EDITOR’S NOTE: Evans’ article includes 28 illustrations which I have grouped together at the bottom of this HTML page. All 28 figures are reproduced in larger sizes (with an additional gloss by Evans) in the companion Gallery exhibit to this e-text. Captions given below for the 28 graphics include links to the related Gallery exhibit as well as the formal entry for each graphic in the she-philosopher.com Gallery catalog.

the geometry of
the mind

§1 Scientific diagrams and medieval thought

Figure 1 is a scientific diagram from a medieval handbook.[1] Figure 2, although formally similar, is nothing of the sort. The text below it describes it as a “representation of our Church”; in fact, it is a summary of Christian ethical and historical theories expressed in a geometrical idiom. The use of diagrammatic illustrations to expound theoretical ideas as well as scientific facts was characteristic of the Middle Ages.[2] Many of these compositions are fine designs, embellished with representational elements and elucidated by erudite inscriptions in verse and prose; but until recently, historians of art and literature alike have neglected them. Figure 2 has not been published before, although it has been associated with one of the best-known mid-thirteenth-century English ateliers; nor are the texts accompanying it indexed in any concordance of initia. This is understandable, as such images are beyond the normal canons of art, and the texts, out of context, are unremarkable. However, it will be argued here that not only do these inscribed geometrical figures merit study in their own right, as examples of medieval design, but also that they provide a valuable commentary on the way medieval thinkers approached some of the problems most important to them, furnishing an insight into thought-processes in the Middle Ages.

Medieval exegesis was particularly suited to, and to some extent influenced by, diagrammatic exposition; so too was medieval logic, because its most characteristic innovations were formalistic rather than epistemological. By the end of the Middle Ages, a thinker like Heimericus de Campo could express his entire philosophical system in a single figure,[3] though these late examples tend to be arcane rather than elucidatory. But during the period to be considered here (mainly the twelfth to fourteenth centuries) geometrical exposition was used in a way that was not only more concise, but also more explicit, than a prose account. It also achieved the status of an abstract art form. Before describing this achievement, and offering an interpretation of the image illustrated in figure 2, it will be appropriate to make a few observations about the inter-relation of art and geometry in the Middle Ages, and about some methodological peculiarities of medieval logic.

§ 2.1 Art and geometry

When the thirteenth-century architect Wilars de Honecort compiled an album of texts and pictures for use as a model-book, he described it as providing help in drawing according to the art of geometry.[4] It included figures based on schematic armatures comprised of circles, squares and triangles. This is an unequivocal indication of the importance of geometry in medieval design. While it has to be admitted that attempts by modern critics to subject the art and architecture of the Middle Ages to geometrical analysis are seldom convincing (points of reference are arbitrary , yet even so correspond only approximately to the superimposed linear grid)[5] it is nevertheless clear that much medieval art is governed by geometrical schemata. In twelfth-century painting, an artificial symmetry is imposed on planar representations of figures and objects; later pictures are less formalistic, but images are still deployed in regular dispositions and framed in geometrically-constructed surrounds. This can be regarded as a corollary to the medieval preoccupation with ordering the universe and imposing a system on every aspect of existence: four elements, five senses, six ages, seven virtues, plus seven vices, arts, planets, and an almost limitless number of other concepts.[6] Some of the most important iconographical themes were depicted in geometrical terms: the conventional representation of the cosmos is a series of concentric circles; that of God shows him in a circular aureole, or one made-up of circular elements; as creator, God wields a geometer’s compasses.[7] He could be defined geometrically as an infinite sphere, its centre everywhere, the circumference nowhere.[8]

§ 2.2 Geometry

Nevertheless, of geometry as understood today, the earlier Middle Ages had little knowledge. Until the translations of Euclid by Adelard of Bath (fl 1119–1142), medieval practical experience of the subject was limited to a curious mixture of fragments of the Elementa and surveying.[9] However, in theory at least, geometry always had a place of eminence in the educational curriculum as one of the seven Liberal Arts.[10] It was, therefore, not only a fundamental subject in medieval schools, but also one of the elements of learning itself, and an essential component of intellectual activity. Hugh of S Victor’s extraordinary statement, that it is “the fount of perceptions and the origin of utterances”,[11] exemplifies the kind of woolly enthusiasm the subject aroused at the time of Adelard’s translations; but proper Euclidian geometry was soon established as a popular subject in schools. Already, by about 1160, Peter of Blois was complaining that pupils were studying it before mastering basic disciplines like grammar.[12] By the early thirteenth century, the science was being advanced for the first time since Antiquity, through the researches of Leonardo of Pisa, also known as Fibonacci (c1170–1250).[13] Geometry was, henceforth, a prominent discipline in scientific theory and pedagogical practice alike.

§ 2.3 Art and argument

Medieval art, too, had its pedagogic aspects. Its defence by the Fathers of the Church as a means of instructing the unlettered need not be taken too literally;[14] it is hard to believe that the real reason for decorating churches was so utilitarian, and an illiterate would be unlikely to encounter an illuminated manuscript; but there were occasions when it was regarded as an aid to memory, and as a tool for clarifying arguments. The Middle Ages was well aware of the value of the visual aid.[15] In book illustration, in particular, pictures could perform a didactic rôle, elucidating and amplifying the text. The layout of word and image was calculated so that the one complemented the other, and the picture was keyed to the argument with a phrase like “as the following figure makes clear” (cf figure 23). Even in the absence of pictures, the text was set out in such a way as to enhance the reader’s understanding of its content. A different size of initial was used to begin book, chapter and verse in the Bible; different grades of script were used to distinguish between text, commentary and nominal gloss. A similar concern for the visual effect of the page is seen in the use of ornamental forms to turn potentially dull, tabulated material (calendars and concordances) into grand decorative designs: worthy frontispieces to books of hours or gospels; but still basically functional tables and indices, clearly set out.

§ 2.4 Argument

Clarity of exposition is fundamental to a philosophical system like scholasticism. The essence of the scholastic method is the dialectical analysis of concepts. Thesis and antithesis are subjected to an examination (interrogatio) against empirical evidence and the opinions of established authorities (auctoritates). After a solution has been reached, the unacceptable arguments are refuted in turn. A necessary prelude to this is the division (distinctio) of concepts into their basic elements. This means, initially, that the argument should be presented as completely and clearly as possible. The distinctio can be set out graphically on the page, so that a topic is visually analysed into its parts and sub-parts by interposing a stratified deployment of terms within the syntax of a sentence (figure 3). Thus, a verbal phrase is broken down into constants and variables. The constants precede and follow the stratified variables, each of which conveys different information; the constants remain unchanged, and provide the context in which the variables are to be understood. Although this is a technique employed by a scribe, not an illustrator, and uses almost entirely verbal material, the result has to be seen to be appreciated. It is no longer a page of text which can be read aloud with equal effect; it is a visual experience.[16]

It is always contentious to draw analogies between the visual arts and other disciplines; even a work as qualified and subtle as Panofsky’s analysis of the development of Gothic architecture in terms of the scholastic method is open to question.[17] However, when, as happens with diagrammatic exposition, graphic means are actually employed for logical demonstrations, such analogies may be more justifiable. Thus, the distinction between variables and constants can be compared with that made by scholastic logicians between the subject and predicate of a proposition on the one hand, and the co-predicates, such as “all”, “some”, “if-then”, on the other. The latter terms, which functioned like the signs for constants in symbolic logic and had no exact equivalent in Aristotelian logic, were called syncategoremata. They provided the formal armature for logical analysis in the Middle Ages; numerous treatises were devoted to them, and they represent one of the major developments in medieval thought.[18] The significance for diagrammatic exposition of this technique of linking different signifiers with an unchanging syncategorema, will be examined later.

As an alternative to a graphic textural layout, the writer could ignore prose exposition altogether, and present the material in the form of a stemmatic analysis (figure 4). The structure and function of these distinctiones becomes more intelligible if the terms employed in them are regarded as not necessarily signifying things, but rather as standing in place of things according to another basic convention of scholastic logic, the theory of suppositio.[19] This proposes that a substantive term need not have ontological status; it may simply be put in place (suppositio) of something, and the question of the exact status of that something can be evaded, as can the problem of universals. A labelled diagram is an ideal medium for this kind of speculation. Thus, “Man” in the Tree of Porphyry (cf section 6.1) stands not necessarily for the concept “humanity”, but for every individual human being. The requirement of completeness of presentation is fulfilled, but in a kind of shorthand. In a similar way, the problem of reasoning by induction is side-stepped: the enumeration of individual entities (either actually, or in the shorthand form offered by suppositio) supersedes it. Again, the affinity of this logical method with diagrammatic exposition is obvious. A term like “being”, which is the subject of the distinctio in figure 4, can now be accepted as non-ontological and purely conceptual. It requires no logical justification (which the stemmatic diagram, being a flat statement, in scholastic terms an enuntiatio, cannot supply) and its nature will be revealed by the exhaustive enumeration of its parts (for which purpose the stemmatic diagram is eminently fitted).

§ 3.1 “Figurae”

The potentially amorphous stemma represented by figure 4 can be given a more definitive form, either in an abstract geometrical framework, or within a figurative image like a tree, a ladder or a wheel. Such designs (called figurae in Latin, schemata in Greek) are more widespread in medieval manuscripts than is usually realised, but difficult to locate through catalogue entries, since their neglect by art historians is generally matched by that of librarians, who omit them from codicological descriptions. Some books consisted entirely of diagrams, with no supporting text; schematic illustrations also provided subjects for monumental art,[20] but usually they formed part of a prose discourse. Some were specially devised for a particular context, but many were taken over from other sources, sometimes undergoing a change of meaning on the way. Thus the figure in Image du monde (cf section 5.2) that illustrates the rotundity of the earth, derives from a picture in the Christian topography of Cosmos indicopleustes, intended to demonstrate the opposite.[21] Occasionally, these diagrams were executed by master craftsmen, and are works of art in their own right; more often, they are unambitious pen-drawings inserted contextually or marginally by the scribe himself. However crude, they are nonetheless integral parts of the text; to omit them, as some editors do, is culpable.

§ 3.2 “Divisio scientiae”

The sequence outlined above, from graphic layout to figurative images, is not a chronological account of the development of schematic illustration, but an ad hoc classification of the main types of visual aid: the typographic, the stemmatic, the geometric and the emblematic. Historically, the elucidatory diagram originated coevally with the illustrated text, in classical Antiquity.[22] It was, with the depiction of narrative, one of the two earliest forms of textual illustration. Aristotle’s works were almost certainly illustrated with diagrams, and while it is unlikely that the Platonic dialogues were, commentaries on them employed figures extensively.[23] Medieval copies of these commentaries may preserve authentic Antique schemata; available evidence suggests that, compared with representational images, diagrammatic designs are transmitted with remarkably little variation.[24] Most such diagrams are geometrical figures or simple linear stemmata; but when early medieval authors took them over to illustrate their own writings, they were embellished with figurative elements: plants, animals and human beings.[25] These may have a mnemonic function; they may, however, be symbolic, or purely decorative. The main purpose of the primitive schemata themselves was analytical: they demonstrated how a subject could be divided into its component parts. Diagrammatic presentation of such information was both more immediate and more economical than a prose account. The divisio scientiae ultimately achieved great complexity, commensurate with the comprehensive analysis required by scholastic method.[26] But its structure was essentially simple, analogous to an organic growth: from nodes representing the main subjects, individual categories issue and are split into their components, as the trunk of a tree divides into branches and twigs. Hence, although these were essentially utilitarian technical diagrams, it was not inappropriate that they should assume the forms of stylised trees. Leaves and fruit represented the individual parts of each “branch” of learning, which stemmed from a trunk inscribed scientia or philosophia.

§ 3.3 “Arbor scientiae”

The transformation of the type of stemma shown in figure 4 into a form that evokes arboreal growth, is easily effected by the addition of fragments of foliate ornament (figure 5). Some of these horizontal trees are invested with considerable botanical detail, despite their orientation;[27] but to get the full benefit of the tree image, it was necessary for the divisio to expand upwards. This inverts the conventional format of the genealogical tree, which is a descending stemma in almost all examples. The most exhaustive divisiones emulate the genealogy, and are non-figurative stemmata deriving from above. The only way such a design could allude to organic growth was for the ramifications to assume the form of roots rather than branches: an expedient only occasionally employed. The one major exception to the descending form for genealogies is the Tree of Jesse, the schematic representation of the ancestry of Christ.[28] It seldom shows the entire 42 generations enumerated by S Matthew, but reduces the number of forebears to a few key figures like David, Solomon and the Virgin. Similarly, the ascending Tree of the Sciences tended to be selective, and its ramifications were emblematic rather than analytical; the seven Liberal Arts stand for the sum of human knowledge, while the label, arbor scientiae, evokes the biblical Tree of Knowledge. Such trees might be regarded as belonging more to the world of allegory than to that of the scientific diagram; but, as will be seen, the two worlds were not discrete, which accounts for the ubiquity and diversity of these designs. When, for instance, the Catalan philosopher, Ramon Lull (c1235–1316), undertook to present a simplified version of his logical Ars generalis (cf section 5.3), he chose to do so in a treatise entitled Arbor scientiae, illustrated with a series of 16 trees, systematically setting out the categories inherent in his philosophy. Preceding them is one vast tree of all the sciences, in which each “branch” is realised literally and analysed in botanical terminology.[29] It is both the tree under which Lull met the monk who prompted him to compile the new redaction of his work, and the conventional form in which to expound it.

§ 3.4 The schematic tree

The stylised arbor was not restricted to the exposition of the categories of knowledge: it was available for any topic that was susceptible to systematic analysis. Some of these diagrams approximate to a tree form without introducing any vegetative features;[30] conversely, others bear no resemblance to an organic growth in their structure, but are nevertheless called arbores and sprout stylised leaves. The Arbor consanguinitatis, for example, is often more like a trellis than a tree; and as a visual realisation of the Table of Kindred and Affinity found at the beginning of the Book of Common Prayer, it does not even embody the principle of analytical ramification found in the divisio. The most common version of this schema is a centralised composition and is to be read from the middle (figure 6), but it is nevertheless often ornamented with foliate decoration.[31] Even when the structure is arboreal, the image need not be; some divisiones employ motifs culled from decorative initials in manuscripts,[32] while others use abstract linear forms derived from no obvious source. The diagram of the ecclesiastical and secular hierarchies that illustrates Gilbert of Limerick’s De statu ecclesiae, is made up of a system of triangles and pointed arches that suggests an architectural elevation; but this late-twelfth-century design predates the use of small-scale architectural drawings.[33] In yet others, the “tree” alluded to is not a botanical image at all, but a religious one, the Tree of the Cross.[34]

Even elaborate descending arbores that emulate no conceivable objective form, employ representational elements like scrolls and columns to provide a structural framework, investing the design with formal grandeur (figure 7). Some are punctuated with small scenes or portraits in medallions, and recall the Roman genealogical trees described by Pliny (Hist nat xxxv, 2) and Seneca (De beneficiis iii, 28, 2): stemmata painted on walls, with ancestors identified either by name or by a portrait mask.[35] In the Middle Ages, similar arbores were used to set out dynastic genealogies, and often included historical events in addition to biographical details.[36] This kind of schema could also be used to present historical information of a less personal nature: not a family tree, but an illustrated time-chart. The late twelfth-century Arbor historiae of Peter of Poitiers is basically an immense genealogy of the Patriarchs of Israel starting from Adam; but it is also a world chronicle, and some versions interpolate a quantity of contemporary secular history.[37] Notable events are recorded not only verbally, but also in laconic illustrations; thus, in figure 7 a marginal urine-flask indicates the discovery of medicine by Apollo shortly before the time of the prophetess Deborah; a hurdy-gurdy stands for the invention of the Greek chorus, nearly contemporary with Gideon.

§ 3.5 “Arbor virtutum”

However, the analogy between vegetable growth and the ramifying subdivision of a topic was often irresistible, particularly when the tree image itself could contribute to the message being conveyed. One such topic was the ethical distinctio of Virtues and Vices. Medieval treatises on morals propounded elaborate analyses of the traits which made up the seven Deadly Sins and their corresponding Virtues;[38] these were obvious material for diagrammatic exposition. The aphorism of Christ about the good and corrupt trees (Matt vii: 17; cf Matt iii: 10) and S Gregory’s dictum that pride is the root of all evil (Moralia xxxi, 45) provided an incentive to use arboreal imagery. Pride and its counterpart, Humility, formed the roots of the trees, the other principal Vices and Virtues were branches, and their subdivisions into individual moral qualities were inscribed on the fruit or leaves (figure 8 verso). The resulting pairs of trees could be distinguished by colour; a green tree for the Virtues, a brown, dead one for the Vices; or form: the leaves and fruit of the good tree turn up, those of the bad one droop.[39] Thus, the innate qualities of natural forms are exploited to enhance the information conveyed by the schema, an advantage the emblematic figure enjoyed over the geometric. A corresponding disadvantage was the congestion created by the attempt to incorporate explanatory inscriptions on the vegetable elements without obliterating them. Exceptionally, the same tree would bear both Virtues and Vices,[40] but generally a tree embodying more than one set of topics would set out kindred, not antithetical, subjects. Trees of this type, like the Arbor sapientiae, which presented the Liberal Arts on one side as a sequence corresponding to the Ages of Man on the other,[41] needed to be read laterally, which is not a natural way of interpreting an organic growth. The design best-suited to this type of visual exposition was not a tree, but a columnar table, which also permitted a clearer disposition of the inscriptions.

§ 4.1 Tabulation

The simplest form of table is a list. Two or more lists intended to correspond laterally are easier to read if they are deployed in a framework that makes the parallels explicit. Tables were in common use in the Middle Ages; every psalter or book of hours had its calendar, for instance, and these were set out on a linear grid embellished with decorative elements. The tabulation of concepts instead of chronological data could be presented in the same way, though the decorative surround was generally made more positive and geometrical so that the parallel terms were clearly delineated. Consequently, the table itself became an abstract work of art; conversely, the representational art of the Middle Ages borrowed the compositional patterns: the design of many stained-glass windows is patently tabular, as is that of a number of medieval picture-books.[42] Although tables are geometrical in form, their principles are arithmetical; the correspondences are not so much spatial or conceptual as based on an equivalent number of parts. This factor was to be illuminatingly exploited in more complex schemata, but some tables present parallels that seem to exist in numerological terms and nothing else. The Arbor sapientiae with its pairing of the seven Arts and the seven Ages of Man is one example; the infant may appropriately learn grammar, but it is ridiculous to suggest that the study of astronomy is the prerogative of old age. Similarly, parallels between the seven Canonical Hours, the seven Acts of the Passion, and the five Senses plus Consent and Free Will (figure 9) are incomprehensible, unless explanatory inscriptions are added to link the topics conceptually as well as visually. The result is a kind of heightened typographical schema, presenting a series of statements of identical syntactical form in a decorative way; a variant, in fact, of the first class of visual aid distinguished above, but sharing some of the characteristics of the geometrical diagram. The most elaborate of these epitomes employ a formalistic logic comparable to the scholastic method, with formulaic phrases linking concepts which lead to (quae ducit ad ...) and counter (quae est contra ...) one another, functioning as syncategoremata (figure 10). The apparent complexity of these tables is, however, belied by the elementary numerological correspondences on which they are founded.

§ 4.2 “Scala virtutis”

Unlike the divisio, the list does not automatically suggest a representational image. However, if it itemised topics in an ascending order, it could aptly be set out on the rungs of a ladder (cf figure 12, lower left) or the steps of a staircase. The principal illustration to the Heavenly Ladder of John Climacus (late sixth century) shows the Virtues inscribed on a ladder stretching from Earth to Heaven.[43] A similar ladder, which appears beside the list of contents at the beginning of the book, reveals one of the limitations of this kind of image: because the rungs correspond exactly to the topics indexed, the ladder reads from bottom to top, the list in the opposite direction. The Ladder of Virtues appears as an illustration to other texts, notably the twelfth-century Speculum virginum, an educational tract made more palatable by its use of schemata and emblematic designs.[44] A passage in Boethius’s De consolatione philosophiae (c523) prompted designers to use the same image to set out the seven Liberal Arts.[45] Like the tree, the ladder offered a motif that re-inforced the message of the inscriptions: in these cases, moral improvement and educational progress. However, an inherently directional image like this had its disadvantages, as the John Climacus example demonstrates. Only the multivalent wheel (cf section 5.1) was sufficiently versatile to fulfil all the requirements of diagrammatic exposition, and this was the most widely-used design in the geometrical repertory. However, a number of other representational forms were occasionally employed with success.

§ 4.3 “Turris sapientiae”

One of these was the schematic building. The biblical reference to the seven pillars of the house built by Wisdom inspired a number of allegorical edifices,[46] some of which are labelled like diagrams. The final picture in the Speculum virginum (figure 11) manages to combine an architectural form with that of a tree: the leaves of the tree are inscribed with the Gifts of the Holy Ghost, and each is associated with seven sets of items including the Petitions of the Lord’s Prayer, the Virtues and the Beatitudes.[47] The allusion to the seven pillars of Wisdom is here made explicit by an inscription; other schematic buildings are independent of the biblical text, and seem to owe more to the architectonic frames that conventionally surrounded concordances at the beginning of gospel books. The Arbor historiae of Peter of Poitiers, for instance, includes some architectural elements, but they function simply as frames and have no allegorical overtones. However, the image called the Tower of Wisdom, which was devised in the later thirteenth century,[48] is a schematic building in which the function of architectural members enhances the meaning of tabulated information (figure 12). The core of the Tower is a tabular grid in the guise of 12 courses of masonry. Each course is inscribed with a Virtue and nine moral injunctions. The building also possesses doors, windows, entrance steps and battlements, each of which is given an allegorical meaning; its foundation is Humility, its four piers, the Cardinal Virtues. The parts of the building are further labelled with the appropriate architectural terms: bases, columns, capitals, etc. The moralised building was a conventional topic for medieval preachers (a fact that has sometimes led critics astray in the interpretation of ecclesiastical structure) and the Tower is obviously influenced by this practice.[49] But it is fundamentally an emblematic tabular diagram, as is revealed by the presence in some examples of an alphabetical key on the left, and a note explaining how the elevation is to be interpreted with the aid of this.

§ 4.4 Text and image

The Tower of Wisdom is an autonomous design, not associated with any particular text. Most of the emblematic diagrams, however, are related to exegetical literature. The dove, the cherub (figure 8 recto) and the four Rivers of Paradise are some of the images that provided the framework for the classification of topics, and each is connected with a text that reads like a commentary on it.[50] Nevertheless, these images are intelligible without the verbal explanation, and sometimes appear independently of it.[51] It is a nice question whether the pictures inspired the text, or vice versa. Although it is orthodox procedure in iconographic studies to try to find a literary text or programme behind a pictorial composition,[52] it is a fallacy to suppose that one necessarily existed. Texts accompanying the trees of Virtues and Vices describe them in detail,[53] but as has been seen these trees derive from the Antique stemmatic divisio, not a written account. They belong to a tradition of visual aids that can function independently of a prose exposition, as well in conjunction with one. Contemporary accounts describe medieval scholars devising just such schematic designs, evidently as autonomous entities.[54] Many of the images discussed here must have originated in this way, either as new creations, like the Tower of Wisdom, or as adaptations of existing diagrams. The figure of the Rivers of Paradise, for example, derives largely from a circular meteorological diagram of the four principal winds and their subsidiaries.[55] Circular diagrams like this were designed initially not as analytical expositions, but as schematic illustrations of cosmic concepts. However, in the Middle Ages they were assumed into the repertory of forms used in the diagrammatic exegesis of philosophical and theological ideas.

§ 5.1 “Rotae”

The value of a circle as a vehicle for expressing concepts pictorially might seem inferior to that of a tree or columnar table. If divided concentrically, it offers only a hierarchy of categories from centre to circumference; if radially, a recurring sequence with no definite starting-point. However, this paradoxical combination of hierarchy and cyclic continuum was fundamental to many medieval theories. The four Elements of which all matter was believed to consist, for instance, were not just an ascending sequence with earth at the bottom and fire at the top; each element shared one of its two qualities with another: both earth and fire are dry, though one is cold, the other hot. Thus they could be set out in either a concentric or radial format.[56]

Furthermore, a circle divided both radially and concentrically presents a schema that is basically a continuous columnar table. Concepts are stratified and aligned, but lack the directional element of the table or tree. Such schemata were used to set out extensive series of numerically-related concepts: the Virtues, the Sacraments, the Works of Mercy, etc (figure 13). Sometimes, a sign marked the point at which the reader should begin, but the format itself was without beginning or end, and could be read clockwise or anticlockwise. The content and context of some of these rotae (theological material in devotional books)[57] suggests that they functioned not only as tabulations of important tenets but also as objects of contemplation, the circular form focussing the concentration like a mandala.

The rota had other advantages: it could partake in the qualities of an actual wheel, either symbolically, as in the Wheel of Fortune or the Wheels of True and False Religion,[58] or practically, insofar as one or more cut-out circles, called volvelles, were attached to the page of a manuscript by a thread of membrane, and could be rotated as a kind of calculating machine (figure 14, cf figure 18). It was also one of the best-established and most widespread diagrammatic designs. It achieved currency in the Middle Ages through illustrations to epitomes of knowledge, of which the medieval archetype was the De natura rerum of Isidore of Seville (560–636); the drawings of wheels formed such a prominent part of the book that it was also known as the Liber rotarum.[59] These and similar rotae figure extensively in school-books and manuals of learning, and in the Middle Ages would have been as familiar a part of the educated man’s visual experience as the graph is of the modern reader’s.[60]

§ 5.2 Astronomy

The original reason for the use of rotae in scientific works is self-evident: they provided the obvious format in which to expound such cosmic concepts as a geocentric universe, the cyclic passage of the months, the zodiac, and the phases of the moon. The basic astronomical diagram is the schema of the spheres, with Earth at the centre surrounded by the planets and the fixed stars. This could be elaborated to include the other Elements between the Earth and the planets, and further spheres beyond the fixed stars (figure 15). Elaboration of a different kind involved the addition of figurative motifs: a hell-mouth in the centre of the Earth, or God and his angels in the Empyrean. Alternatively, further information of a strictly astronomical kind could be incorporated, like the phases of the moon or the duration of the planetary orbits (figure 1). But even when the diagram offered purely scientific data in the context of a technical treatise like Sacrobosco’s Tractatus de sphera (early thirteenth century), it might be adorned with figurative or ornamental motifs that are entirely decorative (figure 15, cf figure 23).[61] In more deliberately popular compilations like the vernacular Image du monde,[62] these embellished schemata received lavish artistic treatment and were executed in expensive pigments and gold. Sometimes this was detrimental to their didactic function, and the results are meaningless ornamental designs; but others, the work of master painters, combine the precision of a scientific diagram with the fantastic beauty of Gothic decorative art, and are unlike any other medieval paintings (figure 16). Even when the draughtsman was content to execute an unadorned linear diagram, technical treatises produced by high-quality workshops contained exquisitely-executed figures, economical of line and elegantly lettered (figure 17). Like early astronomical instruments, they combined utility with beauty; and indeed, a number of them are directly related to scientific apparatus. The revolving discs in manuscripts of the Equatorium of Campanus of Novara (died 1296) are replicas of parts of a vast machine for determining the position of the planets. The instrument was probably never built, but the discs for the individual planets are working components, and function like an analogue computer.[63]

§ 5.3 “Ars demonstrativa”

The rotating disc could also be employed as a logical tool. As has been mentioned, a simplified version of the Ars generalis of Ramon Lull was expounded in a series of trees; but the original ars was inscribed on tables and rotae, some of which revolved.[64] The Lullian ars exists in various forms, but all have a high mathematical valency and are eminently fitted for diagrammatic exposition. The fullest versions employ the 23 letters of the alphabet (I/J and U/V are doublets, W is omitted); A–D signify the Elements; A also denotes a rota and a table; S–Z, too, indicate figures; the remainder of the alphabet stands for one or more concepts and can be presented in combinations which formulate propositions. These letters, B–R, represent initially the 16 absolute principles which are the basis of the ars: the archetypal virtues or “dignities”, of God. They are connected by relative principles, or degrees of relationship, like difference, congruence, greater, lesser, which function as syncategoremata in the Lullian system. To these are added subjects, a comprehensive set of topics open to question or doubt; and rules, which correspond roughly to the Aristotelian categories, and represent ways of doubting. These are deployed on rotae and volvelles drawn in coloured inks. The combination of wheels and letters creates a kind of cypher-machine, providing, in theory, logically consistent answers: it was originally devised as an irrefutable logical system to convert the heathen. The most elaborate forms of the ars employed 16 wheels; followers of Lull attempted to accommodate the essence of the system on a single volvelle of two or three discs, and the degree to which they succeeded indicates the versatility of the revolving rota as a speculative tool (figure 14). The importance of Lull’s philosophy far exceeds that of the mechanics through which he expounded it. If the latter seems to anticipate the computer, it also looks back to the “books of fate”, fortune-telling devices that originated in Antiquity.[65] They employed numerical charts, alphabetical keys and sometimes geared volvelles as an alternative to letter/number equivalents in personal names to predict the future (figure 18). Nevertheless, the figurae of Lull’s ars are essential to it, and it is arguable that their presence influenced the various modifications it underwent from the early Ars compendiosa inveniendi veritatem of c1274 to the final Ars generalis ultima of 1308. No doubt, one of the reasons for reducing the number of “dignities” from 16 to nine was to allow the ars to embody Trinitarian elements; it also facilitated inscription on a rota, and their alignment.

§ 6.1 “Enuntiatio” and “interrogatio”

The example of the Lullian ars demonstrates that although wheels and trees were formally different, they could be used as alternatives to set out the same philosophical system. Trees and tables, too, were sometimes used interchangeably: trees exhibited conspicuous lateral correspondences, and tables assumed arboreal growths (figure 10). This was despite the fact that the two forms are not only structurally distinct, but also that they correspond to two different medieval speculative methods: the scholastic distinctio described above, and typological exegesis. The latter sought not to distinguish the different components of a single topic, but to find analogies between concepts that were recognisably distinct, like the Old and New Testaments.[66] When a schema conflated the tree with the table, it was unable to exploit the characteristics peculiar to each of the forms; it did however make explicit one particular element the two had in common: this was the quality of the information they imparted. In both cases, it was an enuntiatio, an assertive, unsupported statement. But obviously, a visual analogue to a philosophical method that can communicate only in these terms is inadequate; at the very least, something equivalent to the scholastic interrogatio is required. The volvelle, which varied the information supplied by the diagram according to which way the rota was turned, provided this; but it was not necessary to devise mobile schemata to simulate the diagrammatic version of interrogatio: the same thing could be achieved with a static figure if it offered a choice of interpretations of differing validity. A familiar, and very ancient, example of this type of diagram is the labyrinth: it provides a copious variety of choices, but only one leads unerringly through the maze. It is as though in a schematic tree only one of the branches is fruitful; all the others lead, literally, to dead ends. The maze, of course, obfuscates where the tree clarifies; but if in the tree some of the bifurcations are delimitated, while others conduct the reader to further concepts, the result is a kind of labyrinth which compels the entrant to understand where he is going. The simplest figure of this type was the Pythagorean Letter: the two arms of the Y represented the choice between good and evil.[67] A more complex version was the Tree of Porphyry, which set out the relationship between genera and species, and called for a multiplicity of choices (figure 19). It originated as an illustration to commentaries on Porphyry’s Isagoge (late third century), and proceeded from the ultimate genus, substance, to the ultimate species, individual men, by a series of dichotomous divisions.[68] Each bifurcation of the tree contained a positive and negative side. Although it was usually represented as a symmetrical composition, it “grew” only from the positive side, and unlike the divisio was inherently asymmetrical. It also differed from the divisio in being finite: no further division was possible after it had reached the category of individual men, conventionally identified as Socrates and Plato. But the fundamental difference is that the reader was not simply presented with the totality of material; he was given a series of alternatives, and had to chose between them. In a sense, he participated in the diagram, as one participates in a maze.

§ 6.2 Square of Opposition

The participatory quotient is increased in the familiar logical figure called the Square of Opposition. It is still found in modern introductions to logic, but existed in a virtually identical form in classical treatises on the subject.[69] It sets out the relations between propositions in the form of a saltire cross within a rectangle. The relations are not merely stated: they are illustrated for the reader by appropriate sentences which exemplify what contrary, subcontrary, subalternate and contradictory mean. Alternatively, the reader can start with the sentences and use the diagram to discover their relationships. The diagram here is an instrument, like the volvelle, but mobile in the way it can be read rather than the way it can be oriented. It does not simply enunciate truths, but allows the reader to find them. The Square of Opposition could be elaborated with lateral members to cover particular subalternate propositions, or, with a horizontal in the middle and saltire cross above and below it, it became the figure called the pons asinorum, which was used to discover the middle term in a syllogism (figure 20). Late medieval commentaries on scholastic texts, especially on the Summulae logicales of Petrus Hispanus, augmented these logical figures to manic complexity, embracing compound propositions and the so-called ars insolubilis;[70] but the resulting overlapping bands connecting the terms become so intricate that the diagram represents a challenge in itself rather than an aid to a solution.

§ 6.3 “Scutum fidei”

Diagrams like the Square of Opposition had an obvious appeal for theologians: with a little ingenuity, the visual proof provided by a logical figure could be adapted to the presentation of religious dogma. Just as Lull hoped his ars would convince the heathen of the truth of Christianity, so others employed philosophical and scientific visual demonstrations to specifically theological ends. A triangular recension of the Square of Opposition was accordingly used as an illustration of the doctrine of the Trinity and a number of other religious beliefs (figure 21). The device was invented in the mid-thirteenth century; this was the time when heraldry was becoming a strictly codified system, and the earliest examples identify it with the blazon on the Shield of Faith mentioned by S Paul (Ephes vi: 16). But it soon lost its armorial connotations, and later examples present a simple three-sided diagram, sometimes accompanied by instructions on how to read it.[71] A similar example of the theological appropriation of a technical figure is Petrus Alphonsi’s use in his Dialogi (after 1106) of an astronomical figure from Bede’s De natura rerum to show how the Trinity was related to the Tetragrammaton.[72] A fourteenth-century manuscript from Chartres contained an even more determined attempt to implicate science in religion: an illustration of the mathematical properties of Borromini rings was used to demonstrate the relationship between the three Persons.[73] This arrogation of scientific diagrams to religious ends is more than just another aspect of the Church’s willingness to employ visual aids: it can be regarded as a recognition of the irrefutable logic and exactitude of technical illustrations, and an attempt to confer the same qualities on theological propositions.

§ 6.4 Visions

It was not only ecclesiastical doctrines that invoked the scientific diagram as a pictorial exemplar; when ecstatic religious experiences, vatic or revelatory, were represented graphically, they also employed motifs and compositions culled from schemata. The illustrations of the twelfth-century mystic Hildegard of Bingen’s visions derive from cosmic diagrams;[74] the prophecies of Joachim of Fiore (died 1202) are revealed in trees and tables;[75] and Opicinus de Canistris (1296–c1350) utilised recent developments in navigational charts to record his mystical revelations in cartographic form.[76] This does not of course invalidate them: “visionary experience is ... a picture which the mind constructs ... from raw materials already at its disposal”[77] and it has already been noted, how familiar diagrammatic exposition would be to educated persons in the Middle Ages. The connection is not always apparent: it is not immediately obvious that this is the source of a number of pictures in the strange devotional book known as the Rothschild Canticles; but it can be shown that an illustrated copy of the Image du monde was available to the artists as a model, and accounts for some of its extraordinary imagery (figure 22, cf figure 16).[78] The text facing figure 22 refers to Ezekiel’s vision of the wheel within the wheel (Ezek i:15f and x:9f), which provided a biblical precedent for rotae as components of automistic hallucinations. But diagrams in general possessed connotations of authority and certitude which correlate well with accounts of mystical experiences. There was, moreover, a more practical reason why the diagrammatic forms should be so compatible with visions, dependent on one of the graphic peculiarities of the inscribed schema. This was its ability to express, explicitly, several levels of meaning simultaneously: a quality that renders it virtually unique in visual art. Except in very specific instances, a pictorial representation signifies only one thing. The idea that images embody several levels of meaning is one of the myths of modern iconographic studies, unsupported by any evidence.[79] A diagram, however, can do so: the same schema can accommodate a number of concepts, each identified by an inscription. Thus, the same figure that sets out the Elements and the world can be used at the same time to show Man and his Humours and the Year and its Seasons.[80] The mechanics of the analogy are numerological, but the technique depends on the theory of a harmony between Man and the Universe; the one is a microcosm of the other.[81] Every aspect of the physical world has a parallel in Man, either to his bodily organs or to his faculties; and the theory can be represented according either to its anthropomorphic or cosmic content; either one will imply the other. So a single figure, diagrammatic or human, in a kind of visionary intensity, can encompass the whole universe in all its complexity, embracing time, matter, the cosmos, Man and God (figure 23).[82]

§ 7.1 The Five Sevens

This is the context in which the “representation of our Church” (figure 2) must be considered. Its initial resemblance to an astronomical diagram needs no further explanation; it can be readily understood as a tabular rota for devotional use deriving from a long and complex tradition of schematic exposition, of which scientific diagrams formed only one part. Certain aspects of this figure suggest that it conveys a less than perfect version of the original conception. For instance, it embodies, among other ideas, that of a centripetal ascent from sin to salvation; but while the periphery is devoted to the Vices, and an inscription from circumference to centre reads “choose the nearest to God”, the middle of the diagram is occupied, not by the deity, but by an inconsequential rosette. It will, therefore, be helpful to collate it with two other versions of the same image, less well-preserved, but probably reproducing the original more fully. One of these is an early thirteenth-century drawing from France, now in Oxford (figure 24); the other is a slightly earlier French work the present whereabouts of which is unknown (figure 25).[83] Figure 2 is from MS Royal 14 B IX in the British Library: a roll containing the Arbor historiae of Peter of Poitiers, with this figure appended as a colophon. The style of the illustrations in the Royal MS has been associated with the famous scriptorium of St Albans, but this is not endorsed by the palaeographic evidence, and the hegemony of St Albans in this kind of wash drawing has probably been overestimated.[84] However, the roll does appear to be English work, probably dated in the 1270s.

The diagram contains two main components: a circular composition enclosing a large number of smaller circles, both concentric and randomly distributed, articulated by radii; and a rectangular ground into which the circle is set. The rectangle consists mostly of text, and is defined by horizontal and vertical inscriptions in red ink; it is more clearly delineated in other versions of the design (figure 24). The circular part sets out systematically the Vices, the Lord’s Prayer, the Gifts of the Holy Ghost, the Virtues and the Beatitudes, as an ascending sequence. The central part of the top sector makes this clear with a medical analogy: the Vices are debilitating diseases (languores), Man is the invalid, God is the doctor; the Petitions of the Lord’s Prayer are the patient’s sighs, the Gifts of the Holy Ghost are medicine (antidota), the Virtues are health, and the Beatitudes are the joys of the blessed. At the top of the figure, as the inception of this sequence, is a personification of Pride, the source of all evil. In the French examples illustrated here, it is an appropriately prominent figure; but in the Royal MS, it is the same size as the other figures in medallions round the circumference of the circle, which represent in graphic genre scenes, seven other Vices. Each of these is broken down into its components, stemmatically in figures 24 and 25, merely verbally in the Royal MS. The next three sets of septenaries (the Petitions of the Lord’s Prayer, the Gifts and the Virtues) are similar[l]y analysed, and in figures 24 and 25 accompanied by a prose commentary; in the Royal MS, this text appears below the diagram. Thus, the first sector to the left of Pride contains a medallion illustrating the sin of Lust (luxuria), which is analysed into impetuosity, inconstancy, and eight other vicious qualities; the Petition, “hallowed be thy name”, is glossed “through actions, through comprehension, through outward show, through perseverance”; the Gift of Wisdom divides into “from experience” (a sapore), and “from ability” (a sapere); the Virtue of Peace is either “of the heart” or “of eternity”, and the sequence concludes with the essential words of the indivisible Beatitude appropriate to the Virtue: “blessed are the peacemakers: for they shall be called the children of God.”[85]

It will be observed that, if the Petitions of the Lord’s Prayer are to appear in their correct sequence, the diagram must be read anticlockwise; however, the Gifts of the Holy Ghost and the Beatitudes with their associated Virtues only assume their canonical order in a clockwise reading. In exegetical writings on the Lord’s Prayer, these last three topics are sometimes presented inversely, if the author felt this gave a more illuminating set of parallels; alternatively, and most unconventionally, the Lord’s Prayer was discussed back-to-front.[86] It is one of the advantages of diagrammatic exposition that it obviates this need to do violence to the sequence of ideas: it can be read in either direction.

Outside the rota in the Royal MS are four medallions showing different aspects of Christ: uncreated, incarnate, resurrected and as judge. These are linked to form a square by texts in red ink that parallel the four Ages of the World with the festivals of the liturgical year. A clearer version of what is intended is seen in figure 24 and, as far as severe cropping allows, in figure 25. The rectangular figure is not a simple square, but a discontinuous angled strip with a beginning and an end. The fIrst medallion properly shows Adam, the second Moses, and then come the Incarnation, Resurrection and Last Judgement; these are the historical figures or events that demarcate the four Ages described in the text. The quadrants between the circle and the square in the Royal MS contain texts on the Gifts of the Holy Ghost; below the figure is a long text in two columns, which first recapitulates the chronological part of the schema, and then provides a series of commentaries on the Lord’s Prayer. These texts also appear within and above the rota in figures 24 and 25.[87] They are not known from any other source, and were probably composed to accompany the figure.

The design is usually said to illustrate, or derive from, Hugh of S Victor’s treatise, De quinque septenis.[88] Hugh’s work deals with the same subjects as the rota and uses the medical analogy, but there are significant discrepancies between the tract and the image: Hugh discusses only seven Vices, the parallels between the various septenaries do not correspond, and the historical matter is omitted. A number of other examples of the design are, like the version in the Royal MS, pendants to the Arbor historiae, and it is possible that the figure and the texts are the work of Peter of Poitiers.[89] While the Victorine document undoubtedly contributed to it, the rota of the Five Sevens can only be fully understood as an original, autonomous creation: not an illustration to a work of literature, but a sophisticated example of the tradition of visual exegesis through the medium of geometrical design. To use the phrase “geometry of the mind” in connection with this tradition, perhaps endows it with overtones of mathematics and psychology that are unwarranted; and no-one ever pretended that these schemata represented, in any sense, the way the brain functioned.[90] But they do offer a unique commentary on the ratiocination; thought processes and modes of intellectual perception of the Middle Ages.
  
  
  

NOTES

1. On medieval handbooks (sometimes misleadingly called “encyclopaedias”) see W H Stahl “The Systematic Handbook in Antiquity and the Early Middle Ages” Latomus xxiii, 1964, pp. 311-21; M de Gandillac et al La pensée encyclopédique au moyen âge Neuchatel 1966.

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2. The only comprehensive study of this subject is A C Esmeijer Divina quaternitas. A Preliminary Study in the Method and Application of Visual Exegesis Amsterdam 1978. It deals specifically with the period up to c1250, but its extensive notes and bibliography provide a conspectus of research on the subject in general. H Bober’s work on medieval schemata is as yet unpublished, but see his articles cited in notes 56, 60 below, and extracts from his paper on schemata quoted in ME Reeves and B Hirsch-Reich The “Figurae” of Joachim of Fiore Oxford 1972.

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3. Westerlo, archief abdij Tongerlo Liber floridus fol 1r; 550 Jaar Universiteit Leuven (exhibition catalogue, Stedelijk Museum) Louvain 1976, no 467, p. 329.

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4. “... come li ars de iometrie le commande et ensaigne”. The album is Paris BN MS fr 19093; facsimile and commentary in H R Hahnloser Villard de Honnecourt. Kritische Gesamtausgabe des Bauhüttenbuches ms fr 19093 der Pariser Nationalbibliothek, second edition, Graz 1972; for the geometrical figures see pp. 272-79.

T Bowie The Sketchbook of Villard de Honnecourt Bloomington, London 1959, contains all the illustrations in a smaller format with a brief English commentary.

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5. C Bouleau The Painter’s Secret Geometry London 1963, pp. 49–79; G Lesser Gothic Cathedrals and Sacred Geometry London 1957. For an attempt to apply geometrical analysis to medieval poetry, see T E Hart “Twelfth Century Platonism and the Geometry of Textual Space in Hartman’s Iwein: a ‘Pythagorean’ Theory”, Res publica litterarum ii, 1979, pp. 81–107.

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6. For a deft summary of this aspect of medieval thought, see C S Lewis The Discarded Image Cambridge 1964, p. 10; a more detailed account is given in H Meyer Die Zahlenallegorese im Mittelalter (Münstersche Mittlelalter-Schriften xxv) Munich 1975.

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7. J B Friedman “The Architect’s Compass in Creation Miniatures of the Later Middle Ages” Traditio xxx, 1974, pp. 419–29; A Heimann “Three Illustrations From the Bury St Edmunds Psalter and Their Prototypes. Notes on the Iconography of some Anglo-Saxon Drawings” Journal of the Warburg and Courtauld Institutes xxix, 1966, p. 52. Cf the somewhat tentative statement by Cassiodorus (mid-sixth century) that the Trinity employs geometry to bestow species and forms on its creations: “Geometrizat enim, si fas est dicere, sancta Trinitas (some MSS read divinitas) quando creaturis suis, quas hodieque fecit existere, diversas species formulasque concedit”; Institutiones II v 11, R A B Mynors (ed), Oxford 1937, p. 150; Migne Patrologia latina 70, 1212–13. Where practicable, citations of texts will refer to the most recent editions and to Migne, hereafter abbreviated to PL.

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8. “Sphaera infinita cuius centrum est ubique, circumferentia nusquam”; Liber xxiv philosophorum art ii; C Baeumker (ed) Studien und Charakteristiken zur Geschichte der Philosophie insbesondere des Mittelalters (Beiträge zur Geschichte der Philosophie des Mittelalters. Texte und Untersuchungen Bd xxv, Heft 1/2) Münster 1927, p. 208. For other examples of the application of geometrical terminology to theological propositions, see Reeves, op cit note 2 above, p. 42.

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9. B L Ullman “Geometry in the Medieval Quadrivium” Studi di bibliografia e di storia in onore di Tammaro de Marinis iv, Verona 1964, pp. 263–85; on the translations see Dictionary of Scientific Biography New York 1970–, sv Adelard of Bath.

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10. P Abelson The Seven Liberal Arts, a Study in Medieval Culture New York 1906; J Koch (ed) Artes liberales von der antiken Bildung zur Wissenschaft des Mittelalters (Studien und Texte zur Geistesgeschichte des Mittelalters v) Leiden-Cologne 1959. On the teaching of the quadrivium (the mathematical disciplines) see G Beaujouan “L’enseignement du ‘quadrivium’” Settimane di studio del Centro italiano di studi sull’alto medioevo xix, La scuola nell’occidente latino dell’alto medioevo Spoleto 1972, pp. 639–723.

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11. “... fons sensuum et origo dictionum”, Didascalicon II 15, C H Buttimer (ed), Washington 1939; PL 176, 757; see J Taylor The Didascalicon of Hugh of St Victor New York-London 1961, p. 203, n55.

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12. Ep 101; PL 207, 312–313. The letter, although authentic, may not reflect first-hand experience: P Oelhaye “Un témoignage frauduleux de Pierre de Blois sur la pédagogie du XIIe siècle” Recherches de théologie ancienne et médiévale xiv, I947, pp. 329–31.

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13. Dictionary of Scientific Biography New York 1970–, sv Fibonacci.

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14. G Lange Bild und Wort. Die katechetischen Funktionen des Bildes in der griechischen Theologie des 6 bis 9 Jahrhunderts Würzburg 1969; Esmeijer, op cit note 2 above, pp. 1–8.

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15. Esmeijer, op cit note 2 above, passim especially p. 31 n5.

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16. On the practice of reading aloud as a significant factor in medieval literature, see H J Chaytor From Script to Print Cambridge 1945, chapter 2. For a comparable attitude to the layout of the page in printed books, see K J Höltgen “Synoptische Tabellen in der medizinischen Literatur und die Logic Agricolas und Ramus”, Sudhoffs Archiv fur Geschichte der Medizin und der Naturwissenschaften xlix, 1965, pp. 371–90.

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17. E Panofsky Gothic Architecture and Scholasticism Cleveland-New York, 1957.

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18. The earl[i]est usage of the word is by Priscian (sixth century), who restricted the term to prepositions and conjunctions; the medieval system, which in its most extreme manifestation regarded any word affecting the meaning of a proposition that was not its subject or predicate as a syncategorema, is unprecedented. See P Boehner Medieval Logic Manchester 1952, pp. 19–26; H A G Braakhuis De 13de eeuwse tractaten over syncategorematische termen Leiden 1979, (with English summary).

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19. D P Henry Medieval Logic and Metaphysics London 1972, pp. 47–55.

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20. None of the books of diagrams has been published integrally. Very few have even been adequately described; an exception is the first part of Brussels, Bibl Roy MS 9565–9566, which was originally autonomous; see R Derolez “Dubthach’s Cryptogram. Some Notes in Connexion with Brussels MS 9565–9566” L’antiquité classique xxi, 1952, pp. 359–75. An idea of the content and appearance of de luxe volumes of schematic illustrations can be obtained from figures 5, 7–10, 12, 13, and the reproductions of Paris, BN MS fr 9220 in Ligtenberg, op cit note 14 below, figures 1, 13, 14, 22; see Les manuscrits à peintures en France du XIIIe au XVIe siècle (exhibition catalogue BN) Paris 1955, no 66, p. 35. Most of the monumental works have been destroyed (cf note 51 below); survivors include the sanctuary pavement in Westminster Abbey (S Wander “The Westminster Abbey Sanctuary Pavement” Traditio xxxiv, 1978, pp. 137–56); the fresco of the cosmos by Pietro di Puccio in the Camposanto, Pisa (M Bucci, L Bertolini Camposanto monumentale di Pisa, Pisa 1960, figure 99); and the paintings in the crypt of the cathedral at Anagni (L Pressouyre “Le cosmos platonicien de la cathédrale d’Anagni” Mélanges d’archéologie et d’histoire lxxviii, 1966, pp. 551–93). Pressouyre cites further lost examples.

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21. For the Cosmas illustration, see W Wolska La Topographie chrétienne de Cosmos indicopleustès Paris 1962, pl VII and p. 211f; for that to the Image, see R Calcoen Inventaire des manuscrits scientifiques de la bibliothèque royale Albert Ier ii, Brussels 1971, pl XXI.

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22. K Weitzmann Ancient Book Illumination Cambridge Massachusetts, 1959, p. 1.

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23. R S Brumbaugh “Logical and Mathematical Symbolism in the Platonic Scholia” Journal of the Warburg and Courtauld Institutes xxiv, 1961, pp. 45–58.

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24. Thus, while a fifteenth-century copy (Venice Marc MS Cl XIV 35) of a twelfth-century MS (Florence Laur MS S Marco 190) of Martianus Capella’s De nuptiis modifies the pictures of the personified artes stylistically and iconographically, the diagrammatic illustrations are preserved unaltered; for the personifications, see L H Heydenreich “Eine illustrierte Martianus Capella-Handschrift” Kunstgeschichtliche Studien für Hans Kauffmann Berlin 1956, pp. 59–66.

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25. E K Rand “The New Cassiodorus” Speculum xiii, 1938, p. 444, pls C and D.

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26. J Mariétan Problème de la classification des sciences d’Aristote à St-Thomas Paris 1901; L Baur Dominicus Gundissalinus. De divisione philosophiae (Beiträge zur Geschichte der Philosophie des Mittelalters Bd iv, Heft 2/3) Münster 1903.

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27. A Katzenellenbogen Allegories of the Virtues and Vices in Medieval Art London 1939, p. 65, figures 64, 65.

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28. A Watson The Early Iconography of the Tree of Jesse Oxford-London 1934; for roots as divisiones, see F A Yates “The Art of Ramon Lull” Journal of the Warburg and Courtauld Institutes xvii, pls 15, 16.

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29. Yates, op cit note 28 above, p. 144.

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30. See the figurae arborum illustrating theories of rhythm and metre in early fifteenth-century music in MS Vat Barb lat 307; J Smits van Waesberghe Musikerziehung. Lehre und Theorie der Musik im Mittelalter (Musikgeschichte in Bildern, Bd iii, Lfg 3) Leipzig 1969, pp. 172–75.

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31. H Schadt Darstellungen des Arbores consanguinitatis bis zum 4. Laterankonzil 1215 diss Tübingen 1973; Esmeijer, op cit note 2 above, pp. 108–16.

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32. See the schemata illustrating the French translation of Cicero’s Ad Herennium Chantilly, Mus Condé MS 590, fols 9v, 11v; J Folda Crusader Manuscript Illumination at Saint-Jean d’Acre 1275–1291 Princeton 1976, figures 24, 25.

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33. Durham, Cathedral Lib MS B II 35, fol 36v; R A B Mynors Durham Cathedral Manuscripts Oxford 1939, no 47, p. 42, pl 32. On the development of scale drawings of plans and elevations, see R Branner “Villard de Honnecourt, Reims and the Origin of Gothic Architectural Drawing” Gazette des Beaux-Arts ser 6, lxi, 1963, pp. 129–46.

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34. The most spectacular schema of this kind is the lignum vitae of S Bonaventure, familiar from the fresco by Taddeo Gaddi in Santa Croce, Florence; R Ligtenberg “Het lignum vitae van den H Bonaventura in de ikonografie der veertiende eeuw” Het Gildeboek xi, 1928, pp. 15–39.

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35. E H Wilkins “The Trees of the ‘Genealogia deorum’” Modern Philology xxiii, 1925–26, pp. 61–65.

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36. Cf the chronological tables on fols 140r–149v of the Durham MS cited in note 33 above; Mynors op cit pl 44.

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37. The work appears under various titles, and modern scholars prefer Compendium historiae in genealogia Christi; however, it was called an arbor in the Middle Ages, and the term is apt. Text and paradigmatic layout in H Vollmer Deutsche Bibelauszüge des Mittelalters zum Stammbaum Christi mit ihren lateinischen Vorbildern und Vorlagen (Bibel und deutsche Kultur i) Potsdam 1931, pp. 127–88; see also P S Moore The Works of Peter of Poitiers, Master in Theology and Chancellor of Paris (1193–1205) Notre Dame 1936, pp. 97–117; W H Monroe “A Roll-Manuscript of Peter of Poitiers’ Compendium” The Bulletin of the Cleveland Museum of Art lxv, 1978, pp. 92–107.

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38. M W Bloomfield The Seven Deadly Sins Michigan 1952.

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39. Katzenellenbogen, op cit note 27 above, pp. 63–67; for the Gregory text, see note 85 below.

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40. London BL MS Add 30024, fol 2r; the MS is described in M W Evans “Allegorical Women and Practical Men: the Iconography of the artes Reconsidered” Medieval Women (Studies in Church History, Subsidia 1) Oxford 1978, p. 329; and the miniature is illustrated in Lexikon der christlichen Ikonographie iv, Freiburg i Br 1972, sv Tugenden und Laster figure 6.

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41. Eg Paris BN MS fr 9220, fol 16r; illustrated in Ligtenberg, op cit note 34 above, figure 22.

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42. For tabular compositions in glass, see the schemata of compositional forms in M H Caviness The Early Stained Glass of Canterbury Cathedral Princeton 1977, appendix figures 1ff; cf E J Beer Die Rose der Kathedrale von Lausanne und der kosmologische Bilderkreis des Mittelalters Bern 1952. Typological picture-bibles generally employed a tabulated layout: A De Laborde Etude sur la Bible moralisée illustrée Paris 1911–27; Reallexikon zur deutschen Kunstgeschichte Stuttgart 1937–, sv Armenbibel.

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43. J R Martin The Illustration of the Heavenly Ladder of John Climacus (Studies in Manuscript Illumination v) Princeton 1954.

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44. E Greenhill Die geistigen Voraussetzungen der Bilderreihe des Speculum Virginum. Versuch einer Deutung (Beiträge zur Geschichte der Philosophie und Theologie des Mittelalters Bd xxxix, Heft 2) Münster 1962; Esmeijer, op cit note 2 above, p. 62ff; for a different version of the image, see R Green et al Herrad of Hohenbourg. Hortus Deliciarum (Studies of the Warburg Institute 36) London-Leiden 1979, p. 201, pl 124.

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45. The text describes a personification of Philosophy whose robe is marked with degrees like a ladder (“in scalarum modum gradus”; De consolatione philosophiae I pr 1 L Bieler (ed), Turnholt 1957; PL 63, 589); P Courcelle La consolation de philosophie dans la tradition littéraire Paris 1967. Cf Ramon Lull’s use of ladder imagery in a philosophical context: J N Hillgarth Ramon Lull and Lullism in Fourteenth-Century France Oxford 1971, pp. 222, 452–54, pl V.

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46. The house of Wisdom is probably represented by the seven-pillared structure in the initial to Ecclesiasticus in Arras, Bibl de la Ville MS 559, t iii, fol 1r; S Schulten “Die Buchmalerei des 11 Jhs im Kloster St-Vaast in Arras” Munchner Jahrbuch der bildenden Kunst ser 3, vii, 1956, p. 79, figure 50. The busts at the top of the pillars may represent the seven Liberal Arts. The names of six of the Arts are inscribed on a drawing of a tower in Boulogne, Bibl mun MS 40, fol 98r. Towers also appear in astronomical figures; here, however, the allusion is not to the house of Wisdom, but to the importance of Arabic sources in the transmission of scientific ideas to Europe. The Arabic word for zodiacal sign, burj, also means “tower” and was translated, and illustrated, as “turris”; Burnett, op cit note 64 below, p. 81.

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47. A Watson “The Speculum virginum with Special Reference to the Tree of Jesse” Speculum iii, 1928, p. 455. The septenaries also appear in the Tree of Jesse which opens the Speculum.

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48. Its inventor can plausibly be identified with Franciscus Bonacursus, archbishop of Tyre, who died in 1274; F Saxl “A Spiritual Encyclopaedia of the Later Middle Ages” Journal of the Warburg and Courtauld Institutes v, 1942, p. 110.

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49. J Sauer Symbolik des Kirchengebäudes und seiner Ausstattung in der Auffassung des Mittelalters Freiburg i Br 1924, discusses the allegorical interpretation of buildings in medieval exegetical literature. The number of architectural historians who believe that an inclined chancel alludes to Christ’s head sagging on the cross, or that an external door in a nave aisle represents the wound in His side, is smaller than it was, but this kind of interpretation may still be encountered. For an alarming example, see P Fingesten “Topographical and Anatomical Aspects of the Gothic Cathedral” The Journal of Aesthetics and Art Criticism xx, 1961–62, pp. 3–23. R Krautheimer “Introduction to an ‘Iconography of Mediaeval Architecture’”, Journal of the Warburg and Courtauld Institutes v, 1942, pp. 1–33, is a useful corrective.

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50. The dove is associated with the chapters de columba from the bestiary of Hugh of Folieto, PL 177, 15–20; C de Clercq “Le rôle de l’image dans un manuscrit medieval (Bodleian, Lyell 71)”, Gutenberg-Jahrbuch 1962, pp. 23–30. The cherub appears with two texts de sex alis cherubin which were frequently published together; PL 210, 266–280. The probable authors of the treatises are Alan of Lille and Clement of Llanthony; M-T d’Alverny Alain de Lille. Textes inédits (Etudes de philosophie médiévale lii) Paris 1965, pp. 154–55. For the many texts on the Rivers of Paradise, see Esmeijer, op cit note 2 above, pp. 59–67. See also Katzenellenbogen, op cit note 27 above, pp. 62, 69.

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51. The cherub, for example, appears in the psalter mentioned in note 57 below, in many of the related compendia of tables, and in the “spiritual encyclopaedias” described by Saxl (note 48 above). A monumental image of it can just be discerned on a wall of the Westminster Abbey chapter house.

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52. Cf E Panofsky Studies in Iconology New York-Evanston 1962, p. 11: “Iconographic analysis, dealing with images, stories and allegories instead of with motifs, presupposes ... a familiarity with specific themes or concepts as transmitted through literary sources....” It is worth observing that amid this welter of emphases the last two words are not stressed; the number of iconographical studies which pursue potential literary sources with unrelenting vigour would suggest otherwise.

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53. Ps-Hugh of S Victor De fructibus carnis et spiritus PL 176, 997–1006; R Bultot “L’auteur et la fonction littéraire du ‘de fructibus carnis et spiritus’”, Recherches de Théologie ancienne et medievale xxx, 1963, pp. 148–54; the trees are also described and illustrated in the Speculum virginum; see note 44 above.

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54. In 986, Gerbert of Reims described how he made figurae illustrating the art of rhetoric for use in teaching; Ep 92 J Havet (ed), Paris 1889; PL 139, 224. For further educational aids devised by Gerbert, see O G Darlington “Gerbert, the Teacher” The American Historical Review lii, 1947, pp. 456–76; on other early medieval pedagogic devices, see Beaujouan, op cit note 10 above. Such visual aids were used in classrooms throughout the Middle Ages, and Peter of Poitiers is said to have designed arbores of the Virtues and Vices similar to his Arbor historiae; Moore, op cit note 36 above, p. 168.

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55. Esmeijer, op cit note 2 above, p. 60; for illustrations of wind-roses, see J Baltrusaitis “Roses des vents et roses des personnages à l’époque romane” Gazette des Beaux-Arts ser 6, xx, 1938, pp. 265–76.

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56. Compare figures 15 and 23. On the various ways the Elements could be set out, including one explicitly stated to be according to geometrical principles, see H Bober “In principio. Creation before Time” De artibus opuscula XL. Essays in Honor of Erwin Panofsky New York 1961, pp. 13–28; Pressouyre, op cit note 20 above; Esmeijer, op cit note 2 above, pp. 37–41.

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57. For example, the so-called De Lisle Psalter (London BL MS Ar 83) includes tables of the type found in the MSS described in the notes to figures 5 and 9 [these are given separately in the companion gallery exhibit]; for illustrations, see Ligtenberg, op cit note 34 above, figures 5, 6, 17–21, 23. A monograph on Ar 83 is in preparation by L F Sandler.

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58. On the Wheel of Fortune, see E Kitzinger “World Map and Fortune’s Wheel: A Medieval Mosaic floor in Turin” Proceedings of the American Philosophical Society cxvii, 1973, pp. 344–73; Courcelle, op cit note 45 above; on the Wheels of Religion, see de Clercq, op cit note 50 above, and Katzenellenbogen, op cit note 27 above, pp. 70–72.

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59. J Fontaine Isidore de Séville. Traité de la nature Bordeaux 1960 (text and French translation); PL 83, 963–1018. On the illustrations, see Fontaine pp. 15–18.

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60. H Bober “An Illustrated Medieval School-Book of Bede’s De natura rerum”, Journal of the Walter’s Art Gallery xix–xx, 1956–57, pp. 65–97.

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61. L Thorndike The Sphere of Sacrobosco and its Commentators Chicago 1949, provides text and translation but unfortunately omits the figures.

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62. Evans, op cit note 40 above, pp. 314, 321.

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63. F S Benjamin and G J Toomer Campanus of Novara and Medieval Planetary Theory Madison, Milwaukee, and London 1971; on the development of the equatorium, see D J Price The Equatorie of the Planetis Cambridge 1955, chapter 8.

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64. Yates, op cit note 21 above, provides the best introduction to Lull’s work, and illustrates some of the rotae in pls 11, 19; see also R D F Pring-Mill “The Trinitarian World Picture of Ramon Lull” Romanistisches Jahrbuch vii, 1955–56, pp. 229–56; Hillgarth, op cit note 45 above.

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65. T C Skeat “An Early Medieval ‘Book of Fate’: the Sortes XII patriarcharum. With a note on ‘Books of Fate’ in general” Mediaeval and Renaissance Studies iii, 1954, pp. 41–54; Heimann, op cit note 7 above, pp. 39–46; C S F Burnett “What is the Experimentarius of Bernardus Silvestris? A Preliminary Survey of the Material” Archives d’histoire doctrinale et littéraire du moyen âge xliv, 1977, pp. 79–125.

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66. C Spicq Esquisse d’une histoire de l’exégèse latine au moyen âge (Bibliothèque thomiste xxvi) Paris 1944; H de Lubac Exégèse médiévale. Les quatre sens de l’écriture Paris 1959–64; Esmeijer, op cit note 2 above.

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67. W Harms Homo viator in bivio (Medium aevum. Philologische Studien Bd xxi) Munich 1970.

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68. E W Warren Porphyry the Phoenician. Isagoge (Medieval Sources in Translation 16) Toronto 1975; for the Tree, see p. 35; for adaptations of it, see M-T d’Alverny, “Le cosmos symbolique du XIIe siècle” Archives d’histoire doctrinale et littéraire du moyen âge xx, 1953, pp. 31–81; for its essentially asymmetrical form, see Enciclopedia filosofica Florence 1967, sv Albero di Porfirio.

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69. L Pozzi Studi di logica anticae medioevale Padua 1974, pp. 39–61.

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70. W Ong Ramus, Method, and the Decay of Dialogue Cambridge Massachusetts 1958, pp. 74–83; for the pons asinorum, see Pozzi, op cit note 69 above, pp. 63–73.

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71. The design is invariably discussed in terms of the debased version from Simon Vostre’s Paris Horae of 1524, as illustrated by Didron (op cit note 73 below), p. 575, in which its heraldic elements are omitted. A fuller account of the sources and development of the figure will be included in the present author’s forthcoming paper “An Illustrated Fragment of Peraldus’s Summa of Vice” Journal of the Warburg and Courtauld Institutes.

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72. Reeves and Hirsch-Reich, op cit note 2 above, p. 42.

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73. Chartres, Bibl de la Ville MS 233, fol 48v(?); the book was destroyed in the Second World War, and the figure, one of four similar designs, is recorded only in an engraving in A Didron Iconographie chrétienne. Histoire de Dieu (Collection de documents inédits sur l’histoire de France) Paris 1843, p. 569; revised translation Christian Iconography New York 1965, volume ii, p. 46. In both editions, the date and pressmark are wrong; see Catalogue général des manuscrits des bibliothèques publiques de France. Départements xi, Paris 1890, pp. 116, 117; Y Delaporte Les manuscrits enluminés de la bibliothèque de Chartres Chartres 1929, p. 115.

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74. H Liebeschütz Das allegorische Weltbild der Heiligen Hildegard von Bingen (Studien der Bibliothek Warburg 16) Leipzig-Berlin 1930; C Singer “The Scientific Views and Visions of Saint Hildegard” Studies in the History and Method of Science i, Oxford 1917, pp. 1–55.

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75. Reeves and Hirsch-Reich, op cit note 2 above.

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76. R G Salomon Opicinus de Canistris. Weltbild und Bekentnisse eines avignonesischen Klerikers des 14. Jahrhunderts (Studies of the Warburg Institute 1), London 1936; idem “A Newly Discovered Manuscript of Opicinus de Canistris” Journal of the Warburg and Courtauld Institutes xvi, 1953, pp. 45–57; R Glanville “Mapping Realities” AAQuarterly volume 12 number 4.

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77. E Underhill Mysticism London 1911, p. 271.

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78. M R James Description of an Illuminated Manuscript of the XIIIth Century in the Possession of Bernard Quaritch London 1904; Evans, op cit note 40 above, p. 319.

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79. The case against “levels of meaning” is stated in E H Gombrich Symbolic Images. Studies in the Art of the Renaissance London 1972, pp. 15–17.

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80. Esmeijer, op cit note 2 above, p. 37; Bober, op cit note 60 above, p. 84; Gilbert of Limerick’s diagram (note 33 above) similarly superimposes different concepts.

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81. F Saxl “Macrocosm and Microcosm in Medieval Pictures” Lectures London 1957, pp. 58–72.

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82. The degree of complexity such schemata could attain, is illustrated by the famous diagram in Byrhtferth’s Manual (Oxford, S John’s College MS 17, fol 7v). This includes no[t] merely Elements, Humours and Seasons, but, in addition, (reading from the outside inwards) solstices and equinoxes, zodiac signs, months and their duration, winds, the Ages of Man, cardinal points, and the letters A – D – A – M, which not only spell the name of the first Man but are also an acronym of the four parts of the world in Greek. For illustration and bibliography, see C M Kauffmann Romanesque Manuscripts 1066–1190 (A Survey of Manuscripts Illuminated in the British Isles) London 1975, p. 56, and ill 21.

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83. For these two MSS, see A C de la Mare Catalogue of the Collection of Medieval Manuscripts Bequeathed to the Bodleian Library, Oxford, by James P R Lyell Oxford 1971, pp. 254–56; Settimana del libro antico e raro (exhibition catalogue IV Fiera internazionale del libro, Istituto Italiano del Libro) Florence 1932, pp. 4–6, figure 2.

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84. G F Warner and J P Gilson British Museum. Catalogue of the Western Manuscripts in the Old Royal and King’s Collections ii, London 1921, p. 127. The authors compare the drawings and initials with those in a Saint Albans psalter, MS Roy 2 B VI; their date for MS Roy 14 B IX (early fourteenth-century) seems too late.

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85. The development of the exegetical treatment of these themes is described in O Lottin “La doctrine d’Anselme de Laon sur les dons du Saint-Espritet son influence” Recherches de théologie ancienne et médiévale xxiv, 1957, pp. 267–95. The divisio of the Vices is an elaboration of that propounded in S Gregory, Moralia in Job xxxi, 45; PL 76, 621. The analyses of the other topics are unprecedented.

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86. S Bernard Expositio in orationem dominicam PL 184, 811–818.

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87. For incipits, see de la Mare, op cit note 83 above. A similar historical theme is described in the Prologue to Jacobus de Voragine’s Legenda Aurea.

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88. R Baron Hugues de Saint-Victor. Six opuscules spirituels. (Sources chrétiennes clv), Paris 1969, pp. 100–19 (text and French translation); PL 175, 405–410.

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89. De la Mare, op cit note 83 above, suggests that it might be the arbores virtutum et vitiorum cited in note 54 above. The identification is attractive, and there are examples where rotae and trees are conflated: cf the urinary circle-cum-tree illustrated in Calcoen, op cit note 21 above, pl III. It may also be significant that the earliest extant copy of the Arbor historiae contains the rota of the Five Sevens; Monroe, op cit note 36 above, p. 95. At any event, the rota corresponds nicely with Peter’s expository technique as evinced by the Arbor historiae and what is known of his teaching methods; see Esmeijer, op cit note 2 above, p. 98f.

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90. For illustrations that do attempt to portray brain-function, see E Clarke and K Dewhurst An Illustrated History of Brain Function Oxford 1972, especially pp. 29–39.

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Illustrations

^ Figure 2. Rota of the Five Sevens and quadripartite historical schema (circa 1270)

See she-philosopher.com Gallery CAT. NO. 116 (with larger image facsimiles in the companion Gallery exhibit on medieval information design).

^ Figure 1. Schema of the spheres (late 13th century)

See she-philosopher.com Gallery CAT. NO. 115 (with larger image facsimiles in the companion Gallery exhibit on medieval information design).

^ Figure 3. Graphic textual layout (circa 1323)

See she-philosopher.com Gallery CAT. NO. 117 (with larger image facsimiles in the companion Gallery exhibit on medieval information design).

^ Figure 4. Stemmatic analysis of “being” (ens reale) (circa 1323)

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^ Figure 5. Divisio scientiae, setting out the parts of learning (late 14th century)

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^ Figure 6. Tree of consanguinity (1460)

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^ Figure 7 (verso). Schema of world history (late 14th century)

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^ Figure 7 (recto). Schema of world history (late 14th century)

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^ Figure 8 (verso). Tree of Virtues; Cherub (late 14th century)

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^ Figure 8 (recto). Tree of Virtues; Cherub (late 14th century)

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< Figure 9. Tree of the Acts of the Passion (15th century)

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^ Figure 10 (verso). Tables of Christian doctrine (late 14th century)

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^ Figure 10 (recto). Tables of Christian doctrine (late 14th century)

See she-philosopher.com Gallery CAT. NO. 127 (with larger image facsimiles in the companion Gallery exhibit on medieval information design).

^ Figure 12. Tower of Wisdom (15th century)

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^ Figure 11. House of Wisdom (late 14th century)

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^ Figure 14. Volvelle of the Lullian ars (circa 1323)

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^ Figure 13. *Rota* of the Sevens (15th century)

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^ Figure 16. Schema of the Elements, Pure Air and the Planets (late 13th century)

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^ Figure 15. Schema of the spheres (late 13th century)

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^ Figure 17. Astronomical schema (late 13th century)

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^ Figure 18. Geared device for fortune-telling (14th century)

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^ Figure 19. Tree of Porphyry (16th century)

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^ Figure 20. Pons asinorum (the “asses’ bridge”) (17th century)

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^ Figure 21. Shield of Faith (circa 1240–53)

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^ Figure 22. The Trinity and the Vision of Ezekiel (early 14th century)

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^ Figure 24. Rota of the Five Sevens and quadripartite historical schema (early 13th century)

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^ Figure 23. Schema of quaternities (late 13th century)

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< Figure 25. Rota of the Five Sevens and quadripartite historical schema (12th century)

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