by Stephen E. Sachs
Prof. Thomas N. Bisson
May 25, 2000
R. W. Southern began the period of The Making of the Middle Ages in the year 972, when Gerbert of Aurillac pursued the study of logic from Rome to Rheims. Yet had Southern begun a few years earlier, he would have seen Gerbert, the future Pope Silvester II, travel west to northern Spain, “the only place in Latin Christendom . . . where mathematics was taught.” From this time forward, a Europe whose mathematical knowledge had been in virtually uninterrupted decline since late antiquity would witness an “arithmetical dawn,” a rebirth of classical knowledge and the fresh introduction of Arabic innovations. The period from the eleventh century to the thirteenth was one of unprecedented mathematical advance — one of a change, not in degree, but in kind.
Although the cause of this advance has never been fully determined, the quick answer has always been commerce. Buying or selling goods for money relies on familiarity with numbers, and sharing goods or collecting revenues requires division and multiplication; above all, money is quantized rather than continuous and provides a means of exact measure for an abstract quality such as value. Such a ‘countinghouse theory’ of mathematical advance was criticized sharply by Alexander Murray in his work Reason and Society; Murray suggested a more even distribution of causation between commerce and government. The countinghouse theory, however, is more resilient than Murray portrays; the influence of economics, rather than pure trade, should be considered a primary cause of the revival of the mathematical sense in medieval Europe.
It may be difficult to imagine that any society would long be unfamiliar with the basic concepts of arithmetic; yet anthropologists have found cultures with no concept of number greater than ten, and the modern tendency to quantify nature has to be viewed as contingent rather than natural and universal. The West of the Early Middle Ages, while it may have retained numbers greater than ten, certainly seems to be an age in which popular numeracy — the ability to use number, as literacy is the capacity to make use of written expression — was significantly curtailed. Of classical arithmetic, only what Boethius had preserved as part of the quadrivium was still available for teaching, and it does not seem to have been widely taught: in the seventh century, a mathematically reconcilable dispute over the calendar in Northumbria led to the celebration of two different Easters, until an contingent of foreign monks arrived who knew “the Egyptian method” of arithmetic. The primary means of calculation in this period was a system of finger-reckoning, actually no more useful than Roman numerals to multiplication or division; a large multiplication table known as the Calculus of Victorius was passed down jealously through the generations, but its copyists failed to find or correct plain errors in their originals. Aldhelm of Malmesbury, a brilliant scholar of the late seventh century, wrote of the “near-despair” to which the science of calculation had brought him, and he noted that only after much labor had he “grasped the most difficult of all natural principles . . . what they call fractions.” The rarity of mathematical knowledge made the works of those such as Bede all the more remarkable; indeed, in the eighth century his Easter tables were copied and passed for years among churches and monasteries because few others dared try their own hand at the calendar. A brief resurgence of interest during the Carolingian Renaissance, in which Charlemagne made ‘computus’ part of the curriculum for episcopal and monastic schools, faded along with the empire; in sum, the period before the millennium was, as Murray terms it, the “dark age of European arithmetic.”
What followed has been characterized as “an intellectual change as momentous as any” that have happened since: “the evolution of a mathematical sense.” The first signs of change came in the late tenth century, when Gerbert’s arithmetical studies began to exceed mere preservation of Boethius and Bede; he “dazzled” his contemporaries with arithmetical knowledge, and his astronomical instruments, although simple, were judged by others as “almost divine.” Gerbert’s time witnessed a detectable growth in the familiarity of numbers, and new mathematical instruments and techniques began to emerge: the abacus, an improved computus for fixing feast-days, the astrolabe, and the Hindu-Arabic numerals all appeared in the West after the millennium. The eleventh century itself is widely viewed as a watershed; Bloch’s two feudal ages were divided at mid-century, and the latter half saw “many signs of new life” in politics, religion, the economy, and the intellectual scene, with a revival of interest in Roman law and the classics. The twelfth century that followed brought with it a new feeling in some circles that “intellect is the key to effective political power”; the scholars of an expanding Europe hungrily grabbed at any knowledge they could find. The scientific and mathematical knowledge of the Greeks and Arabs began to pour across the borders of Christendom: after 1100, Euclid’s Elements gained increased prominence; in 1126 Adelard of Bath brought Al-Khwarizmi’s trigonometry to the West; in 1145 Robert of Chester translated Al-Khwarizmi’s Algebra; Ptolemy’s Almagest was translated from the Greek in 1160; and in 1202, there appeared the Liber Abaci of Leonard of Pisa (known as Fibonacci), who could solve quadratic and cubic equations and has been said to have had “sovereign possession of the mathematical knowledge of his own and every preceding generation.” Algebra, Charles Haskins records, had advanced so far in this period that there would be no notable gains until the sixteenth century: “the decisive importance of the twelfth century is nowhere more evident.”
Two developments of this period deserve special mention: the abacus and the Hindu-Arabic numeral system, neither of which were known in the West until the tenth century. The counting-table of the Romans had long disappeared and was reintroduced by Gerbert and contemporaries; by the year 1000, knowledge of the abacus had spread substantially. The device consisted of a table or board marked with vertical or horizontal lines, on which numbers were represented by groupings of counters; it allowed for faster and more accurate calculation than contemporary systems of finger-reckoning and could be used to solve practical arithmetical problems resulting from surveying and accounting. A clerk of the English Exchequer wrote a treatise on the abacus with problems as complex as “Divide £800,137 among 1,009 knights” shortly after the device’s adoption there in the early twelfth century. The most important innovation of the abacus, however, was the introduction of place value, by which a counter on one line could represent a greater amount than the same counter on another (usually by factors of ten, although some were made to follow the £.s.d. system). For the eleventh-century West, this was “a radical innovation in the representation of numbers”; Gerbert himself had noted the confusion surrounding place value, asking, “How can the same number be considered . . . now a digit, now an article [in the tens place]?” The introduction of place value allowed the easy and exact representation of arbitrarily large quantities. While Martianus Capella had complained in late Roman times that representing numbers above 9,000 through finger-reckoning required the “gesticulations of dancers,” Gerbert discussed multiplication and division of products up to ten billion. A need soon arose for symbols to record the results of abacus calculations, including the fact that a row could be left empty. In the mid-twelfth century, a mathematician named Ocreatus devised a notation incorporating Roman numerals, place value, and a symbol for zero (either 0 or the Greek letter t) such that MLCCCIX (1089) could be represented as I.0.VII.IX. The time was ripe for the introduction of the newly available Hindu-Arabic numerals.
There has been some question as to whether Boethius was aware of the Hindu-Arabic numeral system; even if he had been, he did not make great use of it, and Gerbert (who did write of the numerals, although without the crucial zero) discussed at length the “iron process” of long division in Roman numerals. The difficulty of such a task shows the importance of the new numerals for the development of mathematical analysis; yet their adoption after seems to have been slow, for institutions such as the Medici bank and the English Exchequer (the pioneer of the abacus) did not adopt them before the sixteenth century. However, some pieces of evidence show an earlier interest: a 1236 poem by Walter of Coincy uses “cipher,” the name of the numeral for zero, to mean a vacuous person — a metaphor which required his non-learned audience (in France, the region from which we have the least evidence of arithmetical advance) to be familiar with the numerals and with the concept of zero. In this period, documents appear that indicate strife between the abacists and the algorists, those who calculated with the Hindu-Arabic numerals. Perhaps the strongest evidence of the numerals’ use is official attempts to block them: a 1299 ordinance of the Florentine bankers’ guild prohibited members from recording accounts in “what is known as the style or script of the abacus” and required them to “write openly and at length, using letters” — the fact that the ordinance had to be repeated three more times meant that by 1299, bankers in Florence had found it faster and more convenient to use the Hindu-Arabic numerals rather than write “at length” in the old script. The 1299 document is the first surviving set of such statutes, so it is entirely possible that the first prohibition of the numerals was considerably earlier. Whatever the rate of their adoption, the move towards symbolic representation that the spread of the numerals entailed spurred progress in other aspects of arithmetic: although Fibonacci infrequently used letters to stand for numbers, his contemporary Jordanus Nemorious did so a good deal, and the introduction of such symbols helped medieval mathematicians to avoid the “long, convoluted, almost Proustian sentences” in which most mathematics had been written to date.
The advances in arithmetic did not remain closeted in their own field; indeed, as the poem by Walter of Coincy reveals, the arithmetical sense could extend into literature and other areas of intellectual life. The twelfth and thirteenth centuries show an increasing amount of quantification in literature; numbers cease to be either nonexistent (such as in the dateless hagiographies) or gross exaggerations (such as the figure of “60,000” repeated for various purposes in Oderic Vitalis); instead, chroniclers such as Salimbene of Parma and Bonvesino della Riva begin to aspire to accuracy and, as in the case of the fourteenth-century Giovanni Villani, occasionally achieve it. A “century’s enthusiasm” for arithmetic resulted in a general consciousness of numbers as “an increasingly salient fact of everyday life”; Fibonacci wrote of numbering that once “science has turned into a habit, memory and mind come to accord with hands and figures to such a degree that all act in harmony together, as if by one impulse and one breath.” By the fourteenth century, it is safe to say that among many Europeans a mathematical sense had, in large part, evolved.
Yet what fueled this evolution? The question of what caused the growth of arithmetic is an old one, and it has an old and simple answer, that arithmetic was revived by the needs of commerce — this is the countinghouse theory of mathematical origins. Salomon Bochner argues, with some aesthetic displeasure, that algebra arose as a response to “the very unlofty and utilitarian demands of counting houses of bankers and merchants in Lombardy, Northern Europe, and the Levant”; he states that it is “strange, and even painful” to record that European arithmetic “owes its origination to countinghouse needs of ‘money changers,’” but “regrettably” cannot suggest an alternative when “economic determinists, from the right, from the center, and from the left, all in strange unison agree.” Florence Yeldham traces the Western use of arithmetic directly to interaction with the financial and commercial interests of the Arab world; Italian merchants adopted the arithmetic in the twelfth century and introduced it to their native lands, where it was taught in the schools. “We traded with the Mediterranean,” she writes, “and our merchants could not fail to bring home some knowledge of these figures.” Theorists of capitalism such as Werner Sombart note that the “beginning and end” of capitalist activities is “a sum of money,” which must be calculated; he places the growth of arithmetic squarely within “those centuries of budding capitalism” in Italy.
A good deal of evidence supports such an interpretation. Merchant networks in the eleventh through thirteenth centuries stretched from Spain to China, and one who buys and sells in foreign markets must obtain “a knowledge of all number systems used in recording prices.” Although Gerbert may have been the first academic to introduce the Hindu-Arabic numeral system, it is highly likely that international traders had been familiar with it for decades before, especially since Gerbert’s numerals seem to be slightly erroneous representations of gobar numerals used in Spanish commerce. Other elements of the revival of mathematics suggest a commercial origin. In the 1187 English legal treatise known as ‘Glanvill,’ the test of legal majority for a burgher’s son was the ability to count money and measure cloth — participation in the adult community was made a function of mathematical thinking, and commercial arithmetic in particular. A substantial portion of the examples in Liber Abaci deal with commercial matters, as their titles indicate: “A Voyage”; “Two Merchants Who Offered Wool in Payment of Shipping Charges”; “The man who went to Constantinople to sell three pearls”; “The two ships which sailed together”; etc. Most tellingly, connections were drawn between commerce and mathematics in the literature of the time; the homilies of Guillame Peyraut, writing circa 1240, equated the “beans” and “little stones” used for counting with “marks of silver”; another thirteenth-century writer suggested changing the name of arithmetic to “aerismetica: ‘the art of money.’”
Such a simple theory for such a general trend should always evoke suspicion, however, and the countinghouse theory has not gone without challenge; Murray explicitly rejects the argument that “mathematics was the child of commercial accounting” or that commerce can serve as the “solitary socio-political explanation of the rise of arithmetic.” Although Murray divides his discussion into two separate arguments, the essence of his criticism lies in the distinction between pure and applied mathematics: academic (or otherwise non-commercial) mathematics and mercantile arithmetic were practiced by different people and covered entirely different material. No known medieval specialist in mathematics, Murray argues, “can be shown to have been a merchant,” nor is there evidence that any medieval mathematician “was directly beholden to a professional trader for the main substance of his knowledge. . . . [M]ercantile arithmetic was a markedly inferior branch of the subject. As for the superior branch, its exponents stood at a distinct distance from practical commerce.” The merchant was neither a “pioneer, nor the patron of pioneers”; the stark discrepancy in popularity between advanced mathematics and mediocre business manuals shows that the leading mathematics of the age was not practiced in countinghouses.
What Murray offers as the “other half” of the mathematical picture is government. At the same time as the commercial boom and the spread of arithmetic, Italy saw the emergence of a “mature city-state” supported by Roman law; indeed, the new mathematics seemed to follow the spread of Roman law across Europe. The first reason was that governments, like merchants, had countinghouses — and likely bigger ones than any in private hands. These larger accounting institutions had correspondingly greater mathematical influence: the word ‘million’ came into use, Murray writes, only after the revenue of the French government topped a million units. The Church, the only state of its time, also fostered mathematics through governance: bishoprics and abbeys had had to measure and distribute produce in the pre-monetary era, and now that the economy used a greater amount of money, the pope was the chief client of the great Tuscan banking houses. Yet the influence of government, Murray alleges, went beyond “a mere extension of the ‘countinghouse’ theory”; rulers had to make larger-scale and longer-term decisions than most — decisions which required more forethought — and thus tended to nourish learning as a practical matter. When “broadly understood,” the stimulus given to arithmetic by government “went beyond accounting . . . . Rulers in their countinghouses cannot explain, any more than merchants in theirs can, the higher flights of arithmetic.”
Yet it is unclear exactly how government counting would promote mathematics at a more complex level than would commercial counting. Murray’s examples of governmental mathematical activity — collecting statistics on people and animals, as in the Domesday book; recruiting soldiers and men of military age; counting votes in a majority-rule election — all deal with simple matters of enumeration, and are unlikely to use as complex a set of operations as, say, compounding interest or paying dividends to shareholders. The only actions of government in which one finds “more lofty mathematics” concern architecture and astrology; the latter is particularly suitable for Murray’s argument, for its calculations “were much more ambitious than other known or surmisable medieval calculations” and its practitioners were generally recipients of government patronage. The typical lay astrologer was generally a courtier, and government employment was a promising career path for those with mathematical interests; a popular manual for government of this era, the Secretum Secretorum (which claimed to be Aristotle’s instructions to his pupil Alexander) advised that political advisors should be “skilled in all sciences; but especially arithmetic, because it is a true science, and is a good proof of intellectual sharpness.”
Yet it would be absurd to say that the mathematical transformation of twelfth- and thirteenth-century Europe was primarily due to astrology and architecture. Perhaps these two fields were the important stimuli to advanced mathematics. However, the more significant change in this period regards numeracy, not cubic equations, and as a social stimulus to numeracy the countinghouse, whether located in private or public hands, had already done its work; the “thousands of practical reckoners” who toiled in countinghouses promoted the spread of numeracy far more than a book of such “massive size and high ambitions” as Liber Abaci, which does not seem to have been copied widely and which should be interpreted more as a high-water mark than a stimulus to further change. Murray’s argument that the demand for mathematics was not restricted to the private commercial economy rings true. Yet rather than attempting to cut the distinction between applied and pure mathematics and into two neat ‘halves,’ commerce and government, it might be more accurate to view the advances in pure mathematics of this era, either astrological or academic, as an outgrowth of a demand for applied mathematics and counting (both public and private) and as having been enabled in significant part by economic changes.
The increasing use of money since the beginning of the eleventh century, for both international exchange and everyday purchases, has been widely recognized as encouraging a quantifying mindset. After all, money comes in individual units that must be counted to be used. Men at the time realized its potential; in the fourteenth century, Walter Burley of Oxford’s Merton College (whose scholars Thomas Bradwardine and Nicholas Oresme broke new ground in attempting to quantify mechanics and optics) wrote that “Every salable item is at the same time a measured item.” Indeed, Joel Kaye argues that money’s use as a “culture-wide instrument of measurement” was critical to the development of the movement towards quantified science. Alfred Crosby argues for the concept of a “New Model,” a “new version of reality” that the West forged from the thirteenth century onward “out of current, often commercial experience,” a model that placed great emphasis on “precision, quantification of physical phenomena, and mathematics.” Murray agrees that the spread of money in a society “is a direct invitation to it to calculate with numbers,” and notes the prevalence of mathematical problems based on the everyday use of small sums; yet he considers this evidence opposed to the countinghouse theory on the belief that countinghouses must represent institutions of international (or at least large-scale) private commerce. An expanded theory could include, rather than exclude, such evidence.
The second influence of economic affairs can be found in the social realm. For one thing, governmental accounting, even when people were counted rather than coins, was in many ways a response to economic pressures: the Domesday Book and its successors owed their origins at least in part to “some ruler’s desire to sponge up liquidity, created by trade,” while the Dialogues of the Exchequer, written circa 1178, called for stricter accounting procedures because “those who condescend to trade” were engaged in “hiding their wealth.” Yet economic affairs had a second social effect in that the vast set of changes that encouraged urbanization (among which economic change played a significant but by no means solitary role) provided the necessary physical environment for the intellectual gains of the eleventh through thirteenth centuries. To begin his discussion of intellectuals in the Middle Ages, Jacques Le Goff exclaims, “In the beginning there were the towns”; Haskins attributes to an “economic and social revolution” the beginnings of “a profound intellectual change.” For instance, the spread of Roman law, which formed a core subject matter of the early universities (and was linked by Murray to the spread of arithmetic) was highly dependent on “the growth of economic and social conditions to which this superior jurisprudence was applicable.” Henri Pirenne wrote that merchants brought mobility to a people “attached to the soil,” exemplifying “a shrewd and rationalist activity” in which success depended only on “intelligence and energy”; it would be natural, then, that more complex and more urban economic activity would carry with it a need for education. For such a life literacy (as well as numeracy) was a necessity rather than a convenience, and the townsmen created lay schools where elementary education was available. The thousand-odd boys who studied the abacus and arithmetic in Villani’s Florence would not have done so had there not been a market for their skills — and not all of them could become astrologers.
The universities that provided further education to an increasingly lay group of students were by no means insulated from such social pressures; rather, the evidence indicates that they taught what mathematics was most socially useful. Despite its place in the quadrivium, there is little evidence that the new mathematics took hold in universities until 1215, when a statute of the Sorbonne mentioned it “only in an incidental way.” Once it did, one of the most popular textbooks was Sacrobosco’s Algorismus, a work which Murray describes as a middle-grade manual for the business public. Commercial and pure academic interest were by no means exclusive: Venice, one of the most innovative cities in the fields of commercial arithmetic and bookkeeping, was also the first municipality in Europe (and perhaps worldwide) to endow public lectures on algebra, and it soon became “one of the best places in the world to study mathematics.”
The final, and perhaps strongest, argument for the primacy of economic interests in the spread of mathematics derives from an investigation into the translated Greek and Arabic works that fueled the new learning. The revival of culture in eleventh-century Italy first appeared in the South, a region that had direct commercial (and sometimes political) contact with North Africa and Constantinople. Intercommunication was both an economic and a cultural phenomenon. A number of those who translated important works from Greek or Arabic were members of merchant colonies — Fibonacci’s father was a Pisan customs clerk in the North African city of Bougie — or were travelers or diplomats encouraged by those colonies’ presence; and others were learned men who followed the trade routes and carried back works of culture. The movement of both goods and ideas were indicators of a civilization that was, in Bloch’s words, “better equipped with antennae.” The linguistic links between Europe and the Arab worlds demonstrate this connection, for they are primarily scientific or commercial; the words ‘algebra’ and ‘cipher’ were learned along with ‘bazaar’ and ‘tariff.’ One can hardly imagine the intellectual growth of the twelfth century without the translations of Adelard of Bath, Robert of Chester, or Gerard of Cremona, through whose hands, Haskins wrote, more of Arabic science passed into Europe “than in any other way”; indeed, Haskins argues, these translations enabled the creation of universities, which “had not existed hitherto because there was not enough learning in Western Europe to justify their existence.” The merchants may not have been translating in their countinghouses; but they attracted the translators and participated in no small way in the dissemination of foreign knowledge.
None of this means that the changing economic life of Europe was the sole and ultimate cause of increasing numeracy and mathematical advance. Without an ideological commitment to some type of determinism, economic or other, a historian’s search for an ‘ultimate cause’ in history is likely to be frustrated; the growth in trade could be said to be dependent on politics or vice versa, and following the question quickly devolves into a chicken-and egg dispute. In any case, in life, as Lynn White wrote of agriculture, “there are usually at least two reasons for doing everything.”
There is some explanatory value, however, in identifying the role that economics could have played in the spread of European numeracy and the advances in European mathematics. Economic trends helped provide the material foundations and initial social impetus to the study of the new medieval arithmetic; such study was by no means limited to economic actors but diffused to successively larger segments of society. Counting (and countinghouses) sowed the seeds; the quantifying mentality and pure mathematics reaped the rewards.
Murray wonders at one point whether peasants in the arithmetical ‘dark ages’ might have had a good deal of undocumented familiarity with mathematics, based on the primitive exigencies” of land-division and the rearing of livestock. He pauses, however, because such assumptions overlook the societies where shepherds know “hundreds of sheep as individuals without having any idea ‘how many’ there are.” Europe before the millennium might have been such a society. By the thirteenth century, however, estate managers like Walter of Henley had begun to recognize the importance of measuring and accounting in order to improve sales to the urban food markets. There can be no better index of the encouragement economics offered to the investigation and the adoption of Europe’s new math.
 R. W. Southern, The Making of the Middle Ages (Yale University Press, 1953), p. 11.
 Alexander Murray, Reason and Society in the Middle Ages (Oxford: Clarendon Press, 1978), p. 158.
 Murray, p. 162.
 Murray, p. 142.
 Charles H. Haskins, The Renaissance of the Twelfth Century, 1927 (New York: Meridian Books, 1957), p. 81; Murray, pp. 146-8. After this time, mathematics seems (at least in England) to have recovered somewhat: in 710 the abbot of Monkwearmouth decided that Easter tables were unnecessary, since “today there are so many people able to calculate, that even in our own church in Britain there are many who know the ancient rules of the Egyptians” (Murray 147).
 Murray, p. 156.
 Murray, p. 148.
 Haskins, Ren., p. 230.
 Murray, p. 151; Haskins, Ren., p. 17; Murray, p. 141.
 Murray, p. 142.
 Haskins, Ren., p. 25. Haskins calls into question the extent of Gerbert’s original accomplishments, arguing that the elementary nature of arithmetical knowledge in the period can be seen from “the extraordinary reputation which Gerbert acquired when he went somewhat beyond these masters” (Haskins, Ren., pp. 310-1).
 Murray, p. 162; Brian Stock, The Implications of Literacy (Princeton University Press, 1983), p. 84.
 Marc Bloch, Feudal Society, ca. 1939, trans. by L. A. Manyon, Vol. 1 (University of Chicago Press, 1961), p. 60; Haskins, Ren., p. 9.
 Murray, p. 121.
 Haskins, Ren., pp. 312-3.
 Haskins, Ren., p. 312.
 J. M. Pullan, The History of the Abacus (New York: Frederick A. Praeger, 1969), p. 36.
 Haskins, Ren., p. 311; Murray, pp. 164-5. Gerbert was, in fact, sufficiently identified with the abacus for a user of the invention to be frequently referred to as a ‘girbercist’ rather than ‘abacist’ (Harriet Pratt Lattin, Introduction, The Letters of Gerbert (Columbia University Press, 1961), p. 19).
 Lattin, p. 6; Charles H. Haskins, Studies in the History of Medieval Science (Harvard University Press, 1924), p. 332.
 Gerbert of Aurillac, The Letters of Gerbert, trans. by Harriet Pratt Lattin (Columbia University Press, 1961), p. 45; Murray, p. 164.
 Murray, pp. 164, 156.
 Murray, p. 167. Pullan, whose history contains little on the Middle Ages, notes that the abacus could also have effects retarding the spread of Arabic numerals; because new rows or columns were occasionally introduced in the abacus to represent the quantity five, it might be easier to record their results with the use of the Roman numerals D, L, and V, functional equivalents to which were lacking in the Arabic system (Pullan, p. 35).
 David Eugene Smith and Louis Charles Karpinski, The Hindu-Arabic Numerals (Boston: Ginn and Company, 1911), p. 65; Haskins, Ren., p. 311.
 Louis Charles Karpinski, Introduction, Robert of Chester’s Latin Translation of the Algebra of Al-Khowarizmi (New York: Macmillan, 1915), p. 13; Alfred W.
Crosby, The Measure of Reality: Quantification and Western Society, 1250-1600 (Cambridge University Press, 1997), p. 116; Murray, pp. 168-9.
 Murray, p. 173.
 Smith and Karpinski, p. 120.
 Murray, pp. 169-71.
 Murray, p. 170.
 David Eugene Smith, History of Mathematics, 1923, 2 vols. (New York: Dover, 1958), p. 2:427; Crosby, p. 41.
 Murray, pp. 174-7, 182-4.
 Murray, pp. 185-6, 166.
 Salomon Bochner, The Role of Mathematics in the Rise of Science (Princeton University Press, 1966), pp. 39, 113.
 Florence A. Yeldham, The Story of Reckoning in the Middle Ages (London: George G. Harrap, 1926).
 Werner Sombart, The Quintessence of Capitalism, 1915, Trans. and ed. by M. Epstein (New York: Howard Fertig, 1967), pp. 125-8.
 Smith and Karpinski, pp. 101-2.
 Smith and Karpinski, pp. 109-114.
 Murray, p. 181.
 Joseph and Frances Gies, Leonard of Pisa and the New Mathematics of the Middle Ages (New York: Thomas Y. Crowell, 1969), pp. 101-2; Murray, p. 190.
 Murray, p. 191.
 Murray, pp. 22, 192.
 Murray, p. 192. The one exception Murray finds to this rule is Giovanni Villani, whose wide-ranging interests “made him a freak anyway” (Murray, p. 192).
 Murray, p. 193-4.
 Murray, p. 194.
 Murray, p. 194.
 Murray, p. 196.
 Murray, p. 196.
 Murray, pp. 197, 210.
 Murray, p. 196.
 Murray, pp. 197-8.
 Murray, pp. 198-9.
 Murray, pp. 200-2.
 Murray, pp. 208, 122, 206.
 Murray, pp. 174, 193.
 Crosby, pp. 70-1.
 Joel Kaye, “The Impact of Money on the Development of Fourteenth-Century Scientific Thought,” Journal of Medieval History 14 (1988), p. 251.
 Crosby, p. 58.
 Murray, p. 191.
 Murray, pp. 195, 190.
 Jacques Le Goff, Intellectuals in the Middle Ages, Trans. by Theresa Lavender Fagan (Cambridge, Mass.: Blackwell, 1993), p. 5; Haskins, Ren., p. 62.
 Haskins, Ren., p. 13-4.
 Henri Pirenne, Medieval Cities (Princeton, 1925), pp. 127-8; quoted in Haskins, Ren., 62.
 Haskins, Ren., 63.
 Sombart, p. 127.
 Smith and Karpinski, p. 131.
 Smith and Karpinski, p. 134; Murray, p. 193.
 Crosby, p. 211.
 Charles H. Haskins, “Arabic Science in Western Europe,” Isis, Vol. VII, No. 23 (Brussels: M. Weissenbruch, 1925), p. 480.
 Murray, p. 192.
 Bloch, p. 103; Smith and Karpinski, p. 104.
 Haskins, Ren., p. 290.
 Haskins, Ren., pp. 287, 368.
 For instance, one might ask which came first, trade or political stability: “There was also a certain amount of political advance... an advance which promoted a certain degree of peace and the travel and communication which go on best in a peaceful society” (Haskins, Ren., 12-3).
 Lynn White, Medieval Technology and Social Change (Oxford University Press, 1962), p. 55.
 Murray, p. 203.
 Murray, p. 203.