John Buridan's Tractatus de infinito. Quaestiones super libros Physicorum secundum ultimam lecturam, Liber III, Quaestiones 14-19: An edition with an ... and indexes (Artistarium: Supplementa) [Bilingual ed.] 9070419300, 9789070419301

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JOHN BURIDAN'S TRACTATUS DE INFINITO

ARTISTARIUM A Series of Texts on Mediaeval Logic, Grammar & Semantics EDITORS L. M. de RIJK &

E. P. BOS Leiden

H. A. G. BRAAKHUIS & C.H.KNEEPKENS Nijmegen

Vol. 1: L. M. de Rijk, Anonymi auctoris franciscani Logica ,,Ad rudium" (edited from the MS Vat. lat. 946), Nijmegen 1981 Vol. 2: Ralph of Beauvais, Glose super Donatum, ed. C.H. Kneepkens, Nijmegen 1982 Vol. 3: L. M. de Rijk, Some 14th Century Tracts on the Probationes terminorum (Martin of Alnwick O.F.M., Richard Billingham, Edward Upton and others), Nijmegen 1982 Vol. 4: Johannes Buridanus, Questiones longe super Librum Perihermeneias, ed. Ria van der Lecq, Nijmegen 1983 Vol. 5: John of Holland, Four Tracts on Logic (Suppositiones, Fallacie, Obligationes, Insolubilia), ed. E. P. Bos, Nijmegen 1985 Vol. 6: Thomas Bricot, Tractatus Insolubilium, ed. E. J. Ashworth, Nijmegen 1986 Vol. 7: L. M. de Rijk, Some Earlier Parisian Tracts on Distinctiones sophismatum, Nijmegen 1988 SUPPLEMENTA to ARTISTARIUM: Vol. I: English Logic and Semantics, from the End of the Twelfth Century to the Time of Ockham and Burleigh, Nijmegen 1981 Vol. II: Mediaeval Semantics and Metaphysics. Studies dedicated to L. M. de Rijk, Nijmegen 1985 Vol. III: Logos and Pragma. Essays on the Philosophy of Language in Honour of Professor Gabriel Nuchelmans, Nijmegen 1987 Vol. IV: Ockham and Ockhamists, Nijmegen 1987 Vol. V: Peter of Spain on Composition and Negation, by Joke Spruyt, Nijmegen 1989 Vol. VI: John Buridan's Tractatus de infinito, ed. J.M.M.H. Thijssen, Nijmegen 1991

ARTISTARIUM SUPPLEMENTA ~~~~-VI~~~~-

JOHN BURIDAN'S TRACTATUS DE INFINITO

QUAESTIONES SUPER LIBROS PHYSICORUM SECUNDUM ULTIMAM LECTURAM, LIBER III, QUAESTIONES 14-19 AN EDITION WITH AN INTRODUCTION AND INDEXES by

J.M.M.H. THIJSSEN

Nijmegen lngenium Publishers

1991

ISBN 90 70419 30 0 Copyright 1991 by lngenium Publishers, P.O. Box 1342, 6501 BH Nijmegen, The Netherlands. All rights reserved. No part of this book may be reproduced or translated in any form, by print, photoprint, microfilm, microfiche or any other means without written permission from the publisher. PRINTED by KRIPS REPRO MEPPEL, THE NETHERLANDS.

PREFACE The present edition of John Buridan's Tractatus de infinito is a revision of the edition that was published as vol. 2 of my Ph.D. dissertation Johannes

Buridanus over het oneindige. Een onderzoek naar zijn theorie over het oneindige in het kader van zijn wetenschaps- en natuuifilosofie.1 The major deviations from my former edition are that in the present edition the text of the base manuscript (C) has been followed more consistently, and that all individual anomalies of witness e (the edition of Paris 1504) have not been included anymore in the critical apparatus. Furthermore, I have taken the opportunity to standardize the spelling of the Latin and to correct a few typographical errors. Besides the supervisors of my thesis Henk Braakhuis (University of Nijmegen) and John Murdoch (Harvard University) many other scholars have been generous with their advice. I wish to acknowledge a profound debt to Onno Kneepkens, who, many years ago, introduced me to the intricacies of the methods of textual edition and who tried to impress upon me his own high standards of precision; to Steve Brown, who agreed to cast his master eye over my edition; and to Monika Asztalos for a careful discussion of my methodology. I wish to thank Faye Getz for patiently functioning as a well-informed sounding-board during the writing of the introduction, and for correcting my English. Finally, I wish to express my gratitude to Jan van Kuppevelt, who at a crucial stage of the material completion of this edition helped me to overcome some serious software problems. None of the persons mentioned above is, of course, to be held responsible for any errors or shortcomings in this edition. As Andre of St. Victor (d. 1175) remarked: "Ab otiosis et in tempore otii et non a discurrentibus et perturbationis tempore sapientia discitur" . Over the course of the years, time and money for research were provided by fellowships from the Netherlands Organization for Pure Scientific Research (N.W.0.) and the Royal Netherlands Academy of Sciences (K.N.A.W.).

1 Published at Nijmegen, 1988.

TABLE OF CON1ENTS Introduction 1.

John Buridan. Life and Works

1.1. 1.2. 2. 2.1. 2.2. 2.3. 3. 3.3. 3.2. 3.3. 3.4. 3.5. 3.6.

Life

xi

Writings

xii

Tractatus de infinito Introduction

xiv

The Text

xvii

Date Composed

xx

The Establishment of the Edition Survey of the Manuscripts

xxii

Editorial Principles

xxiv

Classification of the Manuscripts

xxvi

Selection of the Manuscripts

xxxi

Conclusion

xxxiv

The Edition

xxxiv

Johannes Buridanus, Tractatus de injinito

3

Questio 14:

Utrum est aliquod corpus sensibile actu infinitum

Questio 15:

Utrum est aliqua magnitudo infinita

14

Questio 16:

Utrum linea aliqua girativa sit infinita

23

Questio 17:

Utrum omni numero est numerus maior

34

Questio 18:

Utrum in quolibet continuo infinite sunt partes

46

Questio 19:

Utrum possibile est infinitam esse magnitudinem et in infinitas partes lineam esse divisam

66

Appendices

81

Index of Passages

97

Index of Names Subject Index

99 101

INTRODUCTION

1. JOHN BURIDAN. LIFE AND WORKS 1.1. Life Little is known of Buridan's life.I From the few autobiographical references, university statutes and ecclesiastical documents it can be inferred that Buridan was born sometime around 1300 (but not later than 1304/1305) in the diocese of Arras in Artois and that he died between 1358 and 1361. In any case, the last documentary mention of Buridan occurs on 12 July, 1358, when he appended his signature to an agreement between the Picard and English nations over a controversy over their boundary. The year 1361 marks a probable date ante quem of Buridan's death, for at that date the benefice in Saint-Pol-surTernoise, which formerly belonged to him, went to a new recipient.2 Buridan would have received his early education at the college of cardinal Lemoine, founded in 1302 in the rue St. Victor. A passage in Buridan's own writings indicates that sometime after 1308 he was a socius of this college. By 1328 Buridan must already have incepted as master of arts, for at that date his name first appears, where in the documents he is mentioned as rector of the univerity of Paris. In 1340 he was rector for a second time, although note should be made of the fact that he was no longer rector when the so-called "Ockhamiststatute" of December 1340 was promulgated) Throughout his life Buridan remained a magister at the faculty of arts at Paris, where he belonged to the Picard nation. He was a secular cleric. Several documents attest to the ecclesiastical posts he received for his financial support.

1

The following biographical sketch is based on E. Faral, Jean Buridan. Maftre es arts de

I'universite de Paris (Paris, 1950) and E.A. Mocxly, "Jean Buridan"; reprinted in Studies in Medieval Philosophy, Science and Logic. Collected Papers 1933-1969. (Berkeley, 1975), 441453. The material collected by Faral and Mocxly has been critically evaluated and augmented in B. Michael, Johannes Buridan: Studien zu seinem Leben, seinen Werken und zur Rezeption

seiner Theorien im Europa des spiiten Mittelalters. 2 Teile. (Berlin, 1985). 2 See Michael, Johannes Buridan, Teil 1, 401-402 for this new biographical detail. 3 Cf. W.J. Courtenay and K. Tachau, "Ockham, Ockhamists and the English-German Nation at

Paris, 1339-1341", History of Universities 2 (1982) 53-96 for an investigation of this document and its historical context

xi

1.2. Writings Among the many writings of Buridan are textbooks on logic (Summulae de dialectica; Consequentiae; Sophismata) and a few short polemic treatises. The vast majority of his works, however, are in the form of commentaries (expositio and quaestiones ) on the principal treatises of the corpus Aristotelicum. Several treatises of the Aristotelian corpus were covered twice or even three times during Buridan's teaching career.4 Buridan's works enjoyed considerable success for over two hundred years in almost all the centers of learning in Europe. The universities of Paris, Prague, Vienna, Erfurt, Leipzig, Padua, Bologna and a number of universities of lesser stature all show traces of Buridan's influence in their doctrinal focus, as well as in the possession of copies of his works. In 14th-century Paris there even emerged something of a Buridan "school", which included Nicole Oresme, Albert of Saxony, Themo Judaei and Marsilius of Inghen. Although none of these thinkers was Buridan's student in the strict sense of the word in that they incepted under him, their dependence on Buridan occurs on almost every page they wrote. The transmission of Buridan's thought gained an extra impetus through Marsilius' connections with the university of Heidelberg and Albert's connections with the university of Vienna. Buridan's thought never effectively took root in England and Spain.5 Since the appearance of the studies of Duhem and Maier it has been commonly accepted that Buridan holds an interesting, and perhaps even important place in the history of science. Especially his explanation of projectile motion involving his impetus theory, has been extensively treated in histories of

4 For an exhaustive survey of Buridan's writings, which supersedes all earlier attempts, see Michael, Johannes Buridan, Teil 2. 5 Buridan's influence on European thought is outlined in Michael, Teil 2, 321-398. For

Buridan's impact on Central Europe see especially M. Markowski, "L'influence de Jean Buridan sur les universites d'Europe centrale", in Z. Kaluza and P. Vignaux (eds.), Preuve et raisons a

I'universite de Paris. Logique, ontologie et thiologie au XIVe siecle (Paris, 1984), 149-163. See H. Elie, "Quelques maitres de l'universite de Paris vers l'an 1500", Archives d'histoire doctrinale et litteraire du moyen dge, 18 (1951) 193-243 for Buridan's influence in France. xii

science, sometimes as a background to Galileo's theories of impeto and inertia.6 Buridan's impetus theory has even penetrated contemporary discussions about the growth of scientific knowledge.? Lately, however, research in the field of Buridan-studies seems to focus on the edition and study of his logical and semantic treatises.& Of his many writings in the field of science, only his Qzuiestiones super libris qzuittuor De caelo et mundo are available in a modern edition.9 Buridan's main work, the Questions on the Physics, which in scope is comparable to Ockham's Questions on the Physics and in size even exceeds it, has -perhaps for this last reason- remained unedited.IO The primary goal of the present edition of the Tractatus de infinito, which covers the qzuiestiones 14-19 of Book III of Buridan's Questions on the Physics is 6 Cf. W.A. Wallace, "Galileo and Scholastic Theories of Impetus", in A Maieru and A. Paravicini Bagliani (eds.), Studi sul XIV secolo in memoria di Anneliese Maier (Roma, 1981), 275-299. 7 Cf. P.K. Feyerabend, Realism, Rationalism and Scientific Method. Philosophical Papers. Vol. 1 (Cambridge, 1981), 44-97 and T.S. Kuhn, The Structure of Scientific Revolutions (Chicago, 1970), 119-120. 8 In this field the following works have been edited and/or translated: M.E. Reina, "Giovanni Buridano: 'Tractatus de Suppositionibus' ", Rivista critica di storia dellafilosofia, 12 (1957) 175-208 and 323-352; T.K. Scott, John Buridan: Sophisms on Meaning and Truth (New York, 1966); H. Hubien, Iohannis Buridani Tractatus de Consequentiis (Louvain, 1976); T.K. Scott, Johannes Buridanus: Sophismata (Stuttgart, 1977); G.E. Hughes, John Buridan On Se/fReference. Chapter Eight of Buridan' s 'Sophismata', with a Translation, an Introduction, and a philosophical Commentary (Cambridge, 1982); Johannes Buridanus, Questiones longe super librum Perihermeneias, edited by R. van der Lecq (Nijmegen, 1983); Johannes Buridanus, Quaestiones in Predicamenta ; Herausgegeben von J. Schneider (Miinchen, 1983); P. King, John Buridan' s Logic. The Treatise on Supposition, The Treatise on Consequences (Dordrecht, 1985). 9 Iohannis Buridani, Quaestiones super libris quattuor De caelo et mundo ; edited by EA. Moody (Cambridge, MA, 1941; Kraus reprint, 1970). An Italian translation of this work is provided by A. Ghisalberti, Buridano, II cielo e il mondo (Milano 1983). Also the small polemic treatise on the point (De punctis) has been edited: V. Zoubov, "Jean Buridan et les concepts du point au quatorzieme siecle", Mediaeval and Renaissance Studies, 5 (1961) 43-95. 10 Cf. Guillelmi Ockham, Brevis Summa libri Physicorwn, Swnmula Phi/osophiae Natura/is et Quaestiones in Iibros Physicorwn Aristote/is; edidit S. Brown (St. Bonaventure, N.Y. 1984). xiii

to make accessible Buridan's theory of the infinite. Moreover, the present edition can be considered as a preliminary study and presentation of the text-evidence for a critical edition of the complete Questions on the Physics of John Buridan.

2. TRACTATUS DE INFINITO 2.1. Introduction It is by now a fairly well documented fact that among the many issues in natural philosophy in the fourteenth century, those involving the infinite (and the continuous) prevailed. I I The late medievals produced an impressive amount of material dealing with one aspect or another of the infinite, usually in a manner that was not apparent in the problem as initially stated. And although there is a certain sameness about so many of these writings on infinity, this should not obscure the genuine originality of some of them. Duhem and Maier have even gone so far as to claim that some medieval thinkers developed a notion of the infinite in the sense of a Cantorian transfinite.12 In any case, the background to

Cf. P. Duhem, Etudes sur Leonard de Vinci (Paris, 1909; reprinted 1955) vol. ii, 1-55 and 368-408 and Le systeme du monde (Paris, 1956), vol. vii, 3-157. See now also P. Duhem, Medieval Cosmology, Theories of Infinity, Place, Time, Void, and the Plurality of Worlds; Edited and Translated by R. Ariew. (Chicago 1985), esp. 3-133; N. Kretzmann (ed.), Infinity 11

and Continuity in Ancient and Medieval Tlwught (Ithaca, N.Y., 1982); A. Maier, "Das Problem des Kontinuums in der Philosophie des 13. und 14. Jahrhunderts", Antonianum, 20 (1945) 331-368; A. Maier, "Diskussionen iiber das aktuell Unendliche in der erste Hiilfte des 14. Jahrhunderts", in A. Maier, Ausgehendes Mittelalter. Gesammelte Aufsiitze zur Geistesgeschichte des 14. Jahrhunderts (Roma, 1964),Vol. 1, 41-85; J.E. Murdoch, "Infinity and continuity", in N. Kretzmann, A. Kenny, J. Pinborg (eds.), The Cambridge History of Later Medieval Philosophy (Cambridge, 1982), 564-593; J.D. North, "Eternity and Infinity in Late Medieval Thought'', in G. Toraldo di Francia (ed.), L'infinito nella scienza. (Roma, 1987), 245-255; now also reprinted in J.D. North, Stars, Minds and Fate. Essays in Ancient and Medieval Cosffl()logy (London 1989), 233-243. 12 Cf. Duhem, Etudes, vol. ii, 392, Le systeme, vol. vii, 151 and Maier, "Diskussionen", 45, 55, 69 and 83. I believe that Duhem's and Maier's claims on this point are incorrect

xiv

medieval discussions of infinity was provided by Aristotle's treatment of the subject, especially in chapters 4-8 of Book ill of his Physics. 13 Taking into consideration that the Physics was the most commented upon of Aristotle's natural philosophical texts through the first half of the fourteenth century, it comes as no surprise that the commentaries on this work are an important source for our understanding of the medieval development of the notion of infinity. This is especially so, since in the commentaries on the Physics 1le theory of the infinite is explored in its most appropriate context. Here, there are no diversions provoked by deliberations of some special theological or metaphysical aspects of the infinite, such as the notion of an infinite past time involved in an eternal world, God's infinity, the "infinite values" of meritorious and demeritorious acts of the will, the infinite distance of the various species with one another and with God, and so forth. This is not to say, of course, that these theological and metaphysical settings were not stimulating and even important in furthering the analysis of the infinite, but only that the commentaries on the Physics provide a better context for comparing the medieval offspring with its Aristotelian original. Given the fact that most of our knowledge of the late-medieval discussion of infinity is still based on a study of theological writings in the broad sense of the word, and given the fact that of the fourteenth-century commentaries on the Physics only those of Ockham are available in a modem critical edition, the present edition of the Tractatus de infinito, covering the quaestiones 14-19 of Book III of Buridan's Questions on the Physics, fills a gap in scholarship.14 The designation of the quaestiones 14-19 of Book Ill of Buridan's Questions on the Physics as Tractatus de infinito is given in the manuscripts themselves. The title is a remnant of the traditional division of the eight books of Aristotle's Physics into tractatus, capitula, and in some cases even, (for example in Walter Burley's Commentary on the Physics), into partes principales and 13 Aristotle also dealt with infinity in De Caelo Book I, chapters 5-7. Book VI of the Physics is devoted to a discussion of continuity. 14 The studies of Maier and Murdoch on the late medieval theories of the infinite are mainly

based upon Commentaries on the Sentences and separate theological Questions. An exception is J.E. Murdoch, "William ofOckham and the Logic oflnfinity and Continuity", in N. Kretzmann (ed.), Infinity and Continuity, 165-207 which is based upon a study of Ockham's Commentary

on the Physics. xv

The origin of the abbreviated version of Buridan's commentary must most likely be placed at the faculty of arts of the university of Prague. The author of the accurtatio has not yet been identified, but it is higly improbable that it was Buridan himself,19 The two collections of Quaestiones longae can easily be distinguished, not only because one of them is referred to as secundum ultimam lecturam in most of the manuscripts, but also because the texts of the Quaestiones are substantially different. The differences between the two collections have been outlined by Maier in her study of Buridan's impetus-theory, and are corroborated by my own comparison of those quaestiones in both collections that deal with the problem of the infinite.20 Briefly summarized, both collections differ with respect to the titles of the quaestiones and the line of argumentation.21 The Quaestiones longae secundum ultimam lecturam are slightly longer and definitely develop Buridan's views in greater detail and with more sophistication. Neither Maier's study of Buridan's impetus-theory, nor my own study of Buridan's theory of the infinite revealed any doctrinal divergencies between the two collections of Quaestiones longae. Maier's suggestion that the Quaestiones longae secundum ultimam lecturam represent the authorized version of Buridan's course on the Physics , whereas the other collection of Quaestiones longae is a reportatio of another, earlier course

19 Cf. ibidem, 251-253. 20 For Maier's observations see A. Maier, Zwei Grundprobleme, 370-378. For a comparison of Buridan's treatment of infinity in both collections see J.M.M.H. Thijssen, Johannes

Buridanus over het oneindige. Een onderzoek naar zijn theorie over het oneindige in het kader van zijn wetenschaps- en natuurfilosofie (Nijmegen, 1988) Vol. 1, 7-71. 21 A transcription of the titles of the Quaestiones longae secundum ultimam lecturam is given in Thijssen "The Short Redaction", 240-245. The titles of both collections are transcribed in M. Markowski, "Les Quaestiones super I-VIII libros Physicorum Aristotelis de Nicolas Oresme retrouvees?", Mediaeva/ia Philosophica Polonorum, 16 (1982), 19-43 (pp.37-41). Markowski is, however, under the mistaken impression that Buridan is only the author of the Quaestiones

secundum ultimam lecturam. whereas the other collection is the Commentary on the Physics of Nicole Oresme that was considered lost. There are, however, no arguments that justify the introduction of a new author for this collection. See Thijssen "The Short Redaction", 239 n.6. Moreover, the real Commentary on the Physics of Oresme is preserved in Sevilla, Bibl. Colombina 7.6.30, ff.2r-79v.

xviii

on the Physics seems highly plausible.22 In any case, for the sake of convenience I will refer to the latter collection as the early collection or redaction of Buridan's Quaestiones longae. In Appendix 2 the reader will find a transcription of an extract of Book III, q.12 (utrum in continuo sint partes infinite ) of the early redaction, which illustrates the observed differences between this redaction and the Quaestiones longae secundum ultimam lecturam. In sum, the Quaestiones longae secundum ultimam lecturam are clearly the most important, the "ultimate" collection, so to speak, not only with regard to the date of origin, but also with regard to the quality of the content. Moreover, their historical importance is far greater if judged from the number of surviving manuscripts and printed editions, and from the fact that this collection served as the original for the Quaestiones accurtatae, which became influential at the universities of Vienna and Prague. Buridan is also considered to be the author of two different Expositiones super libros Physicorum, one preserved in Vaticana, Vat. lat. 2162 and the other preserved in Vaticana, Urb. lat. 1489. The Expositiones seem to have served as sources for the Quaestiones longae. Again basing herself on a study of Buridan's impetus-theory, Maier conjectured that the Expositio in Vat. lat. 2162 is connected with the Quaestiones longae secundum ultimam lecturam, whereas the Expositio in Urb. Lat. 1489 has stood at the base of the early collection of Quaestiones longae.23 Maier's conclusion is based on the fact that, in Book VIII of the Expositio in Urb. Lat. 1489, while discussing projectile motion Buridan refers the reader to the quaestiones in Book VII that deal with the same subject. Only the early redaction of the Quaestiones longae, however, treats projectile motion in Book VII. The supposed connection between the Expositio in Urb. Lat. 1489 thus being 22 Cf. Maier, Zwei Grundprobleme, 367 and also Markowski "Les Quaestiones ", 35. Also the brief prologue to the collection of Quaestiones secundwn ultimam /ecturam points in the direction of an authorised redaction: "Propter quod multorum de discipulis seu scolaribus meis precibus inclinatus, ego aliqua scribere presumpsi de difficultatibus libri Physicorum Aristotelis, et hanc illis scripturam conununicare, quia non possent, ut dicunt, multa in scolis audita sine alicuius scripture adiutorio memorie commendare ..... " Cf. Maier, Ausgehendes Mittelalter 1, 255. 23 Maier, Zwei Grundprobleme, 203. XIX

established, Maier linked the other Expositio, Vat. lat. 2162, to the Quaestiones longae secundum ultimam lecturam. By way of corroboration of her thesis she pointed out that in the latter Expositio there is no reference to Book VII, and in the Quaestiones longae secundum ultimam lecturam projectile motion is, as usual, discussed in Book VIII.24 Maier's observations with regard to the connection between the Expositio and the Quaestiones are confirmed by my own study of the relevant passages on infinity. Tue texts of the tractatus de infinito (Book in both Expositiones are almost identical. The Expositio in Vat. lat. 2162 provides a section of text that is lacking in the Expositio in Urb. lat.1489 at the end of Book III. The discussion on infinity in Urb. lat. 1489 closes with the remark that some difficulties still remain, which will, however, be dealt with in the Quaestiones : "et iste difficultates tractabuntur in Questionibus ".The Expositio in Vat. lat. 2162, on the other hand, also adds thirteen theses (conclusiones) "that will be demonstrated later", this "in order not to leave the reader in total ignorance". The theses reflect Buridan's position with regard to the problem of the infinite as developed in the Quaestiones longae. Appendix 1 presents a transcription of the relevant passages of both Expositiones which brings out the similarities and differences of both texts.

rrn

2.3. Date composed It is not possible to attach any precise date to the Quaestiones longae super octo libros Physicorum secundum ultimam lecturam and hence to the Tractatus de infinito contained therein. The work is probably written after 1328 and before 1357. The terminus post quern (1328) is suggested by Maier for the reason that in his Quaestiones (both collections) Buridan shows familiarity with "Bradwardine's rule".25 The date ante quern (1357) is based on evidence gathered by Faral. He drew attention to a passage in Buridan's Commentary on the Meteorologica, according to Faral written around 1357, where Buridan refers to

24 Cf. ibidem. 25 A. Maier, Die VorlaU.fer Galileis im 14. Jahrhundert (Roma, 1949), 100 n.40 and Zwei Grundprobleme, 368 n.9. "Bradwardine's Rule" is explained in the Tractatus de proportionibus, written by Thomas Bradwardine in 1328. xx

the Quaestiones super libros Physicorum (which may or may not be his Quaestiones longae secundum ultimam lecturam ). 26 Other evidence with regard to the dating of Buridan's Commentary on the Physics is provided in the manuscript Urb. lat. 1489, which preserves a copy of the Expositio which mentions the date 1350 in the colophon.27 The date cannot be the date of origin of the codex, because the codex also contains two commentaries of Blasius of Parma -fl. after 1350- written by the same copist. So, perhaps 1350 indeed is the year that Buridan gave his course on the Physics. Furthermore, the ms. Erfurt, Bibl. Ampl. F. 298, which preserves a copy of the early redaction of Buridan's Quaestiones longae is dated on 1352.28 If this date is correct, the early collection of Quaestiones originated before 1352. On the basis of the evidence gathered here, one can tentatively conclude that the Quaestiones longae super octo libros Physicorum secundum ultimam lecturam originated sometime between 1350/52 and Buridan's death, and possibly between 1350/52 and 1357. It should be noted that the dates 1350 and 1352 as termini post quos for the Quaestiones longae secundum ultimam lecturam are only meaningful if one agrees with the relative chronology of Buridan's commentaries on the Physics that have been proposed by Maier: 1) The Expositio preserved in Urb. lat. 1489; 2) the so-called early redaction of the Quaestiones longae super libros Physicorum ; 3) The Expositio preserved in Vat. lat. 2162; 4) The Quaestiones longae super libros Physicorum secundum ultimam lecturam.29

26 E. Faral, Jean Buridan: Maftre es arts, 80: 1,3 "de hoc enim dixi in tertio Physicorum, inquirendo utrum sit corpus sensibile actu infinitum..... " For the date of Buridan's Commentary

on the Meteorologica see op. cit., 87. 27 "Explicit expositio libri phisicorum lecta parisius in vico straminis a reverendo doctore et summo philosopho magistro Johanne Bridans. Anno domini 1350. De ultimo opere." Cf. also Maier, Zwei Grundprobleme, 203 n.5, who is of the opinion that the remark "de ultimo opere" is an error of the copist.

28 Cf. Faral, Jean Buridan: Maftre es arts, 36 and Markowski "Les Quaestiones", 33. See also note 6. 29 Cf. Maier, Zwei Grundprobleme, 368-370.

xxi

3. THE ESTABLISHMENT OF THE EDIDON 3.1. Survey of the manuscripts With the exception of one fragment, contained in the ms. Berlin, Staatsbibliothek cod. lat.fol. 852, ff.136ra-143vb, all known surviving copies of Buridan's Quaestiones super libros Physicorum secundum ultimam lecturam have been consulted for this edition. The fragment does not contain the text that will be edited here.30 Here follows a list of the consulted manuscripts31:

Prologue: "Bonum, ut habetur primo Ethicorum, quanto est multis cornmunius, tanto est melius et divinius. Propter quad multorum de discipulis seu scolaribus meis precibus inclinatus, ego aliqua scribere presumpsi de difficultatibus [a/ii: diffinitionibus] libri Phisicorum Aristotilis, et hanc illis scripturam cornmunicare, quia non possent, ut dicunt, multa in scolis audita sine alicuius scripture adiutorio memorie cornmendare. Super quibus ego peto et supplico de omissis et minus bene dictis obtinere veniam, de inventis autem si qua fuerint convenientia multas habere grates et bonorurn scolarium orationes." incipit : "Circa principium primi libri Phisicorum primo queritur utrum scientia naturalis sit scientia de omnibus rebus. Et arguitur primo quod non, quia nulla scientia est de casualibus et fortuitis, ut habetur secundo huius ....."

30 For a description of the fragment in the Berlin manuscript, which covers Book I middle of q.2 - Book I q.8, where the text abruptly breaks off, cf. Michael, Johannes Buridan, Teil 2, 580-581. 31 Extensive infonnation about the manuscripts that contain a copy of Buridan' s works can be found in Michael, Johannes Buridan, Teil 2 especially pp.578-594 who also refers to the relevant catalogues. It is also profitable to consult the catalogues of medieval Latin commentaries on the works of Aristotle, especially M. Markowsl apostoli sunt duodecim", quoniam sensus istius 3 5 propositionis, scilicet "omnes apostoli sunt duodecim" capiendo signum collective, est /E: f.2lra/ sensus sicut sensus istius propositionis "apostoli 85

preter quos nullus est apostolus, scilicet qui non sit aliquis istorum, sunt duodecim". Unde tantum valet dicere "omnes apostoli" loquendo collective, sicut N: f.48ra/ dicere "omnes apostoli preter quos nullus est apostolus, scilicet qui non sit aliquis istorum". Modo sunt duodecim huiusmodi; ergo duodecim 5 apostoli sunt omnes apostoli. Et ideo concludendum est quod si sit aliquod totum habens partes, non tamen habens partes infinitas, quod de omni tali toto est verum dicere "unum hoc totum est omnes sue partes", vel etiam "omnes partes huius totius sunt hoc totum" loquendo collective. Postea pono conclusionem quod "omnes" capiendo collective, nulle 1 0 partes linee B sunt omnes partes eius, et sic de omni alio toto in quo infinite sunt partes. Probo istam conclusionem, quia contradictio eius est falsa, scilicet ista: si alique partes linee B sunt omnes eius partes, et sic de omni alio toto in quo sunt infinite partes, quia si ipsa esset vera, /H: f.6lra/ tune preter illas nulle essent alie, scilicet que non essent alique illarum. Quod est falsum, quia preter 1 5 quascumque sunt adhuc alie, scilicet que sunt partes illarum. Item. In ducendo in infinitum sine instantia, nulle inveniuntur omnes, quia quotcumque fuerunt, adhuc sunt alie que sunt partes illarum. Item. Distributive omnes partes habent partes alias; igitur sunt preter quas non sunt alie. Igitur nulle sunt omnes collective. 20 Item. Manifestius demonstratur illa conclusio imaginando lineam protensam de puncto A ad punctum B et imaginaretur eius divisio in medietates proportionales incipiendo ab A procedendo versus B Tune dico quod isto modo nulle sunt omnes medietates proportionales talis linee, quia si alique essent omnes, tune ex esset totalis linea, sed hoc est impossibile. Probo, quia 2 5 illa linea est terminata vel infinita per sufficientem divisionem. Sed neutro modo potest dici. Probo primo: non posset dici quod infinita, quia non excedit lineam primam acceptam que vocetur AB, que tamen erat finita; ideo non potest dici quod illa sit infinita. Sed probo etiam quod non sit finita sive terminata, quoniam si esset terminata, tune manifeste imaginandum esset 3 0 imaginari punctum ad quod ipsa terminaretur. Quod non est verum, quoniam non terminaretur ad punctum B. Nunquam enim dividendo lineas per medietates proportionales imaginaretur ad finem eius, sed semper infinitum aliquid restat. Sed etiam dico quod non esset terminata ad punctum imaginatum infinitum B, quia quodcumque punctum infinitum B signatur, 3 5 illud pertransire per divisionem in medietates proportionales. Punctum non est /H: f.6lrb/ enim puncto, et si signetur , tune pertransibitur per divisionem in medietates proportionales; igitur illa linea ita ex illis partibus non esset terminata. igitur quod N: f.48rb/ alique medietates proportionales sunt omnes medietates . Et sit ita, 5 tune multo fortius dicendum est quod nulle partes sunt omnes partes illius linee. Sic igitur habetur quod nullum continuum est omnes sue partes, sive loquendo distributive, sive loquendo collective. Et tamen, hoc non obstante, quod continuum est sue partes, quia est due medietates, vel 1 0 tres tertie que sunt eius partes. Ex quo infero conclusiones de linea girativa circa columnam imaginaretur procedere per medietates proportionales, columne est infinita, quia nulla est infinita si non sit protracta 1 5 per omnes medietates proportionales. Sed hoc est impossibile, quia nulle sunt omnes. partes linee B sunt alique partes , quia per inductionem quecumque accipiantur sunt alique partes , et poterit esse 2 0 instantia. Ideo sequitur alia conclusio quod distributive loquendo : "per omnes proportionales medietates tenditur linea girativa". Quod patet per inductionem, quia etiam per istas duas tenditur linea girativa, et per istas et per istas centum et per mille tenditur linea girativa, 2 5 et sic semper procedendo non invenitur instantia. Sed statim sequitur alia conclusio, quod nulla linea girativa tenditur per omnes medietates proportionales, quamvis accipiamus "omnes" distributive, quoniam si aliqua tenditur per duas, alia per tres, alia per centum, alia per mille, et non alique eadem per omnes, unde differentia 3 0 est inter omnes istas propositiones /H: f.61 va/: "per omnes medietates tenditur linea" et "linea tenditur per omnes", quia in prima propositione "linea" confunditur et in secunda non confunditur. Deinde, quia multi arguunt contra infinitam divisionem continui per potentiam divinam, ideo ad solvendum rationes tales pono conclusiones. Prima 3 5 est quod omnes partes linee B non communicantes, Deus posset ab invicem separare et separatim conservare, et capio "omnes" distributive. Ista conclusio 87

patet per inductionem, quia istas duas et istas mille; , et nunquam enim inducendo invenitur instantia. Eodem modo pono istam conclusionem, quod omnes partes non communicantes possibile est esse separatas per potentiam divinam, et probatur 5 sicut precedens conclusio. Sed postea pono istam, quod omnes esse separatas est impossibile, et etiam impossibile est omnes esse separatas. Sensus enim est hec est impossibilis: "omnes sunt separate". Eodem modo dico quod impossibile est Deum omnes separare, quia hec est impossibilis: "Deus omnes N: f.48ra/ 1 0 separat". Et sic patet differentiam magnam inter compositam et divisam, quia ista divisa est vera : "omnes possibile est esse separatas", et hec composita est falsa: "possibile est omnes esse separatas". Sed restat dubitatio utrum Deus potest omnes separare vel etiam, utrum Deus potest separare omnes. Et est questio querendo /E: f.21rb/ utrum ego 1 5 possum videre omne astrum. Et non est dubium quin omne astrum tu potes videre, ut patet per inductionem. communiter dicitur quod hec est falsa: "tu potes videre omne astrum", quia impossibilis est posita inesse. Hee enim est impossibilis : "tu vides omne astrum", quia semper sunt aliqua astra in oppositum terre que tu non vides. Et quod 20 Deus non potest separare omnes, quia posita inesse /H: f.6lvb/ esset impossibilis, "Deus separat omnes". Tamen contra obicitur, quia sicut tu arguis de posita inesse, ita arguis quod hec est falsa: "omne astrum tu potes videre", quia posita inesse esset impossibilis, scilicet "omne astrum tu vides". 25 hoc solvitur communiter quod non est simile, quia predicatum appellare formam est secundum omnem formam, vel contradictionem predicati debet poni inesse. Sed quia subiectum non appellat formam, non oportet quod ponatur inesse secundum omnem eius formam vel dispositionem. hnmo, sufficit quod universalis ponatur inesse per suas singulares, ut si hec est 3 0 vera: "omne astrum tu possis videre'', tune hec est possibilis: "hoc astrum vides", et ista est possibilis: "illud astrum vides", et sic de singulis. Et non oportet quod ista sit possibilis: "omne astrum vides". Sed quamvis ista dicta probabiliter potes sustineri, tamen non est improbabilis alia via. Unde posset sustineri quod, licet propositio de possibili 3 5 sit vera, tamen non semper oportet quod illa de inesse sit possibilis si predicatum ponatur in propria forma, verbi gratia: demonstrato 88

verum est dicere "iste potest currere in tempore futuro", quia certum est quod iste potest currere et aliquando enim curreret, et non potest currere in tempore aliquo, est aliquid movere nisi in tempore. Et cum iste infans non potest currere in tempore quod est iam preteritum, nee in illo 5 tempore quod nunc est presens, quia non habet instrumenta ad hoc apta; ergo potest currere in tempore quod est futurum. istas duas et 2 5 istas tres et istas centum, et sic sine statu". Deinde de linea girativa N: f.49rb/ pono conclusiones, scilicet quod infinita linea girativa est in continuo, capiendo "infinitum" sincategorematice, quia non est tanta quin maior. dico quod nulla linea in continuo est infinita, quia nulla est omni finita maior. Unde verum est quod omni linea est 3 0 aliqua linea maior, , et hoc est propter differentiam inter suppositionem confusam et determinatam. Sed tu quereres quare nego istam: "linea est infinita", et non nego istam: "in continuo /H: f.63ra/ sunt infinita". Respondeo quod hoc est propter 3 5 suppositionem determinatam quam habet iste terminus "linea". Unde non est eadem per duas giras et per tres et per quattuor. Probatio: illa que est 91

per duas est pars illius que est per quattuor, ideo falsum est dicere quod aliqua linea est infinita. Oportet enim quod ista protensa esset per omnes medietates proportionales, et nulla est talis, ut ante dictum fuit. Sed quando ego dico quod in continuo sunt infinite partes, bene verum est quod iste terminus "continuo" 5 habet suppositionem determinatam. Sed hoc non prohibet veritatem, quia est unum et idem continuum in quo sunt due medietates et tres tertie et sic sine statu. Ideo in ipso sunt infinite. Tune respondendum est AD RATIONES .....

92

INDEX OF PASSAGES

INDEX OF NAMES

SUBJECT INDEX

INDEX OF PASSAGES ARISTOIBLES

De generatione et corruptione

De interpretatione 18 b 1-10: 38,2

I, 3, 318 a 23-25: 3,25 I, 3, 319 a 23 sqq.: 11,17 II, 10, 336 a 24-25: 3,25

Physica

Metaphysica

I, 4, 187 a 16 sqq: 9,30 III, 5, 204 b 6-8: 47,11 III, 5, 205 a 7 sqq.: 14,3 III, 7, 207 b 1-10: 40,11 IV, 3, 210 a 14-23: 8,26 IV, 12, 221 b 14-23: 47, 11 IV, 14, 222 b 21-29: 41,5 VI, 1-2: 25,17; 36,24; 48,15 VI, 1, 231 a 17-25: 69,20 VI, 1, 231 b 10-18: 63,27 VI, 10, 241 b 11-20: 11,9 VII, 4, 248 b 5-10: 44,28

V, 6, 1016 b-1017 a: 40,15; 42,10 X, 3, 1054 a 20-26: 42,9

JOHANNES BURIDANUS

Quaestiones super quattuor libris De caelo et mundo; ed. E.A. Moody (Cambridge MA, 1942) I, 17 (p.77): 5,22

Quaestiones super octo libros Physicorum secundum ultimam lecturam; ed. Venetiis 1509 (Frankfurt a.M., 1964)

De cae/o et mundo I, I, I, I, I, I, I,

1, 268 a 7-268 b 10: 4,24 3, 269 b 18-270 a 13: 5,22 5-7: 14,3 5-7 (272 b 17 sqq): 5,3 7: 10,4 9, 279 a 11: 16,17; 17,17 12, 283 b 12-13: 74,11

I, 9: 56,23 I, 12: 54,17 I, 13: 54,17 III, 19: 64,7

97

SACRA SCRIPTURA

Genesis 1-6: 14,15

Daniel 3-60: 14,16

98

INDEX OF NAMES Albert of Saxony xii, xxxii

Lemoine (cardinal) xi

Andreas de Zbunga (owner) xxxiii

Lohr, C.H. xvii n.16

Ariew, R. xiv n.11

Macken, R. xxv n.33

Aristotle xv, xxii n.31

Maier, A. xiv, xv n.14, xvii n.16, xviii

Aschbach, J. xxx n.37

n.20, xix-xxi

Blasius of Panna xxi

Maieru, A. xiii n.6

Bos, E.P. xxxiii n.45

Markowski, M. xii n.5, xviii n.21, xix

Brown, S. xiii n.10

n.22, xxi n.28, xxii n.31, xxx n.38, xxxi

Cantor, G. xiv

n.40

Caroti, S. xvi n.15

Marsili us of Inghen xii

Chrsitiannus Vrowin de Susato (copist)

Michael, B. xi nn.l, 2, xii n.4, xxii

xxxiii

nn.30, 31, xxx n.39, xxxi n.41, xxxiii nn.

Courtenay, W.J. xi n.3

45,46,47

Dittrich, F. xxix nn.35, 36

Moody, E.A. xi n.1, xiii n.9

Duhem, P. xiv

Mogenet, J. xxv n.33

Faral, E. xi n.l, xvii n.16, xxi nn.26, 28

Moreau, B. xxxii n.42

Feyerabend, P .K. xiii n. 7

Murdoch, J.E. xiv n.11, xv n.14, xvi

Fill, H. xxxii n.44

n.15

Galileo Galilei xiii

Nicole Oresme xii

Gerard of Kalkar xxxiii

North, J.D. xiv n.11

Ghisalberti, A. xiii n.9

Paravicini Bagliani, A. xiii n.6

Grandjean, M. xxxiii n.46

Pattin, A. xxii n.31, xxxiii n.46

Hubien, H. xiii n.8, xxxiii nn.45, 46

Pinborg, J. xiv n.11

Hughes, G.E. xiii n.8

Reina, M.E. xiii n.8

Johannes de Sutha (owner) xxxiii

Renouard, P. xxxii n.42

Johannes de Wulderstorf (owner) xxxiii

Roberts, R. xvi n.15

J!'lrgensen, E. xxxi n.41

Schneider, J. xiii n.8

Kaluza, Z. xii n.5

Scott, T.K. xiii n.8

Kenny, A. xiv n.11

Spirks, H. xxix nn.35, 36

King, P. xiii n.8

Tachau, K. xi n.3

Kink, R. xxx n.37

Themo Judaei xii

Kluxen, W. xvi n.15

Thijssen, J.M.M.H. xvii nn.16, 18, xviii

Kretzmann, N. xiv n.11, xv n.14

nn.19, 20, 21, xxxii n.43

Kuhn, T.S. xiii n.7

Thomas Bradwardine xx

99

Toraldo di Francia, G. xiv n.11 Unterkircher, F. xxxiii n.47 Van de Vyver, E. xxv n.33 Van der Lecq, R. xiii n.8 Vignaux, P. xii n.5 Wallace, W.A. xiii n.6 Walter Burley xv William of Ockham xiii, xv, xvi Wlodek, S. xxii n.31, xxx n.38 Z.Oubov, V. xiii n.9

100

SUBJECT INDEX Actu: infinitus, vide s.v. Infinitum.

est compositum ex indivisibilibus

-- numerate 41,19; 67,5.

25,17; 48,14; Aristoteles negat infinitam divisionem c. 68,14.

Additio 47,7.

Corpus: omne corpus est finitum 4,20;

Arithrnetica 40,8; 40,25; scientia nurnerandi 44,14; geornetria supponit

12,26; 13,18; c. est omniquaque

arithmeticam 44, 17.

extensum vel omniquaque divisibile 4,25; 10,28; c. est omniquaque

Arithrneticus 44,12; supponit unitatem esse indivisibilem 40,10; dividit

distans 10,19; est divisibile in

numeros in pares et in impares

infinitum 3,12.

40,18.

-- celestium 5, 19.

Augmentatio 62,16.

-- Christi 19-21.

Chimera 12.

-- extrinsecum 26, 10; 30, 17.

Categorematice: dictio posita a parte

-- infinitum 3,3; 10,26; 12,2; 54,16;

predicati tenetur categ. et non

63,11; 64,2; 71,9; nullum est corpus

sincategorematice 61,9; 90,34; 91,2;

sensibile actu infinitum 3-13 (q.14).

infinitum categ. sumptum vide

-- medium 6,20.

s. v.Infinitum.

-- naturalis et sensibilis 5,15.

Celum: esse alterius nature 5,22; extra c.

-- sphericum 71, 11.

sit spatium 5; 14-22 (q.15).

Desinit: vide s.v.Incipit.

Circulus: equinoctialis 39,6; 84,11; 84,16.

Distantia 8,5; 18,27; 21,12.

Columna 23-33 (q.16); 70,5; 72,23.

Divisio: continui/magnitudinis 11,14;

Commensurabilis 19-21.

34,4; 48,13; in medietates/panes

Compositio: et divisio vide s.v.Sensus

proportionales 23-33 (q.16); 86,21;

compositus.

87,1.

-- continui vide s.v. Continuum.

-- numeri 40, 14.

Compossibiles 67; 68,4; 76; 78,24; 80,7.

Divisus: vide s.v.Sensus divisus.

Connotare 41; 53,13; 57,9; 78.

Duratio 5,11; 11,5; 11,12; 11,25; 36,16;

Connotatio 53,14.

50,20.

Contiguus 20,17.

Equalis 40,20; et inequalis 38; 90,6.

Continuum: est divisibile in infinitum 36;

Equalitas: infinitorum 50,12; 85,11;

48,13; compositio continui 25,17;

medietatum 47.

34,3; 46-65 (q.18); 84-92; infinite

Etemitas: mundi videtur credidisse

sunt partes in continuo 46,3; 49,20;

Aristoteles 56,9; 73,20.

55,17; 61,6; 62,12; 63-64; nullum c.

101

-- etemaliter fuisse tempus (et mundus)

Inequalitas: partium (proportionalium)

31,19; 51,2; 52,20; 73,19; 74,6;

23,17; 85,4; vide s.v.Equalitas.

74,24.

Infinita: magnitudo: omni maior 34,25;

Exponere 27,16; 27,25; 28; 29,1-2;34,20;

non est possibile per aliquam

49,1; 54,4; 89,3; 89,33.

potentiam esse m. infinitam 73,14.

-- exponens 29,3; 60; 63,9.

-- linea 23-33 (q.16); 45,16; 86.

-- expositio 28,23; 28,29; 49,1; 54,7; 60;

--perfectioDei 17,3-5; 51,24.

61,26.

-- virtus 17,10.

-- expositum 60.

Infinitum: in actu 35,10; 46,25; 47,1-5;

Extensio: infinita 3,13; 5,12; 10,24;

64,l; in a.: repugnat quod sit aliud

11,15; 46,17; 46,20; 63,17.

maius 16,26; in a.: Aristoteles negat 25,12; 35,10; 46,26; 63,29; multis

Falsitas: in componendo conceptus simplices 12, 1.

modis dicatur 5,10; secundum

Fictio: in componendo conceptus simplices

magnitudinis extensionem: corpus

12,7.

extensum sine terminis 5,13; 49,5;

Fornla: predicatum appellatf. 77; 78,12;

sumatur categorematice et

78,18; 88,26; 91,18; subiectum non

sincategorematice: quomodo utroque

appellat f. 88,27.

modo exponatur 49,1; 89,29; infinito

Geometria: supponit arithmeticam 44, 16.

non est comparabile 51,11; infinito

hnaginatio: "secundum imaginationem"

non est maius 85, 11.

4,17; 15,13; 16,1; 18,25; 21,5; 23;

-- categorematice accipiendo 5,14; 23,3;

29,22; 39,13; 69,26; 72,24; 86,20. Immediate 27,18; 27,23; 28,l; 28,26;

29,13; 49-54; 61,23; 64,12; 90; exponeret Arist. per hoc: extensum

29,3.

sine terminis/ extensum non

Incipit: exponendo negative 27,22; 28-29;

terminatum 49,5; 89,31; proprietates

exponendo affirmative 27,25; 28-29.

huius nominis "infinitum" categ.

-- et desinit 29; 39,9; debet exponi per

sumptum 50,13; est privative huius

"primo esse" et "ultimo esse" 29,2;

nominis "finitum" 50,14; 90,6.

39,9.

-- corpus: omniquaque distans sine

Indivisibilis: instans 36,20; 40,11; 40,17;

termino 10,20.

42,13; 42,29; 45.

-- sincategorematice accipiendo 5,14; 29,7; 31,20; 34,18; 45,14; 46,2; 48,23;

-- linea 23,4; 23,19. -- punctus 23,30; 40,17; 46,16; 63,24.

51,21; 54-65; 90; diversis modis

-~

solet exponi 54,4; non tot quin

unitas 36,20; 40,11; 42,13; 42,29; 45.

Inductio 29,21; 47,9; 75,16; 87,18; 87,23;

plura/non tantum quin maius 34,20;

88,1; 88,16.

54,6; 90,1; equivalens expositio:

102

Magnitudo: corpus simpliciter 13; in

omni Best B maius 54,8; 75,21; alia

infinitum divisibilis 3,12; 11,15;

expositio: per carentiam status in rationibus numeralibus 60,24; infinita

36,24; 49-65 (q.18); non composita

esse B significat: duo esse B et tria

ex indivisibilibus 3,17; 11,5. -- infinita 14-22 (q.15); 66; 68,18; 70,24;

esse B et centum esse B et sic sine

73,14; 73,23; 75,19.

statu 60,26; proprietates huius nominis "infinitum" sine. sumptum

-- corporis Christi 19; 20; 23.

63,3; 90,9.

-- grani milii 19,8; 84,10; 84,12. Maius 45,10; 50,21; 51,12; 52,9; 85,11;

-- spatium extra celum 14,8; 21,8. -- tempus 31,16.

alio quod est eo m. secundum

lnfinitus: numerus 34-35; 45; omni maior

magnitudinem corpoream 36, 12; alie

definitiones 36.

34,23; 35,2. Instans 27 ,15; 28,14; 28,15.

Materia: non annihilatur in corruptione

Linea: composita ex punctis 46, 16; partes

11,18.

in I. 46,5; 47,18; 61,11; 62,8; 63,16;

Mathematici 44,26; vide s.v.Imaginatio.

64,20; 66,4; 67,3; 68-69; 75,7;

Minus 51,12; alio quod est eo m.

76,28; 77,14; 78,13; 78,22; 79,19;

secundum magnitudinem corpoream

protracta per omnes panes

36,12; alie definitiones 36.

proportionates 30-31; 69-70.

Minima 35,24; 36,l; 40,12.

-- girativa 23-33 (q.16); 45,16; 54,14;

-- de maximo et minimo 54, 17.

55,6; 61,27; 62,2; 63,8; 87; 91.

Motus: compositus ex indivisibilibus 11,6.

-- recta 23-33 (q.16).

-- circularis 11,2; 11,8; 84,12.

Locus: equalis locato 8, 11; extra celum

-- etemus 52,20.

non est 16, 18; extrinsice continens

-- infinitus 11,10; 31,21; infinitus velox

locatum 9,13.

53-54; 55,7; 59; 61.

-- corporis Christi 19.

Mundus: eternus 52,20; 56,8; 73,19;

-- grani milii 18,29.

73,24; 74,6; 74,23; extra m. alios

-- naturalis 8,25; 9,8.

mundos 15,13; 16,24; 17; 22,11.

Loquendo: collective 86,2; 86,8; 87 ,8.

-- de novo 73,19.

-- distributive 87,8; 87,17; 87,21; 91,23.

-- in loco 21,10.

-- materialiter 12,15.

-- vide s.v.Quantitas.

-- proprie 62,23; simpliciter et proprie

Nomina: sunt ad placitum 49,1.

36,11.

Numerus: est numeratus vel numerabilis

-- significative 12,15.

35,5; 41,l; 41,18; 47,10; est ratio

-- simpliciter 36,11; 75;1,2; s. et de

anime vel ipse res numerate 35,14; 41,19; pro rebus numerabilibus 36,5.

proprietate sermonis 44,4.

103

-- infinitus 35; 45,13; 47,10; 63,1.

Primum: et ultimum 28-29.

-- par et impar 37 ,27; 40, 18; 44; omnis n.

Proportio: 62,15; 65,4; finiti ad finitum 16,23.

est par 44,5. -- partium continui 36,9.

-- naturalis 44,28.

Omnes: sumendo collective sive

-- numeralis 37,24; 59.

distributive 31,49; 32,15; 50; 75,11;

-- vide s.v.Pars.

76,28; 77,l; 79,16; 79,24; 85-86;

Propositio: veritas p. 15,25; 43,15.

duodecim apostoli sunt omnes

-- arithmetica 40,25.

apostoli.50,6; 85,17; 85,33; aliquoti

-- compossibilis (def.) 67,15.

apostoli sunt omnes apostoli 79,25.

-- copulativa 32,7. -- de inesse 77,5; 88,35.

Omnis 60,20; 61; Sor est omnis homo

-- de possibili 66,1; 67,8; 73,25; 76-78;

61,14; 77-78.

80,6; 88,34; 89; 91,17; de p. vera:

Pars: reddat extensionem 3,13; omnis p.

debet correspondere p. de presenti

est divisibilis 25,18.

que erit vera 72,22; de p. si sit vera:

-- continui 34,15; 35,20; 52,17; 55,17; 61-63; partes: sumendo

debet poni inesse salvato predicato

coniunctim/divisim 56,24.

secundum eius totam formam 77 ,5;

-- predicati 61,9; 89,18; 90,34; 91,2;

p. universalis est possibilis cuius

91,21.

omnes singul. sunt. poss. et

-- proportionalis 23-33 (q.16); 47-48; 51;

composs. 67,15.

65.

Proprietas: sermonis 35,12; 44,4.

-- quantitativa 42,26; 84,8; 84,9.

Punctum: non est contiguum puncto

-- subiecti 61,16; 90,33; 91,21.

86,35.

-- totius: omne totum est maius sua parte

-- imaginatum 23,10; 86,30; 86,33.

37,15; 42,25; 56,24.

-- indivisibilis vide Indivisibilis.

Possibile 3,26; 66-80 (q.19); possibili

-- medium 27,13.

posito inesse nullum sequitur

-- ultimum 25,15.

impossibile 76,1; vide

Quantitas 14,9; 44,18; 45,1; 47,2; 47,23;

s. v .Propositio.

60,27; 65,11; 85,5. -- grani milii 50, 27; 84,12.

Potentia: distincta ab actu completo 67 ,6; frustra que non aliquando posset esse

-- hostie 19.

reducta ad actum 47,4; non est ad

-- mundi 50,28; 52,5.

preteritum 74,12; 75,1; vide

Ratio: conceptus anime 41.

s. v .Infinitum.

-- discretiva 42-45; 55; 59; 64,15.

-- divina 74,13; 75,24; 87,34; 88,4.

Res 12,9; 35-36; 41.

-- supematuralis 21,19.

-- extra 13,1.

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55,11; 60,11; 60,14; 60-61; 63,4;

-- vera 12,19. Sensus: virtus passiva 4,12; 18,25; 18,27; intelligi ad s. 31,24.

63,5; 78,12; 90-91; 92,4. Terminus (meta) 25-26; 29; 33,7; 40,12;

-- compositus 74,l; 79,3.

46,10; 49,5; 50,16; 70; 71,14;

-- divisus 57,10; 74,1; 76,4; 79,3; 88,10;

89,32. Theologi: dimitto determinationem dominis

91,14. Significatio 13,1; 38,11; 67,19; 76,23;

t. 17,13.

Totum 47,14; 48,11; 52,6; 65,5; maius

76,24; 78. -- distributiva 61,15; 61,22.

sua parte 37,15; 52,9; partes sunt

-- sincategorematica 61,26. Simpliciter: verificatio 53,16.

suum totum 56,24; 86,7. Ultima: medietas 25-27; 72,22; 73.

-- verum 53,18. -- vide s.v.Loquendo.

-- pars 27,21; 28; 48,11; 49,14; 49,25; 50,3.

Simul 26,22; 28,19; 28,26; 39,9; 46,12;

-- sphera 39,5; 65.

47,14; 50,1; 50,16; 57,21; 57,23;

Ultimum: punctum 25,5.

58,4; 67; 68,4; 68,7; 76,15; 76,22;

-- esse 29,9.

78; 80,7.

Ultimus: terminus 70,5.

Sincategorematice vide s.v.lnfinitum.

Vacuum 18; 22; locus non repletus 12,25.

Spatium: extra celum 14-22 (q.15).

Verificari 50, 17.

-- infinitum: occuparet totum 8,16; 11,9. Sphera 15,12; 16,21; 39,5; 71,9.

Verificatio 53,16. Virtus: sermonis 12,14; 38,15; 38,17.

Successio 11,13; 78,17; 79,22. Suppositio 39; 41,9; 43,15; 50,13; 60. -- confusa 31,13; 60; 91,31. -- determinata 31,14; 32,23; 49,8; 60,19; 62,3; 91,35; 92,5. -- distributiva 60,19; 91,31. Tempus 27-28; 39. -- eternum 74,24. -- infinitum 28,3; 31,17; 49,12; 51-53; 58,25. -- perpetuum 31,17; 85,12. Terminus (conceptus, vox) 12; compositus 12,6; fictus 12,7; (non) connotat 41,16; (non) supponit 15,26; 18,13; 41,9; 40,27; 49,8; 49,23; 53,12;

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