In The Philosophers' Game, Ann E. Moyer invites us to engage with the forgotten chess-like game Rithmomachia ("The Battle of Numbers"), which combined the pleasures of gaming with mathematical study and moral education. Intellectuals of the medieval and Renaissance periods who played this game were not only seeking to master the principles of Boethian mathematics but were striving to improve their own understanding of the secrets of the cosmos.
The Philosophers' Game, which includes a complete, illustrated Elizabethan rulebook, examines the nature and importance of the game's appeal as well as some of the reasons why it faded into obscurity. Rithmomachia enjoyed a last wave of popularity during the Renaissance before the early Scientific Revolution led to its disappearance. The demise of Rithmomachia forms part of the great transformation of fields of learning and the classification of knowledge that marked the final dissolution of the quadrivium among the traditional liberal arts.
The Philosophers' Game will interest anyone who studies the history of science, mathematics, or education in medieval and Renaissance Europe; the intellectual or cultural history of those eras; or the histories of games, sports, and leisure. It will also interest scholars interested in astrology and magic.
Ann E. Moyer is Assistant Professor of History, University of Pennsylvania.
|Publisher:||University of Michigan Press|
|Series:||Studies in Medieval and Early Modern Civilization Series|
|Product dimensions:||6.00(w) x 9.00(h) x 1.00(d)|
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The Philosophers' Game: Rithmomachia in Medieval and Renaissance Europe, With an Edition of Ralph Lever and William Fulke, the Most Noble, Auncient, and Learned Playe (1563)
By Ralph Lever
University of Michigan PressCopyright © 2001 Ralph Lever
All right reserved.
1 - Introduction
Thomas More's Utopians enjoyed their leisure at the end of their six-hour workday. They filled this free time with a number of enjoyable and useful pursuits familiar to More's sixteenth-century audience. They attended lectures; they read books; they engaged in conversation, gardening, and musical performances; and they played games. Even their games were useful as well as entertaining:
They know nothing about gambling with dice or other such foolish and ruinous games. They play two games not unlike our chess. One is a battle of numbers, in which one number plunders another. The other is a game in which the vices battle against the virtues.These games were no mere fictions. Like so many other aspects of Utopian society, they had a lengthy past in the history of medieval clerics. The "battle between virtues and vices" had been offered by Wibold, bishop of Cambrai (d. 965), to local monks as an alternative to games of dice.
The early readers of Utopia would not only have recognized the game of battling numbers as rithmomachia, also known then as the ludus philosophorum, or philosophers' game; if they had spent time at university, they could have played it as well. This game enjoyed a notable wave of popularity during the fifteenth and sixteenth centuries. A number of humanists, mathematicians, and educators-some well known, others less so-wrote and published rithmomachia manuals: John Sherwood, Jacques Lefèvre d'Etaples, Claude de Boissière, Benedetto Varchi and Carlo Strozzi, Francesco Barozzi, Ralph Lever and William Fulke, Gottschalk Eberbach, Abraham Riese, and August Duke of Braunschweig. Boards and pieces were available for purchase at the bookstores stocking these works. Manuals for two related games also appeared: the "astronomers' game," or ludus astronomorum (published as Ouranomachia), and a geometric game called Metromachia.
Such games may hold an intrinsic appeal to people interested in the leisure pursuits of Renaissance Europeans. Yet neither the philosophers' game nor its companions were mere pastimes for idle hours. Rithmomachia was created as an educational board game, designed to teach and exercise principles of Boethian mathematics. Its invention dates to the time when these Boethian texts were themselves being proposed as fundamental parts of a developing Latin liberal arts education, the eleventh century. Despite its many rule books from the age of print, then, it was not originally a Renaissance game at all, but a medieval one. Its playing board and pieces invite comparisons to chess, another game whose pedigree includes medieval Europe.
Chess, however, has remained with us as a modern game, while rithmomachia and its companions have faded into obscurity. Despite its flurry of publications in the middle decades of the sixteenth century, the game's popularity declined forever shortly after 1600. Even during its life span it was distinct in many ways from other games. Texts dedicated to discussing sports and pastimes generally did not mention it. Nor did it find a place in the books of parlor games that began to appear in print. It made its home instead in the schoolroom, in the residences of students, and those of former students. The brief medieval manuals of its rules survive among the manuscript remains of ecclesiastical and university education. Its players could be found among those Europeans who had studied the Latin texts of Boethian arithmetic from which the game drew its basic principles.
This group, mainly clerical and male, composed only a small minority of European society. One might well question the significance of an activity that occupied only a modest part of the leisure of so few. Yet the influence of these men of learning extended, for centuries, far beyond their actual numbers; and rithmomachia reflects their learned interests as well as those of their leisure. The game's own life span encompassed something over five hundred years, an era that saw great changes in both education and society. Such a great continuity of tradition in teaching, learning, and the culture of the learned despite these changes suggests that rithmomachia's principles did indeed hold long-term significance.
Some of the claims made through the years for the value of the game found their basis in general arguments about the value of game playing or recreation in society. Other claims had broader implications; they referred to the importance of arithmetic, and of Boethian arithmetic in particular. They help to impress upon the modern reader just how important were the arguments also made by the authors of textbooks and by educational authorities about the subject's value. Arithmetic deserved study, they asserted, not simply because it served as a useful tool for the solving of computational problems. It improved the character of the person who studied it; further, it offered insight into religious truth. The philosophers' game put these principles into practice. The player benefited by coming to master the calculatory skills needed to win the game (skills that could be applied to other problems outside the game as well). More important, in contemplating and practicing the principles of arithmetic he thereby improved his soul.
To a modern reader, these are unexpected claims about a field that now seems entirely mundane and practical. They call for an examination of the source to which they refer-Boethius and his Arithmetic-and the attitudes about arithmetic found there. Anicius Manlius Severinus Boethius (d. 524) served as a standard textbook author for centuries of medieval education. Not only were his Consolation of Philosophy and other writings used for the teaching of grammar and logic, which, along with rhetoric, composed the cluster of fields in the liberal arts curriculum known as the trivium. His mathematical works, along with some spuriously attributed to him, became the basic textbooks in the other division of the curriculum known as the quadrivium, that cluster of mathematical fields composed of arithmetic, geometry, astronomy, and music. Yet the significance of those quadrivial studies throughout the Middle Ages has often been less than clear.
Historians have differed greatly in their assessments about the importance-as well as the contents-of quadrivial education. To some, the quadrivium has seemed a monolithic and changeless curriculum that persisted for centuries. To them its conservative, inward-looking nature held back innovation in education and scholarship in mathematics and natural philosophy until the very advent of the scientific revolution. Others have portrayed the quadrivium as no less backward, perhaps, but not so long lived, a curriculum limited to the prescholastic era. It formed part of a system of study and classification of subjects that was swept away as early as the twelfth century by the arrival of Aristotelian logic, natural philosophy, and the scholastic disciplinary classifications used at medieval universities. More recent scholarly research, especially that on late medieval and Renaissance universities, has added far more nuance as well as substance to our understanding. This scholarship has noted that the quadrivium did survive to become part of university learning, but also underwent a number of important changes.
Studies of the philosophers' game have echoed, on a smaller scale, our incomplete knowledge about the history of the quadrivium itself. The game has been the subject of several articles, appearing at widely spaced intervals, during the past century. Yet their authors have tended to approach the game almost as a fresh discovery, as a phenomenon unknown to their readers. The early history of rithmomachia and its manuscript tradition have been treated recently with great thoroughness in a German monograph by Arno Borst. Yet the game's rules are still unfamiliar enough to merit describing before turning to its context and significance. Doing so requires a survey in turn of the Boethian texts on which they are based.
Boethius's Arithmetic and Music are the surviving remains of his larger, unfinished project of organizing a Latin liberal arts curriculum. The Arithmetic is essentially a loose translation of writings by the second-century scholar Nicomachus of Gerasa. Much of the Music is probably also based on a text of Nicomachus now lost, with additional material from Ptolemy's Harmonics and other works. Nicomachus, and Boethius in turn, participated in the revival of Pythagorean thought in later antiquity. Pythagorean thought had received a wide early audience in Plato's Timaeus. It was the one Platonic dialogue that was translated into Latin (in part) with a commentary by Chalcidius in the fourth century a.d. These texts would add to the perceived importance of Boethius for medieval Latin readers.
From these sources and a few others, they were taught to think of the creation of the world as the imposition of order upon formless matter. That order was numeric in nature and generated from the smallest units, the numbers one through four. In the Timaeus they were described most clearly in Plato's discussion of the World Soul, composed of the first three powers of two and three, in the double series 1:2:4:8 and 1:3:9:27. Boethius referred to these notions briefly in the Arithmetic and developed them in more detail in the Music. These same proportions can be seen in the distance of the planets from one another, the combination of elements as they form the substances of the physical world, and the organization of the human soul, which itself resembles the World Soul in miniature.
Much of the contents of both Boethian textbooks consists of definitions of types of number and the relationships between them, and discussions of their properties. Significant types include odd and even, prime and composite, perfect numbers, and their products. Relations between numbers (proportions) may be based on equality or inequality. The latter have several types: multiplex (of the form x:1), superparticular [(x + 1):x], and superpartient [(x + 2):x, (x + 3):x, and so on]. These proportions can also have a mean established between them. Boethius identified three means. In the first type, arithmetic, the mean falls at the arithmetic midpoint between the two extremes (mean, or m = (a + b) ÷ 2, for example, 1:2:3). A geometric mean produces similar ratio with the two extremes: (a:m = m:b, for example, 1:2:4). The harmonic mean creates the most consonant proportions (m = 2ab ÷(a + b), for example, 3:4:6). Particular subsets of these proportions constitute musical consonances: multiple and superparticular proportions using the numbers one through four. These terms and proportions are useful for solving some kinds of calculatory problems, such as the construction of a musical scale. Nonetheless, Boethius claims that their ultimate value lies in their ability (through such applications or on their own) to turn the mind of the scholar toward an understanding of the divine.
Boethius's Arithmetic is more difficult to follow for a modern reader than many other ancient mathematical texts, for example, Euclid's Elements of Geometry. Euclid's text proceeds by means of proofs and demonstrations; it sets problems and solves them, and then often uses those solutions in subsequent problems in turn. These sorts of topics and methods are still part of modern expectations about mathematical subjects and how to study them. Indeed, the teaching of geometry is still based to some degree on Euclid's text. Boethius's Arithmetic, on the other hand, seems to devote far too much attention to defining terms that seem arbitrary at best, and far too little to calculation, proof, or problem solving. Boethius's approach to mathematics also differs from the approaches taken by most of the mathematical works introduced into European society from the twelfth century on via contact with the Islamic world. Like Euclid's geometry, they seem a better match for modern expectations about both the style and the substance of mathematics texts: they pose problems both practical and abstract and then solve them, proceeding from simple principles to more complex ones.
Excerpted from The Philosophers' Game: Rithmomachia in Medieval and Renaissance Europe, With an Edition of Ralph Lever and William Fulke, the Most Noble, Auncient, and Learned Playe (1563) by Ralph Lever Copyright © 2001 by Ralph Lever. Excerpted by permission.
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