William Heytesbury: Medieval Logic and the Rise of Mathematical Physics

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The University of Wisconsin PUBLICATIONS IN MEDIEVAL SCIENCE

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PUBLICATIONS IN MEDIEVAL SCIENCE l The Medieval Science of Weights (Scientia de Ponderi­ bus ): Treatises Ascribed to Euclid, Archimedes, Thabit ibn Qurra, Jordanus de Nemore, and Blasius of Parma Edited by Ernest A. Moody and Marshdll Clagett 2 Thomas of Bradwardine: His "Tractatus de proportioni­ bus." Its Significance for the Development of Mathema­ tical Physics. Edited and trans lated by H. Lamar Crosby, Jr. 3 William Heytesbury: Medieval Logic and the Rise of Mathematical Ph Y.sics By Curtis Wilson 4 The Science of Mechanics in the Middle bges By Mars hall Clagett 5 Galileo Galilei On Motion and On Mechanics De Motu, translated with Introduction and Notes by I.E. Drabkin, and Le Meccaniche, translated with Introduc­ tion and Notes by Stillman Drake

William Heytesbury

WILLIAM

HEYTESBURY

Medieval Logic and the Rise of Mathematical Physics

CURTIS

WILSON

Madison, 1960

THE UNIVERSITY OF WISCONSIN PRESS

Published by The University of Wisconsin Press 430 Sterling Court Madison 6, Wisconsin Copyright © 19 56 by The Regents of the University of Wisconsin First printing, 19 56 Second printing, 1960 Printed in the United States of America By Cushing -Malloy, Inc. Ann Arbor, Michigan Library of Congress Catalog Card Number 56- 5888

Forew-ord Continuing its publication 6f critical studies and texts in the history of science, the University of Wisconsin Press now follows Dr. H. Lamar Crosby, Jr.'s edition of Thomas Bradwardine' s ProRortiones with Dr. Curtis Wilson's clear and complete investigation of William Heytesbury's Regule solvendi soRhismata . William Heytesbury, like Thomas Bradwardine, was a member of an influential group of logicians and mathema­ ticians at Merton College,Oxford, in the first half of the fourteenth century. Heytesbury is mentioned in the Mer­ ton College records of 1330, and 1338-39, and he com­ posed his Regule presumably at the College in 133 5 . He was perhaps one of those Merton Fellows who left Merton in 1340 to assist in the foundation of Queen' s College,and there seems little reason to doubt that it is this William Heytesbury who became Chancellor of Oxford University in 13 71, no doubt at an advanced age. Heytesbury's Regule represents, along with the Liber calculationum of Richard Suiseth, also at Merton College in the 1330 's and 1340 's, the high point of the Merton log­ ical and mathematical 4iscussion of physical problems. This whole discussion, in which Heyte sbury played such a prominent part,was of considerable influence through­ out Europe from about 13 50 to the earl¥ years of the six­ teenth century. On the side of kinematics-the analysis of movement in terms of distance and time-Heytesbury's treatise was par­ ticularly important and influential. We can single out a few of its achievements: (I) In it Heytesbury gave a clear­ cut definition of uniform acceleration as a movement in which equal increments of velocity are acquired in any equal periods of time. (2 ) He also gave an analysis {cer­ tainly one of the earliest) and definition of instantaneous velocity. For Heytesbury (as for Galileo later) the meas­ ure of instantaneous velocity is the space which would be vii

FOREWORD

described by a point if it were allowed to move for some given time at the velocity it had at the given instant. Among kinematic descriptions of various forms of movement we can single out finally (3 ) Heytesbury's statement of the so­ called Merton College mean speed law. This "law" (or bet­ ter, "definition" ) held that a uniform acceleration is equiv­ alent, so far as the space traversed in a given time is con­ cerned, to a uniform movement of which the velocity is equal throughout to the instantaneous velocity possessed by the uniformly accelerating body at the middle instant of time. Needless to say, this definition appears as a the­ orem in Galileo's Two New Sciences, and I think there can be little doubt that in kinematics the great Italian master was influenced, at least indirectly, by Heytesbury or his Merton contemporaries. Less influential perhaps but equally as interesting for the history of mathem atics is Heytesbury's subtle and pre­ cise usage of concepts of "limit'' and "infinite aggregate?' Dr. Wilson ' s treatment of this aspect of Heytesbury's thought is most illuminating and original. We can thank Dr. Wilson, then, for a study that exposes a difficult and important segment of medieval science and philosophy. He has brought us one step closer to an ap­ preciation of the contribution of Merton College to the de­ velopment of mathematical and physical thought. Marshall Clagett Professor in the History of Science University of Wisconsin Madison, Wisconsin August, 19 5 5

viii

Prefatory Note I wish here to express my gratitude for a Fulbright Fellow­ ship (1950-51),which provided the leisure and the access to European libraries necessary for the completion of this study. I also owe thanks to Professor Ernest Moody of Co­ lumbia University, who first set me on the track and trace of Heytesbury. I am indebted to Professor Pearl Kibre of Hunter College , New York , and to Professor Paul Oskar Kristeller of Columbia University, for helpful suggestions with regard to bibliography. My chief debt of gratitude is to Professor Lynn Thorn­ dike of Columbia University, from whose teaching and ex­ ample I have learned in the measure of my capacity what is meant by patience and exactitude in historical research. Curtis Wilson St. John's College Annapolis , Maryland May , 1955

ix

Contents Foreword

.

vii

Prefatory Note

lX

I INTROD UCTION .

3

Purpose and General Character of the Regule Date of Composition of the Regule and the Intellectual Milieu Logic Employed in the fugule Sources of the Mathematical-Descriptive Tendency in the Regule Connection between the Logical and Physicomathematical Aspects of the Regule . Relation between Heytesbury 's Metaphysical Position and the Regule . Influence of the Regule upon Later Medieval Thought II DE INCIPIT ET DESINIT . Physical Phase of the Discussion of Beginning and Ceasing . Logical Phase of the Discussion of Beginning and Ceasing Exposition of Heytesbury . III DE MAXIMO ET MINIMO .

4 6 8 18 21 24 25 29 32 38 41 57

Aristotelian Background and Commentary 59 Heytesbury ' s "De maximo et minimo" 69 Suiseth's Liber Calculationum 87 Later Discussions of Maxima and Minima 94 Scholastic Discussion of Maxima and Minima in Relation to Seventeenth-Century Physics . 112 xi

TABLE OF CONTENTS IV

V

.

115

De motu locali De augmentatione De alteratione

. . .

117 128 139

CONCLUSION .

.

148

APPENDIX

.

153

. . .

153 153 163

.

.

169

BIBLIOGRAPHY .

.

206

. . . .

206 208 212 213

.

216

DE TRIBUS PREDICAMENTIS

.

List of Abbreviations A The Sophismata of William Heytesbury B The Sophismata of Richard Kilmington NOTES

A Heytesbury's Works. B Primary Sources Cited. Secondary Works Cited. D Manuscripts Cited

.

INDEX

.

Xll

Willian1 Heytesbury

CHAPTER

I

Introduction

T

HE present study deals with the physical and math­ ematical content of the Regule solvendi so2hismata of William Heytesbury, a logician and theologian of the University of Oxford during the mid fourteenth cen ­ tury. The six chapters of the Regule, intended, as Hey­ tesbury tells us, for the use of the first-year students in logic, were widely known and frequently made the subject of commentary during a period of a century and a half, and thus may be taken as representative of certain of the interests of late medieval thought. We shall attempt to show that, in the solution of problems involving the mathematical continuum, Heytesbury employs the con­ cepts of "limit" and "infinite aggregate'' with consid­ erable precision and subtlety; and that, in treating of variable physical quantities, he approaches the ideal of a purely mathematical description. We wish also to stress the fact that the physical problems dealt with by Heyte sbury are posed secundum imaginationem, and re main without empirical application. It will be apparent that the work of Heytesbury occupies a peculiar position at the intersection of logic, mathematics, and physicsa position which we must try to characterize in relation to the history of medieval and later physical science. As introduction to our explication of the mathematical and physical content of the Regule, we shall discuss: the purpose and general character of the work; its date of composition and the contemporary intellectual milieu; the logic employed in the Regule; the sources of the 3

WILLIAM HEYTESBURY mathematic al-descriptive tendency of the Re gule; the connection between the logical and physicomathemat­ ical aspects of the Regule; the relation between Heytes­ bury's metaphysical position-namely, nominalism-and the Regule; and finally, the influence of the Regule upon later medieval thought.

P

URPOSE and General Character of the �g.uk

The subject matter of the Regule is stated by its title: it is a discussion of rules for the resolving of sophisms, i. e., real or apparent fallacies in argument. The term u sophism" was used in a rather broad sense in me dieval times. It was applied to a proposition supported by an invalid argument which appeared to be valid, or by a valid argument which for some reason appeared to be inval­ id; to a proposition supported by a valid argument of which the premisses were false although seeming to be true, or of which the premisses were true although seeming to be false; to a proposition which, on the basis of different arguments, could be as plausibly affirmed as it could be denied. Essen­ tial characteristics of a sophistical argument were its sub­ tlety, its lack of accord with common sense, its seeming to be what it was not. As will be seen further on, the sophisms treated by Heytesbury belong to a special type and tradition. The purpose of the Regule is stated in the prohemium:* it is intended as a summa of moderate length for the first-year students of logic; it is to deal with the more common sophisms, such as occur in the daily exercitatio ( period of discussion and debate accompanying the lee ture ), and is to avoid the varied and subtle inventions and opinions "which now from day to day shoot up and put forth leaves, and as soon decline:' We may suppose that the problems dealt with in the Regule had been subjected to intensive dialectical analysis over a considerable period; Heytesbury himself disclaims originality. The Regule is divided into six chapters, "in order that the eyes of readers be not fatigued by prolix and unor*Hentisbed de sensu composito et diviso, Regule solvendi !_2P.his­ � ... (Venice: Bonetus Locatellus, 1494), fol. 4va. This incunab­ ulum includes Heytesbury's major works as well as a number of commentaries on them by other authors. Whenever brief reference is possible, folio numbers of this edition are included in the text. Folio numbers in the notes, when not otherwise identified, refer to this edition.

4

INTRODUCTION

ganized narration?' Each chapter deals with a particular species of sophism; proposing rules for the solution of sophisms of this particular type, replying to objections which deny the validity of the rules, and illustrating the rules with applications to particular cases. In brief sum­ mary, the subjects of the six chapters are as follows. The first chapter, "De insolubilibus" (foll. 4va-7rb ), deals with insolubles, i.e., with propositions which, in the very asserting of what they assert, deny their own truth; thus the proposition "l am stating a falsehood;' where the term "falsehood'' refers precisely to the prop ­ osition ( "I am stating a falsehood" ) in which it occurs, is an insoluble. The problem here encountered is that of the vicious circle fallacy, which in recent times has so occupied the labors of Russell, Tarski, and Carnap.1 The second chapter, "De scire et dubitare" (foll. 12va -16va ), deals with propositions involving the terms uto know" and "to doubt:' According to the logic in vogue in Heytesbury's time, these terms are endowed with special logical properties; in particular, they are able to influ ence, within a given proposition, the mode in which the term standing for what is known or doubted refers to what it refers. The way in which a significant term may be interpreted for something, within the context of a proposition, was ·called its "supposition"; the different modes of supposition played a central role in the logica moderna of the thirteenth, fourteenth, and fifteenth cen ­ turies, and are of crucial importance for the understand ­ ing of the Re gule. They will be described farther on in this chapter. The third chapter, "De relativis" (foll. 20ra-2lvb ), treats of propositions involving relative terms, i.e., rela ­ tive or demonstrative pronouns which refer to an antece dent in the proposition in which they occur. Once more the crucial problem is that of the mode of supposition of terms: does the mode of supposition of the relative term diffear from or coincide with the mode of supposition of its antecedent? The fourth chapter, uDe incipit et,desinit" (foll. 23va27ra), treats of propositions involving the terms "to begin" and "to cease:' These terms are given a precise mathematical meaning, and are also assigned special logical properties. The central problem is that of deter ­ mining the circumstances in which a thing may be said to begin or to cease to be, "to begin" and uto cease" being understood in the previously assigned, mathematical sense. 5

WILLIAM HEYTESBURY

The fifth chapter, "De maximo et minimo" (foll. 29va3 3va), is concerned with propositions involving the terms "maximum" and "minimum:• It is essentially a treatise on the setting of boundaries to the range of variable quan­ tities of different types. The sixth chapter, "De tribus predicamentis" (foll. 37ra52rb), is concerned with propositions involving the concepts of velocity and acceleration. It is essentially a treatise on the definitions of velocity in the three categories in which, according to Aristotle, it alone properly occurs: place, quantity, and quality. Despite diversity of subject matter, the chapters of the Regule are characterized by the use of common techniques and modes of logical analysis . Sophisms of a physicomath­ ematical character are analyzed by means of the same logical devices as the sophisms of a more purely logical character. It is the last three chapters of the Regule that the present study is designed to explicate; for in these chapters the mathematical and physical content is pre­ ponderant. We note here that Heytesbury wrote a second work on sophisms, the Sophismata, which consists of lengthy anal­ yses of thirty-two sophisms (foll. 77va- l 70vb). All of these sophisms except the last two are, in primary intent, soph­ ismata logicalia rather than .§_Q_P.hismata E.h_y:sicalia. Com­ parison of the Sophismata with the Regule reveals a close relationship between the two: the Sophismata deals with particular sophisms, while the Regule sets forth the prin­ ciples commonly employed in the analysis of these soph­ isms. Surprisingly enough, the physicomathematical principles stated in the Regule play a major role in the analysis of the sophismata logicalia; on the one hand logic is used in the analysis of mathematical and physical prob­ lems, and on the other hand a kind of mathematical physics is introduced into the analysis of logical problems. Exam­ ination of the Sophismata is helpful in indicating the parentage of Heytesbury's logical concerns, and in showing the manner in which physical and mathematical principles and arguments may be used in the analysis of sophisms which, on first view, appear to be of a purely logical char­ acter.

D

ATE of C omp ositi on of the Re gule and the Intellectual Milieu

Determination of the immediate parentage of Heytesbury' s ideas and methods-the teachers under whom he studied, 6

INTRODUCTION

the contemporary authors he read-is difficult for two reasons. F irst, Heytesbury refers explicitly to none of his contemporaries or immediate predecessors, either in the Regule or in the Sophismata; the sole authorities cited in these works are the Philosopher (Aristotle) and the Commentator (Averroes). Second, although a number of fourteenth-century works can be classed with the Regule as belonging to a single mathematical-logical-sophistical tradition, none of these can be regarded as the immediate predecessor and prototype of Heytesbury's work. According to the explicit of a manuscript at Erfurt, the Regule was written in 1 3 35.2 Of Heytesbury we know that he was a fellow of Merton College, Oxford, in 13 30, and bursar (i.e., recipient of a scholarship) of that college in 1 3 3 8;3 that he is probably to be identified with the "Mr. William de Heightilbury of Merton" who was appointed one of the original fellows of Queen's College, Oxford, in 1 3 40;4 that he was Chancellor of the University of Oxford in 1371.5 The Regule thus be­ longs to the earlier years of Heytesbury's university career: a result we should naturally expect, since the teaching of logic was generally the starting point in a career of university teaching, and the Regule is a text­ book in first-year logic. The date of the Erfurt manuscript permits us to asso­ ci�te the fugule with a number of treatises stemming from the intellectual ambient of Oxford in the second quarter of the fourteenth century-works characterized by a common terminology, a common set of problems and arguments. Thomas Bradwardine's Tractatus P.rOP.Or­ tionum, written in 1328, deals with velocity of local mo­ tion as it depends mathematically on the moving power and the opposing resistance, and also with the kinematical description of motion.6 The Liber calculationum of Richard Suiseth (or Swineshead) is a lengthy and intricate discussion of the variation of real or imagined physical quantities, and of sophisms involving the variation of such quantities to infinity or zero.7 The anonymous Tractatus de sex inconvenientibus, written by an Oxonien or by some­ one who knew the works of the Oxford masters at first hand, deals with the definition of velocity in the various categories of change, and refers to Bradwardine's Trac­ tatus QLQP.Ortionum as well as to the sixth chapter of Heytesbury's Regule, on which it directly depends.8 The ponderous "Summa naturalium" of John Dumbleton (or Dulmeton) deals with many problems of variation of phys­ ical quantities from a mathematical pqint of view, and is 7

WILLIAM HEYTESBURY probably dependent on the work of Bradwardine, Heytes­ bury, and Suiseth.9 The usophismata'' of Richard Kilmington is a work largely physicomathematical in character, and shows a close relationship to the Regule of Heytesbury. 10 The similarities in the method and content of these works suggest that they are reflections or precipitations of a flourishing dialectical discussion of logical and physico­ mathematical problems, carried on in common by the masters of Oxford. 11 Thus the author of the Tractatus de sex inconvenientibus can speak of certain opinions as being held by the whole school of Oxford. 12 None of the above -cited works, however, can be regarded as an immediate predecessor or source of Heytesbury's Regule. If we were to attempt to arrange these works in their probable historical order, it would be necessary to place the fug� second in the list, directly after the .T.!:.fil tatus 2.!..Q_portionum of Bradwardine. The analysis of par­ ticular problems in the treatises of Suiseth, Kilmington, and Dumbleton shows an increased complexity, a multipli ­ cation of arguments and variations on theme, as compared with the treatment of the same problems in the. Regule; and and we must therefore place them at a later stage-histor ical or logical-in the development of the discussion of these problems. The Tractatus de sex inconvenientibus actually refers to the Regule, and depends upon it as a source. On the other hand, the Tractatus QLQ.portionum of Bradwardine deals but briefly with the kinematical problems which are the subject of the sixth chapter of the Regule, and not at all with the physical problems of the fourth and fifth chapters. Heytesbury apparently knew and accepted the conclusions arrived at in Bradwardine's treatise, but they play only a peripheral role in his analysis. 13 The earlier sources of the methods and problems of the Regule are to be sought in two different directions: in treatises on logic in the tradition of the !,Q_gica moderna, and in certain thirteenth-century developments towards a mathematization of Aristotelian problems. TOGIC Employed

L

in the Regule

The logic of the Regule is the 109,ica moderna, a set of devices and distinctions developed in the Faculty of Arts at the University of Paris during the early thirteenth cen ­ tury, in the wake of the dialectic of Abelard. This logic is chiefly concerned with the properties of terms,

8

INTRODUCTION

and in particular with the proper ty known as supP.osition. One of the earliest treatises in the tradition of the logica moderna was the " Introductiones in logicam,, of William of Shyreswood (d. 1249). 14 In succeeding centuries the logica moderna was to be identified with the " Parva log ­ icalia" or seventh treatise of the Summulae logicales of Peter of Spain (d. 1277) , who seems to have been a pupil of William of Shyr eswood. Prior to Heytesbury ' s time , commentaries on the Summulae were written by Robert Kilwardby , who studied at Paris and taught at Oxford (d. 1279) , and by Simon of Faversham , a professor at Paris and Oxford (d. 1 3 06 ). 1 5 It is probable that , when Heyte s bury taught logic at Merton College , the logica rrioderna formed part of the standard curriculum of first ­ year logic. In any case, a large number of the devices and distinctions of the Regule are those of William of Shyre swood and P eter of Spain. We are here concerned with the logica moderna only insofar as it enters into Heytesbury 's analysis of mathe ­ matical and physical problems , and may thus limit our discussion of it to three topics: the doctrine of supposition , the doctrine of exponible terms , and the distinction be tween composite and divisive sense. TH E

DO C TR I N E

OF

S U P P O S I T I ON

The doctrine of supposition is the center point and ful ­ crum of the 12..gica moderna. Supposition is defined as that function of a substantive term , within the context of a proposition , by virtue of which it may be interpreted for some thing or things. 1 6 Supposition differs from , and is logically posterior to , signification or meaning. Significa ­ tion is a property of a vocal sound or written mark , where by it is instituted to designate something other than itself , and so becomes capable of being employed as a term in discourse. Supposition , on the contrary , is a property of an already significant term , as being interpretable , within a given proposition , for some individual or individuals. Three major types of supposition were commonly distin ­ guished: material supP.osition , in which a term is inter preted for itself , as the term " man" in the proposition " Man is a noun" ; simP.le supposition , in which a term is taken for a universal , as the term "man" in the proposition " Man is a species"; 1 7 and personal supposition, in which a term is interpreted for the real individual or individuals to whic h it is able to refer by virtue of its signification or meaning , as in the proposition u M an runs; ' where " man"

9

WILLIAM HEYTESBURY may be taken for Plato, Soc r ates, C icer o, and so on . We ar e her e concer ned only with P.er sonal supposition . Per sonal supposition admits of a number of successive subdivisions . F IRST . - It may be either disc r ete o r common. Discr ete supposition is the acceptation of a pr oper noun in the sin ­ gular, or of a common noun modified by a demonstrative adj ective in the singular , for the single individual r efer r ed to; such is the supposition of the ter m "Soc rates" in the pr oposition "Socr ates r uns ;• or of the ter m "that man" in the pr oposition "That man disputes:• If a common noun 1s not modified by a demonstrative adjective of singular number, its supposition is common . SE C ON D. - C ommon supposition, in tu rn, may be of two kinds, deter minate or confused . Deter minate supP.osition occur s when the acceptation of a common noun is such that it is per missible to descend to all the individuals for which it stands disjunctively; for example, the supposition of the term uman,, in "M an runs" or "Some man r uns" is deter minate, fo r either of these pr opositions implies: " This man r uns o r this man r uns or this man r uns or . . . and so on, ever y existing man being r efer r ed to in one of the disjuncts. Thus for the truth of a p roposition the sub ­ ject of which has determinate supposition it is necessar y and sufficient that ther e be some deter minate individual or individuals for whom the pr oposition is verified. C on­ fused supP.osition occur s when the acceptation of a com­ mon noun is such that it is not per missible to descend disjunctively to the individuals for which it stands. F or example, the supposition of the ter m " man" in the prop ­ osition u Ever y man is an animal,, is confused; for the pr oposition is tr ue only if it can be ver ified in the case of eve r y man, and not mer ely in the case of this man or that man . T HIRD .- C onfused supposition , again, is of two types , distr ibutive and nondistr ibutive or confused only (confusa tantum ) . A term has distr ibutive supposition when it is permissible to descend conjunctively to the indiviciuals for which it stands . Thus the ter m "man" in "Ever y man r uns" has distr ibutive supposition, and the pr oposition means: u This man r uns and this man r uns and this man r uns and .. . . ;• and so on for ever y man. A ter m has non­ distributive supposition �PP.O sitio confusa tantum ) when it is not permissible to descend conjunctively to the indi ­ viduals for which it stands . In some cases of nondistr ibu­ tive supposition it is per missible to descend by a dis10

INTRODU C T I ON

junction of the pre dicate to the individuals for which the common term stands , i. e . , by me ans of the word "or" place d be twe en the individual discrete terms include d in the reference of the common term pre dicate d. For ex­ ample , "Eve ry man is an animal" implie s "Eve ry man is this animal or this animal or this animal or . . . ;• etc. In other case s of nondistributive supposit ion it is not permissible to de sce nd in any manner whate ver to the in­ dividuals denoted by the common term , which is the n said to stand immobiliter. We observe that the supposition of a term de pe nds on the type of proposition in which it occurs , and may be in­ flue nce d by the pre sence or abse nce of other te rms which have the power to alte r , in a particular way , the supposi ­ tion of the term with which they occur . 1 8 In e spe cial , cer­ tain terms have a force o f confounding (vim confunde ndi) the term which follows , i. e . , of causing its supposition to become confuse d; such , for example , is the effe ct of the quantifie rs "e very" and "any" on the te rms which they modify. TH E

DO C TR IN E

OF

E X P ON I B L E

TERM S

A chapte r "De e xponibilibus" is include d in the se v e nth tre atise of the Summulae !,Q_gicale s of Peter of Spain. 1 9 An e xponible proposition is the re defined as a proposition which has an obscure se nse re quiring e xposition , owing to its inclusion of a syncategore matic term or of a term which implicitly involve s a syncate gore matic term. A syn ­ cate gorematic term is one which taken by itse lf has no signification; such are pre positions , adverbs , conjunctions , and the quantifiers "all'' and "some :• A term which implic­ itly involv es a syncate gorematic term is called e xponible. Among the propositions cite d as e xponible in the tre a ­ tise "De e xponibilibus" are exce ptiv es (e.g. , " All teache rs e xcept Socrate s rece ive remuneration" ) , e xclusive s (e . g. , "Only man is capable of smiling" ), re duplicatives (e . g. , "Socrate s insofar as he is a man is an animal" ) , compar isons (e . g. , " You are as strong as any man in the world" ) , and propositions including the exponible terms inciQti (begin) , de sinit (ce ase), and infinitum. Thus the proposition " Only man is capable of smiling" is e xpounde d as follows : man is capable of smiling , and nothing that is not man is capable of smiling. The expositions of such e xponible terms as inciP.it , de sinit , maximum , minimum , and infini­ tum will be give n furthe r on as they be come nece ssary for our e xplication of He yte sbury 's Re gule.

11

WILLIAM HEYTESBUR Y Autho rs later than Peter of Spain (e. g., Richar d B illing­ ham2 0 ) contr asted exponible pr opositions with r esoluble propositions. Whe reas the exposition of an exponible pr op­ osition in vo 1 ves a fuller explanation of meaning, the r e so lution of a r esoluble pr oposition involves a descent fr om a common term to a discr ete ter m, to that which can be demonstr ated by pointing. Indefinite and par ticular pr op­ ositions (i.e. , propositions in which the subject is a com­ mon noun unmodified by a quantifier or modified by the quantifier " some" ) ar e resoluble . Thus the pr oposition " Man smiles" would be r esolved as follows: this smiles, and this is a man. Accor ding to the r ule adopted by most logicians, a pr oposition was to be proved by means of the fir st mediate or pr ovable ter m occur r ing in the sentence which states it; hence a fair ly st r ict distinction could be dr awn between resoluble and exponible pr opositions, on the basis of whether the fir st provable ter m or ph r ase in a pr oposition was resoluble or exponible. 2 1 This distinc­ tion plays a consider able r ole in commentar ies on Heyte sbur y ' s Regule and Sophismata. T HE

DIS T IN C T IO N

BETWEEN

COMPOSITE

A N D D I V I S I V E SEN SE This distinction occurs in the De sophisticis elenchis of Ar istotle. 22 Accor ding to Aristotle, composition and division ar e two of the six kinds of fallacy which ar ise within diction, that is, from the way in which we say what we say. The fallacy of composition occurs when ter ms which should be under stood sepa r ately fr om one another are taken as conjoined. For example, a fallacy of compo sition occur s if the pr oposition " It is possible that the sitting should walk" is under stood to mean that one who is sitting can walk while he is sitting; yet the same pr op­ osition can be true in a divisive sense, that is, when under stood to mean that one who is sitting has a poten­ tiality for walking, or can in fact walk after he is no longer sitting. The fallacy of division occur s when ter ms which should be under stood as conjoined are taken sepa­ r ately . For instance, a fallacy of division occur s if the pr oposition " Five ar e two and thr ee" is under stood to mean that five ar e two, and also that five ar e th r ee ; but the same pr oposition is true in a composite sense, that is, when under stood to mean that five ar e two and thr ee added together . Fallacies of composition and division wer e discussed by thir teenth-centu ry commentator s without mar ked devia -

12

INTRODUCTION

tion from the Ari s tote lian analys i s or e xample s . Thus Albe rtus Magnus in hi s "Comme ntum s upe r duo s libros e le nchorum; ' 2 3 Thomas Aquinas in hi s OP. us culum fallaci ­ arum ,2 4 Dun s Scotu s in hi s ----Que s tione -s ------e le nchorum ,2 5 and ---P e te r of Spai n in the s ixth tre ati s e of hi s Summulae !Qgicale s ( " De fallaci i s" ). P e te r of Spain di s tingui she s two mode s e ach of fallacy of compos ition and fallacy of divi s ion , and Albe rtu s Magnus and Thomas Aquinas di s tingui s h thre e . Late r on , i n the fourte e nth ce ntury , Buridan di s tingui s he s s ix mode s e ach of fallacy of compo s ition and divi s ion , but again without de cide d de parture from the Ari s tote lian anal ­ y s i s .2 6 The di s tinction be twe e n compos ite and divi s ive s e n s e as u s e d in He yte s bury 's Re gule and Sophi s mata i s a doc ­ trine which ha s be e n e xpande d con s ide rably be yond its original Ari s totelian form; s pe cifically , it is a doctrine which has be e n re inte rpre te d in the light of the logica mode rna , and in which propos ition s in compos ite and divi ­ s ive s e n s e are di s tingui s he d ve rbally by s e nte nce orde r and grammatical form. Thi s mode of re inte rpre tation i s to be obs e rve d i n a s hort tre ati s e , "De s e n s u compos ito e t divi s o;' of Richard Billingham.2 7 But our primary s ource for the e xpande d te aching i s a tre ati s e by He yte s bury him ­ s e lf, als o e ntitle d De s e n s u compos ito e t divi so (foll. 2ra4rb). Thi s tre ati s e was wide ly known in fifte e nth-ce ntury Italy, and s e ve ral time s made the s ubj e ct of a comme ntary. 28 He yte s bury di s tingui s he s e ight mode s in which fallacie s of compos ition and divi s ion may occur. According to late r comme ntator s , the phras e ucompos ite and divi s ive s e nse" a s applie d i n the s e e ight diffe re nt mode s i s e quivocal , and no s in � le de finition of i t , valid for all the mode s , can be give n. 9 FIRST.-The fir s t mode of compos ite and divi s ive s e ns e i s caus e d by me an s of the modal te rms ve rum , fal s um , P.O s s ibile , imP. os s ibile , ne ce s s arium , continge ns , or s ome corre s ponding ve rb form s uch as 2ote s t , ,2P.Orte t , contin ­ git. 3 0 A modal te rm i s one which may modify a propo s ition as a whole , s o that it no longe r s tate s a s imple inhe re nce of the pre dicate in the s ubje ct , but rathe r a modifie d kind of inhe re nce , s uch as pos s ible or ne ce s s ary inhe re nce . We are mainly conce rne d he re with the modal P.:Q S s ibile . According to He yte s bury , compos ite s e ns e i s caus e d by a modal te rm whe n the latte r totally pre ce de s the re s t of the propo s i tion in which it occurs ; divi s ive s e ns e i s caus e d whe n the modal te rm me diate s be twe e n the s ubj e ct and the pre dicate of the propos ition. In the firs t cas e the modal 13

W I LL I AM HEYTESBURY ter m is used imper sonal ly ; in the second case it is used per sonal ly. Thus the statements , "It is possib le that white should be black" ( Possible est quod album sit nigr um" ) , "It is possible that you should tr aver se this space" ( "Pos ­ sible est quod tu per tr anseas hoc spacium" ) have a com ­ posite sense ; while the pr opositions , "White can be black" ( "Al bum potest esse nigr um" ) , " You can t r aver se this space" ( " Tu potes per transire hoc spacium" ) , have a div ­ isive sense. The statements with composite sense signify the possibility of an instantaneous identity; possibility is pr edicated of the composite "White is black" or "You tr a ­ ver se this space ;' and hence , in order that the statements should be tr ue, it would be necessar y th at the white thing be black in the same instant in which it is white, and that you tr aver se this space in an instant . The statements with divisive sense , on the cont r ar y , signify the inher ence of a cer tain potentiality in the subject of the pr oposition- a potentiality which may be r ealized in a succession of diver se par ts of time: you have a potentiality for tr aver sing this space ( you can do so only ove r a per iod of time) , and what is white has a potentiality for being black ( it is pos sible for it to be b lack only after it has ceased to be white). It is clear that the above statements with composite sense are false , while those with divisive sense are tr ue.3 1 S EC O N D . - The second type of composite and divisive sense is caused by means of terms which have a vim con ­ fundendi a common noun which fol lows , so that its sup ­ position is r ender ed nondistr ibutive or confusa tantum . Among the ter ms listed by Heytesbur y as having such a vim confundendi ar e the ver bs " to begin" ( incipio) , "to desir e" ( desider o , cupio) , "to owe" ( debeo) , and the adverbs "always" ( semper) , "eter nally" ( eter naliter) , and "immediately" (immediate) . When in a proposition one of these ter ms pr ecedes a common noun , the supposition of the latter is ther eby r ender ed confusa tantum , and the pr op ­ osition is said to have a composite sense. When , on the contr ar y , the term having a vim confundendi does not pr ecede the common noun , the supposition of the latter r emains deter minate or distr ibutive , and the pr oposition then has a divisive sense . Thus the proposition "Always some man wil l be" has a composite sense; while the pr oposition "Some man wil l b e always" has a divisive sense. In the fir st pr oposition the ter m "some man" is pr eceded by the ter m "always ;' and ther efore has supposition confusa tantum , so that it is not per missible to descend disj unctively to the individ -

14

I N TRO DU C TIO N

uals for which it stands. In the second proposition the term usome man" is not preceded by "always ;' and its supposition therefore remains determinate , so that some particular individual is referred to. The first proposition asserts the immortality of men in the aggregate; the second asser.ts the immortality of some particular man. Similarly , the proposition "Body a begins to touch some point of body b" has a composite sense; while the prop ­ osition usome point of body b body a begins to touch" has a divisive sense. In the first proposition the term "some point of body b" is preceded by the term "begins ;' and therefore has supposition confusa tantum; in the second proposition the term "some point of body b" is not thus preceded by the term "begins , " and its supposition is therefore determinate. Such a distinction may play a crucial role in cases where a term which ordinarily has determinate supposition (e.g. , usome point" or "some in ­ stant" ) refers to members of an infinite aggre � ate, the first member of which may not be assignable. 2 THI RD : - The third type of composite and divisive sense is caused by means of relative terms, that is , pronouns (qui , que , quod , ille , illa , illud) which refer to some ante cedent term in the proposition in which they occur. Com ­ posite sense is produced when the relative introduces a dependent clause; divisive sense when it introduces an independent clause. For instance , the proposition uimme diately after the present instant there will be some instant which immediately after the present instant will be" ( "Im ­ mediate post hoc erit instans quod immediate post hoc erit" ) has a composite sense, for the clause "which imme diately after the present instant will be" is dependent. But the proposition "Immediately after the present instant there will be some instant , and it immediately after the present instant will be" ( "Immediate post hoc erit instans , et illud immediate post hoc erit" ) has a divisive sense , for the clause "it immediately after the present instant will be" is independent. In the first proposition the term "some instant" has supposition confusa tantum because it is pre ­ ceded by the term "immediately ;' which has a vim confun ­ dendi; and the relative pronoun "which" that introduces the dependent relative clause has the same type of supposition as its antecedent. In the second proposition , the supposition of the pronoun "it" that introduces the second clause is no longer influenced by the supposition of its antecedent , and is therefore determinate, referring to some particular in ­ stant. Since the instants following the present instant con 15

WILLIAM HEY T ESBUR Y

stitute an in finite aggre gate of whic h the first m e mb e r i s not assignab le , the first proposition is true , while the se c ­ ond is false . 3 3 (This unde rstanding of the nature of instants in time will be analy z e d at gre ate r le ngth in C hapter 2 . ) FO U R TH . - The fourth kind of c omposite and divi siv e se nse is c aused by means of the terms " infinite" (in finitus, infinita, infinitum) and " whole" or "all" (totus, tota, totum), whi c h c an hav e e ither a c ate gore matic or a s ync ategore mati c se nse . " Infin ite" taken in a c ategor e matic sense re fers to an entity without b e ginning or e nd, or in Aristotle ' s terms, to an in finite in ac t . " Infinite" take n in a syn c ate gor e mati c se nse is an e xponib le te rm, and me ans "not so muc h b ut that more ;• or u not so muc h b ut that twic e as muc h, and thre e time s as muc h, and four time s as muc h, and so on in infini ­ tum :' " Whole" or " all" taken in a categore mati c se nse me ans a c omple te e ntity, forme d of the sum of all its parts; taken in a sync ate gore matic se nse it me ans the sam e as "any part :• Propositions in whi c h the se terms are take n c at e gore mati c ally are said to have a c omposite se nse , and propositions in whi c h the y are taken sync ate gore mati c ally are said to have a divisive se nse . Ac c ording to He yte sb ury, the terms " infinite" and " whole" have a sync ate gore mati c sense when the y ar e plac e d at the very start of a se ntenc e ; the y have a c at e gor e matic se nse when the y are pre c e ded by some term whi c h the y modi fy, or whe n the y are plac e d in the pr e di c ate of the se ntenc e . For instanc e, the proposition " In in finitum this c ont inuum c an b e divide d" ( u rn infinitum pote st hoc c ontinuum e sse divi ­ sum" ) has a divisive sense , and m e ans that this c ontinuum c an be divide d into halv e s, thirds, fourths, and so on in infinitum, the pote ntiality for further division n e ve r b e ing e xhauste d . But the propo sition " This continuum c an b e di ­ vide d in infinitum" ( " Hoc c ontinuum pote st e sse di vi sum in infinitum" ) has a c omposite se nse , and me ans that it i s possible for this c ontinuum to b e ac tually divide d- in some instant- into an infinite numb e r of parts. 3 4 FIF TH . - The fi fth kind of c omposite and divisive se nse is produc e d b y me ans of the c onjunc tion "and" ; c omposite sense is produc ed when " and" c onne c ts terms w hi c h are to b e unde rstood c olle c tive ly, and divisive sense when " and" c onn e c ts terms whic h are to b e unde rstood divi ­ sively . Te rms c onn e c te d by " and" are to b e unde rstood c olle c tively whe n it is not p e rmi ssib le to r e solve the prop ­ osition in whic h the y oc c ur into a c onjun c tive p roposi ­ tion ; for instanc e , fr om the p roposition " F iv e a r e two and thr e e" it is not pe rmissible to infe r " Fi ve ar e two and five a r e thre e !' But whe n suc h r e solution is p e r missible ,

16

INTRODUCTION

the terms c onjoined by u and" are to be unders tood divi ­ s ively; for example, in the propos ition "Soc rates and Plato are at Plataea; ' being at Plataea i s predic ated i ndividually of Soc rates and Plato, and it i s permis ­ s ible to infer u soc rates i s at Plataea and Plato i s at Plataea: ' 3 5 SIXTH.- The s i xth type of compos i te and divis ive s ens e i s produc ed by mean s of the di sj unction "or" ; compos ite s en s e i s produced when "or" connec t s term s whi c h are to be under s tood collectively, and divi s ive sense when " or" connect s terms which are to be unders tood divi s ively. Term s c onnected by "or" are to be under s tood c ollec tively when it i s not permi s s ible to res olve the propos ition in whi c h they oc cur into a di s junctive propos ition; for in ­ s tance , from the propos ition "Every integer i s odd or even ;• it is not permis s ible to infer "Every integer i s odd or every integer i s even :• When s uch res olution i s per ­ mi s s ible , however , the term s joined by "or" are to be taken divi s ively; thus the propos ition "Soc rates or Plato runs" ha s divis ive sen s e , and it is permi s s ible to c onc lude: "Socrate s run s or Plato r un s :• 3 6 SEV ENTH: - Compos ite s en s e in the s eventh mode i s pro ­ duced by mean s of the expres s ion s "it was thus that . . . ;• "it will be thus that . . . ;• followed by a propo s ition in the pres ent tens e; the c orres ponding divi s ive sen s e i s pro ­ duced when the propo s i tion i s taken by its elf , its princ ipal verb being in the pas t or future ten s e. For instanc e , the s tatement "It will be thu s that Soc rates i s of s uc h a s ize as Plato i s" ( " Ita erit quod Sorte s e s t tantu s quantu s est Plato" ) ha s c ompos ite s en s e; the c orres ponding s tatement in divi s ive s en s e i s "Soc rates will be of s uc h a s ize as Plato will be" ( "Sortes erit tantu s quantu s erit Plato" ). The s tatement in compo s ite s en s e requires an ins tantane ­ ou s verifi c ation , while that in divi sive s ens e does not. Henc e it may be the c as e that at s ome future ins tant Soc rate s will be of s uch s ize as Plato will be at the same ins tant , so that the s tatement in c ompo s ite s en s e is true; and in the s ame c ase it i s als o po s s ible that , taking future time as a whole into acc ount, Plato will b e of a greater s ize than Soc rates will ever be , s o that the s tatement in divi s ive s en s e is fals e. 3 7 EIGHTH.- The eighth type of c ompos ite and divi s ive s en s e i s caus ed by means of verbs s ignifying an ac t of the intellect or will , s uc h as "I know" ( s cio) , "I unders tand" (intelli gQ), "I doubt" (dubito) , "I perc eive" (perc ipio), "I im ­ agine1• (imaginor) , "I wis h" (volo). These term s behave much in the manner of the modals treated in connection 17

WILLIAM HEYTESB U RY with the fir st type of composite and divisive sense. As they do not enter into Heytesbur y ' s physical and mathematical speculations, we omit consider ation of them her e. 3 8 Heytesbur y ' s distinction s as to wo rd or der and gr am­ matical fo rm will sometimes appear ar tificial o r over ly meticulous; but they a re based on modes of ever yday speech. In defense of them we may refer to the Mad Hat­ ter' s r emar k to Alice : Meaning what you say is not at all the same as saying what you mean .

S

O U R C ES o f the Mathe mati c al - De s c r iptive T e nd e nc y in the R e g ule

Points of or igin for each of the thr ee maj or problems discussed in Chapter s 4 - 6 of the Regule ar e to be found in passages of the PhY. sica and De caelo of Ar istotle . In each case, the essential step fr om Ar istotle to Heytesbur y is that of the mathematization of a physical pr oblem which Ar istotle had left with a passing r ema r k . The Ar istotelian sour ces of the pr oblems of the Regule will be discussed in connection with the exposition of the individual chapter s . The sources of the mathematical ­ descr iptive tendency of the Regule, however , de ser ve pr eliminary attention . In par ticular , we are concer ned with the use of the terms latitudo and gr adus in the attempt to descr ibe mathematic ally the spatial and temporal var iation of a quality or motion . Accor ding to the C ategor ies,39 qualities- or r ather some qualities-admit of more and less; that is, alter ation or change in the categor y of quality may not only be f r om one quality to another , as fr om black to white, but also fr om one intensity of a given quality to another intensity of the same quality, as fr om less white to mor e white. The ontological char acter of v ar iation in intensity of a quality was tr eated by Porphyr y and Simplicius, and in the thirteenth centu r y was the subject of extensive discus si6 n. This development has been traced by Maier; 4 0 we summarize her e the major aspects, for their b ear ing on the pr oblem of a quantitative desc r iption of qualitative change. An or iginal impulse to this discussion was pr ovided by the contr adiction between the obser ved fact of the var iation in intensity of qualities like heat, and the basic principle of the immutability of forms. " Forma est simplici et in­ var iabili essentia consistens." 4 1 Substantial for ms- for example the fo rm of man, that which makes a man what he is- wer e definitely denied the capacity to intend and 18

INTRODUCTION

r e m it .4 2 C an the ob s e r v e d fac t o f v a r i ation in int e n s ity o f ac c i de nt al fo r m s , i . e . , qualiti e s , b e r e c onc ile d w i t h t h e p r inc i p le o f i mmut ab i lity ? Ac c o r di n g to one vi e w , the ob s e r v e d v a r i ation in int e n ­ s ity i s due to g r e ate r o r le s s par tic i pation o f the s ub j e c t o r b o dy i n the qual ity , the latte r b e i ng thu s ab le to r e main inv a r i ab le in it s e l f . T hi s v i e w , s ug g e s te d a s a po s s ibi lity in the C ate go r i e s , wa s g e ne r ally h e ld to b e A r i s totle' s ; it w a s s up p o r t e d with v a r io u s mo dific ati o n s by P o r phy r y , S i m p l i c i u s , B o e thi u s , B on av e ntur a , T ho m a s Aquina s , and E g i diu s R o m anu s . 4 3 T he o p p o s ite vi e w , ac c o r di n g to whi c h int e n s io n and r e mi s s i on ar e to be s oug ht in the quality r athe r th an in the p a r ti c ip ating s ub j e c t , appe ar s in v a r io u s dr e s s . Ar c hyt a s , in c l a s s i c a l time s , h ad he ld that the r e i s a c e r t ain inde t e r min anc y wi thin the qualitie s the m s e l v e s , ac c o r ding to whic h the y admit o f mo r e an d le s s . In the t h i r te e nth c e nt ur y thi s ide a i s t a ke n up ag ain by He nr y o f G h e nt (d . 1 2 9 3 ) , w ho int r o duc e s t h e te r m latitudo t o de s i gn ate the r an g e in whic h the int e n s ity o f the qua lity may v a r y .44 Inte n s i on o f a qual i ty , ac c o r ding to He nr y , c on s i s t s i n app r o ac h t o a c e r t ain te r minu s , i n whi c h the qual ity ac hie v e s it s fu ll p e r fe c tion . 45 C e r t ain autho r s ar gue d that inc r e a s e in int e n s ity c o me s ab o ut thr o u g h addition o f e s s e nti al par t s o f the· quality . T h u s P e t r u s J o hanne s O l i v i ( d . 1 2 9 8 ) . 46 Ac c o r di n g to He r v a e u s N at al i s ,4 7 inc r e a s e in inte n s i ty d o e s no t o c c u r b y addit ion o f one de g r e e o f int e n s ity to ano the r , in the w ay , fo r e x amp le that one c an add s tone s to g e the r to fo r m a pi l e ; b ut the mor e inte n s e fo r m c on ­ tain s the mo r e r e mi s s fo r m v i r tually , a s the s e n s it i v e s o ul c o nt ain s the v e g e t ati v e s o ul .48 A c c o r ding to G ottfr ie d of F ontaine s (d . 1 3 0 3 ) , inte n s i ­ b i l ity b e l o n g s to a qua li ty not b y r e a s o n o f it s qui ddity o r e s s e n c e , b ut in s o f a r a s it e xi s t s i n a n in dividual s ub j e c t . T h e qual ity a s s uc h , i ri ab s t r ac to , d o e s no t admit o f in ­ t e n s io n and r e mi s s io n ; i t do e s s o only in c on c r e to , a s fo r m c on t r ac t e d t o the ind i v i dua l . T hi s vi e w w a s adopte d b y Dun s S c o t u s an d hi s fo l lo w e r s . T hu s fo r the S c ot i s t s it w a s po s ­ s ib l e t o hold that int e n s ion and r e m i s s ion b e l o n g to the qual ity n o t e s s e nti ally but ac c i de ntal ly , as the qu ality e xi s t s in a g i v e n s ubj e c t o r s upP.o s ito . T he Sc o ti s t s al s o he ld t h at inc r e a s e in int e n s ity o c c ur s b y additi on ; the p r ope r an alo gy, ac c o r din g t o Sc otu s , i s the addit i o n not o f s tone t o s tone b ut o f w at e r to w at e r ; the new indiv i dual qual i ty fo r m e d b y s uc h addition c ont ain s the p r e v io u s one .49 Y e t ano th e r v ie w , s up p o r te d b y W a lt e r B ur l e y (d . afte r

19

WILLIAM HEYTESBURY 1 3 4 3 ) and in spec ial form by Durandus de S. P orc iano (d. 1 3 3 4 ) saves the immutab ility of the single c one rete form , while al lowing a latitudo in the _§_pee ies formae. In intension and remission , the partic ipating sub jec t takes up at eac h instant a c ompletely new form , more perfec t or less perfec t than the previous form, altho ugh bel onging to the same spec ies of quality; the new form is produ c ed ex nihilo , the previous form being c o mpletely destroyed. In the latitude within whic h variation of intensity is possib le, it is thus nec essary to admit an infinity of forms whic h differ only in intensity. Ac c ording to Durandus , the unity among these forms whic h suc c eed one another in in ­ tension and remission lies simply in their c ontinuity. 50 Ano ther positio n , not nec essarily in opposition to those previously mentioned , ho lds that intension and remission c omes ab out thro ugh greater or less admixture of a quality with its c ontrary , for example , of hotness with c o ldness. This position has the support of several Aristotelian pas sages , 5 1 and is disc ussed in Burley's treatise and c ommen­ taries thereon. 52 Ac c ording to Maier , the S c ho lastic ism of the fourteenth c entury after the time of Burley and Durandus brought nothing new to the analysis of the onto logic al nature of qualitative variation. Oc kham , and the nominalists gener ally , fo ll owed S c o tus in regarding intension as an additive increase , c harac terized by some kind of unity b etween the original reality and that which is added. Thus Buridan , Alb ert of Saxony , Peter d'Ailly , M arsilius of Inghen . In plac e of the onto l ogic al prob lem , a logic al prob lem c omes to the fore: how to denominate a sub jec t in whic h the intensity of a quality varies from one point to another. This prob lem merges into a mathematic al one: that of describing the different possib le modes of spatial or tem­ poral variation of intensity , and of finding c ertain rules of equivalenc e between one distrib ution of intensities and another . It was in the attempt to pr ovide a quantitative desc ription of suc h intensive variation that the S c ho o lmen developed the c onc ept of a latitude or range of degrees , to represent different intensities whic h a quality may assume. We have seen the term latitudo , as used by Henry of Ghent , meant the range of possib le variation of intensity . Later it was used in addition to apply to a partic ular in ­ tensity , or the qualitative distanc e inc luded b etween zero intensity and a given degree of intensity . U ltimately , with­ out losing its earlier meanings , it c ame to refer to a c on -

20

INTRODUCTION fi g u r at ion o r p a r t i c u l a r mo de o f v a r i at i on o f int e n s ity in s p ac e o r t i m e . 53 T aking the te r m in thi s thi r d s e n s e , the S c ho o lm e n p r o c e e de d t o c o mp ar e di ffe r e nt l atitude s , an d t o s tate c e r t ain r ul e s o f e qui v a l e nc e b e tw e e n the m . A c r u ­ c i a l a s s um p t i o n inv o lv e d in th i s te r mino l o g y a s u s e d b y He yte sb u r y and hi s c o nt e mpo r a r i e s i s that qual i t ative in ­ t e n s it y i s a c e r t ain kind o f quantit y- line ar ly o r de r e d and additi v e . " Que lib e t latitudo finita e s t que d am qu antita s ;• s ay s H e y t e s b ur y . 54 Althou g h the e a r li e r de v e lo p m e nt o f thi s mode o f de s c r i p t i on o f qual it ati v e v a r i ation i s unkno wn to u s , it i s p o s s ib l e t h at i t w a s a r r i ve d a t b y me an s o f an an alo gy with the kine m atic s o f l o c a l moti on , the s tudy o f whi c h w a s b e in g p ur s ue d c onte mp o r ane o u s ly . The kine matic s o f l o c a l motion h a d b e e n di s c u s s e d in the thi r te e nth c e ntur y in a t r e at i s e u De motu ;• p r ob ab ly w r itte n b y G e r a r d of B r u s s e l s ; 55 and b e fo r e the time at whi c h Heyte s b u r y w r o t e hi s te xtb o o k o f l o g i c the l a w o f unifo r mly ac c e l e r ate d m o ti o n s e e m s to have b e c o me c o mmon kno wl e d g e , for He yt e s b u r y s t ate s th at " it c an be p r o ve d ;' without him s e l f o ffe r i ng a p r o o f . 56 Law s s t ating the e qui v al e nc e o f v ar i ­ o u s ly ac c e l e r ate d and de c e le r ate d motion s to uni fo r m m o ti o n s o f t h e s ame dur ati o n may have s e r v e d a s mode l s fo r the r ule s o f e qui v al e nc e b e tw e e n di ffe r e nt l at itude s o f qu a l it y . T he que s ti on o f t h e v a l idity o f .the an alo gy b e twe e n qual ­ i t at i v e inte n s ity and v e l o c ity wi l l b e di s c u s s e d in c o nne c ti o n w i th o u r e xp l i c ati on o f the s ixth c hapte r o f He yte s b u r y ' s R e gule .

C

ONNE C T ION b e tw e e n the Logic al and P hy s i c o m athe m atic al As pe c t s o f the Re gule

T h e c onn e c t i o n b e twe e n the lo g i c al and the phy s i c o m athe m at i c a l a s p e c t s o f the R e gule i s t wo fo ld an d r e c ip r o c al . O n t h e one h an d , phy s i c al and mathe matic a l p r ob le m s ar e analy z e d b y me an s o f th e te c hni que s o f the 12.g i c a mode r na . O n the othe r hand , phy s i c al and m athe m at i c al p r in c i p le s and r ul e s a r e e mpl o y e d in the analy s i s o f the lo g i c al o r s e m antic p r ob l e m o f de no minati o n ; that i s , the p r ob le m o f d e t e r m ining und e r what c o nditi o n s a s ub j e c t c an b e de no m ­ i n ate d s u c h - and - s uc h , s ay " white" o r u r unnin g :' T he fir s t c onne c ti o n w il l b e i l lu s t r ate d thr o u g ho ut o ur e xp o s ition o f t h e R e g u l e . T h e s e c o n d c onne c ti o n m ay b e i l l u s t r ate d he r e b y e xample s fr o m He yte s b u r y ' s SoP.h i s m at a , whe r e i t ap ­ p e ar s in e s p e c i al c l ar ity .

21

WILLIAM HEYTESB U R Y T he s o p hi s m s de alt with in thi s w o r k , w ith the e xc e pti o n o f t h e l a s t two , 57 p r i m a r i ly in v o l v e s e m antic and s ynt ac ­ tic al que s ti o n s whic h w o uld o r dina r i ly b e t r e ate d in t h e c onte xt o f l o g i c r athe r t h an i n that o f p hy s i c s : t h e s up p o ­ s i tion o f noun s an d p r o n o un s , the t i m e - r e fe r e nc e o f v e r b s , the e ffe c t o f one t e r m o f a p r o po s it i o n o n t h e s uppo s i t i o n o r t i m e - r e f e r e n c e o f a n o the r . T h e y a r e , in fac t , s o phi s m s l o n g c u r r e nt in the t r adition o f the !,Q_g i c a mo de r n a ; fo u r t e e n o f t h e thi r ty - two ar e di s c u s s e d i n the " lnt r o duc t i o n e s in lo g i c am,, o f W i lli am of S h y r e s wo o d , and nine (th r e e o f the m di ffe r e nt fr o m tho s e de alt with b y W i l l i a m o f Shy r e s w o o d) in t he Summ ulae !,Q_g i c ale s o f P e t e r o f Sp ain ( s e e Ap p e n d i x A ) . M o r e o v e r , the y a r e s o p hi s m s w hi c h c o ntinue t o be di s c u s s e d in the fo ur t e e nth c e ntu r y in t e r m s of the i r p u r e ly lo g ic al impo r t a s i n t h e S o phi s m at a o f Alb e r t o f S axony . ( T w e nty o f He yt e sb u r y ' s t hi r ty - tw o s o p hi s m s a r e de alt wit h b y Alb e r t ; s e e Appe ndix A . ) It i s He yte s b u r y ' s int r o duc t i o n o f mathe m a t ic a l and p hy s i c al c on s i de r ati o n s int o t h e ana ly s i s o f t h e s e s o phi s m s w hi c h i s h e r e o f in ­ t e r e s t . 58 T he p r ob l e m o f de no min ati o n a r i s e s b e c au s e o u r w o r l d i s a wo r l d o f v a r i ab i lity and c h an g e . Sinc e att r ib ut e s m ay b e a s s um e d b y an indivi dual in v a r y i n g de g r e e s o f int e n s ity or c o m p le te ne s s , the c o nditio n s und e r w h i c h a g i v e n indi ­ v i d u a l m ay b e s ai d to b e the s uppo s itum o f a g i v e n te r m­ i . e . , the c o n d iti o n s unde r w hi c h it m ay b e s ai d t o b e quali ­ fie d b y the att r ibute w hi c h the te r m c o nno t e s - a r e p r ob le m ­ ati c al . He y t e sb u r y s e e k s to e s t ab l i s h a s e t o f c o nv e nt i o n s , m athe mat i c ally p r e c i s e b ut r e l ate d t o t he c o nv e nti o n s o f e ve r y d ay s p e e c h , whic h w i l l de t e r mine h o w a s ub j e c t i s t o b e d e no m inat e d unde r a l l c o n c e i v ab le c i r c um s t anc e s o f c han g e and v ar i at io n . T hu s in S o ph i s m a 5 , " O mni s h o m o qui e s t alb u s c u r r it ;' He yte s b u r y t r e at s o f t he c on d it io n s unde r w hi c h a m an i s e ntitle d to b e c all e d " white" (fo l l . 9 3 r b f f . ) . It i s f i r s t p r o ­ po s e d that a thing s ho u l d b e c alle d " wh i t e " i f an d o n ly i f e v e r y qu ant i t at i v e p a r t o f i t i s whit e . T hi s p r o p o s a l i s r e j e c t e d , h o w e v e r , s inc e i t w o uld e xc l u d e al l me n f r o m the c l a s s o f white t hing s : n e it h e r the f l e s h no r t h e b lo o d i s white . T he s am e o b j e c ti o n h o l d s a g ain s t the p r o p o s a l that a thing b e c al l e d " white" i f mo r e t h an one h a lf o f i t i s whit e . T he c o r r e c t r ule , ac c o r di n g to H e yt e s b ur y , i s t h at a m an i s to b e c alle d " white" i f and o n ly i f the e xt e r n al s u r fac e o f t h e upp e r h a l f o f h i m i s whit e . O r , i f we w i s h t o f a l l b ac k o n t he c onve nt io n s o f e v e r yday s pe e c h , w e w i l l s ay th at a m an i s white i f and o n l y i f t he s ki n o f h i s f a c e

22

I N TRO DU C T I O N

is white , or more precisely , if of every part of his face some part is white. 59 The adoption of this convention leaves the way open to some rather odd inferences. Thus Heytesbury concedes that Socrates can be white and Plato black , although the proportion of P lato, s surface which is white is larger than the proportion of Socrates, surface which is white. This case is realized , for example , if Socrates, face is white and the rest of his skin black , and Plato, s face is black and the rest of his skin white. Since the whiteness of an object may vary not only as to the area which it qualifies but also as to its inte nsity at any point on the surface of the object , it is necessary to decide upon a further convention as to the degree of intensity of whiteness required for denominating an object "white:, Heytesbury holds that the range or latitude of the possible variation of intensity of whiteness is finite , and that the middle degree of this latitude is the maximum degree which does not suffice to confer the denomination "white ;• so that any greater intensity suffices. These two conventions are courageously applied by Heytesbury to the analysis of intricate cases of qualitative variation. For instance , he imagines that , at the start of a certain hour , Socrates, face is partly white and partly black; that during the hour the black area conden � es to z ero quantity while the white part expands so as to occupy the whole face; and that the whiteness of the white part meantime decreases continuously in intensity. If at the end of the hour the whiteness is still of an intensity greater than the medium degree in the latitude of whiteness , it fol ­ lows that Socrates will become white , although the white ness that he possesses continuously decreases in intensity. Another illustration of the problem of denomination may be taken from Sophisma 23 , "Cuiuslibet hominis asinus currit0 (fol. 14lra-14lva). Here the following question is raised : what degree of velocity is sufficient to denominate the movement of Brunellus (an ass) as "running,, ? Heytes ­ bury holds that there is a lower limit to the range of the velocities which are sufficient to denominate a movement as "running ;, and that this lower limit differs for different species of animals. Thus the range of velocities denoted by "running,, differs in the case of the ass and in that of the flea. In the discussion of such problems of denomination , the rules of Chapters 4- 6 of the Regule come into play: the rules of "De incipit et desinit0 when it is a question of the 23

WILLIAM HEYTESBURY

limits of a time inte r val during whic h a sub j e c t is quali ­ fie d by an att rib ute ; the r u l e s of " De maxima e t minima" when it is a que stion of the limits of the r ange of int e nsity of the qua lity by whic h the sub j e c t is qualifie d, o r of the pow e r whic h it posse sse s; the rul e s of " De t r ib us p r e dic ame ntis" when it is a que stion of denominating a motion . D E LATION between Heyte sbury ' a _l� M e t aphy s ic al Po s ition and the .BJ:_gule

" It is to b e note d ;' says Simon de Le nde nar ia, " that He yt e s b ur y was a te r minist, ho lding that line and surfac e ar e not r e alite r distinc t f r o m b o dy , b ut only se c undum r ation e m :• 6 0 And ac c o r ding to Gae tano di T hiene, " [ He yte sbur y] b e li e v e d that motion i s not distinguishe d r e alite r f r om the b ody mov e d, and that instants and tim e ar e not distinguishe d r e alite r f r o m the heaven �• 6 1 And ac c o r ding to H e yte sb u r y himse lf, u . . . many fal se figm e nts c onc e rning instants, time , and mo tion a r e admitte d in the c ommon m o de of sp e e c h, fo r the sake of b r e vity and the e asie r e xpr e ssion of the c onc e pt o f the mind; fo r in the natu r e of things th e r e i s nothing whic h i s an instant as suc h, no r time as suc h, no r motion as suc h ; j ust as So c r ate s is nothing ac c o r ding as he is a w hite man and P lato is nothing ac c o r ding as h e is t o dispute tomo r r o w . . . :• 62 F o r He yte sb ur y, the r e al physic al w o r ld c onsists only o f ob j e c ts; point, line , sur ­ fac e , instant, time , and motion ar e c onc e ptus m e ntis. T h e se affi rmations (o r p e r hap s the y ar e b e tt e r te r m e d " n e ga ­ tions" ) ar e in ac c o r d with the nominalist o r te rmini st posi ­ tion, de v e lo pe d at l e ngth in William of O c kham' s wo r k on the P hysic a . 63 Y e t it is pr e c ise ly with suc h c onc e ptus mentis- math e ­ matic al limits suc h as point, line , su rfac e , and instant ; imaginabilia suc h as time and motion- that the r ul e s of Chapte r s 4 - 6 of the R e gule ar e c onc e r ne d . We shall se e that, w he r e O c kham would apply his r a z o r , simplifying c onc e ptual distinc tions so that the y no mo r e than mir r o r the ob se r vations of e xp e r ie nc e , He yte sb ur y multip lie s suc h _distinc tions to the limit o f c onc e ptual p o ssibility . On what basis do e s He yte sb ur y j ustify this " multipli c ation of c onc e pts b e yond n e c e ssity" ? In the fir st p lac e, He yte sb u r y admits suc h te r ms as u point" and "instant" b e c ause , altho ugh the y do not r efe r to ob se r vab le s, the y ar e ne v e r th e le ss in c ustomar y use . T h e c ommunis mo dus lo quendi is fr e que ntly use d as a c rite r ion by H e yte sb ur y and his c ommentato r s, in the

24

INTRODUCTION

de ci s ion be twe e n dive r s e opinion s . 64 While He yte sbury s e e k s a mathe matical ly pre ci s e de nomination or de scrip ­ tion of hypothe tical phe nome na , hi s point of de parture and re fe re nce i s th e mod e of common s pe e ch . In the s e cond place , He yte sbury wri te s a s a logician , and the provi nce of the logician i s the e ntire range of i m ­ aginab le ca s e s and probl e m s . The phras e "s e cundum im ­ agi natione m" i s fre que ntly us e d in the Re gule . The only re quire me nt for an imaginabl e ca s e or di stinction or probl e m i s that it s hould not involve a formal logical con ­ tradiction; whe the r it i s phy s i cal ly pos s ible or not i s a matte r of indiffe re nce . 6 5 Since i t is the imaginabl e which is the province of He yte s ­ bury' s inve s tigation, we mu st not be surpri s e d if we fai l to fi nd in hi s work that re fe re nce to e xpe ri e nce which ha s s ome time s be e n as sociate d with the s cie ntific te nde ncie s in fourte e nth - ce ntury nominali s m . In fact , the word "e x ­ perime ntum" i s us e d not at al l i n the Re gule , and only twice i n the Sophi s mata. 66 A di s tinction i s drawn be twe e n re alite r or naturalite r or ph y: s ice log ue ndo and logice or s ophi s tice loque ndo: phY. s i ce loque ndo we fol low e xpe rie nce and th e principle s laid down in the natural phi lo s ophy of Ari s totle ; logice or �hi s tice loque ndo we are fre e to introduce whate ve r di stinction s and ca s e s are conveni e nt and imaginabl e . 67 The las t thre e chapte rs of the Re gule deal with point s , line s , s urface s , i n s tants , latitude s of variation , e tc. , which in He yte s bury' s vie w are me re ly imaginab i lia , and which are the re fore prope rly the s ubje ct of a di s cus s ion carrie d on in the conte xt of logic rath e r than i n that of phy s ic s . It i s of s ome inte re s t , the n , that the re ductive te nde ncy in nominali s m- it s te nde ncy to de ny re al e xi s te nce to what i s not ob s e rvable- doe s not ope rate as a pre s cription again s t s pe culation conce rning the imaginabi lia. Quite the re ve r s e : in the di s cu s s ion of hypothe tical phy s ical probl e m s , He yte s bury and hi s conte mporarie s fre que ntly multiply formalitate s in the Sc otian manner. 68 The re s ult i s a kind of mathe matical phys ics which at time s runs stronge ly paral le l to mode rn phy s ics , but which ne i the r s e e k s nor clai m s to have application to the phys i cal world.

I

NF L U E N C E of the Re gule upon Late r M e di e v al Thought

It was on th e Contine nt rathe r than in England that the Re gule e njoye d its gre ate s t and longe s t pe riod of popular -

25

W ILLIAM

HE YTESB UR Y

ity . In the case of England we ar e unable to follow the tr ace of the logical -sophistical -mathematical tradition beyond the end of the fou rteenth century . 69 The fi rst evidence of the influence of the Regule on the Continent is John of Casali's treatise " De velocitate motus alterationis ;' a manuscr ipt of which gives the date 1 3 4 6 . 7 0 This tr eatise follows closely many of the distinctions and problems posed in the thi r d part of the sixth chapter of the fugule . In 1 3 69 a professo r at the Univer sity of P rague , John of Hol land , compiled a tr eatise " De primo et ultimo instanti ;' in which he quotes one of the rules from the fifth chapter of Heytesbury 's Regule .7 1 This same John of Hol land wrote a treatise " De motu ;' which is essential ly an abstr act from the sixth chapter of the Regule and from Suiseth's Liber calculationum .7 2 B y the end of the four teenth centu r y , the Re gule and Sophismata of Heytesbur y had begun to enjoy in Italy a vogue which was to last for a centu r y . The De instanti and the Logica of �etrus de Mantua (d . 1 4 0 0) show the influence of Heytesbur y . 73 Angelo da Fossambr uno , who taught at the University of B ologna from 1 3 95 to 1 4 0 0 and at the University of Padua from 1 4 0 0 to 1 4 02 , wrote in 1 4 0 2 a commentari4 on the first two parts of the sixth chapter of the Regule. 4 Among the fifteenth -centu r y Italians who d r ew from or wrote commentaries on parts of Heytesbur y 's Regule or SoP.hismata , the following may be mentioned : Paul of Venice (d . 1 4 2 9), a brother of the o r der of Hermits of St . Augustine , studied at the Univer sities of O xford and Padua , and taught logic and philosophy at the Univer sities of Padua and Siena: 5 In his Summa natur alium (Venice , 1 4 7 6) he makes use of the conclusions of the fifth and sixth chapters of the Regu le . The fifty sophisms of his SoP.hismata (Venice , 1 49 3 ) are drawn from the Sophismata and Regule of Heytesbur y , and his Logica (Pavia , 1 4 8 0) also shows the influence of Heytesbur y . Gaetano di Thiene , a professor at the Univer sity of Padua from 1 4 2 2 unti l his death in 1 4 6 5 ,7 6 wrote extensive " Recollecte" on Heytesbur y ' s Regule , and also on the fir st thi rty sophisms of Heytesbur y 's Sophismata. (These " Re ­ col lecte" were p r inted with the 1 494 edition of Heytesbur y's Regule and Sophismata , and will be refer red to frequently in what follows. ) These wor ks probably stem f rom the peri ­ od 1 42 2 - 3 0 , during which Gaetano held the chair of logic at Padua .7 7 26

INTRODUCTION

A certain Magister Messinus wrote a commentary on the sixth chapter of the Regule ( "De tribus predicamentis" ).78 The fact that this commentar y is broken off shortly after the start of the section "de motu alterationis ;' and that a completion of it was written by Gaetano di Thiene,79 sug ­ gests that Messinus was connected with the University of P adua.80 Besides the commentary "De tribus predicamentis; ' Messinus also wrote a commentary on John of Casali' s "De velocitate motus alterationis. " 8 1 Simon de Lendenaria� 2 a colleague of Gaetano di Thiene at the University of Padua, wrote a commentary on the fir st seven sophisms of Heytesbury's � P.hismata.83 The Venetian, Paul of Pergola (d. 14 56 ), wrote com ­ mentaries on the first two chapters of the Regule,84 on Heytesbury's De sensu comP.osito et diviso--:SS and on the first seven sophisms of Heytesbury ' s Sophismata.8 6 In 1484 Bernardus Tornius (d. 1 500 ) , a Florentine phy ­ sician and friend of Giovanni Marliani ,8 7 wrote annotations on that part of the sixth chapter of the Regule which deals with local motion.8 8 The extraordinary favor with which Heytesbury ' s logical works were regarded at the University of Padua is shown by the fact that the Venetian Senate in 148 7 decreed that no work other than Heytesbury 's Sophismata should be as signed in logic without consultation of the Senate .8 9 A stat ­ ute of the Faculty of Arts in 1496 named Heytesbury's works among the standard texts for the school of logic, along with the Logica of Paul of Venice.9 0 The teaching of sophistic continued at P adova until 1 560, and it is possible that the use of Heytesbury ' s works as textbooks continued until that time .9 1 The Regule was printed at Pavia in 148 1, and at Venice in 149 1 and in 1494 (see Bibliography); the 1494 edition is particularly widespread. In the early sixteenth century, the works of Heytesbury enjoyed a period of popularity at the College of Montaigu in P aris . The logician and theologian , John Major (d . 1 540), was a leading figure in this movement; in his works he refers to no author more frequently than to Heytesbury.9 2 Major had studied logic under the Spaniard, Geronimo Pardo , whose Medulla dY. alectice s (Paris, 1 50 5 ) shows the influence of Heytesbury. Joannes Dullaert, a pupil of Major, depends upon Chapter 5 of the Regule in his dis cussion of maxima and minima .9 3 After the first quarter of the sixteenth century, however, the vogue of Heytesbury ' s works appears to have undergone 27

W I LL I AM HEYTESB U R Y

a rapid decline. The humanists , Erasmus and Louis Vives, both of whom had studied at the College of Montaigu , at ­ tacked the Latin of the logicians, its subtlety and barbarous technicality 9. 4 The sophistical dialectic of Heytesbury and his followers did not survive the humanist attack.9 5 As Thomas More judiciously remarks , the study of the "small logicals,, is omitted by Utopians.

28

C H AP T E R

2

De i nc i p i t et des init

T

H E c h a pte r o f He y te s b u r y ' s R e gule e nt i t l e d " D e in ­ c i p i t e t de s i n i t " de al s with p r ob l e m s invo l v e d in t h e u s e of the t e r m s inc i P.e r e and d e s in e r e , " t o b e g in " and " t o c e a s e ." T h e s t a r t i n g po int of t hi s di s c u s s i on i s t o be s o u g ht in the P h Y. s i c a o f A r i s to t l e . In B o o k V I o f the 12.h_y� , h av in g a r gue d t h at l e n gt h s , t i m e s , and m o t i on s a r e c o ntinuou s , A r i s t o t le tu r n s to the p r o b l e m o f how a p r o c e s s o f mo t i on o r c h an g e i s l i m it e d t e mpo r a l l y a t i t s s t a r t and e n d . T he " p r i m a r y wh e n " in w hi c h the c o m p l e t i on of c h an g e h a s b e e n e ffe c te d , he h o ld s , mu s t be i n d i v i s ib l e , i . e . , an in s t ant ; fo r the r e i s a m o m e nt i n wh i c h i t i s fi r s t c o r r e c t t o s ay , " T hi s ha s 1 T hu s t he in s t ant in w h i c h a c h an g e i s c o m ­ c h an g e d ." p le t e d do e s not it s e lf b e l o n g to t h e p e r i o d o f c h ang e , b ut fo r m s an e xt r i n s i c l i mi t to it . Si mi l a r ly the p e r i o d o f c h an g e i s b o un d e d e xt r in s i c al l y a t i t s o t he r e xt r e me ; t h e r e i s no s uc h t h in g , A r i s t o t l e s t ate s , a s a b e g inning (fi r s t in s t ant ) of a p r o c e s s o f c h an g e , an d whate v e r i s in m o t i o n mu s t have b e e n i n m o t i on b e fo r e � F o r it the r e w e r e a fi r s t mo me n t o f c h an g e , the mo me nt i m me d i at e ly p r e c e d in g t h e c h an g e an d the mo me nt i n whi c h t h e c h an g e b e g i n s w o u l d b e c o n s e c ut i v e , and mo m e nt s , l i ke po int s o n a l i ne , c anno t b e c on s e c ut i v e o r in c o ntac t . F r o m t h i s a c c o unt it appe a r s t h at the pe r io d in whi c h a c han g e i s a c t u a l l y t ak i n g p l ac e i s b o unde d e xt r in s i c ally at b o t h e nd s . O n e e xc e pt i o n to t h i s c on c lu s i o n i s t o b e n o t e d . A r i s t o t le a l l o w s t h at i n a lt e r ati o n (motion in t h e c at e g o r y o f qu a li t y ) , a s o p po s e d t o l o c a l m o t i o n and c h an g e i n the

29

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HEYTESBURY

c ategory of quantity , the re may be a fi r st , indivisible part whi c h changes. u For in a proc ess of c hange we may distin ­ guish th ree te rms- that whi c h changes , that in wh i c h it c hanges , and the ac tual sub jec t of c hange , e. g. the man , the time , and the fai r complexion . Of these the man and the time are divisible: but with the fai r complexion it is other wise (though they are all divisible ac c identally , for that in whi c h the fai r c omplexion or any other quality is an ac c ident is divisib le): d The assertion that qualities may be essen ­ tially indivisible would permit the assignment of a fi r st instant in a p rocess of alter ation , namely , the instant in whi c h the indivisible is fi rst introduced in that whi c h under ­ goes alter ation. This ac count is amplified in a digression in B ook V III. u lt is also plain that unless we hold that the point of time that divides ear lier from later always belongs only to the later so far as the thing is concerned , we shall be involved in the consequenc e that the same thing is at the same mo ment existent and not existent , and that a thing is not existent at the moment when it has bec ome. It is tr ue that the point is c ommon to both times , the earlier as well as the later , and that , while numer i c ally one and the same , it is theoretic ally not so , being the finishing -point of the one and the starting -point of the other : but so far as the thing is concer ned it belongs to the later stage of what happens to it :' 4 Thus if a body bec omes white as the r e sult of some proc ess of change , ther e will be a fi r st in ­ stant in whi c h it is white , namely , the instant in whic h the process of becoming white is complete ; but ther e will be no last instant in whi c h it is not - white. In gener al the fi r st instant of the existence of a thing (as opposed to a motion) will be ,assignable; but the last instant in whic h it exists will not be assignab le , sinc e its passing away is the result of a c hange whic h is itself te rminated ext r in ­ sic ally b y the instant in whi c h the c hange is complete. This analysis r ests on the under standing that time is not c omposed of consecutive time -atoms or instants ; that the infinite instants in a segment of time , like the infinite points in a line segment , exist there only potentially , ex ­ c ept in the c ase where a parti c ular one is ac tuali zed b y becoming the dividing point between two differ ent motions or between a motion and a state of re st . As a dividing point in time , the instant is both the finishing point of the pr evious time and the starting point of the suc c eeding time; but considered in relation to the dur ation of a part i c ular thing or p roc ess , the instant must be assigned either to 30

DE INC IP IT E T DESINIT the e arli e r or the late r pe riod and not to both , to avoid the conse que nce that a thing would b e e xistent and non ­ e xiste nt in th e same instant. The instant may therefore b e long to the time pe riod which it bounds , or lie just out side it. Thus the distinction b e twe en an e xtrinsic and an intrin sic boundary , as here introduce d by Aristotle for the case of bounde d time inte rvals , has its justification not in an e mpirical observation b ut in the first princi ple of logic , the law of contradiction . In fact , the unive rsal character of se nse ob se rvations and me asuring proce sse s- the ir character of re lative impre cision-e xcludes the possib ility that this distinction should b e verifi e d emp irically. In Aristotle's vi e w , howe ve r , the distinction is not me re ly a logical or mathematical de vice , but is a characte ristic of the physical world , e xe mplifi e d in the boundarie s of p e riods of time. It is of inte rest that a very s imilar distinction (consid­ e re d , however , as pure ly mathematical) has playe d an im ­ portant role in the mathe matics of the nine te e nth and twe ntie th ce nturie s. The pre se nt -day mathematician dis­ tinguishe s b e twe e n two typ e s of e xtre ma of an aggre gate of points on a line or of real numb e rs in the real numb er fie ld: the e xtre mum point or numb e r which b e longs to the aggre gate which it bounds , and the e xtre mum point or numb e r which doe s not so b e long. This distinction has its raison d' �tre in the the ory of functions. A function y= f (x) may b e defin e d for x varying ove r an interval on the numb e r axis, and yet b e undefined whe n � assume s the e xtre mum values in the inte rval , and in the discussion of the function this e xclusion of the e xtre mum values must be spe cifie d. For e xamp l e , the function y = 1 / (1 - x 2 ) is define d in the inter val ( - 1 , 1 ) , b ut not in the end points of the interva l . More fundame ntally , the distinction b e twe e n an intrinsic and an e xtrinsic boundary has playe d a crucial role in the con ­ struction of the real numb er syste m , which underlies the the ory of functions . For Aristotle and the Schoolmen on the contrary , this distinction has its justification in the characte r of re al or imagine d physical proce sse s as analyzed from the standpoint of mathe matical continuity. If we turn now to the work of the me die val Schoolme n , we find that the range of appl ication of this di st inction is pro gr e ssive ly e xte nde d. The discussion de ve lops in two , gene rally se parab le , phase s: one in which , naturaliter loquendo , the Schoolme n sought to apply the distinction dire ctly to the physical magnitude s of the Aristote lian 31

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world- lengths, velocitie s, weights, and indeed whatever physical quantity may be conceived as continuous; the other in which , logice loquendo or sophistice loguendo , and applying the distinction to problems which were im ­ aginable but presumably not capable of physical realiza ­ tion, they arrived at results which are of interest for the mathematical analysis of the continuum and of infinite aggregates. As regards the beginning and ending of time intervals, the two phases of the discussion may be characterized as follows. In commentaries on the Ph Y.sica and in treatises de instanti, the Aristotelian question is pursued: how is the coming-to-be or passing-away of a thing , or the begin ­ ning or end of a process of change, situated in time? In treatises in the tradition of the logica moderna , such as the SoP.hismata of Heytesbury and of Richard Kilmington , the question is generalized to that of the applicability of the terms inciP. it and desinit to whatever may be imagined at some time to be and at some time not to be , including relations as well as things and motions; and in this con ­ text we encounter a verbal analysis of problems involving infinitesimals, limits, the continuum.

P

H Y S ICAL Pha s e o f the Di s c u s s ion of B e ginning and C e a s ing

The physical phase of the discussion of beginning and ceasing merges with the general analysis of the b oundaries of potencies- the subject of our next chapter. We here fol ­ low the development of this phase along some of its way, however, for it is from the discussion in the realm of physics that the discussion in the realm of logic arises , and it is against the background of the form� r that the originality of the latter must be estimated . The commentaries of Averroes and Thomas Aquinas on the PhY.sica reproduce faithfully the Aristotelian pas sages we have cited. W AL T ER B U RLEY

A treatise u De instanti" by Walter B urley (died after 1343) deals with the problem whether a res permanens has a first and last instant of being. 5 A res permanens is defined as a thing all the parts of which can exist at one time, as opposed to a res successiva, such as a motion or time, the different parts of which must exist at different 32

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times. 6 As the tempo r al beginning of existence of a thing, B u r ley notes , it is necessar y to assign either a fir st in­ stant of being (p rimum instans r ei) or a last instant of non ­ being (ultimum instans non esse r ei); the assignment of an intr insic boundar y is incompatible with the assignment of an extr insic boundar y . The same rema r k goes for the end of the existence of a thing ; as boundary it is necessar y to assign either a last instant of being or a fir st instant of nonbeing, and not both .7 The dur ation of a successive thing, B ur ley admits with Ar istotle, is ter minated in either dir ection by an extr insic boundar y . A r es permane ns has a fir st instant of being unless its being and conser vation in being depend on a r es successiva ; in the latter case it begins to be immediately after a last, assignable instant of nonbeing, in the same manne r as the successive pr ocess upon which its existence depends.8 Similar ly, the termination of the dur ation of a r es per manens which is conse r ved in being or destr oyed by a successive pr ocess is, like the ter mina­ tion of the successive pr ocess, extr insic, viz . , a fir st in­ stant of nonbeing. On the othe r hand, the dur ation of a r es per manens will be ter minated by an ultimate instant of being if the thing is an indivisible (such as an instant or the "whe r e" [ubi] occupied instantaneously by a body in local motiort) o r is made up of indivisibles.9 B ur ley gives spee:: ial attention to fo rms which have a latitude of degr ees within which their intensity may var y, r efer r ing her e to his theor y of intension and r emission accor ding to which a change of intensity r esults fr om the production of a totally new individual for m, with the destruction of the pr eviously existing for m ( supr a, pp. 1 9 -20 ). In this case each indi­ vidual fo r m (fo r ma individualis), being an indivisible, has an ultimate instant of being; the for m of the species (for ma _§_P.eciei), which includes an infinite number of individual f o r ms , is ter minated rather extr insically , by a fir st instant of nonbeing. 1 0 J O HN O F

H O L L AN D

J ohn of Holland' s " Tr actatus de pr imo et ultimo instanti ;' 1 1 wr itten in 1 3 6 9 , makes dir ect r efer ence to the tr eatise of B ur ley, and is essentially an amplification of that tr eatise. B ut we may also be sur e that at the time of wr iting this wor k J ohn o f Holland was familiar with the four th and fifth chap­ ter s of Heytesbur y ' s Regule, and also with the Liber cal­ culationum of Suiseth. At the star t, it is stated that the customar y mode of speech r egar ding instants will be assumed, and the question of

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whether instants are or are not, or of what they are if they are , will not be raised. 12 John draws the usual distinction between a res P.ermanens and a res successiva.1 3 With Aristotle, he maintains the principle that a res successiva has neither a first nor a last instant of being. One of the objections to this principle , with John's reply , is worthy of note. We suppose that two drops of liquid , b and �, are falling , and that at some instant d they coalese e to form drop a, in which the motion of fall is continued. It is argued that there is a first instant of motion of the drop a, namely d. For never prior to instant d wa8 drop � in motion , and in instant d it is in motion.1 4 In reply , Johannes denies that in instant d the drop a is in motion. In order to show that a is in mo tion in instant d , it is not sufficient to argue that a in in ­ stant d is in a place in which it previously was not , and that without lapse of time after d it will be in a place in which it is not in instant d. The correct argument would be the following: a in instant d is in another place than it was previous to d , and without lapse of time after d it will be partly or wholly in another place than it is in instant d; therefore a moves in instant d. But in the case posited the first antecedent of this argument is false; for although a in instant d is in a place in which it previously was not , it is not true that a in instant d is in another place than it was previous to d , for this latter statement implies that a existed prior to instant cl- contra casum.1 5 In discussing the beginning and end of the existence of res P.ermanentia , John draws a number of successive dis ­ tinctions. Following Burley , he distinguishes between the res permanens which exists only for an instant , like the ubi in local motion , and the res 2ermanens which exists for a period of time. The first type of res P.ermanens has both a first and a last instant of being , viz. , the single in stant of its existence. Again following Burley , John divides res P.ermanentia which exist over a period of time into tho s.e which depend for their being and conservation in being upon a res successiva , and those which do not so depend. The first type of res P.ermanens has neither a first nor a last instant of being; an example of this type , according to John , is the truth of the proposition usoc rates runs:' The second type of res P.ermanens may be subdivided into the class of � P.er manentia which con sist of a latitude of degrees (e.g. , the whiteness or hot ­ ness by which an object may be qualified) and the class of res P.ermanentia which consist of an indivi s ible form (e. g. , a whiteness of a specified degree of intensity , or

34

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the highest degree of calidity, etc.). A res P.ermanens con ­ sisting of an indivisible form has both a first and a last instant of being. A res permanens consisting of a latitude of degrees has a fir st instant of being but not a last, since the combination of form and subject is corrupted by a suc­ cessive motion, the form passing through every degree of intensity to zero intensity; and the instant in which zero intensity is reached will also be the first instant of the existence of a new form in the subject. With regard to Burley 's view that each different degree of intensity con ­ stitutes a new forma individualis, John holds that it is satis defendibile; and on this view the forma individualis will have an ultimate instant of being, while the forma �eciei will not. 1 6 At this point John introduces a distinction which does not occur in Burley ' s treatise: the distinction between a P.Ositivum and a P.rivatio. Each of the types of entities previously distinguished has its corresponding privation; thus the privation of the res successiva denoted by "run ­ ning" is not - running, and the privation of the res P.er ­ manens denoted by "white" is the not -white. And for the period of duration of the privations of the various types of entities previously distinguished, it is necessary to as ­ sign boundaries of such character as to be compatible with the boundaries of the periods of duration of the cor responding positive entities . For instance, since a res successiva has neither a first nor last instant of being, the corresponding privation will have both a first and last instant; and since a res permanens which consists of an indivisible form has both a first and last instant, the cor ­ responding privation will be limited in its period of dura ­ tion neither by a first nor by a last instant, but rather by an ultimate instant of nonbeing before it exists, and by a fir st instant of nonbeing after it exists.1 7 CO M M E NT A R Y AT T RI B UT E D TO DUN S S COTUS

In a commentary on the PhY. sica attributed to Duns Scotus, but undoubtedly written by a later author, the problem of the existence of natural minima of matter is analyzed on the basis of the rule that the first instant of being of a � P.ermanens is assignable, while the last instant of its period of duration is not. 1 8 In the case of an element (earth, water, air, and fire) the author holds that the form of the element can be introduced into any quantity of matter, arbitrarily small; in order to endure, however, the e1 ement must be in sufficient quantity to resist the

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corrupting action of the surrounding medium, and hence under specific circumstances there is a minimum quantity of matter in which an element can endure without tending to be corrupted. On the other hand, in the generation of an animated heterogeneous mixture, such as an animal or plant, matter which is properly disposed must be fq resent in a certain quantity before the form can enter in. 9 Ac cording to the author of the commentary, the lower bound ­ ary for quantities of matter into which a given form can be introduced must be intrinsic- a minimum amount of matter into which the form can be introduced rather than a maximum amount into which it cannot be introduced. Otherwise, as the quantity of properly disposed matter in ­ creased from insufficiency to sufficiency, there would be assignable a last instant of insufficiency rather than a first instant of sufficiency, and by consequence a last instant of nonbeing rather than a first instant of being of a res P.er manens- contrary to rule. Once such an animated hetero geneous mixture has come into existence, however, there will be no minimum amount of matter in which it can endure, but rather a maximum amount of matter in which it cannot endure; and this conclusion agrees with the as ­ signment of a first instant of nonbeing rather than a last instant of being as the term of existence of a res P.ermanens. P E T E R O F M AN T U A

The De instanti of Peter of Mantua likewise deals with extrinsic and intrinsic limits to periods of time .2 0 Peter holds that all substantial forms are introduced into matter by a successive process, depending on the production of the proper disposition in the matter by some external agent. Since the agent acts successively on different parts of the matter, starting with those nearest by, it follows that no first instant in which the proper disposition is produced in a divisible part of the matter can be assigned, and as a consequence no first instant can be assigned in which the form informs a part of the matter, but only an ultimate instant in which it does not inform a part of the matter. In this connection Peter denies the existence of natural minima, such as a minimum possible quantity of fire, which being introduced instantaneously into matter would allow for the assignment of a fir st instant of existence of form in matter. But with regard to the existence of a com ­ posite, composed of form and of matter which is informed by the form throughout, Peter holds that there is a first instant of being, namely the instant which terminates the

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time during which the form is successive ly introduced into the m atter . An exception to this rule occurs in the case of man , whose subst ant i al form is indivisible or nonextended ; for there is no fi r st inst ant in which the heat of the semen is sufficient to allow the introduction of the form- the cale faction of the semen being a successive process- but rather only an ultim ate instant in which such heat is insufficient , and the indivisible form is introduced immedi ately after this inst ant ; hence we are to assign an ultimum quod non rather th an a primum quod sic instant for the beginning of the existence of a man . In the passing away of a substance , Peter argues , there is an ultim ate instant of being , so th at the final bound ary of the period of duration is intrinsic ; for the form which gives the being of the thing ceases to be in an ultimate instant . M an , being immortal , does not cease to be at all . All qualities being gradual ( u omnis qualitas est gradu ­ alis" ), Peter holds th at qualities are successively rather than instantaneously acquired and lost ; their duration is therefore bounded by extrinsic boundaries . Exceptions arise when the existence of one quality follows immedi ­ ately upon the ceasing -to -be of another ; suppose , for ex ­ ample , th at a certain opinion is accepted with hesitation , and that the hesit ation remits gradually to zero degree , so that the quality uopinion" is fin ally converted into the quality " knowledge" ; the ultimate instant of hesitation will not be assignable , but rather only the first instant in which the hesitation no longer exists ; and the same in ­ stant will be the first instant of knowledge. The duration of res successiva-motions and time- is sim ilarly to be bounded by extrinsic boundaries . A P O L L IN A R I S O F F R E D I

A criticism of Peter of M antua's De instanti was written by Apollin aris Offredi in 1 4 8 2 .2 1 Among the points at which Apollinaris differs with Peter , with regard to the assignment of boundaries to periods of duration , are the following : FIR S T . - Apollinaris believes that a first instant of a man's existence can be assigned , as opposed to an ultimum quod non ; the heat necessary for the introduction of the human soul into the semen may be produced in every part of the semen at once . Peter's opinion that no man ceases to be is characteri zed as a fantasia . S E CO ND . - The fact that a quality is successively ac ­ quired and lost does not necessarily imply that its duration

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is bounded by extrinsic boundaries. Thus the vision of an object , although subject to intension and remission , is produced by the species of color only if the latter have attained a certain minimum intensity; and vision will there fore have a first and last instant of duration , according as this minimum intensity is attained or lost. F ollowing Heytesbury ' s opinion that the whole of a thing is seen be fore the parts are distinctly discerned ,2 2 Apollinaris holds that there will be a first instant in which the whole of a thing is seen , and the same instant will be the ultimate in which the parts are not cle arly discerned. TOG I C AL Phase of the Discussi on of Be ginning and Ceasing W I LL I A M O F S H Y R E S W O O D

L

We turn now to the logical phase of the discussion of beginning and ceasing. The first notice we find of an in ­ terest on the part of logicians in the terms "beginning" and "ceasing" is in the " Introductiones in logicam" of William of Shyreswood.2 3 In this work incipit and de sinit are said to be exponible terms , i . e. , terms having an ob ­ scure sense which requires exposition. It is noted that incipit may be expounded either by a positing of existence in the present instant (per positionem presentis) and denial of existence in the past , or by a denial of existence in the present instant with positing of existence in the future . Similarly desinit may be expounded either by a positing of existence in the present instant and denial of existence in the future , or by a denial of exist ­ ence in the present instant and positing of existence in the past . This is the logician's manner of stating that the bound ­ aries of periods of duration may be either intrinsic or extrinsic. The mode of exposition of incipit or desinit used in a particular case depends upon whether that which comes to be or passes away is a permanent thing (i. e . , a thing of which the parts exist simultaneously) or a successive thing (i. e . , a thing of which the parts do not exist simul ­ taneously). If that which begins to be is a permanent thing , the incipit must be expounded by positing of existence in the present and denial of existence in the past , while the desinit which refers to the passing away of the previous state of affairs must be expounded by denial of existence in the present and positing of existence in the past. Thus " Socrates begins to be healthy" is expounded as follows: 38

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" he i s he althy , and p r e v i o u s ly he w a s not he althy :' And " So c r ate s c e a s e s to b e i l l" h a s the following e xpo s itio n : " he i s no t no w i l l , an d p r e viou s ly he wa s i l l :' I f that whic h b e g i n s to b e i s a s uc c e s s iv e thin g , the inc i2it mu s t b e e xp ounde d b y de ni a l o f e xi s t e n c e i n the p r e s e nt and po s iting o f e xi s te n c e in the fut ur e , whi l e the de s init whi c h r e fe r s to the p a s s in g away o f the p r e vio u s s t ate o f affai r s mu s t b e e xp o unde d b y po s i tin g o f e xi s t e nc e i n the p r e s e nt and d e n i al of e xi s te nc e in the futur e . T h u s " So c r ate s b e gin s to mo v e" i s e xp o unde d a s fo llow s : " So c r ate s do e s not no w m o v e , and afte r thi s he will move :' And " So c r ate s c e a s e s to b e qui e s c e nt" h a s the fo ll owin g e xpo s it ion : " So c r at e s 1 s now quie s c e nt , and aft e r th i s he will not b e quie s c e nt :' F inally , W i l l i am o f Shyr e s wo o d note s that the te r m s inc iP.it and de s init h av e a v i m c onfunde ndi the c o mmon noun whi c h fo l l o w s the m in th e o r de r of the s e nt e n c e in whi c h the y o c c u r . PETER

OF

SP AIN

A s i m i l a r ac c o unt o f " b e g innin g" and " c e a s in g" i s to b e fo un d i n the t r e ati s e " De e xponib i l ib u s" whi c h ac c o mp anie s the Summul ae lo g i c ale s o f P e t e r o f Spain.24 In c i P. it and de s init a r e o nc e mo r e s aid to b e e xponib l e te r m s ; b ut in the " D e e xp o nib i lib u s" " e xponib le" i s h e l d to r e fe r to te r m s whi c h h a v e an ob s c ur e s e n s e b e c au s e o f s o me .§_y:nc ate go r e matic te r m imP.l i c itly c ont ain e d in the m . T he s ync ate g o r e m at i c t e r m w hi c h , ac c o r ding to the " De e xponib i lib u s ;' e nte r s into the e xp o s ition o f inc i 2it and de s init i s " i mme d i a t e ly :' T he aut ho r fi r s t ob s e r ve s th at s o me thin g s c o me into b e ing as a w ho le in s t antane o u s l y ; thi s may be e ithe r b y me an s o f a p r e c e d ing s u c c e s s i ve mutati on , a s in the g e ne r ­ ati on o f m an , o r w itho ut s uc h a p r e c e ding mut ation , a s in the c r e at i o n o f an g e l s . Othe r thin g s c o me into b e in g s uc ­ c e s s i ve ly , one p ar t a fte r anothe r ; s uc h ar e the pe r m ane nt thin g s the de no min ati on o f whi c h de p e nd s o n the do mination of one c o nt r a r y o v e r anothe r , as of white o ve r b l ac k o r c o ld o v e r hot , and al s o all s uc c e s s iv e thing s , e . g . , motion and ti me . S i m i l a r ly , s o m e thin g s c e a s e to be in s tant ane o u s ly, othe r s s uc c e s s i ve ly . Ac c o r din gly , inc i P.it and de s init a r e e ac h e xp o unde d i n t w o d i ffe r e nt way s . F IR S T R ul e .- P r o p o s ition s s t ating the b e g inning o f thing s the b e in g o f whic h i s ac qui r e d in s t antane o u s ly a r e e xp o unde d b y a c opul at i v e , the fi r s t p a r t o f whic h i s an affi r m at i v e in t h e p r e s e nt t e n s e , a n d the s e c o nd p a r t o f whi c h i s a n e g at i v e in the p a st te n s e . F o r e xampl e , " M an b e gin s to b e"

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is expounded thus: M an now is and immediately pr ior to this man was not:' SECOND Rule. -Propositions stating the be ginning of things the being of which is acquir ed successively are expounded by a copulative , the first part of which is a negative in the present tense , and the second part of which is an affirma ­ tive in the future tense. For instance , " Socrates begins to be white" is expounded thus: " Socrates is not now white and immediately after this he will be white:' THI RD Rule. -Propositions stating the passing away of things of which the being is lost instantaneously ar e expounded by a copulative , the first part of which is an affirmative in the present tense and the second part of which is a neg ­ ative in the future tense. For example , " Socrates ceases to be a man" is expounded thus: Socrates now is a man and immediately after this he will not be a man:' FOURTH Rule. -Pr opositions stating the passing away of things of which the being is lost successive ly are expounded by a copulative , the first part of which is a negative in the present tense and the second part of which is an affirma ­ tive in the past tense. For instance , " Socr ates ceases to be white" is expounded thus: " Socr ates is not now white and immed1ate ly prior to this he was white .2 5 0

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These modes of exposition wer e subjected to criticism by later logicians. Thus William of Ockham (d. 1 349 or 1 350 ) 2 6 objects to the having two modes of exposition of of inci@ , accor ding as the thing which begins to be is a res 2ermanens or a r es successiva , and two modes of ex ­ position of desinit , accor ding as the thing which ceases to be does so instantaneously or successively; one mode of exposition each for " beginning" and "ceasing" should suf ­ fice , viz. , that mode by which an intrinsic boundary is as signed. For instance , Ockham would expound " Socr ates begins to be white" by the copulative: " Socrates is white , and immediately before Socr ates was not white:' " Socr ates ceases to be white" he would expound by the copulative: " Socr ates is white , and immediately after he will not be white:' For the duration of all things which begin and cease to be , Ockham would thus assign intrinsic boundar ies.2 7 That Ockham here applies his conceptual r azor is per ­ haps owing to the fact that the distinction between extr insic and intrinsic boundaries of periods of time is not ver ifiable in experience. Pr oceeding yet fur ther towar ds the exper i ­ entially verifiable , Ockham pr oposes modes of exposition 40

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o f inc ipit and de s init whi c h wo uld b e v al i d in e v e r y d ay u s a g e : in c ipit me an s " i t i s , and a little b e fo r e it w a s not ;' and de s in i t me an s " i t i s , and v e r y s o on it will not b e :' The d i s tinc t i on b e twe e n the s e i mp r e c i s e m o de s an d the p r e c i s e o n e s s t at e d p r e v i ou s l y , Oc kham note s , i s s i m i l a r t o Ar i s ­ t o t l e' s di s tinc t i on b e twe e n the t e r m " no w" und e r s t o od in a s t r i c t s e n s e , a s r e fe r r in g to an in di v i s ib l e p o int in time , and the s am e t e r m unde r s t o o d le s s s t r ic tly a s r e fe r r ing to the s ho r t time s u r r ounding the pre s e nt in s t ant- what W i l l i a m J ame s c all s the " s pe c io u s p r e s e nt :' 2 8 ALBERT

OF

SAXON Y

Alb e rt o f S axony (at P a r i s fr om 1 3 5 1 to c a . 1 3 6 2 and fi r s t r e c to r o f V i e nn a in 1 3 6 5 ) in hi s S o P.hi s m at a e xamine s the two t y p e s o f e xpo s ition of inc iP.it and de s init p r o p o s e d b y O c kham and ap p a r e ntly fav o r s the s e c o nd , l e s s p r e c i s e type o f e xp o s it i on . N e v e r the le s s , fo r the analy s i s o f s o ph ­ i s m s , Alb e r t r e ve r t s to the mode s s e t fo r th in the " De e xp o n ib i li b u s" o f the S ummul ae lo g i c al e s ; fi r s t , b e c au s e the s e a r e the m o de s i n c o mmon u s e (among S c ho o lme n p r e s umab ly) , and s e c o nd , b e c au s e we c an e xp r e s s mo r e c e r t ain a n d p r e c i s e me a s u r e s o f time , " i m a g ining a c c o r din g to the s e m o d e s o f e xp o s i tion indi v i s ib le in s t ant s in t i me , alt h o u g h in fac t th e r e a r e none :' S im i l a r 1 y , Alb e r t p o int s o ut , the a s t r onome r i m a g ine s m any c i r c l e s in the he a v e n s , altho u g h in f ac t the r e a r e none th e r e ; and the g e o me te r i m ag ine s indi v i s ib l e p o i nt s , although none s uc h e xi s t . T h e u s e o f s uc h fic tion s fac i l i t at e s the t r an s mi s s i on o f s c i e nc e . It i s c on v e ni e nt to po s it t e r m s l i ke " po int" and " in s tant" - t e r m s whi c h t he anti � b e l i e ve d to s tand fo r r e ally e xi s t i n g indi v i s ib l e s , b ut whi c h a c t u al l y r e fe r to e nt i t i e s e xi s t i n g o n ly s e c undum ima g inati one m- in o r de r to avo i d p r o l i xity o f s pe e c h .29

E

XPOSITION of Heyte sbur y

He yte s b u r y' s " De inc i p it e t de s init " 3 0 b e l on g s t o the l o g i c al pha s e of the d i s c u s s i on o f b e g inn ing an d c e a s ing . It r e p r e s e nt s an a d v anc e on the t r e ati s e s p r e v i o u s ly me ntione d , h o w e v e r , in it s mo r e inc i s i v e analy s i s o f t e r m s , and in the a p p l i c ation o f thi s an aly s i s t o c a s e s of m athe matic al inte r e s t .

E X P O UN D IN G

OF

TERMS

He yte s b u r y b e g in s with a s t ate me nt o f the u s ual mode s o f e xp o unding inc i p e r e and de s ine r e . " B e g inning" m ay b e

41

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e xp o un de d b y P.O s i t i o o f the p r e s e nt ( i . e . , b y the po s it in g o f e xi s te nc e i n the p r e s e nt in s t ant) an d r e m o t i o o f t h e p a s t (i . e . , b y d e n i al o f e xi s te nc e i n the p a s t) , thus : in the p r e s ­ e nt in s t ant it ( a thing , p r oc e s s , o r r e lation , e t c . ) i s , and i mme d i ate l y p r i o r to the p r e s e nt in s t ant it w a s not . O r " b e g innin g" may b e e xp o unde d b y r e motio o f t h e p r e s e nt and P.O s i t i o o f the futu r e , thu s : in the p r e s e nt in s t ant it i s not , and imme d i ate ly afte r the p r e s e nt in s t ant i t wi l l b e . S im i l a r ly " c e a s in g" may b e e xpo unde d in two way s , e ithe r b y r e m ot i o o f the p r e s e nt and P.O s it i o o f the p a s t , thu s : in the p r e s e nt in s t ant it (the thing , p r o c e s s , or r e l ati o n , e tc . ) i s not , and i mme diat e ly p r i o r to the p r e s e nt in s t ant it wa s ; or b y po s itio o f the pre s e nt an d r e m o t i o o f the futu r e , thu s : in the p r e s e nt in s t ant it i s and i m me di ate ly a fte r the p r e s e nt in s t ant it will not b e . 3 1 W e note he r e that Heyte s b u r y' s e xpo s i t i on s o f inc i P. it and de s init we r e c r itic i z e d o n pur e ly l o g i c al g r o und s by P e te r o f M antu a .32 Ac c o r ding t o P e t e r , an e xp on i b l e p r o p o s it i o n mu s t b e s uc h t h at n o t o n l y i s it i mp l i e d b y it s e xp o s ito r y p ro p o s it i on s , b ut that al s o i t i mp l i e s the m . T h e l att e r c ondition d o e s n ot ho l d fo r the s o - c al l e d e xp o s i to r y p r op ­ o s it i o n s o f inc i P. it and de s init , fo r the r e a r e two m o de s o f " e xp o s iti on" fo r e ac h of t h e te r m s , and b oth c annot b e t r ue fo r the s ame c a s e . P e te r p r e fe r s to s p e ak o f " c au s e s o f t r uth" o f the te r m s inc i P.it and de s init r athe r than m o d e s o f e xp o s ition . T hi s c r i t i c i s m i s r e p e at e d b y G ae t an o d i T hi e n e in hi s c o mme nta r y (fo l . 2 7 rb). As the who le o f He yte s b u r y' s d i s c u s s ion tu rn s on the app l i c ab i lity o f the s e mo de s of e xp o s it i o n or u c au s e s o f t r uth" t o p a r t i c ul a r c a s e s , w e m u s t e xa mine the me anin g whi c h he a s s i g n s t o the m mo r e c l o s e ly . In the fi r s t p l ac e , the te r m " im m e di ate ly" i s i t s e lf an e xp onibl e te r m , l i k e " to b e g in" and " to c e a s e :' T hu s P aul o f V e ni c e wo ul d e xpound the s e nte nc e " Im me di at e l y p r io r t o in s t ant a you e xi s t e d" a s fol l o w s : p r i o r to a y o u e xi s te d , and the r e wa s no in s t ant p r io r to a s uc h t h at b e twe e n that in s t ant an d a you d id no t e xi s t .3 3 T he s e nt e nc e " Imme d i ­ ate l y afte r i n s t ant b you will e xi s t :' ac c o r ding to P aul , r e qui r e s the fo l lowing e xpo s i t i on : a ft e r in s tant b y o u w i l l e xi s t , an d the r e w i ll b e no in s t ant aft e r b s uc h t h a t b e twe e n that in s t ant and b y o u w i l l not e xi s t .34 T he s e m o d e s o f e x ­ p o unding " i mme di ate l y" a r e s o p h r a s e d a s t o e mb o dy the Ar i s t o t e l i an doc t r ine th at no two in s t ant s c an b e i mme d i at e to o r i n c ontac t with o ne anothe r ; b e twe e n any t w o in s t an t s the r e mu s t a lway s b e a l e n g th of time . T he pe c ul i a r c ha r ac te r o f the c ont inuum- the fac t t h at

42

D E INCI P IT E T D E S INIT

no two points of a line or instants in a period of time can be immediate to one another, and that every segment of line or time must contain an infinite numb er of points or instants- enters frequently into Heytesbury's discussion in the Regule and SoP.hismata . Thus in Sophisma 29, " Deus erit in quolibet instanti non existens :' the following argu­ ment is posed: God will be imm ediately after the present instant, and there will be no instant immediately after the present instant , hence God will be when He will not be in some instant (fol . 1 5 lvb) . That there can be no instant im ­ mediately after a given instant follows from the fact that two instants cannot be immediate to one another . Heytesbury replies that neither God nor anything else in the world will be immediately after the present instant . A distinction is to be drawn between the statement ( 1 ), "A will be immediately after the present instant : ' and the statement (2 ), " Immediately after the present instant a will be: ' In accordance with the rule that a proposition is to be p roved by reasoning on the first provable term oc curring in the sentence which states the proposition, the second statement is exponib le in terms of the exponible term " immediately : while the first statement is to be analyzed in terms of the supposition of the term "i!:.:' The second statement means: after the present instant a will be, and there will be no instant after the present instant such that between that instant and the present instant a will not be . The first statement, on the contrary, means that a will be prior to all future instants, and hence prior to every future instant, which is impossib le; for nothing will be able to be sooner than some instant will be, since immediately after the p resent instant there will b e some instant-although no instant will b e immediately after the p re sent instant , i . e . , prior to all future instants (fol. 1 5 lvb).3 5 The difference in meaning between the two statements, " Immediately after the present instant some instant will be" and " Some instant will be immediately after the prese nt instant" turns on the fact that the word " immediate :' as previously noted (su2ra, pp . 1 4- 1 5 ), has a vim con ­ fundendi the term which follows it. Hence in the statement " Immediately after the present instant some instant will be : ' " some instant" has confused supposition, and it is not permissible to descend to the individuals for which it stands. In the statement " Some instant will be immediately after the present instant : ' on the contrary, "some instant" has dete rminate supposition , and the statement thus means that

43

W ILLIAM HE Y T ESB UR Y

one of the infinite number of ins tants whic h will be after the pre sent in s tant will be immediately after the pr e s ent ins tant , without lap se of time. The fir s t s tatement is true , the sec ond fals e; for although ther e will be no ins tant after the pr e s ent in s tant s uc h that between it and the pre s ent in s tant there will not be s ome ins tant (and this is the meaning of the fir s t s tatement) , yet none of the ins tants whic h will follow the pr esent ins tant c an be de s ignated as fir s t (as would be required for the tr uth of the s ec ond s tatement). 3 6 In effec t , the ver bal c onvention s used by Heyte s bury and his follower s are bas ed upon an analys i s of the c ontinuum whic h goes a s tep beyond the immediate intuition of c on ­ tinuity as a flux or flow , and s tres s e s its c harac ter as a dense s et of point s or ins tant s .3 7 One further note in c onnec tion with the expos ition of inc iP.it and des init: the s e verbs amplify the s uppos ition of the subjec t of the s entenc es in whic h they s tand , so that in the c as e of the ver b inc ipit the s ubjec t s tand s for that whic h is or begins to be , and in the · c as e of the ver b de s init the s ubjec t s tand s for that whic h is or c eases to be. If this wer e not the c as e , nothing c ould b egin to be by r emotio of the pre s ent and P.O sitio of the futur e , and nothing c ould c ea s e to be by r emotio of the pre s ent and P.O s itio of the pas t; for the s ubjec t of the s entenc e , in refer r ing only to what is , would r efer to nothing .38 M A J OR

A S S E RT I O N S

Following his re s tatement of the modes of expos ition of inc ipit and des init , Heytes bur y makes the following as s er ­ tions (fol . 2 3va): FIRST . - Of whatever kind s omething will begin or c eas e to be , it will begin or c eas e to be s uc h in s ome ins tant. ( "Qualiter c umque autem intelligitur unum vel aliud quale c umque inc ipiet aliquid e s s e vel des inet , ips um in aliquo ins tanti inc ipiet ve 1 de s inet elle tale ." ) SEC OND . - Of whatever kind s omething will be . of whic h kind it is not now, it begin s or will begin to be s uc h . (u Quale _ c umque erit aliquid quale ips um nunc non es t , ip sum inc ipit vel inc ipiet es s e tale : ' ) THIRD .- Of whatever kind s omething was , o f whic h kind it will sometime not be , it c eases or c eas ed or will c eas e to be s uc h. ( "Qualec umque fuerit aliquid , c uius modi ips um aliquando non erit , ip s um vel des init vel des inebat vel de s inet es s e tale: ' ) The fir s t as s er tion , ac c ording to Gaetano, i s bas ed on the under s tanding that every thing whic h will be or was ,

44

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will be or was in some instant; fr om this it f ollows that ev er y thing which begins or ceases to be, begins or ceases to be in some instant. T he second asser tion is demonstr ated as f ollows: if something will be such-and-such, say white, and is not so now, we may consi der the whole time dur i ng wh ich it will be white and the whole ti me dur i ng which i t wil l be not-white; ther e will be a si ngle i nstant which di ­ v ides the two ti mes, and i t f ollows, whether i n that instant it is white or whether i n that instant i t i s not white but i m­ mediately af ter war ds i s so, that i n that instant it begins to be white. T he thi r d asser ti on is demonstr ated i n a sim ­ ilar manner .3 9 AN A L Y S I S

OF

SOPHISMS

T he def ense of these asser tions leads Heytesbur y into an analysis of numer ous sophisms, many of them i nv olv ing aggr egates, limits, and inf initesimals. We li mit the fol­ lowing account to those sophisms havi ng mat hematical signif icance. FIRST . -We suppose that Socr ates knows ten pr oposi ti ons, of which he later ceases to know one. It f ollows that Soc­ r ates knows ten pr opositions and at some ti me h e will not know ten pr opositi ons; but it does not f ollow- so the ob ­ j ection r uns- that he ceases to know ten pr opositions, Heytesbur y' s r ule to the contr ar y notwithstanding. F or accor di ng to the case posited, ther e is only one pr oposition which Socr ates ceases to know (f ol. 23v a). Heytesbur y r epli es denyi ng the v alidity of the ar gument, and maintaining that Socr ates ceases to know ten pr opo­ siti ons. F or if we consi der the pr opositi on which he ceases to kn ow together wi th the other nine which he continues to know, i t f ollows that he ceases to know thes e ten taken col­ lectiv ely; f or i n some instant he knows these ten, and i m ­ medi ately af ter thi s instant- hav ing ceased to know one of the ten- he does not know these ten pr opositions, although one pr oposi ti on i s ev er y pr oposi tion wh ich Socr ates ceases to know. On the other hand, Heytesbur y denies that ten pr oposi­ tions cease to be known by Socr ates. In the statement i n the passi v e v oi ce, " T en pr oposi ti ons cease to be known by Socr ates;' the ex poni ble ter m " cease" i s affi r med of t he plur al ter m " ten pr oposi ti ons;' which stands div isiv ely f or each of the ten. ("Supponit eni m talis ter minus plur alis numer i pr o quolibet ei us supposito r espectu huius ter mini ex ponendi:' ) Hence, although ten pr oposi tions ar e known by Socr ates and no ten pr opositions immedi ately af ter the 45

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HEYTESBURY

pre s ent ins tant will be known by Soc rates , it doe s not fol ­ low that ten propos itions c ease to be known by Soc rate s , unles s eac h of the ten c eas e s to be known by Soc rates . More generally , in order that some individuals s hould begin or c ea s e to be s uc h - and - s uc h , it i s nec es s ary that eac h of the individuals begin or c ea s e to be s uc h. But in the s tatement in the ac tive voic e , usoc rates c ea s e s to know ten propo s ition s ;' the exponible term u c eas e s" i s affirmed of the s ingular term " Soc rates ; ' and Soc rates indeed c eases to know ten propos ition s if immediately after the present in s tant he will know but nine; henc e thi s s tate ­ ment i s to be c onc eded.4 0 Thi s verbal c onvention e s tabli s hed , i t follow s that how ­ ever many propos itions Soc rates knows , if he c ea s e s to know but one of them he c ea s e s to know them all ( " all" being taken here in a c ollec tive s en s e). Indeed , if we s up ­ pos e that Soc rates c an know an infinite number of propo ­ s ition s at one time , it would be pos s ible for him to c ea s e to know all thes e infinite propos itions , although n o partic ular one of them would he c ea s e to know. Heytes b ury does not demon s trate thi s c onc lu sion , but argues by mean s of an analogy. We imagine a rec tangular s urfac e , the end s of whic h are des ignated a and b , and whic h i s marked off s uc c e s s ively along its length in proportional parts in the ratio of one-half , s o that the firs t proportional part ex tend s from b to the middle of ab , and the infinite number of s maller, � uc c eeding proportional parts ( 1/4 , 1/8 , 1/ 16 , etc .) are terminated at a. We s uppo s e further that Soc rates now s ees the whole of thi s s urfac e , and that s ome objec t now begins to c ome between the s urfac e and Soc rates ' eyes , s tarting at a and moving toward s b , and s uc c es s ively ob ­ s c uring more and more of the s urfac e from vi s ion. Heyte s bury admits that , at the ins tant in whic h the motion of the oc c luding objec t begins , there begin s to be oc c luded s ome proportional part from Soc rate s ' vi s ion ( " lam inc ipit aliqua illarum partium oc c ultari a vi s u Sorti s" ); but he denies that any one of the proportional parts begins to be oc c luded from Soc rates ' vi s ion ( " Nulla inc ipit oc c ultari" ). In the fir s t s tatement the term us ome proportional part" follow s the verb " begins ;' whic h has a vim c onfundendi the term whic h c omes after it; and thus it i s not permi s s ible to des c end to the partic ulars for whic h " s ome proportional part" s tand s . In the s tatement " Some proportional part begins to be oc c luded from Soc rates' vi s ion ;' on the c on ­ trary , " s ome proportional part" c omes firs t in the s entenc e , and thus has determinate s uppos ition , referring to s ome 46

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parti cul ar part. The statement i s fal se, because there i s no determi nate proporti onal part whi ch i mmedi atel y after the start of the moti on will be occl uded. F or not onl y i s i t true to say that there i s no determi nate part of ab whi ch i mmedi atel y after the start of the moti on wi ll be traversed -whatever part will be occl uded, the hal f of i t wi ll have been occl uded previ ousl y, and the hal f of the hal f, and so on; but i n fact there i s no fi rst proporti onal part starti ng from a. Therefore i mmedi atel y after the start of the moti on Socrates wi ll cease to see an i nfi ni te number of propor­ ti onal parts, al though no proporti onal part will cease to be seen by hi m. Accordi ng to Heytesbury, a si mil ar argument woul d appl y i n the case of kn owl edge, i f i t i s admi tted that Socrates can know si mul taneousl y an i nfi ni te number of proposi ti ons.4 1 In thi s fi rst sophi sm we observe a passage from the con­ si derati on of the coll ecti ve a nd di vi si ve aspects of a fi ni te aggregate to a si mil ar consi derati on of an i nfi ni te aggre gate, and thence to the consi derati on of the conti nuum. SECOND.-We are to i magi ne that Socrates i s one foot l ong, and Pl ato two feet l ong; and that both Socrates and Pl ato i ncrease i n l ength uni forml y for an hour, Socrates at a faster rate than Pl ato, i n such manner that i n the fi nal i nstant of the hour each of them woul d be three feet l ong i f he then exi sted. But we are further to suppose that at the end of the hour each of them ceases to exi st, so that the fi nal i nstant of the hour i s the fi rst i nstant of thei r nonbei ng. In other terms, three feet of l ength i s the mi ni mum l ength whi ch nei ther Socrates nor Pl ato will attai n i n the course of augmentati on. It i s argued that Socrates will be of such si z e as Pl ato will be, and that he i s not now of such si ze as Pl ato will be, and yet that he nei ther begi ns nor will begi n to be of such si z e as Pl ato will be.42 It i s al so argued i n the same case that Socrates will be equal to Pl ato, and that he i s not now equal to Pl ato, and yet that he nei ther begi ns nor will begi n to be equal to Pl ato.4 3 The fi rst argument is conceded by Heytesbury, and the second deni ed, and the concessi on and deni al are j usti fi ed by reference to the foll owi ng rul e: i f to an affi rmati ve proposi ti on i n the future tense i s annex ed a negati ve prop os i ti on i n t he presen t tens e, i ncepti on foll ows i f the affi rm­ ati ve prop osi ti on i s such that, i n order to be true, i t mus t be veri fi ed for some i nstant; i f i t may be true wi thout b ei ng veri fi ed for any i nstant, i ncepti on doe s not necessari l y fol ­ l ow.44

47

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HE Y T ESB UR Y

An example of a future affirmative whic h for its truth requires verific ation for some instant is , " Soc rates will be white ;' for if Soc rates will be white he wi ll be white in some instant ; and if to this statement is added the negative proposition in present tense , " Soc rates is not white ;' it fol ­ lows from the two propositions that Soc rates be gins or will begin to be white . An example of a future affirmative whic h for its truth does not require ve rific at ion for some instant is " Soc rates will be of suc h a si ze as P lato wil l be :' 4 5 That the latter proposition is true in the c ase posited fol ­ lows from the fac t that , by the time the future hour has run out , Soc rates wi11 have been as 1ong as P1ato wi11 have been ; for eac h will have been of every length less than three feet , and neither will have been of a greater length. But that the proposition is not verifiab le for any instant may be shown as fol lows : If in some instant Soc rates is to be of suc h size as P lato will be , this instant must be the final instant of the hour , or before or after the final instant of the hour . It c annot be the final instant of the hour, since P lato and Soc rates will then no longer exist ; and a fortiori it will not be ab le to be after the final instant of the hour. If it will be prior to the end of the hour , c al l it h. Now ac c ording to the c ase posited , between b and the end of the hour Soc rates and P lato c ontinual ly augment in size , Soc rates at a faste r rate than P1ato; hen c e if in in stant b Soc rates is of suc h a size as P lato , it follows that by the end of the hour three feet will not be the minimum length whic h Soc rates does not attain , but will b e less than his final length- c ontrary to the c ase posited .4 6 Henc e the proposition " Soc rates will be of suc h a si ze as P lato wil l be" is not verifiab le i n the c ase posited for any instant .4 7 More generally , verifiability for some instant is not required for the truth of c omposite c omparisons b y means of the positive degree of the adjec tive in which b oth that whic h is c ompared and that to whic h it is c ompared are of the future or past , and are not limited to a given instant or to a determinate degree or amount of the quantity or quality with respec t to whic h they are c ompared. Examples of suc h c omparisons are : Soc rates wil l be as good as P lato wil l be; Soc rates wil l move as fast as P lato wil l move. Henc e from the propositions , " Soc rates will b e as good as P lato will b e" and "' Soc rates is not now as good as P lato will b e ;' it does not follow that Soc rates b egins or will begin to b e as good as P lato will b e . And from the prop ­ ositions , " Soc rates will move as fast as P lato will move" and " Soc rates does not now move as fast as P lato wil l . 48

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move ;' it does not follow that Soc rates begins or will begin to move as fast as Plato will move .4 8 On the other hand , verifiability for some instant is re quired for the truth of comparisons whic h are made by means of the comparative degree of the adjec tive or c om ­ parisons made by means of the positive degree of the adjec tive in which one of the terms of the comparison is limited to a c ertain instant or to a certain degree or amount of the quality or quantity with respec t to whic h the comparison is made. Thus if the proposition " Soc rates will be whiter than Plato will be" is to be true , it must be true in some instant ; and therefore from the two prop ­ ositions u soc rates will be whiter than Plato will be" and u soc rates is not now whiter than Plato will be ;' it follows that Soc rates begins or will begin to be whiter than Plato will be . Similarly the proposition " Soc rates will be of suc h si ze as Plato is ;• or " Soc rates will be of such size as Plato will be in instant a ;• is true only if there will be some instant in which Soc rates will be of the same si ze as or greater than Plato is at the present instant or in instant a; and therefore from the two propositions " Soc rates will be of such size as Plato is" and " Soc rate.s is not now of such si ze as Plato is ;' it follows that Soc rates begins or will begin to be of such size as Plato is .4 9 . The fac t that the proposition " Soc rates will be of such size as Plato will be" is not necessarily verifiable for an instant does not , ac c ording to Heytesbury , contradic t the rule that of whatever kind a thing will be it will be suc h in some instant . For the rule refers to the qualific ation of a thing by a quality ; but the proposition cited signifies a c omparison or relation (fol . 2 5ra). In Sophisma 3 , " 0mnis homo est to tum in quantitate ;• Heytesbury states that if something was or will be , it was or will be either for a time or for an instant ; but he denies that , from the fac t that Soc rates will be of such size as Plato will be , it fol ­ lows that Soc rates will be of suc h a si ze as Plato will be for a time or for an instant. Similarly from the proposition " This number will be ;' where "this number" refers to two who will never be simultaneously , it does not follow that this number will be for a time or for an instant (fol. 8 7va). We observe that Heytesbury appears to allow a formal existenc e to relations , c lasses , etc . , but denies that they are things. We return to the second argument in the c ase posited , ac c ording to whic h Soc rates will be equal in length to Plato , and is not so now , and yet neither begins nor will

49

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begin to be equal to P lato. Heytesbury denies that Soc rates will b e equal in length to P lato; for in order that a relation of equality should hold , it is necessary that it should hold in some instant , but there is no instant in the c ase posited in which Soc rates will be equal in length to P lato . Against this c onc lusion the fol lowing argument may be raised : Immediately prior to the final instant of the hour (c all the final instant a ) Soc rates will be of suc h size as immediately prior to a P lato wil l be; and immediately prior to a Soc rates and P lato will exist simultaneously ; therefore immediate ly prior to a Soc rates and P lato will be equal in length . Heytesbury denies the consequenc e. Indeed , he holds that immediately prior to a Plato will be larger than Soc rates , and that immediately prior to a Soc rates will be smaller than P lato ( " Immediate ante a erit P lato maior Sorte ; immediate ante a erit Sortes minor P latone" ). For there is no instant prior to a suc h that in that instant P lato will not be larger than Soc rates is in the same in ­ stant . 50 On the other hand , Heytesbury denies that immediately prior to a P lato will be larger than immediately prior to a Soc rates will be (" Et tamen immediate ante a non erit P lato maior quam immediate ante a e rit Sorte-;,, ); as he would also deny the statement that P lato wil l be larger than Soc rates wil l be ("Non se quitur : P lato erit maior Sorte , ergo P lato erit maior quam erit Sortes" ). In these c omposite c omparisons , the repetition of the verb in c on ­ nec tion with eac h term of the comparison permits that verific ation be sought b y independent c onsideration of eac h term of the comparison for any of the time of its future existenc e . B ut sinc e in the c ase posited Soc rates will be of every length less than three feet before he c eases to be , and similarly P lato , either of these state ments is false . 5 1 P rec ision of grammatic al usage thus permits Heytes ­ bury to formulate , without benefit of algeb ra , the c harac ter of a limiting process . THIR D . - We imagine a point a whic h wil l traverse a line b . It is argued that a will traverse line b , and does not now traverse line b , and yet neither begins nor will b egin to traverse line b. Similarly , a wil l sometime traverse line b , and sometime a - wil l not traverse line b- after it has traversed it , but a will not c ease to traverse line b (foll . 2 4vb , 2 5vb ). 50

DE

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E T D E S INIT

Both thes e conclus ion s mus t be conceded , for the prop ­ o s ition " a will travers e line b" i s a future affirmative which cannot be veri fied for any ins tant but only for the whole time during which a will move from the beginning of b to the end of b. Si milarly , Heytes bury denies that point a i s , was , or will be travers ing s ome part of line b ("Et tamen a punctus nee e s t , nee fuit, nee erit pertran ­ s ien s aliquam partem b linee" ); for the grammatical form of the s e propos iti on s (verb "to be" together with an ad ­ jectival participle) neces s itates that they be veri fied for s ome ins tant , which i s impos s i ble. On the other hand , he hold s that line b will begin to be traversed , for there will be a firs t in s tant in which b will have been travers ed by a , and inci2iet will be expounded by pos itio of that ins tant and remotio of the previous time . 52 The s e res ult s appear to be in agreement with the funda ­ mental Ari s totelian premi s s according to which " there cannot be motion in that which i s indivi s ible :' namely , in an ins tant . 53 For Ari s totle the ins tantaneous in time and pos ition i s irrelevant to the analys i s of motion as motion; that which i s i n motion mu st not , in thought , be brought to a s tand while it i s in motion . Otherwi s e we s hould be involved in Zeno' s paradox of the flying arrow: that which i s in locomotion i s alway s occupying an equal s pace at any moment , and i s therefore motionles s .54 It i s to be obs erved , however, that the above res ults follow for Heytes bury from the particular character of the propo s itions concerned , which impos es a particular mode of verification in each cas e , rather than from a general denial of the ins tantaneous in motion . We s hall find that the in s tantaneou s , regarded as a limit , plays a con s ider ­ able role in Heyte s bury ' s analy s i s of movement. TH E

IN S T ANT AN E O U S

AS

A

L I M IT

Following the termini s t pos ition , Heyte s bury hold s that there i s nothing in nature which i s an ins tant as s uch , nor time as s uch , nor motion as s uch; in stants and time are not di s tingui s hed realiter from the celes tial s phere , and motion i s not d i stingui s hed realiter from the body which move s . At the s ame time , Heytes bury admits " according to the common mode of s peech" that everything which i s , whether time or motion or ins tant , i s in an i n s tant , in the s en s e that it i s in s tantaneous ly mea s ured by an ins tant . 55 In the above ca s e of a' s travers ing of line b, Heytes bury admit s the following conclu s ion s P.robabiliter: in any in 51

WILLIAM HEYTESBURY stant � will tr aver se one point of b, and so in a finite numbe r of instants a will trave rse as many points as ther e are instants, although in no finite number of instants will a tr ave rse any pa rt of line b; in all the intrinsic instants of the time of its movement (an infinite number of instants) a will tr averse the whole of b, because it will t r aver se all the points of b; and given any par t of the time of its movement, a will traverse in all the instants of thi s par t all the points of b which cor respond to the given instants; and finally, any point in motion tr ave r ses the quiescent point which it instantaneously touches. 56 These pr opositions a r e presumably conceded only secundum imaginationem . It may be questioned whethe r , with these concessions, Heyte sbur y is not once mo re confr onted with Zeno's dif­ ficulty. Heytesbur y's reply appear s to be that the instant in time and the instantaneous position in movement are always to be regar ded as u limits :' The following r esults, having to do with the instantaneous r egar ded as a limit in movement or in time, are stated by Heyte sbur y and proved by his commentator s in applying the modes of exposition of u immediate : inci2it, and desinit. F IRST . - According to Heytesbu r y , at any instant in its motion point a begins to tr aver se some part of line b, and also ceases to traverse some par t of line J2. ( " P unctus a continue fluens continue incipiet pertransir e aliquam lineam vel spacium, et continue desinet pe r tr ansir e ali ­ quam lineam" ) . " To begin" and "to cease ;' we r ecall, have a vim confundendi the ter m which follows, so that in both clauses " some par t of line b'' stands confuse tantum, and it is not permissible to descend to the particular s to which the phr ase r efers. Thus ua begins to t r aver se some part of line b" means that a does not in the present in stant tr aver se some part of line b (for in no instant does a t r averse any part of line £), and immediately after the pr esent instant a will tr averse some par t of line b. In con­ nection with this case Heytesbur y modifies the r ule accor ding to which the futur e affirmative pr oposition in the exposition of inci2it must r equire ver ification for some instant : inception also follows if the futur e affir mative is verifiable for an arbitrarily small length of time. Although the pr oposition " Immediately after the pr esent instant a will tr averse some part of line b" is not ver ifiable fo r any instant, no particular length of time is requir ed fo r its verification; any length of time, however small, and ap­ pr oaching an instant as a limit, will be sufficient in or der that a will traverse some (indeterminate) par t of line b .

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This result follows from the supposition c onfusa tantum of " some part of line b :' 57 SEC O N D. -A similar proposition is stated in Heytes­ b ury 's " De relativis" (fol. 2 lra) and proved by Gaetano and the author of the " Probationes c onc lusionum" : Now there begins to be traversed some part whic h for the time ending at the present instant is as a whole still to be trav ­ ersed ( " lam in c ipit aliqua pars esse pertransita que per tempus terminatum ad instans quod est presens erit sec undum se totam pertranseunda" ). T he proof proc eeds as follows: (a) Now no part is traversed which for the time ending at the present instant is as a whole still to b e traversed; this statement is ob vious. {b) But immedi­ ately after the present instant there will have been trav­ ersed some part whic h for the time ending at the present instant is as a whole still to be traversed; for there will b e no instant after the present instant suc h that prior to that instant some (indeterminate) part will not have been traversed. From (a) and (b) the required c onc lusion fol­ lows. 58 THlRD . -Another proposition, stated in the " De relativis" (fol. 2 lra) and proved by Gaetano and the author of the " Probationes c onc lusionum ;' has to do with the instants after a given instant c onsidered as a dense set: Now there b egins to be some instant whic h after the present instant will be gin to be. ( " lam inc ipit esse instans quod post hoc inc ipiet esse" ). The term " instant ;' sinc e it follows the verb " begins ;' has supposition c onfusa tantum. The demon ­ stration runs as follows. (a) Now there is no instant whic h after the pre sent instant will begin to be; otherwise the instant now present would beiin to be after the present instant, whic h is impossible. (b) Immediately after the pre sent instant there will be an instant whic h after the present instant will begin to b e. For- and we employ here the usual exposition of " immediately" - there will be no instant after the present instant suc h that between that instant and the present instant there will not be some in­ stant whic h after the present instant will begin to be. 60 From (a) and (b) the required c onclusion follows. FO URTH. - Ac c ording to one of the c onc lusions given in Heytesbury 's " De inc ipit et desinit" (fol. 2 6va) and proved in the " Probationes c onclusionum ;' Soc rates begins to be moved faster than he begins to be moved. ( " Sorte s velo c ius inc ipit mo veri quam ipsemet inc ipit moveri" ). We suppose that So crate s begin s to be moved, his speed i n­ c reasing steadily from zero degree of veloc ity, and the

53

W I LL I AM HEYTESBURY ar gument runs as follow s. (a) Socrate s i s not now moved fa s ter than he begin s to be mov ed, for in the pr e s ent in s tant he i s not moved, but begin s to be moved by remotio of the pre s ent. (b) Imme diate 1 y after the pr e s ent in s tant Socr ate s will be moved fa s ter than he b egin s to be moved. For there wil l be no in s tant afte r the pr e s ent in s tant s uch that between that in s tant and the pre s ent in s tant Soc rate s will not be moved fa s ter than he begins to be moved . For con s ider any in s tant c afte r the pre s ent in s tant . B etween in s tant £ and the pr esent in s tant there is an inter vening time ; and in the po s terior half of thi s time Socr ate s wil l be moved with s ome (aver age) deg ree of vel ocity ; call it d . B etween d deg r ee and z ero de g r ee of velocity there i s a latitude of velocity, and thi s latitude wil l not be acquired in s tantane ­ ou s ly ; hence time will elap s e after the pr e s ent in s tant befor e Soc r ate s acquir e s d deg r ee of velocity . Therefo r e in the po s terior half of the time between £. and the pr e s ent ins tant Socr ate s will be moved fa s ter than he be gin s to be mov ed, and the s ame ar gument hold s whatever in s tant £. i s cho s en. F r om the premi s s e s (a) and (b) the required conclu s ion follow s . 6 1 F IF TH. - Finally, w e note another of Heyte s bur y's con­ clu s ion s which i s pr oved by Gaetano and the author of the "P r obatione s conclu sionum ;' and which concer n s the s ta r t of motion: B oth Socr ate s and Plato infinitely s l ow ly be gin to be moved, and yet Soc r ate s infinitely more s lowl y be­ gin s to be moved than P lato ( " Tam Sor tes quam P lato utr umque in infinitum tar de incipit move r i, et tamen So r te s in infinitum tar diu s P latone" ). Thi s conclu s ion involve s the ter m "infinite" in it s s yncategor matic s en s e, which i s ex­ ponible; thu s " infinitely s lowly" means unot thu s s low ly if not twice a s s lowly, and thr ee times a s s lowly, and s o on in infinitum" ; and " infinitely more s lowly than" mean s " not thu s mor e s lowl y than if not twice mor e s low ly than, and three time s more s lowly than, and s o on in infinitum" ; For the pr oof of the cone lu s ion we s uppo s e that both Soc r ate s and Plato begin to be moved, the velocity of each s teadily increa s ing fr om zero deg ree of vel ocity, P lato' s vel ocity with a con s tant de g ree of acceler ation, and Socrate s' ve­ locity with a rate of acceler ation which it s elf s tar t s at zero deg ree of acceler ation and increa s e s unifor ml y a s Soc r ate s i s moved . (The Schola s tic definition s of unifor m and non­ unifor m accele r ation will be dis cu s s ed in Chapte r 4 .) The pr oof of the fir s t clau s e of the conclu s ion pr oceed s a s follow s : (a) N eithe r of them (Plato and Socr ate s) now with infinite s lowne s s is moved ( " Uter que iUor um nunc non in infinitum tar de movetur" ). 62 F o r neither P lato no r Socrate s

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DESINIT

is moved in the present instant , but each now begins to b e moved. (b) Immediately after the present instant each of them with infinite slowness wi ll be moved (" Immediate post hoc uter que i llorum in infinitum tarde movebitur" ). For immediately after the present instant each of them wi ll be moved , and not thus slowly if not twice as slowly , and three times as slowly , and so on in infinitum ; for each acquires new degrees of velocity successively , so that there is no degree of velocity which either of them wi ll acquire such that he will not ac qui re the half of it fi rst , and one fourth of it previously to that , and so on . From (a) and (b) the re quired result follows . The second clause of the conclusion , acco·rding to which Socrates infinitely more slowly begins to be moved than P lato , is proved as fol lows . If Socrates' rate of accelera ­ tion we re constantly half that of P lato , Socrates would at each instant of his motion be moved with half the velocity of P lato in the same instant ; if Socrates' rate of accelera ­ tion were constantly one fourth that of P lato , Socrates would at each instant of his motion be moved with one fourth the velocity of P lato ; and so on decreasing the ratio in infin itum . But in the case posited Socrates infinitely more slowly begins to accelerate his rate of motion than P lato ( " So rtes in infin itum tardius incipit intende re motum suum quam P lato" ); for he begins to accelerate his rate of motion with a degree of acceleration which itself starts at zero and increases , whereas P lato begins to accelerate his rate of motion with a constant , finite degree of accel ­ eration . Hence the conclusion. 63 In effect , Heytesbury compares two infinitesimals and finds them to b e of different order. In present -day mathe ­ matics , an infinitesimal is defined as a variable which in its variation finishes by becoming and rem aining arbitrari ly small. 64 In the case under consideration both the velocity of P1ato (vp ) and the v e1ocity of Socrates (vs ) are infinites i mals for time .1 approaching zero. Two infinitesimals are said to be of the same order if the limit of their quotient is a finite number , and of different order if the l imit of their quotient is either zero or infinite. 6 5 In the case under con side ration the limit of the quotient v p /vs as !_- 0 is in ­ finite. For in analytic terms , vp = apt where ap is the con ­ stant rate of acceleration of P lato ; and for Socrates a 3 = kt where k is some constant , so that by integration Vs = (kt 2 )/2 ; the quotient of the two infinitesimals vp/vs is (2apt)/ (kt 2 ) = C/t where C is the constant (2ap)/k ; and the l imit of this quotient for diminishing !_ is c learly infinity . Another way of exhibiting Heytesbury's conclusion ana -

55

WILLIAM H EYTESBURY

ly tically is to state the equations for the distances tr aver sed by Plato and Socr ates. F or Plato, s = J vP, dt = J ap tdt = (apt2 )/ 2 ; for Socr ates, s8 = J vsdt = J,P,( (kt2 )1/ 2 )dt = l k t3 )/ 6 . We have to do with two equations of differ ent degr ees, one of the second degr ee, and the other of the thir d degr ee. Accor ding to a r esult of N ewton·, the angle of contact with the !...- ax is is infinitely less in the case of the latter than in the case of the for mer .66 It is per haps fair to conclude that H ey tesb ur y' s analy sis of the instantaneous in motion and time goes about as far as it is possible to go by pur ely ver bal means, and withou t r ecour se to the sy mbolic techniques of the calculus.

56

CH APTER

3 De m a x 1 mo et m 1n1 mo

C

HAP TER V of the Re gule , .. De maxima et minima ;' is concerned with the question of setting a bound to the range of a potency; that is, with such problems as that of setting a bound to the power of Socrates to lift weights, or to his power to see objects at fixed or varying distances. The question has its origin in a passage of Aristotle 's De caelo, and was extensively discussed dur ­ ing the fourteenth and fifteenth centuries. To unclerstand the peculiar character of Heytesbury 's treatment of it , we must follow this development in considerable detail! To the problem of setting a boundary to potencies Aristotle had provided a solution fully in accord with every ­ day use of language: "we may take is as settled that what is, in the strict sense, possible is determined by a limit ­ ing maximum:' 2 Thus when we speak of a man 's being able to lift one hundred talents· , or to walk one hundred stades , we mean that he can accomplish so much or less, but no more; and in so saying we feel we have defined the extent of his power to lift or to walk - u we feel obliged in defining the power to give the limit or maximum:' 3 Confronted with this passage of Aristotle , the School ­ men of the fourteenth century were led on to an analysis of the problem which was at once more rigorous, and less in touch with common -sense experience; an analysis deriving, on the one hand , from Aristotle 's discussion of the continuum and , on the other, from a highly specula ­ tive set of rules and distinctions regarding the nature of powers and resistances . As a preliminary illustration

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of this analysis we may give the proof in the anonymous " Probationes conclusionum" which accompanies Heytes ­ bury 's Regule of the fol lowing proposition : there is to be assigned , not a maximum weight which Socrates suffices to carry , but rather a minimum which he does not suffice to carry. To show that the re cannot be assigned a maximum weight which Socrates suffices to carry , the author uses a reductio ad absurdum. Suppose there were such a weight , and let it be a. N ow since Socrates is able to carry a , his active power must be greater than the resistance of weight a by some excess , for from a ratio of equality no action or motion results. This excess must either be divisible or indivisible ; but it will not be indivisib le , for an indivisible added to a divisible magnitude does not render the latter greater or less (i.e. , there is no such thing as an indivisible magnitude); therefore it will be divisible. Since the excess is divisible , we may consider the hal f of it , and argue as follows. If the power of Socrates exceeded the resistance of a by half the original excess , Socrates would still be able to carry a ; hence by the whole of the original excess Socrates can carry a and more than a; therefore a is not the maximum weight which Socrates can carry- against the original assumption. And as it has been argued con ­ cerning a , thus it can be argued with regard to any weight which is greater than a but of less resistance than the power of Socrates. That there is to be assigned a minimum weight which Socrates does not suffice to carry follows at once. For let the weight which is equal to the power of Socrates be taken, and let it be a. Then Socrates does not suffice to carry a nor any weight greater than a , but he suffices to carry any weight less than a , since his power will exceed the resist ­ ance of any such (or in Scholastic terminology , will have to the resistance of any such a ratio of greater inequality). Hence a is the minimum weight which Socrates does not suffice to carry.4 The proof is rigorous , if two assumptions are granted : (1 ) that the rule according to which no action or motion ac·c rues from a ratio of equality between power and re sistance is correct ; 5 and (2 ) that the sequence of all pos sible weights forms a continuum. 6 From this fir st illus tration , indeed , it is possible to generalize , and to say that the medieval discussion of maxima and minima presents two major aspects of interest : (1 ) a physical or dynamical aspect , involving the relations between powers

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and resistanc es; and (2 } a purely mathematic al aspec t , involving the c onc epts of c ontinuity and limit. As physic ists- or we might say " spec ulative physic ists ;' sinc e the theory here was seldom suc h as to yield c on ­ c lusions admitting of empiric al verific ation- the Sc hool ­ men introduc ed numerous distinc tions between types of powers and resistanc e , distinc tions whic h nec essitated a variation in the manner of assigning limits to potenc ies. For example , how is the power to be bounded if it is ac tive or passive , debilitable (subjec t t o weakening} or indebili ­ table , finite or infinite ? If a power divides a resisting medium , how will the boundary be assigned when the medium is uniform; when it is difform (nonuniform} ? What alterations in the form of the analysis must be introduc ed when the veloc ity and duration of the movement effec ted by a power are taken into ac c ount ? The mathematic al interest in the disc ussion lies , first of all , in the use of two kinds of boundaries for c ontinuous sequenc es: one , in whic h the element serving as a boundary is itself a member of the sequenc e of elements whic h it bounds (maximum quod sic or minimum quod sic}; and the other , in whic h the element serving as a boundary stands outside the range of elements whic h it bounds (maximum quod non or minimum quod non}. In the prec eding c hapter we have noted ( 1 } that this distine tion is not c apable of empiric al verific ation; (2 } but that, in rec ent times , it has proved of importanc e in the theory of func tions and in the definition of irrational numbers; ( 3 ) that the medieval Sc hoolmen , on the c ontrary , were c onc erned with it only in c onnec tion with magnitudes whic h are , at least hypo ­ thetic ally , physic al. Onc e more , however, we find that the medieval disc ussion , partic ularly where it enters the c on ­ text of logic , develops in extremely imaginative direc tions; bec omes , in effec t , a disc ussion of the ways in whic h ag ­ gregates may be bounded , and of the ways in whic h quan ­ tities may vary. The applic ation of the two types of bound ­ aries to finite and infinite aggregates and the inc isive use of the notion of "any" in c onnec tion with variable quan ­ tities permit us to speak of a logic omathematic al interest in the Sc holastic disc ussion . .AR ISTOTE LIAN

fl. Bac kg r ound and

Commenta r y

An original inc entive to the development we are about to trac e was undoubtedly furnished in the Aristotelian text 59

WILLIAM HEY T ESBURY

by ( 1 ) a side remark and ( 2 ) an answer to a possib le ob ­ jection. The aside concerns what is meant by the incapacity of a potency . Having stated that a power to effect the max ­ imum is also a power to effect any part of the maximum , Aristotle proceeds: " B ut the power is of the maximum , and a thing said , with reference to its maximum , to be incapable of so much is also incapable of any greater amount . It is , for instance , clear that a person who cannot walk a thousand stades will also be unab le to walk a thousand and one. This point need not trouble us , for we may take it as settled that what is , in the strict sense , possible is determined by a limiting maximum." 7 In fine , Aristotle states that if a po tency can accomplish so much it can accomplish less , and if it cannot accomplish so much it cannot accomplish more; the question of a precise b oundary between the range of what the potency can accomplish and the range of what it cannot accomplish he does not raise . And indeed this amb i ­ guity would not trouble us i f we were t o think simply of actions in the everyday world , where possible tasks to be accomplished generally present themselves in a discre te series according to magnitude , or , if they admit of a con ­ tinuous variation in magnitude , are not such as to demand or allow the setting of a precise boundary between what a given power can or cannot accomplish. It is when , in the manner of mathematicians , we think of the possible actions which a power can be set to accomplish as forming a con ­ tinuum , that the prob lem arises which was to concern the Schoolmen: as boundary between the range of what a power can accomplish and the range of what it cannot accomplish , are we to assign a maximum quod sic or a minimum quod non ? The possible ob j ection deals with the powers of the senses. Someone might argue that a power is not always to be determined by a maximum , "since he who sees a stade need not see the smaller measures contained in it , while , on the contrary , he who can see a dot or hear a small sound will perceive what is greater: ' Aristotle replies: "This , however , does not touch our argument. The maximum may be determined either in the power or in its ob ject. The application of this is plain. Superior sight is sight of the smaller b ody , b ut superior speed is that of the greater body." 8 Once more , Aristotle' s em phatic meaning is that , all cavils aside , a potency is to b e determined by the utmost it can accomplish i n the direc tion of excellence. The passage nevertheless raises queries . When is the maximum to be determined in the power alone-

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for all finite potencies will have a maximum or greatest excellence of this type- and when in its obj ect as well ? Must the senses , the maxima of which are determined only in the power , be granted a special status in this respect ? And if we wished to measure all potencies in terms of an objective standard , i. e. , a standard outside the potency , would it not be necessary to assign a minimum rather than a maximum in the case of powers like the senses ? A V E RRO E S

The commentary of Averroes , which served as a source for the Schoolmen gene rally , runs very close to the Aris totelian text. Two differences , however , may be noted . The Aristotelian reference to the incapacity of a potency is taken as a definition , and transformed by the use of the term minimum:' Conversely to the termination of the range of action of a potency , its defect or inc apacity is terminated by the minimum which it cannot accomplish.9 Averroes' insistence on the existence not only of a terminus for the range of what a potency can accomplish , but also of another for the range of its incapacity , 1 0 was bound to arouse discussion as to the relation between the two. The other difference occurs in Averroes' treatment of the obj ection concerning the powers of the senses . These powers are indeed terminated by the minimum which they can perceive; they are not on this account unique , but belong to the general class of passive potencies. Every potency is either active or passive , that which acts or that which is acted upon , an agent or a patient. 1 1 While the power of an agent is terminated by the maximum effect it can produce , the "power" of the patient is determined con ­ versely by the minimum agent by which it can be affected. For the patient which can be affected by a given agent can be affected by a stronger , but not necessarily by a weaker agent. 12 Averroes' introduc tion of passive potencies in general thus leads to the following consideration: since agency always accompanies patiency and vice versa , every action - passion can be analyzed from two points of view , that of the agent and that of the patient. For instance , Socrates' active power to lift weights will be terminated by the maximum weight which he can lift , and the passive potency of a weight will be terminated by the minimum active power by which it can be lifted. The fact that the case of vision does not fit with entire success into this schema , however, was to lead the School 0

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WILLIAM HEYTESBURY me n to a more e laborate se t of distinctions conce rning pote ncie s. Following Ave rroe s' practice of assigning a maximum quod sic or a minimum quod sic, we would an ­ aly se the action-passion of vision as follows : The powe r of the e y e is te rminate d by the minimum obj e ct which it can pe rce ive through a give n distance , in accordance with Ave rroe s' inclusion of sight in the class of passive pote n ­ cie s; but the powe r of an obje ct to multiply its u visible spe cie s" through a give n distance - a powe r which by the same classification would have to be conside re d active-is te rminate d not by the maximum powe r of sight but by the minimum or we ake st powe r of sight to which it can make itse lf se e n. If, on the othe r hand, we ke e p to a give n ob ­ j e ct and allow the distance through which it is se e n to vary- a case which Ave rroe s doe s not conside r- the re will be a maximum distance through which it can be se e n by a give n powe r of sight and similarly a maximum distance through which it can multiply its visible spe cie s. B oth powers thus appear active with re spe ct to distance as a passive pote ncy . T HO M A S A Q U I N A S

In Thomas A quinas' comme ntary on the De cae lo, Ave r ­ roe s' interpre tation of the maximum quod sic is adopte d without modification. In the de finition of incapacity, Thomas again make s use of the te rm u minimum" ; 1 3 and the de fini ­ tion is illustrate d by an e xample which could e asily draw atte ntion to the proble m of the re lation be twe e n the maxi ­ mum quod sic and the minimum quod non: the maximum which some one can walk may be twe nty stade s, and the minimum which he cannot walk twe nty-one stade s. 14 What is to be said about the distance be twe e n the twe ntie th stade and the twe nty-first ? In de aling with the obj e ction conce rning vision, Thomas follows Ave rroe s in introducing the distinction be twe e n active and passive pote ncie s. The e xce lle nce of a virtue , as Aristotle had state d, may be d e te rmine d e ithe r in the thing, i. e . , the numbe r of stade s to be walke d or tale nts to be lifte d, or in the virtue . It is de te rmine d e ithe r in the thing or in the virtue whe n the virtue is active ; it is de ­ te rmine d only in the virtue , whe n some thing which doe s not e xce l in quantity neve rthe le ss has an e xce lle nce as a virtue . The latte r situation occurs e spe cially in the case of passive pote ncie s, such as the se nse s. 1 5 Thomas admits with Ave rroe s, howe ver, that the passive pote ncie s may be te rminate d by a minimum quod sic among the obj e cts or age nts by which the y may be affe cte d.

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One other statement in Thomas' commentar y deser ves notice : he relates the pr oblem of defining a potency to what we may cal l a conser vative, antimathematical under standing of for ms or qualities. We deter mine the magnitude of any ­ thing by its maximum, as in defining man we as sign the dif ­ fer ence "r ational" rather than " sensib le;' for it is always the ul timate or maximum of a thing which gives its species.1 6 This tr aditional stand, implicit in the statements of Ar is totle and Aver r oes but made explicit by Thomas, undoubt ­ edly played a motive role in the debate over the maximum quod sic and the minimum quod non. JOH N

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At the time John of Jandun1 7 wr ote his commentary on Ar istotle's P hysica, ther e were appar ently a number of Schoolmen who, on the basis of the text of the De caelo, wer e ar guing that a natur al vir tue must be deter mined by both a maximum in quod P.Otest and a minimum in quod non P.Otest.1 8 What is especial ly interesting is the recog ­ nition that this position implies the existence of indivisi ­ ble magnitudes . John of Jandun is upholding the Aristotelian pr inciple that the continuum is not composed of indivisibles, and the existence of a minimum quod non, as wel l as a maximum quod sic, of a power is pr esented as an objection. This minimum quod non must be greate� than the maximum quod sic, for whoever can lift one hundred pounds can lift one hundr ed pounds or less. It will be greater , then, by a divisible or an indivisible magnitude, say one pound. If the excess is an indivisible magnitude, if foll ows that the con ­ tinuum is composed of indivisibles . If the excess is divisi ­ ble, par t of it may be removed, say one-half pound, and we then ask the following question: is the power able to lift the one hundr ed pounds with the r emaining one-half pound of the original excess or not ? If it is able, then one hundred pounds is not the maximum quod sic of the power , for it can lift mor e . If it is not able, then one hundred and one pounds is not the minimum quod non of the power , since it cannot lift a smaller amount, viz., one hundr ed and one half pounds. Either r esult is contr ary to the or iginal sup ­ position; hence the continuum is composed of indivisibles.19 John of Jandun's answer to the objection is twofold. Fir st, some say that the ar gument supposes what is simply false and impossible, namely that a natural vir tue is ter minated by a minimum in quod non P.Otest. This supposition, mor e ­ over , cannot be supported by an appeal to author ity, for Aris totle states that a potency is deter mined by a maximum 63

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in quod Rotest , and neither he nor his commentator assign a m1n1mum 1n quod non RO test .2 0 . . . A second possible reply-apparently the one favored by John of Jandun-is that the excess b y which the minimum quod non exceeds the maximum quod sic is indeed divisible , in the sense of having one part outside (not in the same place as) another; but it is not divisible in the sense that it can actually be divided. To paraphrase , we may say that as pure magnitude the excess is divisible , but as a magni ­ tude realised in a natural b ody, it forms a natural mi ni ­ mum which cannot be divided.2 1 Thus an atomic or quasi ­ atomic view of natural bodies allows for the existence of both a maximum quod sic and a minimum quod non in those cases in which the boundary of a potency is gi ven as a natural body; in other cases , not considered by John of Jandun , it would be neces sary to regard time and space also as atomic .2 2 B ut the path which leads to a total atomizing of nature was not followed by the Schoolmen generally-as indeed it could not be if an Aristotelian framework was to be main ­ tained. In most of the accounts of maxima and minima we shall examine it is assumed that the active and passive potencie s under consideration are quantities which , within the range of their variation , may vary continuously. J E A N BUR I D A N

Jean B uridan' s discussion of the problem2 3 puts in evi ­ dence a number of new distinctions between types of agency and patiency , most of them deriving directly from the Aristoteli an natural philosophy and not being susceptible to mathematical formulation . The question whether a max ­ imum quod sic can be assigned to a potency , B uridan notes , is extremely difficult because of the diversity of poten ­ cies , both active and passive , and the diverse modes of agency and patiency , motion and transformation.24 The fol ­ lowing cases are considered : FIRST: The divine power. -According to fai th , no maxi ­ mum can be assigned in this case , since the power of God is infinite. According to Aristotle , on the contrary , a maxi ­ mum would be assignable , since the prime mover and that which it moves are immutable; thus the maximum body which God can move would be precisely that which He does move , that is , the ultimate sphere (P. rimum mobile).2 5 SECO ND: The powers of the celestial intelligences.­ Here again i t is probable that Aristotle would assign a maximum body which each intelligence can move , namely

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that sphere which it does in fact move , and also a maxi ­ mum velocity with which each intelligence can move this b ody , namely that velocity with which it does in fact move its sphere. F or the intelligences are finite and immutab le in power , and the bodies which they move unalterable; hence the relation b etween mover and moved will remain constant , and each intelligence in moving its sphere will b e acting according to its maximum power. B ut a doubt arises , suppo sing that God can augment the sphere of the moon , and further that the power of the mover must ex ­ ceed the resistance of what is moved . For the power of the lunar motor will exceed the resistance of the lunar sphere b y a divisib le excess; let God then increase the resistance of the sphere by half this excess , and the lunar intelligence will still be ab le to move it. Hence the prese nt lunar sphere is not the maximum which the lunar intel ­ ligence can move.2 6 B uridan answers that the reasoning is not demonstrative , b ecause the celestial spheres offer no resistance to their movers , b ut have a pure inclination to their proper movement. In such case , although the active power will exceed the passive in nobility and per ­ fection , it is not necessary that its activity be greater than the passivity of the passive.2 7 The distinction b etween pure passivity and resistance is drawn more precisely in the next section. Having considered the powers of God and the Intelligences , Buridan now turns to natural and corporeal powers . THIRD : Light which illuminates a transparent medium. -To b egin with , we must decide whether the medium offers resistance to the light which illuminates it. It would seem that it does insofar as it is opaque , for the less trans parent and more opaque medium is illuminated less intensely and less far b y the same light than the more transparent and less opaque. In opposition , however , it is objected that if opacity constituted a resistanc e , the illumination of the medium would b ecome more intense and extend to a greater distance over a period of time , as we see in the case of heat-action. B ut this is false; the medium is illuminated instantaneously to its maximum potentiality.28 Buridan decides , therefore , that the medium is without resistance; the cause of its being more or less transparent is a greater or less passivity to light. Resistance is through a contrarcf or an inclination to a contrary; b ut light has no contrary.2 It may now b e asked whether there is a maximum dis­ tance to which a given light can propagate its species in a given medium.3 0 If the light be of finite potency , and if

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neither the light nor the medium alter, then the light will illuminate part of the medium and part it will not ; and be tween the two parts there will be only an indivisible or mathematical point . Hence the distance from the light ­ source to this poi nt will be the maximum distance to which the light propagates its species. It will also be the maxi ­ mum distance to which the light can propagate its species; for the illum ination does not improve wi th time, and the light cannot illuminate more than it in fact illuminates. A further consequence is that there is no mini mum distance to which the light cannot propagate its species; for such a minimum quod non could not differ by a divi si ble magni­ tude from the maximum quod sic already assigned, the entire part of the medium beyond the dividing point being uni llumina ted.3 1 Again, as a corollary it follows that there is a maximum space through which a gi ven visible body multiplies the species of color; since the action of light and that of color are entirely similar with respect to the multiplication of species. Thus it is possible for a gi ven colored body to multiply its species to the surface of the eye and no further, or the point which is geometrically at the center of the eye and no further. 3 2 F O UR TH : A given medium to be illuminated by a light. - B uridan now turns to the case of the corresponding passive potency. Given a determinate medium, spherical i n form, is there a minimum or maximum light which can illuminate the whole of it ? B uridan answers that there is a minimum quod sic; and it will be precisely that light the maximum range of which extends throughout the gi ven medium .3 3 F IF TH : Vision . - Here B uridan does not consider, a s had Averroes, the case of different objects seen at a fixed distance, but rather keeps the object fixed and allows the distance to vary. Given a determinate power of vision a, a visible body b, and a uniform medium, is there a maxi­ mum distance through which a can see b ? The common and probable answer, B uridan states, is in the negati ve. F or suppose there were such a distance; through this distance a would see b with a certain intensity of vision (visio ali­ quantae intensionis). N ow a form or quality which is nat ­ urally capable of remission or diminution in intensity is not corrupted all at once from a certain intensity to z ero i ntensity, but remits in a continuous manner to i ts total corruption; hence if the distance between a and b i s in­ creased, a will continue to see b, albeit less and less in­ tensely., until vision is finally reduced to z ero intensity.

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Thus the original distance assigned was not a maximum quod sic , and no such can be assigned.34 And yet , although this reasoning appears demonstrative , a grand difficulty arises , which shows how the problem of assigning limits to potencies could become hopelessly in ­ volved with the problem of the physical nature of particular potencies. It was previously admitted (third section) that there is a rnaximum distance through which the given visi ­ b le ob ject b can multiply its visible species. Suppose this distance extends from b precisely to that point of the eye , say the center of the eye , at which vision becomes pos ­ sible ; will not this be the maximum distance through which the eye can see b ? Thus we have two arguments : one based on the nature of vision as a remissible form , and leading to the assignment of a minimum quod non ; the other based on the mode of propagation of the visible species of an ob ject , as extending a definite maximum distance , and leading to the assignment of a maximum quod sic for vision as well. The two arguments cannot both stand.r;B uridan attempts to extricate himself from the difficulty by means of a distinction. There is a difference between " the space in which b can be seen" and " the space through which b can be seen" ; b is in the first , but not in the sec ond. B uridan then assigns a maximum space in which the body b can be seen ; and a minimum space through which b cannot be seen , equal to the maximum in which it can be seen. Apparently-although B uridan does not say as much - this space is also equal to the maximum distance through which b can multiply its visible species. Thus if b is just outside the given space , it cannot be seen ; if i t approaches the eye by as little a distance as you please , it comes into the range of vision ; and when it is altogether in the given space , it can be seen as a whole.3 6 However ingenious this distinction , it may be questioned whether it meets the cen ­ tral difficulty , which arises from the problem of the nature of light and of vision. The remaining cases which Buridan considers deal with natural agents acting against a resistance , and therefore involve the rule according to which no action results when power and resistance are equal. SIX TH : Motion of an agent against a resistance. - There is no maximum resistance against which a given agent can move or act , but rather a minimum against which it can ­ not move or act ; the proof is altogether similar to that of the u Probationes conclusionum" previously cited. Thus we must assign a medium of minimum density through which

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WILLIAM HEYTESBURY the weight a cannot mo ve downward ; it will be the medium the resistanc e of whic h is equal to the do wnward tendenc y of a. Similarly there is a minimum weight whic h weight a c annot in cline in the balanc e, and this is the weight whic h is equal to a.3 7 In this latter example we c an see c learly the mathemat­ ic al, nonempiric al charac ter of the Sc holastic spec ulations c onc erning potenc ies. The sensitivity of a balanc e is always limited ; so that there is a c ertain range within whic h the sec ond weight c an be smaller than weight a and yet not be tipped up by it in the balanc e. In practic e, if we wished to assign a b oundary between the range of weights whic h a c an inc line and the rang� of weights whic h it c annot inc line, we might imagine ourselves r emo ving weight from the sec ond pan by some c ontinuous proc ess (as pro vided, for example, by a c hain balanc e) until the balanc e tipped ; and at this point it would appear more feasible to as sign a maximum weight whic h weight a c an inc line in the balanc e, rather than a minimum quod non. Suc h c onsiderations, of c ourse, have little to do with the ac tual pro c ess of weighing. B uridan's rule applies not to an actual, but to a mathemat­ ic ally ideal balanc e. SEVENTH : Vel o c ity with whic h an agent c an move a b ody against a resistanc e. -Buridan deals first with natural, involuntary agents, and supposes that the ac tive power as well as the resistanc e remain c onstant, so that the ratio between the t,vo will also b e c onstant. Then-pro vided that the ac tive power exc eeds the resistanc e, for otherwise n o motion o c c urs-there is to be assigned a maximum-minimum vel o c ity with whic h the agent c an mo ve the patient, and it is prec isely that veloc ity whi c h follows from the ratio be tween ac tive power and resistanc e. A different c ase is furnished b y the downward mo vement of a heavy b ody through a uniform medium; for here the ac tive po wer-whether it is the attrac tion of natural plac e o r impetus ac quired through motion or something else-is c ontinuously inc reased, and henc e the veloc ity of move ment is c ontinuously inc reased. Con versely the velo c ity is c ontinually diminished in the violent movement of a b ody away from its natural plac e, or in mo vements c aused b y agents whic h undergo fatigue. 3 8 In these c ases it is not possible to assign a maximum, minimum, or other limit to the velo c ity whatever. The c onclusion of Buridan's disc ussion is thus that a single rule c annot be given for assigning maxima and minima, b ut it is nec essary to have regard for the par -

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ticular exigencies of eac h case . The cases which B uridan considers are distinguished according as the potency is celestial or terrestrial , as it remits continuously to zero degree or not , and as it acts with or without resistance on the part of the medium or passive potency . In the case of the distance through which vision (a remissible form) ex ­ tends and in the case of a potency acting against a resist ­ ance , B uridan assigns a minimum quod non or limit lying outside the range of the potency . He reconciles this as ­ signment with Aristotle's assertion of a maximum by saying that the maximum for Aristotle is either an ordinary maxi ­ mum or a maximum infra quod P.O test , which is the same as a minimum quod non .

H

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" De m axima e t m inima "

If we turn now to the treatise of Heytesbury , we leave be hind the Aristotelian commentary and enter the realm of calculationes and �hismata. The problem of the purely physical and qualitative character of different potencies recedes into the background , and the discussion assumes - what was irnplicit in the notion of the minimum quod non type of boundary to begin with-an extremely hypothetical , logicomathematical character . A set of rules is proposed as covering all imaginable cases of action-passion ; the rules are couched in language borrowed from the logica moderna , and the center of interest appears to have shifted from that of answerin g strictly physical questions to the logicomathematical problem of setting extrema to classes or agg regates . Objections are raised , in the form of case s which do not appear to be in accord with the rules , and are systematically answered. In analy zin g Heyte sbury' s u De maximo et minimo" (foll . 2 9va-3 3va) , we shall make frequent reference to the " Re ­ collecte" of Gaetano di Thiene (foll . 3 3va-3 6vb) and to the anonymous u Probatione s cone lusionum" (foll. 19 3vb- l 98rb) , which though not markedly original often help to clarify a point or to show the arguments to which Heytesbury's state ­ ments led in later discussion. E X P O UNDIN G

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It should first be noted that the terms umaximum" and u minimum" are taken by Heytesbury in an extremely gen ­ eral sense , to cover any kind of bound to a qualitative or quantitative range. Uncle r "maximum" and uminimum" are 69

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thus subsumed such terms as "first" and "last ;' " quickest" and "slowest ;' " strongese' and " weakest ;' "most intense" and "most remiss :' (foll. 3 0rb - 3 0va). Similarly, all potencies considered by Heytesbury are classified as active or passive, even in cases in which it is hard to see how the potencies in question come under one or the other of Aristotle's definitions (see note 1 1 of this chapter). In fact, Heytesbury employs definitions of active potency and passive potency which depend on purely quantitative considerations rather than on meaning. An active potency is one which, inasmuch as it can accomplish a greater amount, can accomplish a less, and not vice versa. 3 9 Thus if Socrates can lift one hundred pounds he can lift twenty, and his capacity to lift weight is therefore an active potency. Conversely, a passive potency is one which, inasmuch as it is susceptible to less or can be affected by less, is susceptible to a greater or can be affected by a greater, and not vice versa.4 For example, if Plato can see a grain of millet at the distance of a mile, a fortiori he will be able to see the Church of the B lessed Mary at the same distance, and therefore his power of vision is a passive potency. From the definitions of active and passive potency it follows that the boundary of an active potency will be an upper boundary, while the boundary of a passive potency will be a lower boundary. The boundaries of potencies are of two kinds (fol. 2 9va): aff irmative (maximum quod sic, minimum quod sic) or negative (maximum quod non, mini ­ mum quod non). An upper boundary, that is the boundary of an active potency, is assigned either by the affirmation of the maximum (maximum quod sic) or by the negation of the minimum (minimum quod non); a lower boundary, that is the boundary of a passive potency, is assigned by the affirmation of the minimum (minimum quod sic) or else by the negation of the maximum (maximum quod non). Be fore deciding in favor of one of these boundaries, in any particular case, it is necessary to know that one of the disjunctions (maximum quod sic or minimum quod non; minimum quod sic or maximum quod non) is valid in that case ; for the establishing of such validity Heytesbury posits the following conditions.4 1

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FIRST . -There must be a range in which the potency can act or be acted on, and another range in which it can ­ not act or be acted on, so that two subcontrary p ropositions 70

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o f the fo l l o w i n g typ e wi ll b e v e r i fi e d : s o me w e i g ht s S o c r at e s c an lift , and s o m e w e i g ht s S o c r ate s c anno t lift .4 2 S E C O N D . - T he s e c ond c o nd i t i o n i s t h at n e ithe r o f t h e s ub c o nt r a r i e s s ho u ld b e ve r ifi e d o n l y fo r infin ite v a lue s . In g e ne r a l , a s G ae t an o n o t e s , infini t e p o te nc i e s like the infinite p o we r of God or the infin ite p a s s ib i l ity o f m ate r i a P. r i ma a r e e xc lude d fr o m the di s c u s s io n , s inc e no te r mi ­ n at i o n c an b e a s s i g ne d to t he m (fo l l . 3 3 v a - 3 3 vb) . T he n e c e s s ity o f t h e s e c on d c ondition a r i s e s , howe v e r , among tho s e w h o admit o ne infinite t o be g r e at e r th an anothe r .4 3 W i th thi s admi s s i o n , an d the fu r t he r s uppo s i tion that Anti ­ c h r i s t w i l l c o m e to b e , b ut that hi s c o ming- t o-b e c an b e infini t e ly di s t ant in futu r e time , t h e fo l l o w ing a r gume nt w o u l d be i n v al i d : by s o m e t i me Ant i c h r i s t' s c o ming- to-b e c an b e di s tant in futu r e t i m e , and b y s o me t i me no t ; t he r e ­ fo r e t h e r e i s a m a xi mum t i m e b y whic h Anti c h r i s t' s c o m i n g- t o-b e c an b e d i s t ant in futu r e t i me o r a mini mum t i m e by w hi c h i t c ann o t be thu s di s t ant . The ante c e d e nt s ub c o nt r a r i e s a r e t r ue fo r tho s e who admit one infinit e t i m e to b e g r e at e r th an ano the r , b ut t h e c on s e qu e nt i s f a l s e , fo r th e r e i s n e ith e r a m axi mum t i m e b y whi c h Ant i ­ c h r i s t' s c o m i n g- to- b e c an b e d i s t ant in the fut u r e (maxi ­ m u m quo d s i c) n o r a minimum t i m e by w hi c h it c anno t b e d i s t ant i n the fut u r e (m inimum quod non). T h e r e i s no m a xi mum quod s i c , fo r no fin i t e time c an b e s uc h b y o u r o r i g in al s up p o s iti o n , a n d n e it h e r c an a n infinite t i me , b e c au s e to s ay t h at an e ve nt w i l l o c c u r in a futu r e infinit e ly d i s t ant invo l v e s a c on t r adic t i o n , n am e ly , that the e ve nt w i l l o c c u r and t h at i t w i l l n e v e r o c c u r . N o r i s t he r e a mini mum quo d no n , fo r i f the r e w e r e s uc h , i t w o u l d h ave t o be an infinite t i m e ; b ut s inc e one infinite t i me c an b e l e s s t h an ano th e r , fo r e v e r y p o s s ib l e c ho i c e the r e c ou l d b e i n d i c at e d a s ho r t e r infinite t i m e b y whi c h Ant i c h r i s t' s c om i n g- t o- b e i s di s t ant in the futu r e .44 B e c au s e o f the dif ­ fic ulty o c c a s ione d b y thi s and s i mi l a r c a s e s , He yte s b u r y e xc l u d e s t h e e xt e n s i v e i n finite , c on s i de r e d a s a po s s ib le b o un d a r y to a p ot e nc y , f r o m h i s d i s c u s s i o n o f m a x i m a and m i ni m a .4 5 T HI R D . - T he thi r d c o nd i t i on fo r in s u r in g the v a l i dity o f t h e di s j un c t i o n o r d i v i s i on in a p a r t i c u l a r c a s e , i s t h a t t h e p o t e n c y wi t h r e s p e c t t o w hi c h the d i v i s i on i s m ade s hould be c ap ab le o f t akin g o n a c ontinuo u s r an g e o f v a lue s be t w e e n z e r o p o t e nc y and the v a l ue whi c h i s to s e r ve a s a b o und a r y , and n o o th e r v alue s . T hu s an ac t i v e p ot e nc y , i f i t i s ab l e t o ac c o m p l i s h a c e r t ain a m o unt , mu s t b e ab l e t o a c c o m p l i s h a n y le s s a m o unt , and i f it i s unab l e t o ac c o m -

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WILLIAM HEYTESBURY plish a certain amount must be unable to accomplish any greater amount. Conversely , a passive potency , if it is passible with respect to a certain agent , must be passible with respect to any agent of greater strength , and if it is not passible with respect to a certain agent must be impassible to any weaker agent. If this condition does not obtain , then the division cannot be admitted simply, but only with certain qualifications. For example , posited that a man can be of a certain si z e , it is not true that a man can be of any less si ze whatever , nor of any greater size whatever , for a man cannot be as small as , say , one foot long or as large as , say , ten feet long. The latitude of possible sizes of men is bounded , therefore , both at the lower and upper extremes; let us sup � ose that it extends from one foot to ten feet exclusively.4 We cannot say simply , with respect to the lower extreme , that there is either a minimum si ze which a man can have or a maxi ­ mum size which a man cannot have , but must add a quali ­ fication , thus: there is either a minimum size under ten feet which a man can have , or a maximum size under ten feet which a man cannot have. Similarly , with respect to the upper extreme , we must say that there is either a maximum size over one foot which a man can have , or a minimum siz e over one foot which a man cannot have.47 F O U RTH.-A final and obvious condition for the validity of the division , not mentioned by Heytesbury but suggeste d by Gaetano (fol. 3 4va) , is that the value which is to serve as a boundary for the potency must not be excluded from the division . Suppose , for instance , that a is a weight equal in resistance to the power of Socrates. Then the fol lowing subcontraries are true: Socrates can lift some wei ghts which are not a nor equal to a , and some weights which are not a nor equal to a he cannot lift . But it does not follow that there is either a maximum weight which is not a or equal to a which Socrates can lift or a minimum such which he cannot lift. We have to do here with a series of weights which is discontinuous at a , whereas the division requir es that it be continuous at a. Having established the conditions under which the division is valid , we may now proceed to the rules which determine which part of the division is to be assigned in a particular case . All potencies are first classified as mutable or immuta ­ ble. This distinction , though crucial , offers some difficulty , since the terms are not define d by Heyte sbury , and the ap ­ plication to particular instanc es is sometimes doubtful.

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DE M A X IM O ET M I N IM O An i mmutab le p ot e nc y appe a r s to b e one whi c h a c t s in a c o mple te ly de te r minate way , s o that it c anno t p r oduc e ano the r e ffe c t than the one it a c tually p r o duc e s ; fo r e xa mp l e , a pote nc y fo r t r a ve r s in g di s tanc e at a de f ­ inite , uni fo r m de g r e e o f ve l o c ity du r in g an hour , o r a p o te n c y fo r t r ave r s i n g di s tanc e a t a ve l o c ity whic h inc r e a s e s un ifo r m ly fr o m a to b de g r e e in an ho ur . A mu tab le pote nc y , on the othe r hand , i s one whic h e ithe r b e c au s e o f it s int r i n s i c c ha r a c te r o r the c onditi o n s unde r w h i c h it a c t s , i s c ap able o f mo r e a n d le s s ; thu s a po t e n c y fo r t r a ve r s in g di s tanc e at a de finite , uni fo r m v e l o c ity dur in g any time g r e ate r than an ho u r i s mutab le on a c c o unt o f the l a c k o f de te r min ation with r e s p e c t to time . It al s o appe a r s that , whene v e r a mut ab le pote nc y i s unde r c o n s i de r ation , the de s c r ip ­ tion o f the s i tuation will invo l v e impli c i tly o r e xplic itly an a p p r o a c h to a l i mit (b o unda r y l yin g o ut s ide the r an g e b o unde d ) , whe r e a s i n t h e c a s e o f i mmutab l e pote nc ie s no s uc h l i mitin g p r o c e s s i s in vo l v e d . In a divi s io n with r e s pe c t t o an immutable pote nc y , we a r e a l w ay s to a s s i gn a s b o unda r y the affi r m at i ve P.a r t o f the div i s io n . T hu s the r e i s a maximum di s t an c e whi c h , c e te r i s P. a r ibu s , c an b e t r ave r s e d b y a de g r e e o f ve l o c ity in an ho u r , r athe r than a mini mum whi c h c anno t be thu s t r ave r s e d ; and the r e is a maxi ­ mum di s tan c e w hi c h S o c r ate s c an t r ave r s e in an ho u r b y a ve lo c ity whic h in c r e a s e s unifo r mly f r o m a to b de g r e e , r athe r th an a mini mum d i s t anc e whic h he c an ­ not t r ave r s e with s uc h v e l o c ity .4 8 Amo n g mutab le p o t e n c ie s , th r e e c a s e s a r e di s t in ­ g ui s he d .

D I V I S I O N B Y C O M M O N I N DE F I N I TE TERM If the d i vi s io n i s made b y me an s o f a c o mmon inde finite te r m and with r e s p e c t to a mut ab l e pote nc y ( " me di ante t e r mino c o m muni inde finito v e l s in g ul a r i r e s p e c tu r e i mut ab ili s qu ate nu s a d s e n s u m ip s iu s c o mpar ati oni s p r op o ­ s it e p e r t ine t" ) , the ne g ative p ar t o f the di v i s ion i s to b e s u s t aine d . B y " c o mmon inde fin it e te r m" i s me ant a c o m ­ mon n o un unmo difie d b y quantifi e r o r de mon s t r ative ad ­ j e c ti v e , a s " ve l oc ity" o r " we i g ht :' T hu s in the divi s ion , " T he r e i s a maxi mum we i g ht whic h Soc r ate s c an lift o r a minimum we i g ht w hi c h he c annot lift ;' the di v i s ion i s made with r e s P.e c t to S o c r at e s ' mutab le p o we r of l i ftin g , and by

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mean s of the common indefinite ter m " weight:' Hence we a s s ign the ne gative par t of the divis ion , or minimum quod non (fol . 3 0 ra ) . Among fur ther ca s e s coming unde r the s ame r ule may be mentioned the following : F IRS T. - The r e i s a maximum par t of a diffo r m medium b which Plato can divide or a minimum par t which he can ­ not divide (fol. 3 0 ra). Two cas e s are di s tingui s hed . ( 1 ) If the medium i s unifor mly diffo rm (i . e. , s uch that it s r e s is tance increa s e s unifo rmly fr om one end to the other) , a minimum quod non i s to be a s s igned; fo r though Plato s hould move eter nally he would never ar rive at the point of the me dium the r e s is tance of which i s equal to his powe r , but any point befor e thi s he would be able to attain; hence the par t o f the medium extending t o this point i s the r equired mini ­ mum quod non. That P lato' s motion would be eter nal i s not proved by Heyte s bur y , but i s pr oved by Gaetano di Thiene a s follow s (fol . 3 6va - 3 6vb). We cons ider a potency of 8 , and a unifo r mly diffor m r e ­ s is tance beginning with zer o deg r ee of re s i s tance and ending at 8 ; we divide the exten s ion of the r e s i s tance into pr o­ por tional par ts accor ding to a ratio of one half , s o that the mo r e r emis s half of the r e s is tance for m s the fir s t pr opor tional par t. Gaetano then s how s that the potency will r e qui r e mo re and mo re time to traver s e each s ucce s s ive pr opo rtional par t , a s it p roceed s towar d s the mor e inten s e extr eme ; and s ince the pr oportional par t s ar e infinite in number , it follow s that the potency will requir e an infinite time to t r ave r s e the whole medium. To pr ove the potency requir e s mor e time to t r ave r s e each s ucce s s ive pr opo r ­ tional par t , Gaetano a s s ume s , fir s t , that the velocity in the motion varie s with the r atio of g reater inequality of powe r to r e s i s tance ( o r mor e precis ely , with the "pr opor tion of p r opo r tion s ;' s o that , fo r example , a velocity which p r o ­ ceed s f r om a r a� / R will be halved when the r atio ha s / 49 been r educed to R ). Second , it i s a s s umed that , given two r e s i s ting media of which one is twice the length of the other , if a potency have to the r e s i s tance of the s ho r te r of the s e a r atio which i s the s ubduplicate of the r atio which it has to the r e s i s tance of the longer , it will trave r s e the two media in the s ame time (if the r atio to the longer 1 s P/ R , that to the s ho r ter will be ✓ P/R ) . This r e s ult in fact follow s from the fir s t a s s umption , s ince the velocity in the longer medium will be twice that in the s ho r ter. F inally , given two r e s i s ting media of which one is twice the length of the other , if a ·

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p ot e nc y h a v e t o the r e s i s t anc e o f the s ho r te r of the s e a r at i o whic h i s l e s s t h an the s ub d up l i c ate of the r atio whi c h it h a s to the r e s i s t an c e of the l on g e r , it will r e qui r e a l on g e r ti me to t r a ve r s e the s ho rte r me dium than the long e r (if the r atio t o the l o n g e r i s P / R , that to the s ho r t e r will be l e s s than �). T hi s is a g ain a r e s ult of the fi r s t a s ­ s umpt i on . N o w G ae tano s ho w s that , to the re s i s t anc e at any po int o f one o f the p r o po rti ona l p ar t s , the pot e nc y o f 8 h a s a r ati o l e s s t h an t h e s ub duplic ate of the r atio it ha s to the r e s i s t anc e of the c o r r e s ponding p o int in the p r e c e d in g pr o p o r t ion al p ar t . F o r the re s i s t anc e of thi s p o int i s p r e c i s e ly the a v e r a g e b e twe e n 8 and the r e s i s t anc e of t he c o r r e s p onding po int in the p r e c e ding p r o p o r tional p a r t ; b ut the a v e r a ge b e twe e n two m a gnitude s i s alway s g r e ate r than the me an p r o p o r t i onal b e twe e n the m ; and the r e fo r e the r ati o of 8 to the r e s i s t anc e of the p oint in que s ti on w i l l be l e s s than the r atio of 8 to the me an p r o p o r tion al b e tw e e n 8 and the r e s i s tanc e of the c o r r e s p onding p o int in the p r e c e ding p r op o r t i o n al p a r t (the latte r r atio b e in g the s ub dup l i c ate of the r atio of 8 to the r e s i s t anc e o f the c o r r e s p ond ing p o int o f the p r e c e ding p r opo r ti onal p ar t) . T he r e fo r e t o the r e s i s t an c e of e ve r y p o int in a g i v e n p r o p o r t ion al p a rt , the p ote nc y h a s a r atio whi c h i s l e s s than the s ub dup l i c ate of i t s r atio t o the r e s i s tanc e o f the c o r r e s p onding point of t he p r e c e ding p r op o r tional p ar t . B ut e ac h p r o po r ti on al p a r t i s j u s t half the le n gth o f the p r e c e d ing p r o p o r ti onal p a r t . Henc e the pote n c y w i l l re qui r e a l on g e r time t o t r ave r s e e ac h s uc c e e ding p r op o r ­ t i o n al p a r t . E r g o , e t c . ( 2 ) If it i s a r gue d that the me d ium m ay b e s uc h t h at the d i v i di n g pote n c y will be b r o u g ht to a stop at an int r in s ic p o int o f it , He yte s b u r y adm i t s the p o s s ib il ity , b ut d e ni e s that the p a r t di v ide d i s a m aximum quod s i c ; fo r P l at o' s p o w e r o f di v i di n g i s d e b i l i t ab l e , and howe v e r litt l e it d e b i l i t ate s it c an d e b i l itate l e s s and thu s d i v i de mo r e than it ac tually d i v i d e s in a p ar t i c u l a r in s tanc e . 50 S E C O N D . - T he r e i s a m ax i mum time du r in g whi c h S o c ­ r at e s c an l i v e o r a m ini mum t i me du r in g whic h he c annot l i v e (fo l . 3 0 r a) . T h e minimum quo d !!Q!!_ i s a s s i gne d and , a c c o r ding to G a e t ano , i s a r r i v e d at a s fol l o w s . W e i m a gine all c i r c um s t an c e s fo r S o c r ate s' e xi s te nc e , s uc h as the r e g u l a r ity of h i s l i fe and the s t ate of the r uling c on s te l ­ l at i o n s , t o b e c o ntinual! y at an opti mum , although it i s not ac tually p o s s ib l e fo r the m a lw ay s to b e s o ; and then the who l e time b e g inni n g fr o m the p r e s e nt in s t ant and t e r mi ­ nating at the fi r s t in s t ant in wh ic h S o c r ate s w i l l have

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ceased to be under these conditions will be the minimum time during which he cannot live. 5 1 T HIR D. - There is a first instant in which Antichrist can be , or an ultimate in which he cannot be (fol. 3 0ra). This alternative , according to Gaetano (foll. 3 4rb - 3 4va), is posed on the assumption that Antichrist does not exist at the present time , but will be , not necessarily but contingently , depending on his productive causes which are not now suf ­ ficient to produce him , but which may be imagined to im ­ prove continual ly until they begin to be able to produce him. The instant in which they begin to be able to produce Antichrist is the ultimate instant in which they will not be able to produce him , and hence the ultimate instant in which Antichrist will be unable to be . The crux and ob ­ scurity of this particular case lie in whatever it means to say the productive causes of Antichrist constitute a mutable rather than an immutable potency . According to Gaetano , if we say that in some future instant these causes will be sufficient to produce Antichrist , and in some not , then the division is with respect to an immutable potency , and the affirmative boundary , or first instant of Antichrist's existence , is to be assigned. B ut if we say that in some future instant these causes will be able to be sufficient to produce Antichrist , and in some not , then the di vision is with respect to a mutable potency , and the negative bound ­ ary , or ultimate instant of Antichrist's nonbeing , is to be assigned. Gaetano does not state what would have to be the character of the productive causes in order to come under one or the other case. In the first case , they are apparently such that they necessarily produce Antichrist in some instant ; and in the second case , they gradually approach the state of being able to produce Antichrist as a limit , and this state having been achieved in some in ­ stant , Antichrist can immediately afterwards be produced. Therefore in this instant Antichrist can begin to be (sensus divisus), although it is not possible that he shoul d begin to be in this instant (sensus compositus). FO U R T H.- There is either a maximum velocity at which Socrates can move weight a for a time , or a minimum velocity at which he cannot move the weight (fol. 3 0ra). Once more the minimum quod non is assigned , since ac cording to Gaetano (fol. 3 4va) it is not possible for Soc rates to move the weight so fast that he cannot move it faster , the latitude of velocities at which he can move the weight being terminated in the more intense extreme by some degree extrinsically , i. e. , by a limit which Socrates can approach but not attain.

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F IF T H. - G iven a uniformly difform latitude , there is either a most remiss degree which is more intense than the medium degree of the whole latitude , or a most intense degree which is not more intense than the medium degree (fol . 3 0ra). The latter alternative is chosen , the medium degree itself forming the required boundary . 52 SIX T H . - There is a minimum visible body wh ich the lynx can see at a given distance , or a maximum visible body which he cannot see (fol . 3 0ra). Once more the negative p art of the division is assigned , in this case a maximum quod non . DI V I S ION B Y C O M M ON DI S TRI B UT E D T E RM H A V IN G IN F INIT E S UP P O S IT A

I f the division i s made by means of a common distrib ­ uted term , having infinite �posita for which it is dis tributed ( "mediante te rmino distributo habente infinita supposita" ) , the affirmative part of the division is to be sustained . B y a common distributed term'' is meant a common noun modified by a quan tifier like "all :' "every ;' or " any:• To say that such a term has infinite supposita means that it stands , in a given proposition , for an infinite number of individuals . T he import of the rule can be clar ­ i fied by an example : there is either a maximum weight which anyone stronger than Socrate s can carry , or a minimum weight such that not every individual stronger than Socrates can carry it ( " Aliquod est maximum grave quod quodlibet fortius Sorte pote st portare , vel minimum quod non" - fol . 3 0ra). The alternative is posed on the as ­ sumption that there are an infinite number of individuals stronger than Soc rates , and that their individual potencies for carrying weight can be of any magnitude whatever so long as it is greater than the magnitude of Socrates' power. " Anyone stronger than Socrates" is thus the common dis tributed term having infinite supP.osita . The affirmative part of the division , i . e. , a maximum weight which anyone stronger than Socrates can carry , is assigned ; for the minimum weight which Socrates cannot carry , namely , the weight which is equal in resistance to his power , can be carried by anyone stronger than Socrates , since its resistance will be exceeded by the power of any suc h . B ut there is no greater weight which anyone stronger than Socrates can carry , since the power of any such can differ from Socrates' power by as little as we please ; and there fore the minimum which Socrates cannot carry will be a maximum which anyone stronger than Socrates can carry . The "term by means of which the division is made" 0

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( " terminus mediante quo fit divisio" ) , we note , refe rs in the example just cited to the potency itself; whereas in the fi rst rule it always refe r red to that on which the po ­ tency acted or by which it was acted upon . It appea rs im ­ possi b le to give a precise technical meaning to Heytes­ bury 's "mediante quo"; thus Gaetano decides that it can int r oduce either ext reme of the compar ison , potency or that on which the potency acts o r by which it is acted upon (fol . 3 5 rb). The te rm with respect to which the division is made ( " terminus respectu cuius fit divisio" ), on the con­ t rary , always denotes the potency for which a boundary is being sought . Heytesbu ry 's description of the cases coming unde r the second rule , then , merely state s that one of the ext r emes of the compar ison is denoted by a dist rib uted term having infinite supRosita . We thus have two types of cases depending on whether the dist r ibuted term refe r s to the potency or t o that t o which the potency is compared . Illu s t rations of the first type a r e the fol lowing: FIRST . -Given a unifo r mly diffor m latitude , there is either a maximum excess by which any degree more in ­ tense than a fixed degree a exceeds the degree which is half of a , or a minimum excess such that not eve ry degr ee mo r e intense than a exceeds the half of a by this excess. The first alternative is chosen, a/ 2 being the r equi red maximum excess. For any degree mor e intense than a ex ­ ceeds the half of a by at least a/ 2 , and there is no exces s g reater than a/ 2 by which any such degree exceeds the half of a , since a degree which is specified only as being mo re intense than a - can differ from a - by as litt le as we 3 5 please . SECO ND . -There is eithe r a maximum distance which can be t r aversed in an hour b y any degree of velocity greater than a , or a minimum distance which cannot b e thus t raversed. The maximum quod sic is assigned; for the distance which can be t r aversed in an hour b y a degree of velocity (say it is one foot) can be t raversed in an hour by any degree of velocity g reater than a; but the re is no distance greater than one foot which can be t r aversed in an hour by whatever degree of velocity greate r than a , since a degr ee of velocity which is specified only as being g reater than a can differ f r om a by as little as we please , and hence can be such that it will t raverse in an hour a distance which exceeds one foot b y as little as we please; hence one foot is the required maximum quod sic . 54 What is char acter istic of each of the examples is that the potency for which a boundary is sought consists of an

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infinite class of pote nci e s of varying stre ngth, bounde d at one e nd of the range by a limiting pote ncy which is not it­ se lf a me mb e r of the class. And the boundary assigne d dif­ fers in kind from those we have pre viously e ncountere d; for although it is affirmative in form (i. e . , a maximum quod sic), unlik e the affirmative boundarie s hithe rto con ­ sidered it is a limit, a boundary standing outside the range bounde d. This can b e se e n, for e xample , in the qi.se j ust cite d: b y any (particular) de gre e of ve locity gre ate r than .a. a gre ate r distance than a foot will be trave rse d in an hour ( "Quolib e t gradu inte nsiori a pe rtransire tur maior quantitas uniformite r in hora quam pe dalis" ). A foot ne ve rthele ss remains the maximum distance which will b e trave rsed by any such in an hour; for no gre ate r distance will be trav­ e rse d in an hour by any and e ve ry de gre e of ve locity more intense than a ( "Nulla maior quantitas quam pedalis pe r transire tur uniformite r in hora a quolib e t gradu intensiori a'-fol. 3 3va). 5 5 A case in which the distribute d te rm refers, not to the pote ncy, b ut to that with which the pote ncy is compare d, is the following: The re is e ithe r a maximum uniformly difform re sistance b e ginning at a de gre e in the more re miss extre me any part of which re sistance Socr ate s can divide , or a minimum such re sistance of which Socrate s doe s not suffice to divide e ve ry part. The maximum quod sic is assigne d, it b e ing the re sistance which te rminate s in the more inte nse ex ­ tre me at a de gre e of re sistance e qual to Socrate s' power. If it is argu e d in opposition that Socrate s could the n divide the whole of this re sistance - a conclusion contrary to the first rule-Heyte sb ury re plies that it is not the same thing to say that Socrates can divide any part of the assigne d re ­ sistance , and to say that he can divide the whole of it, for any part is of less re sistance than the whole , and Socrate s can divide up to the last point e xclusive ly but not inclu­ sive ly . 5 6 In this case as in the pre vious ones, we have to do with an infinite multitude which is b ounde d by a limit at one e nd of the range , so that no last memb e r of the multitude can b e assigne d; namely, the infinite multitude of parts of the re sistance e ach of which b e gins at a de gre e in the re mi $ S e xtre me and te rminate s at some intrinsic point of the me dium b e twe e n a de gre e and the de gre e e qual in re sist­ ance to Socrate s' power. And once more the boundary, though affirmative in form, is a limit lying outside the ran g e bounde d.

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D I V I S I ON B Y C O M M ON D I S T R I B U T E D T E R M H A V I N G F I N I T E S UP P O S I T A

If the division is made by means of a common dist r ib ­ uted te rm having but one or only a finite number of sup ­ Rosita ( " mediante ter mino dist rib uto habente tantum unum vel saltem finita supposita" ), the negative par t of the divi ­ sion is to be sustained. The type of case envisaged b y this r ule may be il lust rated b y the foll owing examples : ( 1 ) The re is eithe r a maximum weight which any man can car r y, or a minimum weight which not ever y man can car r y ( " Aut est da re maximum quod quilib et homo pote st por tare, vel minimum quod non" ) . (2 ) The re is either a minimum visib le body which at a fixed distanc e can be seen by any power of vision, or a maximum visib le body which , a t a fixed distance cannot be seen b y every powe r of vision ("Vel est dare minimum quod quelib et vir tus visiva potest pe rcipere aut maximum quod non" ). As the distr ibuted ter m has but one supRositum or only a finite numbe r of supRosita, one of these must b e the weakest memb e r of the set , and whatever is a b oundar y for this weakest potency wil l b e the boundary for the whole set . Thus in the fir st example cited one of the men denoted b y "any man" wil l b e the weakest; say i t is Soc r ates. F r om the fir st r ule we know that Soc rates' potency for car r ying weight is bounded by the minimum weight which he cannot car r y; b ut the minimum which he cannot car r y is also the minimum which not eve r y man can car r y, since what is possib le for the weakest is possib le for all, and what is not possib le fo r the weakest is not possib le for all. In a similar manner eve r y case coming under the thir d r ule can b e reduced to a case unde r the fir st r ule ; and the r e ­ for e a negative boundary (minimum quod non or maximum quod non) is always to b e assigned. 57 O B J E C T I ON S TO T H E RUL E S AND H E Y T E S B UR Y ' ,S R E P L I E S

The remainder of Heytesbu r y' s tr eatise is taken up with ob jections and r eplies; the ob jections consisting of par tic ular cases to which the r ules do not appear to apply . The cases ar e of two kinds: those in which it seems impossib le to assign any b oundar y at all; and those in whic h an affi r m ­ ative b oundar y seems r equisite where the r ule requir es a negative b oundar y or vice ver sa. ( The fol l owing summar y is c onfined t o a numb er of r epresentative ob jections.) F IRS T Ob jection .- We suppose that point a is distant from a fixed point b b y a foot , and that a can move towar ds

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b with any degree of velocity less than d degree, but not with d degree or any degree greater than d . (The velocity may be uniform or variable, so long as it remains less than d .) This being supposed, it follows that the subcon ­ traries requisite for a di vision are veri fied; for there are some future instants in which a can touch b, and some in which a cannot touch b. According to our opponent, how ­ ever, there is neither a first instant in which a can touch b nor an ultimate instant in which a cannot touch b.58 That there is no first instant in which a can touch h_ is shown as follows: If there were such an instant (call it �, then a could touch b in instant �- Let us therefore imagine a trial run in which a touches b in instant �; since � is the first instant in which a can touch b, in the imagined trial run a could not have reached b at any instant before instant �- Hence a must have moved continually, in each instant of the time during which it moved, as fast as it could. But this conclusion is false, for there is no maximum velocity with which a can move towards b, but rather a minimum velocity d with which it cannot so move. The refore the premiss from which this conclusion was deduced, namely, that there is a first instant in which a can touch b, is also false. That there is not an ultimate instant in which a cannot touch b is argued i n two ways : First, if there were such an instant (call it � ' then i mmediately after �, a could touch b . But in any instant after instant �, a can attain some point beyond b; for the maximum velocity of a can ­ not be assigned, and at whatever velocity it moves it can move at a greater velocity . Therefore as quickly as a can touch b it can touch some point beyond b . But the con ­ clusion (and hence also the premiss from which it is drawn) is false; for any point beyond b is more distant from a than b is distant from a, and a can sooner reach a point that is less remote than one that is more remote, since it trav ­ erses the smaller distance in less time than the greater; hence i t is not the case that a can reach some point beyond b as quickly as it can reach b. The second ar gument is again a reductio ad absurdum. Suppose there is an ultimate instant � in which a cannot touch b, s.o that immediately after �, a can touch b . But immediately after instant �, a cannot touch b . For in in ­ stant � , a touches some point which must either be b or other than b . If it is b, then, contrary to the original as sumption, � is not the ultimate instant in which a cannot touch b but the first instant in which a can touch b. But

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that the point which a touches in instant � cannot be other than b is shown as follows: Two points in a straight line cannot be tangent, but must have a divisible distance be tween them.9J But if the point which a touches in instant � were to differ fr om b by a divisible distance, then, in or der to reach b immediately after instant �' a would have to tr averse this inter vening distance instantaneously- a con ­ clusion which is contr adicted by the tempor al char acter of motion. 6 0 In answering the objection presented by this case, Heytes ­ bury denies that a boundar y cannot be assigned, and ar gues that ther e is an ultimate instant in whie h a cannot touch b. 6 1 As for the fir st objection against assigning such a boundary, namely, that it would follow that as quickly as a could touch b it could touch some point beyond b, Heytesbur y r eplies by means of a distinction between the pr oposition ( 1 ) u As quickly as a can touch b it can touch some point beyond b" ( u lta cito sicut a poterit tangere b poter it ipsum etiam tangere aliquem punctum ultra b" ), and the proposition (2) " It is possible that a touch b as quickly as some point beyond b" ( " Possible est quod a punctus ita cito tangat b sicut punctum ultr a b" ), or in another for m ( 3 ) u A can as quickly touch b as it can touch some point beyond b" ( "A poterit ita cito tanger e b sicut ipsum potest tanger e ali ­ quem punctum ultra b"). The fir st proposition is tr ue. It has a divisive sense (sensus divisus), and means that, differ ent possible trial r uns of a over the given distance being taken into account, as quickly as a can touch b (in one such tr ial r un), it can touch some point beyond b (in another such trial run). The truth of the proposition is evident fr om the extrinsic character of the boundar y which has been as signed: since there is no minimum time in which a can r each b, but rather a maximum time in which it cannot r each b, it follows that before any instant in which a can touch b (in a given tr ial r un), it can touch a point beyond b in another tr ial r un). But the second proposition, in either for m (2 ) or ( 3 ), is false. For it has a composite sense (sensus comP.ositus), and means that, in a single tr ial r un, a can touch b at the same instant of time as it can touch a point beyond b; an impossible r esult because the passage fr om b to a point beyond b r e qui r es time. 62 The second objection is again solved by means of a distinction between a proposition with composite sense and a pr oposition consisting of the same terms but having a divisive sense. The statement "Immediately after instant � ' a can touch b" ( "Immediate post � instans poter it a 82

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tangere b" ) is t r ue , while the statement " It is possible that a immediately after instant � should touc h b" ( .. Possible est quod a immediate post � instans tangat b" ) or .. A c an touc h b immediately after instant e" ( .. A pote st tangere b immediate post � instans', ) is false. The fir st proposition has a divisive sense , and means that , several t r ial r uns of a over the given distanc e being taken into ac c ount , how ­ ever soon after � instant a c an touc h b (in one suc h t r ial r un) , it c an touc h b sooner (in another suc h t r ial r un).63 The sec ond pr oposition , in either for m , has a c omposite sense , and means that the a of a given t r ial run c an touc h b immediately after instant �' that is , without any lapse of time. 64 The ar gument of the objec tion pr oves the impos ­ sibility only of this sec ond pr oposition; while the fir st pr oposition, being expounded , gives the c or r ec t sense of the limit whic h Heytesbur y has assigned. 6 5 SEC OND Objec tion. -We c onsider the infinite c lass of all possible diffor m latitudes or r anges of some suc h qual ­ ity as heat, whiteness, light , or motion , the quality as suming differ ent degr ees of intensity along the length of a body or at differ ent instants of time. Ther e ar e some diffor m r anges of suc h a quality of whic h some par t is mor e intense than a given degr ee a , and ther e are some diffor m r anges of suc h a quality of whic h no par t is more intense than a degree; and yet there is neither a most re­ miss latitude of this kind of whic h some part is more in ­ tense than a degree , nor a most intense latitude of whic h no par t is more intense than a. To under stand this objec tion , it is nec essar y to know the manner in whic h the intensity of a difform latitude is c alc ulated. If the latitude is unifor mly diffor m , it c or re sponds in intensity to its medium degree; for example , if the whiteness of a body var ies unifor mly from an intensity of 2 degr ees at one end of the body to an intensity of 4 at the other end , this latitude of whiteness is r egarded as equivalent to a unifor m latitude of 3 degrees extended over the same length. If the latitude is diffor mly diffor m in suc h manner that eac h par t of it is unifor m , then eac h uniform par t of the latitude c ont r ibutes to the intensity of the whole in propor tion to the amount of the subjec t (a body or length of time) over whic h it extends; for example , if the white ness whic h extends over one- thir d of a body has an intensity of 2 degr ees , and the whiteness of the remainder an inten ­ sity of 6 degrees , the intensity of the whole latitude is 2 ( 1/ 3 ) + 6 (2/3 ) = 1 4/ 3 , or four and two- thir ds. If one of the par ts of a difformly difform latitude is uniformly difform , 83

WILLIAM HEYTESBUR Y thi s part i s con s idered a s equivalent to a unifor m latitude having an inten s ity equal to it s medium degr ee and ex ­ tending over the s ame length; and this equivalent unifor m latitude enter s into the calculation of the inten s ity of the whole latitude as before. With diffor mly diffor m latitude s the ave rage intens ity of which can be dis covered only by mean s of an integr ation- namely , tho se in which the inten ­ s ity varie s in the manner of the ordinate s of a cu r ved line - the Schoolmen we re of cour s e unable to deal . (The p r ob ­ lem of the phy s ical meaning to be attached to the aver age inten s ity of a latitude will be discu s s ed at the end of the next chapter.) F r om the mode in which the inten s ity of a latitude is dete r mined , it follow s that two latitude s may be s uch that any inten s ity po s s e s s ed by some par t of the one is po s ­ s es s ed by s ome part of the othe r , and vice ve r s a , and yet the latitudes diffe r in (aver age) inten s ity. Thu s a latitude, one-third of which has an intens ity of Z degree s and the remainder an intens ity of 6 deg r ees , i s more intens e than a latitude one-half of which has an intens ity of Z degree s and one-half an intens ity of 6 degr ee s , although eve r y in ­ ten s ity belonging to s ome par t of the fir s t al s o belong s to s ome par t of the s econd , and vice ver s a. Moreover , it fol­ low s that one latitude may be mor e intens e than a s econd , and yet have s ome par t which i s mor e remi s s than any par t of the s econd; or one latitude may be mor e r emi s s than a s econd , and yet have s ome par t which is mor e inten s e than any par t of the s econd. It follow s , finally , that no bounda r y can be a s s igned in the ca s e under cons ider ation. There i s no mo s t r emi s s diffor m latitude of which s ome part i s mor e intens e than a degree , becau se any diffo rm latitude of which s ome par t i s mo re intens e than a degr ee being as s igned , a mor e r e­ mi s s diffor m latitude can be found of which s ome par t i s mo re intens e than a degr ee. Similarly , there i s no mos t inten s e diffor m latitude of which no par t i s mor e inten s e than a degree , for any diffor m latitude of which no par t i s mo re intens e than a degr ee being as s igned , a mor e inten s e latitude of this kind can be found of which no part i s mor e intens e than a deg r ee. 66 In r eply , Heytes bu r y concede s that no boundar y can be a s s igned , but denie s that the r ule s for a s s igning boundar ie s a r e the reby pr oved in s ufficient; for the negative s ubcon­ trar y ( " There are s ome diffor m latitude s of which no par t i s more inten s e than a degr ee" ) i s not ver ified in the s en s e requir ed for making a divi s ion. In o r de r that a division

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s hould fo llow , it is not only nece s s ary that there exis t dif ­ form latitudes o f which no part is more inten s e than a degree , b ut als o that the re s hould be no difform latitude which is equally intens e as any o f the s e and which does have a part more intens e than a degree. B ut this condition does not hold , for given any latitude of which no part is mo re intens e than a degree , another equally intens e lati ­ tude can be· found of which s ome part is more intens e than a degree. In effect , the third condition for in s uring the valid ­ ity of the divis ion is here uns atis fied ; namely , that the potency in ques tio n mu st be capab le of taking on a contin ­ uou s range of values up to the value which is to serve as b oundary , and no other value s . 67 THIR D Ob jection.- There are s ome debilitable potencies for carrying which s uffice to carry weight a during a s tretch of time (un specified in length) , and there are s ome s uch potencie s which do not s uffice to carry weight a during a s tretch of time , and yet there is neither a mini ­ mum potency which s uffice s nor a maximum po tency which does not s uffice . There is no minimum potency which s uf ­ fice s , firs t , becau s e it would follo w that a is the maximum weight which this potency can carry-a re sult co ntrary to the firs t rule for mutab le potencies . Moreover , s ince it is mo re difficult to carry a for a greater time than for a les s , no potency which s uffices to carry a for a time can be a minimum , s ince any potency b eing propo s ed as s uch, there will b e a s maller potency which can carry a for a s horter time. Nor is there a maximum potency which does not s uf ­ fice . For by the fir s t rule , any propo s ed a s s uch could not be other than a potency equal to the tendency of weight a to des cend ; s ince no action proceed s from a ratio of equal ­ ity , however , a if s upported b y this potency would not be able to descend , unle s s with infinite s lownes s ; but to de ­ s cend with infinite s lownes s i s the s ame as not des cending at all. Thus a would not des cend , and the potency under con s ideration cannot therefore b e termed a maximum quod non. 68 Heytesbury replies that a maximum quod non is as s ign ­ ab le , and that it is in fact that potency which is equal to the tendency of weight a to de s cend . When it is argued that , b ecau s e of the equality between the downward tendency of a and the po tency , a would not b e ab le to des cend , or at lea s t could des cend only with infinite s lownes s , Heyte s b ury denies the con s equence; the proper co nclu s io n is that with infinite s lownes s a would begin to des cend . For in the fir s t ins tant of the action , the po tency wo uld be equal to a ,

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WILLIAM HEYT E SBURY but i m m e diately afterwa r d s it would debilitate from f..:1tigue, and thu s immediately afterwar d s it s power of r e s i s tance would be le s s than the tendency of a to de s cend, s o that a would in fact de s cend . With r e s pect to the ve r y s ta r t of the action, however , it i s cor rect to say that with infinite s low ­ ne s s a de s cend s , althou gh a doe s not de s cend with infinite s lowne s s ( "Sequitur quod infinita tar ditate de s c endet a g r ave cum illa potentia, et quod a non de s cendet infinita tar ditate" ) . In the fir s t clau s e the te r m " infinite ;' in p r e ­ ceding al l other ter m s , hold s s incategor ematically; and the meaning is thu s that, if at s ome time a will move with a cer tain s lowne s s , at a time clo ser to the s ta r t of it s mo ­ tion it will move with a double s lowne s s , and at a time yet clo s er to the s tar t with a quadr up le s lowne s s , and s o on in infinitum, s o that at the ver y s tar t a move s with a s malle velocity than any a s s ignable. In the s econd clau s e, on the contr ar y, the ter m " infinite" hold s categor ematical ly; and the meaning i s thu s that a move s with a s lowne s s actual ly infinite, that i s , with zer o velocity- which involves the con ­ t radiction that a move s and doe s not move . 69 FO U RTH Objection . - We s uppo s e that a hot body a having the highe s t pos s ible deg r ee of heat and a cold body b having the highe s t po s s ib le deg r ee of coldne s s are p laced in a vacuum, s o that all extr in s ic aid s or hindr ance s to the ac ­ tion of the one on the other ar e r emoved. Then a would s uf ­ fice to cor r upt p r eci s ely half the latitude of col dne s s in b, and b would suffice to cor r upt half the latitude of heat in � and thu s in each ca s e it would be nece s s ar y to a s s ign a maximum quod s ic rathe r than a minimum quod non- a r e ­ s ult contrary to the fir s t r ule for mutable potencie s ( fol. 3 2 vb). In r eply, Heyte s bury concede s that a maximum quod s ic i s to be a s s igned. He ar gue s , howeve r , that the divis ion i s with r e spect to immutable potencie s , s ince thr oughout the action each agent i s continual ly equal to the other ; hence fol lowing the rule for immutable potencie s we a r e to choo s e the affirmative par t of the division. If, on the cont r ary, a pos s e s s ed a mor e inten s e or more r emi s s latitude of heat than b' s latitude of coldne s s , it would have to be con s ider ed a mutable agent, and the negative part of the divi sion be as s igned .7 0 F IF TH Objection.- Final ly, Heyte s bu r y con s ider s an ob ­ jection to a boundar y as s igned in a cas e a l r eady con s id ­ ered : I s the r e a maximum dis tance which can be tr aver s ed unifor mly in an hour by any deg r ee of velocity g reater than a ? T he opponent ar gues that there i s not, becau s e if the r e

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were such a maximum , i t would have to be the distance traversed by velocity a in an hour , and this distance (say it is one foot) is the maximum which would be traversed by any degree of velocity greater than a in less time than an hour . For any degree of velocity greater than a can traverse a foot in less time than an hour , and there is no greater distance which can be traversed by any such in less time than an hour , since there is no ratio in which any degree of velocity greater than a is greater than a , and an hour is required by a degree to traverse the dis ­ tance . Hence this distance is the maximum which would be traversed by any degree greater than a not in an hour , but in less time than an hour (fol l . 3 2 ra - 3 3rb). Heytesbury replies that the same distance is the maxi ­ mum which can be traversed by any degree of velocity greater than a in an hour , and in any less time than an hour , for the re is no ratio in which an hour is greater than any length of time less than an hour (fol . 3 3va). S U M M AR Y

So much for Heytesbury's treatment o f maxima and min ­ ima. Mathematically and logical ly , it is the most interesting of the Scholastic treatises on this subject we have exam ­ ined . While ostensibly concerned with the solution of cer tain hypothetical physical problems , it brings into play a group of ideas and distinctions which bear directly on the logicomathematical problem of setting a bound to finite and infinite aggregates or sets. From a presentday vie w , what i s remarkable about these ideas and distinction is their simi larity to ideas and distinctions which in the nine teen th century we re to have a crucial role in the laying of a logical ly rigorous foundation for mathematical analysis : that the inferior and superior extrema of a numerical ag ­ gregate may belong to the aggregate , or may not so be long ; that a finite set of finite values has a least member, whi le an infinite set of such values may lack a least mem ­ ber ; that the existence of maxima and minima depends on certain conditions , which must be veri fied before an ex ­ tremum value of a set can be assigned .

S

UISE T H ' S Libe r C ale ulationum

Among the treatises on maxima and minima of later date than Heytesbury's , the one most nearly comparable to He ytes bury' s in precision and subtlety- though not in gen -

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erality- is that included in Suiseth's Liber calculationum .7 1 This treati s e deals exclusively with the case of an agent moving against a resistance , and carries the analysis of this problem much further than does Heytesbury. The prin ­ ciple according to which no action results from an equality between agent and resistance becomes crucial. The pri ­ mary distinctions drawn turn on whether the agent is debil ­ itable or indebilitable ,72 the resisting medium uniform or difform . UNIF O R M M E DIA

FIRST: A debilitable or indebilitable agent in compari ­ son with uniform media .- Suppose a is an indebilitable potency; there are some uniform media which a can divide and traverse and some which it cannot. Let £_ be the me dium equal in resistance to the power of a; then a cannot divide £_, but any medium of less resistance than £. it can divide , since to any such a will have a ratio of greater in ­ equality . Moreover, as a can divide some part of any such medium it can divide the whole of it , for it is indebilit ­ able .73 N ow let b be a debilitable potency equal at the start to a , so that a = b = £. . Like a , b cannot divide £., but it can begin to divide any medium of less resistance than £., for to any such it will have at the start a ratio of greater in ­ equality . Indeed , conceiving the mode in which the power of b decreases as undetermined , we may go further and say that .h. can divide the whole of any such medium; for however little it may be supposed to debilitate , it can be such that it will debilitate less , and hence it c an be sup ­ posed to debilitate in such a manner as always to main ­ tain a ratio of greater inequality to any given uniform medium of less resistanc e than c . The conclusion is that £ is a minimum quod non with respect to the debilitable potency b as well as the indebilitable potency a . We note that the indeterminate sense of "debilitable ;' as referring to a power which decreases fro m a given strength but at no definitely assigned rate , is the one con ­ stantly used by Calculator . In present- day terms , the debilitable potencies which are of the same initial strength and are differentiated by their undergoing debilitation at different rates , variable or constant , form an infinite set whic h is not linearly ordered ; they nevertheless find an upper limit in the indebilitable potency to which the y are initially equal , and therefore also in the resistance equal to this indebilitable potency. The limit concept thus comes into operation on both sides of the comparison b etween agent and resistance.

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SECOND : A uniform medium in comparison with debili ­ tab le or indebilitab le agents . - Calculator now turns to the case for the corresponding passive potency , i . e . , the re sisting medium. Here it is necessary to assign a maximum active potency , whether deb ilitab le or indebilitable , which cannot divide a given uniform medium , and this maximum quod non will b e the potency equal to the resistance of the medium . For every greater potency will be able to divide the given medium; this statement is true even if the potency is deb ilitable , for however little it debilitates it can debil ­ itate less , and thus always maintain a ratio of greater in ­ equality to the given resistance until it reaches the end of the medium . (Here we find the class of potencies greater than the given resistance provided with a lower boundary .) THIRD: An active indebilitable potency in comparison to the extent of a given uniform medium which can be traversed in a limited time. - Like Heytesbury , Calculator holds that, in order that maxima and minima can be as signed, it is necessary that two subcontrary propositions be true; for example, in the case considered in the first section , it is necessary that the two propositions " Some uniform media the potency a can divide" and " Some uni ­ form media a cannot divide" both be true , in order to guarantee that the media can be divided into two classes with a boundary between the two. Now of the two subcon ­ trarie s " Some part of a given uniform medium potency a can divide" and " Some part of the same medium a cannot divide ;' · the latter is false; for in the first section we learned that if a debilitable or indebilitable potency has a ratio of greater inequality to the resistance of a uni ­ form medium and can thus divide part of this medium , it can traverse the whole . Both of the subcontraries can be verified , however , if we add a limitation with respect to time , thus: " Some part of the given uniform medium po ­ tency a can divide in time d" and " Some part of the given medium a cannot divide in time d:' With this temporal limi tation , it becomes possible to assign a maximum part of a given uniform medium which can be traversed by a given indebilitab l e potency a in a given time d; and this maximum guod sic (let it b e �) will be that part which a can traverse moving constantly at the greatest velocity it can maintain in the given medium . FOU RTH : An active debilitable potency in comparison with the extent of a given uniform medium which can b e traversed in a limited time. - This case is the same as the previous one, except that the debilitable potency b has been substituted for the indebilitable potency a. As b = a ,

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it follows that the maximum part � of the given medium which a can trave rse in the given time d is the minimum which b cannot trave rse in d time (minimum quod non); for b cannot traverse e , since it deb il itates and thus moves w ith decreasing velocity ; but it can trave rse any smalle r extent of the given me dium , since howeve r little i t deb i li ­ tates it can deb ilitate less , and thus maintain an ave rage velocity sufficient to trave rse any assignab le quantity of the given medium less than � in time d . F IF TH : A uniform medium of specified resistance and length in compar ison with active deb ilitab le or indeb ili ­ tab le potencies , with temporal limitation . - He re the situ ­ ation considered in the thi r d and fou rth sections is viewed from the standpoint of the passive potency . G iven a uni ­ form medium of dete rminate resistance and length ; this medium can be traver sed by some potencies in the speci ­ fied time and not by othe rs. There wil l be a minimum in deb i litab le potency which , moving at the g reatest velocity it is ab le to maintain , can trave rse it in the given time (minimum quod sic) . And the deb i litab le potency equal to this indeb ilitab le potency wi ll be the maximum deb i litab le potency which cannot trave rse the given me dium in the given time (maximum quod non). Fo r this deb i litab le po ­ tency cannot traverse the medium in the given time , since it deb ilitates ; but any greate r deb i litab le potency can do so , since however l ittle any such deb il itates , it can deb i l ­ itate less , an d therefore it can deb i litate in such manner as to maintain an ave rage velocity sufficient to trave rse the whole medium in the given time. SIX TH : Active deb i litab le o r indeb i litab le potencies traversing a uniform medium of specified resistance and length in compar ison with time.- B oth the divi ding potency and the medium are now taken as dete rminate , and a boundary is d rawn in te rms of the time requi red by the given potency to trave rse the given medium. In the case of an indeb i litab le potency (call it � ' there wi l l b e a mini ­ mum time in which , moving at the greatest velocity of which it is capable , it can t raverse the given medium b (minimum quod sic); for in no less time can it t rave rse so much. The same length of time will be the maximum in which a. deb i litab le potency c equal to a wil l not be ab le to traverse b (maximum quod non). This is shown as fol ­ lows. Sine e c = a , and c deb ilitates whi le a does not , _ c cannot traverse as much as a in the given time and hence it cannot traver se b in the given time. B ut whatever time .!_ g reater than the given time be assigned , there is an in -

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debilitable potency less than a which can tra verse b in_!_; and whatever inde bilitable potency p less than a be assig ned, f.. can so debilitate as to maintain an averag e velocity equal to the constant velocity of p; f or however little it debilitates it can debilitate less. H ence in any time g reater than the g iven time, f_ can traverse b.74 D I F F OR M

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Calculator now turns to the cases in which the media to be traversed are diff orm. The resistance of a diff orm me­ dium may be diff orm in every part, or it may contain a part which is of unif orm resistance. In the latter case the part of the whole medium which is of maximum resistance may be the unif orm part, or it may be diff orm. F inally, if the part of the whole medium which is of maximum resist­ ance is diff orm, it can happen that the dividing ag ent will be broug ht to a stop at the point of the diff orm part where the resistance is equal to its power; and it can also happen that the ag ent will never reach such a point, but will move eternally towards it at an ever decreasing velocity.7 5 SEVENTH : A diff orm medium of which the part of g reatest resistance is unif orm, in comparison with debili­ table or indebilitable potencies. -This case reduces to a case previously considered(second section), that of a uni­ f orm medium in comparison with debilitable or indebili­ table potencies. There is thus a maximum potency, debili­ table or indebilitable, which cannot traverse the unif orm part, and which theref ore cannot traverse the whole; namely , the potency equal to the resistance of the unif orm part. EIG HTH : Debilitable or indebilitable potencies in com­ parison with the parts of a diff orm medium when the part of the medium having the g reatest resistance is diff orm. -The part in which the potency beg ins to move may be either unif orm or diff orm. If it is unif orm, there is a max­ imum part of the medium which any debilitable or inde bilitable potency capable of traversing part of the medium can traverse (maximum quod sic); and this maximum is the unif orm part.76 If th e initial part of the medium is dif ­ f orm, it may be such that the potency will be broug ht to a stop at some intrinsic point of it, at a point, that is, in which the resistance becomes equal to or g reater than the potency; or it may be such that the potency will move eter nally without ever reaching this point. In the f irst case, there is a maximum part of the whole medium, starting f rom the po int of inception of motion, which an indebilitable

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potency can traverse (maximum quod sic), namely , the part which it traverses before being brought to a stop; and the same part will be the minimum which a debilitable potency equal to the indebilitable potency cannot traverse (minimum quod non). In the second case , there will be a minimum part , starting from the point of inception of mo tion , which the potency , whether debilitable or indebilitable , cannot traverse (minimum quod non); namely , that part ex ­ tending to the point at which resistance and power become equal , the whole of which the potency , though moving eter nally , can never traverse. NINTH: A totally difform medium in comparison with debilitable or indebilitable potencies , the medium being such that the potency is brought to a stop at the end or at some intrinsic point of it. -The indebilitable potency equal to the maximum resistance of the medium will be the min ­ imum indebilitable potency which can traverse the whole medium (minimum quod sic); for any weaker potency would be brought to a stop before reaching the point of maximum resistance. On the other hand, a debilitable po ­ tency equal to the previous indebilitable potency will be the maximum debilitable potency which cannot traverse the whole (maximum quod non). For since it debilitates , it will be brought to a stop at an intrinsic point of the medium; but any greater debilitable potency will be able to traverse the whole , since however little it debilitates it can debilitate less , and thus maintain a proportion of greater inequality to every point of the resistance. TENTH: A totally difform medium in comparison with debilitable or indebilitable potencies , the medium being such that the potency is never brought to a stop , but moves eternally without reaching the point of resistance equal to its power. -An example of the requisite type of medium is a medium with uniformly difform resistance. W hether the potencies considered be debilitable or inde ­ bilitable , the limit to be assigned will be a maximum potency which cannot traverse the whole medium (maxi ­ mum quod non) , namely that potency which is equal to the maximum resistance of the medium. Any greater potency will be able to maintain continually a ratio of greater in ­ equality to the resistance , and thus traverse the whole. ELEV ENTH: A difform medium in comparison with debilitable or indebilitable potencies , with a limitation as to time. -Given a difform medium of a determinate length , this medium can be traversed by some potencies in a specifie d time and not by others. There will be a mini -

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mum indebilitable potency which , moving continually at the greatest velocity it can maintain , can traverse it in the given time (minimum quod sic). And the debilitable po ­ tency equal to this indebilitable potency will be the maxi ­ mum debilitable potency which cannot traverse the given medium in the given time (maximum quod non). We note that these conclusions are the same as tho�for the cor ­ responding case of a uniform medium (vide supra -- - , fifth section). TW E LF TH : Debilitable or indebilitable potencies in comparison with a difform medium , with a limitation as to time . - This case is the converse of that considered in the eleventh section , and once more the conclusions are the same as those resulting when the medium is uniform (vide supra , third and fourth sections). There is thus a maximum part which an indebilitable potency , moving continually at the greatest ve locity it can maintain , will be able to traverse in the specified time (maximum quod sic); and the same part will be the minimum which an equal debilitable potency cannot traverse in the specified time (minimum quod non). S U M M ARY

I t will be seen that Calculator has furnished a system ­ atic set of rules for assigning maxima and minima in the case of a potency dividing a re sisting medium. Although both the problem of assigning a limit to a potency and that of determininf the relations between a moving power and a resistance 7 are of Aristotelian origin , Calculator's treatment is highly original. In the distinction between cases and in the demonstration of rules , the predominant role is played by considerations of quantity. The limit con ­ cept is expressed with precision ,78 and comes into action on both sides of the comparison between power and re sistance. Particularly ingenious is Calculator's method of handling quantities- debilitable potencies and difform media- which vary in an indeterminate manner , but for which he is able to supply a bound at one end of the range of variation. In such cases , Calculator approaches the abstraction and generality which characterize the modern definition of ulimit:' 79

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TAT E R Di s c u s s ion s

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o f Maxi m a and Minim a

Among the disc ussions of maxima and m1n1ma wr itten in the latter half of the four teenth, in the fifteenth, and in the sixteenth c entu ries, ther e are a few only whic h follow the logic omathematic al distinc tions laid down by Heyte sbur y . In the larger number the c ontext of the disc ussion is onc e mo re the Ar istotelian c ommenta ry, and the analysis is pur sued P.hysic e loquendo rather than 12..gic e loquendo. In par tic ular , two physic al pr oblems- pr oblems whic h often inte rtwine- hold the c enter of inte rest: whether ther e is a minimum possible si ze fo r natur al things suc h as the ele­ ments, and whethe r the dur ation of a thing is bounded ex­ trinsic ally or intr insic ally. We summa riz e b r iefly a .num­ ber of these disc ussions. COMMEN T ARY AT T R IB U T E D TO DUNS S CO T US

The tr eatment of maxima and minima in the pseudo S c otian c ommentar y on the Physic a is similar in some r espec ts to B ur idan' s disc ussion, and in other s to Heytes­ bury' s .8 0 In the assignment of boundaries of potenc ies the author distinguishes fou r possibilities: the potenc y may be ter minated by ( 1 ) a maximum in quod pote st, (2 ) a min­ imum in quod pote st, ( 3 ) a maximum in quod non pote st, or (4 ) a minimum in quod non P.Otest . M aximum in quod potest means "so muc h and no mor e, but any less:' M ini­ mum in quod potest means " so muc h and no less, but any greater :' Maximum in quod non potest means " not so muc h nor any less, but any gr eater :' M inimum in quod non po­ test means "not so muc h nor any gr eate r , but any less :' Types ( 1 ) and (4 ), we note, are upper boundar ies; types (2 ) and ( 3 ) are lower boundaries. An ac tive potenc y takes an upper boundar y; for gener ally speaking, if it c an ac c omplish a c er tain amount, it c an ac c omplish any less amount but not every greater amount. A passive potenc y, on the c ontr ary, takes a lower boundar y; fo r if it c an h e ac ted o n b y a given agent, it c an b e ac ted on b y any str onger agent, but not by ever y weaker agent. Ac tive potenc ies, the author states, may be c ompa r ed with six c i r c umstanc es: ( 1 ) the r esistanc e against whic h they ac t; (2 ) the dur ation of the ac tion; ( 3 ) the veloc ity of the ac tion; (4 ) the distance thr ough w hic h their ac tion o r infiuenc e extends; ( 5 ) the magnitude of the effec t p r oduc ed; ( 6 ) the spac e over whic h they may be suffic ient to move an

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objec t in loc al motion. Eac h of these modes of compari ­ son is taken up separately. FIRST: Potenc ies in comparison with resistance. - To begin with, the author assumes that any excess of potency over resistanc e suffic es for the produc tion of motion; whenever the agent or active potenc y is more able or stronger in producing motion than the patient or objec t ac ted on is able to resist, motion will be produc ed. Certain Schoolmen object to this assumption, conceding that any exc ess of potenc y over resistanc e is sufficient to c ontinue the motion, but denying that any excess is suffi ­ c ient to start the motion. For it is more difficult to start a motion than to continue it. The latter statement, we note, is ac tually a c orrec t observation of everyday experienc e, arising, first, from the nec essity of overc oming the initial inertia of the body to be moved, and sec ond, from the greater magnitude of starting friction as compared with running friction. It is, in fac t, one of the neglec ted obser ­ vations whic h might have led to the modern understanding of the role of forc e, as producing ac c eleration rather than veloc ity. In rebuttal, the author argues that if any exc ess of po tenc y over resistanc e suffic es to continue the motion, it follows that any exc ess suffic es to start the motion. For suppose this were not the c ase. To start a motion c annot be infinitely more diffic ult than to c ontinue it; henc e the starting will be more diffic ult than the c ontinuing in some finite ratio, say 100: I. Let a be an exc ess of potenc y over resistanc e which, for the sake of the argument, we assume to be suffic ient for c ontinuing the motion, but not for starting it. Then suppose b is an exc ess having one hun ­ dredth the value of a . Exc ess b will also suffic e for c on ­ tinuing the motion, since by our opponent' s admission any exc ess suffices for this purpose. But it was assumed that starting the motion is one hundred times more diffic ult than c ontinuing the motion, and exc ess a is one hundred times exc ess b; henc e exc ess a suffices to start the mo ­ tion- c ontrary to the original assumption. The same argu ­ ment may be repeated for any a and for any ratio we c hoose. Therefore if any exc ess suffic es to c ontinue the motion, it follows that any exc ess suffic es to start the mo ­ tion. From a modern point of view, the fault with this proof is that it supposes the resistanc e to be overc ome in starting the motio n is the same as that to be overcome in c ontin ­ uing the motion, whereas the former, b� ing composed of

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starting friction and inertia , is greater than the latter , which consists only of surface fricti on and fricti on of the medium. The term u difficulty; ' as used in the ob jection , remains rather mysterious , si nce it refers to something other than resistance. The proof has a mathematical in ­ terest , however , for it shows the author using the mathe ­ matician's epsi lon as rigorously as any modern. Thus if the ratio of the excess necessary for starti ng the motion to that necessary for continuing it is a/b , then , any arb i ­ trarily small e being chosen , a o may b e found such that e / o = a/b ; b ut since the excess b y which the potency must exceed the resistance in order to continue the motion can be made less than o (say it is x < o) , it follows that the corresponding excess y b y which the potency must exceed the resistance in order to start the motion will b e less than E (y/x = a/b = e / o ) . This assumption estab lished , the author proceeds to determine what type of b oundary is to be assigned for po ­ tencies acting against a resistance. Such potencies ar e active (the resistance b eing the correspondi ng passive po ­ tency) , and ' therefore the b oundary must b e either a maxi ­ mum in quod P.Otest or a minimum i n quod non P.Otest. It cannot be both; for then the minimum in quod non P.Otest would have to exceed the maximum i n quod P.Otest b y a divisib le i nterval , and whatever resistance were to b e found within this i nterval , the potency the b oundary of which is sought would either suffice to act against these resistances or would not suffice , so that in either case one of the proposed b oundaries would have to b e discarded. N or can it b e a maximum in quod P.Otest; the author em ­ ploys the same proof of this as that previously cited from the uProbationes conclusionum:' Hence it is a minimum in quod non P.Otest , namely , the resistance that is equal to the potency the boundary of which is sought. For the fact that Aristotle assigns a maximum in quod P.Otest rather than a minimum in quo d non P.Otest , the author offers three possib le explanations. Aristotle meant to as ­ sign a maximum i ntegral number of measures , e. g. , a 1naximum integral numb er of pounds which someone can carry , without regard for the weights or other resistances whi c h do not correspond to integral measures. Aristotle did not look upon resistances as magnitudes commensur ab le or comparab le with potencies , so that he would not have spoken of a given potency as b eing greater than , equal to , or less than a given resistance. Aristotle may not have

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D E M A XIM O E T M I NI M O distinguished between a maximum i-n quod -- 2otest and a minimum in quod non 2otest. SECO N D : Potencies in comparison with the duration of the action . -A potency to endure, the author holds, is nothing other than a powe r of r esisting a contrary. Thus the heavens, which a r e not qualified by any quality to which there is a contrary quality, cannot be said to have a potency to endure . Among the cor r uptible substances in the subluna r world, none will endure fo reve r. T his conclusion is clea r in the case of mixtur es, which contain within themselve s a prin­ ciple of contr ar iety. In the case of the ele men ts, the au­ tho r holds that eter nal duration is impossible, for the element would then be fr ustrated from attaining the end to which it is o r dained by nature, vi z . , that of enter ing into a mixtur e . Even fire, which in its sphere (just under the lunar spher e) would appear to be exempt from the process of co r r uption, is fo rced by the ruling constellations into the lowe r places, where it enters into mixtures and thus loses its identity . The boundary of the time dur ing which a res naturalis can endu re, the autho r states, cannot be a maximum; whatever the length of time we would assign as such in a given case , the thing or substance could exist for a longe r time under impr o ved ci rcumstances . The remaining pos sibihty is that the boundary should be a minimum time dur ing which the res natur alis cannot endure. In concluding this section, the author states that the magnitude of an active potency cannot be deter mined by r eference to the time of its duration; for we obser ve £re quently that the being or thing of smaller potency endures fo r a longer time than the being or thing of greater po ­ tency . THIRD : Potencies in comparison with the velocity of the action . -No potency in itself, the author maintains, is bounded in any way as to the velocity it can p roduce; fo r velocity follows the ratio of the potency to the resistance, and this r atio can be increased infinitely by diminution of the resistance. If the resistance is fixed, however , the po ­ tency will be te rminated by the maximum velocity it can p roduce against the given r esistance . Since velocity de ­ pends on the amount of resistance as well as on the potency, the velocity a potency is capable of p roducing cannot be taken as an indication of its magnitude. FO U RTH : Potencies in compar ison with the distance thr ough which thei r action extends. -The boundary of the

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distance through which an agent can propagate its influ ­ ence is , according to the author , a minima distantia in quam non P.Otest _E.gere. For this propagated effect (e. g. , the il lumination of a candle) shades off continuously to zero intensity ; so that to whateve r point some of the in ­ fluence of the agent extends , there is a point yet fur the r f rom the source to which some of the influence (a smal ­ ler amount) extends . The boundary is the refore extrinsic rather than intrinsic. The autho r notes that the distance from which a pas sive potency may be affected by an agent , e . g. , as the eye is affected by a colored ob ject , is bounded in both direc tions; thus an object can fail to be seen eithe r because it is too far from the eyes , or because it is too close to the eyes . B oth boundaries are ext rinsic. The magnitude of an active potency cannot be known from the distance through which it p ropagates its influ ­ ence ; for an agent , e.g. , a light-sour ce , which has g reate r extensity but less intensity than anothe r may p ropagate its influence over a greate r distance ; but the magnitude of the potency is to be identified with its intensity rather than with its extensity. F IF TH : Potencies in comparison with the magnitude of the effect p roduced.- The author distinguishes between the potency which p roduces an effect which in tu rn helps the potency in the fur the r production of a greate r effect , as fire p roducing fire is helped b y its product to produce more fire ; and the productive potency which is not thus assisted by its effect , e . g . , a lightsour ce producing light. The fir st type of potency has no boundary ; the effect it p roduces may be augmented infinitely . The second type of potency is bounded by the maximum effect it can pro ­ duce ; for it p roduces p recisely as great an effect as it can p roduce (" Tale agens agit secundum totum conatum suum" ). SIX TH : Potencies in comparison with space t rave rsed. - If -the potency alone is allowed to be the dete rmining factor , the quantity of space which a potency may cause an ob ject to t rave rse is altogethe r unlimited. Hence the magnitude of a potency cannot be known from the quantity of space t raversed. F rom the preceding conclusions it is clear that the magnitud� of a potency is best known or dete r mined b y the resistance which i t suffices t o act against . The author now turns to the p roblem of as signing b ound ­ aries to passive potencies. Among passive potencie s he

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di s t i n g ui s he s b e tw e e n tho s e whic h a r e me r e ly r e c e ptive , l ike mate r i a P. r i m a , and tho s e whi c h r e s i s t the a c t i on o f t h e a g e nt ; t h e fo r m e r typ e c an b e ac te d o n b y an y a g e nt , h o w e v e r we ak o r s t r o n g , and thu s admit o f no te r m inu s . He fur the r di s t ingui s he s b e twe e n p a s s i ve pote nc i e s whic h a r e p a s s i ve only , and tho s e whic h a r e ac tive a s we l l ; thu s s i g ht i s not only a p atie nt with r e s p e c t to the vi s ib l e ob ­ j e c t , b ut al s o c onc u r s ac tive ly in the p r o duc t i on o f vi s ion . Amo n g the r e s i s t an c e s e xe r te d by p a s s i ve p o te nc ie s the autho r di s ti n g u i s h e s th r e e typ e s : tho s e r e s i s t an c e s whi c h d imini s h b o th the v e l o c i ty o f the ac ti on and the int e n s ity o r d e g r e e of the e ffe c t p r o duc e d ; tho s e whic h di mini s h only the ve l o c ity of the ac t i o n ; and tho s e whi c h di mini s h only the int e n s ity of the e ffe c t p r o duc e d . T he l atte r type o f r e s i s t anc e i s e xe r c i s e d b y a m e dium with r e s pe c t to the i llumination c au s e d b y a l i g ht s ou r c e ; the li ght i s p r op ag ate d �n s t antane ou s l y , b ut it s int e n s i ty i s r e duc e d by the r e s i s t ­ anc e o f the me diu m . T he b o unda r y o f a p a s s i ve p ot e nc y w hie h e xe r c i s e s r e s i s t anc e , the autho r a r g ue s , i s not the mini mum o r we ake s t a g e nt b y whi c h t h e pot e n c y c an b e affe c te d . If s u c h we r e the c a s e , it wo uld b e p o s s ib l e to a s s i gn a s l o we s t o r mo s t r e mi s s ac tion , n a me l y , that p r o duc e d b y the we ake s t a g e nt c ap ab l e o f ac tin g on the p a s s i ve pote nc y ; b ut a s l owe s t o r mo s t r e mi s s a c ti on c annot b e a s s i gne d , fo r ac t i on i s a s a c ontinuo u s quantity , div i s ib l e to infin ity . M o r e o v e r , i f the r e we r e a w e ake s t a g e nt b y whi c h a g i v e n p a s s i ve p o ­ t e nc y c ou l d b e a ffe c te d , it would follow that the a c t i on p r o .:­ duc e d b y thi s a g e nt c o ul d b e in s t ant ane o u s ly c o r r upte d b y the ap p l i c ati on o f a c ont r a r y infl ue nc e , and thu s the l a s t in s t ant o f an a c t i on- a r e s s uc c e s s i v a- wo uld b e a s s i gnab l e . T he b o und a r y o f a p a s s i ve p ot e nc y whic h e xe r c i s e s r e ­ s i s t an c e , the autho r n o w c o nc lude s , i s g i v e n b y the m axi ­ mum a g e nt b y w hi c h the pote nc y c anno t b e affe c te d , v i z . , the a g e nt to whi c h i t s r e s i s t anc e i s e qual . T hi s c onc lu s i on i s b ut the r e ve r s e s i de o f o u r p r e vi o u s c on c l u s i o n , that an a c ti ve pot e nc y i s b ounde d b y the minimum r e s i s t anc e w h i c h it i s un ab le to s e t in mo tion , vi z . , the r e s i s t anc e to whi c h i t i s e qual . If we i m ag ine al l the a g e nt s b y whi c h a g i ve n p a s s i v e p ot e nc y c an b e affe c te d , we s e e that the ag ­ g r e g ate o f the m i s b ound e d at i t s l o we r e xt r e me by an e x ­ t r i n s i c b o un d a r y . C o r r e s p ondin g l y , the a g g r e g ate o f p a s ­ s i ve p ote nc i e s o r r e s i s tanc e s whi c h c an b e affe c te d b y a g i ve n ac t i v e pot e nc y i s b o unde d at i t s uppe r e xt r e me b y an e xt r in s i c b o und a r y. In fo c u s in g att e ntion on a g g r e g ate s o f po ­ t e n c i e s , the pr e s e nt di s c u s s ion i s s i mi l a r to He yte s b u r y' s .

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WILLIAM HEYT E SBURY F inally, we should note that the author regards the four ­ membered set of possible boundaries- maximum quod sic , minimum quod sic , maximum quod non, minimum quod non - as applic able to any whole of whic h two opposing qualifi­ c ations are veri fied , so that part of the whole is so- and-so, and the remainder is not so- and- so. Thus if part of a body is hot, and the remainder is c old, there is either a maxi­ mum or minimum part whic h is hot , and either a maximum or minimum part whic h is not hot . Three objec tions to this generaliz ation are examined. F IRS T.- It is objec ted �hat there is neither a maximum weight whic h Soc rates c an sustain, nor a minimum weight whic h he c annot sustain . There is no maximum, bec ause in order to sustain a weight the sustaining potenc y must exc eed the weight by some exc ess- an exc ess whic h c an be made smaller than any value we c hoose to assi gn. Nor is there a minimum weight whic h Soc rates c annot sustain . F or if there were suc h, Soc rates would be depressed to the ground by this wei ght either instantaneously or sue c essively ; but he would not be depressed instantaneously, for no motion is instantaneous; henc e he would be depres sed suc c essively . B ut if Soc rates is depressed to the ground suc c essively, then he sustains the weight for a c ertain time, so that it is not the minimum weight whi c h he c annot sustain. The author replies th at a sustaining potenc y, stric tly speaking , need not be greater than the wei g ht it sustains ; a ratio of equality between the sustaining power and the weight suffic es in order that the latter should not desc end. Thus a sustaining potenc y will be bounded by the maxi­ mum weig ht whic h it is able to sustain ; and this c onc lu ­ sion does not c ontradic t the princ iple that ac tion c annot proc eed from a ratio of equality, for sustaining is not an ac tion or motion. If, on the c ontrary, the potenc y is able not only to sustain the weight but also to move it from plac e to plac e, then it will be bounded by the minimum weight whic h it c annot c arry without being depressed. A potenc y whic h produc es static equilibrium evidently plays the role of a limiting c ase . If one of two potenc ies whic h produc e suc h equilibrium should be au gmented so as to exc eed the other, it would assume the role of an ac tive potenc y, but in the state of equilibrium eac h potenc y merely impedes the other from produc ing motion. S E C O N D.- It is objec ted that the c oming-to-be of Anti ­ c hrist c an be distant in the future by some length of time, and by some length time it c annot be distant in the future; 1 00

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y e t the r e i s n o m a xi m u m t i m e b y whi c h i t c an b e di s t ant in t h e futu r e , n o r any m i n i m u m ti me by wh ic h it c annot b e d i s t an t in the futu r e . F o r Ant i c h r i s t ' s c o m i n g- t o- b e i s a c o nt i n g e n t futu r e e v e nt whic h m ay b e po s tp o ne d inde fin ­ ite 1y .

T he autho r r e p l i e s that the r e i s a mini mum l e n g t h o f ti me b y wh i c h Ant i c h r i s t ' s c o ming- t o-b e c ann ot b e d i s t ant in the futu r e , n a me ly , an in fin ite t i m e . F o r s inc e Ant i c h r i s t w i l l c o me t o b e s o me ti me o r o t h e r , h i s c o ming- to- b e c an ­ n o t b e infini t e l y di s t ant in the futu r e ; b ut b e i n g a c o nt in ­ g e nt e v e nt , it c an b e d i s tant in the futu r e b y any le s s a m o unt o f ti me than an infin i t e t i me . T H IR D . - W e a r e to i m a g i ne a b o dy of whi c h one h a l f i s w h i t e and on e hal f b l a c k ; and it i s s u p p o s e d fu r th e r a s a uni v e r s a l r u l e t h a t a b o d y i s w hi te only i f a g r e ate r p a r t t h an h a l f o f i t i s w h i t e .8 1 T he n i t i s a r g ue d that one p a r t o f t h e b o d y i s white an d t h e r e mainde r i s n o t white , an d y e t t h e r e i s ne i t he r a m axi mum p a r t w hi c h i s white no r a mi n i mum p a r t w h i c h i s n o t w h i te . T he autho r c onc e d e s t h at the r e i s no m ax i mum p a r t w h i c h i s w h i t e n o r any m i n i m u m p a r t w hic h i s n o t white , b ut m a in t ai n s that the b o dy a s a who l e i s a m i n i mum whi c h i s not white .

ROGE R

ROSET US

A t r e at i s e u D e m a x i m a e t m in i ma" b y the E n g li s hm an R o g e r R o s e t u s (o r Lin g ue r i u s S u i s c e p t u s o r R o s e t u s) 82 ap p a r e nt l y s t e m s f r o m the s e c ond h a l f of t he fo u r t e e nth c e n t u r y ,8 3 an d r un s c l o s e t o He y te s b u r y ' s u D e maxima e t m i n i m a" b o t h w i t h r e s p e c t t o t h e p r inc i p l e s e mp l o y e d an d w i t h r e s p e c t to the p a r ti c ul a r c a s e s an a ly z e d . T hi s t r e a ­ t i s e a c tu a l l y fo r m s p a r t o f a c o m me nt a r y on the " S e n ­ t e n c e s " o f P e t e r L o mb a r d , an d i s i nt r o duc e d in the di s c u s s i o n o f t h e que s ti on w h e the r a f r ate r c an b e ob l i g ate d b y a p r e c e p t of t h e p r e l ate to t h e p e r fo r m anc e o f a t a s k - s ay , t h e r e adi n g o f s a c r e d s c r i p tu r e - whi c h i s ag ain s t 84 On e o f the a r g u m e n t s c o nt r a i s t h at the r e h i s c o n s c i e nc e . i s n e i t he r a rn axi m um ac t o f s tudy i n g w hi c h w o u l d c on fo r m t o t h e p r e l at e ' s p r e c e pt no r a m i n i mum ac t o f s tudy i n g w h i c h w o u l d n o t s o c o nf o r m , s i nc e t h e i nt e n s i ty o f t he ac t o f s tu dy i n g m a y b e i n c r e a s e d in i n finit u m . W h e nc e fol l ow s an an a l y s i s o f the c o ndi t i on s unde r wh i c h m a x i m a and m i n i m a a r e a s s i g n ab l e . A m o n g t h e p o int s at w hi c h R o s e tu s ' d i s c u s s i o n c o inc ide s w i t h He y t e s b u r y' s a r e t h e f o l l o w in g : In t h e an al y s i s o f the c on d i t i o n s un de r whi c h the c u s to ma r y d i v i s i o n s (m axi mum

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quod sic o r mini mum quo d non , minimum quo d sic o r m�imum quo d non) a r e assignab l e ,8 5 in the use o f the p r inciple that an active pote ncy must b e of gr e ate r mag ­ nitude than the r e sistanc e against whic h i t acts;8 6 in dis ­ cussion of the case of a po int which i s to trav e r se a gi v e n distance at a v e locity which is specifie d o n l y a s b e ing le ss than a gi ven ve loc ity d ;8 7 in disc ussi on of the p r ob le m of the bo unda r y b e tw e e n th e c lass of de b i litab l e po te ncie s which suffic e to car r y a giv en we ight a pe r se e t Re r te m ­ pus , and the class of de b i litab l e potenci e s which do not suffic e to car r y a ,88 in discussion of th e p r ob le m of the b o unda r y b e twe e n the class of we ights which an yone st ronge r than Soc rate s can ca r r y , and the class of we ights which not e ve r yone st r onge r than Soc rate s can car r y ;89 in discussi on of the pr ob le m of the di vision of a unifo r m ly diffo r m r e sist ­ anc e .9 0 R o se tus , l i ke the autho r of the co mmentar y On the P h :y: sica att r ib ute d to Duns Scotus , ho l ds that Ar i stotl e ' s assignment o f a maximum as the b o unda r y o f potenci e s i s inte nde d as an assignme nt of the maxi mum numb e r of inte g ral m e asu r e s of that which the potency suffice s to o v e rcome o r act upon o r accomplish , rathe r than of the maxi mum amount si mE..!i- ­ cite r . Thi s conclusion , h e adds , follows c l e ar ly fr o m A r is ­ totle's e xamp le s.9 1 ALB E R T O F S A X O N Y

The discussi on of the p r ob l e m b y A lb e r t of Saxony (at Par is fr o m 1 3 5 1 to ca . 1 3 6 2 , and fi r st r ecto r of V i e nna in 1 3 6 5) is si milar to the analysi s in the commenta r y on the P h :y:sica att r ib ute d to Duns Scotus , and also sho ws the in ­ fluenc e of B ur i dan .9 2 C e r tain no v e ltie s in his discuss i on , howe v e r , de se r ve menti on . F IR ST.- In t r e ating the st r ength of pote ncie s in co mpa r i ­ son with the distance th r o ugh which the i r acti on e xtends , Alb e rt int r o duce s a ne w te r mino logy . W e r e cal l that B ur i ­ dan had distinguishe d b e twe en the distanc e in whic h and the di stance th r ough which an ob j ect can b e se e n . Alb e rt d r aws what appe a r s to b e the same distinction in diffe r e nt te r ms: the r e i s a diffe r ence b e twe e n the distance in which or thr ough whic h a p ote ncy can act on an ob j e ct ( " distantia in quam v e l distantia p e r quam fit acti o" ) , on the one han d , and the distance b e Y.ond whic h a p ot e nc y can act on an ob j ect ( " distantia ult r a quam fit actio" ) , on the othe r . In the fi r st case , the obj e ct acte d on is at an int r insic po int o f the distance consi de r e d ; in the second , it i s j ust o utsi de o r e xt r insic t o the distanc e . Alb e rt r e gula r l y use s the ph r ase s 102

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" maxima distantia ult ra quam potest ;' "minima distantia ult ra quam potest ;' " maxima distantia ult ra quam non po test ;' " minima distantia ult ra quam non potest ,, in assigning boundar ies in compar ison with distance . Whe reas Bur idan had int roduced his distinction in order to meet a particular p roblem , Albert's intention in using the te rms "ultra quam,, is not m?-de explic it . The value of such te-rminology appears to be that of specifying the position of an object of some magnitude with respect to a boundary which has no magni ­ tude , that is with respect to a point. We find , however , that Albert applies it also in cases where such considerations of size play no role , vi z . , in cases whe re the magnitude bounded is time. Thus , in dealing with the potency of cor ruptible substances to endu re , Albe rt states that the re wi ll eithe r be a maxi ­ mum time beyond which a given cor r uptible entity can en ­ dure ( " maximum tempus ultra quod ali qua res cor ruptibi lis potest du rare" ) or a minimum time beyond which it cannot endu re ( "minimum tempus ultr.a quod res co r ruptibi lis non pote st durare,, ) . The fi rst alte rnative is excluded , because involving the consequence that the ultimate instant of being of the entity (a res pe rmanens) would be assignable. For sup ­ pose there we re such a maximum tempus ultra quod , and call it a , and let b be the instant which te r minates it ext r insically (that is , which stands just outside the assigned time). Then if a is the maximum time beyond which Socrates can exist , Socra ­ tes can exist until instant b and in instant b , but no longe r ; and in instant b it wi ll be cor rect to say that Soc rates now exists , and immediately afte r this will not exist ; so that b wi l l be the last instant of his existence. B ut that no such ultimate instant of being can be assigned is shown by the following argument : In instant b Soc rates' potency to en dure will be of a divisible magnitude ; hence he will not be cor r upted all at once but only after a lapse of time ; so that b will not have been the last instant of his existence. Albert the refore concludes that the boundar y of the po ­ tency of a thing to endure is given by the minimum time beyond which it cannot endure ; the time , that is , which ter minates ext r insically at the fi rst instant of nonbeing of the thing . The same time is the maximum time in which the thing can endu re . In cases sue h as the p receding , Albert's te rminology would appear to int roduce confusion rathe r than clar ity . SE CON D. - Albert's discussion of the boundary of the potency of a light-source with respect to the intensity of i llumination it is capable of p roducing is particula r ly in -

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te resting , as it shows the influence of te r minist logic . Such a potency , he holds , has for an upper limit the min ­ imum effect which it cannot p roduce . Thus a light-sour ce cannot p roduce illumination as intense as its own light in the su r rounding medium ; but any mo re remiss light being assigned , the light-sour ce can p roduce a more intense light than the assigned one in some medium . This statement does not imply that it is possible fo r a light-sour ce to p ro duce in some medium every intensity of light up to the highest degree of intensity found in the light-sou rce ; for the re will always be a divisible difference between the in ­ tensity of the light-source and the intensity of the most intensely lighted part of the medium , the amount of the difference apparently depending on the relative translu ­ cency of the medium . The conclusion can be for mulated in the paradoxical manne r of the fourteenth-century logi ­ cians as follows : Although light-sou rce a can p roduce any degree of intensity up to the highest in some medium ( " A lucidum quemlibet gradum cit ra summum potest p r o ­ ducere in aliquod medium" ) , it does not follow that it is possible for light-sour ce a to p roduce any deg ree up to the highest in some medium ( " Non sequitur quod possible est a lucidum quemlibet gradum cit r a summum p r oducere in aliquod medium" ) . The fir st statement has a divisive sense , and me ans that , any intensity less than that of the light-sour ce being assigned , there can be found a mediurr1 of such t ranslucency that the light-sour ce will p roduce in it a more intense light than the assigned one . The second statement , with the moda! " possible" entirely p receding the re st of the p roposition , has a composite sense , and means that the light-sour ce can p roduce every degree of intensity up to that of the light-sour ce in some particular, p reassigned medium- which Albert denies . T HIR D . - It has been p reviously noted that , in the com ­ mentary on the Phy:sica attributed to Duns Scotus , cer tain potencies , e . g . , the potency for vision , are r efe r red to as being both active and passive . It is of inte rest that Albert includes vision and the "cognitive" potencies gen e rally in the class of potencies which are both active and passive , and that he makes use of this double characte r of the visive potency in the solving of a difficulty we have p reviously noted : If vision is simply a passive potency with respect to the active potency of the visible body to p ropa ­ gate its visible species , then- just as Socr ates' power of lifting has as uppe r limit the minimum weight which he cannot lift-so the active potency of the visible body ought

1 04

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to be bounded by the minimum potency of vision which it c annot affect , in such manner that while it is un able to act on this potency for vision it can act on any less or weaker potency for vision. In fact , the opposite is the case ; if a visible body cannot act on a given power of vision , much the less c an it act on any weaker power of vision . The rea ­ son for this deviation from the usual pattern for passive potencies , Albe rt holds , is that the potency for vision is active as well as passive , a p artial agent of vision as well as the total recipient of vision ; and on this account the range of impotency among varying powers of vision with respect to a given visible object , like the range of impo ­ tency of other active powers with respect to a given object to be acted upon , is found in the direction of the smaller potencies . FO U R TH . - We note finally that Albert , like B uridan , at ­ te mpts to reconcile the general assignment of a minimum quod non with Aristotle's assertion of a maximum by main ­ taining th at Aristotle means a maximum infra quod 2otest . D O M IN I C U S

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A discussion of maxima and minima by Dominicus de Clavasio , who is named by Michalski as a mid-fourteenth ­ century successor of B uridan at the University of P aris ,93 shows simil arities with the work of B uridan , Albert of Saxony , and the author of the commentary of the Phy:sica attributed to Duns Scotus.9 4 With B urid an , Dominicus admits th at the celestial spheres offer no resistance to their movers . With B uridan , Albert , and the author of the commentary attributed to Duns , he holds th at wherever a potency acts against a resistance , any excess , however small , of po ­ tency over resistance suffices to produce motion . Like Albe rt and the author of the commentary attributed to Duns , Dominicus holds th at the assignment of an in ­ trinsic boundary to a potency would imply th at a res P.er ­ manens has a l ast instant of being. Like the author of the commentary attributed to Duns , Dominicus holds th at there may be a lower as wel l as an upper boundary to the distance through which an active po tency m ay act or a p assive potency be acted on . All finite potencies , according to Dominicus , are bounded by extrinsic boundaries (minimum quod non , max ­ imum quod non). In this connection he states , in opposition to B uridan and in agreement with the commentary attri ­ buted to D uns , that the boundary to the distance through 105

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which a light can illuminate is a minimum quod non ; for the dividing surface between the illuminated and the un ­ illuminated region is itself unilluminated. Like the author of the commentary attributed to Duns , Dominicus reconciles the assignment of extrinsic bound ­ aries with the opposing statements of Aristotle and Com ­ mentator by stating that the latter are concerned with maxima and minima in terms of integers; thus one hundred talents may be the maximum in integers which someone can lift , but this does not imply that he is unable to lift one hundred and a half talents. M AR S I L I U S O F

ING H E N

In his Abbreviationes libri P.hisicorum (Venice , ca. 1 500; no pagination) , Marsilius of Inghen (a teacher at Paris from 1 3 62 until 1382 and first rector at Heidelberg in 1 38 6 ) deals with the question whether natural entities are bounded by maxima and minima . The question is treated under eight sub-headings: (1 ) whether there is a maximum possible size for bodies or a maximum possible length for lines ; (2 ) wheth er there is a maximum possible quickness (velocity) or slowness; ( 3 ) whether there is a maximum quantity of mat ­ ter in which a form can be generated and conserved ; (4 ) whether the possible sizes of natural entities such as man are bounded by maxima and minima; ( 5 ) whether there is a maximum resistance against which a given active power can act; (6 ) whether there is a maximum effect which a given active power can produce; ( 7) whether there is a max ­ imum distance through which or in which a visible body can be seen; and (8 ) whether there is a maximum time through which a natural entity can endure. In a number of cases Marsilius assigns extrinsic boundaries; thus he holds that the possible sizes of a heterogeneous sub stance (such as man) are bounded extrinsically both in the direction of largeness and in the direction of smallness. The entire discussion is undertaken 2.h_y:sice loquendo , and shows little originality. In particular , Marsilius is evi ­ dently dependent upon the work of Buridan; thus he repeats Buridan' s argument with respec t to the equinoctial circle in order to prove that a greatest possible velocity in the universe is not assignable (see note 25 of this chap ­ ter). AN G E L O D A F O S S A M B R UNO

The "Questio de maxima et minima materia" of Angelo da Fossambruno ,9 5 a professor at Bologna and P adua 106

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dur ing the late four teenth and early fifteenth centu ries ,9 6 deals with the problem of whether ther e is a m1n1mum amount of matter in which a substantial for m can be in ­ duced and conserved . Angelo concludes P.r obabiliter that a substantial form cannot be induced in an ar bitrar ily small amount of matter , but r equir es a cer tain quantity of proper ly disposed matter into which it can be induced instantaneously . Angelo holds that the receptive dispo­ sition in the matter can vary within a cer tain latitude , that is , can be better or wor se; he believes- it is not clear why- that this latitude of •• bona dispositio" is bounded at its upper extreme extrinsically (minimum quod non) . If matter having a given degree of .. bona dis positio" be considered , then ther e is a minimum amount of such matter in which a substantial for m can be induced. If the degree of .. bona dispositio" is not pr eassigned , the lower boundar y for quantities of matter in which a sub ­ stantial for m can be induced will be a maximum in quod non. This r esult derives fr om the extr insic termination of the latitude of .. bona dispositio ;' the maximum in quod non amount of matter being such that , in order for a form to be induced , its r eceptive disposition would have to be equal to the minimum quod non which .. bona dispositio" cannot attain; while in any greater quantity of matter a less and therefore attainable degr ee of .. bona dispositio" suffices for the introduction of a for m. Angelo attacks Albe r t of Saxony for holding that the for ma humana alone is induced instantaneously into matter , other substantial for ms being induced successively; for since prime matter can never exist without for m , the human for m must enter in the same manner the for m which previously existed in the matter le aves , and thus all for ms must be induced in ­ to or r ejected from matter either successively or instan ­ taneously. PAUL OF

V ENI C E

P aul of V enice , a teacher at Padua and Siena (d. 1429 ) , gives a summar y of the main principles of Heytesbur y' s treatise , without entering into the mor e subtle sophistic difficulties which Heytesbu r y consider s.9 7 Paul states that the Commentator , in assigning a maximum quod sic , was speaking of an active potency acting under limitation , as to a given degree of velocity; in other ter ms, that Com ­ mentator was only concerned with what Heytesbur y calls immutable potencies. Such an interpretation can har dly bear up in confr ontation with Commentator's text. 107

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PULCHER

Another treatise which follows Heytesbury's maj or dis ­ tinctions is an anonymous " Tractatus pulcher de maximo et minimo super Entisbero ;' probably by some Italian author of the early fifteenth century .9 8 The decision between the extrinsic and the intrinsic types of boundaries of a po ­ tency, the author notes , depends on the question whether any excess, however small , of potency over resistance suf ­ fices to produce motion or whether, on the contrary, a finite excess is required The author concl udes that any excess is sufficient. He gives special attention to the problem of the peculiar status of the senses, which Albert of Saxony, we recall, had regarded as both active and passive, in order to explain how the active power of an object to make it self seen or heard at a given distance can be terminated by a maximum quod non in the power of sense, rather than by the usual minimum quod non of active in comparison with passive potencies. The present author gives a dif ­ ferent solution : a distinction must be drawn between po ­ tencies which act materialiter and those which act ..§ P.iri ­ tualiter. Active potencies of the latter type, as opposed to ordinary active potencies, are terminated by a maximum quod non rather than by a minimum quod non; thus the power of an object to make itself seen at a given distance is terminated by the maximum power of sight which it can ­ not affect. G A E T AN O DI

T H I ENE

Gaetano d i Thiene' s discussion of the subject in his De coelo commentary 99 is, in main part, a restatement of Averroes' commentary. At the end,1 00 however, he raises the doubt as to whether potencies should not be defined by a minimum quod non rather than a maximum quod sic. After noting the solutions of others, for example, that po tencies are to be terminated according to integral num ­ bers, or that Aristotle's maximum is a maximum infra guod potest, he decides- a solution completely in accord with Heytesbury's treatise- that some potencies are ter ­ minated intrinsically and others extrinsically. 1 0 1 Aristotle takes account of potencies which are terminated extrin ­ sically, Gaetano holds, when he defines impotency in terms of a minimum. J O ANN E S D U L L A E R T

The discussion of maxima and minima in the commen ­ tary on the De caelo et mundo of Joannes Dullae rt of G hent 1 08

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(d . 1 5 1 3 ) , a di s c ip l e o f J o anne s Mai o r i s at the Un i ve r s ity of P a r i s , 1 02 s ho w s the influe nc e o f He yte s b u r y and P a ul o f V e n i c e (the wo r k s o f b o th m e n a r e s p e c ific ally c ite d) , a s we ll a s o f B u r i d an and Alb e rt o f S axony . 1 03 Amo n g the p o int s at whi c h Du llae r t' s di s c u s s ion c o inc ide s with He yt e s ­ b u r y' s a r e the fo l l o w ing : F IR S T . - In the an aly s i s o f the c o nditi on s unde r whi c h the c u s to m a r y div i s i o n s (m aximum quod s i c o r mini mum quo d n on , mini mum quo d s i c o r maxi mum qu o d non) a r e a s s i gn ab le . Li ke He yte s b u r y , Dul l ae r t hold s that whe r e o n e o f t h e p r e r e qui s it e s ub c ontr ar ie s i s ve r i fie d only fo r infinite v al u e s , the divi s ion do e s not fo ll o w . L ike G ae t ano , h e h o l d s that , in o r de r fo r the divi s ion to follow , the v alue wh i c h i s to s e r ve a s b o unda r y mu s t no t b e e xc l ude d f r o m the d i v i s i on . S E C O N D . - In di s c u s s i on o f the c a s e o f a b o dy whi c h , mo v in g a g ain s t a d e b i l i t ab l e p o te nc y to whi c h i t s r e s i s t ­ anc e i s e qual , wi th infini te s l o wne s s b e g in s t o de s c e nd , b ut do e s not b e g in to de s c e nd with infi nite s l owne s s . T HI R D . - In di s c u s s i on o f the c a s e o f a p ot e nc y whic h m o v e s th r o u g h a m e dium the r e s i s t anc e o f whic h i s uni ­ fo r mly di ffo r m . H e r e Dul l ae r t g i ve s the s ame p r o o f a s G a e tano o f the c o nc l u s i on t h at the p ot e n c y will r e qui r e an infinit e t i m e t o t r ave r s e th e p a r t o f the me dium whi c h te r mi n at e s i n the mo r e inte n s e e xt r e me a t a de g r e e o f r e s i s t anc e e qual t o i t s p ow e r . F O U R T H . - In di s c u s s ion o f the c a s e o f a p oint a whi c h i s t o t r av e r s e a fixe d di s tanc e a t a v e l o c ity whi c h i s s p e c i fi e d only a s b e ing le s s than d de g r e e o f ve lo c ity . F IF T H . - In di s c u s s i on o f the b oun d a r y o f the a g g r e g ate of p o r t at i ve pote nc ie s whic h a r e s p e c i fie d only as b e ing g r e ate r than the p o r t ati v e pot e nc y of S o c r ate s . B ut he r e Dul l ae r t c r i t ic i s e s t h e s t at e me nt of He yte s b u r y' s r ule , a c c o r din g to whic h , i f the d i vi s ion i s made b y me an s o f a c o m mon di s t r ib ute d t e r m h aving infinite s upP.o s it a fo r whic h it i s di s t r ib ute d , the affi r m at i ve p a r t o f the divi ­ s i on i s to b e c h o s e n . T he c r uc i al c ha r a c t e r i s tic o f the c a s e s c o n s i de r e d as c o ming unde r thi s r u l e , ac c o r din g to D ul l ae r t , i s not th at the c o m mon te r m h a s infinite .§J:!PP.O s i t a , b ut that the ag g r e g ate o f e l e me nt s r e fe r r e d to i s b oun de d e xt r in s i c ally at one e xt r e me . T hi s c r it i c i s m d o e s not appe a r t o b e de c i si ve ; He yte s bu r y' s di s tinc tion of c a s e s is b a s e d on the p r inc iple that a finite s e t o f finite v alue s has a le a s t me mb e r , while an infinite s e t o f s u c h v al u e s m a y l a c k a l e a s t me mb e r , and t h e e xt r in s ic c h a r a c t e r o f the b o un d a r y o f th e infinite a g g r e g at e s c on s ide r e d i s d e t e r m ine d b y the ph r a s e s " any g r e ate r t h an ;' " any 1 09

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part starting at one end :' the grammatic al and logic al anal ­ ysis of whic h is Heyte sbury' s primary c onc ern. In the assignment of maxima and minima generally, Dullaert c onc ludes, regule c erte dari non P.Ossunt; it is nec essary to have regard for the spec ial c harac ter of eac h individual c ase. A C HI L L INI

The disc ussion of Ac hillini, Bolognese and Paduan pro ­ fessor (d . 1 5 1 2 ), 1 04 is c entered on the problem of the exist ­ enc e of a minimum in natural things. He rec ognizes that the assignment of the intrinsic type of boundary in a given c ase exc ludes the assignment of an extrinsic boundary in the same c ase, and vic e versa. 1 05 He ac c epts the rule that a potenc y for c arrying weight, if subjec t to fatigue, is to be terminated by a minimum quod non. 1 06 With r espec t to his princ ipal question, he c onc ludes that every natural thing has a maximum and minimum size, determined by its substantial form. These upper and lower boundaries are to be understood as affirmative or intrinsic in the generation of a thing, for when a thing begins to exist, it is the first instant of its existenc e that is assignable, rather than the ultimate instant of its nonexistenc e; they are to be understood as negative or extrinsic , however, in the c orruption of a thing, for here the ultimate instant of being is not assignable, but rather only the first instant of nonbeing. Although Ac hillini leans heavily on the au ­ thority of Averroes, it is c lear in the present c ase that he understands the differenc e between the extrinsic and intrinsic types of boundary, and the reasons adduc ed by the Sc hoolmen of the two prec eding c entu:ties for assigning one or the other. A G O S T INO NI F O

Agostino Nifo's attempt, in his c ommentary on the De c oelo,1 07 to rec onc ile the position of the iuniores with the statements of Aristotle is of partic ular interest. In an earlier work,1 08 Nifo had maintained that a potenc y for duration is terminated by a maximum quod sic rather than a minimum quod non, basing his position on the argument that c orruption oc c urs through resolution of parts, and that the ultimate parts to be resolved are natural minima whic h are resolved instantaneously. In his De c oelo c om ­ mentary he has c hanged ground, exc using the c hange as follows: "As we have nothing c ertain in natural sc ienc e, I write my thoughts as they c ome into mind; for opinions 1 10

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change with the time , as Empe docle s said:' The rule s of the iuniore s, i. e . , the ir assignme nt of e xclusive boundarie s of the typ e of the minimum quod non, have validity from the point of vie w of mathe matics; Aristotle' s te rmination of an active pote ncy by a maximum, on the othe r hand, is corre ct in natural philosophy, whe re only se nsible magni ­ tude s are conside re d . The mathe matician conside rs what ­ e ve r can be thought se cundum imaginatione m; the natural philosophe r de als only with what is pe rce ive d. Thus any e xce ss of powe r ove r re sistance suffice s to produce a mo ­ tion, but not ne ce ssarily a motion that can be pe rce ive d by the se nse s; for the production of a p e rce ivable motion, the re is re quire d an e xce ss of a ce rtain size , and thus the p ote ncy will be te rminate d by a maximum quod sic ..E-P.Ud se nsile m notam, i. e . , the gre ate st re sistance against which it can produce a p e rce ivable motion. What is e spe ­ cially re markable is that Nifo appe ars to give primacy to the mathe matical point of vie w; thus he state s that the maximum J!.P.Ud se nsile m conside ratione m or se cundum conside ratione m naturale m is not a maximum se cundum re m or se cundum re i naturam-which app e ars to imply that the boundarie s which are valid for the mathe matician se cundum imaginatione m are also valid se cundum re m in se . In any case , with re spe ct to the ge ne ral proble m of the status of the Scholastic discussions of maxima and mini ­ ma, N ifo' s vie w that the rule s of the iuniore s are mathe ­ matically conce ivable but not e mpirically ve rifiable ap ­ p e ars to strike at the he art of the matte r. J E S UIT S

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The Aristote lian comme ntarie s writte n by the Je suits of the Colle ge of Coimbra in the se cond half of the six ­ te e nth ce ntury contain analyse s of proble ms involving in ­ trinsic and e xtrinsic boundarie s . In the ir comme ntary on Book V II I of the P hy:sica, 1 09 the y discuss the proble m of the initial and final boundarie s of the duration of things; the ir distinction of case s he re much re se mble s that in Burle y' s De instanti. In the comme ntary on Book I of the P h y:sica, 1 1 0 the y discuss the proble m of the boundarie s of the possible size s of natural things; like Achillini, the y hold that the uppe r and lowe r boundarie s of size are in ­ trinsic in the way of ge ne ration, and e xtrinsic in the way of corruption. An e xtrinsic boundary of a r ange of magni ­ tude , the y note , e xce e ds the intrinsic quantitie s of the range not colle ctive ly but divisive ly; i . e . , the e xtrinsic boundary e xce e ds any particular quantity of the range by lll

WILLIAM HEY T ESBURY

a divisible excess , but does not thus exceed all the intrin . · . . the aggregate.1 1 1 sic quantities ta ken 1n In their commentary on the De coelo , they discuss the . ,1 1 2 pot enc1es . an d passive . of active . problem of the bound ar 1es and here invoke the r ule accor ding to which action accrues only fr om a ratio of greater inequality betwe en power and resistance. GALILEO

A discussion of maxima and m1n1ma is included in the " Juvenilia" of Galileo1 1 3 - regarded , owing to the state of the codex , as a wor k of tr ansc ription , possibly or iginating from the lectures of Fr ancesco Buonamici at Pisa.1 1 4 As in Achillini' s case , the principal question under discussion is the existence of boundar ies with respect to size for the elements and other natu ral things. At the start the ter ms maximum quod sic , minimum quod sic , maximum quod non , minimum quod non are explained in some detail , and it is noted that an intrinsic and extr insic boundar y cannot both be assigneq in a single case . Paul of Venice is cited as an author ity for the view that the boundar ies of size must be extrinsic , in order to avoid the consequence that the last instant of being of a thing should be assignable; Thomas , Capreolus , Scotus , Occam , Soto, and other s are cited as authorities for var ious views which agr ee in the assign ­ ment of intr insic boundaries. The final solution is simila r to that of Achillini: The elements and other natur al things have boundaries with r espect to lar geness and smallness , which ar e intrinsic in the way of gener ation , and extrinsic in the way of cor r uption. The natural philosopher 's inter est in whether the boundaries are intrinsic or extr insic is justified on the gr ound that he should know how things be gin and cease to be. P otencies , it is stated incidentally , are terminated by a maximum quod sic: the tr aditional ar gu ­ ment for a minimum quod non , accor ding to which the minimum r esistance upon which a potency cannot act is the resistance to which it is equal , has apparently been lost sight of.

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CHO LAS T IC Di s c us s ion of M axima and Minima in Relation to Sevent e e nth - C e nt ur y Phy s ic s

At this point we may r aise the following query: Did the discussion of the differ ent types of boundaries of potencies, originating in the commentar y of Aver r oes and continued until the time of Galileo , enter into the science of Galileo

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and his successors ? The answer is negative , and thus gives what j ustification there is for Duhem's statement that the logical rigor of Albert of S axony will not be re�ained until the advent of nineteenth-century mathematics.1 1 A number of possible reasons for this failure should be noted. As we have pointed out repeatedly , the Scholastic dis tinction between an intrinsic and an extrinsic bound ary of a potency is based on a mathematical understanding of the continuum , and thus would require , for empirical ve rifica ­ tion , measuring instruments which were absolutely pre cise- a requirement which it is impossible to satisfy. Gali ­ leo , on the contrary , however "idealized" the cases he considers , generally requires the possibility of an ultimate , if rough empirical verification . 1 1 6 Duhem has suggested 1 1 7 that the Scholastic considerations are similar to those introduced in thermodynamics in con ­ nection with the idea of a reversible transformation. In ­ deed , reversible transformations involve the effects of infinitesimal differences between the pressure of a gas and that exercised by its container , and also between the tem ­ perature of a gas and that of its container. The gas can thus be considered as constantly in a state of equilibrium in which , alone , its state can be represented by a finite number of co-ordinates. The case , like those of the School ­ men , is purely ideal : in practice it is always necessary to have a finite difference between pressures or temperatures to put the gas in motion with a finite velocity and to over -. come friction. This ideal consideration , however , is useful because many of the results obtained are good with sufficient approximation for re al , irreversible transformations. The concern on the part of Aristotle and the Schoolmen with debilitable potencies- Socrates' power to traverse distance or to lift weights- is entirely left aside by the seventeenth-century thinkers , who were dealing rather with inanimate systems of forces and bodies , in fact with what Heyte sbury had called immutable potencies . The Scholastic conception of the relation between a po ­ tency and a resistance was replaced , in the seventeenth century , by a new understanding of the role of force . Force w as no longer to be something which , by its excess over or proportion to or proportion of proportions to a resistance , produces velocity; instead it is defined as that which pro ­ duces acceleration . To arrive at this result, it was neces ­ sary to neglect temporarily or to discount friction; on the other hand , the second type of "resistance" considered by the Schoolmen- that of the body itself- was no longer to be 1 13

WILLIAM HEYT E SBURY c o mp a r ab le to fo r c e at all ; fo r ma s s i s a quantity o f di f ­ fe r e nt dime n s i on s f r o m fo r c e ; altho ugh it " r e s i s t s" a fo r c e in the s e n s e that the ac c e l e r ation p r o duc e d b y a g i v e n fo r c e i s s m alle r in the s ame r at i o a s the ma s s a g ai n s t whic h i t ac t s i s l a r g e r , i t c an ne ve r b e e qual t o a fo r c e , o r p r e v e nt a fo r c e f r o m p r o d uc in g an ac c e l e r at i o n . T he r e ­ fo r e the e quilib rium b e twe e n a p o te nc y an d the r e s i s t anc e to whi c h it i s e qual l o s t i t s c r uc i al s t atu s f o r the d e te r ­ mi nation o f the m agnitude o f fo r c e s o r p o t e n c ie s whi c h p r o duc e mo tion . 1 1 8 In c e r t a in c a s e s ,, it i s t r ue , two fo r c e s fo r p r oduc ing motion may b e c o mp a r e d to one anothe r s t atic al ly , an d he r e the e quilib r ium s t atu s i s a s i mp o r tant fo r s e v e nte e nth- c e ntu r y me c h anic s as fo r the S c ho l a s tic c on s i de r at i o n s . 1 1 9 F inally , we note that what w a s m athe matic ally inte r ­ e s ting in the type s o f b o unda r i e s di s c u s s e d b y He yte s b u r y h ad not y e t , i n the s e ve nte e nth c e ntu r y , e xe r te d i t s c l aim to i mp o r t anc e within m ath e m ati c s it s e lf . T h at w ou l d o c c u r wh e n the ne e d w a s fe lt fo r a r i g o r o u s fo und ati on o f m athe m atic al anal y s i s on the b a s i s o f the r e al numb e r s y s te m . B ut mathe mati c al an aly s i s , in the s e ve nte e nth and e i g ht ­ e e nth c e ntu r i e s , s e e m s to have b e e n to o b u s y with i t s a s ton i s hing e mpi r i c al t r iumph s to b e c o nc e r ne d w i th l o g i c al r ig o r .

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4 D e tri bus predica m entis

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E Y T E S B U R Y ' S " De t r ib u s p r e dic am e nti s " (fo ll . 3 7 r a - 5 2 rb ) de al s with motion in the th r e e c at e g o r i e s in whic h , ac c o r ding to Ar i s totle , it al one p r o p e r ly o c c u r s : p l a c e , qu antity , and quality .1 M o r e s p e ­ c ific all y , He yte s b ur y i s c o nc e r ne d with the quic kne s s o r s lo wne s s , and uni fo r mity o r diffo r mity o f s uc h motion s ; that i s , w ith th e w a y s in whi c h the v e l o c ity o f s uc h mo ­ tio n s may v a r y , a s me a s ur e d b y the r e s ult p r o duc e d dur ing e qual le ngth s o f time . " De t r ib u s p r e d ic ame nti s " i s thu s a t r e ati s e on kine m a tic s , r athe r than on dynamic s � It i s a t r e ati s e on kine m ati c s , howe v e r , in a s pe c i a l s e n s e : a s tudy o f type s o f motion po s e d s e c undum ima ­ g in ati one m . A s i n the " De maximo e t minimo ;' a c onc e rn fo r ve r ific ati on of c a s e s in the p hy s ic al wo r ld i s lac kin g . An d ju s t a s in the " De maximo e t minimo " the initial aim w a s th at of de fining or de limiting a pote nc y , so he r e i t i s that o f e s t ab l i s hing c e r tain de finition s - de finition s o f ve l o c it y in the thr e e c ate go r i e s o f motion , of s uc h c ha r ac te r a s t o pe r mit the de no mination o f any c onc e i v ­ ab le motion . T he c r ite r i a b y whi c h s uc h de finition s a r e c ho s e n appe a r to b e two : that t h e de finit ion s hould ac c o r d with t h e c ommon mo de o f s pe e c h , and that i t s ho ul d b e f r e e o f m at he m ati c al c o ntr adi c ti o n s . T he Ar i s tote l i an s o u r c e o f the di s c u s s i on o f the de fini ­ tio n o f v e l oc ity appe a r s in Ph Y. s i c a I V (2 1 8b l 5 - l 7 ) , whe r e Ar i s to tle de fin e s the te r m s " fa s t " and " s low " : " ' fa s t ' i s wh at m o v e s muc h i n a s ho r t t i me , ' s lo w ' what mo ve s litt l e in a long time ,, ; b. Length, area, volume, weight, are clear cases of extensive magnitude; the additive pr oper ty, in particular, is verifiable for these magnitudes by easily per for med 144

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phys ical ope r ation s , and thus we cannot only or de r diffe r e nt we ights line ar ly, but having chos e n a unit of we ight, s ay how much one we ight i s gre ate r than anothe r . On the contrary, in a scale of i nte n s ity bas e d on what i s imme diate ly p e rce ivable , the numbe r s lack thi s quantitative s ignificance ; the y indicate s imply pos itions in a line arly orde r e d s e r ie s . Such a scale i s i llu s trate d by the scale of har dne s s of mine r als ; a mine r al to which i s as s igne d a highe r d e gr e e of har dne s s i s able to p roduce a scratch on a mine r al to which i s as s igne d a lowe r de gr e e , but it i s op e rati onally me aningle s s to s pe ak of a de gr e e of har d ­ ne s s of 8 as diffe r ing fr om a degr e e of har dne s s of 7 by the s ame amount as a de gr e e of har dne s s of 5 diffe r s fr om a d e gr e e of har dne s s of 4. The pos s ibility of the cons truction of quantitative scale s of i nte ns ity, in par ticular c as e s , re s ts on a fur the r the o ­ r e tical analy s i s of the par ticular inte n s ity conce r ne d . De n s i ty and p r e s s ur e , for e xamp le , ar e found to have the mathe matical characte r of quoti e nts of e xte n s ive quantitie s (we ight pe r unit volume and force p e r unit ar e a, r e s p ec ­ tive ly); the numbe r s r e p r e s e nting s uch quoti e nts can thus ans we r the que s ti on "how much:' Again, the abs olute scale of te mpe r atur e i s r e fe r r e d to an i de al gas the rmome te r, in which the e xpan s i on of the gas var ie s di r ectly with the te mp e r atur e ; and0 as the e xpan s ion i s he ld to de p e nd on the i nc r e as e i n kine tic e ne r gy of the molecule s of the gas , s o te mpe r atur e come s to cor r e s pond to the ave r age kine tic e ne rgy of the molecule s , i . e . , once mor e , to a quotie nt of e xte ns ive quantiti e s . Similar ly, i nte n s ity of i llumination may be r e fe r r e d to e ne r gy falling upon unit ar e a. We note that, whe r e i nte ns ity var ie s fr om point to point in s pace or from ins tant to ins tant in time , the cor r e s ponding e x ­ p re s s ion for inte n s ity at a point or in an in s tant will be the limit of a quoti e nt of e xte n s ive magnitude s , i . e . , a de r ivative . The o r e tical analys i s of par ticular type s of inte ns ity i nto a quotie nt of e xte ns ive quantitie s , i n the manne r above indicate d, doe s not occur i n the me dieval di scus s ions .9 6 That inte n s ity has the quantitative attr ibute s nece s s ary for the cons truction of a uniform scale of de gre e s of in ­ te n s ity i s s imply as s ume d by He yte s bury and hi s conte m ­ porar i e s . Thus accor ding to Bradwar dine , "s icut ac ­ qui r e ndum p e r motum locale m e s t s pacium, s ic quod acqui r itur p e r motum alte r ationi s e s t latitudo, e t e od e m modo s icut mobile localite r ve lociu s move tur quod plus in e quali te mpor e p e r tr an s i t de s pacio, e ode m modo quod

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WILLIAM HEYTESBURY maiorem latitudinem in aliquo ce r to tempore acqui r it per alte rationem velocius alte r atu r :' 97 And Heytesbury (fol . 4 0 va): " Quelibet latitudo finita est quedam quantitas :' ( 5 ) A fifth postulate appears to be necessar y in orde r that the rules employed by Heytesbu ry rega rding the equiv­ alence between latitudes or configu r ations of intensity- e. g . , the rule acco rding to which a unifor mly diffo r m latitude is equivalent to a uniform latitude having an intensity equal to the medium intensity in the uniformly difform latitude - should have physical meaning . A given dist r ibution of in ­ tensities in space or time must co r respond to a deter min ­ able quantity of some extensive magnitude, so that, in com­ par ing two diffe rent distr ibutions of intensity, we may say that the cor responding extensive magnitude in one case is equal o r unequal to the cor responding extensive magnitude in the other . In the case of velocity, the cor responding ex ­ tensive magnitude is distance t raversed; and in speaking of a unifor mly accelerated motion as being equivalent to a uniform motion which occupies the same time and which has a velocity equal to the average velocity in the unifor mly accelerated motion, we mean that the distances t rave rsed in the two motions are equal. In the case of other inten sities such as tempe rature, a cor responding extensive magnitude is found only where the intensity can be expr es sed as a quotient of extensive magnitudes (e. g. , kinetic energy o r quantity of heat per unit volume in the case of temper ature), or as the limit of a quotient of extensive magnitudes, in which cases the numerator of the quotient is the extensive magnitude sought . For Heytesbury and his contempo raries, however , the r ules of equivalence lacked the char acter of empir ical ap ­ plicability in terms of directly or indi rectly measurable quantities, except in the one impor tant case of local mo ­ tion. The p roposition that a unifor mly diffor m latitude of calidity or whiteness cor responds to its medium degr ee can have had for them one of two meanings : the p r oposi ­ tion may have been regarded as a purely for mal or con ­ ventional rule, useful for the pur pose of denominating motions o r subjects qualified by a quality ; or it may have been held to refer to an equivalence between quantities of intensity, which would then be assumed to have all the char acter istics of an extensive magnitude. That the Scho ­ lastic interpretation of the rules of equivalence was of the latter kind is suggested by Duns Scotus' analogy between inc rease in intensity and the addition of water to water , and by the numerous statements of Heytesbu ry and his

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c onte mpo r a r i e s r e g a r ding the quantitative c har ac te r of inte n s ity .9 8

147

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S Conc l us ion

O

UR study of the mathematical and physical content of H eytesbury' s Regule yields the f ollowing results. In Chapter IV of the Reg ule (u De incipit et desinit" ), H eytesbury shows an appreciation of the v alue of the limit-concept f or the analysis of the instantaneous in time and motion; and by means of the logical exposition of the terms H to begin" and H to cease " he is enabled to deal accurately with simple limiting-processes, including in one case the limit of a quotient of inf initesimals. H e also appears to recognize the v alue of the concept " inf inite aggreg ate" f or the analysis of the continuum . In ChapterV (" De maximo et minimo" ), he applies the limit- concept to the bounding of the ranges of v ariables and aggregates. In the f ir st part of Chapter VI, " De motu locali: he demon strates a knowledge of the manner of calculating the dis tance trav ersed in unif ormly accelerated and decelerated motions. In the second part of Chapter VI, " De augmenta­ tione;' he shows an awareness of the more obv ious prop­ erties of the exponential growth f unction. In the third part of Chapter VI, " De alteratione;' he attempts a mathemati­ cal description of va riation of intensity in space and time, with a success that is only partial, owing to f alse assump­ tions as to the nature of intensity, and owing to the f unction which he assigns to certain arbitrary rules of denomina­ tion. We hav e seen that the physical problems considered by H eytesbury are posed secundum imaginationem, and that 1 48

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the search for empirical exempl ification of conclusions , by means of physical measurement , is absent from the Re gule. W ith the exception of the conclusions re garding distance traversed in uniformly accelerated motion , which were common knowledge in late medieval times , it is not clear that the results set forth in the R egule p layed a role in the formation of the empirical-mathematical science of the seventeenth century . The conc lusions of the Regule with re gard to l imits and ag gregates , however , bear a resemblance to the work of the nineteenth-century mathe maticians Cauchy and Dedekind on the log ical foundations of the calculus. The scientific status which Heyte sbury would attribute to his conclusions remains dubious . B y a rough general ­ i zation , we may say that science proceeds by one of two methods : by the method of classifi c ation as based on ob ­ servation , and by the method of mathematically formulated hypothesis in conjunction with physical measurement . The first method involves a correction and elaboration of every ­ day speech ; the second method employs the esoteric lan ­ guage called mathematics . The first method was dominant in the natural philosophy of Aristotle , and continues to be of importance today in the field of b iology ; the second method assumed a pre-eminent role when Galileo promul ­ gated the dogma that the world is a book written in mathe matical language. In the Regule of Heytesbury , mathe matics is employed in the service of classification , i.e. , of the correction and elaboration of everyday speech. Whereas Aristotle assumes the general adequacy of everyday speech for scientific communication , Heytes bury holds that a mathematical description and distinc tion of conceivable cases is prerequisite to the accurate denomination of variab le things and processes . B ut where ­ as Gal ileo and his successors assume that , uunderlyin g " physical appearances , there are certain universal laws which mathematics alone is able to reveal , Heytesbury views the world as a world of objects for which no mathe matical usubstructure" need be postulated , and restricts the province of mathematics to the realm of the imag ina tion. On the one hand , Heytesbury seeks a correction of the common rr10des of speech in the direction of mathe matical precision ; on the othe r , he appears to deny the ap pl icabi lity of such precise mathematical formulations to the world of sense experience. The intent of the physico ­ mathematical portions of the Regule thus appears to be that of providing a conceptual schema according to which

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all types o f va r i ati on which can be imagined , quite i r re spective o f thei r occ u r rence o r nonocc u r rence in the physic al wo r ld, may be deno min ated as to qu antity and kind .

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He nti sbe r i De s e n s u c ompo s ito et di ­ vi s o , Re gule s ol ve ndi s oP.hi s mata , e tc . V e nic e : B onetu s Lo c ate llu s , 1 4 9 4 . Amb r o s ian Lib r ar y , Mil an

B ib l . N a z . di F i r e n z e

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B o dle i an C anon . Mi s c . Mar c . Latin Ma r c . Z ane tti Latin Pal . Latin

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N ational Lib r ar y of Venic e (Ma r c iana ) , Latin manu s c r ipt National Lib r ary of Ve nic e (Mar c iana ) , Z ane tti Latin manu s c r ipt Vati c an Lib r ar y , Palatine manu s c ript

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A: T H E S O P H I S M A T A O F

W ILLIA M HE Y T ESB UR Y

In the following l i s t o f the thi r ty - two s o phi s ms of He yte s b ur y ' s SoP.hi s mata , w e give the title o f e ac h s ophi s m and the fir s t a r g u -

1 53

APPENDIX

me nt s p r o and c ont r a , toge the r with r e fe r e nc e s to othe r auth o r s who t r e ate d the s ame s o phi s m . T he autho r s who de pe nd on He yte s b u r y' s t r e at me nt a r e me ntione d in the fi r s t p a r ag r aph of r e fe r e nc e s . F o lio numbe r s i mme diate ly fo llo wing the s o phi s m s r e fe r to the 1 4 9 4 e dition of He yte s b u r y' s wo r k s (Ve nic e , B on e t u s Loc ate llu s) , in whic h the Sophi s mata appe a r s at fo ll . 7 7v a - l 7 0 vb . T h i s inc una b ulum al s o inc lude s a numb e r of c o mment a r ie s on the Soph i s m at a ; i n r e fe r e nc e s t o the s e c o mme nta r i e s the e dition i s c i t e d s i mply a s " 1 4 9 4 e d ." 1 O mni s homo e s t o mni s homo . P r ob atur : i s te homo e s t i s te ho mo, e t i s te homo e s t i s te homo , e t s ic de s i nguli s ; e r g o o mni s homo e s t o mni s ho mo . . . . Ad p r i m am c on s e que ntiam r e s pond e tur c o mmunite r duplic ite r : ve l dub itando c on s e que ntiam , qui a dubit atu r an a r guitur e x i m ­ po s s ib ili , ve l c onc e de ndo c on s e que nti am e t ne g ando ante c e de n s . Uncle quoc umque de mon s t r ato e s t hoc fal s um : i s te homo e s t i s te homo e t s ic de s in g ul i s , qui a . . . he c p r opo s itio s ingul ar i s s i gni ­ fic at quad i s te homo s it o mn i s homo , qua d tamen non e s t ve r um ; ide a ne g atur p r ob ab ilite r . . . . [ F oll . 7 7v a - 8 l rb] G ae tano di Thiene , " R e c olle c te :• 1 4 9 4 e d . , fol l . 8 l rb - 8 2 va . S i mon d e Le nde n a r ia , " R e c o ll e c ta :• 1 4 9 4 e d . , fol l . l 7 l r a - l 7 4 r a . P aul o f P e r g ol a , " R e c olle c te : M a r c . Latin V I , 1 6 0 , fo l l . 1 8 l r a 1 84 v a . P aul o f V e n i c e , S ophi s mat a (Ve nic e : B one tu s Lo c at e l lu s , 1 4 9 3 ) , fol l . 2 e t �qq . Alb e r t of S axony , Sophi s m at a (P a r i s , 1 4 9 0 ) , P a r t I , Sophi s ma 1 . P e te r of Spain , Summulae l o g i c al e s , e d . B o c he n s ki { T ur in , 1 9 4 7 ) , p . 1 1 9 . W illiam o f Shy r e s wood , " Intr o duc tione s in l o gic am :• P ar i s B N Latin MS . 1 6 6 1 7 , fol . 2 5v . 2 O mne c olo r atum e s t . P r ob atur : aliquod c o l o r atum e s t , e t nullum e s t c ol o r atum quin ip s um e s t , e r g o e tc . Similite r : Hoc c o l o r atum e s t e t hoc c o l o r atum e s t , et sic de s in g uli s , e r g o e tc . In oppo s itum a r guitu r s ic : ali quod c o l o r atum non e s t , i g itur non o mne c ol o r atum e s t . Ante c e de n s ar g uitur s ic : hoc c ol o r atum non e s t , de mon s t r ato uno non c o l o r ato , e r g o e tc . . . . Ad s o phi s m a r e s ponde tur c onc e de ndo . . . . [ F o l l . 82 vb - 8 4 vb] G a e t ano di Thiene , " R e c olle c te ;• 1 4 9 4 e d . , foll . 8 4 vb - 8 5 v a . Simon de Lende n ar i a , " R e c olle c ta :' 1 4 9 4 e d . , foll . l 7 4 r a - l 7 5 rb . P aul o f P e r g ola , " R e c olle c te :• Ma r c . Latin MS . V I , 1 6 0 , foll . 1 8 4va - 1 8 6 rb . P aul of V e nic e , Sophi s mata {Ve nic e : B on e t u s Loc ate l lu s , 1 4 9 3 ) , foll . 9 e t s gq . 3 Omni s h o m o e s t totum in quantitate . P r ob atur s i c : o mni s h o mo e s t in quantitate , s e d null u s e s t homo in quantitate qui non e s t aliquod totum in e ade m quantitate , i g itu r e tc . A s s umptum p r ob a ­ tur , qui a o mni s homo e s t i n lo c o , e t o mni s l o c u s e s t i n quantitate , e r g o e tc . . . . Si c onc e ditur s o phi s ma s ic ut e s t c onc e de ndum , c ontr a: ali qui s homo in quantit ate non e s t . P r ob atur qui a So r te s in aliqua quanti ­ t ate non e s t , quia in quantit ate o c ul i s ui non e s t S o r te s , e t So r t e s e s t aliqui s homo , i g itur e tc . . . . [ F o l l . 8 5v a - 8 8 v a] G ae tano di T hiene , " R e c o ll e c te ;• 1 4 9 4 e d . , fol l . 8 8 va - 8 9 va .

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S i mon de Le nde nar i a , .. R e c olle c t a ;' 1 4 9 4 e d . , foll . l 7 5rb - l 7 6vb . P aul o f P e r g ola , " R e c olle c te ;' Mar c . Latin M S . V I , 1 6 0 , foll. 1 8 6 rb - 1 8 8 vb . P aul of V e nic e , Sophi s mata , (Venic e : B onetus Loe ate llu s , 1 4 9 3 ) , fo l l . 1 1 e t �qq. P e te r of S p ain , Summul ae logic ale s , ed. B o c he n s ki ( Tu r in , 1 9 4 7 1 p. 124. 4 O mni s homo e s t unu s s olus homo . Quod ar guitur s ic : omni s homo qui e s t , e s t unu s s olus homo ; o mni s homo e s t homo qui e s t ; e r go o mni s homo e st unu s s o lu s homo . Simi lite r : quilib e t homo e s t unu s s o lu s homo , e r g o o mni s homo e s t unu s s ol u s homo . . . . Ad s ophi s ma r e s ponde tur ne g ando so phi s ma . P o s s e t tame n s o ­ phi s ma s o phi s tic e s u s tine ri e t c onc e di . . . . [Foll. 8 9va - 9 0vb] G ae t ano di T hiene , " R e c olle c te ;• 1 4 9 4 e d . , foll . 9 0vb - 9 l r a . S i m o n de Lende n ar i a , " R e c olle c t a ;' 1 4 9 4 e d . , foll . l 7 6vb - l 7 7rb . P aul o f P e r g ola , " R e c olle c te ;• M a r c . Latin MS . V I , 1 6 0 , foll . 1 8 8 vb - 1 8 9v a . P aul of V e nic e , Sophis mata (Venic e : B one tu s Loc ate llu s , 1 4 9 3 ), fol . 1 3 . Albe r t of Saxony , Soph i s m at a (P ari s , 1 4 9 0 ) , P ar t II , Sophi s m a 3 8 . W illiam of Shy r e s wood , . . Intr o duc tione s in lo gic am ;• P ar i s B N Latin MS . 1 6 6 1 7 , fo l l . 2 4 v , 2 6 r . 5 O mni s homo q':!l e s t alb u s c ur r it . Quo d non ar guitur s ic : aliqui s homo alb u s e s t t r un c atu s p e de s , e t null u s tal i s pote s t c u r r e rel, e r g o e tc . Ad o ppo s itum a r g uitur s i c : e x tib i dubio ali qui s homo alb u s c ur r it , e t nullus e s t homo alb u s que rn s c i s non c ur r e r e ; e r go dub it a s an omni s homo alb u s c ur r at ; e r go non habe s ne g a r e quod o mni s homo alb u s c ur r it nulla fac ta ob li g atione . Ide o ad s ophi s m a r e s ponde tur c o mmunite r ip sum dub itando . . . . [Foll. 9 l rb 9 6vb] Gaetano di Thie ne , " R e c olle c te ;• 1 4 9 4 e d . , fol l . 9 7r a - 9 8vb . S i mon de Le nde nar i a , " R e c olle c t a ;• 1 4 9 4 e d . , foll . l 7 7 rb - 1 8 0va . P aul o f P e r g ola , " R e c olle c te ;• Mar c . Latin M S . V I , 1 6 0 , foll . 1 8 9 v a - 1 9 3 vb . P aul o f Ve nic e , Sophi s mat a (Ve nic e : B one tu s Loc ate llu s , 1 4 9 3 ) , fol l . 1 3 e t §_qq. Albe r t o f Saxony , SoP.h i s mata (P ar i s , 1 4 9 0 ), P ar t I, Sophi s ma 2 1 . W illiam of Shyr e s woo d , " Introduc tione s in lo g i c am ;' P ar i s B N Latin M S . 1 6 6 1 7 , fol l . 2 4 v , 2 7v . 6 Ani m a Antic hri s ti ne c e s s ar io e rit . Ponatur quod Antic h r i s tu s non s i t , s e d quo d ip s e e r i t , c onting e nte r tame n , ita quod non s it de te r minatum adhuc quando ip s e e r it. I s to po s ito s o phi s m a non e st ve r um s i c s i gnifi c ando . A r g uitur tame n quod s i c , quia anima Ant ic h r i s ti aliquando ne c e s s ar io e r it , quia ip s a ne c e s s ar i o e rit po s t �- Ponitu r quod � e r it p r imum in stans in quo Anti c h r i stu s e r it , e t ar guitu r tune s ic : anima Antic h r i s ti e r it in � in s t anti , e r g o po s t � in stan s ip s a ne c e s s ario e rit . . . quia po s t � in stan s ip s a e r it pe r pe tua e t inc o r r uptibili s ; e r go po s t � in s tan s ip s a non p o te r it non e s s e , e t s i s ic , e r g o tun c i p s a ne c e s s a r i o e r it . . . . [ F o l l . 9 9 r a - 1 04rb] G ae tano di Thie ne , " Re c o lle c te ;• 1 4 9 4 e d . , fol l . 1 04rb - 1 0 5vb . S i mon de Le nde n a r i a , " R e c olle c t a ;• 1 4 9 4 e d . , foll . 1 8 0va - 1 8 2 vb . P aul o f P e r g ola , " Re c olle c te ;• M ar c . Latin M S . V I , 1 6 0 , foll .

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APPENDIX l 9 4 rb - l 9 8 r a . P aul o f Venic e , Sophi s m ata {Ve nic e : B o ne tu s Lo ­ c ate ll u s , 1 4 9 3 ) , fo l . 3 8 . W i ll i am o f Shy r e s wo od , " Int r odu c ti one s in lo g i c am ;• P a r i s B N Latin M S . 1 6 6 1 7 , fo l . 3 6 v . 7 Omni s p r o po s itio v e l e i u s c o nt r adic to r i a e s t ve r a . P on at u r quo d que lib e t p r opo s itio hab e at c o nt r ad i c to r i um ; e t tune a r guitur quo d s o phi s ma s i t ve rum quia e s t una uni ve r s ali s c ui u s que l ib e t s ing u ­ l a r i s e s t ve r a ; e r g o s o phi s m a ve rum . . . . Ide o s i c o nc e ditur s o ­ phi s m a , c o nt r a : s e qui tu r o mni s p r o po s i ti o v e l e i u s c ont r adi c to r i a e s t ve r a , e r g o tantum ve rum e s t p r o p o s i tio v e l e i u s c o nt r adi c ­ to r i a ; c on s e que n s e s t fal s um , qui a e ti am fa l s um e s t p r o p o s iti o ve l e iu s c ont r adic to ri a , e t nullum fal s um e s t v e r um ; i d e o e tc . l F o l l . 1 0 5vb - 1 0 7 vb] G ae tano di T h i e ne , " R e c o l le c te ;' 1 4 9 4 e d . , fo l l . 1 0 7vb - 1 0 8 r a . Simon de Le nde n ar i a , " R e c olle c t a ;' 1 4 9 4 e d . , fo l . 1 8 3 r a - 1 8 3 v a . P aul o f P e r g o la , " R e c o l le c te ;• M a r c . Latin M S . V I , 1 6 0 , fo ll . 1 9 3 vb - 1 9 4 rb . P aul of V e nic e , S oP.hi s mata {Ve nic e : B one tu s Lo ­ c ate l lu s , 1 4 9 3 ) , fo ll . 1 9 . Alb e r t o f S axony , Soph i s m at a {P ar i s , 1 4 9 0 ) , P a r t I , Sophi s m a 1 0 . W i ll i am o f Shy r e s wo o d , " lnt ro duc tione s in l o g ic am ;• P a r i s B N Latin M S . 1 6 6 1 7 , fo l . 4 3 r . 8 I s ti fe r unt l apide m . P onatur quo d So rte s e t P l ato fe r ant unum l apide m , et quod ne e ip s i ne e aliqui s ip s o r um fe r at ali quid ni s i i llum l apide m v e l p a r te m i s tiu s ; e t s it i s te lapi s que rn fe r unt �­ T une ar guitur s ic : i s ti fe r unt � l apide m , e r go i s ti fe r unt lapide m . E t a r g uitur quo d non , qui a ne ute r i s t o r u m fe r t a li qui d , quo d a r guitur s ic , qui a nul lus i l l o r u m fe r t � n e e al iquam p a r te m � ne e aliqui s i s to r um fe r t aliquid aliud quam � v e l p a r te m �; . . . e r g o neute r i l lo r um fe r t aliquid . C on s e que nti a p ate t e t a s s ump ­ tum ar guitur , qui a ne ute r ip s o r um fe r t � quia s i t � g r at i a e x ­ e mpli it a g r avi s quod nullu s i s to r um s uffi c i at fe r r e . . . . Se d c ont r a hanc r e s po n s ione m a r g uitur s i c : s i nullu s i s to r um fe r t aliquid , e r g o i s ti duo non fe r unt aliquid . . . . Di c itur quo d i s te te r minus " fe r r e aliquid" e s t duple x s i v e e qu i v o c u s ; . . . e t s ic c onc e ditur quo d una mu s c a tantum s uffi c i t po r t a r e s i c ut unu s e quu s s iv e move r e , qui a p a r ti al ite r t antum s uffi c it fac e r e . . . . [ F o ll . 1 0 8 r a - l 1 4 v a] G ae t ano di Thiene , " R e c o lle c te ;• 1 4 9 4 e d . , fol l . l 1 4v a - l 1 6 rb . P aul o f Venic e , SoP.hi s mata {Ve nic e : B on e t u s Lo c at e llu s , 1 4 9 3 ) , fol. 3 5 . Alb e r t o f S axony , S oph i s mata {P ar i s , 1 4 9 0 ) , P ar t I , Soph i s ma 3 9 ; P a r t II , Sophi s ma 7 0 . P e te r o f Sp ain , Summu l a e lo g ic ale s , e d . B o c he n s ki { T u r in , 1 9 4 7 ) , p . 7 5 . W i l l i am o f Shyr e s wo o d , " lnt r o duc ti one s in Lo g i c am ;• P ar i s B N Latin M S . 1 6 6 1 7 , fo l l . 2 6r , 32v - 3 3r . 9 N e ut rum o c u lum hab e ndo tu P.O te s v i de r e . Quod s ic a r guitu r : ali ­ que m o c u lum non hab e ndo tu pate s v i de r e , e t nul l u s e s t o c ul u s quin i l lum non hab e ndo t u p o te s v ide r e ; e r g o que mlib e t oc ulum non hab e ndo tu pate s vide r e , et ult r a : i g itur ne ut r um o c ulum hab e ndo tu pote s v i de r e . . . . [ F o l l . 1 1 6 r b - 1 2 2 r a] G ae tano di T hi e ne , " R e c o l le c te ;• 1 4 9 4 e d . , fo l l . 1 2 2 r a - 1 2 3 v a .

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P aul of V e nic e , S oP.hi s mat a (Ve nic e : B onetus Lo c ate llus , 1 4 9 3 ) , fo ll . 3 6 e t �qq. Alb e rt of Saxony , SoP.hi s mata (P ar i s , 1 4 9 0 ) , Part II , Sophi s ma 2 8 . P e te r o f Sp ain , Summulae logic ale s , e d . B o c he n s ki (Turin , 1 9 4 7 ) , p . 1 2 3 . W i lliam of Shy r e s wo o d , " lntroduc tione s in lo gi ­ c am ;• P a r i s BN Latin M S . 1 6 6 1 7 , fol . 2 8 v . 1 0 Qui lib e t ho mo mo r ie tur qu ando unu s s o l u s homo mo r ietur . Hoc s o phi s m a p rob atu r s ic . Aliqui s homo mo r i e tur quando unus s o lu s homo mo r ie tur , e t null u s e s t homo ne e ali qui s e rit ho mo quin tali s homo mo r ie tur , e tc . , i g itur etc . . . Ad o pp o s itum s ophi s m ati s ar guitu r s ic : qui lib e t homo mo rie tur quando unu s s o lu s ho mo mo r i e tur ; e r g o quando unu s s o l u s homo m o r i e tur quilib e t homo mo rie tur . . . . Ide o ad s o phi s ma dic itur in p r inc ipio ip s um c onc ede ndo . . . . [ F o ll . 1 2 3va - 1 2 4 r al G ae t ano di T hie ne , " R e c olle c te ;• 1 4 94 e d . , fo l . 1 2 4 r a . Paul of Ve nic e , Sophi s mata (Ve nic e : B one tus Loc ate llu s , 1 4 9 3 ) , fol . 2 2 . W illiam o f Shy r e s w o o d , " Intro due: tione s in l o g i c am ;• P a r i s B N Latin MS . 1 6 6 1 7 , fol . 2 4 v . 1 1 T u e s quo dlib e t ve l di ffe r e n s � quo lib e t . P rob atur s ic : t u e s dif ­ fe r e n s a quolib e t , e r g o e tc . Con s e quentia pate t e t a s s umptum ar guitur , qui a tu e s et quo dlib e t e s t , s e d tu non e s quodlib e t , e r g o tu e s diffe r e n s a quo lib e t. Ide o s i c onc e ditur s ophi s ma s i c ut e s t c on c e de ndum c ont r a . . . . [ F o l l . 1 2 4 r a - 1 2 4v a] G ae tano di T hi e ne , " R e c o lle c te ;• 1 4 9 4 e d. , fol . 1 2 4va. P aul of Ve nic e , Sophis mata (Venic e : B one tus Loc ate llus , 1 4 9 3 ) , fol . 44 . P e te r of Spain , Summulae .!.Q_gic ale s , e d . B oc he n s ki (Turin , 1 9 4 7 ) , p . 1 2 2 . W illiam o f Shyr e s wood , " lnt roduc tione s in lo g i ­ c a m ;• P a r i s B N La tin M S . l 6 6 l 7 , fo 1 . 4 4 r . 1 2 I s ti s c iunt s e pte m arte s lib e r ale s ; s uppo s ito quod So r te s e t P l ato de mon s t r ar e ntur et quo d So rte s s c i at t r e s a r te s et Plato s c rat quatuo r arte s , quarum nulla e s t aliqua ar s s c itarum a So r te . Quo p o s ito , i sti s c iunt s e pte m arte s lib e r ale s , quia s e pt e m arte s l ib e r ale s s c iuntur , e t a nullo s c iuntur s e pte m a r te s , ne e ab ali ­ quib u s ali i s s c iuntur s e pte m a rte s lib e r ale s quam a S o r te e t Pl atone ; ponatu r quo d a nulli s alii s quam a So r te e t P l atone s c i antur s e pte m a r te s lib e r ale s . E r go s e pte m arte s s c iuntur a So r te e t P l atone , e t ultr a , e r g o illi s c iunt s e pte m a r te s , de mon ­ s t r ati s illi s duob u s vide lic e t So r te e t P latone . . . . E r go s o ­ phi s ma . Ad oppo s itum a r guitu r s ic : i s ti s c iunt s e pt e m arte s , e r go i s ti duo s c iunt s e ptem a r te s , e t ultr a , i s ti duo s c iunt s e pte m a r te s , e r go i s ti duo s unt s c iente s s e pte m a r te s , . . . e r go i s ti duo s unt duo s c ie nte s s e pt e m arte s , e r go aliqui s i s to r um e s t unu s s c i e n s s e pte m a rte s . . . Se d nullu s unu s s c it s e pte m arte s ut app a r e t s e qui e x c a s u . . . . Ad s o phi s ma po s ito i s to c a s u . . . dic itur c onc e dendo s ophi s ­ m a . . . . [ F oll . 1 2 4va - 1 2 5rb] G ae tano di T hie ne , " R e c olle c t e ;• 1 4 9 4 e d . , fol . 1 2 5rb . P aul of Venic e , SoP.hi s mata (Venic e : B one tu s Loc ate llu s , 1 4 9 3 ), fol . 3 4 . Alb e r t of S axony , SoP.h i s mata (P ar i s , 1 4 9 0 ) , P art I , Sophi s ma

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4 1 . W i lliam of Shy re s wood , " Int roductione s in l o g i c am ;• P a r i s B N Latin M S . 1 6 6 1 7 , fo l . 3 2 v . I s ti p_ggnant ut vinc ant s e . Ho c s o phi s ma p rob atur s ic . P o natur quod So rte s pugne t c um P l ato ne ut vinc at i p s um , et P l ato p u g ­ n e t e c ont r a c um So rte ut vinc at i p s u m , e t non pu gnant i s ti duo c um aliquib u s alii s ne e c u m aliquo ali o . Quo po s ito ar g uitu r s o phi s ma s i c : i s ti pugnant ut vinc ant aliquo s ; aut i s ti pugnant ut vinc ant s e ip s o s aut ut vinc ant alio s quam s e ip s o s . S e d ip s i n o n pugn ant ut vinc ant alio s qua m i p s o s , e r go i s ti pugnant ut vinc ant s e ip so s . . . . Ad oppo s itum ar guitu r s ic : i s ti pugnant ut vinc ant s e , e r go i s ti pugnant ut vinc antu r . C on s e que n s fal s um . . . . Ad s ophi s ma po s ito c a s u i s to dic itur ne g ando ip s um . . [ F o l . 1 2 5rb - 1 2 5vb] G ae tano di Thie ne , " R e c olle c te ;• 1 4 9 4 e d . , fol . 1 2 Svb . Alb e rt of Saxony , Sophi s mata (P ar i s , 1 4 9 0 ) , P a rt I , Sophi s ma 42 . Omni s homo e s t ani mal e t e c onve r s o . Quod s op hi s ma s it ve rum a r g uitur s i c : i s te homo e s t animal et e c onve r s o , et i s te homo e s t animal et e c onve r s o , et s ic de s inguli s ; e r g o o mni s homo e s t animal et e c onve r so . . . . Ad oppo s itum a r guitur s ic : s i o mni s homo e s t animal e t e c onve r s o , e r go o mni s homo e s t animal e t o mne animal e s t homo , quod e s t impo s s ib ile . . . . Ide o ad hoc s ophi s ma di c itu r quo d e s t impo s s ib i l e . . . . [ F o l l . 1 2 Svb - 1 2 6vb] G ae t ano di T hi e ne , " R e c o ll e c te ;• 1 4 9 4 e d. , fo l l . 1 2 6vb - 1 2 7 r a . P aul of V e n ic e , SoP.hi s mata (Ve ni c e : B one tu s Loc ate llu s , 1 4 9 3 ) , fol . 2 0 . P e te r o f Spain , Summulae l o g ic ale s , e d . B oc he n s ki ( T u r in , 1 947), p. 79. B i s duo s unt t r i a e t non P.lu r a . Quod s ophi s ma s it ve r a ar guitu r s ic : duo e t duo s unt t r i a e t non p lur a , e r g o s ophi s m a . C on s e que ntia p ate t , e t ante c e de n s ar guitur s ic , . . . qui a � ' 2.,, .£. s unt he c duo , � e t £_, et he c duo , £_ e t .£_; e r go s unt duo et duo . Ad s ophi s ma dic itur quo d ip s um e st impo s s ib ile . . . . [ F o l . 1 2 7 r a - 1 2 7rb] G ae t ano di T hi e ne , " R e c o lle c te ;• 1 4 9 4 e d. , fo l . 1 2 7rb - 1 2 7 v a . P aul o f Venic e , SoP.hi s mata (Venic e : B one tus Lo c at e l lu s , 1 4 9 3 ) , fol . 3 0 . Omni s homo e t duo homine s s unt t r e s . P r ob atur , qui a aliqui s homo e t duo ho mine s s unt t r e s , e t nullu s e st homo quin ille e t duo homine s s unt tre s homine s ; i g itur omni s homo e t duo h o mi ­ ne s s unt tr e s ho mine s . . . . Si c onc e ditur s ophi s m a s ic ut e st c onc e de ndum , c ontr a : ali ­ qui s homo e t duo homine s non s unt t r e s ho mine s , e r g o non omni s homo et duo homine s s unt t r e s homine s . C on s e quentia pate t e t a s s umptum a r g uitu r , qui a S o r t e s et duo ho mine s non s unt tre s ho mine s , qui a So r te s et P l ato non s unt tre s , et So r te s e t P l ato s unt aliqui s ho mo e t duo homine s . . . . [ F o ll . 1 2 7va 1 2 9 r a] 1 58

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G ae tano di T h i e ne , " R e c o lle c te ;• 1 4 9 4 e d . , fol . 1 2 9 r a - 1 2 9 rb . P aul o f V e nic e , SoP.hi s mata (V e nic e : B one tu s Loc ate llus , 1 4 9 3 ) , fol . 3 0 . Alb e r t o f Saxony , Sophi s mata (P ari s , 1 4 9 0 ) , P art I , Sophi s ma 1 7 . 1 7 O mne s �P.o s toli s unt duo de c im . Qua d s o phi s m a s it ve rum ar gui ­ tur s i c : ali qui s unt o mne s apo s to li , e t non plure s ne e p auc io r e s quam duode c i m ; e r g o duode c i m sunt o mne s apo s toli de i ; e r go s o phi s ma . . . . Ad opp o s itum ar guitur s ic : omne s apo s toli de i s unt duo de c i m ; e r go omne s qui s unt ap o s toli de i s unt duode c i m . Se d P e t r u s e t P aulu s s unt apo s to li ; e r go P e t r u s e t P aulus sunt duode c im . . . . Ide o ad s ophi s ma re s ponde tur di s tingue ndo hoc s ophi s ma , e o quad i s te te r minu s " o mne s" p ote s t tene ri c olle c tive ve l divi ­ s ive . . . . [ F ol l . 1 2 9rb - 1 3 0rbJ G ae tan o di T hi e ne , " R e c oll e cte ;• 1 4 9 4 e d . , fo l . 1 3 Orb - l 3 0v a . P aul of Ve nic e , Soph i s mata (Ve nic e : B one tu s Loc ate llu s , 1 4 9 3 ) , fol . 2 7 . Alb e rt of Saxony , Soph i s mata (P ar i s , 1 4 9 0 ) , P art I , Sophi s m a 4 . W i lliam of Shy r e s wo o d , " lntr oduc tione s in l o gic am ;' P a r i s B N Latin M S . 1 6 6 1 7 , fo l . 2 6 r . 1 8 Infinita s unt finit a . Quo d s o phi s ma s it ve rum ar guitur s i c ; duo s unt finita et t r ia s unt finita et quatuo r s unt finita , et s i c in in ­ finitum , e r g o s ophi s ma . . . . Ide o fo rte suppo s ita illa di s tinctione c o mmuni de i s to te r mino " infinitum :' quod ali quando pote s t tene r i c athe g o r e umati c e e t s inc athe g o r e um atic e , e t limitato illo s e ns u quo tene tur s in ­ c athe g o r e umatic e , tune ut s ic c onc e ditur s ophi s ma s i c ut e s t c onc e de ndum , quia s e n s u s illius s o phi s mati s tune e st i ste : ali ­ qua s unt finita e t non tot s unt finita quin in duplo plura ill i s s int finita , et in t r iplo plur a ill i s s unt finita , et s ic in infinitum . . . . [ F oll . 1 3 0 va - 1 3 4 r a] G ae tano di T hi e ne , " R e c olle c te ;• 1 4 9 4 e d. , fol l . 1 3 4r a - 1 3 5 r a . P aul of V e n ic e , Sophi s mat a (Venic e : B onetus Loc ate l lu s , 1 4 9 3 ) , fol . 3 9 . Alb e rt of Saxony , Sophi s mata (P a r i s , 1 4 9 0 ) , P art I , Sophi s ma 5 3 . P e te r of Spain , Summulae l o gic ale s , e d . B o c he n s ki (Turin , 1 9 4 7 ) , p . 1 2 9 . W i l l i am of Shyre s wo o d , " lntroduc tione s in lo g ic am ;' P ar i s B N L atin M S . 1 6 6 1 7 , fol . 2 6v . 1 9 Qui c quid audito r � S o r te P.rofe rtur � P l atone . P onatur quod P l ato p r ofe r at i s tam p r opo s itione m , "Nullu s ho mo e s t a s inus ;• et quod So rte s nihi l aud iat ni s i i s tam p r opo s itione m , " Homo e s t a s i ­ nu s :• . . . Quo c a s u po s ito ar guitu r s ic : omne quo d auditur a So rte e s t i s ta p r o p o s iti o , '" Homo e s t a s inus" ; . . . s e d omne quod e s t i s ta pro po s itio ve l aliquid il lius e s t pro la tum a Platone ; i g i ­ tur qui c qui d auditur a Sorte p r ofe rtur a P l atone . Ad opp o s itum ar guitur : p r o p o s iti o affir mativa auditur a Sorte , e t nulla p ro p o s iti o affir mativa p r ofe rtur a P l atone . . . . Ad s ophi s ma dic itur quod e s t impo s s ibile qui a . . . non e s t po s s ib ile quod ali qui s p r ofe r at aliquam p r o p o s itione m . . . . [Foll . 1 3 5rb - 1 3 6vb]

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Gae tano di Thiene , u Re c olle c te ;' 1 4 9 4 e d . , foll . l 3 6vb - l 3 7 r a . P aul of Venic e , S02hi s mata (Ve nic e : B onetus Loc ate llu s , 1 4 9 3), fol . 2 4 . Alb e r t o f Saxony , Sophi s mata (Par i s , 1 4 9 0 ) , P ar t I , Sophi s ma 64 . De u s s c it quic quid s c ivit . P rob atu r s ic s o phi s ma : de u s s c ivit ali qui s et null i u s e s t ob litu s quad s c ivit , i gitur e tc . . . . Simili ­ te r in de o non pote s t e s s e aliqua t r an s mutatio ; e r g o in i p s o non pote s t aliqua s c ie nti a t r an s muta ri in non s c ie ntia ne e e c ont r a ; i g itur quic quid fuit aliquando s c ie nti a de i ve l s c ie ntia in ip s o adhuc e s t illud ide m s c ie nti a i n ip s o v e l ip s ius s c i e ntia . . . . Ide o s i c o nc e ditur so phi s ma , c ontr a : de u s s c it quic quid s c ivit et de u s s c ivit aliquod quad iam e s t impo s s ib ile , i g itur deus s c it imp o s s ib ile . . . . Ide a ad s o phi s ma dic itur ne g ando s ophi s ma t anquam i mpo s sibile . . . . [ F o l . 1 3 7r a - 1 3 7vb] Gae t ano di Thi e ne , " R e c oll e c te ;• 1 4 9 4 e d . , fo ll . 1 3 7vb - 1 3 B r a . Paul of Venic e , Sophi s mat a (Venic e : B onetus Loc ate llu s , 1 4 9 3 ) , fol . 3 3 . Alb e rt of Saxony , Sophi s mata (P ar i s , 1 4 9 0 ) , P a r t I , Sophi s ma 6 5 . Tu s c i s gui c guid s c i s . P r ob atu r s ic : tu s c i s aliquid , e r go tu s c i s quic quid s c i s . A s s umptum a r g uitur , e t ponitur quad t u s c ia s ali ­ quam p r opo s itione m que s it �- T une ve l e s t � p r opo s itio multa ve l e s t aliquid . Si aliquid hab e tu r p r opo s itum . Si multa e t non aliquid , tune ar guitu r s ic : tu s c i s � e t � e s t mult a , e r go tu s c i s multa , e t s e quitur : t u s c i s multa , i g itu r multa s c iuntur a te , e r go multa s unt multa s c i t a a te , . . . e r go aliqua duo ve l t r i a e t s ic i n infinitum s unt s c ita a te , e r g o aliquod unum e s t s c itum a te . . . . Ide a p r o i sto dic itur in p r inc ipio ne g ando s ophi s ma po s ito t ali c a s u . . . . [ F o l l . 1 3 8 r a - 1 3 9 va] G ae tano di T hie ne , " R e c olle c te ;• 1 4 94 e d. , fo l . 1 3 9v a - 1 3 9vb . P aul of V e nic e , So2hi s mata (Ve ni c e : B one tus Loc ate l lu s , 1 4 9 3 ) , fol . 3 3 . Quo dlib e t aligu o r um animalium e s t non homo , quo rum quodlib e t e s t homo . Qua d a r guitur s ic : quodlib e t i s to r um t rium ani malium e s t non homo quo r um quodl ib e t e s t homo , e r g o e tc . P on atur quad Sor te s , P l ato , e t C ic e r o habe ant t r e s a s in o s s ib i c o mmune s et quad ip s i non hab e ant aliquod animal ni s i alique m i s to rum t r i u m a s ino rum . Quo po s ito , quo dlib e t i s to rum animalium de mo n s t r ati s So rte e t Pl atone e t C i c e rone e st non h o mo . . . . Ad s ophi s ma dic itur di s tingue ndo illud s e c undum c o mpo s i ­ tione m e t divi s i one m s ic ut e t ill am : o mne ani m al e s t r ational e quad e s t ani mal . E t ute r que s e n s u s e s t impo s s ibile , e r g o s o ­ phi s ma e s t i mpo s s ib ile . . . . [ F oll . l 3 9 vb - 1 4 0vb] G ae t ano di T hi e ne , '" R e c olle c te ;• 1 4 9 4 e d. , fo l . 1 4 0vb . P aul o f Ve nic e , SoP.hi s mata (Ve nic e : B one tu s Loc ate llu s , 1 4 9 3 ) , fol . 2 5 . Cuiu s libe t homini s a s inu s c ur rit ; p o s ito quad quilib e t ho mo habe at duo s a s ino s unum c ur re nte et alium non c u r r e nte . T une p r ob atur s o p hi s ma s ic : alic ui u s h o mini s a s inus c u r r it et nullu s e s t homo quin illius a s inu s c ur r at , e r go e t c . . . . 1 60

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Ad oppo s itum ar guitur s ic : c uius lib e t ho mini s a s inu s c u r r it , e r go non c uius lib e t ho mini s a s inu s non c u r r it . Se d p r ob atur o mnino s ic ut s o phi s m a quod c ui u s lib e t ho mini s a s inu s non c ur ­ rit . . . . Ad s ophi s ma re s ponde tur ip sum c onc e de ndo tanquam s e que ns ex c a s u s uppo s ito . . . . [ F o ll . 1 4 0vb - 1 4 l v a] G ae tano di Thiene , " R e c o lle c te , 1 4 9 4 e d . , fol . 1 4 l v a . P aul of Ve nic e , Sophi s m ata ( V e nic e : B one tus Lo c ate llus , 1 4 9 3 ) , fo l . 1 7 . Alb e rt of Saxony , Sophi s mata (P ar i s , 1 4 9 0 ) , P art I , Sophi s ma 8 . W i lliam of Shy r e s wood , " Int roduc ti one s in logic am ;• P ar i s B N Latin M S . 1 6 6 1 7 , fo ll . 2 Sr , 4 4 r . 2 4 O mne ani mal fuit i n a r c h a N o e ; po s ito c a s u c o mmun i . Ar guitur quo d non , quia non o mne animal fui t in arc ha Noe , qui a tu non 1 fui s ti in a r c h a N o e , et tu fui s ti aliquod animal , e r go e tc . . . . Ide o s i ne g atur s o phi s ma s ic ut e s t ne gandum , ar guitur tune s i c : ani mal fuit in a r c h a N o e , et nullum fuit animal ne e e s t ali ­ quod animal quod s c i s non fui s s e in arc ha N o e ; e r go e x tibi dub io o mne animal fuit in a r c h a N o e . . . . [Foll . 1 4 l v a - 1 4 2 vb] Gae t ano di T hi e ne , " R e c o ll e c te ;• 1 4 9 4 e d. , foll . 1 4 2vb - 1 4 3 rb . P aul o f V e nic e , Sophi s mata (Venic e : B one tus Loc ate llus , 1 4 9 3 ) , fol . 2 3 . Alb e rt of Saxony , Soph i s mata (P a r i s , 1 4 9 0 ) , P art I , Sophi s ma 3 . 2 5 Omne ve rum e t de um � diffe r unt . Quod ar guitur s ic : omni s p r opo s itio e t de um e s s e diffe r unt ; o mne ve rum e s t propo s itio ; e r g o o mne ve r um e t de um e s s e diffe runt . . . . Ide o s i c onc e ditur s o phi s ma ar guitur quod s it fal sum , quia aliquod ve rum e t de um e s s e non diffe r e nt , e r go non omne ve rum et deum e s s e diffe runt. Con s e quentia p ate t et a s s umptum ar gui ­ tur , quia de um e s s e e s t ve r um . . . . Ide o dic itur ad s ophi s ma in p r inc ipio d i s t ingue ndo de i s to te r mino " ve r um ;• eo quod p ote s t ac c ipi p r out s i gnific at p r o ­ po s itione m ve r am , v e l prout s ignific at e s s e ve re . . . . [Foll. 14 3 rb - 1 4 5v a] G ae tano di T hi e ne , " R e c o ll e c te :• 1 4 9 4 e d . , fo ll . 1 4 5v a - 1 4 6rb . P aul of Venic e , Sophi s mat a (Venic e : B onetu s Loc atellus , 1 4 9 3 ) , fol . 2 8 . Albe r t o f Saxony, SoP.hi s mata (Par i s , 1 4 9 0) , Par t I , Sophi s m a 1 5 . 2 6 Omni s fe nix e s t . Quod a r g uitur s i c : omne quo d fuit e s t , fenix fuit , e r g o fe nix e s t , et si fenix e s t , o mni s fe nix e s t , e r go e tc . A s s umptum ar guitur s i c : o mne quo d fuit e s t , quia t antum e n s e s t illud quod fui t . . . . Ad oppo s itum s ophi s mati s ar guitur s ic : e x tibi dubio nul l a f e n i x inc ipit e s s e , e t que lib e t que p r iu s fuit i arn e s t c o r rupta , e r g o nul l a fenix e s t . Ide o po s ito c a s u c o mmuni s c ilic e t quod dub itatur a n aliqua fe nix s it , ve l b r e vite r s ine aliquo c as u po s ito , dub itatur s ophi s ­ ma . . . . [ F o l l . 1 4 6 rb - 1 4 7 r a] G ae t ano di T hi e ne , " R e c olle c t e :• 1 4 9 4 e d . , fol . 1 4 7 r a - 1 4 7rb . P aul o f V e n ic e , Sophi s mata (Venic e : B one tus Loc ate llu s , 1 4 9 3 ) , foll . 5 e t �qq.

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Alb e rt of Saxony , Sophi s mata (P a r i s , 1 4 9 0 ) , P art I , Sophi s ma 2 . P e te r of Sp ain , Summulae l o g i c ale s , e d . B oc he n s ki (T u r in , 194 7 ), p. 1 1 7. Totu s S o rte s e s t mino r S o rte . Qua d ar guitur s ic : que lib e t p a r s So rti s e s t minor Sorte , e r g o totu s S o r te s e s t mino r S o r te . . . . Ad opp o s itum ar guitur s i c : totus Sorte s e s t minor S o r te , e r g o So r te s e s t minor So rte . Ad s o phi s ma dic itur di s tingue ndo de i s to te r mino " totu s ;• e x e o quad pote st s umi c athe g o r e matic e ve l s inc athe g o r e matic e . . [ F o ll . 1 4 7rb - 1 4 8 va] G ae tano di Thi e ne , " R e c o lle c te ;• 1 4 9 4 e d . , fo l . 1 4 8va - 1 4 8vb . P aul of Venic e , Sor.hi s mata (Ve nic e : B one tus Loc ate llu s , 1 4 9 3 ) , fol . 4 0 . Albe rt of Saxony , Sor.hi s mata (P ar i s , 1 4 9 0 ) , Part I , Sophi s ma 4 5 . P e te r of Sp ain , Summulae !Q_gic ale s , e d . B oc he n s ki ( T u r in , 1 9 4 7 ) , p . 1 2 5 . W illiam o f Shyr e s wood , " Int r o duc tione s in lo g i ­ c am ;• P ari s B N Latin M S . 1 6 6 1 7 , fo l . 2 6r . Album fuit di s r.utatu r um . P onatur quad So rte s fue r it alb u s e t quad fue r it dis putatu r u s quando fuit alb u s , e t quad i am non s it alb u s ne e unquam po s t in s tan s p r e s e n s e r it alb us , e t quod ne e di s putavit ne e di s putat , s e d quod e r a s dis putab it g r atia a r g u ­ ment i . I s to p o s ito ar guitur s ophi s ma s ic : alb um fuit di s puta ­ turum , qui a So rte s al b u s fuit dis putatu r u s . . . . Se d a r g uitu r qua d non , qui a s i album fuit di s putaturum e t o mne quad fuit di s putaturum di s put at ve l di s putavit ve l di s puta ­ bit , s e d nul lum album di sp utat ne e di s putavit ne e di s putabit pe r c as um , e r go e tc . . . . Ad s ophi s ma dic itur ip s um c onc e dendo . . . . [ F o l l . 1 4 8 vb l 5 0 r a] G ae tano di T hi e ne , " R e c o ll e c te ;• 1 4 9 4 e d. , fo l . l 5 0 r a - l 5 0 rb . P aul o f V e nic e , Sor.h i s mat a (Venic e : B one tu s Loc ate llu s , 1 4 9 3 ) , fol . 3 1 . Alb e rt o f Saxony , Sor.hi s mata (P ar i s , 1 4 9 0 ) , P a rt III , Sophi s m 9 . De u s e r i t in quolib e t in s tanti non e xi st e n s . P r ob atur s ic : in ali ­ quo in s t anti de u s e r it non e xi s te n s e t nullum e r it ne e e s t in ­ s tan s quin in illo e rit de u s non e xi s te n s , e r g o in quolib e t in s tanti e r it de u s non e xi s te n s . C o n s e que nti a p ate t , et a s s umptum s imili ­ te r , quia in hoc in s t anti de u s e rit non e xi s te n s quo c unque i n s t anti data , . . . quia quoc unque data in s tanti po s t i llud e r it de u s non e xi s te n s in illo . . . . Ide a s i c onc e ditur s o phi s ma . . . c ontr a : de u s e r it in quolib e t in s t anti non e xi s te n s , e r g o de u s e rit s e mpe r non e xi s te n s . . . qua d e s t imp o s s ib ile maxi mum. Ide a dic itu r ad s ophi s ma in p r inc ipio quad ip s um e st me r e i m ­ po s s ib ile . . . . [ F o ll . 1 5 0 rb - 1 5 3 r a] G ae tano di T hi e ne , " R e c o lle c te ;• 1 4 9 4 e d. , fol . 1 5 3 r a - 1 5 3 v a . P aul o f V e nic e , Sor.h i s mat a (Ve ni c e : B one tu s Loc at e l lu s , 1 4 9 3 ) , fol . 3 2 . Alb e r t o f Saxony, Sophi s mata (P a r i s , 1 4 9 0) , P a r t III , Sophi s ma 9 . So r te s de c ir.itur ni s i i.P. s e de c i r. iatur ; p o s it o quad S o r te s s o lum ­ modo c r e dat i s t a duo : Sorte s de c ipitur , e t hoc s e de t , de mon s t r ato

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te s e de nte , que s i c p r e c i s e s i gnific e nt . Quo po s ito p r ob atur quod So rte s de c ipitur ni s i ip s e de c ipi atur , qui a s i ip s e non de c ipitur , e t i p s e c r e dit i s tam p r o po s itione m , " So r te s de c ipitu r ;' que s i c p r e c i s e s ignific at , e r go S o r te s c r e dit p r o po s iti one m que s i gnifi ­ c at p r e c i s e s ic ut non e s t , e r g o ip s e de c ipitur . E r g o in c a su i s to si i p s e non dec ipitur ip s e de c ipitur , e r go de c ipitur ni s i de c ipi atur. Ad o ppo s itum ar guitur s ic : So rte s de c ipitur ni s i ip s e de c ipi atur, e r go S o r te s de c ip itu r s i ip s e non de c ipitur , e r g o s i Sorte s non de c ipitur ip s e de c ipitur , quod e s t impo s s ibile . . . . [ F oll . 1 5 3va 1 5 5 rb] G ae tano di T hiene , " R e c o ll e c te ;• 1 4 94 e d . , fol . l 5 5 rb - l 5 5v a . P aul of V e nic e , Sophi s m ata (Ve nic e : B one tus Loc ate llu s , 14 9 3 ) , fol . 5 2 . 3 1 Ne c e s s e e s t aliquid c onde n s a r i s i al iquid r a r e fi at . Quod ar guitur s ic : non pote st e s s e quin aliquid c onde n s e tur si aliquid r ar e fi at , e r g o s ophi s ma . A s s umptum a r g uitur s ic , quia s i pote s t e s s e quod nihil c onde n s e tur quamvi s aliquid r a r e fi at , p onatur e r go quo d aliquid r a re fiat quamv i s nihil c onde n s e tur , e t s it i ll ud quo d r ar e fit �- . . . E r g o totu s mundu s e s t m ai o r quam p r i u s fuit totu s mundu s , . . . quod e s t impo s s ib ile . . . . Ad s ophi s ma c um p ro ponitur ne c e s s e e s t aliquid c onde n s ari s i ali quid r ar e fi at , dic itur c onc e de ndo s ophi s ma . . . . [F oll . 1 5 5va - 1 6 4 r a] C f. Ar i s totle , P h y.: s ic a IV , 9 , 2 l 6b 2 4 - 2 6 3 2 Impo s s ible e s t aliguid c ale fie r i ni s i aliguid f r i ge fi at . [Foll . 1 6 4r a - l 7 0vbj T hi s s ophi s m de al s with the p r ob le m of " re ac tion" ; fo r He yte s ­ bury' s po s ition he r e and the opinion s o f othe r S c hoo lme n on the s ame s ub je c t s e e Mar s h al l C l a g e tt , Giovanni M a r l i ani and Late Me die val P h y.: s ic s (Ne w Y o r k , 1 9 4 1 ) , C h apte r II , pp . 3 4 - 5 8 .

B: T HE SOP HISM AT A O F R I C HARD K IL M ING TON

T he fo llowing l i s t of the fo rty -nine s ophi s m s o f Kilmington' s "So ­ phi s mata" i s b a s e d o n B ib l . Univ . di P ado v a , M S . 1 1 2 3 , foll . 6 5va 7 9 v a ; B odle i an C anon . M i s c . M S . 3 7 6 , foll . l r a - 2 2 r a [an on .]; and V at . Latin M S . 3 0 6 6 , foll . 1 6 r a - 2 4 vb . Inc . " Ad utr umque dub itare pot e nte s fac ile s pe c ul ab imus e t ve rum et fal s um ut die it Ari s totile p r imo topic o ru m ; ut i gitur in pre s e nti o p e r e ve rum e t fal s um s p e c ule mu r fac iliu s s o phi s matum p e r s c r utando r um ut r a s que p ar te s c ontr adic tioni s intendo di s ­ c ute r e \ . . :• 1 P r i mum i g itu r s ophi s ma e s t hoc : So rte s e s t alb i o r quam P l ato inc ipit e s s e albu s ; po s ito g r atia e xe mpli quod Sorte s s it albu s in s ummo et quo d P lato nun c p rimo inc ipiat e s s e alb u s . . . . Ad s ophi s ma r e s ponde tur e xp onendo l i inc ipit pe r r e motione m de p r e s e nti e t tune s ophi s ma e s t ve rum . . . . 2 S e d fo rte a r g uitur ad illam p r opo s itione m s ic : So rte s e st in in ­ finitum alb io r quam P l ato inc ipit e s s e albu s , quod e s t s e c undum s ophi s ma . . . .

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Ad s o phi s m a c onc e ditur qua d ve r um e s t e xponde ndo l i inc ipit ut p r iu s . . . . S o r te s inc ipit e s s e alb i o r qu am P l ato inc ipit e s s e albu s ; po s ito quad So rte s et P l ato nunc p r imo inc i p i ant e s s e alb i , et s it ne ut r u s ip s o rum nunc alb u s e t intendantur e o rum alb e dine s e qu alite r c e te r i s paribu s . Tune p r ob atur hoc s o phi s ma s i c : S o rte s e r it alb i o r quam P l ato inc ipit e s s e albu s et S o r t e s nunc non e s t alb i o r quam P l ato inc ipit e s s e albu s ; e r g o So rte s inc ipit ve l inc ipie t e s s e alb i o r quam P l ato inc ipit e s s e albu s . Se d nunquam po s t hoc in ­ s t an s inc ipie t e s s e alb i o r quam P l ato inc ipit e s s e albu s ; e r g o So rte s inc ipit e s s e alb i o r quam P l ato inc ipit e s s e alb u s . . . . Ad s o phi s m a c on c e ditur quad e s t ve r um s u mmendo li inc ipit ut prius . . . . So r te s inc ipit e s s e alb i o r quam ip s e me t inc ipit e s s e alb u s , qua d e s t quartum s ophi s ma e t p r ob atur ut p r oximum s ophi s ma . . . . S o r te s inc ipie t e s s e ita alb u s s ic ut ip s e me t e r it albu s ; po s ito qua d tota alb e do in S o rte int e ndatur pe r totam vitam S o r ti s e t v i v at S o r te s diu po s t hoc in s tan s . Tune p r ob atur s o phi s ma s ic : S o r te s e r it ita albu s s i c ut ip s e m e t e r i t albu s e t S o r te s non e s t ita alb u s s i c ut ip s e me t e r it a lb u s , ig itur So r t e s inc i pi e t e s s e i t a a lb u s s i c ut ip s e me t e r it al b u s . . . . Ad s ophi s ma dic i tu r quad e s t fal s um e t ad p r ob ati one m quando a r g uitur Sorte s e r it ita alb u s s i c ut i p s e m e t e r it alb u s , e tc . , ne g atu r . . . . T ame n fo rte dic itur quidam c onc e de ndo qua d S o r te s e r it ita alb u s s ic ut ip s e me t e r it alb u s � e t t une c on s e que nte r ne g anda e s t i s t a c on s e quent i a : S o rte s non e s t ita alb u s s i c ut i p s e me t e r i t albu s , e t S o r te s e r it ita alb u s s ic ut i p s e me t e r it alb u s , i g itur S o r t e s inc ipie t e s s e ita albu s s i c ut i p s e me t e r it alb u s . Cum hoc tame n ve r b o e s t s ine aliquo adiun c to b e ne v al e t c on s e qu e ntia s ic a r gue ndo : hoc non e s t et e r it , i gitu r inc ipit e s s e ; s e d c um mult i s additi s non v al e t c on s e que nti a . . . . S o r te s inc ipie t e s s e ita alb u s s i c ut P l ato e r it alb u s ; po s ito quad S o r te s e t P l at o s int e qualite r alb i et qua d v i v ant pe r e quale m te rn ­ pu s p r e c i s e e t inte n dantur alb e dine s in S o r te e t i n P l atone e qual i ­ te r pe r to tum te mpu s in quo vivent S o r t e s e t P l ato . T une p r ob atur s ic : So rte s e r it ita alb u s s i c ut P l ato e r it al b u s , et So r te s non e s t nunc ita alb u s s ic ut P lato e r it alb u s , ne e inc i pit e s s e ita alb u s s ic ut Plato e r it alb u s , i g itur S o r te s inc ipie t e s s e ita alb u s 'S i c ut Plato e r i t al b u s . . . . Ad s ophi s ma dic o quad fal s um e s t s i c ut p r e c e de n s . E t ad p r o ­ batione m dic itur quad ante c e de n s e s t fal s um vide lic e t qua d S o r te s e r it ita alb u s s i c ut P lato e rit al b u s . . . . S o r te s e r it alb i o r quam P lato e ri t alb u s in ali quo i s to rum ; s up ­ po s ito c a s u p r o xinii s o phi s mati s p r e c e de nti s , e t c um h o c s uppon a ­ tur qua d � s it te mpu s pe r quad vi vent S o r te s e t P l ato e t t une pe r li i s to r um de notantur p a r te s p r opo rtionale s in � te mpo r e . . . . Ad s ophi s ma dic itur quad e s t fal s um . . . . So r te s e r it ita alb u s p r e c i s e s ic ut P l ato e r i t al b u s in ali quo i s to r um ; s uppo s it o quad So rte s e t P l ato s int e qual i te r alb i e t qua d

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p e r � te mpu s inte ndantu r e o r um alb e dine s e qualite r e t in fine � te mp o r i s s it P l ato c o r r uptu s e t tune v i v at S o r t e s . . . . So r te s e r it ita alb u s s i c ut P l a t o de s i ni e t e s s e alb u s ; s uppo s ito c a s u p r i o r i in toto . . . . So r te s e r it in duplo alb i o r qu am P l ato e r it albu s in � ins tanti ; s u ppo s it o i s to c a s u quo d S o rte s e t P l a�o nunc inc i p i ant e s s e alb i e t albe fi ant pe r unam h o r am e t quo d S o r te s alb e fi at in duplo ve l o c i u s quam P l ato , e t s i t S o r t e s mo rtuus in � in s tanti et v i v at P l ato i n � ' e t s i t � in s t an s te r minan s i l l a m ho r am in qua So rte s e t P l at o alb e fi ant . . . . Al i qui d e g it Q g r adum albe dini s ; s uppo s it o hoc c a s u quo d alb e do in P l atone inte nde b a tur pe r unam h o r am p r e c i s e pe r � a g e n s e t non u l t r a , e t s i t n unc fini s i ll i u s ho r e e t vivat P l ato i n hoc in ­ s t anti e t s i t P l ato alb u s in hoc in s tanti 2._ g r adu alb e dini s . T une p r ob at u r s o phi s m a s i c : 2._ g r ad u s alb e dini s e s t actus ab � a g e nte , i g itur � a g e n s e g it , a g it , ve l a g e t Q_ g r adum alb e dini s . Se d � a g e n s non a g it Q_ g r adum alb e d ini s qui a a g e n s nunc c e s s at ab o mni s ua o pe r atione . Et e ade m r atione � a g e n s non a g e t 2._ g r ad ­ um alb e d ini s ; e t pe r c o n s e quenc s e g it Q_ g r adum alb e dini s , e t ult r a i g itur al i quid e g it Q_ g r adum alb e dini s . . . . Ad s o p hi s ma dic itu r quo d e s t fal s um . Ad ar g u me ntum . . . dic o quo d non s e quitu r : Q g r adu s alb e dini s e s t ac tu s ab aliquo , i g itur al i qu i d a g i t Q g r adum . E t c au s a e s t qui a Q_ g r adu s e s t in ­ div i s ib il e e t in t a l ib u s non va l e t c on s e que nti a . . . . So r te s p e r t r an s i vi t a s p ac ium ; .po s i to illo c a s u quod � s p ac ium pe r t r an s e atur a S o rte . T un e ac c ipiatur p r i mu m in s tan s in quo So r te s att in g it ad fin e m � s p ac ium , e t s it nunc i llud i n s tan s g r at i a e xe mp l i . T une p r ob atur s ophi s ma s i c : � s p ac ium e s t pe r ­ t r an s itum a So r te , i g itur S o r t e s pe r tr an s i vit � s p ac ium . . . . Ad s op hi s m a dic itur quo d e s t fal s um . . . . Ad p r ob atione m dic itu r quo d non v al e t i l l a c o n s e qu e n ti a : � s p ac ium e s t p e r ­ t r an s itum a So r te , i g i tu r S o r t e s pe r t r an s ivit � s p ac ium , s e d b e ne s e quitu r : � s p ac i u m e s t pe r t r an s itum a S o rte , i g itur S o r te s pe r ­ t r an s i v i t ve l inc i p i t pe r t r an s i s s e � s p ac ium . . . . S o r te s p e r t r an s ib it � s p ac ium ; p o s ito quo d � s p ac ium e r it p e r ­ t r an s itum a S o r te . T une s i t .Q_ p r i mum in s t an s in quo S o r te s e r it in te r mino vel in fine � s p ac i i . T un e p r ob atu r s o p hi s ma s ic : � s p ac ium e r it p e r t r an s i tu m a S o r t e , e r g o So r te s pe r t r an s ibit � s p ac ium . . . . S o r te s inc ipie t pe r t r an s i r e � s p ac ium e t S o r t e s inc ipi e t pe r ­ t r an s i s s e � s p ac i um , e t non p r iu s i n c i p i e t S o r t e s pe r t r an s i r e � s p a c iu m qua m i n c i p i e t p e r t r an s i s s e � s p ac ium . . . . Hoc p r ob a t u r p e r e xp o ne n s i s t o r u m te r min o r um pe r t r an s i r e e t pe r t r an s i s s e . N a m non p r iu s in c ip i e t S o r t e s e s s e in p e r t r an s e undo � s p ac ium quam ind pi e t fu i s s e in p e r t r.an s e undo � s p ac ium , qui a s ine te mpo r e me dio po s tquam S o r t e s inc ipi e t mo v e r e s upe r � s pac ium e r it i n pe r t r an s e undo � s p ac ium ; e t s ine me dio po s tquam So r te s inc ipie t mo ve r e e r it ve r um quo d S o rte s fuit in pe r t r an s e undo � s pa c iu m . . . . A s p ac ium inc ipi e t e s s e pe r t r an s itum ; p o s ito quo d � s p ac ium dic atur p e r t r an s i tu m quando m ai o r p a r s illiu s fue r it p e r t r an s it a

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et s it a s p ac ium non pe rtr an s itum quando maio r p a r s i s tius fue rit �on pe rtr an s ita , e t ponatur quod aliquid inc ipiat move r i s upe r a s pac ium . . . . A inc ipit e s s e ve rum; p o s ito quo d � s it illa p r o p o s iti o : " A tan ­ git �• ; e t ulte r iu s ponatur quod � s it unum e o rum s pe r i c um mo ve n s s upe r al iquo d planum quod s it _£ e t tune s uppone ndo quod c s it nomen c o mmune p r ime par ti s p r opo rtional i s in � et te rtie pa rti s p r opo rtion ali s et quinte p a rti s p r o p o rtional i s et s e pti me parti s p r oportionali s et s ic in infinitum . . . . A e t Q_ e r unt ve r a . Et suppon atur i s te c a s u s quod � s it illa p r o ­ po s iti o " R e x s e de t" e t Q_ _illa " R e x c u r r it ;• e t quod r e x ne e s e de at ne que c u r r at n e e quod s e debit e t c u r r e t ( s i mul) . . . . A mo veb atu r c ontinue pe r ali quod te mpu s po st Q_ e t � non move tu r ; s umpto i s to c a s u quod b s it hoc in s tan s p r e s e n s e t s it � unum move n s volenta rium quod nunc inc ipit mo v e ri p e r ali quod te mpu s c ontinuum imme diatum Q_ in s tanti p r e die ti , e t quod in quo libet in stanti futu ro pote st c e s s a re a motu s uo . T une s ophi s ma e s t fal s um . . . e t p ote s t e s s e ve rum , e t nullum te mpus e r it lap s um ante quam s o phi s ma pote s t e s s e ve rum . . . . So rte s ita c ito de s ine t move r i s ic ut movebitu r ; po s ito quod So rte s inc ipiat move r i pe r ho r am et quod in quolib e t in s tanti illiu s ho re p o s s it So rte s de s ine r e move r i . . . . So rte s ita c ito e r it c o r r uptu s s ic ut ip s e me t e r it g e ne r atu s . Sit Q in s tan s ad unum annum e t s it So rte s unum inde te r minate futurum s ic ut Antic h r i s t u s qui in Q p r i mo e r it g e n e r atus et qui p o s t ve l ante Q_ pote s t gene r ar i et i g itu r p o s s it ante Q_ ve l in Q. c o r rump i . T une p rob atur s ophi s ma s i c : So rte s pote s t ita c ito e s s e c o r r uptus s i c ut ip s e met e r it g e ne r atu s . A inc ipit intende r e alb e dine m in aliqua p arte Q, e t que lib e t p ar s p r opo rti onal i s in Q_ s ine me dio r e mittitur . . A inc ipit alb e fac e re aliquam p arte m in Q_ e t nulla p a r s in Q_ e r it alb ior quam nunc e s t alb um . . . . A gene r abit albe dine m u s que ad £. p unc tum e t tame n nul l a alb e do e r it imme diata £. punc to ; s umpto i s to c a s u quod Q_ s it unum c o r pu s magnum e t quod � move atur s up e r ip s um Q_ pe r p arte m ante p a r te m c ontinue g ene r ando alb e dine m in Q. s ic ut mov e tur , e t move atur � s upe r Q g r atia e xe mpli pe r unam ho r am unifo r mite r ; tune p o s ito quo d c um � alb e fac i at p r imam par t e m in Q quod g_ s it unum nig re fac ie n s e t inc i p i at move r i s upe r Q_ c ontinue n i g r e do induc e ndo ve lo c iu s quam � mo ve tur . . . . � inc ipie t s imul e s s e di vi s um e t non di vi sum . . . . Supponitur i s te c a s u s quod � s it unum dividen s . . . e t dividat 2. c o r p u s pe dali s quantitati s g r atia e xe mp l i in una ho r a unifor mite r . T une c um divi s e rit me die tate m p ono quo d c s it unum c ontinuan s in ­ c ipie n s c ontinuare . . . e t s uppono qu;-d £. in dupl o ve loc iu s c on ­ tinuet quam � dividit in s e c unda me die tate ho re date . . . . A inc ipie t e s se divi sum a 2._; s upp o s ito i s to c a s u quod � e t 2._ s int due me die tate s alic uiu s c ontinui et quo d £. s it unum divide n s ita l ongum s ic ut � ve l � c uiu s ac utie n s s it s up e r fic i e s non l ine a , e t a s it inc ipiat £. divide r e � e t � pe r p ar te m ante p a rte m quou s que div i s um a 2._.

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2 6 A inc i p i e t e s s e p e r s e al b i u s 2..; po s ito quod ille s it pe r s e al ­ bum quod e s t alb um s e c undum s uam p arte m e t pr opte r hoc quod s ua par s e s t alb a . . . . 2 7 So r te s inc ipiet po s s e pe r t r an s i r e � s pac ium; po s ito quod � s it unum s p ac ium pe rtr an s e undum quo d So rte s non pote s t pe r t !" an ­ s i re , e t augmente tur sua pote nti a quo u s que So rte s po s s e t pe r ­ t r an s ir e !!, s pac ium c omple te e t non ulte r iu s augme nte tur po ­ t e ntia So rti s . . . . 2 8 A s pac ium inc ipi e t e s s e pe rtr an s itum a So rte ; p o s ito quod So rte s move atu r s upe r � s p ac ium in uno die p r e c i s e pe r t r an ­ s e undo � c o mple te e t s it � illa die s in quo pe rtr an s itur � c o m ­ ple te . . . . 2 9 So rte s movebitur s upe r aliquod s pac ium quando non hab ebit potent iam ad move ndum s upe r illud s p ac ium ; s uppo s ito i s to c a s u quod nunc habe at So rte s pote nti am ad pe rtr an s e undum me die tate m � s pac ii p r e c i s e et quo d So rte s p e r pote ntiam quam nunc h abe t in i s to me dia et c um ali i s p a r ib u s non p o s s it move r i ult r a . . . . T un e p ono quod c um Sorte s pe rtr an s e at p r imam me die tate m � s pac ii quo d pote ntia So rti s inc ipiat auge ri quous que So rte s habuit pote ntiam ad pe r t r an s e undum s e c undam me die tate m . . . . 3 0 So rte s p ote s t ita c ito de s ine r e ve lle � s ic ut valet �- . . . 3 1 So r te s movetur in duplo ve loc ior quam P l ato ; po s ito illo c a s u quo d P lato i n p r ima me die t ate huiu s die i move atur i n duplo ve loc iu s quam So rte s et in s e c unda me die tate huiu s die i mo ve a ­ tur So rte s in duplo ve loc ius P l atone , e t s it nunc s e c unda me di ­ e t a s huius die i . . . . 3 2 So r te s e t P l ato inc ipient e que v e l oc ite r move r i ; s uppo s ito i sto c a s u quo d So rte s move atur unifo r mite r pe r unam die m que sit .a. et pe rtr an s e at Q. s p ac ium in � die . T une s it Plato mino r i s po ­ tentie quam So rte s in p r inc ipio hui u s die i e t in p r in c ipio � die i inc ipi at move ri s upe r aliquod s pac ium e qu ale m Q s pac io quod s it £., et pe rtr an s e at P l ato £_ s pac ium in � die c ontinue intende n ­ do motum s uum . . . . 3 3 So r te s non mov e tur v e l o c i u s quam P l ato ; s uppo s ito i s to c a s u quo d S o rte s inc ip i at nunc move r i e t quod move atur s upe r � s p ac i um in aliquo te mpo r e imme di ato huic in stanti quod s it �' et quo d P lato s imilite r inc ipiat move r i pe rtr an s e undo £_ s p ac i ­ um e quale 2_ s p ac io . . . . Similite r pono quod So rte s e t P l ato nunc s unt e qual i s pote ntie g r atia e xe mpli s e d quando So rte s inc ipiat move r i s upe r s pac ium s uum . . . c ontinue deb il iu s ap ­ plic at s e ad mo ve ndum e t e c ontr a de P l atone quod P l ato magi s c ontinue applic at s e ad move ndum . . . . 3 4 So r te s mov e bitur ve lo c iu s qua m So rte s nunc mov e tur ; s uppo s ito i s to c a su quod S o rte s inc ipiat in p r inc ipio huiu s die i move r i s up e r !!_ s pac iu m c ontinue inte nde ndo motum s uum u s que a d hoc ins tan s pre s e n s quod s it Q., e t quod per totum r e s iduum die i po s t Q. in s tan s c ontinue t motum s uum pe r e qual e m applic ation e m. . . . 3 5 P l ato pote s t move ri unifo r mite r p e r ali quod te mpu s e t e que ­ ve l o c ite r s i c ut nunc movetur Sorte s .

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3 6 So rte s inc ipiet po s s e move r i � g r adu ve l o citati s ; s uppo s ito quad � s it unu s g r adu s ve loc itati s uni fo r mi s pe r que rn Sorte s al i ­ quando po s t ho c mov ebitur unifo r mite r e t qua d So rte s non pa s s it modo mo ve ri � g r adu v e lo c itati s ni s i pote nt i a s ua augme nte tu r et aug e atur pote ntia sua pe r ali quo d te mpus ante quam hab e b i t potentiam a d move ndum � g r adu ve loc it ati s . . . . Ad oppo s itum s i c : s i s ophi s ma s it ve rum , ve l igitur s ic qua d al iquod e r it pr imum instan s in quo So rte s hab e b it pote nti am ad move ndum � g r adu veloc itati s ve l s i c quad al iquod e s t ultimum in quo So rte s non habebit pote nti am ad move ndum � g r adu ve l oc i ­ tati s . . . . 3 7 So rte s inc ipie t po s s e move re � l apide m; s uppo s ito qua d � s it unu s magnu s l api s que rn Sorte s non pote s t move re et auge atur potentia Sorti s quou s que So rte s habe at pote nti am ad move ndum � lapide m . . . . 3 8 So rte s pate s t ita c ito habe re pote ntiam ad move ndum � l apide m s i c ut Plato habebit potentiam ad pe rtr an s e undum £ spac ium . 3 9 Pl ato pote s t inc ipe re e s s e fo rti s s imu s hominum qui s unt hie intus . . . . 4 0 A e t sua me die ta s s imul inc ipiunt c o r rumpi a � a g e nte . P ono qua d � s it una r e s inanimata c ui u s imme diate po s t hoc aliqua pa r s c or r umpitu r pe r � age n s . . . . 4 1 In infinitum fac ilius e s t fac e r e £ e s s e ve r um quam fac e r e s! e s s e ve rum; po s ito i s to c a s u quad .f.. s it i s ta p r opo s itio : " Alic uius parti s � que lib e t par s c o r r umpitur ;• et s it s! i s ta p r opo s itio : " Ouelib e t p ar s � c o r r umpitur " ; e t agat � in � c o r r umpe ndo . . 42 � fac iet £ e s s e ve rum; . . . po sito c as u p r e c e de nte . . . . 4 3 In infinitum fac iliu s e s t Q. fac e re quad i s t a p r o po s itio s it ve r a : " Infinite parte s � s unt pe r t r ans ite ;• quam fac e re quad illa s it ve r a : " T atum � e s t pe rtr an s itum" . . . . 44 In infinitum c itiu s e r it � ve rum quam � e r it ve rum; s u pp o s ito quad .f.. s it aliquod s p ac ium pe rtr an s e undum ab al i quo et quad in s! ins tanti inc ipiat aliqui s pe rtr an s i r e £ s p ac ium pe r parte m ante p arte m . . . e t s it � i s ta p r opo s itio : " Ali qua p a r s p r opo r ­ tional i s £ e s t pe r t r an s ita" . . . e t s it Q i s ta p r o po s iti o : " Que lib e t par s p r op ortional i s £. e s t pe r t r an s ita" . . . . 4 5 Tot parte s p r o p o r tionale s in � pe rtr an s ib it S o r te s s ic ut P l ato ; po s ito i s to c a s u quad � s it unum pe rtr an s e undum a So rte c ui u s p r i ma p a r s pr opo rtional i s s it alic uiu s diffic ultati s . . . . 4 6 T u s c i s hoc e s s e o mne quad e s t hoc ; s uppo s ito qua d tu vide s So rte m a te r e motum e t ne s c ia s quad sit Sorte s . . . . 4 7 Tu s c i s hoc e s s e S o r te m ; po s ito quad tu vide a s s imul So r te m e t P l atone m e t quad S o r te s e t P l ato s int o mnino s i mi l e s . . . . 4 8 Tu s c i s r e g e m s e de r e ; s uppo s ito i s to c a s u quad s i r e x s e de at tu s c i s r e ge m s e de re e t quad s i r e x non s e de at quad tu non s c i a s r e g e m s e de r e . . . . 4 9 A e s t s c itum a te ; suppo s ito quad � s it alte r a illarum . . . .

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l S e e B e rtr and R u s s e ll , T he P r inc i P. le s of Mathe matic s (2nd e d . ; N e w Y o r k , 1 9 3 7 ) , p p . 1 0 1 - 7 , 5 2 3 - 2 8 . The mo s t r e c e nt inte re st in the l i a r p a r adox w a s initiat e d by Alf r e d T ar s ki , " Die W ahr ­ he it s be g r iff in d e n for mal i s ie rten Sp r ac he n ;• Studia Rhilo s oRhic a , I ( 1 9 3 5 ) , 2 6 1 - 4 0 5 . On the me die val analy s i s o f the liar paradox , s e e E rne s t A . Moody , T r uth and Con s e q� in Me diae val Logic (Am s te rdam , 1 9 5 3 ) , pp. 1 0 3 - 1 0 . 2 " Loyc a He sb r i valde c o r r e c ta ;• Amp lonian F o l . MS . 1 3 5 , fol l . 1 1 7 . Inc iRi t : " Se c undum philo s o phum in p r e dic ament . . . . " E xRlic it : .. . . . nar r ac ibni s p r ol ixita s , he c dic ta s uffi c iant . Explic it quide m t r ac tatu s optimus datus Oxonie a ma g . W ilhe lmo de Hytthi sbyri a . D . M C C C X X X V :' In the margin o f fol . 1 , in the hand o f the s c r ibe , the r e appe ar s the followin g : " In no mine Domini , ame n ; inc e ptum e s t hoc opu s in f e sta Magdal e ne a . D . M C C CXXXVII , litte r a do mini c al i . . . . "- B e s c h r e ib e nde s Ve r ze ic hni s s de r Amp ­ l oniani s c he n Hand s c hr ift e n -Sammlun g zu E r furt (Sc hum , 1 8 8 7 ), pp. 88-89. 3 G e o r ge C . B r odr ic k , Me mo r ial s o f Me rton C olle g� (Oxfo r d , 1 88 5 ), p . 2 0 7. 4 Anthony W ood , Hi s to r y and Antiguitie s of Oxfo rd , Colle ge s and Hall s , e d. Gutc h (O xfo rd , 1 7 8 6 ) , p . 1 3 9 . 5 B r o d r i c k , Me mor ial s of Me rton �olle g�, p . 2 0 7 . Little e l s e i s kno wn of He yte sbury' s life . W e note th at hi s name ( s pe ll e d indiffe re ntly " He the lbury ;• " He gte rbury ;• and " He gte lbu r y" ) appe ar s s e ve r al time s in the r e po rts on the s c ru ­ tini e s o f M e r ton C o lle ge i n 1 3 3 8 - 3 9 . T he s c r utinie s we r e me e t ­ ing s he ld thr e e time s a' ye ar b y the w a r den and fe llow s t o in ­ qui r e into th e g e ne r al c ondu c t o f the hou s e and the b e havior of it s me mb e r s ; o r dinar ily no minute s we re take n , and the s c r u ­ tinie s o f 1 3 3 8 - 3 9 are the only one s o f whi c h a r e c o r d has b e e n p r e s e r v e d . - J ame s E . T ho r old R o ge r s , A Hi story of _Ag ric ultur e and P r ic e s in E n g l and , II (Oxfo rd , 1 8 66 ) , 6 7 0 - 7 3 . --" Ma g i s te ;-W ille l mo de He ghte rb ury" i s al s o name d a s an e xe c ut o r and be ne fic iary in the will of Simon de B r e don , C anon of C hi c he s te r ; the will was w r itte n in 1 3 6 8 and pr o v e d in 1 3 72 , and He yte s b u r y' s p a rt of the le g ac y include d twe l ve s ilve r s po on s , a numb e r o f l aw b o o k s , and B re don' s b e d .- F . M . Pow -

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ic ke , T he M e die val B oo k s o f Me r ton Colle g� (Oxfo r d , 1 9 3 1 ) , pp . 82 - 8 6 . Se e M ar s hall C l a g e tt , Giovanni Marliani and Late Me die v al P hy s ic s (Ne w Y o r k , 1 9 4 1 ) , pp . 1 3 0 ff. F o r a de s c r i ption o f the c o nte nt of thi s wo r k , s e e Lynn T ho rn ­ dike , A Hi sto ry o f Magic and E xRe rimental S c i e nc e , III (Ne w Yo r k , 1 9 3 4 ) , 3 7 0 ff. It s date o f c ompo s ition c annot b e fixe d , b ut it p r ob ably s te m s fr o m the thi rtie s o r fo r tie s o f the fourte e nth c e ntur y . Cf. Anne lie s e Maie r , An de r G r e n z e von Sc hola s tik und Natu r wi s s e n s c haft (E s s e n , 1 9 4 3) , pp . 3 5 5 - 5 6 . On the p r o b le m of the ide ntity of Sui s e th , s e e C la g e tt , Giovanni Mar liani , Appendix I. T r a c t atu s de s e x inc onvenie ntibu s , M a r c . Latin MS . V III , 1 9 , fo ll . 6 6 - 1 4 5 ; p rinte d at V e nic e , 1 5 0 5 with the Que s tio de Modalib u s of B a s s anus P olitu s . On t h e inc o r r e c t att r ib ution o f thi s wo r k t o J ohn o f Holl and i n the inve nto rie s o f both the V atic an and the M a r c iana , s e e Maie r , An de r G r e n z e von Sc hol a stic und N atur wi s s e n s c haft , p p . 2 6 6 - 6 7 . P ar i s B N Latin M S . 1 6 1 4 6 ; V at . Latin MS . 9 54 ; V at . Latin M S . 6 7 5 0 ; Pal . Latin M S . 1 0 5 6 ; B ib l . Antoni ana d i P adov a , Sc aff. 1 7 , MS . 3 7 5 . Dumb le ton c an be date d at Me rton C olle g e from 1 3 3 1 to 1 3 4 9 . Mo r e p ar ti c ul a r l y t o C h apte r IV , .. De inc ipit e t de s init ." Kil ­ mington (o r Clienton , o r C lide nton , o r Kylve nton , e tc .) w a s a fe llow o f Me rton (Powi c ke , Me die val B oo k s of Me r ton , p . 2 5 ) and De an of St . P aul ' s C athe d r al fr o m 1 3 5 3 - 6 2 . Manu s c r ip t s of the .. Sophi s mata" c ar ry n o date (Vat. Latin M S . 3 0 6 6 , foll . 1 6r a - 2 5 vb ; V at . Latin M S . 3 0 8 8 , foll . 3 7r a - 6 l vb ; B ih l . Uni v . d i P adov a , MS . 1 1 2 3 , fol l . 6 5v a - 7 9 v a ; B o dle i an C anon . Mi s c . M S . 3 7 6 , fo ll . l r a - 2 0 r a [ anon .]; inc o mple te in B odle i an C anon. Mi s c . M S . 4 0 9 , foll . 9 9 r a - 1 0 9 v a [ anon .] , and in Vat . Latin M S . 4 4 2 9 , foll . 4 5 r a - 62 va [anon .] ). A l i s t o f Kilmington' s s ophi s mata is g i v e n in Appendix B . B e s ide s the autho r s j u s t c ite d , Powic ke n ame s M anduit , A s h e n ­ de n , R e de , B re don , and C am s al e a s me mb e r s of the Me rton c i r c le of thi s time .- M e die v al B ooJ{ s o f Me rton , p. 2 5 . In the V e ni c e e dition ( 1 5 0 5 ; no p agination): .. s ic ut p onit tota s c hola auxonie n s i s" ; .. te rtia opinio t e ne t tota s c hola auxoni e n s i s :• W e note that the r e i s a t r e ati s e b y B r adwa rdine e ntitle d .. De inc ipit et de s init'' (Vat. Latin MS. 2 1 54 , fo ll . 2 4 rb - 2 9 v a) whi c h w e have unfo rtunate ly b e e n unabl e t o e xamine , and upon whic h He yte sbury may h ave de pe nde d in w r iting the fourth c h apte r o f the R e gule . P a r i s B N Latin MS . 1 6 6 1 7 . Etie nne Gil s on , La P hilo s oRhie au moy:e n �g� ( 3 r d e d. ; P a r i s , 1 94 7 ) , p . 5 5 5 . "Supp o s itio aute m e s t ac c e ptio te r mini s ub s t antiv a p r o aliquo :• - P e tr i H i s Rani Summulae l o g i c ale s , e d . I . M . B oc he n s ki ( T u r in , 1 94 7 } , p . 5 7. .. Suppo s itio e s t ac c e ptio te r mini in p r o p o s itione p r o a l i quo ve l pro aliquib u s :' - P aul o f V e nic e , Logic a (P avi a , 1 4 8 0 ; no p agination). 1 70

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1 7 B y s o me no minali st s , e . g . , B ur idan ( C o mP.endium lo gic e [Venic e , 1 4 9 9 ; no p agination] ) , s upP.o s itio si mP.le x i s inc lude d unde r s up ­ P.O s itio m ate riali s . P e te r o f M antua � gic a [ Ve nic e , 1 4 9 2 ; no p a g ination] ) de nie s the p ro p rie ty of both s uppo s itio mate r i ali s and s uppo s itio s imple x , on the g r ound that a te r m s tanding in one of the s e mo de s i s not a g r ammatic al part of s pe e c h . 1 8 Ac c o r din g to M oo dy (T ruth and Con s e que nc e in M e diae val Logic , p p . 2 2 - 2 3 ), s up po s iti on i s a s yntac tic al re lation of te rm to te r m and s hould n o t b e c onfu s e d with the s e mantic al r e l ation of the te r m to an e xt r a - lin g ui s tic " ob j e c t" or " de s i gnatum:• 1 9 F o r the te xt , s e e J . P . M ull ally , The Summulae l o g ic ale s o f P e te r of �ain (N otre Dame , Ind . , 1 9 4 5 ) , p p . 1 04 ff . B oc he n s ki (Pe tr i Hi s pani Summulae lo gic ale s , p . xiv ) doub t s P e te r' s au ­ tho r s hip of thi s c hapte r . T he c omme ntato r s of the l ate r middle a g e s , howe ve r , appe ar to have re g ar de d it as authe ntic ; thu s J o anne s de M onte (E xpo s itio s ummula r um [ 1 4 9 3 ] ) and Ve r s o ri u s (Expo s itio s upe r s ummuli s [N aple s , 14 7 7 ] ). Analy s i s o f e xponi ­ ble te r m s i s a l r e ady found in the " Intr oductione s in lo gic am" o f W i lliam o f Shyr e s wood , and i s the r e fo r e p a r t and p a r c e l of the l o g i c a mode rna in its e arlie s t fo r m . 2 0 " T r ac t atu s d e v e r itate e t fal s itate p ropo s itioni s ;• B ih l . N az . di F i r e n z e , C l . V , M S . 4 3 , fo l l . 1 r ff. ; Vat. Latin M S . 3 0 3 8 , foll . Ir - 1 3r. 2 1 T hu s Simon de Le nde na r ia , " R e c olle c ta s up r a s ophi s matibu s Henti s be r i" : " O mni s p r o po s itio e s t p r ob anda r atione p rimi te r ­ mini p r ob ab i li s :' - 1 4 9 4 e dition of He yte sbury , fol. l 74va. C f. G ae tano di T hie ne , " R e c olle c te s upe r s ophi s matib u s He nti sb e r i ;' ib id. , fo l . 9 l r a . 2 2 Se e A r i s tote l i s QP.e r a o mni a , e d . M aur o (Par i s , 1 8 8 5 ) , I , 5 74. 2 3 P a r i s B N L atin MS. 1 4 7 0 8 , fol l . 6 5 r a - l 1 2 vb . 2 4 Imp r e s s um pe r Otinum P apie n s e m , 1 4 9 6 . 2 5 Edition o f 1 4 9 3 ; no typo g r aphic al indic ati on. 2 6 C o mP.e ndium l o g i c e (Venic e , 1 4 9 9 ). l 7 P ar i s BN Latin M S . 1 4 7 1 5 , fol l . 7 9 r a - 8 2 rb . 2 8 F o u r suc h c o mme nt a rie s we re p r inte d at Ve ni c e in 1 5 0 0 - 1 5 0 1 by J ac ob u s P e ntiu s de L e uc o . T he y are : the E xpo s itio of Ale x ­ ande r Se r mone te ; the QP.u s c ulum de s en s u c o mP.o s ito e t divi s o of B e r nar dinu s P e tr u s d e Seni s de Landuc ii s o r dini s C a r me litar ­ um , w r itte n while he wa s gi ving the o rdinary le c tur e s in phi lo s ophy at the Uni ve r s ity o f S i e n a ; the E xP.o s itio of P aul of P e r go l a (d . 1 4 5 6 ); and the E xP.o s itio o f B apti s t a d e F ab r i ano . 2 9 B e rnar dinu s P e tr u s de S e ni s de Landuc ii s , QP.u s c ulum de s e n s u c o mP.o s ito e t divi s o (Ve nic e , 1 5 0 0 ) , fol . 2 vb ; Ale xande r Se r mon ­ e te , E xpo s itio (Ve ni c e , 1 5 0 0 ) , fo l . 5 v a ; B apti s ta de F ab r i ano , E xP-o s itio (Venic e , 1 5 0 0 ), fo l . 4 r a . 3 0 T he que s tion whe the r ve rum and fal s um s ho uld b e inc lude d amon g the modal te r m s c au sing c o mpo s ite and divi s ive s e n s e i s deb ate d in the Que s tio d e modalib u s o f B a s s anus P olitu s ( V e nic e , 1 5 0 5 ). 3 1 He yte s b u r y , De s e ns u c omP.o s ito e t divi s o , foll . 2 rb - 2 v a ; B e r nardinu s P e tr u s d e Seni s d e Landuc ii s , QP.u s c ulum (Venic e ,

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1 5 0 0 ) , fo ll. 3 r a - 7 v a ; Ale xande r Se r mone te , E xpo s itio (Ve nic e , 1 5 0 0 ) , fo l l . 6 r a - 7va ; P aul of Pe r go l a , ExP.o s iti o (Veni c e , 1 5 0 0 ) , fol . l r a - l rb . He yte sbury , De s e n s u c o mP.o s ito e t di vi s o , fol . 2 v a ; B e r na r dinu s Pe tru s de Seni s de Landuc i i s , QP.u s c ulum ( V e ni c e , 1 5 0 0 ) , fo ll. 7vb - 9 vb ; Ale xande r Se r mone te , E xpo s itio (Ve nic e , 1 5 0 0 ) , foll . 7vb - 9 va; P aul of P e r g o l a , Expo s iti o (Venic e , 1 5 0 0 ) , fol . l rb . He yte sbu ry , De s e n s u c o mP.o s ito e t divi s o , fol . 2 va - 2 vb ; B e rna rdinu s P e t ru s de S e ni s d e Landuc ii s , Opu s c ulum (V e nic e , 1 5 0 0 ) , fo ll. 9 vb - 1 2 vb ; Ale xande r Se rmone te-:- ExP.o s iti o (Venic e , 1 5 0 0 ) , fo ll . ' 9va - l l rb ; P aul of Pe r go l a , Expo s itio (Venic e , 1 5 0 0 ), fol . l va - lvb . He yte sbury , De s e n s u c o mp o s ito e t di vi s o , fo ll . 2 vb - 3 r a ; B e r nardinu s Pe trus de Seni s de Landuc ii s , Q pu s c ulum (Venic e , 1 5 0 0 ) , fol l . 1 2 vb - 1 5 rb ; Ale xande r Se r monete , Expo s itio (Ve nic e , 1 5 0 0 ) , fo ll. l lrb - 1 3 r a ; P aul of P e r gol a , E xP.o s itio (Ve nic e , 1 5 0 0 ) , fol l . l vb -2 r a. He yte sbury , De s e n s u c o mP.o s ito e t divi s o , fo l . 3 r a - 3 rb ; B e r na rdinus Pe tru s de Seni s de Landuc i i s , Q p u s c ulum (Ve ni c e , 1 5 0 0 ) , fo l l . l 5rb - 1 7vb ; Ale xande r Se r m one te , E xP.o s itio (Ve ni c e , 1 5 0 0 ) , fol l . 1 3 r a - 1 4 rb ; P aul of P e r gol a , E xP.o s itio (Ve nic e , 1 5 0 0) , fol . 2 r a - 2 rb . He yte sbur y , De s e n s u c ompo s ito e t divi s o , fo l . 3 rb . T he l ate r c o mmentato r s me r g e d type s (e ) and (f ) into one . Se e N ote 3 5 fo r r e fe r e nc e s . He yte sbury , De s e n s u c omP.o s ito e t divi s o , fol . 3 rb ; B e r n a r dinu s P e tru s de S e ni s de Landuc i i s , Q P.u s c ulum (Ve nic e , 1 5 0 0 ) , fol l . l 7vb - 2 Orb � Ale xande r Se r monete , E xP.o s itio (Venic e , 1 5 0 0 ) , fol l . 1 4rb - 1 6r a ; P aul of P e r gola , E xP.o s itio (Venic e , 1 5 0 0 ) , fol . 2 rb -2 va. He yte sbury , De s e ns u c o mP. o s ito e t divi s o , foll . 3 rb - 4 rb ; B e r nardinu s P e t ru s de Seni s de Landuc i i s , Q pu s c ulum (Venic e , 1 5 0 0 ) , fo l l . 2 0 rb - 2 3 vb ; Ale xande r Se r monete , E xpo s itio (Ve nic e , 1 5 0 0 ) , foll . 1 6rb - 2 2 vb ; P aul of P e r go l a , E xP.o siti o (Ve nic e , 1 5 0 0) , fol l . 2 v a - 3 vb . A ninth p o s s ib l e type o f c ompo s ite and divi s ive s e n s e i s di s tingui s he d b y Heyte s b ur y , b ut i s di s mi s s e d a s pe r taining r athe r to fallac y o f fi g u r e o f dic ti on than to fal l ac y o f c ompo s ition and divi s ion.- De s e n s u c omP.o s ito et divi s o , fo l . 2 rb . Chapte r 8 , 1 0b 2 6 e t �qq. T he Ar i s to t e l i an autho r s hip o f the C ate g o r i e s h a s b e e n doubte d . Anne lie s e Maie r , D a s P r ob le m de r inte n s iv e n G r o s s e in de r Sc hol a s tik (Le ipz i� 9 3 9 ). " Libe r s e x p r inc ipio rum" (a s c rib e d to Gilb e rtu s P o r r e tanu s ) , .Qpu s