The Decline of Space: Euclid between the Ancient and Medieval Worlds Michael J. Sauter División de Historia Centro de Investigación y Docencia Económicas, A.C. Carretera México-Toluca 3655 Colonia Lomas de Santa Fe 01210 México, Distrito Federal Tel: (+52) 55-5727-9800 x2150 Table of Contents List of Illustrations ............................................................................................................. iv Acknowledgments .............................................................................................................. v Preface ............................................................................................................................... vi Introduction: The divine and the decline of space .............................................................. 1 Chapter 1: Divinus absconditus .......................................................................................... 2 Chapter 2: The problem of continuity ............................................................................... 19 Chapter 3: The space of hierarchy .................................................................................... 21 Chapter 4: Euclid in Purgatory ......................................................................................... 40 Chapter 5: The ladder of reason ........................................................................................ 63 Chapter 6: The harvest of homogeneity ............................................................................ 98 Conclusion: The deracinated god ................................................................................... 125 Bibliography ................................................................................................................... 128 iii For tribal man space was the uncontrollable mystery. For technological man it is time that occupies the same role. --Marshall McLuhan (1951) 1 1Marshall McLuhan, The Mechanical Bride: Folklore of Industrial Man (New York: Vanguard Press, 1951), 85. vii Chapter 3: The space of hierarchy The good Christian should beware of Mathematicians and all those who make empty prophesies. The danger already exists that the Mathematicians have made a covenant with the Devil to darken the spirit and to confine man in the bonds of Hell. --Augustine of Hippo (354)1 Western thought between 350 and 900 was tinctured by an overlooked intellectual phenomenon, the absence of Euclid’s Elements. This absence is odd, since unlike many classical texts the Elements did not disappear with post-Antiquity’s arrival. When the Roman Empire broke apart in the fifth century AD, both its Latin-speaking western half and Greek-speaking eastern halves maintained contact with Euclid’s work, with the former clinging to incomplete copies of the Elements and the latter complete ones. Despite this accessibility, however, Euclid’s thought entered isolation, as late Antique Christian thinkers were more interested in understanding humanity’s relationship to God and Creation, than in contemplating idealized points, lines, planes, and spheres. Indeed, Christian thought between the fourth century and the end of the twelfth was decidedly post-Euclidian, in so far as its preoccupations were not amenable to the Elements’ space. It was only in the early fourteenth century (and then only in the former Empire’s western half) that Euclid touched religious issues anew. In this chapter and the succeeding one, the world from which Euclid and homogeneous space were conspicuously absent. In this chapter I examine Euclid’s slow departure from Western thought, between the years 350 and 900, and emphasize how the 1Augustine of Hippo, "De Genesi Ad Litteram," in Opera Omnia, ed. A.B. Caillau and M.N.S. Guillon (Parisiis: APud Parent-Debarres, 1835), 218-19. 21 Mediterranean’s growing preoccupation with hierarchy—whether with respect to the theological or philosophical status of being—made medieval European thought impervious to Euclid’s influence. In the next chapter I will pick up the story after the year 1000 and will explain how in the course of the eleventh and twelfth centuries, in spite of profound changes in multiple areas of medieval thought, the resistance to homogeneous space intensified. It was only when the early fourteenth century wrought a theological transformation that the application of homogeneity to every part of Europe’s mental cosmos became possible. Euclid’s medieval absence was foretold by the Mediterranean’s religious ferment of the fourth and fifth centuries. Some of this was Christian, although much of it was not. With respect to early Christianity’s Church Fathers, most of whom were born before 400, none mixed the Elements into their amalgams of Christian theology and pagan philosophy—and this was regardless of whether they wrote in Greek or Latin, the former being Euclid’s native tongue.2 Basil the Great, a Greek speaker from Anatolia who was born around 330, paid little attention to Euclid, and Augustine of Hippo, a Latin-speaking North African who was born in 354, did much the same. Although Augustine dabbled initially in mathematics, he turned away from further study and also chided his co- religionists in On the Meaning of the Book of Genesis not to discuss pagan disciplines that they did not understand.3 It was best, so he thought, for Christians to stick to what they knew, i.e., their sacred texts. Given the Church Fathers’ influence, it is not surprising 2In general, see Etienne Gilson, History of Christian Philosophy in the Middle Ages, The Random House Lifetime Library (New York: Random House, 1955); John Marenbon, Early Medieval Philosophy (480-1150) : An Introduction (London: Routledge & K. Paul, 1983); Colish, Medieval Foundations; Jorge J. E. Gracia and Timothy B. Noone, eds., A Companion to Philosophy in the Middle Ages, Blackwell Companions to Philosophy (Malden: Blackwell Publishers, 2003). 3John Hammond Taylor, ed. The Literal Meaning of Genesis, 2 vols., vol. 1, Ancient Christian Writers (New York: Newman Press, 1982), 39. 22 that for a millennium no one on either side of the Latin-Greek divide applied Euclid to God. Substanceless points, breadthless lines, depthless planes and locationless spheres were afterthoughts for a religion that confronted profound mysteries, such as the Trinity, Creation, and why God became man. Christian disregard of Euclid was, however, only one factor in the Elements’ withdrawal. The ancient Roman world was dissolving in ways that had powerful effects across the Mediterranean basin. In 313, Constantinople was consecrated as a new capital city in the east, inaugurating a process that left the empire divided in into two linguistic halves, with Greek dominating in the east and Latin in the west.4 Moreover, even on the Italian peninsula itself, Rome’s glory as an imperial capital was fading. In the fourth century, the western empire’s capital became Milan, followed in the fifth century by Ravenna—both of which lie in the Italian boot’s north.5 Finally, one byproduct of Rome’s general decay was the western zone’s retreat into monolinguality, as a once cosmopolitan culture limited itself to mere Latin. Since the Elements was composed in (and remained in) Greek, the potential audience in the west for Euclidian space, which had always been small, became even smaller. There was one attempt to swim against these powerful currents. In the mid-sixth century, the Roman Christian Anicius Boethius, who served the Ostrogothic king Theodoric, made the only known late-Antique translation of the Elements into Latin.6 4Ralph-Johannes Lilie, Byzanz: Das Zweite Rom, 1. Aufl. ed. (Berlin: Siedler, 2003). 5A. K. Bowman, "Diocletian and the First Tetrarchy, A.D. 284-305," in The Cambridge Ancient History: Volume Xii, the Crisis of Empire, A.D. 193-337, ed. Stanley Arthur Cook, et al. (London: Cambridge University Press, 1966), 74-78. 6On this history see, Menso Folkerts, Boethius Geometrie Ii: Ein Mathematisches Lehrbuch Des Mittelalters (Wiesbaden: Steiner, 1970); John Murdoch, "Euclid: Transmission of the Elements," in Dictionary of Scientific Biography, ed. Charles Coulston Gillispie (New York, NY: Charles Scribner's Sons, 1971); Menso Folkerts, "Euclid in Medieval Europe," in The Development of Mathematics in 23 This version did not, however, circulate widely.7 Equally important, it was incomplete, as it included only the first five of the original thirteen books.8 (The five in question cover plane geometry, whereas discussions of three-dimensional geometric objects appear only in the final three books.9) Boethius’ Elements dealt purely with two-dimensional geometry (triangles, circles and such) and, thus, presented the reader with a distinctly “flat” Euclid. Even more telling, perhaps, is that Boethius’ translation disappeared, as mere fragments (at best) were preserved in other medieval works on geometry, many of which were informed by Roman-era works on agricultural measurement.10 Although historians have celebrated the medieval echoes of Euclid, the contents of the various manuscripts only underscore three-dimensional space’s absence, since none of them goes beyond the Elements’ fifth book either. The fate of Boethius’ translation epitomizes the slow withdrawal of geometric space from western culture. Euclid’s Elements was part of an ancient tradition of projecting space onto the physical world that dated back, at least, to Pythagoras in the sixth century BC. As a result, many famous ancient thinkers, such as Plato, Aristotle and Democritus not only had access