by Peter Watson
Between the early fourteenth century and 1500 the number of universities grew from about fifteen or twenty to about seventy, though Germany and Spain lagged behind elsewhere.80 Most of the fifteenth-century universities were founded as secular institutions, by municipalities, and were only confirmed by the papacy. They included Treviso (1318), Grenoble (1339), Pavia (1381), Orange (1365), Prague (13471348), Valence (1452) and Nantes (1461). This multiplication enabled far more students to attend a local establishment, which in turn helped confirm the secular nature of the newer universities. Many of the fifteenth-century French studia among them Aix (1409), Dôle (1422), Poitiers(1431) and Bourges (1464) escaped ecclesiastical interference from the beginning, as was the case in Germany, Bohemia and the Low Countries. Vienna (1365), Heidelberg(1385) and Leipzig (1409) were founded by local rulers, whereas Cologne (1388) and Rostock (1419) were sponsored by town authorities. In the main, the northern universities were organised on Parisian lines, as a masters’ university, whereas the southern European establishments were modelled on the Bolognese pattern, as a students’ university.81
Medieval universities had no formal entrance requirements. A prospective student simply had to demonstrate a proficiency in Latin sufficient to understand the lectures. (He was also expected to converse in Latin while inside the university precinct.) There was no obligation to sit a written examination for a degree, but the student was assessed at every point in his academic career. ‘Wastage was higher than today and universities felt no responsibility to shepherd someone to a dubious degree.’82 Apart from attending lectures (obligatory, in the mornings, with no distractions), a student was also required to attend the public disputations which each master delivered once a week in the afternoon. These disputations fell into two types: de problemate, which comprised logical matters; and de quaestione, which related to mathematics, natural sciences, metaphysics and other areas of quadrivium study. Advanced students had to contribute to the magisterial disputations as a requirement of their degree. Even if many of the undergraduates were too young to make much of a contribution to the formal disputations, their very presence at these firework displays helped them transcend the rigid straitjacket of their hitherto ‘authority-dominated’ education. The most liberating of these occasions were the disputations de quolibet.83 At these times, any proposition, regardless of authority, could be argued and any question, ecclesiastical or political, and however controversial, could be considered. They were open to everyone.84
Parallel with the rise of the universities, another major change was overtaking Europe, less coherent, less specific, less sensitive in either religious or political terms, but ultimately just as practical and certainly no less profound. This was the rise of quantification. In the half-century between, say, 1275 and 1325, a whole raft of innovations were made right across the board in Europe that totally changed man’s habits and the way he thought about the world. According to Alfred W. Crosby, ‘there was nothing quite like this half century again until the turn of the twentieth century, when the radio, radioactivity, Einstein, Picasso, and Schönberg swept Europe into a similar revolution’.85 During this narrow isthmus of time, and everywhere one turned, life was becoming more quantified and quantifiable. Some historians see in this a major change, which propelled Europe to advance over China, India and the Islamic world.
Until this point, space and time had been vague. For historical and religious reasons that were ‘obvious’ to Europeans, Jerusalem was the centre of the world, which was divided into four kingdoms derived from a passage in the book of Daniel. Time was still not universally understood as divided into BC and AD. Some preferred a threefold division the Creation to the Ten Commandments, the Commandments to the Incarnation and the Incarnation to the Second Coming.86 It was widely understood that salvation was impossible for those who had lived before Jesus which is why, in The Divine Comedy, Dante places Homer, Socrates and Plato in Limbo, rather than in Purgatory or Paradise. Although ‘hours’ existed, the medieval day was in practice divided into seven canonical ‘hours’ matins, prime, tierce, sext, none (from which the English word ‘noon’ derives), vespers and compline, when prayers were to be said.87 Everything below the heavens was made of the four elements and was changeable. But the heavens were perfect, formed a perfect sphere about the earth and were made of the fifth and perfect element, ‘which was changeless, stainless, noble, and entirely superior to the four elements with which humans were in contact’.88 This idea is reflected in our modern English word quintessence.
Number itself was an approximate notion in the Middle Ages. Recipes for making such things as glass, or the metal parts of organs, rarely included precise numbers instead phrases such as ‘a bit more’ or ‘a medium-sized piece’ were accepted as sufficient. A large group of buildings, such as in the city of Paris, were described as like ‘the stalks in a field’. Roman numerals were still in use, making arithmetic difficult and they were not always written as we understand them: MCCLXVII might be: x.cc.l.xvij. It was the practice to end large numbers with a ‘j’ so that additions could not be fraudulently introduced. Cardinal and ordinal numbers would be represented thus: vo and vm.89 In finger reckoning beyond ten, someone pointed to the joints of his or her fingers for multiples of ten, and for very large numbers for instance, 50,000 one pointed one’s thumb at one’s navel. ‘Complaints were made that the higher numbers required “the gesticulations of dancers”.’90
But, and this is the main point, at the end of the thirteenth century society in Europe changed from one where ideas mainly concerned qualitative perception to quantitative in all aspects. This may have had something to do with population changes–the West’s population at least doubled between 1000 and 1340. Either way, there was introduced what Jacques le Goff has called ‘an atmosphere of calculation’ into European life.91 This too had a great deal to do with the rediscovery of Aristotle. Alfred Crosby draws attention to Peter Lombard’s standard textbook of theology, Summa sententiarum, written in the mid-1100s, which had only three quotations from secular philosophers, amid thousands from the Church Fathers, whereas Thomas Aquinas, in his Summa theologiae, written between 1266 and 1274, had 3,500 quotes from Aristotle alone, 1,500 of them from works unknown in the West a hundred years before.92
It was now, for instance, that literacy surged, partly stimulating and partly caused by changes in writing (the stabilisation of word order–subject–verb–object–was also achieved). The best-known example of this is the change between Innocent III (11981216) who dispatched at most a few thousand letters a year, and Boniface VIII (12941303) who wrote as many as 50,000. M. T. Clanchy reports the extraordinary detail that, on average, England’s royal chancery in the 1220s used 3.63 pounds of wax per week for sealing documents, but that this had risen to 31.9 pounds per week in the late 1260s. At that stage there were few or no divisions between words, sentences or paragraphs (the Romans had abandoned word separation). In general, this meant that reading was difficult and conducted aloud. It was only in the early fourteenth century that the new cursive writing (see above, pages 249250) was combined with word separation, punctuation, chapter headings, running headlines, cross references and other devices we now take for granted (plus some that we don’t, like a half-circle, , to indicate that a word was continued on the next line). Around 1200 Stephen Langton (a future archbishop of Canterbury) devised the chapter and verse system for the books of the Bible, which until then were almost entirely undifferentiated. Libraries had traditionally been organised along religious lines–the Bible came first, then the Church Fathers, with the secular books on the liberal arts last. But beyond this broad agreement the actual order of many texts was arbitrary and unreliable, and so it was now that the scholars introduced alphabetisation. Everyone understood it and the order implied no doctrinal significance.93 In the same way scholars also introduced the analytic table of contents. Each of these innovations changed the experience of reading, in particular from reading aloud to reading in silence. In 1412 Oxford an
d in 1431 Angers introduced the regulation that libraries were to be quiet places hitherto they had been anything but. In the same way, book learning displaced the emulation of charismatic figures as the central feature of education. This was extremely important, in that reading became a private and therefore a potentially heretical act (especially important in fifteenth-century England). There is also evidence that the privacy provided by silent reading led to an increase in erotica.94
The first clocks in towns had no faces or hands but were just bells. (‘Clock’ is related to the French cloche and the German Glocke, which mean ‘bell’.) Bell clocks were very popular from the start. A petition for a city clock at Lyons read: ‘If such a clock were to be made, more merchants would come to the fairs, the citizens would be very consoled, cheerful and happy and would live a more orderly life, and the town would gain in decoration.’95 Many towns, even small ones, agreed to tax themselves so that they could have a clock. The mechanical clock was probably invented in the 1270s (the same decade as spectacles), and Dante refers to clocks in Paradiso, written about 1320. Although China had clocks before Europe, it was the West’s enthusiasm for equal hours that changed perceptions of time equal hours were in general usage in Germany in the 1330s.96 Jean Froissart, historian of the Hundred Years War, began his chronicle using canonical hours, but shifted to equal hours in the course of his narrative. It was not long before the town clock determined when the working day should start and end.
The discovery of perspective (considered later in more detail in Chapter 19, on ideas about beauty), and its relation to mathematics, was another aspect of the quantification of life that took place about this time. We see the first hints of it in Giotto (1266/7 or 1276–1337), then with Taddeo Gaddi (d. 1366) and it was firmly in place by the time of Piero della Francesca (1410/1420–1492). Each of these discoveries and applications complemented one another, so much so that Nicholas of Cusa (1401–1464) was moved to remark ‘God is absolute precision itself.’97 This form of thinking would result in the work of Nicholas Copernicus (1473–1543), which helped start the scientific revolution and made space both much bigger and yet more precise.
Al-Khwarizmi’s book on Hindu numerals, and algebra, was translated into Latin by Robert of Chester in the twelfth century and from then on the influence of the new numerals began to grow (the last mathematics textbook to use Roman numerals was written in1514).98 There was, however, a curious cross-over period when people in Europe used both systems. One writer wrote the year as MCCCC94, that is, two years after Columbus discovered America, while Dirk Bouts dated his altarpiece at Louvain as MCCCC4XVII, which probably means 1447. The operational signs for arithmetic came later. In the last half of the fifteenth century Italians and others were still using for ‘plus’ and for ‘minus’. The familiar ‘plus’ and ‘minus’ signs, + and ?, appeared in print in Germany in1489. Their origins, Alfred Crosby says, are obscure: ‘Perhaps they sprang from the simple marks that warehousemen chalked on bales and boxes to indicate they were over or under weight.’99 In 1542, Robert Recorde in England announced that ‘thys figure +, whiche betokeneth to muche, as this lyne, ? plaine without a crosse lyne, betokeneth to little’. And it appears to be Recorde who, in the sixteenth century, invented the ‘equals’ sign, =, to avoid repetition of the words ‘is equal to’ and because ‘noe 2 thynges can be moare equalle’.100 The × sign for multiplication was not settled for centuries: to begin with in medieval manuscripts it had as many as eleven different meanings. Fractions were a function of trade and, in the Middle Ages, could be very complicated, such as and, in one case, . Decimals existed in embryo but the system was not finally completed for another three hundred years (see Chapter 23).
With the arrival of Hindu-Arabic numerals, algebra was at last capable of development. In the early thirteenth century Leonardo Fibonacci used a letter in place of a number, but never developed this idea. His contemporary, Jordanus Nemorarius, used letters as symbols for known and unknown quantities but he had no signs for plus or minus, or multiplication. It was the French algebraists in the sixteenth century who fully codified this system. Francis Vieta used vowels for unknowns and consonants for knowns, and then, in the seventeenth century, Descartes introduced the modern system, a and b and their neighbours at the beginning of the alphabet for knowns, and x and y and their neighbours at the end of the alphabet for unknowns.101
Alongside these changes in writing and mathematics ran parallel developments in music notation. Gregorian chant, the most famous form of medieval church music, is characteristically nonmensural: the structure of its musical line is determined by the flow of the Latin words. However, by, roughly speaking, the tenth century the number of different chants had grown so much that no one person could remember them all and a system was needed to record them. To begin with they produced what one scholar has called ‘pneumatic notation’–a system of marks to indicate breathing, when the voice should rise in pitch (an acute accent, ´), or drop (grave, `), or rise and fall (circumflex, ˆ). This was improved when the monks lightly traced one and then two or more horizontal lines across the page to make the high and low notes easier to recognise–this was the beginning of the staff or stave. The staff is traditionally credited to the Benedictine choirmaster, Guido of Arezzo, who, whether or not he invented it, certainly standardised it. He famously remarked, about his fellow choristers, ‘we often seem not to praise God but to struggle among ourselves’.102 With the new methods Guido said he could produce a good singer in two years rather than in ten. It was Guido who noticed that in the familiar hymn Ut queant laxis, sung for the feast of John the Baptist, the tones rose as in the staff:
Ut queant laxis Resonare fibris
Mira gestorum Famuli tuorum
Solve polluti Labii reatum
Sancte Iohannes*
The Italic notes above, ut, re, mi, fa, sol and la, became the basis of the elementary methods of teaching notes that all children now learn, with do replacing ut later on, possibly because the t of ut was unsingable.104
The basics of Gregorian chant were the tenor voices (from the Latin tenere, to hold), which formed the cantus firmus (firm song) or, as we might say, the basic drone. From the late ninth century, other higher voices began to break away, though at first they kept in parallel. Later still, they broke away more dramatically and this formed the basis of Western polyphonic music, which also appears to have been the first music to have been specifically composed, written down, in note form, rather than evolved through trial and error with voices. This occurred most of all in Paris, where the profession of musician first emerged. Music was part of the quadrivium, one of the advanced mathematical arts in which all advanced scholars were trained in the Middle Ages. Perotin, of the Notre Dame school, introduced rests (a concept possibly derived from the zero), while Franco of Cologne codified the notation system, determining time values for all notes and rests. He outlined four single-note signs in music notation the double long, the long, the breve and the semibreve, which were exact multiples of each other. The basic unit was a tempus, defined as ‘the interval in which the smallest pitch or smallest note is fully presented or can be presented’.105 The new music polyphonic music, written down, offering far more control over fine detail became known as ars nova, compared with ars antiqua. Not everyone liked the new music, including perhaps inevitably–the pope. In Docta sanctorum patrum, the first papal proclamation dealing with music, he raged against polyphony, which was forbidden in churches.106
The final important element in the growth of quantification was the introduction of double-entry book-keeping and its associated techniques. A continuous record of the books belonging to Francesco di Marco Datini, a merchant of Prato, from 1366 to 1410, shows that Hindu-Arabic numerals began to appear about 1366 and that until 1383 the accounts were kept in narrative form. After that date, however, the practice changed and assets and liabilities began to be kept in parallel columns either on the same page, or on facing pages. From then on it was immediately obvious
, as it had not been obvious before, whether a business was in profit or loss.107 In Tuscany, the technique was known as alla veneziana, in the Venetian manner, suggesting it was in use there earlier. Balancing the books has since become a sacred ceremony of our age but it was an important innovation for the age of discovery, which enabled men to keep control over their enterprises as businesses ventured thousands of miles around the world.
The spread of quantification, no less than the spread of learning, was amplified and accelerated by the invention of printing. In the thirteenth century the majority of students could not afford to buy copies of the texts they studied, at least not without great sacrifice, because of high manuscript price levels. Consequently, the student was very dependent on the reading and expounding of the texts in the university schools. The situation was eased in the later thirteenth century by the growth of cheaper, utilitarian methods of manuscript production, encouraged and then closely controlled by the universities.108 The system was based on the multiple copying of exemplars, which were accurate copies of the texts and commentaries used in teaching. Each exemplar was divided into separate pieces or peciae, usually of four folios each (eight pages), and relating to different portions of the text. Several copyists could therefore work on the same exemplar, each reproducing a different pecia. The system enabled students to buy or hire relatively cheap copies of that particular section. The freer circulation of texts relieved the student of his reliance on the lecturer’s every word, lessened the strain on his memory, and permitted study in a more relaxed and private environment.109
