Why Euclid elements are not a library on geometry.
Picturing Number: Visualizing Quadrivial Concepts in the Central Middle Ages
Orlèans, Médiathèque d’Orlèans, MS 306, p. 10 (detail).
Education changed in the central middle ages. While the arts of grammar, rhetoric, and dialectic continued to be taught as the foundation of all learning, the quadrivium, the four arts of number—arithmetic, music, geometry, and astronomy—received new emphasis. Some of the greatest minds of the era, among them Gerbert of Aurillac (later Pope Sylvester II; c. 940–1003) and Abbo of Fleury (c. 944–1004) gained renown for their numeracy. Teachers both, they educated many of Europe's future abbots and bishops, emperors and administrators, and authored new works on calculation, time-reckoning, geometry, and music. Gerbert and Abbo drew their knowledge of number from late antique and early medieval tracts by such figures as Calcidius, Boethius, Martianus Capella, Macrobius, and Bede. Copied and recopied, the texts of these earlier authors changed little, but the pictures underwent alterations that suggest a significant shift in use and an upsurge in interest. Tenth- and eleventh-century makers of manuscripts provided pictures to elucidate textual passages that made no mention of them; where pictures were an organic part of the tract, they often added more. They experimented with placement, scale, color, and contours in ways that suggest a keen awareness of contemporary notions of materiality, sight, and the limits of representation that were themselves the product of explorations in the domains of mathematics and science.
Nowhere was the quadrivium more vigorously pursued than at the cathedral school of Reims and the monastic school of Saint-Benoît-sur-Loire (Fleury), where Gerbert and Abbo were masters. Picturing Number focuses on the rich material legacy of these major schools, which acted as epicenters of an extensive, pan-European network of exchange that linked monastic, episcopal, and lay institutions. It is grounded in the quadrivial manuscripts that members of these communities made, copied, and used at the turn of the first millennium. Over 600 extant manuscripts have been attributed to the Fleury scriptorium; more than sixty contain quadrivial material. Relatively few manuscripts can be traced with certainty to Reims. Of these, many can be tied to Gerbert, who became tutor to Emperor Otto III in 996 and dispatched a number of deluxe manuscripts—several on quadrivial topics—for the edification of his royal charge. These are today held as a group in the library of Bamberg. Gerbert’s epistles are filled with frequent demands and requests for manuscripts. Scholars have identified many of the exemplars, most now scattered across Europe in small municipal collections, from which Gerbert commissioned copies. Pictures abound in these manuscripts. They appear in the margins adjacent to the text and in breaks within the textblock, and are sometimes woven into the very syntax of a sentence. Elsewhere, they appear apart from the text, spanning entire pages, openings, even quires.
Art historians in other fields have looked to scientific paradigms to explain representational trends, notably the use of perspective in the Renaissance or the desire for verisimilitude in seventeenth-century Dutch painting. Their studies have colored present notions of how scientific thinking did or might manifest itself visually in the past. It is perhaps unsurprising that the characteristics of the Year 1000 styles and the Romanesque—the flattening of forms and space, a taste for un-modeled and patterned surfaces, a-systematic proportion, and the intermingling of textual and graphic elements—have not encouraged similar investigation. Yet, if we turn to quadrivial manuscripts, sources of scientific knowledge during this period, we find these same graphic tendencies in pictures constructed to convey quantitative concepts. Such pictures offer evidence that we are, perhaps, too quick to limit our notion of the visual “scientific” and its impact.
Picturing Number contributes to the study of medieval image theory. Visual and textual material drawn from quadrivial manuscripts permits investigation of period-specific notions on a range of integrated issues: the relations and tensions between word and image, the nature of cognition, and modes of representing both the sensible world and that which was considered to be beyond the reach of the senses.
Picturing as Practice: Placing a Square above a Square in the Central Middle Ages
This essay examines the many pictures of solids added to the margins of Macrobius's fifth-century Commentary on Cicero's Dream of Scipio in light of contemporary geometric practice around the turn of the first millennium. At this time, geometry was less a body of axioms and precepts to be demonstrated and memorized and more a tool that flexed and sharpened the mind, thus heightening the ability to comprehend worldly and divine things. This ability was not considered a native talent; it required teaching and practice. The definitions of geometry's so called elements—points, lines, planes, and solids—in the Commentary and elsewhere, provided opportunities for such practice, that is, for exercising the mind’s eye. Given this, the pictorial annotations added to these definitions are perhaps best understood as goads to the intellect and traces of an otherwise ephemeral activity. Picturing as Practice considers the graphic strategies adopted by annotators along side the didactic tactics and tools employed by medieval masters in the classroom seeking to cultivate the intellectual eye.
The Stoikeioon was repackaged for thes kinds of curricula, but Euclid's original Teaching goal may be surmised from the Title, and its place in the academic institution founded by Plato, supporting Platonic Ideals/Forms