clagett intension-remission qualities

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Richard Swineshead and Late Medieval Physics: I. The Intension and Remission of Qualities (1)Author(s): Marshall Clagett

Source: Osiris, Vol. 9 (1950), pp. 131-161Published by: The University of Chicago Press on behalf of The History of Science SocietyStable URL: http://www.jstor.org/stable/301847 .

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RichardwinesheadandLateMedieval hysics

I. THE INTENSION AND REMISSION OF

QUALITIES (I)

A. The Rise of the QuantitativeTreatment of Qualities.

Certainly one of the most significant evelopments n the riseofearlymodernphysicswas the quantificationfphysics. It hasusually been assumedthat thisquantificationrmathematicizationtook place in the sixteenthand seventeenth enturiesunder theinfluenceof the introduction f the mathematics f ARCHIMEDES

and its use and applicationby GALILEO GALILEI. Now there isno question that it was the spread of the Archimedean works.and spiritwhichdid much to shape the form hatthemathematicalinvestigationfnaturalphenomenatook. But it has becomemoreapparent in the last generationthat the i6th century nterest n

the mathematicalf notexperimentalreatment f natural uestions.is at least in parta productofthe initial nvestigations long thatline undertaken n the Universities of Oxford and Paris in thefourteenth entury. This interest n the quantification of thephysical at these universitieswas manifestedparticularly y thetreatmentfqualitiesquantitatively, treatment hat lso embraceda new interest n the quantitative tudyof local movement. For

the first time in history natural philosophers were seriously

(i) This essay is the first f a series on the role of Swineshead in medievalphysics. It has grown out of an edition of Swineshead's CalculationeswhichI am preparingwiththe gratefullycknowledgedassistance of fellowships1946,1950-I) from the JohnSimon Guggenheim Memorial Foundation. For sectionA. cf. A. Maier's brilliant ie Problemder ntensiven rosse,1939, which read toolate foruse here.







132 M. CLAGETT

thinkingabout kinematics, he study of movement n terms ofthe dimensionsof space and time.

This essay is primarilyconcerned with certain opinions ofRICHARDSWINESHEAD, one of the Oxford logicians who un-hesitatingly ccepted a quantitative reatment f qualities. Butby wayof introductionwe shouldbriefly ote how the quantitativeapproach to qualities arose.

From the latterpart of the thirteenth entury there began tobe discussed and circulated at least fouropinions which hopedto explain differencesn qualitative intensities. Three of these

positions, we shall see, rejected any basic similaritybetweenquantity nd quality,while thefourthnsisted n such a similarity.The discussionof ntensityariationsnqualities,which the school-men called intensionand remissionof qualities,had its medievaloriginin (i) scholasticefforts o discover how the virtue charityincreased in a person,whetheror not it was by the addition ofone charity to another,and kindred questions. On the other

hand, it had its ancient sources in (2) statementsfound in theCategoriesof ARISTOTLE and the Commentaryn the Categoriesof the sixth-century eoplatonist,SIMPLICIUS.

We can conveniently tart our background investigationwithAQUINASwho appears to have been one of thefirst f the Westernschoolmentohavemade use of SIMPLICIUS'Commentarytranslatedby his friend,WILLIAMMOERBEKE) and who therefore s one ofthe first o treat variations n qualitative intensities n the lightof a varietyof opinions fromGreek antiquity.

As the result of his matureconsiderationfoundin the SummatheologiaeST. THOMASadvanced the opinion (I) that intensiveincreases in quality, e.g., where we say somethingis " more"

or " less " white,resultfrom he varying articipation f a subjectin a given,unchanged uality. Thus intension and remissiondonot originatein the quality or form,but in the subject and its

varyingdispositionfora given quality. Since this is so, THOMASgoes on to assert 2) that intensive ncreasesin habits and forms(and thus charity)do not takeplace by the additionof onepartof theformor habit to another, ven thoughextensive increases,e.g., wherewe say " little or " great" whiteness,do take placebyaddition. This theorys very ucidlyoutlined n twoquestionsof the Summa theologiaefrom which I shall select illustrative







RICHARD SWINESHEAD AND LATE MEDIEVAL PHYSICS 133

passages,the first f which appear in his discussionof the increaseof habits 2):

Now the perfection f a form may be considered in two ways: first, ccordingto the formitself: secondly,according to the participation n the form by thesubject. In so far as we consider the perfections f a form according to theform tself, hus the form s said to be littleor great: for nstance,greator littlehealthor science. But in so far s we considerthe perfection f theform ccordingto the participation n it by the subject, it is said to be moreor less: for nstance,more or less white or healthy...

For we said that increase or decrease in formswhich are capable of intensityand remissionhappen, in one way, not on the part of the form tself onsideredin itself,but througha diverse participation n it by the subject. Thereforesuch increases of habits and othersforms s not caused by the addition of formto form,but by the subject participatingmore or less perfectlyn one and thesame form. 3)

Much the same sortof thing s said in a later question wherethe problemof the increasein charity s considered 4)

For since charity s anaccident, its being

(esse)is to be in something. So that

an essential increase of charitymeans nothing else but that it is yet more in itssubject,which mpliesa greater adication n itssubject... Hence charity ncreasesessentially, ot by beginning new, or ceasing to be in its subject,as the objectionimplies,but by beginningto be more and more in its subject...

It follows 5) therefore hat if charitybe added to charity,we must presupposea numerical distinctionbetween them which follows a distinctionof subjects:thus whitenessreceives an increase when one white thing is added to another,although uch an increasedoes notmake a thingwhiter this s the same distinction

above between " little, great" and " less, more.") This, however, does notapply tothe case in point,since thesubject of charitys none otherthan the rationalmind, o that such like an increaseof charity ould onlytakeplace by one rationalmind being added to another; which is impossible. Moreover, even if it werepossible,theresultwould be a great over,but not a more lovingone. It follows,therefore, hat charitycan by no means increase by the addition of charitytocharity, s some have held to be the case.

(2) THOMAS AQUINAS, Summa Theologiae, -II, quaest. 52, art. i (Edition of

the Ottawa Instituteof Medieval Studies, 1941, 982b; translation f the EnglishDominicans, ist partof second part,p. 38).

(3) Ibid., I-II quaest. 52, art. 2 (edit. cit., 984b; transi. it., p. 43).(4) Ibid., Il-Il, quaest. 24, art.4 (edit. cit., I532a; transl. it.,2nd part of second

part, p. 286).(5) Ibid., II-II quaest. 24, art.4, art. 5 (edit. cit., 1533a; transl. it., pp. 287-288).

It should be noted that DUHEM completely misrepresents QUINAS' position onthis question, at least so far as what AQUINAShas said in the Summa Theol.P. DUHEM, Rttudesur Lionard de Vinci, Paris, 1913, vol. 3, pp. 317-3i8.







134 M. CLAGETT

Accordinglycharity ncreases only by its subject partakingof charitymore andmore, i.e., by being more reduced to its act and subject thereto. For this isthe propermode of increasein a formthat s intensified,ince the being of sucha formconsistswholly in its adhering to its subject. Consequently, since themagnitudefollowson its being, to say that a form s greateris the same as tosay that it is more in its subject, and not thatanotherform s added to it ; forthiswould be the case iftheform, f tself, ad anyquantity,nd not in comparisonwith its subject.

The source of THOMAS'theory, t would appear, lies in thestatementsmade in the Categoriesof ARISTOTLE nd the Com-mentary f SIMPLICIUS6). We can be fairlyconfidentof this

because THOMASincludes in the discussion of the increase ofhabits a very close paraphraseof a summarypassage fromSIM-PLICIUS (7):

In thisway, then, therewere fouropinions among philosophersconcerning theintensity nd remission fhabits nd forms s SIMPLICIUS elated nhisCommentaryon the Categories. For PLOTINUS and other Platonists held that qualities andhabits themselveswere susceptible of more and less, for the reason that theywere material,and so had a certain indetermination ecause of the infinity f

matter. Others, on the contrary,held that qualities and habits of themselveswere not susceptibleof more and less; but that the thingsaffectedby themare4aid to be moreand less,accordingto a diversity fparticipation: thatfor nstance,justice is not more or less, but the ust thing. ARISTOTLElludes to thisopinionin the Categories. The third opinion was that of the Stoics and lies betweenthe two precedingopinions. For theyheld that some habits are of themselvessusceptibleofmore and less, for nstance,the arts; and thatsome are not, as thevirtues. The fourthopinion was held by some who said thatqualities and im-materialforms re not susceptibleof more and less, but thatmaterialforms re.

A second opinion relativeto intensive ncreases was that ex-tensively developed by HENRY OF GHENT (8). This view held thatintensive ugmentation f a qualitativeform augmentation formainfusa) does nottakeplace from arying articipationfthesubjectin the quality or fromvariationsof matter,but only fromtheformitself,which has in its essence parts. That is to say, itis intheessentialcharacter fthequality ohavedifferentntensive

(6) Categories, iob-i ia; SIMPLICIUS, In Categ. Chap. 8 (Prussian Academyedition,vol. 8, pp. 283-5).

(7) AQUINAS, op. cit., loc. cit., in note 2.(8) HENRY OF GHENT, Quodlibeta, quodlib. 5, quaest. i8 (Edition of i5i8).

Cf. DUHEM, op. cit., p. 319. I have also used DUNS SCOTUS discussion of HENRY sopinion, In lib. Sententiarum, k. i, distinctio XVII, quaest. 6 (Opera omnia,vol. 5-2, Lyon, i639, pp. 988-995).







RICHARD SWINESHEADAND LATE MEDIEVALPHYSICS 135

parts. The actual intensionresults when each new part passesover frompotentiality o act, or as is sometimessaid, when thenew part s " extracted frompotentiality o act. But thistheory,like the first ne, rejected any increase by an apposition of partto part of the same species, .e., by any strictly artto part addition.

Still a thirdopinion which rejected any part by part additionof formin intension and remission s the theory ntimately s-sociated withthe name of GODEFROID DE FONTAINES 9), and waslater given the authoritative upport of WALTER BURLEY in histreatise De intensione t remissione ormarumio). In the pre-

liminary form of this theory as outlined by GODEFROID it washeld that in the case of an intensively ncreased form or quality,e.g., when there is " more" charity,no identical numerical part-of he preexisting ess intensive uality remains. The preexisting" individual" (individuum) s destroyed nd replaced by a moreperfectindividual which does not contain that preexisting n-dividual as a numericalpart of it and in fact s absolutelydistin-

guished from t. BURLEY embracesthis opinion and develops itfurther. He concludesthat n everyformalmotion, .e., intensiveincrease nqualitativeforms, omething ompletely ew s acquired,and this is a form. And so in such a formal movement, thewhole preceding form from which the movement begins isdestroyedand a totally new form (una forma totaliternova),non-existentn the subject before, s acquired. Since, then, there

is a whole series of distinctforms nvolved in intension,BURLEY

maintained that " no formis intended or remitted,but ratherthe subject is intended and remitted ccording to form." Thiswhole position has obviously been influenced by the Franciscandoctrineof the plurality f substantial forms. It suggestsrathera plurality f qualitative forms.

All oftheprecedingpositions,we have seen, rejected he ncreasein the intensity f a quality by means of a part to part addition.

But as we can note fromthe statements f AQUINAS bove, therewere some people in his day who thoughtcharitycould increase

(9) GODEFROID DE FONTAINES, Quodlibeta,quodlib. 7, quaest. 7, unpublished,but summarizedby SCOTUS, op. cit.,edit.cit.,Bk. I, dist.XVII, quaest. 4 (p. 976).Cf. DUHEM, op. cit., pp. 327-328.

(IO) WALTER OF BURLEY, De intensione t remissioneormarum,Venice, 1496,ff. -i5v, forpassage quoted, f. iov, c. i.







136 M. CLAGETT

by the additionof partto part,formto form. AQUINAS himselfreportsthe common distinctionbetween two kinds of quantity,dimensive rcorporeal quantitasdimensiva,orporalis) nd virtual(quantitasvirtualis) ii). But as we remember,he believedthatwhile corporalquantity,withrespectto a form, ncreasedby theadditionof parts, ntensive r virtualquantitydid not so increaseby addition,but by the varyingparticipationof the subject inthe form. And so AQUINAS rejectedany attemptto treat thesetwo quantities as similar.

RICHARD MIDDLETON, who wrote commentaryn theSentences

probably little fter 28i, retains hedistinction etween orporalquantity,which he calls quantityof mass (quantitasmolis), andvirtual quantity12). The former s measured by the numberofobjectssubmitted otheactionofthepower and thusressembleddiscontinuous quantity,the latter by the intensityof the actproducedin a givenobject,and so resemblescontinuousquantity.In spite of this distinction,however,he believes that intensity

or quantity of force can be increased by addition in a mannersimilarto increase in quantityof mass. He posits that just asadding one quantityof mass to anotherproduces a greatermass,so theadditionof one degree ofa quantity fforce o a preexistingone producessomethinggreater n force.

This doctrineseems to have been supported by DUNS SCO-TUS (I3), who indicates very briefly, fterhaving rejected thedoctrines f GODEFROID and THOMAS and in

thetcoursefrefuting

the opinion of HENRY OF GHENT, that " there is no extraction(of a new part frompotentialityo act)." " But I say thatthereis a new reality added to the preexistingone. This realityis like parts or non-quidditativedegrees, which are individualand existing." One of his latercommentators OANNES PONCIUS

interprets his passage as concludingthat intensiveaugmentation" takesplace bytheaddition fnon-quidditative arts" (I 4). He

goes on to say that " this doctrine of Scotus is commonlyheld

(Ii) AQUINAS, op. cit., edit.cit., 153 Ib, 1532a.(I2) RICHARD OF MIDDLETON, Super quatuor libros Senetentiarum,Bk. T,

dist. XVII, art. 2, quaest. i, Brixiae, 1591, vol. I, p. i62. Cf. DUHEM, Op. Cit.,

PP- 330-331.(I3) DuNS SCOTUS, op. cit., edit. cit., loc. cit. in note 6, p. g9o.(14) Ibid., p. 99i.








byallthosewho aythat he ntensionfa qualitys accomplishedby the addition f degree o degreewhether hesedegrees rehomogeneous r heterogeneous."Certainlyhere s no doubtthat COTUS'S " faithfultudent" JOANNES DE BASSOLIS adoptedtheconcept f intension ydegree o degree dditionIS). Herepeats nd answers n old argument rought p againstpartto part ddition, amelyhat fwe addtepidwater o tepidwaterboth at the same degree of heat, the resultingmixture s nothotter. This,JOANNES nswers,s becausewe areadding ubjectto subjectand gettingn increase n the quantity f mass,but

if we couldput thetwoquantities f heatin the samesubjectso thattherewouldbe no extensivencrease f the quality nthesubject, henwe wouldget an increasen intensity.Manyschoolmenn the 4thandI5th enturiesicked pthisdistinctionbetween " quantity of heat" and " intensity f heat" and usedit in determiningheultimate ffectivenessf a heatagent n aheataction i6).

The doctrine f intension y gradual dditionswas takenup

(I 5) JOHN OF BASSOLS, n quatuor ententiarumibros,Bk.1, dist.XVII, queast. 2;Paris, 1579, ff. 114-117. Cf. DUHEM, op. Cit., pp. 335-339.

(i6) M. CLAGETT,GiovanniMarliani and Late Medieval Physics,New York,194 , PP. 34-39. It is ratherinteresting o note that the distinctionbetweenquantity of mass and virtualor intensivequantitywhich was applied to heatactionsmay, in my opinion,have been the source of the quantitativedescription

of the impetuswhich the schoolmen believed accounted for the continuance ofprojectilemovement. Althoughthe theoryof an impressedforcecontinuing hemovementof the projectile is very old, going back at least to JohnPhiloponusin the sixthcentury,A.D., the first uantitativedescriptionof impetusas varyingwith the quantityof prime matter in the projectile and the velocityimpartedto the projectile is introduced by JEANBURIDAN (See M. CLAGETT, " Some GeneralAspects of Medieval Physics." in Isis, 39 (1948), PP. 40-4i). But what wasthe source of this new definition And why was it expressed in termsof thequantity of prime matter and velocity? My opinion is that it goes back tothe fundamentaldistinctionnoted with respect to qualities, namely quantityofmass and intensivequantity. This is borne out by realizing that when localmotion was treated at the same time as qualitative changes, the velocity oflocal motion was equivalent to intensity n qualitative changes, and both wereexpressed in termsof degrees (gradus). Now the similarity etween quantityofmass (quantitasmole) and quantityof primematter (quantitasprimematerie)is obvious. So just as considerationof both quantityof heat and intensitywerethought o be necessary nmeasuringtheeffectivenessfan agent in a heat action,so the similar factors,quantityof prime matterand velocitywere thought tomeasure the effect f a movingprojectile.







I38 M. CLAGETT

withgreat vigor at Oxford. Nor is this surprising, orOxford,even from the i3th Centurywas known for its nourishment fmathematics, nd this attempt o treatqualitiesquantitatively aspickedup withremarkable lacrityf notalwayswithequal clarity.THOMAS BRADWARDINE' s efforts n dynamicsand kinematics nhis Treatise of the Proportions f Movements n 1328 no doubtconstituted he mostimportantnitialstep in thatmovement, ndthe dependence of later authors on him is evident and acknow-ledged17).

One of the principal leaders at Oxford in the school who re-

cognizedthe value of treating he qualitativeintensionsquantita-tively was RICHARD SWINESHEAD (Suiseth). And while I shallreservequestions of biographyto a later article, it seems fairlycertainthatRICHARD SWINESHEAD was a fellow ofMerton Collegeabout 1340 and that his principal work, later known as theCakulationes,must date from about this time i8). And of allof the English philosophersof Oxford at this time who were

interested n questions of qualities and movements,men such asTHOMAS BRADWARDINE, JOHN DUMBLETON, WILLIAM HEYTESBURY,

the anonymous author of the Six Inconveniences, t. al. (ig),SWINESHEAD seems to havegainedgreatest enown n latercenturiesfor this kind of activity. Fifteenthcenturyschoolmen in Italylike ANGELUS DE FOSSAMBRUNO, GIACOMO DA FORLI, PAUL OF

VENICE, and GIOVANNI MARLIANI, call him ,,the Calculator"with obvious respect, and point to him as the great authorityin this quantitative physics that was growing up. It is alsotrue that Italian humanists of the same century failed toappreciate the subtletiesof his thought and in factappropriatedhis name in coining the derisive term for scholastic inanities,

(17) Thus, forexample, see below whereSWINESHEADmentionsBRADWARDINE.

Most of the treatisesdealing with local motion in the fourteenth entury referto BRADWARDINE. See CLAGETT, GiovanniMarliani, etc.,chap. VI.

(i8) For the essentialpoints n thebiographyofSWINESHEAD consulttheDNB;CLAGETT, GiovanniMarliani, Appendix; and L. THORNDIKE, TheHistoryof Magicand Experimental cience, vol. 3, New York, 1934, PP. 370-385.

(I9) Some of the basic ideas oftheOxfordschoolmen are treatedbyP. DUHEM,

op.cit.,vol. 3, pp. 405-48I. For the talianauthorsmentioned nthe nextsentenceof the text, consult the same work, pp. 48i-5io, and also CLAGETT,GiovanniMarliani, by referringo the index.







RICHARD WINESHEADANDLATE MEDIEVALPHYSICS 139

suisetica 20). But in the sixteenth enturyhe was admired bymen of the caliber of SCALIGER and CARDAN 21), and no lessa figure han LEIBNIZ pleaded forthe editingof the Calculator'swork, t the sametimedescribinghimas the man" whointroducedmathematicsnto scholasticphilosophy (22).

Grantingthat qualities may be treatedquantitatively s manyof the English school did, it became a fundamental uestion asto how to measurethe intension nd remissionof these qualitiesto use scholastic vocabulary,or to put it in anotherway, howto measurethe progressive r regressive lterationn the intensity

of qualitativeforms or the alteration fvelocity n local motion.).This is the questionwhichSWINESHEADraises and seeks to settlein the first ractateof his Calculations. It is this tractatewhichis ordinarilyknown as the Intension nd Remission f Qualities,or along with some of the succeeding tractates, s Intension ndRemissionof Forms 23). It is this first hort tractatewhich Ishould like to analysebrieflyn this article.

Before startingthis analysis, it ought to be noted that theexpression,latitude,which has an interesting istory owardtheend of the thirteenth nd in the early fourteenth entury,nowclearly is used by SWINESHEAD n the sense if a range of alteration

in intensity romone degree of intensity o another,or in thecase of local motiona positiveor negative ncrementn velocity,i.e., an incrementfrom one degreeof velocity equivalent to in-

(2o) ERMALAO BARBARO, Epistolae,orationes,tcarmina, ditedbyVittoreBranca,Florence, 1943, pp. 23, 78. Cf. the statement put in the mouth of NICCOLE

NICCOLI by LEONARDO BRUNI, L. THORNDIKE, UniversityRecords and Life inthe Middle Ages, New York, 1944, p. 269.

(2i) THORNDIKE, op. cit., in note i8, vol. 3, p. 373. Cf. W. G. TENNEMAN,

Geschichte er Philosophie,vol. 8, Leipzig, i8o8, pp. 904-905, with appropriate

references n note 70 to CARDAN and SCALIGER.

(22) THORNDIKE, op. cit., vol. 3, p. 370. Cf. Louis COUTURAT, Opuscules etfragments e Leibniz, Paris, 1903, p. 340: Fuit enim aliquis Johannes Suisset,dictus Calculator, qui circa motus et qualitatum intensiones in media meta-physicorum regione mathematicum sine exemplo agere coepit. See also pp. 177,

i9i, and 330.(23) In all of the various manuscripts the incipits of the first treatise in the

Calculationesreveal that t is concernedwith ntension nd remissionofa quality.However the Paris manuscript (BN, Fonds latin, no. 6558) and later authorsgive the title of Intensio tremissio ormarumo the firstfew treatises which concern

various aspects of intension and remission.







140 M. CLAGETT

stantaneousvelocity)to another. Or, we mightthink fa bodywhere there is a qualitative latitude when the adjacent parts ofthatbodyvary n heat intensity. We would say, thenthatthereis a latitudeof hotnessor calidity n thatbody from uch or sucha minimumdegreeof calidity or zero degree of calidity)to suchor such a maximum degree (gradus summus). Other scholasticterminology ill be explained n the context f thediscussion 24).

B. The Calculations 25); Tractatus I: The Measurementofthe Intensionand Remission of a Quality.

The Calculator informsus at the outset that there are severalopinions as to how intension and remission re to be measured.But beforetakingtheseup he wishes to pointout (z6)

intension s accepted in two ways. In one way it is spoken of as the alterationby means ofwhich a quality is acquired-and speaking thus intension s motion.In the other way it is referred o as the quality by means of which somethingis intended,e.g., a hot body is said to be intendedby calidity.

Though he seems to favourthe second method, as matteroffact the firstmethod of acceptingintension as motionseems tobe impliedon occasions in latersections 27).

(24) Since the question of the terminologyused in the fourteenth enturyphysicaltreatises s oftendifficult can recommend wofourteenthenturyworkswhich offer efinitions:The anonymous ntroduction oundattachedto THOMASBRADWARDINE,Proportionesncluded among the several treatises in B. POuITI,Quaestio de numeromodaliumetc., Venice, 1505, ff.9r-Ior. Compare also inthe same edition,GIOVANNI CASALI, De velocitatemotutslterationis, . 59v et seq.

(25) The Latin textoftheCalculationesused in thesefootnotes as been preparedprimarily rom hefirst ne of the followingmanuscripts nd editions Universityof Pavia, Aldini codex no. 314, ff. I-83 (abbreviated below as A); CambridgeLibrary,Gonville & Caius Ms. 499/268, ff.I65r-2s5r (abbreviated as B); Paris,Bibliotheque Nationale. Fonds latinno. 6558 (abbrev. as C); Rome, BibliotecaAngelica, Inventario no. I963, (abbrev. as D); Editio princeps,Padua ca. 1477

(Abbrev.as EP); Editio secunda,Pavia, 1498 (abbrev.as ES). Since thecompletedtext will be published later the variantreadings have been ommited in all but

a few cases.(26) Ms. A, f. ir : Penes quid habent intensio et remissio qualitatis attendi

plures suntopiniones. Pro quo tamenprimo est notandumquod intensiodupli-citer potest accipi. Uno modo pro alteratione mediante qua qualitas acquiritur.Et sic loquendo intensioest motus. Alio modo dicitur ntensioqualitasmediantequa aliquid est intensumsicut calidum est intensummediante caliditate et sicproportionalitere remissioneest dicendum.

(27) A, f. ir : De intensioneet remissionesecundo modo dictis i) ad presensest locutio. Variant: (i) A seems to have dicta. B, C, D, EP, ES have dictis.







142 M. CLAGETT

use of arguments nvolvingthe infinite. He claims that if weacceptthispositionthen t would follow hatthe maximumdegreeof intensityn any hot body would be infinite3).

This is because if some caliditywere increased in intensity o the maximumdegree, then that caliditywill be a certainamount intense,thendoubly intense,then quadruply intense,and thus to infinity ecause it will be a certainamountnear to the maximum degree, then doubly nearer, quadruply nearer, and thusto infinity, nd proportionally s it will be nearer to the maximum degree, soaccordingto this position twill be more ntense hanany timebefore. Therefore,this caliditywill be intended toward infinity efore the end, and in the end itwill be more intensethan any time before. Hence the maximum caliditywill

be infinitelyntense,which was to be proved.

The substanceof this argumentwill perhapsemergemore clearlyifwe repeat t usingmodern ymbolization. To prove: The firstposition is false. Proof

(i) Assume the first ositionto be true, representingt as follows:(2) Let x = f (i/y),wherex is the intensityndy is themagnitudeofa linedrawn

from the degree of intensity o the maximum degree.

(3) Then as we increase x towards the maximum degree, y becomes smaller,and will, for example, become one-half as long, one quarter as long, gettingas small as you like.

(4) Now y must become zero at the maximum degree.(5) But from 2) wheny is zero, then x must be infinite.(6) Thus the maximumdegree must always be infinitelyntense when we assume

(2) and (4).(7) But since all maximal degrees are not infinitelyntense, the position is false.

SWINESHEADis pointingout, then, thedifficultynvolved n tryingto measure an increase by a decreasingfunctionwhich must bezero when that increase is to some finitemaximum.

A similar type of argument s presented n his next criticismof the firstposition. For he claims that if we accept the first

(3i) A, f. ir: Ex illa [positione] sequitur quod gradus summus est infinite

intensus, quia intendatur i) aliqua caliditas ad sunmmum,unc illa caliditas eritaliqualiter intensa,et in duplo intensior, t in quadruplo (2), et sic in infinitum;quia aliquantulum propinqua erit ista caliditas gradui summo, et in duplo pro-pinquior, et in quadruplo, et sic in infinitum. Et proportionaliter icut eritpropinquior gradui summo, ita iuxta illam positionem erit intensior. Ergo ininfinitum ntensa (3) erithec caliditas ante finem et in fine erit intensiorquamunquamante. Ergo caliditas ummaerit nfinitentensa, uod fuitprobandum 4).Variants: (i) ES adds inhora; (2) B has triplo, P and ES omit; 3) B has intensior;.(4) A omitsquod fuit probandum.








position,then there will not be any degreetwo times less intensethan the mean degree of the latitude. The argument s provedas follows 32):

No degree is distant from the maximum more than double the mean degreebetween themaximum degree and thezero degree,the mean beingequallydistantfrom the extremes. Since no degree is as distant fromthe maximum degreeas the zero degree it followsthat the zero degree is doubly more distant fromthe maximumdegree thanthe mean of the whole latitude. Therefore,no degreeis doubly less intensethan themean degree of the whole latitude. The consequensis false because there is some degree a certain amount intense,and some degreehalf as intense,and some a quarteras intense,and thus to infinity,ust as some

quantity is a certain amount, and some quantity is half as much and anotherquantity s one fourth s much,and thustoinfinity. Thereforetheposition sfalse.

Much of this argument s directed as the previous one at thedifficultieshat arise when we tryto representthe intensitybythe linear distance to the maximum, i.e., when we attempttocorrelate a quantity that rises to a definitefinite imit with amagnitudethat can, as it approaches zero, be any quantity as

small as you wish.

I suggestthatthis second argumentmightbe restated s follows:(i) Assume theposition,x = f (i/y) wherex is the ntensityndy is a linedistance

from ny intensity o the maximum degree of intensity.(2) Accept the common belief that in a latitude of intensity here is maximum

degree and a minimum or zero degree with a mean degree equally distantfromthe extremes.

(3) Assume the lower extreme of the latitude is zero and further ssume that

the mean degree is a distance a fromthe maximum degree.(4) Restating 2), when the intensity s zero, the distance from the maximumwill be 2a.

(32) A, f. ir: Item ex illa positione sequitur quod nullus est gradus in duplominus intensusquam estmedius gradus,scilicet totius atitudinis. Hoc probatursic: Nullus gradusper induplo plus distat gradusummo i) quaam radus mediusintergradum summumet non gradum,eo quod medium est quid equaliter distatab extremis. Cum ergo nullus gradus per tantumdisteta gradu summo sicut

non gradus distat a gradu summo, sequitur quod (2) non gradus per in duploplus precise distat a gradu summo quam distetgradus medius totius latitudiniscaliditatis gradu summo. Ergo nullus gradus est in duplo minus intensusquamestgradusmediustotius atitudinis aliditatis. Consequens est falsum, uia aliquisgradus est aliqualiter intensus, et in duplo minus intensus est aliquis gradus,et in quadruplo, et sic in infinitum, icut aliqua quantitas est aliqualiter magna,et aliqua in duplo brevior, et aliqua in quadruplo brevior, et sic in infinitum.Ergo positio falsa. Variants: (i) B, C have redundant phrases after summo,here omitted; 2) A has quia forsequitur uod.







144 M. CLAGETT

(5) But inassuming i), by thetimethe ntensity is zero,ywill havegone througha whole series of value beyond 2a to infinity.

(6) Now we know that intensity can have all the decreasing values to zero.But according to (4) 2a representsthe maximum value we can accord to y.Thus if we follow i) at the same time that we accept (4), then we cannotgo throughthe decreasing values of intensity o zero because we would verysoon be beyondour limit2a. And ifwe could notgo through alues to x= o,thenthere would not be any degree doubly less intense thanthe mean degree.

(7) It is clear that 4) and (i) are contradictory. But if (4) is accepted as basedon the commonexperienceexpressed n (2), and thus is true, then i) is false,and the position is false. Q.E.D.

The thirdargumentdirectedby SWINESHEAD against the first

position concludes thatout ofthis first osition t follows: (33)

that any degree of motion you wish is of infinity emission,because every degreeof motion is infinitely istant froman infinitedegree of motion. Since thereis no maximum degree beneath the infinitedegree, it follows that any degreeat all is infinitelyistantfrom hemaximumdegree of ts atitude. The consequensis false, and therefore, o is the antecedent.

This argument oes back to the firstrgument gainst hisposition,namely, hat f intension s measuredby nearness o themaximumdegree,then anymaximumdegree s infinitelyntense. Now thisthird argument says that if remissionis measured by distancefromthe maximum degree, then every degree of remission isinfinitely emiss because it must be infinitely istant from aninfinitemaximumdegree.

In the course of the next or fourth argumentSWINESHEAD

introduces he conceptof the infinitelymall (infinitummodicum).Beforetakingup his argument, etus see in whatsense he is usingthe term infinitumn this expression. From at least the timeofPETER OF SPAIN'S Summulae ogicales n the thirteenthentury,it was customaryforthe schoolmento distinguishbetweenwhattheycalled categorematicnd syncategorematicnfinites34). The

(33) A, f. ir: Item sequitur quod quilibet gradus motus est infinite emissionis,quia omnis gradus motus per infinitum istata gradu infinitomotus. Cum ergonullus sit gradus intensissimus itra gradum infinitummotus (I), sequitur quodquilibet gradus motus per infinitum istat a gradu intensissimo sue latitudinissed consequens est falsum; ergo et antecedens. Variant: (i) A omits.

(34) PEr=R OF SPAIN, Summulae logicales, edited by J. P. MULLALLY, NotreDame, Indiana, 1945, P. iI8 : Tertia distinctio est quod " infinitum capiturdupliciter: uno modo capiturcategorematice, ignificativet estterminus ommu-nis et sic significat uantitatem rei subiectae vel predicatae, et signum ut cum







RICHARD SWINESHEAD AND LATE MEDIEVAL PHYSICS I45

categorematic infinite was a generaltermsignifyingn actualquantity without limit or end. For example: " The world isinfinite (i.e., unbounded in extent)." The syncategorematic" infinite was a distributive erm signifying quantity arger(or smaller) than any quantity you please. Thus the expression" the infinitelymall is a part of a continuum signifies hat aquantity smallerthan any quantityyou please is a part of a con-tinuum. There is little doubt that when SWINESHEAD uses theexpression infinitummodicumhe is employing nfinitumn thesyncategorematic ense and thus means by that phrase in the

following nd succeeding arguments a magnitude smaller thanany magnitude you please." But let us turn to SWINESHEAD'S

arguments 35):

dicitur "Mundus est infinitus; alio modo capitursyncategorematice,on proutdicitquantitatem eisubiectaevel praedicativae, ed inquantum e habetsubiectumin ordine ad praedicaturn, t sic est distributio ubjecti et signum distributivum.

Cf. EDWARD STAMM, "Tractatus de Continuo von Thomas Bradwardine," inIsis, vol. 26 (I936), pp. I9-20 whereBRADWARDINE is quoted as saying:" Infinitumcathetice cathegorematice)...est quantum sine fine..., magnum vel multumsinefine eu nonfinitum... Infinitumyncatheticesyncathegorematice)... st quantumfiniturnet maius isto, et finitummaius isto maiori, et sic sine fine ultimoterminante, t hoc est quantum et non tantumquin maius..." The words inbrackets re myown. Cf. also P. DUHEM, op. cit., vol. 2, pp. 22-24, appendix E,pp. 368-408; vol. 3, pp. 274-276. In this astpassage he quotes from he sixteenthcenturyschoolman, SOTO, the example of the syncategorematicnfinite have

given: Infinitapars est pars continui.(35) A, f. ir: Item ex illa positione sequitur quod quelibet caliditas citrasummam est infinite emissionis, quia vel est gradus summus infinite ntensusvel finite ntensus. Si infinite ntensus, ergo omnis gradus citra summam perinfinitum istata gradu summo. Consequentia patetper hoc quod omnisgradusfinitus er infinitum istata gradu infinito. Ergo cum penes appropinquationemgradui summo iuxta positionem habeat intensio caliditatisattendi i) sequiturquod omnis gradus caliditatis sit infinite emissus. Si gradus summus sit finiteintensus, tunc sit a unus gradus remissus. Tunc sic infinitepropinquior estaliquis gradus gradui summo quam est a gradus, quia per infinitummodicum

distataliquis gradusa gradu summo, ut satis patet. Ergo si penes appropinqua-tionemgraduisummo intensioqualitatis vel gradushabeat attendi, equitur quodin infinitumntensiorest aliquis gradus quam est a gradus. Ergo cum quilibetgradus citrasummum et etiam gradus summus sit solum finite ntensus, equiturquod a gradus est infiniteremissus. Sed illud est falsum. Ergo positio falsa.Et quod conclusio sit falsa patet, quia si quilibet gradus foret nfinite emissus,nullus gradus foretaliqualiter intensus et per consequens nullus gradus foretalio intensior,quid est impossibile. Variants: (i) ES adds phrase et penesdistantiam gradu summo emissio.

I0







I46 M. CLAGETT

It also followsfromthatposition thatany degree of calidity hotness) less thanthemaximum s of nfinite emission, ince the maximumdegree s either nfinitelyintense or finitelyntense. If it is infinitelyntensethen every degree beneath

themaximum s an infinite istancefrom he maximumdegree. The consequentiais obviousfrom hefactthateveryfinite egree s infinitelyistantfrom n infinitedegree. Hence, since accordingto thisposition ntensionof calidity s measuredby its proximity o the maximumdegree, it followsthat everydegree of calidityis infinitely emiss.

Now if the maximumdegree is finitely ntenseand a is a remiss degree, thereis some degree infinitely earer to the maximumdegree than is a because somedegree s distantby an infinitemodicumfrom he maximumdegree,as is obviousenough. Therefore, f the intension of a qualityor a degree is to be measured

by itspropinquity o themaximumdegree, tfollows hat some degree s infinitelymore intensethan is a degree. Hence, since any degreeat all beneath the maxi-mum as well as the maximum degree is only finitely ntense, it follows that adegree is infinitelyemiss. But that s false, hence the positionis false.

As in the preceding cases it would, I believe, be of some useto translate this into more modernterminology,rying o get atwhatSWINESHEADmeans. The argumentppearsto be as follows:

(i) Assume as an expressionfor ntension n thefirst osition,x = f (I fy),with xas intensity nd y a line distancefrom the degree intensity o the maximumdegree. But also assume that z = f (u )where z is remission and u is a linedistancefromthe maximumdegreeof intensity.

(2) Now bytheargument hemaximumdegreeof ntensitys either nfinite rfinite.(3) If the maximumdegreeis infinite,hen everyvalue of u will be infinite, ince

everyvalue of u would be measuredfrom terminus n infinite istanceaway.(4) Hence everyvalue of z will also be infinite.(5) But suppose that the maximum degree of intensitywere finite. Then take

any value of intensity less than the maximum.(6) Now therecan alwaysbe some value of ntensitynfinitely earer themaximumthan a because there can be some value an infinitelymall distance frommaximum, infinitely earer and infinitelymaller being used in a syncate-gorematic ense.

(7) If by i) intension s really measuredbyproximity o themaximum nd by (6)there is some value infinitely earer to the maximum than is a, then thereis some degree infinitelymore intense than a.

(8) But since by (5) we assumed thatthe maximum s finite nd thus any degreebelow themaximum s finite, nd yetat the same time by (7) show thatsome

degree below themaximum s infinitely ore ntense han a, hence anydegree amust be infinitely emiss, .e., a is infinitelyemoved from ome degreebelowthe maximumand thus is infinitelyemovedfrom the maximum.

(9) But a is any degree ofcalidity nd is in actualitynot infinitely emiss. Hencethe positionis false. Q.E.D.

It is clear that steps (6) and (7) are the crucial parts of theargument. If we were representingntension by a line drawn








from an initial zero terminus, hen we could not conclude thatbecause there was some degreeinfinitelyearer n a syncategore-matic sense to the maximum,there must be a degree infinitelymore intense than a. This latter conclusion could only followif we are representing ncreasing ntensityby a line drawn tothe maximum, which line is decreasing in length. In such asituationthe line could be reduced to a quantityas small as welike. Hence, insubstance hisarguments similar o the precedingones. The last argument gainstthe first osition s of the samenature and I shall omit any discussion of it here.

Having argued against the firstposition, the Calculator nowproceeds to argue against the second, or at least the second partof the second position, namely that remission s to be measuredfrom he maximum degreeof intensity.

The firstargument against the second position is similar tothe second argument gainstthe first osition 36)

From the second positionit follows that no degree is twice as remiss as the meandegree since no degree is two times farther rom the maximum degree than isthe mean degree, as is clear enough. The consequens s false because there issome degree twice as remiss, some degree fourtimes as remiss,and so on, justas there is some quantity wo times less thanany given quantity,four times less,and so on to infinity.

Although it is difficult o followwith complete assurance theabbreviatedargument presented by the Calculator, I believe his

intention s the following:

(I) Assume z = f (u) where z is remission and u is a line distance from themaximum degree of intensity.

(2) It is commonly accepted that in a given latitude or range of intensity hereis a maximum degree, a minimum degree (either zero or relatable to zero),and a mean degree equidistant fromthemaximum and the zero degrees.

(3) Now suppose thatthemean degree s a distancefrom finitemaximumdegree.The zero degree or lower limit of the latitude then would be a distance of

(36) A, f. ir: Ideo ponatur secunda positio contra quam sic arguitur: Exilla sequitur quod nullus gradus est in duplo remissior i) medio, quia nullusgradus per in duplo plus distat a summo quam gradus medius, ut satis constat.Et consequens est falsum, quia aliquis gradus est in duplo remissior, lius inquadruplo, et sic de aliis, sicut quacumque quantitate data est aliqua quantitasin duplo brevior, t in quadruplo brevior, t sic in infinitum. Variant: i) C addsatio vel.







148 M. CLAGETT

2a from he maximum. And in measuringaccording to (i), the zero degreeof intensitywould be identifiablewith a remission twice that of the mean.Since the zero degree is the lower limit, then according to this position 2a

represents he argestpossible distancefrom he maximum.(4) But in the termsin which remissionis universallyconceived, there can beno limit to the relational comparison of remission, i.e., something can betwice as remiss as a given degree, fourtimes as remiss, and so on to infinity,where presumably ntensitywould be zero.

(5) But if we accept the position assumed in (i), we cannot increase remissionat will to the point where intensity s zero, but would ratherbe limitedbysome arbitrary alue 2a.

(6) Thus since we could not arrive at an intensity f zero by using I), we could

nothave a degreetwiceas remiss s definedby the identificationfzero degreeof intensitywitha degree twice as remiss n step (3).(7) Thus (i) and (4) are contradictory,nd since (4) is generallyaccepted, (I)

must be false. Hence the position s false. Q.E.D.

The onlydifficultyn my interpretationfthe argumentwouldappear to be my assumption that Calculator means in (6) thatit is thedegreetwiceas remiss s themean that must be identifiable

withzero degreeof intension n this positionand yetwhich cannotbe identifiable with it because, accepting the common opinionof remission, we can never reach zero degree using this secondposition. But I believe that this is just what he is trying o say.Let us put it anotherway. Accordingto this second position,the zero degreeof intensitymust be identifiablewiththe distancetwice the mean (step (3)). But at thesame time commonopiniontells us that we can use the expressionstwice as remiss,fourtimes as remiss, and so on as we are approaching the highestpossible remissionwhich s identifiable ith ero intensity.Hence,we cannot reconcile this common opinion which identifies erodegree of intensitywith an unlimited remissionand the opinionof this second position which tries to identify ero degree ofintensitywith twicethe remissionofthe mean degree,but whichnever even yields a degree twice as remiss as the mean degree,

because so long as it accepts any part of the common opinionit cannot reach the zero degree. Then since thecommon opinionis held to be true and since it contradicts he second position,the second positionmust be false.

Omitting wo ofthe arguments gainst the second position,wecan make passing reference o the last of these arguments inceagain we have use of the concept of the infinitelymall. Its








translation ntomodernterminologys quite similarto the fourthargument gainstthe first osition 37)

Let a be a degree of caliditywhich is distant from the maximumby a certainlatitude. This latitude is divided into proportionalparts towardthe maximum.Then by an infinitelyess amount would some partbe distantfrom hemaximumdegree. Since, therefore, emission is to be measured by the distance fromthemaximumdegree, t follows hatsome part s infinitelyess remissthan s a degreeand any part at all is more remiss than the maximumdegree. Therefore,a isinfinitelyemiss. The consequenss false. Hence the positionis false.

Again rearranginghisargumentwe havethefollowing teps

(i) Assume thatremission s measured by distance fromthemaximum,m.(2) Take any degree a. Then from i) its remission s representedby line ma.(3) Now ma can be divided up into proportionalparts 1/2,14, 1/8,... to infinity.(4) Some part will be infinitely earer to m, than is a, infinitely earer in the

syncategore.naticense.(5) Hence thatpartwillbe distantfrom hemaximumbyan infinitelyess amount.(6) Then that part is infinitelyess remiss than a since remissionis measured

by distancefrom he maximum.

(7) Any part at all is more remissthanthemaximum.(8) Hence, any degree a is infinitelyemiss from 6)).

It seemsto me thatwhentheCalculatorproceedsfrom tatement(6) to that noted in (8), he is shifting romthe infinitely mallto the infinitelyarge in an illegitimatemanner,a mistakewhich.he had notmade in the fourth rgument gainstthe first osition,because that positionhad assumedtacitly hat the infinitelyarge

was functionof the infinitelymall.Beforepassingto the thirdpositionwhichtheCalculator ccepts,

I should like to observe that he does not bring out sharplyonefundamental riticism gainst using the maximumas a referencepoint, that namely, the latter must vary from measurementto measurement, romproblem to problem. But of course this

(37) A, f. iv: Sit ergoa unus gradus i) caliditatisqui per certam atitudinemdistet a summo, que latitudodividitur n partes proportionalesversus summum.Tunc per infinitumminusdistataliqua pars a gradu summoquam distet gradus,quia in infinitum ropinquior est aliqua illarum partiumgradui summo. Cumergo remissio habeat attendi penes distantiama gradu surmmo, equitur quodin infinitumminusremissa estaliqua illarumpartium uam est a gradus etquelibetillarumpartiumest remissiorquam est gradus summus. Ergo infinite emissuscst a gradus. Consequens est falsum; ergo positio falsa. Variant: (i) A hasuna latitudo; othersgradus.








intension and remission are numerical reciprocals and from avector or directional tandpoint re reversed. i.e., i800 out).

One of the principal arguments which SWINESHEAD hasoutlined against this position is that if we accept it then agiven maximum degree of intension must be a certain amountremiss. According to the opponents of the third position thiscould notbe. These opponentswould hold that at themaximumdegree of intension,remissionmust be zero, a situationthat thethirdposition provides for only when the maximum degree ofintension s infinitee.i., withz = f (i/y), the z will equal zero

onlywheny is infinite). Hence the argument eeks to showthatsince the maximumdegreeis not at all remiss,a givenremissionmust be instantaneouslyost as it passes to maximum ntension.This is impossiblebecause (39)

to lose remission is nothing else than to acquire intensionas is obvious fromthe factthatremission s relatedprivatively o intension. And a privativebeingacquired is nothingelse than a positivebeing lost nor is a privativebeing lostanything else than a positivebeing acquired. Since, therefore,no latitude of

intension s suddenlyacquired, itfollowsthatno latitude ofremission s suddenlylost...

If then the remissionis not suddenly lost, then according tothe thirdposition the maximumdegree of intensionmust be acertainamountremiss,whichwas thoughtto be impossible.

The Calculator answers the argumentby freely dmitting hataccording to the third position any maximum degree short of

infinity s a certainamount remiss. But he findsthis to be aneasily imaginedpossibility40).

The thirdprincipal position must be supported in light of what has been saidand the conclusion that the maximumdegree is remiss is conceded. But when

(39) A, f. Iv: Sed deperdereremissionemnon est aliud quam acquirere inten-sionem, ut patet eo quod remissio se habet privativerespectu intensionis etprivatum acquiri non est aliud quam positivum deperdi, necque privativum

deperdi estaliud quam positivum cquiri. Cum ergo nulla latitudosubitoacqui-ratur, cilicet ntensionis, equiturnullam latitudinemremissionis ubito deperdirespectu a.

(40) A, f. iv-2r: Pro dictissustinendaesttertiapositioprincipalis et concediturconclusio quod gradus summus est remissus. Et tunc quando arguitur quodnulla caliditas est caliditatesumma intensior, rgo illa non est remissa,negaturconsequentia, quia etsi nulla caliditas sic foret ntensiorde facto,non repugnattamen illi caliditati,quod aliqua foret illa intensioret imaginando caliditatemintensiorem lla foretremissasicutnunc est.







152 M. CLAGETT

it is argued thatno calidity s more intensethan a (given) maximumcalidityandhence thatmaximumcalidity snotatall remiss, heconsequentias denied. Becausealthough no calidity would be de facto more intense, it is not impossible withrespect to that caliditythatthere would be some caliditymore intensethan themaximum. And by imagining more intensecalidity, henas it now is itwouldbe a certain amount remiss.

Assuming that the thirdposition is correct,when we proceedto investigatethe implicationsof it, three furtherpositions oropinions are advanced, the first two of which are not correct,while hethird s 4I).

The firstposits that every degree is just as intense as it is remiss. The secondsupposes thatthere is some degree ust as intense as it is remiss,and conversely,although not every degree is just as intense as it is remiss. The third positsthatno degree is as intense as it is remiss.

He holds thatthese same positionscan be supposed for largenessand smallness and every other latitude which is consideredpositively nd privativelywithrespectto something."

The argumentswhich the Calculator advanced for the firstsupplementaryposition (and which he later refutes) are aimedat provingthat intensionand remission are identical and thusany degree is just as intenseas it is remiss. The first rgumentalong thisline tells us that 42)

Intension ofa degree is measuredby distancefrom he zero degreeand remissionby its nearness to that degree, but everydegree is distantfrom he zero degreeby the same amount it is close to the zero degreebecause themutual propinquityand distance of things do not differwithrespect to something.

In terms of the representations e have made of the Calculator'sargumentsabove, this type of argumentwill not hold up. It

(41) A, f. 2r: Circa tertiampositionemtresversanturpositiones. Prima enimponit quod omnis gradus est ita intensus sicut remissus. Secunda ponit quodaliquis estgradus ta ntensus icutremissus teconverso, ed non quilibet. Tertia

ponit quod nullus gradus est ita intensussicut remissus. Et sicut iste positionesponuntde intensioneet remissione ta etiamponuntde magnitudineet parvitateetde omnialia latitudine ue privative elpositive espectu licuius consideraturI)

vel dupliciterscilicetpositiveet privative onsideratur. Variant: (I) A, C havecontrariatur.

(42) A, f.2r: Intensiogradusattenditur enes distantiam nongraduetremissiopenes accessum ad non gradum, sed omnis gradus equaliter distata non gradusicut ipse est propinquusnon gradui, quia non differuntespectualicuius propin-quitas aliquorum et distantiam nter lla.








is merelypointingout thatthe numericalvalue ofy is the sameforboth intension nd remission,but the intension nd remissionare not the same, but ratherreciprocalswithcontrary irections.

While omitting he details of the next arguments n favorofthe first upplementaryposition,which attemptsto identifyn-tension and remission, wish to point out that in the course ofthese argumentshe clearlystates as a commonlyaccepted partof the basic positionofmeasuring ntensionbydistance from eroand remissionby propinquity o the zero degreethat a " degreeof infinite emission s nothingelse but a zero degree of inten-

sion " (43). This fits the formulation - f (i/y) mentionedabove. Similarlythereciprocalnatureofintension nd remissionis admitted even thoughtthe argumentusing it is consideredunsound by the Calculator. He tells us " that a double degreeof remission is nothingbut a one-halfdegree of intension anda one-halfdegree of remission s nothingbut a double degreeof intension (44). It is true that the Calculator latermodifies

these dentificationsaying hat naddition othefact hat ntensionand remission re reciprocals heydiffer lso in the way we con-sider them directionally. Hence these statements have justquotedwill be trueonlyfor henumerical elationshipsf ntensionand remission.

SWINESHEAD after dvancingthe supporting rgumentsforthe

(43) A, f. 2Rv: Item sequitur eandem realiteresse latitudinem ntensionisetlatitudinemremissionis,quia intensiohabet attendi penes recessum a suo nongradu etremissio penes accesum versusnon gradumsue intensionis, uia remissionon potest attendi penes distantiama non gradu remissionis,quia non gradusremissionisstgradus nfinitentensionis. Sed quilibetgradus per nfinitumistat I)a gradu infinito ntensionis. Ergo si penes huius distantiamhaberet remissioattendi sequitur omnem gradum infinite emissum existere. Ft sic notandumest quod ab omnigraduremissionisusque ad non gradumremissionisest latitudoinfinita t usque ad graduminfinitum emissionissolum, atitudofinitaconsistit,quia nonest aliud gradus nfiniteemissionisuam nongradus ntensionis t ab omni

gradu usque ad non gradum ntensionis st latitudosolumfinita. Ergo et cetera.Variants: (X) B, C, ES add a nongraduremissionisuia per infinitumnistat Note:I have italicized the pertinent ines in the body of the note.)

(44) A, f. zvR: Ex his etiam est notandum quod ab omni gradu remissionisusque ad suum duplum remissionisprivative ccipiendo (i) remissionem,est induplo breviorlatitudo quam inter ipsum et suum subduplum remissionis,quianonest aliud gradusduplusremissionisuaim radus subduplusntensionist gradussubduplusrernissionisuam gradus duplus intensionis. Variant: (I) A has con-siderandode remissions.








the same, i.e., they are not comparable. But SWINESHEAD doesnot yet adopt this line of criticism have suggested,because hehas not yet outlined the basic differences etween intension andremission as far as being contrariesof direction. This he doesin supporting he thirdsupplementary osition. 'But at the veryend of the essay when discussing certain doubts, he notes whatis substantially he criticismwhich I have outlined. Supposingwe have a remissionwith a value of4 degreesand also an intensionof 4 degrees. Are not these two equal ? No, the Calculatortells us, they are not equals, because they are not comparable

terms47).Omittingthe less important riticism hat the Calculator now

employs againstthe second position,we can pass immediately othe thirdsupplementary ositionwhich he accepts. In brief hisargument s: No degree is as intenseas it is remissbecause in-tension and remission re not comparable things. Althoughtheycan be numerically equal, they are directionally ifferent. But

he points out initiallythat remission can be thoughtof in twoways. In one way it reallyis the intension. In the otherwayit is the reciprocalor privativeof intension. The firstway ofspeaking is not often employedand so he will adopt the secondway of speaking 48)

(47) A, f. 4v: Contra hoc obicitur sic: stat quod remissio alicuius gradus utquatuor et sit intensioalicuius gradus ut quatuor. Ergo videturquod ille sintequales. Negatur consequentia, quia non sunt comparabiles adinvicem ut sic,sicut ante planius est argutum.

(48) A, f. 3v: Sequitur ergo tertiapositio ponens quod nullus gradus est itaintensus sicut remissus... Unde illa est positio quam inter ceteras reputo magisveram. Pro cuius intellectuprimo est notandum quod remissio potest dupliciterconsiderari. Uno modo ut est ille gradus intensus vel ut est illa intensio eoquod ideni estrealiter ntensiocum remissioneut est argutum. Alio modo etiamconsideratur emissio ut scilicet estprivativum espectu ntensionis. Primo modo

sunt illa eadem, hoc est sic remissum,hoc est sic intensum. Et in illo modoloquendi de remissionepotestconcedi quod remissiohabet attendipenes id penesquid habet intensio attendi,et ita potestdici omnem gradum ita intensum sicutremissum existere. Contra quem modum loquendi rationes adducte contra pri-mam positionem non procedunt. Ille tamen modus non est multum usitatus.Ideo loquendum est de remissione secundo modo, dicta scilicet prout est quidprivativum espectu ntensionis. Et dicendum est secundum llummodumnulliusgradus intensionem sue remissioni correspondere, sicut argumenta contra illaspositiones probant liquide.







156 M. CLAGETT

(Since the second position has been rejected), the third position which positsthat no degree is as intenseas it is remiss follows... Whence this is the positionI judge to be truer. For its understanding t should be noted in the firstplace

thatremission an be considered n twoways. In one way it is the intensedegreeor is like that ntension, ince in thisway intension eally s the same as remission,as was argued. In the other way, remission s considered as a privative priva-tivum) with respect to intension. In the firstway they are the same. Thisthing is so intense or is so remiss. In this way of speaking about remission,it can be conceded thatremission s to be measured by the same thing s intensionand in thisfashion tcan be said thateverydegreeis ust as intense s it is remiss.The arguments dduced againstthe first ositionare not valid against this methodof speaking. However, thismethodof speakingis not used much. Hence oneought to speak of remissionin the second way as that which is privativewith

respect to intension. And it ought to be said according to this method thatthe ntension f no degree corresponds o itsremission, s thearguments gainstthetwo preceding positions clearly prove.

The Calculatornow answersthe second supplementary ositionin the lightof remission s a reciprocalor privativeof intension.Notice in this passage thathe uses two analogies to explain thedifference etween intension nd remission.

The first f these analogies is with proportions f greater ndlesser inequality. A proportion of greater inequality for thescholastics resulted when the numerator s more than the deno-minator /b >I. Now the Calculator's analogy is simply thisSuppose a and b are the same in both kindsofproportions. Thenthe proportion /b differs romb/a even though the numbers aand b are the same, that is, even thoughthe proportions re thesame accordingto thing (res). The proportions re said by himto differ ccording to " reason" (ratio). This means that theproportionsdiffer s to how we considerthe members,whetherwe are considering heproportion fa to b or of b to a. Similarly,intension nd remissionwhichare reciprocal o differ, .g., y andi/y differ n " reason" (i.e., theyboth vary as a function fybut one as a direct nd the otheras an inverseratio). SWINESHEAD

goes on to tell us thatthisrationaldistinctionn proportionshad

been made in Master THOMAS BRADWARDINE's Tractatus de Pro-portionibus. BRADWARDINE made thedistinction rimarily ecausehe applied proportions o movement. This application of pro-portions being made, it was axiomatic for him that movementcould only arise when the motive power was greater than theresistance, r if a were themotivepowerand bwere the resistance,when aib>I, i.e., was a proportionof greater nequality. And








movement ould neverarise when thatproportion f motivepowerto resistancewas a proportion fequality,a/b= i, or a proportionof lesser inequalitya/b< . BRADWARDINE then adopts the viewthat velocity follows a geometricproportionof the original pro-portion a/b, with a the motive power and b the resistance 49).This formulationpermits him to distinguish clearly betweenproportions of greater inequality, proportions of equality, andproportionsof lesser inequality when applied to movement andthus permitshim to conclude that where motion is concerned,or geometric proportionality, hose three kinds of proportions:

greater nequality, equality, and lesser inequality are not com-parable(50). And so SWINESHEAD follows him and similarly

(49) For a historyof the law of movementin terms of the motivatingforceand resistanceusually associated with the name of ARISTOTLE, consult CLAGETT,GiovanniMarliani, etc. Chapter 6. However, this latter treatment eaves outthe influence fPHILOPONUS' criticism of ARISTOTLE'S formulation, hichcriticisminfluenced the Middle Ages through the work of AVEMPACE as reported by

AVERROES. The traditionalAristoteleanformulation f a dynamicexpressionofmovement in terms of the force and resistance can be represented n modernsymbolization:

V2 P2 R1 where V2 and V1 are velocities,P1and P. are motivepowers,

V1 P1 R2 and R1 and R2 are resistances.Or this formulizationmay be shortenedto

V ocP/R.Now BRADWARDINE and his successors altered this formof the law because ofan apparent inconsistencetherein,namely that if the forceequals the resistance

the formulagives a positivevalue for the velocity,even when experienceshowsthat if force and resistanceare equal no movementarises. Thus BRADWARDINEin his Tractatusproportionummotuum tated categorically hatno velocity couldarise unless P were greaterthan R, i.e., unless the ratio were a proportion ofgreater inequality. He then altered the basic Aristotelean aw which we canratherartificiallyepresent n modernterminologys follows:

P2 _ P n whereP

> i and n = V2

Or rewriting hisformula

V2 P1V = loga (P2/R2),wherea = > I

Consult THOMAS BRADWARDINE, Proportionestc., Venice, 1505, ff. 4r-v; ParisBN Fonds latin 6559, f. 54v.

(5o) He says that they are not comparables because in his formulization,fP,/R, is a proportionof equality,then any otherproportionP21R2which shouldbe greater than the originalproportionmust be equal to one, for no matterhowmany timesP,/R, is taken (i.e., regardlessof what integern is) P2/R2will equal







158 M. CLAGETT

distinguishes ntension nd remission, as not being comparable.The second analogyused by SWINESHEADo clarify he " ratio-

nal" (we might say directional)differencewhich distinguishesintension and remissionand proportionsof greaterand lesserinequality s a kind ofvectoranalogy. When there s a distancerepresentedby line ab, the magnitudeof the line is the sameregardlessof whetherwe proceed froma to b or fromb to a,but the movementsare not the same because the directionsofmovementare different,.e., the spaces from a to b and fromb to a are the same accordingto thing,but are differentationally

since we are consideringthem fromdifferent irectionalpointsof view. This analogy is quite apt, for the Calculator told inthe introductoryines of his treatisethat in one way of speakingintension nd remissionare movements.

So much for brief ntroductiono thesignificant assagewherehe makes the analogy. Here follows the main parts of thepassage (I):

to one when P1/R1= i. Or ifPj/Rj is less than one, thenP2/R2must alwaysbe less thanone, so long as n is an integer. Now we have assumed formovementthat P2/R2 must be greaterthan one, hence when we are implying geometriccomparisons or takingn as a series of integers,we can never have PI/RI equalto or less than one, or to put it anotherway,we can never employ proportionsof equality and lesser inequality. So in this special " geometric way theselatterproportions re not " comparable" to proportionsof greater nequality.

(5 i) A, f. 3v: Ideo pro illo argumento dicitur, posito quod a sit intensum,

quod sua intensionon est maior nec brevior sua remissione, neque etiam sibiequalis accipiendo remissionemprivativerespectuintensionis...ymo sicut diciturde proportione maioris inequalitatis et brevioris inequalitatis que proportionesrealiternon differunt isitantum ecundumrationem icutvia ab a ad b I) econtraa b ad a (2), eadem est secundum rem, differunt amen secundum rationem.Consimiliterhabitudomaiorisad minus et minoris d maius sunteedemsecundemrem, sola ratione differentes,t habitudo maiorisad minus est proportiomaiorisinequalitatis et habitudo minoris ad minus est proportio minoris inequalitatis.Ideo ille proportiones ecundum rationem tantum differunt t proportiomaiorisinequalitatis non est maior neque brevior proportione minoris inequalitatis,

distinguendo proportionemmaioris inequalitatis contra proportionembreviorisinequalitatis, ut venerabilis Magister Thomas Braduardini in suo tractatudeproportionibus liquide declaravit... Et consimiliter sicut dictum est omninode proportionemaioris nequalitatis, ic dicendum est de intensioneet remissionedistinguendo intensionemcontra remissionem accipiendo scilicet remissionemrespectu intensionisprivative... et sic dicendum est quod nullius gradusintensioremissione eiusdem graclus est maior neque brevior neque ei equalis... Et sicsalvunturomnia argumenta que sunt facta contra illam positionem, quia omniafundantursuper hoc, quod illud penes quid attenditur ntensio et illud penes








The answer to the argumentof the second position is as follows: When it hasbeen posited that a is intense,it is remarked that its intension s neithermorenor less than its remission,nor even is it equal to it when we accept remission

privativelywithrespectto intension...We

speakin the same

wayofa proportionof greater nequalityand one of lesser inequality. These proportionsreallydonot differ xcept according to reason, (i.e., the manner in which we considerthem), ust as the way (via) froma to b and the converseway fromb to a arereally the same according to thing (rem, magnitude), but different ccordingto reason (i.e., direction). Similarly,the ratio of a greaterto a lesser quantityand that of a lesser to a greaterare the same accordingto thing i.e., numbersare the same), but only differ ccordingto reason (i.e., theway we consider thenumbers), and the ratio of the greaterto the lesser is a proportionof greaterinequality while the ratio of the lesser to the greateris a proportion of lesser

inequality. These proportionsdiffer ccording to reason and a proportion ofgreater inequality is no more nor less than a proportionof lesser inequality,distinguishing proportionof greaterinequalityfrom one of lesser inequalityas the venerableMaster Thomas Bradwardinehad made completelyclear in hisTractatusdeproportionibus...And similarlyust as it has been statedeverywhereconcerningthe proportionof greater and lesser) inequality so it ought to beremarked concerning ntension and remission, distinguishing ntension fromremission and acceptingremissionprivativelywith respect to intension... Andso it oughtto be said that the intensionof no degree is more than, less than,orequal to, the remission of the same degree.

In concludinghis discussion ofthe third upplementary ositionhe reiterateshat all of thearguments sed against t can be solvedby considering he " rational" difference f intension nd remis-sion when we thinkof remission s a privative f intension.

The tractate s completedbytheadvancement fthree doubts"of statements hat appear to be fundamental n the position of

measuring ntension nd remission dopted by theCalculator 52):

(i) Whether uniform acquisition of intention follows from uniform loss ofremission.

(2) Whether remission s increased equally proportionallynd with equal velocityas intension s decreased.

quid attenditurremissio secundum equalitatem et inequalitatemcomparantur.

Quod tamen (3) remissio penes illud attendituruniversaliter4) est negandum.Variants: (i) A has a ad d et enconverso;2) A omitsab b ad a; (3) ES adds ut;(4) ES has uniformiter.

(52) A, f. 4r : His ergocompletis n illa materiarestatulterius sse dubitandum:primonumquid ex uniformi eperditione ntensionis equituruniformis cquisitioremissionis. Secundo numquid eque proportionalitert eque velocitermaioraturremissio sicut intensiominoratur. Tertio numquid si a non gradu remissioniscontinue eque velociter incipiant aliqua duo acquirere de remissione continuemanebunteque remissa.







i6o M. CLAGETT

(3) Whether two thingswhich begin from zero degree of remissionto acquireremissionequally fast continueto remainequally remiss.

He advances arguments gainst all three of these statements,but the negative argumentsare for the most part based on amisunderstandingf the use ofproportions r on impossiblecases.And so he showsthat hedoubtsare not ustified ndtheaffirmativestatementshold, except in that part of the second one whichdeclares that remission s increasedwith the same velocitywithwhich intension s decreased. He admits that intensionis de-creased and remission ncreasedequally proportionally,.g., whenintension s twiceas less remiss, remission s twiceas great,etc.But since intensionand remissionare not comparable becausethey are rationallydifferentccordingto direction,so then noactual movementof intension positive or negative), i.e., of in-creasing or decreasing intension, can be compared with anypositiveor negativemovementof remission. Hence, we are notjustified in saying that they move with equal velocity simply

because one decreasesproportionallys the other ncreases. Thiscriticismcan be pointed up by a specificpassage fromamongthe concluding paragraphsof the treatise 53):

Therefore,it is said, as before,that if intensionand remission are not mutuallycomparable according to equality and inequality,neither should the intensiveand remissivemovementsbe compared. For if the movements re comparable,then the thingswhich have been acquired by the movements are comparable,and so it is denied that remission is increased equally swiftly s intention isdecreased. However, it is concluded thattheytakeplace equally proportionally,etc.

SWINESHEAD has, then, in this treatisebeen able to examinethe various ways of measuring intensionand remission,firmlyconvincedas he was that qualities can be treatedquantitatively.He examined and showed the logical difficultiesnherent inmeasuringintension by reference o the decreasingdistance to

(53) A, f. 4v: Ideo diciturquod sicut intensio et remissio ut prediciturnonsunt adinvicem comparabiles secundum equalitatem et inequalitatem,sic nequemotus ad intensionemet remissionem debent adinvicem comparari. Si enimsunt motus comparabiles I), acquisita per illos motus sunt comparabilia. Etsic negatur quod eque velociter maioratur remissio sicut minoratur intensioconceditur tamen quod eque proportionaliter t cetera. Variant: (I) A hasequales, othercopies as in text.







RICHARD SWINESHEAD AND LATE MEDIEVAL PHYSICS i61

the maximumdegreeof intension and remissionby the distancefromthatmaximum). Equally untenablewas thepositionwhich,although tmeasured ntension

orrectly,till

ttempted o measureremissionby thedistancefrom hemaximum. He was led there-foreto advance and supportwhat he believed to be the correctposition, namely that intension of qualities must be measuredby the distancefrom the zero degree of intensionand remissionby the distanceto thatdegree. After dvancingthatpositionhefurther larified t by showingthat no degree at all was equallyintense as it was remiss, because intensionand remission are

fundamentally ifferentn thewaywe considerthemwithrespectto direction. It is truethatthey re so related hatgainin remissionis accompaniedby loss in intension, nd thatthis gain and simul-taneous loss take place equally proportionally. But the motionsof loss of intensionand gain of remissioncannot be said to takeplace with equal velocity because this would imply that thesemovements re comparableand thiswould not be true.

(University fWisconsin.) MARSHALL CLAGETT.

clagett intension-remission qualities

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