Crosby 1990 renaissance change cognition by Marcos Virgílio

A Renaissance Change in European Cognition Author(s): Alfred W. Crosby Source: Environmental History Review, Vol. 14, No. 1/2, 1989 Conference Papers, Part Two (Spring - Summer, 1990), pp. 19-32 Published by: Forest History Society and American Society for Environmental History Stable URL: http://www.jstor.org/stable/3984624 . Accessed: 18/06/2014 02:31 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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A Renaissance Change in European Cognition Alfred W. Crosby University of Texas at Austin

It is my belief that the most powerful societies of the last several centuries, those known collectively as "the West," were powerful in part because their people thought about physical reality in a different way than other peoples, and by means of this were able to manipulate that reality with unprecedented success. It is to this contrast that Henry Kissinger referredin the following:' The West is deeply committed to the notion that the real world is external to the observer, that knowledge consists of recording and classifying data-the more accurately the better. Cultures which escaped the early impact of Newtonian thinking have retained the essentially preNewtonian view that the real world is almost completely internal to the observer. I do not entirely agree with Kissinger's interpretation,and I am convinced that the divergence of Western and other cultures in this matterbegan long before Sir Isaac Newton, but the statement provides a good place to begin. This way of thinking about reality, distinctively Western until the last decades of the 19th century, is part and parcel of what we in the industrial nations call common sense, although we know, or should know, that it was not common historicallynor does it always make sense. It provided the conceptual base for all modern science and technology except the most recentand exotic, for business and bureaucraticpracticeto this day, and for most of

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Western art from the Renaissanceto the advent of Cubism. This kind of common sense has been practicedin part through the ages in many places, but I claim that the most critical period and region for its origin was from no later than 1300 to, at the very latest, 1600 and in Western Europe. After that, the periodic and actuarial tables, charge cards, and computer dating were inevitable. This approach to material reality includes four attitudes that I will call Magellan's Postulates. (They could be called Newton's or Andrew Carnegie'sPostulates just as accurately,but I arbitrarilychoose this title for them because they were so undeniably essential to the first circumnavigation). Western Europeanscientists and philosophers of science of the 17th century were among the first to think about them methodically and to state them clearly. The explanation for the lag between these attitudes' debut circa 1300and their first unambiguous formulationis simply that, once again, theory trailed after practice. The explanation for their clear statement in the 17th, rather than some other century, is that the great intellects of that period were selfconsciously revolutionaryand felt obliged to examine and to state their premises, however well-worn in usage they might be. Magellan's Postulates are as follows: 1. God is a reasonablesort of chap. God, in the words of Rene Descartes, "beingthe source of all truth, has not created our understandingof such a natureas to be deceived in the judgementsit forms of the things of which it has a very clear and distinct perception."2

2. Time is homogeneous. Sir IsaacNewton defined time as an "absolute"dimension, which "flows equably without relation to anything external." A minute is a minute is a minute under all circumstances,and there is an absolute time against which all durationscan be measured. 3. Space is homogeneous. Newton's physics was predicated on "absolutespace," which "in its own nature, without relation to anything external, remains always and immovable."3A meter is a meter is a meter under all circumstances,and there is an absolute space, a yardstickin the sky, so to speak, for the measurement of all extension. 4. Physical reality can be usefully described, understood, and manipulated in terms of mathematics. Galileo declared that the book of the universe "is written in the language of mathematics, and its charactersare triangles, circles, and other geometric figures without which it is humanly impossible to

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understand a single word of it; without these, one wanders in a dark labyrinth."4 Thesefour seem truismsto most of us, but duringmost of written history they seemed as contraryto the obvious nature of reality as Einstein's propositions seemed in the first decades of this century. Indeed, the most widely held views in medieval and RenaissanceEurope contradictedMagellan's Postulates: 1. God is reasonable?In human terms? Don't be absurd. How could the mind of the createdpossibly encompass even a whim of the mind of the creator? The Middle Ages and Renaissance abounded in miracles,each one of them an example of God breaking the rules as understoodby human reason. Many of the most influential thinkers of the time recommended against reason. In the 16th century St. Teresa wrote that thepoint of life was union with God, and that everything but supine receptivity to God was beside the point. If the human mind understandsthis experience, she wrote in her autobiography,"it does not understandhow it understands, or at least, cannot comprehend anything of what it understands.... Nor can I myself understandthis."5 2. Time is homogeneous. Not qualitatively, said astrology, with which the medieval and Renaissancemind was drenched. Marriages,births, battles, which would be right in the afternoon or in April, would be intrinsicallywrong at other times of day or in anothermonth. St. Augustine condemnedastrology,but even so he and many others believed time to be divided into seven ages from Creation to the end. Reality within each of these seven ages was different. Before the Flood, for instance, men had lived hundreds of years and had grown to be much largerthan they ever do in our age. St. Augustine had seen an enormous human molar tooth which confirmed this.6 All Christians agreed that time before the Incarnationwas intrinsically different from time after. Before, salvation was impossible, with a very few extremely rare exceptions. After, salvation was possible. What could be a greater example of heterogeneity than that? 3. Spaceis homogeneous. Nonsense. How could you claim that space would have the same qualities everywhere, when you knew, for instance, that Purgatorywas an actual place? Space there was sacred and, therefore, intrinsically different. When Dante's version of Ulysses, sailing south and west from the Gates of Hercules, sighted a high island, a squall arose and sank his galley. The island was Purgatory:he had tried to enter its sacred space,

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and was drowned not for poor seamanship-that could happen anywhere-but for his impudence. When Columbuswas cruising along the coast of Venezuela in 1499 he misread the altitude of the North Star, and his faulty calculation placed him at a higher latitude than where, as a matterof dead reckoning, he was sure he was. Rather than recognize that he had made a mistake quantitatively, he decided that quality had usurped quantity, i.e., that he was coasting along Eden, which was like unto a nipple on a woman's breast, and that the world was not uniformly spherical, but more in the shape of a pear than a ball (which would certainly make cartographya challenging profession).7 As for space in general, there were two kinds. Our kind of earthly and corruptiblespace extended up to the sphere of the moon. Beyond that there was another kind, incorruptible, unchangeable,and perfect,making a common physics for here and there impossible. 4. Reality is mathematical. Most medieval Europeans agreed, but in a way that we find alien. Numerology, not calculus, was their speciality. Six was perfect, being the sum of all the whole numbers that would divide into it evenly, as well as being the numberof the days in which God created the universe. Ten was the number of the Commandmentsand thereforeof law. Eleven, the number of law plus one, was associated with transgressionand sin. 666, of course, was the number of the Antichrist. Quantities, paradoxically, had qualities. The universe was mathematical, yes, but not in ways that made it a reasonable place.8 Obsession with quality made it very difficult to think of material reality quantitively. Aristotle, the intellectual dictator of the high Middle Ages, thought that the universe was a reasonable place, and, therefore, that mathematics was quite useful in understandingsaid universe. But he was also sure that qualities were more important than quantities. Rocks fell and fire rose not because of the first was heavier than the second, a relationship susceptible to mathematical description, but because of their respective qualities. Rocks belonged at the center of the universe, and smoke belongedin the sphere of fire, and each was forever trying to get where it should be-and that effort was not necessarily susceptible to mathematical description.9 The cognitive change that spawned Magellan's Postulates may have first appeared as a change of attitude about God. Certainly such a change was a prerequisitefor modern attitudes

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toward physical reality. Was God a reasonable sort? Would He allow human reason some scope for exercise,or would he always be moving the chess pieces around and even kickingover the board, making it impossible for humans to play the game and silly for them to try? In the 11th century the great Muslim philosopher, AlGhazali, under the influence of Sufi mysticism, plumped for the latter choice, i.e., the belief that God was beyond the comprehension of human reason. Al-Ghazaliprostratedhimself before the God of Islam, who was not only omnipotent(so was the ChristianGod), but also unpredictable.'0 Some scholars of the history of thought have claimed that his choice was decisive for the entire culture of Islam, or at least indicative of its pietistic trend, and that thereafterleadership in science and technology shifted to Christendom by default. I will spend only a few sentences on the contrastingchoice that the medieval Schoolmen of Chn'stendommade, about which a hundred textbooks on the history of Western Civilization exist to inform you. Medieval Westem Europeanthinkerswere faced with a difficult problem: how could Christianity,a cult of steamy mysticism, be reconciled with the cool rationalismof classical Greekphilosophy and science? After much pushing and shoving and hurling of edicts of excommunication,and extravagantexcisions of indigestible pagan material, the feat was accomplished. "Now, although the truth of the Christian faith...," said St. Thomas Aquinas, anticipating Descartes, 'surpasses the capacity of the reason, nevertheless that truth that the human reason is naturally endowed to know cannot be opposed to the Christianfaith."" The cloudburst of miracles diminished to a pelting downpour, and, to pick but one effect,sciencebecamepossible. Possible,that's all I'm trying to say. Thomistic logic-chopping may lead to only more logic-chopping. Furthermore,the path back to mysticism remained open, and indeed still is open. St. Thomas, at the end of his productive years, announced that the mountain of philosophy he had raised was no more than a molehill: "I cannot go on. _All that I have written seems to me like so much straw compared to what I have seen and what has been revealed to me."12 Something more than theology's permission to supplement faith with reason was needed to produce the change in cognition prerequisiteto so much of our civilization. St. Thomas died in 1274,and shortly after somebody or somebodies in WesternEuropeinvented the mechanicalclock,

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which a lot of historians, myself included, claim did more to shape our cognition than any other device, possibly more than any other thing, in the last 1000 years. An appreciableminority of Europeans-eventually to grow into a majority-began to "quantize"time, that is to say, to actively think of it as steady, homogeneous, and divided into durations of equal length. Within two to four generationsevery large and many minor cities throughoutWesternEuropehad huge tower clocks comparablein cost to majorcomponentsof cathedrals.13These clocksannouncing equal hours, equal during day and night throughoutthe year no matter what the actual duration of daylight and darkness. For the first time not the sun, not the moon, not the crowing of the rooster, but a giant machine dictated the activities of large numbers of people. "Techno-stress"begins in the 14th century. The first writing of polyphonic music was another indication of a change in the perceptionof time. WesternEuropeans began to compose music synchronizingtwo, three,and even more voices singing different tones of differentdurations simultaneously. To do that,you need a concept of abstractand uniformlyflowing time. By the beginning of the fourteenthcentury music theorists at the University of Paris and elsewhere in the north of Europe were talking about an abstractunit of musical time called a tempus, and a new kind of song became popular in Paris,the "preciselymeasured song."14

By 1350certainly,by 1300probably,some four centuries beforeNewton, the conceptof absoluteand homogeneous time was in circulationamong WesternEuropeans. In the same period in which Europeansinvented the clock and the "preciselymeasured song," they also began to conceive of space in a new way. Neither the maps nor the pictures of the Early Middle Ages had been depictions of geometric reality, i.e., of objects accurately depicted in their relative sizes and in spatially accuraterelationship to each other. Jerusalem,to give an example, was usually in the centerof most world maps, which is to say, in the center of the Eurasian-Africanlandmass. Jerusalemlay at the center because that is where the Bible and Christiancommon sense dictated that the city of the Crucifixionmust be. In pictures kings were much bigger than noblemen,and noblementhan serfsbecause artists knew that kings were more importantthan noblemen, and noblemen than serfs. In paintings tables often tipped up vertically because artistswanted observersto see the contentsof the soup

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bowls-but the soup did not run out. Artistsnever treatedempty space as something with true dimensions, like solids, because it was-empty! And they usually snubbed the third dimension because there were more importantmattersto worry about in a picture.'5

At the end of the 13th century, within a couple decades of St. Thomas's death, the avant-guarde painters, the most important of whom was Giotto,began to worry about perspective,and began to indicate a third dimension. After a pause of some decades for the Black Death, other Italian painters and theoreticians of painting, most particularlyBrunelleschi(who was also a maker of fine clocks) and Leon BattistaAlberti, worked out the principles and practices of what we call Renaissance perspective. By that term we mean the depiction of objects and of emptyspace,too, in geometrically accuratefashion. In the same years, around 1400, their contemporariesand sometimes they themselves revived Ptolomey's methods of making maps, and began to produce flat maps of the curved surfaceof the world in geometricallyrigorous projections.'6

By 1400or not long aftera concept of space as a dimension that was straight and uniform in all its parts (at the foot of the Cross or in a pig sty) was in circulationin WesternEurope. At the end of the century that 1400began, Columbus,Da Gamaand the other explorers inadvertently shoved Jerusalemout of its central position-in fact, rendered suspect the very idea of geographical centrality. In the next century, the 16th, the astronomerssuggested and then proved that the space above us was not intrinsically different from the space down here. Copernicus'splanets revolved around the sun just like the earth, sapping the belief that their space was markedly different from ours. Tycho Brahe'snova and comets appeared and disappeared, brightened and faded, proving that the heavens above were changable and not at all perfect.'7 A universal physics became feasible. It took a long time for thinking in terms of quantity,an approach of universal applicability, to begin to displace thinking in terms of qualities, an approach which alters with location and time. The new approach'ssources and features(often one and the same) were various. In the Renaissance,Arabicnumerals and the techniques of calculationwe learn in grammarschool today advanced in usage, and Romannumeralsand the countingboardor abacus retreated. Such helpful notations as "+" and "-" and ways to

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expresssquareand cuberootsby number,and knownsand unknowns in algebra by letters were gradually displacing the ancient and confusing system of expressing all mathematicalprocesses by words.18 By the time Descartes was born, mathematicianscould, much of the time, understand each other without recourse to their vernacularsor even to Latin:these might sustain flights of poetic sensibility, but could make a riddle of something as simple as the Pythagorean theorem. Notational improvements enhanced the ability of the Western Europeanmind to think abstractly about physical reality by enabling it to think about it one physical quality, described mathmatically, at a time. If the good old human desire for convenienceurged Western Europeansdown the road toward quantitative thinking, so did the good old human drive for communion with the ultimate mystery. In fact, the Western European intellectuals embraced mystical Neoplatonism while they were still only toying with Arabic numerals. Neoplatonism, too, had a strong influence on mathematicsduring the Renaissance. Many of the same men who bought spring-drivenclocks for their homes and lauded the new techniques in painting and cartography,spent their treasureon translationsof texts of Plato obtained from Byzantium, crumbling before the Turkishadvance. Marsilio Ficino produced translations of all Plato's dialogues, founded a Platonic academy near Florence (and, incidentally, invented the concept of Platonic love), and indulged in near worship of numbersand geometricalforms as the clearest evidences available to mortal senses of ultimate reality and God. He probablyknew Piero della Francesca,one of the greatest painters of the quattrocentoand also a leading mathematician. His paintings, most famously "The Flagellation of Christ,"are simultaneously as geometrically accurate as if they were made with a theodolite, and chockful of Neoplatonism manifested as Platonic solids and numerical relationships.19In this he was kindred spirit to JohannKepler, the astronomerand mathematician, a century and a half later. But mathematics, especially the Neoplatonic kind, is just as likely to lead nowhere as the most enraptured contemplation of one's navel. While Neoplatonists pursued pure abstraction,two other groups were applying mathematics directly to tangible reality. Artisans and merchants mixed arithmetic,geometry and material reality together in the same bowl. The artisans built palaces and cathedrals, water- and windmills, bridges, catapults

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and, after a while, cannons, utilizing mathematicsmore and more as their creationsbecame bigger and more complicatedin structure and function. Even more important,I think, were the merchants and bankers, the bourgeoisie, who lived in "an atmosphereof calculation"(Jacquesle Goff's phrase).20 Double-entrybookkeeping-an accountantto the Fuggers called it "a magic mirrorin wlich the adept sees both himself and others"21-appeared in the 14th century, enhancing the merchants' tendency to transformeverything pertinentbut the hope of salvation into numbers. WernerSombart,an early 20th century political economist, described the medieval businessman working over his double-entry books as not really thinking about corn or wool or cotton or cloth or the cargoesof ships, or tea or pepper. These, (the true realitiesof commerce)become mere shadows, they become unreal and the apparent reality seems to lie in bookkeeping ciphers.22 To Sombart'slist we can add time, as well. Intereston a loan is payment for time. When the North Italian money-changers invented the basic mumbo jumboof modernbanking, interestcame to mean an accruingdebt for the right to call one's own for a period of time something that might or might not actually exist at a given moment, i.e., gold in a strongboxin Florenceor, say, Venice. All of it, the whole mind-boggling schmier, happened not in reality, but in numbersin minds and on paper. The medieval and Renaissance bourgeoisie played with mathematical abstractions as continually as do our theoretical physicists. By the time Galileo and Kepler and Descartes were born, a lot of people in Europepossessed convenientsystems of mathematicalnotation, and were comfortable thinking about reality in terms of quantity, i. e., of thinking about it in terms of mathematical abstraction. Galileo, Kepler, and Descartes were as much the end productof a momentousdevelopmentin human thought as they were a beginning. By the last decades certainly of the 16th century, and probably by the end of the 15th century, a significantly large and growing numberof WesternEuropeanswere gazing fixedly on a fresh and remarkablevision of reality, of time and space. To them God was still hugely important-they were ready to kill in His name at the merest provocation-but God, being outside of time and

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and space, was really not in this vision. What was present (permit me to wax poetic for the sake of brevity) was time and space as a plane stretching in all directions without feature except for carefully numbered lines drawn equally distant from each other and crossing each other at right angles. It was a Euclidian stage (it smacks as much of Chiricoor Dali, I think), a stage which was, so to speak, empty of everything except the observer. The belief that time and space are each undeviatingly uniform and accuratelydescribablein terms of mathematics enormously enhanced our abilities to gather and to control data about material reality, to understandor at least to believe that we understandour universe. It understandablyproduced intellectual arrogance,as was clearly enunciated by JohannKepler400 years ago:23 For what is there in the human mind besides figures and magnitudes? It is only these we can apprehend in the right way, and if piety allows us to say so, our understandingis in this respect of the same kind as the divine, at least so far as we are able to grasp some of it in our mortal life. Only fools fear that we make men godlike in doing so; for God's counsels are impenetrable,but not his material creation. Pride, which is usually listed first among the Seven Deadly Sins, often leads to problems. For instance, the belief that time and space are, respectively,homogeneous has encouraged our faith that there is nothing which, by its very nature, is hidden from us. We believe that we can understand that which in fact is very different,even alien, from what we know well. On the one hand, this belief has stimulated the glorious hubris of the cosmologists, who discourse arrogantlyand harmlessly on black holes and the Big Bang. On the other hand, it encourages those who are confident that we understand the long range effects of clear cutting a forest. It provided the premise for Americanforeign policy after World War II, which Henry Kissinger described in 1974 as "basedon the assumption that technology plus managerial skills gave us the ability to reshape the international system and bring about domestic transformationin the 'emergingcountries,"'an assumption which he granted "has proved too simple."24 Those who make a creed of Magellan's Postulates are obliged to emphasize the quantifiable at the expense of the

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unquantifiable. Lord Kelvin, a loyal celebrant of the Postulates, wrote that when you can measure what you are speaking about and express it in numbersyou know somethingaboutit;but when you cannotmeasureit, when you cannotexpress it in numbers,your knowledge is of a meagreand unsatisfactory kind...

Thus defended against the distractions of the unquantifiable,Lord Kelvin became one of the greatest of scientists of all history. (It may be obliquely pertinent here to note that until recently the scientific establishmentignored turbulenceas a subjectfor mathematicalanalysis. It is much more common than linear flow, but it is also devilishly difficult to quantify.) The mathematician's creed, when applied to matters of greater complexity than Kelvin's thermodynamics,can lead to farce. Petrus Camper, the anatomist, sought in the 18th century a statistical definition of human loveliness and found one. Facial beauty, he decided, was a matter of the angle between lines drawn from the ear opening to the base of the nose and from the upper lip to the brow: "Whatconstitutes a beautiful face? I answer a disposition of traits such that the facial line makes an angle of 100 degrees with the horizontal."26There is also Kenneth L. Pike's heroic attempt in 1967 to supply the social sciences with a basic unit like the physicist's atom or the linguist's phoneme, indispensable to mathematical analysis. He called his invention "the behavioreme,"and offered as examples the church service or the football game.27 Humorless faith in mathematics can lead to the worst kind of reductionism, as in the extension of cost benefit analysis to the exploitation of natural resources. Attempts to place a monetary value on disruptions of an ecosystem are quite as insane as maunderingsabout "harmonicconvergence,"and a greatdeal more dangerous.28Faithin numbersrepresentinghuman events and emotions can lead to tragedy. JamesWilliam Gibson, in his book, The PerfectWar:Technowarin Vietnam,points to numbers as chief among the hallucinogens of the American leaders in the 1960s, numbersthat became "thebase for the generationof more numbers," numbers that were in their apparentprecision so clear and objective that they possessed the power to "outlive verbal commentary that expresses doubt as to their validity." By early 1967 American

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military intelligence in southeast Asia was printing 1400 pounds of reports a day, most of them based on numbers mistaken for reality and expressed crisply in numerous graphs and charts.29 In the last page or so we have discussed that which, in 20/20 retrospection,were clearly faulty applications of Magellan's Postulates. Perhaps we have been beating the proverbial dead horse: no sophisticated person thinks that Camper was right about facial beauty or that Vietnamese peasants under stress will necessarily react in ways that Americans define as reasonable. It is worthwhile, however, to point out that the falseness of the conclusionsdrawn by Camperand the Americanpolicy makersgrew not so much out of the insufficiencyor inaccuracyof data as out of inappropriateness. If these investigators had possessed a hundred times as much data, and all of it precisely accurate,to boot, they would still would have made fools of themselves. They had taken their Flatland postulates and their Flatland yardsticks into a universe with three and maybe more dimensions. Such silliness continues:some of us continueto believe we can representhuman intelligence accurately with a number. The work of Albert Einstein is especially pertinent to the point of this paper about Magellan'sPostulates, because he proved that at least two of them are false. Neither time nor space is homogeneous; they are subjectto all kinds of stretching,shrinking, and warping. His teachings about the elasticity of "where-when" has meant little of immediate practical difference to even most scientists,much less policy makers,but should be acknowledged whenever any of us think about applying the Postulates, acknowledged the way some Christians,no matterhow much in a hurry, bend at least a perfunctoryknee as they pass in front of the altar. As for physical reality being neatly predictable, our contemporarypioneers in the study of chaos seem to have knocked that into a cocked hat. As for physical reality being mathematical in nature, it was BertrandRussell, not an enemy but a friend of science, who said, 'Thysics is mathematicalnot because we know so much about the physical world, but because we know so little: it is only its mathematicalproperties that we can discover."30 Kepler was very close to right when he asked, "Forwhat is there in the human mind besides figures and magnitudes?"but that may well be more a matter of our inadequacy than of the simplicity of reality. The West to which Kissinger referred in the quotation at the beginning of this paper has not been wrong in "recordingand

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classifying data-the more accurately the better," but perhaps has been so in forgetting that there are important things which resist and even defy recordingand classifying. As for the non-Western cultures, their "essentially pre-Newtonian view" (Kissinger's words) may include perceptions of a subtler reality than that which Magellan's Postulates presuppose. Such might lead to insights and adaptations that the West would find useful. When Alice fell down the rabbit hole, she found herself in a universe she did not understandand so she had an identity crisis. Alice, a very modern young lady, tried to find her mental bearings by reciting multiplication tables:31 I'm I, and-oh dear, how puzzling it all it! I'll see if I know all the things I used to know. Let me see, four times five is twelve, and four times six is thirteen,and four times seven is-oh, dear!-I shall never get to twenty at this rate. Magellan's Postulates were sometimes, but not always, adequate guides for Alice in her Wonderland,nor are they for us in ours. They dictate the questions that occur to us to ask, they provide at least half of each answer, and they include nothing about humility, which all the sages and many physicists tout as indispensable.

'Henry A. Kissinger, American Foreign Policy, ExpandedEdition (New York: W. W. Norton & Co., 1974), 48. 2Rene Descartes, Discourse on Method and the Meditations, trans. F. E. Sutcliffe (Harmondsworth: Penguin Books, 1968),180. 3lssac Newton, Mathematical Principles, trans. Andrew Motte and Florian Cajori (Berkeley: University of California Press, 1934), 6. 4Discoveries and Opinions of Galileo, trans. Stillman Drake (Garden City: Doubleday and Co., 1957), 237-8. 5The Life of Saint Teresa of Avila by Herself, trans. J. M. Cohen (Harmondsworth: Penguin Books, 1957), 127. 6Saint Augustine, ne City of God, trans. Marcus Dods (New York. The Modern Library, 1950), 489-90, 867. 7Dante Aligheiri, The Inferno, Canto XXVI, lines 94-131; Samuel Eliot Morison, Admiral of the Ocean Sea, A Life of ChristopherColumbus (Boston: Little, Brown and Co., 1942), 557-8.. 8St., Augustine, City of God, 374-6, 508; The Convivio of Dante, trans. Philip H. Wicksteed (London: J.M. Dent and Sons, 1912), 120-1. 9A. C. Crombie, Medieval and Early Modern Science (Garden City: Doubleday and Co., 1959), HI, 36-7.

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10MarshallG. S. Hodgson, TheVentureof Islam (Chicago:University of Chicago Press, 1974),II, 180-92. l1MedievalPhdosophy,SelectedReadingsfrom Augustineto Buridan, ed. Herman Shapiro (New York.TheModernLibrary,1964),346. 12W.A. Wallaceand J. A. Weisheipl,'ThomasAquinas,"New CatholicEncyclopedia(New York: McGraw-Hill,1967),XIV,109. 13David S. Landes, Revolutionin Time:Clocksand the Makingof the ModernWorld (Cambridge:HarvardUniversityPress, 1983),53-82. 14Johannes de Grocheo:ConcerningMusic, trans. Albert Seay (Colorado Springs: Colorado College Music Press, 1973),21, 22; F. AlbertoGallo, Musicof theMiddleAgesII, trans.Karen Eales (Cambridge:CambridgeUniversity Press, 1985),11-12;SourceReadingsin MusicHistory, I, AntiquityandtheMiddleAges, ed. OliverStrunk(New York,W. W. Norton, 1965),140, 142. 15H. T. Kimble,Geography in theMiddleAges (London:Methuenand Co., 1938),241-4;Leo Bagrow, Historyof Cartography,trans. D. L. Paisey, rev. R A. Skelton (Chicago:Precedent Publishing, 1985),49; MiriamS. Bunim, Spacein MedievalPaintingand the Forerunners of Perspective(New York:AMSPress, 1940), 127-35;John Beckwith,EarlyChristianand Byzantine Art (Harmonsworth:PenguinBooks,1979),241-85. 16SamuelY. Edgerton,Jr. TheRenaissanceRediscovery of LinearPerspective(New York:Basic Books, 1975),passim. 17Crombie,Medievaland EarlyModernScience, II, 167-9. 18FrankJ. Swetz, Capitalismand Arithmetic:TheNew Mathof the 15th Century (La Salle, Il.: Open Court, 1987),11-2;David. EugeneSmith, Historyof Mathematics(New York:Dover Publications,1958),11,397-409. 19'Platonism and neo-Platonism,"A ConciseEncyclopaedia of the ItalianRenaissance,ed. J. R. Hale (New York:OxfordUniversityPress, 1981)256-7;MarilynA. Lavin,PierodellaFrancesca, theFlagellation(London:Allen Lane e PenguinPress, 1972),12-3,23, 24, 45-8,53-82. 20Swetz, CapitalismandArithmetic,passim; Jacquesle Goff, 'The Town as an Agent of Civilisation, 1200-1500,"TheFontanaEconomicHistoryof Europe:the MiddleAges (Glasgow: William Collins Sons and Co., 1972), 91.

21Michael Baxandall, TheLimewoodSculptorsof RenaissanceGermany (New Haven: Yale UniversityPress,1980),136. 22TheHistoricalDevelopment of Accounting,A Selectionof Papers , ed. Basil S. Yamey (New York ArnoPress,1978),53-4. 23Crombie, Medievaland EarlyModernScience, R, 188. 24Kissinger,AmericanForeignPolicy, 57-8. 25WilliamThomson,PopularLectures andAddres (London:MacmillanAnd Co., 1891),I, 80. 26 StephenJay Gould, '"etrus Camper'sAngle,"NaturalHistory,XCVI(July1987),12-16. 27KennethL. Pike, Languagein Relationto a UnifiedTheoryof the Structureof HumanBehavior (OheHague:Moutonand Co., 1967),121. Forfurtherdiscussion,see pages 120 through150of this book. 28Se, for examples,Colin W. Clark,"Clear-CutEconomies:Should We HarvestEverything Now?" TheSciences, XXIXqan.-Feb.1989),16-19. in Vietnam(Boston:AtlanticMonthly 29JamesWilliamGibson, ThePerfectWar,Technowar Press,1986),152,153,158. 3OBertrand Russell,An Outlineof Philosophy(Cleveland:The WorldPublishingCo., 1960),163. 31Lewis Carroll,7he AnnotatedAlice, ed. MartinGardnerCNewYork:ClarksonN. Potter,1960), 38.

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