 Hello everyone, good evening and welcome. I'm Xerxes Mazda, head of Collections and Curation here at the British Library and I'm really delighted to welcome you to the second of three lectures in the 37th annual series of Penitzi Lectures. The Penitzi Lectures were founded about 40 years ago by Catherine Devers, a long time lover of both books and of the British Library, and were named after Sir Anthony Penitzi, who served the British Museum for nearly 40 years and was its principal librarian from 1856 to 1866. Thanks to the kind support of Jonathan Hill Bookseller, we're able to provide the live stream of the lectures, the Penitzi Lectures, and I know that we have many more people watching tonight, the live stream that we have with us here today. So this year, we're delighted to welcome Professor Jeffrey Hamburger, Kuno Franke, Professor of German Art and Culture at Harvard University. Through a long and celebrated career teaching at Harvard for over 20 years, he has specialized in many aspects of medieval art and his numerous books have focused on medieval diagrams, illumination, and the use and meaning of the visual in medieval culture more generally. So, without further ado, Professor Hamburger, over to you for the codex in the classroom. Well, to those of you who are returning for a second helping of Hamburger, welcome back. And to those of you who are here for the first time, thank you so much for coming. I should say in advance that since I delivered the first lecture, I seem to have come down with a terrible cold, so if my voice is raspy, I hope you'll excuse me. For those of you who did miss the first lecture, I might just quickly say that the basic premise of that introduction was that the history of diagrams from classical antiquity right up until the present day is essentially a history of theories of cognition, and that's a theme to which I'll return in the course of this evening's lecture. Of all the ideas and institutions the modern world has inherited from the Middle Ages, the university remains among the most durable. This space, no longer the British Library's auditorium, is hardly a classroom, let alone a medieval one, but for our purposes it is an adequate substitute. Still more venerable, and one hopes no less resilient, even in the age of COVID, is the interactive dialogue enacted in a space shared by students and a teacher, a venerable form of pedagogy exemplified by the dialogues of Plato. According to a legend attested no earlier than the sixth century, more than a millennium after the philosopher had died, the inscription above the door to his academy read, let no one ignorant of geometry enter here. Geometry provided not only a means of demonstration, it provided a pathway to transcendental truth, a tradition on which philosophers and theologians would continue to draw deep into the Middle Ages. In his treatise De Ordine, seen here in a ninth century Harley manuscript from Lorsch, the Church Father Augustine writes about geometry as part of a program of the liberal arts that leads the mind to God. Reason stepped out onto the domain of the eyes. Surveying the earth and the heavens, it found that nothing but beauty pleased it. Within beauty forms, within forms proportion, within proportion number, it asked itself where in the real world this or that straight, curved, or other lines as conceived by the intelligence might be. It found reality far inferior. Nothing real stood comparison with what the mind could see. It analyzed these forms one by one and arranged them into a discipline which is called geometry. Augustine describes a process by which the intelligible informs the sensible world, which in turn can serve as a ladder leading back to invisible truths. Among images, diagrams seemed uniquely suited to mediating between two realms. Not only are they hybrid in form combining words and images, they also chart a middle ground between the visible and the invisible, serving as intermediary objects between the sensible and intelligible. As representations of rational thoughts, they fall far short of the forms, yet they serve to represent them in substantial terms. Diagrams need not be platonic, they can also be pragmatic. Medieval manuscripts are full of such diagrams. For example, legal, here showing the degrees of affinity and consanguinity permitted in marriage according to canon law. Therapeutic, charting interrelationships among the bodily humors thought to govern human health. Calendrical, at center for determining the date of Easter, according to the Domenical Letters and Golden Numbers, a zodiac man to the left and at right, a set of vovelles linking days of the month to different liturgical chants, about which more later in this lecture. Exegetical, as in this copy of Nicholas of Lyra's popular commentary on the Bible in which architectural diagrams rooted in Victorine exegesis of the 12th century, here of the Temple of Solomon to emphasize the literal sense of Scripture. Or astrological, here a horoscope entered in 1349 when due to the Black Death the need for such a device no doubt was keenly felt along with the foretelling of a new sect in the margin of a 12th century copy of the world chronicle by Sigebert of Jean Bleu. The truth claims of diagrams do not need to be accepted at face value. Postmodern philosophers have subjected diagrams to scathing suspicion if for reasons different than those marshaled by late 19th century mathematicians such as Frege, Pash, and Hilbert, who following Descartes converted geometry into algebra. Writing on such postmodernists as Gilles Deleuze, whose concept of the rhizome reaffirms diagrammatic thinking even as it deconstructs rational traditional hierarchies, John Malarkey has declared and I quote, There is no truth in diagrams, nothing sacred in geometry. The flexuous line is not an intimation of the divine. Its eminence, its materiality keeps it at some distance from the infinite lines, pure circles, and perfect triangles of Nicholas of Cusa, unquote. And it's Nicholas of Cusa's paradigmatic diagram expressing the interpenetration of divinity with contracted creation in terms of intersecting pyramids of light and dark of which you here see a copy that passed through the cardinal's hands. Our critic continues and I quote again, Such infinities, purities, and perfections smack of the virtual, the transcendental, the vision of God, unquote. Yes, but that was precisely their purpose. The iconoclastic critique of the diagram which casts it not as an instrument of reason but rather as a rhetorical device opens its seemingly cogent demonstrations of irrefutable logic to accusations if not necessarily of mystical irrationality, then at least a fiction which in turn makes them instruments of ideological persuasion. Some medieval diagrams, for example those in the so-called Ars Notoria of Solomon, a magical handbook, were intended to function as operative instruments of mystification as well as memorization. The Notoria of the title has nothing to do with notoriety but refers rather to the note, notations or diagrams that constitute its essential instruments here with a view to theological knowledge. Others deal with grammar, dialectic, rhetoric, arithmetic, geometry, logic, in short the traditional program of the seven liberal arts, not nearly as exciting as black magic. Postmodernists are not alone in having rejected the link between diagrams and divinity. As William Blake declared, and he didn't mean it as a compliment, the gods of Greece and Egypt were mathematical diagrams. For Blake, reason and its instruments were idols that serve in the way of imaginative spiritual vision. Blake's Newton, personification of a man limited by reason, represents a reaction to the enlightenment celebration of reason. The image shows the scientist seated on rocks, his eyes fixed not on the heavens but rather on the diagram unfurled on a scroll in front of him. The pejorative purpose of Blake's image, which served as a model for Eduardo Paloazzi's sculpture, past which you had to walk on your way to this lecture, has always made the latter seem a strange choice to serve as an emblem of the British Library. In the sculpture, which converts the scientist into an automaton with mechanical joints, Newton naked except for the straps around his thighs and abdomen remains earthbound, an uninspired prisoner of scientific thought. Sorry to interrupt, but I wonder whether the noise coming from behind me is as loud for you as it is for me. Okay, I will try to ignore it, but up here it's quite distracting, I have to say. In endowing Newton with calipers, Blake invokes the instrument that medieval artists assigned to the creator himself, most famously in the 13th century, Bible-Malise, a vast biblical picture book, imagery on which Blake drew in creating his ancient of days, Horizon, who as the imposer of reason and laws is a negative figure. Medieval theologians and scientists also looked to the heavens, if within an eye very different than Blake's. A 15th century Parisian copy of The Image of the World, the mid-13th century encyclopedic poem by Gauthier de Metz, portrays God as a geometer who ordered all things in measure and number and weight. The rubric reads, on the completion of the world. Other miniatures remake God and the image of man, specifically a cleric who either holds a geometers square or holds in his hands a diagrammatic image of the division of the world into three continents on the model of a TO map of a kind found elsewhere in the manuscript. Such maps, which reduce the known world to the most basic of schemas, trace their ancestry to the 7th century scholar Isidore of Seville. The repeated motif responds to the full title of the work, the book of the clergy, which is called The Image of the World, which Gauthier adds, is translated from Latin into French and contains 55 chapters and 28 figures, without which the book cannot be easily understood. The illustrations to Gauthier's verse encyclopedia frame diagrams as utilitarian instruments of instruction and as reflections of divine order. The same held true for Plato, for whom diagrams demonstrated the truth of the transcendent forms. The episode in the manoe, seen here in the oldest late 10th century copy, which lacks the relevant diagram, and in a somewhat later version in which it appears amongst the marginal skolia, evokes a slave boy, that is someone without any formal education, to introduce the concept of anemesis, knowledge as recollection. Socrates interrogates manoe's slave concerning the proportions of the square to prove that he can understand geometrical problems on the basis not of teaching, but rather recollection. Socrates concludes, if the truth of all things always existed in the soul, then the soul is immortal, wherefore be of good cheer and try to recollect what you do not know or rather what you do not remember. In Socrates' scenario, diagrams function as tools for thinking in concrete spatial terms. In the Republic, however, Socrates argues that as visual images, diagrams are a means, not an end. Dialectic, the highest human science, is superior in that unlike geometry, it does not rely on visible images, including those that can be imagined, rising instead to intelligible objects, that is, the forms. Ironically, however, this scale of knowledge known as the divided line also came to be illustrated by diagrams, which attempted to resolve contradictions in the text regarding the lengths of its different segments, each of which stood for a different kind of knowing, in this case employing arcs with the appellation arbalus shoemaker's knife on account of their shape. Already at the beginning of a long tradition, the stage is set for a contentious debate about the relative merits and truth value of visual versus verbal representation. In the 12th century, the manoeuvre was translated into Latin by Henrique's Aristopus of Calabria. It was only much later, however, in the translation of Marchilio Ficino, that the larger corpus of Platonic texts became available in copies such as this one, made after a printed edition for Ferdinand I of Aragon, King of Naples. The manoeuvre thus remained essentially unknown to medieval thinkers who had access only to the Timaeus, a work of cosmology, in the translations first of Cicero and later in the fourth century of Calcidius, whose commentary was complemented by a set of diagrams. In this idiosyncratic work regarded as representative of Plato's thought, seen here in a manuscript made around the millennium, the philosopher argued that everything could be analyzed in terms of five perfect solids, four of which corresponded to the four elements, with the fifth matching the universe as a whole. Calcidius's commentary was often paired with that of Macrobius on Cicero's dream of Scipio, which also included illustrations, among them a lambda diagram, so named after its shape, a depiction of the numerical harmonies informing the world's soul. On its left side, multiples of three, on the right multiples of two, in all the seven numbers that govern the relationship of the microcosm to the macrocosm. In the 12th century copy on the right, the world's soul herself inhabits the diagram. Each manuscript further includes a diagram of the planets, demonstrating how they traverse the zodiac, which however in the copy on the left shows the heavenly spheres not in the Platonic sequence propounded by Macrobius, but rather the alternative arrangement argued by Pliny, with Mercury and Venus between the moon and the sun, an order introduced by Dungal, an Irish monk, astronomer, and poet of the 9th century at the northern monastery of Bobio. Also included are Macrobius's so-called Silly Rain diagrams. These are amongst my favorite diagrams. The Silly Rain diagram designed to prove to flat earthers of the time that bodies with weight, in this case raindrops, fall towards the center of the world rather than right past it. A diagram of the terrestrial climate zones, sorry, that one. Another that maps correspondences between the heavens and the terrestrial zones defined by a zodiacal line joining the tropics of capricorn and cancer. And finally, a map not of the entire world, and hence not a mappa mundi, but rather of the eastern hemisphere in which the inhabited oikumene in the upper half, whose people were in principle open to conversion, is matched by its uninhabited mirror image in the lower. These late antique commentaries continue to be read to the end of the Middle Ages. The Harley manuscript belonged to and was annotated by the greatest philosopher of the 15th century, the aforementioned Nicholas of Cusa, whose writings stand in the platonic tradition of theological speculation expressed in diagrammatic terms. Another manuscript belonging to Nicholas, written at Oxford in the third quarter of the 13th century, contains works by Aristotle on the natural world, including On the Heavens, which opens with an image of Christ as creator. The incoate form of four fused faces at the center of the cosmos denotes the chaotic state of the four elements, earth, air, fire, and water, before the imposition of order upon them in the medieval equivalent of modern physicist's grand unified theory. Diagrams demonstrating the properties of circular versus linear motion adjoin Aristotle's demonstration that, quote, the body which moves in a circle is not endless or infinite, but has its limit, unquote. In keeping with Greek mathematics, Aristotle designates the points or entities whose movements he analyzes with letters. To these, however, the illuminator lends bodies in the form of expressive heads akin to those found in the margins, in addition that lends unaccustomed impetus to these diagrams on motion. In keeping with the platonic wellsprings of his philosophy, Nicholas saw the human capacity to grasp the mathematics as a reflection of the divine. In his own words, quote, in the mirror of mathematics there shines forth truth. The intellect, he adds, is, quote, as an inscribed polygon is to the inscribing circle. The more angles the inscribed polygon has, the more similar it is to the circle, unquote. Anticipating the concept of a limit, he adds, again, even if the number of its angles is increased at infinitum, the polygon never becomes equal to the circle. It was perhaps in the context of the elusive quadrature of the circle that he added an autograph diagram, it's one of only two that survive in his hand, this one dealing with the trisection of angles to a 12th century German compilation of astronomical works by the astrologer Abu Mashar Jafar, active at the Abbasid court in Baghdad in the 9th century. Such diagrams found their way into the formal presentation of Kuza's finished works, as seen in this Italian manuscript of his twin treatises, complementary mathematical and complementary theological considerations. The two topics were complementary insofar as both treated of truth. In Kuza's words, that which must be affirmed in mathematics mathematically must without doubt be affirmed in theology, theologically. God himself was the consummate mathematician. So popular was Microbius that he can be called one of the school masters of the Middle Ages, and of modernity. The 17th century polymath Robert Flood, who I discussed briefly in my first lecture, certainly knew his writings along with their illustrations. Floods engraving of the harmony of the spheres harks back to a Microbian diagram, in this instance added to a mid-13th century French manuscript of Abbasid's metamorphoses. Kuza was likely the intermediary between Flood and medieval tradition. On the right you see Kuza's universal diagram, universal, not in the sense that it is a representation of the cosmos of the universe, but in that it includes or embraces all things. A universal diagram of three triune distinctions leading from the darkness of the sublunary sphere to the brightness of the divinity. In printed editions, which post-date his death, color was eliminated. But in this manuscript made for the cardinal himself, it retains its essential role in lending his diagrams not only dynamism and depth, but also the immediacy of the process of perception that they sought both to structure and to simulate. In raising the soul to God, a thesis, perception, and math thesis, mathematical comprehension, went hand in hand. Microbius was not the only late antique school master who had a lasting impact on European intellectual history. Equally significant were Boethius and Cassiodorus, both of whom transmitted to the Middle Ages the learning of antiquity in condensed form. In this process diagrams played an instrumental role. In addition to specific bodies of knowledge about logic, music, mathematics, rhetoric, or grammar, each supplied diagrammatic techniques that bore fruit for centuries to come. Boethius's diagrams on harmonic proportions provided the cornerstone of medieval music theory. No less influential were the stomata or tree diagrams that Cassiodorus deployed in his institutions to lay out divisions of knowledge. The figural imagery attached to these stomata, usually in the form of animals, mixes metaphor with geometry, thereby making them memorable. A mid-12th century theological miscellany from Canterbury employs similar techniques to subdivide philosophy and mathematics, rooting the branches of moral philosophy in Abraham, those of natural philosophy in Isaac, and those of the contemplative sciences in Jacob. Appropriately tangled is the web of vices in a Gloss Leviticus written in 1176 at the Cistercian Abbey of St. Mary, Bilbas in Shropshire. In contrast, the virtues are more symmetrically disposed and in some of Canterbury's similitudes, a treatise on the vices as well as the virtues in which this diagram accompanies the ramifications of faith, wisdom, and virtue with small exemplary images. In each of these cases, no less important than the diagram's content is the technique by which that content is laid out. Such techniques were transferable from one body of knowledge to another. For example, in a contemporary copy of Augustine's On Christian Doctrine from the Augustinian Abbey at Rochester, Stemma mapped the different branches of doctrine in grammatical terms. In the medieval classroom, geometrical know-how and theological speculation overlapped. We can enter the medieval classroom through the codex. A drawing in the lower margin of an English manuscript of Aristotle's Logica Nova shows two Greek mathematicians of the fifth century BCE, Hippocrates of Chios and Bryson of Heraklion, wielding diagrams much like speech bubbles as instruments of dialogue and debate. The manuscript contains such staples of the medieval university curriculum as the sophistical refutations in introduction to deductive reasoning, the topics on the art of dialectic, which in part addresses the theory of the syllogism, and the Analytica priora at posteriora, another work on syllogistic. All parts of Aristotle's Organon, his collected works on logic. The marginal glosses concretize the oral arguments that characterized classroom debate. The diagrams held by Hippocrates and Bryson exemplify the classic conundrum of how to square the circle, a topic on which both wrote, although Aristotle held their supposed proofs up to ridicule lending the image a satirical note. Pricking holes left by the compass, as well as faint traces of the diagonals defining the squares, testify to the precision the artist sought to achieve. The precision of the geometrical figures matches that of the overall misempage, which is ruled to allow for layer after layer of commentary, marginal, and interlinear. The challenge of squaring the circle linked to the problem of pi was no ordinary puzzle. Problem 50 in the British Museum's Rhined Mathematical Papyrus, which dates to the mid-sixth century BCE, asks, actually mid-sixteenth century BCE, as, quote, a circular field has a diameter of nine ket, about 50 meters. What is its area, unquote. The answer entails a remarkably precise approximation of pi. As an irrational number, pi figures as an emblem of impossibility. This problem introduces the double-sidedness of the diagram in the Middle Ages and of its extraordinary, even paradigmatic appeal to medieval thinkers and artists. On the one hand, as an instrument of logical demonstration, the diagram represents reason. Very often the inclusion of a diagram in medieval manuscripts serves less to map out a specific argument than as an invocation of authority whose presence on the page lends credence to a set of propositions. Many diagrams make their case less by logic than by metaphors, in turn translated into figuration, which by definition, the word comes from the Greek metaforafum metaferin, to transfer or carry over, it involves less lines connecting one point, whether mathematical or logical, to the next, than leaps, including leaps of faith. Does history or do concepts really unfold like the ramifications of a tree? Whereas in the role history unfurls in a line of descent, the trees which symbolize fertility grow upward. How organic are the degrees of consanguinity mapped out in this late 12th century version of the diagram from the etymologies of Isidore of Seville, whose form, crowned by a cross, is more reminiscent of a metalwork shrine? What difference does it make in the same manuscript to envelop another such diagram with elements appropriated from a tree of Jesse? In such cases, visual metaphors have the virtue of lending the disposition and dependencies of concepts a poetic persuasiveness that were they simply laid out in expository prose they might otherwise not possess. In this double character of the diagram, its liminal status at the boundary between the physical and the metaphysical, the rational and the irrational, the visible and the invisible, lay its appeal to medieval exegetes and artists. The very appearance of diagrams, their elementary abstraction and etiolated forms stamped them as exemplifications of thought that stood at the boundary between what is visible and what is not. Medieval philosophy, moreover, did not recognize the distinction between separate realms of reason and faith. Geometry itself could prove paradoxical. Sylogistic logic was applied to the mysteries of the Trinity. Nicholas of Cusa argued that a triangle is analogous to the Trinity. No ordinary triangle, however, the paradoxical triangle to which he referred had three right angles. God, moreover, according to a widely circulated comparison, was like a circle whose center is nowhere and whose circumference is everywhere. Augustine had sought to ban the application of such metaphors to theological truths on account of their misleading character. Metaphor, however, which extends knowledge of what is known to that which is not or cannot be defined, had the virtue of lending the paradoxical concepts at the heart of the Christian credo in apparent logic. Within this scheme of things the medieval diagram participated in a dynamic dialectic of revelation and concealment, one in which the viewer was invited to participate. To look at a diagram was to be instructed, but it was also to be initiated. Let us return to the classroom by turning to a mid-12th century Parisian manuscript of the Isidogue, an introduction to Aristotle's categories by the Greek logician Porphyry that's circulated in a Latin translation by Boethius. This Parisian manuscript, still in its original soft binding, dates to the middle of the 12th century, which makes it more or less coterminous with the founding of the University of Paris. Its illustration places the reader in the midst of a scholastic disputation. Lady dialectic, a towering figure reminiscent of Lady Philosophy as she appears to Boethius in a dream at the beginning of the constellation of philosophy, stands at the center, holding in her left hand a snake, a symbol of shrewdness. In her right hand she holds the no less serpentine porphyrian tree of genera and species named after the third century Neoplatonic philosopher that became a classroom classic. To call this diagram a tree is to enter the realm of metaphor in ways that are problematic in that unlike any tree in nature, this one grows not from the ground up, but rather from the top down, suspended from the tree of the cross, which here serves to ground it, again a metaphor, in a form of truth higher than that of nature. At the top of the tree substancia occupying the central roundel sprouts two branches named the corporeal and incorporeal. The subsequent set of terms divides corpus, body, between the animate and inanimate. The final division distinguishes the rational soul between man and god. As noted by Umberto Echo, the porphyrian system employs the dictionary model of semantic representation as opposed to the encyclopedic. Rather than attempting to describe a given thing or term as it exists in the world with all of its accidents, it analyzes it linguistically by taking into account only those properties necessary and sufficient to distinguish that particular concept from others. Those are Echo's words. The dangling vine-like tree arranges all of creation into a ladder-like set of hierarchical divisions that ultimately lead to god. That the differences defined by the tree are accidents and that accidents can be multiplied indefinitely is a problem of categorization that the diagram conveniently allows. The image, however, offers more than the diagram alone, which would have sufficed for didactic purposes. It stands at the center of an exemplary dialogue as if to show how it should be used performatively. At the top, Plato engages Aristotle in dialogue. Below Socrates and Magister Adam likewise converse. Perhaps identifiable with the 12th century Parisian master Adam of the petit pont, Adam is also the exemplar of all humans, one of the subdivisions of the diagram's final set of categories. Adam does not simply represent the category of humankind, however, as the one who had assigned names to the animals he represents the quintessential philosopher and logician. His act of discrimination and denomination underwrites the use of inscriptions in medieval images. In effect, the discussants in the corners supply the learned disputation in which the reader is expected to participate. Some learned graffiti scratched on the south tower wall of the Romanesque church at Bro in Gotland, Sweden, most likely during the 13th century, brings us right inside the medieval classroom. The incised diagrams are perhaps unique in permitting us to witness these tools for thinking in statu nascendi, that is, in the process of being used. They effectively present us with a medieval blackboard that was never erased. A school teacher inscribed into wet plaster, a syllogism, and several versions of the square of opposition, along with some associated terminology, as well as a cosmological diagram of the four elements whose relations and transformations, as we saw in the first lecture, could be explained in terms of the square. Also present is a summary of the threefold nature of statements, a topic that accompanies the presentation of the square in logic textbooks. The first damaged square at the right displays the diagram derived from the categorical syllogisms by Boethius, seen here in a 13th century scholastic miscellany, including variants of the proposition regarding the just man derived from chapter seven of Aristotle's On Interpretation. These read, all men are just, a universal affirmative, no man is just, a universal negative, a certain man is not just, a particular negative, and a certain man is just, a particular affirmative, the very same syllogism scratched into the plaster at Brood. The second diagram represents a modal square diagram constructed around four propositions regarding necessity and possibility, which are supplemented by the technical terms contradictory, contrary, subcontrary, and subaltern. I think those people up there are having much more fun than we are. Logic stood at the core of the new learning. The square I just showed you comes from an exceptionally elegant scholastic miscellany made between 1309 and 16 for Francescus Caracciolo of Naples, Chancellor of the University of Paris. The manuscript could not have a more distinguished provenance. It passed to Robert of Anjou, King of Naples, Pope Gregory XI, the Antipope, Clement VII, and then the great bibliophile Jean du Tiberi. In addition to Pritian, Cicero, and Pseudo-Cicero, Ptolemy, the latter and Gerard of Cremona's translation from the Arabic, it also includes set texts by Boethius, Aristotle, Euclid, and Adelard of Bath, a natural philosopher and translator of Greek and Arabic science. These texts between them cover the gamut of the liberal arts. In short, it offers a curriculum between two covers. For some subjects, for example, music and astronomy, diagrams occupy the margins. In other instances, however, we see diagrams brandished by personifications of the arts. Introducing Pritian's institutions of grammar, Lady Grammar holds a tree that diagrams the parts of speech divided into congruence and incongruous constructions. To the right, a boy who has failed to learn his lessons holds out his hand to receive the rod, in this case at the hands of the beetle, the university's disciplinary officer. Another tree introducing Cicero on rhetoric lays out the five faculties of invention, among them disposizio, disposition, the order and distribution of matter, and memoria, memory, both functions to which diagrams were ideally suited. A third tree diagram illustrating the branches of dialectic introduces Porphyry's Isogoge, an introduction to Aristotle's categories, one of many texts on logic central to the university curriculum. Two students engage in a lively dialectical debate. Yet another diagram in the form of a flowering plant introduces Aristotle's Analytica priora, part of this same curriculum, and it identifies the four leaves with four different kinds of propositions, demonstrative simple dialectical and sophistic, and the flower itself with the resulting syllogism. For medieval students, whether in Paris or at Peru, diagrams did more than merely illustrate ideas. They provided tools with which to think, argue, and create. This initial from Caraccio Lo's miscellany, prefacing Euclid, Euclid's elements in the translation by Adelaide of Bath, depicts Lady Geometry using calipers and T-square with which to draw explanatory figures such as those in the right margin produced using the very same instruments. Dynamic generative geometry and the diagram's experimental utility went hand in hand and remain critical to understanding how they work, no less now than then. Diagrams far from simply illustrating ideas or serving as handy aid memoir serve to generate new knowledge. Diagrams do more than package predetermined content. By mimicking the process of ratiosination itself, they recapitulate and reinforce the processes of thought by which that knowledge is generated in the first place, and at the same time engender fresh patterns of thought. Each time a diagram is drawn or its lines are retraced in the mind's eye, a thought process is reenacted step by step. Diagrams, whether medieval or modern, do not simply represent a set of ideas or even map the relationships among them. Like machines for the mind, they ideally have to be recapitulated, starting from scratch each time they are used, as if in making use of them any given reader had to retrace the connections they establish. Diagrams thus have an inherent temporal aspect. Moreover, they embody technique. They help the user remember the terms of the discussion. They also exemplify and generate rules that in turn can be applied to other questions. In this context, the dead metaphor of drawing conclusions takes on fresh meaning. Even if one doesn't draw a diagram, one still has to recapitulate in one's mind the steps it represents. Diagrams thus have a processual aspect. Operative thinking of this kind is hardly unique to the Middle Ages. It also characterizes the conceptualization of problems in computer software. Here, to quote a software engineer, syntactic possibilities extend into time and space as they define transposition operations that facilitate the dynamic exploration of information. Information is entered or played against time. The analyst, the computer analyst, enters the diagrammatic space and what was formerly a snapshot now becomes a flow of information, unquote. In the Middle Ages, no less than today to draw a conclusion was to enter into such a flow. The Middle Ages was familiar with this way of thinking about diagrams from Aristotle's treatise on memory and reminiscence. In this copy, which belonged to Nicholas of Cusa, part of a compendium of writings attributed to the Staggerite and made in Paris at the time when his influence on the curriculum was at its height, Aristotle contemplates the figure of man. All the illuminator needed to paint this initial were the schematic prompts of which one can still see traces in the lower margin. Aristotle distinguishes between memory common to mankind and animals and remembering, which in his view is uniquely human, which is why the initial portrays him in contemplating a naked man. Diagrams, it turns out, are critical to the distinction he wishes to draw. Having affirmed, and I quote, that it is not possible to think without an image, this is a famous passage, Aristotle adds, for the same effect occurs in thinking as in drawing a diagram. Aristotle does not say that in order to think one has to draw or even the drawing diagrams helps with thinking, although it does. Rather, as part of his effort to distinguish thought from imagination and memory and humans from animals, he draws his reader's attention to the effect that thinking and generating a diagram have in common, the requirement to work things through step by step. In comparing cognition to the process of drawing, Aristotle goes beyond asserting that a diagram represents the content of thought. Rather, the procedure of producing the diagram resembles and even enables the process of thought. Both in his view involve the method of defining and drawing relationships that point towards particular conclusions. If drawing a diagram in the process of thinking represent two sides of the same equation, then it is not simply a matter of thought generating diagrams, but also of diagrams generating thought. Rather than a representation, the diagram structures the patterns according to which one thinks. The diagram is more than a mere representation, it is active or operative. The closest we get to any medieval text to a recounting of the moment at which a diagram first was formulated occurs in Hugh of St. Victor's Dieta Scalicon, a manual on teaching, alternatively titled De studio Legendi on the study of reading, a treatise on the seven liberal arts written in the late 1120s, of which this 13th century manuscript, which opens with a sermon on the vision of Daniel, includes a substantial excerpt. The manuscript mixes prose and poetry, Latin and vernacular, Anglo-Norman, as well as religious and secular texts. Unlike Daniel, Hugh looks for inspiration not to the heavens, but rather to the earth, stating, quote, I dare to affirm before you that I myself never looked down on anything which had to do with education, but that I often learned many things which seem to others to be a sort of joke or just nonsense. Often I proposed cases and when the opposing contentions were lined up against one another, I diligently distinguish what would be the business of the rhetorician, what of the orator, what of the sophist. I laid out pebbles for numbers and I marked the pavement with black coals and by a model placed right before my eyes, I plainly showed what difference there is between an obtuse angled, a right angled, and an acute angled triangle. Whether or not an equilateral parallelogram would yield the same area as a square when two of its sides were multiplied together. I learned by walking both figures and measuring them with my feet. These were boyish pursuits to be sure, yet not without their utility for me, nor does my present knowledge of them lie heavy upon my stomach, unquote. Perhaps the key element in Hugh's account consistent with the weight he gives to observation and experience is his reliance on visual aids. Rather than thinking through problems in his head, he externalizes them, not simply to enable the teaching of others, but to master the problem for himself. The scene he describes takes us all the way back to that of Greek philosophers arguing over drawings scratched in the sand. What reveal nets consistent with theories of the extended mind has characterized as intersubjective objects. Hugh's ambitious theological treatise, The Mystic Arc, evokes a diagram of enormous complexity, comprehending the whole of salvation history. Addressing his readers, Hugh speaks of the figure of this spiritual building that I'm going to present to you, namely the Ark of the title. I depict the Ark as an object, he states, so that you may learn outwardly what you ought to do inwardly, and that once you imprint the form of this example in your heart, you will be glad that the house of God has been built inside of you, unquote. Once again, architecture is linked to edification. No trace of any such drawing survives in any of the manuscripts, and the degree of detail mapped out by Hugh defies depiction. Did Hugh's didactic diagram perhaps take some other form, if not codified in a book, than perhaps as a set of parchment sheets hung on a classroom wall? That such objects existed, we know from Gerber of Oriac, a polymath, later pope Sylvester I, who in the late 10th century writes, it is the best scholars that last autumn, for whom I drew up a diagram of rhetoric on 26 leaves of parchment sewed together, and forming in all two columns side by side each of 13 leaves. It is without doubt a device admirably adapted for the ignorant and useful to the studious scholar in order to help him understand the subtle and obscure rules of rhetoric and to fix these in his memory. The British Library holds a miscellany of works on geometry, astronomy, and mathematics with diagrams illustrating works by, among others, Boethius and Gerber. There is however no trace of the figure of rhetoric. Whatever its format, two single columns each 13 leaves long represents a great deal of real estate. It is hard to imagine that the principal purpose of such an object was simply to provide material for rote memorization. One can more easily imagine it as a conversation piece unfurled in a classroom, whether on a wall, a table, or a tablet, such as this set from the second century. Consisting of bound wooden leaves, such tablets represent the origin of the codex per se. The word codex, from which the term codex derives, means a block of wood. On the first tablet, a teacher wrote two iambic maxims by the poet Menender on two ruled lines. A student then copied them twice between ruled lines. And on the second, the teacher wrote out multiplication tables on the left, whereas on the right, a student employed a grid to break down five words according to their syllabic structure. Leafing through scholastic manuscripts, such as this collection of biblical study aids, one quickly becomes aware of the extent to which the parchment page, once ruled for writing, offered not the proverbial blank slate, but rather like graph paper, a field that invited the creation of geometrical forms, whether blocks of text divided into columns and differentiated by size, or images that took advantage of this predetermined linear armature. Into the overall mise en page, words and pictures created a flexible matrix governing the relationship of text to text, text to image, or image to image. Before a word was written, the page is prepared by describe established patterns of interpretation that in turn, through memory, fostered corresponding habits of mind, a diagrammatic mentality. Once mastered, the skills required to encode and decode this specialized visual vocabulary could be applied to a broader range of texts and images. Traces of diagrammatic habits of mind structure both the original layout and the annotations of readers witnessed the stomatic loss in the lower margin on the verso, becoming dominant in the collection of theological diagrams conjoined with the biblical commentaries, a firework of diagrammatic invention. With an excerpt from a text known as The Morals of the Philosopher's Dogma, a 12th-century ethical treatise apparently known to Chaucer, a reader resorted to diagramming the gloss by using a compass to generate intersecting arcs and circles. For whatever reason, inscriptions, if ever intended, were never added. If diagrams themselves were generative in nature, so too was the geometry by which they were produced. In modern computational mathematics, generative geometry investigates the algorithms, rules, or logical grammars that govern the creation of certain shapes and patterns. In the Middle Ages, similar, if simpler, sets of rules were used to generate complex designs, whether architectural plans, pinnacles, or vault designs. It is therefore hardly surprising that the geometrical armatures on the last page of a 13th-century English manuscript of Peter of Poitier's Compendium have been described as designs for a rose window and a round-headed lancet window, respectively. It is not impossible that whoever produced them took architectural forms as his point of departure, but the armature resembling a lancet window was clearly intended to serve as a table in two columns. That architecture provided one of the principal metaphorical matrices for diagrams should not come as a surprise. Diagrams constituted cornerstones in the edifice of edification. Among the most famous diagrams of the Middle Ages, that depicting the liberal arts in the Hortes de Lisiarum, or Garden of Delights, of Herod, Abbas of Hohenberg, and Alsace, in addition to providing an encapsulation of an idealized monastic curriculum and the foundation of philosophy, whose personification sits enthroned at the center, draws on the idealized geometry of a rose window to depict the seven liberal arts occupying the biblical house of wisdom defined by its eight pillars. In contrast, the designer of another incomplete diagram had in mind eight sets of ten, all radiating from the center. Still more complex in its multiplication of elements, 61 in all, is the calendrical and cosmological program of the window in the south transept of Lausanne Cathedral, designed by Pierre de Haas, and dated around 1225. Viad on Angkor, like Pierre de Haas from Picardie, and most likely not an architect himself, but someone who was clearly interested in contemporary architecture, includes in his famous portfolio a variant of the window generated by the rotation of squares, which can only be affiliated with Lausanne on the basis of inscription, coming to a conclusion. Diagrams chart movements of the mind. In thinking them through, we effectively retrace the movements of the draftsman's hand. As in floods, microcosmic man, hand, eye, and mind are coordinated. Some diagrams have actual moving parts in the form of vovelles, a device derived from the astrolabe so designated after the Latin term vulvora to turn. In this astronomical miscellany from the collection of Sir Hans Sloan, the eminent physician whose bequest of 71,000 items to the nation constituted the cornerstone of the British Museum and Library, four vovelles, one in the form of a human figure, occupy two openings. An early modern cabalistic digest by Samuel Gallico of Moses Cordervo's orchard of pomegranates, aptly named the juice of pomegranates, incorporates ten vovelles, one for each of the sephirote or emanations of the godhead. Such sets convert diagrams into combinatorial machines akin to calculators. In the end, however, all diagrams entail similar motions of the mind. Those with moving parts only actualize what remains latent in all diagrams, which invite the viewer to make connections and link data sets according to prescribed pathways. A set of diagrams added at the front of the Toledin tables originally assembled in the 1080s by a group of scholars in Toledo working from Arabic materials offers an example of this phenomenon. The uppermost diagram, animated by an angel and two Franciscan friars, in fact, consists of instructions for the use of the other two. The friars point to the circumference consisting of a circular zigzag that divides the circle into legible segments. Back to back, they hold a tablet on which are inscribed instructions on how the underlying diagram derived from the calculations of Dionysius Exiguus, who invented the familiar system of dating from the year of the Lord, the notation A.D., should be used to determine the number of years since Christ's birth. Beneath the tablet, the right hand of God emerges from a cloud to point to the diagrams below, thereby identifying the diagram as a divine dictate. The two scrolls held by the friar on the left identify the surrounding circles as indicators of the solar cycle and the dominical letters respectively, whereas that held by the angel indicates the current year. The gestures are not precise. Instead, they perform how the diagram should be put into action. The diagram, indeed, all diagrams, enact motions of the mind, bringing it to life in the hands of the viewer. The Catalan philosopher, poet, and mystic, Raymond Lull, who died in 1316, introduced the vovel to the Latin West. Lull applied his combinatoric art to topics ranging from law and theology, mixing mystification and elucidation in equal degrees. Gottfried Wilhelm Leibniz's calculating machine, the first of its kind, derives in part from Lull's diagrams and shared with them their ambition of being able to solve all the world's problems. When Leibniz published his dissertation De Arte Combinatoria in 1666, it was to Lull that he turned. Leibniz's method was modern, but his frontispiece descends directly from medieval diagrams of the Pythagorean tetrad of the four elements. His grandiose premise was that just as words are combinations of letters, so too all concepts represent combinations of a small number of basic terms, in effect an irreducible alphabet of so-called primitive ideas. Combinatorial logic diagrammatically expressed, could he held, generate all possible combinations of subjects and predicates, in short, all that could be thought. The fascination of diagrams, I hope to have persuaded you, is not only that they too encompass much of what has and can be taught, but also, and more important, that they hold a key to the nature of thought itself. Thank you. I realized my lecture was a tad on the long side tonight, but if there are questions either from those of you who are here or from those of you who are listening online, I'd be happy to try and address them. I have a question from the internet. I'm interested in the role of Franciscans may have played in education. Are they well known for their use of diagrams in education? Well, that's an interesting question. I'll confess that I haven't tried to break down diagrams by religious affiliation or order. One of the manuscripts that I showed, this one, was clearly made for or at least altered by Franciscans. This is a manuscript that originated in Spain, but to judge from the style of these drawings ended up with Franciscans in Central Europe, perhaps in Prague. I strongly suspect that the use of diagrams was more or less universal. To take one famous Franciscan example, Bonaventure, the cardinal and famous theologian, some of his treatises are predicated on diagrammatic structures. His treatise, The Tree of Life, for example, is essentially the exposition of an imagined diagram, which in turn was translated into images, whether in the form of drawings and manuscripts or full-blown mural paintings, panel paintings also. So that's a long-winded way of saying, yes, Franciscans, Dominicans, I think diagrams were used across the curriculum and certainly any order that had a hand in university education was well versed in diagrams. Please. Thank you. That was absolutely fantastic. I'm trying to get my head around how much do we know about how the practicalities of how the diagrams were used in the classroom. So you've kind of hinted at things. You've talked about unfurling diagrams and then with the bro example you showed sort of essentially a diagram that was copied. Did people turn up with their own manuscripts? Well, this is a very good question and one to which I wish we had more of an answer. All we have are the diagrams that have been transmitted to us. What makes the graffiti in that church so extraordinary is that it suggests that we are witness to the creation of these diagrams much in the way that one might watch, say, a physicist or someone today inscribe formulas or diagrams on a blackboard. But that's a unique survival as far as I know. Textual references to the use of diagrams are also relatively rare in the same way that references to how images in general were used in the Middle Ages are quite infrequent. But I do think that we have to imagine these diagrams not simply as illustrations of ideas, but as tools for thinking that were used to elaborate the ideas in the first place. Very similar to the way in which historians of Greek mathematics have invoked the scenario of a group of mathematicians gathered in a circle drawing diagrams in the dust or in the sand thinking something through together by way of demonstration. I wish we knew more. It's one of the things I love about Hugh of St. Victor's account. It's very cunning because he's writing, if not for students, then for those who will be instructing novices. And he evokes his own experience, which is a very characteristic move on his part. He wants to show that he too was once a beginner, but he talks about that very process of working things out through these external objects, whether pebbles or chalk drawings, charcoal drawings, in a very evocative way. Gerber of Oriac, whom I mentioned, talks about using diagrams that unfortunately don't survive. There's a common rule when it comes to medieval art. I sometimes tell my students that if we have a text, we don't have the image, and if we have an image, we don't have the text. There are exceptions to that rule, but in most cases, we have this fractured mosaic, and so we have to try and put the pieces together. Yes. I think that after we're done here, we should all go up and join the party upstairs. They're having a wail of a time. We'll come to you, Anand. Okay. Fantastic diagrams. I'm wondering whether you've observed any trends in a relation of painting color or pigment or font in relation to the topic that diagrams were trying to put forth. Are there any interesting trends that you noticed? Well, color in diagrams is a topic that interests me also in relationship to the history of printing, because color, aside from Gutenberg's initial efforts to introduce colored initials into the first choir of the Gutenberg Bible, a process that proved so complicated or so costly or inefficient that he gave up on it, printing could easily accommodate different colors until well into the second half of the 15th century. And indeed, as far as I know, the first diagrams that include two colors, red and black, come in a calendar associated with the writings of Reggio Montana, a famous mathematician, and the different colors are used to explain or illustrate eclipses. In the case of Nicholas of Cusa, which is one example that I've given a lot of thought to, in really quite novel ways, he makes colors central to his diagrams, not simply to make them more legible or for that matter to make them prettier, but because the diagrams effectively set out to demonstrate the very process of perception to which color is critical. They're epistemological instruments, as well as ontological illustrations. And so he devises a novel way of coloring his diagrams, and we know that he had a hand in it because he left handwritten instructions in his own hand for the painters. And I find it fascinating that if you look at the very large body of scholarship on Nicholas of Cusa, it's all predicated on the Edizio Prinkeps, the first edition of 1488, well after his death, a generation after his death, in which his volumetric diagrams, which in keeping with his dialectical thought, have darkness in the light and light in the darkness, are reproduced in black and white. So effectively, the whole history of Nicholas of Cusa's thought, which is diagrammatic to its core, think of his complementary treatises on mathematics and theology, has in some ways been predicated on a false assumption in so far as it leaves color out. I'll just add one more thing to a perhaps too complicated answer. Nicholas of Cusa learned a good deal about color from one of his teachers at the University of Cologne. He had given up his career as a canon lawyer and switched to theology under the influence of this fellow Hemerick of Kemp, who taught theology at the University of Cologne. And Hemerick of Kemp used diagrams as exemplifications and of instruments of knowledge in his treatises, of which Nicholas had copies in his personal library, half of which survives here in the British Library, thanks to Robert Harley's agents, and the other half is still in Cua's near Trier. But Hemerick of Kemp used color in a much more traditional way. He used it extensively, but he used it symbolically, green for life, blue for the heavens, red for the passion, and so on and so forth. So I think it's in part from there that Nicholas of Cusa derives his fascination with the color diagram, but by introducing modulations of light and shade, he does something much more radical, and I think there he must have been looking at contemporary painting, particularly Netherlandish panel painting in which we know he took a keen interest because he mentions it in some of his other treatises. So that's just one example, I guess to sum up, I would say that color can play a symbolic function in medieval diagrams, but as far as I know it's not till Nicholas of Cusa in the 15th century that it begins to take on a more, for lack of a better word, epistemic function within the processing of the diagram itself. Very stimulating lecture, thank you. Contrary to the theme, are there examples in which the power of a diagram actually impedes the development of thought? I'm thinking here of the extraordinary persistence of the Ptolemaic sphere. That's a very good question, and I think diagrams do serve a rhetorical function as I tried to suggest, and there are instances, it's a little bit like the way I suppose some people today brandish statistics in order to impress upon their readers the validity of an argument that may not in fact be so readily verified or demonstrated, and there are instances in which diagrams are used in that sort of hand-waving kind of way, and your question raises interesting issues in terms of how diagrams might have been tested, and certainly a great deal of medieval thought was predicated on received knowledge, not that people didn't think critically and experimentally. A manuscript I could have shown here in the collection, I believe there's a wonderful manuscript here in the collection of the British Library of some of the manuscripts of Bacon, the 13th century philosopher who was so interested in optics, but authority plays a tremendous role, and the Ptolemaic system, its persistence is an excellent example of that. So in a word I would say yes, there are cases in which a diagram perpetuates what we, with the benefit of hindsight, might identify as a false view of the world, but as an historian I have to say I'm just as interested in why people might have continued to believe what they believed. It's very easy to look back and hold these past systems of thought up to scorn, but they have their own integrity and their own reason that, but thank you for an interesting question. Thank you, I'm being told that we should draw things to a close. I'd happily stay all night, but the party seems to be over, so I guess it's time to go.