pertransita,dum tamen non fuerit totaliter pertranseunda, sicut, si dicetur tempus ab aeterno,praeteritum ponetur infinitum per exclusionem primi tantum; et ideo non repugnaret

sibi esse pertransitum. Futuro autem repugnaret esse pertransitum, si ponetur infinitumper exclusionem ultimi; eodem modo de magnitudine. Dico igitur quod, si motus fuis-set ab aeterno, magnitudo infinita per exclusionem primi, non per exclusionem ultimi,fuisse potuisset pertransita.”




shows that he was not troubled by the possible or hypothetical existence

of infinities. This indicates that his atomism does not result solely from

his incapacity to deal with the notion of infinity.More important for my present purpose, however, is another question

on infinity entitled: “Whether the universe could be infinitely large”.33

The answer that Odo gives here is negative. In itself this is not surprising

since most philosophers at that time would deny it. What is surprising,

however, is the way in which Odo responds to the well known argument

that God in his omnipotence could create an infinite magnitude and

also both an infinite multiplicity and intensity.34 For, as the argument

runs in Odo’s version of it, all that God can create in a single day

he can create in a single moment; and all he creates, he can sustainafterwards. In this way God can create an infinite multiplicity and an

infinite magnitude since each day he could create a single stone and

therefore in an infinity of days he could create an infinite number of

stones, which he can also join together to create an infinite magnitude.

And if he could create this infinity in an infinite number of days, he

could also create it in an infinite number of moments.

There are several ways of dealing with such an argument, for

example by pointing out that a physically realized infinite magnitude is

an incoherent notion, since it would occupy a place, and place implies

limitation. But Odo takes a very different and unique route as can be

seen in the following fragment:

To this I say that just as an act is incompatible with an infinite magni-tude, so it is incompatible with an infinite multitude; hence there cannot

33 Sentences II, dist . 44, q. 1 (Ms Valencia, Cab. 200, ff. 97vb–98va): “Utrum machina

mundialis possit esse infi

nita magnitudine”.34 Ibid. (Ms Valencia, Cab. 200, ff. 98rb–va): “Sed in oppositum arguitur quia:magnitudo infinita et multitudo infinita et virtus infinita sunt possibiles; ergo mundialimachinae inquantum magnitudo est non repugnat infinitas; quare ipsa poterit esseinfinita magnitudo. Consequentia est evidens. Sed antecedens probo quia: quaecumqueet quantacumque et quocumque Deus potest [potest] producere in diebus infinitis,<potest> producere in instantibus infinitis, et illa producta conservare, supposito quodquodlibet illorum sit possibile in instanti. Hoc enim patet, quia quod Deus possetfacere uno die, posset facere in instanti, supposita conditione instantanea factionis.Sed Deus posset facere infinitum magnitudine et multitudine in diebus infinitis. Quodpatet, quia quolibet die posset facere unum lapidem, et per consequens in infinitisdiebus infinitos lapides, ex quibus resultaret multitudo infinita. Posset etiam omnia illa

copulare continuatione, et esset magnitudo infinita. Posset etiam quolibet die producereunum calorem, et in infinitis infinitos, qui in unum redacti facerent virtutem infinitam.Quare Deus in instantibus infinitis posset producere infinitam multitudinem, infinitammagnitudinem et infinitam virtutem.”




be an infinite number of instants as the minor premiss of the argumentsupposed. To the argument I say that the minor is false. Neither is theproof valid, because I do not concede that whichever continuum is com-posed of an infinite number of parts, nor that it is divisible in parts thatare always further divisible. On the contrary, I say that it is composedof indivisibles and is resolved in a finite number of indivisibles, not aninfinite number.35

Odo denies the existence of an infinite number of moments! Even if

God in each moment creates a stone or a degree of heat and preserves

and joins them, the result will always be finite, since the number of

moments available for this creation is always finite. The impossibility of

an infi

nite multitude is deduced from the non-existence of an infi

nitenumber of moments, that is, from the atomistic structure of time. This

means that even from the perspective of God, time has an atomistic

structure. And if even God is limited by the atomistic structure of time,

interpreting Odo’s form of atomism as solely the result of an analysis

of the Aristotelian concept of continuous quantity seems too limited. In

this context, Odo uses his atomism as a description of real space and

as a partial explanation of the fact this real space cannot be infinite.

That is, his atomism seems to be a physical atomism.

3.2. Intension and Remission

The second context I want to discuss is that of the intension and

remission of forms. The relevant questions occur in Odo’s Sentences

commentary, but also circulated as a (very long) separate tract, just

as the questions on the continuum. Here, I will use the separate tract

found in Ms Madrid, Biblioteca Nacional, 4229.

In the first question of this tract, Odo treats the intension of light.36

He proceeds byfi

rst stating the opinion of a certain (unnamed) doc-tor, almost certainly Walter Burley, and then giving his own solution.37

35 Sentences II, dist. 44, pars II, q. 1 (Ms Valencia, Cab. 200, f. 98va): “Ad illud dicoquod sicut repugnat actus infinitae magnitudini, sic repugnat infinitae multitudini; quarenon possunt esse infinita instantia, ut minor rationis supponebat. Ad principale dicoquod minor est falsa. Nec probatio valet, quia non concedo quod quodlibet continuumsit compositum ex partibus infinitis, nec sit divisibile in semper divisibilia, ymo dicoquod componitur ex indivisibilibus et resolvitur in indivisibilia finita, non infinita.”

36 De continuo (Ms Madrid, Bibl. Nac., 4229, ff. 133ra–135vb): “De augmento formae:

utrum lumen augeatur per adventum novae partis ad priorem, utraque manente.”37 Walter Burley ( c. 1275– c. 1346) was a contemporary of Odo and was also (from1310–1326) connected to the University of Paris. His commentary on the Physics contains an extensive criticism of Odo’s atomism. Also, Burley was one of the most




The opinion of Burley, as described by Odo, is that the intensionof light does not occur by the addition of light to light, where both

would remain. The argument that is used to disprove such an additiontheory is that of a light source continuously moving toward an objectin which light is caused. For the continuous movement would meanthat in an infinite number of moments an infinite number of parts oflight is added; and since all the parts would remain, the result wouldbe a light of an infinite intensity. Just as in the text on the continuumthe objection is introduced that the parts exist only potentially. And thisobjection is again dismissed on the grounds that the resulting intensityis actual and so the composing parts must be actual as well.

Ultimately, Odo accepts Burley’s position, that light is not intensifiedby an addition of part to part, but not without some critical remarks.Odo says that although Burley’s solution is true and provable, his argu-ment is only ad hominem and not ad rem.38 Now why is the argumentonly ad hominem? We can infer the reason from the following passage:“Secundam conclusionem, scilicet quod iste doctor demonstraveritconclusionem istam ad hominem concedentem quod continuum sitdivisibile in infinitum, probo.”39 The argument is directed against aposition that accepts the infinite divisibility of the continuum, and, sowe may add, therefore assumes an infinite number of moments in atime-interval. The argument cannot be ‘ad rem’ since according to Odothere is only a finite number of moments.

After discussing the special case of light, in the next question Ododiscusses the intension and remission of qualities. Here Odo’s use ofatomism is even clearer. In this question Odo wants to prove, among

prominent defenders of the so-called succession theory in the debate about the intensionand remission of forms; a position that Odo will ascribe to the doctor in the rest ofhis tract. Burley defended this position in a tract called De intensione et remissione formarum[Venice, 1496], which is dated in the 1320’s shortly before Odo’s commentary on theSentences. Finally, the scribe of the Ms Madrid, Bibl. Nac., 4229 inscribed the nameGal Burley in the margin on f. 142r. For Burley as a defender of the succession theoryand for the date of his tract, see Dumont, “Intension and Remission of Forms fromGodfrey to Burley” (forthcoming).

38 Tractatus de augmento formae: utrum lumen augeatur per adventum novae partis ad priorem,utraque manente (Ms Madrid, Bibl. Nac. 4229, f. 134va): “Nunc tertio pro evidentia solu-tionis rationum doctoris pono quinque conclusiones. Prima est quod conclusio sua estvera et demonstrabilis ad rem. Secunda quod est demonstrata per eum ad hominem.

Tertia quod non est demonstrata per eum ad rem. Quarta quod, dato etiam quoddemonstraret, demonstratio sua non esset ad propositum. Quinta quod non respondetad argumentum factum.”

39 Tractatus de augmento formae . . . (Ms Madrid, Bibl. Nac. 4229, f. 134vb).




other things, that a form is not intensified by addition of degree todegree in an infinite number of moments. As evidence he gives an

example of hot water heating iron.40

According to the addition-theory,at every moment a degree of heat will be caused in the iron. Since theheat in the water is unnatural, the water will continuously cool down,until finally it causes a last and minimal (!) degree of heat in the ironand then the heating stops. Now, if there were an infinite number ofmoments, and if in each of these moments the water caused a higherdegree of heat than the last minimal degree, then according to Odothere would have to be an infinite number of degrees of heat in theiron when the heating stops. At this point he has already reached his

desired conclusion: that there will be an infinite intensity of heat in theiron and that therefore one of the premises must be wrong.

What is interesting is that the argument continues. In the final stepof the argument Odo once again reduces these degrees to equal parts,by saying that an infinity of degrees, all of which are higher than thelast minimal degree, must also imply an infinity of equal degrees. Forthe greater, says Odo, is always equal to the smaller plus somethingextra. Therefore there must be an infinite number of equal and mini-mal degrees. Note that this final step has only one function, and thatis to return to an atomism of minimal parts composing a continuum.

Odo also states the argument in the formal way of the syllogism:

I argue as follows: every form that includes an infinite number of degreesof the same quantity is infinite; this <form> is such; therefore it is infinite.The conclusion is false. Therefore one of the premisses <is false>. Not

40 Tractatus de augmento formae . . . (Ms Madrid Bibl. Nac. 4229, f. 139rb): “Tertiam

conclusionem, scilicet quod forma non intendatur per additionem gradus ad gradumper infinita instantia, probo sic: ponatur aqua calida quae calefaciat ferrum vel aliquamrem neutram, quae non magis sit frigida quam calida; secundum te in quolibet instantiaddit gradum alium; et cum aqua habeat calorem innaturaliter et violenter continueminoratur in calore, quia omnis motus violentus fortior est in principio quam in fine;minuitur ergo continue in calore, et per consequens in virtute calefaciendi; et ita inducitprimo maiorem gradum, et tandem in fine non auget amplius calorem; ergo est deveniread gradum quem ultimo inducit; et iste est minus. Modo si in quolibet instanti induciturmaior gradus, et sint infinita instantia inter primum instans et ultimum, sequitur quodinfiniti erunt gradus maiores isto ultimo, qui est minimus; sed ubi est dare infinitosgradus maiores, ibi est dare infinitos gradus aequales, ut patet per Philosophum, quartoPhysicorum, quia omne maius est divisibile in partem aequalem et in partem excessus;

et sic erit dare infinitos gradus eiusdem quantitatis praecedentes gradum minimum. Ethis premissis, arguo sic: omnis forma includens infinitos gradus eiusdem quantitatis estinfinita; ista est huiusmodi; ergo est infinita. Conclusio est falsa.”




the maior premiss. Therefore the minor premiss, that follows from this:that between the first and last <degree> there are an infinite number ofdegrees; this follows from another, namely this one: that a form is enlargedby the addition of degree to degree throughout an infinite number ofmoments; hence this is false.41

The major premise: every form including an infinite number of degrees

of the same quantity is itself infinite; the minor: this form (of heat

in the iron) is such; and the conclusion: so this form is infinite. The

conclusion is false, so therefore one of the premises, namely the minor.

This minor, that the form of heat in the iron has an infinite intensity,

follows itself from the premise that between the first and last degree

there is an infinite number of other degrees. And this in turn followsfrom the form being intensified by addition of degree to degree in an

infinite number of moments. This last premise then, is false. There is

only a finite number of moments.

Again Odo’s analysis is consistent with his views on the continuum

in De continuo. He consistently analyses concrete physical continuous

processes as having a finite number of mutata esse, and as occurring

in a finite number of moments. The topic of the tract in which these

passages occur is not continua, but intension and remission. The intro-

duction of atomism in this tract must therefore be interpreted as anexplanation and description of the structure of physical reality.

3.3. God’s Omnipotence

Perhaps it could be objected at this stage that it might well be the

case that Odo uses atomism in contexts that are very closely related to

De continuo, like infinity and intension and remission, but that claim-

ing that it is important for the whole of his views, or even the whole

of his natural philosophical views, is making too large a claim. Tostrengthen my conclusions, let me therefore give one final example of

how deeply rooted Odo’s atomism actually is. The title of the ques-

tion I want to discuss, taken from book I of his Sentences commentary,

41 De continuo (Ms Madrid, Bibl. Nac., 4229, f. 139rb): “. . . arguo sic: omnis formaincludens infinitos gradus eiusdem quantitatis est infinita; ista est huiusmodi; ergo estinfinita. Conclusio est falsa. Ergo aliqua premissarum. Non maior. Ergo minor, quae

sequitur ex isto quod inter primum et ultimum sint infiniti gradus; illa vero ex alia,scilicet ista quod forma intendatur per additionem gradus ad gradum per infinitainstantia; quare hoc est falsum.”




is: “on the assumption that God’s omnipotence is infinite, is it infinitein respect to an infinite number of acts or in respect to an infinitely

intensive act?”42

There is no need to discuss the arguments given for both types ofinfinity, but it is important to point out that we have already seen allthese arguments in the previous texts I discussed. One of the arguments,claiming that possibilities that have no order among themselves can allbe actualized at the same time, is even found in the De continuo tract asone of the six positive arguments in favour of atomism.

As we have already encountered these arguments, it is clear whatOdo’s answer must be if he is to be consistent. And indeed he denies

both forms of infinity saying the only infinity that pertains to God’sactions is an infinity in the sense that no matter how many acts heperforms, he can always do more.43 And after explaining this last formof infinity, Odo concludes with the following passage:

I see what can be answered. For all hold in common that in whicheverinstant the luminous body is in another place. Secondly, that in whicheverinstant there is another ray, and it is certain that God can preserve andunite all these rays; hence, because there are an infinite number of them,one infinite ray will result. For that reason it seems to me that we must

say that in a magnitude there are not an infinite number of points, norin motion an infinite number of having changeds, nor in an hour aninfinite number of moments.44

There is no compelling reason to introduce atomism in a question onGod’s infinity. Odo, however, examines the (hypothetical) results of Godusing his omnipotence in nature in a way similar to that in which he

42

Sentences I, dist. 44, q. 2 (Ms Sarnano, Bibl. Com., E. 98): “Supposito quodomnipotentia Dei sit infinita fundamentaliter, utrum sit respectu actuum infinitorummultitudine vel unius intensive infiniti.”

43 Ibid. (Ms Sarnano, Bibl. Com., E. 98, f. 115ra): “Pro solutione quaestionis, ponotres conclusiones. Quarum prima est quod omnipotentia non potest elicere actus mul-titudine infinitos. Secundo quod nec actum aliquem extensive vel intensive infinitum.Tertio quod omnipotentia potest aliquo modo infinita, quia quotiscumque et quan-tiscumque datis adhoc, potest in plura et in maxima.” The Ms gives ‘ in fi nite’, which Ihave corrected to ‘in fi nitum’.

44 Ibid. (Ms Sarnano, Bibl. Com., E. 98, f. 115rb): “Video quod possit responderi.Omnes enim communiter tenent quod in quolibet instanti corpus luminosum est inalio situ. Secundo quod in quolibet instanti est alius radius, et certum est quod Deus

potest conservare et unire omnes illos radios; quare cum sint infiniti, resultabit unusradius infinitus. Ideo videtur mihi dicendum quod in magnitudine non sunt infinitapuncta, nec in motu infinita mutata esse, nec in hora infinita instantia.”




examines normal natural processes. Nowhere in this question is thisapproach qualified by a statement like ‘I only speak here as a natural

philosopher’ or something of the sort. It seems to me that we can onlyconclude that Odo was convinced of the real atomistic structure ofcontinua, and that he intended his atomism to be an explanation andaccurate description of physical reality.

4. Conclusions

In conclusion, Odo’s atomism turns out to be a constant factor in his

philosophy and theology. He consistently uses his atomism to explainreality, and the application of this atomism to God’s power and to theinner structure of continuous physical processes is not provoked byany mathematical argument. On the contrary, mathematics only playa role in countering the common arguments against atomism, as Ododoes in De continuo. The main role of Odo’s atomism is to explain whatoccurs in concrete continuous processes. Given that Odo considers theseprocesses to take place by addition of part to part, where each partcontributes to the resulting quality of the whole continuum, it is not

surprising he should be convinced that there must be a finite numberof these basic indivisible parts in each given continuum.These observations notwithstanding, Murdoch’s general description

of atomism in the 14th century as using both mathematical atomsand mathematical relations between them, still is substantially appli-cable to Odo. His atoms are similar to mathematical points, and therelations between them are at least semi mathematical. This results ina tension between a mathematical model and a physical function; atension that for example becomes clear in the strange and awkward

physico-mathematics he uses to counter some of the arguments againsthis atomism.

The interpretation of Odo’s atomism that I propose also gives riseto the following observation. We know that in Paris another atomist,Nicolas Bonetus, takes substantial parts of his treatment of the con-tinuum from Odo, but sides himself explicitly with Democritus.45 Wealso know that Nicholas of Autrecourt defends a very physical version

45 Cf. Zubov, “Walter Catton, Gérard d’Odon et Nicolas Bonet,” p. 267.




of atomism.46 From this perspective it would be interesting to see towhat extent their positions could be explained as putting more and

more emphasis on the physical and ontological side of atomism; a sidealready clearly present in Odo.

46 Cf. Grellard, Croire et savoir: les principes de la connaissance selon Nicolas d’Autrécourt , inparticular the second part.



NICHOLAS OF AUTRECOURT’S ATOMISTIC PHYSICS

Christophe Grellard

In the second chapter of his book Identity and reality, Emile Meyerson,

the early twentieth-century French philosopher, provides a picture of

what he considered to be some of the perennial characteristics of any

atomistic physics.1 His aim is to show that the mechanist conception

of natural phenomenona always existed in the history of science.Nicholas’s of Autrecourt figured prominently in the list of medieval

thinkers who ushered in a new-wave of atomism. Carefully separat-

ing Gerard of Odo’s mathematical atomism from Nicholas’s physical

atomism, Meyerson’s basic intuition was sound.2 Unlike most medieval

atomists, Nicholas did not deploy his atomism in order to solve problems

associated with the concept of the continuum.3 His aim was to find a

mechanical and reductionist answer to the problem of generation and

corruption and, finally, to the question of the eternity of things in the

world.4

For Nicholas, problems connected with the void and continuumare subordinated to his mechanism. I have claimed elsewhere that

Nicholas’s atomism is quite unique, original, especially when compared

to other medieval thinkers.5 In this paper, I will demonstrate how atom-

ism functions within Nicholas’s physics and how he believes it can resolve

some of the basic problems of medieval natural philosophy.

1

Meyerson, Identité et Réalité , p. 90. For a comprehensive study of Meyerson’s con-ception of the history of science, and especially atomism and mecanism, see Fruteaude Laclos, La philosophie de l’intellect d’Emile Meyerson. De l’épistémologie à la psychologie, pp.42–47 & pp. 62–66.

2 Nevertheless, perhaps should we moderate the mathematical label in the case ofGerard of Odo. See S. de Boer’s contribution in this volume.

3 The mathematical dimension of medieval atomism (seen as a purely intellectualreaction to Aristotle’s conception of continuum) is Murdoch’s main thesis. See Murdoch,“Naissance et développement de l’atomisme au bas Moyen Age latin” and his contribu-tion in this volume. Nevertheless, the cases of Odo, Crathorn, Autrecourt, Wyclif, andeven Burley, seem to suggest that we should be more cautious on this point.

4 Cf. Kaluza, “Eternité du monde et incorruptibilité des choses dans l’ Exigit ordo de

Nicolas d’Autrécourt”.5 Cf. Grellard, “Les présupposés méthodologiques de l’atomisme: la théorie ducontinu chez Nicolas d’Autrécourt et Nicolas Bonet”; “Le statut de la causalité chezNicolas d’Autrécourt.”



108 christophe grellard

Ultricurian atomism begins with the explanation of the eternity of

the world. The Exigit ordo’s first chapter deals immediately with this

problem and relies on an atomistic conception of nature. Atomismallows Nicholas to criticize Aristotle’s definition of generation and cor-

ruption as the movement from being to non-being. Nicholas’s atomism

is primarily physical and only secondarily mathematical. In the chapter

dedicated to the continuum (called De indivisibilibus by O’Donnell and

De continuo by Nicholas himself ),6 Nicholas makes use of the theory of

atomic motion attributed to Mutakallimun, according to whom atoms

jump from one place to another and atomic velocity is related to the

amount of time an atom rests. While it is not useless to study the

behaviour of individual atoms or indivisibles, Nicholas is much moreinterested in giving an account of generation and corruption in terms

of the local motions of atoms as they aggregate and disaggregate:

Thus in the natural things there is only local movement. When thismovement results in an assembly of natural bodies which gather togetherand require the nature of a subject, this is called generation. When theyseparate, it is called destruction. When through local movement atomicparticles are joined to a certain subject, particles of such a kind that theirarrival seems unrelated both to the movement of the subject and to what

is called its natural functioning, that is called alteration.7

Nicholas could have found this theory attributed either to Democritus

in Aristotle’s writings, or to the Mutakallimun in Maïmonide’s Guide for

the Perplexed .8 As we shall see, the Lorain was probably influenced by

6 Cf. Exigit ordo, p. 219, 27, in O’Donnell, “Nicholas of Autrecourt” (The title ofthe work will be abbreviated as EO). All translations of Autrecourt’s work are fromUniversal Treatise [trans. Kennedy e.a.].

7

Universal Treatise, p. 63, EO, pp. 200, 48–201, 6: “Sic ergo in rebus naturae nonest nisi motus localis; sed quando ad talem motum sequitur congregatio corporumnaturalium quae colliguntur ad invicem et sortiuntur naturam unius suppositi diciturgeneratio; quando segregantur, dicitur corruptio, et quando per motum localem corporaatomalia <conjunguntur> cum aliquo supposito quae sunt talia, quod nec adventusipsorum fieri videtur ad motum suppositi, nec ad illud quod dicitur operatio naturalisejus, tunc dicitur alteratio.”

8 See, Moses Maimonides, Dux seu director neutrorum sive perplexorum [ Venice, 1516],p. 1a, c. 72, f. 32r: “Mundus universaliter, scilicet omne corpus quod est in eo estcompositum ex partibus valde parvis, que non habent partes prae nimia sui parvitate,et nullo illarum particularum habet quantitatem ullo modo, et cum una coniunctafuerit aliis, compositum ex eis habebit quantitatem et tunc erit corpus et si coniu-

guntur duae de particulis illis, tunc utraque pars esset corpus et fierent duo corporasecundum quosdam illorum. Omnes autem illae particulae sunt similes sibi invicem etnulla diversitas est inter illas: et dixerunt quod nullum corpus potest esse [om., ed.] nisicompositum ex istis particulis similibus positione et loco. Et secundum eos, generatio



nicholas of autrecourt’s atomistic physics 109

both texts, even if he modifies and adapts them. It is worth noting that

this claim is central to Nicholas’s physics. It allows him to develop a

comprehensive account of nature that is both mechanist and reduction-ist. All natural phenomenona (even psychological phenomena) can be

explained by atomic movements.9 I will try here to give an account of

the principles (or elements) of Nicholas’s natural philosophy and the

modes of atomic compositions.

1. Matter and Atom

The theory of atoms, in Nicholas’s mind, is a weapon against Aristotle’stheory of generation and corruption. Indeed, the atomistic conception

of natural change is linked to the denial of the matter-form couple.

Nicholas’s aim is first to show that the matter-form distinction is a kind

of metaphysical construction useless for natural philosophy.

1.1. Matter as an Atomic Fluxus

The elaboration of Ultricurian atomism relies mostly on the usual claims

attributed by Aristotle to the antiqui . Nicholas never tries to identify

them, as if all natural philosophy before Aristotle were atomistic. It

is enough for him that the antiqui should offer a means of criticizing

Aristotle. Hence, from a careful reading of the first book of the Physics

Nicholas retains two important notions: minima naturalia and materia prima.

Adapting the former to his atomism allows him to reject the latter.

Nicholas often evokes the notion of prime matter as Aristotle’s solu-

tion to the problem of generation and corruption. But Nicholas judges

this alleged solution to be a mere fiction, one that cannot claim to be

better than the solution the antiqui offer. Indeed, Nicholas contends thatAristotle has no necessary argument to prove the simple generation,

the passage from non-being to being:

From this conclusion, indeed, it can be concluded that the assertions ofAristotle in various places are false, and something in certain places thereis only fiction. For what he says about prime matter is neither relevant

est congregatio et corruptio segregatio neque nominant corruptionem et generationem,

sed segregationem et congregationem, motum et quietem.”9 The scope of this study will not allow to give a detailed account of Nicholas’spsychological atomism. See the fifth chapter of my book, Croire et savoir. Les principes dela connaissance selon Nicolas d’Autrécourt , pp. 121–149.




nor true, because his basis in that investigation is that things pass frombeing to non-being, and vice versa. Observe that Aristotle has not at allremoved the reason for the ancients’ hesitation.10

For Nicholas, the last remark is most important. Once it is demonstrated

that Aristotle’s system is not evident, but only probable, it is legitimate

to search for alternative solutions.11 After he examines and rejects the

arguments in favour of the prime matter, Nicholas will elaborate his

own conception of atomism.

A first argument, called ratio aristotelis, is based on a comparison

between accidental and substantial change:

But there is no necessity for <prime> matter to exist. For this necessitywould result chiefl y from two arguments. The first would be Aristotle’s,as it seems: A substantial change is comparable to an accidental change,but in the latter there must be something acting as subject to the terminiof the change. For example, if something changes from whiteness toblackness, there is given a surface which acts as subject to both whitenessand blackness.12

If an accidental change requires a material substratum, we can conclude

that substantial change (that is, generation and corruption) also requires

a substratum. When Socrates’s hair whitens as he ages, Socrates is thesubstratum of the change. The concrete individual substance, that is

the material substance, remains the same through modification, and

allows these accidental modifications. But can we conclude that there

is a substratum which remains the same when Socrates passes from

non-being to being? Nicholas denies it. Indeed, the only evidence for

accidental change is the Aristotelian definition of accident as what

10 Universal Treatise, p. 68, EO, p. 204, 14–19: “Ex hac siquidem conclusione possuntconcludi de dictis Aristotelis in diversis locis esse falsa, et interdum in quibusdam estsolum fictio. Non enim habent locum nec sunt vera quae dicta sunt ab eo de materiaprima quia fundamentum in illa inquisitione est quod res transeunt de esse <ad> nonesse et e converso. Et videte quod nullo modo Aristoteles removit causam dubitandiantiquorum.”

11 On this topic, see Grellard, Croire et savoir . . ., chap. 4 (pp. 94–113) & 7 (pp.192–210); Kaluza “La convenance et son rôle dans la pensée de Nicolas d’Autrécourt,”pp. 83–126.

12 Universal Treatise, pp. 48–49, EO, p. 192, 12–17: “Nunc vero non est necessariumesse materiam <primam> quia hoc esset maxime propter duas rationes quarum prima

esset ratio Aristotelis ut videtur: sicut est in transmutatione accidentali, sic est in trans-mutatione substantiali; sed in illa est dare aliquid quod subicitur terminis mutationis,ut si aliquid mutatur de albedine in nigredinem est dare superficiem quae subicituret albedini et nigredini.”




cannot exist by itself, but only in a subject. Even if we accept this defini-

tion, accidental change still differs from substantial change in a signi-

ficant way:

But, admitting that in accidental change a subject is necessary, this argu-ment requires the positing of matter only because accidents, accordingto Aristotle, are beings only in a relative sense, so that they can haveno independent existence. It does not follow from this that the sameholds true in substantial generation. For Aristotle also in Book 7 of the

Metaphysics, seems to mean that accidents are beings only because theybelong to a being.13

Nicholas attributes the second argument in favor of substantial change to

Averroes.14 It consists in a reductio ad absurdum: if we deny the existence ofprime matter understood as the substratum of change, we must accept

either that a thing is changed without changing, or that change rests

upon on non-being. We are left with two untenable positions. Either

there is no change and then we cannot explain how a thing passes from

non-being to being or there is a change without prime matter, in which

case there is either no substratum or, if there is, the substratum is one

of the termini, that is to say the terminus a quo or the terminus ad quem. All

these possibilities are absurd, Averroes contends, therefore we need to

posit prime matter as a subject. Nicholas’s criticism is methodological.

Averroes, Nicholas contends, presupposes as evident ( per se notum ) the

passing from non-being to being, but this fact is denied by those who

defend the eternity of the world. Hence, prime matter is not necessary

in order to explain substantial change, since the proposition “this being

is changed into a substance” only means “this being exists (that is, is

appearing) and didn’t exist before (that is, was not appearing).” Nicholas

wants to show that atomism can give an account of this proposition in

13 Universal Treatise, p. 49, EO, p. 192, 18–22: “Sed ista ratio cogit ponere materiamquia, si in transmutatione accidentali requiratur subjectum, hoc non est nisi quia acci-dentia secundum Aristotelem sunt entia secundum quid ita quod non sunt nata existereper se; non ex hoc sequitur quod sic sit in generatione substantiae; unde idem Aristotelesin 7 Meta. Videtur intendere quod accidentia non sunt entia nisi quia entis.”

14 EO, p. 192, 23–31: “Alia ratio ad probandum materiam primam esse videturesse Commentatoris; nam si non esset materia prima sequeretur alterum duorum, velquod transmutatum esset sine transmutatione, vel quod transmutatio fundaretur innon ente quia vel est transmutatio, vel non est; si non sit, et certum est quod aliquid

est transmutatum de non esse in esse; ergo transmutatum erit sine transmutatione; sisit transmutatio, vel ergo habet subjectum non ens, et sic alterum inconveniens; velterminum a quo vel terminum ad quem; et utrumque est falsum quia isti sunt terminitransmutationis; ergo aliquid praeter illa, et illud vocatur materia vel subjectum.”




a more economical way than the Aristotelian metaphysical construc-

tion does. The only fundamentum in ente needed is the atom. Nicholas

can answer both arguments by opposing the movement of atoms tothe notion of prime matter. Qualitative atoms gather and separate

themselves by a local movement, and there is a mutual compensation

of atomic movements in the universe taken as a whole:

For in their view, when something is said to decay, there seems to be acertain withdrawal of atomic particles; when it is generated, there is again of others in addition. So they said that nothing decays into non-being, nor is anything generated from non-being, as is reported in Book 1of the Physics, and Book 1 of On Generation.15

Nicholas claims to give the real meaning of the antiqui ’s thesis ex nihilo

nihil fi t which simply means that there is a general order in the universe

such that atomic movements mutually compensate for each other. When

one compound is generated, another is corrupted.16 In order to explain

this fact, Nicholas uses the model of light. When light is corrupted

(i.e. by night), light does not pass into the state of non-being, it only

passes to the other hemisphere. The notions of being and non-being,

act and potentiality, which justify the need for first matter, are senseless.

Nicholas explains the appearance and disappearance of objects by astrictly mechanist and reductionist theory. Finally, Nicholas accepts the

thesis attributed to Democritus by Albert the Great that generation is

merely the passing of an object from being hidden to being visible.17

Beyond this adhesion to a kind of Democritean atomism, Nicholas’s

theory presents some original aspects. Unlike Democritus, he defends

a qualitative conception of the atom as an answer to the ratio aristotelis:

since Aristotle makes a comparison between accidental and substantial

changes, it is enough to demonstrate that all sort of change is merely

accidental. This reduction requires Nicholas to lodge a significant chal-

15 Universal Treatise, p. 68, EO, p. 204, 21–24: “Quando aliquid dicitur corrumpi secun-dum eos, videtur quidam recessus corporum atomalium; quando generatur accessusetiam aliorum, et ideo dicebant quod nihil corrumpitur in non ens nec aliquid generaturex non ente ut in 1 Phys. Et 1 De Gen. recitatur.”

16 EO, p. 193, 1–5: “Si antiqui per hanc propositionem voluerunt denotare ordinemnaturalem qui est inter entia, nam quando unum ens generatur aliud corrumpitur etita nihil generatur quin praecesserit aliquid ad quod illud quod fit habebat ordinemnaturalem in fieri; sic intellectus eorum esset verus secundum illam opinionem.”

17

Albert the Great, De Caelo et mundo [Hossfeld], III, 2, 6, p. 233: “Et si quidamdicant eam actu esse intus, tunc ponunt latentiam formarum et generationem nihilesse nisi exitum occulti actu existentis ad apertum, sicut Democritus et Empedocleset Anaxagoras dicunt.”




lenge against Aristotle. According to Aristotle accidents cannot exist by

themselves. For Nicholas accidents are verae entitates, that is to say atoms,

and can exist by themselves. However they can only be perceived whencompounded or combined with other atoms.

1.2. A Qualitative Atom

What is the origin of this notion of qualitative atoms? It seems that

Nicholas derives this notion from the Aristotelian text itself, by assimilat-

ing Democritus’s atoms with Aristotle’s minima naturalia and Anaxagoras’s

homeomeron. Indeed, when Nicholas first alludes to something like an

atom in the Exigit ordo, he does not use the term “atoms,” but “minimanaturalia,” as if Nicholas wanted to make the atomist’s most basic

assumption acceptable to an Aristotelian:

For natural forms are divisible into their smallest units in such a way thatthese, when divided off from the whole, could not perform their properaction. And so, though they are visible when existing in the whole, theyare not visible when dispersed and divided or separated. For this is trueeven according to the mind of Aristotle when he says that natural beingshave maximum and minimum limits.18

For Aristotle, the quantity of a substantial form is determined betweentwo limits, one minimal, the other maximal, since for any given sub-

stance there are physically impossible sizes.19 Nicholas adopts this theory

but gives it a new and different meaning. Aristotle does not admit the

existence of discrete corpuscles and, more importantly, according to

him, minima only have a potential existence. These minima cannot

exist separately and have no physical existence in act.20 Nevertheless, the

Aristotelian theory remains embryonic and limited. The Stagirite does

not use it to explain natural phenomenona and he seems to have the

refutation of Anaxagoras as its main goal. Nicholas takes advantage of

this unfinished nature of the theory. He uses the concept of minima to

18 Universal Treatise, p. 60, EO, p. 199, 42–46: “Nam formae naturales sunt itadivisibiles in minima quod seorsum divisa a toto non possent habere actionem suamet ita licet ipsa existentia in toto videantur, dispersa tamen et divisa seu segregata non

videntur. Hoc enim veritatem habet etiam secundum intellectum Aristotelis dicentis:entia naturalia esse terminata ad maximum et minimum.” It’s the first occurrence if

we accept Kaluza’s reconstruction which puts the end of the first prologue in the firstchapter. See Kaluza, Nicolas d’Autrécourt. Ami de la vérité , pp. 160–161.19 See Physics, I, 4, 187b 14–21.20 See Pyle, Atomism and its Critics. From Democritus to Newton, pp. 214–217.




introduce two main features of the atom; the atom can act only inside

an atomic compound and, when separated, the atom is not visible.

Hence, the atom can be known only by its action inside a compound.Aristotle never gave such a property to the minima.

The assimilation of atoms to minima is not unique to Nicholas and

can already be found in the writings of Albert the Great. When Albert

presents the antiqui ’s theories about the number and nature of the ele-

ments, he curiously mixes Anaxagoras’s homeomeron and Democritus’s

atoms. Indeed, he credits Democritus with the thesis according to which

(1) atoms are the smallest elements of natural entities, (2) atoms are

similar to these entities (atoms of flesh or bone are similar to flesh or

bones), and (3) if we try to divide the atom, the compound will not beable to act.21 This account of atomism is interesting since it stresses the

atom’s capacity to act inside a whole and the incapacity of this whole

to act when atoms are subtracted (that is, when we try to divide them).

This explicit assimilation of atoms to minima naturalia is an important

link to understand how we can pass from the Democritean unqualita-

tive atom to the Ultricurian qualitative atom.

Nicholas is not at all a Democritean when he claims that atoms are

qualitative.22 Nevertheless, Nicholas is never explicit on this point. The

21 Cf. Albert the Great, De generatione et corruptione [ Hossfeld], p. 120, 44–55: “Demo-critus autem videbat quod omnia naturalia heterogenia componuntur ex similibus sicutmanus ex carne et osse et huiusmodi, similia vero componuntur secundum essentiam exminimis quae actionem formae habere possunt, licet enim non sit accipere minimum inpartibus corporis, secundum quod est corpus, quod autem non accipi minus per divisio-nem, tamen est in corpore physico accipere ita parvam carnem qua si minor accipiatur,operationem carnis non perficiet, et hoc est minimum corpus non in eo quod corpus,

sed in eo quod physicum corpus, et hoc vocavit atomus Democritus.” Minima have therole of the form according to Albert, but this problem is not solved in Nicholas’s theory.The assimilation of atoms and minima can also be red in Nicholas Bonetus, OFM, DeQuantitate, in Nicholae Bonetti vir perspicassimi quattuor volumina: Metaphysicam videlicet, naturalem philosophiam, praedicamenta, necnon theologiam naturalem [Venise, 1505] f. 81rb: “Respondettibi Democritus quod non, sed omne continuum actu finitum potest resolvi usque adindivisibilia simpliciter, et in hoc videtur concordare Aristoteles de continuo naturaliquoniam in primo physicorum 3 commento et 2 de anima 42 dixit quod est devenireusque ad minimam carnem.”

22 The other ancient atomism known to medieval philosophers, the Epicureanone, also claims that atoms have no quality. See Cicero, De natura deorum, [Van denBruwaene], vol. 2, II, 94, p. 119: “Isti autem quemadmodum adserverant ex corpusculis

non colore, non qualitate aliqua quam poiotèta Graeci vocant, non sensu praeditis sedconcurrentibus temere atque casu mundum esse perfectum, vel innumerabiles potiusin omni puncto temporis alios nasci, aliso interire. Quod si mundum ef ficere potestconcursus atomorum cur porticum, cur templum, cur domum, cur urbem non potestquae sunt minus operosa et multo quidem faciliora.”




first time he deals with this question, he simply asserts that atoms have

different natures:

If the question is raised whether the atoms are of the same kind or are ofdifferent kinds one must say, of differents kinds. But the means of provingthe different kinds will perhaps become apparent later.23

Unfortunately, Nicholas never fully explains what this means. Only

once he repeats his assertion without offering further details.24 On at

least two occasions, he explicitly describes two varieties of qualitative

atoms: white atoms and fire atoms.25 Moreover, when dealing once

again with the question of atomic differences, Nicholas provides some

precious interpretative keys:It should be observed that the eternity of things could be understoodin one of two ways. One is that they would always remain integral assome composite whole, just as they are now. For example, in Socratesthere are many realities of different natures joined together, such as flesh,bone, and soul.26

23 Universal Treatise, p. 69, EO, p. 205, 15–20: “Si autem quaeratur de illis atomali-

bus, an sint unius rationis vel alterius, dicendum quod alterius; sed ex quibus prob-etur diversitas rationum inferius forsitan apparebit.” We may assume that atoms bythemselves have no quality but when put in the compound, they change their natureand become active.

24 Cf. EO, p. 251, 5–8: “Unde de qualibet re non erit nisi punctus et sicut torchiafit non nisi sicut aggregatum ex pluribus diversarum rationum, sic lumen est aliquidaggregatum ex pluribus diversarum rationum.” We must admit that it is hardly imagin-able that light is compounded from atoms different by nature. This text would attest aDemocritean atomism. But the following is not at all Democritean.

25 Cf. EO, p. 189, 26–29: “Sic hic habeo media satis probabilis ad concludendumquod conclusio de aeternitate rerum est probabile, sed quia non possum ostendere illasmodiculas albedines ad modum granorum ire et venire, aliqui forsan discrederent; non

tamen propter hoc est negandum.”; p. 259, 3–8: “Propter quod sciendum est quod utinnui aliqualiter supra virtus calefactiva perfectissima copulatur quantum ad operaricum igne perfectissimo, ita quod operatur ubi est ignis perfectissimus, si sit materiadisposita quam corpora ignita subintrant, et quanto magis subintrant, ei penetrant,supposita identitate materiae, tanto virtus calefactiva perfectior operatur.”

26 Universal Treatise, p. 141 (translation modified), EO, p. 251, 44–48: “Et adver-tendum quod secundum alterum duorum posset intelligi aeternitas rerum vel quodremanerent semper sub integritate alicujus totius copulati sicut nunc sunt, ut verbigratia, in Socrate sunt multae realitates diversarum rationum copulatae ut caro, oset anima”. The second part of the alternative opened by the ‘vel’ is the following,p. 252, 6–10: “Alius esset modus intelligendi aeternitatem in rebus secundum viamsegregationis ut nullicubi esset Socrates per modum totius copulati, sed alicubi esset

albedo ejus causata, alibi ejus virtus et sic de aliis, et tandem revoluto circulo magniorbis, iterum fieret congregatio; et horum hororum recipe probabiliorem”. The problemunder discussion here is about eternity. See Kaluza, “La récompense dans les cieux.Remarques sur l’eschatologie de Nicolas d’Autrécourt,” p. 90, particularly the note 15which deals with these questions.




We already know that the soul is an atom. This was also the case for

Democritus.27 What is new is the existence of atoms of flesh and bone.

In other words, Nicholas identifies atoms with homeomeron, as we havealready said, and this offers a key for understanding the shift toward a

qualitative atomism.

We have seen how Albert in his commentary on De Generatione et

corruptione dealt with atoms of flesh and bone as little particles of flesh

and bone ( parva caro ) and this presentation of Democritus’s ideas no

doubt prepared the way for the Ultricurian confusions.28 Nicholas even

uses the same example, attributed by Albert to Democritus, of atoms

of flesh and bones:

This is true of a simple effect but not of one composed of things of dif-ferent natures. In a complex effect, it would be otherwise, because thenin truth there are different beings there, as in Socrates there are bones,flesh, soul, blood, etc.; and so one must posit different causes there. 29

Nicholas’s equivocal notion of thing ( res ) complicates and undermines

this text. But here, we may assume that Nicholas is dealing with atoms

and compounds because of the use of being ( entia ). Of course, atoms

are the real beings ( vera entia ).30 Similarly, this qualitative conception of

the atom could have been influenced partly by the Mutakallimun, whodiscussed atoms in terms of substance and accident. The Mutakallimun

even claim that atoms of snow are white. Something similar crops up

in the Ultricurian classification of atoms.31

27 For Democritus, see Aristotle, De anima, I, 2, 404 a 1–10.28 See the quotation in the note 21.29 Universal Treatise, p. 148 (translation modified), EO, p. 256, 21–24: “Et hoc est

verum de effectu simplici, non de composito ex rebus diversarum naturarum, secusesset in alio, quia tunc quantum ad veritatem ibi sunt diversa entia sicut in Socrateossa, caro, anima, sanguis etc.; et ideo ibi oportet ponere diversas causas.”

30 Cf. EO, 225, 43–46: “Sic hic verum est quod omnis entitas vera, quae in me estnunc, semper fuit et semper erit, sed non erunt secundum indistinctionem subjectivamut nunc sunt.” On this equivocal notion see Kaluza, “L’éternité . . .,” pp. 234–238, “Larécompense dans les cieux . . .,” p. 90, n. 15. Also, Dutton “Nicholas of Autrecourtand William of Ockham on Atomism, Nominalism and the Ontology of Motion,”p. 66, n. 8.

31 Moses Maimonides, Dux seu director . . . [ Venice, 1516 ], f. 32v: “Sed tale accidens,secundum ipsos, invenitur in quolibet atomorum ex quibus componitur illud corpus,

verbi gratia, albedo partis nivis non invenitur in suo universo solummodo sed in

qualibet substantiarum illius nivis est albedo et idcirco invenitur albedo in compositoex illis.” There is nevertheless an important difference since for Mutakallimun atomsare substances which can have accidents. Nicholas tries to go beyond the accident-substance dichotomy.




It seems clear that Nicholas’s atomism draws from both Democritus,

as transmitted and deformed by the Aristotelian tradition, and from the

Mutakallimun. It is important to note, however, that Nicholas’s goal isto build a consistent atomistic theory and not simply to set Democritus

against Aristotle. He is looking for the best explanation of natural phe-

nomenona and he believes that the best explanation is atomistic. And

this means that he must go beyond the few clues left in the writings of

the antiqui in order to complete his theory.

1.3. The Unity of Atomic Flow: Principles of Atomic Composition

The interpretation of matter as an atomicfl

ow is tied to the critiqueof the metaphysical notion of form and matter, or act and potentiality.

By rejecting prime matter as a substratum, Nicholas is also rejecting

the idea that the form is an actualization of matter and, a fortiori , he is

rejecting the idea of substantial form. The problem Nicholas then faces

is to explain what gives unity to atomic compounds. Since Nicholas

admits the possibility of monsters in nature, understood as inconsistent

( disconveniens ) atomic compounds,32 he must provide an account of what

produces consistent compounds. The answer is two-fold. On the one

hand, Nicholas defends an atomistic mereology that relies on the par-ticular nature of different atoms. On the other hand, his answer relies

on the astral causality that determines the atomic composition.

Since accidents are the result of corpuscles, it is necessary to find a

way of allowing a distinction between accidental and essential proper-

ties. Accidents are atoms which superficially modify the compound from

which they can be subtracted. Essential properties are necessary to the

existence of the compound. In other words, after giving an atomistic

account of matter, Nicholas must give the same account of form.

Hence, he introduces a distinction between atoms which are essentialto the whole, such that their separation leads to the destruction of the

compound, and atoms which separation does not alter the nature of the

compound. Atoms of the first kind allow an operation or a movement

(as is the case for the soul).33 Atoms of the second kind are nothing

but qualities like whiteness:

32 EO, p. 205, 43–44: “In rebus materialibus extra, propter disconvenientiam

<et convenientiam> in congregatione dicuntur interdum monstra interdum benecomposita.”33 On the atomic status of the soul, see Democritus, quoted in Aristotle’s De anima,

I, 2 (see note 27) and the Mutakallimun, quoted by Moses Maimonides, Dux seu




On the basis of the foregoing discussion it might be said that theseaccidents are only certain atomic particles, and that they are not in thesubject except as a part in the whole, but a part, one must understand,that is essential and necessary to the whole. Still more can these parts besayed about the substance of the subject. Upon the departure of theseatoms in the substance, what is called the functioning of the thing, andthe movement which previously appeared in the thing, cease to appear.Upon the departure of the other atoms they do not go away. These oughtmore properly to be called accidentals of the subject; yet they too are asa part in the whole.34

Among atoms which are parts of a whole, we have to make a distinc-

tion between necessary parts, such that the whole cannot exist without

them, and accidental parts, such that the whole can exist without them.Since each atom is qualitative, this distinction is similar to the one

between essential and accidental properties. Essential properties allow

the compound to operate. Generation is explained by the gathering of

such atoms, and alteration by the separation of atoms of the second

kind. With this distinction, Nicholas claims to be able to dissolve all

the false Aristotelian problems linked to the notions of quiddity, form,

and so on.35

But it might seem odd that Nicholas still uses the notion of sup-

positum at all. What does this Aristotelian notion mean in an atomistic

context? Nicholas is not very clear on this point and, at first glance, it

seems that he should give up such concepts as species and genus. On at

least one occasion, he explains what an atomistic conception of form

and species might mean in the case of human beings. After reducing

generation and corruption to local movement, he writes:

Perhaps there is something there which connects and retains the indivisibles in this union, as a magnet does with iron. The stronger the force

director . . . [ Venice, 1516], f. 32v: “Quidam autem eorum dicunt quod anima componiturex atomis quae sine dubio conveniunt in accidente per quod sunt anima et quidamdixerunt quod ipse virtutes et ipse substantie coniuguntur cum substantiis corporeis etnon evadunt quin rationem anime ponant accidens.”

34 Universal Treatise, pp. 68–69, EO p. 204, 38–45: “Secundum praedicta dicereturquod talia accidentia non sunt nisi quaedam corpora atomalia, nec sunt in suppositonisi sicut pars in toto; verumtamen intelligendum quod est quaedam pars essentialis etnecessaria toti; et istae magis possunt dici de substantia suppositi. Et ista sunt atomaliaquibus abeuntibus non apparet amplius illud quod dicitur operatio rei, nec motus qui

prius appareret in re; alia sunt quibus abeuntibus non abeunt; ista magis debent diciaccidentalia suppositi; sic tamen sunt sicut pars in toto.” This text is examined byKaluza, “Les catégories dans l’ Exigit ordo de Nicolas d’Autrécourt,” pp. 101–102.

35 EO, p. 204.




of this thing, the longer the subject survives as a subject. If there werea force of this kind, it would be called the quasi-formal principle of thething.36

When local movement leads to the gathering of mutually consistent

atoms, they produce a suppositum which, according to Nicholas, simply

means a totality of essential atoms. Nicholas then adds that one of

these essential atoms plays the role of a magnet and attracts the other

atoms. The gathering of the atoms in the compound depends on the

strength of this atom, which he refers to as a “quasi-formal principle.”

When it ceases to act, the other atoms spread out the whole.

We may assume that this atom is not the only vital principle in liv-

ing beings, since the death of animals is not the dispersion of atoms,but the departure of some essential parts. Nicholas is more probably

dealing with the problem of decomposition. In each compound, there

is an atom whose function is to attract atoms and keep them together.

As long as it continues to operate, the compound exists as a perceptible

whole. When it stops operating, the compound is disintegrated. The

fifth proposition of the Mutakallimun, as reported by Maimonides,

could well have influenced Nicholas here.37 The Ultricurian innovation

is the link between this formal atom and an astral causality. Indeed,

the last level of Ultricurian atomism relies on the interaction between

lunar and sublunary worlds.

2. From Atoms to World: The Nature of Atomic Compound

Nicholas still must explain how his atomism provides a convincing

account of natural phenomenona. More specifically, the challenge he

faces is to explain how consistent atomic compositions can be possible.

36 Universal Treatise, p. 63, EO, p. 201, 9–11: “Et forsan sicut adamas ferrum, ita estibi unum quod connectit et retinet in tali colligatione ipsa indivisibilia, et secundumhoc quod est majoris vigoris magis durat illud suppositum in ratione suppositi; et illud,si sic esset, diceretur quasi principium formale rei.”

37 Moses Maimonides, Dux seu director . . . [ Venice, 1516], f. 32v: “Idem dixerunt incorpore mobili quod quaelibet substantiarum suarum separatarum movetur et idcircototum movetur. Similiter secundum ipsos vita invenitur in qualibet atomo vivi, sic etsensus invenitur in qualibet substantia separata sentientis, quia sensus et intellectus etscientia secundum ipsos sunt accidentia sicut albedo et nigredo, sicut explanavimus

in eorum opinio sed obiectum fuit eis in anima. Convenerunt autem in hoc quod estaccidens inventum in una substantia separata de universitate substantiarum ex quibuscomponitur homme et illa universitas dicitur habere anima qua substantia separataest intra ipsam universitatem.”




As we have seen, Nicholas reduces all change to local movement, but

what are the principles behind local movement and what is the role

of matter?

2.1. Matter, Disposition and Virtus

A constant thesis in the Exigit ordo is that matter, understood as an

atomic flow, is designed to prepare or to receive a formal atom called

virtus which is an essential part. This atom constitutes the principle of

unity and operation of the compound, in other words its form.

Nicholas gives a clear example of his theory when he discusses com-

bustion.

38

How isfi

re produced? It needs a virtus ignitiva, in additionto the heat that is produced by a virtus calefactiva. If one wants to escape

the standard criticisms of virtues, summed up in Molière’s joke about

the “vertu dormitive de l’opium,” it is necessary to explain how they

operate.39 While these virtues or forces are a particular kind of atoms,

they never act alone. According to Autrecourt, these virtues are trig-

gered by superior agents, the celestial bodies. Atomic action requires a

copulatio between the star and the atom. The force found in the atomic

compound is a formal effect of the same force found in the appropriate

star.40

Light can heat because the atoms which compose it follow themovements of the stars. Hence, the celestial body is an ef ficient cause

that acts upon an atom which is the formal cause in the compound.

Still this is not enough. How can we explain why a bellows, when used

gently, can stir up a fire, while using it to create a vigourous wind might

well extinguish the fire? What is the role of wind in the generation of

fire? According to Nicholas, it provides a kind of help. When the wind

is gentle, it expels from the compound atoms liable to delay combustion

(that is, atoms that would delay the copulatio between the atomic and

celestial virtues). Conversely, a strong breeze will expel atoms liableto facilitate the copulatio, those hot atoms which are the condition for

38 EO, p. 257, 3–9: “Potest tamen dici quod illa virtus inexistit alicui superiori agentiet haec virtus habet quamdam copulationem in operari cum effectu formali, ita quodquando approximatur ignis alicui calefactibili, saltem <si> ignis talis sit perfectus, tunccalefacit illud calefactibile, et secundum hoc quod est calefactibilius, copulatur sibi virtuscalefactiva perfectior; et secundum hoc est falsum dicere quod lumen calefacit vel quodmotus nisi quia ista assequuntur virtutes quaedam calefactivae.”

39

For a contemporary outlook on this topics, see Gnassounou & Kistler (eds.), Lesdispositions en philosophie et en science.40 For a more detailed account of the celestial causality, see Grellard “La causalité

chez Nicolas d’Autrécourt.”




the forces to act. These atoms, it would seem, dispose or prepare the

composite towards certain behaviors, in other words, they are similar

to material causes.41

From this example, Autrecourt applies the model of a copulatio be-

tween stars and atomic virtues to different natural phenomenona, and

first of all to generation and corruption. Hence, he rejects spontaneous

generation which he assimilates to normal generation. In both cases,

we have an equivocal generation, that is, the reception of atomic vir-

tues into a prepared matter. The atoms of a worm can be received

in a putrefied matter; similarly, the sperm of a man or a donkey is

not the cause of the generation of another man or donkey but only a

requisite part. Sperm is the material condition, the atomic virtus in theman is the formal cause and this formal cause still requires the ef ficient

action of the star.42 Usually, if these material and formal conditions are

lacking, the star cannot act on the compound. Nevertheless, we may

41 EO, p. 257, 9–14: “Ventus autem facere videtur ad generationem ignis ut in sibillooris vel in sibillo artificiali, praecipue quando est moderatus; et hoc est quia cum tali

vento recedunt aliqua corpora quae impediebant receptionem ignis intali subjecto, ita

quod virtus ignitiva perfecta non copulabatur sibi; quando vero est immoderatus, tuncetiam removentur illa quae disponebant subjectum ad igneibilitatem.”42 EO, pp. 257–258: “Ex illis regulis suprapositis quod unus effectus non precedit

nisi ab una causa etc., sequitur quod anima unius hominis qualiscumque sit non estproducta ab alio homine quia seqitur: ab eo producta, ergo praecise producta; igituralterum non est ignobilius quia causa numquam potest excedere suam perfectionemin producendo ut dictum est; neque aeque perfecta quia, ut dictum est supra nonest dare duos effectus aequales in una specie; igitur erit perfectior; et hoc est falsum,immo plerumque imperfectior reperitur quantum ad omnes virtutes et operationes.Producitur igitur ab aliquo agente superiori; non dico ab agente perfectissimo nisi siteffectus perfectissimus ut probatum est superius. Et si homo, qui diceretur producenssit perfectior, hoc accidit qua tantum videtur facere imperfectior circa generationem

hominis sicut magis perfectus. Et sicut dictum est de homine sic intelligendum de asinoet omnibus aliis, et per consequens nulla erit productio univoca quia vel causa essetimperfectior quod non potest esse, vel aeque perfecta quod improbatum est supra, velperfectior, et hoc accidit sibi in quantum causa effectus ut statim dictum est. Homotamen bene potest esse aliquorum praerequisitorum ad quae forma hominis habetordinem ut seminis et aliorum; homo, id est, aliqua virtus inexistens homini. Et con-siderandum est juxta praedicta quod illa est falsa, quod virtus aliqua producit effectumperfectiorem in subjecto perfectiori et imperfectiorem in imperfectiori quia virtus aliquanon habet <nisi> unum effectum ut dictum est supra, et ideo vel illum producit velnullum. Sed verum est quod, nisi subjectum sit conveniens ad recipiendum illum, nonproducet in illo subjecto, sed cum illo subjecto copulabitur in operari virtus imperfec-tior. Subjectum enim indispositum nihil immutat de natura agentis quin, si producat,

producat secundum convenientiam suae naturae.” On the sperm as a preparation, seethe quotation in the following note. For a different account of the role of the sperm,according to the medical theories, see Jacquart, “L’influence des astres sur le corpshumain chez Pietro d’Abano”.




assume (even if Nicholas does not explicitly say so) that monsters are

produced when one of the material or formal requisites are partially

lacking in such a way that the stars are still capable of acting on thecomposite, but the result will not be normal. Hence, all atomic changes

are the result of a celestial ef ficiency into a formal or accidental atom

in a preexistent atomic flow, materially disposed to receive this atom.

However, within this celestial causality, material conditions have a role

to play in defining the resulting species. At first glance, Nicholas seems

to assume that such a notion is useless. He first underlines that the

trans-temporal identity of the same individual is never complete. The

young Socrates and the old one are not exactly the same individual.

This person is continuously modified by the movements of the atoms,and is never identical to himself between moment t and moment t+1.

If a child could become an old man instantaneously, we would not

be able to identify him as the same person. But since these changes

are continuous and since the general order of the atoms remains the

same, we can speak about these various atomic compositions as the

same person.43 From this, we may conclude that according to Nicholas

all sorts of form, even the substantial form, are reduced to the figures

43 EO, pp. 251–253: “Nunc posset sic intelligi quod Socrates sic esset aeternus quodsic semper esset sicut nunc est, sic intelligendo quod, cum Socrates non sit omninoidem sibi puer et senex, immo aliquo modo variatu de hora in horam, intelligeturquod quando desineret esse sub una dispositione utpote sub dispositione pueritiae, alibiesse sub eadem dispositione et postea alibi et sic semper usque ad circulum, donecfuisset ubique et non solum secundum horas vel dies, sed etiam secundum momenta,ut statim cum hic desineret esse sub una dispositione, inciperet esse alibi sub eadem.(. . .). Et ex praedictis potest dici, si quaeratur an puer sit homo, si appellatio recipiatura figura magis in genere, tunc quia habet figuram ad modum hominis, caput sursum,pedes deorsum, ut sic deberet dici homo. Sed appellationem ex hoc recipere non

est conveniens, sed magis ex operatione et virtute inexistente; nunc sicut non habetoperationem, ita posset dici non habere virtutem et solum deberet dici quod habet virtutem ut operatur. Et ita de puero unius diei, si moriatur, potest dici quod nec habuit virtutem ridendi neque ratiocinandi, licet bene habuerit aliquas praeparationes remotasad hoc sicut est in spermate. Consuetum tamen est dici quod homines et <illi sunt>ejusdem speciei; et causa consuetudinis forsan est ut homines magis compatiantur eis.Et si puer non habet virtutes hominis, conveniens est inquirere an virtus existit suoquiete, verbi gratia quando aliquis non movet lapidem vel aliquod aliud grave, utrumhabeat virtutem movendi lapidem. Videtur quod non, quia tunc ejas esse esset otiosumpro illo tempore. Videtur contra, quia tunc nullus deberet eligere quietem nec virtusnaturalis cum fatigat naturam, natura non deberet naturaliter ad hoc inclinari, cum inhoc consisteret ejus destructio; et ideo posset satis dici quod remanet sub quiete, nec

est otiosum cum in hoc accidit conservatio ejus in supposito, alias causaretur alibi ethic desineret esse et loco ejus esset imperfectior. Unde corpus caeleste cum quibusdamconcurrentibus spritibus causat talem virtutem qui continuatio opere utpote motionisgravis recedunt, et loco eorum veniunt magis imperfecti.”




or to the general appearances of a thing. If somebody has a head at

the top and two feet at the bottom, we can call him a man. Nicholas,

however, concedes that such a conception of form is not enough. Rather,he defines form as an operation or virtus (that is an atom) which allows

us to classify individuals into species and genus. Moreover, it is not

only this operation that allows us to classify individuals into species,

but also the tendency of these types of atoms to act in certain ways.

Otherwise, a stillborn child could not be called a human being, since

he would lack the ability to laugh and reason. If we do not want to

call it a human being out of mere custom, we have to assume that

the appropriate virtue is already present, if latent, inside the material

preparation (the sperm) and that this virtue is just waiting to be movedby an astral causality.

We can see that from an external point of view, atomic movements

depend upon astral movements. But there is also an internal condition

to be considerered, the inter-atomic void.

2.2. Void as a Condition of the Atomic Movements

The treatise De vacuo is one of the best known sections of the Exigit ordo.44

Against Aristotle, Autrecourt argues for the possibility of the void. Nota separate void, but an inter-particulate void, that is, a void internal to

the compound. Aristotle attributes this doctrine to Democritus in the

fourth book of the De Caelo. Nicholas uses it mostly to give an account

of the phenomenon of condensation and rarefaction.

It is well known that Nicholas, like the Mutakallimun,45 explains

atomic movements by the motion of an atomic body through atomic

places. In De vacuo, Nicholas demonstrates that these atomic places

are empty. Contrary to the scholastic doxa, motion in a void does not

44 See specifically, Grant, “The Arguments of Nicholas of Autrecourt for the Exist-ence of Interparticulate Vacua,” pp. 65–69; Kaluza, “La convenance et son rôle,” pp.100–103; Grellard, Croire et Savoir . . ., pp. 211–218.

45 Moses Maimonides, Dux seu director , f. 32r: “Dixerunt quod motus est mutatiosubstantiae separatae de numero atomorum unius scilicet singularis substantiae adaliam substantiam propinquam et ex hoc sequitur quod non est unus motus velocioralio. Et secundum hanc positionem dixerunt quod vides duo mobilia pervenire ad duosterminos diversos in remotione in eodem tempore, non est quia unus motus est velocior

alio sed causa eius est quia mobile cuius motus dicitur tardior habet plures quietes inspatio suo quam illud quod dicitur velocius. Et ideo dixerunt in sagita fortiter ab arcuemissa quod sunt quietes in spatio motus ipsius.” See Murdoch, “Atomism and Motionin the Fourteenth Century;” Grellard, “Nicolas d’Autrécourt.”




imply instantaneous velocity since every movement is a relative one,

and depends on the time during which each atom rests. Moreover, each

atom moves place after place, by a local movement. From these assump-tions, Nicholas explains condensation, that is, the quantitative alteration

of bodies, by the local movement of atoms inside the compound. The

classical example of the rarefaction of new wine is explained not by

the generation of a new quality (which would mean the arrival of new

corpuscles), but by the separation of parts already present in the body

and the concomitant introduction of a void between these atoms. In

the same way, water seems denser than air because there is more void

between its parts. Hence, in the case of liquid and gas, the mutual

proximity of the parts give an account of density, and their distanceexplains the rarity.46 In the case of solid bodies, Nicholas offers two

different explanations: alteration is caused by an increase or decrease

of the void, or by the arrival or the departure of new atoms.47

Against Aristotle, Nicholas asserts that nothing is absolutely heavy or

light. These concepts are relative and depend upon the inter-particulate

void and the proximity of atoms to one another. Atoms can move within

a composite because of the presence of an inter-particulate void, other-

wise two bodies could be in the same place at the same time, which

is universally believed to be false.48 This is how the antiqui explained the

possibility of a local increase of a body. How else can we explain, for

example, how our limbs grow in size, if not by the addition of atoms?

Nicholas gives the following explanation of this phenomenon. Let us

assume that a body is composed of four atoms, abcd . These four atoms

are indivisible. Each of them has a specific quantity since each has a

situalitas and they are mutually apart from one another. Let us assume

also that between each atom there is an empty space.49 Body abcd will

grow as additional atoms come tofi

ll in spaces that grow between abcd .For example, atom e will insert itself between a and b to produce a

new body, aebcd . The ability to receive new atoms depends on a kind

of dilation which creates empty spaces within the composite. But there

is no modification of the atoms themselves, that is, no modification of

46 EO, p. 218, 11–31.47 EO, p. 217, 33–37: “Nam non dicimus quod densum sit per generationem alicujus

novae qualitatis quae prius non erat, sed solum est densum per recessum corporum ut

in lana, vel quia partes coeunt, id est, quod magis propinque se habent quam prius. Etrarum non erit nisi quia partes illius corporis magis distant quam prius, et ita densum vel rarum non est nisi per solum motum localem partium.”

48 EO, p. 221, 42–222, 7.49 EO, p. 208, 10–18.




their own properties and quantity. The compound appears to grow

because it comes to be made up of additional atoms, atoms that, on

their own, are imperceptible.50

Nicholas may have John Buridan’s argument against atoms in mind.

In his Questiones super De Caelo, Buridan wants to prove that an atom

must be heavy and, therefore, must be divisible.51 His purpose is to

show that any quantified body is liable to increase or decrease. If we

can show that an atom is a body which can sometimes be bigger and

heavier, and sometimes smaller and lighter, Buridan believes he can

prove his point: an atom possesses quantity and is divisible. On this

point, Buridan refutes an adversary who claims that elements (water,

air, earth, fire) are composed of atoms of different weights, such thatatoms of air are less heavy than atoms of earth. Buridan begins with an

Aristotelian argument. Imagine a body composed of three atoms abc. If

this body condenses, it will be smaller, but it will still be composed of

these same three atoms. We must conclude that the individual atoms

have lost weight. Secondly, if we examine the status of quantitative

external parts outside each other, and if we assume that an earthly

body is composed of three atoms abc, to which we add a fourth similar

part d , we must claim that abcd is more extended and heavier than abc.

Thirdly, if we examine the phenomenon of calefaction, we are faced

with three assumptions: either, 1.) calefaction is produced by the com-

pression or condensation of the parts such as they become closer to each

other; 2.) calefaction is produced by the addition of a qualitative degree

of hotness (as in the theory of intension and remission of forms); or,

3.) calefaction is produced by the addition of an atom of hotness, that

is, a quantitative part. Against this last possibility (the other two do not

interest him in this case), Buridan claims that a hotter body should be

heavier and more extended. This thesis of the addition of hot corpuscles

50 EO, p. 221, 12–24: “Pone corpus aliquod compositum est ex indivisibilibusquattuor a b c d ; dico quod inter ista indivisibilia sunt vacuitates; perpone inter a et b per unum indivisibile; dico quod ad adventum rei convenientis quasi dilitabant se, itaquod quodammodo elongabant se, puta a b per vacuitatem duorum indivisibilium, etibi recipient [se] intra se illud quod est conveniens suae naturae, et sic fiet augmen-tatio. Et intellectus hujusmodi propositionis illius: quaelibet pars aucti est aucta, noncomprehendit omnino partes primas corporis aucti ut sit intellectus quod indivisibileaugetur; sed intellectus est quod post augmentationem de qualibet parte composita

demonstrabili ad sensum comparata ad talem, quae antecedebat augmentum, verumest dicere quod est major, ut puta manus post augmentum est major quam ante aug-mentum, et sic in aliis.”

51 John Buridan, Expositio et Quaestiones in Aristotelis De Caelo [Patar], L. III, q. 1, pp.513–518.




is indeed Nicholas’s thesis.52 But for Nicholas, the link between exten-

sion and heaviness is not necessary as in Buridan’s argument, because

Nicholas contends that a body can be more extended and lighter, if it iscomposed of more empty places. Hence, the arrival of atoms of hotness

neither increases the composite’s extension nor its weight. Likewise, the

modification of a body’s size does not imply any modification of the

atoms, but only of the empty places which separate them.

3. Conclusion

From the celestial bodies, which simplicity renders them eternal, tothe sublunary world composed of atoms, Nicholas offers a completely

atomized cosmos. In his attempt to offer a non-Aristotelian answer to

the twin problems of eternity of the world and of generation and cor-

ruption, Nicholas constructs an atomistic theory using parts he could

have drawn from Aristotle’s own writings in natural philosophy and from

the Mutakallimun. He then attempts to elaborate a non-metaphysical

physics founded on very few principles: qualitative atoms, local motion,

and celestial causality. In this context, atomic flows provide an account

of the make-up of these compounds and the possible changes theymay undergo in the sublunary world. But at the same time, Nicholas’s

atomism testifies to the dif ficulty that medieval natural philosophers had

to face when they sought to escape Aristotle’s most basic frameworks.

Indeed, by attempting to rewrite Aristotle’s theses in atomistic terms,

Nicholas effectively preserves Aristotle’s framework, a framework that

would remain essentially unchallenged for at least another 200 years.

Due to the condemnation of 1347, it is not possible to know how

Nicholas may have contributed to the rejection of the Aristotelian

model. But Nicholas’s ingeniousness in the development of an alterna-tive physics deserves, in the words of Edward Grant ,“a tribute to his

courage and intellectual acumen.”53

52 EO, p. 257: “Supra in tractatu de aeternitate rerum dixi quod quando ignis diciturproduci quantum ad veritatem non est nisi adventus aliquorum corporum calidorum,et frigidditas per recessum illorum et adventum corporum frigidorum ut in aqua calidaquae sibi derelicta revertitur ad naturam priorem, scilicet frigiditatem, per recessum

calidorum corporum et accessum frigidorum.” On the Ultricurian approach of intensionof forms, see Grellard, “L’usage des nouveaux langages d’analyse dans la Quaestio deNicolas d’Autrécourt. Contribution à la théorie autrécurienne de la connaissance.”

53 Grant, “The Arguments . . .,” p. 68.



WILLIAM CRATHORN’SMEREOTOPOLOGICAL ATOMISM

Aurélien Robert

Little is known about Crathorn’s life and career, except that he was a

Dominican friar who lectured on Peter Lombard’s Sentences in Oxford

around 1330–32—his only surviving work.1 He’s often considered by

recent scholars as a secondary witness to the most important debates infourteenth-century philosophy, but rarely as an inventive thinker. None-

theless, isolated arguments have been carefully examined, notably his

solution to scepticism2 and his criticism of William of Ockham’s theory

of mental language.3 But, concerning the question of the existence

of indivisibles in a continuum, he’s generally mentioned as a simple

doxographer, even if John E. Murdoch recognized him as a defender

of an original theory of speed in such a context.4 Contrary to this

common reading, we would like to show how singular and interesting

is Crathorn’s atomist position, which can neither be reduced to that ofhis Oxonian predecessors Henry of Harclay and Walter Chatton, nor

identified with his Parisian contemporaries Gerard of Odo, Nicholas

Bonet and Nicholas of Autrecourt.

Indeed, Crathorn doesn’t limit himself to mathematical analysis of

the divisibility of a continuum, but puts forth the foundations of a

genuine atomist physics. In this theory, an indivisible is not conceived

as a purely mathematical point anymore, but rather as the ultimate

component of reality, from which a conception of motion rivalling the

one defended by Aristotle in his Physics can be derived. At the heartof Crathorn’s physics, the indivisible thus acquires a new ontological

status: it is a thing ( res ), actually existing ( existens in actu ) and not only

1 The Questions on the book of the Sentences have been edited by Hoffmann, Quaestionenzum ersten Sentenzenbuch. For biographical elements, cf. Schepers, “Holkot contra dictaCrathorn: I. Quellenkritik und biographische Auswertung der Bakkalaureatsschriftenzweier Oxforder Dominikaner des XIV. Jahrhunderts;” Courtenay, Adam Wodeham: An Introduction to his Life and Writings. For a general survey, see our “William Crathorn.”

2

Cf. Pasnau, Theories of cognition in the Later Middle Ages.3 Cf. Panaccio, “Le langage mental en discussion: 1320–1335;” Perler, “Crathornon Mental Language.”

4 For example, Murdoch, “Atomism and Motion in the Fourteenth Century.”



128 aurélien robert

potentially, which possesses a kind of extension ( quantitas dimensiva ) and

a certain nature or perfection ( quantitas perfectiva ). Moreover, much of

Crathorn’s effort was directed towards demonstrating that the numberof indivisibles in nature is finite, if not countable. Therefore, as he will

conclude, the indivisibles are real parts of a continuum.

Our aim in this paper is to show how Crathorn brought out an impor-

tant turn in his natural philosophy, from the indivisible considered as a

mathematical point to the atom conceived as a physical entity. Needless

to say that he isn’t the only one who endeavoured to conceptualize the

nature of indivisibles in a more physicalist fashion, but in some respect

he’s probably one of the most systematic atomistic thinker of the first

half of the fourteenth century even if, as we shall see, some theoreticalhesitations still persist in his view.

To understand Crathorn’s originality, it should be remembered that

his ontological analysis of the indivisible is based on two distinctive

features: it depends on a special conception of the part-whole relation;

and it is supported by a systematic use of the notions of place ( locus )

and position ( situs ). For this reason, we may call Crathorn’s theory a

“mereotopological atomism”.5 The notion of place is so central that

it will serve as a tool for reconstructing the physical nature of atoms,

that is to say their quantity (dimension and perfection); for, as we shall

see, indivisibles are primarily defined by the place they occupy in the

world. Consequently, this mereotopological strategy will also allow him

to avoid falling into some of the traditional blind alleys of indivisibilism,

e.g. the explanation of the composition of a quantity from extensionless

points and the problem of contact between indivisibles.

Before going through Crathorn’s arguments, we must mention how

important and recurrent is the topic of indivisibilism in the Questions

on the Sentences.

5 We borrow the term “mereotopology” from contemporary science and phi-losophy, using it in a slightly different way. For a contemporary definition, see Smith,“Mereotopology: A Theory of Parts and Boundaries,” p. 287: “Mereotopology . . . is

built up out of mereology together with a topological component, thereby allowingthe formulation of ontological laws pertaining to the boundaries and interiors ofwholes, to relation of contact and connectedness, to the concepts of surface, point,neighbourhood and so on.”



william crathorn’s mereotopological atomism 129

1. The Importance of the Continuum Question in

Crathorn’s Writings

The most significant occurrence of our topic is to be found in q. 3 of

the Questions on the Sentences after a long discussion about the nature of

natural knowledge and language (q. 1 and q. 2). As a consequence,

one shouldn’t be surprised by the gnoseological pretext of this first

long digression: does the wayfarer understand cum continuo et tempore.6

The few commentators who have studied the continuum question in

Crathorn have usually restricted their reading to this sole passage, but

many others are relevant in the Quaestiones.

As early as q. 1, Crathorn adopts atomist explanations of naturalphenomena as the diffusion of light when explaining the multiplication

of species through a medium.7 In q. 4, when questioning the possibility

of knowing God’s infinity, our topic reappears in the explanation of the

infinite.8 But the most important passages on atoms and continua can

be traced in Crathorn’s discussion of Aristotle’s categories, especially

in his long developments in q. 14 on quantity, and in q. 15, entirely

dedicated to the quantity of indivisibles.9 Finally, after developing his

own interpretation of Aristotelian categories, Crathorn turns to the

nature of time and motion in q. 16,10 where he defines more precisely

the central elements of what could be the foundations of an atomist

physics. Therefore, for a complete reconstruction of Crathorn’s theory,

one must at least take into account all these passages in which indi-

visibilism is at stake.

To be absolutely exhaustive, one should also look at the series of

forty-two quodlibetal questions found in a partially unedited manuscript

conserved in Vienna (Cod. Vindob. Pal. 5460).11 Most parts of the text

have been inserted in the Questions on the Sentences, but the manuscriptalso contains some interesting developments devoted to indivisibles, as

6 In Sent. q. 3, pp. 206–268: “Utrum viator intelligat cum continuo et tempore.” 7 In Sent. q. 1, p. 111. 8 In Sent. q. 4, pp. 289–293. 9 In Sent. q. 15, pp. 426–441: “Utrum aliqua res omnino indivisibilis possit esse

longa, lata et profunda.”10

In Sent. q. 16, pp. 442–459: “Utrum tempus sit aliquid positivum reale vel aliquares producta a deo vel ab aliquo.”11 For a brief description of the manuscript, cf. Richter, “Handschriftliches zu

Crathorn.”




another version of the question on the continuum12 and some discus-

sions on the infinite.13

In this short paper, we’ll limit ourselves to the edited Questions on theSentences, without ignoring the aforementioned relevant passages. To

begin with, we must examine the mereological principles on which the

atomist edifice is built.

2. The Mereological Composition of the Continuum

The gnoseological excuse for discussing the divisibility of continuous

quantities helps Crathorn to reveal afi

rst apparent paradox in Aristotle’sanalysis. If a continuum were composed of an infinite number of parts,

shouldn’t we accept that the wayfarer’s intellect should understand

the infinite when thinking of a continuous body? If not, could we still

say, as Aristotle himself asserts in the De memoria et reminiscentia,14 that

the intellect knows cum continuo et tempore?15 Though it may seem sophis-

tical, the formulation of the problem chosen in this context is not

totally innocent and already indicates his finitist presuppositions (that

a continuum is composed of a finite number of parts) and his reduc-

tive mereology (that a continuum is nothing but its parts). It is alsonoteworthy that when these two claims are combined, parts cannot be

just potential ones, but rather are actual components of the whole as

such. This is the reason why Crathorn presupposes that the wayfarer’s

intellect could know ( de jure ) all the actual parts of a continuous being.

12 In Sent. q. 8: “Utrum continuum componatur ex indivisibilibus, id est ex punc-tis.” This question has been partially edited in Wood, Adam de Wodeham, Tractatus de

indivisibilibus, pp. 309–317.13 In particular q. 9 on the nature of instants (ff. 39va–40rb: “Utrum instans secun-dum substantiam maneat idem toto tempore”) and two others implying the notionof infinite (q. 22 ff. 55vb–57va: “Utrum ex infinitate motus extensiva possit concludiinfinitas virtutis intensiva in primo motore”; q. 23 ff. 57va–59ra: “Utrum causa primapossit producere extra se aliquem effectum actu infinito”).

14 Aristotle, De memoria et reminiscentia, 1, 450a8–9.15 In Sent. q. 3, p. 206: “Utrum viator intelligat cum continuo et tempore. Quod

non videtur. Nullum continuum est intelligibile a viatore. Igitur viator non intelligitcum continuo et tempore. Consequentia patet. Probatio antecedentis, quia si aliquodcontinuum posset intelligi a viatore, sit illud a; aut est finitum aut infinitum. Si infinitum,non potest intelligi. Si finitum, cum continuum non sit aliud quam partes, sequitir quod

viator intelligens a intelligeret omnes partes eius et per consequens intelligeret infinita,quia omne continuum divisibile est in infinitum, igitur continet in se infinitas partes.Ad oppositum est Philosophus in primo libro De memoria et reminiscentia, ubi dicit quodintelligimus cum continuo et tempore.”




This leads Crathorn to a distinctive mereology, according to which a

whole is nothing more than the sum of its parts.

The main argument supporting this assertion is the following: if, froman ontological point of view, a whole were different from its parts, one

should consider it as a distinct entity; but two absolute things can exist

separately without any contradiction, at least thanks to God’s absolute

power. It would follow from these premises that God could conserve the

whole as existing while destroying its parts, which seems self-contradic-

tory.16 It looks rather as if the whole had no ontological existence, for

a whole must be identified with the mere continuous addition of its

parts.17 This is therefore the first principle of Crathorn’s mereology:

P1: a whole is nothing but the sum of its parts

Philosophers have traditionally opposed to this principle the aporia

of diachronic identity. Indeed, shouldn’t we conclude from P1 that a

whole is not the same after losing or gaining one of its parts? Socrates’s

essence would have changed with one of his hairs being torn off.

Anticipating this kind of objection, Crathorn envisaged the existence

of essential parts.18 Taking essential parts into account, one should be

able to distinguish the identity of a whole W in a given time t , and the

identity of a whole W through time, i.e. through a series of instants

16 In Sent. q. 3, p. 206: “. . . totum est suae partes. Hoc probo sic: si totum non estsuae partes et totum est aliquid, igitur totum ponit in numerum cum partibus. Con-sequentiam probo, quia sit a multitudo partium, sit b totum, tunc sic: b non est a necaliqua pars ipsius a. Igitur si b est vera res, b ponit in numerum cum a et cum qualibetparte illius a. Igitur per potentiam dei b totum posset esse non existente multitudinealiqua nec aliqua pars illius, quod falsum est.” As usual in the fourteenth century, God’s

absolute power is considered as a logical principle in such a context, because God’somnipotence is only limited by the principle of non-contradiction.17 In Sent. q. 3, p. 217: “Propter praedicta videtur mihi quod totum non est alia res

ab omnibus suis partibus sibi invicem continuatis.”18 Crathorn often uses the notion of partes essentiales and though he strongly criticized

the Aristotelian categories—notably the very idea of substance—he nonetheless seemsto maintain the general distinction between essence and accident. Cf. In Sent. q. 13,pp. 386–402. For a use of the notion of essential part in the continuum debate, see In Sent. q. 3, p. 207 et 223. Essential parts are sometimes called naturae coextensae. Cf. In Sent. q. 13, pp. 386–387: “Istud nomen ‘substantia’ derivatur ab isto verbo ‘substo’‘substas’; unde illud proprie vocatur ‘substantia’ quod stat sub alio vel aliis; sed nihilest in isto ligno de quo proprie possit dici quod stet sub aliquo alio, quod est in ligno.

Licet enim in ista re sunt multae naturae coextensae, tamen una illarum non est magissub alia quam econverso. Igitur nulla illarum potest proprie dici substantia.” (italicsmine). This precisely means that some parts are essential, but also that all parts havean essence as we shall see.




t, . . ., tn. In a given time t , the identity of W is nothing but the totality

of its parts, as an instantaneous ontological photograph, but the iden-

tity of W through time is the subset of its essential parts only. But thisdoes not evacuate P1, Crathorn contends, because even at the level of

essential parts, one cannot conceive the whole formed by the essence

as distinct from its parts.19 Then, even if one wanted to distinguish the

essence of a thing from its accidental parts, it should be recognized that

P1 works in the same way for the whole set of essential parts. What

is at stake here is the generalization of the mereological principle P1,

which is applicable to every kind of continua, and not only to natural

substances and artefacts. As we shall see, it can also be applied to time

and space and in general to all continuous quantities.Among the several arguments offered by Crathorn,20 one must be

carefully examined, because it contains the keystone of the theory,

namely, that the composition of spaces is parallel to the composition

of things existing in space.

3. The Mereological Composition of Space

To convince his readers, our Oxford master invites them to think aboutP1 within the category of place. How could we conceive of the place

of a whole and its parts if not in a mereological fashion? When a

body is divided into its parts, the total place of the whole body is also

divided into proportional parts of space. For example, half of a body

occupies half of the place of the whole body, and so on if we could

divide it again several times. As a corollary of that point, one has to

consider that the division of places is strictly parallel to the division of

the things existing in those places.

Further, it might be noted that this new element strengthens the argu-ment for P1. This new argument runs as follows: if a whole W were

different from its parts, then the place occupied by W must be distinct

from the place occupied by its parts—for something always exists in

19 In Sent. q. 3, p. 207: “Deus potest conservare omnes partes istius ligni non destru-endo totum lignum. Sed forte aliquis posset dicere quod quia istae partes sunt de essentiaistius ligni, ideo non potest destruere istius ligni, ideo non potest destruere partes istius

ligni, nisi destruat totum lignum. Contra: eadem ratione non potest conservare partesistius ligni, nisi conservaret totum lignum; sed partes ligni sunt de essentia ligni.” Cra-thorn declines this argument in many different versions. Cf. p. 207.

20 There are twenty-eight different arguments.




a place; but this sounds nonsensical, otherwise my house could be in

London and its parts in Paris.21 Likewise, if we were to divide W in

actu, each part would occupy a distinct place and once gathered theywould occupy again the whole space of W.

This is an important theoretical move from the composition of a

continuum to the composition of space.22 Crathorn thus holds a second

mereological principle:

P2: the space occupied by a whole thing is composed of parts of space

corresponding to the thing’s parts

This is an important shift that will serve as the basis of Crathorn’sattacks against the divisibilists.23 Indeed, if it is conceivable that a thing

is infinitely divisible, it is far more dif ficult to understand an infinite

division of space. Without determining the number of parts in a con-

tinuum for the time being, Crathorn assumes that P1 and P2 imply a

third principle concerning the proportional parts in division.

21 Ibid., p. 212: “Si continuum non est suae partes sed res distincta, implicaret con-tradictionem unum corpus esse in loco, nisi plura et distincta corpora essent in eodemloco. Et hoc probo sic: in eodem loco est corpus et suae duae medietates, sicut hoclignum et suae medietates, et hoc loquendo de toto loco totius ligni. Sed totum lignumest aliud a suis medietatibus. Igitur in eadem loco sunt distincta corpora occupantiatotaliter locum illum, scilicet hoc corpus et suae medietates, quia certum est quod totumlignum occupat suum totum locum et duae medietates occupant eundem locum totum.Igitur impossibile est quod aliquod totum occuparet totaliter totum locum, nisi aliqua

alia occuparent totaliter eundem locum. Dicitur quod duae medietates non occupanttotum locum, sed partes loci, scilicet duas medietates totius, quia sicut totum locatumest alia res a suis medietatibus, sic totus locus non est suae medietates. Et ideo licetduae medietates totius occupent suas medietates loci, non potest concedi ex hoc quodoccupant totum locum, sed illud quod occupat totum locum est totum componitur exduabus medietatibus locati, et non sunt duae medietates . . . Igitur totus locus non estaliud quam suae duae medietates, et per consequens omne corpus locatum est suaeduae medietates. Igitur eadem ratione omne totum est suae partes.”

22 We use indifferently ‘space’ and ‘place’, because as we shall see later, Crathorndefines the place of a thing as the space it occupies. Cf. In I Sent. q. 14, p. 417. Wewill turn to this point in the forthcoming sections.

23 Here, it may be suggested that the main source of this theory is the Liber sex

principiorum, which is often mentioned by Crathorn, and where a similar approach tospace can be found. Cf. Liber sex principiorum [Minio-Paluello] pp. 46–47: “Ubi autemaliud quidem simplex aliud vero compositum; simplex quidem est quod a simplici locoprocedit, compositum autem quod ex coniuncto.”




4. The Proportions in Division

According to Crathorn, it follows from P1 and P2 that there should bethe same number of parts in the two halves of a thing, and that there

shouldn’t be as many parts in a quarter than in the rest of that very

thing for example.24 Generally speaking, the number of proportional

parts—be it finite or infinite—is conserved through division. It also

works out when comparing two different continuous magnitudes: if a

piece of wood is twice as big as another, the former has double the

number of parts of the latter.25 The third principle is therefore the

following:

P3: if a whole W is naturally composed of n parts, if we divide W in x

parts, then each of the x parts will be composed of n/x parts

Considered separately, these three principles do not support a particular

version of atomism, for the number n of natural parts could still be

finite or infinite. Furthermore, Crathorn has not yet given an answer to

Aristotle’s anti-atomist critiques. At least, we may assume that if there

were an infinite number of indivisibles in a continuum, one should

accept according to P3 that there are unequal infinities.26 Implicitly,

Crathorn seems to reject this claim, because he uses P3 in the series

of arguments tending to prove that there is a finite number of indi-

visibles.27 Indeed, he seems to consider that parts should be countable

in some way (at least by God, it might be said). But this is not enough

24

For example, In Sent. q. 3, p. 225: “Secunda conclusio est quod non sunt totpartes in medietate continui quot in toto continuo. Primo quia totum continuum estduplum respectu suae medietatis, sed totum continuum est multitudo omnium partiumcontinui et medietas totus est multitudo partium medietatis, igitur multitudo partiumtotius continui est dupla respectu multitudinis partium suae medietatis . . .” and p. 226:“Quarta conclusio est quod multitudo omnium partium medietatis continui est aequalismultitudini omnium partium alterius medietatis.”

25 Ibid. p. 226: “Quinta conclusio quod generaliter qualis est proportio continui adcontinuum, talis est proportio multitudinis partium unius continui ad alterius. Unde siunum continuum sit duplum ad aliud, multitudo partium illius continui est dupla admultitudinem partium alterius et sic de aliis proportionibus.”

26 Henry of Harclay, for example, accepts such a possibility. Cf. Murdoch, “Henry

of Harclay and the Infinite.” For the medieval debates on infinity, see Côté, L’in fi nitédivine dans la théologie médiévale (1220–1255) and Biard & Celeyrette, De la théologie auxmathématiques. L’in fi ni au XIV e siècle.

27 In Sent. q. 3. pp. 226–227.




to prove the finite constitution of the continuum. Hence, in support of

his finitist view, Crathorn adds another stone to the edifice:

Everything which is a part of a continuum is either something actual( actualiter aliquid ) or not. One cannot say rationally that it is not some-thing actual, because one cannot understand that what is an actual partof a continuum and belongs to its essence is not something actual ora certain thing. Therefore one must say that a part of a continuum issomething actual.28

Indeed, how could a thing be composed of non-things? There are no

potential parts in a continuum, but only actual ones, according to P1,

because parts constitute the whole as its essence. This fourth principle

could be summed up as follows:

P4: the parts of a whole W are all actual parts

The terms ‘composition’ and ‘part’ should thus be taken in a strong

sense. The parts in question are the components of reality and can, in

principle, exist independently:

But the same things that are called different [pieces of ] wood when they

are discontinuous, these very same are called one wood and one wholewhen continuously joined together, in such a way that the expressions ‘thewhole wood’ and ‘one wood’ signify nothing more than the essence of theparts, except from the continuation of these things with each other.29

If we add the compositionality of spaces (P2) to that claim, each

part of a continuum is an actually existent thing occupying a single

place. Indivisibles are parts of bodies ( partes indivisibiles ) and not only

mathematical points.30 Therefore, as we shall see in much more detail

further down, parts must be conceived in a very strong sense as actual

28 Ibid. p. 227: “Omne id quod est pars continui, vel est actualiter aliquid vel non.Non potest dici rationabiliter quod non sit actualiter aliquid, quia istud non est intel-ligibile quod id quod est actu pars continui et de essentia continui, non sit actualiteraliquid nec res aliqua. Igitur oportet dicere quod pars continui sit actualiter aliquid.”

29 Ibid. p. 217: “Sed illae eaedem res numero, quae dicuntur plura ligna, quandosunt discontinuata, illae eaedem dicuntur unum lignum et unum totum quando sibiinvicem continuantur, ita quod isti termini ‘totum lignum’ vel ‘unum lignum’ nihil aliud

significant ultra essentiam partium nisi continuationem illarum rerum ad invicem.”30 Ibid., p. 237: “Septimo conclusio est quod indivisibile est pars corporis con-tinui . . . Nullum corpus continuum finitum est divisibile in infinitum; igitur cuiuslibetcorporis continui finiti est aliqua pars indivisibilis.”




things, with a certain quantity equal to the part of space in which they

are located.

On this point, Crathorn parts from his colleague Walter Chatton,who strongly rejects the existence of indivisibles in actu. Chatton’s opin-

ion is particularly well expressed in the Reportatio of his Commentary on

the Sentences, where he criticizes a point of view similar to Crathorn’s.31

According to Chatton, a continuum can’t be composed of actual

indivisibles, for if it is the case, then the whole wouldn’t be a totum per

se unum but a mere aggregate of indivisible parts.32 In Chatton’s view,

indivisible parts exist are only potentially in the whole, but they would

be actual if really divided.

On the contrary, the existence of actual indivisible things in thecontinuum is one of the main arguments of Crathorn’s finitist theory,

for one cannot understand an infinity of actual parts in a finite con-

tinuum.

5. The Number of Indivisibles

From the elements previously posited, Crathorn infers his critique of

Aristotle’s view about the infinite divisibility of a continuum, as wellas Henry of Harclay’s atomist version of it.33 Crathorn clearly asserts

that “no finite continuum can have an infinite number of proportional

parts”.34 Considering P1 and P4, it follows that if a finite continuum

were composed of an infinite number of parts, it would be a self-con-

tradictory claim, and the continuum should be actually infinite, for its

31 Walter Chatton, Reportatio super Sententias [Etzkorn e.a.] II, d. 2, q. 3, pp. 22–23:“Modo volo ego declinare ad hoc quod continuum componatur ex indivisibilibus inpotentia, non ex indivisibilibus in actu.”

32 Ibid. p. 126: “Dico quod continuum componi ex talibus in actu includat contra-dictionem, quia eo ipso quod continuum et contiguum differunt oportet quod partescontinui uniantur et faciant per se unum, quod si non, non facerent continuum sedcontigua esse tantum.” Chatton ascribed the position he is challenging to Democritus(cf. ibid. p. 125).

33 Harclay accepts the existence of indivisibles, but they are infinite in a continuum.

Crathorn frequently refers to Harclay by name. For example, In I Sent. q. 3, p. 259.34 In Sent. q. 3, p. 226: “Nulla multitudo finita est infinita, sed si aliqua multitudo par-tium alicuius continui esset infinita, aliqua multitudo infinita esset finita vel aliqua mul-titudine finita esset infinita; igitur nullius continui multitudo partium est infinita . . .”




parts are always actual according to P4.35 Of course, no Aristotelian

philosopher would accept the existence of such an actual infinite.36

In attempting to explain this argument, Crathorn recalls that follow-ing P1 and P2, “there are as many parts of extension as [parts] of the

extended thing”,37 which means that if a continuum were composed

of an infinite number of parts, it would not only be composed of an

infinity of places, but it would also consist in an infinite extension.38 As

a corollary to this first philosophical consequence, it should be noticed

that to accept the infinite division of places would imply the idea of an

infinite number of places inside the continuum according to Crathorn,

i.e. the existence of an infinite place by composition and addition.39

Moreover, one must also admit the possible existence of an infinitebody, for “if there were an infinite number of proportional parts in a

finite continuum, this finite body according to the dimension of place

would stretch to the infinite”.40 Admittedly, such an infinite body can’t

exist according to the Aristotelian cosmology.41 In such a context, it is

clear that P2 and the notion of locus are the most important require-

ments for Crathorn’s position.

The aforementioned strategy consists in showing that from a mereo-

topological point of view, the only conceivable infinity of parts is actual

35 Ibid.: “Si igitur in quolibet continuo sint actualiter infinitae partes, quod oportetdicere si in quolibet continuo sint infinitae partes, sequitur quod quodlibet continuumest actualiter infinitae partes. Igitur eadem partes continui sunt actualiter finitae, quiasunt continuum finitum, et sunt actualiter infinitae, quia in quolibet continuo suntinfinitae partes . . . igitur eadem partes sunt finitae et non finitae.”

36 Aristotle’s critique of the possibility of an actual infinity is well-known and it isnot necessary to review it in this context. Cf. Physics, book III and VI in particular. Forthe medieval background, see Murdoch, “Infinity and Continuity.”

37

In Sent. q. 3, p. 227: “Tot sunt partes extensionis quot rei extensae.”38 Ibid. p. 227: “. . . sicut tota res extensa correspondet toti extensioni, sic medietas reiextensae medietati extensionis, et secunda pars proportionalis rei extensae correspondetsecundae parti proportionali extensionis, et sic de aliis partibus proportionalibus. Igitur sicontinuum est infinitae partes, est infinitae partes extensae. Igitur si continuum finitumest infinita secundum multitudinem, ista est infinita secundum extensionem.”

39 Ibid. p. 228: “Aliter potest dici quod partes continui sunt infinitae secundum multi-tudinem sed finitae secundum locum, quia sunt in loco finito. Sed contra: locus non estaliud quam partes loci; sed partes loci sunt infinitae secundum se si partes rei locatae sintinfinitae, quia medietas totius locati est in medietate totius loci et medietas medietatislocati est in mediate medietatis loci; igitur quot sunt partes proportionales illius locatitot sunt partes proportionales illius loci. Sed multitudo partium proportionalium illius

locati est infinita; igitur multitudo partium proportionalium illius loci est infinita.”40 Ibid. p. 229: “Si in aliquo continuo finito essent partes proportionales infinitae,illud corpus finitum secundum omnem dimensionem loci extenderetur in infinitum.”

41 See for example, Aristotle, De caelo, I, 5 et 7.




and not just potential, because if we start thinking from the point

of view of the part-whole relation we cannot consider parts as mere

potential units. Here, the most original feature of the argument restson the use Crathorn makes of the notion of place. Indeed, beyond

his adherence to the four principles P1 to P4, it follows that to each

indivisible part corresponds a single place and a single position ( situs

punctualis ), and therefore a kind of quantity, as we will see in the next

chapters. If not, the arguments of the infinite extension and the infinite

body would fail.

It is illuminating to compare Crathorn’s solution to similar argu-

ments given by some of his contemporaries. As an example, Gerard of

Odo in Paris, who wrote his Commentary on the Sentences and other smalltracts on the continuum a few years earlier than Crathorn, also uses

the argument of actual infinities for his own finitist theory.42 Its form

is indeed quite the same: if we suppose that a continuum is composed

of an infinite number of parts, this continuum should be itself actually

infinite.43 Odo’s followers and critics will also report this argument as a

major one. For example, it is to be found in John the Canon’s Questions

on the eight books of the Physics 44 and in Gaetano of Thiene’s Collection on

the eight books of the Physics.45 Even infinitists such as Nicholas of Autre-

court and Nicholas Bonetus used this argument.46

Whether or not Crathorn is the first to have used this argument, he

nonetheless has a particular place in this story, for it isn’t likely that he

was influenced by Gerard of Odo or John the Canon, who both were

42 See Sander de Boer’s contribution in this volume for the context of the argument.43 Gerard of Odo, De continuo, Ms. Madrid, Bibl. nac. 4229, ff. 179rb–va (quoted by

S. de Boer in this volume): “Omne totum compositum ex magnitudinibus multitudine

infi

nitis, sicut componitur cubitus ex duobus semicubitis, est magnitudo actu infinita.Sed non est dare magnitudinem actu infinitam. Ergo nullum continuum est divisibile

in infinitum.”44 John the Canon, Quaestiones super octo libros Physicorum Aristotelis [Venice, 1520],

f. 59vb: “De totum compositum ex partibus vel magnitudinibus multitudine infinitis,sicut componitur cubitus ex duobus semicubitus, est magnitudo infinita; sed nullumcontinuum est magnitudo actu infinita.”

45 Gaetano of Thiene, Recollectae super octo libros Physicorum [Venise, 1496], f. 38vb:“. . . si continuum componitur ex semper divisibilibus ipsum in infinitum excedit aliammagnitudinem et est actu infinitum . . .”

46 For example, Nicholas of Autrecourt, Exigit ordo [O’Donnell], p. 212, ll. 29–42:“Dicerent autem contra hoc forsan: si punctum additum puncto faciunt majus et

extensionem quamdam et tres faciunt majus quam duo et sic semper et ibi sint infinita,sequitur quod ibi sit infinita extensio.” On this point, cf. Grellard, “Les présupposésméthodologiques de l’atomisme: la théorie du continu de Nicolas d’Autrécourt etNicolas Bonet.”




in Paris at that time, or by Nicholas of Autrecourt who wrote his Exigit

ordo at the very same time. It seems more likely that Crathorn and his

contemporaries were inspired by their Oxonian predecessors Henry ofHarclay and Walter Chatton.

The Chancellor of the University of Oxford, Henry of Harclay,

af firmed that a continuum is composed of an infinite number of indi-

visibles, but he accepted only one sort of actual infinite: the straight

line.47 But according to him it is not rational to extend this for a sur-

face or a body. As a consequence, Harclay considered indivisibles as

potential parts and would probably have refused Crathorn’s mereo-

logical principles. Nevertheless the attack on actual infinitism already

existed in embryonic form in Harclay’s position, and Crathorn probablyfound his starting point in it, adding to this intuition his mereological

principles.

More probable still is Walter Chatton’s influence on Crathorn,48 for

he will use a similar argument based on the actual infinite, although

he didn’t attach as much importance to the notions of locus and situs.49

One of the reasons for this difference of opinion is that indivisibles are

still considered as potential parts of the whole continuum in Chatton’s

view.50 They cannot exist separately, therefore they cannot have a single

place on their own.

From this point of view, Crathorn’s position is quite original in the

atomist family. While using a common matrix of arguments, he is

the only one who emphasizes both the mereological composition of

47 On this point, cf. Murdoch, “Henry of Harclay and the Infinite” (in particularp. 233).

48 It might be noticed that Crathorn doesn’t agree with Chatton on several points.

For references to Chatton, see In Sent. q. 3. pp. 261 and 265.49 Walter Chatton, Quaestio de continuo, in. Murdoch & Synan, “Two questions onthe continum: Walter Chatton(?), O.F.M. and Adam Wodeham, O.F.M.,” p. 258:“. . . responsio ad primum: dico quod sunt finita. Ad philosophum dico quod continuumesse divisibile in infinitum potest <dupliciter> intelligi: vel quod in continuo sint actuinfinite partes, quarum quelibet est extra aliam et nulla est alia, que possunt dividi adinvicem, et hoc est falsum et contra rationem . . .”

50 See the texts quoted above at the end of the previous section. See also WalterChatton, Quaestio de continuo, in. Murdoch et Synan, “Two questions on the continuum:Walter Chatton(?), O.F.M. and Adam Wodeham, O.F.M.” p. 246: “Istis suppositis,teneo 3 conclusiones. Prima: quod non componitur continuum ex indivisibilibus inactu, quia inter terminos est contradiccio ‘continuum’ et ‘indivisibile in actu’ quia, si

sit continuum, igitur partes eius sic se habent quod nulla est in actu per se existensseparata ab alia; et si sit indivisibile in actu, est per se existens separatum ab alioeiusdem rei. Oppositum dicit Democritus ponens continuum fieri ex athomis, tantumper congregationem quandam continuatis . . .”




things and spaces as well as the actuality of parts.51 But his singularity

comes from his use of the notion of locus. Indeed, the arguments of

the actual infinity of parts and infinite extension require P2 and thecoincidence between indivisible parts and single places for its ef ficacy.

Moreover, Crathorn contends that ‘whole’ and ‘part’ are only imposed

to signify the local conjunction of indivisible places.52 In this respect,

he is probably one of the more consequent atomists, because if an

indivisible were not considered as an indivisible part, with a kind of

extension resulting from the minimal place it can fill, all the previous

argumentation would fail. Gerard of Odo, for example, whose theory

is very close to Crathorn’s, is not consistent when he seems to consider

indivisibles as unextended.53

Here again, Crathorn is indebted to Harclay’s analysis of contact

between indivisibles when he tries to define contiguity and continuity

of atoms thanks to the central notion of place.

6. Contiguity and Continuity

Aristotle’s well-known paradox formulated in the sixth book of the

Physics arises when asking the atomist how two atoms can generatean increase in size.54 It raises the problem of the contact between

indivisibles, for two things can touch together parts to parts, whole to

whole, or parts to whole, Aristotle said. Having no parts, atoms can

51 Except from Crathorn, there were no other philosophers at Oxford, as far as Iknow, who held this view about the actuality of indivisible parts of a continuum. InChatton’s Reportatio, the reportator ascribes this position to some contemporaries. Cf.Walter Chatton, Reportatio super Sententias [Etzkorn e.a.] II, d. 2, q. 3, p. 136: “Sed

secundum usum modernorum, potest componi ex actu indivisibilibus, quia vocat actu talequod est tale extra animam et extra causam, quantumcumque non sit separatum abalio; componitur ergo ex actu indivisibilibus, id est ex non habentibus partes.” (ital-ics mine) It’s impossible to know to whom Chatton was refering in this text (writtenaround 1322–23). If we believe Gregory of Rimini, the finitist position, with actualor potential indivisibles, was common among his contemporaries. Cf. Gregory ofRimini, Lectura super secundum Sententiarum [Trapp], t. II, d. 2, q. 2, p. 278: “Nec potestdici, sicut communis dicunt tenentes huiusmodi compositionem ex indivisibilibus, quodcomponatur ex finitis tantum indivisibilibus.” We don’t know whether Gregory hadChatton, Odo or Crathorn in mind, or some other unknown philosopher. At the endof the fourteenth century, Wyclif held a position very similar to Crathorn’s own thesis.For Wyclif, see the references in the next sections.

52

In Sent. q. 3, p. 224: “. . . hoc nomen ‘totum’ et hoc nomen ‘pars’, . . . imponuntur alocali coniunctione rerum, quae coniunctio non est aliud quam loca vel partes loci.”53 As it is revealed in the texts quoted by Sander W. de Boer in this volume.54 Aristotle, Physics VI, 1, 231a29–b6.




only touch whole to whole, and this makes no increase in size but a

mere superposition; now, superposition non facit maius.55

When replying to this famous objection, Henry of Harclay claimsthat Aristotle missed the point, because the Stagirite used to consider

indivisibles in one and the same position ( situs ), but according to him,

indivisibles can touch according to different positions ( secundum distinctos

situs ) and this way they can cause an increase in size.56 Henry of Haclay

doesn’t develop this point further, but Crathorn will explain this point

very clearly.57

To my mind, the interesting idea that emerges from these topological

elements is the idea of a reference landmark, thanks to which atoms

can be located by their respective positions. This is not totally extra-neous to Aristotle’s cosmology, since similar tools can be found in the

De caelo for example.58 Moreover, Aristotle sometimes defines a point

as a substance with a position, as in the Posterior analytics.59 But the

appeal to locus and situs of indivisibles is problematic for philosophers

55 On this argument and the consecutive medieval debates, see Murdoch, “Super-

position, Congruence and Continuity in the Middle Ages.”56 Henry de Harclay, Quaestio de in fi nito et continuo, Mss Tortosa Catedral 88, f. 89rand Florence, Biblioteca Nazionale, Fondo principale, II. II. 281, f. 98r–v (quoted inMurdoch, “Henry of Harclay and the Infinite,” p. 244): “Sed, licet hec responsio suf-ficeret ad hominem, non tamen est realis responsio. Et ideo pono aliam et dico quodindivisibile tangit indivisibile secundum totum, sed potest hoc esse dupliciter: vel totumtangit totum in eodem situ, et tunc est superpositio sicut dicit Commentator, et nonfaciunt infinita indivisibilia plus quam unum. . . . Et ideo dico quod non propter indi-

visibilitatem quod unum indivisibile sic additum indivisibili non facit maius extensive,sed quia additur ei secundum eundem situm et non secundum distinctum situm. Sitamen indivisibile applicetur immediate ad indivisibile secundum distinctum situm,potest magis facere secundum situm.”

57

Gerard of Odo, for example, will follow Harclay’s text to the letter. Cf. Gerard ofOdo, De continuo, Ms. Oxford, Bodleian Can. Misc. 177, f. 230v (quoted in Murdoch,“Superposition, Congruence and Continuity . . .,” p. 435): “Dico quod totum indivisi-bile tangit totum aliud indivisibile, non tamen secundum omnem differentiam situs,sed secundum unam tantum, scilicet secundum ante vel secundum retro et sic de aliis.Unde si unum indivisibile tangeret aliud indivisibile secundum omnem differentiamloci, scilicet secundum ante et retro et secundum alias differentias omnes, tunc benesequitur quod indivisibilia non essent loco discreta nec constituerent aliquod maius.Sed si unum tangit reliquum secundum unam differentiam loci, ideo sunt loco discretaet constituunt aliquod maius.”

58 Cf. Aristotle, De caelo, II, 2.59 Aristotle, Posterior analytics, I, 27, 87a36. As Rega Wood points out in her chapter

in this volume, this surprising assertion by Aristotle has been ignored by a majorityof medieval philosophers, although Robert Grosseteste paid some attention to it inhis commentary. Cf. Robert Grosseteste, Commentarius in Posteriorum analyticorum libros,1, 18 [Rossi], p. 258.




such as Henry of Harclay or Gerard of Odo, who do not conceive of

indivisibles as occupying places. In fact, how could indivisibles occupy

a place if they are neither extended nor actual things?Again, Aristotle himself raised this point explicitly. When criticizing

the atomists in De generatione et corruptione (I, 2) Aristotle repeated the

argument from the Physics saying that it is not possible for a quantity

to come from non-quantitative things.60 He immediately added: “Fur-

thermore, where will the points be?”61 When he turned back to the

problem of the continuum in chapter 6 of book I, Aristotle insisted on

the notion of contact, thanks to which continuity can be conceived. But,

as he contended, the notion of contact cannot be understood without

the notion of position, which in turn relies on the notion of place.

Nevertheless, ‘contact’ in its proper sense belongs only to things whichhave a ‘position’, and ‘position’ belongs to those things which have alsoa ‘place’.62

Touch always occurs between two situated things because things in

contact need to be in a certain spatial relation. These touching things,

Aristotle said, also need to have a certain discrete magnitude, for if not

they cannot touch.63 So, when reading the De generatione et corruptione, the

medieval philosophers could find the key to the problem of contigu-ity and continuity: to ascribe a place and a certain magnitude to the

indivisibles. But other sources could be used, as the Liber sex principiorum

which is sometimes cited by Crathorn and where one can find a similar

notion of indivisible places for points and body’s minima.64

Due to the need to consider indivisibles as occupying a single place,

Crathorn is able to reconstruct contact between things and their con-

tinuity from the notions of locus and situs. As he told us in a passage

of his Questions on the Sentences:

60 Cf. Aristotle, De generatione et corruptione, I, 2, 316b5.61 Ibid.62 Aristotle, De generatione et corruptione, I, 6, 322b30–323a5.63 Ibid.64 Liber sex principiorum [Minio-Paluello] pp. 46–47: “Locus autem simplex est origo

et constitutio eius quod continuorum est, locus vero (ut dictum est quidem) compositushabet particulas quidem ad eundem terminum copulatas ad quem et corporis parti-

cule coniunguntur, corporis vero partes ad punctum. Loci ergo partes iuxta punctumnecesse fieri; erit itaque locus simplex in quo punctum adiacere constabit, loci veroparticule soliditatis particulas claudunt; etenim loca quidem simplicia minimi corporisoccupativa sunt.”




What continuity in fact is will become clear below when asking whetherGod is ubiquitous (q. 15). There it will be said, indeed, what is place,and once the nature of place is understood, one can easily see what iscontinuity and discontinuity, and what is contiguity.65

John E. Murdoch has already noticed that the notion of situs used by

Harclay implies a physicalist version of the indivisible, even if Har-

clay doesn’t develop his view in this way.66 If Harclay didn’t take the

plunge, Crathorn seems to have thought of indivisibles in this physi-

calist fashion.

Crathorn’s goal is to redefine the problematic notions in Aristotle’s

critique of atomism. If two things are continuous, according to Aristotle,

their limits have to be one, and if they are contiguous, the extremitiesof the things must be together.67 The strongest attack against atomists

comes from this characterization of continuity and contiguity. Therefore,

Crathorn suggests an alternative definition:

[contiguity] has to be defined in this way: “contiguous things are situatedand located things, between which there is no intermediary place or posi-tion,” and this definition suits for bodies, surfaces, lines and points.68

It should be objected to this definition that contiguity never explains

continuity.69 But since Crathorn adds the clause of non-existence ofinterparticulate spaces between indivisibles, contiguity becomes con-

tinuity, because they produce a new thing without any vacuum in it.

Therefore, properly speaking, a continuous thing is full of matter, there

is no empty space in it, no void. This is true, in slightly different senses,

for every kind of entity, including time.

65

In Sent. q. 4, p. 218: “Quid vero sit continuitas patebit infra, cum quaeretur utrumdeus sit ubique; tunc enim dicetur quid est locus, et intellecto quid est locus, cito potest videri quid est continuatio et quid dicontinuatio et quid contiguitas.”

66 Murdoch, “Henry of Harclay and the Infinite,” p. 244: “It is clear that in claim-ing that indivisibles can touch according to distinct positions Harclay was consideringthese indivisibles not as absolutely extensionless entities they really were, but as if theywere physical things.” It is not absolutely evident that Harclay was not aware of thisphysicalist consequence. Nevertheless, it would be incompatible with his infinitism. Onthe contrary, it is clear enough that Crathorn endorsed consciously these physicalistimplications.

67 Aristotle, Physics, VI, 321a21–25.68 In Sent. q. 3, p. 255: “Sed debet sic definiri: ‘contigua sunt situata vel locata, inter

quae non est locus vel situs medius’, et ista definitio competit corporibus et superfi-ciebus, lineis et punctis.”69 We find this argument in Chatton’s Reportatio for example. Cf. Reportatio super

Sententias [Etzkorn e.a.], d. 2, q. 3, p. 126.




For a thing to be continuous is nothing but its parts being joined togetherin a local or a temporal way, without any intermediary place or time[between them], and these parts being joined, they hold together ormutually succeed in place or in time without subtraction of place ortime. It follows that the continuity of a body, of a line or a surface, is thecontinuity of the parts of place, because parts are said to be continuouslylocated according to the continuity of the parts of place . . .70

From the beginning, Crathorn presupposes that an indivisible occupies

a single place and that it must consequently have a certain quantity.

This is implied by the argument from actual infinity and also from the

aforementioned definition of contiguity and continuity. Aristotle himself

thought that position and place should imply weight or lightness in somesense.71 Indeed, Aristotle’s target was mainly the possibility of touching

unextended points, but not of touching things (i.e. something with a

place, a position and a certain extension). Crathorn’s reappraisal of

contiguity and continuity seems to work as follows: Aristotle was right

in claiming that extensionless points cannot touch, but if points or

indivisibles were given a certain place and extension, they could touch

according to their contiguous positions. Crathorn’s theory is in some

sense a reinterpretation of Aristotle, paying attention to the physical

possibilities of his cosmology.The keystone of Crathorn’s atomist theory is thus to have consid-

ered that indivisibles may have a sort of quantity and extension. How

are the notions of place, extension and quantity linked in Crathorn’s

thought?

7. Extended and Qualified Atoms

In a long discussion on the category of quantity (q. 14) and on thequantity of indivisibles in particular (q. 15), Crathorn incidentally

70 In Sent. q. 16, pp. 456–457: “Rem esse continuam non est aliud quam partesillius rei sibi invicem coniungi localiter vel temporaliter sine loco vel tempore medio ettales partes sic coniunctas simul teneri vel sibi invicem succedere vel loco vel temporesine interceptione loci vel tempori. Unde continuitas corporis vel lineae vel superficieiest continuitas partium loci, quia continuitate partium loci dicuntur partes locataecontinue . . .”

71 Aristotle, De generatione et corruptione, I, 6, 323a10: “Now, since position belongs to

such things as also have a ‘place’, and the primary differentiation of ‘place’ is ‘above’and ‘below’ and other such pairs of opposites, all things which are in contact withone another would have ‘weight’ and ‘lightness’, either both of these qualities or oneof them.”




examines the nature of place.72 This addition is not fortuitous as we

shall see.

The nature of quantity was probably one of the most debated topicsin the fourteenth-century natural philosophy. The question usually was

to know whether quantity is a distinct and real category or if it is reduc-

ible to substance and/or quality for example. As an example, William

of Ockham endeavoured to reduce the ten Aristotelian categories to

mere names and concepts, but he thought that there really were exist-

ing singular substances and singular qualities in the world, which are

signified by terms from the ten categories. Substance terms (as ‘homo’)

signify only singular substances, quality terms (as ‘albedo’) signify only

singular qualities; the other eight categories signify only substancesand qualities and are distinguished by the way they refer to them, i.e.

by their connotations. On the contrary, other authors such as Walter

Burley will accept other categories as real. The medieval positions are

so numerous that it is impossible to mention them all here.73

Concerning categories in general, Crathorn holds a strong and radical

view, even more than Ockham’s own position, for he thinks that the very

same thing can be either a substance, a quality or a quantity.74 It would

be too long to detail Crathorn’s analysis of the aristotelian categories

in this paper,75 but his aim is to show several points: that there are only

atoms and that no one can be called properly a substance with respect

to the others; that quantity is nothing real except the situated atoms;

that quality also flows from the very nature of the atoms; that place

is nothing else than the local organization of atoms; etc. On quantity

in particular, Crathorn invites us to distinguish between two kinds of

quantitas: according to dimension ( quantitas dimensiva ) and according to

perfection or value ( quantitas perfectiva ).

To begin with the quantitas perfectiva, Crathorn is quick to admit thatit is not something different from the thing itself, because the value or

perfection of some entity is nothing else than its very nature or essence.

Nonetheless, though things have a natural perfection, they are not

72 In Sent. q. 14, p. 411: “Hic interponitur una questio de loco . . . Utrum locus proprieloquendo de loco sit aliquid reale productum a Deo vel a creatura.”

73 For an overview of some of these positions, cf. Biard & Rosier-Catach (eds), Latradition médiévale des catégories; and Lamy, Substance et quantité à la fi n du XIII e et au début

du XIV e

siècle.74 In Sent. q. 17, p. 462: “Secunda conclusio est quod eadem res numero est sub-stantia, quantitas, qualitas . . .”

75 I am preparing a paper on this topic.




equivalent to one another, for they can be compared. Two equivalent

portions of different essences are not equivalent: a bit of gold is not

equivalent to a bit of lead, even if they have the same weight anddimension. And this is perfectly appropriate for indivisibles as well. An

indivisible is an actual thing, it has a natural perfection, which can be

compared with others: an atom of gold is not equivalent to an atom

of lead.76

As a consequence, indivisibles are not the undifferentiated atoms

of Democritus or Epicurus. There are atoms of gold, of lead, etc.

This sounds as a modern chemical theory of elemental atoms, but

Crathorn prefers to call ‘points’ or ‘indivisibles’ the minimal parts of

things rather than ‘atoms’. After accepting that there are points of gold,could he admit that those indivisibles have any quantity, in the sense

of an extension or a dimension? Once again, Crathorn still presup-

poses that indivisibles have a kind of extension when explaining what

he understands by ‘points of gold’, but he never explicitly says that

they are extended.77

Turning then to the quantitas dimensiva, Crathorn refuses, as for quan-

titas perfectiva, to admit that this kind of quantity should be something

different from the thing itself. Quantity does not have an independent

mode of being. But Crathorn also refuses to identify dimension with the

thing itself. Of course, quantity depends on the indivisibles and their

order, but cannot be identified with them, because a thing can increase

or decrease in size without changing its nature.78 A compressed sponge

remains the same sponge in essence. Then, how can dimension be

apprehended? Crathorn’s answer is not surprising: quantity according

76

In Sent. q. 14, p. 405: “Secunda conclusio est quod res omnino indivisibilis, quaescilicet non habet partem extra partem nec partem inexistentem parti, est quantasecundum perfectionem et valorem, cuiusmodi sunt deus et angeli, sicut communiterponitur. Similiter talis res est punctum auri omnino indivisibilis, quia scilicet non habet partemextra partem nec partem inexistentem parti, et hoc patet sic: Talis res essentialiter excedit aliam etaequivalet plures res sicut punctum auri praedicto modo indivisibile essentialiter excedit punctum plumbiconsimiliter indivisibile et aequivalet plura puncta plumbea praedicto modo indivisibilia. Igitur talesres indivisibiles sunt essentialiter quantae et quantitates illo modo, quo res dicuntur quantae secundum perfectionem et valorem.” (italics mine).

77 In Sent. q. 14, p. 405: “Quod autem sint talia puncta aurea, quae sunt partesauro omnino indivisibilis, quae puncta non habent partem extra partem nec parteminexistentem parti, probatio: quia sit a unum punctum aureum habens partem extra

partem, quod opportet concedere de quolibet auro finito scilicet quod habeat partemsui, que pars non habet partem extra partem; aliter aurum finitae longitudinis essetinfinitae longitudinis, sicut probavi supra . . .”

78 Cf. In Sent. q. 14, pp. 406–407.




to dimension is nothing other than the place or space occupied by the

quantified thing.79 Here we are, the notion of locus is back.

Now, how to define place? First of all, place is immobile, it is not areal thing and it is not the limits of a body.80 What does it mean when

it is said that place is not a real thing produced by God? It means

that place is not a real category, no more than quantity. When a part

of space is not occupied by something, it is empty, that is to say it is

void, which means that place is firstly a purum nihil .81 Therefore, the

classical arguments of impenetrability of bodies and place don’t work

here, because space itself is nothing real. Thus, the same space can be

called either void or place, and even quantity or dimension. Place is

a part of space (according to P2) filled by something having a certainquantity and a certain position. It functions as a system of measure or

as a natural cadastre with no ontological reality, and it simply allows us

to determine the position of each indivisibles. The quantitas dimensiva of

a thing is thus defined by the space it occupies, i.e. by its place in the

79 In Sent. q. 14, p. 411: “. . . quantitas dimensiva vel dimensio rei dimensionataeest dimensio spatii, in quo est res, et partes dimensionis sunt partes spatii, ita quodlongitudo aeris non est aliud quam longitudo spatii, in quo aer est, et latitudo aerisest eiusdem spatii latitudo, et profunditas eiusdem spatii profunditas. Et idem est intel-ligendum de longitudine, latitudine et profunditate cuiuscumque rei longae, latae etprofundae, et istud patet quasi ad sensum, quia rem esse longam nihil aliud est quamrem esse in longo spatio . . .”

80 In Sent. q. 14, p. 412–413: “Ad cuius intellectum primo probo istam conclusionemquod de ratione loci est quod sit omnino immobilis . . . Secunda conclusio est quod locus

non est aliquid positivum reale . . . Tertia conclusio est quod locus non est ultimumcorporis continentis loquendo proprie de loco.”81 In Sent. q. 14, p. 417: “Ad quartum dicendum quod quando nihil est in spatio,

id est, quando nulla res positiva est in spatio, tunc spatium non est locus sed vacuum;quando autem aliquid est in spatio vel ponitur de novo in spatio, tunc id idem, quodprius fuit vacuum et non locus, fit plenum et locus. Vacuum esse non est impossibile,sed necessarium, quia extra caelum est vacuum infinitum, in istis autem inferioribus perpotentiam Dei posset esse vacuum et est pro aliquo tempore . . . Ad sextum dicendumquod locus nihil est, et concedo quod locatum est in illo quod est pure nihil, et potestponi in illo quod est pure nihil, sicut Deus posset ponere hominem in spatium extracaelum, et tamen illud spatium est pure nihil”. Aristotle already pointed out that, accord-ing to his adversaries, the notion of void implies the notion of a place deprived of a

body. Cf. Physics IV, 1, 208b26. Cf. Grant, Much Ado about Nothing , pp. 9–13. Elsewhere,Crathorn defines ‘place’ as an imaginary or purely intellectual thing: “Locus vero nonest aliqua res accipiendo hoc nomen ‘res’ primo modo, sed tantum est res intelligibiliteret imaginative proprie loquendo de loco, ut infra ostendetur.” (q. 3, p. 224).




general world’s landmark.82 It is worth noticing that this conception of

space is fairly original in the Middle Ages.83

On several occasions, Crathorn seems to admit that an indivisible canoccupy a punctual place, for he always uses P2 and its finitist corollary

in this context,84 and accepts the existence of locus punctualis.85 Unfortu-

nately, it is never specified whether punctual place is either one, two, or

three-dimensional. But if punctual place were one or two-dimensional,

this wouldn’t help the demonstration of the existence of indivisibles,

for the aporia of touching points would still be threatening with regard

to places. How could unextended places touch each other? Anyway,

from P2 and from the conclusion that the number of indivisibles is

finite, we may assume that punctual places can be at least in some casesthree-dimensional—in bodies for example. In q. 15, even if he is more

concerned with the cases of angels and souls, Crathorn concludes:

To the question I say that it doesn’t seem to me irrational to say thatsomething absolutely indivisible be essentially long, large and deep, ifsomething absolutely indivisible could be essentially in a long, large anddeep space.86

It is important to note that this does not mean that indivisible parts

and places are extended in the same way as a body, because indivisibleshave no configuration in space. They are qualified, extended in some

way, but positions cannot be distinguished inside an indivisible, that is to

say that points do not have sides, orientation, nor angles. Nevertheless,

as a locus punctualis is contiguous to another and both cannot coincide,

82 In Sent. q. 14, p. 419: “Locus et locatus sunt aequalia secundum dimensionem . . . etisto modo res locata dicitur aequalis suo loco vel spatio in quo est, quia locus vel spa-

tium est dimensio vel dimensiones ipsius rei locatae.”83 For an overview of medieval doctrine of space, see Grant, “Place and Space inMedieval Physical Thought” and idem “The medieval Doctrine of Place: Some Fun-damental Problems and Solutions”. Let us note that Crathorn’s description of spaceseems far closer to the one professed by Francesco Patrizi (1529–97) than to the viewsof his contemporaries.

84 In Sent. q. 14, p. 419: “. . . res locata dicitur aequalis suo loco vel spatio, in quo est,quia ipse locus vel spatium est dimensio vel dimensiones ipsius rei locatae. Isto habitoarguo sic: quanta est multitudo partium spatii vel loci, tanta est praecise multitudopartium rei locatae situ et loco distinctarum, neque maior neque minor.”

85 For example, In Sent. q. 16, p. 456. We will turn to this notion with more detailsin a subsequent section when examining the nature of motion (section 8).

86

In Sent. q. 15, p. 440: “Ad quaestionem dico quod non apparet mihi irrationabiledicere quod aliqua res omnino indivisibilis essentialiter sit longa, lata et profunda,si aliqua res omnino indivisibilis essentialiter possit esse in spatio longo, lato etprofundo.”




three points or three indivisible places may form a continuum and each

part will be situated relatively to the other with a certain orientation,

and with angles.87

Therefore, what he calls ‘points’ should have a certain quantity or

dimension, even though they are indivisibles. Crathorn’s originality is

thus to have developed some intuitions concerning the importance of the

notion of place already present in some other philosophical texts.88

Now that the nature of indivisibles has been reconstructed, the last

points still to be dealt with are the principles of Crathorn’s atomist

physics we announced at the beginning of this paper. How could one

explain natural phenomena from this minimal analysis of the ultimate

components of reality? Let us turn first to the explanation of motion,from which the rest will follow.

8. The Explanation of Motion

As a consequence of the continuum’s mereotopological structure,

motion will be defined as a local motion of atoms. Just as the continuum

is composed out of indivisibles, motion is made of atoms of motion.89

If we analyse in detail the nature of motion, it is composed of partswhich are atoms covering a certain space in a determinate time. In

other words, Crathorn says:

What is moved was in a certain place, in which it is not for the timebeing, and will be in a certain place, in which it is not for the time being,

87 In Sent. q. 3, p. 259: “. . . licet nihil sit rectum vel obliquum nisi divisibile et ideopunctum non est rectum nec obliquum, quia est indivisibile, tamen unum punctum

potest continuari alteri puncto secundum situm rectum . . .”88 Chatton however was close to discovering a similar solution when he wrote: “Adsecundum: quid vocas esse quantum? Si quod habeat partes eiusdem racionis, dico quodindivisibile non habet partes, nec est quantum, quia includit contradiccionem; si quod sitpars quanti, vel quod sit talis res que cum alia re eiusdem racionis componit quantum,concedo.” ( Quaestio de continuo [Murdoch & Synan], p. 259). Nicholas of Autrecourt willalso have this kind of intuition in the Exigit ordo [O’Donnell], pp. 207–208: “. . . vel perquantum intelligis habens propriam situalitatem et esse circumscriptive et sic concedoquod punctum potest dici quantum, et tunc non poteris secundum hoc concludere exnon quanto fieri quantum.” See also the brief description of Wyclif ’s atomism in thesection 10 of this chapter. For more details on Wyclif, see Emily Michael’s chapter inthis volume.

89

In Sent. q. 16, p. 443: “Circa primum sciendum quod sicut supra tenui quodcorpus continuum componitur ex indivisibilibus, ita teneo quod motus componiturex mutationibus subitis.” Here Crathorn doesn’t use the vocabulary of mutata esse, butthe idea is quite similar.




speaking of places of this space which are covered by the mobile throughthis motion.90

Of course, this general definition of motion is suitable for bodies aswell as for indivisibles.91 As indivisibles can be contiguously situated,

all that is necessary for them to be moved is to pass from a punctual

place to another contiguous punctual place in a given time made of

a certain number of instants. This conception of motion requires the

existence of void, for an atom must arrive in an empty punctual place

when moving, and we have seen that Crathorn accepts this point with-

out any kind of hesitation.

How to understand the continuity of motion from this general

account of local motion? Here, one must always have in mind the

idea of a cadastral system. As Crathorn wrote, “one must know that

time is to motion what place is to body”.92 The parallelism of space

and time goes much further, because as place is nothing really existing

but only empty space that could be filled out by an atom or a body,

time and instants too are nothing but a measure of length, they are

not things ( res ).93 “As place is the measure of a continuous body, time is

[the measure] of a continuous motion”.94 Each point occupies a distinct

position in space and can change from one place to another in one ormore instants of time. Therefore, a continuous motion will be the one

by which a mobile passes from one place to another contiguous one

in one instant—i.e. passing from an instant to another one contiguous

to the former. But this theory of motion raises some new problems, in

particular for the understanding of variations of speed.

90

In Sent. q. 3, p. 255: “. . . illud quod movetur fuit in aliquo loco, in quo modo nonest, et erit in aliquo loco, in quo modo non est, et loquendo de locis illius spatii quodmobile pertransit illo motu.”

91 In Sent. q. 3, p. 255: “Ad quartum dico quod punctum potest moveri et moveturquandocumque corpus movetur . . .”

92 In Sent. q. 16, p. 455: “Sciendum igitur quod tempus se habet ad motum sicutlocus ad corpus.”

93 In Sent. q. 16, p. 450: “Instans vero se habet ad mutationem sicut locus punctualisad punctum. Sicut enim locus punctualis non est punctum vocando punctum aliquidreale indivisibile situaliter nec est aliquid reale positivum distinctum realiter a puncto,sed spatium indivisibile, in quo est tale punctum reale positivum, sic instans non estmutatio vocando mutationem aliquid positivum reale subito et indistincte acquisitum

alicui nec est aliquid positivum reale distinctum ab illo, sed est duratio indivisibilis, inqua duratio talis acquiritur.”94 In Sent. q. 3, p. 242: “Sicut locus est mensura corporis continui, sic tempus motus

continui.”




9. The Analysis of Speed

An indivisible cannot pass from a punctual space to another one whichis not contiguous without passing through all the punctual spaces in

between. It cannot jump from one place to another distant one.95

Variations of speed can only be explained by the variation of time a

mobile can take to move from one place to another. This rather puz-

zling assertion entails another strange conclusion: given the fact that

punctual spaces and instants are the indivisible units for measuring

motion, if a mobile goes from one punctual space to the next contiguous

one in one instant, the ratio of space and time will always be equal to

one. Moreover, the speed of any continuous motion will always be thehigher speed ever reachable by a mobile, because motion cannot take

less time than one instant to cover one punctual space.

If a point is moved over three points, it will be moved continuously overthis space in a time composed of three indivisibles. And it doesn’t mat-ter how much the motive power is increased, it cannot be moved faster,because continuous motion is the fastest.96

Crathorn doesn’t deny that there can be variations of speed, but they

must be understood by discontinuity of motion. As an example, anindivisible can move over three punctual spaces through six instants.

There are times of rest during the changing of places that sensitive

cognition cannot grasp.97 Several examples are given, and Crathorn

seems to be very proud of this thesis about speed when af firming that

95 This sort of theory existed in the Arabic tradition. For a brief overview of theseArabic atomist theories, cf. Pines, Beiträge zur islamischen Atomenlehre; Wolfson, The philosophy

of the Kalam; Baf fioni, Atomismo e Antiatomismo nel Pensiero Islamico; Jolivet, La théologie et les Arabes. On the extension of atoms in the Mutakallimun, see Dhanani, The physical theory

of Kalam. On al-Ash’ari in particular, cf. Gimaret, La doctrine d’al-Ash’ari , I, 1 et III.96 In Sent. q. 3, p. 256: “Si punctum moveatur super trium punctorum, movebitur

continue super illud spatium in tempore composito ex tribus indivisibilibus. Et quan-tumcumque virtus motiva augeatur, non potest moveri velocius, quia motus continuusest velocissimus.”

97 Ibid. p. 256: “Si vero ponatur quod punctum motum moveatur per spatium triumpunctorum discontinue, hoc potest contingere multipliciter: uno modo sic quod quiescatin quolibet puncto spatii per duo instantia, alio modo per tria, alio modo per quattuoret sic de aliis numeris, et talis motus potest sic velocitari. Unde si esset virtus motivain tali gradu quod moveret punctum per spatium trium punctorum in sex instantibus,

alia virtus quae esset dupla respectu primae, moveret id punctum per spatium triumpunctorum in tribus instantibus . . . Sed apareat nobis quod multi motus sunt continui,cum tamen non sunt continui propter quietes interceptas, quas non percipimus.” Cf.also p. 258.




he has proved this point during the year of redaction of his Questions

on the Sentences.98 Actually, we must limit Crathorn’s self-conceit about

this paternity. Indeed, it might be noticed that this theory of speed isnot Crathorn’s privilege and that he could have been aware of some

version of it from elsewhere.

Very early in the Middle Ages, the Arabic theologians of the Kalam

developed several different atomist theories. Some of them were really

similar to Crathorn’s own view with regard to the importance attached

to place, position and extension in the definition of atoms.99 More strik-

ing is the fact that some of them had the very same idea about the

speed of a continuous motion: variations of speed have to be explained

by times of rest in motion.100 Of course, this kind of analysis of speeddirectly flows from the atomist view of time and space, whoever devel-

ops it. But could Crathorn have known such an Arabic source? It is

dif ficult to decide, but we know that this theory was available in the

Latin West quite early thanks to the Latin translation of Maimonides’s

Guide of the perplexed , in which he criticized in detail the Mutakallimun’s

analysis of speed.101

In the Dux perplexorum, Maimonides summed up the atomist theory of

the Mutakallimun in twelve points, among which the existence of simple

98 In Sent. q. 16, p. 456: “Ex dictis patet evidenter una conclusio, quam probaviisto anno, scilicet quod omnis motus vere continuus est velocissimus et quod implicatcontradictionem unum motum vere continuum esse velociorem alio motu vere con-tinuo.” Crathorn probably refers to a university lecture different from his Questions onthe Sentences or to some still undiscovered text.

99 See footnote 95 for bibliographical references.100 Cf. Ibrahim & Sagadeyef, Classical Islamic Philosophy, pp. 88–94. D. Gimaret gives a

detailed description of this theory ( op. cit., pp. 114–115): “De ce que le parcours d’une

distance est compris comme le franchissement obligé de toute la série de ‘lieux’ consti-tuant cette distance, tous ‘lieux’ d’égale dimension (il s’agit, en fait, d’atomes); de ce qu’àcette série uniforme de ‘lieux’ correspond une série pareillement uniforme d’instantsd’égale durée (atomes de temps), chaque ‘franchissement d’un lieu’ correspond à unnouvel instant; de tout cela il résulte que tous les mouvements sont quantitativementégaux. Il n’y a pas de mouvement plus lent ou plus rapide qu’un autre, tout mouvementest déplacement d’un point-atome de l’espace au point-atome contigu, et cela dans letemps d’un instant-atome . . . Si l’un est plus lent que l’autre (c’est-à-dire parcourt dansle même temps une distance moindre), c’est que ses mouvements sont entrecoupésd’arrêts imperceptibles à l’œil, pendant que l’autre mobile continue d’avancer.”

101 On the reception of Maimonides in the Latin West, Cf. Kluxen, “Maimonidesand Latin Scholasticism” and idem “Maïmonide et l’orientation de ses lecteurs latins”.

For Aquinas’s reading of this Latin version, see Anawati, “Saint Thomas d’Aquin etles penseurs arabes: les loquentes in lege Maurorum et leur philosophie naturelle”. On theLatin version of the text itself, see Freudenthal, “Pour le dossier de la traduction latinemédiévale du Guide des égarés.”




substances, of void, of instants of time, of qualified atoms. Atoms are

thus considered as small parts of the thing.102 Contrary to Crathorn,

they seem to have held that atoms are unextended, but the conclusionconcerning the variations of speed are all the same.103

Crathorn never quotes Maimonides, but we know that he was exten-

sively used by other English scholars since Adam Marsh and Thomas

of York.104 Interesting enough is the fact that we also find the same

explanation in Nicholas of Autrecourt whose dependence on Mai-

monides has already been suggested by Andrew Pyle and Christophe

Grellard105 and also in John Wyclif ’s Logica.106 In any case, Crathorn

was not influenced by Autrecourt nor by Wyclif, since it is not possible

historically speaking. Although we cannot decide if Crathorn’s has beeninfluenced in some way by this Arabic theory, notice should be taken,

however, of the existence of a matrix of arguments which cannot be

reduced to the reappraisal of Aristotle’s critiques, nor to the simple

reconstruction of Democritus through Aristotle. John E. Murdoch

states that this position is shared by almost all the indivisibilists,107 but

in 1330 Autrecourt and Crathorn were probably the first ones to put it

so clearly. Moreover, this kind of theory requires the mereotopological

structure of the continuum previously studied, which wasn’t theorized

by all the indivisibilists.

102 Maimonides, Dux seu director neutrorum sive perplexorum [Paris, 1520], I, 73, f. 32v(collation with the Ms. Sorbonne 601 (S), f. 38ra): “Ratio igitur primum [the separatesubstances exist] est: dixerunt siquidem quod mundus universaliter, id est omne corpusquod est in eo, est compositum ex partibus valde parvis que non habent partes . . .”

103 Maimonides, Dux seu director neutrorum sive perplexorum [Paris, 1520], I, 73, f. 33r(collation with the Ms. Sorbonne 601 (S), f. 38rb): “Dixerunt quod motus est mutatiosubstantiae separatae de numero atomorum unius, scilicet singularis substantiae, ad

aliam substantiam propinquam et ex hoc sequitur quod non est unus motus velocioralio. Et secundum hanc positionem dixerunt quod vides [S: in] duo mobilia perveniread duos terminos diversos in remotione in eodem tempore non est quia unus motussit velocior alio, sed causa eius est quia mobile cuius motus dicitur tardior habet pluresquietes in spacio suo quam illud quod dicitur velocius . . . Quod autem credis moverimotu continuo aliquid contingit ex errore sensuum et brevitate in apprehendendo . . .”Note that in the text the substantiae separatae correspond to simple substances, i.e. toatoms, and not to angels in this context.

104 See Kluxen, “Maimonides and Latin Scholasticism” and idem, “Maïmonide etl’orientation de ses lecteurs latins.”

105 Cf. Pyle, Atomism and its Critics, pp. 336–337 and pp. 687–688; Grellard, “Lesprésupposés méthodologiques de l’atomisme: la théorie du continu chez Nicolas Bonet

et Nicolas d’Autrécourt.”106 John E. Murdoch has once remarked on the similarity between the three authors.Cf. Murdoch, “Atomism and Motion in the Fourteenth Century,” p. 52.

107 Cf. Murdoch, “Atomism and Motion in the Fourteenth Century,” ibid .




Let us now direct our attention to another chapter of physics in

Crathorn’s thought: rarefaction and condensation. Unfortunately,

generation and corruption are never explicitly analysed, nor alteration,even though we can imagine what could have been such an analysis in

an atomistic fashion. Nonetheless, one can find detailed but surprising

developments on condensation and rarefaction.

10. Condensation and Rarefaction: The Limits of

Crathorn’s Mereotopology

As we have seen, the role attributed to the notion of place is far moreimportant in Crathorn’s physics than in Odo’s, Bonet’s or Autrecourt’s

atomistic philosophy. The true finite divisibility of a continuum, accord-

ing to Crathorn, is the finite divisibility of space: there are indivisible

places where atoms can be located. One of the conclusions that should

be posited from what has been said is the coincidence of an indivisible

place with an indivisible thing. One atom, one single place. This can be

traced in the different steps of his argumentation, for it is presupposed

in the analysis of contiguity, continuity, motion and speed. Consider-

ing this mereotopological framework, there are two possible ways ofexplaining condensation and rarefaction: these natural phenomena

can be reduced to some kinds of alteration or can be understood as

a change in the arrangement of atoms that occurs thanks to vacuum

(empty spaces newly filled out, or in the contrary, new empty spaces

in a body that cause an increase in size). Of course, one should hold

a combined theory with both elements.

The first way has been followed by John Wyclif, a few years after

Crathorn,108 on the same mereotopological basis. It is now little known

that Wyclif subscribed to an atomist theory of matter, combined withan Aristotelian hylomorphic world view.109 Unfortunately, recent scholars

didn’t examine closely his principles, which are basically the same as

Crathorn’s. Let us compare Wyclif ’s strategy to Crathorn’s in order to

evaluate both of their solutions.

The first element to be noticed is that Wyclif treats the problem of

the continuum in a logical treatise (the Logicae continuatio, tractatus tertius,

108 Around 1363. Cf. Thomson, The Latin Writings of John Wyclif , p. 6.109 Cf. E. Michael’s chapter on Wyclif in this volume. See also Kretzmann, “Continua,

Indivisibles and Change” and Pabst, Atomtheorien des lateinischen Mittelalters.




ch. 9), when he turned to the study of local propositions, as ‘Socrates

runs where Plato runs”. A few lines later, he abandoned this proposi-

tional analysis for a long digression (132 p.) on the locus of the world andthe situs of its atoms. According to Wyclif, there are punctual places110

as well as instants of time, and therefore atoms of matter and atoms

of time. As for Crathorn, locus is nothing but the thing located,111 and

thus quantity is only determined by its place. As a corollary to this, it

is suggested that points occupy punctual places and can be located this

way. But Wyclif ’s theory of situs is a bit more precise than Crathorn’s,

for he defines the place of an atom within a definite landmark formed

by the centre of the world and its poles.112 Wyclif ’s atoms resemble

Crathorn’s indivisibles because they occupy a single place in the worldbut also because they are considered as qualified entities.113 In his

Logicae continuatio, Wyclif even imagines, against a sceptical theologian,

that no one would refuse the possibility for God to create a punctual

substance in each punctual space; therefore, the existence of punctual

substances is possible.114 Moreover, these punctual substances exist in

a finite number in the universe.115

This description of atoms seems very similar to Crathorn’s, as

well as for its physical implications: contiguity and continuity can be

defined in terms of contiguity of places116 and a continuous motion

110 Wyclif, Logicae continuatio [Dziewicki], p. 2: “Et sic, quamvis species situs punctualissit principium integrandi omnem situm divisibilem, tamquam minimum metrum illiusgeneris, tamen totalis situs mundi est nobis mensura cognoscendi alios sitos particu-lares . . .”

111 Ibid. p. 3: “. . . omnis situs est aliquid situari.”112 Ibid. p. 4: “Ideo, sicut in natura omne motum vel mobile innititur alicui fixo,

sic non est possibile nos locum cognoscere, nisi in comparatione ad aliquod fixum.

Sicut ergo mundus ad eius motum situalem presupponit polos et centrum quieta, sicpresupponit ad eius situacionem eadem . . .”113 Actually, as Emily Michael shows in her chapter in this volume, there are dif-

ferent ontological levels of atoms: unqualified and unextended indivisibles, elementalatoms and minima naturalia.

114 Logicae continuatio [ Dziewicki], p. 33: “Similiter, ut credo, nullus theologus negaretquin Deus de potentia absoluta potest facere substantiam punctualem . . .; et tunc patetquod punctualitas vel punctus que est substancia huiusmodi esse punctualis, est actuspositivus [in] illa substancia . . . Punctus ergo potest esse. Nec dubium quin situs essentcorrespondenter iuxtapositi, cum situs sit subiectum situari.”

115 Ibid. p. 36: “Unde impossibile est quod aliquis numerus substanciarum vel punc-torum vel aliud preter deum sit simpliciter infinitum.”

116

Ibid. pp. 30–31: “Similiter de immediacione ubicacionum vel situum indivisibilium,patet quod est dare tales immediatas. Nam est dare duo puncta immediata, ut patet decorporibus tangentibus, sic ubicaciones vel situaciones eorum sunt immediate. Et cumilli situs manent expunctantes alia puncta, patet quod quandocumque alter eorum erit




has the higher speed ever reached by a mobile.117 There is at least one

important difference between Crathorn and Wyclif, because the latter

seems to deny the existence of vacuum. Moreover, Wyclif is not clearabout the quantity or extension of atoms, saying sometimes that they

are unextended and sometimes that they occupy a place and are real

parts of things118. What about condensation and rarefaction now?

Wyclif ’s answer is astonishing. The world is totally full of matter:

It is to be noticed that the world is composed of certain atoms, and canneither be increased nor diminished, nor locally moved in a straight line,nor be figured in some other way (. . .).119

This point requires some explanations. There are three premises tothis conclusion:

1) there is no vacuum

2) there is a finite number of atoms and places in the world

3) two things (even indivisibles) cannot be in the same place at the same

time (i.e. two atoms cannot exist in the same punctual place)120

The first conclusion to be drawn is that all possible punctual places are

filled by atoms.121 The second is that a body cannot be increased ordiminished in size without losing or gaining parts.122 The quantity of

occupatus aliquo punctuali intra corpus, reliquus erit occupatus punctuali sibi imme-diato. (. . .) Similiter de instantibus; videtur quod erunt immediata . . .”

117 Ibid. p. 97: “Ad illud dicitur quod impossibile est aliquod indivisibile velociusmoveri localiter quam continue in quolibet instanti dati temporis describere situmsuum punctualem.”

118 See Michael’s chapter for the texts.119

Ibid. p. 1: “. . . notandum mundum componi ex certis athomis, et nec posse maiorarinec minorari nec moveri recte localiter vel aliter figurari . . .”120 Ibid . p. 42: “ad tertium dicitur quod impossibile est multa puncta vel substancias

punctuales esse simul in eodem situ indivisibili.”121 Before being informed by substantial forms and arranged by aggregation, the

prime matter is composed of indivisibles which fill all possible places. Cf. Logicae continua-tio [Dziewicki], p. 119: “. . . ymaginandum est igitur unam essenciam corpoream, inprincipio productam, esse ex indivisibilibus composita, et occupare omnem locum pos-sibilem, nec esse secundum eius partem aliquam corruptibilem, nisi forte per divisionem

vel separacionem unius partis a reliqua.” On the notion of prime matter in Wyclif, cf.Kaluza, “La notion de matière et son évolution dans la doctrine wyclifienne.”

122 Ibid. p. 69: “Ex istis colligitur quod nullum corpus potest esse maius aut minus

quam prefuit, nisi propter adquisicionem aut deperdicionem materie, quamvis putaturquidlibet rarefactum esse maius quam prefuit, ignorando situs quod perdit intrinse-cus, sicut et ignoratur commutacio situum extrinsecorum pro intrinsecis in partibuscondensati.”




matter always remains the same. Indivisible parts of the whole world

can change from one punctual place to another by an extrinsic force,

but without occupying an empty space. A sponge, when wrung, has lessair than before, but air is not vacuum according to Wyclif.123

Even if he rejected assumption 1), Crathorn should have contended

that the quantity of matter is constant in the world for he at least

admits 2), that the number of atoms is finite and that to one punctual

space must correspond one indivisible. He thus could have explained

condensation and rarefaction by the absence of interparticulate spaces

in an absolutely dense body and, on the contrary, the presence of some

empty punctual spaces in a rarefied body (the second way previously

described).124 He also could have thought, as Wyclif did, that conden-sation and rarefaction are kinds of alteration. But Crathorn did not

agree either with Wyclif ’s solution or with the second one just posited,

as Nicholas of Autrecourt did for example. He attempted to overturn

these argumentations by denying 3) and claiming—with some appar-

ent self-contradiction—that several points can be in the same punctual

place.

In this view, a rarefied body will have a few parts existing in one and

the same place; on the contrary, a dense body will have a lot of parts

existing in the same place.125 In this case, to what extent can a body be

diminished or increased? Crathorn’s answer is quite radical:

A point can be rarefied, because there is nothing more in the notion of apoint than to be something located and situated, having no parts outsideof parts, which is compatible with having parts existing in parts and beingcomposed out of several things really and essentially distinct, thoughthese are not distinct according to place and position. Thus: everythingthat has several parts, indistinct according to position and place, can berarefied; therefore it is not incompatible for a point to have several parts

indistinct according to place and position; therefore it is not incompatiblefor a point to be rarefied . . . A point can be rarefied into a body, for it isposited that a point has a hundred parts indistinct according to position,in such a way that it is composed of a hundred parts: such a point can

123 Cf. p. 63.124 Nicholas of Autrecourt held such a theory. Cf. Christophe Grellard’s chapter

in this volume.125

In Sent. q. 14, p. 421: “Corpus esse rarum non est aliud quam illud corpus nonhabere partes aliquas loco et situ indistinctas vel habere paucas partes situaliter vellocaliter indistinctas . . . Unum vero corpus esse densum non est aliud quam illud corpushabere multas partes localiter indistinctas vel in eodem loco et situ.”




be rarefied into a body, because with rarefaction, the parts of the pointcan be distinguished according to their position . . .126

This also works the other way around for condensation. If parts aredistinguished according to their position in a body, after condensation

they will not be further differentiated by their position.127

This new element is threatening for the coherence of Crathorn’s

view, because all his arguments rest on the correspondence between

an indivisible and a punctual place. As we have seen, the position of

an indivisible in space implies the notion of a single place ( locus punc-

tualis ), but place also determines the quantity of the located thing; and

then points or indivisibles must have in some way a certain quantity.

Crathorn presupposed this view of indivisibles throughout his analysis

of the part/whole relation, contiguity, continuity, motion and speed.

Moreover, his finitist view of the number of indivisibles seems to imply,

as in Wyclif, the principle of correspondence between punctual places

and indivisibles. Indeed, from this consideration about rarefaction and

condensation, the finite number of atoms seems to be postulated rather

than demonstrated by Crathorn, for nothing seems to prevent infini-

ties of indivisibles if more than one indivisible can occupy a punctual

place and rarefaction could be in principle infi

nite.128

So, according tothis view, the world could be densified into one point or rarefied into

an infinite space, which is contradictory with what we have posited

before.

Although Crathorn endeavoured to restrict his analysis to finite cases,

always giving examples with a finite number of indivisibles,129 his finit-

126 Ibid. p. 423: “Punctum potest rarefieri, quia non est plus de ratione puncti, nisi

quod sit aliquid situatum vel locatum non habens partem extra partem, cum quostat quod habeat partem inexistentem parti et quod sit quid compositum ex pluribusrealiter et essentialiter distinctis, licet omnia ista sint loco et situ indistinctas. Tunc sic:omne id quod habet plures partes situaliter vel localiter indistinctas potest rarefieri; sednon repugnat puncto habere plures partes situ et loco indistinctas; igitur non repugnatpuncto rarefieri . . . Punctum potest rarefieri in corpus, qui ponatur quod punctumhabet centum partes situ indistinctas ita quod componatur ex centum partibus; talepunctum potest rarefieri in corpus, quia partes talis puncti possunt per rarefactionemsitu distingui . . .”

127 Ibid. p. 424: “. . . corpus potest condensari in punctum. Si enim punctum possetrarefieri in corpus, eadem ratione idem corpus condensari potest in punctum, etc.”

128 It is a commonly held argument in the Middle Ages against indivisibilism. See,

for example, its use in the dispute between John Buridan and Michel of Montecalerio(cf. Celeyrette’s chapter in this volume).129 In Sent. q. 14, p. 424: “Vigesima prima conclusio est quod punctum rarefit

naturaliter per actionem agentis naturalis, et hoc patet sic: sicut alias probavi, corpus




ist physics becomes impossible to prove. What is more, indivisibles are

no longer quantified, if they can be a hundred in the same indivisible

place, and they also cannot be parts, because P1 and P2, which arethe basic claims of Crathorn’s indivisibilism, require that indivisibles

be considered as parts of the continuum, and thus as real and actual

things occupying a punctual place. Wyclif saw the dif ficulty and strongly

af firmed the principle of equivalence between the number of punctual

places and atoms.130 So, we might say that he is incoherent or that he

cannot escape from the mathematical and infinitist view of indivisibles

considered as mere unextended mathematical points. The second alter-

native wouldn’t fit well with many of his developments. Therefore, it

seems more likely that he is a little incoherent.On the other hand, we may assume that the main reason for this

departure from the basic mereotopological principles is motivated by

theological reasons. Indeed, we must keep in mind that Crathorn not

only treats the issue in q. 3, but also in the more theological context of

other sorts of indivisibles, as God, Angels and souls (q. 15 for example).

The questions of God’s ubiquity and Angel’s motion were the most

frequent pretext in the Middle Ages when dealing with indivisibles and

their location.131 In this respect, Crathorn was forced to af firm that the

componitur ex superficiebus, superficiebus ex lineis, linea ex punctis, et ita omne corpusfinitum componitur ex punctis finitis. Si igitur a multitudo maxima et finita omniumpunctorum unius corporis situ distinctorum et totalium et suppono quod illud corpusrarefiat, quousque secundum omnem dimensionem habeat quantitatem dimensivam,quae sit centupla respectu quantitatis dimensivae primo habitae, et suppono quodomnes partes illius corporis ante rarefactionem sint uniformiter et aequaliter densae etquod uniformiter et aequaliter rarefiant. Istis suppositis sequitur quod facta rarefactionemultitudo punctorum ipsius corporis rarefacti accipiendo puncta modo praedicto, scilicet

quorum quodlibet situ distinguitur ab alio et quorum nullum est pars alterius puncti,multitudo talium punctorum est centupla respectu multitudinis punctorum consimiliteracceptorum ipsius ante rarefactionem. Igitur multa puncta, quae facta rarefactionesunt localiter distincta, fuerunt ante rarefactionem indistincta localiter et tamen persuppositum quaelibet pars vocando partem totam rem occupantem partem spatii anterarefactionem sit aequaliter rarefacta.”

130 Wyclif, Logicae continuatio [ Dziewicki], pp. 42–43: “. . . impossibile est multa puncta vel substancias punctuales esse simul in eodem situ indivisibili . . . Unde argumentahominum volencium destruere quotlibet talia puncta in eodem situ indivisibili petuntpro fundamento quod non sit possibilis composicio continui ex non quantis.”

131 The continuum is frequently treated in the second book of the Sentences whendealing with angels. John Duns Scotus, for example, took this pretext in his different

versions of his commentary on the second book of the Sentences (e.g. Ordinatio [Balice.a.], d. 2, q. 5, pp. 296: “Utrum angelus possit moveri de loco ad alium locum motucontinuo”). All the same, in his Reportatio super Sententias [Etzkorn e.a.], II, d. 2, q. 3,Walter Chatton asks “Utrum motus componatur ex indivisibilibus” and introduces




soul can be everywhere in the body, though it is indivisible. Same here

for God and Angels, who are purely indivisible but can be in different

places, either extended or punctual. Finally, it is plausible that Crathorntried to find a general definition of condensation and rarefaction that

would suit all sorts of indivisibles, material and spiritual. Incidentally,

he did not seem to notice the incoherence of his views on density and

rarity with the basic principles of his atomism.

11. Conclusion. A Theory of M INIMA N ATURALIA?

In most of the medieval debates on the nature of the continuum andthe existence of indivisibles, one has to choose between a geometrical

view of indivisibles—which does not require such a detour through

mereotopology—and a more physical atomism, which requires at some

point to consider indivisibles as sharing some physical properties (mini-

mal quantity, qualities or nature, position in space, etc.). If one chooses

the second option of the alternative, as Crathorn did with many others

in the fourteenth century, it cannot be asserted at the same time that

several indivisibles can occupy the same punctual place and that indi-

visibles are in fi ne defined by the place they occupy as parts of something.Beyond the simple topological intuitions of Crathorn’s atomism is the

thesis that indivisibles are actual parts of the continuum. The idea of

minimal actual parts was already in Aristotle when he accepted the

existence of minima naturalia. Indeed, he accepted both claims, that a

continuum is infinitely divisible in potentia, but finitely divisible in actu.

The result of an actual division of a continuum into its parts would

be a minimum of flesh or bones in the case of human beings. If we

continue the division, parts won’t exist any more: they won’t have any

nature, nor quantity, nor virtue or action, etc. But isn’t this idea whatCrathorn claims about indivisibles? They have a nature, a quantity and

his development as follows: “Et quia non potest sciri de motu angeli utrum sit con-tinuus vel discretis in motu nisi sciatur utrum motus et alia continua componanturex indivisibilibus, ideo quaero propter motum angeli utrum quantum componatur exindivisibilibus sive permanens sive succesivum.” Gerard of Odo’s discussion aboutthe continuum occurs in the context of God’s and angels’ simplicity. Cf Super primumSententiarum, dist. 37, Mss Naples, Bib. Naz. VII. B.25, ff. 234v–244v; Valencia, Cated.

139, ff. 120v–125v: Ad quorum evidentiam querenda sunt quatuor . . . Tertium utrummotus angeli habeat partem aliquam simpliciter primam. See also Gregory of Rimini, Lectura super secundum Sententiarum [Trapp], d. 2, q. 2, pp. 277–339: “Utrum angelus sitin loco indivisibili aut divisibili.”




they can move and act, as actual parts of the continuum. Moreover,

they can exist separately as Aristotle’s minima naturalia. Of course, the

last considerations about condensation and rarefaction tend to make thedescription of indivisibles quite confused. Finally, are there two distinct

theories in Crathorn? A theory of atoms conceived as kinds of minima

naturalia on the one hand, taking up some of Aristotle’s intuitions, and

a theory of extensionless points on the other hand.

John Wyclif, for example, who held a very similar theory of indi-

visibles, clearly distinguishes between the two levels. According to him,

there are unqualified and unextended indivisibles and, at a higher

level, they are minima naturalia, which are composed of such primitive

indivisibles, already joined in elemental atoms.132 The problem is thatCrathorn makes no distinction between unextended indivisibles (as

points) and qualitative atoms (kinds of minima naturalia ). Moreover, he

always uses a geometrical vocabulary, as the majority of his contem-

porary fellows. For example, he sometimes refers to “points of gold”

that occupy a punctual place. Consequently, it becomes impossible to

distinguish between the level of points and the level of minima naturalia,

because both seem to have a nature and a kind of extension.133 Crathorn

would probably have to admit the Wyclifian theory. In any case, except

for theological contexts, Crathorn’s atoms are a sort of mix between

Aristotle’s minima naturalia and Democritus’s atoms.

In this paper, we tried to show how systematic is the use of mereol-

ogy—relations of parts and wholes—and topology—notions of place

and position—in Crathorn’s atomism. In fact, his whole atomistic

conception of the world rests on these notions. We then compared

him with some of his contemporaries in order to demonstrate his

originality, although his point of departure was probably Harclay’s

and Chatton’s positions. Indeed, Crathorn was probably thefi

rst realatomist in Oxford who considered indivisibles as real and actual enti-

ties. As we have already mentioned, the origin of his atomism could be

132 See Michael’s chapter in this volume.133 When Crathorn distinguishes between mathematical and corporeal figures, he

seems to assert that they share the same properties. Sent. I, q. 3, p. 239: “Dicetur fortequod lineae istae, quas nos facimus et videmus, sunt corpora; et ideo ex coniuctione etdistantia talium linearum, quae veraciter non sunt lineae, non potest argui coniunctio

vel distantia verarum linearum. Contra: istud non est bene dictum, quia si istud esset verum, numquam geometria fuisset inventa nec unquam posset doceri; non enimpotest doceri sine talibus protractionibus.” We thus may infer that points and corporealindivisibles have the same properties.




traced back to his interpretation of some of Aristotle’s physical works,

and notably of the De generatione et corruptione and the De caelo, and also

to his readings of other ancient sources as the Liber sex principiorum, the Dux perplexorum of Maimonides or the De elementis of Isaac Israeli. A

thorough examination of his sources would shed light upon some other

points in the theory, for Crathorn knew a lot of authors to whom he

does not always refer.134

The general conclusion that can be drawn from this reading of

Crathorn’s Questions on the Sentences is that there are no hints of math-

ematical concern in such a physical theory. Of course, Crathorn

endeavoured to respond to the rationes mathematicae that he knew from

John Duns Scotus, but only as a kind of formal obligation. Crathorn’sgoal was firstly to find some physical principles to describe natural sub-

stances and in the process to find an alternative ontology to Aristotle’s

metaphysics. This attitude becomes evident if one turns to his critique

of Aristotle’s categories, where it is af firmed that the distinction of the

ten categories does not make sense. Thus, it may be helpful to read

Crathorn’s texts on indivisibles in the light of his metaphysical claims, in

order to understand what he took and what he rejected from Aristotle.

In any case, Crathorn’s is a good example of an atomist theory which

is not merely geometrical, as John Murdoch used to say about all the

indivisibilist positions of the fourteenth century. On the contrary, he

attests the existence of different atomist traditions and of the use of

different kinds of ancient sources, not only aristotelian ones.

134 We may assume that Isaac Israeli’s De elementis, which has been translated intoLatin by Gerard of Cremona at the end of the twelfth century, is one possible sourcefor this kind of mixed theory that has been described in the previous paragraph.Crathorn refers to this text in another context ( In Sent. q. 4, p. 280: “Quid autem sitrisus, dicit Isaac libro suo De elementis.”). In the second part this book, Isaac Israelipresents is position against the atomists. He asks whether Galen’s assumption that

there must be minimal parts in bodies, Aristotle’s assertion that there must be minimanaturalia, Democritus and the Mutakallimun atomist theories, could fit together in aunified conception. Cf. Isaac Israeli, De elementis. In Isaac Israeli opera omnia (Lyon, 1515),f. 7D sqq. This point would deserve another paper.



AN INDIVISIBILIST ARGUMENTATION AT PARISAROUND 1335: MICHEL OF MONTECALERIO’S

QUESTION ON POINT AND THE CONTROVERSY

WITH JOHN BURIDAN

Jean Celeyrette

The debate concerning the nature of point and the divisibility of the

continuum between John Buridan and a master named M. of Monteca-lerio1 has been known for a long time.2 It is described as such in one of

the tables of the ms. BNF Lat. 16621 (f. 195r)3 in which both masters’

texts appear. This scientific manuscript comes from Etienne Gaudet’s

collection.4 It is the only surviving witness of Montecalerio’s question

and it is particularly dif ficult to transcribe. This is why, despite several

attempts, the text still remains unpublished, even though Buridan’s

text, which appears in another manuscript, has been published by

Zubov.5 Although Bernd Michael considered any number of possiblities

concerning the identity of Buridan’s oppponent, until recently he was

1 In the manuscript, the master is called Montescalerio. Following W.J. CourtenayI will designate him as Montecalerio (see footnote 5).

2 K. Michalski thought that Buridan’s text was directed against Walter Burley. Cf.Michalski, La physique nouvelle et les différents courants philosophiques, p. 120. Indeed, we willse below that Montecalerio’s position is close to Burley’s.

3

“Deinde sequitur determinatio de puncto, videlicet utrum sit alica res addita linee,Magistri Johannis Bridani contra magistrum de Montescalerio que durat circuiter per7 folia [. . .] deinde determinatio magistri M. de Montescalerio de puncto, videlicet anper divisionem continui corrumpatur aliqua res, contra magistrum Johannem Bridanper XI folia usque in fine sisternum.” (BNF Lat. 16621, f. 195r).

4 On Etienne Gaudet’s collection, see Kaluza, Thomas de Cracovie, Contribution àl’histoire du Collège de Sorbonne.

5 Zubov, “Jean Buridan et les concepts du point au XIVe siècle.” The edition isbased on the manuscripts Paris, BNF Lat.16621, ff. 196r–202v and 203v, and Paris,BNF Lat. 2831, ff. 123r–129v. Zubov’s edition is reliable even if a comparison withMontecalerio’s text enabled me to make some corrections. The quotations hereafterwill be either drawn from Zubov’s edition, noted QP, or from the manuscript Paris,

BNF Lat. 16621 noted EG. In this last case, for reasons of clarity, I indicate by (M.M.)or (Bur.) the author of the text which I quote. I will use Montecalerio’s presentationof Buridan’s arguements and the references will be thus with EG; I will also indicatethe corresponding passages in QP.



164 jean celeyrette

very much a mystery.6 In two recent papers, William Courtenay shed

a new light on Michel of Montecalerio, master of arts of the French

Nation around 1342 and still active in 1346.7

Most notably, we learnthat he was of outstanding importance, an element that could explain

why Buridan devoted such attention to his writings.

A comparative reading of both texts reveals that the controversy,

which seems to be quite severe by the tone of their respective retorts,8

should be situated in the context of the Parisian discussions about

Ockhamist physics. We know that Buridan does not share Ockham’s

view about the nature of point in the two known versions of his

commentary on the Physics.9 Nevertheless, he accepts it in the text we

are commenting upon in this chapter. Indeed, like Ockham, Buridanmaintains that “point” is a privative name and that a point, as every

privation, is nothing real: as an example, a terminative point ( punctum

terminans ) corresponds to the fact that a line goes until there and not

beyond. But this means nothing else than the word “point” is the signifi-

cate of the proposition “a line goes until this limit and not beyond”.10

It seems that Buridan shares this view in this context, even if he does

not in others.

On several occasions Montecalerio refers to this Buridanian position

to reject it but his detailed refutations are lost: the end of his question

which was certainly devoted to this refutation has disappeared. How-

ever, it should be noted that what remains of Montecalerio’s text fills

9 folios while 11 folios are announced in the table at the f. 195r. So

the lost refutation was at most two folios long. As Buridan’s exposition

6 Michael, Johannes Buridan: Studien zu seinem Leben, seinen Werken und zur Rezeption seinerTheorien im Europa des späten Mittelalters, pp. 451–452.

7

Courtenay, “The University of Paris at the Time of Jean Buridan and NicoleOresme” (especially pp. 8–10) and idem, “Michael de Montecalerio: Buridan’s opponentin his quaestio de puncto.” The text of Montecalerio, that of Buridan’s question mentionedin footnote 12, and other unedited texts from the ms Paris, BNF Lat.1 6621 are in thecourse of publication by Zénon Kaluza and myself.

8 See, for example, Montecalerio’s reply: “Et miror de isto doctore sic experto quodistius <quod> exposuit fuit immemor.” EG (M.M.) ff. 222v–223r.

9 Cf. Celeyrette, “La problématique du point chez Jean Buridan”. We find a clearstatement of Ockham’s position in his Quaestiones in libros Physicorum, q. 63 and in hisTractatus de quantitate. The detailed quotations and references appear in the above-men-tioned paper, from note 27 to 30, pp. 93–94.

10 About the notion of complexe signi fi cabilia, see Gabriel Nuchelmans, Theories of the

Propositions. Ancient and Medieval Conceptions of the Bearers of Truth and Falsity. On Buridan’srefutation of the Ockhamist position in his Questions on the Metaphysics and his Sophismata,see pp. 243–250. See also Biard, “Les controverses sur l’objet du savoir et les complexesigni fi cabilia à Paris au XIVe siècle.”



an indivisibilist argumentation at paris around 1335 165

of his own opinion takes only 1 folio out of 7, that is to say 5 pages

out of 32 in Zubov’s edition, we can be sure that we still have access

to most of the debate.

1. The Controversy

Before going through the analysis of Montecalerio’s texts, we must

make clear who is responding to whom. At first sight, it seems that each

responds to the other; the arguments of a venerabilis doctor are mentioned

or quoted in Buridan’s text and said to be contra me, all of which are

also found in Montecalerio’s question and conversely.However, a more careful reading allows us to settle this point. In

Montecalerio’s question all of Buridan’s arguments—or nearly all of

them—are cited, repeated, disproved, and usually in the same order

that they appear in Buridan’s text, so that we can reasonably think that

Montecalerio’s text is responding to Buridan’s. As to Buridan’s quota-

tions of Montecalerio, it can be noted that they are less close to the text

than in the latter’s quotations of Buridan. Then to what do they cor-

respond? Montecalerio probably supplies the answer, for he refers—at

least six times—to an alia determinatio (or alia quaestio ) that he claims tohave written in reply to Buridan’s dicta, the existence of which is con-

firmed by the incipit of Buridan’s question: Montecalerio is mentioned

in this text as a doctor who was opposed to some Buridanian teachings

on the nature of point.11 Buridan was not the only master aimed at,

for concerning his opponent Buridan talks about “those against whom

he is disputing.”12 We can thus think that in the text we have in hand,

Montecalerio quotes—sometimes textually—Buridan’s refutations of

the arguments taken from the “other determination,” that he gives his

contra-objections and confirms his preceding arguments.Thus, the controversy may be reconstructed in this way: first, teach-

ings by Buridan, and very likely others, defending positions close to

Ockham’s; then, Montecalerio’s determination against those teachings,13

not written or lost; then, Buridan’s response as copied by Etienne Gaudet

11 “Doctor unus venerabilis obviavit quibusdam dictis meis de puncto multum

subtiliter.” QP p. 63, l. 1–2.12 “illi contra quos ipse disputat” QP p. 73, l. 25.13 Montecalerio indicates that he is also the author of a treatise De motu which is

lost.



166 jean celeyrette

which would be the text edited by Zubov; finally, Montecalerio’s text,

the one we are discussing here, also copied by Etienne Gaudet.

Nothing here allows us to maintain that this controversy corre-sponds to a publicy head disputatio.14 Moreover, the absence of ritual

polite formulas and the lively tone assumed by both men makes such

a hypothesis improbable.

The different steps of the controversy cannot be precisely dated. I

hold the weak hypothesis that both texts are contemporaries of another

question disputed by Buridan and copied by Etienne Gaudet in the

same manuscript, and to which they can be compared. It is entitled

“On the possibility that one and the same thing exist and do not exist

at the same time” ( De possibilitate existendi eandem rem et non existendi simulin eodem instanti ) and is dated from 1335.15 William Courtenay, in the

above-mentioned paper, shows that Montecalerio was master of arts

around 1340; therefore, it is reasonable to think that the controversy

took place between 1335 and 1345.

2. Montecalerio’s Text

Montecalerio’s text comes as a quaestio the plan of which is the following:

Question: is a thing corrupted by the division of the continuum

(Utrum per divisionem continui corrumpatur aliqua res: EG ff. 214r–223v)

First Article

First Part of the First Article ( ff. 214r–214v)

Buridan’s arguments corresponding to QP pp. 63–73.

– A continuing point ( punctum continuans ) would be neither in potency

nor in act. (f. 214r)

– The introduction of terminating points ( punctum terminans ) is in vain.

(f. 214r)

– A point would be neither a substance nor an accident. (f. 214v)

14 Cf. Weijers, La ‘disputatio’ dans les Facultés des arts au moyen âge, pp. 46–47.15

EG (Bur. De possibilitate existendi . . .) ff. 233v–237r. The question ends as follows:“Explicit questio de possibilitate existendi rem eandem et non existendi simul in eodeminstanti determinata per magistrum Bridam anno domini millesimo CCC°XXX°V°.Deo Gratias.”




– The introduction of points distinct from bodies is useless, since all

appearances can be saved without them. (f. 214v)

Second Part of the First Article ( ff. 214v–217v)

Answer to the four preceding arguments and confirmation of the

arguments from the alia determinatio with a series of propositions and

corollaries.

– Reply to the first argument (ff. 214v–216v). I study this answer

below.

– Reply to the second argument (f. 216v): points are the cause of the

line. – Third reason (ff. 216v–217v): the existence of points is not naturally

impossible.

– Fourth reason (f. 217v): points do exist and this is supported by the

classical example of the sphere tangent to the plane.

Second Article

In this section Montecalerio’s three reasons in support of his thesis—i.e.

that a point is a real accident—are presented. These are the three rea-sons that Buridan refutes.16 It follows that they have been put forward

in the alia determinatio.

First Part of the Second Article ( ff. 217v–220v)

Defence of the first reason: if a point didn’t exist, it would follow that

the division of a continuum would amount to a local motion of this

continuum (the parting of its sections), a position that has been refuted

in the alia determinatio.17

– Refutation of a Buridanian thesis maintaining that the division of a

continuum is an intermediary indivisible change between rest and

local motion formed by the parting of sections (ff. 217v–218v). I

present this argumentation below.

– Division consists in a change.

16 “Tunc volo defendere rationes meas quibus probavi punctum esse accidens quas

iste doctor multum reprobat et non veraciter impugnat.” EG (M.M.) f. 217v.17 “Arguebam enim primo sic quod sequeretur quod divisio seu discontinuatio essetmotus localis huius continui, et falsitatem consequentis probavi diffuse in alia determi-natio.” EG (M.M.) f. 217v.



168 jean celeyrette

– Division is neither an alteration nor an increase in size, nor a decrease

in size, nor a generation (f. 218v).

– Division is not a local motion. Montecalerio then concludes thatdivision is a corruption, for it cannot be another type of change

(argument of suf ficient division) (ff. 218v–220r).

– Refutation of Buridan’s objections against this position (f. 220v).

Second Part of the Second Article ( ff. 220v–222v)

Defence of the second reason: if a point didn’t exist it would follow

that a non-continuous thing would be continuous.18

– Presentation of Buridan’s arguments: the cause of a body’s divisioninto parts is not the surface but the nature of the body (f. 220v).

– Refutation of these arguments and contra objections (ff. 220v–

222v)

I give a general survey of this argumentation below.

Third Part of the Second Article ( ff. 222v–223v)

Defence of the third reason: the surface of a body exists because it is

what prevents us from seeing through it.

Last Part of the Second Article ( f. 223v)

Discussion of Buridan’s position according to which “point” is a

privative name. This discussion is interrupted immediately after the

presentation of Buridan’s position.

It is obviously impossible to give an exhaustive presentation of

such a lengthy and complexly stuctured discussion. We will here

restrict ourselves to select some of the arguments indicated above. AsMontecalerio’s text is the last in the series, the examples of arguments

presented here will always be preceded by those from Buridan’s to which

Montecalerio is replying and which, for that matter, he expounds very

thoroughly. Of course, it is impossible to know precisely what appeared

in the two preceding steps.

18 “Secunda ratio mea talis est: sequeretur quod non continuum esset continuum(ms: contiguum) etc.” EG f ° 220v. The correction from «contiguum» to «continuum»is justified by Buridan’s text: “non continuum esset continuum” QP p. 82, l. 1–2. ForBuridan, this reason is one of the main arguments against his adversary.




Reply to the First of Buridan’s Arguments, that “a Continuing Point can neither

be in Potency not in Act” (Second Part of the First Article)

As every potency can be actualized, the argument would amount to

denying the possibility for a continuing point to be actualized. This is a

classical topos of the anti-indivisibilist argumentation and its discussion

is one of the longest in Montecalerio’s text (ff. 214v–216v).

Presentation of Buridan’s Argumentation ( f. 214r)

Montecalerio’s presentation of the Buridanian argumentation—espe-

cially of the objections that he has made against some arguments of

the alia determinatio —is particularly detailed. Maybe this is due to thefact that the initial reasoning, the one that appeared in the preceding

step, that is to say in Buridan’s dicta, was marred by rough error and

that Buridan maintains the same erroneous reasoning in QP.

Buridan formulates four suppositions assumed, he says, by his adver-

saries:

– If a continuing point is in act, it is the case for all the others.

– The continuing point really exists in act.

– Points are ordered in a continuum in such a way that none of them

is equally distant from the terminating point.

– The terminating point can be separated by imagination, or even in

reality, from the rest of the continuum.

This being stated Buridan reasons as follows. Let us separate this first

point, which is possible by the fourth supposition. Either there exists

a first point or not.

– If it is the case, this new first point is in act according to the second

supposition and there is no intermediate point between the first

and the second point; and from the first supposition every other

point of the continuum is in act. Applying the same reasoning to

this new point, it appears that there is another point equally in act

immediately after it.

– If it is not the case, one should admit that in a finite space there are

several things existing in act and ordered by their position without

one being before the other. Now, he briefl y says, that there is aninfinity of things does not prevent that there should be a beginning



170 jean celeyrette

in existing things, ordered in relation one to the other, and, moreover,

it would be contradictory that there be none.19

The same reasoning is taken up again in various classical forms, in

particular that of the point of contact of a sphere rolling along a line

on a plan; it is shown that after the first point of contact there is a

second, etc.

The adversary is then led to admit that immediately after each point

there is another, which is denied in Physics VI20 and refutes the initial

assumptions.

The second part of Buridan’s reasoning (“si non etc.”)21 constitutes

one of the many alternatives of an argument whose paternity is usu-ally allotted to Harclay, that is to say to an indivisibilist. It is worth

noting that in this context this argument is used to refute the existence

of indivisible points. With regard to the form of the argument, the

existence of a first point in the multitude of points in the continuum

from which the terminating point has been removed appears in Harclay

and his successors or critics as the result of the fact that God, who

sees the totality of the points of the multitude, necessarily sees the first

of them.22 In Buridan’s argument, no allusion is made to the divine

absolute power. This is why the parallel is perhaps more relevant with

another indivisibilist text: Gerard of Odo’s Quaestio de continuo.23 Gerard

thinks in the same way as Buridan, but he considers the multitude of

19 “His suppositis probatur consequentia quia: vel est aliquid ipsorum ante omnia,aut nullum. Si aliquid, tunc inter ipsum et primum punctum nullum est medium punc-tum et sic erunt proxima. Si nullum est ante omnia alia, hoc est inymaginabile quodcum in alico spacio finito sut plura ordinata secundum situm et actum ibi existentia

et tamen nullum sit ante alia. Non apparet ratio, ymo contradictionem implicat, quiainfinitas in mediis non tollit primitatem in ordinatis extra se invicem existentibus ergoetc.” EG (M.M.) f. 214r; cf. QP p. 64, l. 15–27.

20 Aristotle, Physics VI, 231a21–231b18.21 The argument runs as follows: if not, one would have to accept that there are

many things in act and ordered by their respective positions in an infinite space withoutone being before the others.

22 “Certum est quod Deus modo intuetur omne punctum quod possit signari incontinuo. Accipio igitur primum punctum in linea inchoativum linee; Deus videt illumpunctum et quodlibet aliud punctum ab isto in hac linea. <Tunc a primo puncto>usque ad illum punctum immediatiorem quem Deus videt intercipit<ur> alia lineaaut non. Si non, Deus videt hunc punctum esse alteri immediatum. Si sic, igitur cum

in linea possint signari puncta, illa puncta media non erant visa a Deo.” Henry ofHarclay, “utrum mundus poterit durare in eternum a parte post,” quoted in Murdoch,“Henry of Harclay and the Infinite,” p. 228, note 24.

23 On this text, see Sander W. de Boer’s contribution in this volume.




the parts of the continuum, not the multitude of points. He then shows

the existence of “first” parts, thus indivisible, of which the continuum

is made up. However, it is necessary to introduce some shade into whatprecedes because it is not completely obvious that Gerard considers

that his reasoning must be applied to magnitudes as magnitudes and

not to natural things.24

Buridan, on the other hand, says it clearly. Moreover, considering

that the points are infinite in number, he af firms very clearly that this

infinity does not prevent the existence of a first point: the formula he

uses, reported twice by Montecalerio, is in fi nitas non tollit primitatem. The

error is all the more remarkable because if I have reconstructed the

polemic correctly, it appeared in his first teaching and his adversary hadthen replied to it: QP refers to the objections of Montecalerio.25

3. The Polemic about Buridan’s Reasoning

It does not appear that Montecalerio appreciated the full implications

of his critique of Buridan and there is no evidence that he posed theo-

retically the question of the existence of an infinite multitude and that

of a possible first element. His principal objection is built on a parallelwith parts: if one considers the proportional parts of a continuum,

starting from the first point and having them ordered according to

the time a mobile would take to go through them, Buridan’s inference

24 Gerard of Odo, Quaestio de continuo, Ms. Madrid, Bibl. Nac. 4229, f. 180rb.: “Tertiaratio est haec: in omni ordine essentiali est aliquid simpliciter primum respectu cuiusomnia dicunt posteriora, quia alias non esset ordo. Sed totius ad suas partes est ordoessentialis, quia essentialis dependentia et in via constitutionis et in via resolutionis.

Ergo in illo ordine est aliquid simpliciter primum, puta aliqua pars vel partes, quaesic est pars quod non est totum. Continuum autem est quoddam totum consitutum expartibus suis. Ergo in ipso est aliqua pars quae sic est pars quod non est totum. Et sicipsa erit indivisibilis. Quare continuum componetur ex primis partibus indivisibilibus,quod est propositum.”

25 “Contra istam rationem obicit reverendus doctor dicendo quod per consimilemrationem probabit de partibus continue proportionalibus . . .” QP p. 65, l. 16–17. It isknown that this mode of reasoning, in any case in the form which Harclay gives to it,is refuted by Wodeham. One of Wodeham’s refutation consists in saying that when thefirst point of a continuous line is removed, it remains a line without a first point: “Illaratio est bona ad improbandum puncta, sicut tangetur in questione proxima. Tamen,sustinendo puncta indivisibila, dici haberet quod destructo solo puncto terminante

lineam, linea manet interminata, nata tamen terminari, et ideo <linea> privareturtermino, id est puncto. Nec tamen propter hoc esset infinita positive, sed solum privative,id est carens fine nato haberi.” Adam de Wodeham, Tractatus de indivisibilibus, [Wood],p. 108, l. 5–10.



172 jean celeyrette

seems to apply; and if this inference were true, one should admit the

existence of a first proportional part, which is obviously false. The

argument, presented as a resumption of the alia determinatio, appears in various forms. It seems indisputable and Buridan’s response becomes

even more surprising.

Buridan essentially challenges the analogy between points and pro-

portional parts. He af firms that this parallel doesn’t work, because the

first point is reached in first position after the terminus, whereas the first

of the proportional parts, if it existed, would be reached in last posi-

tion by successive divisions starting from the whole.26 The only analogy

which he regards as valid is the one between points and parts in which

a mobile would move successively, and in the same order, starting fromthe terminus, exactly as it would do for the points; then, he says, it is true

to af firm that one part is closer to the terminus because if the continuum

is divided into two halves, one will be closer than another; into four

quarters, one quarter will be closer than the others, etc.27

In a lengthy response to these arguments, Montecalerio distinguishes

between two types of parts: those in which a part is always the part of

another, and those in which no part is part of another, in other words, consecutive parts.28 His refutation of Buridan’s replies and the confirma-

tion of his former argumentation are based on this distinction.

Moreover, Montecalerio uses his adversary’s argument in a polemical

way to establish the existence of points; in fact, he uses approximately

the same reasoning as did Gerard of Odo. It is enough, Montecalerio

says, to consider the sum of the consecutive parts: there is an infinite

number of parts in relation to a terminus. Buridan’s inference, Monte-

calerio contends, demonstrates that given this infinity of parts, there

would be one part closer to the terminus. This part cannot be divisible.

If it were, then one of its halves would be closer than the other. It must,

26 “Et addit iste doctor quod non similiter arguitur hic (pour les parties proportion-nelles) et de punctis quia in istis propter ordinem concluditur primum ex ea parte quaproceditur in infinitum, sed in punctis concluditur primum ex ea parte qua incipitur.”EG (M.M.) f. 214r; cf QP p. 67, l. 30–34.

27 “Sed inter quartas est una prior omnibus aliis quartis, et inter octavas est unaprior omnibus aliis quia prima, ergo sic debet esse in punctis quia primo est aliquid

propinquius omnibus aliis.” EG (M.M.) f ° 214r; cf QP p. 65, l. 21–66, l. 3.28 The length of the development on the two families of parts is a priori surpris-ing. But, apparently, at this time in the Parisian milieu these considerations were notregarded as trivial. It is notably the case in Burley.




therefore, be indivisible, i.e. a point. Montecalerio then turns Buridan’s

argument against its author.

4. Montecalerio’s Position

Montecalerio specifies that the existence of points of this type, i.e.

indivisible parts, does not correspond to his position. For him, the point

is not a part, because points and parts have different natures. After a

point there is always a part which is a line, and not another point,

just as immediately after a line there is a point which is its terminus.

Of course there is no point closer to the terminus, for this would implythat the continuum is composed of indivisibles, a position he does not

admit. On the other hand, he does admit that a whole is different from

its parts, the word “parts” meaning for him divisible parts.29

In his conclusion to this first article, Montecalerio states his position

without really developing it, a position that will be demonstrated more

precisely in the following article: for him, to be divided or undivided is a

property that belongs only to quantity. Therefore it cannot be said that

the subject is divided by itself, it is divided through quantity. To explain

this Montecalerio uses the example of extension (this parallel appearedin the other determination): if a subject is extended, it is through quan-

tity; it is not extended by itself, for its extension comes from a process

of rarefaction, i.e. by an alteration; it is therefore extendable. Finally,

the point is an accident in the category of quantity, it is indivisible, it

exists subjectively in a substance that is undivided, unextended in act

but naturally divisible.30 Montecalerio is particularly clear about this.

There is no punctual substance, no indivisible atom, and the fact of

being punctual and without extension is only accidental. Of course,

Buridan retorts that it supposes the possibility of an infinite rarefac-tion, a point which is conceded by Montecalerio. Let us notice that

29 It is interesting to make a link between this aspect of Montecalerio’s position andsome conclusions to the first questions in the book VI of Burley’s In Physicam Aristotelisexpositio et quaestiones.

30 “Et hoc voluit Aristoteles primo Physicorum quod divisio et indivisio primo conveniunt quantitati et aliis non nisi postea, et quia punctum est de genere quantitatis etest simpliciter indivisum et inextensum, ideo non habent a subjecto, sicut nec quantitas,

divisionem, sed ipsum dat subjecto suo indivisionem et inextensionem actualem, sicutquantitas divisionem. Per hoc potest dici ad formam rationis, sicut prius, quod punc-tum est accidens indivisibile existens subjective in substantia actu indivisa et inextensa,divisibili tamen naturaliter.” EG (M.M.) f. 217r.



174 jean celeyrette

the objection traditionally opposed to the composition of a continuum

by indivisibles, according to which a point added to a point does not

make something larger, does not impact this position.

5. The Argument of the Infinite Multitude

Let us go back to the arguments concerning infinity. One can only be

struck by their awkwardness. Neither of the two adversaries seems to

consider that there should be the slightest dif ficulty in speaking of an

infinite totality. There is no trace of a logic of the infinite: expressions

as “categorematic infi

nite” and “syncategorematic infi

nite” are missing.More surprising is Buridan’s stubbornness in the maintening a bad argu-

ment despite of the acceptable objections of his adversary, even if it is

also used by other Masters, like Gerard of Odo, to obtain a indivisibilist

conclusion immediately. A way of saving Buridan’s argument would

be to imagine that Buridan wanted to turn against his adversaries an

argument which he knew to be false. But nothing in his presentation

nor in Montecalerio’s reply supports this charitable interpretation.

However, one can certainly take into account the fact that developing

refutations or conter-objections at this point in his question, Buridandeviates from his own problematics. This could explain the lightness of

his answers before an adversary that he sometimes—and more likely

for polemical purposes—pretends not to take seriously.

Nevertheless it is indisputable that in this text Buridan does not at

all seem in control of the concept of infinite: he does not express any

hesitation when speaking about an infinite multitude and he regards it

as a totality having the same properties as a finite totality. We are very

far from the treatise on the infinite in the last version of his Physics.31

6. The Demonstration of the Existence of Points

(First Part of the Second Article)

This demonstration apparently figured in the alia determinatio, so that

there too Montecalerio’s argumentation is mingled with answers to

Buridan’s objections to the former series of arguments.

31 As an example, question 17: “utrum omni numero est numerus maior suppositosemper quod nullum continuum est compositum ex indivisibilibus.” ed. J.M.M.H.Thijssen, Johannes Buridanus over het oneindige, pp. 55–69.




His reasoning is as follows. On the basis of the fact that division, into

two for example, is a change in the subject, he establishes successively

that this change is neither an increase, nor a decrease, nor an altera-tion, nor a generation, nor a local motion. From this, he deduces that

it can only be a question of corruption which is instantaneous. This

enables him to af firm that there is a thing which is corrupted, a thing

which is the point of continuity.32

This demonstration is preceded by the refutation of the Buridanian

position on the division of a continuum. I will outline this refutation,

because several of the arguments given in response to Buridan are

taken up later to establish that division is not a local motion, which is

in fact the node of his reasoning.33

7. Buridan’s Argumentation, its Refutation by Montecalerio

(ff. 217v–218v) and the Q UAESTIO DE P OSSIBILITATE E XISTENDI . . .

Buridan’s position about the division of a continuum is the following:

division is indeed a change but it is nothing more than a relative local

motion of parts, one of which remains motionless while the other one

is moving, or both of which are moving in contrary directions. Thischange, like any local motion, is not instantaneous but temporal. The

dif ficulty of this position is that since the body was continuous before its

separation, no distinction is made between continuous parts and con-

tiguous parts, and thus between rupture of continuity ( discontinuatio ) and

rupture of adjacency ( discontiguatio ). In order to overcome this dif ficulty

Buridan considers that between the rest of an undivided continuum

and the local motion of separation of parts, or more generally between

a rest and a local motion, both temporal, there is an indivisible change

which makes the parts which were continuous become contiguous parts.But Buridan specifies at once that when one speaks about this indivis-

ible change one is only saying that at first the parts were not divided

and that afterwards they are divided.34

32 A similar reasoning can be found in Burley, notably in his Tractatus de formis [Down Scott], pp. 70–71. For the similitude between Montecalerio and Burley aboutthe nature of point, see Alice Lamy, Substance et quantité à la fi n du XIII e siècle et au début

du XIV e

siècle. L’exemple de Gautier Burley, p. 270.33 Cf. QP pp. 74–78.34 “Ad formam igitur rationis dicit quod mutatio divisionis est indivisibilis quia iste

terminus «divisio» significat «partes prius immediate fuisse non divisas et immediate



176 jean celeyrette

Montecalerio, who considers that division is not a local motion and

must refute Buridan’s position, shows that between rest and motion

there is no indivisible change. He does this with three proposals. Thispart of the controversy, like the preceding one, is paradoxical. What

exactly is the indivisible change introduced by Buridan? Either it does

not have any reality and there is no longer difference between contigu-

ous and continuous anymore, or it is real and then it seems that one

is led to admit the existence of instants. Now it is Montecalerio, the

indivisibilist, who refutes a position seeming to imply the existence of

indivisible instants, whereas this position is supported by Buridan, for

whom points and instants do not exist.

The need for the introduction of an indivisible change between twoincompossible temporal things is the subject of Buridan’s question de

possibilate existendi . . . to which I have already referred,35 and which can

be regarded as a treatise on the first and the last instant of motion.

Let us clarify. Imagine two incompossible things: a black body that is

whitening and the same body that is simply maintaining its whiteness,

i.e., that is at rest. The intermediate instant will be characterized by

the fact that the body will be at the same time white and not white,

therefore two contradictories will be true at the same time. However

it will be in the sense given above, i.e., immediately before the body

was black, and immediately after it will be white. At the end of the

question Buridan gives an interpretation of this simultaneity in terms

of measures with an example in which he supposes that a non-white at

rest is transformed into white and then remains white. It is necessary

then, he says, to suppose that there exist fi ve measurements, namely

three moments and two instants: the three moments measure the rest

of the white and of the non-white and the motion of whitening, the

two instants measure the intermediate instants.36

It is worth noting

post istas esse divisas et separatas—dans QP: discontinuas-. Et idem est quod signifi-cat «unam partem non moveri sine altera immediate prius, et immediate post unammoveri sine alia vel ambas moveri motibus contrarie» [. . .] Secundum solvit quodsecundum veritatem mutatio divisionis non est indivisibilis, scilicet generatio vel cor-ruptio indivisibilis, sed est motus localis divisibilis quia una pars totius removetur abaltera secundum profunditatem et hoc non potest esse in instanti.” EG (M.M.) f. 218r;cf QP p. 77, l. 12–20 et p. 78, l. 10–11.

35 Cf. footnote 15.36

“Et illorum oportet esse quinque mensuras, scilicet tria tempora et duo instantia:primum tempus mensurat non esse albi in sua quiete et secundum tempus in suo fluxuad esse tempus mensurans generationem albi, tertium tempus mensurat esse albi in suaquiete; instans quod est medium inter tempus generationis et tempus quietis precedentis




that Buridan gives here a certain reality to the instant of intermediate

change, since he regards it as a measurement, that of the instant during

which the two incompossibles are simultaneous.This last question of Buridan being dated, as we said, from 1335,

it can perhaps give us an indication for dating this controversy. In

his presentation, Buridan indicates that he argued elsewhere that two

incompossibles are possible at the same instant and that many people

had been surprised by this and that it was for this reason that he wrote

his question.37 It is tempting to connect it with Montecalerio’s observa-

tion made in his text:

And my third reason, given in the other question [the alia determinatio ],proves this point, namely that it is only because of a succession of timethat a proposition can be true now and its contradictory be true afterwards,and this doctor [Buridan] does not formally reply to that.38

The succession of times being opposed to the succession times/instants,

the drafting of the question de possibilitate existendi . . . could be, inter alia,

Buridan’s response to Montecalerio, which could give a terminus ante

quem to our controversy.

Even if one does not accept this conjectural reconstruction, it is worth

noting that the problematics, the references and a number of argumentsthat appear in the question de possibilitate existendi . . ., are coherent with

what is found in the polemical text of Buridan. That all these questions

were composed at roughly the same time is very probable.

8. The Common Surface of Parts as a Formal Cause of their

C ONTINUATIO (Second Part of the Second Article)

The continuity of the parts of a line consists in having a commonextremity, the continuity of the parts of a body in having a common

surface. As Montecalerio considers that the extremity point and the

mensurat momentum a quo incipit motus vel generatio, et instans medium inter tempusgenerationis et quietem sequentem mensurat ultimum momentum generationis.” EG(Bur. qu. de possibilitate existendi . . .) f. 236v.

37 “Multi mirantur qualiter alias dixi quod possibile sit idem esse et non esse in eodeminstanti, propter quod intendo dare medium per quod non solum ostendetur hoc esse

possibile sed necessarium.” EG (Bur. qu. de possibilitate existendi . . .) f. 233v.38 “Et hoc probat ratio mea tertia quam feci in alia questione, scilicet unum contra-dictorium non potest nunc esse verum et post aliud nisi ratione successionis temporum,ad quam iste doctor in forma non respondet.” EG (M.M.) f. 222r.



178 jean celeyrette

common surface are accidents in the category of quantity, he allots

logically to this point or this surface the status of cause, formal but

non material, of the continuity of the corresponding parts.39

Buridan’sobjection is, therefore, that if the cause of the continuity of the parts

is an accident in the category of quantity, one cannot explain why it

is easier to divide a piece of cheese than a stone of the same quantity,

i.e. of the same volume, or why it is easier to put in continuity two

parts of water than two stones of same size, etc. And then, Buridan

af firms that the cause of setting two things in continuity ( continuatio ) is

not in the category of the common quantitative accident which would

be a common surface, it can be only qualitative and/or substantial.

Many examples are given: that fluid bodies are easily put in continu-ity, especially when their moisture is not viscous, whereas it is not so

easy for others, which are not fluids. Buridan explains this by the fact

that the parts of the fluid body interpenetrate easily whereas those of

the other objects do not completely touch.40 Buridan’s argumentation

takes a very physical turn, with the evocation of experiments which

are not only thought experiments, but experiments that he describes

as essential for philosophical knowledge. He even invokes the activity

of the alchemist.41

To this formidable objection Montecalerio answers with a succession

of four propositions, and reaf firms that the causes of the continuity of

a body are indeed its common parts and their surfaces and not some

agreement ( convenientia ) between substantial or qualitative natures.42

His arguments do not take the same quasi experimental form; they

are repeated in various forms, and remain very general:

1) In the Eucharist the separated quantity is continuous, and yet there

is no substantial or qualitative nature present.

39 There too a comparison with Burley is relevant.40 “Et ideo causa continuationis <non> est nisi substantialis vel qualitativa vel

u traque: quia corpora bene fluxibilia faciliter continuantur et specialiter si non sithumiditas viscosa, sed alia non fluxibilia non continuantur, et causa est quia primabene subintrant se invicem, secunda nullo modo.” EG (M.M.) f. 220v; cf QP p. 81,l. 1–15.

41

QP p. 81, particularly l. 19–34.42 “Quarta propositio est quod causa suf ficiens continuationis <1°> non est conve-nientia in natura substantiali; <2°> quod nec convenientia in qualitatte vel similitudo;3° quod nec ambo sunt causa suf ficiens continuationis.” EG (M.M.) f. 221r.




2) If a quality causes a continuity, it must be humidity, because certain

humid bodies, continuous, heated, are transformed into ashes, which

are discontinuous. However, humidity is not the cause of continuitysince fire is continuous.

3) The sky is continuous and yet it does not have a corruptible quality.

4) Contrary to what Buridan says, when parts of water are brought

closer they are not immediately in continuity, but are only contiguous.

They interpenetrate at once and because of this interpenetration,

surfaces are corrupted, and a new common surface is generated

which ensures continuity.

5) A piece of dead wood can be in continuity with a piece of alive

wood although these two pieces of wood have different substantialnatures.

In conclusion, to confirm the position he had expounded in the other

determination, he adds somehow maliciously that when Buridan con-

tends that no power could divide continuum without a local motion or

without an alteration, he contradicts the divine absolute power.43

Taking into account, however, the arguments of his adversary, Mon-

tecalerio introduces the distinction between immediate formal cause of

the continuation or division of parts of a continuum, and causes that

make the continuation or division easier. He maintains that the surface

(or the line or the point) common to the parts is the formal cause of

continuation, and that the two parts ready to be put in continuity are

its material cause. On the other hand, what makes continuation or

division more or less easy varies according to the nature of the bodies

we consider. Adapting certain arguments of Buridan he admits that

this cause is sometimes qualitative: in the case of liquids, a nonviscous

humidity can make easier the continuation or division of the parts; in

43 Buridan: “Ad illud autem quod adiungitur de potentia divina forte dicereturquod sine motu locali aut aliqua facta mutatione humidi continuantis Deus non possetdiscontinuare partes lapidis. De quo tamen nihil asserere intendo.” QP p. 82, l. 32– p. 83, l. 2. Montecalerio: “Sed respondendo ad alias, specialiter ad secundam, dicitquod per nullam potentiam poterat fieri de continuo non continuum nisi facto motu

locali vel alteratione, et quia salva eius reverentia hoc est contra potentiam divinam etforte contra naturam alico modo, ut probavi superius in defensione prime rationis, ideoratio manet concludens necessario quod punctum est, et per consequens omnes rationesmee—celles de l’autre détermination-manent in suo vigore.” EG (M.M.) f. 222r.



180 jean celeyrette

some cases, humidity can only make easier the division, and in other

cases, it cannot cause either of them, etc.44

9. Conclusion

One will certainly feel frustrated after having read this text—or rather

these texts. It teaches us Montecalerio’s position, but in an incidental

way, and we do not really have an exposition in due form: only the

answers to Buridan’s criticisms remain. The alia determinatio was supposed

to give such a presentation, and perhaps it would have allowed us to

know the nature and the extent of Montecalerio’s debt with regard toBurley. Given the present state of the texts, such a debt is only probable

because of our very rudimentary knowledge of Montecalerio on the

one hand, but also because of the disparate character and the lack of

coherence of Burley’s argumentation on the other hand.

In addition, as it was pointed out earlier, the criticism of the

Buridanian position, a position very close to Ockham’s, is missing in

Montecalerio’s text. Such a criticism would have perhaps provided expla-

nations regarding the point of view adopted at that time by Buridan

on the status of quantity. It is clear that he departs from Ockham onthis topic in the two known versions of his physics, but more clearly

in the ultima lectura than in the tertia lectura.45 However this question

becomes central in our debate on the nature of points since one of

44 “Patet quod alia est causa continuationis et divisionis prima et immediata et aliaest causa facilitatis vel dif ficultatis continuationis et divisionis. Quia continuationis causaformalis et immediata est superficies vel linea vel punctus communis, sed causa materialis

immediata sit quanta vel quantitates nate habere idem ultimum. [. . .] In aliquibus esthumiditas fluxibili, hoc est non coagulata bene nec bene permixta sicco, ut patet inliquidis, et hoc est causa facilitatis divisionis quia talis humiditas et licet possit tenerepartes unitas in eodem ultimo, tamen faciliter cedit dividenti [. . .] In aliquibus autemtalis humiditas non est causa continuationis nec satis discontinuationis et divisionissicut in igne.” EG (M.M.) f. 222r.

45 In the tertia lectura which is generally dated from the beginning of the years1350 Buridan seems to reply in a conservatory way in accordance with the via antiqua(quantity is an accident distinct from the substance) but without engaging clearly withthe question utrum omnis res extensa est magnitudo (qu.I-7). On the other hand in thecorresponding question of the ultima lectura (qu.I-8), after giving arguments favorableto the ockhamist position (the quantity is not really distinct from the substance), and

refuting them, he declares that these refutations are insuf ficient and gives new oneswhich are presented as personal and which enable him to determine against Ockhamthat nulla substantia est magnitudo. On the two versions of Buridan’s physics, see mypaper cited in footnote 9.




the protagonists considers that the point is an accident in the category

of the quantity. Alice Lamy highlighted this point concerning Burley’s

polemic against Ockham.46

Still, it remains that we have here, even incompletely reported, the

example of a Parisian academic debate on Ockhamist physics, a debate

which shouldn’t be neglected, all the less so since Buridan, a Master

whose importance cannot be disputed, here defends the Ockhamist

point of view. Moreover, it must be stressed that no other text on the

continuum and no other text of Buridan’s than those I mention here

appear in Etienne Gaudet’s collection.

What are the formal characteristics of this polemic? First of all, a

great promptness of tone. An example has been given at the begin-ning of this article,47 but there are many other instances of this sort:

Buridan does not hesitate to write that his adversary answers very

weakly ( valde debiliter ); on several occasions he ironically admires the

arguments of his adversary, while the other retorts that Buridan does

not have to admire them, but that quite to the contrary it is he who

admires—ironically—what Buridan claims to demonstrate, etc. The

apparently respectful tone of the incipit is completely out of phase with

the general tone of the two texts.

One is also struck by the lack of theoretical reference to the infinite,

which was pointed out earlier, and by the remarkably restricted place

occupied by the logical considerations in Montecalerio, but also in

Buridan who is known for having written logical treaties in the years

1320. I will not try to explain this fact, but will simply notice that when

studying the divisibility of continuum, putting aside the initial argu-

ment, the two protagonists call upon considerations on the rarefaction

of the air or fire, the transparency of the sheets of gold, but make no

reference to the refi

nements of the supposition theory, of the compositeand divided sense, etc.

Last but not least, mathematics are mostly absent. They are men-

tioned only once, when Buridan, indeed in a rather clumsy way,

asserts that mathematics treat natural things from the point of view

of the quantity and that if points—quantitative accidents of natural

things—existed, there should exist mathematical points. The argument

46 Alice Lamy, Substance et quantité à la fi n du XIII e siècle et au début du XIV e siècle, l’exemplede Gautier Burley, pp. 228–277.

47 See footnote 8.



182 jean celeyrette

is weak since points indeed exist in mathematics. Actually Montecalerio

does not take the trouble to answer it. One finds no allusion, even to

challenge their relevance, to the rationes mathematicae very fashionableamong English scholars since Duns Scotus.

All these features of our controversy make it very different from

English polemics like the one which opposed Chatton and Wodeham,48

but in the current state of our knowledge we cannot say up to which

point it is characteristic of the Parisian debates of the years 1330.

48 See, for example, Murdoch & Synan, “Two Questions on the Continuum: WalterChatton (?), O.F.M. and Adam Wodeham, O.F.M.”



JOHN WYCLIF’S ATOMISM1

Emily Michael

The fourteenth century evangelical doctor, John Wyclif (1320–1384),2 a

prominent,3 if controversial, Oxford master, was an atomist. Like other

scholastics of his time, Wyclif adopted a hylomorphic ontology, and he

accepted the Aristotelian view that all corporeal things are composed

of and reducible to four elements (earth, air,fi

re, and water). Still,unlike his contemporaries, he also maintained that the body of the

total universe, and each individual body within it, is composed of and

reducible to a finite number of elemental atoms.4

John Wyclif is dubbed the evening star of scholasticism and the

morning star of the Reformation by H.B. Workman, and by such

contemporary commentators as Kenny and Spade.5 In another paper,6

I examined the question of whether he can with some justice also be

dubbed the morning star of a reformation in science, for the fact is that

his distinctive atomism as well as his mind-body dualism anticipates, insome respects, developments of early modern natural philosophy. The

final section of this chapter will provide a discussion of this distinctive

1 I gratefully acknowledge that research for this chapter was partially supportedby a grant from the Research Foundation of the City University of New York. Alltranslations are my own, unless otherwise noted.

2

The best biographical study of Wyclif ’s life is by Workman, John Wyclif, A Studyof the English Medieval Church; Kenny, Wyclif , provides a clear and concise introductionto Wyclif ’s thought.

3 Conti, “Analogy and Formal Distinction: On the Logical Basis of Wyclif ’s Meta-physics,” p. 133, says that Wyclif “was one of the most important and authoritativethinkers of the late Middle Ages.” For an excellent bibliography, see: Thomson, The Latin Writings of John Wyclif: An Annotated Catalog .

4 See also, for discussion of Wyclif ’s atomism: Pabst, Atomtheorien des lateinischen Mittelalters; Zubov, “Walter Catton, Gérard d’Odon et Nicolas Bonet;” Kretzmann,“Continua, Indivisibles, and Change in Wyclif ’s Logic of Scripture.”

5 A point reflected in Lewis Sergeant’s title John Wyclif, Last of the Schoolmen and Firstof the English Reformers. Workman, John Wyclif, A Study of the English Medieval Church i,

4: “The first of the reformers was, in fact, the last of the schoolmen.” Kenny, Wyclif ; John Wyclif, On Universals, [trans. Kenny], introduction by Paul Vincent Spade. Seealso Milton, Areopagitica [Bohn], ii, 91.

6 Michael, “John Wyclif on Body and Mind.”



184 emily michael

atomism. I will begin with an exploration of several other questions

that Wyclif ’s atomism raises.

First, we might note that this distinctive and interesting atomism isnow little known.7 What is more, though Wyclif was well known in his

time, currently there is no evidence of discussion or influence of his

atomism in the natural philosophy of his immediate successors. One

might well ask why, though this fourteenth century atomistic view is

clearly a striking development in natural philosophy, it received so little

attention and had so little impact in its time and why this early atom-

istic account, which is clearly a dramatic development in the history

of thought, is now still so little known. That is:

Q1: Why the neglect of John Wyclif ’s atomism?

Second, one might reply to this query that the uniqueness and neglect

of Wyclif ’s atomistic matter theory is easily understood in the light

of the seeming lack of necessity for and perhaps inconsistency of an

atomistic account in conjunction with a hylomorphic ontology. One

might add that scholastic resistance to a Wyclifian corpuscular matter

theory was also supported by Aristotle’s strenuous rejection of atom-

ism. This raises a second question, namely, what is the rationale for

and the framework of an atomism that is coupled with a hylomorphic

ontology. This, in fact, gives rise to two questions:

Q2: How is a corpuscular matter theory possible in conjunction with

the Aristotelian view that prime matter and substantial form are

the fundamental principles of corporeal things?

Q3: Why is a corpuscular matter theory needed in conjunction with

the Aristotelian view that prime matter and substantial form arethe fundamental principles of corporeal things?

In the following section, I will examine the first two questions. In the

subsequent section, in response to the final question, I will consider, first,

the scholastic context of Wyclif ’s matter theory and, second, motivation

for his commitment to an atomistic account. In the final section, I will

7 See Spade, On Universals, introduction, p. 1: “John Wyclif ’s philosophical viewsare almost entirely unknown to modern scholars . . .” This is especially true of Wyclif ’snatural philosophy.



john wyclif’s atomism 185

further consider the rationale for and I will investigate the nature of

John Wyclif ’s atomism.8

1. The Impact and Framework of Wyclif’s Corpuscular

Matter Theory

1.1. Q1: Why the Neglect of John Wyclif ’s Atomism?

One might well ask, why, though Wyclif ’s fourteenth century atomistic

view is clearly a striking development in natural philosophy, it seem-

ingly generated so little interest among Renaissance philosophers and

is now so little known. The fact is that though our Oxford master wasan innovative thinker, a prolific writer, and extremely influential in

his time, his philosophical views generally are little known even to his

fifteenth century successors. His biographer, Herbert Workman, tells us

that, in Wyclif ’s lifetime, “his influence is beyond dispute. The source of

this influence is clear. As a schoolman he was the acknowledged leader

among his contemporaries at Oxford;”9 and Henry Knighton, though

an implacable adversary, describes his foe as “the most eminent doctor

of theology of his time, in philosophy, second to none, in the training

of the schools without a rival.”10 Kenny correctly claims that Wyclif ’s

philosophical views are as interesting and well-reasoned as those of

Ockham or Scotus, and that his influence in his time was at least as

extensive. This seems to argue that there was an interest in and influ-

ence of Wyclif ’s natural philosophy in his lifetime. What then is the

reason for Wyclif ’s seemingly short shelf life and his current obscurity?

One reason would seem to be the following.

Thirty-one years after Wyclif ’s death, the Council of Constance

[1414–1418] declared him a heretic, and (after condemning theologicaland philosophical propositions attributed to him)11 ordered that his body

8 Works of Wyclif relevant to this study include: De Compositione Hominis [Beer],hereafter cited as DCH; De Materia et Forma [Dziewicki], hereafter cited as De M&F;Tractatus De Logica [Dziewicki] (Wyclif ’s matter theory is raised in his discussion ofpropositions of place to explain his distinctive notion of place, in vol. III, LogicaeContinuatio, tract 3, ch.9, which will be hereafter cited as LC 3.9); De Ente [Dziewicki];Trialogus [Lechler], also printed in 1525 and 1753, hereafter noted as Tr .

9 Workman, John Wyclif, A Study of the English Medieval Church., vol. i, p. 3.10

Ibid ., vol. ii, p. 151.11 Conciliorum Oecumenicorum Decreta [Alberigo e.a.], p. 427. “Cum itaque in sacrogenerali concilio nuper Romae celebrato decretum fuerit, doctrinam damnatae memo-riae Ioannis Wicleff damnandam esse, et libros eius huiusmodi doctrinam continentes



186 emily michael

be exhumed and that his corpse and all his works be burned.12 The

Council dogmatically proclaimed that it forbids the reading, teaching,

expounding and citing of the said books (they mention Dialogus, Trialogus,and the same author’s other books): “It forbids each and every Catholic

henceforth, under pain of anathema, to preach, teach, expound, or hold

the said articles or any one of them”13 and it “orders local ordinaries

and inquisitors of heresy to be vigilant in carrying out these things . . .”.14

Jan Hus, the Evangelical Doctor’s influential Czech follower, though

granted safe conduct to these proceedings by the Pope and the Emperor

Sigismund, was condemned by the same Church council and burned

at the stake.15 Nonetheless, Wyclif ’s works (copied, for example, by his

Czech admirers and carried back to their homeland) survived, andone, his Trialogus, was printed in 1525.16 This provides some evidence

fore tamquam haereticos comburendos, et doctrinam ipsam damnatam et libros eiustamquam doctrinam insanam et pestiferam includentes, combustos fuisse cum effectu;et huiusmodi decretum huius sacri auctoritate concilii fuerit approbatum . . .”

12 For an excellent discussion of this condemnation and history of Wyclif ’s reputa-tion among Catholics, see Kenny, “The Accursed Memory: The Counter-ReformationReputation of John Wyclif”. See also Leff, “John Wyclif: The Path to Dissent,” and

idem, Heresy in the Later Middle Ages. The focus of an of ficial Church Council on thecondemnation of an individual and the establishment of this condemnation as Church

dogma is unusual.13 Conciliorum Oecumenicorum Decreta [alberigo e.a.] p. 422: “. . . inhibens omnibus ex

singulis catholicis sub anathematis intermination, ne de cetero dictos articulos autipsorum aliquem audeant praedicare, dogmatizare, offerre, vel tenere.”

14 Ibid.: “Super quibus exsequendis et dibite conservandis mandat praedicta sanctasynodus ordinariis locorum ac inquisitoribus haereticae pravitatis vigilanter intendere,prout ad quemlibet pertinet, secundum iura et coanonicas sanctiones.”

15 The Council declared that “John Wyclif, of accursed memory, by his deadlyteaching, like a poisonous root, has brought forth many noxious sons, not in Christ

Jesus through the gospel, as once the holy fathers brought forth faithful sons, but rather

contrary to the saving faith of Christ, and he has left these sons as successors to hisperverse teaching. This holy synod of Constance is compelled to act against these menas against spurious and illegitimate sons, and to cut away their errors from the Lord’sfield as if they were harmful briars, by means of vigilant care and the knife of eccle-siastical authority, lest they spread as a cancer to destroy others.” Ibid., pp. 426–427:“(Sententia contra Ioannem Huss) . . . vir damnatae memoriae Ioannes Wicleff suamortifera doctrina non in Christo Iesu per evangelium, ut olim sancti patres fideles filiosgenuere sed contra Christi salutarem fidem, velut radix virulenta, plures genuit filiospestiferos, quos sui perversi dogmatis reliquit successores. Adversus quos haec sanctasynodus Constantiensis tamquam contra spurios et illegitimos filios cogitur insurgere,et eorum errores ab agro dominico tamquam vepres nocivas, cura pervigili et cultroauctoritatis ecclesiasticae resecare, ne ut cancer serpant in perniciem aliorum.”

16

This work may have been known, for example, by J.C. Scaliger, who providesa rudimentary hylomorphic corpuscular matter theory in his Exotericae Exercitationes [Lyon, 1557], and, in turn, influenced such seventeenth century hylomorphic atomistsas Daniel Sennert and Sennert’s student and advocate, Johannes Sperling.




of a role for Wyclif during the Renaissance. Nonetheless, the Council

of Constance succeeded very well in suppressing Wyclif ’s views. All

but one of Wyclif ’s massive array of philosophical works remained inmanuscript form, unpublished until the late nineteenth century. Many

remain unpublished to this day.

It might with merit be claimed that the evangelical doctor’s legacy

was preserved, first, through the martyrdom of Hus and his subsequent

persistent and devoted following, and, second, through the propositions

condemned. Some of these were clearly never asserted by Wyclif, for

example, among the 45 recorded in session 8 of the Council, “God

ought to obey the devil.” and, among the 58 recorded in the fifteenth

session, “Every being is everywhere, since every being is God.” Othersdistort features of his philosophic views. Of particular importance to

the current study is the condemnation of such propositions as:

P1: “Any continuous mathematical line is composed of two, three or

four contiguous points, or of only a simply finite number of points;

and time is, was and will be composed of contiguous instants.”17

P2: “It must be imagined that one corporeal substance was formed

at its beginning as composed of indivisibles, and that it occupies

every possible place.”18

P3: “God cannot annihilate anything, nor increase or diminish the

world.”19

This is a direct condemnation of foundations of Wylcif ’s atomism.20

Wyclif was declared a heresiarch by (as described by the Council,

Session 15, 6 July 1415) “[t]his most holy general synod of Constance,

17 Conciliorum Oecumenicorum Decreta [Alberigo e.a.] p. 426 “51. Linea aliqua math-ematica continua componitur ex duobus, tribus, vel quatuor punctis immediatis, autsolum ex punctis simpliciter finitis; vel tempus est, fuit, vel erit compositum ex instan-tibus immediatis.”

18 Ibid.: “52. Imaginandum est, unam substantiam corpoream in principio suo ductamesse ex indivisibilibus compositam, et occupare omnem locum possibilem.”

19 Ibid.: “49. Deus nihil potest annihilare, nec mundum majorare vel minorare . . .”20 It might be conjectured that these propositions were condemned because of the

dif ficulties atomism raises for an account of the Eucharist, in particular, in regard tothe doctrine of transubstantiation. Of particular importance in Wyclif ’s downfall is thesubject of fi ve condemned propositions, viz., his view of the sacrament of the Eucharist,

which analysis was influenced by his atomistic account of the physical world. See, forexample, Wyclif, De M&F (which is an early work), in the context of his discussionof substantial change, p. 189: “De conversione autem panis in corpus Christi, quamecclesia vocat transubstanciacionem, est longus sermo, et mihi adhuc inscrutabilis.”



188 emily michael

representing the catholic church and legitimately assembled in the holy

Spirit, for the elimination of the schisms, errors and heresies.”21 The

lack of influence of his corpuscular matter theory can be explained bythe general acceptance, among his contemporaries, of hylomorphism

and of Aristotle’s arguments opposing atomism. But this does not

explain the absence of all discussion of and seeming lack of knowl-

edge about Wyclif ’s atomism. It seems likely that the repudiation and

extensive condemnation of Wyclif ’s views, the prohibition “under pain

of anathema” of the reading, teaching, expounding and citing of his

works, and the order that his books, treatises, volumes and pamphlets

be publicly burned had an impact on subsequent knowledge about his

philosophical views and on his legacy generally.

1.2. The Ontological Structure of Material Substances

We turn now to the second question, that is, how can atomism be con-

sistent with Aristotelian hylomorphism.22 Like other scholastics of his

time, Wyclif accepted the view that prime matter and substantial form

are the fundamental ontological principles of each material substance.

What I wish to show is that Wyclif adopted a distinctive hylomorphic

account, to be identified in what follows as scholastic pluralism, andthat scholastic pluralism is consistent with atomism.

Some commentators claim that Aristotelian hylomorphism is neces-

sarily inconsistent with atomism. For example, Van Melsen says that, in

Aristotle’s view, “If one admitted that some body was a compound, the

logical conclusion had to be drawn that it possessed only one form.”23

But, in a hylomorphic atomism (like Wyclif ’s) “the original forms of

the elements are said to remain [in the compound]. This impaired the

unity of the compound in virtue of the rule: one substance means one

form, and a plurality of forms means a plurality of substances.”24 Dijk-sterhuis, discussing Daniel Sennert’s hylomorphic atomism, similarly sees

Aristotelian hylomorphism and atomism as inconsistent. He says that

21 Conciliorum Oecumenicorum Decreta [Alberigo e.a.], p. 421: “Sacrosancta Constantiensissynodus generalis, ecclesiam catholicam repraesentans, ad extirpationem schismatiserrorumque et haeresium in Spiritu sancto legitime congregata, auditis diligenter etexaminatis libris et opusculis damnatae memoriae Ioannis Wicleff per doctores et

magistros studii generalis Oxioniensis.”22 In what follows, hylomorphism will refer solely to Aristotelian hylomorphism.23 Van Melsen, From Atomos to Atom: The History of the Concept of Atom, p. 126.24 Ibid ., p. 125.




Sennert, “to reassure his philosophic conscience, introduces subordinate

and higher forms, which had always been unanimously rejected by the

scholastic philosophers.”25

What I wish to show is that these commenta-tors misrepresent the development of scholastic hylomorphism.

The fact is that from the inception of scholasticism in the thirteenth

century until its demise in the late seventeenth century, Aristotelians

were divided into two camps in their analyses of how prime matter and

substantial forms compose material things. One hylomorphic approach,

influenced by Thomas Aquinas and to be identified as scholastic monism,

is very familiar. It is this interpretation of form and matter that Van

Melsen and Dijksterhuis identify as Aristotle’s own view. The other now

little known approach, namely, scholastic pluralism,26 was associatedin the thirteenth century with, for example, archbishops of Canter-

bury, Robert Kilwardby and John Pecham (mentioned in this regard

by Wyclif ),27 and, in the fourteenth century, was the interpretation of

Aristotle’s view presented not only by Wyclif, but also by such prominent

and influential figures as Scotus, Ockham, and John of Jandun.

Aquinas and his followers (i.e., scholastic monists) support the fol-

lowing theses:

TT1: Prime matter is pure potentiality.

TT2: Each substantial form is an absolute and immutable actuality,

determining a complete substance.

TT3: An individual substance can have no more than one substantial

form, which inheres directly in prime matter.

TT4: The one substantial form of a human being is a rational soul.

The scholastic pluralists maintain instead:

25 Dijksterhuis, The Mechanization of the World Picture, p. 283. Daniel Sennert was aninfluential seventeenth-century physician, who adopted atomism and a hylomorphicontology.

26 For discussion of scholastic pluralism, see Zavalloni, Richard de Mediavilla et lacontroverse sur la pluralité des formes.

27

Wyclif, DCH , p. 74, cites Robert Kilwardby and John Pecham, respectively aDominican and a Franciscan of the previous century, both of whom, as archbishopsof Canterbury, issued condemnations in which the unicity of forms of Aquinas wasseen as a heretical doctrine.



190 emily michael

TP1: Prime matter has an actuality of its own.

TP2: Substantial forms are of two sorts, subordinate forms, which

determine a part of a substance or an incomplete substance,and supervening forms, which determine a total substance and

place it in its species.

TP3: An individual substance can have a plurality of substantial

forms.

TP4: An intellective soul is the ultimate supervening substantial form

of a human being.

We find here, in these respective theses, two very different interpreta-

tions of Aristotle’s fundamental principles, prime matter and substantialform, and this, in turn, as might well be expected, had implications for

a variety of other doctrines from embryology to immortality. So, for

example, Aquinas and his followers maintain that each human being

has a simple ontological structure. Each is wholly determined only by

a rational soul that inheres in prime matter. Here prime matter is pure

potentiality, incapable of existing apart from form, and since forms are

the object of cognition, no idea of matter is possible, even for God.

A substantial form gives a substance being, activates it, and makes it

what it is. Socrates’s one substantial form is a rational soul, and his

intellect is but one power of this soul, along with powers of nutrition,

sensation, local motion, and appetite.

Scotus and Ockham support instead the view each human being has

a pluralistic and hierarchical structure. From the viewpoint of both,

Socrates is composed of prime matter, which has a positive reality of its

own,28 and a plurality of kinds and grades of substantial forms.29 Both

would agree that prime matter is the subject of the form of Socrates

body, and that Socrates body, in turn, is the subject of a superveningsubstantial form, namely, Socrates human soul, which places him in

his species and makes him what he is. But Scotus and Ockham each

provide a different account of the corporeal and psychic substantial

forms that determine a human being. In Ockham’s view, Socrates body

28 See, for example, John Duns Scotus on the reality of matter in Liber II Sententia-rum, dist. 12, q. 1, 2 [Vives], hereafter cited as Sent . See, also, Ockham’s discussion ofmatter, in William of Ockham, Summula Philosophiae Naturalis [Brown], OPh VI, pp.

181–195; and McCord Adams, William Ockham, pp. 680–690.29 See, John Duns Scotus, Sent . [Balic e.a.], lib. IV, dist. 11, q. 3. See, also, Ockham, Quodlibeta Septem [Wey], 2nd Quodlibet, q. 11. See translation in Ockham, QuodlibetalQuestions [trans. Freddoso & Kelly], vol. I, pp. 136–139.




(an incomplete substance) is determined by one subordinate substantial

form. Scotus claims instead a plurality of corporeal substantial forms of the

parts of Socrates’s body, e.g., of blood, bones, flesh and the like. Further,each provides a different analysis of the relation between vegetative,

sensitive, and intellective psychic principles. Scotus claims one soul in

which these psychic principles are formally distinct. From Ockham’s

viewpoint, Socrates must have two really distinct souls, an organic soul

and mind. Jandun, taking yet another view, supports three really distinct

souls, two inhering souls, vegetative and sensitive, and a mind, which,

following Averroes’s monopsychism, is a separated substance that is one

for all human beings. Scholastic monists adopted a single view, namely,

that of Aquinas; scholastic pluralists developed a plurality of competinganalyses of the structure of corporeal things.

Among Aquinas’s thirteenth-century contemporaries, the common

view was the pluralistic one that at least some substances have more

than one substantial form and that prime matter has a positive reality

of some sort. Aquinas’s principle rationale for his opposition to plural-

ism was a metaphysical claim that each substance must be one per se.

Thomas argues: each entity is a single substance, but, since the actuality

or nature of each substance is determined by its substantial form, if

one subject has two substantial forms, it will have two actualities, and,

therefore, it will be two substances.30 So, for example, Aquinas con-

tends: if a man is a living thing by one form, the vegetative soul, and

an animal by another form, the sensitive soul, and a man by another

form, the intellective soul, it would follow that a man is not absolutely

one.31 That is, a human being would, in fact, be three distinct things,

a plant, for the vegetative soul is the plant soul, and a brute animal, as

well as a human being. From this Thomistic viewpoint, distinct forms

inform, i.e. determine, distinct substances. But a single substance mustbe one unified entity; a substance cannot be composed of substances.

Therefore, a single substance must have a single substantial form.

Aquinas therefore claims:

TT5: The one substantial form of a substance is the principle that

makes it one per se.

30 Aquinas, Summa Theologica, q. 76, art. 4.31 Aquinas, Summa Theologica, q. 76, art. 3.



192 emily michael

It was thereafter incumbent upon pluralists to respond to this meta-

physical argument and demonstrate how a substance with a plurality

of substantial forms is nonetheless one per se.Aquinas’s rigorous and systematic monistic theory (TT1–TT5) was a

controversial interpretation of Aristotle’s view,32 and there were numer-

ous attacks against it. Perhaps the most important attack was William de

la Mare’s 1279 Correctorium Fratris Thomae,33 a compilation of errors from

the works of Aquinas in which William prominently includes TT1–TT5,

as well as Aquinas’s view that the principle of individuation is signated

matter.34 William ascribes Thomas’s complaint to an erroneous view of

forms, in particular, to a denial of the relation of “grades of forms”

within a single substance, and attributes to Aristotle the view that asubstance in which there are various grades of forms is unified by an

ultimate form. William and subsequent pluralists maintained:

[TP5] The ultimate or supervening specific substantial form of a

substance is the principle that makes it one per se.35

Further, the pluralists supported their claims by a vast arsenal of oppos-

ing arguments, logical, metaphysical, physical, and theological.

The text of William de la Mare, a Franciscan, was adopted by the

General Chapter of Franciscans in Strassburg on May 17, 1282, and

was ordered, by the same Chapter, to be read along with the works of

Aquinas as part of the text.36 This Franciscan endorsement of William

de la Mare’s Correctorium further established, as common Franciscan

32 See Zavalloni ( Richard de Mediavilla et la controverse sur la pluralité des formes, pp.383–419) for a careful study of pre-Thomistic positions. Zavalloni provides convincing

evidence that, prior to Aquinas, “la doctrine de la pluralité des formes . . . est univer-sellement acceptée en ce qui regarde le corps ou, du moins, le composé humain.”(p. 405)

33 This work appears in Correctorium corruptorii “Quare” [Glorieux].34 In January of 1279, John Pecham, an influential Franciscan theologian, was

appointed Archbishop of Canterbury, and, late in the year 1279, William de la Marecompleted, possibly under the patronage of Archbishop Pecham, a compilation oferrors from the works of St. Thomas. William’s Correctorium Fratris Thomae [Glorieux]includes 117 excerpts from Aquinas’s works.

35 William says, for example: “Ad illud tamen quod multitudo formarum est contrarationem unitatis, dicendum quod res in qua est multitudo et gradus formarum, estuna per formam ultimam, ut expresse dicit Philosophus VIII Metaphysicae et Com-

mentator . . .,” Correctorium Fratris Thomae [Glorieux], p. 396.36 For discussion of this Franciscan endorsement of William de la Mare’s Correcto-rium, along with detailed consideration of the response of followers of Aquinas, seeRoensch, Early Thomistic School .




theses, the actuality of matter37 and the plurality of substantial forms,

supporting thereby TP1–TP5, and it contributed to the entrenchment

of these theses, and to the division of Franciscans and Dominicans onphilosophical grounds.38 A complex of doctrines, viz., TP1–TP5, along

with, for example, the view that matter is not the principle of individua-

tion, and the rejection of the Thomistic distinction between essence and

existence, remained associated with the distinctive and quite long-lived

approach that I have referred to as scholastic pluralism.

Wyclif argues that, from the Thomistic viewpoint, the composition

of a human being is as simple as that of an element, which is absurd.

Like his Franciscan predecessors, Scotus and Ockham, he maintains

that human beings and other compound substances have a pluralisticstructure. He claims, for example, that “many substantial forms of

different species are together in the same composite, however one

will be subordinate to another, as is clear of a compound.” and “one

form that is more general and another more specific, which are in the

same suppositum, are ordered together, as forms of different species, as

is manifestly clear of bone, of flesh, of nerve and other heterogeneous

parts in man.”39 So, in Wyclif ’s view, Bossy the cow is composed of a

plurality of grades of matter and of substantial forms. First, prime mat-

ter is the subject of the substantial forms of the elemental atoms,40

and differing combinations of the elements, i.e., of earth, air, fire, and

37 Among William’s arguments for his position, as especially important, are argu-ments that focus upon God’s omnipotence. For example: (1) The Thomistic view thatmatter is pure potentiality, with no positive reality, implies that God cannot makematter without form ( Correctorium, p. 409, art. 108).; that God cannot cause matterto precede its form in time ( ibid., p. 113, art. 27), and that God can have no idea ofmatter simpliciter ( ibid., p. 326, art. 79; p. 389, art. 97; see also articles 4, 80, 81). But

this is a denial of God’s omnipotence, and so must be false. The last point, of course,also questions divine omniscience.38 Various lengthy replies to William’s Correctorium by Dominican Thomists appeared

in short order. Four, entitled the Correctorium corruptorii , have been identified respectivelyas “Quare,” “Circa,” “Sciendum,” “Questione,” and the first three, with much dis-agreement among scholars, have been respectively attributed to Richard Knapwell,

John Quidort, and William of Mackelsfield. See Glorieux, Le Correctorium corruptorii“Sciendum,” vol. II, pp. 12–19.

39 Wyclif, Trialogus, p. 92: “. . . multas formas substantiales dispares specie esse in eodemcomposito dum tamen una sit subordinata alteri, ut patet de mixtis. Unde sciendum,quod sicut una forma generalior et alia specialior, quae sunt in eodem supposito, adinvicem ordinantur, sic formae disparium specierum, ut manifeste patet de osse, de

carne, de nervo et caeteris partibus etrogeneis in homine.”40 Wyclif ’s distinctive account of prime matter and his analysis of elemental atoms,each of which is a composite of prime matter and a substantial form, will be examinedin following sections.



194 emily michael

water, in turn, are the matter of the substantial forms of the minima

naturalia (the compound corpuscles) that compose Bossy’s blood, bones,

flesh and the like.41

Next, Bossy’s body is the matter of a superveningsubstantial form, i.e. Bossy’s cow soul. Scholastic pluralism provided

an approach to corporeal substances that made possible an atomism

that is consistent with a hylomorphic ontology. His pluralistic analysis,

therefore, provided the framework for Wyclif ’s atomistic account as well

as for his dualistic account of the human body and mind.

2. The Final Question

2.1. The Context of Wyclif ’s Atomism

The final question posed above is:

Q3: Why, in Wyclif ’s view, is a corpuscular matter theory needed inconjunction with the Aristotelian view that prime matter and substantialform are the fundamental principles of corporeal things?

To respond to this question, we turn first to Wyclif ’s context.

In the fourteenth century, the nature of prime matter became a

hotly debated philosophical problem. As noted above, Aquinas’s viewprovided grounds for identifying prime matter with pure potentiality.

He, therefore, says that prime matter “does not have being (existence)

per se.”42 He explains:

It is sometimes under one form, and sometimes under another. Butthrough itself matter can never exist, because—since in its nature itdoes not have any form, prime matter does not have existence in actual-ity, since existence in actuality is only from a form. But [prime matteris] only in potentiality. And so whatever is in actuality cannot be called

prime matter.43

41 Wyclif used the term “minima naturalia” to refer to specific secondary particles,composed of atoms. Pierre Gassendi (1592–1655) is generally credited with the firstuse of the term “molecule” to identify secondary or compound particles.

42 Thomas Aquinas, De Principiis Naturae [trans. Bobick], p. 31: “Sed per se numquampotest esse.”

43

Ibid.: “Quandoque enim est sub una forma, quandoque sub alia. Sed per senumquam potest esse, quia—cum in ratione sua non habeat aliquam formam, nonhabet esse in actu, cum esse in actu non sit nisi a forma. Sed solum in potentia. Etideo quidquid est in actu, non potest dici materia prima.”




Therefore, although it is the persistent substratum, in which all sub-

stantial forms inhere and which itself inheres in nothing else, prime

matter is pure potentiality.Scotus, a thoroughgoing scholastic pluralist, sought to make clear

inadequacies of this account of prime matter. He distinguishes subjec-

tive potentiality from objective potentiality:

For something can be in potentiality in two ways: In one way as an end;in the other way as a subject that is in potentiality toward an end (. . .) sothat the subject existing is said in subjective potentiality; and the same inrespect to the agent is said to be objective; they can be said to be sepa-rated, as in creation, where it is objective potentiality and not subjective,

because here something is not a subject.44

His analysis and arguments raised consciousness in regard to the claim

that matter, to serve as a principle of generation and corruption, must

have subjective potentiality ( potentia subjectiva ).45 That is, prime matter

must be an actual subject to receive substantial forms.46 This means,

Scotus concludes, that prime matter must actually exist, and, what is

44 John Duns Scotus, Sent ., lib. 2, dist. 12, q. 1 [Vivès], p. 556: “Aliquam enim potestdupliciter: Uno modo ut terminus; alio modo ut subjectum, quod est in potentia adterminum . . . ita quod subjectum existens dicitur in potentia subjectiva; et eadem utrespicit agens, dicitur objectiva; possunt tamen separari, ut in creabili, ubi est potentiaobjectiva et not subjectiva, quia ibi non subjicitur aliquid.”

45 Ibid., pp. 556–558. The Scotistic notions of subjective potentiality and objectivepotentiality are clarified by a late Renaissance Scotist, Fillipo Fabri, who explains thataccording to Scotus: “Something can be in potentiality in two ways, one way as the endof the action of an agent, so that is said to be in potentiality that is in itself nothing,but is in the power ( virtute ) and the potentiality of an agent, as the world before it was

created was in potentiality and in the power ( virtute ) of the agent God; but in itself itwas nothing, and this potentiality he names objective potentiality; another way thatsomething is in potentiality is in respect to a capacity ( capax ) of a subject and the recep-tion of whatever forms, and this is subjective potentiality.” Cf. Fillipo Fabri, Philosophianaturalis Joan Duns Scoti ex quatuor libris Sententiarum et Quodlibetis collecta [Venice, 1616],p. 126: “. . . aliquid potest esse in potentiae dupliciter, uno modo, ut terminus actionisagens, et sic dicitur esse in potentia illud quod ex se nihil est, sed est in virtute, etpotentia agentis, ut mundus ante quam crearetur erat in potentia et virtute agentis Dei;sed in se ipso nihil erat, et hanc potentiam appellat potentiam objectivam, alio modoaliquid est in potentia quatenus est subjectum capax, et receptivum alicuius formae,et haec est potentia subjectiva.”

46 Ibid ., p. 558: “Dico igitur, quod materia est per se unum principium naturae,

ut dic Philosophus primo Physicorum . . . Quod est per se subjectum mutationum sub-stantialium, quinto Physicorum . . . Quod est terminus creationis; igitur sequitur quodest aliquid, non in potentia objectiva tantum, . . . sed oportet tunc quod sit in potentiasubjectiva existens in actu . . .”



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more, must be capable of existing independently, apart from form.47

He further justifies his contention that prime matter has an “actus” (an

act or actuality) of its own. Aristotelians distinguished two fundamen-tal kinds of acts that a substance has, first the act of simply being (or

existence) and second those acts that follow from its nature, e.g., fire

moving upwards or earth falling downwards. Substantial form is the

principle of these two acts in each substance. Prime matter, though,

has just the former act, an entitative act (an act of being), that gives it

actuality (actual existence).48

Scotus objects that in Aquinas’s view, prime matter is accorded only

“objective potentiality” ( potentia objectiva ). That is, prime matter has

reality only in regard to some ef ficient cause and is merely the poten-tiality to become some x (a future object that does not yet exist), as

the universe has objective potentiality in regard to God as its ef ficient

cause, before it is created by Him. Aquinas’s prime matter, therefore,

has no reality in itself (it is, in effect, nothing) and, therefore, cannot

serve as the continuous and stable subject of substantial forms required

for substantial change.49 The subtle doctor’s arguments had a dramatic

effect upon fourteenth century matter theory. The intuition that prime

matter must have a robust reality to serve as a material cause in sub-

stantial change inspired a variety of subsequent theories.

A further problem was that of making sense of the extension natural

to a body composed of prime matter and substantial form, if neither

has extension, an issue raised by Averroes. His claim, in his De substantia

orbis,50 that prime matter has indeterminate dimensions, in the light

of Scotus’s opposition to all attribution of quantity to prime matter,

focused attention upon this issue. The intuition that the extension of

bodies must have its foundation in the nature of the matter of a sub-

stance inspired a variety of fourteenth-century interpretations of thisAverroist view.

47 John Duns Scotus, Sent ., lib.2, dist. 12, q. 2, p. 576: “Sed forma est causa secunda,quae non est de essentia materiae . . .; ergo Deus sine illa potest facere materiam.”

48 John Duns Scotus, Ibid ., p. 589: “. . . et sicut forma est actus formalis, quia potestinformare per receptionem ipsius, ita etiam materia est actus entitativus et positivus

vere receptivus ipsius formae, et nihil amplius habens.”49 John Duns Scotus, Sent ., lib. 2, dist. 12, q. 1, pp. 559–561. See also discussion

of Scotus’s notions of subjective potentiality and objective potentiality by Cross, The

Physics of Duns Scotus: The Scienti fi c Context of a Theological Vision, pp. 17–20; and McCordAdams, William Ockham, pp. 642–643.50 Averroes, Sermo de substantia orbis [Venice, 1562], p. 4: “. . . hoc subjectum recipit

primus dimensiones interminates.”




We find here stimuli that inspired two hotly debated fourteenth

century issues in matter theory:

MQ1: Does prime matter have an actuality [distinct reality] of its own

(as Scotus claims)?

MQ2: How does the extension of bodies have its foundation in prime

matter?

Ockham provided a distinctive response to these queries. Like Scotus,

Ockham supports the dictum: prime matter has an actuality of its

own. Ockham says that “matter is a certain thing actually existing in

the nature of things, which is in potentiality towards all substantialforms;”51 but this potentiality is unlike “the way in which a future white

is just in potentiality.”52 This potentiality is not simply the absence or

a lack of some objective state, i.e., the non-existence [or privation]

of a future white which can be said to be in potentiality (this lack or

absence is like Scotus’s objective potentiality). Instead, prime matter is

a thing that is in potentiality towards a form that it lacks (as Socrates

now tan is potentially white).

Further, Ockham maintains that “potentiality is not some existing

thing in matter, but it is matter itself,”53 because this potentiality does

not cease when matter receives a form. Matter is potentiality in respect

to substantial forms generally. It may be in potentiality to one particular

form or other, but even when this form is received, the potentiality in

respect to substantial forms generally is not corrupted. The prime matter

of air is still able to be [posse] air, but it is also potentially fire or water.

Hence, potentiality is not merely a property of matter. Instead:

O1: Prime matter is potentiality.

So Ockham says that potentiality, and so prime matter, remains

along with form. Further, prime matter can neither be generated nor

51 William of Ockham, Summula philosophiae naturalis [Brown], p. 179: “Materiaest quaedam res actualiter exsistens in rerum natura, quae est in potentia ad omnes

formas substantiales.”52 Ibid ., p. 179: “. . . modum quo albedo futura est tantum in potentia”.53 Ibid., p. 185: “. . . potentia non est aliqua res existens in materia, sed est ipsa

materia.”



198 emily michael

destroyed, so it has no potentiality to not exist. It therefore must be

something that is “truly actual from itself.”54 He also argues:

That which is not can be the part or principle of no being; but matteractually is a part and a principle of a composite being. Therefore, it itselfis actually a being in actu [act or actuality].55

So prime matter actually exists; it “is truly a being [ens] in actu.”56

Ockham concludes:

O2: Prime matter is some thing [res] actually existing and distinguish-

able from form.

Further, Ockham makes his prime matter a yet more robust “being in

actuality.” Claiming that each entity has its individual prime matter

(I have my prime matter and you have yours), he also supports the

following view:57

[I]t is impossible that matter should lack extension, for it is not possiblethat matter should exist without having one part at a distance from anotherpart. (. . .). [T]he parts of matter can never be in the same place. There-fore, matter always has one part at a distance from another part.

The Venerable Inceptor here argues that for matter to be divisible

into parts, extension (length, breadth, and depth) is essential to matter.

Distinctive of Ockham’s account of matter is his claim:

O3: Quantity or extension is not anything apart from prime matter.

Prime matter essentially has quantity or extension, which is here ana-

lyzed as parts outside of parts. Matter is not different from its parts, nor

is it therefore different from extension or quantity or dimensions.58

54 Ibid., p. 180: “. . . vere actu ex se ipsa”.55 Ibid., p. 186: “. . . illud quod non est, nullius entis potest esse pars vel principium;

sed materia actualiter est pars et principium entis compositi, igitur ipsa est actualiterentitas in actu.”

56 Ibid.: “. . . sit vere ens in actu.”57 Ibid ., p. 191: “. . . impossibile est quod sit materia sine extensione: non enim est

possibile quod materia sit nisi habeat partem distantem a parte . . . numquam partesmateriae possunt esse in eodem loco. Et ideo semper materia habet partem distantema parte.”

58 Ibid., pp. 191–192.




Wyclif finds unsatisfactory Ockham’s response to these two questions.

In regard to MQ1: Is prime matter an actual entity?, Wyclif agrees

with Ockham’s contention (O2) that prime matter has a positive realityof its own. But how can a potentiality have actuality? If, as Ockham

claims, prime matter is potentiality simpliciter (O1), how can prime mat-

ter have a positive reality? Still, Wyclif says (Tr, p. 87), citing Aristotle,

that prime matter has no quiddity, no quality, no character at all.59 He

takes this claim very literally. This raises the problem: How can that

which has no character whatsoever have a positive reality?

In regard to MQ2: in which way does extension (or quantity) have

its source in prime matter? Wyclif claims, as noted above, that prime

matter has no quantity or quality. Prime matter, therefore, cannot essen-tially be identified with extension, as Ockham claims (O3). But Wyclif,

nonetheless agrees that prime matter is the source of extension. This

raises the perplexing question: How can prime matter, independently of

any quantity or characteristics in itself, be the source of extension?

Interestingly, Wyclif ’s distinctive atomism contributes to his response

to these quandaries. His atomism, therefore, is not only consistent with

his hylomorphism (scholastic pluralism), i.e., with his scholastic matter/

form theory. His atomism enables him to resolve problems in matter

theory that, in his view, confounded his Aristotelian predecessors and

contemporaries.

We turn first to MQ1 (Is prime matter is an actual entity?). In his

Logicae continuatio, Wyclif provides his prime matter with a palpable

reality. He contends that prime matter is:

composed from indivisibilia [indivisible points], and occupying every pos-sible place, not corruptible according to any of its parts, except acciden-tally by the division or separation of one of its parts from the rest.60

Likewise, he explains, in his Trialogus that time is a continuum composed

of indivisibilia, i.e., instants, and so too prime matter (“occupying every

possible place” in the world) is a continuum composed of indestructible

indivisibilia (indivisible points). [Tr, p. 88] Although each indivisible

point has no quantity, no quality, no character at all, indivisibilia make

his prime matter entitative. Hence, prime matter has actuality.

59 Wyclif, Trialogus [Lechler], p. 87: “Aristoteles autem dicit septimo Metaphisicae,

quod nec est quid nec quale nec aliquid aliorum entium . . .”60 LC3.9, p. 119: “. . . esse ex indivisibilibus composita, et occupare omnem locum

possibilem nec esse secundum eius partem aliquam corruptibilem, nisi forte per divi-sionem vel separacionem unius partis a reliqua.”



200 emily michael

In response to MQ2 (In which way does matter have quantity?),

Wyclif ’s contends that indivisibilia, constituting prime matter, also

provide it with extension. He claims: “the world is composed fromcertain atoms, and cannot be increased or diminished, or arranged in

a straight line or another figure, for the number of atoms [ multitudinem

athomorum ] is a cause from which such a continuous quantity and such

a figure naturally immutably follows.”61

In his view, the totality of indivisible points or atoms defines the

matter of the world. Further, each indivisible point has its position

relative to fixed points at the centre and poles of the world, and no two

points can occupy the same position. As a result, indivisibilia, though

unextended, are, in effect, impenetrable, so they cannot overlap orinterpenetrate each other, nor can there be an actual place where there

is no point (because every line is actually composed of points). As a

result, indivisibilia must be located contiguously, one next to the other.

This entails, in Wyclif ’s view, that the totality of contiguous corporeal

indivisibilia define the total size and shape of the world, and, thereby,

its extension. Although Wyclif rejects Ockham’s view that extension

belongs to prime matter, extension is produced by the contiguous indi-

visibilia that constitute prime matter.62

Wyclif, therefore, responds to these fourteenth-century concerns

by employing indivisible points (identified as primitive atoms) both to

provide matter with a robust reality (actuality) and to account for how

matter provides extension to corporeal things. We might, however,

note that though Wyclif ’s atomistic account is effective in resolving

some problems in matter theory, this utility of his indivisibilia does not

entail the existence of atoms. One might argue that this provides some

rationale for accepting indivisibilia. But it does not explain the grounds

and motivation for Wyclif ’s atomism. We are still left with the questionof why Wyclif adopted an atomistic account, especially in the light of

the almost universal rejection of atomism and arguments against it

provided by his contemporaries.

61 LC3.9, p. 1: “. . . mundum componi ex certis athomis, et nec posse majorari necminorari nec moveri recte localiter vel aliter figurari, ita quod tantam multitudinem

athomorum consequitur tanta quantitas continua et talis figura, propter causas immu-tabiles naturales.”62 It should be noted that, although Wyclif adopts an atomistic account, he clearly

departs from the Ancient atomists in his rejection of a void.




Second, we should also note that Wyclif goes beyond indivisibilia

theory. He develops a full blown atomistic account that presumes

elemental corpuscles and, in turn, compound particles. This also raisesthe question of the reason for his commitment to an atomistic account.

Following is a further exploration of this question. What I wish to show

is that his atomism was motivated and inspired by his distinctive meth-

odology, which he identifies as the “logic of scripture” ( logica scripture ).

Although this too will not provide us with a proof that atoms exist, it

will provide further insight into the rationale for Wyclif ’s atomism.

2.2. Why did Wyclif Develop a Corpuscular Matter Theory?

Wyclif’s Methodology

Wyclif adopted a distinctive methodology for the acquisition of truth.

In his De Veritate Sacrae Scripturae, he contends that all truth is in Scrip-

ture.63 As such, we turn to Scripture as the source of all truths. This

includes truths of natural philosophy. Further, Wyclif contends that,

though Scripture provides only truth, the language of Scripture is

frequently figurative. This requires appropriate interpretation. We are

aided in this endeavour, in particular, by what Wyclif calls the logic of

scripture,64

which he explores and expounds in his lengthy Tractatus delogica. From this viewpoint, logic, properly employed, follows and clari-

fies Scripture. In turn, the logic of Scripture enables us to understand

how to retain consistency of interpretation throughout Scripture, to

analyze conceptions, including those involving equivocation, to identify

the various well-formed structures of terms and correctly interpret

these, to draw valid consequences from inter-connected propositions, to

formulate sound arguments in support of interpretations of Scripture,

and, thereby, to make correct judgements in interpreting Scripture and

in supporting interpretations of Scripture.Intensifying Wyclif ’s confidence in his view is his Augustinian com-

mitment to the foundation of truth in faith, which he claims is cor-

rectly given only by Scripture. Further, reason supports what we here

are taught by faith. Reason plays a significant role for Wyclif. He takes

seriously the Augustinian claim that reason enables us to understand

63

De Veritate Sacrae Scripturae [Buddensieg], hereafter cited as DVSS. For an excellentabridged translation, see John Wyclif, On the Truth of the Holy Scripture [trans. Levy].64 Wyclif explains ( Tractatus De Logica, Proemium): “Motus sum per quosdam legis

dei amicos certum tractatum ad declarandam logicam sacre scripture compilare.”



202 emily michael

what we believe. Through the logic of scripture, reason provides clarifi-

cation and confirmation of what we believe by faith. Therefore Wyclif

presumes the following methodological principle:The authoritative source of all truth is scripture. The correct inter-

pretation of scripture is guided by the logic of scripture.65

His logic of scripture would seem to have its foundation in a method

for the acquisition of all knowledge, natural and spiritual, much like

the following:

1. Guided by logic, i.e., an understanding of literal and figurative lan-

guage, the logical role of terms and of propositions, and the like:

formulate an interpretive thesis of a scriptural truth.2. Use logic (rules of the relations of propositions, rules of infer-

ence, etc.): to test for internal consistency with other propositions

of scripture; to test for support from interpretations provided by

reliable commentators, e.g., Wyclif mentions Augustine, Jerome,

Ambrose, Lincoln,; to test for consistency with evidence of sense

and reason.

3. Use logic to: draw conclusions from the thesis presumed.

4. Use logic to: test conclusions for consistency with other scriptural

truths; with evidence of sense and reason.

5. If inconsistencies (contradictions) result, reject the interpretive thesis.

If it is consistent and well supported, retain the thesis.

In Wyclif ’s view, his logic of Scripture provides the principles of cor-

rect reasoning that aid us in interpreting scripture accurately and in

understanding what we believe by faith. Scripture, thereby, provides

the “matter of knowledge” ( materia de scire )66 and is the focus of his

discussion of logical proofs.Our Oxford master, therefore, explains in his discussion of prime

matter that “ignorance of true physics makes moderns give too little

weight to Scripture and to impose upon the sacred doctors, ignorance

of logic or philosophy . . .”.67 Instead, “Scripture will be the model . . .,

65 Wyclif, DVSS , esp. part 1, ch. 3.66 Wyclif, Tractatus de logica, Proemium.

67 Wyclif, De M&F , p. 218: “Ignorancia huiusmodi veritatum physicarum facitmodernos parum pendere scripturam et imponere sanctis doctoribus ignoranciamloyce vel philosophie . . .”




not only in respect to right living, but also in speaking the truth and

philosophical wisdom. Now however . . . we make our logic the rule cor-

recting scripture, . . . when, however, it ought to be on the contrary.”68

From this viewpoint, there are no foolproof authorities, but some early

doctors of the Church, especially Augustine, are a more reliable guide

to truth because they are so well attuned to the meaning of scripture.

So, in response to some objections, Wyclif says: “For a solution to these,

one ought to assume the grammar and logic of the holy doctors, which

they elicit from Scripture.”69 He also tells us that other or secular (in

Wyclif ’s words, ‘foreign’) logics are numerous (as many as there are

logic masters) and shortlived (“a foreign logic endures for barely twenty

years”).70 “The logic of Scripture, however, stands eternally, becauseit has been established by the indestructible truth . . .”.71 Nonetheless,

Wyclif contends that “Aristotle’s logic is correct for the most part, and

consistent with the logic of Scripture.”72 Still, Aristotle, though a great

philosopher, has committed errors. [DVSS, p. 29] Aristotle correctly

understood the logical investigation of language, but logic cannot be

secure independently of a foundation of truth that is its subject, and

this foundation of truth is the words of Scripture.

Using this method, Wyclif explains the account of prime matter that

he learned from Scripture by interpreting the “true proposition, Genesis 1:

‘In the beginning God created heaven and earth’.”

“In the beginning God created heaven and earth.” That is: In the wordGod created a spiritual and a corporeal creature. From which, orderedalso with admirable subtlety common to all, he calls the same creaturethe said corporeal essence (under the idea that is matter) earth, water, andabyss, because common people cannot understand the corporeal natureunder the idea that is matter; so it is necessary to have these express thenames of sensible things, that are most unformed. Nor is it falsely named,

as is said after; and towards showing that it is unformed, he calls it void, vacuum, and darkness. These said privations however are nothing except

68 Ibid ., p. 219: “. . . scriptura fuit exemplar . . ., nedum ad recte vivendum, sed ad vereloquendum et philosophice sapiendum. Nunc autem . . . constituimus loycam nostramtanquam regulam rectificantem scripturam, . . . cum tamen debet esse e contra.”

69 Ibid ., p. 235: “Pro solucione istorum oportet supponere grammaticam et loycamsanctorum doctorum, quam eliciunt ex scriptura.”

70 DVSS , p. 54: “. . . vix durat una aliena logica per viginti annos . . .”71 Ibid.: “Logica autem scripture in eternum stat, cum fundatur independenter a

fama vel favore hominum infringibili veritate.”72 DVSS , p. 47: “. . . logica Aristotelia, que ut plurimum est recta, sit logica scrip-

ture . . .”



204 emily michael

this unformed matter. After this (nature not time) is, in fact, the orderof the first day, light, in substantial form in the first instant of time andmatter naturally, not temporally, before that same instant.73

Wyclif formulates an interpretive thesis of the scriptural truth provided

in Genesis 1, namely: “In the word,” i.e., in His act of creation, God

created two kinds of creature, corporeal and spiritual. And, alterna-

tively, he says: In the beginning, “the first essence (i.e., God) before all

substance created two essences, . . . namely, a corporeal essence and an

incorporeal essence.”74 Further, looking at Genesis 1 beyond this pas-

sage (“earth, water, abyss,” “void, vacuum, darkness”), and considering

Augustine’s exegesis,75 Wyclif analyses the latter ( essencia incorporea ) as

angels (immaterial substantial forms that exist per se ) and the former( essencia corporea ) as unformed prime matter (matter without any inhering

material substantial form). He further explains this passage.

Wyclif describes the “incorporeal essence” as “pure per se stans,”76

“a substance that is indivisible in respect to mass and intelligible in

respect to operation, which are creatures most proximate to God, which

philosophers in respect to their innate action call intelligences, and in

respect to their duty to us they call these angels.”77 The incorporeal

essence consists in separately existing individual created Minds, intel-

ligibilia, created immediately by God, with powers of intellect, will, and

memory. Wyclif adopted the common scholastic view that each angelic

substance is indivisible, because what is immaterial is not extended, and

only extended entities are divisible.

73 De M&F , pp. 209–210: “In principio creavit deus celum et terram; id est: In verbocreavit deus spirtualem creaturam et corpoream. Unde ordinate et mirabili subtilitate

communiter eandem creaturam vocat dictam essenciam corpoream (sub racione qua estmateria) terram, aquam et abissum, quia rudis populus non suf ficiebat comprehenderenaturam corpoream sub racione qua materia; ideo necesse habuit illam exprimerenominibus rerum sensibilium, que maxime accidunt ad informitatem. Nec false nomi-nat, ut post dicetur;. et ad testandum eius informitatem dicit eam inanem, vacuam ettenebrosam. Dicte autem privaciones non sunt nisi informitates materie huius. Post hoc(natura non tempore) facta est, prima die ordinis, lux in forma substantiali in primoinstanti temporis et materia naturaliter, non temporaliter, ante idem instans.”

74 DCH , p. 24: “. . . prima essencia ante omnem substanciam creavit duas essencias, . . .,scilicet essenciam corpoream et essenciam incorpoream.”

75 De M&F , p. 209: “Et ille est sensus Augustini, 12 de Confessione, ubi, priusquamdeclarat angelum et materiam priman esse creaturam dei . . .”

76

De M&F , p. 177.77 Ibid.: “. . . substancie quoad molem indivisibiles et intelligibiles quoad operacionem,que sunt creature deo proxime, quas philosophi ab innata accione vocant intelligencias,et nostri ab of ficio vocant eos angelos.”




Further, bodies are composed of prime matter and material substan-

tial forms. In Wyclif ’s view, unlike spiritual substantial forms, which exist

per se and can never inhere in prime matter, material substantial formsinhere in prime matter and cannot exist apart from prime matter.78

Prime matter, however, has a robust reality. First, prime matter is

the corporeal essence. That is, this essence (prime matter) provides the

material foundation that makes my cat Jake, my cherry tree, my gold

ring each a corporeal thing. Further, as noted above, Wyclif describes

prime matter as “composed from indivisibilia” [ LC 3.9, p. 119]. And,

likewise, in his Trialogus, he says: “componitur ex suis partibus quantitativis

usque ad sua indivisibilia . . .” [ Tr , 88].

As such, Wyclif ’s interpretation of Genesis 1 has the consequence that:In the beginning, God created two kinds of indivisibilia, spiritual and

corporeal. So, Wyclif explaining the grades of simplicity in the created

world, tells us, first, that the highest grade of simplicity is “the divine

nature,”79 which, existing in eternity (outside of the instants of time

and all other created indivisibilia) excludes all possibility of variation,

succession, or of composition from parts. “In the second grade,” i.e.,

the first grade of simplicity of the created world, “are created spirits and

other indivisibilia without quantitative parts, such as points, instants,

and the like.”80 The latter are the fundamental parts of material things,

i.e., of prime matter , from which, along with substantial form, are com-

posed elemental corpuscles, and these, in turn, combine to form compound

particles (minima naturalia), which are identified by Wyclif with subsequent

higher grades of simplicity respectively.81 These grades of matter will

be explored in greater depth in the following section.

78 Tr , p. 87: “Non autem intelligo formam substantialem materiae primae essealiquid, quod potest per se existere . . .”

79 De M&F , p. 199: “In summo igitur gradu simplicitatis est natura divina, exclu-dens possibilitatem ad quamcunque composicionem ex paratibus, vel variacionem inaccidentibus . . .”

80 Ibid.: “In secundo gradu sunt spiritus creati et alia indivisibilia quoad partesquantitativus, ut punctis, instans, et similia.”

81 Ibid.: “In tercio gradu sunt materia et forme substanciales vel accidentales, nonhabentes partes quantitativas disparium naturarum.” (matter and form consideredseparately). “In quarto gradu sunt quatuor elementa, quorum quelibet pars quan-titativa est eiusdem nature cum toto.” (elemental corpuscles, and collections of the

same kind of elemental corpuscle); and “in isto gradu sunt multa omogenia, quorumquelibet pars quantitativa per se sensibiliter est eiusdem nature cum suo toto, ut caro,os, nervus, et cetera” (compound particles, and collections of the same kind of com-pound particle).



206 emily michael

Wyclif ’s atomism further explains the account of creation in Genesis 1

and is consistent with his interpretation of Scripture more generally.

Further, his indivisibilia help him to resolve a variety of objections thatfeatures of his interpretation can raise.82 Consideration of this role of

Wyclif ’s indivisibilia is a complex project and beyond the scope of this

chapter. We might just note that the consistency with a variety of pas-

sages in scripture and the explanatory role of indivisibilia in relation to

scripture is important to Wyclif. In addition, he cites Scripture in direct

support of indivisibilia.

In his Trialogus, Wyclif raises the question of whether a body or

whatever continuum is composed from parts that are indivisible and

not extended, or divisible and extended.83 If the second, he says, thenall parts are always further divisible into other parts, so the number of

parts assigned to A in the beginning is not the total number of parts

in A.84 He responds, citing Gen 1:31 (“God sees all that He made.”)85

in defense of indivisibilia: “I suppose with Augustine and the faith of

Scripture, that just as God sees all that he made, so he distinctly under-

stands all parts of whatever continuum, so that no further or different

components of a continuum can be given. And on that basis the rea-

soning seems plainly to proceed.”86 Wyclif here supports the view that

82 See, for example, his discussion ( De M&F , pp. 187–88) of how an explanationconsistent with nature can be provided for such seeming miracles as the change of Lot’swife to a pillar of salt by understanding this as a rapid relocation, possible for God,of salt particles from other parts of her body to her surface. Likewise, explaining thatthe change of water to wine is not the creation of a new substance ( De M&F , p. 188):“Facillimum namque est auctori nature capere minucias elementorum vel inordinatesparsas vel noviter generatas, et armonice componere illas, ut forma serpentis vel vini,

vel quecunque alia de potencia materie educibilis, statim resultat; cum nichil ibi creatur,sed vel generatur pure naturaliter, vel prius generatur aliter situaliter.”83 Wyclif, Tr , p. 83. “Capiatur A corpus vel quodcunque continuum, et noto omnes

ejus partes, ex quibus componitur; et quaero ulterius, utrum sunt indivisibiles et nonquantae, vel divisibiles atque magnae.”

84 Ibid.: “Si secundo modo, tunc omnes illae partes et quaelibet earum est divisibilisin partes ulteriores, ergo numerus illarum assignatus in principio non est numerustotalis omnium partium A signati.”

85 See also Kenny, Wyclif , p. 62: “But the work frequently repeats that scripture con-tains all truth, and in one passage Wyclif even offers to prove his own atomic theoryout of it, from the beginning of Genesis and from Matthew 10:30.” This citation fromMatthew is as follows: “But the very hairs of your head are all numbered.”

86

Wyclif, Tr , p. 83: “In ista autem responsione suppono cum Augustino et fide scrip-turae, quod sicut Deus vidit cuncta quae fecerat, sic distinctissime intelligit omnespartes cujuscunque continui, sic quod non est dare ulteriores vel alias ipsum continuumcomponentes; et sic videtur ratio plane procedere.”




God’s distinct understanding of “all parts of whatever continuum”87

entails a fixed total number of primitive parts that are not further

divisible (“no further or different components of the continuum canbe given”).88 This interpretation of Scripture led him to the twofold

consequence of supporting finitism and atomism. Wyclif contends

that only God, existing in eternity (outside of the instants of time and

points of space), is infinite. He says: “It is impossible that any substance

or points or any other thing besides God is simply infinite.”89 Wyclif,

therefore, asserts the following principle of finitude: whatever actually

exists is finite.90 Consistent with this principle, he concludes that every

continuum, including the created universe as a whole, must have mini-

mal parts that cannot be further divided. This entails that prime mattercannot be infinitely divisible. Instead prime matter is, in Wyclif ’s words,

composed from indivisible points or atoms [Tr, p. 88].

He further supports this claim by citing Ecclesiastes 1891 and 192 which

he interprets to mean first, that God created a complete world, every

instant of time and whatever exists in time, however simple or complex,

each at its proper time and place.93 What is more, this entails that God

eternally has a distinct Idea of every part of His Creation.94 S.H. Thom-

son points out, as a foundation for Wyclif ’s atomism, Wyclif ’s identifica-

tion, in De Materia et Forma95 of the possible with God’s knowledge of his

creation (that is, whatever has real possibility is in God’s mind because,

87 See preceding footnote.88 See preceding footnote.89 See, for example, LC 3.9, p. 36: “Unde impossibile est quod aliquis numerus

substanciarum vel punctorum, vel aliud preter deum sit simpliciter infi

nitum.”90 For Wyclif ’s support for finitism, see also LC 3.9, pp. 37–38. Wyclif contends ( LC 3.9, 37): “Omnem ergo numerum qui excedit ingenium nostrum ad aptandum sibiterminum specificum naturalem vocamus infinitum.”

91 De M&F , p. 222: “Qui vivit in eternum creavit omnia simul.”92 Ibid.: “. . . nihil novi sub sole.”93 See, also Wyclif ’s claim that, at the beginning of time, God created all things

simultaneously, Tr . p. 86: “Deus creavit omnia simul;” “. . . qui vivit in aeternum, creavitomnia simul,” and also LC 3.9, p. 56: “. . . Deus ordinat istos propter melius ordinisuniversi.”

94 See, for example, LC 3.9, p. 36: “Ymmo deus satis noscit quomodo omne quadra-tum per se sensible integratur ex partibus minimis et principiis eorum indivisibilibus

cumulatis. Et sic dicitur de qualibet alia figura principiata ab indivisibilibus priminumeri, ut figurati. Novit eciam in qua proporcione quicunque numerus punctorumse habet ad alium . . .”

95 De M&F , pp. 234–235.



208 emily michael

as Wyclif says: “God sees all that he has made.”).96 Thomson, explain-

ing the rationale for Wyclif ’s atomism, says: “Increase or decrease in

the magnitude of the universe is likewise unthinkable, for whatever hasbeing already exists in God’s thought . . . That is to say, that whatever

is known to God has reality, and the smallest possible unit of space or

time, the point or the instant becomes quite real.”97

Kretzmann similarly says: “Why, then, was Wyclif an indivisibi-

list? . . . All that we have seen that might have prompted such a belief

on his part is certain theological considerations, especially concerning

the requirements of omniscience, and I think there can be no doubt

that they are indeed the bedrock of his indivisibilism.”98 He explains

this claim:

God alone knows the detailed composition of things out of indivisibles,but in Wyclif ’s view that sort of knowledge necessarily includes knowingthe precise number of the indivisible constituents of the world and ofeach thing in it. If there are literally infinitely many points in a line, then,it is logically impossible that anyone, even omniscient God himself, canknow the number of its points. And the principle reason why it will notdo to say simply that God knows that there are infinitely many of themis, I think, that since they are real and natural, God made them; and

omniscient God must know each of his creatures individually. As Wyclifputs it more than once, quoting or paraphrasing Genesis 1:31, ‘God seesall the things that he has made.’99

Like Democritus and Epicurus, Wyclif claims that his primitive atoms

(i.e. points) are indivisible, but, quite unlike Democritus and Epicurus,

interestingly enough, Wyclif ’s fundamental motivation for this view

is theological. It is his interpretation of Scripture, in particular, that

inspired Wyclif ’s atomism. As required by his methodology, this inter-

pretation of Scripture is further supported by reason and experience.

These supporting arguments and the atomistic view that he developedwill be considered in the final section.

96 Wyclif says, at this point in De M&F ( Ibid. ): “Et talis veritas est racio vel exemplarquod deus necessario videt, et videndo illud quod est essencialiter divina essencia, videtomnes creaturas, si sint in tempore suo.”

97 Thomson, “The Philosophical Basis of Wyclif ’s Theology,” p. 112.98 Kretzmann, “Continua, Indivisibles, and Change in Wyclif ’s Logic of Scripture,”

p. 50.99

Kretzmann, Ibid., p. 45. See Kretzmann ( Ibid ., pp. 51–63) for Wyclif ’s interestingdiscussion of the terms “incipit” and “desinit,” and, in this context, Wyclif ’s indivisibilistanalysis of motion. Kretzmann discusses, in particular, LC 2.14 (a chapter in Wyclif ’s Logicae Continuatio that examines the terms “incipit” and “desinit”), pp. 191–195.




3. Wyclif’s Atomism

While their hylomorphic ontology led Wyclif ’s predecessors to rejectatomism, he used his hierarchical and pluralistic analysis of composite

substances to justify his atomism. From Wyclif ’s viewpoint:

1. Prime matter is constituted by a finite number of indivisible unex-

tended atoms ( indivisibilia ).

2. Indivisibilia, united by an appropriate elemental substantial form, are

extended elemental atoms of earth, air, fire, and water.

3. Elemental atoms, in different proportional arrangements, compose

minima naturalia (compound particles), which are the fundamen-tal particles of compound bodies. Each compound particle has a

supervening substantial form.

4. Homœomerous compound bodies, e.g. blood, bones, flesh, nerves

(each with its distinctive kind of compound particle), compose animal

bodies, which are animated by a supervening organic soul.

5. An appropriate animal body united with a spiritual human soul, a

human mind, composes a human person [a rational animal].

6. The total set of indivisible atoms, which are the minimal units

composing all individual material substances, define the shape and

size of the world.

Wyclif cites Democritus and Plato as authorities in support of his

atomistic view. [ Tr , pp. 83–84; LC 3.9, P. 132].100 He, like Democritus,

contends that the fundamental building blocks of natural things are

atoms. But, unlike Democritus, Wyclif claims that there are several

grades of atoms, all of which were formed by God at Creation. The

simplest atoms, like the atoms of Democritus, are indivisible, impen-etrable, immutable, indestructible. But unlike the atoms of Democritus,

Wyclif ’s indivisible atoms are unextended. Further, in Wyclif ’s view,

extended elemental corpuscles (which, though relatively stable, are divis-

ible) are composed of a structure of these unextended indivisible atoms

and an elemental substantial form. Next, compound particles (called

minima naturalia ), in turn, are composed of a structure of elemental

corpuscles and a supervening substantial form. Compound particles

100 See also LC 3.9, 61, where Wyclif speaks of the ancient view of fi ve simple corpo-real figures associated with the four simple elemental bodies and a neutral element.



210 emily michael

themselves are substances, that, like living things, have a hierarchical

and pluralistic structure. Wyclif ’s grades of atoms (i.e., his indivisible

points, elemental corpuscles, and compound particles) are the subjectof what follows.

3.1. Wyclif ’s Indivisible Atoms

Like in other works of his time, in Wyclif ’s Logicae Continuatio the term

‘indivisibilia’ refers to points of a continuum, in particular, of space, time,

or motion. Indivisibilia are unextended entities, and are therefore not

physical atoms. But, Wyclif also argues, unlike his contemporaries that

bodies are in fact composed of physical atoms, which are themselvescomposed of indivisible and unextended points (i.e., of indivisible and

unextended atoms).101 This requires explanation.

The common view at Wyclif ’s time was the Aristotelian view that

a continuum is not composed of actual indivisibilia, but rather is infi-

nitely divisible. Wyclif sided, instead, with those who maintained that a

continuum is composed of actual points.102 His chief adversaries were

the nominalists, the most prominent of whom was medieval giant, Wil-

liam of Ockham, who maintained that the term “indivisibilia,” refers to

nothing actual; these “beings of reason” have a negative connotation,namely, a lack of extension, like, for example, the limit of a line.103

The most common view of the opposition was that points are merely

potential, not actual.

Wyclif ’s divisibilist opponents maintained:

TD1: There are no actual indivisibilia.

TD2: Indivisible and unextended points cannot be arranged contigu-

ously to produce a continuous line.

Wyclif maintains instead:

TI1: A finite and fixed number of actual indivisible points determine

the extension and figure of the universe.

101 See, for example, LC 3.9, 1.102 For an excellent discussion of the status of these views at Wyclif ’s time, see

Kretzmann, “Continua, Indivisibles, and Change in Wyclif ’s Logic of Scripture.”103 William of Ockham, De sacramento altaris [Birch]. For an excellent discussion ofOckham’s view, see McCord Adams, William Ockham, pp. 201–212.




TI2: Space, time, and motion are all composed of parts or points.

TI3: Indivisible points are the prime matter of all bodies in the world.

The evangelical doctor argues that, as Aristotle claims, points are

naturally prior to a line, and so are a necessary condition for a line to

be. Therefore, points cause a line. But they are not an extrinsic cause.

They are therefore an intrinsic cause and are consequently part of a

line. [ LC 3.9, p. 30] He contends that Aristotle maintains that points

compose a line, lines a surface, and surfaces a volume. Further, parts

always precede the whole of thing. So points must be actual, as the

component parts that compose a line. In addition, if all the points are

removed from a line, surely nothing remains; so a line must be com-posed of actual points.

He also argues, citing Aristotle, that an instant is the principle of

time and unity of number and each is required for these to be. Like-

wise, a point is the principle of a line and therefore actual points are

required for a line to be. Otherwise the points contained in a line could

be successively removed without reducing the length of the line. So all

points could be removed and it could remain the same line. But that

which can be removed without changing a subject is an accident of

that subject. So that which is the principle of a line will be an accident

of that line, which is absurd [ LC 3.9, p. 30].

Wyclif cites Lincoln, i.e. Grosseteste, in support of actual points in

a line,104 but also disagrees with this predecessor, who claims a line

is composed of an infinite number of actual points. The evangeli-

cal doctor, who rejects the possibility of any actual infinite in nature,

agrees instead, for example, with Chatton105 and Crathorn,106 that a

continuum must be composed of a finite number of points, which are

immediately contiguous to one another. He argues: that which is infi

-nite cannot be known. But we know continua of space, motion, and

time, so the component smallest parts of these, though they cannot be

apprehended by our senses are indirectly known by us, and, further,

they must actually be known by God.

104 LC 3.9, p. 35.105

For discussion of Walter Chatton, see V. Zubov, “Walter Catton, Gérard d’Odonet Nicolas Bonet.”106 For discussion of Crathorn’s view, see Robert, “William Crathorn’s Mereotopo-

logical Atomism” in this volume.



212 emily michael

This leads Wyclif to a distinctive account of space or place and

of the matter that composes what exists. Points, visible to God, are

the component parts of space and the matter of what exists. [ LC 3.9,p. 2ff.]107 Space is the locus of points, and the place of a particular point

is its site relative to certain fixed points, those at the centre and poles.

Wyclif identifies the contiguity of two points, A and B, with the absence

of anything between A and B. Two points compose a line, three a sur-

face, four a volume. In his view, a finite number of points defines the

extension of space,108 and, in accordance with God’s design at Creation,

fixed numbers of points, united by an appropriate elemental substantial

form, compose the four elemental atoms, those of earth, air, fire and

water, which, in turn, are the matter of all compound bodies.Clearly Wyclif ’s indivisible atoms are not like those of Democritus,

whose atoms have size, shape, and motion. Wyclif ’s indivisibilia or points

have no extension, no qualities, no characteristics at all. Nonetheless,

their contiguity produces the size and shape of the world and of the

elementary corpuscles that compose that world. Further, Wyclif ’s world

is a plenum, so he, also, rejects Democritean motion of atoms in a void.

Still, his primitive atoms can move relative to fixed points at the center

and the poles by one point successively replacing another.

3.2. Wyclif ’s Elemental Atoms and Minima Naturalia

In the second part of section one above, we found two conflicting

analyses of the ontological structure of material substances, that is, two

107 See Wyclif ’s interesting argument, LC 3.9, pp. 33–34, supporting his view thatthe world is composed of contiguous points. He argues: God has the power to make a

substance the size of a point (“nullus theologus negaret quin Deus de potentia abolutapotest facere substanciam punctualem”), and that if God can do this, he can likewise juxtapose points contiguously (“Nec dubium quin, si Deus potest unum punct[u]aleproducere, potest et quodlibet juxtaponere.”). It is clear that, from such points, Godcan make a single extended substance (“Et ultra patet quod Deus potest ex talibusnon quantis facere unum quantum)”. Suppose God creates at every point in theworld such a punctual substance, and annihilates all continuous substance, preservingpunctual substances. (“Creet Deus ad omnem situm punct[u]alem mundi unam sub-stanciam punct[u]alem, et annichilet post omnem substanciam continuam, servandopunct[u]ales substancias inmotas”). Wyclif proceeds to argue that this would changenothing. Further, this is possible. So, it cannot be concluded that this is not so in fact.(“Nec dubito quin, admisso hoc pro possibili, omnes philosophi mundi non haberent

infallibilem evidenciam ad concludendum quod non est sic de facto”).108 See LC 3.9, p. 42 for Wyclif ’s argument that because the number of pointscomposing the world entail its size and shape, so it cannot be larger or other than itis, this nonetheless does not detract from God’s infinite power.




renditions of how a substance is a composite of prime matter and

substantial form, namely, scholastic monism and scholastic pluralism.

Scholastics were also divided in their analyses of the physical structureof compound substances. Wyclif here examines the vexing scholastic

problem of how, in the generation of a new substance, the elements

remain in a mixed (i.e., compound) body.109 He distinguishes three

basic sorts of analyses, identified by Wyclif as those of the moderns,

of Averroes, and of Avicenna [ LC 3.9, pp. 75–79].

Wyclif cites Aristotle’s definition of a “mixtio” (compound): “mixtio

est miscibilium alteratorum unio” (A mixed substance is the alteration of

the mixables (the elements) so that they will be one) [ LC 3.9, p. 74].110

His scholastics contemporaries, following Aristotle, commonly agreedthat the four elements (earth, air, fire, and water) are simple bodies,

that all other material substances are compound bodies, and that the

four elements are the fundamental physical principles in the genera-

tion of each compound body. This was also Wyclif ’s view. Further, he

explains the process of generation. In the dissolution of compound

substances, there is a fine division of elements into small parts, and

these elemental particles interact, and are mixed together in a regular

quantitative proportion, which gives rise to a compound substantial

form (also a common scholastic view). Further, Wyclif agrees with

his peers that this in fact produces a single compound substance. But

scholastics disagreed about how, as Aristotle claims, the elements remain

potentially in a mixed body. The evangelical doctor presumes that the

traditional three analyses of the moderns, Averroes, and Avicenna are

the choices available to reason, and he argues that only one of these,

which supports an atomistic account, is a viable alternative.

Wyclif tells us that most of his contemporaries take the modern view

that generation of a compound is caused by the interaction of theelements, but, in fact, no elements actually remain when a superadded

compound form arises [ LC 3.9, p. 76]. For example, the followers of

Thomas Aquinas claim that each compound substance is a composite

of prime matter and just one substantial form, so, in a new mixed

109 For a detailed discussion of the scholastic problem of how the elements remainin the mixed, see Maier, An der Grenze von Scholastik und Naturwissenschaft , pp. 1–140.

110 Aristotle, On Generation and Corruption, 1, ch. 10, 327b–328b. See also Wyclif, Tr.,

pp. 88–89: “Mixta autem concipio sic componi, cum secundum Aristotelem mixtio sitmixtibilium alteratorum unio; elementa enim secundum partes indivisibiles vel saltemnobis insensibiles ad invicem commiscentur, et formae superadditae resultant, habitaproportione mixtionis suf ficienti ad illam formam superadditam principiandum.”




An alternative account, ascribed to Averroes116 is the view that each

element actually remains in a compound substance, but the substantial

form of each is reduced in intensity [ LC 3.9, p. 75]. In the words ofCambiolus of Bologna, a fourteenth-century Averroist: “These four

forms are refracted in a certain median form, which informs one mat-

ter.”117 Here, just as when blue and yellow interact, each is reduced

in intensity and a new form, that of green, is produced, so too when

the contrary elements interact, their forms are reduced in intensity to

produce a single new median compound form. The elements therefore

remain in a compound body, but they remain in an altered state.

Wyclif maintains that this view is also inconsistent, in this case with

the nature of substantial forms. Socrates can become more hot in thesummer, less hot in the winter, more tanned by the sun, less corpulent

in old age, but he cannot become more or less of a human being at

any stage of his life. Humanity, unlike hotness or brightness, does not

admit of degrees. In Wyclif ’s view, by parity of reason, no substantial

form can admit of more or less; fire cannot become more or less fiery

nor air more or less airy.118 So the Averroist account also will not do.

Wyclif concludes: “I believe (. . .) that the elements are really in a

compound, according to their own proper forms and places, as Avicenna

says [ primo causarum ca. 3 and many other places].”119 Some Aristote-

lians adopted the Avicennist view that the elements actually remain in

a compound. For example, both Richard of Middleton (before Wyclif )

and, after Wyclif, Paul of Venice (c. 1369–1429) maintain that the

116 Wyclif, LC 3.9, p. 75: “Patet ista posicio 3 de celo 67.” See Aristotle, De Caelo, Book 3, folio 67, Aristotelis Opera cum Averrois Commentariis [Venice, 1562–1574], vol. 5,p. 227: “Dicemus quod formae istorum elementorum substantiales sunt diminutae

formis substantialibus perfectis et quasi esse est medium inter formas, et accidentia, etimmo non fuit impose ut formae eorum substantiales ad miscerent, et proveniret excollectione earum alia forma, sicut, cum albedo et nigredo admiscentur, fiunt ex eismixti colores medii.”

117 Cambiolus of Bologna, “Utrum elementa maneant in mixto secundum propriasformas aut solum secundum esse virtuale,” Ms. Vat. Ott. 318, ff. 4v–8r, edited by Kuk-sewicz, Averroïsme Bolonais au XIV e Siècle, p. 151: “. . . iste forme quatuor sunt refracte inquandam formam mediam, que informat unam materiam.”

118 Wyclif, LC 3.9, p. 75 “. . . essencia que est forma substancialis, non suscipit magiset minus pocius de substanciis elementaribus quam de mixtis; ut sicut nichil est reliquomagis homo, sic nec aliquid est reliquo magis ignis.”

119 Wyclif, LC 3.9, p. 79: “Credo 3am sentenciam in hac parte; scilicet, quod elementa

sunt realiter in mixto secundum situs and formas proprias, ut dicit Avicenna, primocausarum, ca 3, et alibi multis locis.” So Wyclif says ( LC 3.9, p. 80) that the juxtaposi-tion of corpuscles (i.e., elemental atoms) constitutes a true compound (“juxtaposiciocorpusculorum, ceteris requisitis, constituunt vere mixtum”).



216 emily michael

elemental forms remain extended uniformly throughout a compound

substance. In Paul’s description, the elemental forms remain as inhering

but not informing [i.e., not determining] forms.120

But Wyclif also findsthis way of interpreting Avicenna’s view unacceptable. This homoge-

neous distribution of the elements throughout a substance will not do,

because substantial forms of the same grade cannot be in the same

part of matter at the same time. First, Wyclif contends, a substantial

form cannot inhere in a substance without informing it. Second, earth

and water cannot occupy the same place at the same time.

Instead, Wyclif recommends a solution that he well knew virtually all

of his Aristotelian contemporaries found unacceptable. He maintains

that the elements remain as minuscule atomic parts, too small to beperceived, so that all compound bodies are in fact heterogeneous. Like

seventeenth century atomists, he here distinguishes appearance and

reality. The problem that his contemporaries found with the view that

he accepted was that the interaction of elements would not produce a

compound substance. The claim here is that if air, earth, fire, and water

actually remain “according to their own proper forms and places”121

(as corpuscles or atoms), this would produce a mere mixture of many

substances, a collection or a heap. The result would be not a single

compound substance ( mixtio ) which must be homogeneous, but merely

an aggregation of corpuscles.

Wyclif explains this objection: “First, it is seen that it is not properly

mixed, but is a juxtaposition of corpuscles; that thus, one having the

eyes of Lynceus will see in which way whatever element would be in

a place separately; and in this way likewise humans and all genera of

bodies are mixed in the world, and there would be no superadded sub-

stantial form; since no compound would be truly something one.”122 He

120 Paul of Venice, Expositio super libros De genertione et corruptione [Venice, 1498], f. 63a,explains of the elements: “They do not remain as a composition of form with matter,but rather do they remain as forms inhering in matter . . .” (. . . non manent secundumcompositione forme cum materiae; manenteni bene inherentia forme ad materia); and,f. 63b–c: “The form of the mixed itself . . . informs and inheres. However other forms,namely of the elements, inhere and do not inform.” (Ipsa forma mixti . . . informat etinheret; alie autem forme ut elemententorum inherent et non informant.”)

121 Wyclif, LC 3.9, p. 79.122 Wyclif, LC 3.9, pp. 79–80: “Primo, videtur quod proprie non sit mixtio, sed iux-

taposicio corpusculorum; quod sic, habens occulos linceos videret quomodo quodlibetelementum foret seorsum positum; et sic per idem homines et omnia genera corporeumessent commixta in mundo, et nulla foret forma substancialis superaddita; cum nullummixtum foret vere aliquod unum.”




rejects this conclusion: “To that is denied the first consequent (i.e., no

superadded substantial form), because the juxtaposition of corpuscles,

with other requisites, constitutes a true compound.”123

This is becausewhen cats or cows or human beings are mixed together with others in

the world, this is “not towards an end that results in a substantial form

constituting a substantial compound of a different species.”124 When

elemental corpuscles are mixed together, it is towards such an end.

Hence:

1. Fido, Mungojerry, and Dumbo cannot combine to compose a new

substance that has its own specific substantial form.

2. But all agree that the elements (earth, air, fire, and water) cancombine in an appropriate proportional relation to produce a new

compound substance with its own substantial form.

3. Therefore, elemental corpuscles can do so.

Dogs and cats and elephants are merely collections or aggregates of

entities. The elemental corpuscles are not. Unlike a mere aggregate of

entities, elements, in an appropriate proportional relation, can compose

a distinct compound substance with a higher grade of substantial form.

So, Wyclif contends, elemental corpuscles can act “towards an end

that results in a (distinct superadded) substantial form constituting a

substantial compound of a different species.”125

Wyclif, therefore, claims that in substantial change, the elemental

corpuscles interact and, towards the end of generating a new sub-

stance, they are mixed together in a regular quantitative proportion.

This process produces what Wyclif calls minima naturalia (analogous to

molecules),126 and, in each of these, different proportional relations of

the elemental particles are naturally associated with and so give rise todifferent compound substantial forms. The apparent homogeneity of a

golden globe is the sameness of a supervening compound form in each

compound particle of that substance. He concludes: “whoever speaks

123 Wyclif, LC 3.9, p. 80: “Ad illud negatur prima consequencia, cum iuxtaposiciocorpusculorum, ceteris requisitis, constituunt vere mixtum.”

124 Ibid.: “Et sic conceditur homines commisceri ad invicem cum aliis, et propor-cionaliter de ceteris partibus huius mundi, sed non ad finem quod forma substancialis

resultet constituens mixtum substanciale disparis specie.”125 Ibid.126 Ibid.: “. . . quod philosophi secundum gradum minimum vocant minimum natu-

rale.”



218 emily michael

truly of a compound [mixtio] ought to concede either small bodies

[corpora parva] or the matter of bodies to be juxtaposed, and to be

continued by a superadded form.”127

Wyclif here claims a twofold consequence of his Avicennist view, first,

there must be orderly structures of contiguous ( juxtaposed) elemen-

tal atoms, and second, a mixed body, i.e., each natural minimum (or

compound particle), must be determined by a distinct super-added

substantial form. This leads him to a view in which, each compound

material substance, organic and inorganic, is composed of a plurality of

substantial forms. Gold or salt, blood, bones or flesh are each a single

mixed substance because each is composed of homoeomerous minima

naturalia. Wyclif ’s world is therefore composed of orderly structures ofparticles and organized by collections of like substances. Air is each

air atom and any collection of these atoms; salt is a natural minimum

composed of a particular proportional organization of elemental atoms

and a supervening salt form, and salt is any collection of such compound

particles. A human being is a particular composite of body and mind,

and humanity is the collection of all such substances.

At Creation, God created material indivisibilia, and, in turn, from

combinations of these, produced the size and shape of the elemen-

tary corpuscles (of earth, air, fire and water) that compose Wyclif ’s

world. This is a world composed of elementary units, and, in turn, of

compound units and orderly collections and combinations of these. It

seems that Wyclif thought it necessary to accept, on the basis of reason

and observation, that the elements fundamental to the formation of a

compound body must remain in that body, and he was willing to accept

as logical consequences of this view, enduring elemental corpuscles and

a plurality of substantial forms in every compound substance. Further,

in accordance with his methodology, this conclusion of reason is con-sistent with his finitist interpretation of Scripture and his consequent

commitment to an atomistic account of the natural world.

4. Conclusion

John Wyclif has long been cast in the role of an innovator, and, in

particular, as the morning star of the Protestant Reformation. What I

127 Ibid.: “. . . cum omnes vere loquentes de mixtione oportet concedere vel corporaparva, vel materias corporum, iuxtaponi, et per formam superadditam continuari.”




have tried to show is that this is a limited picture of the innovations of

this now little known figure. In the context of his Aristotelian ontology,

Wyclif provides an interesting atomistic account of the natural world.Further, here too he anticipates some later views, though his particular

theory as a whole remains distinctive. For example, like seventeenth

century atomist Daniel Sennert, Wyclif maintains that the fundamental

building blocks are indivisible atoms and both of these atomists cite

Democritus in support of their atomistic views.128 But both depart

significantly from Democritean atomism.

Unlike Democritus, both claim that there are several grades of atoms,

all of which were formed by God at Creation. Both assume elemental

corpuscles (of earth, air, fire, and water), and both claim that eachelemental corpuscle is a substance that is a composite of prime matter

and a substantial form. Both also claim that combinations of elemental

units, in turn, constitute compound particles, which are hierarchically

composed of a structure of elemental corpuscles and a supervening

substantial form. Both develop atomistic accounts in the context of

Aristotelian hylomorphism and scholastic pluralism (the view that a

compound substance can have a plurality of substantial forms).

Sennert, however, praises Democritus, because this ancient atomist

realized that since bodies cannot come to be from nothing nor from

points, there must be smallest physical particles.129 Sennert’s primitive

indivisible units are extended elemental atoms. Wyclif adds a simpler

grade of matter. Our Oxford master, rejecting that bodies cannot be

composed ultimately of points, makes that very claim. His indivisible

atoms, cast as the fundamental building blocks of extended bodies, are

not blocks at all. That is, they are not extended physical units; they

have no length, breadth, or depth. According to Wyclif, elemental cor-

puscles are themselves composed from some structure (known to God)of unextended indivisible atoms, and these elemental corpuscles are the

extended atomic units fundamental to all bodies in the world.

To conclude, I have here argued that Wyclif ’s atomism was moti-

vated not simply by the ancient atomism of, for example, Democritus

or Plato, authorities he cites [ Tr p. 84; LC 3.9, p. 132], nor simply by

128 For Daniel Sennert’s atomistic theory, see Daniel Sennert, Hypomnemata Physica,

“De Atomis, & Mistione,” [Lyon, 1650], pp. 156–167. See also Michael, “Sennert’sSea Change: Atoms and Causes.”129 Sennert, ibid ., p. 158: “Cum enim videret Democritus, corpora naturalia neque ex

nihilo, neque ex punctis fieri, necessario statuit, ea ex minimis corpusculis componi.”



BLASIUS OF PARMA FACING ATOMIST ASSUMPTIONS

Joël Biard

Fourteenth-century arts masters offered better and more thorough

accounts of atomism than their predecessors, even when they intended

primarily to criticize it. The masters knew Democritus only indirectly

through Aristotle’s various critical works, especially De generatione et cor-

ruptione. But, as Aristotle linked the Democritean conception of elementswith the notion of the indivisible,1 the whole study of continuum and

indivisibles that has been developed in the thirteenth and fourteenth

centuries, both in physical and mathematical contexts, invites us to

confront atomism.2 Not only did Thomas Bradwardine provide an

overview of the various solutions to problems associated with the con-

tinuum,3 but Walter Chatton, Nicholas Bonetus, and Gerard of Odo

(if we assume that Nicholas of Autrecourt had no direct influence on

those debates) made much more consistent the atomist hypothesis,

whether one adheres to it or rejects it.In his various writings on natural philosophy Blasius of Parma often

develops issues which are not strictly Aristotelian. A marginal comment

in a manuscript copy of the second draft of his commentary on the

Physics, raises the question of Blasius’s acquaintance with the atomist

thesis:

This question is dealt with in the first book On the Heavens, you alsofind it determined by Gerard of Odo in a small powerful book, at the

1 See De Generatione et corruptione, I, 1, 314 a 21–24.2 Hence, Democritus is quoted twice in Nicole Oresme’s Questions on De generatione

and corruptione. Concerning the definition of the element, he mentions the Aristoteliandefinition according to which the element is that in which the body is reduced. Hedistinguishes three possible interpretations: “. . . pro cuius expositione est sciendum quodtriplex est dissolutio: quedam in partes integrales, et sic intelligit Democritus corporaresolvi in elementa, id est athoma . . .” (Nicole Oresme, Questiones super De generatione et

corruptione, [Caroti], pp. 199–200).3 The text of the Tractatus de continuo has been edited by John Murdoch, in Geometryand the Continuum in the Fourteenth Century, pp. 139–471. Some extracts have been translatedinto French in Sabine Rommevaux, “Bradwardine: Le continu,” pp. 89–135.



222 joël biard

beginning of which one can find the Questions on the Physics by Thomasthe English.4

This marginal note is probably not from Blasius, but rather from thecopyist.5 Nevertheless, it invites us to compare the various authors, a

comparison that, unfortunately, I won’t be able to carry out in this

short chapter. Still, it suggests that atomism constitutes the horizon

for these questions about the continuum. At any rate, it makes it clear

that atomism cannot be eliminated from the discussion. It is hard to

say much else about this remark insofar as it would require a better

understanding of Blasius overall conception of matter. Unfortunately,

the lack of a critical edition of the Questions on the Physics makes it

extremely dif ficult to get any comprehensive picture of the whole set

of his conceptions, inasmuch as it is quite possible that Blasius changed

his mind on this topic.6

Blasius’s attitude towards atomism, or rather his use of atomist argu-

ments, is related to two main problems. The first one concerns the void,

a problem I will not deal with in detail in this chapter. On this point,

Blasius defends an original position. He denies the real existence of a void while accepting at the same time a hypothetical void as fecund

for studying the nature of motion, both in his Questions on Bradwardine’sTreatise on Proportions as well as in his Questions on the Physics. Concern-

ing the first treatise, he utilizes the assumption that a void exists when

discussing Bradwardine’s rule in order to determine the motion and

speed of a body in the absence of a resisting medium.7 This approach

allows him to dissociate the temporal factor from the form accomplished

in and by a motion (either a qualitative form, in the case of an altera-

tion, or distance, in the case of a local motion). He scrutizines this

hypothesis much more extensively in the Question on the Physics. Here,

4 “Tangitur etiam hec question I° Celi; habes etiam ipsam determinatam perGerardum Odonis in libro parvo viridi, in cuius principio sunt Questiones Physicorumsecundum Thomam Anglicum,” in Questiones Physicorum, VI, q. 2, Ms. BAV, Vat. Lat.2159, f. 164va ( in margine ), quoted according to Federici Vescovini, Astrologia e scienza,p. 340. For the Questiones de Celo, see bk. I, q. 8, in Ms. Milano, Biblioteca AmbrosianaP 120 sup.

5 Bernardus a Campanea di Verona (See Federici Vescovini, loc. cit .; and Caroti, I codici di Bernardo Campagna. Filoso fi a e medicina alla fi ne del sec. XIV ).

6 This is Graziella Federici Vescovini’s claim ( op. cit .), who argues that Blasius modified

or smoothed out his theses between the first and the second redaction of his Questionson the Physics after the condemnation of 1396.7 See Blasius of Parma, Questiones circa tractatum proportionum magistri Thome Bradwardini

[Biard & Rommevaux], pp. 164–166 and p. 174.



blasius of parma facing atomist assumptions 223

an entire question is devoted to the possibility of motion in the void.8

It seems likely that he takes as a starting point the traditional opin-

ion—since Ibn Baja—that motion would be instantaneous in a voidsince, resistance being equal to zero, the ratio of force/resistance would

be infinite. Denying the possibility of an infinite speed, he transforms

this objection against a void into a useful hypothesis. He admits that

a motion in the void would be successive. It can therefore be assigned

a certain speed and be used in order to compare motion in the void

with motion in a given medium.

However interesting this question concerning a hypothetical void

may be, it does not seem explicitly connected with atomism. The

situation is different in the case of the continuum. It should be noted,however, that by way of the first article in question 1 of book 6 of the Physics (devoted to the continuum) Blasius introduces a problem, albeit

quite briefl y, concerning the motion of two mixed bodies—unequal in

quantity but similar with regard to their mixture—in the void. While

it might simply be a purely scholastic exercise, without any link to the

question,9 it may suggest a strong connection between the problem of

the void and that of the continuum. Dealing with the continuum, Blasius

develops a moderate and complex position on indivisibles.

1. The Composition of the Continuum

Blasius of Parma examines the composition of the continuum in ques-

tions 1 and 2 of book 6 of the second version of his Physics commentary.

I will deal with some related passages from the first version of this text

later.10 In the first question, Blasius asks whether one may conclude

with valid reasons that a continuum is composed of indivisibles. The

second question tackles the problem from the opposite perspective,asking whether a continuum is infinitely divisible.

8 Questiones Physicorum (second redaction), IV, 5, Ms. BAV, Vat. Lat. 2159, ff. 122vb– 126rb.

9 In some textbooks, we may find sophisms introduced as exercises, but withoutany link to the topic; this is the case in the Questions on the Logical Tracts, according to ausual practice in Northern Italian Universities. See A. Maierù, “I commenti bologneseai Tractatus di Pietro Hispano.”

10

The first version corresponds to the lectures given in Padua between 1382 and1388; we only have the two first books. The second version, of which we have a com-plete text, corresponds to the lectures given in Pavia in 1397. See Federici Vescovini, Astrologia e scienza, p. 430 and pp. 433–434.



224 joël biard

In the first question, Blasius adopts an attitude that can be found in

other texts which makes it particularly dif ficult to figure out his posi-

tions. In the determinatio of the second article of the question, he seemsto embrace the opposed views (the same attitude is found, for example,

in the lengthy Question on intension and remission of forms ).11 This procedure

obviously makes it dif ficult to assess which one corresponds to his own

position: does he set forth the various possible answers because of his

inability to settle the issue? Is he subtly more inclined towards one of

the answers without making it explicit (whatever may be his reasons

to do so)? Does he implicitly oppose each point of view? With these

interpretive problems in mind, here is how he introduces the second

article:

I begin the second article, in which I will answer the question in a nega-tive way, and then I will make you understand that the question can beaf firmatively defended.12

The refusal to consider the continuum as composed of indivisibles is

based on a very traditional style of argumentation, which from the

outset relies on geometry. Indeed, Blasius demands that one accepts a

Euclidean principle according to which any line can be divided into two

equal parts. Such a claim, as we well know, is particularly problematic if a line is made up of points, above all if they exist in a finite number.

The argument unfolds in three quick steps.

The first part (arguments 1 and 2) considers the number of points

into which a line can be divided: it can be neither even nor odd. The

second part claims that a line can neither be made up of an infinite

nor of a finite number of points (conclusions 3 and 4). The assump-

tion of a finite number is invalidated by the preceding arguments. If

the number is infinite, then the line is either finite or infinite. It cannot

be infinite, so it should be finite. Again, it is asked whether its half is

composed of a finite or an infinite number of points. Finally, arguments

5 and 6 proceed to a quick generalization concerning the composition

of a surface from lines, and a body from surfaces.

11 This question has been edited by Federici Vescovini, “La Questio de intensione

et remissione formarum de Biagio Pelacani di Parma,” pp. 432–535.12 “Transeo ad secundum in quo determinabo questionem pro parte negativa ettandem dabo tibi intelligere questionem posse sustentari pro parte af firmativa,” Ms.BAV, Vat. Lat. 2159, f. 162va.




All of this is nothing but ordinary reasoning against the composition

of a continuum from indivisibles and we may even say that he goes far

less deeply into the details than did English scholars at the beginningof the fourteenth century. But it is interesting that he introduces—only

as a defensible position for the time being—the assumption that the

continuum is composed of indivisibles: “Now I present some conclusions

which can be defended without contradiction from the other point of

view”.13 Blasius then opts for the infinitist solution outlined in the first

fi ve conclusions.

1. “Even if a line were made up of points, we could not infer that it

is made up of an odd or even number of points”.14 The purposeis to set aside the aforesaid contradiction between an odd and even

number of points, of which the line would be made up in this

case. How to proceed from here? At this moment, Blasius takes up

the proportional parts of a continuum, which are neither even nor

odd.

2. “Even if a line were made up of points, we could not infer that it

is made up of a finite number of points”.15 This is a consequence

of the former conclusion.

3. “If a line were made up of points, it would be made up of an infi-

nite number of points”.16 Indeed, if it is made up of points, it will

be a finite or an infinite number of points. Yet it cannot be a finite

number, according to the preceding conclusion.

4. “That any line is made up of an infinite number of points does

not include any contradiction”.17 The rationale is very weak: the

opposite cannot be demonstrated, and no contradiction follows.

But Blasius significantly adds that many learned people support

this claim: “multi sapientes tenuerunt illud”.18

He will come back tothis problem later: “To say that a line is made up of points is not

13 “Nunc pono conclusiones pro alia parte defensibiles a contradictione,” ibid .,f. 162vb.

14 “licet linea esset ex punctis composita, non ex hoc posset determinari illa esse expunctis paribus vel imparibus compositam,” ibid . ff. 162va–163ra.

15 “licet linea esset composita ex punctis, non ex hoc determinari posset illam expunctis finitis esse compositam,” ibid . f. 163ra.

16 “si aliqua linea est composita ex punctis, ipsa composita esset ex punctis infinitis,”

ibid ., f. 163ra.17 “quod omnis linea sit composita ex punctis infinitis contradictionem non includit,”ibid ., f. 163ra.

18 Ibid ., f. 163ra.




clearly wants to preserve an infinitely divisible continuum for math-

ematical objects.

More interesting is the fourth dif ficulty. The preceding assumptionsbeing accepted, Blasius asks whether points are immediately adjacent

to one another. Here, two alternative answers are indicated: it may be

said that a point is immediately adjacent to another, but since it has no

part, it touches it as a whole and not by one of its parts (this relates to

the problem of contact, something which Blasius paid great attention

to in other texts),22 or it may be said that contact only concerns mag-

nitudes, in which case a point cannot be said to be touching another,

nor to be distant from another.

The following dif ficulties and their solutions deal with the notion oflimit: firstly the limit of a line, and then, quite lengthily, the limit of

a quality that would be exclusively limited by another one. I leave out

these developments, which would suppose a detailed examination of

the status of a quality and of the theory of the intension of forms.

The second question returns to the main problem, but tackling it

from the opposite direction. It is asked whether a continuum is infinitely

divisible. It is here that the copyist mentions Gerard of Odo. But the

reciprocity between the two questions is only apparent. The first ques-

tion stuck with simple mathematical examples (or, in the end, with some

considerations on the limit of a quality). Here, we actually find a long

set of similar arguments, at the beginning of the part quod non. Indeed,

in the second set of arguments (introduced by “Secundo ad principale”),

fifteen are of a rather mathematical sort, and the last three deal with

motion. But it is preceded by another shorter series that immerse us in

a different clime made of purely physical examples: a wire fence resist-

ing a pulling force; water freezing in a clogged vase (curiously enough

such a phenomenon is supposed to create void); a ray refracted whenpassing from one medium to another with a different density.

The question’s determinatio provides a set of terminological clarifica-

tions, which are taken from the first draft of the text, which I will have

to deal with later. The first details concern the categorematic and syn-

categorematic meaning of the infinite, that is to say, its determination

22 See the question “Utrum duo corpora dura possunt se tangere,” mss. Venice,

Bibl. Marc. Lat. cl. VI, 155 (3377) = codex 18, Valentinelli IV, 230, ff. 105ra–112ra;Bologna, Bibl. Univ. 2567; a copy in Oxford Canon. Misc. 177, ff. 155ra–158vb givesa different text; but we find the same text in the Questiones de anima, Ms. BAV, Vat.Chig. O. IV. 41, ff. 195ra–197rb, and Naples, Bibl. Naz., VIII. G. 74, ff. 128r–136r.



228 joël biard

according to the place of the term in a sentence and the various defi-

nitions that can be given to it. These topics had become almost com-

monplaces from the second quarter of the fourteenth century, after theeighteenth sophisma of William Heystesbury’s Sophismata and after John

Buridan’s, Gregory of Rimini’s and Albert of Saxony’s work.23

Blasius next provides some more details about the terms “divisible”

and “indivisible”. What is important here is the distinction between

two meanings of “indivisible”:

In one sense, something is said to be divisible because it is naturally ableto be divided, and conversely, something is said to be indivisible becauseit is not naturally able to be divided. In another sense, something is said

to be divisible because, although it is not naturally able to be divided,one can nevertheless think that it is divisible without any contradiction.Hence, the heavens are actually divisible, as well as anything that hasparts located one apart from the other.24

In question 10 of book 1 in the first version of his Physics, Blasius

makes the same distinction, with some additional details. A one foot

piece of earth is used as an example of a thing able to be divided. It

is not impossible to divide it into two or three pieces, etc. (by means

of an actualis divisio ). The sun, the heavens and the eighth sphere are

examples of the second sense. But Blasius also presents this distinction

between the two senses by opposing the “real division” and the “intel-

lectual division”.25

After a last terminological clarification concerning proportionality,

the second question continues and establishes conclusions concerning

the ratio of the whole to its parts. These conclusions will be put to the

test in the third article.26 Curiously enough, Blasius is so concerned

about establishing that a line is infinitely divisible, mostly by means of

a sequence of divisions into proportional parts, that he seems to recon-sider what he has established in the preceding question concerning the

23 On this topic, texts from John Buridan and Gregory of Rimini can be read inthe textbook De la théologie aux mathématiques, respectively pp. 253–279 and 197–219. Seealso, Joël Biard, “Albert de Saxe et les sophismes de l’infini,” pp. 288–303.

24 “Uno modo dicitur aliquid divisibile quia illud est aptum natum dividi, et peroppositum aliquid dicitur indivisibile quia non est aptum natum dividi. Alio mododicitur aliquid divisibile quia licet non sit aptum natum dividi potest tamen sine con-

tradictione intelligi quod sit divisibile. Et sic celum bene est divisibile, et omne habenspartem situaliter extra partem”, Ms. BAV, Vat. Lat. 2159, f. 166vb.25 Ms. BAV, Vat. Chigi O IV 41, f. 245ra.26 Some passages examine in particular the summons of proportional parts.




composition from points. Conclusions 1, 2, and 3 assert that a line is

not made up of points, neither finite nor infinite in number.27 However,

this contradiction should be taken as relative. In the first question, oneconsiders composition and makes the assumption that a line is composed

of an infinity of points. In the second question, one considers division

and insists on the infinite divisibility, from a geometrical point of view

as well as from a physical one, as we will see.

Now, about this point, we find once again an argument Blasius had

used in question 10 of book 1 in the first draft. Before considering

infinite division into proportional parts, he sets out as a preliminary

condition that a line is not composed of points, neither a finite nor an

infinite number. In the first draft, Blasius’s position was not limited tothe theses supported in this question: question 11 asks whether there

are limits to the greatness and smallness of natural bodies.

2. Indivisibles and Minima Naturalia

The first version of Blasius’s Physics, question 10, concerns the division

of a continuum. This is immediately followed by a question about the

limits that one should assign to the greatness and smallness of naturalbodies. Even if the conceptual context is quite different, we should take

the proximity of these two questions quite seriously. Before considering

this problem, it is necessary to examine another text, one of those where

Blasius explicitly deals with the indivisible.28 This text is question 15

of book 1 On Generation and Corruption where it is asked: “Can an indi-

visible be altered?”29 The question has undoubtedly a twofold motive.

First, having no parts, it seems that an indivisible cannot move, neither

locally nor qualitatively. Again, the question of the relation between

a quality and a subject is raised, whether the subject be imagined asindivisible (as a point) or as something indivisible informing a material

and divisible subject (as the soul). For this reason, the text supplements

other passages where the soul is conceived as an indivisible.

27 Ms BAV, Vat. Lat. 2159, f. 167ra.28

Blasius also deals more or less directly with the indivisible in his Commentary on the Heavens, and his Questions on the Soul .29 “Consequenter queritur utrum indivisibile possit alterari” ( Questiones de generatione

et corruptione, I, q. 15, ms. BAV, Vat. Chigi O IV 41, f. 23rb sqq.)



230 joël biard

The terminological clarifications given in the first article will be

retained here: a point can be conceived in two different ways. On this

occasion, Blasius uses the verb “ymaginari”:

First article: I notice that we can imagine points in a continuum in twoways. In the first sense, they have a position in a continuum; in the second they do not. An example of the first acceptation: if a line were made upof points. An example of the second acceptation: the intellective soul,which is in the human body although it does not have a position and aplace in the human body.30

This distinction goes beyond the difference between the two meanings

of the division presented in the second redaction of the Physics, namely

the real and the intellectual one. There, Blasius argued that the Heavens

could not be divided, although one could distinguish in them some dif-

ferences in position. Here, it is about a reality that is not susceptible of

division, neither real nor imaginary, since by itself, without considering

the body it animates, it has no parts, in whatever sense. Blasius makes

it clear that this meaning will not be relevant in the remainder of the

question: “. . . this distinction is made insofar as the question only deals

with an indivisible which has a position in a continuum”.31 Therefore, in

this question, he does not really consider whether a spiritual indivisible,

such as the intellective soul, can be altered—even if he incidentally men-

tions the classic problem of whether a soul could suffer in the infernal

fire.32 Following G. Federici Vesconvini’s analysis, it seems that in the

Questions on the Heavens, Blasius considers the soul as infinitely divisible

from a physical point of view. The only relevant point for our present

matter is once again Blasius’s claim that everything is divisible from a

physical point of view:

30 “Pro primo articulo noto <quod> duplicter possumus ymaginari puncta in con-tinuo. Uno modo quod habeant positionem in continuo, secundo modo quod non.Exemplum de primo si linea componeretur ex punctis. Exemplum de secundo sicutde anima intellectiva que est in corpore humano licet non habeat positionem et situmin corpore humano,” Questiones de generatione et corruptione, I, q. 15, Ms. BAV, Vat. Chigi

O IV 41, f .

23rb.31 “ista distinctio posita est pro tanto quia questio querit solum de indivisibile habentepositionem in continuo,” ibid ., f. 23rb.

32 Ibid ., f. 24rb–va.




Given any thing that is one, it is divisible since it is a natural thing, andthe intellective soul is no exception since, from a physical point of view,it is not indivisible.33

On the other hand, the mathematical concept of the indivisible may

be applied mutatis mutandis to the soul which is compared to a point

in the De generatione et corruptione. Like a prime number, the intellective

soul would be only divisible by itself or by the unit. For this reason, it

would be considered as a ratio of equality ( ratio equalitatis ).

The above-mentioned question from De generatione et corruptione tends

to show the contradiction that might result from the possibility of a

physical indivisible:

Secondly I notice that, for the question asked, in no way one shouldimagine that indivisibles have a position in a continuum. However, wewant to seek and examine some dif ficulties that would follow from suchan thought experiment ( tali ymaginatione ).34

Does it mean that Blasius denies any form of physical minima? It is well

known that since the thirteenth century minima naturalia have sometimes

been distinguished from indivisibles. This notion finds its origin in a

passage from Aristotle’s Physics where he objected to Anaxagoras’s idea

that the constituent parts of a natural whole cannot be of any size what-soever.35 Averroes developed the idea and Roger Bacon and Albert the

Great introduced it to the Latin world. It continues to appear up to the

Renaissance, especially in texts of Italian natural philosophy. The natural

minimum is determined by the nature of each substance. However, the

status of this minimum can vary: sometimes it is the minimum capable

of producing an effect, sometimes it is the minimum perceptible by the

senses. As a consequence, the concept of a natural miminum opened

itself to various kinds of philosophical issues. Duns Scotus denied the

opposition between quantum and naturalia, and he criticized those who

33 “quacumque re data que est una, illa est divisibilis quia est una naturalis res, necfiat instantia de anima intellectiva quia, physice loquendo, ipsa non est indivisibilis,”See Questiones de Celo, I, qu. 9, Ms. Milano, Ambrosiana P 120 sup, f. 23ra—accordingto Federici Vescovini, op. cit., p. 338, n. 32.

34 “Noto secundo quod nullo modo propter questionem propositam ymaginandumest quod aliqua indivisibilia habeant positionem in continuo. Tamen volumus querereet videre aliquas dif ficultates sequentes ex tali ymaginatione,” Questiones de generatione et

corruptione, I, 15, Ms. BAV, Vat. Chigi O IV 41, f. 23va.35 Aristotle, Physics, I, 4, 187 b 13–188 a 5. On the medieval history of this notion,see John E. Murdoch, “The Medieval and Renaissance Tradition of minima natura-lia,” pp. 91–131.



232 joël biard

accepted some physical minima, while admitting at the same time their

incompatibility with mathematics.36 As for Blasius, he will not set the

mathematical point of view against the physical point of view. As wehave already seen, matter is infinitely divisible. On the other hand, he

introduces the idea of a necessary proportions among natural beings,

along with the idea of limits beyond and below which a natural being

ceases to exist. This is the subject matter of question 11 of book 1

of Blasius’s Physics in its first redaction: “Eleven, we ask whether any

natural body is limited by its greatness and shortness”.37

In this question, Blasius does not talk about indivisibles but only about

natural limits. We are leaving the field of mathematical indivisibilism,

but on the other hand, if the term ‘atom’ is not used in this context,we come very close to the idea of the smallest part of a body that can

exist, including the elements. The problem is raised from the beginning

of the first article:

As regards the first article, here is a first remark: that a body be limitedin greatness and smallness is equivalent to the fact that in relation to itthere should be two limits, of greatness and smallness, which cannot benaturally exceeded.38

Beyond the results expressed in such and such a conclusion, it is thegeneral thought process that is interesting. The limit can be considered

from the point of view of greatness and smallness or from the point

of view of active and passive powers. Greatness and smallness clearly

imply the consideration of matter or subject, and on the other hand,

the powers imply the consideration of the relationship between matter

and form, and this from two different points of view: either from the

point of view of the introduction of a form into matter, or from the

point of view of the conservation of a substantial form. In the latter

case, Blasius still further differentiates the case of a substantial formeduced or likely to be educed from matter, and the case of the intel-

lective soul coming to animate matter.

36 See Ordinatio [Balic], II, dist. 2, p. 2, qu. 5, pp. 305–306.37 “Queritur undecimo utrum omne corpus naturale sit in magnitudine et parvi-

tate limitatum,” Quaestiones Physicorum (1a redactio), ms. BAV, Vat. Chigi, O IV 41,

f. 250ra.38 “Quantum ad primum articulum sit primum notabile quod corpus esse inmagnitudine[m] <et> parvitate limitatum est idem quod respectu eius sint duo terminimagnitudinis et parvitatis quod egredi naturaliter non contingat,” ibid ., f. 250va.




Here, I only reconstruct the main theses regarding the minimum. The

general tendency is to deny a minimum to simple elements, considered

in themselves. This is the case in the fifth and sixth conclusions dealingwith the example of fire. Blasius will later contend that this holds for

any element:39 “there is no minimum of fire since for any given fire, its

half will be smaller”.40 Here we do not consider the existing fire, but

a conceivable or imaginable one. On the one hand, it seems infinitely

divisible. On the other hand, there exists a fire smaller than any other:

“There is a minimum of fire existing by itself and separately”.41

More interesting is the position of a minimum fire between certain

limits in a given medium. Here we catch a glimpse of the motives behind

Blasius’s thought: he wants to introduce ratios between a minimumof matter, a medium, and, possibly, relations of power and resistance.

“There is a minimum of fire that can subsist in such a way and such a

time in this medium”.42 The conclusions are reminiscent of the various

cases according to which such an element can subsist or be corrupted,

in such and such a medium and such and such a time.

However, we still remain within the framework of infinite divisions.

The case is different for mixed bodies which, of course, constitute the

most important part of our world. In this case, the connections to be

kept gain more and more importance. We will first consider unanimated

mixed bodies. They can be treated as simple bodies from the standpoint

of permanency and decay: “In no time a mixed and unanimated whole

can subsist in a medium”.43 The aim is actually to prevent permanency.

But a difference should be noted between unanimated mixed bodies

and animated mixed bodies. Paradoxically, the first ones are modified

or destroyed if we remove a part. For example, if we remove a part of

a stone, we will have another stone or no stone at all. The second ones

can subsist if we remove a part, for example, if we remove Socrates’shand. They are destroyed when the whole is destroyed. Hence, in the

case of animated bodies or animals, there is a given maximum and a

39 “et notetis quod in huiusmodi conclusionibus nominamus ignem sed quicquid estdictum de ignis, intelligatis de quocumque alio simplici corpore,” ibid . f. 251rb.

40 “Non datur minimus ignis quia quocumque dato eius medietas est minor,” ibid .41 “Datur minimus ignis per se seorsum existens,” ibid .42

“Datur minimum ignis qui in hoc medio potest sic et taliter permanere pertantum tempus,” ibid .43 “Per nullum tempus potest aliquod mixtum totum inanimatum in aliquo medio

permanere,” ibid .



234 joël biard

given minimum. But there is no absolute minimum if we consider that

a part of the animal is animated.

3. Conclusion

The sequence of questions in the first redaction of the Questions on the

Physics is not meaningless. After studying the mathematical division of

the continuum, Blasius of Parma wonders whether there are physical

minima. It is dif ficult to class him within a predefined grid of positions

about indivisibles. Blasius is certainly not an atomist in the strict sense,

since matter is infi

nitely divisible according to him. Nevertheless, likesome of his contemporaries, he cannot avoid problems connected with

indivisibility. He addresses these problems in various places, defending

some positions that at first glance may even seem contradictory. In fact,

his goal is to present various points of view. But unlike his habits on

other occasions, the aim is not only to contrast the mathematician’s

and the physicist’s differing points of view. His distinctions cross these

two disciplinary fields.

From a mathematical point of view, the composition and the division

of a continuum are to be treated differently. Concerning composition,two different claims have been developed. The first one takes up the

traditional geometrical criticism of indivisibles. The other one is more

original and admits composition from an infinite number of indivisibles.

This non-contradictory assumption is acceptable in mathematics pro-

vided that one redefines some concepts, such as the concept of whole

and parts. Concerning division, the infinite division of mathematical

objects must be accepted.

From a natural point of view, the infinite divisibility of matter must

be admitted without reserve. But natural bodies are composed of matterand form. Some connections are established and must be maintained

for the generation and permanency of such and such natural bodies.

These connections are not only the more relevant and interesting ele-

ments to be studied, they also define the limits within which any such

being is conceivable. It is all the more true if the form in question is

a soul, since it is indivisible when it is considered in its simple con-

nection to itself, and divisible when it is considered as the form of a

natural body.



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Murdoch, J.E. & E. Synan, “Two Questions on the Continuum: Walter Chatton (?),

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INDEX OF ANCIENT, MEDIEVAL ANDRENAISSANCE AUTHORS

Adam Buck field 58, 236, 245Adam of Marsh 153Adam of Wodeham 16, 40, 88, 127,

130, 139, 171, 182, 236, 238, 240,244

Albert of Saxony 18, 23, 25, 228, 235,236

Albert the Great 5, 42, 112, 114, 116,235, 236Alexander of Aphrodisias 22Al-Ghazali (Algazel) 7, 19, 66, 68, 87,

236, 245Anaxagoras 15, 112, 113, 114, 231Aristotle 2, 3, 5, 6, 7, 8, 9, 10, 11, 12,

15, 17, 18, 19, 21, 22, 24, 25, 27, 30,32, 33, 34, 35, 37, 38, 39, 40, 41, 42,44, 45, 47, 48, 50, 51, 52, 53, 57, 58,63, 64, 73, 77, 83, 85, 88, 89, 107,108, 109, 110, 111, 112, 113, 114,

116, 117, 123, 124, 126, 127, 129,130, 134, 136, 137, 140, 141, 142,143, 144, 147, 153, 160, 161, 162,170, 184, 188, 190, 192, 199, 203,211, 213, 215, 221, 231, 236

Augustine 4, 14, 53, 55, 202, 203,204, 206

Averroes 26, 111, 191, 196, 213, 215,231

Avicenna 10, 19, 87, 213, 215, 216,236

Bede the Venerable 3Benedict Hesse 84Blasius of Parma 7, 14, 221–234Bonaventure 22, 23, 236

Cambiolus of Bologna 215Campanus of Novara 35, 36, 74Cicero 3, 6, 114, 236

Daniel Sennert 186, 188, 189, 214,219, 220, 236

David of Dinant 5

Democritus 1, 2, 3, 5, 6, 7, 12, 17, 18,27, 69, 90, 105, 108, 112, 113, 114,116, 117, 123, 136, 139, 146, 153,161, 162, 208, 209, 212, 219, 221

Diogene Laertius 34

Epicurus 3, 4, 5, 12, 17, 34, 90, 146,208,

Etienne Gaudet 162, 165, 166, 208Euclid 10, 19, 20, 21, 22, 26, 35, 36,

67, 70, 72, 83, 237

Filipo Fabri 195Franscesco Patrizi 148

Gaetano of Thiene 138Galileo Galilei 15, 23, 39, 237, 243Gerard of Cremona 162Gerard of Odo 6, 7, 10, 11, 16, 17,

18, 25, 26, 31, 34, 85–106, 107, 127,138, 140, 141, 142, 160, 170, 171,174, 183, 211, 221, 222, 227, 235,240, 245, 246

Giuseppe Veronese 25, 237Godfrey of Fontaines 36, 37, 101, 237Gregory of Rimini 23, 31, 37, 140,

160, 228, 237

Henry of Harclay 7, 9, 10, 14, 16, 17,19, 20, 22, 23, 25, 26, 27, 28, 30, 31,35, 40, 96, 127, 134, 136, 139, 140,141, 142, 143, 161, 170, 171, 235,244

Iacopo Zabarella 220, 237

Ibn Baja 223Isaac Israeli 162, 237Isidore of Seville 3, 15

Jan Hus 186 John Buridan 12, 13, 15, 20, 24, 25,

33, 40, 84, 85, 125, 126, 158,163–182, 228, 235, 237, 240, 245,246

John Dorp 40 John Duns Scotus 7, 10, 17, 19, 46,

66, 67, 68, 69, 70, 82, 87, 159, 162,

190, 195, 196, 231, 237, 240, 245 John Gedeonis 16, 86 John Major 84, 237 John Quidort 193



John of Jandun 189 John of Mirecourt 37, 244 John of Ripa 37, 235, 237

John of Salisbury 4 John Pecham 189, 192 John Tarteys 18, 235 John the Canon 16, 31, 87, 138, 235,

237 John Wyclif 6, 7, 13, 16, 34, 153,

154, 161, 183–220, 237, 242, 243,245

Julius Caesar Scaliger 186, 237

Lactantius 4, 237Leucippus 3, 69

Lucretius 3, 4, 5, 6, 12, 17, 22, 237

Marbode of Rennes 4, 237Marc Trivisano 86Michel of Montecalerio 7, 12, 13,

158, 163–182, 235, 240Moses Maïmonides 12, 108, 116, 117,

119, 123, 152, 153, 162, 235, 237,242

Nicholas of Autrecourt 1, 6, 7, 11, 14,16, 34, 86, 105, 106, 107–126, 127,

138, 149, 153, 157, 221, 238, 240,241, 244

Nicholas Bonetus 16, 18, 86, 87, 105,107, 127, 138, 238

Nicole Oresme 18, 23, 25, 37, 64,221, 235, 238, 240, 243

Parmenide 15Paul of Venice 215, 216, 238Peter Abelard 8, 238Peter Ceffons 37, 243, 244Peter Lombard 41, 81, 87, 127

Peter of John Olivi 64, 238Peter of Spain 32, 238Philoponus 22, 238Plato 2, 3, 7, 10, 14, 18, 27, 76, 209,

219Plutarch 22Poggio Bracciolini 3Prisician 3Proclus 22

Rhaban Maur 3Richard Fishacre 49, 238

Richard Fitzralph 66, 246Richard Kilvington 7, 32, 65–84, 235,

241, 242

Richard Knapwell 193Richard of Middleton 189, 192, 214,

215, 246Richard Rufus 3, 7, 9, 39–64, 235,

238, 242Robert Grosseteste 5, 23, 40, 47, 50,

53, 141, 238Robert Kilwardby 32, 189Roger Bacon 5, 69, 231, 238Roger Roseth 16, 83, 84, 238Roger Swinshead 37

Theophrastus 18Thomas Aquinas 5, 13, 152, 189, 190,

191, 192, 194, 196, 213, 214, 238,239, 240

Thomas Bradwardine 7, 16, 18, 19,20, 23, 24, 26, 27, 32, 33, 34, 36, 37,66, 75, 82, 87, 221, 222, 235, 236,238, 240, 241, 243, 244, 245

Thomas of York 153

Urso of Salerno 5

Vincent of Beauvais 4

Walter Burley 7, 16, 22, 31, 66, 100,145, 163, 172, 173, 175, 178, 180,181, 238, 239

Walter Chatton 7, 10, 16, 17, 19, 20,86, 87, 88, 105, 127, 136, 139, 140,159, 182, 183, 211, 221, 238, 239,246

William Crathorn 6, 7, 12, 33, 34,127–162, 211, 235, 238, 241

William Heytesbury 32, 66, 228, 246

William de la Mare 192, 193, 238William of Alwick 16, 27, 28, 96, 235William of Champeaux 8William of Conches 2, 3, 5, 15, 238William of Ockham 16, 23, 24, 30,

40, 56, 66, 67, 68, 70, 71, 75, 82,116, 127, 145, 190, 196, 197, 210,238, 240, 241, 243, 244

William of Mackesfield 193

Zeno (of Elea) 10, 30, 83

248 index of ancient, medieval and renaissance authors



Ebbs, G. 61Elders, L. 239, 240

Federici-Vescovini, G. 222, 223, 224,230, 231, 240

Frede, M. 42, 236Freudenthal, G. 152, 241Fruteau de Laclos, F. 107, 241

Gauthier, R.-A. 236Gimaret, D. 151, 152, 241Glorieux, P. 192, 193, 236, 238Gnassounou, B. 120, 241Goddu, A. 75, 241Gorlaeus, D. 214Grant, E. 19, 96, 123, 126, 147, 148,

241Grellard, C. 6, 11, 106, 107, 110, 120,

123, 126, 138, 153, 157, 241, 242

Hallamaa, O. 84

Ibrahim, T. 152

Jacquart, D. 5, 121 Jolivet, J. 151 Jung-Palczewska, E. 66

Kaluza, Z. 107, 110, 113, 115, 116,118, 123, 156, 163, 164

Kennedy, L. 108Kenny, A. 183, 185, 186, 206Kistler, M. 120Kluxen, W. 152, 153Knuuttila, S. 32Kretzmann, B. 18, 66Kretzmann, N. 10, 18, 66, 154, 183,

208, 210Kuksewicz, Z. 215

Lamy, A. 145, 175, 181Lasswitz, K. 1Lawlor, K. 61Lear, J. 40, 41, 42, 45

Leff, G. 186Leijenhorst, C. 243Lévy, T. 245

Allard, G.H. 239, 244Alliney, G. 239, 242Anawati, G.C. 5, 152, 239Annas, J.E. 39, 239Argerami, O. 23, 240Ashworth, E.J. 40, 239Asztalos, M. 239

Bachelard, G. 2, 239Baf fioni, C. 239Bakker, P.J.J.M. 85, 239Barnes, J. 41, 236Bazan, C. 236Bentley, R. 23Biard, J. 14, 134, 145, 164, 222, 228,

236, 239, 241, 245Boas, G. 86, 239Boehner, Ph. 29, 239Bostock, D. 42, 236Braakhuis, H.A.G. 245Brancacci, A. 6, 239Burns, S. 61

Caroti, S. 66, 221, 222, 231, 239, 240,241, 243

Celeyrette, J. 7, 12, 13, 134, 158, 164,239, 240, 241, 245

Chandler, B. 240Cohen, I.B. 23, 240Conti, A.D. 183, 240Cosman, M. 240Courtenay, W.J. 66, 127, 163, 164,

166, 240Cote, A. 134, 240Cova, L. 239, 242Crombie, A.C. 240Cross, R. 196, 240

Dales, R.C. 23, 240De Rijk, L.M. 8, 32, 85, 238, 240Dhanani, A. 151, 240Dijksterhuis, E.J. 188, 189, 240Donati, S. 41, 240Duhem, P. 15, 86, 87, 240

Dumont, S. 101, 240Dutton, B. 116, 240Dziewicki, M.H. 185, 237

INDEX OF MODERN AND CONTEMPORARY AUTHORS



History of Science

and Medicine Library

Medieval andEarly Modern Science

Subseries Editors : J.M.M.H. Thijssen and C.H. Lüthy

1. FRUTON, J.S. Fermentation: Vital or Chemical Process ? 2006.

ISBN 978 90 04 15268 72. PIETIKAINEN, P. Neurosis and Modernity. The Age of Nervousness inSweden. 2007. ISBN 978 90 04 16075 0

3. ROOS, A.M. The Salt of the Earth . Natural Philosophy, Medicine, andChymistry in England, 1650-1750. 2007. ISBN 978 90 04 16176 4

4. EASTWOOD, B.S. Ordering the Heavens . Roman Astronomy and Cosmologyin the Carolingian Renaissance. 2007. ISBN 978 90 04 16186 3 (Publishedas Vol. 8 in the subseries Medieval and Early Modern Science )

5. LEU, U.B., R. KELLER & S. WEIDMANN. Conrad Gessner’s Private Library.

2008. ISBN 978 90 04 16723 06. HOGENHUIS, L.A.H. Cognition and Recognition: On the Origin of Movement .Rademaker (1887-1957): A Biography. 2009. ISBN 978 90 04 16836 7

7. DAVIDS, C.A. The Rise and Decline of Dutch Technological Leadership. Technol-ogy, Economy and Culture in the Netherlands, 1350-1800 (2 vols.). 2008.ISBN 978 90 04 16865 7 (Published as Vol. 1 in the subseries KnowledgeInfrastructure and Knowledge Economy )

8. GRELLARD, C. & A. ROBERT (eds.). Atomism in Late Medieval Philosophyand Theology. 2009. ISBN 978 90 04 17217 3 (Published as Vol. 9 in thesubseries Medieval and Early Modern Science )

9. FURDELL, E.L. Fatal Thirst . Diabetes in Britain until Insulin. 2009.ISBN 978 90 04 17250 0

Published previously in the Medieval and Early Modern Science book series:

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2. THIJSSEN, J.M.M.H. & J. ZUPKO (eds.). Metaphysics and Natural Philosophy

of John Buridan. 2001. ISBN 978 90 04 11514 93. LEIJENHORST, C.The Mechanization of Aristotelianism. The Late AristotelianSetting of Thomas Hobbes’ Natural Philosophy. 2002.ISBN 978 90 04 11729 7



4. VANDEN BROECKE, S. The Limits of Influence . Pico, Louvain, and theCrisis of Renaissance Astrology. 2002. ISBN 978 90 04 13169 9

5. LEIJENHORST, C., C. LÜTHY & J.M.M.H. THIJSSEN (eds.). The Dynamics of Aristotelian Natural Philosophy from Antiquity to the Seventeenth Century.

2002. ISBN 978 90 04 12240 66. FORRESTER, J.M. & J. HENRY (eds.). Jean Fernel’s On the Hidden Causes

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