Tractatus de Proportionibus

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THOMAS OF BRADWARDINE HIS TRAC-­ TATUS DE PROPORTIONIBUS

Thomas of Bradwardine · His 'Tractatus de Proportionibus Its Significance for the Development of Mathematical Physics

Edited and Translated by

H. LAMAR CROSBY, JR.

The University of Wisconsin Press MADISON

-

1961

Published by the University of Wisconsin Press 430 Sterling Court, Madison 6, Wisconsin Copyright © 1955 by the Regents of the University of Wisconsin Second printing , 1961 Printed in the United States of America by Cushing-Malloy, Inc., Ann Arbor, Michigan Library of Congress Catalog Card Number 54-6740

Ofttd

CARO PATRI qui primus mihi orbis Latinae portas aperuit

FOREWORD The rehabilitation of medieval mechanics first started by Pierre Duhem after the turn of the century and successfully carried on by the recent studies of Annaliese Maier now founders on one main obstacle, namely, poor or inadequate texts. Recently the Univer­ sity of Wisconsin Press published a corpus of medieval statical works edited by Ernest Moody and myself, The Medieval Science of Weights. It now undertakes to publish one of the most widely influential works in medieval dynamics and kinematics. Bradwardine's work performed a crucial service to the develop­ ment of mechanics, for in it we find the juncture of two important traditions of mechanics, the philosophical and the mathematical. It is generally recognized that Bradwardine stands in the Aristo­ telian scholastic tradition present at Oxford in the early fourteenth century in the giant figures of Duns Scotus and William Ockham. Less clearly recognized is the fact that Bradwardine draws from the revived Hellenistic mathematical tradition, represented in me­ chanics by the statical corpus of Jordanus and Gerard of Brussels' kinematic work entitled Liber de motu. It was a re-examination on mathematical grounds of Aristotle's rules relating forces with distances and times that stimulated the writing of Bradwardine's Treatise on Proportions. As the result of that re-examination Bradwardine arrived at a new mathematical way of relating the variables involved in movement. As Dr. Crosby shows, Bradwardine related velocity exponentially to the ratio of force and resistance producing the movement. He did this prima­ rily to save Aristotle's rules from what appeared to be mathemat­ ical inconsistencies. From the standpoint of empirical science Bradwardine's "law" was without even approximate verification. But his law had great consequences, for it appears to have been the first attempt to present a mechanical equation which repre­ sented instantaneous changes rather than completed changes as are involved in Aristotle's rules. It brought into prominence the idea of instantaneous velocity and led to rapid developments in kin­ ematics by his colleagues and students at Merton College. It led particularly to a sound description of acceleration and a celebrated rule for representing uniform acceleration by its mean speed, so far as the traversal of distance in a given time is concerned. This rule, perhaps known by Bradwardine himself, was given manifold expression during the 1330's and 1340's by his junior contempo­ raries at Merton College: William Heytesbury, Richard Swineshead, Vll

Vlll

FOREWORD

and John Dumbleton. It was, of course, later used by Galileo and applied to the motion of falling bodies. It is well known that Bradwardine's treatise was widely read at Paris, and, in fact, throughout Europe in the second half of the four­ teenth and in the whole of the fifteenth century. In his stimulating introduction Dr. Crosby suggests that Dumbleton interpreted the Bradwardine dynamic formulation in a manner not unlike Newton. This I believe to be incorrect, but I urge the reader to examine the evidence for himself. Considerable thanks are due to Dr. Crosby for this edition. The number of manuscripts of the text is large and presented a real problem in the establishment of a sound text. The inclusion of a translation makes the text accessible for those readers competent to evaluate the "scientific" content of the text, but unable to get be­ hind the medieval Latin. It is hoped that Dr. Crosby's efforts will encourage students of medieval mathematics and mechanics to undertake the publication of other monuments in the still little known area of medieval sci­ ence. Marshall Clagett Professor of the History of Science University of Wisconsin

ACKNOWLEDGEMENTS

I wish, first of all, to express my deep indebtedness to Profes­ sor Ernest A. Moody, of Columbia University, who not only con­ ceived the plan of this study, originally, but whose selfless and untiring help in every department of its development alone made the fulfillment possible. Sincerest thanks are also due Professor Paul 0. Kristeller, of Columbia University, to whom I owe what­ ever knowledge I have acquired of mediaeval paleography and whose meticulous criticism of the Latin text and variant readings was of particular assistance. My thanks too,to Professor Marshall Clagett, of the University of Wisconsin, for his many useful criti­ cisms of both thought and expression, and to Mrs. Loretta Freiling for her expert and painstaking typewriter composition of this manu­ script for photographic reproduction. Not least, I wish to thank my wife, Louise; her part has been so all-embracing that it would not be improper to say that this book is as much hers as mine.

CONTENTS INTRODUCTION 8

Editions and Manuscripts The General Significance of the De Pro2ortionibus

11

Descriptive and Critical Analysis

18

THOMAS BRADW ARDINI TRACTA TUS PROPORTIONUM SEU DE PROPOR TIONIBUS VELOCIT ATUM IN MOTIBUS

57

Prolegomena

57 61

The Text The Translation

64

Proemium

65

Introduction

66

Capitulum Primum

67

Chapter One

86

Capitulum Secundum

87

Chapter Two

110

Capitulum Tertium

111

Chapter Three

124

Capitulum Quartum Chapter Four

12 5

Sigla

145 146

Variant Readings

179

Appendix

180

Notes

195

Bibliography

201

Index

Xl

INTRODUCTION

INTRODUCTION I.

THOMAS OF BRADWARDINE

As is so frequently true of mediaeval authors, very little is known of the life of Thomas Bradwardine.1 Yet he was a man whose con­ temporary prominence as a mathematician and philosopher was even exceeded by his fame as a theologian, churchman , and mem­ ber of the King of England's retinue during the many years of Edward Ill's endless campaigns against the French. Though his birthplace has been variously assigned as Bradwar­ dine, Hertfield, and Cowden, he himself says he was born in Chi­ chester.2 There seems to be no good reason to doubt his statement, and it may have been at Chichester that Bradwardine first began what was to become a most fruitful acquaintance with Richard of Bury (later Bishop of Durham), who held a prebendal stall at Chi­ chester Cathedral early in the fourteenth century. 3 The date of his birth is yet more obscure; it is usually assigned by his biographers as c. 1290. In any event, one of the earliest gen­ uine records establishes him as having attained the rank of Proctor at Merton College, Oxford, in the year 1325. • This record tells of what now seems an amusing squabble be­ tween the mayor and the chancellor of Oxford, commencing in 1325 and lasting into 1326. The dispute was concerned with the moving of a certain pillory without the chancellor's consent, and (the record relates) in January, 1326, the mayor appeared before the chancel­ lor's commissary and the proctors, begging pardon for moving the pillory and appealing the excommunication laid on him by the chan­ cellor. Thereupon, all the litigants (presumably Bradwardine among them), together with a great crowd of townspeople, went to the pil­ lory, and decision was taken to place it six feet nearer the North gate than the mayor had placed it. "At that time, the proctors of the University were William of Harrington and Thomas of Brade­ wardyn. And [the recording notary adds] the pillory was moved within fifteen days:' 5 It seems perhaps ironic, and yet somehow not altogether deplorable, that this earlie st attested fact in the life of a great man should touch upon a matter at once so trivial and at the same time so perennially human, in its careful recording of official foolishness. Galhardus de Mora was archdeacon of Oxford at that time, and Bradwardine again took part, in behalf of the University, in a lengthy litigation with him concerning the question of where the spiritual authority over students at the University properly belonged. E� ward III finally submitted the case to English judges and obtained

4

INTRODUCTION

autonomy for the University from episcopal interference . 6 During his residence at Oxford, Bradwardine became a promi­ nent figure through his work in mathematics, astronomy, geometry, moral philosophy, and theology, and this appears to have been the period of composition of all his works. 7 His magnum opus, the De causa Dei, though presented in its final form in 1344 8 (when Brad­ wardine was no longer at Oxford), presumably represents an elal:r oration upon the theological lectures he gave while at Merton College. 9 It was for this last work (a volume of some nine hundred folio pages in Sa vile's edition) that Bradwardine was best known to his own and succeeding generations, and the De causa Dei long remained a touchstone of authority for Augustinian and Calvinist theologians. 10 It is also recorded that during Bradwardine's years at Oxford, "at the time w hen John Bacondorpius returned from Paris,he [Brad­ wardine] held with him a famous disputation concerning foreknowl­ edge and freedom of the will. But that, out of reverence for the old theology, and already many years versed in studies of that kind , he broke off the argument and condescended to agree with his [Bacond­ orpius'] point of view. Neither doctrine was contrary to faith, both being freely permitted by the Church:' That Bradwardine not only attained considerable reputation as a theologian while still within the University, but later became known to the world at large is in­ terestingly attested by a pas sage from Chaucer's Canterbury: Tales. In his Nun's Priest 's Tale, composed some time toward the close of the fourteenth century, Chaucer flatteringly couples Bradwar­ dine's name with those of Boethius and St. Augtistine .12 In remarking the distinction between "simple" and "conditional" necessity and the difficulty of understanding it, the Nun 's Priest speaks as fol­ lows: But I ne kan nat bulte it to the bren, As kan the hooly doctour Augustyn, Or Boece, or the Bisshop Bradwardyn As might be expected, Bradwardine 's contemporary reputation as a mathematician and natural philosopher is better shown by the widespread influence which his treatises exerted in succeeding generations (especially in the work of the English "Calculatores"1� than by the vague generalities of annals. It is interesting to note , however, that the significance of the De proportionibus was widely enough felt for it to have been made required reading for the B.A . degree at the University of Vienna and at Freiburg University by the close of the century.1 4 The date of Bradwardine's departure from the University is not known, but presumably it was not much before 1335, in which year he was called to London by Richard of Bury.15 Bury made him his own chaplain, obtaining for him the chancellery of St. Paul 's Ca-

THOMAS OF BRA.DWARDINE

5

thedral, together with the prebend of Cadington Minor,these being in addition to the prebendal stall at Lincoln Cathedral which Brad­ wardine had accepted only two years previously.16 Bury, who continued to play a not inconsiderable part in Brad­ wardine's fortunes, seems to have been a powerful and picturesque figure. He had been madeBishop of Durham in 1333 and chancellor in 1334,17 enjoyed having a large retinue of scholars about his per­ son, and tookBradwardine into his household as a protege. William de Chambre, in a contemporary account, speaks of Richard of Bury as the "learned Bishop of Durham;· a great bibliophile and patron of the arts, "and his house so full of books that a person had scarcely space to stand in or enter his room without kicking a­ gainst them:· 18 Richard, he tells us, died in 1345, after a full and gaudy career. "He was much pleased with a multitude of clerics and always had several in his menage. Among these were Thomas Bradwardyn (later Archbishop), Richard Fitz Raufe (later Bishop of Armagh) , W�lter Bury, John Maudyt , Robert Holcote and Richard Kilwington-all Doctors of Theology.... On any day there was or­ dinarily a reading at the table,unless perchance it was prevented by something important, and after dinner he used to hold disputa­ tion with selected clerics and others of his house, on appointed days�'19 Such chance scraps as the above are all that remain to us in record of Bradwardine's personal life,yet they do breathe some individuality into a figure now so remote. It was again Bury who, together with the archbishop, Stratford, obtained forBradwardine the post of chaplain-confessor to the king. He joined Edward Ill's court in Flanders,and, on August 16, 1338, was in the king's company in his progress up the Rhine to confer at Coblenz with his brother-in-law, Lev.is of Bavaria. At Cologne, Bradwardine is said to have reminded Edward that Richard Coeur­ de-Lion had there given thanks for his escape from the Duke of Austria and, the present cathedral being then in the process of con­ struction ,persuaded the king to contribute the sum of £1,500 (mod­ ern) to the project . 2 0 Contemporary opinion was that Bradwardine' s holy presence and virtuous influence greatly assisted the English army in its vic­ tories at Crecy, Calais, and Neville's Cross, after which he also served as peace emissary to King Philip of France.21 From this same period of Bradwardine's service with the armies dates the one extant letter of his,addressed to his friends in Lon­ don; it lends an authentic touch to the history of these years when Bradwardine was rounding out the political and military phase of a varied and illustrious career. In July, 1346, he writes: You must know that on the twelfth of July we made a good attack on a certain Norman port, called "le Hoghes;' near Barflete. There my lord, the king, together with many armed men disembarked and

6

IN TRODUC TION

bestowed military orders on his son (making him chief lord), on Lord Roger Mortimer, Lord William Montague, and a host of others. He also, as head of the army, bestowed afterwards the belt of Knight­ hood on many more. Our exceedingly small force won, thereafter, frequent victories from the enormous multitude of the enemy, killing and capturing many, seizing quite a large amount of booty, and win­ ning the battle completel y; so that in the circumad iacent countryside, for a distance of twenty miles and more, there was not a man left who stood against us. In this same place where we attacked we stayed till the following Monday, that is, St. Kenelm's Day. On that day de­ cision was taken in the king's council to retreat the following day and turn against the major Norman states and thence, God leading us, to France at last. Written at Hoghe s, St. Kenelm's Day. 22 Presumably this letter went to his friend and patron, Bury, who, in turn, relayed this news from the fighting front, as he had before in a letter of his own, dated July 3, 1340. On that occasion, writing to the prior and convent of Durham, he tells of just receiving news from Bradwardine , telling of another great victory of the English armies . 23 From this point onward the records connected with Thomas Brad­ wardine's life increase in number. In 1347 he was made archdeacon of Norwich and in 1348 was elected archbishop of Canterbury to fill the post left vacant by Stratford's death on August 23 of that year. Owing, apparently, to his anger that the clergy should have pro­ ceeded to elect an archbishop in his absence and without his approv­ al, rather than to any dislike of Bradwardine, Edward III thereupon requested Pope Clement VII to appoint John Ufford, then a very old man, in his stead. This Clement did (lending , by the act, additional pungency to the incident atBradwardine' s subsequent consecration). But Ufford died of the black plague, which 1?Y then ha.cl reached its height in England, and Bradwardine was reappointed with royal sanction and consecrated at Avignon on July 19, 1349, Clement is­ suing a bull in confirmation of the election. 24 William de Dene, in his Hist_g_ria Roffensis,Z 5 gives a contem­ porary account of these most confusing events . After recording of John of Ufford that he was a man �__e_t P-araliticus who died elected but before his consecration, he writes as follows: At that time the Canterbury chapter elected "Magister Thomas Bradewardyn" ... for a second time, having previously elected him shortly after the death of John of Stratford, and even before the Pope heard of the election he gave the Archepiscopacy of Can­ terbury to Magister Thomas in the accustomed manner, not wish­ ing to have less thanks than would the electors. The king ... who formerly had opposed it, endorsed the election, but, sad to tell, the archbishopric was thus destroyed by so many dispensations, by an evil guardian, and by the actions taken by the Roman Curia. So that even today the harm is not thought to be reparable

T H O M A S O F B R ADW A RDI N E

7

T he accounts contained in the lndiculus de Successione Archepis­ coporum Cantuariensium and Birchington' s Vitae Archiq�iscopor­ um do not differ substantially from the above . 2 6 Birchington' s account of the insult offered to Bradwardine and to the English king , at the time of the farmer's consecration as arch­ bishop, is, however, interesting, providing a documentary note on the tension and ill will existing between England and the Avignon papacy at that time. " In the same year (13 49), during the Vigil of St . Margaret, Thomas being at Avignon, was there consecrated in the church of the Minorites. On the day of consecration, however, Hugo, cardinal of Tulle, and blood brother of the pope, while the archbishop [ Thomas] was seated at table, brought great shame on him by someone riding in on a donkey, at which the rest of the car­ dinals were wickedly pleased :' 2 7 Presumably this clum sy joke against T homas was intended to indicate the opinion that Edward, military and political master of northern Europe, could send whom he would to the pope for consecration, even a donkey. At any rate , apologies were made to Bradwardine for the unhappy incident . 2 8 Birchington ' s narrative concludes by telling how Bradwardine, returning to England to take up his new duties, was struck down by the plague, scarcely one month after his consecration . " Then, as is customary , having had a visit with the pope and cardinals, and embarking on his journey to England, he arrived on Wednesday the nineteenth of August at Dover and on that day was a guest at the castle, going forth from there by way of Chertham and , in the fol­ lowing days, coming through Dartford and , on Saturday of the Vigil of the Assumption of the Blessed Mary, coming to the king at Eltham, where he received the temporalities. The same day arriving at the manor of the bishop of Rochester at Lambeth, he was a guest there during Sunday , the Feast of the Assumption of the Blessed Mary, Monday, Tuesday and the ensuing Wednesday ; on which Wednesday he died there, and on Saturday, the Feast of the Beheading of St. John the Baptist, was buried in the church at Canterbury. So it fell out that the church of Canterbury was three times vacated in one and the same year, excepting five days [i.e., by Stratford, Ufford and Bradwardine] ." 2 9 The chronicles for the year 13 49 are full of vivid and horrifying desc riptions of the Black Death that was then devastating all of western Europe and in which the lives of so many of the great men of the time were simultaneously cut short. 3 0 It should b e of some comfort t o scholars that, in spite of Brad­ wardine' s prominence as a man of affairs, he was esteemed more especially by his own and succeeding generations for his intellec­ tual achievements. Several honorifics have been bestowed on his 3 name : " Sacrae 2aginae 2rofessor solem2nis:• 1 " �gregius theolo­ gus :• 32 " Magnus logicus :• 33 and, most characteristically, " Doctor 2rofundus :• 34

INT ROD UC T ION

8 2..

E D I T IONS AND MANUS C R IPTS

With the exception of hi s De cau sa Dei ( 16 1 8 ) and De prae scientia �P-raede stinatione (1935 ), none of Bradwardine ' s works have ap­ peared in any but the earlie s t edition s . The Geometria S P-eculati va, De quadratura circuli, Arithmetica s peculativa, and De proportion­ ibus ( s ome of thes e in many edition s ) were printed a ver a period extending from 149 5 to 1536 and in such variou s place s a s Pari s, Venice, Vienna, Valenc ia, and Wittenberg. If the complete ab sence of Bradwardine' s theological writing s among thi s li s t of early edi­ tion s i s any criterion, it would seem that h i s work in natural phi­ lo sophy wa s of con s iderably greater intere st than h i s theological writing s at the beginning of the s ixteenth century . Copie s of the se early printed edition s of Bradwardine ' s scien­ tific writing s are relatively scarce, but an examination of some of the more important holding s of European libraries reveal s large number s in manu script form . No s y s tematic attempt was made to collect information regarding manu script s other than tho se of the De proportionibu s, yet con s iderable number s of them were encoun­ tered in pas s ing, and, in the ca se of the De P-rOP-Ortionibu s itself, the location of thirty widely scattered manu script s wa s di scovered w ith little difficulty . It would seem quite po s s ible that at lea s t double that number might s till b e in exi s tence, and one i s again im­ pre s sed with the con s iderable e s teem in which Bradwardine' s math­ ematical and philo sophical work s were held in the year s before the advent of printing . Becau s e of the relatively meager amount of modern scholarly s tudy thu s far devoted to the literature of the fourteenth century, bib liographical information concerning s uch a figure a s Thoma s Bradwardine i s difficult t o obtain and yet m ore difficult to sub stan­ tiate . The following information i s, therefore, pre sented , in sofar a s pos s ible, in the order of it s certainty . 3 5 E D I T I O N S

De cau sa Dei contra Pelagium et de v irtute cau sarum, ad suos Mertonens e s . Ed . Sir Henry Savile, London , 16 1 8. Thi s work, of more than nine hundred folio page s , contains in the editor' s Preface the earlie s t modern biography of Bradwardine. The work it s elf, falling into three main div i s ion s, develop s deduc­ tively and in mathematical style the the s i s that God i s the imme­ diate s u s taining and moving cau se of every being and every action.36 Variant title s : Summa theologica and, po s s ibly, Sum ma scienti­ arum. 37 Geometria s peculativa . Ed. Petru s Sanchez Cirvelu s, Pari s, 149 5 : Guy Marchan t , 2 2 fol s . , fofio ? 8 ed . Thoma s Dura , Valencia , 1 5 03 : I. Iofre , 1 4 fp l s. , folio . 3 9

EDITIONS AND MANUSC R IP T S

9

In four pa rt s : ( 1 ) Ste llate d P-ol y: g o n s : For mulae e s tabl i s he d in e xt e n s ion of Campanu s' wo r k ; (2 ) I s oP-e ri m e t r i c fi gur e s : De mon­ s t r ate s , among othe r thing s, that the c i r c le ha s the maximal a r e a o f e ve r y i s ope r im e tr ic fi gur e . B ra dwa r dine' s s our c e p r obably wa s the ano nymo u s De i s oP-e r im e t r i s , ba s e d upon Z e no do r u s' o r i ginal wo r k on the s ubje c t , whic h had b e c ome popula r in the thi r te e nth c e ntury ; (3 ) T he o r y___Q_f_P-rOP-o r tion s : A s tudy o f r ati onal and i r ra­ tional qua ntitie s (he r e c al l e d c o mm unicante s , inc ommunic ante s , c om m e n s ur abile s, i r rationale s and a s s i m e t r i) : (4 ) So lid g e ome try: T he final propo s ition s o f thi s s e c tion are taken from the �he r i c s o f The o do s i u s o f B ithynia (le . B C ). • 0 Var iant title s : G e o m e t r i c a P-rinc i2ia , 4 1 Geome tria the o r e ti c a . 42 De P-rOP-O r tionib u s . Pa r i s , unot b e fo re 1 4 8 1 " : Ge o ffr e y de Ma rne f, 24 fol s . , folio ; 4 3 V e nic e , 1 5 0 5 , fo lio . 44 Fo r de s c ription o f the c ontent s o f the De P-rOP-O r tionib u s , s e e P a r t I V o f thi s Int r o duc tion . Va r iant title s : P r o P-o r t io ne s; T r ac tatu s P-rOP-O r t ionum ; De ve loc i­ tate motuum ; 4 5 T r a c tatu s b r e vi s (o r �P-itomatu s) gg__propo r tioni­ 6 b.fil . 4 A r i thm e ti c a s pe c ulativa . E d . Pe tru s Sanche z C i r v e lu s, Par i s, 1 4 9 S : Guy Mar c hant , 1 6 fol s . , 4to ; 41 e d. Tho ma s Dura , Vale nc ia., 1 5 0 3 : I . Io fr e , 1 4 fol s . , fol io ; 48 Pa r i s 1 5 1 5 o r 1 502 : M . L e s c le nche r, 4to . 49 Thi s wor k i s de s c r ib e d by Sa r ton a s b e ing o f the " p r a c tic a l , B o e thian type" and , if e quivalent t o the e dition at the Co lumb ia Unive r s ity Lib ra r y , 50 i s v e r i fi e d to be no m o r e than an ab r i d g e­ m e nt o f B o ethi u s' Arithm e tic a : p r e s umab ly de s i gne d fo r u s e a s a te xt b o o k . Var iant title s : Ar ithm e ti c a ; 5 1 A rithm e ti c a pra c ti c a . 52 Utrum Deus habeat P-rae s c ie ntiam futur o r um c ontinge ntium a d utrum l ib e t . E d . B a rto m e u Ma . Xib e r ta (0 . Carm .). " Fr a gments d' una que s tio ine dita de Tom a s B r adwa r dine;• Ma rtin G r abmann Fe s t s c hr i ft , (Mun s t e r i . W. , 1 9 3 5 ) , pp . 1 1 6 9- 8 0 . 53 Thi s i s the Quae s tio a l r e a dy r e fe r r e d to in the b i o g raphic a l p o r­ tion o f the p r e s e nt chapte r a s having b e e n di spute d b y B ra dwa r dine and B a c o ntho rpe . It i s , of c ou r s e , one of the ke y p r ob l e m s o f the De c au s a De i , al s o . 54 Var iant title s : D e P-ra e s c i e ntia et prae de s tina tione , De futur i s . c o ntinge nt!. b u s . 55 MA N USC R I P TS °

T ra c tatu s de c o ntinuo . MS. R 4 2 , Stadtbib li othe k , To r un (Tho r n) ; MS . CA4 3 8 5 , fol s . l 7 r- 4 8 r, Sta dtb uc he r e i , E r fur t . O f a l l the wo r k s o f B ra dwa r dine still e xi s ting in manu s c r ipt fo r m o nl y , thi s i s c e r ta i nly the m o s t inte r e s ting t o hi s t o r ia n s o f philo-

10

IN TRODUCTION

sophy and mathematics . Rediscovered only in modern ti mes, it has thus far been examined by Curtze, Cantor, and Sta m m, Sta m m having promised an edition of it in his article, " Tractatus de continua von Thomas Bradwardine :• 56 From references within the text , Stam m takes the attribution to Bradwardine to be sufficiently established , and the contents of the work, as described by him , would certainly seem consistent with the strongly Aristotelian position developed in the De P-roportioni­ bus. Progressing from " definitiones" to "suppositiones" to "conclu­ siones :• the De continua exhibits the same general form as the De Rroportionibus . The theme is mathematical and physical continuity ( "de compositione continui quantum ad sua essentialia" ) , and , a­ gainst the "atomism" of Grosseteste, Bradwardine maintains that a continuum can only be decomposed into continua similar to itself , and not into "atom S:' Actual and potential infinities are distinguished ( "categorimatice" and "syncategorimatice" ) ,� and also "con.tinuum permanens" (e . g . , a l�ne or surface) and "continuu m successi vum" (e .g. , ti me or motion). 58 Ars memorativa MS . 3 744, Sloan Collection , British Museum . This work is described as a plan for aiding the memory by the association of places and ideas.59 Variant title : De memoria artificiali. 60 " Propositiones ;• MS . Vat . Lat. 3102. This manuscript is of the PersP-ectiva communis of John Peck­ ham (second half of the twelfth century) , and is said to contain four propositions added by Bradwardine , which show him to have been familiar with "tangent" and "cotangent" and their reciprocal relation . 6 1 Commentarii in quattuor libros sententiarum. MS . 505, B iblioth�que de Troyes . (Catalogue Generale des Manuscrits des B iblioth� ­ QUes Pub liq�P-ar Departements , Paris 18 55, I I, 222 . ) Insolubilia . Erfurt , Stadtbucherei, Amplonian Collection : MS. F l20 (frag. ) , MS. F 135 (wrongly attributed to Roger Swine shead) , MS. Q 1 7 6 (frag. ) , MS . Q 76. (Besch. Verzeichn . d . AmP-lon . Handschr. Samm, z, Erfurt.) De fallaciis tractatus. Erfurt, Stadtbucherei , Amplonian Collection : MS . F 297. (I bid. Such a work would seem appropriate to a " mag­ nus logicus:• ) Liber metricus rithmimachie, id est de pugna numerorum . Erfurt , Stadtbucherei , Amplonian Collection : MS. F3 13. (Attributed by Thorndike and Kibre , QP-, cit. , Col. 2 84.) Quaestiones physicae. Vatican , Palatine MS . 1049. (Attributed by Thorndike and Kibre , QP-• cit. , Cols. 284, 758 . ) Quaestiones de velocitate motuum et de proportionibus velocitatum . Erfurt , Stadtbiicherei , A mplonian Collection : MS . F 3 13 . ( Besch , Verzeichn . d . Am plon, Hand schr. Samm. z . Erfurt. )

G E N E R A L S IG NI F IC A NC E

11

For the following works no manusc ripts have been located and either their existence or attribution must therefore remain a mat­ ter of some doubt . Tabulae astronomicae . (Mentioned by Savile , Fabric ius , and Uber­ weg .) De sanc ta trinitate . (Mentioned by Pits , Hahn_ 1 and Uberweg .) P lac ita theologica . (Mentioned by Pit s, Hahn, Uberweg, and Fabric ius.) De praemio sal vandorum . (Mentioned by Pits , Hahn, and Uberweg.) Sermones. (Mentioned by Pits, Hahn, Uberweg , and Fabricius.) Meditationes. (Mentioned by Hahn, Uberweg , and Fabricius.) De quidditate pec c ati . (Mentioned by Hahn and Uberweg.) 3.

T H E GENERAL SIG N I F ICAN CE O F THE DE PROPORT IONIB US

Before embarking upon the detailed and tec hnical portion of the present study, it may first be well to c onsider more generally the signific anc e and interest which B radwardine 's De 12ro12ortionibus possesses for the c ontemporary student of the history of western thought. Of the entire period of intellec tual development in the West, those c enturies which have thus far rec eived the least attention are the fourteenth and fifteenth. From the standpoint of the theological in­ terest which has, quite appropriately , motivated so much of the study of scholastic philosophy, this period has been traditional ly c onsidered as one of regret table decay after the high tide of the great thirteenth c entury . More rec ently, a reawakened interest in the logic of William of Ockham, and in the more truly Aristotelian , or at least the more truly modern, spirit which he introduced into scholastic ism, has c arried historical studies forward into the early fourteenth c entury, but even this period (within which falls the work of Thomas B radwardine) remains largely unexplored. It was, however, during this first half of the fourteenth century that the studies then being c arried out by B radwardine and his suc­ c essors were, for the first time sinc e antiquity, making real prog­ ress in the wedding of mathematics to natural philosophy. On both sides of the channel the laying of foundations for what is now the c harac teristically modern treatment of the natural sc iences moved ahead. The English (B radwardine, Hentisberus, Dumbleton , Swine s­ head, and others) were concerning themselves with the more strictly mathematical treatment of problems in physics; at the University of Paris, John B uridan was making a more philosophic al and Nicolas Ore sme a more geometric approach to these problems . Almost the only published work thus far to recognize the signal im portance of Thomas Bradwardine's De pro2ortionibus has been

12

I NTRODUCT I ON

tha t o f Anne l ie s e Mai e r , who s e b r i e f b ut c le a r- s ighte d e s timate s c anno t b e to o highly p r a i s e d . 62 D uhe m , with what may be pa r donab l e human p r ide in the b r il­ lianc e o f the Unive r s ity o f Pa ri s (to who s e g r e a te r g l o r y hi s Etude s s u r Le ona r d de Vinci a r e qui te e xplic itly c o mpo s e d) give s s c ant t r e atm e nt to ac hie v e m e nt s on the othe r s i de o f the c hanne l . In g e n­ e r a l , he a c c u s e s the E n g li s h o f thi s pe r io d o f b e in g un r e g e n e r a te A r i s tote lians in the i r s tudie s o f d yn ami c s , po s s ib l y o f s o me s i g­ nifi c anc e in the de v e l o p m e nt o f mathe mati c s, b ut fo r the m o s t pa r t c onc e r ne d with b a r r e n l o g i c a n d the o l o g i c a l quibb l e s . B r a dwa r dine, he c r iti c i ze s s ha r ply (and , a s we s ha l l s e e, witho ut muc h j u s ti fi c a­ tion) fo r having take n the e n d r athe r than the m i dpoint o f a moving r a diu s a s the me a s u r e o f its v e l o c ity 6 3 and pa s s e s o ve r , a lm o s t without m e ntion , the c e nt r a l c o ntr ib ution s o f the D e P-r O P.o r tionib u s : it s e mploym e nt o f a c ompl e x m ath e m atic a l func tion in the e xp r e s­ s ion o f a phy s ic a l la w a nd it s c l e a r di s tinc tion b e twe e n v e lo c ity c on­ c e ive d a s an in s tantane o u s " quality" o f a motion and ve loc ity c on c e ive d a s to tal di s tanc e t r a ve r s e d pe r totai time e la p s e d . The c la r ifi c ation o f thi s l a tte r di s tinc tion and B r a dwa r dine ' s a s s o c ia­ tion o f hi s fo r m ula in dynam i c s with v e l o c ity c o nc e ive d a s an in­ s ta ntane o u s rate po i nte d , as we s hall s e e , 6 4 to the ine v itab l e c onc lu s io n that c o n s tant fo r c e s p r o duc e c o n s ta nt ac c e le r atio n s r a the r than c on s ta nt v e l o c itie s . B r adwa r dine' s s uc c e s s o r s at M e r­ ton C o l l e g e the r e upon c ontinue d to the l o gic a l de duc tion o f the c o r­ r e c t kinematic law r e la ting time e lap s e d to di s tanc e t r a ve r s e d in a unifo r mly a c c e l e rate d motio n . B r a dwa r dine • s De P.r ogo r tio nib u s i s, inde e d , one o f the ke y wo r k s in t h e hi s t o r y o f the de ve lopme nt o f m o de rn s c i e nc e , having b e e n the fir s t to a nno un c e a g e ne r a l l a w o f phy s i c s who s e e xp r e s s i on c al l s fo r anything mo r e than the mo s t r udi m e nta ry math e ma ti c s . The anc i e nt s c ie nc e o f hydr o s tatic s , fo r e xa m pl e , did not de mand ma the mati c s m o r e c ompl e x tha n that o f s imple func tion s, or dir e c t p r o po r tionality . The we i ght o f any s ub s tanc e i n wa te r divi de d by it s we i ght in air is c on s tant fo r a ny c o n sta nt vo lume of the same s ub s ta nc e , a nd thi s c on s tant , in turn , va r i e s in dir e c t propo r tion to the de n s iti e s of diffe r e nt s ub s ta nc e s {i . e ., we ight in wate r -;- w e i ght in a i r = K ). Indepe ndently o f phy s i c a l the o r y , mathemati c s had r e a c he d , e ve n in anc i e nt tim e s, a high pitc h of de ve lopm e nt , A r c hi­ m e de s, him s e lf , ha ving b e e n fa miliar wi th the funda me ntal p r inc iple of wha t late r c a me to be de ve l ope d as the the o r y o f l o g a r i thm s . It wa s a ppa r e ntly in the De gr oP.o rtionib u s o f Thoma s B r a dwa r dine , howe ve r , that what may b e c a ll e d a lo g a r ithm i c , e xp o ne ntia l , or g e o m e tric func tion fi r st c ame to b e appli e d in the e xp r e s s ion of a phy s i c a l the o r y . Hi s the o r y o f t h e r e lat ion o f a fo rc e a n d t h e re s i s tanc e impe ding it to the r e s ultant v e l o c ity may be e xpr e s s e d thu s :

G E NE R A L SIGNI FIC ANC E

13

(!) n V = (ii)

V = log n o r, to avoid an anachronistic use of the term, " log" That this fo rmulation required alteration and a further clarifica­ tion o f the terms, " vel ocity ;• " fo rce � and " resistance" before it could become a part o f modern dynamics goes without saying. As we shall see in o ur further exploration o f the Tractatus, this theo ry was primarily an ingenious so lution to a specific problem which faced those who were attempting to uphold the Aristotelian laws, or axioms, concerning natural motion. As such , it has its own his­ toric purpose and c onsequent limitations. However faulty B radwardine 's application o f the above formula to his o wn problem , it is nevertheless o f considerable importance that he introduced this particular function to the consideration o f those who were attempting t o express physical relationships with the accuracy and generality o f mathematics. Not only has the log­ arithmic function since become an indispensable tool for the ex­ pression o f such phenomena as those in which growth is a function o f size, but in Bradwardine 's own day it provided a considerable stimulus to the development o f mathematical analysis because o f its capacity t o overcome certain logical paradoxes t o be encoun­ tered in the employment o f simple functions for the expression o f physical relations . The serious nature o f these paradoxes and the obstacle they presented to fourteenth-century mathematical physics will be evident from our subsequent anal:rsis o f Chapter II , The­ o ry III, o f the present treatise. Those fol lowing immediately in Bradwardine 's footsteps at Ox­ ford (o ften known as Calculatores and consisting for the most part o f fellow Mertonians) were quick to realize the power of this gen­ eral mathematical relation as a means for expressing an enormous variety o f physical processes, and, in their hands , Bradwardine's exponential function came to be applied no t only to " uniformly dif­ fo rm motion" (acceleration) but to qua litative and quantitative al­ teration and even to problems in psych o logy, ethics and theology. HW herever one quantity is definable in terms o f a relation between two others, and there is no express reason to the contrary, it is taken for granted [by the Calculatores] that Bradwardine's function expresses the relation :• 6 5 To appreciate the magnitude o f the reorientation wo rked by the Ik._gro P-ortionibus upon the character o f theorizing within the field of natural phil osophy, one need only cast a backward glance at the type of treatment accorded problems concerning mo tion during the preceding century. The existence o f these problems and the neces­ sity of dealing with them had been made quite evident to scholastic

14

IN T ROD UC T ION

phi lo s ophe r s by th e Arabian com m e ntato r on A r i s totl e ' s wo r k s , Ave r ro e s , who p r i ncipa l l y in h i s di s cu s s ion of what wa s known to m e diae val s cho lar s as " T e xt 7 1 " of th e Phy: s ic s (IV , viii) had fo­ cu s s e d atte ntion on ce rtain appar e nt paradoxe s e ntai l e d by Ar i s ­ tot l e' s s canty r e mark s conce r ning dynam i c s . W hat , i n the fi r s t p lace, i s to b e con s i de r e d the e s s e ntial cau s e o f the fact that s om e chan g e s (i . e. , tho s e p r ope r ly to b e cal l e d " mo­ t ion s'' ) occupy ti m e ? If , a s i s indicate d in the Phy: s ic s the s ucce s­ s ive char acte r o f motion is att ributabl e to an oppo s ition betwe e n moving an d r e s i s tin g powe r s , what of the motion o f the heave n s , whe r e p r e s um ably no r e s i s tance i s to be e ncounte r e d ? In s p ite of the fact that the s cho la s tic s tude nt s o f Ari s totle' s natu r al phi lo s o­ phy we r e p r e par e d to ag r e e i n the di s tinction b e twe e n two kinds of natu ral motion (t he one r ectilinear and s ublunar, the othe r ci r cular and ce l e stial) , th e y we re s ti ll incl ine d to be l i e v e that a s in g l e p r i n­ ciple s houl d e xp lai n the s ucce s s ive cha racte r of both . S ince , fur the r mo r e , Ar i s totl e had claim e d that ve locity var i e s acco r di n g to the p ropo r tion b e twe e n th e powe r of the move r and that o f the th ing move d , what wa s to be con s i de r e d th e co r r e ct in­ te r p r e tation of the te r m, o. v gX. o:y_{g or " p r opor tion" ? T he wo r d, " p r oportion ;• ha s today a clar ity of m e aning which it ha s not al­ way s po s s e s s e d; pe r hap s the mo r e g e n e r al m e aning of the wo r d, " function ;• b e tte r p r e s e rve s the s e n s e of th e anci e nt an d m e diae val u s age . ' Av a"A oyl._g or grogo r tio , m i g ht m e an : "a function of the ar ithmetic di ffe r e nce ;• "a function of th e ratio ;• or any one of a ho s t of othe r function s . The p r ob l e m o f choo s i n g betwe e n the s e alte r na­ tive s i s the s e cond p r incipal one po s e d by Av e r roe s' co mme nta r y. In hi s own e ffo r t s to fi nd s olution s fo r the s e two p robl e m s we find Ave r ro e s adhe r i n g as faithful l y as po s s ibl e to what s e e m s m o s t cl e a r l y to b e Ar i stotl e ' s g e n e ral point o f v i e w. R e garding the que s tion of what i s to be vie we d a s th e e s s e ntial and g e n e r i c cau s e o f the s ucce s s ive char acte r o f motion, h e ar gue s that thi s i s n e c­ e s s a r i ly the p r e s e nce o f a fo r ce r e s i stant to th e moving powe r ; the r e fo r e the t e m po ral duration of ce l e s tial motion s mu s t indicate the p r e s e nce of a r e al r e s i s tance o ffe r e d by th e s phe r e s to the powe r s which move th e m . As to the othe r p roble m (that o f e stab­ l i s hin g the tr ue functional r e lation ship be twe e n s p e e d, fo rce and r e s i s tance) it i s cl e ar that Ave r roe s r e pudiate s th e the s i s that thi s function i s one of ar ithm etic di ffe r e nce. To m ention only one of hi s ar gum e nt s : If it we r e true that s p e e d var ie s with the di ffe r e nce b e ­ twe e n fo r ce an d r e s i s tance , the n i t would not fol low that halving the r e s i s tance would doubl e the s p ee d (a s Ari stotl e had clai m e d it would) ; mo r e impo r tant s til l , i f s pe e d i s to b e unde r s tood a s va rying with the di ffe r e nce betwe e n oppo s in g force s , not only doe s thi s com mit u s to the un- Ar i s tote l ian con c lu s ion that motion s in vacuo ar e po s s ible but that all the so- call e d " natural" m otion s s tudi e d in phy s ic s a r e not r e ally natural at al l-the trul y " natural" motion s

G E NE R A L SIGNIFIC ANC E

15

being the idea l speed s with which bodie s would move in the ab sence of any re s i s tanc e. Although Aver roe s a r gue s conclu sively that, on Aris totelian ground s, speed c annot be a function of the a rithmetic difference between force and re sis tance, it i s not too c lea r ju s t wha t he took the true func tion to be. If, according to the u s ual interp retation of A ri s totle, he under s tood this func tion to be one of simple propor­ tionality, then his po sition i s open to the deci s ive ma thema tical objection s which Bradwa r dine di s c u s s e s in hi s refuta tion of The­ o r y III in the second c hapter of hi s trea ti se . Whether or not the po sitive por tion o f Aver roes' di scu s sion of thes e problem s wa s altogether sati s fa c tory, hi s inclu sion of an ad­ ver se c ritici s m of the view s of an ea rlier Arabian, lbn Badja (Avem­ pa c e), w a s of momentou s influenc e for thir teenth-century schola s ­ tic i s m . T aking a s hi s point o f departure c ele stial rather than ter res trial phenomena, lbn Badja had a rgued tha t, since the heav­ enly bodies move at determinate rates without encounterin g any appa rent re s i s tanc e, every r e s i s tance should, therefore, be under­ s tood a s dimini shing what would otherwi s e be a determinate, ab­ s olute speed. Fundamentally, thi s view re solve s it self into a simple equation of forc e and speed, and the rea s on why finite fo r c e s do not p roduce in s tantaneou s transla tion s mu s t, therefore, be that it i s impo s sib le for a force of any given magnitude to act, even with­ out opposition, ins tantaneou sly. It i s a s though empty s pace it self mu s t prevent the in s ta ntaneou s t r an s la tion of a body from one place to another. A s a matter of fac t, Ibn Badja i s quoted a s s aying j u s t thi s , and the s chola s tic s who puzzled over the problem during the thir teenth c entury developed the idea a t length. Not only i s the di scu s sion of p r oblem s concerning motion that went on during thi s century of intere s t in the c ontra s t exhibited between it s dialectic al method and the mathematic al method which la r gely s upplanted it during the pos t-Bradwa r dinian period, it i s al so o f some intere s t a s providing yet one more example o f the ea s y freedom with which s c hola s tic philo sopher s depa r ted from A ri s totelian tea chings whenever tha t seemed to them the more rea sonable c our s e.66 The bulk of the great figure s of the thir teenth c entury quite emphatically preferred the ab s oluti s t, rather Pla­ tonic, solution s to the s e problem s s ugges ted by Ibn Badja to the relational and typically Ari s totelian approach provided by Aver roe s ; Thoma s Aquina s, Roger Bacon, Pier re Jean Olivi, and Duns Scotus were all, in one way or another, adhe rent s of Ibn Bad ja' s view s . 67 Their c omplete neglec t of mathema tical con sider ation s-not only a neglect which failed to deal with problems entailed by Ibn Ba dja's dynamic s but one which failed even to make c apital of the manife st difficultie s inherent in the dynamic formula to which Aver roe s ap­ pa rently s ub s c ribed- s tand s in s t riking cont ra s t to the treatment

16

INTROD U CTION

of the s e i s sue s in Bra dwardine' s De 2ro2ortionibu s . Among the fir st to re-e s tabli s h the reputation of the Averroi s t- Ari stotel ian po s ition at the beginning of the fourteenth century, hi s cour se wa s not to attack fir st the general problem of what i s to be con s idered the e s s ential cau s e of succe s s ive motion but rather to dea l directly with the prob lem of developing an interpretation of the dyna m ic function who s e mathematical impl ication s would be in harmony with the general axiom s concerning both motion and re st which had been current s ince antiquity . Hi s handling of thi s que s tion i s , a dm ittedly , pre- modern in its fa ilure to extend or refine the empirica l ob servation s upon which the verification of any hypothe s i s nece s sarily depends ; thi s latter methodological innovation had to wa it for Gal ileo . What he di d ac­ compl i s h wa s , in the fir st place, to ma ke clea r once and for all the ine scapable impo s s ibi lities enta iled in Ibn Badja' s view that speed i s a function of the difference whereby a moving power exceeds one that i s oppo sed to it. The general character of the s e difficultie s ha d already been indicated by Averroe s' reply to that theory but apparently had fa llen on deaf ear s s o far a s mo st of the earl ier s chola s tic s were concerned. Rejecting al so (becau s e of the math­ ematical impo s s ibilitie s to which it al so leads) the theory tha t the function in que stion i s one of s imple proportional ity, Bra dwardine' s second con s iderable achievement wa s the succe s sful reinterpre­ tation of thi s relation s hip in term s of a logarithmic function which would s ati sfy con s i stently the accepted empirical genera lizations rega rding motion and re st. It i s, indeed, no s mall honor to Ari stotle that mediaeva l natural philo sopher s were able to proceed (on the a s sumption that the ba s ic theoretical s tructure and s imple empi rical content of Ari stotle' s writing s were correct) to the development of a scientific edifice which in many direction s far outreached it s original . What may , however, s trike a contemporary a s a s toni s hing i s that such an e s­ s entially deductive approach to the phy s ical s cience s s hould have met with such succe s s . Yet , Buri dan' s theory of "impetu s" (which fore shadowed Galileo' s " impeto" and De scarte s' " quantity of mo­ tion" ) , Nicola s of Ore s me's di scu s s ion s of the diurna l rotation of the earth and hi s development of the element s of coordinate geom­ etry, and the Mertonian proof of the correct law of uniform accel­ eration , 68 all were developed at thi s ti me and without benefit of that experimentation which the modern i s inclined to regard a s in­ di spen sable to any advance in the science s . Bradwardine' s De proportionibu s repre sent s, at any rate , a s ig­ nificant advance in two very important a spects of the s cientific enterpri se : fir s t , it move s forward with the ta s k of developing mathematical formulae for the expre s s ion of phy scial law s who se enta iled con sequence s do not contradict other generally accepted law s or ob servation s - in other words the ta s k of achieving sel f-

GENE RAL SIGNIFICANC E

17

c on s i s tenc y in phy s ic s through mathemati c s ; second , through the introduc tion of mathemati c al analy s i s , it set s the s tage for the quantitative measurement of phys ical proc e s s e s and , hence , that typi cally mo dern phy s ic s whi c h wa s to appear with Gal ileo' s wed­ ding of mathemati c s and experimenta l ob servation . B radwardine u s ed mathemati c s for the s y s tematic and general expre s s ion of theory; Galileo u sed it for the s ys temati c general ization of experi­ mental ob servation . A s ha s been already briefly noted, one of the intere s ting points of contra s t between B radwardine' s approac h to the problem s of dy­ namic s and the approac h which was mo s t prevalent during the pre­ c eding century i s that between h i s own Ari s toteliani s m and the qua s i- Platon i s m of these predeces sor s . Quite the contrary to what one m ight expec t, on the bas i s of that general contemporary opinion which would see in Platoni s m rather than in Ari stotel iani s m the true paternity of modern s c ience, we find that, in philo sophi c con­ s ideration s, B radwardine al ign s hi m s elf c on s i ste ntly with Ari s totle. Not only i s h i s preferenc e for a c orrelative rather than an ab s o­ lute ac count o f the nature and c au s e of veloc itie s clearly Ari stote­ lian , the c areful c oncern with which the legitimac y of expre s s ing phy s ic al c orrelation s in mathematical term s i s treated (in the di s­ tinction s drawn in Theory I V of C hapter I V, and el s ewhere) ari s e s explic itly o n the ba s i s of Ari s totle' s treatment of thi s problem in Ph y.: s ic s V I I, Chapter 4 . Perhap s yet more strikingly, in hi s De continua , B radwardine bases hi s po s ition (in whic h, ac c ording to Cantor, he fore s ha dow s the theory of tran s finite number s) on a de­ fens e of the Ar i s totelian conc eption of a continuum again s t the atomi s m of Gro s sete s te . Though B radwardine ha s long been known a s one of the reali s t s, there i s perhap s a greater degree of agreement between hi s phi lo­ sophic view s and tho s e of William of Oc kham than thi s contra st s ugge s t s . The c entral doc trine of the De cau s a Dei , that God i s the immediate and omnipotent effi c ient cau se of every effec t , i s s ub­ s tantially the same a s that of Oc kham and an obvious ly appropriate premi se for the sub sequent c ontention (by the " Oc kham ite s ;· Holkot and Mirecourt) that the will of Go d i s the effic ient c au se of human s in . Though thi s return to an emphas i s on the c entrality of the doc­ trine of God ' s omnipotenc e, c haracteri s tic of the theological trend at the beginning of the fourteenth c entury, need hardly indicate that B radwardine s hared with O c kham a nominal i stic theory of know­ ledge, both the De P-rOP-Ortionibu s and Oc kham ' s Philo soP-hia natur­ al i s do s how the s e men to be returning to a le s s Platoni stic interpretation of Ari stotle. The foregoing s houl d , then , indi cate something of the con s i der­ able range of intere st which Bradwardine ' s De P-TOP-ortionibu s po s se s s e s , not only a s a turning point in the development of the s c ienc e s , but a s an example of s c hola stic natural philo sophy in a

18

IN T RODUC T ION

pe rio d fro m which the re survive ve ry fe w readil y availabl e te xts. It was, as is being increasingly bette r recogniz e d, a richly crea­ tive pe rio d, and , in the inte llectual scope and energy of its great innovators, truly co mparable to the age of Galil e o, De scarte s and N ewton. 4.

DESC R IPT IV E AND C R IT ICAL ANALYSIS

The De proportionibus co m m e nces by taking as axio matic Aristotl e ' s state m ent that motions and the ir ve locitie s e xhibit a propo rtional relationship. The re appears to have be e n littl e dis­ agre e m e nt re garding the truth of this by any of the scho lastics ,6 9 dive rge nce of opinion ce nte ring , instead, on the question o f ( 1) the nature of such a pro portion , and (2 ) the factors prope rly to be unde rsto o d as invo lve d . Bradwardine quite rightly points to the fact that in spite o f the conse quence that a study of proportions is thus necessarily in­ volve d in a study of ve l ocities, no one had as yet carrie d out the task. He com m e nce s his tre atise, the re fore , with an e xpo sition of the fundam e r1tal nature and prope rties of proportion , base d prima­ rily on Bo e thius' Arithmetica and Campanus de N ovara' s Com m e n­ tarium suger �uintum librum Ele m e ntorum Euclidis. The fact that he considers such an e xposition necessary se e ms indicative not only of the mathe matician 's de sire for co mpletene ss in argumen­ tation, but also of the almost complete ne gl e ct of mathe matical conside rations in the formulation of theorie s of m otion which is so cl early manifeste d by those scholastic precursors of Bradwar­ dine already me ntione d. He be gins the De P-ro gortionibus, the re­ fore, with a live ly aware ness that he is e mbarking on a ne w course and yet one which had, from the b e ginning , be en indicate d as logi­ cally nece ssary fro m the ve ry nature of the mate rial involve d (i. e., the axiom that velocitie s are , in som e sense , proportional to fo rces}. The tone of Boe thius' intro duction to the Arithmetica is strongly ne o-Platonic and ne o-Pythagorean, being full of re fere nc e s to the Timae us and Re gublic in which numbe rs are spo ke n o f a s " real" and as "gene rate d" in a most lite ral sense . The idea is pe rvasive ly de velope d that the cosmos e xhibits , in its fo rm and behavior, nu­ m e rical prope rtie s, and what strikes the m o de rn reader a s truly re markable is that s uch a ke e n appreciation of the possibilitie s inhe re nt to a mathe matical physics , whe reby the pre dominantly cla ssificatory and qualitative physic s of Ari stotle might we ll be furthe r e xplo re d, s hould have re maine d s o l ong unuse d. The first step s toward the deve lopment of such a science had long be e n the prope rty of the weste rn mind as e xemplifie d in the scie nce of har­ monic s , yet nothing furthe r had be e n done.

D E S C RI P T I V E AND C RI T I C A L AN A L Y S I S

19

Many e xplana ti o n s may b e m a de fo r th i s ne g l e c t o f th e po s s ib il­ i t i e s o f a ma them atic a l phy s i c s , no t the l e a s t o f whi c h i s that A r i s to t l e, the dom inant fi gur e o n the s c ho la s t i c s c e n e, had him s e l f done s o l ittl e i n thi s di r e c t i on , having b e e n p r i m a r ily c onc e rne d w i th the e r e c tion o f a s c ie nc e o f c l a s s ifi c a t i on and ana l y s i s r a th e r than o n e o f m e a s ur e m e nt a n d s ynthe s i s . Pe r ha p s an e qua l l y po we r­ ful de te r r e nt , howe ve r , wa s la c k o f ma the ma tic a l s ymbo l i s m fo r the e a s y manipula tion o f al g eb r a i c e qua tion s . The Arabic de c imal s y s te m and no tation we r e i n c om m o n u s e s o m e time b e fo r e the c o m po s ition of the De P-r O P-O r tionib u s , b ut a l though L e o na r do o f Pi s a (c . 1 2 2 5 ) ,7 0 t h e g r e a te s t w r i t e r o n al g eb ra dur ing the m i d dle a g e s , had c o mme nc e d the intr o duc tion of ope r a tiona l s i gn s into we s te r n u s a g e, the i r e mp loyme nt and i m p r o v e ment by o the r wr ite r s wa s l o n g de l aye d . W e find , fo r e xa m pl e , B ra dwa r dine s till u s ing the " na m e s" fo r di ffe r e nt p r opo r ti o n s, which ha d b e e n hande d down in a t r a dition o f B o e thian a r ithm e ti c in whi c h the s e name s we r e t h e r e s ul t o f an e ffo r t t o de v e lo p a s c i e n c e o f g e ne r al fra c ti o n s a t a tim e whe n n o g o o d s ymbo l i s m for tha t p u rp o s e e xi s te d . Th e di f­ fi c ultie s p r e s e nte d in a ttempting to r e nde r al geb r ai c e qua tio n s w i thout the a i d o f o p e r ationa l s ymb o l s i s ab undantly i l l u s trate d th r oughout the c o ur s e o f B r a dwa r dine • s tr e ati s e . The fi r s t di s tinc tion d r a wn , at the be ginning o f Chapte r I o f the De P-r OP-o r tio nib u s, i s that b e tw e e n the g e ne r a l and the s t r i c tly a c­ c u r a te, the e quivo c al a nd the univoc a l , me a ni n g o f p r op o r t i o n . B e­ twe e n any two thing s whic h a r e c om pa r ab l e a p r opor tion may b e s a i d to e xi s t in r e s p e c t to t h e fa c to r c o mpa r e d . B ut , ac c u ra t e l y s pe a kin g , a p r o p o r tion c a n o nl y b e b e twe e n quantitie s " o f t h e s a m e kind :' The phr a s e , " e i u s de m g e ne r i s : • i s a n impo r tant one . Campanu s (who s e de finitio n o f p r opo r tion B ra dwa r dine quote s) 7 1 whe n drawing the di s t inc tion b e twe e n c om m e n s u r ab le and inc o m­ m e n s u r ab le qua ntitie s, di s tingui she s the ob j e c t s o f a r i thm e t i c and g e o m e t r ic a na ly s i s fr o m tho s e not c a pab l e of the a r i thm e t i c b ut only o f the g e o m e t r i c . It i s , thu s , c l e a r that the te r m , " qua ntity ;' d o e s not a pply s i mpl y to numb e r but a l s o to g e om e tr ic c ontinua . T h e te r m , " quantity� i s y e t m o r e e xp l i c itly b r o a de ne d in Ar i s totle' s u s a g e . In the Cate g o r ie s, (Chapt e r 6 ) h e di s ti n g ui s he s phy s i c a l c o n­ tinuiti e s ( s uc h a s l ine s and s u r fa c e s) fr o m the di s c o ntinuity o r di s­ c r e te ne s s o f numb e r . The s e c o n s titute , fo r Ar i s to tl e , diffe r e nt g e ne r a o f quantity, and , a s s uc h , a r e he l d to b e inc o m m e n s u r ab l e . In Phy: s ic s (V I I , iv) he fu rthe r po i nt s out that not only mu s t qua n­ titi e s b e o f the s am e g e nu s b ut a l s o o f th e s a m e s p e c i e s , i f th e y a r e t o b e uni vo c a ll y c o mme n s u rable . Ac c o r ding to thi s v i e w, the r e fo r e , the r e c o ul d only b e a pr o p o r,.. t i o n (P-r o pr i e dic ta) b e twe e n numbe r s o r b e twe e n v e lo c itie s , b ut n o t b e twe e n a numb e r and a v e lo c ity , s inc e the o ne i s di s c o n tinu­ o u s and the o the r c o ntinuou s . No r c ou l d th e r e b e s uc h a univocal p r op o r t ion e ve n b e twe e n c o ntinuo u s mo tio n s, if th e y b e o f diffe r e nt

INT ROD UC T ION

20

genera (e. g . , locomotion and a lteration) or di fferent species (e.g. , rectilinear and circular locomotion). In what sense, then , is a math­ ematical formulation of physical process justifiable ? The answer to this question is given at the commencement of the second sec­ tion of this cha pter of the De P-rO P-Ortionibus, wherein is discussed the general theory of proportiona lity; in the meantime, the strictly mathematical aspects of proportions remain to be described . Proportions, Bradwardine continues , are of two orders or de­ grees , a " first order proportion" being called "rational0 and de­ fined as that which is immediately denominated by some number (e .g . , " double proportion ;' denominated by the number 2 , ••triple proportion ;' denominated by the number 3 , etc . ). A "second order proportion ;' called " irrational ;' is one which is not immed iately denominated by a number , but mediately (i . e. , by mediation of a first order proportion) . Bradwardine uses the example of " half a double proportion" (medietas du2lae P-ro2ortionis) which, he says , is the proportion of the diagonal of a square to its side. This is not expressible as a single, simple proportion of integers but may be expressed by two such immediately denominated integra l pr� portions, the one being denominated by the othe r--i . e •

.(})½

Here , for the fir st time in the De P-rOP-Ortionibus , we encounter roots and powers , and this is only the first of severa l cases in which a terminology, which to the modern reader must seem v ague and a mbiguous, poses an i mportant problem of interpreta tion. 1n this instance , it is obvious, from the nature of the example given , that we must understand a proportion of two integers to be "denom­ inated" by another proportion , or number , in a geometric rather than arithmetic , exponential rather than factorial, sense. The pr� portion ,

f,

cannot be multiplied by an integral arithmetica l factor

in order to yield tha t of the diagonal to the side of the square . It can be so multiplied , or denominated , by an exponential factor, however. Therefore , the " denomination" of one proportion by an­ other is equivalent to raising it to a given power or extracting from it a given root. A power would be equiva lent to denomination by a proportion of "greater inequali ty" (i . e . , a proportion in which the denominator is 1 a nd the numerator is any integer greater than 1 ) , and a root would be equivalent to denomination by a proportion of " lesser inequality'' (i .e . , a proportion in which the numerator is I and the denominator is any integer greater than l ) . I t is o f considerable importance to rea l i ze that this denomina­ tion of one proportion by another (or by a number , i .e. , a propor­ tion the denominator or numerator of which is 1 ) may be easily confused with si mple arithmetic multiplication and div ision . Brad­ wa rdine, or at least the contemporary copyists of his treatise, fail to m a ke a clear-cut terminological distinction between these two

D ESCRIPTIVE

A ND

C R I T I C A L A NA L YSIS

21

proc esses, the one exponential and the other fac torial. Consider, fo r example, the term " double" (duQlum) . If A is said to be duQlum B, it means " double" or " twic e" in the sense that A = B + B, or 2B. A B A . A B . I f, on the othe r hand , = C , then - 1s duQlum - or - m the sense B C B C 2 2 - -- _ A B . . A A A that C = , or ( ) . This 1s tantamount to saying that = - • -, B C C B B B B or C . C and therefore we may draw the general distinc tion that,

(A)

in modern parlance, a fac torial frac tion or integer, as applied to intege rs, indicates arithmetic multiplication and, as applied to propor tions, indicates the raising of that propor tion to the given power, or the ext rac tion from it of the given root. Campanus 72 avoids this confusion by a cir c umloc ution . Instead of saying that � is dupla ad � ' he says that Qr O QO r tio � = 2r 0Qortio � dY.Qlicata, thereby using our modern way o f stating that � = �

" squared" (i. e . , b y using the past-par tic ipial for m). Bradwar dine, if he had used our terms, "square" and " squared;' would have said that � is the "square" of �, but sinc e in his time the equivalent of both " square" and " twic e" was duplum, the c onfusion whic h we have been desc ribing was an ever-present danger. Giovanni Marliani,73 an important Italian theorist in the centur y suc c eeding that o f Bradwar dine, may have fallen into this ver y pit­ fall of misinter pretation in his attempt to refute the main thesis of Bradwardine 's De 12ropor tionibus, and the moder n reader need not be too sur prised at the difficulty which this ambiguity presents today. That the question of which is the proper reading of such te rms as duP-lum , t ri12lum, etc., c an be resolved with c er tainty in any par tic ular c ase is not, however, to be doubted. The c ontext of argument, together with the axioms and theorems c ited in support of the statement in question, provide sure c riteria of choic e be­ twzen the two possible readings, simply on the internal grounds of supposing the author's reasoning to be logically consistent. Continuing his definition of the difference between rational and ir rational propor tions, Bradwardine goes on to point out that ra­ tional propor tions are only found in c ommensurable quantities (those quantitie s for whic h there exists an exa c t c ommon measure in the form of an aliquot par t, or fac to r). Ir rational propor tions are found only in incommensurable quantities (i.e., those possessing no such aliguot fac tor, of which each would be an exac t multiple) . Rational propor tions are found in number s and al so in all other kinds of quantities (commensurable ones) ; ir rational propor tions are not found in numbers (exc ept "mediately � as explained above) but are found in all other kinds of quantities. Therefore, rational propor tions belong b oth to arithmetic and to other b ranc hes of

22

IN T R O D UC T IO N

mathematics, whereas irrational proportions belong to all branches of mathematics other than arithmetic.This last observation would seem to indicate clearly that Bradwardine includes, as branches of mathematics , such "exact" physical sciences as that of music , which had been presented by Boethius in a wholly mathematical guise. Such a broad usage of the term , u mathematics;' is at least as old as Plato, who, in the &.P-ublic (V II, 52lc- 5 31c ; also Laws 8 1 7 E), includes plane and solid geometry, astronomy and music in his classification of mathem atical studies, as well as arithmetic. The remaining portion of the first part of Chapter I is given over to an exhaustive description of variou s types of proportion which may be generated from the variation and permutation of no more than three interrelated terms. Since the text is both difficult to fol­ low and also possesses a certain interest for the history of number­ theory, a schematic presentation in modern notation is given, below. Progortio aegualitatis : Progortio inaeguali tati s : a ) major. b ) minor:

n n n m m n

[where n is greater than m]

Progortio maioris inaegualitatis : a ) multiglex : n [where n is any integer greater than l ] : b ) suP-ergarticular:

c ) suP-e rgartient :

i. (m varied) : ii. (n var ied): iii. (m &: n varied) :

2 (®P-1.a ) ; 3 (tri p.la). . .

T

T

3 (sesguialte ra) , 4 {se sguin+ l 2 3 n tertia), . . . [where n is any integer greater than 1] n +m [where n and m are intege rs n greater than one, and whe re n is greater than m ] n+ 2 {sugerbipartiens), n+ 3 (su�n n triP-a rtiens). . . 3+m (suP-eri�artiens te rtias) 4+m -43 (s UP-e rP-artien s g ua rtas), . . . 3 +2 (s� P-erbiP-artiens te r tias, o r -3S U P-e rbitertia) . 5+ 2 (suP-e r bigartiens -5guintas, or suP-e rbi q uintas), . . . 4 + 1 4

D E S C RI P T I V E A N D C RI T I C A L A N A L Y S I S

23

(su2erte q�artiens guartas , or suger­ te rquartas}, . . . d } multiglex sugergarticularis: m n+ l [where m and n are intP-­ n g e rs greater than l ] i . (m varie d) : 2n+ l (du2le x suge q�articularis} ,

n

3n+ 1 n ii . (n varied}:

(triP-le x sugergarticularis} , . . .

m 2 + 1 (multi2l e x se sguialtera} , m 3 + 1 2 3 (multiglex sesguitertia}, . . .

iii. (m & n v aried} :

2 · 2 + 1 (dugla se squialtera} , 2 · 3 + 1 (du2 3 gla sesguitertia} , . . . 3 · 2+ 1 (trigla ses2 guialtera), 3 · 3 + 1 (triP-la sesquiter3 . ),. . . tia e ) multiplex suge rP-a rtiens : kn+ m [whe re k, m, and n are inten g e r s greater than one ] i . (k varied):

11 .

(m varie d) :

iii . (n varie d):

1v . (m & n varied}:

v.

(k varie d, m sp ecified} : (m varie d, k specifie d) :

2n+m (duP-lex S UP-e q�artiens) , 3 n+m n n (trigle x su2e q�artie ns) , . . . kn+Z ( multiP-lex suP-erb iP-artiens) , n kn+ 3 (multi2le x S UP-e rtrigartiens), . . . n k3 +m (multi2le x suP-er2artiens ter3 tias), k4+m (multiglex su2e r2artie ns 4 quartas), . . . k 3 +2 (multiP-lex S UP-e rb iP-artie ns ter­ 3_ tias or multi2l e x su2erbite rtia) , k4+3 4(multi2le x sugertrigartie n s gua rta s or multi2le x supe rtriquarta) , . . . 2n+ 2 (duplex su2erb ipartiens), 3 n+ 2

n

n

(triplex su2erbigartiens} , . . . 2n+2 (duP-lex superbiga r tiens), 2 n+ 3

n

(duglex suP-ertrigartiens), . . .

n

24

INT ROD UC T ION

vi. (k varied , n spec i fied) :

2- 3 +m (duglex sugergartien s tertias) , 3 3· 3+m (triglex suP-erga !" tiens ter3

(n varied , k specified):

vii. (k varied , n & m specified): (m varied , k & n specified) :

tias) , . . . 2· 3+m (duP-lex sugergartiens tertias), 3 2· 4+m (duglex suP-erP-artiens quar4 tas) , . . . 2· 3+2 (duP-lex suP-e rbite rtia) , 3 · 3+2 3 3 (tri P-lex sUP-er bitertia), . . . 2· 3+2 (duP-lex suP-erbitertia) , 2 · 3+3 3 3 (duP-lex sugertritertia), 2· 3+4 (duglex 3 sugertr iguarta), . . .

Re marks 1. The above schema does not include an analysis of propor­ tions of lesser inequality, this would be the inverse of the ana lysis of P-rO P-ortio maioris inaequalitatis . 2 . It should a lso be noted tha t the break-down from genus to species is not carried out ful l y in all possible cases. Such a task would be infinite . Within· the genus, proP-ortio, however , we are given the pro­ gressive ana lysis, through the sub-genera : inaegualitas, maior, and m ultiP-lex, to the species: duP-la . Progortio maioris inaequalitatis duP-la would be made up of the following individual proportions: 2/ 1, 4 /2 , 6 /3, . . . . Another case in which the analysis from genus to species is completed is that of P-rOP-ortio maioris inaegualitatis triP-la sesquitertia , the break-down bein g a s follows: (genus): P-ro­ P-Ortio ; (suh-genera , differentiae): inaequalita s, multiplex and su™­ partic ular ; (species): triP-la sesquitertia . In this case the individ­ uals would be : 10/3, 20/6, 30/9, . . . . The plan is, at any rate, carried out in sufficient detail to show that each species may be individually named by qu a ntifying the final sub-genus. 3. Aside fro m the more obvious meanings of such terms as: P-ro­ P-Ortio, inaegualita s, etc. , i t is worth noting that mul tiplex means " m ultiplied by an integer � suP-erparticular means " with a n al iguot part added ;' a nd suP-ergartient means " with more than one a l iguot part added� With the above meani n g s in m i nd , it i s not i mpossible to thread one' s way throu gh the m a ze o f de fini tion s .

DESC R IPTIVE

A ND

C R ITICAL

A NA L Y S IS

25

Ha v i n g e s tab l i s h e d the de finiti onal nam e s o f and th e m e a n s o f g e n e r a t i n g p r o p o r t i o n s in g e n e r a l , B r a dwa r din e now tur n s to a n­ o the r a s p e c t o f th e s ub j e c t , tha t o f " p r o po r ti ona lity :• P r o po r tio n­ a l ity , a s di s tin g ui s he d fr om wha t i s s i mply c a ll e d " p r o p o r ti on s � i s c onc e r ne d with c e r ta in s p e c i fi c s e r i e s o f te r m s and the s e r i e s o f p r o p o r tion s wh ic h ma y b e c on s tr uc te d fr o m tho s e t e r m s, whic h po s s e s s , a s s e r i e s , fo r ma l p r op e r t i e s o f inte r r e latio n s hip fr o m whi c h va r i o u s p o s tulate s a n d c o nc lu s i on s m a y b e de duc e d . P r o p o r­ ti onal i ty may, th e r e fo r e, b e un de r s t o o d a s r e fe r r ing to the fo r mal a s p e c t s of a s e r i e s and the de duc ib l e l a w s of inte r r e latio n sh ip b e­ twe e n te r m s o f th at s e r i e s , a s o r i g inally de fine d. It wil l b e s e e n t h a t any g iv e n t y p e o f p r o po r tio na l i ty ha s t wo a s­ pe c t s : ( l) the t yp e o f int e r r e la ti o n s h ip po s s e s s e d b y the ba s i c s e r ie s o f inte g r a l te r m s , a n d (2 ) the typ e o f inte r r e la t i on s hip p o s s e s s e d b y p r o po r t i o n s c o mp o s e d o f tho s e te r m s . Alluding t o the typ e s o f p r o po r tional s e r i e s de s c r ib e d i n the A r ithm e t i c a , B ra dwa r dine s a y s that , of the t e n that B o e th i u s di s c u s s e s , th r e e a r e of i m p o r­ tanc e fo r the ta s k in h an d : a r i thme t i c , g e o m e tr i c , and h a r m oni c . T he fun da m e nta l c ha r a c t e r i s t i c o f the a r ithm e ti c s e r i e s i s that the a r i thm e t i c a l di ffe r e n c e s b e twe e n the s u c c e s s iv e t e r m s a r e e qual (e . g . , the s e r i e s , 1, 2 , 3 , . . . , i n whi c h the diffe r e n c e i s 1, o r the s e r i e s , 1, 4 , 7 , 10 , . • • , in whi c h the di ffe r e n c e i s 3 ). The funda m e n­ ta l c ha r a c te r i s t i c o f g e o m e t r i c s e r i e s , on th e o the r hand , i s that p r o p o r t io n s b e twe e n s uc c e s s i ve te r m s a r e e qua l (e . g . , the inte g r al s e r i e s , 1, 2 , 4, 8 , . • . , is a g e o m e t r i c one , b e c au s e the p r opo r ti o n s

1 2 4 . . . b e twe e n it s s uc c e s s i ve te r m s a r e e qua 1 , i . e . , = 4 = 8 . . . ) . Ha r2

m onic s e r ie s ha s, a s it s fun da m e n ta l c ha r a c te r i s ti c , th e p ro pe r ty o f e qua l i ty (a m o n g t h r e e t e r m s , c on s t ituting th e inte g r a l s e r i e s) o f the p r o p o r ti o n s o f the e xt r e me te r m s and o f th e diffe r e nc e s b e­ twe e n the fi r s t and s e c o nd a n d s e c o nd and thi r d te r m s (e . g . , the inte g r a l s e r i e s, 6 , 4 , 3 , i s a ha r m on i c one , b e c a u s e the p r o po rtion of the fi r s t te r m to the la s t is e qua l to the p r o po r ti o n of the di f­ fe r e n c e b e twe e n the fi r s t a n d s e c o nd , to the di ffe r e nc e b e twe e n the

6

6-4

= 4_ 3 . 3 W h i l e a r ithm e tic a n d g e o m e t r i c s e r i e s may b e p r oduc e d i nd e f­ inite l y, ha r m onic p r op o r t i on s a r e l i m i te d to th r e e te r m s . Fu r the r, the fi r s t two type s o f pr opo r ti o n may be e i the r c on tinuo u s or d i s­ c ontinuou s , whe r e a s the thi r d c an only be c ontinuou s . C o ntinuity in a r ith m e tic p r o p o r t i onality m e a n s that the e qua l diffe r e nc e s po s­ s e s s c o m m o n te r m s (a s i n the a r i thme tic p r o po r ti on s , 3 -2 = 2 - 1 , o r , 8 - 6 = 6-4 = 4-2 ). Di s c o nt i nuity m e a n s that the e qual di ffe r e nc e s do not po s s e s s c o mmon te r m s (a s i n , 6 -4 = 3 - 1) . C o ntinuity i n g e o­ m e t r i c p r o p o r tionality m e a n s that the e qual pr o p o r t i o n s po s s e s s s e c on d a n d thir d :

4 c o m mon te r m s {a s i n the g e om e t r i c p r o po r ti. o n s , 2

=

2 ) ff1 s c o ntlT .

26

I N TROD U CT I ON .

14

nuity m e a n s that the y do not po s s e s s c o m m o n te r m s (a s m 7 =

6 I

=

2 T).

B r ad wa r di ne' s d i ffe r e ntiation of c o ntinuou s fr o m di s c ontinuo u s p ro p o r tio n s , inv o l v in g , a s it d o e s , the Ari s to t e l ian di s t inc tion b e­ twe e n c o ntinuo u s and di s c o n tinuo u s qua nt i t i e s and the inc o m m e n­ s u r abi lity o f quantiti e s o f d i ffe r e nt g e n e r a , int r oduc e s a c r uc ial pa s s a g e in the De pr o go r tionib u s . Ho w c an nu m e r ic a l p r opo r t io n s , c o m po s e d o f di s c r e te entiti e s , e xpr e s s phy s i c a l p r o p o rtion s , c o m­ po s e d o f c o ntinuo u s e ntitie s ? How c an a p r o po rtion o f fo r c e s b e a c c u ra t e l y r e l a t e d t o a p r o p o r t ion o f v e l o c itie s , who s e t e r m s m u s t b e o f a di ffe r e nt g e nu s ? T o th e fi r s t que s t ion B r adwa r d ine g i v e s no a n s w e r, p r e s umab l y c ont e nt w i th the A r i s tote l ian do c t r ine that c ont inua a r e nume r ab l e , tho ugh not num e r a te d . To the s e c ond qu e s­ t i o n , he r e p l i e s by c iting , a s a utho r i ty, th e EP-i s t o la de P-rO P-O r ti o ne QP-r O P-O r tiona l itate of Ahmad Ibn Ju s u f . 74 In the E p i s tola i t i s c l ai m ed th a t , tho ugh c o ntinuo u s p r opo r ti o n­ a l ity c a n only ho ld b e twe e n te r m s o f the s a m e g e n u s (b e c au s e o th e r­ wi s e it would not c ontain c o m m o n te r m s) , di s c o ntinuo u s p r o po r ti on­ a l ity may ho l d b e twe e n p ro po r ti o n s o f h e t e r o g e n e o u s kind s . Why A r i s to tl e had no t , h i m s e l f , made thi s r e s e r v a t i o n i t i s , pe r hap s , i d l e to s p e c ulate , b ut the le giti m a c y o f s uc h p r o p o r tion s wa s c e r­ tainly known to hi m . B r adwa r d ine' s e xa m p l e o f the di ve r g e nc e o f g e n u s pe r m i s s ib l e b e tw e e n te r m s i n di s c o ntinuo u s p r opo rti o n i s d r awn fr o m the c la s s i c s c ie nc e o f ha r m on i c s : a s i s the p r opo rtion b e twe e n the l e n g th s of two v ib r ating s t r i n g s , s o is the pr opo r tion b e twe e n the tone s whi c h the y, r e s p e c t i v e l y, p ro d uc e . In l i ke ma nne r, add s B r adwa r dine : a s i s the p r o p o r tion b e twe e n two m o t i on s , s o i s the p r o po r ti o n b e twe e n the i r r e s p e c t i v e v e l o c itie s . It d o e s not g r e atly matte r that we may fi nd it d i ffi c ult to unde r s ta nd the e xac t s e n s e in whi c h a m o ti on d i ffe r s i n g e nu � fr o m it s v e l o c ity ; i t i s at l e a s t s uffi c i e ntly c le a r that the ve l oc ity of a motion is a phy s i c al l y a s s o c i ate d a s pe c t o f the m o tion qua m o t i o n in the s a m e s e n s e that the tone p r o d uc e d b y a v ib r a t i n g s t r i n g i s a s s o c ia t e d with the v i­ b r ation o f tha t s t r i n g . In the c a s e o f B r adwa r dine' s func t i o nal e q ua­ t i o n o f fo r c e s to v e l o c iti e s, d e v e lo p e d in Chapte r III o f the T r ac tatu s , the j u s ti fi c a t ion l i e s in the fa c t that the two s ide s o f the e quation r e p r e s e nt di ffe r e nt a s pe c t s o f the s a m e m o t i o n , o n e s i d e e xp r e s s in g it s ti m e- di s tanc e , o r kine m ati c , a s p e c t , t h e othe r e xp r e s s i n g i t s fo r c e- r e s i s ta nc e , o r d yna m i c , a s pe c t . In hi s a r gum e nt s in fa v o r o f t h e ju s ti c e o f s uc h e quation s , Ah m ad ib n Ju s u f e xpa nd s thi s d i ffe r­ e ntiation of g e n u s to a c on s id e rab l e e xtent , a d m i t t i n g the e qua tion of p r o p o r t ion s o f : l i n e s to l i n e s , to a r e a s to a r e a s , to vo lu m e s to volume s

.

.

to t i m e s to ti m e s

.

1.e . ,

r,L ,

may

=

A

V

T

A ' == V ' - T ' · He r e a g ain , the r e fo r e , we e nc o unte r the c l a s s i c c onfide nc e in the m a the mati c a l c ha r a c te r of phy s i c a l pr o c e s s e s to whi c h a tte n-

DE S CRI P TI V E A ND CRI TIC A L ANA L YS IS

27

tion has al ready been d rawn . The heritage of a tradition which in­ cluded both number and physical objects within the concept of quantity and included music and astronomy among the exact sci­ ences became, inevitably, a powerful impulse to the development of a mathematical physics : a heritage to which today we stand in great debt, however much such an unhesitating confidence in the commensurability of the conceptual and the actual , the mathemat­ ical and the physical, may occasionally be deplored as a philosophic view . At any rate, it is of no small significance that, when B rad­ war dine speaks of propo rtion, he finds it legitimate to think of them as composed of existential terms, and that, consequently, his problem is to justify the equation of different genera of such terms, rather than to justify (as the contemporary theo rist often feels he must) the commensu rability of the mathematical ideal with the physical reality . The second par t of Chapter I closes with the definition of six proper ties of geomet ric propor tions which will be of use later in the argument . Basically, they follow definitiones given in Campanus' " Euclid � Book V (given, the geomet ric series : 8, 4, 2, 1 ). 8 4 · · · 1 y" · = , then "permu tat ive 1 . Permutatim P-r OP-O r tlona 1 ia · : since 2 T (by interchanging the consequent of one propor tion with the antecedent of the other) ¾ = } . (Cf. Campanus, Definition 1 2. ) 4 l · · · proP-or hona , th en " contran· 1 y" (b y 1 ia: · since 2 • Econt rario = 8 2 inter changing antecedents and consequents in each propor tion .

¾ � }.

separately) (Cf . Campanus, Definition 1 2. ) 8+4 4+2 . . . . . . ) -, t hen "d 1s: since 1 itas · 3. D isiuncta ( simP-1 ex�roP-orhona =2 junctively" (by eliminating the value of each consequent' added to the antecedent) ¾ :; } . 7 5 ( Cf . Campanus, Definition 1 4.) 76 8 4 . · t 1· ve1 y" = , t hen "conJunc · · P-ropor h· ona 1 1tas : since 4 . C oniuncta 4 2 8 4 4 2 • = ; (Cf . Campanus , Definition 1 3.) 8 +4 4+2 . . . = -- t hen, " conversely,. : since 5 . C onversa P-r O P-O r tlona 1 1tas 2 {by substituting the antecedents, 8 and 4, for the consequents, 8 4 4 2 4 and 2 ) = : . (Cf. Campanus , Definition 1 5 , in which the

!

!

---:r-

--:r-

term, eversa, is used to denote what is here called conver sa . ) 6 . Aequa proP-or tionalitas : given the series 3 , 2 , I and 6 , 4 , 2, these are in "equal propor tionality" because propor tions between cor4 6 2 3 responding te rms in the two series are equal : = , = , 2 2 4 T } = } . (Cf. Campa nus, Definition 1 6. )

28

INTRODU C T I ON

The third and final portio n of Chapter I co n tai n s the heart of the mathematical material o n which Bradwardine's own theo ry, to be expounded in Ch a pter III, depends. It co nsists of the setti n g up of eight axioms or sup-P-ositiones (hypotheses, in the Greek sense) and in the deduction, from them, of eight conclusio n s . Expressed in modern notation, the axioms are as follows : 1. All proportions are equal whose denomina tions are equal. 2. Given A and C, positing a m ean , B, such that

A

B -

B then A =

c'

A

c

B

3. Given A and N, positing means , B, C, M, ... , such that = B - C C M A A .B . C M ' t hen ·.· .. . . . M N N B C M N A = B C C 4. If A = B . then , given C , A = B and C C . (Cf.Campanus, Proposition 7 . ) 5. If A

f B,

a nd

given C, then , if A > B, � > � , a nd if A < B , � < � ­

i-�,

(Cf. Campanus, Proposition 8 . )

6 . If � = � , then A = B , or if Proposition 9 .)

then A = B. (Cf . Campan us,

A B C A B 7. If - = - = - ' then - = - . (Cf. Campanus , Pro positio n 25. ) B C C D D

A

B

C

= C = 0 , and A > B , then A+D > B+C . (Cf. Campanus, Pro pB ositio n 25.)

8. If

Axioms 4 to 8, inclusive, follow closely the i ndicated Propositio n s in Campanus' Euclid, to which the text of the De P.rOP.ortio n ibus, itself, gives accurate reference . The first three, however, present something of a mystery . T hey are not contained in Euclid , in the form give n by Bradwardine . He cites, as authority for them, a wo rk called De pro2ortionibus , which is unknown at presen t.77 Axiom 1, being very simple and more a matter of definitio n than of mathem a tica l postulatio n , it is, perhaps, n ot surprisin g that n o autho rity is cited fo r it . Axioms 2 a nd 3 co rrespo nd rather closely to C ampa nus' Defin itio ns 10 and 11, but have a ge nerality n ot pos­ sessed by Euclid 's version . Campan us, himself , remarking on this lac k of generality, i n his comment on Definition 11,78 explai n s Eu­ clid 's failure to generalize the axiom beyo nd four terms by the f a c t that th ree dimension al solids, which represe nt a natural limit to geometric abstractio n, are denominated by no more tha n fo ur terms. In his commen tary, Campanus, i ndicates what the pro jec­ tion to " n" terms would yield ,79 but Bradwardine appa re ntly was acquai nted with some other work on the subject in which the ge n-

D E S C R I P T I V E A N D C R I T I C A L A NA L Y S I S

29

e ral

form of the axiom was dire ctl y e xpresse d and which would, th e r e fore, pro v id e a more co n v e nie nt r e fere nce. Th e eight theorem s d e veloped from th e pre c e ding axiom s ar e as fol lows. 8 0 A 2 . A B A Gi v e n Th e ore m I . , then = = B B C C ( ) � = � (Axiom 1 )

Proof.

. A A B (Axiom 2 ) = C B · C

... � = � .

�

=

(

�y

= (�)

3 A B •' and ' gi· v e n G i· ven � = � = � then � B C D' D B B - CC D L then (i· f " n" = the num b e r o f t e rms in .. D = L . = M' n1 A A th e series} M = ( ) B

Theorem II.

Proof as for Theorem I, plus the definition s of triRla, QUadruP-la, etc . T heorem III . Proof .

Given A

> 2B and B = 2C, then � < (�)'

Posit D, such that � = �, then D D

lC

(Axiom 4 } an d

t C (Axiom 5 and ex hY-P-Oth

.". D < C

e si}

� > � (Axiom 5 } and C D � =

(�Y

(Theorem I )

. · . � < (�)2 Theorem IV .

Given A = 2B and B > 2C, then �
(�)'

Proof as for The orem III . Theore m VI .

Given A = 2B and B < 2C , then �

Proof as for Th e ore m III .

> @J

30

IN T ROD UC T ION

B A B B J. A Theorem VIL If B > A, then A ..i 1 A and A i- A and A B < A ' then � i-l � and � 11 � J. � and � 1 A A A A A A

Proof .

Po s it that { =

({)2 and that (?Y

= { , the n

4, 2, l are continuou s proportional s

. ° . ! = (�) 1 2

2

} = · 1

• •T

=

and ! = y'""4 and � = I 2 T

y-:rT

(ex hy_P-othe s i)

T and

l = 2 (impo s s ible) 8 1

2

Theorem VIII . If A > B, then

A

B

-1.. -r

B

A

and

A

B

f AA ; a nd

y-:rT

?

. 1f

= } a nd

(Theorem I )

B 1' A

J.

Proof a s for Theorem V II . The foregoing axiom s and theorem s pre sent n o particular diffi­ culty, a side from Theorem s V I I and V I I I . The se a re of crucia l im­ portance to Bradwardine· s attempt to s ave the Peripatetic the ses, that no motion re sul t s either from an equilibrium of force s or from a ca se in which re sistive force is greater than motive force . The intrica te argument which Bradwardine submit s in their proof i s reducible, in the final analy s i s , t o a demon s tration o f the discon­ tinuity of proportion s of le s ser inequa lity, equality , and grea ter inequa lity. He show s them to con stitute di screte genera in a math­ ematical sen s e analogou s to the phy sically di screte genera of force­ re sis tance re la tion s con s tituted by Ari stotle' s three dicta = ( l ) that no motion re sult s from a proportion of force s in which re sis tance is greater than motive force, or (2 ) in which re si stance equa l s motive force, and that ( 3 ) a proportion of force s in which motive force exceed s re s i s tive force re sults in a motion proportional to the force s . To put the principle on which the s e two theorem s depend in the fa miliar language of Aristotle' s argument concerning the void : ju st as there can be no motion in a void, because there is no proportion between the zero re sis tance of a void and a motive force having a po s itive value, however s mall, ju st s o there can be no proportion by which any proportion exceed s a proportion of equality, becau se a proportion of equality pos s e s s e s a zero a mount of inequa lity, and a ny exces s over a proportion of equa lity will be in an infinite pro­ portion (vi z . , no propo rtion at all) to it . Thi s i s not to argue that 2 is not greater than l but that } is not greater than + by any given

prc,portion (i.e . , root or power) ; there can be no exponent which ,

D E S C R I P T I V E A ND C R I T I C A L A NA L Y S I S

Jl

when applied to } , wil l yield } . 82 So long as vel ocities are taken (as they are by Bradwardine) to vary according to the " proportion of proportions" and not simply according to a mathematical excess it would certainly fo l l ow that an equilibrium of forces would pro­ duce zero velocity, that no disequilibrium of forces would bear any proportion to an equilibrium, and that (insofar as the mathematical argument is concerned) there would be no proportion expressing the relation between equal and unequal proportions. Having established the mathematical foundations of the particu­ lar theory of vel ocities which is to be put forward in Chapter III {b y the simple expedient of formulating certain properties of the proportional interrelationship of terms belonging to a geometric series), Bradwardine commences Chapter I I , more Aristotelis , with the destructive criticism of rival theories. These are four in number, each one based on a different interpretation of the meaning of Aristotle's statement, that velocities are proportional to the mo­ tive and resistive forces invo lved . An examination of Bradwardine's mediaeval precursors in this field of theoretical physics makes it quite apparent that (with the exceptio n of Avempace· s 83 ) none of the theories advanced are for­ mulated in such a way as readil y to suggest a strictl y mathemat­ ical statement . To do these men justice, it must be reali zed that this was no t their endeavo r . Instead of attempting such a course , they were more interested to decide the nature of the constitue nt principles on which the successive character of motion is gro unded . Any mathematical laws implied by their various conclusions re­ mained unformulated, and it is, therefore, to some extent misleading for a later writer so to formulate and crystallize them . With Bradwardine, the case is quite the opposite . Appro aching the pro blem of e stabli shi ng an adequate law of motion fro m the standpoint of a mathematician rather than that of a phil osopher, it is the mathematical formulae which appear explicitly and the phil­ osophical theory on which they are ultimately grounded , that re­ mains implicit. It is indeed, quite conceivable that Bradwardine might have been almost wholly unconcerned with the phil osophical issues invo lved , but simply interested in de m onstrating the math­ ematical fallacies impl ied in previous theories and in showing how a proper understanding of the mathematics invo lved in the manipu­ lation of expo nential series could solve any objections against the c lassie notion that velocities are pro portio nal to proportions o f forces. In any event, this co ncern to eradicate mathem atical fal­ lacies and to demonstrate a law of motion mathematical ly consist­ ent with the classic axioms is what most clearly appears to the reader o f the De P-rOP-ortionibus . As has already been pointed out , there had been , genera lly speak­ ing, onl y two fundamental ly divergent tendencies i n the attempt to

32

INTRODUCTION

fo rmulate an ad e quate the ory of the r e lation of ve loc itie s and fo rce s . Fi r s t , the r e wa s the v i e w which, larg e ly on logico-mat hemat ical grou nd s , s ought to account fo r ve locity in te r m s of an ab so lute . S e cond , the r e wa s the te nde ncy to s e e ve locity a s a r e lative func­ tion . The for me r te nde d to vi e w re s i s tive force a s s omething to be s ubt r acted from motive fo rce (th e re s i duum be ing the cau se of what­ e ve r ve locity re maine d) ; th e l atte r e nvi sage d v e locity a s b e ing s imp ly the d i re ct r e sul tant of t he i nt e raction of oppo s e d force s . Whethe r through th e conc e p t of " impe tu s ;• employed by B uri dan , or th rough th e concept of th e l ogi co-mathe matical re s i s tance of pur e d i me n s ion, it s e lf, th e form er th eory wa s committ e d to the imagination of an ab so lute ve locity . Of the fou r " e rroneou s the orie s" which Bradward ine treat s i n hi s s e co nd chapte r, the fi r s t i s of the ab soluti s t s o r t of wh ich w e have ju s t no w b e e n s p e aki ng. T h e s e cond r e p r e s e nt s a compromi s e b e tw e e n th e ab soluti st and the r e lativi s t . The fou r th th e ory re pre ­ s e nt s th e v i e w of tho s e who, on the ground thc1.t fo rce s and re s i st­ ance s are " inte n s ive" magnitude s, argu e d that a p roport ion of fo rce and r e s i stance is not to be ta ken a s mathe mati cal ly re latabl e to a ve locity (d i s tance an d time) , s i nce ve l oci ty i s an " e xte n s ive" mag­ nitud e . The thi r d th eory, on th e other hand, re pre s e nt s one fo rm of the s e cond main te n d e ncy in th e d e v e l opme n t o f a the o r y of ve locit y . T hi s for m of the th eo ry (that commonly attribute d to Ar i s totl e ) make s ve l ocity the dire ct o r s imple function of a proporti o n o f fo rce s . It i s in s olving the pa radoxe s con s e que nt upon thi s form of the the ory, by cha ng ing the di rect functi o n to a ge ome tri c on e , that B rad wa rd i ne achi eve s hi s re i nte r pretation, out l i n e d in Chapte r III of th e D e groP.ortion ibu s . T he ory_l. - Afte r fir st citing t e x t s from Ari stotle and Ave r r o e s whi ch might be ta ke n a s s up porting th e th e o ry that " th e p roportion of ve loc iti e s in moti on s follows the e xce s s of th e force of th e move r ove r that of the t hing mov e d ;' B radwardine rai s e s s e ve n d e s truc­ t ive argume n t s again s t it . In doi ng s o, h e fo llow s , a s e l s e where , the cus tomar y p r actice of givi ng, fir st, th e argument s which mak e t he i r ap pe al primarily to t he ne e d for a co n s i ste nt r e n d e ri ng of the " author iti e s" and of the n ad ding oth e r argume nt s wh ich appeal more ind e p e nde n tly to re a son and e xp e r i e nce . In the p r e s e nt ca s e, re fu­ tati on s 1- 3 ar e bas e d d ire ctl y on ce rtain pas sage s from A r i s tot l e a nd Av e rroe s , r e futation s 4- 6 e xp o s e furth e r impo s s ib l e con s e ­ que n c e s , and re futation 7 appeal s t o commo n human e xp e rie nce . For the sake of d iagrammatic s impl i city, we s hal l e x p r e s s th e p r e s e nt the ory by th e formul a , k V = F - R where F = mot iv e force, R = r e s i sti v e fo r ce, o r mob i l e , and k = a conve r s ion co n s tant . M 1 . Given the above theo r y , it fo l low s that if F move s R through S ( s pac e ) in T (ti me) , � will not move � th rough S i n T , but on ly 2 2

D E S C R I P T I V E A ND C R I T I C A L A NA L Y S I S throu gh

zS .

Fo r , i f kV = F - R then

33

F R z z - 2. kV

=

Thi s c ont radic t s F A ri s totl e' s s ta t e m e nt that , i f F m ove s R th rough S in T , the n 2 R thro ugh S in T . The r e fo r e , thi s i s a fal s e the o ry . move s 2 Thi s re futa tion re s t s , o f c o ur se , o n the g r ound that a con si s te nt inte r p re tation o f Ar i s totle i s re qui r e d by both partie s . Tho u gh s uc h ha r dy s pi r it s a s Olivi we re qui te c apab le o f s aying c r ue l thing s about the " Phi lo s oph e r' s" the o r i e s , 8 5 i t doe s appe a r g e ne rally t r ue that s uc h ob s e rva tion s a s that "halving both the fo r c e s a nd the r e ­ s i stanc e s doe s n o t c hange t h e ve loc ity" w e r e c o n s ide r e d t o be suf­ fi c i e nt ly ob vio u s to s e r ve a s axio m s fo r all . 2 . On the ba s i s of thi s the o r y, i t a l s o fol lo w s that , if F m ove s R thr ou gh S in T , the n two s uc h fo rc e s t og e the r ( F+F ) will not move two s uc h r e s i s tan c e s (R+R ) , thro ugh S in T, but thr o u gh 2S . Fo r, i f kV = F - R , the n 2 kV = 2 F - 2 R . T hi s r e sult i s fa l s e , b e c au s e it , to o , c ontr ad ic t s A r i s to tl e' s s ta te m ent that the c o mb ine d fo r c e s and r e ­ s i s ta nc e s will p r oduc e the sa me ve loc ity . 3 . On the ba s i s o f thi s int e rpr e ta ti o n o f Ar i stotl e , i t al so fo llow s that , fro m ge o me tr i c p r o p o r tion (i . e . , e qual ity o f ' pr op o r ti o n s) the re do e s no t fo llow e quality o f ve loc ity , fo r the r e i s no e qua lity of e x c e s s e s . Fo r e xample : } = }, but 6 - 3

f 2- 1.

Thi s r e sult c o n­

t radi ct s Ar i stotle' s state ment that e quality of v e lo c ity depe nd s o n e qua lity in the pr opo r tion o f move r s t o mob ilia , a nd i s , the r e fo r e , al so fa l s e . No r c an it b e a r g ue d that Ar i s to tl e he r e inte nd s " a r ithme tic" r a the r than " g eo me t ric" p ropo r tion (i . e . , e quality o f diffe ren c e s rathe r than e qua lity o f pr opo r tions). Suc h an int e r p r e tati on wa s s hown to be fa l s e in r e futatio n s 1 and 2 . In the c o ur s e o f the pre sent r e futation , a s e l s e whe r e , B radwa r­ d ine a l s o c i te s Ave r r oe s' a rg u m e nt s ; Ave r r o e s had , a s we hav e s e e n , al s o r e c o gni z ed the inc on s i s t enc ie s c on s e que nt upon the sup­ po s i t ion that A r i s to tle wa s s pe aking o f " a r ith me tic" p ro po r tion (i . e . , nume ric al diffe r e nc e s). 4. It would al s o fo llow that a given " mixed body ;• p o s s e s s ing in­ te r nal r e s i s ta nc e , would move fa s te r in a ple num than in a void . For l e t A = s u c h a mixed body, fal ling in a m e dium , B , and l e t C = a body o f pure ea r th , s uc h th at the m otive fo r c e o f C i s l e s s tha n the r e sulta nt motive fo r c e of A . The n l e t B b e r a r i fi e d until C m ove s in B with the s a me ve loc ity a s A move s in a void . Then l e t A be pla c e d i n the s a m e m e dium a s C . I t wil l m o v e fa s te r , s inc e it ha s a g r ea t e r re s ultant m otive fo r c e , and , s ine e C now mo ve s in B wi th the s a me ve loc ity a s that o f A in a vo id , A wil l move fa ste r in thi s Rle num than in a void . It wi l l be ea sily s e en , upon the mo st c ur s o r y e xa mination , that

34

IN T RODUC T ION

i t i s impo s s ib le to s peci fy a medium o f s u fficiently low r e s i s tance so tha t ( g iven the fo rm ula , kV = F - R ) A will move at the s a me vel ­ ocity i n it a s C move s in the void , wi tho ut ma kin g the re s ul ta nt fo r ce of A equal to the for ce of the s imp le b ody , C . B r a dwa r dine , hims el f , wa s do ubtl e s s awa re o f thi s ma thema tical di lemma , fo r he conclude s thi s fou r th refuta tion by poi nting ou t that Ar i s totle po s i t s a n unl imited increa s e of vel ocity co r ;e s pondent to a n equiv­ a lent decrea s e o f the r e s i s ta nce of the med i um . Ther efo r e , in the l a s t ana ly s i s , thi s r efuta t ion , l i ke the p r eceding one s , r e s t s upon a demon s t r a ti on of the theo r y' s incon s i s tency with a given d ictum of Ar i s totle , r ather than on a ny impo s s ib il ity to which i t lead s , i n i t s o wn ter m s . Thi s type of a r g umen t typefie s , in a mo s t i ntere s tin g way , the s cho la s tic att itude towa r d Ar i s to tle . The Phi lo s opher wa s not ta ken to have wo rked out the deta i l s of a l l hi s theor ies no r to have been in fa l lib l e , � P-r ior i , in a ll hi s s ta tement s . It wa s , o'n the othe r hand , felt t ha t he wa s e s s en t ia l l y cor r ect in hi s natu r a l ph ilos ophy, and that one co uld p r ope r ly check the va l id ity of a contempor ar y theo r y b y seein g whethe r i t wa s a n i nte rpr eta tion con s i s tent with Ar i s ­ totle' s obse rvation s , a s fa r a s t ho s e ob se rvation s went . 5 . It wou ld a l s o fol low t hat, if F - R < F ' - R ' , then V < V ' , which i s vi sib ly fa l s e . For (l etti n g r = the i nte r na l r e s i s ta nce of a body, a nd m = the r e s i sta nce of a g iven medi um) let F - r > F ' - r ' , and let m be s uch that ( F - r) -m = ( F ' - r ' ) -m ' . Then the velocitie s a re equal , but the d i fference s a re unequa l , whi ch i s cont ra r y to the hypothe s i s . 6 . It follow s a l so that, i f F = Z R or > Z R , the velocity ca nnot be doub led in a no the r med ium, for, in tha t ca se , doub l e the di fference between F a nd R would be equal to F . Fo r example , i f F = 8 a nd R = 4 , prod uce a veloci ty , V, then to doub le the velocity (a cco rdi n g to the p re s ent theor y) one wou l d have t o double the " exce s s ;• o r d i ffe r ence . The "exce s s" o f F ove r R i s 4 ; doubli n g it wou ld ma ke it 8 (the l imit ing velocity which F can p ro duce, when R = 0) , a nd , t herefo r e , the fo rce i n que s t i on can­ not move any fa s ter tha n thi s , even thr ough a vo id . Thi s wa s s hown to be i ncon s i s tent with Ar i s totle' s po s ition , in re futa tion 4 . 7 . Since i t fol l ows tha t , i f F - R > F ' - R ' , then V > V ' , i t a l so fol­ l ow s tha t t he g reater the di fference between F a nd R, the fa s te r w i l l be the motio n , a nd , s ince a s t ron g man exceed s the powe r o f wha t h e moves by a gr eate r exce s s tha n a fl y exceed s anything tha t i t move s, he s hould move it fa s ter. Thi s i s ea s ily seen to be fa l se , b y e xpe r i m e n t . B r adwa r di ne , the re fo re , concludes tha t the propor tion of v eloc­ itie s does not fol low the " exce s s" of power of mover over mob ile a nd a dd s that what Ari s to tle a nd Aver roe s meant , when they s a i d that vel oci ty " fol low s" s uch an exce s s , wa s simply that mot ion re­ s ult s from a p ropo r tion o f g r ea ter i nequa lity in which F , of nece s­ s ity , exceed s R .

DESCRIPTIVE

AND

CRITICAL

A NA L Y S IS

35

Th e o r y....!L -T he s e c o nd th e o ry o f v el o c i ty whi c h B r a d wa r di n e c r it­ F R 86 i c i z e s may be e xpr e s se d thu s : kV = In none of the p r e ­ � . B r a dwa r dine the o r i s t s who m w e have e xa mine d do e s thi s c o mb i n e d va r i ant o f The o r ie s I a nd I I I r e c e iv e a ful l d e ve lopm ent ; fr o m th e v e r y c u r s o ry tr e at m e nt a c c o r de d i t , we m i ght , ind e e d , b e t e m pt e d t o b e li e v e t ha t n o o n e had r e a ll y c on s id e r ed it s e r i o u s ly. The o r y I I i s , howe v e r , s u g g e s te d b y Ave r r o e s' C o m m e nt 3 6 , Ph y.: s i c s V I I (whic h B ra dw a r d in e c i te s a s i t s autho r i ty) a nd wa s , the r e fo re , n o t t o b e o v e r l oo k e d b y any o n e w r iting a t r ea ti s e o n t h e the o r y o f ve l­ o c i ty . It i s int e re s ti n g to not e that thi s the o ry wa s , at la s t , s e r i­ o u s l y ta ke n up by Gio vanni Ma r lian i , in the fi ft e e n th c e ntur y . 1 . The fi r s t o f B r adwa r d in e' s t wo r e futa ti o n s o f thi s vie w i s mo s t inge nio u s . He c l ai m s that , i f F - R = R , then "'n o m ov e r c a n m ov e a ny m ob il e fa s t e r o r s lo w e r than that mo t i o n :' T hi s fo ll o w s , b e ­ c au s e {in T he or e m V I I , Chapte r I ) i t h a s b e e n d e m o n s t ra t e d that no p r opo r ti o n c an b e g r e a t e r o r s ma l l e r than a p r o po r ti o n o f e qualF-R . . · At th e ity , a nd , i f F - R = R , th e n -y 1 s a p r o p o rt ion o f e qua 1 1ty. s a me ti m e , the c a s e in whi c h F - R = R would n o t b e one o f e quil i­ b r ium { s inc e F e xc e e d s R) a nd , henc e , wo uld r e p r e s e nt a d e t e r mi­ nate ve l oc i ty. In b r i e f , i f T he o ry II w e r e t r ue , e v e r y thi ng that mov e d would m o v e with the sa me ve l o c ity . 2 . A m o ving fo r c e , m o r e o v e r , move s the who l e m ob i l e wi th it s who l e fo r c e a n d no t by an e xc e s s o f fo r c e ; 8 7 the r e fo r e , a mo ti o n a nd i t s ve l oc i ty , p r i ma r il y a nd e s s e nti a lly , fo llow the p r o po r ti o n o f the who l e m o v e r to th e who l e m ob i l e . A s no t e d ab o v e , B r a dwa r dine d o e s not de v e l o p a g r and r e futa tion of thi s t he o r y. As a the o ry o f a r ithm e t i c p r o p o r t i o na l i ty i t ha d b e e n s uffi c i e ntly s hake n i n the c r i t ic i s m o f The o r y I ; a s a the o r y o f ge o­ m e t r ic pr o po r t io na l ity , it wa s too und e ve lop e d to de s e r ve sp e c ia l t r e at me nt . Th e o r y I II . -B r adwa r dine' s r e futa t i on o f T he o ry III i s hi g hl y s i g­ nific ant fo r an unde r s ta nding of th e o n e whic h he , h i m s e l f , adduc e s i n Chap t e r III o f the D..�..:p r ogo r t i on ib u s . Thi s thi rd the o r y i s the one whi c h s e e m s mo s t ob vio u s l y i m pl i e d b y A r i s to t l e' s s c a nty t r e at­ m e nt o f the s ub j e c t : the the o r y that v e l o c it y i s a di r e c t p r opo r t i on al fun c t io n o f fo r c e s (i . e . , kV

= � ).

0 s te n s ibl y, thi s i s t h e the o r y he ld

b y th e m e d ia e val Pe r ip a te tic s . Tha t ve l oc i ty should b e a s i mple func tion o f the p r o po r t i on o f o ppo s e d fo r c e s l e a d s , ho we ve r , t o c e rtain i n c on s i s te n c i e s with o the r Ar i s t o t e lia n a xi o m s c on c e rning mo tion , on th e b a s i s o f whi c h B ra d wa rd i ne i s fo r c e d t o r e j e c t t h e th e o ry . O f the two g e ne ral c r itic i s m s rai s e d a g ain s t The o r y II I , the fi r s t i s that i t la c k s g ene ral ity , s i nc e i t po s i t s tha t e i the r F o r R m u s t b e c o n s ide r e d a s c on stant , i f two v e l o c it i e s ar e t o b e c o mpa r e d .

36

I N T RO DUC T I O N

Such a criticism might seem unwarranted, if Theory II I is accu­ rately expressed by our formula : k V = � . but, as will be easily understood from a reading of the text of the De proP-ortionibus, translation into modern notation of verbally expressed mathemat­ ical theories may easily misrepresent , to some extent, the exact (and sometimes the inexact) meaning of the original . Keeping in mind that Theory III is more accurately expressed by the two formulae, kV =

!:'C

{where c = constant resistance) and k V = � (where

R = constant motive force) it is, perhaps, permissible to return to c the less cumbersome k V = � . since all the criticisms , except that

concerning lack of generality, apply equally well to the latter fo r­ m ulation. The second general criticism (divided into three main arguments) is that Theory III leads to false consequents . 1 . The first false consequent is that any force, however small , k can move any resistance, however large . For if kV = � ' then : =

kV F F . . Since , however large F may be , it will be possiand � = ZR nR ble to posit n such that n R will be larger than F, this theory is seen to be a violation of the universally accepted Aristotelian axiom, that F must exceed R, if there is to be any motion . F 2. The second false consequent is similar. For if kV = ' then R F kV because , any mobile can be moved by any force. = n nR 3. Furthermore, sense experience teaches the opposite of this view, for we see, for example, that if one man can scarcely move a heavy rock, two men working together can move it much more than twice as rapidly. The same principle is illustrated in the case of clock weights; to double a weight may more than double the speed of its descent. F . Therefore , Bradwardine concludes , k V

fR

T he remainder of his discussion of Theory III is concerned with an atten:1p� to show that Aristotle , Averroes and the author of the

.

F

D

Tractatus de ponder1 b us could not have meant that kV = R = T (D = distance , T = time), but that , instead, V varies as the proportion of V F F D ' . ' 's . .1s, of course , a prev1ew ofBrad ward 1ne R or T ( 1.e . , n = R ) . Th1s own theory, as it appears in Chapter III of the De P-rogo rtionibus , but he goes on to point out that , although Aristotle , Averroes, and the conclusion concerning velocities contained in the De gonderibus V had pointed toward the n = ; formula, no one had proved it by

D E S CRI P T I V E A N D CRI T IC A L A N A L Y S I S

37

adducing mathemati cal princ iples (such as those which Bradwar­ dine sets forth in Chapter I ). Theory_l_Y.-The fourth erroneous theory takes , as has already been pointed out , a non-mathematic al view of the relation of veloc ­ ity to forces, claiming that velocities vary neither as a proportion of forc es nor as an arithmeti cal excess of motive fore e over resis­ tive force, but vary , instead, by a "natural relation of mover to moved:' This view is supported, in the first place, by the contention that force is not a body, that it therefore has no magnitude, and that c onsequently it c annot be made proportional to anything . Sec ond , the objection is raised that motive and resistive forces are of dif­ ferent genera and thus cannot be compared. Thirdly, it is objected that to suppose a proportion of greater inequality to be involved would nec essitate that this proportion be divided into " what is ex­ c eeded" and the " exc ess" ; forces, being incorporeal , c annot be thus divided. The groundwork nec essary for a reply to these objections has already been laid by Bradwardine's remarks on " proportionality" in Chapter I , and in his present rebuttal, he argues that, if there were no proportion between fore es bec ause they are not quantities , by the same argument , there would be no proportion between mu­ sical pitches. The very existence of the scienc e of harmonics is, therefore, taken as suffic ient disproof of this ob jec tion . Moreover, Theory IV holds that a mover "dominates" its mobile , and suc h a dominanc e must be in ac c ordance with some proportion : Aristotle and Averroes in s ist on this at many points. Therefore, some pro­ portion of motive and resi stive forces exists in every c ase . The arguments in favor of Theory IV are refutable : first , be­ cause it was not c laimed that opposed forces were proportionate univocally and _P-roprie, but only equivoc ally and communiter ; sec ­ ond , because anything exc eeding another thing is not , in itself, di­ vided into exc ess and what is exceeded , but only in c omparison to that other thing. (It is interesting that Bradwardine , himself , in­ vokes this princ iple , that fore es are not to be considered as divided in their ac tion, in his sec ond re futation of Theory II. ) Bradwardine' s argument against Theory IV is of intere st , not only as involving the whole problem of j ustifying a mathematic al physics in the face of that inc ommensurability between phenomena of different genera upon whic h Aristotle had insisted , but also as a step toward the establishment of force as a quantity and, there­ fore , as something measurable. This is not, of course , to maintain that force is a body. It is to support Oc kham's position (the keynote of his Philosophia naturalis) that suc h thing� as natural motions and forc es have no existence apart from bodies , but , on the con­ trary, are no more than the quantifiable behaviour of those bodies . From suc h a viewpoint it bec omes foolish to argue that , sinc e forc e

38

INT ROD UC T ION

i s no t a bo dy, it c a nnot b e me a s ur e d ; fo r c e s and ve l o c iti e s , by vir­ tue of th e i r r el a tive ly s i mpl e s pat ial b e havio u r , be c o m e , ind e e d , the me a s ur ab l e phe no m e na , 2a r e x c e l le nc e . T he old e r vie w , r e p­ r e s e nte d b y T he o ry IV, tha t l oc a l moti o n s a r e a forma fl uen s ,88 in­ c o r p o r e al , c ea s e le s s l y ge n e r ate d and d e s t ra ye d , and unm ea s urable, had to be o v e r c o m e b e fo r e real he adway c o uld b e ma de in d e v e l oping a mathe mati c al p hy s i c s . The g rand o pe ni n g o f C ha pte r I II , " Hi s e r go igno rantiae neb uli s d e mo n s trationum fl atibu s e ffug ati s , s up e r e s t ut l um ine s c ie ntia e r e spl ende a t v e r i ta s :• int r o du c e s B r adwa r d in e' s o wn s o luti on o f the p rob le m : " Th e p ropo rtion of the ve loc iti e s o f m otion s follo w s the pr o p o r t ion of the fo rc e of the mo ve r to that o f the m o ve d :' 89 B rad­ wa r dine c o m m e nc e s hi s e xe g e s i s b y quoting A r i s totle and Ave r r o e s in g e ne r al s uppo rt o f hi s vie w , a ft e r whic h he la unc he s d i r e c tl y into hi s twe l v e th e o r e m s c o nc e r nin g v e l o c ity. V ({) o r n = { n Thi s i s the mode r ni z e d s tate me nt o f the ge n e ra l the o r y quote d abo v e .9 0 I n a c e r ta in s e n s e , i t i s a n axio m r a th e r than a the o r e m , s inc e it s phy s ic a l c o m po n e nt s c annot , o f c o ur s e , be ju s ti fi ed by a math e mati ca l p r oo f . It i s , howe v e r, a the o re m (o r, in B radwa rdine' s wo r d s , a c o nc lu s io) in the s e n s e tha t it i s taken to have b e e n p r o ve d b y t h e re futation o f the fo ur pr e c eding the or i e s . T h e fo l l o win g e le v­ e n th eor e m s a r e , howe ve r, math e m ati cal ly d e m on s t r�b le fr o m the c o mb ina tion of T h e o re m I with th e ma the matic a l m ate r i a l s e t fo r th in Chapte r I .

=

The o r e m I .

V

The o r e m I I .

If F

Pr o o f.

log

=

2 R , the n

ZF

R

Let F = Z R and F '

! ' (f/ =

= 2V .

=

Z F , the n

(The o r e m I , Cha pte r I )

T he n , if { = V ,

F' 2F - (o r - ) = 2 V R R The o r e m I I I .

If F

=

2 R , the n

F fR

=

2V.

Proo f a s fo r The o r e m I I . It w i l l be s e en that Th e o r e m s I , II , a n d I II a r e de s i g ned pri ma r i l y t o vindic ate what A r i s totle h a d s a id c o ne e r ning the r e l at ion of ve­ l o c ity to a pro po rtio n of fo rc e s . Spe c i fi c al ly , the y s upp o r t A r i s­ totl e' s Phy sic s (VII , v , 2 5 0a 2 , and De cae lo , I , vi , 2 7 4a 1 - 2 ) in the c laim that v e lo c ity va r i e s with a pr opo r tion o f fo r c e s , that to doub l e the fo rc e moving a giv en r e s i s tanc e w i l l doub l e t h e v e l o c ity , and that to halve the r e s i stanc e will al so do ub l e the v e l o c ity.

D E S CR I P T I V E AND CR I T I CA L ANA L Y S I S

39

F 2 ZF If - > - then - < Z V. l' R R Proo f by Theo rem IV, Chapter I, and Theorem I, Chapter III.

Theorem IV.

F 2 If !' > , th en < 2 V. 1 R +R Proof by Theorem III, Chapter I, and Theorem I, Chapter I II .

Theo rem V.

F 2 ZF If - < - then - > 2 V. R R 1 ' Proof by Theorem VI, Chapter I, and Theorem I, Chapter I II.

Theo rem V I .

Theo rem VII.

F If R


2V.

Proof by Theorem V, Chapter I, and The orem I, Chapter III. Theorems I V, V, V I,91 and V II show that Theorems I I and III rep­ resent a special case ; if the original force producing a mo tion is not exactly twice tha t of the resistance which it moves, then, wheth­ er that force is do ubled or the resistance is halved, the velocities will in no case be d oubled . It also follows tha t, if the force is halved or the resistance do u­ bled , the velocity will no t be halved. For example : 2 F V F . G1 ven n = R let R = T , 2 V V Then - = - but - � - . 2 T 2 1 ' 2 Altho ugh Bradwa rdine does not give this Theorem , it follows from the basic fo rmula and provides a sa tisfactory defense of Aristotle 's dictum (Physics, V I I , v , 2 50a 10 - 1 1 ) that to do uble the resistance is no t to halve the velocity. It is, perhaps, idle to ask why Brad­ wardine does not draw this specific conclusion , but the awkward­ ness of indicating mathematical ro ots suggests itself as an obvious reason . Such a theo rem w ould , a t any ra te, provide a nea t solution to the dilemma posed by the twin Aristotelian dicta : tha t an equi­ librium of forces produces no mo tion , and that velocity is the func­ tion of a proportion of forces . 1 1 F Theo rem V III. If R = T or T+ , then V = 0 . Pro o f by Theorems VII and VIII, Chapter I, and Theorem I , Chapter III .

, y'z ,

This theorem is designed t o deal , in another way, with the axiom that no mo tion arises from an equilibrium of forces. This foll ows mathematical ly, on the ground that there is no proportion which expre s ses the relation of a proportion of equality to one of inequall+ 92 1+ 1 ity, i . e ., no exponentia l function of T which will yield T o r T . Bradwardine sees the pro of o f Theorem VIII as resting on this es-

40

INTRODUCTION

s e ntial di s c ontin uity , thi s la c k o f pr opo rti ona l r e la t i on b e twe e n p r o p o r t ion s o f g r eat e r ine qua l i ty and tho s e o f e qua li ty. Sinc e The o­ r e m I and the ge ne r a l Ar i s to t e l ian axiom s ta t e that m o tion s a r e p r o p o r t ional t o e a ch othe r, th e r e fo r e , s i n c e the r e i s n o p r o po r t i o n in thi s c a s e , and s inc e i t i s a l s o axio m a tic that m o t i o n i s p r odu c e d i n a l l c a s e s i n wh ic h F e xc e e d s R , no m otion c an a r i s e fr o m a p r o­ po rti on of e qua l i ty . Sinc e , a fo r tio r i , the r e i s a l s o a n e s s ential d i s c o nt i nuity b e twe e n p r opor t ion s o f g r eat e r and l e s s e r ine qua l ity, and , g i v e n t h e axi o m s that m o ti on s ar e r e latab l e b y a pr opor t ion and t hat m o t i o n d o e s a r i s e fr o m a pr o po r ti o n o f g re a te r ine quality, the r e fo r e , s inc e any moti on s uppo s e d to ar i s e f r o m a p r o po r tion o f l e s s e r ine qua l ity w i l l not be p r opo r t i onally r e latab l e to o n e a r i s in g fr o m a p r o po r­ tion of g r e ate r i ne qual ity , i t i s i mp o s s ib l e fo r the r e to be s uch a m otion .

F

I+

then V = +. If - = R 1 ' P r o o f b y The o r e m s I a nd V I I I , Cha pte r I II .

T he o r e m I X .

Th i s the o r e m fo ll ow s , o f c our s e , fr om Th e o r e m V I I I b y e l i m in­ a tio n and with the addition o f the g e n e ral axiom that , i f F e xc e e d s R , th i s i s s u ffic i ent to p r o duc e m o ti on . T he o r e m X .

F o r any va l ue o f V, p r o po r ti on s o f F to R c a n b e found whic h wi l l y i e l d 2 V a nd

fv .

P r o o f b y T h e or e m s I and IX , Chapte r I I I .

V

The o r e m X shows an i mpo r ta nt c o n s e que nc e o f the n

F

= R fo r-

m ula . Si nc e ve l o c itie s v a r y as the p r opo r t ion be twe e n the i r p r o po r­ t i on s o f fo r c e to r e s i s t anc e , any g i v e n ve loc ity ma y b e d oubl e d b y s qua r i n g o r ha lve d b y e xt r a c t in g the s qua r e r o o t o f , th e p r o po r ti o n a s s o c i a te d with t hat ve l o c ity. Theo r e m X , the r e fo re , p r e s e nt s t h e unive r s al r ul e fo r t he do ub l ing and ha lving o f v e l o c it i e s , o n whi c h the t r uth o f The o r e m s I I and I I I d e p e nd s , and o f whi c h T he o r e m s I I a nd I I I a r e s pe c ia l c a se s . Thi s the o r e m fi r m ly e s tab l i s h e s the pow e r of t he n V = � fo r m ula , not only to e xp r e s s any de s i r e d ve lR oc ity by m e a n s o f a p ro po rtion o f fo r c e s , but to r e late thi s ve loc ity to any m u ltiple or fr ac tion of it , by m e an s of anothe r p r opo r t ion (i . e . , an e xpone ntia l one). T he o r e m X I .

F

F'

For any c o n s tant va lue o f R , R may b e : > R ,

F'

F'

= R ' or < R .

D E S C R I P T I V E AND C R I T I C A L A NA L Y S IS Proof.

41

Let F represent a mixed body (�), and let F ' represent a pure body . and F' f let - = - . R r f F' Then , and < R+r R F F' - < - and V < V ' (Theorem V, C hapter III ) R R F' f Let - < -- and R R -r• V > V ' ( Theo r e m I , Chapter III ) f F' Let - = -- and R R-r' V = V ' (Theorem I . C hapter III )

In the above theo r em is illustrated a further special area for V application of the n = { formula , namely that of dealing with

com pound motions involving a body possessing intrinsic resistance arising from its mixed composition. An important idea embodied in Bradwardine 's development of this theorem is that , in motions involving mixed bodies, the internal resistance is to be added to any external resistance which is present , i .e . , i f a mixed body may be represented by

!..r . then the velocity

of such a body through a

medium possessing the resistance R, is to be represented as V = , and not V = (� . Therefore , though velocity remains the func­ R�r tion of a proportion of forces . coacting forces are to be added to eac h other rather than made proportionate ; the resultant motive force is proportionate to the resultant resistive force .

Th�orem X II. If R = 0 , then , if £ = � . V = V ' , and if f > f ' , and r r f' f f f' f then , if - is balanced against , , - will move r r r r

f'

Proof.

r' '

f

f'

Let f b e > f ' and - = , • then f+r' > . r+f ' . (Theorem r r VIII , Chapter I ) . and since f+r' operates against r+f ',

r will move ?f . 9 3

• f

I

This theorem should not , of course , be taken as an argument either in favor of or against the existence of a void. It serves , in

42

INT ROD UCT ION

fa c t , to point o ut the s e c onda ry i mpo r tanc e o f s uc h a c on s ide ratio n . He r e , a ga i n , B r a dwar d ine t r eat s Ar i s to tl e' s a r gum e n t a ga i n s t the e xi st e nc e o f a voi d (on the g ro und that m o ti o n in it wo uld be in s tan­ tane o u s and , the r e fo r e , not a m o ti on a t a l l) a s a s p e c ial c a s e o f th e m o r e g ene ral th e o r y that an o ppo s it i on o f mo ti ve a nd r e s i s ti v e fo r c e s , o f wha te v e r so rt , i s wha t i s e s s e ntial t o m otion . If a vo id w e r e to e xi s t , the n a mixed body , in whi c h o ne of two e l e m e nt s te ndi n g in oppo s ite di re c ti o n s po s s e s s e d a p r e po nd e r ant fo rc e , would m o v e a t a d e t e r m in ate v e loc ity dep e nde n t o n th e p r o po r tion of tho s e oppo s e d fo r c e s . N e e dl e s s to s ay, m i xe d b o d i e s po s s e s s i n g e quivale nt oppo s iti o n s o f fo r c e to i nte r na l r e si s ta nc e woul d m ov e with e qual v e l o c i ty. The s e c o nd pa rt o f T he o re m X I I i s pa r ti c ul a r l y inte r e s tin g i n t h e c onn e c tion it e s tab li she s b e twe e n B r adwa rdine' s g e ne r al law of v e l o c itie s a nd the m e di a e v al s c i e nc e of w e i g ht s . T ho ugh m ixe d b odie s who se mo tive fo r c e s a r e p r o po rtionate ly e quiv al e nt to the i r inte r nal r e s i s tanc e s woul d , in a vo id, m o ve with e qual v e lo c ity, th e body po s s e s s i n g the g r e at e r m otive fo r c e would , i f the two b o di e s we r e balan c e d o n a s c ale , d e s c e nd . T hu s a the o re m c onc e r nin g s tati c s i s de d uc e d fr o m a la w o f mo tion wh ich i s e s s e ntia lly dy­ na mi c ; the c o n s i de r ati on o f we i ght i s intr o duc e d , and " wo r k" b e­ c o me s r e la te d to ve loc ity. The s e c o nd pa rt o f The o r e m X I I de p e nd s , m o r e ove r, o n a n e ve n m o r e g e ne ral i z e d fo r m o f th e c ombination o r '" add iti on" o f fo r c e s a l r e ady all ude d t o . In the pr e s e nt c a s e i t i s p r i m a r i ly the m o ti o n o f a balanc e b a r tha t i s invo lve d r a the r than the m ot i on o f t h e two b odie s the m s e l ve s ; c on s e qu e ntl y , we find that the mot ive for c e o f o ne body i s t o b e add e d to th e r e s i stive fo r c e o f the othe r . Thi s the o r e m make s it a ppa r e nt , the re fo r e , tha t , in an e v e n m o re g e n­ e ral s e n s e , m o t io n s a re the func tion o f r e sultant o ppo s i tion s o f fo r c e s , wh eth e r the fo rc e s added a r e like o r unli ke . Se e n in thi s l i ght , the di s tinc tion o f motive a nd re s i s t ive fo r c e b e c o me s la r ge ly o ne whi c h i s r e l ativ e t o a g i ve n c a s e . In the two g ra via m i xta to whi c h B ra dwa r d ine r e fe r s , it r e mai n s , of c o ur s e , true tha t the downwa rd fo r c e o f the hea vy c o m po ne nt s po s s e s s e s a na tur al di s­ tinc t ion fro m the downwa r d fo rc e of th e l i ght c o mpon ent s , b ut it i s heavine s s add e d to l i g htne s s th at produc e s the c o mbined m o t i o n o f the two bodie s . B r adwa r dine' s de ve l op m e n t o f the s e la st two the o re m s c l e a r l y indic ate s the re ma r kab l e s c ope o f hi s dyna mic l a w . By i t s appli­ c ation t o p r ob l e m s o f s tat ic s and me c hanic s i s fo r e shad owe d onc e a gain the e no r m o u s s ynthe s i z i n g powe r o f ma the matic s in the s tudy o f na tu r al pro c e s s e s : it s po we r to b r in g to ge the r s e pa r ate fi eld s o f phy s i c al the o r y in te r m s inc r e a singly mo r e g e ne r a l . On e la s t po int wh ic h i s wo rth rai s i n g i n c onne c tion with T he o­ r e m s X I and X I I i s an ob j e c tion to th e m b a s e d o n the m e dia e val c onc eption o f " m ixed b odie S:' Su c h b odi e s we r e c o m monly c onc e i v e d

D E S C R I P TI V E A ND C R ITI C A L A N A L Y S I S

43

a s be ing what in mode rn parlance are cal l e d chemical compound s . T hat i s to say, they we re not simply thou ght of a s ag glome ration s o f dive r s e e l e me nt s , b ut as homo g e ne o u s bodie s who s e b e havior, or "natural motion ;• wa s to be determinabl e in some way, a s the re s ultant of their compone nt e le me ntal s ub s tance s . If the s e mix e d bodie s are to be conce ive d a s natural compound s, the n it b ecome s hig hl y que s tionab l e whether they can properl y be thoug ht of as po s s e s sin g " inte rnal re s i s tance :• If the y do not po s­ s e s s , in actu, such an oppo sition o f force s, the n, o f cour s e , Brad­ wardine' s two final the orems fail on the ground of faulty premis e . Without actual oppo sition of force s, all mixed bodie s would move in s tantaneou s ly in a void . This s e e ms too obvious an ob jection to have b e e n ove rlooke d b y any compete nt me diae val the ori st, and one can only concl ude tha t Bradwardine eithe r conceive d corpora mixta a s containing a dis tinction of compone nt e l e m e nt s in actu, or el s e i s he re spe aking not o f natural compound s but o f ag glome ration s o f dive r s e e l e me nts . The further ob jection the n s ug g e s t s its e l f that, s uppo s in g the compound to be compo s e d of wate r and earth and to be fal lin g throu gh a vacuum contained within the re alm of air, Bradwardine would s e e the two compone nt s a s oppo sin g e ach oth e r and, thereby, givin g ris e to a de te rminate velocity. Though Bradwardin e doe s not hims e l f e l ucidate the matte r, it s e em s reasonab l e that in such a cas e he would hol d tha t since both e l emP. nts tend in the s ame d i re c­ tion and since e ach , s e parate ly, would have an instantaneous ve loc­ ity in the po site d vacuum , the whole compound woul d move at an in finite vel ocity and without inte r nal re s i s ta nce . We mu s t , there ­ fore , concl ude that, in both Theore m s xr and X I I, whe n a mixe d body i s said to pos s e s s a pote ntia re s i stiva, thi s is the ca s e onl y whe n the natural motion of that compone nt i s in a direction oppo site to that of the component d e signated as pote ntia motiva. This doe s not s e em in any s en se to di stort th e meanin g which one a s s ociate s with the re lative te rm s, motiva and re s i s tiva, and doe s s e em to pre s erve the s e n s e of the two th eorem s in que stion . We have s e en that a mixe d body, each of whose compone nts te nd in th e s am e direction, would move in s tantaneou s ly in a vacuum . In T heorem X I, whe rein motion o f a compound through a P.l e num woul d V b e expre s s ib l e by th e formula, n = __£__ , it i s a s s ume d that r re p-

R+r

re s e nt s a compone nt who s e natural motion is oppo s e d to tha t of F. If the compound is s o locate d and s o compo s e d that both of its e l e ­ ment s tend in the same direction, th e n th e formula s hould simpl y b e al te red to read V n The remaining portion of C hapte r III i s d evote d to re futa tion o f two further ob jections to Bradwardine' s law o f motion which had not

44

IN TROD U CTIO N

previously be en raised. The se cond se ems of lesse r inte re st; it raise s the o bj e cti on that a magnet (which should, acco rding to Brad­ wardine ,move a small pie ce of iron m ore rapidly than a large one ) actually m ov e s b oth at the same spe ed. Since the objecti on rests solely on the gro und that a magnet in m oti on, with the pi ece s of iron in que sti on adhe ring to it, would produce e qual ve locities in both pie ce s of m e tal, it does not seem to damage Bradwardine•s the ory particularly . The velocity in this case is not the functi on of a pro porti on of fo rces betwe en magne t and me tal but betwe en magn e t and whatever moves the magne t or obstructs its natural moti on . It is conce ivable, of course, that the m otion of the ma gn e t might involve a force component of the m etal carri ed with it, but the the o ry of magne ts is too compl e x a topic to e nte r upon in this brief discussi on. The f irst of the two obj ections raised is, howe v e r, of conside r­ ably gre ate r sign ificance. It is argued that if e qual velocities are produce d by e qual pro portions of forces, then, if a large quantity of earth be ars the sam e pro portion to a large quantity of air that a small quantity of earth bears to a small quantity o f air, the ve ­ locities of the two bits of earth through the ir re spective m edia should (by Bradwardine• s account) be e qual . But they cannot be, b e ­ cause the large r pi ece o f earth, i n trave rsing its medium i n the same time that the smalle r trave rse s its own medium must go farthe r (b e cause the m edium is bigge r in e xtent, or quantity), and in such an event the velocitie s would not be e qual . This dilemma thus raises the problem o f re lating force to distance, dynam ic to kinematic functi ons, and brings out the ambiguity inherent in Aristotl e•s remark that the "wei ght" of a body is a facto r of its ve locity in fre e fall. 94 Bradwardine offe rs, in solution of the above dilemma, a distinction be twe en " qualitative" and " quantita­ tive" propo rti ona lity as applied to moti on. Qual itati ve l y, the moving force be ars the sam e proportion to any and all fracti ons of the im­ peding m edium, and this qualitative proportion de te rmines quali­ tative velocity. Quantitati vely, howeve r, the pro po rtion is betw e e n the tim e s of the two m otions . Ve locity as an " instantaneous" quality o f motion is there by cl e arly distinguished from veloci ty as a simple functi on of tim e and dis­ tance . N e e dl e ss to say, ve locity of any sort must be thought of in te rms of distance and tim e ,but the distincti on which Bradwardin e he re draws be twe e n quantitative and qualitative velocitie s is actu­ ally the distincti on which may be re nde red, in m ode rn parlance , D as that be twe e n V = � and V = 9.. _ Thus, in Bradward ine•s law, T d T dD which is primarily dynamic, it must be unde rsto od that V = dT D . 1y T " rathe r than s1mp T he treatm ent of this distincti on betwe en the " quality" and " quan-

DESCRIPTIV E

AND

C R I T I C A L A N A L Y SIS

45

ti ty" of motion i s extre mely brief and does not attain a great gen­ erali ty of express ion . For exa mple, though this distinc tion i s made with regard to resistances in order to overcome the objection that V F . . . D . n = R 1s mco mpat l ble with the fact that V = ' it is not made wi th T

:-egard to mot ive bodies or forces . It is however, the outline of an idea of "force per unit volu me" essentially not unlike that contained in Archimedes' principle of specific gravity (which was fa miliar to mediaeval writers) and provides a theoretical basis for clarifying the above mentioned ambiguity of Aristotle some two and a half centuries before Galileo is said to have actually gone to the length of dropping things from the tower of Pisa in order to prove that different quanti ties of the same substance will fall at the same rate . Chapter IV of the De RrO Rortionibus deals with circular mot ions and commences, in strict mathematical plan, with a series of def­ initions and a xioms from which various conclusions are drawn . Squares, similar surfaces, and quadrilaterals are defined from the first, si xth, and first books of E uclid , respectively. The axioms are as follows : 1 . All right angles are equal . (Euclid I ) 2. Of any two similar polygons, the proportion of areas equals the proport ions of any two corresponding sides, squared. ( Euclid VI) 3. Of any two circles, the proportion of their areas equals the pro­ portion of the squares of their diameters . ( Euclid X II ) 4. Of any two circles, the proportion of their circumferences equals the proportion of their diameters . (Archimedes De curvis suP-er­ ficiebus, Theorem V , actually Theorem III, see p. 1 94 below) 5. Of any two spheres, the proport ion of their volumes equals the proportion of their diameters, cubed. (Euclid XII ) 6. The surface of any sphere is equal to that of a rectangle whose opposite sides are equal to the diameter and greatest circum ­ ference o f the sphere. (Archimedes De curvis S UP-erficiebus, Theorem V II I, actually T heorem V I , see p . 1 94 below) From the above definitions and axioms, the following six con­ clusions are drawn : 1 . Of any two circles, the proportion of their areas equals the pro­ port ion of their diameters, squared. 2. Of any two circles, the proportion of their areas equals the proport ion of their circumferences, squared . 3 . Of any two spheres , the proportion of their volumes equals the proport ion of their greatest circumferences, cubed. 4. Of any two spheres, the proportion of their surfaces equals the proport ion of their diameters, squared. 5. Of any two spheres, the proportion of their surfaces equals the proport ion of their greatest circumferences, squared . 6. Of any two spheres, the proport ion of their volumes equals the proport ion of their surfaces, to the l½ power .9 5

46

INTRODUCTION

In the second part of this fourth chapter, Bradwardine proceeds to a further so lution of the problem of the relation of velocity con­ sidered as a function of a proportion of forces , to velocity consid­ ered as a function of distance and time , which had already been treated in terms of a distinction between qualitative and quantita­ tive vel ocity at the conclusion of Chapter I I I. The first suggestion concerning the relation of velocity to space , or distance, in local motion he rejects as false ; namely, that the proportion of velocities equals the proportion of the volumes of space described by the respective moving bodies. That it is false fol lows from the fact that if it were true , then any body would move twice as fast as the half of it (since it would traverse double the volume of space) . And, if the who le body moved one foot in an hour and hal f the body moved half a foot in an hour , this theory would have to suppose the velocities equa 1. A second theory, that the proportion of vel ocities equals the pro­ portion of the surfaces of the volumes of the contained spaces , is rejected on the same grounds as the first theory. The third theory , that of Gerard of Brussel s 9 6 as contained in his treatise De P-rOP-ortione motuum et magnitudinum , is that cir­ cular velocities are proportional to the arcs and areas traversed by a given radius . Bradwardine thinks that this earliest known ac­ count of " angular velocity" 97 is a considerable improvement upon the two theories already mentioned , but he objects to it , neverthe­ less , because it posits that any part of a rotating radius would move at the same speed as the radius' midpoint . If Bradwardine's quotation of the theory in question is correct , one might wel l agree with his ob jection . After all, speed is a prop­ erty belonging to bodies , and bodies are moved as rapidl y or as slowly as any of their parts. It may be correct to speak of the av­ erag� speed of a rotating radius as equa l to that of its midpoint , but this does not support the the sis that this speed of the midpoint is that of any and al l parts of the radius . Moreover, in the case of celestial motions , the midpoint of their radius to the earth would not even lie within the body whose motion it is supposed to express . What we are faced with in this c ase is evidently a question of interpretation , and whether Bradwardine misinterpreted the theory in question it is impossible to say . Those , at any rate , who later upheld Bradwardine's defi nition of vel ocity as being the maximum attained by any part of an object , felt no hesitation in coupling this definition with the thesis that when velocity varies through time , the distance traversed by the motion as a whole is equivalent to that which would be traversed by a body moving uniformly at the mean degree of that velocity . On the supposition , at any rate , that circular vel oc ity may be properl y measured by the motion of the circumferential locus , Bradwardine sets down three further physical axioms in addition to the preceding geometrical ones :

D E S CRI P TI V E A N D CRITIC A L A N A L Y S I S

47

1 . T h e v elocity of any local mot ion is to be me asure d by th e ma x­ i m u m d istance tra v ers e d by any point of th e mov ing ob j e ct . 2 . T h e proportion b etwe e n th e v e locitie s of any two local motions e quals the proportion b e twe e n th e maxi m u m lines descr ibe d by two poi nts on the two mob ili a . 3 . O f t h e circular planes conta i ne d i n a sph ere , that whi ch passe s through the center of the sph ere is th e larg e st . Proposing now to de monstrate the opposite of som e of the con­ clusi ons to be found in th e De grogortione motuu m et magnitudi­ num, Bradwardin e launche s into h is e xpos ition of the following si x theore ms drawn fro m the abov e geom e tric and p hysical axioms and basic theore ms : 1 . Of any two points de scribi ng the circu mfe re nc e s of circles uni ­ formly and in e qual ti m e s , the proportion of v elocit i e s e quals th e proportion of di a m e te rs . 2 . Of any two dia m e te rs or radi i de scribing circle s uniformly and in e qual ti me s , the pro po rtion of v elocities e quals th e propor­ ti on of di a me te rs or radi i . 3 . Any two circu mfe rence s o f c ircle s de scribe d uniformly and i n e qual t i me s (wh e th e r de scribe d s i m ply a s circles , or on th e surface s of sph e res , or th e one as a circle a nd the other on th e s urface of a sphere) ar e proportional to the ir ve lociti es . 4 . Of any two c ircles re vol ving un i formly and in e qual ti m es (wh eth­ er s i mply de scribi ng circle s , or descr ibing sph ere s , or one describing a circle a nd the oth er a sphe r e ) th e proportion of v elociti e s e quals th e s quare of th e proportion of are as . 5 . O f a n y two spherical surf ace s mo ving unifor mly and i n e qual ti m e s o n the ir axe s , the proportion of v e lociti e s e qu als the s quare of the proportion of are as . 6 . Of any two sph e res re volv ing u niformly and in e qual ti m e s on th e ir axes, th e proportion of velociti es e quals the cube of th e proportion of volu m e s . T h e fi nal portion of the tre atis e is de vote d to Bradwardine's the ­ ory of the si z e of ele m e ntal sphere s a nd , not b e longi ng to a study of v e locitie s , is includ e d si m ply be ca use he fe els tha t here is ye t a nother case in which analysis by ge om etric seri es is j ustifiable . T hat the the ory t urn e d out to be false was unfortuna te, for it ca m e to b e widely known and acce pte d . It does ha v e a n int e re st for th e historian of sci e nce , howe ver, in its e x e mplifi cation of that p e re n­ nial hop e of th e mathe matici a n-physicist to find a pplica tion for a gene ral type of function in what m ight o utwardly app e a r to be to­ tally div erse pheno m e na . 98 The intere sting thing is not so m uch that this particular application of th e ge om etric function was off the m ark but that Bradwardin e's instinct in se e king such an appli­ cation h as turne d out to be so dra m ati cally j ustifie d in the history of m ode rn mathe ma tical p hysi cs.

48

IN T ROD U C T ION

B r adwa rdin e c o m m e nc e s wit h thre e a s s umpti on s ; the fi r s t i s s upp o r t ed by two c o m m e n t s o f Ave r r oe s on A r i s t otl e' s De c a e l o , the s e c ond by the e s s e nti a l c ha r ac t e r o f the theo r y o f sub lunar e l e ­ m e nt s c o m mo nly he ld by all s c ho l a s t i c s , and the thi rd by the au­ tho r i ty o f Thabit ib n Qur r a and Al- Far g hani , two e a r l y A r abian a s t ro no me r s . 99 T he y a r e as fo l l ow s : 1 . The fo ur e l e m e nt s a re joine d in c o nt inuo us pr o por tional ity. 2 . The fou r e le me nt s oc c upy ( o r nat urally s houl d o c c upy) the who l e c or r upt ib l e s phe r e {i . e . , the s ub l una r). 3 . The r ad i u s of the wl:ole c or r upt ib l e s phe r e i s e qual to 3 3. 3 ti m e s th e r ad i u s o f e a r th. A lpha r ganu s , in D i ffe r e ntia 21 of hi s E le m e n t s , i s not c onc e rne d , a s i s B radwa rdi ne , t o de ve l op a the o ry o f the di st r ib ution o f the s ub l una r e l e m e n t s b ut , r a the r , to e s tab l i s h the r e lative di s tanc e s b e tw e e n the s e ve ral c e l e s tial s phe r e s . B radwa rdine s i mply ma ke s u s e o f A lpha r g anus' e s t i m ate o f the r a tio b e twe e n the radiu s o f the s phe re of the m o on and the radiu s o f the e a r th a s the e m pi r ic c on­ t e nt of h i s the o r y o f th e p r op o r tionality o f the e l e m e nt s . Alphar­ ganu s e stimat e s tha t th i s ratio is 3 3 . 3 a nd , s in c e the di a m e te r o f the e a r t h i s e st i m at e d a s 6 , 5 0 0 m i l e s , the d iame t e r o f the t o ta l s ub luna r s phe r e wo uld be 2 1 6 , 4 5 0 m il e s . B r a dwa rd ine d oe s not c onc e r n hi m s el f with wor k ing o ut hi s p r o po r ti o na t e d i s tr ib u tion o f the s ub l una r e l e m en t s i n mil e s . E mpl oyi ng the pr ope r ti e s o f the p r o por tional s e r i e s e stabl i s he d in Chapte r I , to g e th e r with t h e the o r e m s c onc e rning sphe ri c al g e ­ o m e t r y gi v e n i n C ha pte r IV, fo ur the or e m s c onc e rni ng the e l e m e nt s a r e d e v e l o pe d . 1 . T he pr o po r tion o f a ny l a r g e r e le m e nt to the e l e m e nt i m m e diate l y le s s e r i s g r e ate r than the pr o p or t ion o f 3 2 t o l a nd l e s s tha n tha t o f 3 3 to 1 . 2 . T he pr opor tion b e twe e n the s phe r e o f fi r e and that c o m po s ed o f th e thre e r e maining e l e m ent s i s g r e a te r tha n the pr o po r tion o f 3 l to 1 . 3 . The d i s tanc e o f t he out e r sur fac e o f the ai r fro m the c e nt e r o f th e e a r th i s mo r e than 1 0 and l e s s than 1 1 t i m e s the r adiu s o f the e a r th . 4 . A point hal fwa y b e twe e n the inne r s u r fa c e o f the heave n s and th e c e nte r of the e a rth wi ll be muc h ab o v e the out e r su r fa c e o f th e a i r . B e fo r e ta kin g fina l le ave o f the De grogo r t ionib u s , i t may b e we l l t o a s k o ur s e l v e s onc e a g a in wha t i mpo r tanc e th i s wo r k had fo r the fu r the r d e v e l o p m en t o f phy s ic s . A s id e fr o m wha t s i gn i fic anc e it po s s e s s e s as a m a j or st e p fo r ward in the mathe mati c a l t r e a t m e nt o f pr obl e m s in d yn a m i c s , the c l ear di stinc tion wh ich i s drawn b e ­ twe e n th e " qua li tati v e" and the " quanti tati ve" m e a ni n g s o f the te r m

DESC RIPTIVE

AND

C RITICAL

ANA L Y SIS

49

" v e lo c i ta s :• to get he r with a c o ne e ntra tion u pon the fo rme r, pr o v ide d a mo st fr u itfu l point o f de p a r tu r e fo r the wo r k o f th e fo l l owing g e n­ e r ati on at Me rto n Coll e g e . Owing , p e r h ap s , to a fa i l u r e to r e c o g ni z e that n o und i ffe r e nti ate d fo rm u lati o n o f a fun c ti on be twe e n fo r c e , r e s i s tanc e , and v e l o c ity c an ade quate l y e xp r e s s b oth th e moti on s invo l ve d in s tati c s and tho s e invo l v e d in th e kine ma tic s o f i n e r tia l re s i s ta n e e , or " fr e e ma s s ;• th e pa rti c ul a r fo r m o f B r a dwa r d ine ' s e q u ati on tu r ne d o ut to be e r r one o u s . If s u c h d i s tinc tion s b e twe e n th e va r io u s kind s o f " r e s i sta nc e" a r e not made , th e d i ffi c u l ti e s whi c h fo r c e d B r a dwa r ­ d i n e t o r e j e c t a s imply dynamic fu nc tion in fa v o r o f a l o g a r ithmi c one s ti l l pe r s i st . To s e e the tr uth of thi s , on e ha s me r e l y to c om­ pa r e , in s o fa r a s tha t i s p o s s ib l e , the B r adw a r din ian to th e N e w­ tonian dynami c s . Ha ving d i s c ove r e d , i n o u r a na ly s i s o f Cha pt e r II I , that B r a dwa r ­ d ine int e nd s hi s fo rmula t o apply t o " qua l itative" r athe r tha n to " qua ntitati ve" ve l o c ity, it i s c l e a r that , in the fi r s t pl a c e , the pro­ p o r ti on o f fo r c e to r e s i sta nc e is c o nc e i v ed to v a r y with what find s it s mod e r n c o unte rp a r t in th e c onc e pt o f " in s ta ntane o u s ve l o c ity :' In mode rn dyn amic s it i s , of c o u r s e , r e c o g ni z e d that s uc h ve lo c ­ iti e s , wh e n a s so c iated with a c on s ta nt p r o po rt i o n o f fo r c e t o ma s s , a c c umu l ate u ni fo rmly thr o u gh time and a r e h e n c e e qu i va l e nt to r ate s o f a c c e l e r a ti o n . If we then c o n s ide r that Ne wto ni a n "ma s s" i s to be me a s ur e d i n te r m s o i i t s " we i gh t ;• we ha v e a l s o a c e r tai n d e g r e e of e qu i va le n c e b e twe e n B radwa rd i ne ' s " r e s i s tanc e" and N e wton' s "ma s s :• B r adwa r din e' s d i ffi c u l ty n ow be c ome s appa r e nt , fo r F = ma i s s ub j e c t to e xa c tly the s a me c r itic i s m s wh i c h a r e r a i s e d a ga i n s t th e fo r mu l a , � = V- the the o ry whi c h B r a dwar dine i s at pa in s to r e f ute i n Chapte r II. Un l e s s th e r e s e r vati o n i s ma d e that Ne wton' s fo r mu l a m u s t b e u nde r stood a s applying o n ly t o " fr e e ma s s' o r " ine rtia � the impo s s ib il ity fo l l o w s tha t a fo r c e l e s s tha n that exe rte d (i n the fo r m o f w e i g ht) by a g i v e n ma s s c a n p r o d u c e a p o s itive a c c e l e ration u pon th a t ma s s . F u nda me nta l ly , th e r e fo r e , th e r e a s oning whi c h unde r lay B rad­ w a r d ine ' s obj e c tion to r e nde r i n g po s itive motion of a ny s o r t in te r m s o f a s impl e fu nc ti on of the p r o p ortion o f fo r c e to r e s i s ta nc e appe a r s s o und e v e n tod a y. Wha t wa s r e qu i r e d w a s a f u r th e r c l a r i­ fi c a tion o f the di s ti nc tion s b e tw e e n v a r i ou s fo r m s o f r e s i sta nc e (e . g . , th o s e o f fe r e d by the d e n s i ty o f a m e d i um , by i n e rtia , a nd by

weight).

B r adw a r dine' s di s tinc ti on o f the " q ua l it ativ e" me a ni ng of v e l o c ity had , howe v e r , mo s t impo r tant c on s e q ue nc e s . I f th e p r o po r ti o n o f fo r c e t o r e s i stanc e w a s to b e unde r s to o d a s p r od u c i ng a g i v e n " qual ­ i ty" of moti on , h o w wa s thi s to be r e l a ted to the " qua ntity" o f a moti on po s s e s s ing that " qua l ity" ? how , in s ho r t , wa s th e d i s tanc e t r a v e r s e d dur ing a time i nte r v a l of qual ita ti ve l y uni fo r m motion

50

IN T R OD U C T ION

to be c alculated ? In the work of John Dumbleton1 0 0 and the other Merton College Calc ulatores, of who m mention has already been made, the ba sis for an answer to this problem wa s soon developed. In dealing with the question o f how physic al magnitudes should be understood to inc rease or diminish , Dumbleton argues with great skill against two earlier theories whic h had enjoyed con sid­ e rable prestige . The one c laimed that a quality c an, indeed, be said to increase, but that the inc rease c annot be treated additively , a s a sum o f qualitative parts, and the other that a quality cannot really increase, but that instead , at eac h moment of an apparent inc rease , the prec eding quality is annihilated and one of a greater degree substituted for it . 1 0 1 In combatting the former the sis, Dumbleton is c onc erned to put the analysis o f qualitative c hanges upon a quan­ titative basis amenable to mathem atic al manipulation and , in op­ posing the latter, to avoid the paradoxes attendant upon an atomic treatment of c ontinua . Any .. intensive magnitude" (suc h a s a degree of heat) , just a s any "extensive ma gnitude" (suc h as a distanc e) , was to be c on ceived a s c ontaining a c ertain ''latitude :' o r quantitative span , extending from a zero a mount of the given quality to the maximum degree of that quality whi c h terminates its latitude. Moreover, just as a linear distance, or " latitude" is equivalent to the sum of the parts into whic h it may be divided , so also a qualitative latitude may be di­ vided into part s of whic h it , too , is the sum. An analogy having thus been established between the quantifiable a spect of both "extensive" (dimensional) and "intensive" (qualita ­ tive) magnitudes, Dumbleton goes on to point out tha t , just a s an "extensive magnitude" (e .g. , a given distanc e) must be traversed suc c essively, so also an "intensive magnitude" must be ac quired part by part . The alternative would be that of instantaneous trans­ lation, or transmutation (in other words, not a true " c hange" a t all). There is , however, a critical differenc e between the ways in whic h "intensive" and "extensive" ma gnitudes are ac quired . A distanc e is traversed by a moving body only by its leaving�_hind the suc ­ cessive parts o f that distanc e , whereas , in the a c quisition of suc ­ c essive parts of an "intensive" magnitude , the intensive parts must , of c ourse, be a c c umulated by the body in question. Otherwise , what­ ever part of the intension could be produc ed by a given agent work­ ing through a given time would have to be rec reated from zero a fter eac h part, and (sinc e these parts are infinitely divisible) thus no c hange at all c ould take pla c e. The foregoing provides a brief summary of Dumbleton' s position regarding the quantific ation and the manner of generation o f quali­ tative c hanges . In Part III ( C hapters 2- 7 ) of his _Summa logic ae et P-hilosoghiae naturalis he then picks up the theme of Bradwardine' s treatise, and , after rehearsing muc h the same arguments as Brad-

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war d ine' s again s t th e the o ri e s o f an a r i thm etic o r a s i mple ge o­ m e t r ic pr o po rtiona lity b e tw e e n ve loc iti e s and the i r c or r elati v e p r oport ion s o f fo r c e t o r e s i stanc e , p roc e e d s to a n e lab ora tion o f B radwa rdi ne' s dyna mic s . Sin c e the m e aning o f " ve l oc ity" in B r ad wa r dine' s tre ati s e ha s b e e n ide nti fi e d a s that o f a qual itative , o r " int en s ive" magnitude , v Dumble ton po int s out tha t , i f { = n , the n we m u s t unde r sta nd thi s

to m ean that the "lati tude s" betwe e n d i ffe r e nt val ue s o f V a r e r e­ l ated to the e quivale nt "latitude s" b etwe e n d i ffe rent value s o f � i n the manne r indi c at e d . S inc e t h e quality i n que stion i s tha t o f a n in­ s tantane ou s ve loc ity, thi s por tion o f Dumb l e ton' s the s i s i s a c tually deali n g with the c o r r e l ati on betwe en change s in the propor tion o f fo r c e t o r e si s tanc e a nd the c o r r e s ponding c han g e s i n r ate s o f qual i­ tati v e o r in s tantane ou s ve loc ity. It no w cl e a r ly fo llow s that , i f a g i ve n p r opo r ti o n o f fo r c e to r e­ s i s tanc e pr oduc e s a g i ve n inte n sion , or "de gr e e ;' o f ve loc ity in a g iven ti me , the n , i f thi s p ro p o rtion r e ma in s c on s tant , in a s e c ond such time the velo c i ty will b e d o ubl ed , a ft e r the third int e r val tre­ b l e d , e tc . Thi s is inevitab le , on the a s sumption that " inte n s ive lat­ itude s" ar e ac cumul ate d a nd tha t c o n s tant c au s e s p r o duc e c on s tant e ffe c t s . Dumb l eton doe s not , hi. m s e l f , gi ve an e xplic it s ta te m ent o f thi s c on se quenc e , b e ing oc c upie d with the m o r e ab s t ru s e pr obl e m s to b e enc o unt e r e d in c a s e s whe r e the value o f the { p r o po r ti on fluc tua t e s within a g ive n m otion . Thi s i s quite appropr iate to the portion of hi s s t ud y in whic h he i s r e main1ng str ic tly within the t e r m s o f the B ra dwar d inian func ti on , but in the fo llowin g c hapte r (Chapte r 8 ) he wo r k s o ut e xplic i tly the re lation o f ti m e e lap s e d to d i s ta nc e trave r s ed in a uni fo r m ly ac c e l e rat e d moti on. Dumble ton' s de mo n s tration c o nc e r ning thi s r e la tionship c ente r s on the t he s i s tha t the di s tance trave r s e d i n a ny m o tion in whic h the re i s a uni for m " inten s ion" o f ve loc ity (c o m m e nc ing fro m z e r o and e nd in g a t any given de gr e e) m u s t b e e qual t o the d i s ta nc e whi c h w ould b e trav e r s ed b y a b od y mo ving uni fo r mly at one ha lf the te r­ mina l d e g r e e of that ac c e l er a te d m otion. In o the r wo r d s , S = f v · t (whe re V i s the te r minal veloc ity). Thi s i s , o f c our s e , s t r i c tl y 2

e quivale nt t o the mo re fa m i l ia r , S = ½ a t , a nd o f c on side rable in­ t e r e s t a s havi ng b e e n a mong the ve r y fi r st s uc c e s s ful a tte m pt s to r elate t i me e l ap s e d to di s tanc e trave r s e d in a uni fo r ml y a c c e le r­ a ted moti on. Dumbl eton' s pr o o f i s a pur e l y fo r mal one , by no me an s d e p e nd e nt upo n exp e ri me ntati on o r involving the the s i s tha t any natural motio n s a r e , in fa c t , unifo r m ly ac c el e rated . We have al­ r e ady s e en that s uc h a the s i s i s , in fa c t , e nta il e d in the s uppo s i tion of c o n s tant p r opo rti ons o f fo r c e to r e si s tanc e ac c umulatin g e qua l

52

INTRODUCTION

latitude s o f ve loc ity in e qual tim e s , b ut it wa s appa re ntl y not unti l s o me time late r that thi s c on c l u s i on wa s made e xpl ic it . The c onc ept of " i mpetu s ;· by which John Buridan sou ght to ex­ plain the ac c u mulati ng ve l oc i ty o f b odi e s in fre e fal l , s e e m s c l e ar l y to ha ve b e e n e ithe r in s pi r ed o r antic ipated b y thi s e a r l i e r the o r y o f t h e manne r in whic h inte n sive qua l itie s a re ac qui r ed , wh ich we have ju s t b e en exa m ining . In the pa s s age in wh i c h he s u gg e s t s the " i mp e tu s" the o ry of a c c e l e rated fall , 1 02 Bu ridan n e ve r the l e s s ma ke s no u se o f the mathe matical anal y s i s re lating ti me and d i s ta nc e whi c h Dumble ton ha d wo r ke d out , a nd i t i s , o f c ou r s e , c o nc e ivab l e that B ur idan wa s a c tually unfa mi lia r with thi s wo r k o r with the the o r y o f i nt e n s i on whi ch und e r lay its c o r r e lation of ve l oc i ty, di s­ tanc e and tim e . Whate ve r the a n s we r to thi s que s tion , it i s none­ the l e s s of con side rabl e inte r e st to note that not o nly wa s the suc­ c e s s ful kine matic ana l ys i s of unifo r m ac c e l e ra tion wo r ked o ut by the mathe mati c ia n-phys ic i s t s o f Me rton Col l e ge qui te ind e p e ndentl y o f the hypothe si s o f any e ntity such a s the fo rc e o f " i mpetu s" po s ited by B ur idan , b ut that the kine matic s o f John Dumb leton g r e w natu­ rally fr om the e a rli e r anal y s e s in dyna mi c s whic h we r e s um m e d u p and g ive n a c l e ar, mathe mati c a l d e finiti on b y B ra dwardine . B rad­ war d ine ' s c on c l u si o n , tha t a p r o po rtio n of fo rc e to re s i s tanc e mu s t b e unde r s tood a s c o r r e late with a n ins tantan e o u s ra te , r athe r than with the tra ve r s a l of an e xt e nd e d di s tanc e in an extend ed ti me, wa s , o f c o ur s e , to b e c ouple d wi th the stand a r d a s s umpti on tha t c on s tant c aus e s pro duc e c on s tant e ffe c t s . Dumb l e t on' s the o ry of the manne r in which an " i nte n s i ve" ma gnitude may b e said to inc r ea s e the n l e ad s ine v itab ly, not only to hi s c onc lus io n s r e ga r ding the r e lation of di stanc e to time in a uni fo r mly ac c e l e ra te d moti on , b ut to the r e ali zation that a c on s tant propor tion of fo r c e to re s i s tanc e mu s t produc e j u st s uc h a motio n . I t i s , o f c our s e , t r u e that ne ithe r B ra dwardine no r Dumb l e ton a r e h e r e di s c us s in g the pr oble m s o f pro j e c til e m oti on ; the y a re de a l i n g s o l e l y with c a s e s in which an ac ting fo r c e i s r e qui red fo r the pro­ duc tion o f a moti on , and the que stion o f whethe r a moti on c an c on­ tinue in the ab s e nc e o f an a c ting fo r c e i s ne ve r rai s e d . One may , pe rhap s , do ub t that the y envi s a g ed the po s s ib ility o f the c o ntinua nc e of a motion in th e ab s enc e o f a pr ope l ling fo r c e , but one ma y b e c e r tain that Buridan di d not . Hi s c onc e pt o f " i mp etu s" a s " a c e r­ tain ene r gy (vi s} capab le o f mo ving th e p r o j e c tile in the d i re c ti on in which th e m o v e r s e t it in mo tion" i s c e rtainly m o r e c l e ar l y an e xpla na tion in te r m s o f c ontinuing fo rc e than in tha t of " ine rtia" o r "natu ra l motion :• It i s , if anything , the M e r tonia n kine matic s , t r e a ting .. ve l o c i ty" s i mply a s a qual ity induc ed in a b ody by an a c ting powe r and no t e nta iling the notion of s om e additional " fo r c e � whic h s e e m s c l o se r to the mode rn point o f vi e w . Pi e r r e Duhe m ove r l oo k s e nti re l y th e po s sib il ity that Buridan' s " i mpetu s" the ory may we ll have o r i g inated in the r e c o g nition , by

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Bradwa rdine and Dumbleton , that it i s in s tantaneous velocity which s hould be under s tood a s p ropor tiona te to a proportion of force to r e s i s tance and that such a quali ty, acc um ulating th rough ti me, will produce an accelera ted motion . It is , in fact , to Albe rt of Saxony that he attributes the recognition of the impor tant di s tinction be­ tween 'the dyna mic and the kine matic a s pect s of motion,1 03- th i s in s pite of the fact tha t the T r acta tu s P-r OP.or tionu m of Albe r t i s s o clea rly dependent upon Bradwa rdine' s treatise in both i t s organi­ z ation and content . Dumbleton' s further development of Br adwa r­ dine' s dyn a mic s Duhem take s to be no more than an explora tion of the way in wh ich velocity would va ry (ac c ording to Bradwa rdi ne' s function) i f the moving force we re to increa se . 1 04 A careful exa m­ ination of B radwa rdine' s T racta tu s and Dumbleton' s Sum ma show s that what i s actually being com pa r ed is not an increa se of force a s socia ted with an inc rea s e of velocity thr ough time, b ut rather an inc rea s e of force a s sociated with an increa se of instantaneou s r ate. Fortunately, the e r r or s of Duhe m' s wo rk and his unr ea s onable hostility to the natur al philo sopher s of Merton have, to a la rge ex­ tent, been cor rected by the fu rther re searches of Anneliese Maier. Some few ove r s ight s neverthele s s rem ain . Mi s s Maier ha s, for exa mple, per petuated Duhe m' s att ribution of the di s tinction be­ tween dyna mic s and kinematic s to Albe rt of Saxony . 1 0 5 T hi s we have found already quite clea rly developed in Bradwa rdine' s reply to po s si ble objections aga in s t h i s theory ; the d yn amic r elation of force to r e s i s tance i s the re explicitly related to an in stantaneou s r ate, and thi s meaning of velocity i s s ha rply di s tinguis hed from the kinema tic relation of ti me ela p s ed pe r di s tance t raver sed . Mi s s Maier m ake s also the fur ther cla i m that it wa s axiomatic for s chola stic phy s ic s that a con s ta nt propor tion of force to re­ s i sta nce pr oduce s a con sta nt velocity and tha t the " impetus" theory of John Bur ida n i s the clea re st expre s s ion of thi s kind of mechanic s . 1 06 S uch a theor y i s , a s s he say s, the inevitable outcome of an a ttempt to account for accelerated motion in te r m s of the two sup­ position s t hat ( 1 ) force is requi red for the continuance of mo tion and that (2 ) a given force working on a given re s i stance produce s a deter m inate veloc ity. 1 07 Bur ida n' s "impetus" theory is, the refore, s een to be a development of Ari s totle' s theory of "con tactual" cau­ s ation into one entailing the notion of a cau se-effect relation ship inter nal to the body in motion, such a mechanic s being unknown, either to ancient or to modern phy sic s . The a bove view s seem es sentially sound, es pecially a s an inter­ preta tion of Buridan' s thought, but they, unfor tunately, leave out of a ccount the i mportant line of development in phy sical theory whos e his tory we have ju s t now been a ttempting to t r ace . It may well be that the ma jor ity of the schola stic s thought of a given proportion of force to re s i s tance a s a s s ociated with a "con stant veloc ity ;• but it

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doe s not ne c e s s a r i ly fo ll ow that " c o n s tant ve loc ity" had the s a m e una m b i guou s m e a ning whic h it po s s e s s e s toda y . In th e m athe m atic al portion o f B r adwa rd ine' s t r e a ti s e we have al r e a dy s e e n the a mb i g uity (fro m a c o nte mpo r a r y vi e wpo int) o f the te r m , " pr o go r tio :• A c a r e ful e xa m inatio n of the me ani n g of thi s te r m s h o w s it to b e much mo r e c lo s e ly a kin to that o f th e mod e r n te r m , " fun ction ;• than t o t hat o f the t e r m , "pr opo rtion ;• fo r i t in­ c lude d (in i t s sc h o la stic s e n s e) not o nly the r e lation ship o f an e qual­ ity of a r ithm e tic ratio s b ut a l s o that of an e qua l ity of a r ithm e ti c d i f­ fe r e n c e s . In the c a se o f the te r m , " vel o c ita s ;• anoth e r a mb i g uity wa s at l e a s t l ate nt . " Ve loc i ta s" mi ght r e fe r s i mply to a given di s­ tanc e tr ave r s ed in a g i ve n ti m e , or it m i ght re fe r to the s pe e d o f the m ov ing b ody a t a g i v e n m o m e nt dur ing it s t ra v e r s al o f that d i s ta nc e . Onc e B r adwa r dine had fo r mulat e d c le a r ly thi s di stinc tion and had p o inted out tha t it i s t o i n s tantan e o u s and " qual itati ve" ra the r than to tota l a nd " quantita ti ve" ve l oc i ty that pr oportion s o f fo r c e to r e s i s tanc e a r e to b e r e lated , it fo l lowed tha t the " c o n s tant ve­ l o c ity" t o be a s s o c ia ted with s uc h a p r o po rtion would be what wo uld now b e c al led an " in s tanta ne o u s rate :• D umbl e ton the n s ho w s that s uc h a rate (inte r p r e te d a s a " quality" inte n s i v e l y a c quir e d) g i ve s r i s e to an ac c e le rat e d " quantitative :• o r total , v e l o c i ty . The ups hot o f t hi s de v e l o p m ent i s tha t for e e b e c o me s uncle r s to o d p ri ma r i ly a s re l ate d t o an in s ta ntane o u s qua l i ty o f mot ion whic h , c on s ide re d quant itati ve ly, give s ri s e to a n a c c e l e rated m o ti o n r ath­ e r than one of unifo r m ve loc ity . The l o g ic a l c onc lu s ion o f s u c h a r e ali zation i s no t th at in the ab s e n c e o f fo r c e the re i s no m o t ion , b ut that in the ab s e n c e o f fo r c e the r e i s no ac c e l e ration . It wo uld , admitte dly, b e going too fa r to c l a i m tha t thi s r e v i s i on in the unde r­ s tand ing of ho w fo r c e is re l ate d to mot ion wa s wo r ke d out c o m­ plete ly by the M e r ton phy s i c i s t s , but it s houl d a t l e a s t b e c le a r that i t i s the i r po int o f vi ew , r a the r than that o f the advoc ate s o f B ur i dan' s " i m pe t u s" the o r y , whi c h appe a r s c lo s e r t o the s pi r it o f m ode r n phy s i c s . On the b a s i s o f D umble ton' s w o r k in kine mati c s , one m u s t , at a l l e ve nt s , di s a g r e e with M i s s Maie r' s state m e nt that the noti o n o f ve­ l o c ity a s a quali tative m a gnitude shut o ut the po s s ib il ity of a ny p ro g re s s i n the quanti tat ive analy s i s o f m otio n . 1 08 It i s , in fa c t , thi s v e r y vi e wpo i nt whic h ha s b e e n shown to lead d ir e c tly to a c o r­ r e c t und e r sta nd i ng o f the r e la tion ship b e twe en dyn a m ic s and kine­ mati c s .

THOMAE BRADWARDINI TRACTATUS PROPORTIONUM SEU DE PROPORTIO, N IBUS VELOCITATUM IN MOTIBUS

PROLEGOMENA THE TEX T

The following text of Tho mas of Bradwardine's Tractatus de proP-ortionibus, based upon a collation of four fourte e nth-century manuscripts, was developed for the purpose of making available to the modern re ade r an i mportant work at pre se nt e xtant only in manuscript form and in the relatively scarce copie s of the late fif­ teenth- and early sixte enth-c entury editions already mentioned in the bibliographical section of the pre sent study . The Venice edition of 1 505 has not be en examined by the pre se nt editor , but on the basis of a comparison of the earlie r Paris edi­ tion 1 with se ven yet e arlier manuscripts it was fe lt that the most satisfactory procedure for the preparation of a te xt (which , for many reasons, could not be carried out with critical comple te ness) would be to disre gard the early editions alto g e the r and to conce n­ trate , instead , on the collation of what appe ared to be the be st avail­ able early manuscripts . The Paris edition ( Chapter I of which was care fully com pared to the manuscripts me ntioned) was found to suffe r not only from a numbe r of minor omissions approximately e qual to that found in single manuscripts but also from frequent instance s in which its highly abbre viated manuscript source s were quite clearly misread , or from passage s (apparently corrupt in the source s e mploye d by the early editor) which we re badly recon­ structed . During the course of an exa mination of some of the more impor­ tant printe d library catalogues and other bibliographical source s the location o f some thirty manuscripts of the �proportionibus was discovere d , and it would appear quite possible that at least as many more are still in e xiste nc e . From among the se , twenty-thre e were eliminated on the grounds of their probable dating , prove ­ nanc e , comple te ness , and similar conside rations, and (in the case of the very promising Erfurt manuscripts) because of their inac­ ce ssibility. Enlarge m ents were then made of microfil m of the se ven re maining manuscripts, and , on the basis of the above-mentioned collation of C hapter I , three of the le ss reliable manuscripts were set aside for use only in those passage s in which some doubt con­ cerning the correct re ading of the te xt might not be re solved by the othe r four . Of the four basic manuscripts e mployed, the Vatican 1 76 and Bruge s 500 are the only two which manife st any clear interdepend­ e nce not attributable to the ir com mon dependenc e upon the original source . Their fre que nt concurre nce in the matter of omissions and 57

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addition s i s the s trongest indic ation that they form a manus c ript group. A large number o f the se c oncurrenc e s again st the reading s o f the other two manu s c ript s are to be found in c a s e s where the text i s repetitive in it s struc ture and the eye o f the c opyi s t might ea sily lo se it s plac e, and the Bruge s manu sc ript (whic h, o f the four, appears to be the mo s t c learly a produc t o f a pro fe s sional c opyi st) manife st s s everal errors of thi s kind in addition to tho s e c ontained in the Vatic an c opy . In the variants whic h they c ontain , the other two manu s c ript s (the Bibliotheque N ationale 1 4 576 and Bodleian 228 ) appear to be independent b oth o f eac h other and o f the Vatic an-Bruge s group . The B . N . 1 4576 i s o f s ome intere s t as inc luding the date o f publi­ c ation o f the De P-ro portionibus at the c lo s e .2 TheBodleian 228, on the o ther hand, is notable as frequently giving readings whic h are grammatic ally c orrec t agains t grammati c al error s in whic h the other three manu s c ript s agree. A s ide from the few qua lification s noted above, h owever, all four manu s c ript s s how remarkably little di sagreement among eac h other, and, in general, eac h ha s been given equal weight in e stablis hing the pre sent text. The oc c a s ion s upon whic h it bec ame nec e s s ary to draw upon the three auxiliary manu s c ripts were very few in number, and there s eem s little rea­ s on to expect that a full c ritic al editi on o f the work would intro­ duce any s ub stantial change s . Glanc ing through the rather large apparatu s o f variant reading s whic h ac c o mpani e s the present text, the reader may perhap s won­ der whether the above remark s c oncerning the high degree o f agreement between the manu s c ript s are ju s tified . The large nur11ber o f variant s i s, h owever, not inc on s i s tent with thi s claim . In the fir st place, perhap s all but one o f the manu s c ript s employed were not the work o f pro fe s sionals , and, since they were c opie s pre sumably intended primarily for the private u s e o f the c opy­ i st, there wa s little need to ob serve a c areful ac curac y in the re­ c ording o f the c orrect inflectional ending s. In the sec ond plac e , the policy of the pre sent editor ha s been to rec ord a s ac c urately a s po s sible the reading s o f the manu s c ript s in exac tly the fo rm in whic h they appear to be s et down. Thi s undoubtedly doe s violenc e to the intention s o f the original c opyi st s, who were obviou sly not s triving fo r a meticulou s ac c urac y either in grammar or in the u se made o f abbreviation s and who relied upon the reader' s gra s p o f the c ontext t o c arry him s afely over letters and word s incor­ rectly o r illegibly spelled . The c on s iderable dangers to be enc oun­ tered in an attempt to distingui s h between c a ses in whic h one might legitimately tran s c ribe what the c o pyi st "must have intended" and tho se in whic h he ha s s et down a true va riant dic tated, however, a polic y o f unmodified tran s c ription . In the primary apparatu s to the pre s ent text have been rec orded all variant reading s with the ex cepti on s o f si mple tran spo sition s

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a nd tho s e pe c uliar itie s o f s p e ll in g whi c h a re g r a mmatic a lly and s e ma nti c al ly i r re le vant . Variation s in the s p e ll ing of p rope r na me s have , howe ve r, b e en inc lud e d . Though the e mplo y m e nt of c la s s ic a l o rtho g r aphy i n t h e p r e s e nt e dit ion may , pe rha p s , b e o p e n t o c e r­ tain c r i tic i s m s , i t w a s fe l t that in the inte r e s t s o f le g ibi lity fo r the mode rn r eade r and of sho r tening the te xtual a ppa r a tu s (whi c h would othe r wi se ha v e b e c o me unm ana g e ably la r g e) thi s po li c y wa s suf­ fic i e ntly we ll ju s tifie d . The or tho g raphy o f the manus c r ipt s i s , m o r e o v e r, by no me a n s c on s i s te nt enough to indi c ate c l ea r ly wha t would b e a l e giti mate polic y in atte mpting to e mplo y m e diaeval s pe l l ing s . It s hould al s o be m e nt ioned that the e xpan s ion o f c e r ta i n abb r e viat i o n s ha s b e e n unavoidab l y a rb i tr a r y (i s te a nd ill e , e r gQ a nd igitur , for e xa m ple , b e in g o fte n ind i stin gui shab l e in the ir ab­ b r e viate d for m s) . The s e c ond a ry a ppa rat u s c o mpri s e s the ve rifi c a tion o f r e fe r­ e nc e s to oth e r wo r k s , whe r eve r s uc h ve rifi c ation ha s b e e n po s s ibl e . The a r ra ng e m ent , punc tua ti o n , and the d ivi s ion o f the te xt into s ente nc e s and pa rag raphs i s , o f n e c e s s ity, alm o s t who lly the wo r k o f the p r e s e nt e d it o r. The manu s c r ipts show n o pa rtic ula r c ons i st­ e nc y o r c a r e in the s e matte r s , and the r e a de r s hould be war ne d that i t i s oc c a s i o na l ly po s s ibl e to c on st rue c e rta in pa s sa g e s i n way s s l ightly di ffe r e nt fr om tho s e ind ic a ted . Punc tuation ha s b e en e m­ plo y e d in a s o m e what m od e rn fa s hion . T he r e having b e e n intro­ duc e d into the body of the te xt no inte r polatio n s o f any kind , pa r e n­ the s e s ha ve b e en e mplo y e d s o le ly fo r the purpo s e s o f s e tting apa rt mate r ial which m i ght w e ll be tr eated i n the s a m e way nowaday s and fo r mar king o ff c laus e s within lar ge r c laus e s s o a s to make a n int r ic ate s e nten c e s t r uc tur e m o r e r e adable . The title s o f wo r k s r e fe rr ed to have b e e n unde r s c o r e d , and und e r s c o ring ha s l ikewi s e b e e n e mployed t o s e t o ff te r m s whic h a r e b e ing d e fine d , de finition s, a nd c e rta in o f the m o r e i mpo r tant c onc lu s ion s . T he fou r b a s i c manus c ript s for the te xt a r e a s fo llow s : 1 . Par i s . Bib liothe que Nationa le , M S . lat . 1 4 5 76 , fol s . 2 5 5v- 2 6 2r. Se e Bib liotheque de l ' e c o le d e s c ha r te s (Pa ri s , 1 8 3 9-1 9 44 ) , XXX , 26 . 2 . R o m e . Bib liote c a Vati c ana , M S . Ottob . lat . 1 76 fo l s . 9 2 r-9 8r . Se e R e v. Philothe u s B o e hne r, 0 . F . M . , The T ra c ta tu s de P rae­ d e s tina tione et d e P r a e sc ie ntia De i et de Futur i s Continge nti­ b u s of Will ia m O c kha m { Fran c i s c an In stitute Public ation s No . 2 , St. B onave ntu r e , N ew Yo rk , 1 9 4 5 ) , p . i x . 3 . B ruge s . B ib lioth e que Pub lique de l a Vill e de B ruge s , MS . 5 0 0 , fol s . 1 5 8r- 1 7 2 v . Se e Cata log� ge ne ral de s manu s c rits de s bibliotheque s d e B e lgique , II: Ca talo gue d e s manu s c r it s de la B ibl iothe que de l a Ville de B rug e s {Ge mb loux , 1 9 3 4 ), pp . 5 8 1- 5 8 4 . 4. O xfo rd . B odl eian Lib r ary, MS . Digby 2 2 8 , fol s . 5 6 r- 6 l r .

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P RO L E G OM E NA See Catalogus Codicum Manuscriptorum Bibliothecae Bod­ leianae (Oxford , 18 8 3 ), Part IX : Digby MSS . , pp . 23 9-41.

The three auxiliary manuscripts are as follows : 5. Oxford . Bodleian Library, MS. Digby 76 , fols. 1 10r- 120r. See ibid . , pp. 82- 8 3 . 6 . --. MS. Canonici 17 7, fols . 16 4r- 17 lv. See ibid . , Part III : Greek and Latin , Canonicianos, col . 16 4 . 7 . Rome . Biblioteca Vaticana , MS . lat . 1 108 , fols . 6 9r- 8 lr . See Cod . Vat . Lat . (Vat ican Library , 19 3 1 ) , Vol . I I , Part I , pp . 7 19-2 5 . The following manuscripts , not employed i n the establishment of the present text , const itute the remainder of those discovered . 8 . Vienna . Biblioteca Palatina , MS. 495 1 (Univ. 3 20 ), fols . 260r2 7 lv. See Tabulae codicum manuscriptorum in Biblioteca Palatina Vindobonensi (Vienna, 18 10 ) . IV, 4 4 4 . 9 . Bruges . Bibliothe que de la Ville de Bruges, MS. 4 9 7 , fols. 59v6 4r. See Catalo � ge ne ral des manuscrits des biblioth�que s de Belgi_que , II : Catalogue des manuscrits de l aBibliotheque de la Ville de Bruges (Gembloux , 19 3 4 ) , pp . 5 78- 8 0. 10. Cambridge . Corpus Christ i College Library, MS. 24 4 (24 5 ) . fol s . 7 8 r-8 3 r . See A De sc riEtive Catalogue of the Manuscripts in the Library of Corpus Christi Colleg�, Cambridg� (Cambridge, 19 12 ) , I, p . 54 3 . 1 1. 0 xford . New College Library, MS. Cha under 28 9 , fo 1s . 6 3 v-6 8 r. See Catalogus codicum manuscri P-torum qui in Collegiis Au­ lissue Oxoniensibus hodie adservantur (Oxford , 18 52) , Part I , pp . 103-5. 12. Oxford . University College Library , MS . 26 , fols . 146v- 150r. See ibid . , pp . 7-8 . 13 . Oxford . Corpus Christi College L ibrary, MS. 228 , fols . 4r- 8 r. See ibid . , Part II . pp. 9 2- 9 3. 14 . Paris . Bibliothe que Nat ionale, MS . lat . 62 5, fols. 59v-6 2r. SeeBiblioth e que Nat ionale nouvelles ac qui sitions du Departe­ ment des Manuscri ts (Paris, 1890 ) , pp . 9- 10. l 5 . --. MS. lat . 1662 1, fols. 1 10v- 2 12v. See B.iblioth�sue de l ' e cole des cha rtes (Paris , 18 70 ) . XXX I, 155. 16 . Erfurt . Stadtbucherei, MS. Amplon . F l 3 5, fols . 20v- 2 5r. See Beschreibendes v� � zeichniss der Amplonianischen hand­ schriften- Sammlung z u Erfurt (Berlin , 188 7 ), pp . 8 8 - 8 9. 17 . --. MS. Amplon. F 3 13 , fols . 166r- 190r . See iJilil . , pp . 2 16 - 17 .

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1 8. --. M S . Amplon. F 3 8 0 , fol s . 4 9 r- 5 8 r . Se e ibid . , pp. 2 6 6 - 6 8. 1 9. --. MS. Amplon. Q 3 2 5 , fol s . l S 0r - 1 6 7 r . Se e ibid. , pp. 5 5 9- 6 1 . 2 0. --. MS. Amplon . Q 3 4 8 , fol s . l 3 r- 2 2 r . Se e ibid . , pp . 5 8 1 - 8 3 . 2 1 . --. M S. Ampl o n. Q 3 4 9 , fols. 4 8 r- 5 5v. Se e ibid . , pp. 5 8 3 - 8 7. 2 2. --. MS. Amplon . Q 3 8 5 , fols. l r- 1 6 v. Se e ibid. , pp . 6 4 1 - 4 4. 2 3. --. MS. Amplon Q 3 8 7 , fols. lv- 8r . Se e ibid . , pp. 6 4 8- 4 9 . 2 4. V enice . Bibliote ca Mar ci ana , MS . V I . 6 2 , fol s . 9 9- 1 1 1 . Se e B ibl ioteca Nazional e Mar ciana, B ibl iothe cae Manu s c r i p-=. tor um ad. S. Mar ci ve netia r um Codice s manuscr i P-to r um latini (Venice , 1 8 6 8- 1 8 7 3 ) , I V, 2 3 3-3 4. 2 5. -. MS. V I. 1 5 5 , fo 1 s . 9 2- 1 0 5 . Se e ibid. , pp . 2 2 9- 3 1 . 2 6 . --. M S . V III . 3 8 (Val e ntine l l i X I. 1 4) , fols. 1 - 8. Se e ibid . , pp. 2 2 5- 2 6 . 2 7 . -. M S . V I II. 7 7 , fols. 1- 1 4. Se e ibi d . , pp. 2 5 7- 5 8 . 2 8 . New Yo r k . Columbia Unive rsity Lib r ar y , MS. Plimpton 1 8 1 , fols. 1 0 1 - 1 5. Se e C ensu s of Mediae va l and R enaissance Manuscr iP-t s in the Unite d States and Cana�a (N e w Yo rk , 1 9 3 5- 4 0) , II , 1 7 86 . 2 9. London. B r itish Mu s e um Lib rary , MS . O ld Royal 8 AX V III , ite m 6 . Se e B r i ti sh Mus eum C atalogue of We ste rn Manu scr ipt s in the Old Royal and King s Col le ctions (O xfor d , 1 9 2 1 ) , I, 2 1 5 . 3 0. --. MS. Har le i an 2 6 7 , fo 1 s . 2 2 8- 3 0 . Se e B r iti sh Mu s e um D e partm e nt o f Manu s criP-t s , Catalogue of the Har l e ian Manuscrip.!.§_ ( London , 1 8 0 8 ) , I , 1 0 1 . T HE T R ANSLAT ION In making the fol l owing tran s lation of B radwardine's T r actatu s de P-r O P.o rtioni bus the p rima r y inte ntion has be e n to prov i de for th e Engl ish- sp eaking r e ade r a s clear and flue nt a v e r s ion of th is wo r k a s po s s ibl e . Not only hav e both the punctuation and ar r angeme nt be en chos e n with th is e nd in v i e w, but , s ince a Latin te xt i s avail­ abl e for pu rpose s of compar ison , it has al s o s e emed pe rmis s ibl e to ma ke occa s ional departur e s f rom a s tyl e that i s fr e que ntl y ove r­ cumb rous to the mode r n taste . T h e t ran s lation o f te chn ical te rms has , of cour s e , pr e sented it s s pe cial diffi c u lti e s , and the only ge ne ral pol icy adopte d has , again , b e e n simpl y that of att empti ng to gain as high a degr e e of inte l li-

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gib i l ity a s p o s s ib le . In the ab s e n c e o f E n g l i sh e quiva l e nt s fo r Latin mathe mati c a l te r m s , th e t r an s la ti o n of Chapte r I ha s had to r e so rt to ne w a nd , it i s fe a r e d , ra the r b a rba r o u s c o i na g e s . The r e a d e r s ho u ld , ho we v e r , e xpe ri e nc e no p a r t i c ula r d i ffi c ul ty wi th th e s e , s inc e th i s fi r s t c hapte r i s l a r g e ly c onc e r ne d with th e i r de finit ion , anyway. E v e n th e u s e o f th e t e r m , " p r opo r ti on ;• fo r e xa mp l e , i s no longe r mi s l e a din g , if on e pa y s c l o s e atte nti o n t o th e ma nne r in whi c h it i s de fine d . Othe r t e r m s , how e v e r, have p r e s e nt e d mo re o f a p rob le m . " Mo­ b ile,, ha s , p e rhap s , s uffic i e n t c u r r e nc y to s t and unt r a n s l a t e d o r t o b e unde r s t o o d wh e n r e n d e r e d a s " th e m o v e d" o r " the m o v e d b o dy :• " Ve l oc i ta s" ha s b e e n r e nd e r e d a s " s p e e d :• r athe r tha n " ve loc ity ;• in o r de r to a v oi d the e r r o ne o u s infe r e nc e that a v ec t o r quantity i s t o b e und e r s t o od a s i nv o lv e d . " P o t e n tia" ha s b e e n t ra n s l at e d a s " powe r :• r a the r than .. fo r c e ;• fo r the r e a s on tha t " P-o t e nt ia" i s c o n­ c e ive d mo s t funda me nta l l y a s a c apa c it y to do wo r k . It i s , o f c o u r s e , t r ue tha t i m po r ta nt a mb i g ui t ie s r e ma i n , the d i s tinc tion de v e l o pe d b e tw e e n " t ot a l" a nd " in s tanta n e ou s,, v e lo c it y b e in g , fo r e xample , o ne o f th e i mp o r t ant c o nt r ib ution s o f th e �IU.QP-O r tionibu s . Ne ve r­ the le s s , the te r m s , " s p e e d ;' and , " powe r ;• s e e me d th e b e s t a v ail­ ab l e , and the c a r e ful r e ade r s ho uld e xp e r i e nc e no g r e a t e r di ffic ulty in un d e r s tand in g the s e o r o the r u s a ge s than he wo uld i n unde r­ s tand ing the Latin t e r m s , the m s e l v e s . Fo r a ve r i fi c a t ion o f o the r w o r k s a l l ud e d to in th e De P-r o P-o r ti o ni­ b u s , the r e a d e r i s r e fe r r e d to the s e c o nd a r y app a r a tu s to the Latin te xt .

TEXT AND TRANSLATION

THOMAE BR ADWARDINI TRACTATUS PROPORTIONUM S EU DE PROPORTIONIBUS VELOCITATUM IN MOTIB US P ROEM IUM 1 Omne m motum s uc c e s s i vum a l te r i in v e loc ita te p r o po r ti o na r i c onti n gi t ; qua p r op t e r phi lo s ophia na turali s , qua e d e m o tu c on­ s ide r at , p r opo r ti one m mo tuu m e t v e l o c i tatum in motibus i g no r a r e non d e be t . E t quia c o gn i tio i l l i u s e st ne c e s s a r ia e t m ul t um diffi c il i s , 5 n e e i n a l i qua pa r t e phil o s o phiae tra dita e s t a d p l e n u m , id e a de p r op o r ti one ve l o c ita tu m in m otibu s fe c i m u s i s t ud opu s . E t quia , te s­ tante B o e thio , pr i mo Ar ith m e ti c ae s ua e : Qui s qu i s s c i e nti a s ma the ­ mati ca l e s p ra e te r m i s e rit , c on s tat e u m o m ne m philo s o phiae pe r ­ d idi s s e d o c tr ina m , - i d e o m ath e m atic alia qui bu s ad p ro p o s itum i ndi g e m u s p r ae m i s im u s , ut sit d o c t rina fac ilio r e t p r o mpti o r 1 0 inqui r e nti . E t p ropte r m a io r e m p r o mptit ud i ne m e t fac i li ta te m doc t r ina e , i s tud ne gotium in quattu o r di ffe r e nti a s s e u c api tula s e pa ratur . Quo r u m p r i m u m ma the mati c a lia , quib u s ad p r o po s i tu m ind i g e ­ m u s , p r oponit ; quod in t r e s pa r te s d i viditur . Qua r u m p r i ma pr o- 1 5 p o r ti oni s d e fi niti one s , di v i s io ne s e t c e te r a s p r o p r ie tate s o s te ndit . Se c u nda , s i m i li m odo , de p r o po r tional itate d e te r m inat . T e rtia v e r o qua s da m s uppo s i ti o ne s adiun git , e x quibu s qua s dam ma the mati c a s c onc l u s ione s de m o n s t r a t . Ca pit ul u m a ute m se c und um di sputat quattu o r opinione s , s e u s e c- 2 0 ta s e xo rta s d e p r opo r tione ve l o c ita tu m i n m o tib u s ; quad e tia m , s e c undum nu m e rum o pinion um il la r um , i n pa rte s quattuo r e s t d i vi s um . Te r ti um c ap itulum ve r a m s e nt e ntiam d e p r o p o rtione ve loc ita tum in m otib u s , in c o mpa ra tione ad m o ve ntium e t m ot o r u m pot e nt ia s , 2 5 mani fe s tat ; quad etiam i n dua s p a r t e s e s t d i vi s um . Qua r um p r i m a qua s da m c o nc lu s ione s d e p r o po r ti o ne v e l o c itatu m i n m otib u s d oc e t e t d e te r minat . Se c unda ve ro c on t ra e a s d e m ob ic it e t s o lvit . Capitul u m aute m qua rtum d e p r o p o r t io n e v e l o c i ta tum in motibu s , in c o mparatione ad m o t i e t s pa tii pe r tr a n s iti quantitate s, pe rt r ac tat , 3 0 e t s p e c ialit e r ad m o tu m c i r c ul ar e m d e s c endit ; quod i n parte s t r e s s i mi lite r e s t pa r titum . Qua r um pr ima quae da m ma the matic a l ia , ad i l l ud ne c e s sa r ia , p r i mo do c e t . S e c unda ve r o qua edam op inio ne s d e pr oport ion e ve loc ita tu m in m otibu s , in c o mpa ra tione ad ma gni­ t udin e s moto r u m e t s patio r u m p e r tr a n s i to r um , r e da r guit , e t ve ri- 3 5 tate m o s te ndi t . Te rtia a ute m c i r c a pr opo rti one s el e m e nto rum qua s da m la te ntia s manife s ta t . Igitur a d pr opo s it u m t r an s e a m u s .

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THOMAS OF BRADWA R DINE'S TR EA TIS E 0� PR OPORTIO:\"S, OR ON THE PROPORTIO�S OF THE S PEEDS Of MOTIO:\"S INT ROD UC T IO N Sin c e e a c h suc c e s s ive motion i s proportionable to another with r e spect to spe ed , natur al philo s ophy , which studie s motion , ought not to ignore the pr oportion of motion s and the ir spe ed s , and , be­ c au s e an unde r s tanding of thi s is both ne c e s s a ry and extre me ly diffic ult , nor ha s a s yet be en tre ated fully in any b ranch of phi lo s­ ophy, we have accordingly c om po s e d the following wor k on the sub­ ject. Sinc e , more ove r (a s B oe thiu s point s out in B oo k I of hi s Arith­ metic), it is a gre ed that whoeve r omits m athe matical s tudie s ha s de str oyed the whole of philosophic knowle dge , we have commenc e d by s etting for th the mathe matic s ne e de d for the ta s k in hand , i n orde r to make the s ubj e c t ea sie r and more ac ce s s ible to the s tu­ dent. For the sake of thi s sa me ea s e and a c c e s sibil ity , the wor k i s al so divide d into fo u r s ections , o r chapte r s . The fir s t of the s e , s etti ng for th the ne c e s sar y mathe matic s , i s subdivided into thr e t; part s , of which the fir st take s up the defini ­ tion s , type s and other prope r tie s of p roportion . The s e c ond deal s with proportionality in a si m ila r fas hion. The thi rd add s c e r tain axiom s , fro m whic h seve ral mathe mati_c al c onc lu s ion s are drawn. Chapte r two , on the othe r hand, argue s against four opinion s , or s c hool s of thought , whic h have ari sen c onc e rning the proportion b etween the s pe e d s of motion s and , following the numb e r of thos e opinions , i s divided into four part s . Chapte r thr ee make s c lear the corre ct und e r sta nding o f the pr� portion betwe en the spe ed s of motions , with r e s pe c t to both moving and r e s i sting powe r s , and thi s al so is divided into two pa rts . The fir st of the se de velops se ve ral theorem s conc e r ning the propo rtion b etween the s pe e d s of motions , and the s ec ond rai s e s and s ettl e s objec tions to the m . Chapte r four tr eat s of the p ropor tion b etween the s pe e d s o f m o­ tions with r e sp e c t to the quantitie s of the moved body and the inte r­ val t rave r s e d , and inc lude s a spe c ial d i s c us s ion of cir cular motion . It i s divided into thr e e part s , the fi r st of which comm enc e s by e stablishing the re qui s ite mathe matic al mate ria l . Part two unde r­ take s the r e futation of s eve ral opinion s c onc e r ning the proportion between the spe e d s of motion s, with re spe ct to the magnitude g both of m ove d bodie s and of interval s trave r se d , and s e ts for th the cor­ r e c t acc ount. The thir d , finally , di sc los e s c e r tain hidden truth s c onc e rning the proportion s b etwe e n th e e l e m ent s . Le t u s then pa s s on to the ta s k in hand . 65

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CAPIT ULUM P R IM UM Omnis pro portio vel est dicta communite r , ve l proprie est acce pta . Comm uni t e r in omnibus quae aequale , vel maius vel minus , seu e tiam simile , ve l e tiam magis m i nusve , suscipiunt rep eritur . Ide o , in quibuscumque po te st aliqua comparatio fi e ri , est inve nta . Quae sic pot est de finiri : Proportio est duorum comparato rum , in aliquo in quo comP-arantur, unius ad alt e rum habitudo . Proportio aute m quae propri e e st accepta , i n solis quant itatibus r e p e ritur . Quae de fi nitur hoc modo : ProP-o rtio est duarum guanti­ tatum e iusdem g e n e ris unius ad alte ram hab itudo. Et hae c e st duple x ; nam rationalis , e t in primo gradu propo rtio nalitatis , est illa quae imm ediate de nominatur ab aliquo num e ro : sicut propo rtio dupla , e t tripla , e t sic de ali is. S ecundum v e ro g radum illa t e n e t quae irratio nalis vocatur , quae no n imme diat e de nominatur ab aliquo nume ro , se d m e diate tantum (quia imme diate de nominatur ab aliqua pro porti o ne , quae imm e diate de nominatur a num e ro : sicut m e di e tas duplae proport ionis , quae est proportio diam e tri ad costam , et m e dietas sesquioctavae proport ionis , quae to ni m e di e­ tat em co nstituit) . Proportio aute m rationalis di ffe rt a p ro portione irrati onali , quia haec tantum in quantitatibus comme nsurab ilibus se u rationalib ilibus re pe ritur ; i lla in solis quantitatibus i ncomm e nsurab ilibus seu i rrationalib ilibus invenitur . Quantitate s communicante s s eu com­ m e nsurabile s seu ratio nales sunt quibus est una me nsura communis , illarum quamlib e t prae cise m e nsurans : sicut linea b ipe dalis e t linea tripe dalis , quarum utramque line a pe dalis praecise m e nsurat . Quantitates autem non- communicantes seu incomm e nsurab ile s sive i rrationale s sunt quibus non e st aliqua me nsura communis , quamlib e t illarum praecise me nsurans : cuiu smodi sunt diam e t e r et costa quadrati . Proportio autem rationalis reperitur in nume ris et aliis quanti­ tat ibus quibuscumque ; proportio vero irrationalis non in num e ris , se d in omnibus aliis quantitat ibus , pot est e sse . Haec aut em arith­ m e ticae e t ali is mathematicis pe rtine t ; illa v e ro non arithm e ticis , sed om nibus aliis mathematicis , dignosci tur p e rtine re . Pro portio aut em mag is pro pri e dicta , quae arithme tico p e rtine t , ab ari thm e ticis dividitur isto modo : Proportio quae dam e st aequal­ itati s ,_quae dam inaequalitatis. Pro portio aequalitatis est duarum quantitatum_ _� egualium adinv icem habitudo . Proportio aut em inaequalitatis est duarum guanti tatum inaequalium adinvicem habitudo . Et haec e st duple x : quae dam e nim est maioris inae qualitatis , e t quae dam minoris . Quarum prima est habit udo quantitat is mai oris ad minore m ; secunda ve r o est hab itudo mino ris quantitatis ad mai or e m . Harum aut em prima species hab e t quinque . Quarum tres sunt sim plice s : scilice t multiple x , mp�articularis , e t fillp�parti e ns . Duae vero re siduae sunt compositae e x prima e t duabu s ali is

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C HA P T ER ONE Every proportion is to be taken e ithe r in the g eneral or in the stri c t s ense . In the g e ne ral sense i t i s found to e xist b e twe e n all things c ap­ able of be ing te rm e d e qual, greater, and less e r or s i m ilar, greater, and lesser. It is , in fact, found to e xist b e twe e n whatever things are open to some c om parison an d may be de fine d as follows : " A proportion is the relation of one thing to another, with resp e c t to that in whic h they are c om pare d :' In the str i c t s e ns e , howev e r, proportion is found only b e tw e e n quanti­ ti e s an d is de fine d as follows : "A proportion is the relation b e twe e n two quantiti e s of the sam e kind :' T he r e are two typ e s of such proportions, for one that is .. rational" and in the first order of proportionality is im­ m e d iately d e nom inate d by a g iven numbe r (as is the c ase with the pro­ portions of two to one , thre e to one, and so forth) . T he se c on d or de r c om pri z e s those proportions whi c h are c alle d " irrational :' These are not i m m e diately but only m e diately denom inate d by a g ive n number, for the y are i m m e diately de nom inate d by a g iven proportion, whi c h is, in turn, im m e diately denom inate d by a numbe r . Of this sort is the s quare root of the proportion of two to one , whi c h is the proportion o f the diagonal of a s quare to its si de, an d the s quare root o f the propor­ tion of nine to e ight, whi c h c onstitutes a m us i c al half-tone . R ational di ffers from irrational proportion, moreover, in that the form e r e xists only b e twe e n c om m e nsurable or rational quantiti e s, where as the latter is foun d to e xist only b e twe e n inc om m e nsurable or irrational quantiti e s . " Com muni c ative :• " c om m e nsurable ;• or " ra­ tional" quantities are thos e for whi c h there e xists a c om mon m easure whi c h m e asures each of the m e xac tly (as i s the c ase with a two foot line and a thre e foot line, both of whi c h a one foot line m e asures e x­ ac tly) . " Non- c om m uni c at ive � " in c o m m e nsurable :• or .. irrational'' quantitie s are those for whi c h the re doe s not e xist suc h a c om mon m e asure (of whi c h sort are the diagonal and s i de of a s quare). Further, a rational proportion may e xist both be twe e n numbers and betwe e n quantiti e s of any other kind, whereas an irrational proportion c annot e xist b e twe e n numbers but c an betwe e n all other kinds of quan­ tity . The for m e r belongs both to arithm etic and to the other mathe ma­ tic al studi e s ; the latter doe s not b elong to arithm e ti c but do es to all oth e r branches of mathematic s . Proportion in its stri c te r s e nse (as p e rtaining to arithm e ti c ) is sub­ divi de d by ar ithm e ti c ians as follows : A proportion may be e ith er one of .. e quality" or one of " ine quality." A p roportion of e quality is the mutual relation o f two e qual quantiti e s . A proportion of ine quality, on the other hand, i s the mutual relation o f two un e qual quantiti e s an d may b e o f two type s : e ither one of " gre ate r ine quality" or one of "lesser ine quality :• T he first of thes e is the re­ lation of a larg e r quantity to a smalle r one , an d the s e c ond is that of a smaller quantity to a lar g e r. O f the s e two typ es the first has five furth er sorts . Thre e of th ese are simple, nam ely : " m ultiple ;• "sup e rparti c ular" and .. supe rparti e nt :' The

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simplicibu s : sc ilicet multiP-lex S UP-eq�artic ularis et multiP-lex suP-er­ P-artiens. 50 M ultiP.lex vero P-T OP-ortio est habitudo quantitatis maioris ad minorem, illam multotiens continentis . Et haec ulterius in species infinitas partitur . Si enim maier bis minorem contineat, duplex sive duP-la P.roportio nominatur; si autem ter , triplex sive tr.iplg ; et sic in infinitum proceditur . 55 Superpartic ularis autem P-roportio est habitudo quantitatis maioris ad minorem, illam semel et eius partem aliquotam con­ tinentis . Pars autem aliquota est illa quae, ali quotiens sumpta , reddit aequaliter suum totum . Pars vero non-aliquota est illa quae nullatenus , aliquotiens sumpta , reddit aequaliter suum totum : ut binari�s respec tu quinarii . Haec autem proportio infinitam rec ipit 60 sec tionem, quia, si maier quantitas semel minorem et eius medie­ tatem contineat, sesquialtera et hemiolia proportio nuncupatur : ut est proportio trium ad duo . Si autem maier quantitas semel minorem et eius tertiam partem contineat, sesg_uitertia dic itur : sicut quattuor ad tria se habent . Et sic , in infinitum , speci � s producun65 tur . SuP-erpartiens vero P.rOP.Ortio est habitudo g_uantitatis maioris ad minorem, illam semel et aliquot eius P-artes aliquotas (ex quibus non sit una P-ars aliquota) continentis . Et haec, sicut aliae, in spec ies infinitas secatur . Et hoc triplic iter : Uno modo ex parte numeri 70 partium praedic tarum ; secundo modo ex parte denominationis partium eorumdem ; et tertio modo mixtim ex ambobus . Et ideo , si maier quantitas semel minorem et eius duas tales partes (ut quinque , tria) c ontineat, dic itur ista proportio S UP-erbipartiens ; si tres, suP-ertriP-artiens; et sic non est status . Si autem maior quan- 75 titas semel minorem et aliquot tales partes eius contineat , quae sunt tertiae totius , S UP.erpar tiens tertias vel supertertia P-rOP-Ortio nun­ cupatur ; si istae partes sint quartae, suP-erpartiens g_uartas ; et sic semper procedit . E x mixtione harum specierum aliae infinitae generantur . Si 80 enim maier quantitas semel minorem et eius duas tales partes c ontineat, quae sunt tertiae totius, suP-erbipartiens tertias vel §JJP.erbitertia P-roportio dici debet : c uiusmodi est proportio quinque ad tria . Et si istae partes sint quintae totius, superbipartiens guintas vel superbiquinta dicetur : qualis est .proportio septem ad 8 5 quinque . Et processus huius nullatenus terminatur . Si autern maier quantitas semel minorem c ontineat et tres tales partes eius , quae sunt quartae totius , §.Yperte q�artiens quartas vel fila:l.perterquarta P-rO P-ortio appellatur : qualis est septem ad quattuor . Et sic, utroque 90 modo , sine termino sit proc essus . Proportio autem multiplex superparticularis est habitudo quanti­ tatis maioris ad minorem illam multotiens et eius partem ali­ quotam continentis ; quae in species infinitas tripliciter est partita . Primo , ex pa rte multiplicitatis : ut duplex su�particularis, triP-lex superparticularis , et sic semper proc editur . Sec undo , ex 95 parte superpartic ularitatis : ut multiP-lex sesquialtera, m!!l1.iP-lex

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two r e ma 1mng a r e fo r m e d b y c o mpounding the fi r s t o f the s imple sp e­ c i e s with the o the r two , na m e ly , " multiple supe r p a r tic ul a r" and " m ulti­ ple supe r pa r ti e nt :• Now , a " m ultipl e p r opor tion" i s the r e lation o f a la r g e r quantity to a s ma l le r one in whi c h the la r ge r c o nta in s the s ma l l e r a numbe r o f time s , and thi s type i s o p e n t o di v i s ion into a n infinite nu mb e r o f fur­ the r s ubc la s s e s ; if the la r ge r quantity c o ntain s the s ma l l e r twic e , it is c al l e d a "doub l e ;' or " dupl e x ;• p r o po r tion , i f thr e e t i me s " t r eb l e � o r " tr iple x ;• a nd s o on , inde finite ly. A " s upe r pa r tic ular pr opo r tion � on the o the r hand , i s the r e la tion o f a l a r g e r quantity to a s malle r one in whic h t h e la r ge r c o nta ins the s ma ll e r one onc e , plu s an a l i quot pa rt of it . An "aliquot pa r t" of a qua ntity i s that whic h , take n a given numb e r o f ti m e s , wi l l r e c o n s titute the who l e quantity. A " non- a li quot pa r t ;' howe ve r, i s one whi c h , no matte r what numb e r of ti me s it be take n , wi l l not r e c on s titute the o r i g­ ina l quantity e xa c tly, a s i s the c a s e with two in r e lation to five . T he ab o v e type o f pr opor tion i s a gain o p e n to infinite s ub divi s i o n . If th e la r g e r quantity c o ntain s t h e s ma l l e r o ne o nc e , plus a hal f o f it , it i s c al l e d a " s e s quialte rna te" o r " he mio lian" p r o po r tion ( a s i s the p r o po r tion of thr e e to two). If the la r g e r c o nta in s the s ma l le r one onc e , plu s a thi r d of it , it i s c a ll e d " s e s quite r tian" (a s fo ur i s r e la t ed to thr e e ) . Thu s , additiona l s ub typ e s may be p r o duc e d , ad infinitum . A s fo r " s upe r pa rtie nt p r o po rtio n ;• it i s the r e lation of a l a r g e r qua n­ tity to a s ma l l e r one , in whic h the la r g e r c o ntain s the s malle r one onc e , plus m o r e than one o f it s a l iquot pa r t s . Thi s , like the othe r s , i s divi s ib le into infinite sub type s , and that i n thr e e way s . In one way , a c c o rding to the numb e r o f the ab ove me ntione d pa r t s ; in a s e c ond way, a c c o rding to the d e no mination o f tho s e pa r t s ; and in the thi r d way, by a c o mb inati on of the two . If, fo r e xa mple , the lar g e r quantity c onta in s the s malle r one o nc e , p l u s two s uc h pa r t s of it (a s five c ontain s thr e e ) , the p r opo rtion i s c a l l e d " s upe rbipar tie nt" ; i f thr e e pa r t s , " s up e r t r i­ pa r tie nt" ; a nd s o fo r th , e ndle s s ly. If , o n the othe r hand , the la r g e r quan­ tity c o nta ins the s ma l l e r one onc e , plu s a give n numb e r of s uc h pa r t s a s a r e thi r d s o f i t , i t i s c al l e d a " supe r pa r ti e nt te r tian" o r, " s upe r te r­ tian" p r o po r tio n ; i f the s e pa r t s a r e four th s , the n " s up e rpar ti e nt qua r­ tan" ; and s o on , ind e finite ly. F r o m a c o mbina tion o f the p r e c e ding typ e s . an infinite numb e r o f o the r s a r e g e n e rated . If , fo r e xa mple , t h e l a r g e r quantity c ontain s the s malle r one onc e , plu s two s uc h par t s a s a r e thi r d s of the who l e , it i s t o b e c al l e d a " s up e rb ipa rtient te r tian ;• o r " s upe rb ite r tian ;• pr opor tion ( o f whi c h s o r t is the p r o po r tion of five to thr e e). If the s e pa r t s a r e fi fths of the who le , it is c a l l e d " s up e rb ipa r ti e nt quintan ;• o r " s upe rb i quinta n" ( of whi c h s o r t i s the p r o po r tion o f s e ve n to five). Thi s s e r ie s ha s no e nd . If, o n the o the r hand , the l a r g e r quantity c onta i n s the s malle r one onc e , plus thr e e s u c h par t s a s a r e fo ur th s o f the who le , it i s c a ll e d a .. sup e r­ t r ipa r tie nt qua r tan" or " s up e r t ri qua r ta n" p r o p o r ti on ( o f whic h the p r o­ po rtion o f s e ve n to four i s a n e xa mple . Thu s , in both way s , the s e r ie s i s without e nd . " Multi ple s upe r pa r tic ula r p r o po rtion" i s the r e lation o f a la r g e r

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s e sguite rtia , e t s i c s i ne fin e . T e r ti o , m i xtim e x ut ri s que : ut dupl a s e s guialte ra , dupla s e squite rtia , t r ipla s e squialte ra , t r i2la s e s qu i­ te r tia , e t s i c p r o c e di tur s ine fine , qu o r um de fini tione s e t e xe m pla 1 00 ex p r a e di c t i s appa r e nt . P r o po r t io aut e m m u ltipl e x s upe rpa r ti e n s e s t hab i tudo qua ntitati s m aio r i s ad m i no r e m , i l l a m m ul t ot i e n s et a l iqu ot e i u s pa rte s a liquota s (e x quib u s non fit una pa r s e i u s a l i quota) c ontine nt i s . Ha e c aute m s e pt e m m o di s divi di tur i n i nfinitum . Pr i m o , e x pa rt e m ultip l i c itati s , s ic d iv i d e n do : dupl e x s upe rpa r ti e n s , t r ip l e x S UP.e r- 1 0 5 pa r tie n s , e t s i c s in e fine . S e c undo , e x pa r t e s upe rpa rti e nti s , div iditur t r ib u s m o di s ; p r im o s i c : multi P.l e x s uP.e rbipa r tie n s , m ultipl e x s upe r t r ipa r ti e n s ; s e c undo s ic : m u ltiP.l e x s upe rP.a rti e n s t e rtia s , m ultipl e x s uP.e rpa r ti e n s qua rta s ; t e r t io s i c : m u ltipl e x s upe rbi­ 1 10 P-a r tie n s t e rtia s v e l m ul t ipl e x s upe r b it e r t i a s , m ultipl e x s uP.e r t r iP.a r ti e n s qua rta s v e l m ultipl e x s uP.e r t r iqua r ta s ; e t s i c s i ne a l i quo nume r o . Ite r um , e x m i xti on e p r i m a e div i s i oni s c um t r ib u s m e mb r i s divi s ioni s s e que nti s , t r ib u s m o di s pr o c e ditu r ab s qu e fine . P r imo s i c : dupl e x s upe rbipa r ti e n s , t r i pl e x s upe r t r i pa r ti e n s , e t 1 15 c o nve r s i m , s upe rbipa r t i e n s dupl e x , s upe rbipa r t i e n s t r ipl e x . S e c undo s ic : dupl e x s upe q�a rti e n s t e r ti a s , t r ipl e x s upe rpa r t ie n s te r tia s , e t c o nv e r s im , dupl e x s upe r pa rt ie n s qua r ta s . T e r t i o s i c : dupl e x s upe rb i te r tia , t r ipl e x s upe rb i t e rtia , e t e c ontra , dupl e x s upe rbite rtia , dupl e x S UP.e rb iqua r ta . E t s i c i s to r um num e r o r um 120 inte r m inab i l i s e s t p r o c e s s u s , quo r u m o m nium de finitione s e t e xe mpla e x p r a e c e dentib u s s at i s l i qu e n t . P r opo rtio m i no r i s ina e qualitati s e s t hab itu do m i no r i s qua nti­ tati s ad m aio r e m ; c ui u s t ot s unt s pe c i e s quo t p r opo rtione s ina e­ qua l itati s m a io r i s et e i s de m n o m i nib u s app e l lantur (ha c p r e po s itione , " s ub ;' a ddita ; ut s ub m ultipl e x , s ub su�pa r ti en s , e t ita de 1 25 e iu s o m nibu s a l i i s s p e c i eb u s) . Et ho r um o mnium de finitione s c um e xe mp li s s a ti s e x p r i o r ibu s innot e s c unt . E x i s t i s notandum quo d o mn i s p r o p o r t io tantum inte r duo s t e r­ m i no s r e p e r i tu r . S c i a s e ti a m quo d quanta e s t p r o p o r t i o uni u s quantitati s a d a l i a m , tanta e s t i l l a a d r e l i quam : ut s i dup l a , dupla : 1 3 0 e t s i s ub dupla , s ub dupla . Et quanta e s t una quantita s a d a l ia m , ta nta e s t p r opo r tio e iu s a d i ll a m . E s t e ti a m a dv e rt e ndum quo d hae c p r a e di c ta p r i m o e t p r o p r i e in s o la pr opo rtione p r op r i e acc e pta e xi s tant , c o m m uni t e r aute m e t s e c undum t r an s latio n e m in p r opo rt i o ne dic ta c om m unit e r s unt r e p e rta . Igitu r de divi s i one 1 35 e t definitione pr opo r t i oni s e t s p e c i e r um s ua r um s i c pate t .

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quantity to a s ma l ler, in which the larger contain s the s ma ller more than once , plu s an aliquot part of it. Thi s type i s infinitely divis ible in three way s . Fir s t , with re s pect to multiplicity (e .g. , "double s uperpar­ ticular ;• "treble s uperparticular ;• etc . ) . Second , with re s pect to s uper­ particularity (e. g. , " m ultiple s e s quialtern ;• " multiple se s quitertian ;• etc . ) . Third , by a combination of the two other s (e. g . , "double s e s qui­ a ltern ;• "double ses quitertian � ��treble s e s quialtern " "treble ses qui­ tertian ;• etc. ). Further definition s and exa mples will be evident from the foregoing. " Multiple s uperpartient proportion" i s the relation of a larger quan­ tity to a s ma ller, in which the larger contains the s ma ller m ore than once , plu s more than one of it s aliquot part s . Thi s type is indefinitely divisible in s even way s. Fir s t , with re spect to multiplicity , it i s to be divided thu s : " double s uperpartient ;• "treble s uperpartient ;• etc . Sec­ ond , with re s pect to being s uperpartient , it i s divided in three way s : ( I ) "multiple s uperbipartient ;• "multiple s upertripartient ;• etc. , (2) " mul­ tiple s uperpartient by third s :• "multiple s uperpartient by fourth s :• etc. , ( 3) " multiple s uperbipartient by third s" (or "multiple s uperbitertian" ) , " m ultiple supertripartient by fourth s" (or "multiple s upertriquartan" ) , etc . Again , by combining the fir s t method of divis ion with the three branches of the s econd method , three more infinite serie s are gener­ ated : ( I ) "double s uperbipartient ;• " treble s upertripartient ;• etc . - and conver sely-" s uperbipartient duplex ;• " s uperbipartient triplex ;• etc . , (2) "double s uperpartient by third s :• "treble s uperpartient by third s :• etc .- and conver sely-"double s uperpartient by fourth s � etc. , ( 3 ) "double s uperbitertian :• "treble s uperbitertian;• etc . - and the opposite-"double s uperbitertian ;· "double superbiquartan ;' etc . Thu s , the s eries of the s e number s i s without end , a l l the definitions and examples o f which may be ea s ily worked out from the foregoing. A proportion of les ser inequality i s the relation of a s maller to a larger quantity and this type po s s e s ses a s many s ubdivis ion s a s that of greater inequality . They are a l so identified by the s ame names , by adding the prefix , " s ub" (e .g . , " s ubmultiple ;• " s ub superpartient ;• etc . ) . The definition and exemplification of all the s e is s ufficiently well under­ s tood from the above . From what ha s been s aid it i s to be noted that each proportion con­ s i s t s of only two term s . It will al so be recognized that , whatever the m agnitude of the proportion whic h one quantity bear s to another, that i s the m agnitude o f the one quantity a s related to the other; if the propor­ tion is double , the fir st quantity i s twice the s econd -i f the proportion i s s ubdouble , the first quantity i s hal f the second . However great one quantity i s in comparison to another, j u s t s o great i s the proportion be­ tween them . It i s al so to be kept in mind that the foregoing s tatements apply pri­ marily and s trictly only to proportion taken in the s trict sen s e ; they apply generally and by tran s ference , however, to proportion ta ken in the general sen s e . T hu s the situation i s clear regarding the divis ion and definition of proportion and it s s ubtype s.

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Modo sequitur secunda pars huius capituli , quae de proportion­ alitate determinat. Scias ergo quad differentia sive excessus unius quantitatis ad aliam est illud quo minor quantitas exceditur a maiori . Proportionalitas , quae secundo Arithmeticae Boethii et 1 40 sec undo Musicae eiusdem medietas nominatur , propter sui multi­ plicitatem in sua communitate definitio nem unam non habet , sed in dee em memb ra dividitur , ut secundo Arithmeticae Boethii satis patet ; quorum tantum tria sunt famosa et ad veterum locutiones intelligenda utilia. Ideo septem residuis praetermissis , tria prima 1 45 membra remanent declaranda . Quorum primum P-roportionalitas seu medietas arithmetica appellatur ; secundum 2roportionalitas seu medietas geometrica appellatur ; tertium nroportionalitas sive medietas harm onica nominatur. Prima est aequalitas differentiarum : scilicet , quando 1 50 quarumlibet duarum quantitatum comparatarum adinvicem dif­ ferentiae sunt aequales : ut tria , duo , unum. Et hoc arithmeticis pertinet. Secunda autem quae geometricas speculationes concernit , quinto E lementorum Euclidis isto modo definitur : " Proportionalitas est 1 5 5 similitudo proportionum"; quando scilicet quarum libet duarum quantitatum co mparatarum adinvicem proportiones sunt similes vel aequales : ut quattuor , duo , unum . Tertia est , in tribus terminis , aequalitas seu similitudo propor­ tionum extremorum et differentiarum ; quando , scilicet , in tribus 1 6 0 terminis proportio primi ad ultimum est similis vel aequalis pro­ portio ni differentiae primi et medii ad differentiam medii et ex­ tremi : ut sex , quattuor, tria . Senarii enim ad ternarium est dupla propo rtio ; et binarii , quae est differentia sex et quattuor, ad uni­ tatem , quae est differentia quattuor et tria , etiam dupla proportio 1 6 5 reperitur. Et differunt istae me dietates abinvicem , quia duae primae in tribus terminis reperiuntur ad minus , sed nullus est maximus numerus terminorum in quo existunt . Tertia autem tantum in tribus terminis reperitur. In alio etiam differunt , quia primae duae 1 7 0 possunt esse continuae et discontinuae , tertia autem semper continua repe ritur . Est autem tam medietas arithmetica quam geometrica duplex : quaedam co ntinua , quaedam discontinua. Medietas autem arithmetica continua est aequalitas diffe rentiarum per com munem ter- 1 7 5 minum medium , vel per co mmunes terminos medias , copulata ; per unum terminum , ut sic dicendo : sicut tria ad duo , ita duo ad unum ; per plures sic : sicut quattuor ad tria , ita tria ad duo , et duo ad unum. Discontinua est aequalitas differentiarum per nullum terminum communem , nee co mmunes terminos copulata : ut dicen- 1 8 0 do sicut sex ad quattuor, ita tria ad unum . Geo metrica vero medietas continua est similitudo proportio num per co mmunem terminum medium , vel per communes terminos

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TWO

T he r e now fo l l o w s the s e c o nd pa r t o f thi s c hapte r, whi c h i s c onc e r n e d with " p r o p o r tiona l i ty:' It i s to be r e c o g ni z e d , in the fi r s t plac e , that the di ffe r e nc e , or e xc e s s , of o ne quantity o ve r ano the r i s that by whic h the s m al le r quantity i s e xc e e de d by the la r ge r . Pr opo rtiona l ity {c a l l e d " m e di e ty" i n B o o k II o f B o e thi u s' Ar ithm e tic and a l s o i n B o o k II o f hi s M u s ic) ha s no g e ne r a l de finiti o n , owing to i t s multipli c ity , b ut is divi d­ e d into t e n c la s s e s (a s i s c l e a r e no u gh in B o o k II o f B o e thius' A r i thme­ tic). Sinc e only thr e e of the s e are we l l known and u s e fu l fo r an unde r­ s tanding of the anc ie nt s , o m i tting the s e ve n othe r s , we wil l take up the thr e e p r im a r y c la s s e s . The fi r s t o f the s e i s c al l e d " a r ithm e tic" p r opo r ti o nal i ty (o r me di­ e ty) ; the s e c ond i s c al le d " g e o m e t r i c " p r op o r tio nal ity (o r me die ty) ; the thi r d i s c al l e d " ha r m onic" pr opo r tiona lity (o r m e di e ty). The fi r s t c on s i s t s i n an e qua l i ty of di ffe r e nc e s , nam e ly, whe n the di ffe r e nc e s b e twe e n a ny two pa i r s o f quantiti e s a r e e qua l to e a c h othe r (a s i s t r u e o f the s e r ie s : th r e e , two , one). Thi s c la s s b e l o ng s to a r ithm e ti c . T he s e c o nd , whi c h c onc e r n s g e o m e t r i c s p e c ul ation s , i s de fine d , in B o o k V of E uc l i d' s E l e m e nt s , a s fo ll o w s : " P r opo r ti onality c o n s i s t s in a s im i la r i ty o f p r o po r ti o n s ;• whe ne v e r, na m e ly, the p r o po r ti o n s b e ­ twe en a n y t w o pai r s o f quantiti e s a r e s i m i l a r, o r e qual , to e a c h othe r (a s i s t r u e o f the s e r i e s : fo ur, two , one). The thi r d c on s i s t s in an e qua l ity, or s im ila r ity, a m o n g thr e e te r m s , b e twe e n the p r opo r ti o n o f th e e xt r e m e s a n d the p r o po r ti o n o f th e dif­ fe r e nc e s-wh e n , nam e ly, the p r opo r ti o n o f the fi r s t to the l a s t o f thr e e t e r m s i s s im il a r, o r e qua l , t o the p r opo r tion o f the di ffe r e nc e b e twe e n t h e fi r s t a n d middl e t e r m s to t h e di ffe r e nc e b e tw e e n t h e mi ddl e a n d l a s t te r m s (a s i s the c a s e with the s e r i e s : s i x , four, thr e e) . He r e the p r opo r tion of s i x to th r e e i s a doub l e one , and that o f two (the di ffe r­ e nc e b e twe e n s i x and fo u r) to one (the diffe r e nc e b e twe e n fou r and thr e e) is l ikewi s e found to be doub l e . The s e c l a s s e s o f m e di e ty a l s o di ffe r fr o m e a c h o the r i n that , tho ugh the fi r s t two inv o lve at l e a s t thr e e te r m s , the r e i s no uppe r l i m it to the numb e r o f te r m s whi c h may b e s o r e late d , whi l e the thi r d is c on­ s ti tute d b y thr e e te rm s o nl y . T h e y di ffe r in y e t a nothe r r e s pe c t in that , whi l e the fi r s t two may b e e i the r c ontinuou s or di s c ontinuou s , the thi r d mu s t a lway s b e c ontinuo u s . The fa c t i s that b o th a r ithme tic and g e o m e t r ic m e di e ty may b e o f the s e two kinds : c o ntinuou s and di s c o ntinuou s . " C ontinuou s" a r i th­ m e tic me die ty c on s i s t s i n an e quality o f di ffe r e nc e s unit e d by a c omm o n , m e an te r m , o r te r m s (by a s in g l e te r m in the e xampl e : a s th r e e i s to two , s o two i s t o one- and b y mo r e than o ne a s i n . the e x­ a m p l e : a s fo ur i s to th r e e , s o thr e e i s to two , and two i s to o ne). " Di s c o ntinuous" a r ithm e tic me di e ty c on s i s t s in an e quality o f di ffe r­ e nc e s no t uni te d by a c o mmon te rm , o r te r m s (a s in th e e xampl e : a s s i x i s to fo ur, s o th re e i s to on e) . " C ontinuou s" g e o m e t r i c me die ty, on the othe r h and , c o n s i s t s in a

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media s , copulata: per unum, ut s icut quattuor ad duo , ita duo ad unum : per p lure s, ut sicut octo ad quattuor, ita quattuor ad duo , 185 et duo ad unum . Di scontinua autem e s t similitudo proportionum per nul lum terminum comm unem nee aliquo s termino s commune s copulata : ut sicut s ex ad tria , ita duo ad unum . I s ta autem dicuntur de i s ti s medietatibu s non primo nee proprie, s ed secundum tran s sumptionem a quantitate continua (cuiu s parte s 1 90 ad communem terminum copulantur) et di screta (cuiu s parte s ad nullum communem terminum coniunguntur) , ut in praedicti s apparet . Et haec etiam differunt , quia medieta s continua in tribu s termini s et quotlibet pluribu s reperitur, di scontinua vero in quattuor termini s ad minu s et in quotlibet pluribu s reperitur . 195 Aliter autem et notabiliter differunt s ecundum Ametum filium lo se phi in epi s tula sua , De P.roportione et P.rOP.Ortionalitate, in hoc qua d in p r opo r tionalitate continua oportet omne s termino s in genere convenire, s ed in di s continua sive di siuncta po s sunt aliqui ter­ mini in genere diver s ari . V erbi gratia , sicut chorda ad chordam 200 ita sonu s ad sonum ; et s icut moven s ad moven s , ita velocita s uniu s motu s ad velocitatem alteriu s motu s . E t quia proportionalia a pro­ portionalitate dicuntur, po s s unt , per relationem ad p roportionalita­ tem in sua communitate, per unam definitionem analogam , non autem 205 univocam , definiri hoc modo : Proportionalia sunt quae in aliqua P-rOP.ortione conveniunt . Et quia proportionalita s dividitur decem modi s, et proportionalia similiter dividuntur, quorum tantum tre s p r im i modi ad prae sen s sunt tractandi , igitur proportionalia propor­ tionalitate arithmetica sunt il la quorum differentiae sunt aequales . Proportionalia autem proportionalitate geometrica sunt illa quo r- 2 1 0 um proportione s sunt aequale s vel simile s, vel quorum est una proportio, ut placet Euclidi . Proportionalia autem proportionalitate harmonica sunt illa quorum extremorum et differentiarum propor­ tione s sunt simile s seu aequales . Haec omnia , cum sui s exemplis, patent per praedicta. Ex i sti s patet quod ternar iu s est pauci s si2 15 mu s numeru s termino r um in quibus proportionalita s repe ritur, et quod nullu s e s t maximu s numeru s terminorum in quibus exi stit . Permutatim P.rO P.ortionalia P.rOP.ortionalitate geometrica sunt illa quorum proportionalium proportionalitate geom etrica , sicut 2 20 anteceden s uniu s proportioni s ad anteceden s alteriu s proportioni s , sic con sequen s illiu s ad cons equen s al teriu s . V erbi gratia , octo , quattuor, duo , unum sunt proportionalia proportionalitate geornetrica ; e s t enim eadem proportio octo ad quattuo r sicut duo ad unum, et etiam sicut octo ad duo , ita quattuor ad unum ; et idea illi quattuor termini permutatim proportionales exi stunt . 225 Econtrario P.rOP.Ortionalia P.roportionalitate geometrica sunt illa quorum proportionalium proportionalitate geometrica , sicut con sequen s uniu s propo r tioni s ad anteceden s illiu s , ita reliquum con sequen s ad prop rium anteceden s. Verbi gratia , sicut octo ad quattuor, ita duo ad unum , et econt rario sicut quattuor ad octo, ita 2 3 0 unum ad duo .

T R E A T IS E O N P R O PO R T IONS s im il a r i ty o f p r opo r ti o n s unit e d b y a c o m m o n , m e a n t e rm , o r t e r m s (b y a s in g l e t e r m in the e xample : a s fo u r i s to two , s o two i s to o ne­ a nd b y mo r e tha n o ne in the e xampl e : a s e i ght is to fo u r, s o fo u r i s to two , and two i s to o ne). A s im i l a r ity o f p r opo r tio n s no t unit e d b y a c om m o n te r m , o r te r m s , i s " di s c ontinuo u s " (a s i n the e xampl e : a s s ix i s to th r e e , s o two i s to one) . The fo r e g o in g i s no t p r e dic ate d o f the s e c l a s s e s o f m e die ty i n the p r im a r y o r s tr i c t s e n s e , b ut o nl y by t r a n s fe r e nc e fr om c o ntinuo u s qua ntity (who s e pa r t s a r e j o in e d b y a c o mmon t e r m) a n d di s c r e t e qua ntity (who s e pa r t s a r e not s o j o ine d) , a s i s c l e a r fr om what wa s s ai d e a rl i e r . C o ntinuo u s al s o diffe r s fr o m di s c o nt inuo u s m e di e ty i n that th e fo rm e r m a y b e e xh ib it e d in thr e e t e r m s o r mo r e , whe r e a s the l a tt e r r e qu i r e s a t l e a s t fo ur t e r m s . Ac c o r ding to Ahm a d ib n J u s uf (in hi s l e tt e r, O n P-r O P.O r tion and pr o­ P-O r tiona l i ty) th e y di ffe r i n y e t anothe r a n d m o s t i m p o r tant way i n tha t , i n t h e c a s e o f c o ntinuou s p r opo r ti o na l i ty, a l l th e t e r m s m u s t b e o f the s a m e kin d , whe r e a s i n di s c ont i nuo u s , o r di s j unc t , p r o po r ti o n­ a l i ty, s o m e o f the t e r m s m ay b e o f di ffe r e n t kinds . Fo r e xampl e : a s the l e n gth o f o n e m u s i c a l s t r in g i s to tha t o f ano the r, s o i s the p i t c h o f t h e o n e to th e pitc h o f t h e o the r - a n d a s o n e m ov i n g p o we r i s to an­ o the r, s o i s the s p e e d of the o n e m o tion to the s p e e d o f the o the r. In­ a s m u c h a s p r o p o r tional s a r e c la s s i fi e d ac c o r di n g to p r o p o r t io na l ity, the y c an b e de fi ne d by the i r r e la t i o n to p r opo r t i o na l ity in g e n e r al i n a s in g l e , a nal o g o u s (though no t u n ivo c a l) de finit i o n : Tho s e quantiti e s a r e p r opo r tional s whi c h a g r e e i n a g i v e n p r op o r tio n . Sine e p r o p o r­ tiona l ity i s div i de d into t e n c l a s s e s a n d p r o po rtiona l s a r e s im i l a r l y div i de d (the thr e e p r im a r y fo r m s b e ing o u r o nl y c o nc e r n at the mo­ m e nt) , th e r e fo r e tho s e quantit i e s who s e di ffe r e nc e s a r e e qual a r e p r o po r ti o na l s b y a r ithm e ti c p r opo r t i o na lity. Tho s e qua nti ti e s who s e p r o po r t io n s a r e e qua l , o r s im i la r, o r whic h have a s i n g l e pr opo r ti o n (a s Euc l i d ha s it) a r e p r o po r ti ona l s b y g e o m e t r i c p r op o r tional ity. Tho s e qua ntiti e s the p r o po r ti o n s b e twe e n who s e e xt r e me s a n d di f­ fe r e nc e s a r e s im i l a r, o r e qua l , a r e , i n tur n , p r o po r tional s b y ha r­ m onic p r o po r ti o na l ity. A l l thi s , t o g e the r with e xample s , i s e v ident fr om the fo r e g o in g , fr om whi c h is a l s o e v i de nt that the s ma l l e s t numb e r o f t e r m s b e tw e e n whi c h p r o p o r t i o na l i ty m a y b e fo und i s th r e e , a n d that the r e i s n o upp e r limit t o the numb e r o f s u c h te rm s . Tho s e t e r m s a r e " p e r mutative l y" p r opo rt i o na l s b y g e o m e t r i c pr o­ por tionality fo r whic h the a nte c e de nt of the one p r o po r tion i s to the ant e c e de nt o f the othe r, a s the c o n s e quent o f the one i s to the c o n s e ­ que nt o f the othe r. Fo r e xample , the te r m s : e i ght , fo ur, two , one , fo r m a s e r i e s o f p r o p o r t io na l s by g e o m e t r i c p r o po r tiona l ity, fo r the p r opo r tio n o f e i ght t o fo u r i s the s a m e a s that of two to o ne , a n d e i g ht i s al s o to t w o a s fo ur i s to one . Thu s the s e fo ur te r m s a r e p r o­ po rt io nal s pe rm utati v e l y . Tho s e t e r m s ar e " c ont r a r ily" p r o po r ti o na l s b y g e o m e t r i c p r o po r­ tiona l ity fo r whi c h th e c o n s e qu e nt o f the o n e p r o p o rt i o n i s to it s ante c e de nt a s the c o n s e que nt of the o the r i s to it s r e s p e c t ive ant e c e d-

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Disiunc ta sive simP-le x P-roportionalitas ge ome tri c a e st quorum­ lib et ant e c edentium separatim ad sua c onsequentia proportionum aequalitas; ut sicut oc to ad quattuor, ita duo ad unum. Coniunc ta P.roportionalitas geometric a e st quorumlib et ante c e- 2 35 dentium c um suis c onsequentibus , in disiunc ta siv e simplic i pro­ portionalitate , unius ant e c e dentis illorum c um suo c onsequente ad idem c onsequens, et c uiuslib e t alterius ante c e dentis illorum c um suo conseque nte ad proprium c onsequens, similitudo proportionum; ut sic ut oc to et quattuor, ad quattuor, ita duo e t unum ad unum. 240 Eversa P-T OP.Ortionalitas geome tric a e st quorumlibe t ant e c e de n­ tium c um suis c on s equentibus , in disiunc ta sive simplici propor­ tionalitate , uniu s ante c e dentis c um suo c onse quente ad idem ante c e dens, e t c uiuslibe t alterius ante c e de ntis illorum c um suo c onsequente ad ipsum ante c e de ns, proportionum similitudo; ut 245 sic ut oc to e t quattuor ad oc to , ita duo et unum ad duo. Aequa P.roportionalitas geome tric a e st duabus multitudinibus quantitatum propositis, quarum qua e cumque duae proxima e unius ad duas sibi c orre sponde nt e s alterius geome tric e proportionale s e xistant , primorum ad sua ultima proportionum similis habitudo; 250 ut , istis multitudinibus propositis: tria , duo , unum : se x , quattuor, duo-sicut tria ad duo ita sex ad quattuor, e t duo ad unum , sic ut quattuor ad duo ; et sic ut unum primum ad suum ultimum , ita re li­ quum ad suum : sic ut enim tria ad unum , ita sex ad duo se habent . Ha e sex de finitione s e x quinto Elementorum Euclidis, lic et 255 obsc ure , poterunt apparere . C onsimile s e tiam de finitione s pro­ portionalitati arithme tic ae e x proportionibus ista proportionalitate attendendo aequalitatem differentiarum poterunt adaptari , ut , istis intelle c tis , facile e st vide re .

T E R T I A

P A R S

C A P I T U L I P RI M I

Iam supere st tertia pars huius c apituli , quasdam suppositione s pra emittens. Quarum ha e c e st prima : Omne s proportione s sunt aequale s quarum denominationes sunt e ae dem vel a equale s. Se c unda e st ista : Quibusc umque duobus e xtremis , interposito media , c uius ad utrumque e st aliqua proportio , e rit proportio primi ad tertium c omposita e x proportione primi ad se c undum et proportione se c undi ad tertium . T e rtia e st ista : Duobus vel quotc umque me diis inte rpositis duo­ bus e xtre mis , proportio primi ad extre mum produc itur ex proportione primi ad se c undum et se c undi ad tertium et tertii ad quartum et sic de inc eps usque ad extremum. (Ha rum autem duarum prima e st se c unda De P.rOP-Ortionibus , se c unda v e ro te rtia e st eiusde m.) Quarta est ista : Si duae quantitate s a equale s ad tertiam quaml ib e t c ompare nt ur, e arum ad istam erit una proportio , iterum i llius ad amb as eadem e st proportio.

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e nt. For e xamp l e , a s e i g ht i s to fo ur, s o two i s to o n e , and, c o ntrari ly, a s fo ur i s to e i g ht, s o o n e i s to two . " Di s junct" or " s impl e'' g e o m e tri c proportio na lity i s e qua l ity o f the pro porti o n s b e twe e n e a c h ante c e de nt, s e parate ly, a n d it s c o n s e.­ que nt . For e xa m p l e , a s e i g ht i s to fo ur, s o two i s to o n e . " Co n j un c t'' g e om etric pro porti o na l ity ( g i v e n proportio n s i n di s­ j un c t, or s im p l e , proporti o n ality) is e qual ity b etw e e n th e proporti o n s o f o ne o f the ante c e de nt s , p l u s it s c o n s e qu e nt, t o th at c o n s e qu e nt and another a nte c e de nt , p l u s it s c o n s e qu e nt, to it s re s p e c ti v e c o n s e qu e nt . For e xa m p l e , a s e i ght p l u s fo ur a r e t o four, s o two pl u s o ne a r e to o ne . " E v e r s e'' g e o m e tric pro portiona lity ( g i v e n pro portio n s in di s j unct, or s im p l e , pro portio nality) i s e qual ity b e twe e n th e pro portio n s o f o n e o f th e ante c e de nts, p l u s its c o n s e qu e nt, t o tha t ante c e de nt a n d a noth e r ante c e de nt, p l u s it s c o n s eque nt,. t o it s re s p e c ti v e a nte c e de nt. F o r e x­ a m p l e , a s e i g ht pl u s fo ur are to e ig ht , s o two p lu s o ne ar e to two . Giv e n two s e rie s o f quantiti e s, for whi c h th e proportio n s b e twe e n s uc c e s s iv e p air s o f term s i n the o n e are g e o m etri c proporti o na l s o f c orre s po ndi n g p a ir s i n th e oth e r, u e qual" g e o m e tric pro porti o n­ a l ity i s the s im i lar re l atio n o f th e pro portion s b e tw e e n the fir s t and the la s t term s . For e xa m p l e , p o s iti n g th e two s e ri e s : thr e e , two, o ne , and s ix, fo ur, two- a s thre e i s to two, so s ix is to fo ur, and two i s to one a s fo ur i s to two- and a s the fir s t term of th e one s er i e s i s to the l a s t te rm o f that s erie s . s o i s the fir s t te rm o f the oth e r to it s re s p e c ti v e l a st te rm, for a s thre e i s to o n e , s o s i x i s to two . Altho u g h ob s c ur e l y e xpre s s e d, th e s e six de finiti o n s wil l b e fo und in B o o k V of Eu c l i d' s El e me nt s . Sim i l ar de finiti o n s may a l s o be de­ v e l o p e d fo r arithm eti c proportio na lity on the b a s i s of pro porti o n s o f the pr e s e nt typ e , s im p ly b y ta kin g into a c c o unt a n e quality o f di f­ fer e nc e s (a s m a y b e e a s i ly s e e n, o nc e the fore g o in g ha s b e e n gra s p e d). C H A P T E R O N E , P A R T

T H R E E

T h e r e now re main s p art thre e o f th e pre s e nt c ha pte r, c om m e nc ing with c erta in a x i o m s . T he fir st i s that a l l pr op ortio n s are e qu a l who s e de nom inatio n s are the s am e , or e qua l . T h e s e c on d i s that, g iv e n two e xtr e m e term s, and i nterpo s in g an inte rm e diate term po s s e s s in g a g iv e n proportion to e a c h, th e propor­ ti o n o f th e fir s t to th e third will b e the pro duc t o f the pro porti o n s o f the fir s t t o th e s e c ond a n d the s e c o nd t o th e third. The th ird i s that, g i ve n two or more inte rm e diate term s p l a c e d b e ­ twe e n two e xtre m e s, the pro portion o f the fir s t to the la s t will b e th e pro du c t o f th e prop orti on s o f th e fir s t to th e s e c ond, th e s e c o n d to the th ird, the third to th e fo urth, and s o o n, to th e la s t te rm . (O f the latter two a xio m s , th e fir s t i s Axiom 2 o f th e De P-ropor­ tio nib u s and th e s e c o nd i s A xio m 3 o f th e s a me work.) The fo urth is th at, i f two equal qua ntiti e s are c o mpare d to a third, the y w i l l b e ar the s a m e pro porti o n to it and it to th e m .

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Quinta est ista : Si duae quantitates inaequales ad unam quanti­ t atem proportionentur, maior quidem maiorem, minor ve ro min­ orem obtinebit proportionem ; illius vero ad ambas, ad minorem quidem proportio m aior, ad maiorem vero minor erit. 280 Sexta est ista : Si fuerit aliquarum quantitatum ad unam quanti­ tatem proportio una, ipsas esse aequales, si vero uniu s ad eas una e st proportio , aequale s esse necesse est . Septima est ista : Si fuerint quattuor quantitates proportionale s, permutatim proportionales erunt . 2 85 Octava est i s ta : Si fuerint quattuor quantitates proportiona les et fuerit prima illarum m axima et ultima minima, primam et ultimam pariter accepta s ceteris duabus maiores esse necesse comprobantur. Istarum quinque ultimarum suppositionum, prima e s t septima 290 quinti Euclidis, secunda autem octava, tertia vero nona, quarta quidem sexta decima, quinta vero ultima est eiusdem . Prima conclusio : Si fuerit proportio maiori s inaequalitatis primi ad secundum ut secundi ad tertium, erit proportio primi ad tertium praeci se dupla ad proportionem primi a d secundum 295 et secundi ad tertium . Hane probes ostensive hoc modo : Eaedem vel similes sunt denom­ inatione s proportionum primi ad secundum et secundi ad tertium ; igitur, per primam suppo sitionem , istae sunt ae qua les et , per s ecundum suppositionem, proportio primi ad tertium componitur 300 praecise ex illis. Igitur, per definitionem dupli, ista est praecise dupla ad utramque illarum . Et hoc e s t quod o s tendere volebamu s. Secunda est ista : Si fuerint quattuor termini continue propor­ tionale s, proportio primi ad ultimum, cuiuslibet proportionis a licuiu s illorum terminorum ad proximum sequentem est tripla . 305 Si quinque, quadrupla, et sic in infinitum semper uno minus, ita quod s emper denominatio proportioni s sit unitate minor numero terminorum . Sic istam per suppositionem primam et tertiam , a diuncta definitione " tripli" et kquadrupli" et sic de aliis, ut prox310 imam demon strabi s . Tertia conclu sio e s t i s ta : Si fuerit primum maiu s quam duplum s ecundi et fuerit s ecundum aequaliter duplum tertii, erit propor­ tio primi ad tertium minor quam dupla proportioni s primi ad s ecundum. Hane s imiliter o stensive demon strabis: Sit enim A prim um 3 15 maius quam duplum s ecundi, quod sit B, et s it B aequaliter dupl um tertii, quod s it C, et sit B ad D sicut A ad B. Tune D non est a equale C , propter quartam suppo sitionem , nee e s t maiu s propter quintam s uppos itionem cum hypothe s i ; igitur D est minu s C. lgitur per quintam suppo sitionem , proportio A ad D e s t maior 320 proportione A a d C, et per primam conclu sionem, proportio A ad D e s t dupla ad proportionem A ad B; ergo proportio A ad C est minor quam dupla ad proportionem A ad B. Haec eadem potest aliter demons trari per nunc probata : C est maiu s D et inter i s ta est proportio maiori s inaequalitati s; igitur, 325

T R E A T I S E O N P RO P O R T I O NS

The fifth i s that , i f two une qual quantiti e s a r e made p r opo r tionate to a thi r d , the la r ge r will b ea r a l a r g e r p r opo rtion to it and the s mall e r a s mall e r pr opo r tion ; the pr opo r tion of that thi rd quantity wil l be g r e at e r to a smalle r quantity and l e s s e r to a la r g e r one . The s i xth i s that , if the r e i s a s in g l e p r opo r ti o n o f a numbe r of quantitie s to s o me othe r quantity, or of it to the m , it fo llows ne c e s­ s a r i ly that the s e quantitie s a r e e qual . The s e v e nth i s that , i f four quantiti e s a r e p ropor tional s , the y will be p e rmutativ e l y p ropo rtio nal s . The e ighth i s that , i f four quantitie s a r e propo r tiona l s and the fi r s t the lar ge s t and the la s t the s m alle s t , the s um o f the fi r s t and la s t will b e ne c e s s a r ily found g r ea te r than t h e s um o f t h e othe r two . (Of the s e five latte r axiom s , the fi r s t i s Axiom 7 o f B oo k V o f Euc li d , the s e c ond i s Axiom 8 , the thi r d i s Axiom 9 , the fou r th i s Axiom 1 6 , a n d the fifth i s the la s t axiom o f the b oo k .) The o r e m I : If a p r opo rtion o f g r e ate r ine quality b e twe e n a fi r s t and a s e c ond t e r m i s the s ame a s that b e tw e e n the s e c ond and a thir d , the p r o po rtion o f the fi r s t to the thi r d will b e e xac tly the s quar e o f the p r opo rtion s b e twe e n the fi r s t and the s e c on d , and the s e c ond and the thi r d . Thi s y o u may p r o v e c onc l u s i v e l y a s follow s : T h e denom ination s o f t h e p r opo rtions b e twe en the fi r st and s e c ond a n d the s e c ond an d thi r d a r e the s ame , o r s imila r . T he r e fo r e (by Axiom 1) the s e a r e e qual , and (by A xiom 2) the pr o�o rtion o f the fir s t t o the thir d i s the i r e xa c t p r o duc t . The r e fo r e (by the d e finition o f " s qua r e" ) thi s p r opo rtion i s e xac tly the s qua r e o f e a c h o f the othe r s , and thi s i s what we wi s he d t o s ho w . T he o r e m I I : If four te r m s a r e c ontinuo u s ly pr opo rtional , the p r o­ portion o f the fir s t to the la s t will b e the c ub e o f the p r opo rtion b e­ twe e n any o f the m to the one s uc c e e ding i t . If the r e a r e five te rm s , it will b e to the fourth powe r, and s o fo r th , a d infinitum , in s u c h a way that the denomination o f the propo rtion i s alwa y s one l e s s than the numb e r of te rm s . Thi s , like the p r e c e ding the o r em , may be dem­ o n s trate d by Axio m s 1 and 3 , and with the additiona l de finition s o f " c ub e d ;' " t o the fourth powe r ;• etc . T he o r em III : If the fir st t e r m i s m o r e than twic e the s e c ond and the s e c ond i s e xac tly twi c e the thi r d , the p r op o rtion of the fi r s t to the thir d will be l e s s than the s qua r e o f the pr oportion of the fir s t to the s e c ond. Thi s may al s o be c onc l u s iv e ly de m o n s trate d : Fo r l e t A b e a fir s t te rm mo r e than twic e a s e c ond t e rm , B , and l e t B b e e xa c tly twic e a thi r d te rm , C , and l e t B be to D a s A i s to B . Then D i s not e qual to C (by Axiom 4 ) nor i s it g r e ate r (b y Axiom 5 and by hypothe s i s) ; the r e fo r e , D i s le s s than C . Then (by Axiom 5 ) the propo r tion o f A to D i s g r e a te r than the propo r tion of A to C , and (b y The o r em I ) the p r opo r tion o f A to D i s the s quar e o f the p r o po rtion o f A to B , and , the r efo r e , the p r opor tion o f A to C i s l e s s than the s quar e o f the p r o­ por tion of A to B . Thi s s ame the o r e m c an b e demon s t rate d othe r wi s e b y m e a n s o f

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A et D duobus extremis , interponatur C medium . Tune , per see un­ dam suppositionem , proportio A ad D e omponitur ex proportioni­ bus A ad C et C ad D . Igitur proportio A ad C est minor proportione A ad D , et ista est aequaliter dupla proportioni A ad B , ut pr ius est probatum . Igitur proportio A ad C est minor quam dupla ad 330 proportionem A ad B. Et hoc est quod quaesivisti . Quarta e onclusio est haec : Si fuerit primum duplum secundi fueritque secundum maius quam duplum tertii , erit proportio primi ad tertium minor quam dupla proportionis secundi ad tertium. 335 Haee cum proxima similem omnino demonstrationem sortitur. Quinta : Si fuerit primum minus quam duplum secundi fueritque secundum aequaliter duplum tertii , erit proportio primi ad tertium maior quam dupla ad proportionem primi ad secundum . Haec , ut tertia , dupliciter demonstratur. Primo capiatur D , ad 3 4 0 quod se habet B sicut A ad B , et probetur quod D sit maius C , et ex hoc ut in probatione tertiae conclusionis propositum concluditur. Secundo , sumatur D praecise duplum B et ostendatur D esse maius A , et ex illo demonstratur propositum sicut in secunda ostensione conclusionis tertiae erat factum . 345 Sexta conclusio : Si fuerit primum duplum seeundi fueritque secundum minus quam duplum tertii, erit proportio primi ad ter­ tium maior quam dupla proportionis secundi ad tertium . Haee cum proxima similem demonstrationem habebit . Septima conclusio : Proportione aequalitatis nulla proportio est 3 5 0 maior vel minor . Nam nulla proportione aequalitatis aliqua proportio aequalitatis est maior vel minor propter primam suppositionem . Nee aliqua proportio maioris inaequalitatis est maior vel minor proportione aequalitatis , quia tune proportio aequalitatis secundum aliquam 355 proportionem maioris inaequalitatis ab alia proportione maioris inaequalitatis excederetur ; et cum secundum aequalem proportionem aliqua proportio maioris inaequalitatis exe edatur ab illa proportione maioris inaequalitatis , sequitur, per sextam supposi­ tionem , proportionem aequalitatis et proportionem maioris inae- 360 qualitatis esse aequales ; et tune , per eandem suppositionem, sequitur maius et minus invicem adequari . Verbi gratia , ponatur proportionem quadruplam esse in duplo maiorem proportione aequalitatis , et e apiatur proportio maioris inaequalitatis in duplo minor proportione quadrupla , quod fieri 365 patest si , duo bus extremis quorum maius est quadruplum minoris , interponatur medium quad se habeat ad minus extremum sie ut maius ad ipsum : quarternario enim et unitati interponatur binarius . Tune isti sunt tres termini continue proportionales ; igitur per primam e onclusionem , proportio primi ad ultimum est dupla pro- 3 7 0 portionis primi ad secundum . Igitur proportio primi ad see undum est subdupla proportionis p r imi ad ultimum ; igitur proportio dupla est subdupla proportionis quadruplae. Et , secundum falsigraphum, pr oportio aequalitatis est subdupla proportionis quadruplae , igitur,

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what has now be e n prove d. C is gre ate r than D and be twe e n the m the re e xists a proportion of gre ater ine quality . Therefore , taking A and D as two e xtre m e s and interposing an int e rm e diate t erm, C , then (by A xiom 2 ) the proportion of A to D is the product of the pro­ portions of A to C and C to D . Therefore, the proportion of A to C is less than the proportion of A to D, an d the latte r is also the square of the proportion A to B (as was prove d above). T h e re fore , the pro­ portion of A to C is less than the square of the proportion A to B, and this is what we were after . Theore m I V : If the first term is twice the s e cond and the second is more than twice the third, the proportion of th e first to th e third will be le ss than th e square of the proportion of the s e cond to the third. T his may be de monstrate d by a proof altoge th e r similar to that of the pre ce ding the or e m . Theore m V : If the first te rm is less than twice the secon d and the s e cond is also twice the third, the proportion of the first to the third will be gre at e r than the square of the proportion of the first to th e s e c ond. T his, like T h e or e m III, m ay b e de monstrat e d in two ways . First, let a D be posite d such that B is relate d to it as A is to B , and l e t it be prove d that D is greater than C ; from this, as in the proof of T h eore m III, the conclusion is arrive d at . S e condly, le t a D be pos­ ite d that is e xactly twice B, an