THE ORGANIZATION OF KNOWLEDGE Disciplines and Practices · THE ORGANIZATION OF KNOWLEDGE...

THE ORGANIZATION OF KNOWLEDGE

Disciplines and Practices

Joan Cadden

Carved on the west facade of Chartres cathedral is the most familiar anddurable representation of the learned disciplines in the Middle Ages theseven liberal arts Along with the allegorical figure of Grammar (who deploysa switch against two sleepy little boys) the six other branches of systematicknowledge appear accompanied by their founders or main authorities ndashGeometry with Euclid for example Sculpted in the mid-twelfth centurythese figures express at once the broad cultural acceptance of this particularpicture of how learning was organized and also some of the problems asso-ciated with taking such cultural consensus at face value On the one handthe cathedralrsquos school famous for its academic excellence since the earlytwelfth century continued to associate the seven arts with the curriculumfor beginning students ndash first the three verbal disciplines (grammar rhetoricand logic) and then the four mathematical disciplines (arithmetic geometryastronomy and music) (Figure ) On the other hand this template hadnever entirely fit the shape of scientific enterprises in the early Middle Agesand by the time the portal was carved changes within and beyond the schoolwere making the taxonomy obsolete The Chartres portal to the contrarynotwithstanding medieval disciplines were not written in stone Both thefluidity of disciplinary divisions over time and their flexibility at any givenmoment pose problems for constructing an overview of the borders andcompartments of medieval science By their very nature however theseuncertainties and variabilities do provide opportunities for understanding

Philippe Verdier ldquoLrsquoiconographie des arts liberaux dans lrsquoart du moyen age jusqursquoa la fin du quinziemesieclerdquo in Arts liberaux et philosophie au moyen age Actes du quatrieme Congres International dePhilosophie Medievale Universite de Montreal Montreal Canada 27 aoutndash2 septembre 1967 (MontrealInstitut drsquoEtudes Medievales Paris Librairie Philosophique J Vrin ) pp ndash

On this figure see John E Murdoch Album of Science Antiquity and the Middle Ages (New YorkCharles Scribnerrsquos Sons ) fig p

The author is grateful to the Max-Planck-Institut fur Wissenschaftsgeschichte Berlin to the membersof the ndash Abteilung II research group and to its Director Lorraine Daston for support andadvice

available at httpswwwcambridgeorgcoreterms httpsdoiorg101017CHO9780511974007011Downloaded from httpswwwcambridgeorgcore University of Warwick on 03 Feb 2017 at 135656 subject to the Cambridge Core terms of use

The Organization of Knowledge

Figure Allegorical representations of quadrivial arts with attributes Thequadrivium or four mathematical arts appears in a ninth-century copy of atract on arithmetic accompanied by identifying objects From left to right Musicholds a stringed instrument Arithmetic has a number cord in her right hand anddisplays a technique of finger-reckoning with her left Geometry holds a measur-ing rod or radius and looks down at a tablet inscribed with geometrical figuresand above Astronomyrsquos head are the stars Moon and Sun Because of differencesamong authorities as well as different readings or misreadings of texts or earlierimages such depictions did not follow set formulas The column in this illustrationmay reflect Martianus Capellarsquos description of geometry or may be an allusion tothe role of geometry in architecture the torches held by Astronomy have not beenexplained By the end of the Middle Ages new symbols were available Arithmeticsometimes carries an abacus board and Astronomy sometimes has an astrolabeor an armillary sphere By permission of Staatsbibliothek Bamberg MS HJIVfol v


Joan Cadden

how the various sciences were both delimited and related and the extent towhich natural science constituted a coherent endeavor in the Middle Ages

The purpose of this chapter is to investigate the boundaries and relationsamong medieval disciplines dealing with the natural world Since medievalintellectuals themselves sought to organize the knowledge they inherited orproduced about the natural world their own views serve as a point of depar-ture The systems of classification they articulated reflected at once a respectfor the programs of their ancient sources an attentiveness to the problems ofcoordinating various traditions and a concern for the ways in which learn-ing could be used In their prefaces or in the arrangement of their worksmedieval authors named defined and diagramed the relationships amongthe disciplines that embodied what we have come to regard as ldquosciencerdquoYet alongside the formal and explicit mapping of knowledge other linesof organization often informal and unspoken emerged Understanding thetaxonomy of the sciences therefore requires placing them in the context ofmedieval scientific practices that is in terms of the ways medieval peopleacquired transmitted and applied ideas about nature

Given the changes over time and the slippage between theory and prac-tice the result is not a clear and fixed map of the sciences but rather a setof perspectives from which to approach the question ldquoWhat was medievalsciencerdquo The first section of the chapter surveys general notions about dis-ciplines and their relations to one another as they were laid out before thetwelfth century For scholars of that period retaining and transmitting theoutlines of received wisdom was often a difficult task In such an environ-ment however scholars were free to try out various strategies and new usesfor old knowledge The second section sketches some of the changes thatrendered the older formulations obsolete Starting in the late eleventh cen-tury new social conditions for learning and the translation of Greek andArabic texts introduced not only new subject matter but also new methodsand even new goals for the sciences Finally the third section deals with theways in which these changes shaped and were shaped by new conditionsespecially the organization of learning within the university between thethirteenth and fifteenth centuries

THE ERA OF THE LIBERAL ARTS FIFTH

TO TWELFTH CENTURIES

The Latin terms ars disciplina and scientia all signified elements ofphilosophia and as such were manifestations of ordered thought They werefrequently associated with specific definitive texts and with characteristicrules by which they operated ndash that is both with what was to be knownand what was to be done When arts disciplines and sciences were distin-guished from each other they usually formed a hierarchy of abstraction or



of certainty For example the encyclopedist Bishop Isidore of Seville (candash) assigned the terms scientia and disciplina to what was known withcertainty They are about things that cannot be other than they are Artsin contrast ndash including tenets of natural philosophy such as the belief thatthe Sun is larger than the Earth ndash were the domain of mere opinion Suchdistinctions were not however either fixed or enforced An author indebtedto Isidore reported a variant Disciplines deal with what can be produced bythought alone whereas arts such as architecture are expressed in materialmedia And Isidore himself went on to conflate disciplines and arts sayingldquoThere are seven disciplines of the liberal artsrdquo

Medieval authors often employed one of these three terms which will beused interchangeably here to designate the principal divisions of ldquophiloso-phyrdquo as the recognized body of systematic learning was persistently calledBut just as the meanings and relations of ldquosciencesrdquo and ldquoartsrdquo varied sodid their membership and order ndash and indeed the principles upon whichthey were arranged Medicine for example might be located according toits subject matter (eg the maintenance of health) according to the typeof study it represented (eg a practical art) or according to texts in whichits substance was contained (eg Galenrsquos Art of Medicine) Furthermoreboth architecture and medicine were classified sometimes as mechanical orpractical and sometimes as liberal or theoretical arts Even the familiar namesof individual disciplines could be problematic ldquoastronomiardquo and ldquoastrologiardquowere sometimes synonymous and sometimes quite distinct

This tangle of terms suffices to illustrate some of the issues involved inconcepts about the constellation of knowledge The structures were notsimple the articulations of them were not formulaic The utterances of anauthority like Isidore or the representations of a source like the cathedral atChartres were only a part of what was involved but they convey some of thedifficulties of drawing a map of natural knowledge in the early Middle Ages

the liberal arts and their sisters

Medieval authors did draw such maps however Divisions of the scienceshave a long and intricate history borrowing from a variety of traditions andreflecting the dynamics of the intellectual scene Even before the influence

Hugh of Saint Victor The Didascalicon of Hugh of Saint Victor trans Jerome Taylor (New YorkColumbia University Press ) bk II chap p

Isidore of Seville Etymologiarum sive originum libri XX ed W M Lindsay vols (ScriptorumClassicorum Bibliotheca Oxoniensis) (Oxford Oxford University Press )vol bk II chap sec bk I chaps and see also bk III chap p All translations are mine unless otherwisenoted See also Boethius De trinitate in The Theological Tracts The Consolation of Philosophy edE K Rand (Loeb Classical Library ) (Cambridge Mass Harvard University Press )pp ndash at Prologue p

The basic treatments of the medieval disciplines and their classification are Richard William HuntldquoThe Introduction to the lsquoArtesrsquo in the Twelfth Centuryrdquo in Studia mediaevalia in honorem admodum


Joan Cadden

of Arabic science and the wholesale introduction of Aristotlersquos natural workssome basic elements of what was to be a continuing medieval conversationabout disciplines were already present The most important of these were() the seven liberal arts and sometimes their stepsisters the mechanical arts() the distinction between theoretical and practical sciences with its subdi-vision of the theoretical into divine mathematical and natural sciences and() the schema of physical logical and ethical knowledge

A highly influential work by Martianus Capella (fl ca ndash) enu-merated seven liberal arts and offered an introduction to (as well as a per-sonification of ) each including the four ldquomathematicalrdquo arts (later namedthe ldquoquadriviumrdquo) arithmetic geometry astronomy and music For earlymedieval authors these illuminated nature in various ways Mathematicalrelations represented the essence of the created world the subject of mathe-matical sciences was quantity separated in thought from the (natural) matterin which it actually inhered and the quadrivium had functions and usesrelated to material objects The particulars of arithmetic geometry andastronomy are treated in other chapters of this volume (see North Chap-ter Molland Chapter ) Against its persistent inclusion by medievalauthors historians of science have generally declined to treat music seri-ously in this context Its claim to a place among the mathematical sciencesrests on its central concern with intervals and thus with ratios In additionthrough such notions as harmony or proportion which applied not onlyto sounds but also to the macrocosm of the heavens and the microcosm ofthe human body the discipline of music sometimes incorporated significantnatural-philosophical as well as mathematical material

The other group of arts the ldquotriviumrdquo ndash grammar rhetoric and logic ndashbore virtually no formal relation to the pursuit of natural knowledge in itsmedieval or modern senses In practice however the verbal sciences wererelevant in three ways First medieval authors used literary skills representedby grammar and rhetoric to analyze the natural questions contained inauthoritative texts from the book of Genesis (in which the six days ofCreation became a traditional site for discussions of the natural world) tothe Timaeus of Plato In addition literary sources contained valuable wisdomVirgilrsquos Georgics contained agricultural information and one twelfth-centuryauthor referred to Hesiod as a ldquoteacher of natural sciencerdquo Finally althoughit took on its full prominence only later the discipline of logic became

reverendi patris Raymundi Josephi Martin Ordinis Praedicatorum S Theologiae Magistri LXXumnatalem diem agentis (Bruges De Tempel []) pp ndash James A Weisheipl ldquoClassification ofthe Sciences in Medieval Thoughtrdquo Mediaeval Studies () ndash and James A WeisheiplldquoThe Nature Scope and Classification of the Sciencesrdquo in Science in the Middle Ages ed DavidC Lindberg (Chicago History of Science and Medicine) (Chicago University of Chicago Press) pp ndash See also A J Minnis The Medieval Theory of Authorship nd ed (PhiladelphiaUniversity of Pennsylvania Press ) especially chap

Brian Stock Myth and Science in the Twelfth Century (Princeton NJ Princeton University Press) pp



relevant as a subject and as a method bearing on such questions as howmuch certainty a science could attain

Close to but always in the shadow of the liberal arts stood what cameto be called the mechanical arts also often numbered seven though theirexact membership varied The usual candidates included textiles armscommerce agriculture hunting medicine theater architecture and sports(Later enumerations included navigation alchemy and various forms ofdivination) They played two roles in the conceptualization of medieval dis-ciplines The first was negative In contrast to the liberal arts the mechanicalor ldquoadulteraterdquo arts engaged the body as well as the mind and their subjectwas ldquomerely human worksrdquo The superiority of the former was reinforced bythe social distinctions between those who work with their hands and thosewho do not ndash ldquothe populace and sons of men not freerdquo in contrast to ldquofreeand noble menrdquo More positively construed the mechanical arts supple-mented the liberal arts particularly with respect to their engagement withthe natural world This very involvement with objects which placed themoutside the domains of philosophy (they were regularly denied the statusof ldquodisciplinerdquo) made them potentially useful for expanding the systematicunderstanding of nature Some of the links between the mechanical andliberal arts manifested themselves in practices and instrumentation Thusa pair of compasses not only regularly accompanies the allegorical figure ofGeometry but also appears as an emblem of stonemasons

traditions of classification

Although the notion of the seven liberal arts was the most widely knownbasis for classifying knowledge including that concerned with the naturalworld it coexisted with other persistent schemata The existence of alter-natives invited scholars to choose combine or modify their elements inways that suited them The second major framework distinguished betweentheoretical (or speculative) sciences and practical (or active) sciences Thisdivision was most influentially articulated in the Latin works of Boethius(ca ndash) who depicted the Lady Philosophy with the Greek letterstheta (for ldquotheoryrdquo) and pi (for ldquopracticerdquo) on her garment The so-calledpractical sciences however concerned not the efforts of artisans but ratherthe responsibilities of the aristocracy ndash ethics household management (ldquoeco-nomicsrdquo) and politics Under the influence of Platonism and Christianityhowever the contemplative enjoyed a higher value than the active

Elspeth Whitney ldquoParadise Restored The Mechanical Arts from Antiquity through the ThirteenthCenturyrdquo Transactions of the American Philosophical Society no () ndash at chap

Hugh of Saint Victor Didascalicon bk I chap p Ibid bk II chap p

Boethius De consolatione philosophiae in Rand The Theological Tracts The Consolation of Philosophypp ndash at bk I prosa ll ndash p


Joan Cadden

The three constituents of ldquotheoryrdquo likewise formed a value hierarchy The-ology was concerned with a subject that existed independently of mattermathematics with the formal relations abstracted from their material sub-jects (eg dimensions abstracted from the land they measured) and physica(that is natural philosophy) with the properties of material objects Just asassociation with manual labor devalued the dignity of the mechanical artsvis-a-vis the liberal arts so association with matter placed mathematical andnatural sciences in descending order below theology Such a ranking whichaccording to Boethius corresponded to different ways of knowing suiteda Christian sensibility that emphasized the triumph of immaterial spirit overmaterial flesh It was not however static In the intellectual as in the spiritualrealm the mundane could be a stepping stone to higher levels thus lendingdignity to natural and mathematical sciences

The third and less influential arrangement of the disciplines derived froman ancient Stoic tradition and was passed on by Isidore of Seville It dis-tinguished ethics (that is the active sciences of the second scheme) physica(including the quadrivium) and logic (including the trivium) WhereasBoethius had separated the mathematical disciplines from natural philos-ophy here mathematics is part of it Indeed this arrangement sometimesalso included in the category of physica the more practical arts of astrol-ogy mechanics (meaning certain kinds of craft production) and medicine

Although it too found expression within monastic schools this taxonomywas less hierarchical than the previous one and less closely associated witha program of spiritual ascent In these respects it placed a higher and moreindependent value on at least some natural and verbal sciences

Throughout the early Middle Ages and beyond tensions among the vari-ous schemata along with the variety of traditions that nourished them gaverise to a fluid and eclectic outlook on the divisions and relations of scientificdisciplines The work of individual scholars often represented compromisesamong the various options For example the abbess Herrad of Landsberg(ca ndash) represented Philosophy as a queen encircled by figures of theseven liberal arts wearing a crown with figures of ethics logic and physics

Such reworkings have contributed to the perception that ldquonobody knewwhat to make of lsquophilosophyrsquo or lsquosciencersquordquo and scholars have suggested

Boethius De trinitate chap p Weisheipl ldquoClassification of the Sciences in Medieval Thoughtrdquo pp ndash Manuel C Dıaz y Dıaz ldquoLes arts liberaux drsquoapres les ecrivains espagnols et insulaires aux VIIe et

VIIIe sieclesrdquo in Arts liberaux et philosophie au moyen age pp ndash Murdoch Album of Sciencefig p Hrabanus Maurus De universo in Opera Omnia ed Martin Mabillon (PatrologiaLatina ) (Paris J-P Migne ) vol cols ndash at bk II chap col

Herrad of Hohenbourg Hortus deliciarum reconstructed with commentary by Rosalie GreenMichael Evans Christine Bischoff and Michael Curschmann with T Julian Brown and Kenneth(Levy Studies of the Warburg Institute ) vols (London The Warburg Institute Leiden Brill) vol pl The manuscript was destroyed in the Franco-Prussian War of On thisfigure see Murdoch Album of Science fig p



that physica remained a virtually empty category until the assimilation ofAristotlersquos natural works starting in the twelfth century In one sense thisis true in that unlike the liberal arts each of which was regularly linked toa basic text (Porphyry on logic Boethius on music and so forth) naturalphilosophy had no standard introductory authority But this perspectiveignores not only the extent to which other kinds of texts ndash Genesis andPlatorsquos Timaeus ndash provided textual grist for the natural-philosophical millbut also the extent to which subject matter was imported from a varietyof other categories Latin authors not only arranged and rearranged butalso added to the list of disciplines a process that further illustrates themalleable and living nature of medieval classifications In the ninth centuryan encyclopedic work by Hrabanus Maurus (ca ndash) full of informationabout natural philosophy included under the heading of physica not onlyarithmetic geometry astronomy and music but also such ldquopracticalrdquo orldquomechanicalrdquo arts as astrology medicine and mechanics This realignmentreflects a process by which information and ideas migrated across putativeboundaries

Especially in an environment in which authoritative texts were scarcescholars often appealed to learning in one domain to illuminate anotherMedicine in particular was a resource for those seeking to explore the prin-ciples of nature Isidore of Seville had likened medicine to philosophy itselfbecause it drew upon all of the liberal arts In the course of the earlyMiddle Ages standard medical texts mentioned the constituents of both thebody and the environment materia medica spoke of plants animals andstones and tracts on obstetrics touched on principles of reproduction as wellas practical advice The intellectual cross-fertilization suggested by the per-mutations of classification is confirmed by material evidence For examplebook owners bound medical mathematical and natural-philosophical textstogether in the same manuscript books

The absence of specialization enhanced these processes The VenerableBede (ndash) wrote on geographical subjects in addition to mathematicaldisciplines Practitioners of medicine might be socially distinguishable butits content was accessible to others After giving a stranger some advice on hishealth for example Gerbert of Aurillac (ndash) offered this disclaimerldquoDo not ask me to discuss what is the province of physicians especiallybecause I have always avoided the practice of medicine even though I have

Weisheipl ldquoNature Scope and Classification of the Sciencesrdquo p See also Weisheipl ldquoClas-sification of the Sciences in Medieval Thoughtrdquo p and Whitney Paradise Restored p andn

Hrabanus Maurus De universo bk V chap col See Dıaz y Dıaz ldquoLes arts liberaux drsquoapresles ecrivains espagnols et insulairesrdquo p nn and Murdoch Album of Science fig p Whitney Paradise Restored

Isidore of Seville Etymologie bk IV chap cf Bruce S Eastwood ldquoThe Place of Medicine in aHierarchy of Natural Knowledge The Illustration in Lyon Palais des Arts ms f r from theEleventh Centuryrdquo Sudhoffs Archiv no () ndash


Joan Cadden

striven for a knowledge of itrdquo Taken together this diverse body of evidencebears witness to the gradual formation of a loosely associated body of knowl-edge about the constituents causes and arrangements of the natural worldrather than the scholarly void that has been suggested

cultural functions of disciplinary ideals

Both the attempts to define and arrange specific disciplines and the condi-tions that moved or eroded boundaries manifested themselves in the varioususes to which medieval authors put the sciences During the early MiddleAges even the reiteration of fixed names and definitions could serve a varietyof cultural religious and political functions For example in her drama aboutthe conversion of a prostitute the abbess Hrotswitha of Gandersheim (candash) has the saintly Paphnutius name the quadrivial arts and define thediscipline of music as he explains to his students the harmony of the elementsin the human body Both for Hrotswitha whose own familiarity with theliberal arts was extensive and for her protagonist the preservation of learnedtraditions was a significant project in itself Similarly something as simple asa shared terminology facilitated more complex scholarly exchanges as whenGerbert the future Pope Sylvester II sought help from correspondents acrossEurope in acquiring old and new works on astrologia

Such cultural reproduction played a role in social and political develop-ments such as the construction of the Carolingian Empire and the evolutionof clerical power The prominence of the seven arts in the early Middle Agesis as much a product of a political agenda as it is a reflection of the intellec-tual projects and practices of the time Charlemagnersquos biographer Einhardemphasized that he had his sons and daughters educated in the liberal arts (ofwhich the rulerrsquos own favorite was astronomy) A classicizing curriculumlike a classicizing biography suited Carolingian claims to be successors tothe Roman Caesars and protectors of the Roman Church Alcuin of York(ca ndash) master of Charlemagnersquos palace school and Bedersquos intellec-tual heir was thus advancing a broad cultural and political program aswell as following his own scholarly trajectory when he gave the subjectsof the quadrivium a respectable (though not a prominent) place in thecurriculum

Gerbert of Aurillac The Letters of Gerbert with His Papal Privileges as Sylvester II trans HarrietPratt Lattin (Records of Civilization Sources and Studies) (New York Columbia University Press) no p cf no pp ndash

Hrotswitha von Gandersheim ldquoConversio Thaidis meretricisrdquo [or ldquoPafnutiusrdquo] in Opera ed HHomeyer (Paderborn Ferdinand Schoningh ) pp ndash scene lines ndash

Gerbert of Aurillac Letters of Gerbert no pp ndash cf no pp ndash Einhard Vita Karoli Magni The Life of Charlemagne ed and trans Evelyn Scherabon Firchow and

Edwin H Zeydel (Miami University of Miami Press ) chap pp ndash and chap ppndash



Hrotswitha of Gandersheim illustrates how command of the terminol-ogy and substance of scientific disciplines conveyed and even constitutedclerical superiority over the laity In her allegorical Latin drama on the mar-tyrdom of virgins named Faith Hope and Charity one of their persecutorsinquires about the girlsrsquo ages Their mother Wisdom asks ldquoDoes it pleaseyou my daughters that I should exhaust this fool with an arithmetic dis-putationrdquo and she proceeds to overwhelm him with a long and learnedexposition on numbers derived from Boethius Although the classical dis-ciplines are not as powerful as the Christian virtues the allegorical figureWisdom and the abbess Hrotswitha wield the two sets of weapons in closecoordination appropriating and thereby according dignity to the arts As thecases of Hrotswitha and Alcuin (a Benedictine monk) suggest the namingand arrangement of the disciplines belonged first and foremost in the earlyMiddle Ages to monastic environments that played a central role in thetransmission and validation of the disciplines

beyond disciplinary ideals

The political and cultural uses of the scientific disciplines depended in part ontheir clarity stability and links with recognized authority To that extent theywere conservative ndash in tension with the dynamics by which the definitionsand materials of individual disciplines and natural knowledge more generallywere shifting and expanding Gerbert expressed an awareness of preciselythis problem as he sought to enhance the texts and practices available toa student of arithmetic by laying out rules for the use of an abacus ndash atool of practical calculation ldquoDo not let any half-educated philosopherthink that [these rules] are contrary to any of the arts or to philosophyrdquo

The ldquohalf-educatedrdquo purists did not prevail An eleventh-century tract ongeometry incorporated not only passages from Euclid but also discussions ofthe abacus land measurement map making and land tenure

The practices discussed so far even those relating to calculation and cartog-raphy were textual in nature They involved the transmission and elaborationof written traditions whether associated with an ideal curriculum or withmore immediate and mundane matters As Gerbertrsquos apparently contestedinterest in the abacus suggests we have evidence of nontextual practicesinscribed in sources ranging from the geometrical artifacts of stonemasonsto records of the heuristic methods used in schools A monastic teacher ofthe twelfth century took his pupils out in front of the church in the middleof the night extending his arm and using his fingers to show them how to

Hrotsvitha von Gandersheim ldquoPassio sanctarum virginum Fidei Spei et Karitatisrdquo [or] ldquoSapientiardquo in Opera pp ndash scene secs ndash on pp ndash

Gerbert of Aurillac Letters of Gerbert no p see also pp ndash n John E Murdoch ldquoEuclidrdquo in Dictionary of Scientific Biography IV cols ndash at cols andashb


Joan Cadden

observe the course of the stars Gerbert himself illustrates the comfortablecoexistence of textual and manual practices He was famous for the swiftnesswith which he could do calculations using an abacus and he described to astudent how to construct and use a tube for astronomical observations

Determination of the date of Easter generated both active and contem-plative science It was a source of perennial concern (as well as sectariandiscord) and required the use of astronomical data and mathematical calcula-tions The problem being solved was essentially liturgical in that its purposewas to answer not a question about nature but rather one about ritual obser-vance seen through the lens of natural phenomena Thus the fixing of Easterbore a limited relationship to quadrivial and natural-philosophical disciplinesas formally defined Nevertheless just as artists depicted both stonemasonsand the allegorical figure of Geometry with a compass so copyists andlibrarians perceived some link when they copied and bound these texts oncalendrical calculation along with a variety of materials treating quadriv-ial and natural-philosophical subjects The abacus and the astrolabe mayoften have been instruments more of intellectual fascination than of practicalapplication but they were understood and even used to illustrate principlesand perform specific operations

CULTURAL CONFLUENCES AND TRANSFORMATIONS

OF THE ARTS TWELFTH CENTURY

The existence of a variety of tools adds complexity to our picture of early-medieval practices suggesting not only a manual but probably also an oraldimension to the pursuit and transmission of natural knowledge ndash fromeclipse prediction to surgery from numerology to divination Yet the mostpowerful scientific instrument in the Middle Ages remained the book Andwithin the book though illustrations and diagrams played a variety of impor-tant roles written words did the lionrsquos share of the work The period fromthe late eleventh to the early thirteenth centuries witnessed a proliferation oftext-based analytical and argumentative techniques These accompanied theformation of the universities which dominated the intellectual scene in thethirteenth and fourteenth centuries The enrichment of the substance meth-ods and taxonomies of the sciences during this transition in the Latin Westdepended on two closely related processes the selection translation adap-tation and incorporation of Greek and Arabic learning and the expansion

Philippe Delhaye ldquoLrsquoorganisation scolaire au XIIe sieclerdquo Traditio () ndash at p n Gerbert of Aurillac Letters of Gerbert letter of Richier quoted at no p n no pp ndash Stephen C McCluskey Astronomies and Cultures in Early Medieval Europe (Cambridge Cambridge

University Press ) especially chap Delhaye ldquoLrsquoorganisation scolaire au XIIe sieclerdquo



of literary and philosophical activity associated with the ldquoRenaissance of theTwelfth Centuryrdquo

converging traditions

European access to Greek Arabic and Hebrew learning was concentratedin southern Italy and Spain ndash two multicultural crossroads of Mediterraneansocieties Constantine the African (fl ndash) for example a convertedMuslim who became a monk at Monte Cassino both carried Arabic medicalbooks from North Africa to Italy and rendered them intelligible to a Latinaudience Consolidating a huge body of learning from the Aegean WestAsia and North Africa Arabic works by philosophers and physicians espe-cially fostered the adoption of Aristotlersquos ideas and methods and providedinterpretations of the Aristotelian natural world With respect to practicesLatins learned about specific instruments such as the astrolabe and the zeroand an array of ways to treat and order texts ndash from structures for medicalformularies to modes of philosophical commentary With respect to the orga-nization of knowledge Europeans confronted a number of serious challengesthat opened new areas of inquiry and revivified old ones Arabic authors notonly proposed their own versions of how disciplines were constituted andarranged but also made massive highly developed substantive contributionsto subjects that had commanded little or no place in older Latin systemsAreas such as optics or alchemy hardly discussed in early Latin schematafor dividing the sciences became impossible to ignore Natural science hadbecome more important while the substance of its diverse parts had becomericher and even harder to map

In ways that varied with local conditions many twelfth-century scholarsnot only welcomed but actively pursued the lush profusion of possibilitiescontained in newly available texts In some areas of Southern Europe forexample medicine was the intellectual seed around which natural questionscrystallized At Salerno and Monte Cassino in the late eleventh centuryand later in northern Italy and southern France it reshaped Latin inquiriesinto the natural world First medicine as received from Greek and Arabicsources offered explicit models for the relation between theory and practicein the arts Second as medical writers sought to elaborate and strengthen thetheoretical foundations of their knowledge they directly addressed the formand content of natural philosophy In doing so authors like Constantinethe African not only shaped medicine but also conveyed to natural philos-ophy a flood of material that was to be put to many uses The theory ofthe four elements for example which finds no specific place in the oldertaxonomies of natural knowledge occupied a pivotal position between the

Paul Oskar Kristeller ldquoThe School of Salerno Its Development and Its Contribution to the Historyof Learningrdquo Bulletin of the History of Medicine () ndash


Joan Cadden

constitution of the world in general and the physiological principles of med-ical science Along with the body of knowledge textual and pedagogicalpractices developed Prominent among them were loosely organized seriesof short queries that came to be known as ldquoSalernitan Questionsrdquo Thesebear the marks of a method in which rote learning was coupled with medicalapprenticeship As they were disseminated the questions acquired writtenanswers which in turn became more elaborate not only incorporating morenatural-philosophical material but also making room for the seeds of debateFor example one such text summarizes Hrabanus Maurusrsquos explanation forthe deadly look of the basilisk then states that it is not the creaturersquos lookbut rather its ability to poison the air that makes it dangerous

By the late twelfth century such Salernitan Questions flourished in thevery different cultural climate of the Ile-de-France Normandy and Englandwhere natural philosophy had previously drawn much of its content and itsmethods from literary studies Indeed northern learning about the naturalworld owed more to the practices associated with the trivium especiallygrammar and rhetoric than to those associated with the quadrivium Inparticular the glosses produced at Chartres and elsewhere not only on Platorsquoscosmogonical myth the Timaeus but also on the works of Macrobius (flearly fifth century) and Martianus Capella brought to the intellectual stagesuch powerful concepts as prime matter and the four elements Scholarsin this environment applied a variety of textual techniques to topics suchas the emergence and differentiation of the cosmos William of Conches(ca ndash) for example after writing a formally conventional gloss onthe Timaeus produced a work that combined aspects of Platorsquos account ofnature with Salernitan material Whereas some of Williamrsquos contemporariesmustered the quadrivial and natural-philosophical material to serve literarypurposes he struggled to define give shape to and legitimize the disciplineof natural philosophy (physica) as ldquothe true understanding of what existsand is seen and of what exists and is not seenrdquo And whereas Williamdrew upon the Northern academic culture of grammar and rhetoric othersmost notably Peter Abelard (ndash) advanced the third member of thetrivium logic

The variety of academic practices (question-and-answer and textual expli-cation mythological poetry and practical prose) and the diversity of interests(medical and cosmological natural-philosophical and metaphysical) formed

Brian Lawn The Salernitan Questions An Introduction to the History of Medieval and RenaissanceProblem Literature (Oxford Clarendon Press )

Brian Lawn The Prose Salernitan Questions Edited from a Bodleian Manuscript (Auct F310) AnAnonymous Collection Dealing with Science and Medicine Written by an Englishman c 1200 with anAppendix of Ten Related Collections ed Brian Lawn (Auctores Britannici Medii Aevi ) (Londonfor the British Academy by Oxford University Press ) B p and P p

William of Conches Philosophia ed Gregor Maurach (Studia ) (Pretoria University of SouthAfrica ) bk I chap sec p See Stock Myth and Science in the Twelfth Centuryespecially pp ndash and Helen Rodnite Lemay ldquoGuillaume de Conchesrsquo Division of Philosophyin the Accessus ad Macrobiumrdquo Medievalia () ndash



one axis of the twelfth-century legacy the wholesale importation of textsformed the other These changes in turn produced new challenges andopportunities for European intellectuals seeking to order knowledge andorganize education

a new canon of the arts

One of the earliest and most influential Latin treatises to reflect the chang-ing intellectual climate was On the Division of Philosophy by DominicusGundissalinus Active in Toledo (Spain) in the late twelfth century hehad participated in the translation efforts that brought so much previouslyunavailable scholarship into the West His classification of the sciences reflectslasting reorientations in Western thinking about the scientific disciplines() direct indebtedness to Arabic ideas about the arrangement of system-atic knowledge () adjustment to the introduction of massive new materialand even new sciences and () involvement of classification in fundamentalquestions about the order of nature and the path to secure knowledge

Gundissalinusrsquos work drew heavily upon a treatise of al-Farabı (ca ndash) which he had translated into Latin not only with respect to theenumeration of specific branches of learning but also with respect to thenature and order of the world that natural science sought to describe Severalof Gundissalinusrsquos moves were far from revolutionary as the improvisationsof the early Middle Ages attest He expanded mathematics beyond thetraditional quadrivium (arithmetic music geometry and astronomy) toinclude the science of weights (statics) and the science of engines (ieusing natural bodies and mathematical principles to some end) Theselast two were among the areas virtually unexplored by earlier Latin authorsand amply developed within the Arabic tradition In addition though ona much more modest scale he offered subdivisions of physica or as hecalled it ldquonatural sciencerdquo medicine omens necromancy magical imagesagriculture navigation optics (ldquomirrorsrdquo) and alchemy The strong presenceof sciences of divination and control is an indication of the powerful influenceof new intellectual appetites and materials

Beginnings of deeper structural changes also appear in Gundissalinusamong them the organization of the sciences around Aristotelian texts Hiseight-part division of natural philosophy bypasses his own list of subdisci-plines just mentioned and sets up a sequence of subjects ranging from thestudy of bodies in general through the more particular properties of min-erals plants and animals He names a text newly available in Latin as anelement in his characterization of each subdivision of science Although

Dominicus Gundissalinus De divisione philosophiae ed Ludwig Baur Beitrage zur Geschichte derPhilosophie des Mittelalters fasc ndash (Munster Aschendorff ) pp ndash

Ibid pp ndash


Joan Cadden

the specific texts did not all remain the same this was the way in whichnatural philosophy came to be structured in European universities

In some respects the way Gundissalinus presents the relations among thesciences reflects ideas about pedagogical process ndash one must learn grammarbefore turning to more complicated subjects ndash but this sequence also mirrorshis ideas about the ranking of objects of knowledge and the ways in whichthey are known In Gundissalinusrsquos day long-familiar sources ranging fromPlato to Augustine and new works by Arabic authors like al-Farabı and IbnSına (Avicenna ndash) lent a strong Platonic color to Latin philosophyone aspect of which was the conceptualization of a hierarchy of substancesand thus of the academic subjects treating them Physics studies the generalprinciples of change without reference to any particular bodies cosmologystudies change of place in otherwise changeless bodies generation and cor-ruption studies the changes of bodies coming to be and passing away andthe lowest subjects are concerned with the specific properties and operationsof particular bodies in the elemental world This ladder of value resonateswith some of the older classifications such as Boethiusrsquos view that theology ismore exalted than mathematics and that mathematics is higher than naturalphilosophy At the same time it seems to undermine the status Gundissal-inus lent to medicine alchemy and other arts concerned with the materialand the particular rather than the formal and the general

Gundissalinusrsquos attempts to sort out the subjects relations and valuesof knowledge about nature like those of earlier medieval authors bespeakthe variety flexibility and mobility of the disciplines and reflect an activeintellectual scene Furthermore older textual practices from the examinationof etymologies to the preparation of compilations continued to play a rolein scientific learning By the early thirteenth century however much hadchanged in the substance methods and conditions of the sciences The workdone by Isidore of Seville or Hrotswitha of Gandersheim simply to namedefine and iterate the fundamentals of the disciplines was no longer calledfor in a world in which thousands of students traveled from one Europeancity to another to hear masters lecture call out questions at disputationsand purchase and annotate books

THE ERA OF THE FACULTIES OF ARTS THIRTEENTH

TO FOURTEENTH CENTURIES

The works of Aristotle whose titles became metonymic for many disciplinesin the later Middle Ages had displaced (though not entirely erased) the

Weisheipl ldquoClassification of the Sciences in Medieval Thoughtrdquo pp ndash George Ovitt Jr The Restoration of Perfection Labor and Technology in Medieval Culture (New

Brunswick NJ Rutgers University Press ) pp ndash



liberal arts as critical landmarks on the map of learning More technicalspecialized and advanced they never took on the iconographic status oftheir predecessors ndash arithmetic with her number cord or astronomy with herquadrant Some general classificatory principles however persisted Writerscontinued to distinguish in principle between ldquosciencesrdquo and ldquoartsrdquo Accord-ing to Thomas Aquinas (ca ndash) the former (eg metaphysics andphysics) involve ldquoonly knowledgerdquo whereas the latter (eg logic whichconstructs syllogisms and astronomy which calculates planetary positions)ldquoinvolve not only knowledge but also a work that is directly a product ofreason itselfrdquo or in the case of nonliberal arts (eg medicine and alchemy)ldquoinvolve some bodily activityrdquo As in the earlier period however these dis-tinctions were not widely enforced in the language or institutions of the lateMiddle Ages thus students in the ldquoartsrdquo faculties of universities attendedlectures on both physics and logic

The world in which knowledge about nature was shaped and transmit-ted had also changed considerably by the early thirteenth century Thegrowth of towns for example had created demand for higher levels ofpractical knowledge in such areas as calculation and medicine and newforms of political administration had created demand for training notonly in law but also in astrology With support from civil or ecclesiasti-cal authorities (or both) universities took shape Through the formula-tion of curricula the support of advanced investigation and the positionof natural sciences within the larger institutional structure they providedboth opportunities and constraints for defining and pursuing scientificdisciplines

arts and methods

Questions about curriculum and pedagogy challenges associated with theprofusion of disciplines and debates contained in the works of newly avail-able authorities all contributed to a sense of urgency about the methods ofthe sciences Did each have its own rules of investigation forms of argu-mentation and degree of certainty Boethiusrsquos assertion that divine math-ematical and natural sciences were known differently no longer sufficedfor thirteenth- and fourteenth-century scholars interested in the distinctionbetween ldquonatural philosophyrdquo and ldquomathematicsrdquo The latter had once meantthe quadrivium but mathematical developments in the Islamic world notonly revolutionized old categories such as arithmetic but also introducednew ones In particular the distinction between mathematical and natural

Thomas Aquinas The Division and Methods of the Sciences Questions V and VI of His Commentaryon the De Trinitate of Boethius trans Armand Maurer rd ed (Toronto Pontifical Institute ofMediaeval Studies ) q art p

Boethius De trinitate chap p


Joan Cadden

knowledge unsettled even in the early Middle Ages (see Figure ) recededwith the incorporation of what came to be called ldquomiddle sciencesrdquo Opticsthe science of weights and astronomy (the last of these once housed in thequadrivium) dealt with specific properties of natural objects but employedmathematical representations and demonstrations Some of the issues raisedby these changes were formal Do the middle sciences actually constitute asubcategory of mathematics Others were epistemological What degree ofcertitude can astrology or medicine attain

Theoretical debates on the relation of subject to method took a numberof forms From one perspective the crux of the matter was what kind ofdemonstration each group of sciences could muster The conviction thatgeometry (as represented by Euclid) could produce airtight proofs and henceincontrovertible explanations enjoyed wide acceptance as did the comple-mentary view that natural philosophy insofar as it dealt with material objectsand was thus burdened by the attendant contingencies could not aspire togive a complete and certain account Disagreement nevertheless aboundedFor some scholars such as Albertus Magnus (ca ndash) the physicalworld in which form and matter were actually inseparable posed questionsto which mathematical methods could offer only partial solutions becausethey treated just a small number of properties abstracted from the actualnatural body For others such as Roger Bacon (ca ndashca ) naturalobjects could not be properly understood without mathematics

Such disagreements illustrate the extent to which classification of thesciences had become implicated in debates about the nature of scientificthought itself Yet when scholars were working on specific problems thetheoretical divisions often blurred Albertus Magnus for example was com-mitted in general to clarifying the independence of natural philosophy andmathematics When discussing the generation of a surface by the motion ofa line however he saw number not only as located in the mind of the math-ematician but also as inhering materially in numbered things ConverselyRoger Bacon articulated a strong theoretical program for the subordination

Murdoch Album of Science fig p Edward Grant ldquoNicole Oresme on Certitude in Science and Pseudo-Sciencerdquo in Nicolas Oresme

Tradition et innovation chez un intellectuel du XIV e siecle ed P Souffrin and A Ph Segonds (Scienceet Humanisme) (Paris Les Belles Lettres ) pp ndash and Michael R McVaugh ldquoThe Natureand Limits of Medical Certitude at Early Fourteenth-Century Montpellierrdquo Osiris nd ser

() ndash Weisheipl ldquoClassification of the Sciences in Medieval Thoughtrdquo pp ndash David C Lindberg

Roger Bacon and the Origins of Perspectiva in the Middle Ages A Critical Edition and EnglishTranslation of Baconrsquos ldquoPerspectivardquo (Oxford Clarendon Press ) Introduction pp xxxviindashxlivand David C Lindberg ldquoOn the Applicability of Mathematics to Nature Roger Bacon and HisPredecessorsrdquo British Journal for the History of Science () ndash

A G Molland ldquoMathematics in the Thought of Albertus Magnusrdquo in Albertus Magnus and theSciences Commemorative Essays 1980 ed James A Weisheipl (Texts and Studies ) (TorontoPontifical Institute of Mediaeval Studies ) pp ndash at pp ndash



Figure Combined divisions of philosophy This twelfth-century diagram illus-trates the mix-and-match character of medieval maps of scientific knowledge Thetop half divides philosophy into theory and practice with the former (on the left)constituted of (from left to right) natural mathematical and divine sciences Thecircle of the mathematical sciences contains the quadrivium from the liberal artsarithmetic music geometry and astronomy The bottom half starts with a three-part division (articulated by Augustine and attributed to Plato) natural moraland rational sciences (left to right) To the right of each of these almost-circles ascribe has carried on the tradition of associating disciplines with their foundersinserting the names of Thales of Miletus Socrates and Plato respectively Littlecircles containing the members of the quadrivium are here clustered around naturalscience or physica rather than belonging to a separate mathematical division as inthe top half On the right rational science is flanked by circles for dialectic (logic)and rhetoric Six of the seven liberal arts are thus represented with grammar themost elementary omitted Reproduced with the permission of the President andScholars of St Johnrsquos College Oxford MS fol r

of natural philosophy to mathematics but his accounts of specific phenom-ena sometimes contained elements that were not reducible to mathematicsThus his treatment of refraction while deeply mathematized depended nev-ertheless upon his understanding of the physical properties of light and upona metaphysical principle of uniformity

Lindberg Roger Bacon and the Origins of Perspectiva in the Middle Ages Introduction pp lndashlii


Joan Cadden

Debates about the relation between mathematics and natural philosophywere among the most heated in the late Middle Ages Their urgency wasenhanced by the tension between the qualitative cosmology of Aristotleand the quantitative astronomy of Ptolemy and by the mathematization ofmore and more fields from pharmacology to the study of local motion

Even within the Aristotelian tradition which had traditionally bypassed themiddle sciences classification involved ideas about the order and value ofthe entities studied and about the methods proper to each or common to alldisciplines

Indeed the Aristotelian perspective on what constituted appropriate andsecure demonstration was at the heart of one of the most striking disciplinaryrearrangements of the period the elevation of logic as the most importantpreparation for the study of philosophy as the source of critical methodsfor the pursuit of systematic knowledge and even as a subject for advancedresearch in its own right The privileged position of logic had earlier prece-dents but it acquired new meaning and force through the availability ofthe full body of Aristotlersquos logical writings The curricula of universities aswell as the declarations of natural philosophers and learned physicians tes-tify to this reconceptualization of the starting point for higher learning AsThomas Aquinas said citing first Aristotle and then Ibn Rushd (Averroesndash) ldquoWe must investigate the method of scientific thinking beforethe sciences themselves And before all sciences a person should learnlogic which teaches the method of all the sciences and the trivium concernslogicrdquo Logic precedes natural philosophy not because its subject matter ismore exalted but because it offers tools necessary for the pursuit of the othersciences

Collections of texts and university curricula embodied the methodologicalprinciple that logic comes before the sciences but at the same time theysubscribed to two other ways of ordering knowledge the principle that higherbeings have precedence (and power) over lower beings and the principle thatone should move from the general to the particular The placement ofAristotlersquos On the Heavens before his Generation and Corruption reflects thefirst of these for the celestial subject matter is more exalted than the earthlyBut the placement of On Vegetables before On Animals reflects the secondfor plants are not superior to animals Rather they embody the definingfundamentals of life ndash nutrition growth and reproduction

The priority of logic and the high value placed on what was general didnot preclude either a role for sense experience or attention to the partic-ulars of nature From the early Middle Ages onward there is evidence of

Murdoch Album of Science especially pts and Michael R McVaugh ldquoThe Developmentof Medieval Pharmaceutical Theoryrdquo in Aphorismi de gradibus vol of Arnaldi de Villanovaopera medica omnia ed Luis Garcıa-Ballester et al (Seminarium Historiae Medicae Granatensis)(Granada np ) pp ndash

Aquinas Division and Methods of the Sciences q art p



the purposeful examination of natural phenomena With a simple tube thatsheltered the eye from ambient light the curious could focus their attentionon a star or a planet with a complex astrolabe keyed to the local latitudethe trained observer could make measurements and calculations relating tothe same object In Latin and Hebrew compilations of herbal remedies terseexpressions of approval follow some recipes ndash but not all ndash with phrases likeldquoThis has been testedrdquo Occasionally observations are singular ndash reports ofspecific events or conditions at a particular place and time Some astronomi-cal data including those incorporated into ldquonativitiesrdquo or horoscopes are ofthis kind as are autopsy reports which proliferate in the late Middle AgesHowever first-person accounts were not necessarily based on singular expe-riences In late-medieval Italy compilations of clinical reports by prominentphysicians became a genre of scientific literature and it is likely that someof the cases recorded were encapsulations of medical theory or more generalclinical experience Nevertheless the existence of such works is evidencethat experience had a certain status in the profession as was the fact thatmedical students at the University of Paris received bedside training as wellas lectures Particular disciplines such as astronomy were more orientedthan others toward seeking and using data directly related to the questionsaddressed

Most often the observations invoked in natural philosophy are of a generalcharacter even if they may have been built on personal and perhaps hands-on experience Albertus Magnus for example in his explanation of thephenomena of growth makes use of the fact that lower creatures are able toregenerate more of their bodily parts than higher creatures Furthermoremost works with significant empirical content blended material from ancientauthorities contemporary informants and personal experience as was thecase with the book on hunting with birds compiled by the Holy RomanEmperor Frederick II (ndash) Occasionally however the context andwording of an appeal to experience strongly suggest a specific observationor series of observations deliberately and personally undertaken Such is thecase when Roger Bacon gives detailed instructions for constructing and usingan apparatus to demonstrate the phenomenon of double vision

Observation served a number of functions Reports of anomalous occur-rences especially those regarded as ldquomarvelsrdquo excited wonder and gave riseto reflections about what could and what could not be brought withinthe fold of natural sciences On the other end of the spectrum everyday

Albertus Magnus De generatione et corruptione in Opera omnia ed August Borgnet (Paris LouisVives ) vol bk I tract chap pp ndash

Roger Bacon ldquoPerspectivardquo in Lindberg Roger Bacon and the Origins of Perspectiva in the MiddleAges pt distinction chap pp ndash and lxindashlxii

Bert Hansen Nicole Oresme and the Wonders of Nature A Study of His ldquoDe mirabiliumrdquo (TorontoPontifical Institute of Mediaeval Studies ) and Lorraine Daston and Katharine Park Wondersand the Order of Nature 1150ndash1750 (New York Zone Books )


Joan Cadden

experiences ndash whether directly relevant or in the form of analogies ndash oftenserved heuristic and persuasive purposes Observations were frequently madeand called upon to confirm or illustrate preexisting knowledge This was thecase with the practice of human dissection when it first became integratedinto university curricula Although conservatively framed such practicessometimes slid from illustration to clarification to revision to critique asoccurred in the field of anatomy In addition experience often occupied aplace within the structure of an argument For example the size of a humanbody increases either because material is added to it or because its originalmaterial gets rarified but we see that a manrsquos flesh is denser not rarer thana boyrsquos therefore growth occurs by the addition of material Although thepractice was not common more specialized observations could be similarlyinvoked to confirm or rule out a theory or to choose among competingpremises To establish that refracted rays of light are involved in visionBacon offers the evidence of a thin straw held close to the face against adistant background The straw does not block our perception of the back-ground which it would if only direct rays were involved The invocationsof experience are too varied to constitute a single scientific method but theprofusion of observations and attentiveness to the particulars of nature attestto the seriousness with which scholars approached the phenomena that theirdisciplines undertook to record and explain

In spite of the diverse roles played by experience the differences amongthese Aristotelian hierarchies and the disagreements about the role of math-ematics late-medieval sciences achieved a certain coherence when it came toscholarly practices As in the earlier Middle Ages these were first and fore-most textual Now however the new bodies of knowledge the lessons fromGreek Arabic and Hebrew scholarship and especially the development ofthe universities contributed to the creation of far more varied and techni-cally sophisticated ways of dealing with the corpus of authoritative texts andgenerating a corpus of modern texts Some modes of university teachingand research such as the explication of an authorityrsquos literal meaning wereindebted to older habits of exegesis Others such as the public debate of dis-puted questions (often in a raucous environment) were unique to the newconditions Masters had to be able to take and defend positions on a varietyof topics philosophers on whether the Earth is always at rest in the middle

Albert of Saxony Questiones de generatione et corruptione in Questiones et decisiones physicales (ParisIodocus Badius and Conrardus Resch ) bk I question fol ra

Bacon ldquoPerspectivardquo bk III distinction chap pp and lxivndashlxv John E Murdoch ldquoFrom Social to Intellectual Factors An Aspect of the Unitary Character of

Late Medieval Learningrdquo in The Cultural Context of Medieval Learning Proceedings of the FirstInternational Colloquium on Philosophy Science and Theology in the Middle Ages September 1973 edJohn E Murdoch and Edith D Sylla (Boston Studies in the Philosophy of Science SyntheseLibrary ) (Dordrecht Reidel ) pp ndash and Jole Agrimi and Chiara Crisciani EdocereMedicos Medicina scolastica nei secoli XIIIndashXV (Istituto Italiano per gli Studi Filosofici HippocraticaCivitas ) (Naples Guerini )



of the heavens or whether it can be moved physicians on whether men orwomen experience greater pleasure in sexual intercourse and so forth In anintricate structure a master preparing responses had to present argumentsfor and against each proposition raise objections to the arguments andprovide responses to the objections as well as muster the relevant evidencefrom authoritative texts Many such practices were widespread deployednot only in a variety of disciplinary areas from mathematics to meteorol-ogy but also in all parts of Europe at institutions that differed in otherrespects

uses of the arts

University students encountered these patterns of scholarly inquiry in thefaculty of arts where all began their education with Aristotelian logic andnatural philosophy Although in the early Middle Ages scientific ideas andpractices had fulfilled a number of social functions from the calculation ofEaster to the enhancement of cultural prestige in the changing demographiceconomic and political scene of the late Middle Ages people with scientificknowledge became more common and more prominent Some went on toadvanced degrees in theology medicine or law (civil or canon) others movedmore quickly into opportunities available to this literate elite

As the new class of university-trained men pursued a variety of newlydeveloping careers not only in the professions but also in the managementof secular and ecclesiastical government the old distinctions between theoryand practice underwent radical revisions Boethius had distinguished the-ory (theology mathematics and natural philosophy) from practice (ethicseconomics and politics) encyclopedists had valued the liberal arts abovethe mechanical arts because the latter had involved the use of the handsNow texts converged with social conditions to produce a growing respectfor action in the world including the mechanical arts Under the influ-ence of Arabic traditions Westerners began to take seriously the idea thateach art had a theoretical and a practical part More important those sametraditions had been the source of significant bodies of ldquopracticalrdquo learningin such areas as mathematical calculation observational astronomy magicand medicine Signs of this shift appear in specific institutional changes Forexample what had originally distinguished the university-trained physicianfrom other medical practitioners was his mastery of classic Latin texts Bythe end of the Middle Ages however surgery ndash the most manual branchof medicine ndash had acquired a place within the university curriculum itselfManuscripts of astronomical tables abounded in the libraries of princes as

Pamela O Long ldquoPower Patronage and the Authorship of Ars From Mechanical Know-Howto Mechanical Knowledge in the Last Scribal Agerdquo Isis () ndash and Whitney ParadiseRestored pp ndash

Hunt ldquoIntroduction to the lsquoArtesrsquo in the Twelfth Centuryrdquo pp ndash


Joan Cadden

well as those of schools and enough horoscopes survive to indicate that theywere not there just for show A passing reference to ldquoalchemistsrsquo booksrdquo ina letter from Christine de Pisan (fl ndash) to a member of the Frenchcourt suggests that people were ready to put these texts to use ldquoSome readand understand them one way and others completely differently And onthis basis they open and prepare ovens and alembics and crucibles and theyblow hard for a little sublimation or congelationrdquo

The mushrooming of the middle sciences the enrichment of the applieddimensions of theoretical sciences the articulation of institutions dedicatedto the development and transmission of learning and the multiplicationof social functions for scientific knowledge all contributed to a situationin which the most advanced study in many fields was highly technicalThese changes too are reflected in late-medieval divisions of the sciencesAccording to a diagram in one fifteenth-century manuscript for examplemathematics has eleven distinct parts some of which are parts of partsof parts The intricacies of such divisions and subdivisions reflect a realsituation in which not only the specificity but also the sophistication ofadvanced scientific work is inscribed Few students or even masters in thefaculties of arts or medicine actually read Ptolemyrsquos Almagest Likewise inother fields works of comparable complexity (if not always of comparablestature) were accessible to only the most advanced scholars This situationgave rise to a degree of specialization and thus a hardening of disciplinarylines The commentators on Ibn Sınarsquos Canon of Medicine typically did notexpound theories of the rainbow Gerbert who in the tenth century hadaccess to a very modest collection of texts had busied himself producingtextbooks and instruments for teaching rhetoric astronomy and music andenjoyed a reputation for his astonishing calculational abilities By contrastAlbertus Magnus theldquoUniversal Doctorrdquo of the thirteenth century had avail-able a vastly larger library but produced little to suggest proficiency in themathematical sciences At the same time as Arabic arithmetic techniquesgrowing academic interest in mathematics and flourishing urban commerceall converged new systems of calculation joined if they did not entirelydisplace Gerbertrsquos counting method By the thirteenth century for examplescholars in Paris did what were recognized as Arabic ldquoalgorithmsrdquo dealing

Christine de Pisan ldquoA maistre Pierre Col Secretaire du roy nostre sirerdquo in Christine de PisanJean Gerson Jean de Montreuil Gontier Col and Pierre Col Le debat sur le Roman de la rose edEric Hicks (Bibliotheque du XVe Siecle ) (Paris Honore Champion ) no pp ndash atp

On this figure see Murdoch Album of Science fig p M J E Tummers ldquoThe Commentary of Albert on Euclidrsquos Elements of Geometryrdquo in Weisheipl

Albertus Magnus and the Sciences pp ndash and Molland ldquoMathematics in the Thought ofAlbertus Magnusrdquo Compare David C Lindberg ldquoRoger Bacon and the Origins of Perspectivain the Westrdquo in Mathematics and Its Applications to Science and Natural Philosophy in the MiddleAges Essays in Honor of Marshall Clagett ed Edward Grant and John E Murdoch (CambridgeCambridge University Press ) pp ndash at pp ndash



with remainders and carrying by writing and erasing digits in sand on atable whereas their predecessors had used tokens marked with numerals ona board laid out as an abacus

Although much of what medieval scholars did when they applied orenacted their knowledge is inaccessible to us some evidence points to livelyeconomic social and even mechanical activities The construction of clockscalled upon both mathematical knowledge and mechanical know-how

The horoscopes and other forms of astrological counsel offered for a feeby university mathematicians of fifteenth-century Vienna represented atonce expert calculations and useful products Similarly the consilia orcase histories written down by physicians constituted not only texts forinstruction but also representations (if not always transparent) of their careersas medical practitioners

the arts and the body of medieval science

Although late-medieval classification schemes were mainly concerned withthe internal structure of systematic learning ndash with the functions and rela-tions of its parts ndash they also served to delineate what constituted the bodyof legitimate knowledge as a whole Whether explicitly or implicitly thetaxonomies marked off what might (or had to) be excluded from considera-tion The same diagram that so intricately parsed mathematics also dividedastronomy into two parts the study of heavenly motions and the study oftheir effects The second of these is bluntly divided into ldquoprohibitedrdquo (withno further elaboration as to the subjects and texts implicated) and ldquonotprohibitedrdquo (see Figure )

Medieval authors did not always agree about which inquiries were licit butwherever the line was drawn some ways of knowing and dealing with naturewere left outside of a boundary that thus defined the proper domains ofnatural science in general Distinctions between permitted and prohibitedor proper and improper were not limited to astrology Medical treatisesfor example reflect controversies about what aspects of sexual experiencea physician ought properly to consider Much of the excluded material

Guy Beaujouan ldquoLrsquoenseignement de lrsquoarithmetique elementaire a lrsquouniversite de Paris aux XIIIe

et XIVe siecles De lrsquoabaque a lrsquoalgorismerdquo in Homenaje a Millas-Vallicrosa vols (BarcelonaConsejo Superior de Investigaciones Cientificas ) vol pp ndash at pp ndash

Richard of Wallingford Tractatus horlogii astronomici in Richard of Wallingford ed and trans JohnD North vols (Oxford Clarendon Press ) vol pp ndash vol pp ndash vol pp ndash

Michael H Shank ldquoAcademic Consulting in Fifteenth-Century Vienna The Case of Astrologyrdquoin Texts and Contexts in Ancient and Medieval Science Studies on the Occasion of John E MurdochrsquosSeventieth Birthday ed Edith Sylla and Michael McVaugh (Leiden Brill ) pp ndash

Jole Agrimi and Chiara Crisciani Les consilia medicaux trans Caroline Viola (Typologie des Sourcesdu Moyen Age Occidental ) (Turnhout Brepols )

Joan Cadden ldquoMedieval Scientific and Medical Views of Sexuality Questions of ProprietyrdquoMedievalia et Humanistica ns () ndash


Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -

Figure Division of the mathematical sciences fifteenth century This schemaof the parts of mathematics could only have been drawn in the late Middle Ageswhen texts for and branches of inquiry devoted to subjects such as optics andweights had become established The diagram indicates that certain unspecifiedareas of astrology are prohibited the licit portion includes horoscopes (ldquonativitiesrdquo)By permission of Basel Offentliche Bibliothek der Universitat MS FII fol r



was what opponents labeled ldquodivinationrdquo or ldquosorceryrdquo from the casting oflots to the manipulation of images to achieve specific results Many of theworks associated with these arts were (or were purported to be) from ArabicHebrew ldquoChaldeanrdquo or other exotic traditions making them both moreinteresting and more suspect Curricular statutes and learned arguments aswell as rhetorical attacks acted to contain the pursuit of such sciences butthey were by no means successfully suppressed or even marginalized Theirsurvival was due not only to the wealth of texts but also to their perceivedutility In the early thirteenth century the Holy Roman Emperor receiveda commentary on a work supposedly written by Aristotle for Alexander theGreat It included material on judging a personrsquos character from physicaltraits ndash physiognomy ldquothe science of which should really be kept secretbecause of its great effectiveness It contains secrets of the art of nature thatmeet the need of every astrologer [A]mong other things of which youshould be mindful is the science of good and evilrdquo An array of evidenceattests to diverse flourishing learned and occasionally highly technicalactivity in precisely the domains targeted such as geomancy and chiromancysuggesting the futility of medieval (and modern) attempts to exclude thesesubjects from the canon of medieval natural knowledge

In addition to such hotly contested lines of demarcation other signs pointto the ambiguous relationships of individual sciences to the central body ofscientific knowledge This situation was intensified by the newness of somesubjects and texts for scholars in the Latin West For example works onphysiognomy of which there was hardly a trace in the early Middle Ageswere sometimes enshrined with the Aristotelian natural corpus and adornedwith learned commentaries sometimes copied into manuscripts containingmedical or magical texts and sometimes reproduced in the company ofreligious and moral writings The boundaries of exclusion and inclusionwhether indefinite (as in the case of physiognomy) or contested (as in thecase of certain branches of astrology) thus manifested the same sorts offlexibility and fluidity as the internal lines dividing the constituent parts ofnatural knowledge from each other

CONCLUSION

Divisions and classifications (whether explicit or implicit) reflected embod-ied or activated but did not determine the ways in which knowledge aboutnature was received created shaped and transmitted Even in the earlierpart of the period alternative models and cheerful syncretism left authors

Michael Scot De secretis secretorum cited in Charles H Lohr ldquoMedieval Latin Aristotle Commen-taries Authors Johannes de Kanthi-Myngodusrdquo Traditio () ndash at p no

The greatest mass of evidence is contained in Lynn Thorndike A History of Magic and ExperimentalScience vols (New York Columbia University Press ndash)


Joan Cadden

much freedom to rearrange the components of the intellectual map to suittheir purposes The dyad of theory and practice and the triad of theologymathematics and natural philosophy both intersected with the seven lib-eral arts Later pluralism and outright conflict prevented the hegemony ofany particular system For example scholars categorized questions about themotions of the heavenly bodies differently depending upon whether theyarose from Aristotlersquos On the Heavens or texts on mathematical astronomyMore important at no time did the most favored taxonomies encompass allof the activities that medieval scholars themselves called ldquosciencesrdquo and thatthey associated with the objects and operations of the created world In theearly Middle Ages the theory of the four elements did not occupy a secureposition in the later Middle Ages the proper place of physiognomy wasunclear For these reasons not only the internal organization but also theexternal boundaries of natural knowledge were flexible and fluid contestedand contextual in the Middle Ages

Changing material institutional and intellectual conditions from urban-ization to the accessibility of Arabic science added a chronological dimensionto this variability After the twelfth century the number of areas of investi-gation that were candidates for the denominationldquosciencerdquo had multiplieddramatically as had the kinds of issues that denomination raised Not onlydid new subjects such as alchemy challenge the boundaries of the nat-ural and mathematical sciences and new texts such as the Optics of Ibnal-Haytham test the capacities of individuals and even curricula to reachthe most advanced levels in all fields but new questions concerning thefoundations of knowledge about the world such as the role of mathematicsdemanded increased attention to how sciences were conducted In this intel-lectual and social environment the stabilizing and conservative functions ofdividing and classifying the sciences characteristic of the early Middle Agesgave way to more dynamic functions such as creating institutional space forcompeting bodies of knowledge and providing a medium for debates aboutsubstances and methods

No matter how differently scholars construed and used the arrangementsof the various fields of natural knowledge before and after the changescentered on the twelfth century notions of disciplinary distinctions andorder played certain continuing roles throughout the Middle Ages Firstthey provided a vocabulary with which to express successive attempts toorganize not just concepts but also books curricula and activities Sec-ond they highlighted even as they circumscribed certain persistent dis-tinctions that precluded a simple static and unified science of nature ndashdivisions between mathematics and natural philosophy for example orbetween theory and practice Systems of classification thus brought orderto a diverse set of activities and helped to create a foundation and a mapfor a wider range of knowledge and practices At the same time because



of the many purposes they served because of the variety of traditions andoutlooks they encompassed and because they were neither complete nor con-sistent with each other their lacunas tensions and fissures constituted anaspect of the productive open-ended environment in which medieval sciencethrived



Figure Allegorical representations of quadrivial arts with attributes Thequadrivium or four mathematical arts appears in a ninth-century copy of atract on arithmetic accompanied by identifying objects From left to right Musicholds a stringed instrument Arithmetic has a number cord in her right hand anddisplays a technique of finger-reckoning with her left Geometry holds a measur-ing rod or radius and looks down at a tablet inscribed with geometrical figuresand above Astronomyrsquos head are the stars Moon and Sun Because of differencesamong authorities as well as different readings or misreadings of texts or earlierimages such depictions did not follow set formulas The column in this illustrationmay reflect Martianus Capellarsquos description of geometry or may be an allusion tothe role of geometry in architecture the torches held by Astronomy have not beenexplained By the end of the Middle Ages new symbols were available Arithmeticsometimes carries an abacus board and Astronomy sometimes has an astrolabeor an armillary sphere By permission of Staatsbibliothek Bamberg MS HJIVfol v


Joan Cadden


















Joan Cadden
















Joan Cadden


















Joan Cadden



















Joan Cadden
















Joan Cadden















Ibid pp ndash


Joan Cadden













arts and methods





Joan Cadden















Joan Cadden

















Joan Cadden









uses of the arts






Joan Cadden





















Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -






CONCLUSION





Joan Cadden








Joan Cadden


















Joan Cadden
















Joan Cadden


















Joan Cadden



















Joan Cadden
















Joan Cadden















Ibid pp ndash


Joan Cadden













arts and methods





Joan Cadden















Joan Cadden

















Joan Cadden









uses of the arts






Joan Cadden





















Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -






CONCLUSION





Joan Cadden


















Joan Cadden
















Joan Cadden


















Joan Cadden



















Joan Cadden
















Joan Cadden















Ibid pp ndash


Joan Cadden













arts and methods





Joan Cadden















Joan Cadden

















Joan Cadden









uses of the arts






Joan Cadden





















Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -






CONCLUSION





Joan Cadden

















Joan Cadden


















Joan Cadden



















Joan Cadden
















Joan Cadden















Ibid pp ndash


Joan Cadden













arts and methods





Joan Cadden















Joan Cadden

















Joan Cadden









uses of the arts






Joan Cadden





















Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -






CONCLUSION





Joan Cadden
















Joan Cadden



















Joan Cadden
















Joan Cadden















Ibid pp ndash


Joan Cadden













arts and methods





Joan Cadden















Joan Cadden

















Joan Cadden









uses of the arts






Joan Cadden





















Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -






CONCLUSION





Joan Cadden

















Joan Cadden
















Joan Cadden















Ibid pp ndash


Joan Cadden













arts and methods





Joan Cadden















Joan Cadden

















Joan Cadden









uses of the arts






Joan Cadden





















Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -






CONCLUSION





Joan Cadden















Joan Cadden















Ibid pp ndash


Joan Cadden













arts and methods





Joan Cadden















Joan Cadden

















Joan Cadden









uses of the arts






Joan Cadden





















Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -






CONCLUSION





Joan Cadden















Ibid pp ndash


Joan Cadden













arts and methods





Joan Cadden















Joan Cadden

















Joan Cadden









uses of the arts






Joan Cadden





















Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -






CONCLUSION





Joan Cadden








Joan Cadden













arts and methods





Joan Cadden















Joan Cadden

















Joan Cadden









uses of the arts






Joan Cadden





















Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -






CONCLUSION





Joan Cadden











arts and methods





Joan Cadden















Joan Cadden

















Joan Cadden









uses of the arts






Joan Cadden





















Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -






CONCLUSION





Joan Cadden








Joan Cadden















Joan Cadden

















Joan Cadden









uses of the arts






Joan Cadden





















Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -






CONCLUSION





Joan Cadden













Joan Cadden

















Joan Cadden









uses of the arts






Joan Cadden





















Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -






CONCLUSION





Joan Cadden
















Joan Cadden









uses of the arts






Joan Cadden





















Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -






CONCLUSION





Joan Cadden










uses of the arts






Joan Cadden





















Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -






CONCLUSION





Joan Cadden








Joan Cadden





















Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -






CONCLUSION





Joan Cadden






















Mathematical

Sciences

(a)

(b)

Discontinuous

quantity

Particular

= music

General

= arithmetic

General

= geometry

(Euclid)

Particular

Continuous

quantity

Weights ([ps-

Jordanus de

Nemore] De

ponderibus)

Optics

Astronomy

Motions

of

heavens

Practical with

astrolabe

Theoretical

Expository

([anonymous]

Theory of the

Planets and Al-

Farghani Book

of Distinctions)

Demonstrative

(Ptolemy

Almagest)

Expository

(Masharsquo allah)

Demonstrative

(Ptolemy The

Planisphere)

Effects of

motions of

heavens

Prohibited

Not prohibited

(Ptolemy

Tetrabiblos)

Revolutions

Nativities

Elections

Interrogations

- -

- - -






CONCLUSION





Joan Cadden











CONCLUSION





Joan Cadden








Joan Cadden








THE ORGANIZATION OF KNOWLEDGE Disciplines and Practices · THE ORGANIZATION OF KNOWLEDGE...

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Transcript of THE ORGANIZATION OF KNOWLEDGE Disciplines and Practices · THE ORGANIZATION OF KNOWLEDGE...