 Hello and welcome. My name is David Pearson. I'm the Chairman of the Selection Council of the Paneetsey Foundation, and I am delighted to welcome you here for the 37th series of the annual Paneetsey lectures. We're in a slightly strange space this year. We're usually in the British Library auditorium, but unfortunately they had a big electrical fire in the summer. But the events team very cleverly came up with the suggestion to do it here, and we are very grateful to the events team for all their working making this possible. And I must say I rather like the idea of being here. It does give a sense of being right at the front and the heart and the centre of the British Library, which is very much where the Paneetsey lectures ought to be. And we can once again welcome the whole world because through the wonders of modern technology and contemporary live streaming, we're hugely grateful to Jonathan Hill, the New York Bookseller for sponsoring the costs of that live streaming. And I gather that there are about 500 people across the world as well as you good folks here tonight. So wherever you're tuning in from around the world, you are indeed very welcome. The Paneetsey lectures were founded about 40 years ago by Catherine de Vasse, who was a longtime lover of both books and of the British Library. And the series was named after one of the library's great heroes, Sir Anthony Paneetsey, who served the British Museum for nearly 40 years and was its principal librarian in the 1850s and 1860s when the British Library was back in the British Museum Library. Paneetsey was a moving force in creating the great cultural institution which is the British Library that we know today. He created the library's catalogue, he built its famous round reading room and he turned it into the biggest library in the world. The lectures build on his achievements by using the library's resources to advance knowledge and understanding in any field associated with the history of the book. We're always keen to draw in as wide an audience as possible to be excited to be inspired by discovering more perhaps in unexpected ways about the history of books and about the British Library's collections. So since 1985 that brief has been fulfilled by a distinguished series of scholars and it's the challenge and the privilege of the Paneetsey selection council to sustain that roster. So this year we are delighted to welcome Professor Jeffrey Hamburger who is the Cuno Frank Professor of German Art and Culture at Harvard University. Through a long and distinguished career teaching at Harvard for over 20 years he's specialised in many aspects of medieval art and his many books have focused on medieval diagrams, illumination and the use and meaning of the visual in medieval culture more generally. His various fellowships and memberships take in the Medieval Academy of America, the American Philosophical Society, the American Academy of Arts and Sciences and he was recently awarded the Gutenberg Prize of the International Gutenberg Society. The CV which he sent us ran to over 20 pages so I could go on but when he sent it he did laudably ask for a short and sweet introduction to save time for substance. I think we can all applaud lecturers who take that approach and it is without a doubt that it's him you came to hear, not me. His theme is Medieval Diagrams and so I am delighted to clear the stage for his first lecture which is Maps of the Mind, Diagrams, Medieval and Modern. Jeffrey Hamburger. Thanks David. Is the mic live? Clearly it is. Well first of all thank you David and my thanks also to the selection committee for giving me this marvelous opportunity and thanks to you all for coming this evening. The pre-Socratic philosopher Thales of Miletus Apuleus tells us, quote, was easily the first among the famous seven wise men for he was the first discoverer of geometry among the Greeks. The most accurate investigator of nature, the most skilled observer of the stars. He used small lines to find the greatest truths, the circuit of the seasons, the direction of the winds, the course of the stars, the portentious crash of thunder, the oblique paths of the planets, the annual revolutions of the sun, and similarly the phases of the moon as it waxes when new, wanes when old or disappears when eclipsed, unquote. To use small lines to find the greatest truths, this phrase aptly characterizes the function of diagrams in the Middle Ages, whether in scientific or religious discourse. Diagrams constitute an omnipresent feature of medieval art and thought. From antiquity onwards the forms and procedures of geometric reasoning held a privileged place in the pursuit of truth, the understanding of which remained closely linked to the ideals of beauty and perfection. Drawing on the collections of the British Library, whose holdings embrace all ramifications of the diagrammatic tradition, these lectures examine practical, theoretical, and aesthetic dimensions of medieval diagrams as matrices of meaning and patterns of thought informing diverse areas of medieval culture. This alone explains why I chose diagrams as the subject of this series. Diagrams have always been and continue to be maps of the mind. This illustration to Robert Flood's Utrius Que Cosme, The Two Worlds, published between 1617 and 1621, the full title of which, in translation, is the metaphysical, physical, and technical history of the two worlds, namely the greater and the lesser, could be said to mirror my mind, and perhaps yours too, by the time I was born. I am these lectures are over. Deeply indebted to medieval traditions, Flood's image contains much of what I wish to convey, and I will spend much of this first lecture unpacking its contents. The title page to Flood's Magnum Opus doubles as a stymatic table of contents, itself a diagrammatic form. Flood's work aspired to comprehend all that there was to be known, theoretical and practical. Volume one is divided into two parts of which the first deals with the metaphysical dimensions of the macrocosm. The second with the physical, including the arithmetical, musical, geometrical, and perspectival, as well as the pictorial and the military arts, the science of time and motion, also cosmology, astrology, and geomancy, or divination. I too will touch on most, if not all of these topics, and like Flood, I will do so using diagrams without, I trust, setting your head spinning as Father Time does to the diagram in the images lower half by pulling on a chord that unwinds the cosmos as if it were a giant mechanism. Diagrams exhibit an uncanny ability not only to map the mind, but also to set it in motion. Flood's image of the processes of thought founded in sense perception, but aspiring to the heavens, serves not simply to illustrate, but further to demonstrate the intimate integration of the microcosm, that is man in particular his inner life understood in terms of the faculties of the intellect, reason, and the imagination, and the macrocosm, that is exterior creation. The macrocosm extends from the sensible world represented by the concentric circles in front of the man's forehead, as detected by the five senses, each embodied by one of the five lines leading respectively to his hand, lip, nose, eye, and ear, to the suprasterestial spheres constituted not by the elements themselves, but rather their likenesses, or shadows as Flood calls them, which reach from the earth to the heavens in the form of fire, in short their forms is perceived by the soul, in particular the imagination, and finally further still to the world of the intellect, connected to the mind by the hierarchy of angels wherein dwells the highest reality, namely the trinity. All in all, the image crams in a great deal. Flood's concatenation of interconnected diagrams testifies to his confidence in the ability of images to convey, and no less important, create knowledge. It also manages to map out his entire philosophy, which combines neoplatonized Christianity with Renaissance Hermeticism, astrology, alchemy, as well as Rosicrucian and Kabbalistic lore. Flood's diagrammatic image is at least superficially reminiscent of the illustration to Fritz Kahn's Das Laban Dismension, The Life of Human Beans, published in 1924, here in an English edition, which combined figurative realism with Bauhaus functionalism. It also brings to mind the mechanized bodies of Duchamp and Picabia, in which however the optimization of the body's inner operations deconstructs desire. Rather than imitating the body, the body imitates the machine. We are presented with two radically different visions of the mind and its relationship to the world, the mind-body dualism that dominates Western philosophy, of which Flood's version embodies the idealist version, according to which physical states are really mental, and of which Picabia's, to the contrary, embodies the materialist, according to which mental states are really physical. What the two conceptions have in common, however, is an understanding of the mind that is captured and conveyed through diagrams, a way of thinking that can be traced back to the Middle Ages and beyond to antiquity. Flood's diagram maps the workings of memory. It also makes them memorable. To that extent, it is remarkably self-referential in ways that make one think of one of the perpetual motion machines in whose invention Flood invested so much energy. In acting on the diagram, our own mind set that of its subject in motion, just as the motions of his mind act in turn on ours. Diagrams possess this uncanny capacity to allow ourselves to see our minds at work. Flood's image draws on the visual rhetoric of anatomical sections such as this one in a medical miscellany, which depicts the seven tunics and three humours of the eye and skull. In contrast, Flood dissects the spiritual as opposed to the corporeal man. In like fashion, I would like to unpack the layers of Flood's diagram to unfold the genealogy and typology of medieval diagrammatic method. I am more interested in the how than in the what of diagrams. In other words, while the content of the diagrams is critical, I'll be focusing on what makes them tick. Flood's worldview likely strikes most of you as utter nonsense, but as a diagram, it is ingenious in ways that lay claim to our attention. Once again, we live in a diagrammatic age. Think algorithms, machine learning, artificial intelligence, graph theory, and neural networks. The latter defined as, quote, a series of algorithms that endeavor to recognize underlying relationships in a set of data through a process that mimics the way the human brain operates, unquote. Scientists debate the extent to which computers can mimic brain and mind function, whether by means of algorithms, flow charts that map step-by-step processes incorporating feedback loops, or by means of neural networks, collections of interconnected nodes. It is not just that diagrams continue to perform critical roles as instruments of explanation, but rather that today, as in the past, they provide compelling analytical models for how both the mind and nature floods two worlds work. Over the past 50 years, diagrams have enjoyed a renaissance in fields as diverse as cognitive and computer science to physics and the philosophy of mathematics that has rescued them from the disdain in which they were held by mathematicians, philosophers, and physicists alike from approximately the middle of the 19th century. Those without a training in higher mathematics, and that includes me, might well assume the geometry in Euclid, represented by this early 14th century Italian manuscript of his elements, in a translation by the 12th century English natural philosopher Adelhard of Bath, are synonymous. But in 1882, the German mathematician Moritz Pasch declared, quote, that if for the grasp of a proof the corresponding figure is indispensable, then the proof does not satisfy the requirements that we imposed on it. In any complete proof, the figure is dispensable, unquote. In a similar vein, in his Foundations of Geometry published in 1894, David Hilbert stated that, quote, a theorem is only proved when the proof is completely independent of the diagram, unquote. To take an extremely simple example, a circle with a radius r can be represented by means of either a diagram or a sentential equation. To the question of which is better the visual or the algebraic representation, Pasch and Hilbert would have answered unequivocally the latter. More recently, however, diagrams have been formalized and utilized in computer science, and nowhere have they more famously staged a comeback than in physics. Feynman diagrams are not just nifty representations of how subatomic particles interact, but tools that capture in mathematically exacting terms the nature of those interactions in ways that are both precise and predictive. Penrose diagrams figure among contemporary cosmologists critical tools. The debate or better dialogue between figures and formulas continues. Today diagrams play a central role in fields as diverse as combinatorics, set theory, and graph and network theory, all of which are used by neuroscientists to try to unlock the mysteries of the human mind. Perhaps we are not so different from flood after all. At issue is not just a critical chapter in the history of the diagrammatic method, but further still the nature of the diagram itself. Are diagrams representations? If so, of what precisely? Or are they rather instruments, tools for thinking that generate knowledge? If that is the case, do they mimic the actual structure of cognition? Speaking of cognition, why in keeping with theories of extended cognition do we find such externalized devices so useful? Are diagrams simply spatial metaphors, or are they models? To what extent are their forms culturally specific? These are just a few of the questions that present themselves. Let's see to what extent the Middle Ages provides some answers. To begin is promised with the genealogy of Flood's diagram, which offers a medieval world view in modern guise. I show you a page from an astrological, medical, and magical compilation of the late 15th century. One can imagine such a book having passed through Flood's hands. Its constellation of diagrammatic images invites comparison to Flood, who constructs his arguments visually as well as verbally. At the center of the large circle is Hell. Proceeding upward and outward, we progress from Earth through the outer elements, through the other elements, water, air, and fire, to the circuits of the Moon and the other planets. Mercury and Venus are located between the Moon and the Sun, an arrangement that originated with the late antique commentator on Cicero's dream of Scipio, Microbius. There follows the various firmaments, a top of which sits the throne of God. Joining the deity around the circumference of the circle are the nine orders of angels. The conception of God is the summit of a ladder of all creation, culminating in the orders of angels, up which one can climb, but down which one can also tumble is rooted in the thought of the Pseudodionysius, a sixth-century Syrian theologian who, more than any other figure, transmitted Neoplatonism to the medieval west. It is further reflected in diagrams such as those found in the Howard Salter, which pairs the orders of angels with eight rewards of heaven and the eight beatitudes to supply the viewer with a lattice for contemplative ascent. Turning to the peripheral diagrams, they include, at upper left, a TO map of the world with Jerusalem at its center and various astronomical diagrams of eclipses derived in part from the treatise on the Spears by Johannes de Sacrobosco, or John of Hollywood, a 13th-century astronomer who taught at the University of Paris. In another manuscript of this popular work, Sacrobosco shares space with his contemporary, the Bishop of Lincoln, Robert Großtest. A fancier version of the scheme, based on John of Peckham's Tractatus de Sfera, appears in the Delisle Salter paired with the Howard Salter, indicating that such images had a devotional as well as a didactic function. In the Middle Ages, cosmology comprehended the religious as well as the scientific sphere. We do not know the extent of Flood's library. His diagram, however, depends on a conception of the mind that had been popularized in works such as the Margarita filosofica or Pearl of Philosophy by the Carthusian monk Gregor Rhaish, first published in 1503. Rhaish's treatise offers an introduction to the seven liberal arts, in short, a digest of the medieval curriculum, not unlike what one finds expanded in Flood's compendium. Its image of the human head encapsulates the cell doctrine of brain function, according to which a front ventricle served as the seat of the vis communis or common sense. Not what we think of it as being, but rather the place where, as in Flood's diagram, sense impressions are gathered. The middle ventricle, that of the vis cognativa, the imagination, and the vis estimativa, the faculty of estimative or rational thought, and the rear ventricle, that of the vis memorativa or memory. In this terminology, vis simply means force or faculty. The ventricles are connected by passages that on account of their appearance meant by the term vermes or worm, tilt your head back and you'll be able to collect your thoughts. The roots of this model lie with Aristotle. Sometimes, as in this diagram from an early 13th century English manuscript of the pseudo-Augustinian tract on the spirit and the soul, the number of ventricles or cells is expanded to a full five. These five, labeled common sense imagination, fantasy, estimation, and memory, correspond to the five familiar senses, making a total of ten. Five inner and five outer. In Flood's case, the reduced number of three permitted him to match three triads of which the uppermost, of course, is the trinity. This number-driven matching often determines the symmetrical structures of diagrams. Of Flood's three-fold sets, the first consists of three faculties located within the brain, each represented by a kind of Venn diagram, itself a triad. The first at the forehead combines the sensitive and imaginative faculties, the second atop the cranium, the cognitive and the estimative, the third at the back, the memorative, that is memory, and the motive, that is motor functions. The second set resembles a set of planetary diagrams, the first again at the front of the sensible world, the second of the imaginable world, the third, the components that govern memory. At both levels within and without the brain, each part of each triad is itself three-fold in nature, and all such triads in turn take their form from the ultimate triad, that of the trinity, represented at the apex of the entire composition, where Father, Son, and Holy Ghost converge, surrounded by the nine orders of angels, themselves three times three in number. Everything falls into place, it all makes sense, which is what one expects of a cosmological diagram. Cosmos, of course, simply means order. Cosmological diagrams make sense of the world, and in this case, of how we make meaning out of what the senses make available to us. I just wanted to pause to allow you to take in the beauty of this particular diagram. A diagram in a miscellany assembled in Leipzig in the 1470s by the historian Johann Lindner of Munchberg brings us closer still to the patterns of thought that inform Flood's own way of thinking. It prefaces the marvelous philosophie naturalis, or little natural philosophy, here attributed to Albertus Magnus, but in fact by Peter of Dresden, who was born in 1365 and shares many elements with Flood's diagram. The five senses connected by lines, in this case to the first ventricle of common sense, followed by the cells of imagination, fantasy, estimation, and finally memory. The abbreviation S-I-F-E-M, inscribed at the center, makes it easier for the viewer of the diagram to remember the sequence of its component parts. The upper part of the page has been described as a kind of scratch pad used for all kinds of diagrams and pen tryouts. There is, however, much more at stake. At the center of the upper portion of the page is a configuration of words arranged in a square with connecting diagonals, a diagram in disguise, and one more over that is intimately connected, no less than the floating diagrams at the top of Flood's image to the skull beneath it. No medieval student with a training in the liberal arts, in the art of logic, would have failed to recognize this configuration as a variant of the square of opposition, a logic diagram included among the manuscript's illustrations, as well as in printed editions of the work. The miscellany is a collection of classic texts on logic, precisely the kind of textbook with which Flood likely had been schooled two centuries later. The earliest of all logic diagrams, the square was hardly a medieval invention. Rooted in Aristotle's distinction between contradiction, a pair of affirmative and negative statements, and contriety, mutually exclusive propositions, it first appears in the second century of the common era in the peri hermeneuas of the Neoplatonic philosopher Apuleus of Madhau. It occurs repeatedly among the skolia or glosses in the margins of a 15th-century Greek compilation of Aristotle's work on logic. Applied to standard topics as the relation of propositions, the construction of syllogisms, and the mathematics of musical intervals, the square constituted a cornerstone of medieval pedagogy supporting the entire syllabus. Four conceptions or propositions are placed at the corners, the lines making up the square and its diagonals map the logical relationships among them. Note that in this modern version, the logical content of different positions on the square is indicated, not only by logical shorthand, but also by Venn diagrams. The square of opposition constituted part of every medieval student's toolkit. In our diagram, it expresses the relationship of elements within natural philosophy. In the classic square of opposition, just as fire would serve as the contrary of water, so to earth would stand as the contradiction of water. Here, however, we are dealing not with the four elements per se, but rather qualities associated with them. At the top, from left to right, the terms calidos, fiery, and humidos, humid, and at bottom, frigidos, cold, and secus, dry. As a result, the diagonals link heat to dryness and cold to humidity, thereby expressing qualities shared by various elements as seen in this schematic representation. By using the Pythagorean tetrad within the matrix offered by the square of opposition, Lindner lends expression to the means by which they can be converted, one into the other. Whereas in the canonical square of opposition, intersecting diagonals define the antagonism between, on the one hand, fire and water, and on the other, air and earth, which stands simultaneously for the forces of corruption and degeneration that push the primordial elements apart. In Lindner's diagram, the diagonals, rather than denoting contradiction, represent a consequent, or entailment, the second half of a hypothetical proposition. A topic indirectly derived from Boethius' commentary on Aristotle. In this configuration, heat and cold as active qualities generate dryness and humidity, respectively. The sides of the square defined by the associated qualities of heat, humidity, dryness, and cold can join the otherwise disparate elements in dynamic harmony. Other sets of terms scribbled in the surrounding space, associate the four elements with other quotanities, or sets of four. At the upper right, the four humours. At the lower right, the four elements. Below to the left, the four ages of man. At the far right, the four winds. And immediately below the central square, the four seasons. Other notations work out possible pure mutations of these sets. In short, as in Flood's image, Lindner, far from making random jottings, mapped out to use Flood's terminology, the correspondences between inner and outer worlds. The key point here is that the upper part of the page, far from being incidental in its relationship to the diagram beneath it, is integral to its operations. It constitutes a diagram of both how the mind works and what it works on. Another way of putting this might be, whereas the square of opposition stands too core for the power of reason, the terms that fill it stand for the world of sense impressions that constitute the material, the stuff on which reason and hence memory go to work. The terms that surround and extend the square could easily be substituted for those that already occupy it. In effect, they function almost like alternate sets of punch cards that can be plugged into a simple computer, which is how we might think of the square in the first place, a machine for thought and the mechanism that allows the mind to be thought of as a kind of machine in the first place. We find the same information more efficiently and elegantly integrated into a wheel diagram or rota in a collection of calendrical and astronomical treatises from the second half of the 13th century. Its petal-like forms resemble those of a rose window. It is but a small step from such diagrams to modern debates about the relationship of diagrams to the nature of cognition, and whether diagrams, which have the uncanny ability both to mimic and engender mental processes accurately express or embody mental representations. These questions are not only modern, they constitute among the oldest debates in the history of philosophy. We find the square of opposition applied in the same way in this early 15th century scholastic handbook from Oxford, the only book of its kind to have survived, such is its ephemeral nature. Written by a Cistercian monk from Lincolnshire, its contents cover the trivium, quadrivium, natural philosophy, ethics, and metaphysics. Here, too, the square has been adopted to the dynamic tetrad of elements. We have only to look up the road to the Warburg Institute, where the Mundus Anna's Homo Diagram of Isidore of Seville crowns the entrance to trace the ancient provenance of such diagrams and the wisdom they embody. A sum of all knowledge, the diagram expresses relationships among not only the four elements, but also man, the microcosm, and the year represented by the four seasons. From a Christian perspective, Isidore's image has the advantage of having inscribed within it the ultimate cosmological cryptogram, the cross. For all its apparent esotericism, informed by the Paracelsan doctrine of alchemical triads, Flood's diagram remains rooted in diagrammatic techniques common to the medieval classroom, the focus of my second lecture. It is no less indebted to medieval theology and theories of vision. For example, in finding parallels between a tripartite division of the powers of the soul and the three persons of the Trinity, Flood is ultimately indebted to the church father Augustine, who in day Trinitate saw on the faculties of memory, understanding, and will, the image of God in whose likeness and similitude man was made. Likewise, the third division of vision into the sensible, imaginative, and intellectual harks back to Augustine's hierarchy, elaborated in De Ganesi Adlitaram of corporeal spiritual and intellective vision. For Flood, as for Augustine, both neoplatonists, geometrical images provided the most reliable point of departure for the soul's journey to God. In this view of things, as articulated by David Albertson, quote, the mathematicianization of nature is not an accident of the history of science, but rather a necessary consequence of theological anthropology as human mathematicians reflect their divine exemplar, having established the medieval genealogy of Flood's famous diagram, let us now turn to the typology of its various component parts, which reveal him to have been deeply versed in textbook traditions of diagrammatic imagery. Here you see an image of microcosmic man from an astrological medical miscellany, including a set of calendrical tables known after their compiler as John Somers Calendarium, assembled in 1380 for Lady Joan Holland, the mother of Richard II, a work that has been posited as a source for some of the astrological passages in Chaucer. It is easy enough to see the kinship between this relatively prosaic diagram and Flood's diagrammatic derivation. At the centre stands homunculus with upraised hands. He represents the element earth, the other elements water, air, and fire rise above him. The surrounding circles trace the motions of the planets and at a greater remove the signs of the zodiac, in short the motions of the macrocosmic bodies that govern the human microcosm. Such images are closely related to bloodletting diagrams such as that found in this 15th century medical miscellany, which happens to be on exhibit in the room next door at this very moment. It comes from northern England, probably York, in which on a recto various body parts are linked to instructions, then overleaf to the signs of the zodiac. The facing page centres on a vovel, a rotating disc, which permits the correlation of different sets of information. Once again, the signs of the zodiac, the months, and the days. At the top, the two Saint Johns, evangelist and Baptist, engage in dialogue, whereas at the bottom, the physician saints, Cosmas and Damien, one holding a urine flask, the other a medicine box, consult. Turning the page, personifications of the four humours surround the bust of Christ. The holy face at the centre functions like a period at the end of a sentence. It is the point to which all else tends. An atropopaedic image of Christus Medicus, an ancient concept of the saviour, a sacred healer, it serves also as an image of the perfect man and the macrocosm to which man, the microcosm is related. The image is a diagram in disguise. The ancient roots of such conceptions manifest themselves and among the most familiar of all medieval diagrams, that of birthfet of Ramsey, who lived in the late 10th and early 11th century. Three 12th century manuscripts, including this one from Peterborough, transmit the diagrammatic apparatus that accompanied his commentary on the compotus, the art of reckoning time, whose importance in an age prior to the invention of mechanical clocks can hardly be underestimated. The proper reckoning of time depended on determining the date of Easter, a movable feast. If this image reminds you of a medieval Christ in Majesty, you are not mistaken. That is precisely what you are supposed to see amidst the configuration of diamonds, circles and semicircles into whose forms are fitted the letters A, D, A and M, the name of the first man who stands for the microcosm, but which also in Greek represent the first letters of the four points of the compass. Drawn into this cosmic constellation by the outermost semicircles are two additional quaternities, the four elements and the four seasons. In short, what we are confronted with is a diagrammatic apparatus permitting the manipulation and correlation of much of the same information as that found in a more crudely drawn illustration in the welcome collection of treatises on logic, stamped with the message that in the Platonic sense, Christ, the new Adam, not only created but exemplifies the entire cosmos. A diagram from an Austrian manuscript dated around 1300 expresses the same idea. Christ embraces the entire configuration just as in the so-called map salter made in London in the second half of the 13th century, he embraces the world, represented in the form of a TO map of the three continents, Asia, Europe and Africa. Even as in keeping with the words of Psalm 91, he, quote, walks upon the asp and the basilisk, unquote, an image of his triumph over evil. In the manuscript from Peterborough, a second Christ in majesty on the page facing the diagram of quaternities, presents him amidst the heavenly host of four evangelists, the twelve apostles and corresponding prophets. Without actually depicting them, it is a figure without figuration and points the way to the topic of my third lecture, namely the ways in which medieval artists use geometry to structure the making and meaning of images. Other diagrams in the same manuscript are also called Epileon Sphere, used for medical prognostication. Together aptly, with personifications of life and death, and on the facing page, a second Epileon Sphere, in this case, Logen's shape. Then, over leaf, a wind rose, which correlates the winds, depicted as heads with wings, with the points of the compass, together with an unfinished diagram of the lunar months. Then, over leaf, again, a diagram of the divisions of philosophy and an acrostic poem by Berthferth's teacher, Abbawe Fleury, of a type known as a Carmen consulatum, in which letters within the diagrammatic forms do double duty and are read in multiple directions, a reminder that diagrams can be constituted not just by lines, but also lines of verse. Then, following the Christological diagrams, Amapamundi, another way of linking the micro to the macrocosm, and finally, following astronomical tables, a table of the phases of the moon. As in Flood's cosmos, this material all fits together. Everything is connected to everything else. No less instructive is another miscellane, this one combining works on the compotus and astrology. Both arts depend on counting, hence the inclusion of an illustration, accompanying beads on the nature of things and the calculation of time. The demonstrates absent a calculator how to count to 9,999 and determine the date of Easter in relation to the 19-year lunar cycle using only one's hands and fingers. This is counting by hand taken to the next level. The figures in the lower register indicate how to show 10, 20, and 30,000 respectively, part of a system that extends as far as a million. Among the manuscripts' illustrations, we again find the Apuleian prognostic spheres as well as a diagram associated with Abo of Fleury, tracing the relative motions of the sun, moon, and planets vis-a-vis the signs of the zodiac, an example not of graph paper but of graph parchment. We have seen that flood made use of overlapping circles in the manner of Venn diagrams, named after the English mathematician, logician, and philosopher of the 19th century to denote the properties of the soul. Similarly constructed diagrams also constitute a legacy from the Middle Ages. Perhaps the earliest extant examples come from a 12th century musical treatise, the Book of Music, known from two manuscripts of which one is here in the British Library. Among the treatises, many diagrams are a pair filling an opening and depicting the disposition of the modes according to the diapsan, one of the three ways according to Boethius, in which music is related to mathematics. Given flood's interest in connecting the microcosm and the macrocosm through evidence of the senses, he was fascinated by the music of the spheres. His treatise portrays the spheres as a monocord, finely tuned by the hand of God, grafted onto a diagram of the sub-emperian and celestial spheres. The proportions of the chords themselves derive from the most influential of all medieval treatises on the subject de institutione musica by Boethius, seen here in a manuscript of the last quarter of the 12th century. Boethius' treatise has little to do with actual performance. Rather it expresses, as in his view does music itself, the underlying harmony of the cosmos. Similar chordal relationships between the micro and the macrocosmus echo throughout flood's treatise. Another of flood's diagrams combines Boethian chordal arcs representing the spiritual bodily and divine diapsans with the tetragrammaton, the Hebrew name of God, long a locus of mysticism and mystification, and a trinitarian triangle that outlines the connections between three worlds, the elemental, spiritual and angelic. Here again there is little that is new. The kernel of the idea, labeled a geometrical figure, appears in a mid-12th century manuscript of Petrus Alfonsi's dialogue against the Jews. If in flood's conjunction of inner and outer worlds two-part Venn diagrams define the powers of the soul at its summit, as if by a higher order three such vectors representing Father, Son and Holy Spirit overlap in a single unity of persons expressed in geometrical terms. Theology and diagrams both deal with abstractions, the one lends itself to expression in terms of the other. Theological diagrams predicated on logical principles enable the mode of thinking elaborated in such treatises, which applied syllogistic logic to the mysteries of the faith. From Boethius to the end of the Middle Ages, medieval thinkers sought to reconcile the principal paradox of the Christian credo with terms of debate inherited from Aristotle's categories. Ontology needed to be squared with logic, a daunting challenge considering trinitarian doctrine. Among the logical categories at stake in such debates were sameness and difference, simplicity and composition, relation and essential and accidental predication. If trinitarian doctrine had to be true and if, as Achimus maintained, all truths were compatible with other truths, then any fault had to lie in logic itself, a conclusion that prompted attempts to modify its rules. Rather than a stumbling block, trinitarian theology compelled Christians to hone and refine their techniques by inventing fresh forms of syllogistic logic that, so they claimed, were compatible with doctrine. If faith were to be reconciled with reason, then the laws of logic dictated it should be possible to represent the mystery in diagrammatic form. In this respect, diagrams were not just an instrument of argument, they were also assertions of authority. The shield of faith here appended to Peter of Poitiers' diagrammatic compendium of the genealogy of Christ encapsulates the mystery of three and one by standing simultaneously as a three-sided triangle and a single geometric figure. The diagram seeks to lend the trinity the same self-evident character as a set of loctite propositions. The phrase non-est is not that links each of the three corners and the word est is that in turn links the corners to the word deus at the centre. We cast trinitarian theology in terms of propositional logic resulting in 12 separate statements of biconditional equality relations. Read in combination with one another, the propositions generate a Christian creed. The shield of faith assumed apotropaic agency, not unlike that asserted by the diagrams found in manuals of necromancy. In an English theological miscellany that includes paralysis' treatise on the vices, a Christian knight wheels the sword of the seven virtues against the seven vices arrayed in tabular stymatic form on the facing page. He further protects himself with the trinitarian diagram. Utility of diagrams as instruments of both spiritual and physical protection added to medical manuals. This folding physician's almanac opens with the calendarium of John Somer including diagrams of eclipses followed by the zodiac man, an essential diagnostic and prescriptive instrument. It concludes with mnemonic hand diagrams which employ the digits and joints to chart interrelated concepts of a pastoral as well as a medical nature for handy reference. Linking morals to medicine are trees of the virtues and vices in which the flourishing life-giving virtues contrast with the drooping desicated branches of sin. To these are added the articles of faith, the ten precepts and plagues of Egypt and septonaries of the petitions of the paternoster, gifts of the Holy Spirit and the virtues. Less expected, the almanac also includes a diagrammatic epitome of biblical history derived from Peter of Poitier's historical compendium of the genealogy of Christ which in this instance also incorporates a diagram of the Ptolemac universe. Just as diagrams themselves lent themselves to unexpected correlations of disparate materials, so too diagrammatic compendia allowed for the easy mixing of different types of content. In the case of the almanac, the combination of historical, moral and medical material was no doubt prompted by manuscripts of Peter's work manifesting the same mixture. In these two English examples, the paternoster diagram adopts a more elaborate form derived from 12th century French models. A scholastic theologian who taught for more than 30 years at the University of Paris, Peter also served as its chancellor. Techniques of representing logical and genealogical relationships overlap in his representation of the complicated line of descent from Solomon in which his task was to harmonize the contradictory accounts of Christ's ancestry in Matthew and Luke. According to the church historian Eusebius, Matthew recounts the biological Luke, the legal ancestry. The square of opposition provided a way to rationalize this obstacle to understanding. To clarify these tangled relationships, which amount to what according to Deuteronomy is a leverate marriage in which the brother of a deceased man is obliged to marry his brother's widow, and to make them easier to remember, Peter incorporates into his genealogy a diagram based on the square. Matan and Melchee, the two husbands in that order of Esta placed at the center, occupy the roundels constituting the upper right and left corners of the square, whereas the sons of each of those unions, Jacob and Eli respectively, occupy those at the lower right and left. Matan and Jacob represent the direct line of descent indicated by the thick line that connects Matan and his ancestor Solomon above and his descendant Joseph below. In other versions of the same diagram, a horizontal bar here missing, connecting the two fathers, Melchee and Matan, reads, quote, connected because killed, quote, a reference to Melchee having succeeded Matan following the latter's death as Esta's husband. Mapt onto the square of opposition, their relationship is contrary in that they could not have been married to the same woman simultaneously. The bar connecting Eli and Jacob, the two sons, reads, uterine brothers identifying them as stepbrothers by the same mother. Mapt onto the square in the same fashion, their relationship analogous to that of their fathers is subcontrary. The vertical lines connecting fathers and sons read carnal sons, the equivalent in the square to subalternation. Finally, the diagonals linking the two fathers via Esta to the two sons read conjugal son and represent the lines of contradiction connecting the corners in that Matan is not the father of Eli while Melchee in turn is not the father of Jacob. The adoption of Peter's application of the square of opposition to contradictions within biblical genealogy in the 1568 Bishops Bible testifies to its continued utility as an instrument of elucidation, clarification, but also no less justification in that it papered over an apparent problem with a familiar and hence acceptable tool for thinking. The boundary between magic and medicine, both instrumental arts was porous. Medical manuals are full of charms many accompanied by diagrammatic images. A diagrammatic image of necromancy that is the art of summoning spirits in a 15th century manuscript of the pilgrimage of man attributed to John Liddgate, a monk and poet at Burieson Edmonds, shows the pilgrim's encounter with a necromancer standing within a magic circle with the treasures brought to him by the devil on the right, whose visage has been partially rubbed out by a reader. Magic circles are essentially a form of diagram. Here you see an example with the names and characters of the planets. Across within a circle suffice to suggest to suspicious souls that conjuring of demons was afoot. The symbols arrayed within such circles included crosses, solemonic pentacles, along with other geometrical motifs, astral signs, and exotic letters, real and imagined. A similar circle features prominently in one of the so-called Ripley Scrolls, an alchemical manuscript of the late 16th century or early 17th century named for George Ripley, a canon at Bridlington Priory in Yorkshire, who supposedly had studied alchemy in Italy and at the University of Louvain. Diagrams were central to alchemical practice in which they not only aided in various forms of manipulation, but also of mystification. Here you see a mosaic of horoscope diagrams in Thomas Norton's poem, The Ordinal of Alchemy, dated 1477. Such books provide a way of coming full circle to Robert Flood, for whom diagrams served not just as illustrations of esoteric ideas, but also as potent instruments. Flood's mnemonic alphabet derives from the visual vocabulary of late medieval conjuring. The immediate source for Flood's alphabet was that of Jacobus Publitsius, a 15th century rhetorician and physician, whose art of memory, the first of its kind to be printed, was published in 1475. Among its illustrations is a diagram with a vovel at its centre in the form of a rotating serpent designed to enable not simply the recollection, but also the generation of new ideas. This kind of paper machine took as its point of departure the combinatorial art of Raymond Lull, which, as we will see in a subsequent lecture, found further fruit in the work of the greatest polymath of the 17th century, Gottfried Wilhelm Leibniz. The worm at the centre of the wheel evokes the worm-like valves that were thought to control the passage of spirits among the brain's ventricles. To manipulate the diagram was in effect to invoke its power as an operative instrument to perform a kind of experiment on one's own brain. I hope that my lectures can serve as a similar sort of experiment. Not painful, I trust, but at least somewhat simulating. Thank you. I believe there's time for a few questions, and I'm going to be impolite and simply point, as in most cases I don't know your names, but if you have a question and will alternate with questions from the online audience, please just raise your hand. Yes. I have a question from the online audience here. I wonder to what degree medieval diagrams are predominantly teaching tools, and if so, are there kindness for students who, as St Augustine says, find it easier to learn by looking at things rather than just reading about a topic. In essence, are diagrams bottom-up pedagogical kindnesses rather than top-down organisations of knowledge? That's a good question. The short answer would be no. Certainly some diagrams serve as pedagogical aids, but as I hope I was at least able to intimate in my remarks, diagrams really suffuse the whole of the curriculum at all levels, and they're used in esoteric contexts as well as in introductory catechetical or basic material. So there's really no part within the spectrum of human knowledge and human activity that does not employ diagrams in the Middle Ages, and I'm tempted to say the same holds true today. Penrose diagrams and Feynman diagrams are not for the uninitiated, and so Augustine was indeed prejudiced against images, and that's something that art historians have never forgiven him for, but in practice diagrams are used as much as instruments of initiation as they are as instruments of introduction. The evening to a close. Thank you very much for coming.