Ann E. Moyer. 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) (Studies in Medieval and Early Modern Civilization.) Ann Arbor: The University of Michigan Press, 2001. vi + 232 pp. illus, bibl. index. $57.50. ISBN: 0-472-11228-7.

Students at medieval universities studied a curriculum based upon the seven ancient liberal arts: the trivium of linguistic arts (grammar, rhetoric, and dialectic, or logic) and the quadrivium of mathematical ones (arithmetic, geometry, music, and astronomy). The trivium and its relationship to scholastic philosophy have been the subject of many studies. The teaching of the quadrivium has by no means been neglected, either, and Moyer provides an excellent bibliography of the recent scholarship on the subject. But some of the most interesting questions have hardly been explored. Most scholarship focuses on such accomplished mathematicians such as Bradwardine or Oresme; but what were ordinary students, with no great mathematical aptitude, supposed to take away from the quadrivial curriculum? Why should trainee clerics devote themselves to studying arithmetic? And how was it taught?

Moyer provides answers to many of these questions in her close study of rithmomachia. This fantastically complicated game first appeared in the cathedral schools of the eleventh century (though many handbooks attributed its invention to Pythagoras). It was played on a long rectangular checkerboard with counters of various shapes on which were inscribed numbers, calculated according to a complicated algorithm. The two players in this "battle of the numbers" (which is what "rithmomachia" means) marshaled odd numbers against even, moving and capturing pieces according to complex rules. In the rithmomachia manual printed at the end of the volume, seven ways of capturing an enemy piece are given: for instance, if an enemy piece has a higher number on it than your own, you may take it if the number of empty spaces between the pieces multiplied by the number on your piece is equal to the number on his piece. Movement and capture is, however, only the first phase of play. The game is won when a player manages to array the captured pieces into various progressions illustrating arithmetical, geometrical and harmonic proportion. Rithmomachia makes its great rival chess seem positively simpleminded.

Moyer shows that the fortunes of rithmomachia were closely connected with the teaching of Boethian arithmetic, which was preoccupied with the classification of ratios, and required the memorization of endless types of arithmetical, geometrical, and harmonic proportion. The art had little practical use; the exceptional figures, such as Oresme and Bradwardine, who did find fruitful applications, were hindered more than they were helped by its clumsy and pedantic terminology. Such an arithmetic was, one presumes, difficult to teach and tedious to learn. Rithmomachia made the tasks of both the teacher and the pupil a little easier.

The game also provided an opportunity to practice Boethian arithmetic--probably the only one most students would ever have. At times, it seems that rithmomachia was identified as the final goal of arithmetic; no longer just a useful tool for imparting the theory of proportion, it was what arithmetic was) for. Nicholas of Cusa, for instance, wrote that rithmomachia was to arithmetic as the monochord was to music, and the author of the doggerel poem De vetula (ascribed, incongruously, to Ovid in the Middle Ages) described it as the "flower and fruit of arithmetic," (41) not just an aid to its comprehension.

Some introductions to the sciences, however, argued that arithmetic (in its Boethian form) led upwards to the contemplation of divine harmonies and proportions (and Moyer provides a thorough and extremely useful survey of such arguments). Rithmomachia shared in arithmetic's elevation, and was often praised as an honorable--even spiritual--form of entertainment, unlike chess (although, as Moyer shows, rithmomachia players and chess players were often one and the same).

The game enjoyed its heyday in the Renaissance when (paradoxically, one might think) humanists promoted the study of Boethian arithmetic in the schools; its decline in the seventeenth century equally mirrors the abandonment of the old arithmetic at the university in favor of practical, applied arithmetic and algebra.

For half a millennium, rithmomachia occupied scholars in their studies and students in their pubs. This book is a thorough survey of the history of a game, and also a perceptive study of its symbiotic partner, the medieval and Renaissance scientific curriculum. My main criticism of the book is its layout. There ate very few "signposts" to the reader: chapter 3, for instance, contains not a single paragraph or section heading in its thirty pages. This makes her argument, which draws on many sources and often digresses into related subjects, difficult to follow at times. The book is often repetitive, in a way that suggests it has not been edited carefully enough (Oresme's spurious association with the game, for instance, is mentioned on pp. 48, 68, and 90, each time as if it were the first; similarly with Clichtove's work on number symbolism [88, 93]). These flaws can detract from the author's very real contributions to scholarship: a study of a little-known, but fascinating cultural phenomenon, and an insightful history of the sciences at the medieval and Renaissance university.

ROBERT GOULDING

University of Notre Dame